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p¯ L = L From these results, it follows immediately that M Lp = Lp¯ ,
¯ M Lp− = Lp+ ,
¯ M Lp+ = Lp− .
Notice that, if we define f , g to be multiplicable whenever f g ∈ L1 , then the space of multipliers M (f ) of a given element f is more complicated, but we still have M Lp = Lp¯, etc., as follows from [19]. As for the multiplier topologies, we also have that • ρLp is the Lp norm topology; • ρLp− is the Fr´echet projective topology on Lp− ; • ρLp+ is the DF topology on Lp+ . For both I and F , the smallest space is L∞ = M L1 , and it is dense in all the other ones. The involution f 7→ f¯ is of course L1 -continuous. The multiplication is continuous from L∞ × L1 into L1 . In fact it is not only separately, but even jointly ¯ into L1 , thanks to continuous, and similarly from Lp × Lp¯ and from Lp− × Lp+ H¨ older’s inequality and the fact that all topologies are either Fr´echet or DF [38]. Since this result is general, we state it as a proposition. Proposition 4.1. Let A[τ ] be a partial *-algebra with locally convex topology τ, and I R a generating family. Assume that : (i) τ is a norm topology and A[τ ] is a Banach space. (ii) Each space N ∈ I L , with the topology λN , and each space M ∈ I R , with the topology ρM , is a Banach space. Then the multiplication is jointly continuous from LM × M into A, for every M ∈ I R , and one has kabk 6 kakLM kbkM ∀ a ∈ LM, b ∈ M .
(4.3)
The proof of (4.3) essentially reduces to the principle of uniform boundedness. Indeed, for fixed a ∈ LM, the map a 7→ TaL is continuous from LM into the space of bounded operators on M, which gives kabk 6 ckakLM kbkM for some constant c > 0. The latter may then be eliminated by renormalizing all norms by a factor c. Notice that (4.3) is strongly reminiscent of a H¨ older condition. In fact it reduces to the latter in the case of Lp considered as a topological partial *-algebra, as discussed in Sec. 4 below. A partial *-algebra that satisfies the conditions of Proposition 4.1 may be called a Banach partial *-algebra, since the relation (4.3) is the analogue of the characteristic property of Banach algebras. A similar result holds if one of the spaces M, LM is a Fr´echet space and the other a DF-space, with A[τ ] itself a Fr´echet space.
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In conclusion, the topological structure, the PIP-space structure and the multiplier structure of I all coincide, and we have a tight topological partial *-algebra. By the same token, we can consider every space Lp , as a topological partial *-algebra, simply by replacing the partial multiplication (4.1) by the following one: f ∈ M (g) ⇔ ∃ r, s ∈ [p, ∞], 1/r + 1/s = 1/p, such that f ∈ Lr , g ∈ Ls .
(4.4)
This amounts exactly to replace I or F by the (complete) sublattice indexed by [p, ∞]. The rest is identical. R , dx) 4.2. The spaces Lp (R We turn now to the Lp spaces on R. If we consider the family {Lp (R) ∩ L1 (R), 1 6 p 6 ∞}, we obtain a scale similar to the previous one (except that the individual spaces are not complete), which may be used to endow L1 (R) with the structure of a tight topological partial *-algebra. However, the spaces Lp (R) themselves no longer form a chain, no two of them being comparable. We have only Lp ∩ Lq ⊂ Ls , ∀ s such that p < s < q . Hence we have to take the lattice generated by I = {Lp (R, dx), 1 6 p 6 ∞}, that we call J . The extreme spaces of the lattice are, respectively: \ [ X Lq , and VJ = Lq = Lq . VJ# = 16q6∞
16q6∞
16q6∞
Here too, the lattice structure allows to give to VJ a structure of topological partial *-algebra, as we shall see now. The lattice operations on J are easily described [2, 5, 22, 25]: • Lp ∧ Lq = Lp ∩ Lq is a Banach space, with the projective (topology corresponding to the) norm kf kp∧q = kf kp + kf kq . • Lp ∨ Lq = Lp + Lq is a Banach space, with the inductive (topology corresponding to the) norm kf kp∨q = inf (kgkp + khkq ) , g ∈ Lp , h ∈ Lq . f =g+h
• For 1 < p, q < ∞, both spaces Lp ∧Lq and Lp ∨Lq are reflexive and (Lp ∧Lq )0 = Lp¯ ∨ Lq¯. At this stage, it is convenient to introduce a unified notation: ( p L ∧ Lq , if p > q , (p,q) = L Lp ∨ Lq , if p 6 q .
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1/q L(∞,1) = L∞ ∩ L1
6
Lp ∩ L1
•
L1
•
•
,
L∞ ∩ Lq
, Lq , •, ,
Lp ∧ Lq = L(p,q)
•
•
A A A A
, ,
,
•
L(1,q) = L1 + Lq
,
2 , AL , A• , , A A Lp • , • A , p q (q,p) A L ∨L =L , AA , •
, ,
, , ,
•,
Lp¯ ∨ Lq¯ = (Lp ∧ Lq )0
•
L∞
L(p,∞) = Lp + L∞
•
-
1/p
L(1,∞) = L1 + L∞
Fig. 3. The unit square describing the lattice J.
Thus, for 1 < p, q < ∞, each space L(p,q) is a reflexive Banach space, with dual ¯ q) . The modifications when p, q equal 1 or ∞ are obvious. L(p,¯ Next, if we represent (p, q) by the point of coordinates (1/p, 1/q), we may associate all the spaces L(p,q) (1 6 p, q 6 ∞) in a one-to-one fashion with the points of a unit square J = [0, 1] × [0, 1] (see Fig. 3). Thus, in this picture, the spaces Lp are on the main diagonal, intersections Lp ∩ Lq above it and sums Lp + Lq below. 0 0 The space L(p,q) is contained in L(p ,q ) if (p, q) is on the left and/or above (p0 , q 0 ). Thus the smallest space L(∞,1) = L∞ ∩ L1 corresponds to the upper left corner, the largest one, L(1,∞) = L1 + L∞ , to the lower right corner. Inside the square, duality corresponds to (geometrical) symmetry with respect to the center (1/2, 1/2) of the square, which represents the space L2 . The ordering of the spaces corresponds to the following rule: 0
0
L(p,q) ⊂ L(p ,q ) ⇔ (p, q) 6 (p0 , q 0 ) ⇔ p > p0 and q 6 q 0 .
(4.5)
For ∞ > qo > 1, consider now the horizontal row q = qo , {L(p,qo ) : ∞ > p > 1}. It corresponds to the chain: · · · ⊂ Lr ∩ Lqo ⊂ · · · ⊂ Lqo ⊂ · · · ⊂ Ls + Lqo ⊂ · · · (∞ > r > qo > s > 1) ,
(4.6)
sitting between the extreme elements L∞ ∩ Lqo on the left and L1 + Lqo on the right. The point is that all the embeddings in the chain (4.6) are continuous and have dense range.
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The same holds true for a vertical row p = po , {L(po ,q) : 1 < q < ∞}: · · · ⊂ Lpo ∩ Ls ⊂ · · · ⊂ Lpo ⊂ · · · ⊂ Lpo + Lr ⊂ · · · (1 < s < po < r < ∞) .
(4.7)
Combining these two facts, we see that the partial order extends to the spaces L(p,q) (1 < p, q < ∞), inclusion meaning now continuous embedding with dense range. Now the set of points contained in the square J may be considered as an involutive lattice with respect to the partial order (4.5), with operations: (p, q) ∧ (p0 , q 0 ) = (p ∨ p0 , q ∧ q 0 ) (p, q) ∨ (p0 , q 0 ) = (p ∧ p0 , q ∨ q 0 ) (p, q) = (¯ p, q¯) , where, as usual, p ∧ p0 = min{p, p0 }, p ∨ p0 = max{p, p0 }. The considerations made above imply that the lattice J generated by I = {Lp } is already obtained at the first generation. For example, L(r,s) ∧ L(a,b) = L(r∨a,s∧b) (see Fig. 4), and the latter may be either above, on, or below the diagonal, depending on the values of the indices. For instance, if p < q < s, then L(p,q) ∧ L(q,s) = Lq , both as sets and as topological vector spaces.
1/q
6
r
a
,,
1
L(r∨a,s∧b) = L(r,b) = Lr ∧ Lb b
, , ,
s
∞
,,
,
,,
•
•
,, L , , , L La
•
b
•
,,
•
,, , Lr •
Ls
, , ,
•
(a,b)
= La + Lb
•
L(r,s) = Lr + Ls
•
•
r
a
Fig. 4. The intersection of two spaces from J.
1
1/p
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283
The conclusion is that, using this language, the only difference between the two cases {Lp ([0, 1])} and {Lp (R)} lies in the type of order obtained: a chain I (total order) or a partially ordered lattice J. From this remark, the lattice completion of J can be obtained exactly as before, using the results of [25]. This introduces again Fr´echet and DF-spaces, all reflexive if we start from 1 < p < ∞, and in natural duality as in the previous case. In particular, for the spaces of the first “generation”, it suffices to consider intervals S ⊂ [1, ∞] and define the spaces \ [ Lq , LI (S) = Lq . LP (S) = q∈S
q∈S
Then: • If S is a closed interval S = [p, q], with p < q, then LP (S) = Lp ∧ Lq = L(q,p) and LI (S) = Lp ∨ Lq = L(p,q) are Banach spaces. If S is a semi-open or open interval, LP (S) is a non-normable Fr´echet space and LI (S) a DF-space. q : q ∈ S}. Then • Let S ⊂ (1, ∞) and define S = {¯ (LP (S))0 = LI (S) ,
(LI (S))0 = LP (S) .
A special rˆ ole will be played in the sequel by the spaces LI corresponding to semiinfinite intervals, namely: [ Ls = Lp + L∞ , which is a nonreflexive Banach space . L(p,∞) = LI ([p, ∞]) = p6s6∞
L(p,ω) = LI ([p, ∞)) =
[
Ls , which is a reflexive DF-space .
p6s<∞
As for the lattice completion F , one can essentially repeat the argument of [2, Example 3.B] and build an “enriched” or “nonstandard” square F , exactly as in the previous section. Take first 1 < q < ∞, that is, the interior Jo of the square J. The extreme spaces of the corresponding complete lattice Fo are \ [ X Lq , and Vo = Lq = Lq , Vo# = 1
1
1
with their projective and inductive topologies, respectively. All embeddings are continuous and have dense range. Thus the space Vo , together with either of the two lattices Jo = {Vα , α ∈ Jo } or Fo = {Vα , α ∈ Fo }, is a PIP-space, with the usual L2 inner product and (Vα )# = (Vα )× = (Vα¯ ). Similar results are valid when one includes L1 and L∞ , except for the obvious modifications concerning duality. The extreme spaces of the full lattice F then are 1 ∞ and VJ = Lρ = L1 + L∞ , with their projective and inductive VJ# = L# ρ = L ∩L norms, which make them into non-reflexive Banach spaces (none of them is the dual of the other). Notice that the space Lρ , originally introduced by Gould [44], contains strictly all the Lp , 1 6 p 6 ∞. We turn now to the partial *-algebra structure on VJ . Again we start from the lattice J , which is generating for the PIP-space structure. The basic fact is
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H¨ older’s inequality, which says that pointwise multiplication is continuous from Lp × Lq into Lr , where 1/p + 1/q = 1/r. From this we can compute the multipliers of all the elements of J in several steps (as usual we write p ∧ q = min{p, q}, p ∨ q = max{p, q}): • Ls ⊂ M Lp iff p¯ 6 s 6 ∞. Thus M Lp =
[
¯ Ls = L(p,∞) .
s>p¯
• Let p > q, so that L(p,q) = Lp ∧ Lq . Then ¯ ¯ ∨ L(¯q,∞) = L(p,∞) . M L(p,q) = M Lp ∨ M Lq = L(p,∞)
• Let p < q, so that L(p,q) = Lp ∨ Lq . Then ¯ ∧ L(¯q,∞) = L(¯q,∞) . M L(p,q) = M Lp ∧ M Lq = L(p,∞)
• Thus in all cases ¯ q,∞) = L(p∨q,∞) . M L(p,q) = L(p∧¯
(4.8)
If one does not want to include L∞ , one simply replaces (4.8) by ¯ q,ω) = L(p∨q,ω) . M L(p,q) = L(p∧¯
(4.9)
Applying the rule (4.8) or (4.9) twice, one gets immediately M M L(p,q) = L(p∨q,∞) , resp. M M L(p,q) = L(p∨q,ω) . In conclusion, the generating family for multiplication is the set J M = {L(p,∞) , 1 6 p 6 ∞}, corresponding to the bottom side of the square J in Fig. 3, and it is a chain of Banach spaces, exactly as in the case of the Lp spaces over [0, 1]. Thus we write the partial multiplication on L1 + L∞ as q = 1, f ∈ M (g) ⇔ ∃ q ∈ [1, ∞] such that f ∈ L(q,∞) , g ∈ L(¯q,∞) , 1/q + 1/¯
(4.10)
and that on V as q = 1. f ∈ M (g) ⇔ ∃ q ∈ (1, ∞) such thatf ∈ L(q,ω) , g ∈ L(¯q,ω) , 1/q + 1/¯
(4.11)
Finally, we can immediately conclude that the complete lattice F M is the “enriched” chain F M = {L(p,∞) , p = p−, p or p+, 1 6 p 6 ∞}, and similarly with ω instead of ∞. Exactly as in the case of a finite interval, we may restrict the generating spaces to {Ls , p 6 s 6 ∞), which amounts to take a subsquare of J. The rest is obvious. Another interesting structure of partial *-algebra may be given to the spaces Lρ or V , simply replacing multiplication by convolution. According to Hausdorff– Young’s inequality, convolution maps Lp × Lq continuously into Lr , where 1/p + 1/q = 1 + 1/r. From this we can compute the multipliers of all the elements of J as in the previous case (to avoid confusion, we use here the notation M∗ ):
TOPOLOGICAL PARTIAL *-ALGEBRAS: BASIC PROPERTIES AND EXAMPLES
285
• Ls ⊂ M∗ Lp iff s 6 p¯. Thus M∗ Lp =
[
¯ Ls = L(1,p) = L1 + Lp¯ .
s6p¯ ¯ , and for p < q, M∗ L(p,q) = L(1,¯q,) . • For p > q, M∗ L(p,q) = L(1,p) • Thus in all cases ¯ q) = L(1,p∨q) . M∗ L(p,q) = L(1,p∧¯
(4.12)
Again these multiplier spaces constitute a chain, this one corresponding to the right-hand side of the square J. 4.3. Amalgam spaces The lesson of the previous example is that an involutive lattice of (preferably reflexive) Banach spaces (that is, a PIP-space of type B or H [3]) turns quite naturally into a (tight) topological partial *-algebra if it possesses a partial multiplication that verify a (generalized) H¨ older inequality. A whole class of examples is given by the so-called amalgam spaces first introduced by N. Wiener [43] and developed systematically by Holland [29]. The simplest ones are the spaces (Lp , `q ) (sometimes denoted W (Lp , `q )) consisting of functions on R which are locally in Lp and have `q behavior at infinity, in the sense that the Lp norms over the intervals (n, n + 1) form an `q sequence (see the review paper [27]). For 1 6 p, q < ∞, the norm ( kf kp,q =
∞ Z X n=−∞
n+1
q/p )1/q |f (x)| dx p
n
makes (Lp , `q ) into a Banach space. The same is true for the obvious extensions to p and/or q equal to 1 or ∞. Notice that (Lp , `p ) = Lp . The spaces (Lp , `q ) have many interesting applications, for instance in the context of various Tauberian theorems. New ones have been found recently in the theory of frames (nonorthogonal expansions) [21]. These spaces obey the following (immediate) inclusion relations, with all embeddings continuous: • If q1 6 q2 , then (Lp , `q1 ) ⊂ (Lp , `q2 ). • If p1 6 p2 , then (Lp2 , `q ) ⊂ (Lp1 , `q ). From this it follows that the smallest space is (L∞ , `1 ) and the largest one is (L1 , `∞ ), and therefore: • If p > q, then (Lp , `q ) ⊂ Lp ∩ Lq ⊂ Ls , ∀ q < s < p. • If p 6 q, then (Lp , `q ) ⊃ Lp ∪ Lq . Once again, H¨older’s inequality is satisfied. Whenever f ∈ (Lp , `q ) and g ∈ (Lp¯, `q¯), then f g ∈ L1 and one has kf gk1 6 kf kp,q kgkp,¯ ¯q .
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Therefore, one has the expected duality relation: (Lp , `q )0 = (Lp¯, `q¯) ,
for 1 6 q, p < ∞ .
The interesting fact is that, for 1 6 p, q 6 ∞, the set J of all amalgam spaces {(Lp , `q )} may be represented by the points (p, q) of the same unit square J as in the previous example, with the same order structure. In particular, J is a lattice with respect to the order (4.5): 0
0
0
0
0
0
0
0
(Lp , `q ) ∧ (Lp , `q ) = (Lp∨p , `q∧q ) (Lp , `q ) ∨ (Lp , `q ) = (Lp∧p , `q∨q ) , where again ∧ means intersection with projective norm and ∨ means vector sum with inductive norm. We turn now to the partial *-algebra structure of J . At first sight, the situation becomes different, because whereas L1 is a partial *-algebra, `∞ is an algebra under componentwise multiplication, (an )·(bn ) = (an bn ). The Lp component characterizes the local behavior. Hence, M (Lp , `q ) ⊃ (Lp¯, `∞ ), ∀ q , and since the latter are totally ordered, we obtain, exactly as in the cases of the Lp spaces: M (Lp , `q ) = (Lp¯, `∞ ) . Thus the natural partial multiplication on J reads: f ∈ M (g) ⇔ ∃ p ∈ [1, ∞] such that f ∈ (Lp , `∞ ) and g ∈ (Lp¯, `∞ ) .
(4.13)
The rest is as before, including the identification of the complete lattice F with the “enriched” interval [1, ∞]. Since the amalgam spaces (Lp , `q ) obey the same Hausdorff–Young inequality as the Lp spaces, we may obtain here too another structure of partial *-algebra with 0 0 the convolution as partial multiplication. Let f ∈ (Lp , `q ) and g ∈ (Lp , `q ), with 00 00 1/p + 1/p0 > 1, 1/q + 1/q 0 > 1, that is, p0 6 p¯, q 0 6 q¯. Then f ∗ g ∈ (Lp , `q ), with 1/p00 = 1/p + 1/p0 − 1, 1/q 00 = 1/q + 1/q 0 − 1. By the same arguments as in the previous section, we obtain ¯ q ) = (L1 , `p∨q ) M∗ (Lp , `q ) = (L1 , `p∧¯
(4.14)
As before, these multiplier spaces constitute a chain, corresponding to the righthand side of the square J. 5. Examples 2: Topological Partial *-algebras of Operators 5.1. Operators on a lattice of Hilbert spaces Our first example is the partial *-algebra of operators on a lattice of Hilbert spaces (LHS), also called indexed PIP-spaces of type (H) [3]. By this we mean a vector space V together with a family of subspaces VI = {Hr , r ∈ I}, where
TOPOLOGICAL PARTIAL *-ALGEBRAS: BASIC PROPERTIES AND EXAMPLES
287
P (i) V = r∈I Hr ; (ii) the index set I is an involutive lattice with order-reversing involution r ↔ r (that is, p 6 q implies q 6 p and p = p) and a unique element o such that o = o; (iii) each Hr is a Hilbert space with norm k · kr and Hr = Hr× , the anti-dual of Hr ; in particular, Ho = Ho× = Ho ; (iv) the family VI is an involutive lattice under set inclusion and lattice operations • Hp∧q = Hp ∩ Hq , with the projective norm kf k2p∧q = kf k2p + kf k2q , • Hp∨q = Hp + Hq , with the inductive norm kf k2p∨q = inf f =g+h (kgk2p + khk2q ), (g ∈ Hp , f ∈ Hq ) (we use squared norms in these definitions in order to get Hilbert norms for the projective and inductive ones); (v) The inner product of Ho extends to a partial inner product h·|·i, that is, a Hermitian sesquilinear form defined exactly on dual pairs Hr , Hr . T It follows that (V, h·|·i) is a PIP-space, and V # = r∈I Hr . We assume the partial inner product to be nondegenerate, that is, (V # )⊥ = {0}, which means that hf |gi = 0, ∀ f ∈ V # , implies g = 0. This entails that (V # , V ), as well as every pair (Hr , Hr ), is a dual pair in the sense of topological vector space theory [38]. Note that the Mackey topology τ (Hr , Hr ) on Hr coincides with the original norm topology. Once again the topological and lattice structures coincide: q < p implies Hq ⊂ Hp and the embedding is continuous with dense range. Similarly, Hp∧q and Hp∨q are dual to each other. Moreover, V # is dense in every Hr , r ∈ I. Typical examples of LHS are: (i) Hilbert scales Many examples of Hilbert scales, discrete or continuous, appear in applications. For instance: • The scale built on the powers of a positive self-adjoint operator H > 1 : Hn = D(H n ), with the graph norm kf kn = kH n f k, for n ∈ N, and H−n = Hn× . • In particular, the scale of Sobolev spaces Ws2 (R), s ∈ R, where f ∈ Ws2 (R) if its Fourier transform fb satisfies the condition (1 + |.|2 )s/2 fb ∈ L2 (R). The norm is kf ks = k(1 + |.|2 )s/2 fbk, s ∈ R. Of course, similar considerations hold for the Banach scale {Wsp (R), s ∈ R}, 1 < p < ∞, but here we restrict ourselves to the Hilbert case p = 2. We will come back to these two examples at the end of this Sec. 5. (ii) Weighted `2 sequence spaces Given a sequence of positive numbers, r = (rn ), rn > 0, define `2 (r) = P∞ {x = (xn ) : n=1 |xn |2 rn−1 < ∞}. The lattice operations read: • Involution: `2 (r) = `2 (r)× , r n = 1/rn ; • Infimum: `2 (p) ∧ `2 (q) = `2 (r), rn = min(pn , qn ); • Supremum: `2 (p) ∨ `2 (q) = `2 (s), sn = max(pn , qn ).
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As for the extreme spaces, it is easy to see that the family {`2 (r)} generates the space ω of all complex sequences, while the intersection is the space ϕ of all finite sequences. (iii) Weighted L2 function spaces Instead of sequences, we consider locally integrable (i.e. integrable on bounded sets) functions f ∈ L1loc (R, dx) and define again weighted spaces: I = {r ∈ L1loc (R, dx) : r(x) > 0, a.e.} Z L2 (r) = {f ∈ L1loc (R, dx) : |f (x)|2 r(x)−1 dx < ∞}, r ∈ I . Then we get exactly the same structure as in (ii): • Involution: L2 (r) ⇔ L2 (r), r = 1/r; • Infimum: L2 (p) ∧ L2 (q) = L2 (r), r(x) = min(p(x), q(x)); • Supremum: L2 (p) ∨ L2 (q) = L2 (s), s(x) = max(p(x), q(x)); • Extreme spaces: \ [ L2 (r) = L1loc , L2 (r) = L∞ c , r∈I
r∈I
L∞ c
is the space of (essentially) bounded functions of compact where support. The central space is, of course, L2 . An interesting subspace of the preceeding space is the LHS Vγ generated by the weight functions rα (x) = exp αx, for −γ 6 α 6 γ(γ > 0). Then all the spaces of the lattice may be obtained by interpolation from L2 (r±γ ), and moreover, the extreme spaces are themselves Hilbert spaces, namely Vγ# = L2 (R, e−γx dx) ∩ L2 (R, eγx dx) ' L2 (R, eγ|x| dx) Vγ = L2 (R, e−γx dx) + L2 (R, eγx dx) ' L2 (R, e−γ|x| dx) . This LHS plays an interesting role in scattering theory [13]. Actually the whole construction goes through if one takes for Hr a reflexive Banach space, as in interpolation theory [22]. In this way one recovers the families {`p } or {Lp } (1 < p < ∞) discussed in Sec. 4. For simplicity we restrict the discussion to a LHS. Let VI = {Hr , r ∈ I} be a LHS. The whole idea behind this structure (as for general PIP-spaces) is that vectors should not be considered individually, but only in terms of the subspaces Hr , which are the building blocks of the theory. The same spirit determines the definition of an operator on a LHS space: only bounded operators between Hilbert spaces are allowed, but an operator is a (maximal) coherent collection of these. To be more specific, an operator on VI is a map A : D(A) → V , such that: S (i) D(A) = q∈D(A) Hq , where D(A) is a nonempty subset of I. (ii) For every q ∈ D(A), there is p ∈ I such that the restriction A : Hq → Hp is linear and bounded (we denote it by Apq ∈ B(Hq , Hp )).
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(iii) A has no proper extension satisfying (i) and (ii). The bounded linear operator Apq : Hq → Hp is called a representative of A. Thus A is characterized by two subsets of I:a D(A) = {q ∈ I : there is a p such that Apq exists} , I(A) = {p ∈ I : there is a q such that Apq exists} . We denote by J(A) the set of all such pairs (q, p) for which Apq exists. Thus the operator A is equivalent to the collection of its representatives A ' {Apq : (q, p) ∈ J(A)} .
(5.1)
D(A) is an initial subset of I: if q ∈ D(A) and q 0 < q, then q 0 ∈ D(A), and Apq0 = Apq Eqq0 , where Eqq0 is the unit operator (this is what we mean by “coherent”). In the same way, I(A) is a final subset of I: if p ∈ I(A) and p0 > p, then p0 ∈ I(A). Figure 5 illustrates the situation in the case of a Hilbert scale (I totally W ordered). Notice that, even then, the extreme elements qmax = q∈D(A) q, resp. V pmin = p∈I(A) q need not belong to D(A), resp. I(A), since I is not a complete lattice in general. Also J(A) ⊂ D(A) × I(A), with strict inclusion in general. We denote by Op(VI ) the set of all operators on VI . Since V # ⊂ Hr , ∀ r ∈ I, an operator may be identified with a sesquilinear form on V # × V # . Indeed, the
6p
p0 > p
J(A) I(A) 0
q
pmin I D(A)
-
p qmax
q
I Fig. 5. The various sets characterizing the operator A (in the case of a scale). a The set I(A) was denoted R(A) in [3], but this obviously conflicts with the space of right multipliers, to be defined below.
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restriction of any representative Apq to V # × V # is such a form, and they all coincide. Equivalently, an operator may be identified with a linear map from V # into V . But the idea behind the notion of operator is to keep also the algebraic operations on operators, namely: (i) Adjoint A∗ : every operator A ∈ Op(VI ) has a unique adjoint A∗ ∈ Op(VI ), defined by: hA∗ x|yi = hx|Ayi, for y ∈ Hr , r ∈ J(A) and x ∈ Vs , s ∈ I(A) , that is, (A∗ )rs = (Asr )∗ (usual Hilbert space adjoint). This implies that A∗∗ = A, ∀ A ∈ Op(VI ): No extension is allowed, because of the maximality condition (iii). (ii) Partial multiplication: AB is defined iff there is a q ∈ I(B) ∩ D(A), that is, iff there is continuous factorization through some Hq : B
A
Hr → Hq → Hs ,
i.e. (AB)sr = Asq Bqr .
Notice that here, contrary to the case of a general PIP-space, the domain D(A) is automatically a vector subspace of V . Therefore Op(VI ) is a partial *-algebra (which means, in particular, that the usual rule of distributivity is valid). Now we turn to the spaces of multipliers. Our building blocks are the sets: Opq = {A ∈ Op(VI ) : Apq exists} .
(5.2)
Clearly we have: • • • •
S LOpq ≡ Lp = {C ∈ Op(VI ) : p ∈ D(C)} = s B(Hp , Hs ) ' End(Hp , V ); S ROpq ≡ Rq = {B ∈ Op(VI ) : q ∈ I(B)} = t B(Ht , Hq ) ' End(V # , Hq ); RLOpq = RLp = Rp ∈ F R ; LROpq = LRq = Lp ∈ F L .
(in these relations, End(X, Y ) denotes the space of all linear maps from X into Y ). From this we deduce immediately, using the fact that L, R are lattice antiisomorphisms: Lp ∧ Lq = Lp∨q
Lp ∨ Lq = Lp∧q
Rp ∧ Rq = Rp∧q
Rp ∨ Rq = Rp∨q .
In particular, q 6 q 0 implies Rq ⊂ Rq0 and Lq ⊃ Lq0 . Thus I L = {Lp } is a sublattice of F L , I R = {Rp } is a sublattice of F R , and both are generating (except that they do not contain the extreme elements in general, see below). In addition I L , I R consist of matching pairs (Rq , Lq ). Indeed, given A ∈ Op(VI ), we may rewrite D(A) = {q ∈ I|A ∈ Lq } ,
I(A) = {p ∈ I|A ∈ Rp } ,
(5.3)
and therefore A ∈ L(B) ⇔ ∃ p ∈ I such that A ∈ Lp , B ∈ Rp .
(5.4)
TOPOLOGICAL PARTIAL *-ALGEBRAS: BASIC PROPERTIES AND EXAMPLES
From (5.4), we deduce individual multiplier spaces: _ _ Lp = Lpmin , R(A) = L(A) = p∈I(A)
Rq = Rqmax .
291
(5.5)
q∈D(A)
Note that these two subsets do not belong to I L , resp. I R , in general, but to the complete lattice generated by the latter. In the same way, we obtain \ Rr , (5.6) R Op(VI ) = {A|I(A) = I} = r∈I #
which may be identified with the space End(V ) of all linear maps from V # into itself. Again, R Op(VI ) 6∈ I R . Similarly, _ Lp ' End(V ) . (5.7) L Op(VI ) = {A|D(A) = I} = p∈I
The final point concerns topologies on spaces of multipliers. As a consequence of the identification (5.1) of an operator with the set of its representatives, the partial *-algebra Op(VI ) itself has the structure of an inductive limit of Banach spaces: [ B(Hq , Hp ) . (5.8) Op(VI ) ' q,p∈I
T P One may also consider the extreme spaces V # = r∈I Hr , V = r∈I Hr . On V , the inductive limit topology coincides with the Mackey topology τ (V, V # ), but on V # , the projective topology may be coarser than the Mackey topology τ (V # , V ). This gives another possibility of giving a topology to Op(VI ), by identifying an operator with a continuous linear map from V # into V (each of them endowed with its own Mackey topology), that is Op(VI ) ' L(V # , V ) .
(5.9)
These various possibilities may be different in general, which makes the problem quite involved. Instead we will consider several simpler cases. (1) First, suppose that the extreme spaces V # and V are themselves Hilbert spaces, as for the LHS Vγ described above, or, in the Banach case, the lattices {`p }, {Lp [0, 1]}, {Lp (R)}, {(Lp , `q )}. In that case, the relation (5.9) gives immediately the identification Op(VI ) ' B(V # , V ), with its usual norm topology. Similarly, R Op(VI ) ' B(V # ) and L Op(VI ) ' B(V ). More generally Lq = ind lim B(Hq , Ht ) ' B(Hq , V ) , t
Rq = ind lim B(Hs , Hq ) ' B(V # , Hq ) , s
and these norm topologies coincide with the topologies λ, resp. ρ, on Lq , resp. Rq . Finally the involution is clearly continuous on Op(VI ), so that Op(VI ) is a topological partial *-algebra. However, tightness is open in general. (2) The situation is still simple, and most of the results of (1) survive, when VI consists of a scale (either continuous or discrete) of Hilbert spaces. Then, indeed,
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I contains a countable subset J, stable under the involution, and coinitial to I, which means that, for each r ∈ I, there exists q ∈ J such that q 6 r (J is then automatically cofinal to I : ∀ r ∈ I, there exists p ∈ J such that r 6 p). As a consequence, the projective topology tI , defined by VI , is equivalent to that defined by VJ = {Hs , s ∈ J}. In this case V# =
\
Hs ,
s∈J
[
V =
Hs ,
(5.10)
s∈J
and hence V # is a reflexive Fr´echet space and V is a reflexive DF-space. Thus the projective topology on V # coincides with the Mackey topology τ (V # , V ), and no pathology arises. The space L(V # , V ) of Mackey continuous operators coincides exactly with the space of all linear maps from V # into V which are continuous from V # [tI ] into V [t0I ], where [t0I ] denotes the strong dual topology. In this situation, the space L(V # , V ) provides an example of a quasi*-algebra of operators [33, 42] and the usual theory applies. In particular, in addition to (5.9), we have the identifications: R Op(VI ) ' L(V # ) ,
L Op(VI ) ' L(V ) ,
(5.11)
where L(V # ) is the space of all continuous operators from V # [tI ] into itself and L(V ) and the space of all continuous operators from V [t0I ] into itself (both these spaces can be identified with subspaces of L(V # , V )). Similarly one gets Lq ' L(Hq , V ) ,
Rq ' L(V # , Hq ) ,
q∈I.
(5.12)
Of course, these results remain valid if I is not a scale, but a lattice containing a countable subset J = J, coinitial to I : V # is Fr´echet and V a DF-space. Topologies on L(V # , V ) can then be introduced following [33]. The most interesting seems to be the uniform topology defined by the set of seminorms A ∈ L(V # , V ) 7→ sup |hf |Agi| , f,g∈M
where M is a bounded subset of V # [tI ]. Then there are several possible ways of turning L(V # , V ) into a partial *-algebra, in such a way that one always has, as in (5.11): (5.13) RL(V # , V ) = L(V # ) , LL(V # , V ) = L(V ) . Since the involution and the multiplications are continuous with respect to the uniform topology, L(V # , V ) becomes a topological partial *-algebra, no matter how many Hilbert spaces we use to define (by composition) the multiplication (provided that the relations (5.13) are satisfied). The simplest possibility, usually adopted in the theory of quasi *-algebras, consists in considering none of them: this choice yields very poor lattices of multipliers (for instance J R contains only RL(V # , V ) and L(V # , V ) itself). With this trivial lattice of multipliers, L(V # , V ) is a tight topological partial *-algebra for well-behaved spaces V # , typically a Fr´echet space
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whose topology is the projective topology generated by an O*-algebra. In that case indeed, both L(V # ) and L(V ) are uniformly dense in L(V # , V ) [32, 39]. But this was clearly not what we had in mind when we considered a LHS! We were, in fact, interested in finding a larger (and possibly the largest) lattice of multipliers, making use of the factorization via the spaces {Hp , p ∈ I} (this corresponds, of course, to the possiblility of getting the largest possible set of multiplicable pairs). As said before, in all these cases, L(V # , V ) is a topological partial *-algebra, but tightness is still to be proven. It is interesting to notice the analogy of this procedure of “enrichment” of the lattice of multiplier spaces with the similar operation of refinement or coarsening of a compatibility relation, which also leads to the construction of suitable lattices of subspaces, either containing, or contained in, the corresponding lattice as a sublattice (see [2, 5] for a systematic discussion). One should also beware of possible pathologies linked to associativity, discovered by K¨ ursten [31]. One of the most interesting cases for applications is that of the Hilbert scale built on the powers of a self-adjoint operator H > 1. That is, I = {Hs , s ∈ I ≡ R or Z}, where Hs = D(H s ), for s > 0, with the graph norm, and H−s = Hs× , T P V # = s∈I Hs = D∞ (H), V = s∈I Hs . The partial multiplication in Op(VI ) ' L(V # , V ) is defined by continuous factorization through some Hs : A · B is defined whenever there exists s ∈ I such that B ∈ L(V # , Hs ) and A ∈ L(Hs , V ). The spaces of multipliers themselves, given in (5.12), form scales: I L = {Ls = L(Hs , V ), s ∈ I} ,
I R = {Rs = L(V # , Hs ), s ∈ I} .
(5.14)
In the case of a discrete scale, I = Z, the lattices I L , I R are already complete. For T instance, if K is a subset of Z, bounded from above, then n∈K Rn = RnK , with nK = max K. For a continuous scale, I = R, this is no longer the case, but the lattice completion is obtained exactly as in the case of the Lp spaces described in Sec. 4, by “enriching” the line R. For instance, \ [ Hr , Hs+ = Ht . Hs− = r<s
t>s
With their projective, resp. inductive topology, Hs− is a reflexive Fr´echet space and Hs+ is a reflexive DF-space. The rest is as before, duality relations and lattice completions. Then we have the following result: Proposition 5.1. Let I = {Hs , s ∈ I ≡ R or Z} be the Hilbert scale built on P the powers of a self-adjoint operator H > 1, with V # = D∞ (H), V = s∈I Hs . Then, with partial multiplication defined by continuous factorization through the spaces Hs , Op(VI ) ' L(V # , V ) is a topological partial *-algebra with respect to the uniform topology. The proof is almost immediate, for conditions (i) and (ii) of Definition 3.5 are clearly satisfied. Concerning tightness, it is true that R Op(VI ) ' L(V # ) is uniformly dense in Op(VI ), but to show that it is dense in all the multipliers spaces Rs = L(V # , Hs )
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probably requires additional conditions on H. So far, we can prove it only for s < 0, but that is not sufficient. Here instead, let us consider the two examples already mentioned (in both cases, tightness is open): (i) The Hilbert scale around L2 (R, dx) built on the powers of the self-adjoint d2 2 operator H = 12 (− dx 2 + x ) (this is the Hamiltonian of a quantum mechanical harmonic oscillator in one dimension). Going to the limits n → ±∞ yields \ X Hn = S(R) and V = Hn = S 0 (R) , V# = n∈Z
n∈Z
Schwartz’ spaces of smooth fast decreasing functions and tempered distributions, respectively. In fact, this scale may be used for a simpler formulation of the theory of tempered distributions, called the Hermite or N -representation [36]. This example, which has been studied also in [37], illustrate the usefulness of considering Op(VI ) ' L(S, S 0 ) as a partial *-algebra. (ii) The Sobolev scale {Ws2 (Rn ), s ∈ R} is also of this type, with H = 1 − ∆, acting in L2 (Rn , dn x) (∆ is the n-dimensional Laplacian). The operators on this scale are the building blocks of the theory of partial differential operators. Again the point of view of a topological partial *-algebra might be useful in applications. Notice that, if we take together the scale {Ws2 } d2 }, we recover the Schwartz spaces S, S 0 as and its Fourier transform {W s extreme spaces. In the general case, where I does not contain a countable coinitial subset (or sublattice) J, things get quite involved. Standard examples are the full LHS of weighted `2 or L2 spaces described above. It is probably pointless to treat the problem in such generality. 5.2. Partial O*-algebras Let H be a complex Hilbert space with inner product h·|·i and D a dense subspace of H. We denote by L† (D, H) the set of all (closable) linear operators X such that D(X) = D, D(X*) ⊇ D. The set L† (D, H) is a partial *-algebra with respect to the following operations: The usual sum X1 + X2 , the scalar multiplication λX, the involution X 7→ X † = X*D and the (weak ) partial multiplication X1 X2 = X1 † *X2 , defined whenever X2 is a weak right multiplier of X1 , X2 ∈ Rw (X1 ), that is, iff X2 D ⊂ D(X1 † *) and X1 *D ⊂ D(X2 *). It is easy to check that X1 X2 is well defined iff there exists C ∈ L† (D, H) such that hX2 f |X1 † gi = hCf |gi ,
∀ f, g ∈ D ;
(5.15)
in this case X1 X2 = C. When we regard L† (D, H) as a partial *-algebra with those operations, we denote it by L†w (D, H). A partial O*-algebra on D is a *-subalgebra M of L†w (D, H), that is, M is a subspace of L†w (D, H), containing the identity and such that X † ∈ M whenever
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X ∈ M and X1 X2 ∈ M for any X1 , X2 ∈ M such that X2 ∈ Rw (X1 ). Thus L†w (D, H) itself is the largest partial O*-algebra on the domain D. A partial O*algebra M is said to be self-adjoint if \ D(A∗ ) . D = D∗ (M) := A∈M
The sets RL†w (D, H) of universal right multipliers of L†w (D, H) and LL†w (D, H) of universal left multipliers of L†w (D, H) can be described as follows [6]: RL†w (D, H) = B ∈ L† (D, H) : B ∈ B(H), BD ⊂ D∗ , where D∗ ≡ D∗ (L† (D, H)) and
LL†w (D, H) = B ∗ : B ∈ RL†w (D, H) .
If D = D∗ , then L†w (D, H) is self-adjoint and both RL†w (D, H) and LL†w (D, H) are algebras (albeit not *-algebras in general). In order to introduce a topology on L†w (D, H), it is convenient to endow D with a topology which makes each A ∈ L†w (D, H) continuous. This can be done by defining the topology on D by the following family of seminorms: f 7→ kAf k ,
A ∈ L†w (D, H) .
This topology will be denoted in what follows by tL† . Clearly, tL† is the projective topology defined on D by L† (D, H) and for this reason each A ∈ L†w (D, H) is continuous from D into H. We will now define several topologies on L†w (D, H) and check whether L†w (D, H) is a topological partial *-algebra with respect to them. These definitions can be adapted to a general partial O*-algebra. The multipliers to be used in that case are, of course, the internal ones, for instance, RM = R(M) ∩ M, and the whole lattice structure is the same as usual. 5.2.1. Quasi-uniform topology, τ∗ It is defined by the set of seminorms A 7→ sup (kAf k + kA† f k) , f ∈N
N bounded in D[tL† ] .
By definition, the map A 7→ A† is continuous for τ∗ . If M ∈ F R , then the corresponding topology ρ∗M , as defined in Sec. 3, is defined by the set of seminorms A ∈ M 7→ sup (k(X A)f k + k(A† X † )f k) , f ∈N
X ∈ LM, N bounded in D[tL† ] .
We use the notation ρ∗M to remind the dependence on τ∗ . Analogously, if N ∈ F L , its topology will be called λ∗N . The following lemma, proved in [6, 7], shows that if L†w (D, H) is self-adjoint, the first two conditions of Definition 3.5 are fulfilled if L†w (D, H) is endowed with τ∗ .
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Lemma 5.2. If L†w (D, H) is self-adjoint, then the maps A 7→ X A and A 7→ A Y are τ∗ -continuous for all X ∈ LL†w (D, H) and Y ∈ RL†w (D, H). Thus L†w (D, H)[τ∗ ] is a topological partial *-algebra. By Lemmas 3.1 and 5.2, it follows that the topologies ρ∗L† (D,H) and λ∗L† (D,H) w
w
both coincide with τ∗ , a fact already proved in [6]. The following result was also proved, in a slightly different form, in [6]: Proposition 5.3. L†w (D, H) is complete in τ∗ . If M ∈ F R , then M is complete in ρ∗M . Similarly, if N ∈ F L , then N is complete in λ∗N . As for the third condition of Definition 3.5, the question as to whether RL†w (D, H) is ρ∗M -dense in each M ∈ F R (i.e. the tightness of L†w (D, H)[τ∗ ]) is still open. Let now M be an arbitrary partial *-algebra, and assume it is self-adjoint (which implies that L†w (D, H) is also self-adjoint, since D ⊂ D∗ ⊂ D∗ (M)). If the space of right multipliers RM contains only bounded operators (and then so does LM), the argument of [6, Proposition 5.4] goes through and shows that the three topologies ρ∗M , λ∗M and τ∗ coincide on M. It follows that the maps A 7→ X A and A 7→ A Y are τ∗ -continuous for all X ∈ LM and Y ∈ RM, and therefore: Proposition 5.4. Let M be a self-adjoint partial *-algebra, such that RM contains only bounded operators. Then M[τ∗ ] is a topological partial *-algebra. 5.2.2. Strong* topology, τs ∗ With an obvious generalization of the case of bounded operator algebras, the strong* topology on L† (D, H) is defined by the set of seminorms A 7→ kAf k + kA† f k ,
f ∈ D.
This topology plays a fundamental role in the study of unbounded commutants [12]. Here again, the map A 7→ A† is continuous by definition. As for the partial multiplications, if L†w (D, H) is self-adjoint it is readily checked that the maps A 7→ X A and A 7→ A Y are τs ∗-continuous for all X ∈ LL†w (D, H) and Y ∈ RL†w (D, H). Therefore, in this case, L†w (D, H)[τs ∗] is a topological partial *-algebra. In addition, the close connection between the strong* topology and commutants allows to get an interesting density theorem. We will show, in fact, that the set B = {B ∈ L† (D, H) : B ∈ B(H); BD ⊆ D} is dense in L†w (D, H)[τs ∗]. This result is a consequence of the following stronger statement. Proposition 5.5. The *-algebra F generated by the identity operator and the set F (D) of all finite rank operators in D is dense in L†w (D, H)[τs ∗] 00 , the weak unbounded Proof. By [35, Proposition 9], the τs ∗-closure of F is Fσσ bicommutant of F . For this reason it is enough to prove that Fσ0 consists only of multiples of the identity operator. Let X ∈ Fσ0 ; then X commutes (weakly) with
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each Pφ , φ ∈ D where Pφ ψ = (φ, ψ)φ. Therefore, Xφ =
1 (φ, Xφ) 1 XPφ φ = Pφ Xφ = φ. 2 2 kφk kφk kφk2
Now starting from two elements φ1 , φ2 ∈ D such that (φ1 , φ2 ) = 0 and using the linearity, it is easy to show that the coefficient (φ,Xφ) kφk2 does not depend on φ. Now, if L†w (D, H) is self-adjoint, then one has B = RL†w (D, H). In addition, it is proved in [35] that L†w (D, H)[τs ∗] is complete. We summarize the previous discussion in the following: Proposition 5.6. If L†w (D, H) is self-adjoint, then L†w (D, H)[τs ∗] is a complete topological partial *-algebra, and the sets LL†w (D, H) and RL†w (D, H) of left and right multipliers, respectively, are dense in L†w (D, H). If M is a general partial O*-algebra, then, even in the self-adjoint case, it is not necessarily a topological partial *-algebra when endowed with the topology τs ∗, since the multiplications may fail to be continuous. For this reason it is worth introducing a new topology that helps to overcome this difficulty. 5.2.3. Quasi strong* topology, τqs ∗ Let M be a partial O*-algebra and A ∈ M. For X ∈ RM, Y ∈ LM and f ∈ D, we put pf,X (A) = k(A X)f k + k(A† X)f k , Y
pf (A) = k(Y
A)f k
+ kY
† A fk .
The locally convex topology τqs ∗ defined by the set of seminorms {pf,X ,Y pf ; X ∈ RM, Y ∈ LM, f ∈ D} will be called the quasi strong*-topology on M. If M is self-adjoint, then both RM and LM are algebras and no problem arises with the associativity. Therefore we get Proposition 5.7. Let M be a self-adjoint partial O*-algebra. Then M[τqs ∗] is a topological partial *-algebra. 5.2.4. Weak topology, τw It is defined, on L†w (D, H), by the set of seminorms A→ 7 |hf |Agi| ,
f, g ∈ D .
In this case also, it is readily checked that the map A 7→ A† is continuous. If M ∈ F R , then the corresponding topology ρw M is defined by the set of seminorms A ∈ M → |hf |(X A)gi| , X ∈ LM, f, g ∈ D . It is very easy to prove the following:
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Lemma 5.8. If L†w (D, H) is self-adjoint, then the maps A → X A and A → A Y are τw -continuous for all X ∈ LL†w (D, H) and Y ∈ RL†w (D, H). From this lemma, we easily deduce: Proposition 5.9. If L†w (D, H) is self-adjoint, then L†w (D, H)[τw ] is a topological partial *-algebra. More generally, every self-adjoint partial O*-algebra M is a topological partial *-algebra with respect to the weak topology. The last statement results from the obvious fact that the weak topology on M is nothing but the topology induced on M by the weak topology of L†w (D, H). We will consider now the density condition of Definition 3.5 for the maximal partial O*-algebra L†w (D, H). In order to get results in this direction it is useful to have at hand some information on the τw -continuous functionals on L†w (D, H). In the very same way as in the case of weakly continuous functionals of B(H) (see e.g. [40, Chap. I]), we can prove the following: Proposition 5.10. For each τw -continuous linear functional F on L†w (D, H) there exist elements f1 , . . . , fn , g1 , . . . , gn in D such that F (X) =
n X
hfi |Xgi i ,
X ∈ L†w (D, H) .
i=1
Furthermore the vectors f1 , . . . , fn , g1 , . . . , gn can be chosen so that hfi |fj i = δij kfi k2 and hgi |gj i = δij kgi k2 . Making use of this result and of Lemma 3.6, we get easily that Proposition 5.11. Let M ∈ F R . Then, for each ρw M -continuous linear functional F on M there exist elements f1 , . . . , fn , g1 , . . . , gn in D and operators A1 , . . . , An in LM such that F (X) =
n X
hfi |Ai X)gi i ,
X ∈ M.
i=1
We now prove the following: Proposition 5.12. RL†w (D, H) is τw -dense in L†w (D, H). Proof. Were it not so, there would exist a non-zero τw -continuous linear functional F on L†w (D, H) which is zero all over RL†w (D, H). By Proposition 5.10, there exist elements f1 , . . . , fn , g1 , . . . , gn in D such that F (X) =
n X
hfi |Xgi i ,
X ∈ L†w (D, H) .
i=1
We choose the vectors f1 , . . . , fn , g1 , . . . , gn so that hfi |fj i = δij kfi k2 and hgi |gj i = δij kgi k2 .
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The finite rank operator X defined by Xϕ =
n X
hfj |ϕigj ,
ϕ∈D
j=1
clearly belongs to RL†w (D, H). Then we have * n + n n X X X kfi k2 kgi k2 = 0 . gi hfi |fj igj = F (X) = i=1
i=1
i=1
This implies f1 = · · · fn = g1 = · · · gn = 0. Therefore F = 0 and this contradicts the assumption. Unfortunately, the argument used in this proof cannot be adapted to show that R RL†w (D, H) is ρw M -dense in each M ∈ F , so that the tightness of this topological partial *-algebra remains to be proven. Other topologies can be introduced on a partial O*-algebra M mimicking the construction of the σ-weak and the σ-strong* topologies of bounded operator algebras. The only difference is that one should start by considering sequences {fi } P∞ of elements of D such that i=1 kfi k2 < ∞. The construction then goes through easily. For instance, the σ-strong* topology is defined by seminorms of the following type: !1/2 ∞ ∞ X X 2 † 2 , fi ∈ D , kfi k2 < ∞ . kAfi k + kA fi k A 7→ i=1
i=1
The results obtained in this case are, as for bounded operators, easy adaptations of those discussed above and we do not write them explicitly. In view of the results of Sec. 4, one might hope that abelian partial O*-algebras would be topological partial *-algebras, possibly even tight ones. However, this is not the case, as shown by the following counterexample. Let T be a maximal symmetric operator and D = D(T n ), n < ∞. Then [9, 11] the partial O*-algebra generated by T [1] = T D is the set Pn (T [1] ) of polynomials of degree at most n, powers being defined as T [n] = T [1] T [1] · · · T [1]. This is an abelian, finite dimensional partial O*-algebra. The partial multiplication is the usual weak multiplication and P1 P2 is well defined iff deg(P1 ) + deg(P2 ) 6 n. Thus, if Pj has degree j, M (Pj ) = Pn−j (T [1] ), so that the set of multiplier spaces is the finite scale P0 ⊂ P1 ⊂ · · · ⊂ Pn , Pj ' Cj+1 . In particular, RPn = P0 = C, which of course cannot be dense in any Pj . Thus Pn (T [1] ) is a (trivial) nontight topological partial *-algebra. Additional examples of the same nature may be found in [11]. Remark. Once we have endowed D with the topology tL† , it is natural to consider L† (D, H) as a subspace of L(D, D0 ) where D0 is the conjugate dual of D endowed with the strong dual topology t0L† . In this case if A, B ∈ L† (D, H) then the product A · B always exists in L(D, D0 ). Indeed, each A ∈ L† (D, H) has an
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b (the transposed map of A† ) which is continuous from H into D0 . Then extension A b A · B is defined by A · Bf = A(Bf ), f ∈ D. The definition of the multiplication · comes directly from the duality. This fact, together with Eq. (5.15) shows that if A B is also well defined, then necessarily A B = A · B. (This is reminiscent of the notion of weak derivative in L2 : given f ∈ L2 , its derivative exists always as a tempered distribution f 0 ∈ S 0 , but f belongs to the (Hilbert space) domain of d/dx only if f 0 ∈ L2 .) One can go one step further if D = D∞ (H), for some self-adjoint operator H > 1. Then one may interpolate between D and D0 by the Hilbert scale {Hn , n ∈ Z}, as discussed in Subsec. 5.1. The result is the same, the partial multiplication on L(D, D0 ) defined by continuous factorization through the spaces Hn coincides again with the weak partial multiplication . 6. Outcome The definition of topological partial *-algebra that emerges from this study looks quite natural, and fits well with all the examples we have given. In the abelian cases, where the partial multiplication is pointwise multiplication or convolution of functions, one even gets tight topological partial *-algebras. In the more interesting case of partial *-algebras of operators, the definition still works, but the validity of the tightness condition is generally open. It is satisfied for the “nicest” infinite scale, namely that built on the powers of a self-adjoint operator, but it is not for a finite scale in general. In fact, it is not clear how much this condition is needed. It will obviously play a role in the definition of representations by a suitable version of the GNS construction [10]. When it is satisfied, it may offer interesting approximation procedures, following the standard pattern of functional analysis. Of course, many open questions remain, in particular for partial O*-algebras. However, as we emphasized in the introduction, this paper is only a first step toward a general theory. Our aim was to find a structure suitable for as many significant examples as possible, and that has been obtained. But presumably the resulting framework is too general, and one ought to specialize it to particular cases. Clearly, more experience in this direction is needed before significant progress can be made. Acknowledgements This work was performed in the Institut de Physique Th´eorique, Universit´e Catholique de Louvain, and the Istituto di Fisica dell’ Universit` a di Palermo. We thank both institutions for their hospitality, as well as financial support from CGRI, Communaut´e Fran¸caise de Belgique, and Ministero degli Affari Esteri, Italy. We also thank the referee for his constructive criticisms, which have notably improved the paper, in particular concerning the contents of Sec. 5.2. References [1] J.-P. Antoine and A. Grossmann, “Partial inner product spaces. I. General properties. II. Operators”, J. Funct. Anal. 23 (1976) 369–378, 379–391. [2] J.-P. Antoine, “Partial inner product spaces. III. Compatibility relations revisited”, J. Math. Phys. 21 (1980) 268–279.
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[3] J.-P. Antoine, “Partial inner product spaces. IV. Topological considerations”, J. Math. Phys. 21 (1980) 2067–2079. [4] J.-P. Antoine and K. Gustafson, “Partial inner product spaces and semi-inner product spaces”, Adv. in Math. 41 (1981) 281–300. [5] J.-P. Antoine and W. Karwowski, “Interpolation theory and refinement of nested Hilbert spaces”, J. Math. Phys. 22 (1981) 2489–2496. [6] J.-P. Antoine and W. Karwowski, “Partial *-algebras of closed linear operators in Hilbert space”, Publ. RIMS, Kyoto Univ. 21 (1985) 205–236; Add./Err. ibid. 22 (1986) 507–511. [7] J.-P. Antoine and F. Mathot, “Partial *-algebras of closed operators and their commutants. I. General structure”, Ann. Inst. H. Poincar´ e 46 (1987) 299–324. [8] J.-P. Antoine, F. Mathot and C. Trapani, “Partial *-algebras of closed operators and their commutants. II. Commutants and bicommutants”, Ann. Inst. H. Poincar´ e 46 (1987) 325–351. [9] J.-P. Antoine, A. Inoue and C. Trapani, “Partial *-algebras of closable operators. I. The basic theory and the abelian case”, Publ. RIMS, Kyoto Univ. 26 (1990) 359–395. [10] J.-P. Antoine, A. Inoue and C. Trapani, “Partial *-algebras of closable operators. II. States and representations of partial *-algebras”, Publ. RIMS, Kyoto Univ. 27 (1991) 399–430. [11] J.-P. Antoine, A. Inoue and C. Trapani, “On the regularity of partial O*-algebras of generated by a closed symmetric operator”, Publ. RIMS, Kyoto Univ. 28 (1992) 757–774. [12] J.-P. Antoine, A. Inoue and C. Trapani, “Partial *-algebras of closable operators: A review”, Reviews Math. Phys. 8 (1996) 1–42. [13] J.-P. Antoine, “Quantum mechanics beyond Hilbert space. Applications to scattering theory”, in Quantum Theory in Rigged Hilbert Spaces — Semigroups, Irreversibility and Causality, pp. 3-33; eds. A. B¨ ohm, H. D. Doebner and P. Kielanowski, Lect. Notes in Physics, vol. 504 Springer, Berlin, 1998. [14] J.-P. Antoine, F. Bagarello and C. Trapani, “Extension of representations in quasi*algebras”, Ann. Inst. H. Poincar´ e 69 (1998) 241–264. [15] F. Bagarello and C. Trapani, “States and representations of CQ*-algebras”, Ann. Inst. H. Poincar´ e 61 (1994) 103–133. [16] F. Bagarello and C. Trapani, “CQ*-algebras: Structure properties”, Publ. RIMS, Kyoto Univ. 32 (1996) 85–116. [17] F. Bagarello and C. Trapani, “The Heisenberg dynamics of spin systems: A quasi *-algebras approach, J. Math. Phys. 37 (1996) 4219–4234. [18] F. Bagarello, A. Inoue and C. Trapani, Standard CQ*-algebras (in preparation). [19] F. Bagarello and C. Trapani, “Lp spaces as quasi *-algebras”, J. Math. Anal. Appl. 197 (1996) 810–824. [20] F. Bagarello and C. Trapani, “CQ*-algebras of operators in scales of Hilbert spaces” (in preparation). [21] J. J. Benedetto, C. Heil and D. F. Walnut, “Differentiation and the Balian–Low theorem”, J. Fourier Anal. Appl. 1 (1995) 355–402. [22] J. Bergh and J. L¨ ofstr¨ om, Interpolation Spaces, Springer, Berlin, 1976. [23] H. J. Borchers, “Decomposition of families of unbounded operators”, in RCP 25 (Strasbourg) 22 (1975) 26–53; also in Quantum Dynamics: Models and Mathematics, ed. L.Streit, Acta Phys. Austr. Suppl. 16 (1976) 15. [24] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I, II, Springer-Verlag, Berlin, 1979. [25] H. W. Davis, F.J. Murray and J.K. Weber, “Families of Lp spaces with inductive and projective topologies”, Pacific J. Math. 34 (1970) 619–638; “Inductive and projective limits of Lp spaces”, Portug. Math. 31 (1972) 21–29.
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[26] G. Epifanio, T. Todorov and C. Trapani, “Complete sets of compatible nonself-adjoint observables”, Helv. Phys. Acta 65 (1992) 1–10; “Complete sets of compatible nonselfadjoint observables: An unbounded approach”, J. Math. Phys. 37 (1996) 1148–1160. [27] J. J. F. Fournier and J. Stewart, “Amalgams of Lp and `q ”, Bull. Amer. Math. Soc. 13 (1985) 1–21. [28] R. Haag, Local Quantum Physics, Springer-Verlag, Berlin, 1993; 2nd. ed. 1996. [29] F. Holland, “Harmonic analysis on amalgams of Lp and `q ”, J. London Math. Soc. 10(2) (1975) 295–305. [30] S. S. Horuzhy and A. V. Voronin, “Field algebras do not leave field domains invariant”, Commun. Math. Phys. 102 (1988) 687–692. [31] K.-D. K¨ ursten, “The completion of the maximal Op*-algebra on a Fr´ echet domain”, Publ. RIMS, Kyoto Univ. 22 (1986) 151–175; “On topological linear spaces of operators with a unitary domain of definition”, Wiss. Z. Univ-Leipzig, Math.-Naturwiss. R. 39 (1990) 623–655. [32] G. Lassner, “Quasi-uniform topologies on local observables”, Mathematical Aspects of Quantum Field Theory, Acta Univ. Wrat. No. 519 (Proc. Karpacz 1978), pp. 43-60; eds. A. P¸ekalski and T. Paszkiewicz, Wroclaw, 1979. [33] G. Lassner, “Topological algebras and their applications in Quantum Statistics”, Wiss. Z. KMU-Leipzig, Math.-Naturwiss. R. 30 (1981) 572–595. [34] G. Lassner, “Algebras of unbounded operators and quantum dynamics”, Physica 124A (1984) 471–480. [35] F. Mathot, “Topological properties of unbounded bicommutants”, J. Math. Phys. 26 (1985) 1118–1124. [36] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I: Functional Analysis, Academic Press, New York and London, 1972. [37] A. Russo and C. Trapani, “Quasi *-algebras and multiplication of distributions”, J. Math. Anal. Appl. 215 (1997) 423–442. [38] H. H. Schaefer, Topological Vector Spaces, Springer-Verlag, Berlin, 1971. [39] K. Schm¨ udgen, Unbounded Operator Algebras and Representation Theory, AkademieVerlag, Berlin, 1990. [40] S. Stratila and L. Zsido, Lectures on Von Neumann Algebras, Abacus Press, Tunbridge Wells (England), 1979. [41] W. Thirring and A. Wehrl, “On the mathematical structure of the B.C.S.-model. I, II”, Commun. Math. Phys. 4 (1967) 303-314; 7 (1968) 181–189. [42] C. Trapani, “Quasi *-algebras of operators and their applications”, Rev. Math. Phys. 7 (1995) 1303–1332. [43] N. Wiener, “On the representation of functions by trigonometric integrals”, Math. Z. 24 (1926) 575–616; “Tauberian theorems”, Annals of Math. 33 (1932) 1–100. [44] A. C. Zaanen, Integration, 2nd. ed., Chap. 15; North-Holland, Amsterdam, 1961.
FORMAL AND ANALYTIC RIGIDITY OF THE WITH ALGEBRA∗ LUCA GUERRINI Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA Received 20 March 1998 A family of deformations Wf of the Witt algebra W parametrized by the space E of even polynomials with vanishing constant terms is defined. The existence of an isomorphism c where b refers to suitable completions of W, is proved. A relation between Wf cf w W, W and Krichever–Novikov algebras of genus 0 and 1 is given.
1. Introduction One of the more prominent and persuasive trend in mathematics has to do with deformations. Intuitively a deformation [3, 4] of a mathematical object is a family of the same kind of objects depending on same parameter(s). Suppose we have a Lie algebra G (over C) which is rigid. This means that for any family (Gt1 ,...,tk ) of deformations of G (for t1 = · · · = tk = 0, Gt1 ,...,tk = G) there is a formal isomorphism Gt1 ,...,tk w G[[t1 , . . . , tk ]] over C[[t1 , . . . , tk ]]. It makes sense to ask if this is also true at the analytical level, that is, if it can be shown that at least for small |t1 |, . . . , |tk |, Gt1 ,...,tk w G over C. If this is true, then we may say Gt1 ,...,tk is holomorphically rigid. The question whether rigidity implies holomorphic rigidity deserves more attention than it has received. In this paper, we examine various aspects of this question when G = W (the Witt algebra) and for certain special families of deformations (Wf ), where f varies over an infinite dimensional linear space. c where b refers to suitable completions of W. In Sec. 2, cf w W, We prove W we discuss this isomorphism in the case of formal completion. In Sec. 3, in the case of analytic completion. In Sec. 4, we show how our family (Wf ) is related to Krichever–Novikov algebras [6] (KN -algebras) of genus 0 and 1. An application to the case of genus 0 is given. ∗ This
paper is substantially the thesis of the author done in U CLA (Spring 1998) under the supervision of Prof. V. S. Varadarajan. 303 Reviews in Mathematical Physics, Vol. 11, No. 3 (1999) 303–320 c World Scientific Publishing Company
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2. Formal Theory We are going to introduce a family of Lie algebras Wf parametrized by the space E of even polynomials f with vanishing constant terms: f = f1 z 2 + f2 z 4 + · · · + fN z 2N
(fj ∈ C) .
The Witt algebra W is known to be rigid; this fact is mentioned (with only some indications of proof) in [2]. One can thus expect to reduce Wf to W by a formal transformation. We shall show this explicitly here and build the formal (equivalence) transformation as a product :
∞ Y
(1 + Tk ) :
k=1
where : : means the product is taken as (1 + T1 )(1 + T2 ) · · · . QN Our requirement heuristically is that : k=1 (1 + Tk ) : reduces Wf to a Lie algebra WfN whose bracket is ≡ bracket in W modulo term of degree ≥ N + 1 in f and f 0 . The rigidity of W will guarantee that the choices of Tk are always possible. But the Tk are not unique and it is essential, when discussing convergence Q∞ properties of : k=1 (1 + Tk ) : that the Tk have to be chosen carefully. The bulk of the work in this section is devoted to a specific explicit choice of the Tk . It will c of W and W cf however turn out that we have to work with formal completions W of Wf to get the expected isomorphism. Definition 1. Let W be the Lie algebra of complex valued functions on the circle expandable into a finite Fourier series with the bracket [f, g] = f g 0 − f 0 g , where 0 denotes differentiation with respect to the angle parameter θ on the circle. d to the function f (z), Denote z = exp(2 π i θ). If we assign the vector field f (z) dz then the above bracket gives an isomorphism of W with the Lie algebra of vector d , f (z) ∈ C[z −1 , z]. The elements b li = z i+1 , i ∈ Z, constitute a basis fields f (z) dz lj ] = (j − i)b li+j . W is usually in W, with the bracket given by the formula [b li , b called the Witt algebra by mathematicians and the centerless Virasoro algebra by physicists. c be the space of all formal Laurent series with finitely Definition 2. Let W c (same many negative powers of z. The Lie algebra structure of W extends to W definition). c is an adic completion of W. [., .] is continuous in the adic Remark 3. W topology. Note that if g is any polynomial in z without a constant term, (1 + g)−1 ∈ c W.
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c Lemma 4. For u, v, g, h ∈ W, [ug, vh] = uv[g, h] + [u, v]gh .
Proof. Straightforward.
c i.e. Eb is the space of all fb ∈ W c of Let us define Eb to be the closure of E in W, the form fb = fb1 z 2 + fb2 z 4 + · · · . Definition 5. Let with bracket given by [g, h]f [g, h]f [g, h]f
cf ) be W (resp. W) c b Let Wf (resp. W f ∈ E (resp. f ∈ E). = (1 + f )[g, h]
(g, h odd) ,
= [g, h]
(g, h even) ,
1 = (1 + f )[g, h] + f 0 gh 2
(g even, h odd) .
cf are Lie algebras is straightforward using The verification that Wf and W Lemma 4. c× W c→W c such that Definition 6. Let ω1 be the continuous bilinear map W (g, h odd) , ω1 (g, h) = f [g, h] (g, h even) , ω1 (g, h) = 0 1 ω1 (g, h) = f [g, h] + f 0 gh (g even, h odd) . 2 It is an easy calculation that ω1 is a cocycle. We shall see that there is a c→W c such that δT1 (g, h) = ω1 (g, h) for all g, h ∈ W. c continuous linear map T1 : W c→W c be defined by Proposition 7. Let T1 : W (g odd) , T1 g = −f g T1 g = − 1 f g (g even) , 2 and extended by continuity. Then δT1 (g, h) = ω1 (g, h),
c g, h ∈ W.
Proof. Straightforward. Note that if φ1 = 1 + T1 , then [g, h]1f = φ−1 1 [φ1 g, φ1 h]f = [g, h] + ω2 (g, h) + · · ·
c g, h ∈ W
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where the term ω2 consists of elements of degree 2, and . . . of degree ≥ 3 in f and f 0 . This method can be continued to all orders. The point of what follows is that certain specific explicit choices of Tk have to be made, which will be essential in the analytic part of this paper. Theorem 8. One can determine scalars (ak )k>1 and (bk )k>1 uniquely from the following power series identities (in the variable z) (1 + z)
∞ Y
(1 + ak z k ) = 1 ,
(1)
k=1 ∞ Y (1 + bk z k )2 = 1. 1 + ak z k
(2)
k=1
c→W c be the continuous linear map such that Then, ∀ k > 1, let Tk : W ( ak if g is odd , k Tk g = ck,g f g, ck,g = bk if g is even . ck be the Lie algebra structure of W c defined inductively Let φk = 1 + Tk and let W f in k by k−1 [g, h]kf = φ−1 k [φk g, φk h]f
= [g, h] + ωk+1 (g, h) + · · ·
c, g, h ∈ W
where ωk+1 (g, h) denotes all the elements of degree k+1 in f and f 0 in the expression [g, h]kf . Then Tk+1 satisfies the equation δTk+1 (g, h) = ωk+1 (g, h)
c. g, h ∈ W
Proof. The proof is by induction on k. It needs some preparation which we formulate as lemmas. Lemma 9. (i)
ak =
X k (−1)k k d + add (−1) d k k
∀k > 1,
16d
(ii)
bk =
X k k d (−1)k + bdd (−1) d 2k k
∀k > 1.
16d
Proof. (i) (resp. (ii)) easily follows by taking logs on both sides of (1) (resp. (2)).
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c Let φ1 = 1 + T1 . Let W c1 We shall prove that δT2 (g, h) = ω2 (g, h)∀ g, h ∈ W. f c be the Lie algebra obtained by giving the bracket formula on W as [g, h]1f = φ−1 1 [φ1 g, φ1 h]f
c. g, h ∈ W
We shall start proving the following lemma. Lemma 10. We have (i)2 (ii)2 (iii)2
[g, h]1f = (1 + a1 f )(1 + f )[g, h] (1 + b1 f )2 [g, h] [g, h]1f = 1 + a1 f [g, h]1f = (1 + a1 f ) (1 + f ) f0 (a1 − b1 ) f 0 + × [g, h] + gh 2(1 + f ) (1 + a1 f )(1 + b1 f )
(g, h odd), (g, h even),
(g even, h odd).
Proof. The proof is straightforward. It uses the fact φ1 preserves parity, Lemma 4 and Definition 5. Here and in what follows, all the infinite sums are convergent in the adic topology since f s → 0 as s → ∞. We should also keep in mind the remark made earlier that if k(z) = k1 z 2 + k2 z 4 + · · · , then (1 + k)−1 is a c well-defined element of W. Corollary 11. Let ω2 denote the terms of degree 2 in f and f 0 in the expression [g, h]1f = φ−1 1 [φ1 g, φ1 h]f
c. g, h ∈ W
Then (i) ω2 (g, h) = −a2 f 2 [g, h] (ii) ω2 (g, h) = (a2 − 2b2 )f 2 [g, h] (iii) ω2 (g, h) = −a2 f 2 [g, h] + (−a2 + b2 )(f 2 )0 gh
(g, h odd), (g, h even), (g even, h odd).
Proof. (i) and (ii) follow from Lemmas 9 and 10. From (iii)2 of Lemma 10, we have f0 (a1 − b1 ) f 0 gh + gh . [g, h]1f = (1 + a1 f ) (1 + f ) [g, h] + (1 + a1 f )(1 + b1 f ) | {z } 2(1 + f ) {z } | {z } | (a) (b)
(c)
Using Lemma 9, we have that the terms of degree 2 in f and f 0 of this expression are given by 1 1 −a2 f 2 [g, h] − (f 2 )0 gh + (−a2 + b2 )(f 2 )0 gh + (f 2 )0 gh 4 4 = −a2 f 2 [g, h] + (−a2 + b2 )(f 2 )0 gh . Therefore ω2 (g, h) = −a2 f 2 [g, h] + (−a2 + b2 )(f 2 )0 gh.
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c Let now assume It is now straightforward that δT2 (g, h) = ω2 (g, h) g, h ∈ W. that the statement of the theorem is true for k. We shall prove it is true for k + 1. ck be the Lie algebra obtained by giving the Lemma 12. Let φk = 1 + Tk . Let W f c as bracket formula on W k−1 [g, h]kf = φ−1 k [φk g, φk h]f
Then (i)k (ii)k (iii)k
[g, h]kf = [g, h]kf = [g, h]kf = (
Qk r=1
Qk
r=1
Qk
r=1
× [g, h] +
c. g, h ∈ W
(1 + ar f r ) (1 + f ) [g, h] (1 + br f r )2 1 + ar f r [g, h] (1 + ar f r ) (1 + f ) k X (as − bs )(f s )0 f0 2(1 + f )
s=1
(1 + as f s )(1 + bs f s )
(g, h odd) , (g, h even), !
) gh
(g even, h odd).
Proof. The proof will use the inductive hypothesis, the fact φk preserves parity, Lemma 4, Definition 5 and the following fact k−1 [g, h]kf = φ−1 k [φk g, φk h]f −1 −1 = φ−1 k φk−1 · · · φ1 [φ1 · · · φk−1 φk g, φ1 · · · φk−1 φk h]f # " k k k Y Y Y −1 = (1 + Tr ) (1 + Tr )g, (1 + Tr )h . r=1
r=1
r=1
f
(i)k and (ii)k are straightforward. The proof of (iii)k easily follows from the following established fact # " k k k Y Y Y r r (1 + br f ) g, (1 + ar f ) h = (1 + f ) (1 + ar f r )(1 + br f r ) r=1
r=1
( ×
f
X (as − bs ) (f s )0 f0 + 2(1 + f ) s=1 (1 + as f s )(1 + bs f s ) k
[g, h] +
r=1
!
) gh
.
Corollary 13. Let ωk+1 denote the terms of degree k + 1 in f and f 0 in the expression k c. g, h ∈ W [g, h]kf = φ−1 k [φk g, φk h]f Then (i) ωk+1 (g, h) = −ak+1 f k+1 [g, h] (ii) ωk+1 (g, h) = (ak+1 − 2bk+1 )f k+1 [g, h] (iii) ωk+1 (g, h) = −ak+1 f k+1 [g, h]+(−ak+1 +bk+1 )(f k+1 )0 gh
(g, h odd), (g, h even), (g, even, h odd).
FORMAL AND ANALYTIC RIGIDITY OF THE WITT ALGEBRA
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Proof. (i) It follows from (i)k of Lemma 12 that [g, h]kf =
k Y
(1 + ar f r ) (1 + f ) [g, h]
r=1
= [g, h] + ωk+1 (g, h) + ωk+2 (g, h) + · · · where ωr (g, h) is of the form Pr [g, h], with Pr = pr f r , pr ∈ C. Now taking logs on both sides we have k X
log(1 + ar f r ) + log(1 + f ) = log(1 + Pk+1 + · · · ) .
r=1
The elements of degree k + 1 in f of the left-hand side are given by (−1)k + k+1
X
(−1)
k+1 d −1
k+1
ad d
16d
d k+1 f k + 1
i.e. by −ak+1 f k+1 (Lemma 9). Therefore we have ωk+1 (g, h) = −ak+1 f k+1 [g, h]. (ii) It follows from (ii)k of Lemma 12 that [g, h]kf
k Y (1 + br f r )2 = [g, h] 1 + ar f r r=1
= [g, h] + ωk+1 (g, h) + ωk+2 (g, h) + · · · where ωr (g, h) is of the form Pr [g, h], with Pr = pr f r , pr ∈ C. Now taking logs on both sides we have k X
2 log(1 + br f r ) −
r=1
k X
log(1 + ar f r ) = log(1 + Pk+1 + · · · )
r=1
the elements of degree k + 1 in f of the left-hand side are given by
X
(−1)
k+1 d −1
k+1 k+1 d d 2bd − ad
16d
d k+1 , f k + 1
i.e. by (ak+1 − 2bk+1 )f k+1 (Lemma 9). So ωk+1 (g, h) = (ak+1 − 2bk+1 ) f k+1 [g, h]. (iii) It follows from (iii) k of Lemma 12 that [g, h]kf
k 1 f 0 gh X (as − bs ) (f s )0 gh = (1 + ar f ) (1 + f ) [g, h] + + s )(1 + b f s ) | {z } 2 1 + f (1 + a f s s r=1 | {z } |s=1 (a) {z } k Y
r
(b)
= [g, h] + ωk+1 (g, h) + ωk+2 (g, h) + · · ·
(c)
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where ωr (g, h) is of the form pr f r [g, h] + qr f r−1 f 0 gh, pr , qr ∈ C. Since we know from (i) that k Y (1 + ar f r )(1 + f ) = 1 − ak+1 f k+1 + · · · r=1
we have that the terms of degree k + 1 in f and f 0 in [g, h]kf are given by X k+1 k+1 k+1 k+1 k k1 0 d d [g, h] + (−1) f f gh + b d − ad (−1) d d f k f 0 gh −ak+1 f 2 16d
that is −ak+1 f
k+1
(−1)k [g, h] + (−1) f f gh + (k + 1)(−ak+1 + bk+1 ) − f k f 0 gh 2 2 k
k1
0
by Lemma 9. Therefore ωk+1 (g, h) = −ak+1 f k+1 [g, h] + (−ak+1 + bk+1 )(f k+1 )0 gh. c So the It is now straightforward to show δTk+1 (g, h) = ωk+1 (g, h) g, h ∈ W. statement of Theorem 8. Theorem 14. If Sf =:
∞ Y
(1 + Tk ) :
k=1
c→W cf is an isomorphism. then Sf : W Proof. We shall prove it is a Lie morphism. From k−1 [g, h]kf = φ−1 k [φk g, φk h]f
it follows
[g, h]kf ≡ [g, h] mod (z 2(k+1) ) " k # k k Y Y Y k (1 + Tr )[g, h]f = (1 + Tr )g, (1 + Tr )h r=1
r=1
r=1
f
Letting k → ∞ Sf [g, h] = [Sf g, Sf h]f , where Sf = lim (1 + T1 ) · · · (1 + Tk ) (limit in the adic topology) . k→∞
Write Sf =:
∞ Y
(1 + Tk ) :
k=1
Then
c w W cf . Sf : W
FORMAL AND ANALYTIC RIGIDITY OF THE WITT ALGEBRA
311
Remark 15. The case f = fb2 z −2 + fb4 z −4 + · · · is analogous. For this the adic completion of W will be the set of all Laurent series with only finitely many positive powers of z. There is no change in any of the arguments. 3. Analytic Theory cf parametrized In the previous section we introduced a family of Lie algebras W b by the space E of holomorphic Laurent series with no constant terms and proved c Everything was formal. cf → W. the existence of an isomorphism W Now we define the algebra ( ) X c∞ = C ∞ (S 1 ) = g = gn z n , |gn | = O(|n|−M ) for all M > 0 W n∈Z
with bracket defined by [f, g] = f g 0 − f 0 g . c∞ as the algebra having the same underlying vector space, Similarly we define W f c∞ , but with Lie algebra structure defined by namely W [g, h]f = (1 + f )[g, h] [g, h]f = [g, h] [g, h]f = (1 + f )[g, h] + 1 f 0 gh 2
(g, h odd) , (g, h even) , (g even, h odd) .
c∞ which are even. We c∞ are now defined for all f ∈ W Note that the algebras W f prove now that the absolute value of the coefficients ak , bk of the maps Tk defined in Theorem 8 are bounded above by 1. Lemma 16. Let (ak )k>1 be defined by (1). Then −1 ak = 1 0
if k = 1 , if k = 2r , r > 1 , if 1 < k 6= 2r .
In particular, |ak | 6 1 for all k > 1. Proof. It follows from Lemma 9 (i) and induction. Lemma 17. Let (bk )k>1 be defined by (2). Then 1 , (i) If k is odd, then bk < 0 and |bk | 6 2k (ii) If k is even, then bk > 0 and |bk | 6 k1 .
In particular |bk | < 1 for all k > 1.
312
L. GUERRINI
Proof. In both cases we use Lemma 9 (ii). Let k be an odd number. The proof 1 if 1 6 d < k, d|k. Note that k/d is is by induction. Assume bd < 0 and |bd | 6 2d an odd number. Therefore bk = −
X k d 1 + |bd | d . 2k k 16d
Moreover since
X
k
|bd | d
1 d < k 2k
16d
Let now k be an even number, i.e. k = 2r , r > 1. Induction on k, the fact that if d|k then k/d is an even number imply bk =
X kd 1 1 1 + k + bdd . 2k 2 k k 26d
This is clearly > 0. It also follows that |bk | 6 k1 . Let now k = 2r l, l odd > 1. Again the proof is by induction. First of all, if d|k then d = 2s e 0 6 s 6 r, 1 6 e 6 l, e|l, (s, e) 6= (r, l). Therefore we have bk = where A=
X
l
(b2r e ) e
16e
1 l e
,
bk = 1 2k ,
C=
X 06s
We can write
Since A, C are > 0, A <
1 −A+C, 2k r−s l e
(b2s e )2
1 2r−s el
.
1 −A+C. 2k
it follows bk 0. We also have bk 6 k1 .
We need some functional analytic lemmas. Lemma 18. Let B be a Banach space, Tk : B → B bounded linear maps such that X ||Tk || < ∞ . k
313
FORMAL AND ANALYTIC RIGIDITY OF THE WITT ALGEBRA
Then ∃ S : B → B, a bounded linear operator, such that
N
Y
(1 + Tk ) : −S → 0 (N → ∞) ,
:
k=1
where :
N Y
(1 + Tk ) : = (1 + T1 )(1 + T2 ) · · · (1 + TN ) .
k=1
We write
∞ Y
S=:
(1 + Tk ) :
k=1
Proof. Let tk = ||Tk ||. We have :
N Y
(1 + Tk ) : = 1 +
∞ X
) L(N n ,
n=1
k=1
where for n ≥ 1
X
) = L(N n
Tk1 Tk2 . . . Tkn .
1≤k1 <···
Note that if n > N, Ln ) ||L(N n ||
= 0. X
≤
tk1 . . . tkn
k1 <···
So the series
X
tn ≤ n!
t=
X
! tk
.
k
) L(N n
n
converges uniformly with respect to N in the norm topology. Hence we may pass to the limit under the summation sign, as N → ∞ X tn (N ) Tkn . . . Tk1 . ||Ln || ≤ Ln → Ln = n! k1 <...
This proves :
N Y
(1 + Tk ) : → 1 +
∞ X
Ln
n=1
k=1
in the norm topology.
Corollary 19. Suppose each Tk analytically depends on an additional parameter s, s ∈ neighborhood of origin of Cn . Assume t = sups ||Tk,s ||. If X tk < ∞ k
then S = Ss defined above is analytic in s.
314
L. GUERRINI
Proof. Since everything converges uniformly with respect to s, this is a consequence of Morera’s theorem. Lemma 20. Let notations and assumptions be as in Lemma 18. Suppose in addition that 1 + Tk is invertible for each k. Then S above is invertible and S −1 = lim (1 + TN )−1 . . . (1 + T1 )−1 N →∞
the limit being in the norm topology. Proof. Observe that tk = ||Tk || < 1 if k > k0 . Since each 1 + Tk is invertible we can omit the first k0 factors. In other words we may assume that tk < 1 for all k. Write UN = (1 + TN )−1 . . . (1 + T1 )−1 .
SN = (1 + T1 ) . . . (1 + TN ), We have
S N U N = U N SN = I . It is thus sufficient to prove that UN is convergent in the norm topology. If U = lim UN , then N
SU = U S = I . We now have UN = (1 − TN + TN2 − · · · ) · · · (1 − T1 + T12 − · · · ) ∞ X
= 1+
Mn(N ) ,
n=1
where
X
Mn(N ) =
(−1)kN TNkN . . . (−1)k1 T1k1
k1 +···+kN =n ki ≥0
So X
||Mn(N ) || ≤
tkNN . . . tk11
k1 +···+kN =n ki ≥0
X
≤
. . . tk22 tk11 = τn say .
k1 +···=n ki ≥0
But 1+
∞ X
1 < ∞. τn = Q (1 − tk ) n=1 k
Hence
X n
Mn(N )
FORMAL AND ANALYTIC RIGIDITY OF THE WITT ALGEBRA
315
converges in the norm topology uniformly with respect to N . So we can pass to the limit N → ∞ under the summation. X . . . (−1)k1 T1k1 = Mn say . Mn(N ) → k1 +···=n ki ≥0
||Mn || ≤ τn So 1+
X
Mn(N ) → 1 +
n
∞ X
Mn = U
n=1
in the norm topology.
Corollary 21. Let conditions and assumptions be as in Corollary 19 and Lemma 20. Then Ss−1 is analytic in s. Let F be a Frechet space with topology determined by seminorms || · ||(m) , || · ||(1) ≤ || · ||(2) ≤ · · ·
(B (m) = completion of F by || · ||(m) ).
We shall assume that the || · ||(m) are actually norms; this is not a big loss of generality and it is sufficient for the application we have in mind. We can identify B (n) inside B (m) if n > m. So F ⊂ · · · ⊂ B (n) ⊂ B (n−1) ⊂ · · · ⊂ B (1) F=
∞ \
B (n)
n=1
Let Tk : F → F be a continuous linear operator. We make the following assumptions. (a) Tk extends to a continuous map B (m) → B (m) and 1 + Tk : B (m) → B (m) is invertible. P ||Tk ||(m) < ∞ for each fixed m. (b) k
Lemma 22. Under these conditions S =:
∞ Y k=1
(1 + Tk ) :
=def lim (1 + T1 ) . . . (1 + TN ) N →∞
(limit in the strong operator topology) is a continuous invertible map S:F →F.
316
L. GUERRINI
Proof. Let SN = (1 + T1 ) . . . (1 + TN ) . By (a), (b) and Lemma 20, for fixed m SN : B (m) → B (m) converges in norm to a continuous linear map S =:
∞ Y
(1 + Tk ) : B (m) → B (m) .
k=1
In other words, for f ∈ F, Sf ∈ B (m) ∀ m, hence Sf ∈ F. So S:F →F is a continuous linear map. Injectivity: Since S : B (m) → B (m) if f ∈ F and Sf = 0, ||f ||(m) = 0 ∀ m. So f = 0. Surjectivity: Let g ∈ F. ∃! hm ∈ B (m) such that Shm = g. For m < n, we have B (n) ⊂ B (m) and h(n) = h(m) under this map. This means h(1) = h(2) = h(3) = · · · , i.e. h = h(1) ∈ F. So ∃ h ∈ F such that h(n) is the image of h in B (m) . But then Sh = g. Corollary 23. Let conditions and assumptions be as in Lemma 22. Then Ss and Ss−1 are both analytic in s. Let F = C ∞ (S 1 ). For f ∈ C ∞ (S 1 ) X X |n|r |fb(n)|, ||f ||(m) =
||f ||(0) = ||fb||1
0≤r≤m n
P
(f = n pn z n means fb(n) = pn ). All the conditions are satisfied. The || · ||(m) impose the same topology as m d sup m f (eiθ ) (m = 0, 1, . . .) . dθ This is a consequence of Sobolev’s classical result. Now X |fcg(n)| ||f g||(0) = n
=
X
|(fb ∗ gb)(n)|
n
6
X n
|fb(n)|
X n
6 ||f ||(0) ||g||(0)
|b g(n)|
(l1 is a Banach algebra)
FORMAL AND ANALYTIC RIGIDITY OF THE WITT ALGEBRA
317
Lemma 24. For any f ∈ C ∞ (S 1 ) and any r > 1, ∃ a constant Ar (f ) > 0 such that ||Dr f k ||(0) 6 Ar (f )k r (||f ||(0) )k−r ∀ k > r . Proof. We have, for any M functions u1 , . . . , uM , X r! (Ds1 u1 ) . . . (DsM uM ) . Dr (u1 , . . . , uM ) = s1 ! . . . sM ! s1 +···+sM =r sj ≥0
[This is easy by induction on M ] In particular, X Dr (f k ) = s1 +···+sk =r sj ≥0
r! (Ds1 f ) . . . (Dsk f ) . s1 ! . . . sk !
We assume k > r. Then in each (s1 , . . . , sk ), at least k − r of the sj have to be 0. So X r! (||f ||(r) )r (||f ||(0) )k−r ||Dr (f k )||(0) 6 s1 ! . . . sk ! s1 +···+sk =r sj ≥0
6 k r (||f ||(r) )r (||f ||(0) )k−r Theorem 25. The map Sf =
Q∞
k=1 (1
c∞ → W c∞ is an isomorphism + Tk ) : W f
if (i) sup |f | < 1 (ii) ||f ||(0) < 1. Proof. We recall that
( k
Tk g = ck,g f g,
ck,g =
ak
if g is odd ,
bk
if g is even .
where |ak | 6 1, |bk | < 1. We shall verify that the Tk satisfy the conditions discussed earlier. (i) Since multiplication by any element of C ∞ (S 1 ) defines a bounded operator on C ∞ (S 1 )(m) , each Tk is a bounded linear map C ∞ (S 1 )(m) → C ∞ (S 1 )(m) . In fact if g ∈ B (m) , then ||f g||(m) 6 L(f )||g||(m) for a suitable constant L(f ).
(0)
X r
r (0) t r−t
D u · D v ||D (uv)|| = t
06t6r
≤ 2r ||u||(r) ||v||(r) .
318
L. GUERRINI
Hence ||f g||(m) 6 2m (m + 1)||f ||(m) ||g||(m)
(3)
(ii) 1 + Tk is invertible for each k. We shall verify that 1 + Tk is multiplication by a gk ∈ C ∞ (S 1 ) with g1k ∈ C ∞ (S 1 ). Then (1 + Tk )−1 is multiplication by g1k , hence a bounded operator by (i). For this we need only check that gk is never 0 on S 1 . We shall show (4) inf |gk (eiθ )| > 0 θ
Since multiplication by f k preserves parity, we can check (4) separately for the odd and even case. For either of these gk = (1 + ck f k )
|ck | 6 1
and |gk (eiθ )| = |(1 + ck f k (eiθ )| > 1 − sup |f k (eiθ )| θ k
> 1 − (sup |f |) > 0 (since sup |f | < 1) In view of Lemma 18, 20 and 22 it remains only to check X ||Tk ||(m) < ∞ for each m. k
Now Tk g = ak f k P g + bk f k Qg (P (resp. Q) is the projection on the odd (even) parts). Therefore ||Tk g||(m) 6 ||f k P g||(m) + ||f k Qg||(m) 6 2m (m + 1)||f k ||(m) (||P g||(m) + ||Qg||(m) ) (by (3)) g(eiθ ) − g(e−iθ ) (P g)(eiθ ) = 2 iθ −iθ (Qg)(eiθ ) = g(e ) + g(e ) 2
However
so that ||P g||(m) 6 ||g||(m) ,
||Qg||(m) 6 ||g||(m)
Hence ||Tk g||(m) 6 2m+1 (m + 1)||g||(m) ||f k ||(m) so ||Tk ||(m) 6 2m+1 (m + 1)||f k ||(m) . Hence, for fixed m X X X ||Tk ||(m) 6 2m+1 (m + 1) ||Dr f k ||(0) < ∞ . k
06r6m k
We are done in view of Lemma 24. ∞
Let N = {f ∈ C (S ) : f even, sup |f | < 1, ||f || 1
(0)
< 1}.
FORMAL AND ANALYTIC RIGIDITY OF THE WITT ALGEBRA
319
Theorem 26. Suppose F is a finite dimensional space of even elements of C ∞ (S 1 ). Then f 7−→ Sf is an analytic map from F ∩N into the space of continuous linear operators C ∞ (S 1 ) → C ∞ (S 1 ). Proof. It follows from the previous corollaries.
4. Application A generalization of the Witt algebra was given by Krichever and Novikov [6] for compact Riemann surfaces of genus higher than 0. They studied the algebra of meromorphic vector fields with poles only at fixed points P± (called markings or punctures) on a compact Riemann surface of genus g. These algebras are nowadays called Krichever–Novikov (KN) algebras. The generalization of their concept for more than two markings of a surface was given by M. Schlichenmaier [8, 9]. It was shown [1, 7] that restricting the points P± to special values makes the structure of the Krichever–Novikov algebras of genus 1 more transparent and the connection to the Witt algebra can be investigated. In fact this algebra turns out to be a deformation of the Witt algebra W and simultaneously this algebra itself is embedded into a two parametric family of Lie algebras, denoted by Lp,q . The Lie algebra structure is given by l ] = (m − n) (b lm+n + pb lm+n+2 + qb lm+n+4 ) [b l ,b n m p,q b b b b [lα , lβ ]p,q = [lα , lβ ] b b lα+n + (n − α + 1) p b lα+n+2 + (n − α + 2) q b lα+n+4 [lα , ln ]p,q = (n − α)b where b li = z i+1 , i ∈ Z, α, β denote odd indices, and n, m even indices. The KN-algebras are obtained for p = 3e1 , q = (e1 − e2 )(e1 − e3 ), where 1 1 1 , e 2 = ℘τ (1 + τ ) , e3 = ℘τ τ e 1 = ℘τ 2 2 2 and τ denotes the normalized period of the torus. ln ), Wf reduces to Remark 27. If we take f = pz 2 + qz 4 and use the basis (b Lp,q . Let consider now Krichever–Novikov algebras of genus 0. Let Z α denotes the algebra of complex functions on P1 which are holomorphic outside the points α, −α and ∞(α 6= 0, ∞). The bracket structure can be written as [10, 11] l ] = (m − n) b lm+n + α2 b lm+n−2 [b l ,b n mα [b lγ , b lβ ]α = [b lγ , b lβ ] b b lγ+n + (n − γ − 1) α2 b lγ+n−2 [lγ , ln ]α = (n − γ)b where b li = z i+1 , i ∈ Z, γ, β denote odd indices, and n, m even indices.
320
L. GUERRINI
Remark 28. If we take f = α2 z −2 and use the basis (b ln ), Wf reduces to Z α . cα ∞ denote the C ∞ -completion of the algebra Z α . Let Z cα Theorem 29. If |α| < 1, then Z isomorphism is analytic in α.
∞
c∞ . Moreover the is isomorphic to W
Proof. This follows from Theorem 25 and 26 in which we take f = α2 z −2 . Acknowledgments I would like to thank my advisor Prof. V. S. Varadarajan for his invaluable help, guidance and constant encouragement throughout all these years. To him my sincere gratitude. References [1] T. Deck, “Deformations from Virasoro to Krichever–Novikov algebras”, Phys. Lett. B251 (1990) 535–540. [2] A. Fialowski, “Deformations of some infinite-dimensional Lie algebras”, J. Math. Phys. 31(6) (1990) 1340–1343. [3] M. Flato and D. Sternheimer, “Deformations of Poisson brackets, separate and joint analyticity in group representations, nonlinear group representations and physical applications”, Harmonic Analysis and Representations of Semisimple Lie Groups, eds. J. A. Wolf, M. Cahen, M. De Wilde; Mathematical Physics and Applied Mathematics Vol. 5, D. Reidel, Dordrecht (1980), pp. 385–448. [4] M. Gerstenhaber, “On the deformation of rings and algebras”, Ann. Math. 79 (1964) 59–103. [5] L. Guerrini, Ph.D thesis, Univ. of California, Los Angeles, 1998. [6] I. M. Krichever and S. P. Novikov, “Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons”, Funct. Anal. Appl. 21 (1987) 126–142. [7] A. Ruffing, T. Deck and M. Schlichenmaier, M., “String branchings on complex tori and algebraic representations of generalized Krichever–Novikov algebras”, Lett. Math. Phys. 26 (1992) 23–32. [8] M. Schlichenmaier, “Krichever–Novikov for more than two points”, Lett. Math. Phys. 19 (1990) 151–165. [9] M. Schlichenmaier, “Krichever–Novikov for more than two points: explicit generators”, Lett. Math. Phys. 19 (1990) 327–336. [10] M. Schlichenmaier, “Central extensions and semi-infinite wedge representations of Krichever–Novikov algebras for more than two points”, Lett. Math. Phys. 20 (1990) 33–46. [11] M. Schlichenmaier, “Degenerations of generalized Krichever–Novikov algebras on tori”, J. Math. Phys. 34(8) (1993) 3809–3824.
REMARKS ON DECAY OF CORRELATIONS AND WITTEN LAPLACIANS II ANALYSIS OF THE DEPENDENCE ON THE INTERACTION BERNARD HELFFER UA 760 du CNRS, D´ epartement de Math´ ematiques Bat. 425, F-91405 Orsay C´ edex, France Received 12 January 1998 Revised 13 April 1998 This is the continuation of previous notes on the subject referred as [10] and devoted to the analysis of Laplace integrals attached to the measure exp −Φ(X) dX for suitable families of phase Φ appearing naturally in the context of statistical mechanics. The main application treated in [10] was a semi-classical one (Φ = Ψ/h and h → 0), and the assumptions on the phase were related to weak non-convexity. We analyze here in the same spirit the case when the coefficient of the interaction J possibly is large and give rather explicit lower bounds for the lowest eigenvalue of the Witten Laplacian on 1-forms. We also analyze the case small J by discussing first an unpublished proof of [2] and then an alternative approach based on the analysis of a family of 1-dimensional Witten Laplacians. We also compare our results to those by Sokal’s approach. In the last paper of this series [11], we shall analyze, in a less explicit way, but in a more general context, applications to the logarithmic Sobolev inequality.
1. Introduction Our aim is to analyze Laplace integrals corresponding to the measure exp −Φ(X) dX in the case when Φ (which is attached to a subset Λ ⊂ Zd ) has the form, for X ∈ RΛ , X J X Φ(X) = φ(xj ) + |xj − xk |2 (1.1) 2 j∈Λ
j,k,j∼k
where, in the case of our toy model, which actually plays an important role in Quantum Field Theory (see for example [5]), the one-particle phase φ takes the form 1 1 λx4 + νx2 . (1.2) φT M (x) = 12 2 The parameters λ and ν satisfy λ > 0, (1.3) and we mainly want to analyze the possibility of having ν < 0: the consideration of non-convex phases is indeed our main interest. Here j ∼ k means that j and k are nearest neighbors in Λ considered as a discrete torus. The sum is over the nonoriented pairs of Λ × Λ. Finally, J is usually assumed to be positive, but all the results we shall give under the condition that J is small enough could be extended to the case when |J | is small enough. 321 Reviews in Mathematical Physics, Vol. 11, No. 3 (1999) 321–336 c World Scientific Publishing Company
322
B. HELFFER
Our main problem is to analyze the properties of the measure Z dµ := exp −Φ(X) dX/ exp −Φ(X) dX
(1.4)
RΛ
and more precisely the covariance associating to (f, g) Cov (f, g) = h(f − hf i)(g − hgi)i ,
(1.5)
where h · i denotes the mean value with respect to the measure dµ. As it is now established,a an elegant approach is given by the analysis of a Witten Laplacian on 1-forms. The aim of this article, in continuation of [10], is to show that this approach leads also to very explicit criteria in the non-convex case which we compare in particular to Sokal’s approach. 2. Lower Bound for the Spectrum of the Witten Laplacian Let us recall that the Witten Laplacian on C0∞ (RΛ ) 1-forms is defined as,b W1 := W0 ⊗ I + Hess Φ ,
(2.1)
where W0 is
X 1 1 W0 := −∂/∂xj + ∂Φ/∂xj ∂/∂xj + ∂Φ/∂xj . 2 2
(2.2)
j∈Λ
ostrand The strict positivity of the Witten Laplacian W1 was first observed by J. Sj¨ [16] for a somewhat narrow class of Φ, which does not contain the example above. Let us recallc for reference the extension recently obtained by J. Johnsen [13]. Proposition 2.1. Let us assume that the phase Φ satisfies the following assumptions.d β (H1) |DX ∇Φ(X)| ≤ Cβ h|∇Φ|i(1−ρ|β|)+ (2.3) for some ρ > 0, and for all β. There exists δ > 0 and a constant C such that, for all X such that |X| ≥ C, we have 1 (2.4) (H2) X · ∇Φ(X) ≥ |X|(1+δ) . C Then W1 is essentially self-adjoint and its self-adjoint extensione as an unbounded operator on L2 1−forms with respect to the standard Lebesgue measure has compact resolvent and is strictly positive. a See [16] or [10] for historical comments on the origin of the method. b It was denoted by ∆(1) in [16]. Φ c For simplicity, we give a stronger condition than in [13], which is sufficient for our purpose. p d For u ∈ Rm , we use the notation: hui = 1 + ||u||2 . e Consequently, the Friedrichs extension.
REMARKS ON DECAY OF CORRELATIONS AND WITTEN LAPLACIANS II
...
323
We observe that these assumptions are satisfied for the model (1.1) with φ satisfying, for some strictly positive constants δ, ρ, C, (Ck )k∈N , (h1) |φ(k+1) (t)| ≤ Ck hφ0 (t)i(1−ρk)+ ,
(2.5)
and, for |t| ≥ C,
1 (1+δ) |t| . (2.6) C It is immediate to see that the phase φ = φT M introduced in (1.2) satisfies (h1) and (h2) with ρ = 13 and δ = 3. The aim of this section is the proof, as a typical example of the method, of the following: (h2) tφ0 (t) ≥
Theorem 2.2. Let Λ ⊂ Zd and let Φ(Λ,T M) = ΦT M be the phase on RΛ given by ΦT M (X) =
X
φT M (xj ) +
j∈Λ
J X |xj − xk |2 , 2
(2.7)
j∼k
where φT M satisfies (1.2) and (1.3). Then if (λ, ν, J ) satisfy λ > 0, ν < 0, J ≥ 0,
(2.8)
1 1 λ2 + √ ν > 0 2
(2.9)
and 1 1 ν J d < √ λ2 + 2 2 Jd<
λ 4|ν|
√ 1 ν if − √ < λ 2 ≤ − 2ν 2 if −
√ 1 2ν ≤ λ 2 ,
(2.10)
then there exists ρ1 > 0 such that, for all Λ and all u in C0∞ (RΛ ; RΛ ), hW1 u|uiL2 ≥ ρ1 kuk2 .
(2.11)
The basic fact here, in comparison to Proposition 2.1, is that we obtain a strictly positive lower bound which is independent of Λ. Remark 2.3. • The method of proof is not limited to this example, and we can surely analyze other cases (see [2] or Antoniouk–Antoniouk [1] for the type of assumptions which can be considered). • We also recall that semi-classical results — that is, when Φ is replaced by Φ h and the limit h → 0 is considered — were presented in [10] (see also [12, 15, 16, 2]). • The strictly convex case ν > 0 does not create any problem and no condition except J ≥ 0 appears. • Finally it is interesting to compare this approach with the results by A. Sokal [17]. This will be done in Sec. 3.
324
B. HELFFER
This theorem has interesting consequences in statistical mechanics. We get for example the following: Corollary 2.4. Under the same assumptions, there exists C and κ > 0 such that, for any Λ and (i, j) ∈ Λ × Λ such that f d(i, j) ≤
1 inf d(i, Zd \ Λ), d(j, Zd \ Λ) , 4
(2.12)
we have Cov (xi , xj ) ≤ C exp −κd(i, j) .
(2.13)
We say in this case that the pair correlations have exponential decay. The proof of this corollary is similar to the proof given in [10] (or in Naddaf– Spencer [14]) and based on a Combes–Thomas distorsion argument ([3], see also [12] for a study based on the Maximum Principle in the strictly convex case and [15] in the weakly convex case). Proof of Theorem 2.2. This proof follows the same lines as in [10] (in which the semi-classical case was considered). Letting 1 Xj = ∂j + ∂j Φ , 2 we start from hW1 u|uiL2 =
X
kXk uj k2 +
j,k
XZ j,k
(2.14)
∂2Φ uj uk dX . ∂xj ∂xk
(2.15)
We first omit the terms kXk uj k2 with k 6= j and get Z X ∂2Φ 2 kXj uj k2 + hW1 u|uiL2 ≥ uj dX ∂xj ∂xj j +
XZ j6=k
∂2Φ uj uk dX . ∂xj ∂xk
(2.16)
Fixing j, we analyze the term Z (1)
hwj uj |uj i := kXj uj k2 + (1)
∂2Φ 2 u dX . ∂xj ∂xj j
(2.17)
The operator wj , initially defined as an unbounded operator on L2 (RΛ ), can also ˆ j := (x` )`∈Λ\{j} , of one-particle Witten Laplacians be seen as a family, indexed by X (relative to the effective phase φj at j ∈ Λ) X |t − xk |2 , (2.18) φj (t) = φ(t) + J k∼j f d is the euclidean distance.
REMARKS ON DECAY OF CORRELATIONS AND WITTEN LAPLACIANS II
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325
and defined on L2 (R) by d d (0) := wj + φ00j (t) , t, t, dt dt
(2.19)
d2 d 1 1 := − 2 + |φ0j (t)|2 − φ00j (t) . t, dt dt 4 2
(2.20)
(1) wj
and (0) wj
ˆ j (and actually only on the xk We observe that φj depends on the variables X (k ∼ j)) which are now considered as parameters. Note that φ0j (xj ) = (∂xj Φ)(X) ,
(2.21)
φ00j (xj ) = (∂x2j Φ)(X) ,
(2.22)
and ˆ j ). with X = (xj , X This effective one-particle phase φj at the point j ∈ Λ appears also in Dobrushin’s approach (see [1]). We introduce the following decomposition of the phase Φ Φ := Φd + Φi X Φd (X) := φ(xj )
(2.23)
j∈Λ
Φi (X) :=
JX |xj − xk |2 2 j∼k
Easy computations give (see [10]) that, for any satisfying 0 < ≤ 1,
(2.24)
Z Z 2− 1 ∂ 2 Φd 2 ∂ 2 Φi 2 (1) uj dX − uj dX hwj uj |uj i ≥ k∂xj uj k2 + 2 ∂xj ∂xj 2 ∂xj ∂xj Z ∂ 2 Φi 2 u dX . (2.25) + ∂xj ∂xj j Here we have introduced, for any , a new 1-particle operator sj := −∂x2j +
2 − ∂ 2 Φd 1 ∂ 2 Φi − , 2 ∂x2j 2 ∂x2j
(2.26)
which is now independent of the variables (x` )`∈Λ\{j} , and the inequality (2.25) takes the form Z ∂ 2 Φi 2 (1) hwj uj |uj i ≥ hsj uj |uj i + u dX (2.27) ∂xj ∂xj j
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B. HELFFER
Coming back to (2.16) and using the convexity of the interaction Φi , we get X kuj k2 , hW1 u|uiL2 ≥ σ1 (, J ) (2.28) j
where σ1 (, J ) is a lower bound for the spectrum of the operators sj seen as selfadjoint operators on L2 (RΛ ) or a lower bound for the family of operators also denoted by sj defined on L2 (R) but depending on the parameters xk with j ∼ k. The existence of ρ1 > 0 follows, in turn, from the existenceg of a strictly positive (and independent of Λ) (2.29) θ1 := sup σ1 (, J ) > 0 . ∈ ]0,1]
We now have to find suitable lower bounds for specific examples and to optimize w.r.t. . The case when = 1 corresponds to the approach presented in [9]. We observe that our operator sj actually depends only on the variable xj and takes the simple form d , (2.30) sj = s xj , dxj with
d2 d 2 − 00 := − 2 + φ (y) − J d . s y, dy dy 2
Let us start the analysis of our specific model, introduced in (1.2). We have to consider d2 d 2− 2 2− := − 2 + λ y + ν − J d. s y, dy dy 2 2
(2.31)
(2.32)
For fixed , we get that the lowest eigenvalue σ (which we take as our σ1 ) of s is given by 1 1 2− 2− 2 2 − J d. (2.33) +ν σ(J , ) = λ 2 2 q √1 and +∞. With Let us introduce the parameter η = 2− 2 which varies between 2 this new variable, we get 1
σ(, J ) = σ ˆ (η, J ) = λ 2 η + νη 2 − J d . Then the maximum (assuming ν < 0) is obtained for 1
η=− if 1
λ2 +
λ2 , 2ν
√ 2ν > 0 ,
(2.34)
g The lower bound ρ whose existence is claimed in the theorem will actually be obtained, once 1 the strict positivity of θ1 is proved, as sup∈]0,1] (σ1 (, J )).
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or for 1 η=√ . 2
(2.35)
In the first case (2.34), we get sup η∈[ √1 ,+∞] 2
σ ˆ (η, J ) =
λ − J d, 4|ν|
(2.36)
and the condition of strict positivity for this supremum is Jd<
λ . 4|ν|
(2.37)
In the second case (2.35), we get sup η∈[ √1 ,+∞] 2
1 1 ν σ ˆ (η, J ) = √ λ 2 + − J d , 2 2
(2.38)
and the condition of strict positivity becomes 1 1 ν J d < √ λ2 + . 2 2
(2.39)
As necessary condition for the strict positivity, we have finally obtained the condition 1 1 (2.40) λ2 + √ ν > 0 2 √ √ 1 1 and, combining the two cases − √ν2 < λ 2 ≤ − 2ν and − 2ν ≤ λ 2 , we get the conditions (2.10) for getting the strict positivity. This finishes the proof of Theorem 2.2. Remark 2.5. If we want to get a more abstract version of the previous result, we can see in the proof that our criterion says more generally that, if there exists d2 2− 00 ∈ ]0, 1] such that the operator s := − dy 2 + 2 φ (y) − J d is strictly positive, then there exists ρ1 > 0 such that (2.11) is verified for any Λ ⊂ Zd . The direct one-dimensional approach leads to the criterion (2.29) with = 1. This is not R optimal if R φ00 (y)u1 (y)2 dy > 0 (where u1 is the first eigenfunction of s for = 1). We can indeed analyze, by the Feynmann–Hellmann formula, the sign of the first derivative with respect to η of the lowest eigenvalue of s . In principle, this allows for the treatment of more general cases in which φ00 is negative in a sufficiently small region. The question that we have treated very explicitly in the case of the harmonic oscillator can also be formulated in more general terms as follows: Given a real regular potential v tending to ∞ at ∞, a constant σ0 and η0 > 0, under which condition on v can we find η ≥ η0 such that the unbounded operator on d2 L2 (R), P η = − dy 2 + ηv(y) − σ0 , is strictly positive?
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B. HELFFER
Remark 2.6. The proof is given in the case of periodic boundary conditions but could be extended to other conditions after small modifications. By looking at these other conditions and verifying some uniformity, one would get a proof of the Log–Sobolev inequalities as suggested by T. Bodineau referring to recent results by B. Zegarlinski [18] (Theorem 5.1). We shall analyze this point in detail in [11]. 3. Comparison to Sokal’s Results Let us compare our results to those proved by A. Sokal [17]. Under the assumption that the one-particle phase φ satisfies some Lebowitz inequality, that is that φ is an even differentiable function such that φ0 is convex on ]0, +∞[ ,
(3.1)
(and this is the case for the above example), A. Sokal obtains the decay of the correlations in a rather general framework by introducing some kind of effective quadratic potential associated to φ. More precisely, starting from φ, we first consider φJ (t) = φ(t) + J d t2 . We then consider the moment of order 2, Z t2 exp −φJ (t) dt . cJ = ZR exp −φJ (t) dt
(3.2)
(3.3)
R
Then A. Sokal proves: Theorem 3.1. If φ satisfies the Lebowitz inequality (3.1) and the condition 2J d cJ < 1 ,
(3.4)
then we have the exponential decay of the correlations in the sense of the Corollary 2.4. Let us first verify that, when φ(t) = α2 t2 with α > 0 (convex case), we recover the property that there is no condition except J > 0. We immediately get cJ =
1 , α + 2J d
and see that (3.4) is satisfied. The main idea in Sokal [17] is that the correlation functions are controlled by the new potential 1 eff − J d t2 (3.5) φJ (t) = 2cJ The condition (3.4) simply means the strict convexity of φeff J .
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329
The first thing to observe is that this condition is always satisfied when J is small enough. Indeed we get Z t2 exp −φ(t) dt R > 0. (3.6) lim cJ = Z J →0 exp −φ(t) dt R
As we already said, this can be applied to our previous toy models φT M . But it may be interesting to make the condition that we get explicit and to understand the links between the criteria obtained by the various methods. In particular, what is going on when J is large? Let us consequently consider Z t2 exp −φJ ,λ,ν (t) dt R , (3.7) c(J , λ, ν) = Z exp −φJ ,λ,ν (t) dt R
with φJ ,λ,ν (t) =
1 4 λt + 12
1 ν + J d t2 . 2
(3.8)
For fixed λ > 0 and ν < 0, let us compute the asymptotics of c(J , λ, ν) as J → +∞. An easy computation based on the Laplace integral method (in dimension 1) gives Proposition 3.2. As J tends to +∞, we have the following asymptotic behavior ν 1 1 1 1 − 2 2 +O . (3.9) c(J , λ, ν) = 2d J 4d J J3 Indeed, we first observe that Z 3 ! 12 Z ν 1 1 2 2 +J d exp −s ds + O . t exp −φJ ,λ,ν (t) dt = 2 ν + 2J d J (3.10) Then the renormalized denominator of c(J , λ, ν) has the following asymptotical behavior ν 12 Z +J d exp −φJ ,λ,ν (t) dt 2 3 ! Z −2 Z λ ν 1 2 4 2 +J d , s exp −s ds + O = exp −s ds − 12 2 J and we get ν 2
12 Z +J d exp −φJ ,λ,ν (t) dt Z =
3 ! −2 3λ ν 1 +J d . +O 1− exp −s ds 48 2 J 2
(3.11)
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B. HELFFER
Taking the quotient, we obtain c(J , λ, ν) =
1 ν + 2J d
1+O
1 J
2 !! ,
(3.12)
which gives the proof of (3.9). This shows that (3.4) is not fulfilled for large J . This is what we expect, because we hope to see a phase transition when J is large enough. On the other hand for fixed J > 0 and ν < 0 fixed, one easily gets, using the 1 change of variable t = λ− 4 s in (3.3), the Proposition 3.3. For fixed J > 0 and ν < 0, we have lim c(J , λ, ν) = 0 .
(3.13)
λ→+∞
This gives one case when Sokal’s theorem could be applied. We can give more explicit sufficient conditions in the following way. We do not try to be optimal but just want to sketch how it can be done. We observe first that tφ0J ,λ,ν (t) = (2J d + ν)t2 +
λ 4 t . 3
We can, for example, start from the inequality t2 ≤
1 + J d t4 , 4J d
(3.14)
and get, under the condition that 2J d + ν > 0 ,
(3.15)
the estimate t2 ≤
3J d 0 1 + tφJ ,λ,ν (t) . 4J d λ
(3.16)
This leads immediately, after an integration by parts, to the Proposition 3.4. Under the condition (3.15), we have the c(J , λ, ν) ≤
3J d 1 + . 4J d λ
(3.17)
We then obtain as a corollary the exponential decay of the correlation pair function under the conditions (3.15) and λ > 12J 2 d2 . The inequality (3.18) implies, using (3.15) and (3.17), the condition (3.4).
(3.18)
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Remark 3.5. This is of course very rough and can surely be improved by considering instead of (3.14) the inequality t2 ≤
Jd 4 α + t , 4J d α
for a strictly positive α, and using also an improvement of (3.16) leads to ht2 i ≤
1+
3J d 2J d + ν) · α λ
−1
3J d α + 4J d αλ
.
The question is then to determine under which condition one can find α such that the r.h.s is strictly less than 2J1 d . Then one could compare with the condition appearing in (2.10). This is left to the reader as an exercise. 4. The Case of Small J : Perturbative Approach Sokal’s result allows one to prove, for a large class of ferromagnetic examples that in any case — in particular, for highly non-convex situations — we have exponential decay of the correlations, for J small enough. It is natural to try to understand also this property in the framework of the Witten Laplacians. This was one of the goals of a recent analysis presented by Bach, Jecko and Sj¨ ostrand [2]. Let us describe their approach in a rather special situation which we shall also analyze differently in the next section. The basic idea is that when J = 0, the Witten Laplacian on 1-forms is clearly strictly positive (uniformly with respect to the subset Λ of Zd ). It is indeed diagonal. Consequently we have to justify a perturbation argument for small J . The only small difficulty is the control with respect to the dimension. Let us start again from (2.15), where we now introduce J explicitly in our notation, X XZ ∂2Φ kXk uj k2 + uj uk dX . (4.1) hW1J u|uiL2 = ∂xj ∂xk j,k
(0;J )
j,k
(1;J )
and wj the one-particle Witten Laplacians attached to the We denote by wj site j and the phase X |t − xk |2 , (4.2) φj;J (t) := φ(t) + J k∼j
on respectively functions or 1-forms. We analyze first the case J = 0 X (0) X (1) hW10 u|uiL2 = hwk uj |uj i + hwj uj |uj i .
(4.3)
j
j6=k (1;0)
If ρ1 (0) is the lowest eigenvalue of wj , which is strictly positive according to Proposition 2.1, we first get as before X (1;0) hW10 u|uiL2 ≥ hwj uj |uj i ≥ ρ1 (0)kuk2 . (4.4) j
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B. HELFFER
The new point is to use the other terms (which were neglected before by positivity, see (2.15)–(2.16)) in order to get some control on the perturbation when J is (0;0) takes simply the form, different from 0. The Witten Laplacian wj d2 d 1 1 = − 2 + |φ0 (t)|2 − φ00 (t) , w(0;0) t, (4.5) dt dt 4 2 (0;0)
with t = xj , wj
(1;0)
d = w(0;0) (t, dt ), and wj the form, d2 d 1 1 = − 2 + |φ0 (t)|2 + φ00 (t) , w(1;0) t, dt dt 4 2
(4.6)
(1;0)
d = w(1;0) (t, dt ). with t = xj , wj Under conditions (h1) and (h2) on φ, we observe that (with s = 0 or 1) Z Z Z d 0 2 2 (s;0) 2 f (t) f (t) dt + w t, φ (t) f (t) dt ≤ C f (t) dt . dt R R R
(4.7)
We just apply this inequality for the pairs j, ` and get the following regularity property, for a suitable constant C, kφ0j;0 u` k2 ≤ C(hwj
(0;0)
u` |u` iL2 + ku` k2 ) .
(4.8)
We also obtain for another constant C, that, for (j, k, `) ∈ Λ with j ∼ k, (0;0)
k(xj − xk )u` k2 ≤ C(hwj
(0;0)
u` |u` iL2 + hwk
u` |u` iL2 + ku` k2 ) .
(4.9)
These estimates are not sufficient. The detailed proof is more tricky and is given in [2]. One of the points is to introduce a parameter 0 < q < 1 and to write the following inequality: X XZ ∂2Φ J dΛ (j,k) 2 2 q kXk uj k + uj uk dX , (4.10) hW1 u|uiL ≥ ∂xj ∂xk j,k
j,k
where dΛ (j, k) is the distance between j and k on the cube Λ. The introduction of this q transforms an error of order |Λ| · O(J) into an error of order O(J ) · P ( k∈Λ q dΛ (j,k) ), that is actually of order O(J ). We then obtain, for a suitable C, hW1J u|uiL2
≥ (1 − CJ )
X j6=k
(0) q d(j,k) hwk
uj |uj i +
X
(1) hwj
j
uj |uj i − CJ
X
kuj k2 .
j
(4.11) So we have proved the existence of another Λ-independent constant C such that X J 2 kuj k . hW1 u|uiL2 ≥ (ρ1 (0) − CJ ) (4.12) j
This is summarized in:
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333
Proposition 4.1. If φ satisfies (h1) and (h2) and Φ is defined by (1.1), there exists J0 > 0 such that, if 0 ≤ J ≤ J0 , then there exists ρ1 > 0 such that, for any cube Λ ⊂ Zd , (4.13) W1J ≥ ρ1 . Remark 4.2. This proof could probably work also for other boundary conditions (periodic, free, . . .). But the condition that |J | is smaller than some J0 may depend on the choice of these boundary conditions. This will be analyzed in [11]. Let us also observe that the condition on the sign of J is unimportant in the proof. Sokal’s approach is more related to ferromagnetic systems. 5. The Case J Small: A 1-Dimensional Approach In the preceding section, we were suggesting that in the highly non convex case there was a problem to find a strictly positive lower bound uniformly with respect (1;J ) seen as a family of operators to the parameters xk (k ∼ j) for the operator wj 2 on L (R). We presented in Sec. 2 one way which was efficient in the case when the non-convexity was not too strong. In Sec. 4, we explained the trick used by Bach, Jecko and Sj¨ ostrand [2] consisting in the use of the non-diagonal terms for controlling the perturbation and also eliminating any assumption of weak convexity. The study in this section will present another way, less perturbative in spirit, for treating the difficulty to have uniform control of the lower bound with respect to J . The problem is indeed the following. We want to analyze !2 X d2 d 1 1 (1;J ) 0 := − 2 + φ (t) + 2J d t − 2J t, xk + (φ00 (t) + 2J d) . w dt dt 4 2 0∼k
(5.1) The criterion given in Sec. 2 is (when taking = 1) actually related to the assumption that the operator d d (1;J ) (1;J ) := w − 2J d , t, t, p dt dt that is, p
(1;J )
d2 d 1 := − 2 + t, dt dt 4
0
φ (t) + 2J d t − 2J
X 0∼k
!2 xk
1 + (φ00 (t) − 2J d) , 2 (5.2)
is uniformly stricty positive (with respect to xk and J ). The previous criteria correspond to taking the lower bound obtained by dropping P the term 14 (φ0 (t) + 2J d t − 2J 0∼k xk )2 and comparing to the lowest eigenvalue d ), but this works only when the later operator is strictly positive. This is of s(t, dt not satisfactory for J small. In this regime, it is sufficient (the two operators differ
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B. HELFFER
d each other by 2J d), to consider w(1;J ) (t, dt ) and, changing slightly the various parameters, to analyze the following family depending on J and α: d2 1 1 d (5.3) w t, , α, J := − 2 + (φ0 (t) + 2J d t − α)2 + (φ00 (t) + 2J d) , dt dt 4 2
with α = −2J
X
xk .
(5.4)
0∼k
We will then analyze the uniform strict positivity of this family with respect to (α, J ) ∈ R × [−J0 , +J0 ] which is better because we can forget the relation (5.4). For any fixed pair (α, J ), the operator is strictly positive by Proposition 2.1, and a standard perturbation argument permits us to show (under assumptions (h1), d , α, J ) is continuous with (h2), and with δ ≥ 1) that the lowest eigenvalue of w(t, dt respect to J and α. So it is a problem of uniformity as α = ±∞. Let us treat our model where φ(t) = φT M , using a semi-classical approach. Changing ν, we may forget J although remembering that our argument has to be locally uniform in ν. So we have finally to verify the following: Lemma 5.1. There exists τ > 0 such that for |α| ≥ τ the operator 2 d2 d 1 λ 3 1 t + νt − α + (λt2 + ν) s t, , α := − 2 + dt dt 4 3 2
(5.5)
2 1 d s t, , α ≥ |α| 3 . dt τ
(5.6)
satisfies
The problem is symmetric in α. It is consequently sufficient to treat the case α > 0. 1 Let us introduce the scaling t = α 3 y. We then obtain the following inequality 2 2 d α2 λ 3 − 23 d − 23 y + να y − 1 q y, , α := −α + dy dy 2 4 3 1 2 1 2 2 + α 3 (λy 2 + α− 3 ν) ≥ |α| 3 2 τ to be proved. In order to get a more semi-classical form we can write it as d d 2 q y, , α := α q˜ y, h ; h; µ , dy dy
(5.7)
(5.8)
with d q˜ y, h ; h; µ := dy
1 d2 −h + 2 dy 4 2
λ 3 y + µy − 1 3
2
! 1 2 + h(λy + µ) , 2
(5.9)
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335
where h = α− 3 ; µ = να− 3 = O(h 2 ) . 4
2
1
(5.10)
This is a non-degenerate one-well semi-classical problem near the point y = 1 and d this leads, for the operator q˜(y, h dy ; h; µ), to a lowest eigenvalue which is asymp2
totically equal to hλ. We finally have obtained a behavior like λα 3 for the lowest d , α) as α → +∞ uniformly with respect to ν and consequently eigenvalue of q(y, dy d of s(t, dt , α). This proves the lemma. We have consequently given an alternative proof of Proposition 4.1, in the case of the toy model φ = φT M . Remark 5.2. We shall discuss the uniformity of the argument with respect to other boundary conditions in [11]. The price to pay is that the estimates given in this section will be much more implicit. 6. Conclusion As we have seen, the lowest eigenvalue of the Witten Laplacian plays an important role. A natural question is the understanding of the phase transition through this approach. The answer is not clear. Let us just observe here that the results obtained on the transition of phase by other methods like infrared estimates or Peierls argument (see [5] for a presentation of these methods) will give an upper bound on the splitting between the two lowest eigenvalues of the Witten Laplacian on the functions or an upper bound of the smallest eigenvalue of the Witten Laplacian on the 1-forms. We recall that in the infrared estimates (see [7] for a recent presentation and a bibliography) one gets (but under the condition d ≥ 2), a uniformh lower bound δ0 > 0 of the mean value of the variance Cov (f, f ) of the function X 7→ f (X) = 1 P |Λ| ( j∈Λ xj ) with respect to the measure exp −Φ(X) dX and this gives an upper bound of the smallest eigenvalue, and consequently of the splitting, by δ01|Λ| . Acknowledgements We are grateful to V. Bach, T. Jecko and J. Sj¨ostrand for useful discussions (and in particular for communicating before publication the results of [2]). Many thanks to T. Bodineau for discussions around possible applications to logarithmic Sobolev inequalities and to J. Johnsen for useful comments on various versions of these notes. The constructive remarks of the referee have also contributed to the improvment of the final manuscript. We thank also the Fields Institute in Toronto where one part of this work was done and the European Union which partially supported this research through the TMR Programme FMRX-CT 960001 of the European Commission – Network Postdoctoral training programme in partial differential equations and application in quantum mechanics. h With respect to Λ.
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References [1] A. V. Antoniouk and A. V. Antoniouk, “Decay of correlations and uniqueness of Gibbs lattice systems with nonquadratic interaction”, J. Math. Phys. 37 (11) (November 1996). [2] V. Bach, T. Jecko and J. Sj¨ ostrand, “Part I Witten Laplacians: Perturbative approach”, and “Part II Witten Laplacians: Semiclassical approach”, in preparation. [3] J. M. Combes and L. Thomas, “Asymptotic behavior of eigenfunctions for multiparticle Schr¨ odinger Operators”, Commun. Math. Phys. 34 (1973) 251–270. [4] J.-D. Deuschel and D. Strook, “Hypercontractivity and spectral gap of symmetric diffusions with applications to the stochastic Ising models”, J. Funct. Analysis 92 (1990) 30–48. [5] J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, Springer, 1987. [6] B. Helffer, “Spectral properties of the Kac operator in large dimension, Proc. Mathematical Quantum Theory II: Schr¨ odinger Operators”, (August 1993), eds. J. Feldman, R. Froese, and L. M. Rosen. Proc. Canadian Math. Soc. annual seminar on mathematical quantum theory held in Vancouver, Canada, August 4–14, 1993. CRM Proc. Lect. Notes. 8 (1995) 179–211. [7] B. Helffer, “Splitting in large dimension and infrared estimates II — Moment inequalities”, to appear in J. Math. Phys. (1997). [8] B. Helffer, “Witten Laplacians and Antoniouk’s results”, preliminary notes. [9] B. Helffer, “Dobrushin’s criteria and Antoniouk’s results”, preliminary notes. [10] B. Helffer, “Remarks on decay of correlations and Witten Laplacians — Brascamp– Lieb inequalities and semi-classical analysis”, J. Funct. Analysis 155 (1998) 571–586. [11] B. Helffer, “Remarks on decay of correlations and Witten Laplacians. III — Application to logarithmic Sobolev inequalities”, preliminary notes (October 1997) and final version in preparation, to appear in Annales IHP (Section Prob–Stat.) (1999). [12] B. Helffer and J. Sj¨ ostrand, “On the correlation for Kac like models in the convex case”, J. Statist. Physics 74 (1-2) (1994) 349–369. [13] J. Johnsen, “On the spectral properties of Witten Laplacians, their range projections and Brascamp–Lieb’s inequality”, preprint, December 1997. [14] A. Naddaf and T. Spencer, “On homogeneization and scaling limit of some gradient perturbations of a massless free field”, Commun. Math. Phys. 183 (1997) 55–84. [15] J. Sj¨ ostrand, “Ferromagnetic integrals, correlations and maximum principles”, Ann. Inst. Fourier, Tome 44 (2) (1994) 601–628. [16] J. Sj¨ ostrand, “Correlation asymptotics and Witten laplacians”, preprint Ecole Polytechnique (December 1994). St Petersburg Mathematical J. 8 (1997) 160–191. [17] A. Sokal, “Mean-field bounds and correlation inequalities”, J. Statist. Phys. 28 (3) (1982) 431–439. [18] B. Zegarlinski, “The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice”, Commun. Math. Phys. 175 (1996) 401–432.
ON SOLITON AUTOMORPHISMS IN MASSIVE AND CONFORMAL THEORIES ∗ ¨ MICHAEL MUGER
Dipartimento di Matematica Universit` a di Roma “Tor Vergata” Via della Ricerca Scientifica 00133 Roma Italy E-mail : [email protected] Received 6 April 1998 For massive and conformal quantum field theories in 1+1 dimensions with a global gauge group we consider soliton automorphisms, viz. automorphisms of the quasilocal algebra which act like two different global symmetry transformations on the left and right spacelike complements of a bounded region. We give a unified treatment by providing a necessary and sufficient condition for the existence and Poincar´ e covariance of soliton automorphisms which is applicable to a large class of theories. In particular, our construction applies to the QFT models with the local Fock property — in which case the latter property is the only input from constructive QFT we need — and to holomorphic conformal field theories. In conformal QFT soliton representations appear as twisted sectors, and in a subsequent paper our results will be used to give a rigorous analysis of the superselection structure of orbifolds of holomorphic theories.
1. Introduction Solitons in massive quantum field theories in 1+1 dimensions continue to attract the interest of quantum field theorists. Primarily this is due to the fact that they constitute topological excitations of a QFT which are non-trivial yet amenable to thorough understanding. They occupy a prominent position in the analysis of exactly soluble classical and quantum models, and there are strong indications [38] that soliton sectors are the only interesting sectors of massive QFTs in 1 + 1 dimensions. The first rigorous approach to the study of solitonic sectors in the framework of general QFT was given by Roberts [45], and a very general analysis of soliton sectors and their composition structure has been provided by Fredenhagen [22, 23]. As is well known, massive QFTs with a spontaneously broken symmetry group give rise to inequivalent vacua and thereby to soliton representations. (Of course, spontaneous symmetry breakdown, which occurs only for discrete groups [12], is not the only possible origin for the existence of inequivalent vacua.) Rigorous constructions of soliton sectors and soliton automorphisms for several models have been ∗ Supported
by EU TMR Network “Noncommutative Geometry”. 337
Reviews in Mathematical Physics, Vol. 11, No. 3 (1999) 337–359 c World Scientific Publishing Company
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given by Fr¨ohlich [24, 25] relying on methods from algebraic and from constructive quantum field theory. One aim of this work is to exhibit and exploit the similarities of soliton representations of massive and conformal quantum field theories in 1 + 1 dimensions with a global gauge group. Since spontaneous breakdown of inner symmetries is impossible in conformal theories due to the uniqueness of the vacuum [44] and since left and right spacelike infinity coincide, the role of solitons in CQFT is necessarily different. Picking a Minkowski space within the conformal covering space [35] and restricting the theory to this Minkowski space, the soliton condition O O ) = π0 ◦ αg F (WLL ), π F (WLL O O π F(WRR ) = π0 F (WRR ),
(1.1)
O O where WLL , WRR are the left and right spacelike complements of the double cone O, makes sense. It can be shown [40] that for g 6= e such a representation cannot be unitarily equivalent to π0 since it is not locally normal at infinity. Since positive energy representations of CQFTs are normal on every double cone [10], thus also at infinity, soliton representations do not constitute proper superselection sectors of the theory F . In restriction to the fixpoint theory A = F G , however, the discontinuity at infinity disappears, π A being localized in O:
π A(O0 ) = π0 A(O0 ) .
(1.2)
For this reason the solitons, better known as twisted representations, of the field theory F are relevant for the superselection structure of the fixpoint theory A, cf. [13]. The results of this work will be used in a subsequent paper [40] for giving a rigorous analysis of such ‘orbifold models’ (of holomorphic models) and clarifying the role of the Dijkgraaf–Witten 3-cocycle ω and of the twisted quantum double Dω (G). Since the (chiral) Ising model [36, 4] obviously is not covered by the analysis in [13] although it is an Z2 orbifold model, there must be an implicit assumption in the latter analysis. Our reconsideration of orbifold models was partially motivated by the desire to clarify which properties the triple (F , G, α) must possess in order to lead to the results of [13]. As it turns out this is just the existence of ‘twisted sectors’ in the guise of soliton automorphisms, not just of soliton endomorphisms [4]. We briefly recall the framework of local quantum physics [30, 33]. We consider a QFT to be given in terms of a net of algebras, i.e. a map K 3 O 7→ F(O), where K is the set of all double cones in Minkowski space and F (O) is a C ∗ algebra. This map being inclusion preserving O1 ⊂ O2 ⇒ F (O1 ) ⊂ F (O2 ) we can S k·k define the quasilocal algebra as the inductive limit: F = O∈K F (O) . There are commuting automorphic actions on F of the Poincar´e group P and of a locally compact symmetry group G such that αΛ,x (A(O)) = A(ΛO + x) ∀ (Λ, x) ∈ P and αg (F(O)) = F(O) ∀ g ∈ G. When considering observables we require locality, i.e. [A(O1 ), A(O2 )] = {0} whenever O1 , O2 are spacelike to each other. In the presence of fermions we assume the the usual Bose–Fermi commutation relations
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w.r.t. a Z2 grading given by the automorphism α− = αv , where v is an element of order two in the center of G. The quasilocal algebra being automatically simple, all representations are faithful. All representations π we consider are required to be Poincar´e covariant, i.e. there is a unitary representation of P with positive energy such that π ◦ αx (A) = Uπ (x)π(A)Uπ (x)∗ . We assume the existence of a vacuum state ω0 such that the algebras π0 (F(O)) ⊂ B(H0 ) in the GNS representation π0 on the Hilbert space H are weakly closed. Thus F (O) is a W ∗ -algebra and αg is locally normal. The group of unbroken symmetries (w.r.t. ω0 ) is defined as G0 = {g ∈ G | ω0 ◦ αg = ω0 } .
(1.3)
If we assume some form of split property (see below) then G0 is automatically compact when topologized with the strong topology in the representation π0 [17], 0 and [18, Theorem. 3.6] implies F ∩ F G0 = C1. Vacuum representations of local nets are usually required to satisfy Haag duality π0 (A(O)) = π0 (A(O0 ))0
∀O ∈ K,
(1.4)
which for fermionic nets is replaced by twisted duality π0 (F (O))t = π0 (F (O0 ))0 where X t = ZXZ ∗ with Z = 1+iV 1+i , V being the unitary implementer for α− = αv (v ∈ G0 is automatic [43]). For many purposes it is sufficient to replace (twisted) Haag duality by wedge duality R0 (W )t = R0 (W 0 )0 ∀ W ∈ W, where R0 (W ) = π0 (F(W ))00 and W is the set of all wedges, i.e. translates of WR = {x ∈ R2 |x1 ≥ |x0 |} and the spacelike complement WL = WR0 . We will be interested in soliton automorphisms of F , viz. automorphisms ρO g,h which coincide with αg on the left spacelike complement of a double cone O O O ) and with αh on the right complement (WRR ). For g, h ∈ G0 and assuming (i.e. WLL the existence of a vacuum representation π0 satisfying Haag duality and the split property for wedges (SPW) such automorphisms can easily be constructed using disorder operators, cf. [37] and the next section. Besides the defining properties of soliton automorphisms the ρO g,h obtained in this way have the following remarkable properties: (a) The map G × G 3 (g, h) 7→ ρO g,h ∈ Aut F is a group homomorphism. = α ∀ g. (b) ρO g g,g O (c) αk ◦ ρO g,h = ρkgk−1 ,khk−1 ◦ αk ∀ g, h, k. (This property follows from the first two.) In particular, if G is abelian then the soliton automorphisms commute with the global symmetry. It is easy to see that families of two-sided soliton automorphisms can be equivalently O characterized using only left-handed soliton automorphisms. Write ρO g := ρg,e . Then properties (a–c) imply: O (A) αk ◦ ρO g = ρkgk−1 ◦ αk ∀ g, h. (B) The map G 3 g 7→ ρO g ∈ Aut F is a group homomorphism. O O Conversely, defining ρO g,h ≡ αh ◦ ρh−1 g = ρgh−1 ◦ αh , g, h ∈ G one verifies that properties (a–c) follow from (A) and (B).
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Postulating the existence of soliton automorphisms with the above properties Rehren [42] has recently derived the modular theoretic assumptions of [41], where a general proof of the cyclic form factor equation is announced. Since these results are quite interesting it seems important to understand better when soliton automorphisms exist. Our aim will thus be to find conditions for the existence of soliton automorphisms ρO g without appealing to the SPW or to g, h ∈ G0 , preferably in such a way that the above properties (A), (B) are valid. The paper is organized as follows. Before turning to the general analysis, we give two results on solitons in massive theories (characterized by the SPW). In particular, in Subsec. 2.2 we reconsider the dual net obtained as in [37] from a massive theory with unbroken abelian symmetry and show that it possesses soliton automorphisms with all desired properties, which allows to reconstruct the original net. Section 3 is the core of the paper and contains the proof of our criterion for the existence of soliton automorphisms and the proof of their Poincar´e covariance. While the existence part is not entirely new, our proof of Poincar´e covariance is and relies on the uniqueness of soliton sectors up to unitary equivalence, proved in [38]. In Sec. 4 we show that the results of Sec. 3 apply to all QFT models which possess the local Fock property, e.g. the P (φ)2 and Y2 theories. 2. Solitons in Massive Theories 2.1. Soliton automorphisms from soliton sectors Given two vacuum representations π0L , π0R , a representation π is said to be a soliton representation of type (π0L , π0R ) if it is translation covariant and L/R F (WL/R ) , π F (WL/R ) ∼ = π0
(2.1)
where WL , WR are arbitrary left and right handed wedges, respectively. Clearly, a (π0L , π0R )-soliton representation is locally normal w.r.t. π0L and π0R . In [47] it was shown that for every pair of mutually locally normal vacuum representations π0L , π0R there is a soliton representation of type (π0L , π0R ) if the vacua satisfy Haag duality and the split property for wedges (SPW). Recall that a graded local net F with twisted duality satisfies the SPW if for every double cone O there is the following isomorphism of von Neumann algebras: O O t O O t ) ∨ R(WRR ) ' R(WLL ) ⊗ R(WRR ) . R(WLL
(2.2)
(This isomorphism is automatically spatial, i.e. unitarily implemented.) For free massive scalar and Dirac fields the SPW is satisfied, and it will be assumed in this subsection and the next. Considering now the case of a broken symmetry let πg be the GNS representation corresponding to ωg = ω0 ◦ αg . Due to πg ∼ = π0 ◦ αg all πg ’s satisfy Haag duality and the SPW if π0 does. Assuming the existence of an irreducible soliton representation of type (πg , π0 ) we show that this implies the existence of soliton automorphisms of the (abstract) C ∗ -algebra F , restricting ourselves to the case of a local net.
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Proposition 2.1. Let F be a local net and let π be an irreducible soliton representation of type (πg , π0 ), where one of the (thus both) vacuum representations satisfies Haag duality and the SPW. Then for each double cone O there is an autoO O O O morphism ρO g of F such that ρg F (WLL ) = αg and ρg F (WRR ) = id. Proof. By [38, Theorem 4.3] π automatically satisfies Haag duality, and the SPW carries over to π by [38, Theorem 5.1]. Thus also wedge duality holds [38, Proposition 2.5]. Let O ∈ K. By the soliton criterion there is a representation ρ1 O 0 on H0 equivalent to π such that ρ1 F (WRO ) = π0 , where WRO = (WLL ) . Then we have (2.3) R(WRO ) = ρ1 (F (WRO ))00 = π0 (F (WRO ))00 = R0 (WRO ) , O O and wedge duality implies R(WLL ) = R0 (WLL ). Now, by the soliton criterion ρ1 is equivalent to π0 ◦ αg ∼ on every left wedge. Thus there is a unitary U on H0 π = g such that O ) = π0 ◦ αg . (2.4) Ad U ◦ ρ1 F (WLL O This formula shows that Ad U maps the ultraweakly dense subalgebra F (WLL ) O of R(WLL ) onto another such algebra, and by ultraweak continuity U acts as an O ). The SPW for π gives rise to a spatial isomorphism automorphism on R(WLL O O O O ) ⊗ R(WRR ) implemented by a unitary between R(WLL ) ∨ R(WRR ) and R(WLL O O∗ O ˜ Y . As in [37] we define U = Y (U ⊗ 1)Y which by wedge duality is contained O ) as U . Define in R(WLO ) and has the same adjoint action on R(WLL
˜ ◦ ρ1 . ρ2 = Ad U
(2.5)
O O ) = ρ1 F (WRR ) = π0 , ρ2 F (WRR
(2.6)
˜ we have By the localization of U
whereas by construction we have O ) = π0 ◦ αg . ρ2 F (WLL
(2.7)
ˆ whenever ˆ = π0 (F (O)) Since π satisfies Haag duality we easily see that π(F (O)) ˆ O ⊃ O. (This was already observed in [25, p. 403].) Now the soliton automorphism −1 is obtained by ρO g = π0 ◦ ρ2 . Remark. Since the proof involves the SPW it is evident that the interpolation region O between π0 and π0 ◦ αg cannot be eliminated. In [24] a soliton representation for the Z2 symmetric (φ4 )2 theory in the broken phase was constructed by a doubling trick, and the procedure given in [47] is an abstract version of the former. In both references, however, the irreducibility of the constructed soliton representation is left open, such that the above theorem cannot be used. Therefore an alternative approach to the construction of soliton automorphisms will be developed in the next section. But before doing so we will give an instructive direct proof of the existence of soliton automorphisms satisfying
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conditions (A) and (B) for an interesting special class of models considered first in [37]. 2.2. Soliton automorphisms for dual theories We start by recalling some results of [37]. Again we assume F to satisfy twisted duality and the SPW. Let G be a group of unbroken, i.e. unitarily implemented symmetries. By the split property G must be strongly compact and second countable [17], and the Hilbert space H is separable. For g, h ∈ G we define disorder operators by ULO (g) = Y O∗ (U (g) ⊗ 1)Y O , URO (h) = Y O∗ (1 ⊗ U (h))Y O ,
(2.8)
where Y O implements the spatial isomorphism (2.2). One easily verifies Ad ULO (g) O O ) = αg and Ad ULO (g) F (WRR ) = id and similarly for Ad URO (g). Since, as F(WLL ˆ O ˆ ⊃ O into itself we can obtain already observed in [25], Ad ULO (g) maps π0 (F (O)), soliton automorphisms by −1 O O ρO g,h = π0 ◦ Ad UL (g)UR (h) ◦ π0 .
(2.9)
Using the definition (2.8) and Y O U (g) = (U (g) ⊗ U (g))Y O one can verify that the automorphisms (2.9) satisfy the properties (a–c). This construction clearly relies on the existence of the global implementers U (g) of αg , which is due to invariance of ω0 . Since these soliton automorphisms are unitarily implemented in the vacuum representation, they may seem uninteresting from the point of view of superselection theory. But, as discussed in [37, 38] they are quite useful for elucidating the structure of the fixpoint net A = F G , which violates Haag duality, and its dual net. To this purpose one introduces a nonlocal extension of the net F : Fˆ (O) = F (O) ∨ ULO (G)00 .
(2.10)
ˆ Lemma 2.2. F(O) is isomorphic to the crossed product F (O) oαL G, where O αL (g) = Ad UL (g). Proof. Recall the result [46, Appendix] according to which the action of G on ˆ The same a fortiori holding for each F(O) has full spectrum: Γ(α F (O)) = G. O the wedge algebra R(WR ) and the latter being factorial [20], R(WRO ) oα G is a factor by [27, Corollary 6]. But then [31, Corollary 2.3] gives us R(WRO ) ∨ U (G)00 ' R(WRO ) oα G. Since F(O) is unitarily equivalent to R(WRO ) ⊗ R(WLO ) and ULO (g) to U (g) ⊗ 1 we are done. Remark. In [37] this result was obtained only for finite groups, but see also [11]. Restricting now to the case of abelian groups G, Takesaki duality gives us conˆ on all algebras Fˆ (O). These actions being tinuous actions α ˆ O of the dual group G
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ˆ on the quasilocal algebra Fˆ which is compatible they give rise to an action of G ˆ χ 6= ω0 ∀ χ 6= eˆ. The dual symmetry α ˆ commuting spontaneously broken: ω0 ◦ α ˆ with the action α of G it acts on the fixpoint net A(O) = Fˆ (O)G , the restriction of which to the vacuum sector H0 was shown to be just the dual net of the fixpoint net A H0 . The net Aˆ H0 = (A H0 )d satisfies Haag duality and the SPW and ˆ is countable. we are in the scenario introduced above. The discrete abelian group G Definition 2.3. A local net B satisfying Haag duality and the SPW with a completely broken countable abelian symmetry group K is called dual if it arises from a net with unbroken compact abelian symmetry by the above construction. Remark. This notation is consistent since B is the dual net in the conventional sense of B = (F G H0 )d . In [38] it was shown that the representation of Aˆ on the charged sectors Hχ ⊂ H is the representation of the (unique up to unitary equivalence) soliton interpolating ˆ −1 between the vacua ω0 and ωχ ≡ ω0 ◦ α χ . But we can do better: Theorem 2.4. Dual nets admit soliton automorphisms satisfying the properties (A), (B). Proof. Let F be the field net with unbroken symmetry G from which B arises. By [46, Appendix] the action of G on each F (O) has full spectrum, i.e. ∀ O ∈ ˆ ∃ ψχ ∈ U(F(O)) : αg (ψχ ) = χ(g)ψχ ∀ g ∈ G. Due to the split property K ∀χ ∈ G H is separable and G is second countable (called separable by many authors). Since the fixpoint algebra A(O) is properly infinite due to the Borchers property we can apply [48, Proposition 20.12] due to Connes and Takesaki which tells us that ˆ → U(F (O)) such that there is a strongly continuous homomorphism s : K = G ˆ ˆ αg (s(χ)) = χ(g)s(χ) ∀ g ∈ G, χ ∈ G. (Since G is discrete in our case continuity is trivial, but the homomorphism property is not.) Due to the defining properties of disorder operators ρχ = Ad s(χ) implements an automorphism of AˆL which is the O O and α ˆ −1 identity on WLL χ on WRR , thus a right soliton automorphism. Property (B) ˆ ξ . The soliton is fulfilled by construction, and (A) is true since ρχ commutes with α automorphisms are clearly transportable with intertwiners in A = F G = B K . Given soliton automorphisms satisfying properties (A), (B) one can construct, along the lines of [14], a “crossed product theory” acted upon by the quantum double D(K). For K abelian this was sketched in [42]. Applying this construction to a dual net in the above sense and restricting to net of K fixpoints one re-obtains ˆ In fact, it seems likely the original theory with unbroken symmetry under G = K. that every Haag dual net B(O) with completely broken countable abelian symmetry and with soliton automorphisms satisfying (A) and (B) is a dual net in the above sense under some additional conditions. In particular, this should be true if B satisfies the SPW (and thus by [38, Proposition 4.1] also property 4 of the following section). We refrain from going into details since this would lead us to far away from the main subject of the present investigation.
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In this section we have written B, K in order to avoid confusion with the original net with from which B arises as the dual net. From now on we return to F and G. 3. General Approach to Soliton Automorphisms 3.1. Assumptions and preliminary results In our considerations of soliton automorphisms we will allow for graded-local nets since solitons appear in the Yukawa2 model, and since also for the consideration of conformal orbifold theories the fermionic case is quite interesting. Our assumptions on the field net F in the vacuum representation are the following: 1. 2. 3. 4.
(Twisted) Haag duality. Split property (for double cones). The local algebras F(O), O ∈ K factors. Minimality of twisted relative commutants, i.e. 0
ˆ ∧ F(O)t = F (O1 ) ∨ F(O2 ) F(O)
ˆ, ∀ O ⊂⊂ O
(3.1)
ˆ as in Fig. 1. where O1 , O2 are related to O, O 5. The automorphisms α− F (O) are outer for all double cones O. (In the pure Bose case condition 5 and the twists in condition 1 and 4 disappear.) We give a few motivating remarks for these conditions. As to massive theories, all the above properties follow from (twisted) duality and the SPW, as was shown for 5 (only for unbroken symmetries) and 3 in [37] and (in the local case) for 2 and 4 in [38]. We refrain from giving the easy proofs that the latter properties follow from the SPW also in the fermionic case. Free massive scalar and Dirac fields satisfying (twisted) duality and the SPW, they fulfill our assumptions 1–5. Unfortunately, up to now the SPW has not been proven for any interacting theory, but as we will see in Sec. 4, assumptions 1–5 hold in all models with the local Fock property. In the case of local conformal fields local factoriality [6, 26] and condition 5 [40, 44] are automatic. The split property is a very weak assumption since it follows [26] from finiteness of the conformal characters. It is well known that condition 1,
@@
@@ @@ @@ @@ @@ @@ @@ @@O @@O @@O @@ @@ @@ @@ @@ ˆ O
2
1
Fig. 1. Relative spacelike complement of double cones.
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i.e. duality on Minkowski space (or the line) is equivalent to strong additivity, which will be shown below to follow from the split property and property 4. So we remain with the latter property which is quite restrictive since (in the local case) it implies the absence of DHR sectors [21, 38]. Twisted duality on the conformal spacetime and local factoriality will be taken for granted also in the fermionic case since they can be shown by reconsidering the arguments in [6, 26]. In [38] it was shown that Haag duality and the SPW imply strong additivity. Since instead of the latter we assume only the conditions 1–5 it is reassuring that we still have the following. Lemma 3.1. The conditions 1–5 imply strong additivity, i.e. F (O1 )∨F(O2 ) = ˜ whenever the double cones O1 , O2 are spacelike with a common boundary F (O) ˜ is the smallest double cone containing O1 , O2 . point, and O ˆ We will prove Proof. Consider O ⊂⊂ O. ˆ , ˆ ∧ F(O)t 0 ) ∨ F(O) = F (O) (F (O)
(3.2)
ˆ The first which due to condition 4 is equivalent to F (O1 ) ∨ F (O) ∨ F (O2 ) = F (O). two algebras being contained in the algebra of the smallest double cone containing O1 and O, strong additivity follows. The split property provides us with spatial isomorphisms (implemented by the same operator Y Λ ) F (O) ∼ = F (O) ⊗ 1 , ˆ 0. ˆ 0∼ F (O) = 1 ⊗ F (O)
(3.3)
With F(O)t ∼ = F(O)+ ⊗ 1 + F (O)− V ⊗ V we compute 0 ˆ ∧ F(O)t 0 ∼ ˆ + + F (O)0 ⊗ F (O) ˆ −. F(O) = F (O)t ⊗ F (O)
(3.4) 0
Condition 3 implies F(O)∨F(O)0 = B(H), and condition 5 entails F (O)∨F(O)t = 0 B(H) as a consequence of F(O) ∧ F(O)+ = C1. Thus (3.2) follows. Remark. Note that Haag duality has not been used. To the contrary, in the conformal case the above result implies Haag duality on Minkowski space when combined with conformal duality. Since the group acts locally normally we have also in the broken symmetry case: ˆ O ⊂⊂ O ˆ and g ∈ G there is a bosonic Lemma 3.2. For every Λ = (O, O), Λ ˆ unitary implementer Xg ∈ F (O) for αg F (O). Proof. Since the vacuum vector is cyclic and separating for F (O), the algebra is in standard form and there is a unitary representation Xg of G on H which implements α F(O), cf. [9, Sec. 2] (the construction given there coincides with Haagerup’s canonical implementation [48, p. 41]). The vacuum state being invariant under α− [43] there is also a GNS implementer V , with which Xv is easily seen to
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coincide. Since v is in the center of G, the Xg commute with V = Xv , i.e. are bosonic. Using the split property for double cones we can define a representation ˆ [8] by XgΛ = Y Λ∗ (Xg ⊗ 1)Y Λ which still implements α F (O). Due of G in F(O) to Y Λ V = (V ⊗ V )Y Λ also XgΛ is bosonic.
In theories with spontaneously broken symmetries the analog of the preceding result for wedges is false. This fact is responsible for additional difficulties in the treatment of soliton automorphisms as compared to sectors which are localizable in double cones or (left and right) wedges. Lemma 3.3. Assuming G to be compact, broken symmetries αg , g ∈ G − G0 act non-normally on the algebras of wedges. Proof. Let A(O) = F (O)G and B(O) = F (O)G0 . As shown in [43, Proposition 9] B(O) ⊂ A(W )00 whenever O ⊂ W and thus B(W )00 = A(W )00 . (In the case G0 = {e} this implies that the subnet A still satisfies wedge duality on the Hilbert space H.) Assuming that αg acts normally on F (W ), it acts triviality on B(W ) since by definition it is trivial on the ultraweakly dense subalgebra A(W ). Then by translation invariance αg acts trivially on B. Now by [18, Theorem 3.6(b)] every automorphism of F which acts trivially on B is a gauge automorphism αg with g ∈ G0 . Remarks. 1. We emphasize that [18, Theorem 3.6] is true also in 1 + 1 dimensions with the possible exception of twisted duality for F , which we do not need to prove anyway since it is one of our axioms. 2. The only step where compactness of G is used is the argument in [43, Proposition 9] leading to B(O) ⊂ A(W )00 for O ⊂ W . The latter is seen without difficulty to work also for locally compact abelian groups acting integrably on the algebras F(O). This is due to the fact [48, Corollary 21.3] that F (O) is generated by the operators transforming under the action of G by multiplication with a character. Finally, since H is separable due to the split property, the integrability property follows (for separable G) by an application of [48, Proposition 20.12] if we assume ˆ there is a field operator in F (O) transforming according to γ. that for every γ ∈ G While for non-compact groups the argument in [46, Appendix] does not work, this assumption is physically very reasonable. It is satisfied, e.g., in the sine-Gordon model where G = Z. Lemma 3.3 will not be used in this paper. The considerations in the sequel are considerably more transparent in the case of purely bosonic, i.e. local nets. For this reason I prefer as in [39] to treat the pure Bose case first in order to avoid confusion by the inessential complications of the general Bose/Fermi situation. 3.2. Existence of soliton automorphisms: Bose case The following is a version of a well-known result from [16]. It follows by plugging condition 4 into [39, Proposition 4.2], but incorporates simplifications using strong additivity. We state the proof mainly for reference purposes in the next subsection were we treat the fermionic case.
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Lemma 3.4. Let O 7→ F (O) fulfill conditions 1–4. Then every locally normal endomorphism ρ of the quasilocal algebra F satisfying ρ F(O0 ) = id for some O ∈ K is an inner endomorphism of F , i.e. a direct sum of copies of the identity morphism. Proof. Let ρ be localized in O and choose a double cones K fulfilling O ⊂⊂ ˆ Thanks to the split property there exist type I factors M1 , M2 such that K ⊂⊂ O. ˆ . F(O) ⊂ M1 ⊂ F (K) ⊂ M2 ⊂ F (O)
(3.5)
˜ into itself whenever O ⊂ O, ˜ in particular ρ(M1 ) ⊂ By Haag duality ρ maps F (O) 0 ˆ ⊂ F (O)0 ∩ F (O) ˆ = F(K). Being localized in O, ρ acts trivially on M1 ∩ F (O) 0 ˆ F(O ∩ O ), where we have used condition 4. This implies ˆ 0 ∩ M2 ⊂ (M 0 ∩ M2 )0 ∩ M2 = M1 , ρ(M1 ) ⊂ (M10 ∩ F(O)) 1
(3.6)
the last identity following from M2 being type I. Thus ρ restricts to an endomorphism of M1 . Now every normal endomorphism of a type I factor is inner [34, Corollary 3.8], i.e. there is a (possibly infinite) family of isometries Vi ∈ M1 , i ∈ I P with Vi∗ Vj = δi,j , i∈I Vi Vi∗ = 1 such that ρ M1 = η M1 , where X Vi A Vi∗ (3.7) η(A) ≡ i∈I
is well-defined on B(H), the sums over I being understood in the strong sense. Again ˆ which implies Vi ∈ (F (O)0 ∩ M1 )0 ∩ ρ acts trivially on F(O)0 ∩ M1 ⊂ F (O)0 ∩ F (O), M1 = F(O) ∀ i ∈ I. Therefore in addition to ρ = η on F (O) we have ρ = η = id on F (O0 ). But now local normality of ρ and strong additivity, cf. Lemma 3.1, imply ρ = η on all local algebras, thus on F . Now we can identify a necessary condition for the existence of soliton automorphisms. ˆ and let Proposition 3.5. Let F satisfy the assumptions 1–4. Let O ⊂⊂ O O1 O2 O1 , O2 be as in Fig. 1. Let ρg , ρg be locally normal soliton automorphisms. Then O2 −1 1 ˆ unique up to a phase, such that ρO there is a unitary UgΛ ∈ F(O), = g ◦ (ρg ) Λ Λ Λ Ad Ug . Furthermore, Ad Ug F (O) = αg and Ad Ug leaves F (O1 ) and F (O2 ) stable. Proof. It is obvious from the definition of soliton automorphisms that γ = O2 −1 1 ˆ 0 ) and as αg on F (O). Now by Lemma 3.4 acts trivially on F (O ρO g ◦ (ρg ) ˆ this implies that γ = Ad UgΛ where UgΛ ∈ U(F (O)). UgΛ is unique up to a phase O1 by irreducibility of F. Now, ρg acts as an inner symmetry on F (O2 ) and leaves Λ 2 F(O1 ) stable due to Haag duality. Arguing similarly for ρO g we see that γ = Ad Ug leaves F (O1 ) and F(O2 ) stable. Remarks. 1. This result is the analog in the broken symmetry case of [37, Lemma 2.3] which stated the uniqueness of disorder operators up to localized unitaries.
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2. Given a family of soliton automorphisms we see that transportability is automatic. This fact will play a crucial role in our proof of Poincar´e covariance. By Lemma 3.2 local implementers exist also in the case of broken symmetry. ˆ ∩ F (O)0 = Whereas a local implementer XgΛ acts as an automorphism on F (O) F(O1 ) ∨ F(O2 ), the above property of leaving F(O1 ) and F (O2 ) separately stable is stronger. We will now show the converse, viz. the existence of local implementers with this additional property implies the existence of soliton automorphisms. Proposition 3.6. Let F satisfy assumptions 1–4. If for some g ∈ G and all ˆ such that Ad U Λ F (O) = αg and ˆ O ⊂⊂ O ˆ there is U Λ ∈ U(F (O)) Λ = (O, O), g g Λ such that Ad Ug restricts to automorphisms of F (Oi ), i = 1, 2 then there are locally normal left soliton automorphisms ρO g ∀ O ∈ K. O1 ˆ2 ). Let O Proof. Choose two double cones O1 and O2 O1 (i.e. O2 ⊂⊂ WLL 0 0 ˆ be the smallest double cone containing O1 and O2 , and let O = O2 ∩ O1 ∩ O2 . Then ˆ2 ) implementing αg on F (O) and implementing autothere is a unitary z2 in F (O morphisms of F(O1 ), F(O2 ). Our aim will be to construct a soliton automorphism which acts like Ad z2 on F(O1 ) and like αg on the left complement of O1 . Consiˆ3 the smallest double cone dering thus another double cone O3 O2 we denote by O ˜ containing O1 and O3 (thus also O2 ) and by O3 the smallest double cone containing O2 and O3 . Again there is z˜3 implementing αg on the double cone between O1 ˆ3 ) and O3 and acting on F (O1 ), F (O3 ) by automorphisms. Now, X = z˜3∗ z2 ∈ F (O ˜ commutes with F(O), which by condition 4 implies X ∈ F(O1 ) ∨ F (O3 ). This ˜3 ) and the adjoint algebra being isomorphic to the tensor product F (O1 ) ⊗ F (O action of X implementing an automorphism of F (O1 ), the lemma below implies ˜3 )). Defining z3 = z˜3 X1 we X = X1 X3 where X1 ∈ U(F (O1 )), X3 ∈ U(F (O have Ad z3 F(O1 ) = Ad z2 F (O1 ) and z3 still implements αg on the double cone between O3 and O1 . Let now On be a sequence of double cones tending to left spacelike infinity. More precisely, for every O ∈ K there is a N ∈ N such that On < O ∀ n > N . In the above way we can construct operators zn , n ∈ N implementing αg on the double cone between O1 and On and such that Ad zn F (O1 ) = Ad z2 F (O1 ) ∀ n. Defining ∗ 1 ρO g (A) = k · k − lim zi Azi , i→∞
1 it is clear that ρO g is a locally normal soliton automorphism.
(3.8)
Lemma 3.7. Let M1 , M2 be factors and let U ∈ U(M1 ⊗ M2 ) be such that U (M1 ⊗ 1)U ∗ = M1 ⊗ 1. Then U = U1 ⊗ U2 where Ui ∈ U(Mi ). Proof. Due to factoriality we have 1 ⊗ M2 = (M1 ⊗ M2 ) ∩ (M1 ⊗ 1)0 . Thus α = Ad U stabilizes also 1 ⊗ M2 and factorizes: α = α1 ⊗ α2 . α being inner, the same holds for α1 , α2 by [48, Proposition 17.6]. Thus there are unitaries Ui ∈ Mi such that α = Ad U1 ⊗ U2 . Since we are dealing with factors the inner implementer is unique up to a phase and U1 , U2 can be chosen such that U = U1 ⊗ U2 .
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Remark. Obviously the proof of the above proposition is in the spirit of Roberts’ local cohomology theory. Up to now have established a necessary and sufficient criterion for the existence of arbitrarily localizable soliton automorphisms. We conclude this subsection by giving sufficient criteria in terms of the localized implementers UgΛ for the soliton automorphisms to satisfy conditions (A), (B). ˆ there are localized implementers Proposition 3.8. If for every Λ = (O, O) Λ which besides the properties required in Proposition 3.6 satisfy αk (UgΛ ) = Ukgk −1 then there are soliton automorphisms satisfying property (A). If there are UgΛ ’s such Λ then there are soliton automorphisms satisfying property (B). that UgΛ UhΛ = Ugh Λ If there are Ug ’s satisfying both conditions then there are soliton automorphisms fulfilling (A) and (B).
UgΛ
O O Proof. By the definition of soliton automorphisms it is obvious that ρO g ρh = ρgh O O 0 and αk ◦ ρg = ρkgk−1 ◦ αk are satisfied in restriction to F(O ). By strong additivity and local normality of the soliton automorphisms it suffices to prove the above rela1 tions for F(O). But since there are soliton automorphisms such that ρO g F (O1 ) = Λ Ad Ug for some Λ, the claimed implications are obvious consequences of the assumptions.
Remark. The conditions given above are stronger than necessary, since the 1 which is constructed adjoint action of UgΛ on O2 does not leave traces in the ρO g in the theorem. For our purposes the above result is sufficient. Anyhow, a detailed investigation of when properties (A), (B) can be satisfied, part of which will be found in [40], would have to be cohomological. 3.3. Existence of soliton automorphisms: General case In the local case condition 4 immediately gives us the relative commutant of two double cone algebras. In the general case we instead have the following: Lemma 3.9. With the notation of Fig. 1 we have ˆ ∧ F(O)0 = (F(O1 ) ∨ F(O2 ))+ + (F (O1 ) ∨ F(O2 ))− U Λ , F(O) v
(3.9)
ˆ implementing α− on F (O). where UvΛ is an arbitrary Bose unitary in F (O) ˆ + ∧F (O)0 = Proof. The Bose part of the left-hand side in (3.9) is given by F (O) t0 t ˆ ˆ ˆ F(O)+ ∧F (O) , where we have used (F (O)+ ) = F (O)+ . Using condition 4 we see that for the Bose parts (3.9) is correct. As to the Fermi part we know by Lemma 3.2 that a UvΛ as needed exists, and it is easy to see that (F (O1 ) ∨ F (O2 ))− UvΛ ⊂ ˆ − ∧ F (O)0 then XY ∈ F (O) ˆ + ∧ F (O)0 for ˆ − ∧ F(O)0 . Conversely, if X ∈ F (O) F(O) Λ 0 ˆ a unitary Y ∈ (F(O1 ) ∨ F(O2 ))− Uv . Since F (O) ∧ F(O) is an algebra it contains also X.
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In the sequel an automorphism will be called even if it commutes with α− , i.e. respects the Z2 grading. We begin by reconsidering Proposition 3.6, assuming the assumptions made there plus condition 5. All geometrical notions are as in the proof of Proposition 3.6 and let z2 be as defined there, but in addition we need to assume that Ad z2 acts on F (O1 ), F (O2 ) by even automorphisms. By the split property F(O1 ) ∨ F (O2 )t is isomorphic to the tensor product, thus a factor. That also F(O1 ) ∨ F (O2 ) is factorial is proved by exactly the same argument as in [37, Corollary 3.13]. (Here condition 5 is used.) By Lemma 3.2 there is a Bose implementer V1 of α− F (O1 ) which commutes with F (O2 ). Using the split property it is easy to see that Ad V1 acts as an automorphism on (F(O1 ) ∨ F(O2 ))+ with fixpoint algebra F (O1 )+ ∨ F(O2 )+ . Since Ad zi F (O1 ) is an even automorphism, V1 XV1∗ X ∗ commutes with F (O1 ) and the same holds trivially for F(O2 ). Thus by factoriality of F (O1 ) ∨ F (O2 ) we have V1 XV1∗ = ±X. In case the minus sign occurs we replace z2 by z2 Y1 Y2 with Yi ∈ F (Oi ), i = 1, 2 Fermi unitaries. Since Y1 Y2 is bosonic the required implementation properties of z2 are not affected. We may thus assume that z2 commutes with V1 , and ˆ3 )+ ∩ F (O)0 , the the same holds for the z˜n ’s. Reconsidering X = z˜3∗ z2 ∈ F(O ˜ Bose part of Lemma 3.9 implies X ∈ (F(O1 ) ∨ F (O3 ))+ . By the above we can ˜3 )+ . The assume that X commutes with V1 which yields X ∈ F(O1 )+ ∨ F (O latter algebra being isomorphic to a tensor product the rest of the proof works 1 as in the Bose case and (3.8) defines an even soliton automorphism ρO g . Since O1 all the zn ’s commute with V1 , we have ρg (V1 ) = V1 . Returning to our stan1 is bosonic in the dard notation for localized implementers this means that ρO g following sense. Definition 3.10. Let ρ be an endomorphism of F which is localized in O in ˆ + be a localized bosonic implementer the usual or solitonic sense. Let UvΛ ∈ F (O) for α− F (O). Then ρ is called bosonic if ρ(UvΛ ) = UvΛ . Remarks. 1. This definition is independent of the choice of UvΛ since for another ˆ + we have U ˜ Λ U Λ∗ ∈ F (O) ˆ + ∧ F(O)0 = F (O ˆ ∩ O0 )+ , on which ρ ˜ Λ ∈ F(O) choice U v
v
v
acts trivially. 2. If ρ is implemented (on a bigger Hilbert space) by a multiplet of field operators Ψi then ρ is bosonic iff the Ψi commute with UvΛ , i.e. are Bose fields. O2 1 Let conversely the soliton automorphisms ρO g , ρg considered in Proposition 3.5 O1 2 −1 clearly is a bosonic in addition be even and bosonic. Then γ = ρg ◦ (ρO g ) ˆ even endomorphism localized in the double cone O of Fig. 1. As shown in [39] ˆ into itself and in Lemma 3.4 we have evenness implies that γ still maps F (O) γ(M1 ) ⊂ F(K). Another change in the latter Lemma is due to the fact that in the ˆ appearing in the proof case with fermions the relative commutant F(O)0 ∩ F (O) 0 ˆ is not given by F(O ∩ O ), but instead as in Lemma 3.9. (Note that the O in ˆ wheras the O ˆ appearing there has nothing Lemma 3.4 corresponds to the above O, to do with that above!) Yet γ still acts trivially on the relative commutant, since it is bosonic and thus acts trivially on the UvΛ in Lemma 3.9. Thus the conclusion
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ˆ such that γ = Ad U Λ . of Lemma 3.4 holds and we have a Bose unitary UgΛ ∈ F (O) g Λ Clearly Ug implements αg F (O) and even automorphisms of F(O1 ), F (O2 ). We summarize the results of the preceding discussion in the following: Theorem 3.11. Let F satisfy conditions 1–5. Then the following are equivalent (i) There are [bosonic even] soliton automorphisms ρO g ∀ g ∈ G, O ∈ K. ˆ O ⊂⊂ O ˆ there is a [bosonic] unitary imple(ii) For every g ∈ G, Λ = (O, O), ˆ for αg F (O) whose adjoint action implements [even] menter UgΛ ∈ F(O) automorphisms of F(Oi ), i = 1, 2. (In the pure Bose case omit the words within square brackets.) Proposition 3.8 remains true in the Bose/Fermi case. 3.4. Poincar´ e covariance Up to now we have seen that under certain conditions there are even soliton automorphism which are transportable with Bose intertwiners. In [29, Theorem 5.2] it was proved that every transportable sector which is localizable in wedges and has finite statistics is Poincar´e covariant provided certain conditions on the net are satisfied. Unfortunately this result cannot applied since soliton automorphisms typically are non-normal on the algebra of the wedge in which they are localized, cf. Lemma 3.3. We will thus adopt another approach, always assuming the conditions 1–5 on the net F. Lemma 3.12. Let ρO g be a bosonic even soliton automorphism. Then for every ˆ containing O and ΛO + x the automorphism β(Λ,x) (Λ, x) ∈ P and a double cone O defined by −1 O α(Λ,x) ◦ ρO g ◦ α(Λ,x) (F ) = β(Λ,x) ◦ ρg (F )
∀F ∈ F
(3.10)
ˆ which is determined up to an arbitrary is implemented by a unitary Z(Λ, x) ∈ F (O) phase. The map −1 ◦ α(Λ,x) P 3 (Λ, x) 7→ αρ(Λ,x) ≡ β(Λ,x)
(3.11)
is a group homomorphism. −1 Proof. α(Λ,x) ◦ρO g ◦α(Λ,x) is a bosonic even soliton automorphism for g localized in ΛO + x. Thus by Lemma 3.4 and the discussion in the preceding subsection ˆ with O ˆ as above, such that (3.10) holds with there is a unitary Z(Λ, x) in F (O) β(Λ,x) = Ad Z(Λ, x). By irreducibility of the quasilocal algebra Z(Λ, x) is unique −1 O −1 ◦ α(Λ,x) = ρO , and the group property up to a phase. Now β(Λ,x) g ◦ α(Λ,x) ◦ (ρg ) ρ of α is obvious.
Proposition 3.13. The automorphisms αρ(Λ,x) of the preceding lemma extend uniquely to B(H). The action P 3 (Λ, x) 7→ αρ(Λ,x) ∈ Aut B(H) is continuous.
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Remark. The natural topology on the automorphism group of a von Neumann algebra A is the u-topology, viz. the topology defined by norm convergence on the predual, since it turns Aut A into a topological group. For a type I factor B(H) this topology coincides with the p-topology, which is the restriction to Aut B(H) of the pointwise weak topology, cf. [48, pp. 41–43]. Proof. The extendibility statement is obvious since αρ(Λ,x) is unitarily implemented, and uniqueness follows since F is irreducible. It clearly suffices to prove ˆ continuity for a neighbourhood V ⊂ P of the unit element e = (1, 0). Let O, O ˆ be double cones such that ΛO + x ⊂ O ∀ (Λ, x) ∈ V. Then by the lemma we have −1 O −1 β(Λ,x) = α(Λ,x) ◦ ρO g ◦ α(Λ,x) ◦ (ρg )
(3.12)
ˆ Thus the β(Λ,x) , (Λ, x) ∈ V are with β(Λ,x) = Ad Z(Λ, x) and Z(Λ, x) ∈ F(O). ˆ Furthermore, for every inner automorphisms of every algebra which contains F (O). localized F the map (Λ, x) 7→ β(Λ,x) (F ) is strongly continuous since ρO g is locally ˜ ⊃⊃ O ˆ be another double cone and normal and α is a continuous action. Let now O ˆ ⊂ M ⊂ F (O), ˜ the existence of which is let M be a type I factor such that F (O) guaranteed by the split property. By the above β(Λ,x) , (Λ, x) ∈ V acts continuously ˜ thus on M and trivially on M 0 ⊂ F (O) ˆ 0 . But B(H) = M ∨ M 0 ' M ⊗ M 0 on F(O), implies that β(Λ,x) , (Λ, x) ∈ V acts continuously on B(H). (We have used that, w.r.t. the u-topologies, αι → α implies αι ⊗ id → α ⊗ id and that M , M 0 , B(H) are type I, such that the above remark on the topologies applies.) Since Aut B(H) is a −1 ◦ α(Λ,x) of P on B(H) is topological group also the action (Λ, x) 7→ αρ(Λ,x) = β(Λ,x) continuous for (Λ, x) ∈ V, thus for all of P. e covariant, i.e. in Theorem 3.14. The soliton automorphism ρO g is Poincar´ there is a strongly continuous representation U ρ (Λ, x) of the the sector π0 ◦ ρO g O Poincar´e group with positive energy such that Ad U ρ (Λ, x) ◦ π0 ◦ ρO g = π0 ◦ ρg ◦ α(Λ,x) . Proof. By the preceding results we are in a position to apply a result of Kallman and Moore [48, Theorem 15.16] according to which every continuous one parameter group of inner automorphisms of a von Neumann algebra M with separable predual is implemented by a strongly continuous unitary representation in M . (Since B(H) is a factor the proof by Hansen, reproduced in [48, p. 218], is sufficient for our purposes.) Thus we have ρ
αρΛ = Ad eiΛK ,
ρ
αρ+,a = Ad eiaP+ ,
ρ
αρ−,b = Ad eibP− ,
(3.13)
where αρ± are the lightlike translations and K ρ , P+ρ , P−ρ are self-adjoint operators on H. From now on we omit the superscript ρ. The unitary implementer of an automorphism being unique up to a phase the commutation relation αΛ ◦ α+,a = Λ α+,eΛ a ◦αΛ in P together with (3.13) implies eiΛK eiaP+ = c(a, Λ) eie aP+ eiΛK , where c is a continuous phase-valued function satisfying c(a1 + a2 , Λ) = c(a1 , Λ)c(a2 , Λ)
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and c(a, Λ1 + Λ2 ) = c(a, Λ1 )c(eΛ1 a, Λ2 ). Together with continuity the first equation implies c(a, Λ) = eiad(Λ) , where d satisfies d(Λ1 + Λ2 ) = d(Λ1 ) + eΛ1 d(Λ2 ). The lefthand side of this equation being symmetric in Λ1 , Λ2 we have d(Λ1 ) + eΛ1 d(Λ2 ) = d(Λ2 ) + eΛ2 d(Λ1 ), which for Λ2 6= 0 gives d(Λ1 ) = (eΛ1 − 1)d(Λ2 )/(eΛ2 − 1) = A(eΛ1 − 1). Proceding similarly for the other commutation relations we thus have Λ
eiΛK eiaP+ = eiAa(e
−1)
−Λ
eiΛK eibP− = eiBb(e
Λ
eie
−1)
aP+ iΛK
e
−Λ
eie
,
bP− iΛK
e
(3.14) ,
eiaP+ eibP− = eiCab eibP− eiaP+ ,
(3.15) (3.16)
where A, B, C ∈ R are independent. By differention we obtain the commutation relations i[K, P+ ] = P+ + A1 ,
i[K, P− ] = −(P− + B1) ,
i[P+ , P− ] = C1 ,
(3.17)
which we need not make precise. Now, the generators K, P+ , P− are determined by (3.13) only up to addition of a multiple of 1. Using this freedom we can replace P+ Λ by P+ +A1, which removes the factor eiAa(e −1) from (3.14). This fixes P+ uniquely and achieves that the Lorentz group acts on P+ as dilatations: eiΛK P+ e−iΛK = eΛ P+ . Thus the spectrum of P+ is one of the sets {0}, [0, ∞), (−∞, 0], R. In any case 0 ∈ Sp(P+ ). P− is treated similarly. For K there is no prefered normalization, since shifting its origin does not affect the relations (3.14–3.16). (This is just the fact that the Poincar´e group in 1+1 dimensions has non-trivial one dimensional representations (Λ, a) 7→ eiCΛ , C ∈ R.) (3.16) is not affected by shifting P± , thus the constant C cannot be changed. (This reflects the fact that the Lie algebra cohomology H 2 (P, R) is one dimensional.) We have a true representation of the Poincar´e group iff C vanishes. Now we observe that C 6= 0 implies Sp(P+ ) = Sp(P− ) = R, cf. e.g. [2]. (For λ ∈ Sp(P− ) the differentiated commutation relation eiaP+ P− e−iaP+ = P− + Ca1 implies λ+ Ca ∈ Sp(P− ) ∀ a ∈ R. Since Sp(P− ) cannot be empty C 6= 0 implies Sp(P− ) = R.) Before we know that C = 0 it makes, of course, no sense to consider the joint spectrum of P+ and P− . We will however prove that P+ ≥ 0, P− ≥ 0. By the above this then entails C = 0 and positivity in the usual sense: Sp(P µ ) ⊂ V + . We consider only P+ since the argument for P− is the same. The proof is modeled on the one [15] for DHR sectors. There are simplifications since we are dealing with soliton automorphisms, thus ρ¯ = ρ−1 is a α+ -covariant soliton automorphism, too. Yet, we spell the proof out since there we have to use lightlike instead of spacelike clustering. The spectra of P+ρ , P+ρ¯ containing 0, positivity of P+ρ , P+ρ¯ follows from positivity in the vacuum sector if we prove Sp(P+ρ ) + Sp(P+ρ¯ ) ⊂ Sp(P+π0 ) .
(3.18)
Let N1 , N2 be arbitrary open sets in R intersecting Sp(P+ρ ), Sp(P+ρ¯ ), respectively. Then there is a vector Ψ1 6= 0 with P+ρ -support in N1 and, by [15, Lemma 5.1], a ρ B ∈ A such that Ψ2 = BΩ 6= 0 has P+ρ¯ -support in N2 . Now Ψa = ρ(B)e−iaP+ Ψ1
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has P+π0 -support in N1 + N2 for all a ∈ R and we are done if there is an a such that Ψa 6= 0. With ρ
ρ
0 kΨa k2 = (Ψ1 , eiaP+ ρ(B ∗ B)e−iaP+ Ψ1 ) = (Ψ1 , ρ(απ+,a (B ∗ B))Ψ1 )
(3.19)
such an a exists if the second step in the computation 0 (B ∗ B))Ψ1 ) = kΨ1 k2 · ω0 (B ∗ B) = kΨ1 k2 · kΨ2 k2 6= 0 lim kΨa k2 = lim(Ψ1 , ρ(απ+,a
?
(3.20) is justified for a → +∞ or a → −∞. The cluster result [19, Proposition 1.2] for lightlike translations gives us weak convergence: 0 (B ∗ B) = ω0 (B ∗ B) 1 ∀ B ∈ A . w − lim απ+,a
|a|→∞
(3.21)
(This result uses that P+π0 has half-sided spectrum and the Reeh-Schlieder theorem in the vacuum representation.) Assume that the soliton automorphism ρ is localized in a left wedge. Then it acts normally on the algebras of all right wedges. If B is localized in a bounded region then there is a right wedge W such that α+,a (B ∗ B) ∈ 0 (B ∗ B))Ψ1 ) A(W ) ∀ a ≥ 0 and (3.20) holds for a → +∞. The maps B 7→ (Ψ1 , ρ(απ+,a being uniformly bounded and the stricly local operators being norm dense in A, (3.20) holds for all B ∈ A and a → +∞ and we are done. If ρ is right-localized then let a → −∞. 4. The Solitons of Theories with the Local Fock Property As an application of Theorem 3.11 we will give in this section a new construction of the soliton sectors of the superrenormalizable quantum field theories with the local Fock property, like the P (φ)2 theory (the polynomial P is assumed even in order to have Z2 symmetry) or the Yukawa2 model. We briefly indicate the main steps of the hamiltonian approach to the construction of these models. One begins with a finite number of massive free fields and a formal interaction hamiltonian HI . We are interested in the case where the non-interacting theory has a group G of inner symmetries which leave HI formally invariant and which survives the renormalization (i.e. there are no anomalies) but which may be spontaneously broken. Furthermore, we assume that G commutes with the Poincar´e action on the free and the interacting theory. In the rigorous construction of the models, which we sketch on the example of the P (φ)2 model, one first obtains the interacting field φ˜ on the Fock space of the free field φ0 (as a quadratic form) via ˜ t) = U I (t) φ0 (x, 0) U I (t)∗ φ(x,
(4.1)
for x in an interval I, where U I is a propagator obtained by smoothly cutting off ˜ 0) = φ0 (x, 0).) One can show that the interaction outside I. (In particular φ(x, ˜ the field φ(x, t) is independent of the form of the cut off of the interaction provided (x, t) is contained in the double cone with basis (I, t = 0). φ˜ carries an action of the Poincar´e group, but the action is not unitarily implemented and there is no
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invariant vacuum vector. Still there is an invariant vacuum state ω0 , and by GNS construction one obtains the physical representation π of the quasilocal algebra F and a unitary representation of the Poincar´e group. In restriction to arbitrary double cones ω0 is normal which implies local normality of π. This in turn implies the existence of the physical field φ(x, t). Quantum field theories which can be constructed in the above way (including renormalizations of the hamiltonian where necessary) are said to possess the local Fock property [28]. This property is the only piece of information on these models which we will need since (4.1) in conjunction with the local normality of π allows to carry over many properties from the free fields to the interacting ones. Lemma 4.1. The free massive fields permit local implementers UgΛ of the unbroken symmetries with the property required in Theorem 3.11. The UgΛ can be Λ Λ Λ Λ chosen such that αk (UgΛ ) = Ukgk −1 and Ug Uh = Ugh . Proof. Unbroken symmetries are unitarily implemented, and the split property for wedges implies the existence of disorder operators [25, 37] ULO (g) implementing the αg on the left complement of O, acting trivially on the right complement and implementing an even automorphism of F (O). The latter is even since the disorder ˆ and let O1 , O2 be as in Fig. 1. operators commute with V . Now let O ⊂⊂ O ∗ ˆ 0 , implements αg on O and maps Clearly, U Λ (g) = ULO1 (g)ULO2 (g) acts trivially on O ˆ 0 )0 = F (O). ˆ the algebras F(Oi ), i = 1, 2 into themselves. By duality, U Λ ∈ F (O Oi Oi Λ Λ ∗ −1 Furthermore, αk (Ug ) = Ukgk−1 follows from U (k)UL (g)U (k) = UL (kgk ), i = 1, 2 [37] and we have UgΛ UhΛ = ULO1 (g)ULO2 (g)∗ ULO1 (h)ULO2 (h)∗ = ULO1 (g)ULO1 (h)ULO2 (h−1 g −1 h)ULO2 (h−1 ) ∗
Λ , = ULO1 (gh)ULO2 (gh) = Ugh O1 00 ) . where we have used UgO2 ∈ F (WLL
(4.2)
Remark. The unitaries UgΛ can be shown to satisfy and to be determined ˆ 0 . Here η is the unique vector in by UgΛ ABη = αg (A)Bη ∀ A ∈ F (O), B = F (O) ˆ 0 , Ω) implementing the state ωη (AB) = ω0 (A)ω(B), A ∈ F (O), B = P \ (F(O)∨F(O) 0 ˆ F(O) , where ω in turn is the product state (existing due to the SPW) which ˆ ˆ O O ) and on F (WRR ). Recall that the usual localized implerestricts to ω0 on F(WLL menter [8, 17] is obtained by replacing the product state ω by ω0 . We summarize the properties of the interacting theory obtained as indicated above. Lemma 4.2. Theories with local Fock property satisfy the assumptions 1–5 of Sec. 3 in their vacuum sector. Proof. All these properties are fulfilled by the free scalar and Dirac fields of non-zero mass [1, 7, 49] as well as by theories of finitely many such fields, since
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they follow from twisted duality and the split property for wedges [38] which is known to be fulfilled. (Apart from perhaps condition 5 all these properties have been known before.) The last four properties are of a purely local nature, thus they carry over immediately to the interacting theory by the local Fock property. The nontrivial fact that also duality, which is a global property, carries over to the interacting theory has been proven in [21] for local nets and in [49, Theorem 2.8] for the twisted case. Lemma 4.3. The theories with local Fock property permit local implementers of their inner symmetries with the property required in Theorem 3.11 and satisfying Λ Λ Λ Λ in addition αk (UgΛ ) = Ukgk −1 , Ug Uh = Ugh . Proof. By Poincar´e covariance and the fact that G and P commute it suffices to prove the existence of soliton automorphisms for double cones O which have as ˆ with respective bases I ⊂⊂ basis an interval I in the line t = 0. Let O ⊂⊂ O ˜ ˆ I. In view of F0 (O) = F (I) = F (O) a local implementer UgΛ be the free field (provided by the Lemma 4.1) also implements αg on F˜ (O). On the Fock space the group G is unitarily implemented, and since the disorder operators for the free field behave covariantly, the localized implementers have all desired properties also in the physical representation π. Putting things together we obtain our main result. Theorem 4.4. Irreducible quantum field theories in 1 + 1 dimensions which have the local Fock property w.r.t. a finite number of free massive scalar and/or Dirac fields admit Poincar´e covariant locally normal soliton automorphisms which can be chosen even and bosonic and satisfying properties (A) and (B). Proof. In view of Lemmas 4.2 and 4.3 the existence of even bosonic soliton automorphisms follows from Theorem 3.11 and the Poincar´e covariance from Theorem 3.14. Properties (A) and (B) are a consequence of Proposition 3.8. Remark. After we proved the above theorem we discovered that the essential idea of the existence part is already contained in [25, pp. 402–404]. Still, our results go beyond those of [25] in several respects. Theorem 3.11 provides a convenient sufficient and necessary condition for the existence of soliton automorphisms, which also applies to the twisted sectors of holomorphic conformal theories. Furthermore, in our proof of Poincar´e covariance we do not appeal to any result of constructive QFT except the local Fock property, which, to be sure, is a rather deep result. Finally, the fact that one finds soliton automorphisms with the covariance (A) and homomorphism (B) properties is new. 5. Summary and Outlook Theorem 2.4 completes the abstract treatment of the duality between massive theories (satisfying twisted duality and the SPW) with unbroken compact abelian
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and broken abelian symmetry groups, respectively, which was begun in [37]. The central theme, however, of this work was the observation that soliton automorphisms of massive quantum field theories in 1 + 1 dimensions and twisted sectors of holomorphic conformal field theories can be treated on equal footing. The crucial property shared by these apparently unrelated classes of models is the condition 4 in Sec. 3. Section 4 finally complements and extends the earlier rigorous works on solitons [22–25, 47] by establishing sufficiency of the local Fock property for the existence of Poincar´e covariant soliton automorphisms. The results of that section make plain that contrary to a widespread belief a profound understanding of the free fields is quite useful, if not necessary, in the study of interacting models, at least of those with the local Fock property. Our analysis relies on several deep results from the theory of automorphisms of von Neumann algebras. The theorems of Connes/Takesaki and Kallmann/Moore, used in Theorems 2.4 and 3.14, respectively, make statements on the existence of continuous unitary group representations inside von Neumann algebras. Crucial were furthermore the facts that unital normal endomorphisms of type I factors are inner, and that inner tensor-product automorphisms factorize into inner automorphisms. As emphasized the results of Sec. 3 apply also to conformal theories. This will be the basis for a rigorous analysis of conformal orbifold models in [40]. As mentioned in the Introduction the present analysis was partially motivated by the desire to understand why the chiral Ising model [36, 4] does not fit into the analysis of orbifold models given in [13]. As will be discussed further in [40] this is due to the fact that real fermions on the circle do not admit soliton automorphisms whereas the existence of the latter — called “twisted sectors” — is implicitly assumed in [13]. (In [32] the existence of twisted sectors is proved for a class of WZNW models.) A chiral theory of complex fermions or, more generally, of an even number of real fermions does possess soliton automorphisms which is why for these theories the fusion rules of the Z2 orbifold theory [5] are given by Z2 × Z2 or Z4 in accordance with [13]. As to solitons in massive models, it would be very interesting to have a proof, to the largest possible extent model independent, of bounds on the soliton mass of the sort proven in [3]. Furthermore, one should try to extend Theorem 4.4 to more general models which do not possess the local Fock property. Acknowledgments I would like to thank K. Fredenhagen, D. Guido, R. Longo, K.-H. Rehren and J. Roberts for useful discussions concerning this work. I am particulary indebted to D. Guido for pointing out an error in the proof of Theorem 3.14 in an earlier version of this paper. Note Added in Proof I thank D. Buchholz for drawing my attention to a small problem in the proof of Theorem 3.14 in the case of massless theories. Namely, in 1+1 dimensions Driessler’s
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lightlike cluster theorem can be proved only assuming a mass gap. As also noted by ρ Buchholz, the problem is easily circumvented considering Ψa = ρ(B)e−iaP1 Ψ1 and the limit a → +∞ (if ρ is localized in a left wedge). The point is that commutatively of P+ρ and P1ρ is not needed. Then the usual spacelike cluster theorem applies and the argument goes through otherwise unchanged. References [1] H. Araki, “A lattice of von Neumann algebras associated with the quantum theory of a free Bose field”, J. Math. Phys. 4 (1963) 1343–1362. [2] K. Baumann, “Quantum fields in 1 + 1 dimension carrying a true ray representation of the Poincar´e group”, Lett. Math. Phys. 25 (1992) 61–73. [3] J. Bellissard, J. Fr¨ ohlich and B. Gidas, “Soliton mass and surface tension in the (φ4 )2 quantum field model”, Commun. Math. Phys. 60 (1978) 37–72. [4] J. B¨ ockenhauer, “Localized endomorphisms of the chiral Ising model”, Commun. Math. Phys. 177 (1996) 265–304. [5] J. B¨ ockenhauer, “An algebraic formulation of level one Wess–Zumino–Witten models”, Rev. Math. Phys. 8 (1996) 925–947. [6] R. Brunetti, D. Guido and R. Longo, “Modular structure and duality in QFT”, Commun. Math. Phys. 156 (1993) 201–219. [7] D. Buchholz, “Product states for local algebras”, Commun. Math. Phys. 36 (1974) 287–304. [8] D. Buchholz, S. Doplicher and R. Longo, “On Noether’s theorem in quantum field theory”, Ann. Phys. 170 (1986) 1–17. [9] D. Buchholz, S. Doplicher, R. Longo and J. E. Roberts, “A new look at Goldstone’s theorem”, Rev. Math. Phys. Special Issue (1992) 49–83. [10] D. Buchholz, G. Mack and I. Todorov, “The current algebra on the circle as a germ of local field theories”, Nucl. Phys. B (Proc. Suppl.) 5B (1988) 20–56. [11] F. Ciolli, “Simmetrie di gauge spontaneamente rotte: Tracce osservabili e strutture matematiche associate”, unpublished diploma thesis, Rome 1997. [12] S. Coleman, “There are no Goldstone bosons in two dimensions”, Commun. Math. Phys. 2 (1966) 259–264. [13] R. Dijkgraaf, C. Vafa, E. Verlinde and H. Verlinde, “The operator algebra of orbifold models”, Commun. Math. Phys. 123 (1989) 485–527. [14] S. Doplicher, R. Haag and J. E. Roberts, “Fields, observables and gauge transformations II”, Commun. Math. Phys. 15 (1969) 173–200. [15] S. Doplicher, R. Haag and J. E. Roberts, “Local observables and particle statistics II”, Commun. Math. Phys. 35 (1974) 49–85. [16] S. Doplicher, “Local aspects of superselection rules”, Commun. Math. Phys. 85 (1982) 73–86. [17] S. Doplicher and R. Longo, “Standard and split inclusions of von Neumann algebras”, Invent. Math. 75 (1984) 493–536. [18] S. Doplicher and J. E. Roberts, “Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics”, Commun. Math. Phys. 131 (1990) 51–107. [19] W. Driessler, “Comments on lightlike translations and applications in relativistic quantum field theory”, Commun. Math. Phys. 44 (1975) 133–141. [20] W. Driessler, “On the type of local algebras in quantum field theory”, Commun. Math. Phys. 53 (1977) 295–297. [21] W. Driessler, “Duality and absence of locally generated superselection sectors for CCR-type algebras”, Commun. Math. Phys. 70 (1979) 213–220. [22] K. Fredenhagen, “Generalizations of the theory of superselection sectors”, in [33].
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[23] K. Fredenhagen, “Superselection sectors in low dimensional quantum field theory”, J. Geom. Phys. 11 (1993) 337–348. [24] J. Fr¨ ohlich, “New super-selection sectors (‘Soliton-states’) in two-dimensional Bose quantum field models”, Commun. Math. Phys. 47 (1976) 269–310. [25] J. Fr¨ ohlich, “Quantum theory of non-linear invariant wave (field) equations”, or “Super selection sectors in constructive quantum field theory”, in Invariant Wave Equations. eds. G. Velo and A. S. Wightman Proc. Erice, 1977. [26] F. Gabbiani and J. Fr¨ ohlich, “Operator algebras and conformal field theory”, Commun. Math. Phys. 155 (1993) 569–640. [27] P. Ghez, R. Lima and J. E. Roberts, “The spectral category and the Connes invariant”, J. Oper. Theor. 14 (1985) 129–146. [28] J. Glimm and A. Jaffe, “The (Φ4 )2 QFT without cutoffs III. The physical vacuum”, Acta Math. 125 (1970) 204–267. [29] D. Guido and R. Longo, “Relativistic invariance and charge conjugation in quantum field theory”, Commun. Math. Phys. 148 (1992) 521–551. [30] R. Haag, Local Quantum Physics, 2nd ed., Berlin, Springer Verlag, 1996. [31] Y. Haga, “Crossed products of von Neumann algebras by compact groups”, Tˆ ohoku Math. J. 28 (1976) 511–522. [32] V. Kac and I. T. Todorov, “Affine orbifolds and rational conformal field theory extensions of W1+∞ ”, Commun. Math. Phys. 190 (1997) 57–111. [33] D. Kastler, ed. The Algebraic Theory of Superselection Sectors. Introduction and Recent Results. World Scientific, 1990. [34] R. Longo, “Simple injective subfactors”, Adv. Math. 63 (1987) 152–171. [35] M. L¨ uscher and G. Mack, “Global conformal invariance in QFT”, Commun. Math. Phys. 41 (1975) 203–234. [36] G. Mack and V. Schomerus, “Conformal field algebra with quantum symmetry from the theory of superselection sectors”, Commun. Math. Phys. 134 (1990) 139–196. [37] M. M¨ uger, “Quantum double actions on operator algebras and orbifold quantum field theories”, Commun. Math. Phys. 191 (1998) 137–181. [38] M. M¨ uger, “Superselection structure of massive quantum field theories in 1 + 1 dimensions”, hep-th/9705019, to appear in Rev. Math. Phys. [39] M. M¨ uger, “On charged fields with group symmetry and degeneracies of Verlinde’s matrix S”, hep-th/9705018, to appear in Ann. Inst. H. Poincar´ e B (Phys. Th´ eor.). [40] M. M¨ uger, “On conformal orbifold models and the twisted quantum double”, in preparation. [41] M. R. Niedermaier, “A derivation of the cyclic form factor equation”, Commun. Math. Phys. 196 (1998) 411–428. [42] K.-H. Rehren, “Spin-statistics and CPT for solitons”, hep-th/9711085. [43] J. E. Roberts, “Spontaneously broken gauge symmetries and superselection rules”, in Proc. International School of Mathematical Physics, Camerino, 1974, ed. G. Gallavotti. [44] J. E. Roberts, “Some applications of dilatation invariance to structural questions in the theory of local observables”, Commun. Math. Phys. 37 (1974) 273–286. [45] J. E. Roberts, “Local cohomology and superselection rules”, Commun. Math. Phys. 51 (1976) 107–119. [46] J. E. Roberts, “Localization in algebraic field theory”, Commun. Math. Phys. 85 (1982) 87–98. [47] D. Schlingemann, “On the existence of kink-(soliton-) states in quantum field theory”, Rev. Math. Phys. 8 (1996) 1187–1203. [48] S. Strˇ atilˇ a, Modular Theory in Operator Algebras, Abacus Press, 1981. [49] S. J. Summers, “Normal product states for Fermions and twisted duality for CCR- and CAR-type algebras with application to the Yukawa2 quantum field model”, Commun. Math. Phys. 86 (1982) 111–141.
TWO-DIMENSIONAL COULOMB INTERACTIONS IN A MAGNETIC FIELD∗ JINGBO XIA Department of Mathematics State University of New York at Buffalo Buffalo, NY 14214 USA Received 21 March 1997 Revised 10 April 1998 We consider the 2D Hamiltonian of N charged particles in a magnetic field. We show how the Coulomb interactions cause the first Landau level to split in the essential spectrum.
1. Introduction Consider N particles with masses m1 , . . . , mN and charges s1 Z1 , . . . , sN ZN confined to the two-dimensional plane R2 , where sj = ±1 and Zj > 0, j = 1, . . . , N . Suppose that there is a homogeneous magnetic field perpendicular to R2 with magnitude β > 0. Discounting the (possible) spins of the particles, this system is represented by the Hamiltonian !2 N X 1 β ∂ 2 ˜ = −i + H 1 − sj Zj 2 xj 2m ∂x j j j=1 +
X 1≤i<j≤N
∂ β −i 2 + sj Zj x1j ∂xj 2
!2
si sj Z i Z j , |xi − xj |
where we write xj = (x1j , x2j ) ∈ R2 . The two terms on the right-hand side, Hmag and VCoul , respectively represent the interaction of the particles with the magnetic field and the Coulomb interactions among the particles themselves. The magnetic Hamiltonian Hmag is well understood: It is essentially self-adjoint on Cc∞ ((R2 )N ), its spectrum consists of Landau levels λ1 < λ2 < · · · < λn < · · · , and each λn has infinite multiplicity. Due to the singular nature of VCoul , however, ˜ = Hmag + VCoul is not nearly as well understood. In fact, because |xi − xj |−1 H is not square-integrable on compact sets in R2 × R2 , the multiplication by VCoul does not even map Cc∞ ((R2 )N ) into L2 ((R2 )N ). This contrasts sharply with threedimensional interactions [2–5]. ∗ Research
supported in part by National Science Foundation grant DMS-9400600. 361
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˜ as a self-adjoint operator, one must start with the quadratic form To realize H Z ¯ 2N , ϕ, ψ ∈ C ∞ ((R2 )N ) , VCoul ϕψdm [ϕ, ψ]H˜ = hHmag ϕ, ψi + c (R2 )N
which is known to be bounded from below. Thus one obtains a self-adjoint opera˜ F is ˜ F from [·, ·] ˜ through the well-known construction of Friedrichs [6,8]; H tor H H ˜ commonly referred to as the Friedrichs extension of H. In this paper we will study how the Coulomb interactions represented by VCoul ˜ F near the first Landau level λ1 . To be more precise, we affect the spectrum of H would like to know if the addition of VCoul causes λ1 to split in the essential spectrum ˜ F . In the simplest case where N = 2, m1 = m2 = me and s1 Z1 = s2 Z2 = −e, of H i.e. in the case of a pair of electrons, one can easily check that such splitting does occur. The purpose of this paper is to show that the splitting of λ1 occurs quite generally. ˜ F , in Sec. 2 we unitarily transform To facilitate the study of the spectrum of H ˜ F Z = (β/2)HF , where ˜ H so that its magnetic part is “standardized”. That is, Z ∗ H HF is the Friedrichs extension of r r N X 2 X si sj Z i Z j 2 s V = H µj Dj j + + H= 0 −1/2 −1/2 β β xi − Zj xj | j=1 1≤i<j≤N |Zi s
with µj = Zj /2mj and Dj j = (−i∂/∂x1j − sj x2j )2 + (−i∂/∂x2j + sj x1j )2 . In the polar coordinates (x1j , x2j ) = (rj cos θj , rj sin θj ), H0 has a natural core Ω = · · · + Ω−` + · · · + Ω0 + · · · + Ω` + · · · , where Ω` is the linear span of the states with total orbital angular momentum `. The quadratic form associated with H decomposes into orthogonal sum of forms with the Ω` ’s as their cores. Accordingly we have L∞ the orthogonal decomposition HF = `=−∞ HF,` . At this point we divide our investigation into two cases. In Sec. 3 we consider the case where all charges have the same sign. Without loss of generality, we may assume s1 = · · · = sN = −1. Since V is now nonnegative, we have σ(HF ) ⊂ [λ1 , ∞). (By λ1 we now mean the first Landau level of H0 .) We will show that λ1 actually lies in the essential spectrum of HF . When s1 = · · · = sN = −1, ker(H0,` − λ1 ) 6= {0} only if ` ∈ Z+ . Here H0,` is the closure of the restricted operator H0 |Ω` . Furthermore, for ` ∈ Z+ , the dimension `+N −1 = (` + N − 1)!/`!(N − 1)!. We will of the space Ω1` = ker(H0,` − λ1 ) is CN −1 `+N −1 th show that there is a c = c(m1 , . . . , mN , Z1 , . . . , ZN , β) > 0 such that the CN −1 P` `+N −1 m+N −3 ((1 + CN −1 − m=0 CN −3 )th if N ≥ 3) eigenvalue of HF,` is at least λ1 + c. This shows how the Coulomb interactions cause the states of H0 corresponding to λ1 to split. Whereas the proof that λ1 ∈ σess (HF ) requires an upper-bound estimate, this result requires a lower-bound estimate, which has to be handled more carefully. When the magnetic field is astronomically strong, that is, when (2/β)1/2 is small, it follows from the above eigenvalue estimate that σess (HF ) ∩ (λ1 , λ2 ) 6= ∅. Section 4 deals with a pair of oppositely charged particles. Here the Coulomb potential is attractive. We show that in this case the Coulomb interaction causes the bottom of the essential spectrum of HF,` to shift below λ1 .
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In the generality that we consider in this paper, the particles are possibly distinguishable by masses and/or charges. Therefore we can only state our results on L2 ((R2 )N ) = L2 (R2 ) ⊗ · · · ⊗ L2 (R2 ). In the case where the particles are actually indistinguishable, or at least when certain subsets of the particles are indistinguishable, our results are still valid on physical subspaces of L2 ((R2 )N ) which possess the right symmetries. Similarly, spins can also be accommodated. 2. Friedrichs Representation of Form Sum Suppose that there is a constant magnetic field of magnitude β > 0 in the direction perpendicular to R2 , where particles of masses m1 > 0, . . . , mN > 0 and charges s1 Z1 , . . . , s1 ZN are confined. Here, Z1 > 0, . . . , ZN > 0 are the absolute values of the charges and sj = ±1, 1 ≤ j ≤ N , are the signs of the charges. We assume N ≥ 2 for the rest of the paper. The magnetic Hamiltonian is then given by the formula !2 !2 N X ∂ 1 β β ∂ + −i 2 + sj Zj x1j −i 1 − sj Zj x2j . (2.1) Hmag = 2mj ∂xj 2 ∂xj 2 j=1
The Coulomb potential of this system is given by the formula X si sj Z i Z j , (x1 , . . . , xN ) ∈ (R2 )N . VCoul (x1 , . . . , xN ) = |xi − xj |
(2.2)
1≤i<j≤N
˜ = Hmag + VCoul . It is well The Hamiltonian of the entire system is, of course, H ∞ 2 N ˜ is bounded from below known that, as a form sum on the domain Cc ((R ) ), H ˜ can be realized as a self-adjoint operator H ˜ F through Friedrichs’ [11]. Therefore H construction [6,8]. Furthermore, it is elementary to deduce from [11] the norm estimate (2.3) k |VCoul |1/2 (Hmag + b)−1/2 k ≤ 1 P if b > b0 = 1≤i<j≤N (N − 1)(Zi Zj )2 max{mi , mj }. To facilitate our further in˜ in the following form which has a “stanvestigation, let us put the Hamiltonian H dardized” magnetic part. For s = ±1 and 1 ≤ j ≤ N , let us denote !2 !2 ∂ ∂ s 2 1 + −i 2 + sxj . (2.4) Dj = −i 1 − sxj ∂xj ∂xj Let Z : L2 ((R2 )N ) → L2 ((R2 )N ) be the unitary operator ! N r r r Y Zj β Z1 β ZN β x1 , . . . , xN . (Zf )(x1 , . . . , xN ) = f 2 2 2 j=1 It is easy to see that Z ∗ Hmag Z = (β/2)H0 where H0 =
∞ X j=1
s
µj Dj j
with
µj = Zj /2mj .
(2.5)
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Similarly, Z ∗ VCoul Z = (β/2)1/2 V where V (x1 , . . . , xN ) =
X
si sj Z i Z j −1/2 −1/2 xi − Zj xj | 1≤i<j≤N |Zi
.
(2.6)
Let HF be the self-adjoint operator obtained from the form [ϕ, ψ]H = hH0 ϕ, ψi + (2/β)1/2 hV ϕ, ψi ,
ϕ , ψ ∈ Cc∞ ((R2 )N ) ,
(2.7)
˜ F Z = (β/2)HF . It through the Friedrichs construction. Then, of course, Z ∗ H 1/2 1/2 −1/2 k ≤ 1. This must hold in follows from (2.3) that k |(β/2) V | ((β/2)H0 + b) P the special case β = 2. Hence we conclude that for any b > b0 = 1≤i<j≤N (N − 1)(Zi Zj )2 max{mi , mj }, k |V |1/2 (H0 + b)−1/2 k ≤ 1 .
(2.8)
˜ = Hmag + VCoul , we will only be ¿From now on, instead of the Hamiltonian H concerned with (2.7) and the self-adjoint operator HF obtained from it through the Friedrichs construction. Moreover, the quantity (2/β)1/2 in (2.7) will play the role of a parameter. Let Ds be the same operator as defined by (2.4) except we drop the subscript j. In the polar coordinates x = (x1 , x2 ) = (r cos θ, r sin θ), we have Ds = −
1 ∂2 1 ∂ ∂ 1 ∂ r − 2 2 + 2s + r2 . r ∂r ∂r r ∂θ i ∂θ
This operator can be decomposed as follows. Let R denote the subspace of L2 (R2 ) which consists of all the radial functions, i.e. all the functions which only depend on the radial variable r = |x|. It is easy to see that the orthogonal projection from L2 (R2 ) to R is given by the formula Z 1 f (uθ x)dθ , f ∈ L2 (R2 ) , (P f )(x) = 0
where uθ is the counterclockwise rotation of R2 about the original by an angle 2πθ. Obviously P maps Cc∞ (R2 ) into itself. Let W0 = P Cc∞ (R2 ). For each m ∈ Z\{0}, let Wm denote the collection of functions of the form ϕ(r, θ) = eimθ η(r) on R2 , where η is any function in Cc∞ (0, ∞). As linear subspaces of L2 (R2 ), the Wm ’s are orthogonal to each other and the linear span of these subspaces is dense in L2 (R2 ). Since P commutes with Ds and since Cc∞ (R2 ) is a core for Ds , on the domain W0 , the restricted operator Ds |W0 is essentially self-adjoint in R. Let Wm be the closure of Wm in L2 (R2 ). For each m ∈ Z, let Ym : Wm → L2 (0, ∞) be the unitary operator defined by the formula (Ym ϕ)(t) = (2πt)1/2 f (t) if ϕ(r, θ) = eimθ f (r) . Thus Ym Wm = Cc∞ (0, ∞) if m 6= 0. (But Y0 W0 6= Cc∞ (0, ∞).) Let Y =
∞ M m=−∞
Ym .
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We have Y Ds Y ∗ =
∞ M
Lm ,
m=−∞
where Lm
d2 1 1 2 =− 2 + m − + t2 + 2sm . dt 4 t2
For each m ∈ Z\{0}, it is elemetary that Lm is in the limit point case at both ends of (0, ∞). Thus Lm is essentially self-adjoint on Cc∞ (0, ∞) when m ∈ Z\{0}. It follows from the preceding paragraph that Y0 W0 is a core for the self-adjoint operator L0 . Hence W = · · · + W−m + · · · + W−1 + W0 + W1 + · · · + Wm + · · · is a core for the self-adjoint operator Ds . The spectrum of Lm is the set {2|m| + 2sm + 2 + 4j : j ∈ Z+ }. Indeed it is well known that for each j ∈ Z+ , there is a polynomial p|m|,j of degree j such that v|m|,j (t) = (2π)1/2 t|m|+1/2 e−t
2
/2
p|m|,j (t2 )
is an eigenfunction of Lm corresponding to the eigenvalue 2|m| + 2sm + 2 + 4j. See [1]. If p|m|,j is normalized in such a way that kv|m|,j k = 1, then {v|m|,j : j ∈ Z+ } is an orthonormal basis in L2 (0, ∞). Define the function ϕm,j (r, θ) = eimθ r|m| e−r
2
/2
p|m|,j (r2 ) .
(2.9)
In particular, we may require ϕm,0 (r, θ) =
1 |m|!π
1/2
eimθ r|m| e−r
2
/2
.
(2.9.0)
Because Ym ϕm,j = vm,j , we have Ds ϕm,j = (2|m| + 2sm + 2 + 4j)ϕm,j .
(2.10)
Furthermore, {ϕm,j : m ∈ Z, j ∈ Z+ } is an orthonormal basis in L2 (R2 ) and σ(Ds ) = σess (Ds ) = {2 + 4j : j ∈ Z+ }. It follows from this and (2.5) that σ(H0 ) = σess (H0 ) = {(2 + 4j1 )µ1 + · · · + (2 + 4jN )µN : j1 , . . . , jN ∈ Z+ } . The values in σ(H0 ) are usually called Landau levels, the first two of which being λ1 = 2(µ1 + · · · + µN ) ,
λ2 = λ1 + 4 min{µ1 , . . . , µN } .
(2.11)
For each ` ∈ Z, we define Ω` to be the linear span of {ϕn1 ,j1 (r1 , θ1 ) . . . ϕnN ,jN (rN , θN ) : n1 + · · · + nN = ` , j1 , . . . , jN ∈ Z+ } . Let H` be the closure of Ω` in L2 ((R2 )N ). Then M {ein1 θ1 . . . einN θN f : f ∈ L2 ((0, ∞)N , r1 . . . rN dr1 . . . drN )} . H` = n1 +···+nN =`
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One can think of ` as the “total orbital angular momentum” for the particles. Obviously each Ω` is invariant under H0 and Ω = · · · + Ω−` + · · · + Ω0 + · · · + Ω` + · · · is a core for H0 . For each ` ∈ Z, the restricted operator H0 |Ω` is essentially selfadjoint in H` . We will denote the closure of H0 |Ω` by H0,` . Because of (2.8), Ω is a form core for [·, ·]H as defined in (2.7). It is easy to see that Ω` ⊥ Ω`0 if ` 6= `0 . Note that for any c, c0 ∈ C, |crj eiθj − c0 rk eiθk | depends not on eiθj and eiθk individually but on ei(θj −θk ) only. Now H` is invariant under multiplication by any ein(θj −θk ) , 1 ≤ j, k ≤ N , n ∈ Z. Using this fact it is easy to check that if ϕ ∈ Ω` , ψ ∈ Ω`0 and ` 6= `0 , then hV ϕ, ψi = 0. For each ` ∈ N, let [·, ·]H,` be the form obtained from [·, ·]H by restricting it to Ω` × Ω` . Then [·, ·]H L∞ can be written as an orthogonal sum `=−∞ [·, ·]H,` in the following sense: Suppose P P ϕi , ψi ∈ Ω`i , i = 1, . . . , n, where `i 6= `j for i 6= j. Then [ ni=1 ϕi , ni=1 ψi ]H = Pn i=1 [ϕi , ψi ]H,`i . Thus each [·, ·]H,` produces a self-adjoint operator HF,` and HF =
∞ M
HF,` .
`=−∞
At this point we divide our investigation into two cases: In Sec. 3 we consider the case where all charges have the same sign. In Sec. 4 we consider a pair of oppositely charged particles. 3. When s1 = · · · = sN = −1 We assume that H0 , V , HF are the same as in Sec. 2. It goes without saying that the equality in the title is assumed throughout the section. This assumption determines the natural of the spectrum of HF . Proposition 3.1. When all the charges have the same sign, the operator (HF,` + z)−1 is compact for every ` ∈ Z and z ∈ C\R. Consequently HF has no continuous spectrum. Proof. By (2.8) and the usual expansion √ of resolvent, it is easy√to show that ((β/2)HF + b)−1 = (((β/2)H0 )1/2 − i b)−1 X(b)(((β/2)H0 )1/2 + i b)−1 for sufficiently large b, where X(b) is a bounded operator b. Hence it √ p depending on p suffices to show that (((β/2)H0 )1/2 + i b)−1 |H` = 2/β((H0,` )1/2 + i 2b/β)−1 is a compact operator. But this only requires a counting of the eigenvalues of H0,` . Given an integer k > 0, let us count how many eigenvalues of H0,` lie below 4kµ, where µ = min{µ1 , . . . , µN }. Since s1 = . . . , sN = −1, according to (2.10), if the eigenvalue t of H0 corresponding to ϕn1 ,j1 (r1 , θ1 ) . . . ϕnN ,jN (rN , θN ), does not exceed 4kµ, then n1 ≥ −k, . . . , nN ≥ −k and j1 ≤ k, . . . , jN ≤ k. If we now impose the restriction n1 + · · · + nN = ` required by the definition of Ω` , then
TWO-DIMENSIONAL COULOMB INTERACTIONS IN A MAGNETIC FIELD
367
nj ≤ (N −1)k +`, j = 1, . . . , N . Hence the number of tuples (n1 , . . . , nN , j1 , . . . , jN ) satisfying these conditions is at most (N k + |`|)N × (k + 1)N . This completes the proof. The assumption s1 = · · · = sN = −1 means that V ≥ 0. Hence σ(HF ) ⊂ [λ1 , ∞), where λ1 is the first Landau level for H0 (see (2.11)). With this in mind, we have: Theorem 3.2. There is a constant C > 0 which depends only on N and {(Zi /Zj )1/2 : 1 ≤ i < j ≤ N } such that for ` ≥ 2, r 2 X 3/2 Zi Zj `−1/2 log ` . inf{λ : λ ∈ σ(HF,` )} ≤ λ1 + C β 1≤i<j≤N
Consequently λ1 = inf{λ : λ ∈ σess (HF )}. The proof requires several steps, all of which are quite elementary. Lemma 3.3. There is a C > 0 such that for any 0 ≤ ρ < 1, Z π 1 1 dθ ≤ C 1 + log . iθ 1−ρ −π |1 − ρe | Proof. Note that |1 − ρeiθ | = (1 − 2ρ cos θ + ρ2 )1/2 ≥ (1 − r cos θ)1/2 , where r = 2ρ/(1 + ρ2 ). We only need to consider the case 1/2 ≤ ρ < 1. Hence r ≥ 4/5. We have 1 − r cos θ = 1 − r + (rθ2 /2)(1 − (θ2 /3 · 4) + · · · ). There are c1 , δ ∈ (0, 1) such that (1 − (θ2 /3 · 4) + · · · ) ≥ c1 if |θ|√≤ δ. With c2 = c1 /10, we have |1 − ρeiθ | ≥ √ √ (1 − r + c2 θ2 )1/2 ≥ ( 1 − r + c2 |θ|)/ 2 when |θ| ≤ δ. Hence √ Z δ Z δ dθ 2dθ √ ≤ √ iθ | |1 − ρe 1 − r + c2 |θ| −δ −δ √ Z √ √ 1+ 1−r dt 2 2 δ c2 √ ≤ C3 log √ , = √ c2 0 1−r+t 1−r which is at most C4 (1 − log(1 − r)). Since log(1 − r) = 2 log(1 − ρ) − log(1 + ρ2 ), the lemma follows. Lemma 3.4. For any α > 0, there is a C = C(α) > 0 such that Z ∞ Z ∞ 1 tn tn log dt ≤ C(1 + log m) dt α 2 m+3/2 2 1− t (t + 1) (t + 1)m+3/2 α α for any m ∈ N and 0 ≤ n ≤ 2m. Proof. Define Z 1 tn log I1 (E) = 2 t2 + 1)m+3/2 (α 1 − E
Z 1 t
dt ,
I2 (E) = E
tn dt . (α2 t2 + 1)m+3/2
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Then the two integrals in the inequality are αn+1 I1 (1, ∞) and αn+1 I2 (1, ∞) respectively. Now I1 (21/m , ∞) ≤ log((1 − 2−1/m )−1 )I2 (21/m , ∞). And log((1 − 2−1/m )−1 ) = m−1 log 2 + log m + log((1/m)/(21/m − 1)) . Since log((1/m)/(21/m − 1)) ≤ log(1/ log 2), we have I1 (21/m , ∞) ≤ C1 (1 + log m)I2 (21/m , ∞) . On the other hand I1 (1, 21/m ) ≤
2n/m 2 (α + 1)m+3/2
2n/m ≤ 2 (α + 1)m+3/2
Z
21/m
log
1 1−
log
2 dt . t−1
1
Z
21/m
1
R 21/m
1 t
dt
s=(21/m −1)
2 log t−1 dt = (21/m − 1) log 2 − (s log s − s)|s=0 = (21/m − 1) Now 1 (log 2 + 1 + log m − log((21/m − 1)/(1/m))). Since n ≤ 2m, 2n/m ≤ 4. Therefore
I1 (1, 21/m ) ≤ C2 (1 + log m)(21/m − 1)/(α2 + 1)m+3/2 .
(3.1)
Now I2 (1, 2
1/m
21/m − 1 21/m − 1 ) ≥ 2 1/m ≥ (α 2 + 1)m+3/2 (α2 + 1)m+3/2
1
m+3/2 .
21/m
(3.2)
A comparison of (3.1) and (3.2) completes the proof. Lemma 3.5. There is a C > 0 such that for each pair of p, q ∈ N, Z ∞Z ∞ 1 1 2p+1 2q+1 −t2 −s2 1 , min s e dt ds ≤ C(p + q)−1/2 . t p!q! 0 t s 0
Proof. Because min{1/t, 1/s} is symmetric in the two variables, we may assume that p ≥ q. Writing I for the integral to be estimated, since min{1/t, 1/s} ≤ 1/t, we have √ √ Z ∞ Z ∞ (2p)! 1 π (2p)! q! π 1 2p −t2 2q+1 −s2 · · 2p · = · 2p t e dt s e ds = . I≤ p!q! 0 p!q! 2 2 p! 2 4 2 (p!)2 0 By Stirling’s formula k! =
√ 2πk k+1/2 e−k eσ(k)/12k ,
0 < σ(k) < 1 ,
(see [1,12]) and the assumption p ≥ q, I ≤ C1 (2p)2p+1/2 e−2p /22p (pp+1/2 e−p )2 = √ √ √ 2C1 / p ≤ 2C1 / p + q. For each integer ` ∈ N, let Φ` (r1 , θ1 , . . . , rN , θN ) =
N Y j=1
1 (nj (`))!π
1/2
n (`) inj (`)θj −rj2 /2
rj j
e
e
,
TWO-DIMENSIONAL COULOMB INTERACTIONS IN A MAGNETIC FIELD
369
where the integers n1 (`), . . . , nN (`) are defined by the following formula: If ` = nN + k, where n ∈ Z+ and 1 ≤ k ≤ N , then n1 (`) = · · · = nk (`) = n + 1 and, in the case k < N , nk+1 (`) = · · · = nN (`) = n. We have n1 (`) + · · · + nN (`) = ` and nj (`) + 1 ≥ `/N for every j. Lemma 3.6. There is a C > 0 which depends only on N and {(Zi /Zj )1/2 : 1 ≤ i < j ≤ N } such that, for every ` ∈ N, X
hV Φ` , Φ` i ≤ C
3/2
Zi
√ Zj (1 + log `)/ ` .
1≤i<j≤N
3/2
Proof. It suffices to consider a single Zj Zk W where W = |rj eiθj − (Zj /Zk )1/2 rk eiθk |−1 . To simplify our notation, let p = nj (`), q = nk (`) and α = (Zj /Zk )1/2 . We set M1 (t, s) = min{1/t, 1/αs} and ( M (t, s) =
(1/t) log(1/(1 − (αs/t)))
if t > αs
(1/αs) log(1/(1 − (t/αs)))
if t < αs
.
Then by straightforward integration and an application of Lemma 3.3, Z π Z π Z ∞Z ∞ 1 dθdγ 2p+1 2q+1 −t2 −s2 t s e dt ds hW Φ` , Φ` i = i(θ−γ) | p!q!π 2 0 0 −π −π |t − αse Z ∞Z ∞ 2 2 C1 t2p+1 s2q+1 e−t −s (M (t, s) + M1 (t, s)) dt ds . ≤ p!q! 0 0 Now M1 (t, s) = min ≤ min
1 1 , t αs
= min
1+α 1+α , αt αs
α 1 , αt αs
1+α min = α
1 1 , t s
.
Therefore, in view of Lemma 3.5 and the fact that p + q + 2 ≥ 2`/N , to complete the proof, it suffices to show that there is a C2 = C2 (α) > 0 such that Z 0
∞
Z
∞
t2p+1 s2q+1 e−t
0
2
Z
≤ C2 (1 + log `) 0
−s2
∞
Z
M (t, s) dt ds ∞
t2p+1 s2q+1 e−t
2
−s2
M1 (t, s) dt ds .
(3.3)
0
RR RR Writing each integral as t>αs + t<αs and making the substitutions t = sr, s = u/(r2 + 1)1/2 and s = tr, t = u/(1 + r2 )1/2 in the two terms respectively, we have
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J. XIA
Z
∞Z
0
∞
t2p+1 s2q+1 e−t
2
M (t, s) dt ds
0
Z
∞
=
+ ∞
0
Z
∞
0
Z
1 α
∞
1/α
t2p+1 s2q+1 e−t
2
∞
= 0
Z
(Z
∞
1 r2p log 2 + 1)p+q+3/2 1 − (r α ) 1 r2q log 1 dr , (r2 + 1)p+q+3/2 1 − αr
u2(q+p+1) e−u du 2
0
Z
−s2
−s2
α r
dr
M1 (t, s) dt ds
u2(q+p+1) e−u du 2
(Z
×
∞
α
1 r2p dr + 2 p+q+3/2 α (r + 1)
Z
∞
1/α
r2q dr (r2 + 1)p+q+3/2
Thus (3.3) follows from these identities and Lemma 3.4. proof.
) .
This completes the
Proof of Theorem 3.2. We have Φ` ∈ Ω` and H0 Φ` = λ1 Φ` according to (2.9.0), (2.10) and (2.11). Of course Φ` belongs to the form domain of [·, ·]H,` . Since kΦ` k = 1 and D(HF,` ) is a form core for [·, ·]H,` , we have r 2 hV Φ` , Φ` i inf{λ : λ ∈ σ(HF,` )} ≤ [Φ` , Φ` ]H,` = hH0 Φ` , Φ` i + β r 2 hV Φ` , Φ` i . = λ1 + β
Thus Lemma 3.6 provides the desired estimate.
For each ` ∈ Z+ , let Ω1` denote the linear span of all ϕn1 ,0 (r1 , θ1 ) · · · ϕnN ,0 (rN , θN ) where n1 ≥ 0, . . . , nN ≥ 0, n1 + · · · + nN = `, and let Ω2` denote the linear span of ϕn1 ,j1 (r1 , θ1 ) · · · ϕnN ,jN (rN , θN ), where n1 + · · · + nN = `, either nk < 0 for some k or jk0 > 0 for some k 0 . Then Ω` = Ω1` + Ω2` . By (2.10), we have Ω1` = ker(H0,` − λ1 ) ,
hH0 ϕ, ϕi ≥ λ2 kϕk2
if ϕ ∈ Ω2` .
Furthermore, elementary combinatorics tells us that dimΩ1` =
(` + N − 1)! `+N −1 = CN . −1 `!(N − 1)!
For a self-adjoint operator A, let λj (A) =
sup u1 ,...,uj−1
inf{hAϕ, ϕi : ϕ ∈ D(A) , kϕk = 1 , ϕ ⊥ {u1 , . . . , uj−1 }} .
TWO-DIMENSIONAL COULOMB INTERACTIONS IN A MAGNETIC FIELD
371
If A is obtained from a form [·, ·]A which is bounded from below, then λj (A) can be alternately expressed as λj (A) =
sup u1 ,...,uj−1
inf{[ϕ, ϕ]A : ϕ ∈ domain of [·, ·]A , kϕk
= 1, ϕ ⊥ {u1 , . . . , uj−1 }} . See [9]. If A has at least j eigenvalues below the lowest point in its essential spectrum, then λj (A) is the jth eigenvalue of A counting multipicity and in ascending order. In this notation, λ1 (H0 ) equals the first Landau level λ1 given in (2.11). In fact, since λ1 ∈ σess (H0 ), λn (H0 ) = λ1 for all n ∈ N. We will also write λ(A; j) instead of λj (A) if the subscript involves multiple characters. Thus, in particular, for each ` ∈ Z+ , `+N −1 ) = λ1 , λ(H0,` ; CN −1
`+N −1 λ(H0,` ; 1 + CN ) = λ2 . −1
This provides the background for our next result. Theorem 3.7. (a) There is a constant c > 0 which depends only on m1 , . . . , mN , Z1 , . . . , ZN and β such that, for any ` ∈ N, `+N −1 ) ≥ λ1 + c . λ(HF,` ; CN −1
(b) Suppose that N ≥ 3. There is a constant c > 0 which depends only on m1 , . . . , mN , Z1 , . . . , ZN and β such that, for any ` ∈ N, ! ` X `+N −1 m+N −3 CN −3 ≥ λ1 + c . λ HF,` ; 1 + CN −1 − m=0
The second term on the right-hand side is due to the contribution of the Coulomb interactions between the particles. In other words, given that the total orbital angular momentum equals `, the Coulomb interactions elevate the energy of the (` + N − 1)!/`!(N − 1)!th state by at least a fixed amount. We start the proof of Theorem 3.7 with some technical preparations. Suppose α > 0. For each ` ∈ N, define 2 α (` − 1) , `(α) = 1 + α2 the integer part of α2 (` − 1)/(α2 + 1). Lemma 3.8. For any given α > 0, there are positive numbers C(α) and c(α) such that `(α) ` ` X X X α2j Cj` ≤ α2j Cj` ≤ C(α) α2j Cj` c(α) j=`(α)+1
for every ` ∈ N.
j=0
j=`(α)+1
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Proof. It suffices to consider those ` ∈ N for which `(α) ≥ 2. Writing out the factorials in the binomial coefficients, we have ` ` = C`(α) C`(α)+j
j−1 ` − `(α) Y ` − `(α) − i `(α) + j i=1 `(α) + i
` − `(α) ` = C`(α) `(α) + j for 1 ≤ j ≤ ` − `(α). (Here ` C`(α)−j
Q1−1
` − `(α) `(α)
i ` − `(α) , i 1+ `(α)
j−1 j−1 Y 1− i=1
. . . means 1.) Similarly,
i=1
`(α) ` = C`(α) ` − `(α) + j
`(α) ` − `(α)
j−1 j−1 Y i=1
i `(α) i 1+ ` − `(α) 1−
if 1 ≤ j ≤ `(α). Since `(α) = [α2 (` − 1)/(1 + α2 )], (` − `(α))/`(α) = α−2 {1 + (θ/`(α))}/{1 − (θ/(` − `(α)))} , where 0 ≤ θ ≤ 2. Hence there are 0 < c1 (α) < C1 (α) < ∞ such that j−1 j−1 `(α) `(α) ` − `(α) ` − `(α) = α−2j ξj , = α2j ζj `(α) + j `(α) ` − `(α) + j ` − `(α) with ξj , ζj ∈ [c1 (α), C1 (α)]. If we set j−1 Y i i 1+ , aj = 1− ` − `(α) `(α) i=1 bj =
j−1 Y
i 1− `(α)
i=1
1+
i ` − `(α)
,
` ` ` ` = C`(α) α−2j ξj aj and C`(α)−j = C`(α) α2j ζj bj . That is, then C`(α)+j ` X
X
`−`(α) 2j
α
Cj`
=
`(α)
j=0
α
` C`(α)+j
=
` C`(α) α2`(α)
j=1
j=`(α)+1
X
X
`−`(α) 2(`(α)+j)
X
`(α)
α2j Cj` =
ξj aj ,
j=1
X
`(α) ` ` α2(`(α)−j) C`(α)−j = C`(α) α2`(α)
j=0
ζj bj ,
j=0
where ζ0 = b0 = 1. Since {ξj } and {ζj } are comparable in magnitude, to complete the proof, it suffices show that X
`−`(α)
c2 (α)
j=1
X
`(α)
aj ≤
X
`−`(α)
bj ≤ C2 (α)
j=0
where 0 < c2 (2) ≤ C2 (α) < ∞ depend only on α.
j=1
aj ,
(3.4)
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TWO-DIMENSIONAL COULOMB INTERACTIONS IN A MAGNETIC FIELD
Note that aj /bj =
j−1 Y
1−
i=1
i ` − `(α)
2 ! ,
1−
i `(α)
2 ! .
(3.5)
Let us first consider the case where 0 < α ≤ 1. In this case `(α) ≤ `/2, i.e. `−`(α) ≥ `(α). This implies aj ≥ bj for 1 ≤ j ≤ `(α). Since a1 = b1 = b0 = 1 and since ` − `(α) ≥ `(α), we see that the second half of (3.4) holds in the case α ≤ 1 if C2 (α) ≥ 2. To establish the first half of (3.4), let T be the smallest integer such that T `(α) ≥ `−`(α), i.e. T /(`−`(α)) ≥ 1/`(α). We have T ≤ 1+(`−`(α))/`(α) = `/`(α). If j ≥ 1 is such that 1 + jT ≤ ` − `(α), then QjT Qj−1 i iT i 1− jT 1 − i=1 1 − i=1 Y ` − `(α) ` − `(α) ` − `(α) ≤ ≤ bj . ≤ ajT +1 = i Qj−1 Qj−1 i i i=1 1+ 1+ 1+ i=1 i=1 `(α) `(α) ` − `(α) Suppose that ` − `(α) = nT + m, n ∈ Z+ , m ∈ N, 1 ≤ m ≤ T . By the choice of T , Pn−1 n ≤ `(α). Since ai ≥ aj whenever i ≤ j, we have (with j=0 . . . = 0 when n = 0) `−`(α) T n−1 m n n X XX X X X ai = ajT +i + anT +i ≤ T a1 + ajT +1 ≤ T bj . i=1
j=0 i=1
i=1
j=1
j=0
This establishes the first half of (3.4) in the case 0 < α ≤ 1. We now consider the case α > 1. Since α2 /(1 + α2 ) > 1/2, there is an n(α) ∈ N such that `(α) > `/2 if ` ≥ n(α). And by the nature of the lemma, these are the only `’s we need to consider. Since now ` − `(α) < `(α), it follows from (3.5) that bj ≥ aj for 1 ≤ j ≤ ` − `(α). That is, in the case α > 1 and ` ≥ n(α), the first half of (3.4) holds for any 0 < c2 (α) ≤ 1. Now if S is the smallest integer such that S(` − `(α)) ≥ `(α), i.e. 1/(` − `(α)) ≤ S/`(α), then Qj−1 Qj−1 iS i i 1− jS 1 − i=1 i=1 1 − Y `(α) ` − `(α) `(α) = aj . ≤ ≤ bjS+1 = i Q Q i i 1 + `−`(α) j−1 j−1 i=1 1 + 1 + i=1 i=1 ` − `(α) `(α) Again, bi ≥ bj when i ≤ j. Also S ≤ `/(` − `(α)). Writing `(α) = kS + p with 1 ≤ p ≤ S and k ∈ Z+ , we have X
`(α)
i=0
X
`(α)
bi ≤ 2
bi = 2
i=1
S k−1 XX
bjS+i + 2
j=0 i=1
p X i=1
bkS+i ≤ 2S
k X
bjS+1 ≤ 4S
j=0
Lemma 3.9. Given any α > 0, there is a c(α) > 0 such that X j=0
for every ` ∈ N.
aj .
j=1
This completes the proof.
Z
`(α)
k X
Cj` 0
1/(1+α2 )
√ (1 − x)j x`−j dx ≥ c(α)/ `
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J. XIA
Proof. It suffices to consider ` ∈ N such that `(α) ≥ 1 and ` − `(α) ≥ 2. Define P`(α) `−1 and (` − j)Cj` = f` (x) = j=0 Cj` (1 − x)j x`−j . It is easy to verify jCj` = `Cj−1 `Cj`−1 . Differentiating term by term and applying these formulas, we have `−1 (1 − x)`(α) x`−1−`(α) , f`0 (x) = `C`(α) `−1 (1 − x)`(α)−1 x`−2−`(α) ((` − 1 − `(α)) − (` − 1)x) . f`00 (x) = `C`(α)
(3.6)
Since (` − 1 − `(α))/(` − 1) ≥ ((` − 1) − (α2 (` − 1)/(1 + α2 )))/(` − 1) = 1/(1 + α2 ), f`00 ≥ 0 on the interval [0, 1/(1 + α2 )]. That is, the graph of f` is concave upward over this interval. It follows from Lemma 3.8 and the binomial expansion formula that there is a c1 (α) > 0 such that f`
1 1 + α2
X
`(α)
=
Cj`
j=0
α2 1 + α2
j
1 1 + α2
`−j
`(α) X 1 = C ` α2j ≥ c1 (α) . (1 + α2 )` j=0 j
Let L be the tangent line to the graph of f` at the point (1/(1 + α2), f` (1/(1 + α2 ))), which lies below the graph of f` over [0, 1/(1 + α2 )] because of the concavity of f` . It suffices to consider the case where L intersects the x-axis in the positive half. Let T be the triangle in the xy-plane bounded by the vertical line x = 1/(1 + α2 ), the x-axis, and L. By the concavity of f` and elementary calculus, Z 1/(1+α2 ) f` (x)dx ≥ area of T = (f` (1/(1 + α2 )))2 /2f`0 (1/(1 + α2 )) . 0
Hence√the proof will be complete if we can find a c2 (α) > 0 such that f`0 (1/(1+α2 )) ≤ c2 (α) `. Substituting 1/(1 + α2 ) for x in (3.6) and applying Stirling’s formula to the `−1 , we find that there is a C3 > 0 such that, when ` − 1 − `(α) ≥ 1, factorials in C`(α) f`0
1 1 + α2
` ≤ C3 p `(α)
(α2 )`(α) (1 + α2 )`−1
`−1 ` − 1 − `(α)
`−1−`(α)
`−1 `(α)
`(α) . (3.7)
˜ Let `(α) = α2 (` − 1)/(1 + α2 ). Since ˜ θ ` − 1 − `(α) =1− , ` − 1 − `(α) ` − 1 − `(α)
˜ `(α) θ =1+ `(α) `(α)
˜ with θ ∈ [0, 1], we see that there is a C4 (α) > 0 such that if we replace `(α) by `(α) in the fractions inside the last two (. . .) in (3.7) and multiply the right-hand side by C4 (α), the inequality still holds. But with this replacement, the p product of the last three factors in (3.7) is 1. Hence f`0 (1/(1 + α2 )) ≤ C3 C4 (α)`/ `(α) as promised.
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TWO-DIMENSIONAL COULOMB INTERACTIONS IN A MAGNETIC FIELD
We need to introduce a substitute for the Coulomb potential. For each α > 0, define ( (1/r1 )Re{(1 − (αr2 /r1 )ei(θ2 −θ1 ) )−1 } if r1 > αr2 Qα (r1 , θ1 , r2 , θ2 ) = i(θ1 −θ2 ) −1 (1/αr2 )Re{(1 − (r1 /αr2 )e ) } if r1 < αr2 . The most important feature of Qα is that it satisfies the inequality 0 ≤ Qα (r1 , θ1 , r2 , θ2 ) ≤ |r1 eiθ1 − αr2 eiθ2 |−1 . Next we define for each pair p, q ∈ Z+ the function ξp,q (r1 , θ1 , r2 θ2 ) =
2 2 1 rp rq e−(r1 +r2 )/2 ei(pθ1 +qθ2 ) . (p!q!)1/2 π 1 2
In the space L2 ((R2 )2 ) = L2 (R2 ) ⊗ L2 (R2 ), ξp,q = ϕp,0 ⊗ ϕq,0 (see (2.9.0)). Lemma 3.10. Let ` ∈ N and let p, q, p0 , q 0 ∈ Z+ be such that p+q = ` = p0 +q 0 , p > p0 , and q 0 < q. Then hQα ξp,q , ξp0 ,q0 iL2 ((R2 )2 ) is real. Furthermore, Z
0
hQα ξp,q , ξp0 ,q0 iL2 ((R2 )2 ) ≥
αp−p (p!q!p0 !q 0 !)1/2
1/(1+α2 )
0
Z
0
∞
xq (1 − x)p dx 0
u2`+2 e−u du. 2
0
Rπ Proof. Note that p − p0 = q 0 − q. Since −π eikθ Re(1/(1 − ρeiθ ))dθ = πρ|k| for k ∈ Z\{0} and 0 ≤ ρ < 1, we have Z π Z π 0 0 Qα (t, θ1 , s, θ2 )ei((p−p )θ1 +(q−q )θ2 ) dθ1 dθ2 −π
−π
( = 2π
2
0
(1/t)(αs/t)p−p
if t > αs
p−p0
.
if t < αs
(1/αs)(t/αs) Thus hQα ξp,q , ξp0 ,q0 i ≥
2 0 (p!q!p !q 0 !)1/2 0
2αp−p = (p!q!p0 !q 0 !)1/2
Z Z
0
t>αs
Z
∞
1 αs p−p p+p0 +1 q+q0 +1 −t2 −s2 t s e dt ds t t
Z
0
∞
0
0
s2q +1 t2p e−t
2
−s2
dt ds .
αs
After the substitutions t = sr and s = u/(1 + r2 )1/2 , we obtain 0
hQα ξp,q , ξp0 ,q0 i ≥ Now Z ∞ α
0
r2p dr ≥ (1 + r2 )`+3/2
Z
2αp−p (p!q!p0 !q 0 !)1/2
∞
α
0
Z
∞
α
1 r2p rdr = (1 + r2 )`+2 2
0
r2p dr (1 + r2 )`+3/2
Z
∞
α2
0
Z
∞
u2`+2 e−u du . 2
0
1 wp dw = (1 + w)`+2 2
Z
∞
1+α2
0
(y − 1)p dy . y `+2
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0
0
Since (y−1)p /y `+2 = (1−y −1 )p (y −1 )q +2 , the lemma follows upon the substitution x = y −1 . Lemma 3.11. Given any α > 0, there is a c3.11 = c3.11 (α) > 0 such that the following holds true: For any ` ∈ Z+ , there is unit vector ϕ` ∈ L2 ((R2 )2 ) which is a linear combination of {ξj , ` − j : 0 ≤ j ≤ `} such that hQα ϕ` , ϕ` i ≥ c3.11 . Proof. The case ` = 0 is easy: c00 (α) = hQα ξ0,0 , ξ0,0 i > 0. Now suppose ` ∈ N. Set
X
`(α)
ξ=
(Cj` )1/2 αj ξj,`−j ,
ξ0 =
j=0
` X
(Cp` )1/2 αp ξp,`−p .
p=`(α)+1
Since {ξj,`−j : 0 ≤ j ≤ `} is an orthonormal set in Ω1` , we have X
` X
`(α)
kξk2 =
Cj` α2j ,
kξ 0 k2 =
j=0
Cp` α2p .
(3.8)
p=`(α)+1
By Lemma 3.10, for any `(α) + 1 ≤ p ≤ `, hQα ξp,`−p , ξi is real and not less than X
`(α)
(Cj` )1/2 αj
j=0
Z
√ (2(` + 1))! αp−j π · 2(`+1) · 1/2 2 2 (p!(` − p)!j!(` − j)!) (` + 1)!
1/(1+α2 )
x`−j (1 − x)j dx
× 0
√
2 Z `(α) π ` 1/2 p (2(` + 1))! X ` 1/(1+α ) `−j (Cp ) α 2(`+1) C x (1 − x)j dx . = 2 2 (` + 1)!`! j=0 j 0
√ By Stirling’s formula, there is a c1 > 0 such that (2(` + 1))!/22(`+1)(` + 1)!`! ≥ c1 `. Thus, hQα ξp,`−p , ξi ≥
c2 (Cp` )1/2 αp
Z X √ `(α) ` ` Cj j=0
1/(1+α2 )
x`−j (1 − x)j dx .
0
Lemma 3.9 now implies that hQα ξp,`−p , ξi ≥ c3 (α)(Cp` )1/2 αp . Hence hQα ξ 0 , ξi ≥ c3 (α)
` X
Cp` α2p = c3 (α)kξ 0 k2 .
p=`(α)+1
By (3.8) and Lemma 3.8, there is a c4 (α) > 0 such that kξ 0 k ≥ c4 (α)kξk. Hence hQα ξ 0 /kξ 0 k, ξ/kξki ≥ c3 (α)c4 (α). By the usual polarization formula for hQα ϕ, ψi, we see that if we set c(α) = c3 (α)c4 (α)/2, then there is a unit vector ϕ` in the linear
TWO-DIMENSIONAL COULOMB INTERACTIONS IN A MAGNETIC FIELD
377
span of ξ and ξ 0 such that hQα ϕ` , ϕ` i ≥ c(α). Taking the case ` = 0 into account, we find that c3.11 = min{c00 (α), c(α)} will do. Lemma 3.12. There is a c > 0 which depends only on {(Zi /Zj )1/2 : 1 ≤ i < j ≤ N } such that the following holds true: For any ` ∈ N, there is a linear subsapce K` ⊂ Ω1` , K` 6= {0}, such that inf{hV ϕ, ϕi : ϕ ∈ K` , kϕk = 1} ≥ c Furthermore, dimK` =
P`
m+N −3 m=0 CN −3
max
1≤i<j≤N
3/2
Zi
Zj .
if N ≥ 3.
Proof. Since H0,` and Ω1` are invariant under the permutation of the indices 3/2 3/2 1, . . . , N , we may assume that Z1 Z2 = max1≤i<j≤N Zi Zj without loss of 3/2 generality. Set α = (Z1 /Z2 )1/2 . Then 0 ≤ Z1 Z2 Qα (r1 , θ1 , r2 , θ2 ) ≤ V (r1 eiθ1 , . . . , rN eiθN ). Hence it suffices to find a c > 0 and a K` ⊂ Ω1` with the desired dimension for each ` ∈ N such that inf{hQα ϕ, ϕi : ϕ ∈ K` , kϕk = 1} ≥ c . When N = 2, this is an immediate consequence of Lemma 3.11 since we only require dimK` ≥ 1 in this case. Suppose now that N ≥ 3 and let ` ∈ N be given. For each 0 ≤ m ≤ `, let ϕm be the linear combination of {ξj,m−j : 0 ≤ j ≤ m} with kϕm k = 1 such that hQα ϕm , ϕm i ≥ c3.11 as provided by Lemma 3.11. Define Um = linear span of {ϕm ⊗ ϕn3 ,0 ⊗ · · · ⊗ ϕnN ,0 : n3 , . . . , nN ∈ Z+ , n3 + · · · + nN = ` − m} . QN Here, ϕm ⊗ ϕn3 ,0 ⊗ · · · ⊗ ϕnN ,0 is the function ϕm (r1 , θ1 , r2 , θ2 ) j=3 ϕnj ,0 (rj , θj ) and ϕn,0 is given by (2.9.0). Obviously Um ⊂ Ω1` . Note that for any ϕ ∈ Um , we have (3.9) hQα ϕ, ϕiL2 ((R2 )N ) = kϕk2 hQα ϕm , ϕm iL2 ((R2 )2 ) ≥ c3.11 kϕk2 . `−m+N −3 . It is easy to see that By elementary combinatorics, dimUm = CN −3 L` Um ⊥ Um0 if m 6= m0 . Hence if we define K` = m=0 Um , then its dimension P` P` `−m+N −3 m+N −3 = C . Furthermore, it is easy to see that equals m=0 CN −3 m=0 N −3 0 0 0 if m 6= m , ϕ ∈ Um , and ϕ ∈ Um0 , then hQα ϕ, ϕ i = 0. Thus (3.9) implies that hQα ψ, ψi ≥ c3.11 kψk2 for every ψ ∈ K` .
Proof of Theorem 3.7. Let us write V˜ = (2/β)1/2 V . By Lemma 3.12, there is a c1 > 0 which depends only on {(Zi /Zj )1/2 : i < j} with the property that for any ` ∈ N, there is a subspace K` ⊂ Ω1` such that r 2 3/2 ˜ max Z Zj = c01 . (3.10) inf{hV ϕ, ϕi : ϕ ∈ K` , kϕk = 1} ≥ c1 β 1≤i<j≤N i
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J. XIA
P` m+N −3 Furthermore, dimK` ≥ 1 when N = 2 and dimK` = m=0 CN when N ≥ 3. −3 1 2 0 Recall that Ω` + Ω` = Ω` . Now let Ω` be the orthogonal complement of K` in Ω1` and let Λ` = K` + Ω2` . Then Ω0` is the orthogonal complement of the closure ˜ ` denotes of Λ` in H` . Since Ω` is a form core for [·, ·]H,` and dimΩ0` < ∞, if Λ 0 ˜ the form closure of Λ` , then Ω` + Λ` is the form domain for [·, ·]H,` . Thus, by the min-max principle (see [9]), it suffices to produce a c > 0 which depends only on m1 , . . . , mN , Z1 , . . . , ZN and β such that ) h(H0 + V˜ )(aϕ + ψ), (aϕ + ψ)i ≥ λ1 + c . (3.11) if ϕ ∈ K` , kϕk = 1, ψ ∈ Ω2` , kψk2 = 1 − |a|2 , |a| ≤ 1 Let ϕ, ψ and a be as in (3.11). If E denotes the spectral measure for H0 , then ψ ∈ E[λ2 , ∞)L2 ((R2 )N ). Thus if we let C3 = (2/β)1/2 supt≥λ2 (t + b0 )/(t − (λ2 + λ1 )/2), P where b0 = 1≤i<j≤N (N − 1)(Zi Zj )2 max{mi , mj }, then it follows from (2.8) that r 2 ˜ h(H0 + b0 )ψ, ψi ≤ C3 h(H0 − ((λ2 + λ1 )/2))ψ, ψi hV ψ, ψi ≤ β ≤ C3 h(H0 − λ1 )ψ, ψi .
(3.12)
Since (t − λ1 )/(λ2 − λ1 ) ≥ 1 for t ≥ λ2 , we also have the inequality kψk2 ≤ (λ2 − λ1 )−1 h(H0 − λ1 )ψ, ψi .
(3.13)
Denote c(ϕ) = hV˜ ϕ, ϕi. Then c(ϕ) ≥ c01 . It also follows from (2.8) that c(ϕ) ≤ (2/β)1/2 h(H0 + b0 )ϕ, ϕi = (2/β)1/2 (λ1 + b0 ) . (3.12) implies that |hV˜ ϕ, ψi| ≤ kV˜ 1/2 ϕk kV˜ 1/2 ψk ≤ (C3 c(ϕ)h(H0 − λ1 )ψ, ψi)1/2 .
(3.14)
Let us first consider the case where h(H0 − λ1 )ψ, ψi ≤ min{(λ2 − λ1 )/2, c01 /64C3 } .
(3.15)
Since |a| ≤ 1, by (3.14) and (3.15), |2Re{ahV˜ ϕ, ψi}| ≤ 2|hV˜ ϕ, ψi| ≤ (c(ϕ)c01 )1/2 /4. Also, (3.13) and (3.15) imply that 1 − |a|2 = kψk2 ≤ 1/2, i.e., |a|2 ≥ 1/2. Since (H0 − λ1 )ϕ = 0, we have h(H0 + V˜ )(aϕ + ψ), (aϕ + ψ)i = λ1 + h(H0 − λ1 )ψ, ψi + |a|2 hV˜ ϕ, ϕi + hV˜ ψ, ψi + 2Re{ahV˜ ϕ, ψi} ≥ λ1 + (c(ϕ)/2) + 2Re{ahV˜ ϕ, ψi} ≥ λ1 + (c(ϕ)/2) − (c(ϕ)c01 )1/2 /4 ≥ λ1 + c01 /4.
(3.16)
Hence (3.11) holds with any c ≤ c01 /4 in the case of (3.15). Now suppose that (3.15) does not hold. Denote the right-hand side of (3.15) by 2c4 . Since h(H0 − λ2 )ψ, ψi ≥ 0, we have h(H0 − ((λ1 + λ2 )/2))ψ, ψi ≥
379
TWO-DIMENSIONAL COULOMB INTERACTIONS IN A MAGNETIC FIELD
h(H0 − λ1 )ψ, ψi/2 ≥ c4 . Pick a δ1 > 0 such that δ1 (2/β)1/2 (λ1 + b0 ) < (λ2 − λ1 )/2. If 0 < δ ≤ δ1 , then λ1 + δc(ϕ) ≤ (λ2 + λ1 )/2. With (3.12) in mind, for such a δ, h(H0 + V˜ )(aϕ + ψ), (aϕ + ψ)i = λ1 + δc(ϕ) + (1 − δ)|a|2 hV˜ ϕ, ϕi + hV˜ ψ, ψi +2Re{ahV˜ ϕ, ψi} + h(H0 − λ1 − δc(ϕ))ψ, ψi
√ 1 − δ)Re{ahV˜ ϕ, ψi} √ ≥ λ1 + δc(ϕ) + h(H0 − λ1 − δc(ϕ))ψ, ψi − 2(1 − 1 − δ) ≥ λ1 + δc(ϕ) + h(H0 − λ1 − δc(ϕ))ψ, ψi + 2(1 −
× {(2/β)1/2 (λ1 + b0 )C3 h(H0 − λ1 − δc(ϕ))ψ, ψi}1/2 √ √ ≥ (h(H0 − λ1 − δc(ϕ))ψ, ψi)1/2 ( c4 − 2(1 − 1 − δ) × {(2/β)1/2 (λ1 + b0 )C3 }1/2 ) + λ1 + δc(ϕ) . √ √ Hence if 0 < δ2 ≤ δ1 is such that c4 − 2(1 − 1 − δ2 ){(2/β)1/2 (λ1 + b0 )C3 }1/2 ≥ 0, then h(H0 + V˜ )(aϕ + ψ), (aϕ + ψ)i ≥ λ1 + δ2 c(ϕ) ≥ λ1 + δ2 c01 . Combining this with (3.16), c = c01 min{δ2 , 1/4} will do.
Theorem 3.13. If β > 2((λ1 + b0 )/(λ2 − λ1 ))2 , then σess (HF ) ∩ (λ1 , λ2 ) 6= ∅. Proof. Let c be the constant provided by Theorem 3.7. For each ` ∈ N, since dimΩ1` = (` + N − 1)!/`!(N − 1)!, there is a unit vector η` ∈ Ω1` which is orthogonal to the subspace spanned by the first (` + N − 1)!/`!(N − 1)! − 1 eigenvectors of HF,` . By Theorem 3.7, this means E` (−∞, λ1 + c)η` = 0 .
(3.17)
Here, E` denotes the spectral measure for HF,` . On the other hand, by (2.8) and the condition η` ∈ Ω1` , [η` , η` ]H,` = hH0 η` , η` i + (2/β)1/2 hV η` , η` i ≤ λ1 + (2/β)1/2 (λ1 + b0 ) , P where b0 = 1≤i<j≤N (N −1)(Zi Zj )2 max{mi , mj }. This implies that E` (−∞, λ1 + (2/β)1/2 (λ1 + b0 )]η` 6= 0. Combining this with (3.17), we have E` [λ1 + c, λ1 + (2/β)1/2 (λ1 + b0 )]η` 6= 0 for every ` ∈ N. On the other hand, the condition β > 2((λ1 + b0 )/(λ2 − λ1 ))2 implies [λ1 + c, λ1 + (2/β)1/2 (λ1 + b0 )] ⊂ (λ1 , λ2 ). This concludes our proof. As is well known, (1/β)1/2 = 1 corresponds to a magnetic field of 2.43 × 109 Gauss. Therefore the requirement that β be large in Theorem 3.13 is not a laboratory condition. We would like to stress, however, that Theorem 3.13 is the only result in this paper which needs any assumption on β.
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J. XIA
4. Oppositely Charged Particles In this section we make the following assumptions: (i) N = 2, (ii) s1 = −1, s2 = 1, (iii) Z1 = Z2 = Z. The masses of the particles, m1 and m2 , however, are still arbitrary. While we believe that the main result of this section should still hold true without (iii) and even without (i), at the present our methods do not cover such generalities. For ` ∈ Z, we continue to denote ker(H0,` − λ1 ) by Ω1` . But it is now a different space because we have a different set of sj ’s. In fact, according to (2.10), Ω1` now is the linear span of {ϕk,0 (r1 , θ1 )ϕ`−k,0 (r2 , θ2 ) : k ≥ max{0, `}}. It follows from (i)–(iii) that V (x1 , x2 ) = −(Z 5/2 )/|x1 − x2 | . The goal of this section is to show that V causes the bottom of the essential spectrum of HF,` to shift below λ1 by at least an amount proportional to 2−|`|(2/β)1/2 Z 5/2 . Theorem 4.1. Under the assumptions (i), (ii) and (iii), there is a c > 0 which is independent of m1 , m2 , Z, β such that for any ` ∈ Z, inf{λ : λ ∈ σess (HF,` )} ≤ λ1 − c2−|`| (2/β)1/2 Z 5/2 . We start the proof with the following simple observation: Proposition 4.2. Let H be a Hilbert space and let D be a dense linear subspace in H. Suppose that [·, ·]A is a symmetric quadratic form on D × D which is bounded form below. Let A be the self-adjoint operator obtained from [·, ·]A through Friedrichs’ representation. If {xn } is a sequence of unit vectors in D which converges to 0 weakly (with respect to h·, ·i), then inf{λ : λ ∈ σess (A)} ≤ lim inf [xn , xn ]A . n→∞
Proof. Let E denote the spectral measure for A and let c = lim inf n→∞ [xn , xn ]A . Suppose that the proposition were false. Then there would be an > 0 such that dim(E(−∞, c + ]H) < ∞. Let yn be the orthogonal projection of xn on the finite dimensional subspace E(−∞, c + ]H. Then the weak convergence xn → 0 implies kyn k → 0. Of course each yn belongs to the form domain of (the closure of) [·, ·]A and so does zn = xn − yn . If a ∈ R is a lower bound for [·, ·]A , then [yn , yn ]A ≥ −|a| kyn k2 . Now zn ∈ E(c + , ∞)H. Hence [zn , yn ]A = hzn , Ayn i = 0 and, by a standard approximation argument, [zn , zn ]A ≥ (c+)kzn k2 . And kzn k → 1 because kyn k → 0. Therefore [xn , xn ]A = [zn , zn ]A + 2Re[zn , yn ]A + [yn , yn ]A ≥ [zn , zn ]A − |a|kyn k2 ≥ (c + )kzn k2 − |a|kyn k2 → c + , which is a contradiction. This proves the proposition.
TWO-DIMENSIONAL COULOMB INTERACTIONS IN A MAGNETIC FIELD
381
Proof of Theorem 4.1. Given an ` ∈ Z, we need a sequence of unit vectors {Ψn } in Ω1` which weakly converges to 0 and for which the numerical sequence {[Ψn , Ψn ]H,` }∞ n=1 provides the desired estimate for the bottom of σess (HF,` ). We will only consider the case ` ≥ 0; the treatment for the case ` < 0 is completely similar. Thus ` ≥ 0 is assumed for the rest of the proof. For each k ∈ Z+ , let 1/2 (` + k)! ϕ`+k,0 (r1 , θ1 )ϕ−k,0 (r2 , θ2 ) ηk (r1 , θ1 , r2 θ2 ) = k! =
1 `+k k −(r12 +r22 )/2 i(`+k)θ1 −ikθ2 r r2 e e e , k!π 1
which belongs to Ω1` because (` + k) + (−k) = ` and because of (2.10). For each Pn n ∈ N, define Gn = k=0 ηk and Ψn = Gn /kGn k. Since kηk k = ((` + k)!/k!)1/2 Pn and kGn k = ( k=0 (` + k)!/k!)1/2 , it is clear that the sequence {Ψn } converges to 0 weakly. Since Ψn ∈ Ω1` , we have hH0 Ψn , Ψn i = λ1 . Let ( (1/r1 )Re{(1 − (r2 /r1 )ei(θ2 −θ1 ) )−1 } if r1 > r2 . Q(r1 , θ1 , r2 , θ2 ) = (1/r2 )Re{(1 − (r1 /r2 )ei(θ1 −θ2 ) )−1 } if r1 < r2 As we have mentioned before, 0 ≤ Q(r1 , θ1 , r2 , θ2 ) ≤ |r1 eiθ1 − r2 eiθ2 |−1 . Thus [Ψn , Ψn ]H ≤ λ1 − (2/β)Z 5/2 hQΨn , Ψn i. By Proposition 4.2, to complete the proof, it suffices to produce a c > 0 which is independent of m1 , m2 , Z, β and ` such that hQΨn , Ψn i ≥ c2−` for every n ∈ N. Equivalently, it suffices to show that hQGn , Gn i ≥ c2−`
n X (` + k)! k=0
k!
,
n ∈ N.
(4.1)
By a computation in the proof of Lemma 3.10, we have hQηk , ηk0 i ≥ 0 for all k, k ∈ Z+ . Therefore 0
hQGn , Gn i =
X
hQηk , ηk0 i ≥
0≤k,k0 ≤n
n m(1) X X
hQηm−k , ηk i .
(4.2)
m=0 k=0
Here, m(1) = [(m − 1)/2] = [12 (m − 1)/(1 + 12 )] as defined in the previous section. By a computation similar to that performed in the proof of Lemma 3.10, since (1 − x)` ≥ 2−` for x ∈ [0, 1/2], we have Z 1/2 Z ∞ 1 2(`+m+1) −u2 u e du (1 − x)`+k xm−k dx hQηm−k , ηk i ≥ (m − k)!k! 0 0 √ (2(` + m + 1))! π (` + m)! · · 2−` Ckm ≥ m! 2 22(`+m+1) (` + m + 1)!(` + m)! Z 1/2 × (1 − x)k xm−k dx . 0
√ By Stirling’s formula, the {. . .} in the above is at least c1 m. Thus it follows from Pm(1) Lemma 3.8 that there is a c > 0 such that k=0 hQηm−k , ηk i ≥ c2−` (` + m)!/m!. Therefore (4.1) follows from this and (4.2).
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J. XIA
References [1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965. [2] J. Avron, I. Herbst and B. Simon, “Schr¨ odinger operators with magnetic fields. I. General interactions”, Duke J. Math. 45 (1978) 847–884. [3] J. Avron, I. Herbst and B. Simon, “Separation of center of mass in homogeneous fields”, Ann. Phys. 114 (1978) 431–451. [4] J. Avron, I. Herbst and B. Simon, “Schr¨ odinger operators with magnetic fields. III. Atoms in homogeneous magnetic field”, Commun. Math. Phys. 79 (1981) 529–572. [5] T. Kato, “Fundamental properties of Hamiltonian operators of Schr¨ odinger type”, Trans. Amer. Math. Soc. 70 (1951) 195–211. [6] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1976. [7] N. Naimark, Linear differential operators, II, Linear differential operators in Hilbert space, Unger, New York, 1968. [8] M. Reed and B. Simon, Methods of Modern Mathematical Physics, II, Fourier Analysis, Self-adjointness, Academic Press, New York, 1975. [9] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV, Analysis of operators, Academic Press, New York, 1978. [10] B. Simon, “Schr¨ odinger semigroups”, Bull. Amer. Math. Soc. 7 (1982) 447–526. [11] B. Simon, “Maximal and minimal Schr¨ odinger forms”, J. Operator Theory 1 (1979) 37–47. [12] E. Whittaker and G. Watson, Modern Analysis, Cambridge Univ. Press, London, 1940.
ASYMPTOTIC COMPLETENESS IN QUANTUM IN FIELD THEORY. MASSIVE PAULI FIERZ HAMILTONIANS ´ J. DEREZINSKI Department of Mathematical Methods in Physics Warsaw University Ho˙za 74 00-682 Warszawa Poland
´ C. GERARD Centre de Math´ ematiques URA 169 CNRS Ecole Polytechnique 91128 Palaiseau Cedex France Received 24 April 1998 Spectral and scattering theory of massive Pauli–Fierz Hamiltonians is studied. Asymptotic completeness of these Hamiltonians is shown. The proof consists of three parts. The first is a construction of asymptotic fields and a proof of their Fock property. The second part is a geometric analysis of observables. Its main result is what we call geometric asymptotic completeness. Finally, the last part is a proof of asymptotic completeness itself.
1. Introduction Our paper is devoted to a class of Hamiltonians used in physics to describe a quantum system (“matter” or “an atom”) interacting with a bosonic field (“radiation”). K and K are respectively the Hilbert space and the Hamiltonian describing the matter. The bosonic field is described by a Fock space Γ(h) with the one-particle space h = L2 (Rd , dk), where Rd is the momentum space, and a free Hamiltonian of the form Z dΓ(ω(k)) = ω(k)a∗ (k)a(k) dk . The function ω(k) is called the dispersion relation. The interaction of the “matter” and the bosons is described by the operator Z V = a∗ (k)v(k) dk + hc , where Rd 3 k → v(k) is a function with values in operators on K. Thus, the system is described by the Hilbert space H := K ⊗ Γ(h) and the Hamiltonian H = K ⊗ 1l + 1l ⊗ dΓ(ω(k)) + V . 383 Reviews in Mathematical Physics, Vol. 11, No. 4 (1999) 383–450 c World Scientific Publishing Company
(1.1)
384
´ ´ J. DEREZINSKI and C. GERARD
The class of such Hamiltonians is very common in the physics literature. It is also quite natural from the mathematical point of view, as in particular we will see in our paper. Nevertheless, it does not seem to have a generally accepted name. We will call the Hamiltonians of the form (1.1) Pauli–Fierz Hamiltonians. In the thirties, Pauli and Fierz wrote a paper on nonrelativistic quantum electrodynamics [41], where a Hamiltonian of the form (1.1) was obtained, and since then the name Pauli–Fierz Hamiltonian has been occasionally used in this context (see for example [8]). Let us describe some typical examples of Pauli–Fierz Hamiltonians. If dim K = 1, then they are exactly solvable — by a Bogolyubov transformation they are equivalent to a quadratic bosonic Hamiltonian. If dim K = 2, K = σz and v(k) = g(k)σx , where σz , σx are Pauli matrices and g(k) is a real function on Rd , then the Hamiltonian H goes under the name of a spin-boson Hamiltonian. In a sense, it is the simplest non-trivial example of a Pauli–Fierz Hamiltonian. After a certain approximation (dropping interaction terms quadratic in the fields) nonrelativistic quantum electrodynamics can also be put in the form (1.1). In this case ω(k) = |k| and K is a Schr¨ odinger Hamiltonian (see [9, 5]). If the bosonic field describes√a relativistic particle of mass m, then the dispersion relation is of the form ω(k) = m2 + k 2 . Various branches of physics, such as solid state theory and quantum optics, furnish more examples of Hamiltonians of the form (1.1). The bosonic field may describe effective quasiparticles, e.g. phonons. ω(k) is then a phenomenological dispersion relation and can be, to a large extent, an arbitrary function. The matter Hamiltonian K and the interaction V may also vary depending on the model. Therefore, from the physical point of view, it seems natural to consider the class of Pauli– Fierz Hamiltonians under as broad conditions as possible. Let us now describe the assumptions that we will impose on the Hamiltonian H in our paper. First of all, we will assume that the function vˆ(x) decays sufficiently fast in the space variables. We call this the short-range condition. Physically, it means that the interaction is well localized. This assumption is needed to prove the existence of asymptotic fields. Note, however, that the results about the location of spectrum (our analog of the HVZ theorem) and the Mourre estimate hold under weaker decay condition on vˆ(x). Secondly, we will assume that the dispersion relation is positive and bounded away from zero, that is inf ω(k) := m > 0 . (1.2) The number m is sometimes called the mass of the field and (1.2) is the positive mass assumption. Besides, we will make some other technical assumptions on ω(k) (which in general can be relaxed): we will assume that zero is the only critical point of ω(k), all the derivatives of ω(k) are bounded and lim|k|→∞ ω(k) √ = +∞. Thus, a typical dispersion relation satisfying our assumptions is ω(k) = m2 + k 2 . Unfortunately, due to the assumption (1.2), the dispersion relation ω(k) = |k| is
ASYMPTOTIC COMPLETENESS IN QUANTUM IN FIELD THEORY
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not covered by our paper. We hope that our results, appropriately modified, can be extended to this case – under suitable conditions on the decay of v(k) as k → 0. The assumption (1.2) means that there is no “infra-red problem”. This assumption plays an important role in our considerations and relaxing it will entail additional technical difficulties. Finally, we assume that the matter Hamiltonian K has a compact resolvent. Physically, this means that the Hilbert space K is supposed to describe a confined system, e.g. K is finite dimensional or K = − 12 ∆ + W (x) with lim|x|→∞ W (x) = ∞. Note also that this assumption plays a role only in the so-called HVZ theorem, the Mourre estimate and its consequences, and in the last stage of the proof of the asymptotic completness. The existence of asymptotic fields, the Fock property of wave operators and the geometric asymptotic completeness are true without this assumption. In Sec. 3 we describe some general properties of the Pauli–Fierz Hamiltonians. We prove the self-adjointness of these Hamiltonians and some other technical properties. In Sec. 4 we impose the condition that the resolvent of K is compact. Under this condition, we show an analog of the HVZ theorem. This theorem says that the essential spectrum of H equals [E0 + m, ∞[ where E0 is the infimum of the spectrum of H. This clearly implies the existence of a ground state. This theorem is well known [22, 5, 3] (although the proofs found in the literature seem to be more complicated). We show also the Mourre estimate for Pauli–Fierz Hamiltonians. Its proof mimicks the proof of its analog from the case of N -body Schr¨ odinger operators. One of the key new ingredients is the induction with respect to the energy interval: in the nth step, the theorem is proven for the energy in [E +(n−1)m, E +nm[. Note that the proof breaks down if m = 0. An immediate consequence of the Mourre estimate is the local finiteness of the pure point spectrum away from the threshold set. The remaining part of our paper is devoted to the scattering theory of Pauli– Fierz Hamiltonians. The first step of scattering theory for such Hamiltonians is the existence of the so-called asymptotic fields. They are defined as the limits on a dense domain of the usual fields in the so-called interaction picture: a],+ (h) := lim eitH a] (ht )e−itH , t→∞
where a] (h) equals either a∗ (h) – the creation operator — or a(h) – the annihilation operator, and ht := e−itω(k) h. The asymptotic creation and annihilation operators satisfy the canonical commutation relations (CCR). Let the Hilbert space K+ be defined as the space of the states annihilated by asymptotic annihilation operators a+ (h). Physically, it can be understood as the space of asymptotic (“dressed”) matter — it contains states with no asymptotically free bosons. Define H+ := K+ ⊗ Γ(h) — the full asymptotic Hilbert space. Then, there is a natural definition of an isometric operator Ω+ : H+ → H interwining the usual and the asymptotic fields: Ω+ a] (h) = a],+ (h)Ω+ .
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The operator Ω+ can be defined as a wave operator by the formula Ω+ := s- lim eitH Ie−itH , +
t→∞
(1.3)
where H + = K + ⊗ 1l + 1l ⊗ dΓ(ω(k)) is a (non-interacting) asymptotic Hamiltonian defined on H+ and I : H+ → H is a certain naturally defined “identification operator”. Note that the above results about scattering theory for massive Pauli–Fierz Hamiltonians, possibly in a weaker form, can be extended to the mass zero case. Using the positive mass assumption one can show that the operator Ω+ is unitary. This means, in particular, that the representation of the CCR given by the asymptotic fields a+ (h) is of the Fock type. Note that, in the case of a zero mass, depending on the assumptions on v(k), the unitarity of Ω+ may be violated, which means that the asymptotic fields may have non-Fock components. It may even happen that the space K+ is reduced to {0}. The construction of asymptotic fields and of the wave operator use rather straightforward methods and has been essentially known for a long time. Up to technicalities related to the unboundedness of field operators, it follows by the socalled Cook’s method. Very similar results, including the fact that the positivity of mass implies the Fock property, are contained in a series of papers by Høegh–Krohn [28–30]. After the asymptotic fields are defined, it is natural to ask how to characterize the space of asymptotic matter K+ , and its analog for t → −∞, K− . A property, which is physically desirable is the equality K− = K+ .
(1.4)
This property implies in particular the unitarity of the scattering operator S := Ω+∗ Ω− . It is easy to show that Ran 1lpp (H) ⊂ K− ∩ K+ , where Ran 1lpp (H) denotes the space of bound states of H. Thus, it is natural to expect that, if the matter system K is not to large, then K+ = K− = Ran 1lpp (H) .
(1.5)
Clearly, (1.5) implies (1.4). We call the property (1.5) asymptotic completeness. The remaining part of our paper is devoted to proving this property. The eighties and the early nineties were a period when a substantial progress was reached in our understanding of scattering theory for N -body Schr¨ odinger Hamiltonians. In papers [16, 43, 24, 14, 50] efficient techniques have been developed, which made it possible to prove asymptotic completeness for long-range systems with an
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arbitrary number of particles. A natural next step was to apply these techniques to Hamiltonians of quantum field theory. This was the idea behind the work of one of the authors [21], where asymptotic completeness for the spin-boson Hamiltonian with a particle number cut-off was proved. In Sec. 6 we show a number of propagation estimates for Pauli–Fierz Hamiltonians. These estimates are very similar to the analogous estimates from the case of N -body Schr¨ odinger Hamiltonians. This section can be viewed as a technical introduction to the next section, were more conceptual results will be given. Section 6 can be skipped on the first reading. Section 7 is devoted to a proof of asymptotic completeness for massive Pauli– Fierz Hamiltonians. Most of the section is devoted to a proof of an intermediate result called geometric asymptotic completeness. In order to formulate this result one needs observables such as Γ(q( xt )), with q ∈ C0∞ (Rd ) and q = 1 in a neighborhood of zero, which localize in space. Using such observables, we construct a certain projection P0+ projecting onto the states that for a large time do not spread faster than o(t). The precise statement of geometric asymptotic completeness is Ran P0+ = K+ .
(1.6)
The proof of geometric asymptotic completeness has a number of ingredients known from N -body Schr¨ odinger operators, such as propagation estimates and asymptotic observables. One of the main new ideas, is the use of certain natural operators Pk (f0 , f∞ ). The operator Pk (f0 , f∞ ) describes the states with exactly k bosons multiplied by f∞ , and the rest multiplied by f0 . Using asymptotic observables constructed with help of such operators, we construct mutually orthogonal projections Pk+ , which project onto the states with exactly k asymptotically free bosons. The projections Pk+ form a partition of unity on the space H, that is, their sum is the identity. We show that Ran Pk+ is the range of the wave operator Ω+ restricted to k-particle states. The reader familiar with the scattering theory of N -body systems, as described in [14, 13], will note a very close analogy. In the proof of the asymptotic completeness of N -body Schr¨ odinger Hamiltonians, one of the important steps is the following: using asymptotic observables one constructs certain projections 1lZa (P + ) that form a partition of unity on the Hilbert space. Then one shows that Ran 1lZa (P + ) equals the range of the wave operator Ω+ a. The proof of geometric asymptotic completeness does not use the assumption of the compactness of the resolvent of K. This assumption is needed in Subsec. 7.8, where we show asymptotic completeness itself. Here, the basic tool is the minimal velocity estimate, which is a consequence of the Mourre estimate. We show that states spreading not faster than o(t) are exactly the bound states, in other words Ran P0+ = Ran 1lpp (H) .
(1.7)
Now (1.6) and (1.7) imply asymptotic completeness (1.5). Note that all these arguments are very close to the arguments used in the scattering theory of N -body Schr¨ odinger operators.
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Our paper is essentially self-contained. In Sec. 2 we describe all the concepts related to Fock spaces that we need. We recall some basic constructions such as the operators Γ(q) and dΓ(b) [7, 44, 42]. We introduce also a number of definitions that seem to be new in the literature. They were very useful in our paper and we think that they may find an application outside of our work. In particular, let us mention the operators Qk (f0 , f∞ ), which have very interesting properties playing an imporant role in our proof of geometric asymptotic completeness. Physically, asymptotic completeness means that for large times states evolve according to a simpler evolution. In particular, it implies that the usual formalism of scattering theory involving a unitary scattering operator is justified. The scattering operator is one of the central objects of quantum field theory, usually introduced in a formal, perturbative way. Our article shows that, at least for a certain class of relatively simple but nontrivial models, the usual physical formalism is well-founded. Let us mention another physical consequence of asymptotic completeness. Let us assume additionally that the interacting Hamiltonian has only one bound state, which can be shown in some cases, at least for small coupling (see [40, 5]). Then, as noted in [31], asymptotic completeness implies the property of return to equilibrium. This property plays an important role in statistical physics [6]. We believe that our result is just one of initial steps of a mathematical study of scattering in quantum field theory. Quantum field theory is a vast subject with diverse models and various interesting problems [17, 27, 23, 7, 25, 49]. From the point of view of scattering theory one can distinguish certain natural classes of models. First of all, one should distinguish: (1a) models with a localized interaction; (1b) models with a translation invariant interaction. Secondly one should make the following distinction: (2a) models conserving the number of particles; (2b) models changing the number of particles. Clearly, models with the property (1b) or (2b) are more difficult than models with the property (1a) or (2a) respectively. Pauli–Fierz Hamiltonians are models with a localized interaction, but they do not conserve the number of particles — they are of type (1a, 2b). We hope that the methods of our article can be extended to treat the scattering theory of other models of this type. For example, after minor modifications, one can extend our results to the interactions containing a term quadratic in the fields with a sufficiently small coupling constant. Likewise, instead of bosonic fields one can study fermionic fields. The extensively studied [44, 30, 22, 23] P (φ)2 model with a spacial cutoff also belongs to the type (1a, 2b) — it would be interesting to study asymptotic completeness also in this case. Scattering theory for translation invariant models (1b) is more difficult. There exists however one case where this problem seems to be well understood — it is the class of models considered in [15]. These models are of type (1b, 2a), they are however quite special — they conserve the number of particles of each species and they are Galilei-covariant, which is also a severe restriction. In the case of these
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models, the Hilbert space can be split into sectors and within each sector they are described by an N -body Schr¨ odinger Hamiltonian. There exist also some partial results in the case of relativistic quantum field theory. The Haag–Ruelle theory (see [25] and references therein) and its continuation due to Buchholz and Fredenhagen [4] allow us to define asymptotic fields in an axiomatic local quantum field theory. One can also show asymptotic completeness for low energies and small coupling constants in the λφ42 model [10, 33]. A lot of research was devoted to Hamiltonians of quantum field theory in the sixties and the early seventies. Let us mention in particular the book by Friedrichs [17], which in a mathematically rigorous way described the perturbative approach to quantum field theory, papers of Høegh–Krohn [28–30], early papers on the constructive field theory (see [22] and references in [23, 44]), papers of Fr¨ ohlich on translation-invariant models [18, 19] and the work of Davies on the weak-coupling limit for Pauli–Fierz-type Hamiltonians [11, 12]. It seems that in the late seventies and the eigthies there was a long period when little research on this subjects was performed (see however [1, 2, 40, 37, 46]). The Euclidean [23, 44] and the axiomatic [25] approaches replaced the Hamiltonian approach to quantum field theory. It was also a period of a considerable progress in the study of Schr¨ odinger operators, especially the N -body Schr¨ odinger operators [16, 43, 24, 14, 50, 13]. In the recent years one can see a renewed interest in Hamiltonians of quantum field theory, at least in the Pauli–Fierz Hamiltonians. Let us mention the paper of Huebner and Spohn [31] where wave operators for the spin-boson Hamiltonian were shown to exist and the problem of asymptotic completeness for such operators was discussed. Note in particular, that the formula (1.3) comes from this paper. Other results on the bound states and resonances of Pauli–Fierz Hamiltonians were given recently in [32, 5, 3, 34–36, 47, 48, 45]. 2. Basic Constructions in Bosonic Fock Spaces 2.1. Introduction In this section we describe various general constructions related to bosonic Fock spaces, which we will use in our paper. In Subsecs. 2.2–2.8 we recall various wellknown objects and their properties, such as field operators, the operators dΓ and Γ. In the remaining part of the section we introduce concepts that seem not to belong to the standard tools used in the literature, but nevertheless we think that they can be useful outside of our work. Among the constructions that we present let us mention the operators Qk (f ) and Pk (f ), used to define certain partitions of unity on the Fock space Γ(h), which have very useful positivity properties. Their use is one of the key ideas of the proof of the geometric asymptotic completeness, presented in Sec. 7. ˇ We also describe operators Γ(j), which map the Fock space Γ(h) into the doubled ˇ Fock space Γ(h)⊗Γ(h). The operators Γ(j) are easily defined using the usual functor Γ and the identification of the spaces Γ(h) ⊗ Γ(h) and Γ(h ⊕ h). One of the main tools used in the “geometric approach to scattering theory” is calculating the so-called Heisenberg derivative. It is therefore useful to introduce
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ˇ k), which arise when one computes certain operators dΓ(q, r), dQk (f, g) and dΓ(j, ˇ respectively. the Heisenberg derivative of Γ(q), Qk (f ) and Γ(j) Subsections 2.8 and 2.9 are devoted to the operators Γ(q) and dΓ(q, r). Subsections 2.10 and 2.11 are devoted to the operators Qk (f ) and dQk (f, g). Subsecˇ ˇ k). In our exposition, tions 2.13 and 2.14 are devoted to the operators Γ(j), dΓ(j, we tried to present the properties of these objects stressing their analogies. 2.2. Bosonic Fock spaces
Nn Let h be a Hilbert space, which we will call the 1-particle space. Let s h denote the symmetric nth tensor power of h. Let Sn denote the orthogonal projection of Nn Nn h onto s h. We define the Fock space over h to be the direct sum Γ(h) :=
n ∞ O M n=0
h.
s
Ω will denote the vacuum vector — the vector 1 ∈ C = N is defined as N |Nn h := n1l .
N0 s
h. The number operator
s
The space of finite particle vectors, for which 1l[n,+∞] (N )u = 0 for some n ∈ N, will be denoted by Γfin (h). 2.3. Creation and annihilation operators If h ∈ h, we define the creation operator a∗ (h) by setting a∗ (h) : Γ(h) → Γ(h) , a∗ (h)u :=
√ n + 1 Sn+1 h ⊗ u ,
u∈
n O
h.
s
a(h) denotes the adjoint of a∗ (h), and is called the annihilation operator. Both a∗ (h) and a(h) are defined on Γfin (h) and can be extended to densely defined closed operators on Γ(h). By writing a] (h) we will mean both a∗ (h) and a(h). Note the canonical commutation relations: [a(h1 ), a∗ (h2 )] = (h1 |h2 )1l , [a(h2 ), a(h1 )] = [a∗ (h2 ), a∗ (h1 )] = 0 . It follows from the boundedness of [a(h), a∗ (h)] that a(h) and a∗ (h) have the same domain. Lemma 2.1. (i)
n n
Y Y
p ] −p− n 2 ≤ C a (hi )(N + 1) khi k ,
(N + 1) n,p
i=1
i=1
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(ii) the map hn 3 (h1 , . . . , hn ) 7→ (N + 1)p
n Y
a] (hi )(N + 1)−p− 2 ∈ B(Γ(h)) n
i=1
is norm continuous. (iii) If w − limi→∞ hij = 0, and hij ∈ h are uniformly bounded, then p
s- lim (N + 1) i→∞
n Y
a(hij )(N + 1)−p− 2 = 0 . n
j=1
2.4. Field operators We define the field operator 1 φ(h) := √ (a∗ (h) + a(h)) , 2
h ∈ h.
The operators φ(h) are essentially selfadjoint on Γfin (h) and can be extended to self-adjoint operators on Γ(h). We have 1 a∗ (h) = √ (φ(h) − iφ(ih)) , 2 1 a(h) = √ (φ(h) + iφ(ih)) , 2 [φ(h1 ), φ(h2 )] = iIm (h1 |h2 ) . The following proposition is useful when one tries to reconstruct creationannihilation operators from field operators. Proposition 2.2. If q, p are self-adjoint operators on a Hilbert space H satisfying [q, p] = i1l in the sense of forms on D(q) ∩ D(p), then the operators 1 a∗ := √ (q − ip) , 2
1 a := √ (q + ip) 2
defined on D(q) ∩ D(p) are closed. Proof. We have 1 1 1 (kquk2 + kpuk2 ) = ka∗ uk2 − = kauk2 + . 2 2 2 D(q) is complete with the norm kquk and D(p) p is complete with the norm kpuk. Hence D(q) ∩ D(p) is complete with the norm kquk2 + kpuk2 .
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Lemma 2.3. (i)
n n
Y Y
p −p−n/2 φ(hi )(N + 1) khi k .
(N + 1)
≤ Cn,p
i=1
i=1
(ii) The map hn 3 (h1 , . . . , hn ) 7→ (N + 1)p
n Y
φ(hi )(N + 1)−p−n/2
i=1
is continuous for the norm topology. 2.5. Weyl operators We introduce also the Weyl operators: W (h) := eiφ(h) . Note the identities: [φ(h), W (g)] = Im (g|h)W (g) , W (g)φ(h)W (−g) = φ(h) − Im (g|h) ,
(2.1)
W (h)W (g) = e−i 2 Im (h|g) W (h + g) . 1
Proposition 2.4. (i) For 0 ≤ ≤ 1 k(W (h) − 1l)uk ≤ C k|φ(h)| uk . (ii) The map R 3 s 7→ W (sh)(N + 1)− 2 1
is C 1 in the strong topology and the map R 3 s 7→ W (sh)(N + 1)− 2 − 1
is C 1 in the norm topology. More precisely, lim sup s−1 k(W (sh) − 1l − isφ(h))(N + 1)−1/2− k = 0 .
s→0 khk≤C
(iii)
k(W (h1 ) − W (h2 ))uk
≤ C kh1 − h2 k ((kh1 k2 + kh2 k2 ) 2 kuk + k(N + 1) 2 uk) . Proof. (i) follows from the spectral theorem and the inequality |eis − 1| ≤ C |s| .
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(ii) follows from Lemma 2.3. To show (iii) we note that i W (h1 ) − W (h2 ) = W (h1 ) 1l − e− 2 Im (h1 |h2 ) + e− 2 Im (h1 |h2 ) W (h1 )(1l − W (h2 − h1 )) . i
We note also that 1 − e− 2i Im (h1 |h2 ) ≤ C |Im (h1 |h2 )| , p 1 |Im (h1 |h2 )| ≤ √ kh1 − h2 k kh1 k2 + kh2 k2 , 2 and by (i)
k(1l − W (h2 − h1 ))uk ≤ C k|φ(h2 − h1 )| uk ≤ C kh2 − h1 k k(N + 1) 2 uk . 2.6. Operator dΓ If b is an operator on h, we define the operator dΓ(b) : Γ(h) → Γ(h) , dΓ(b)|Nn h := s
n X j=1
1l ⊗ · · · ⊗ 1l ⊗ b ⊗ 1l ⊗ · · · ⊗ 1l . | {z } | {z } j−1
n−j
An important example is the number operator N = dΓ(1) . Lemma 2.5. (i) Heisenberg derivatives: d dΓ(b) = dΓ dt
d b dt
,
[dΓ(b1 ), dΓ(b2 )] = dΓ([b1 , b2 ]) . (ii) Commutation properties: [dΓ(b), a∗ (h)] = a∗ (bh) , [dΓ(b), a(h)] = −a(b∗ h) , [dΓ(b), iφ(h)] = φ(ibh), if b = b∗ 1 W (h)dΓ(b)W (−h) = dΓ(b) − φ(ibh) + Re (bh|h) 2 (iii) If b1 ≤ b2 , then dΓ(b1 ) ≤ dΓ(b2 ). Moreover,
−1
N 2 dΓ(b)u ≤ dΓ(b∗ b) 12 u .
if b = b∗ .
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2.7. Tensor product of Fock spaces Let hi , i = 1, 2 be Hilbert spaces. Let pi be the projection of h1 ⊕ h2 onto hi , i = 1, 2. We define U : Γ(h1 ⊕ h2 ) → Γ(h1 ) ⊗ Γ(h2 ) , by UΩ = Ω ⊗ Ω , U a (h) = (a] (p1 h) ⊗ 1l + 1l ⊗ a] (p2 h))U , ]
h ∈ h1 ⊕ h2 .
(2.2)
Since the vectors a∗ (h1 ) · · · a∗ (hn )Ω form a total family in Γ(h), and since U preserves the canonical commutation relations, we see that U extends as a unitary operator from Γ(h1 ⊕ h2 ) to Γ(h1 ) ⊗ Γ(h2 ). Moreover one has the following identity: b1 0 = (dΓ(b1 ) ⊗ 1l + 1l ⊗ dΓ(b2 ))U . (2.3) U dΓ 0 b2 Nn It is easy to check that on s (h1 ⊕ h2 ), U is given by s n X n! p 1 ⊗ · · · ⊗ p1 ⊗ p2 ⊗ · · · ⊗ p2 . U |Nn (h1 ⊕h2 ) = {z } | {z } (n − k)!k! | s k=0
n−k
k
2.8. Functor Γ Let hi , i = 1, 2 be Hilbert spaces. Let q : h1 7→ h2 be a bounded linear operator. We define Γ(q) : Γ(h1 ) 7→ Γ(h2 ) Γ(q)|Nn h1 = q ⊗ · · · ⊗ q . s
The Γ functor has the following properties: Lemma 2.6. (i) Relationship with dΓ: assume h1 = h2 . Then edΓ(b) = Γ(eb ) . (ii) Intertwining properties: Γ(q)a∗ (h1 ) = a∗ (qh1 )Γ(q) , Γ(q)a(q ∗ h2 ) = a(h2 )Γ(q) ,
h1 ∈ h1 , h2 ∈ h2 .
(iii) Commutation properties: assume h1 = h2 . Then [a∗ (h), Γ(q)] = a∗ ((1 − q)h)Γ(q) , [a(h), Γ(q)] = −Γ(q)a((1 − q ∗ )h) . (iv) If kqk ≤ 1, then kΓ(q)k ≤ 1 .
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Let us note some additional properties in the isometric and unitary cases. Lemma 2.7. (i) If q is isometric, that is q ∗ q = 1, then Γ(q)a] (h1 ) = a] (qh1 )Γ(q) , Γ(q)φ(h1 ) = φ(qh1 )Γ(q) . (ii) If q is unitary, then Γ(q)a] (h)Γ(q −1 ) = a] (qh) , Γ(q)φ(h)Γ(q −1 ) = φ(qh) .
2.9. Operator dΓ(q, r) Let q, r be operators from h1 to h2 . We define dΓ(q, r) : Γ(h1 ) → Γ(h2 ) , dΓ(q, r)|Nn h1 = s
n X j=1
q ⊗ ···⊗ q⊗r ⊗ q ⊗ ··· ⊗ q . | {z } | {z } j−1
n−j
Lemma 2.8. (i) Relationship with dΓ and Γ: dΓ(1, r) = dΓ(r) , dΓ(r, r) = N Γ(r) . If q is invertible, then dΓ(q, r) = dΓ(rq −1 )Γ(q) = Γ(q)dΓ(q −1 r) . (ii) Heisenberg derivatives of Γ(q): dΓ(b2 )Γ(q) − Γ(q)dΓ(b1 ) = dΓ(q, b2 q − qb1 ) , d d Γ(q) = dΓ q, q . dt dt (iii) Intertwining properties: a(h2 )dΓ(q, r) = dΓ(q, r)a(q ∗ h1 ) + Γ(q)a(r∗ h1 ) , dΓ(q, r)a∗ (h1 ) = a∗ (qh1 )dΓ(q, r) + a∗ (rh1 )Γ(q) . (iv) Commutation properties: assume h1 = h2 . Then [a(h), dΓ(q, r)] = −dΓ(q, r)a((1 − q ∗ )h) + Γ(q)a(r∗ h) , [a∗ (h), dΓ(q, r)] = a∗ ((1 − q)h)dΓ(q, r) − a∗ (rh)Γ(q) .
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(v) If kqk ≤ 1 then we have the following estimate:
1 1 |(u2 |dΓ(q, r2 r1 )u1 )| ≤ dΓ(r2 r2∗ ) 2 u2 dΓ(r1∗ r1 ) 2 u1 . (vi) If kqk ≤ 1 then kN − 2 dΓ(q, r)uk ≤ kdΓ(r∗ r) 2 uk . 1
1
Proof. Let us indicate the proof of parts (v) and (vi), the other being elementary. For an operator r acting on h, we set rj := 1 · · ⊗ 1}l ⊗ r ⊗ 1|l ⊗ ·{z · · ⊗ 1}l , |l ⊗ ·{z j−1
For ui ∈
Nn s
acting on
n O
h.
s
n−j
hi , we have: |(u2 |dΓ(q, r2 r1 )u1 )| ≤
n X 1 1
(r2 r∗ ) 2 u2 (r∗ r1 ) 2 u1 , 2 j 1 j j=1
since kqk ≤ 1. By the Cauchy–Schwarz inequality, we have 12 12 n n n X X X
1
1
(r2 r2∗ ) 2 u2 (r1∗ r1 ) 2 u1 ≤ (u2 |(r2 r2∗ )j u2 ) (u1 |(r1∗ r1 )j u1 ) j j j=1
j=1
j=1
1 1 = dΓ(r2 r2∗ ) 2 u2 dΓ(r1∗ r1 ) 2 u1 , N which proves (v). To prove (vi), we have for u ∈ ns h: kdΓ(q, r)uk ≤
n X
1 1 krj uk ≤ n 2 dΓ(r∗ r) 2 u ,
j=1
again by the Cauchy–Schwarz inequality.
2.10. Operators Pk and Qk Let f0 , f∞ be operators from h1 to h2 . Let f := (f0 , f∞ ). We define the operators Pk (f ) = Pk (f0 , f∞ ) and Qk (f ) = Qk (f0 , f∞ ) for k ∈ N by setting Pk (f ) : Γ(h1 ) → Γ(h2 ) , X Pk (f )|Nn h1 := f1 ⊗ · · · ⊗ fn , s
]{i|i =∞}=k
Qk (f ) : Γ(h1 ) → Γ(h2 ) , X Qk (f )|Nn h1 := f1 ⊗ · · · ⊗ fn , s
]{i|i =∞}≤k
where i = 0, ∞. The following properties of Qk (f ), Pk (f ) can be verified by direct inspection.
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Lemma 2.9. (i) P1 (f ) = dΓ(f0 , f∞ ) , Qk (f ) =
k X
Pj (f ) ,
Pk (f ) = Qk (f ) − Qk−1 (f ) ,
j=0
P0 (f ) = Q0 (f ) = Γ(f0 ) , Pk (qf ) = Γ(q)Pk (f ) ,
Qk (qf ) = Γ(q)Qk (f ) .
(ii) Intertwining properties (we set Q−1 (f ) = 0): Qk (f )a∗ (h1 ) = a∗ (f0 h1 )Qk (f ) + a∗ (f∞ h1 )Qk−1 (f ) , ∗ h2 ) . a(h2 )Qk (f ) = Qk (f )a(f0∗ h2 ) + Qk−1 (f )a(f∞
(iii) Commutation properties: assume h1 = h2 . Then ∗ [a(h), Qk (f )] = −Qk (f )a((1 − f0∗ )h) + Qk−1 (f )a(f∞ h) ,
[a∗ (h), Qk (f )] = a∗ ((1 − f0 )h)Qk (f ) − a∗ (f∞ h)Qk−1 (f ) . (iv) Assume h1 = h2 . If 0 ≤ f0 , 0 ≤ f∞ , f0 + f∞ ≤ 1, then 0 ≤ Qk (f ) ≤ Γ(f0 + f∞ ) ,
0 ≤ Pk (f ) ≤ Γ(f0 + f∞ ) .
Proposition 2.10. Let f = (f0 , f∞ ) and f˜ = (f˜0 , f˜∞ ) and f˜0 f∞ = 0. Then Ql (f˜)Pk (f ) = 0 ,
l < k,
(2.4)
˜ k (f ) = Pk (f˜)Pk (f ) = Pk (f˜0 f0 , f˜∞ f∞ ) , Qk (f)P Ql (f˜)Qk (f ) = Ql (f˜0 f0 , f˜∞ (f0 + f∞ )) , Pl (f˜)Qk (f ) = Pl (f˜0 f0 , f˜∞ (f0 + f∞ )) ,
l ≤ k.
(2.5) (2.6)
2.11. Operator dQk (f , g) For f = (f0 , f∞ ) and g = (g0 , g∞ ) we define dQk (f, g) : Γ(h1 ) → Γ(h2 ) , dQk (f, g)|Nn h1 := s
n X
X
f1 ⊗ · · · ⊗ fj−1 ⊗ g0 ⊗ fj+1 ⊗ · · · ⊗ fn
j=1 ]{i|i =∞}≤k
+
n X
X
j=1 ]{i|i =∞}≤k−1
f1 ⊗ · · · ⊗ fj−1 ⊗ g∞ ⊗ fj+1 ⊗ · · · ⊗ fn .
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Lemma 2.11. (i) dQ0 (f, g) = dΓ(f0 , g0 ) . (ii) Heisenberg derivatives of Qk (f ): dΓ(d2 )Qk (f ) − Qk (f )dΓ(d1 ) = dQk (f, d2 f − f d1 ) , d d Qk (f ) = dQk f, f . dt dt (iii) Intertwining properties: ∗ a(h2 )dQk (f, g) = dQk (f, g)a(f0∗ h2 ) + dQk−1 (f, g)a(f∞ h2 ) ∗ + Qk (f )a(g0∗ h2 ) + Qk−1 (f )a(g∞ h2 ) ,
dQk (f, k)a∗ (h1 ) = a∗ (f0 h1 )dQk (f, g) + a∗ (f∞ h1 )dQk−1 (f, g) + a∗ (g0 h1 )Qk (f ) + a∗ (g∞ h1 )Qk−1 (f ) . (iv) Commutation properties: assume h1 = h2 . Then ∗ h) [a(h), dQk (f, g)] = −dQk (f, g)a((1 − f0∗ )h) + dQk−1 (f, g)a(f∞ ∗ + Qk (f )a(g0∗ h) + Qk−1 (f )a(g∞ h) ,
[a∗ (h), dQk (f, g)] = a∗ (1 − f0 )h)dQk (f, g) − a∗ (f∞ h)dQk−1 (f, g) − a∗ (g0 h)Qk (f ) − a∗ (g∞ h)Qk−1 (f ) . (v) If h1 = h2 , 0 ≤ f0 , 0 ≤ f∞ , f0 + f∞ ≤ 1, g0 , g∞ are selfadjoint, then
1 1 |(u2 |dQk (f, g)u1 )| ≤ dΓ(|g0 |) 2 u2 dΓ(|g0 |) 2 u1
1 1 + dΓ(|g∞ |) 2 u2 dΓ(|g∞ |) 2 u1 . (vi) If h1 = h2 , 0 ≤ f0 , 0 ≤ f∞ , f0 + f∞ ≤ 1, then we have the estimates
−1 1 ∗
N 2 dQk (f, g)u ≤ dΓ(g0∗ g0 + g∞ g∞ ) 2 u . Proof. As for Lemma 2.8, we content ourselves to indicate the proofs of parts (v) and (vi), the rest of the lemma being easy to check. To prove (v), we write dQk (f, g) =
n X
Mj,0 g0,j + Mj,∞ g∞,j ,
j=1
where Mj,0 =
X
f1 ⊗ · · · ⊗ fj−1 ⊗ 1l ⊗ fj+1 ⊗ · · · ⊗ fn ,
]{i|i =∞}=k
Mj,∞ =
X ]{i|i =∞}=k−1
f1 ⊗ · · · ⊗ fj−1 ⊗ 1l ⊗ fj+1 ⊗ · · · ⊗ fn .
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Since f0 + f∞ ≤ 1, we have kMj,0 k ≤ 1, kMj,∞ k ≤ 1. Then we argue as in the proof 1 1 of Lemma 2.8, writing g = g2, g1, for g1, = |g | 2 , g2, = sgn g |g | 2 . A similar argument gives the proof of (vi), following the proof of Lemma 2.8 (vi). 2.12. Partitions of unity In this subsection we further study the operators Pk , Qk under the additional assumption h1 = h2 = h ,
f0 + f ∞ = 1 .
Lemma 2.12. (i) If h1 = h2 = h, 0 ≤ f0 , 0 ≤ f∞ , f0 + f∞ = 1 then the operators Pk (f ) form a partition of unity on Γ(h): s- lim Qk (f ) = 1l , k→∞
s−
∞ X
Pk (f ) = 1l .
k=0
(ii) Intertwining properties: a(h)Qk (f ) = Qk (f )a(h) − Pk (f )a(f∞ h) = Qk−1 (f )a(h) + Pk (f )a(f0 h) , Qk (f )a∗ (h) = a∗ (h)Qk (f ) − a∗ (f∞ h)Pk (f ) = a∗ (h)Qk−1 (f ) + a∗ (f0 h)Pk (f ) . (iii) Commutation properties: [a(h), Qk (f )] = −Pk (f )a(f∞ h) , [a∗ (h), Qk (f )] = a∗ (f∞ h)Pk (f ) . Finally, the operators Pk (f ) and Qk (f ) have other special properties, which will play an imporant role in our geometric analysis of scattering. Proposition 2.13. Let f0 + f∞ = 1, f˜0 + f˜∞ = 1. (i) Let f˜0 f∞ = 0. Then for l ≤ k Ql (f˜)Qk (f ) = Ql (f˜) , Pl (f˜)Qk (f ) = Pl (f˜) . (ii) If 0 ≤ f0 ≤ f˜0 ≤ 1l, then Qk (f ) ≤ Qk (f˜).
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Proof. (i) follows from Proposition 2.10. Let us prove (ii). Note that if f = (f0 , f∞ ) satisfies f0 + f∞ = 1, and depends on some parameter s then d d f0 = − f∞ , ds ds
[b, f0 ] = −[b, f∞ ] .
We observe now that the operator dQk (f, g) under the condition g∞ = −g0
(2.7)
has a simpler form: n X
dQk (f, g)|Nn h = s
X
f1 ⊗ · · · ⊗ fj−1 ⊗ g0 ⊗ fj+1 ⊗ · · · ⊗ fn
(2.8)
j=1 ]{i|i =∞}=k
Clearly, (2.8) is nonnegative if f0 ≥ 0, f∞ ≥ 0 and g0 ≥ 0. Now to prove (ii), we set f s := (1 − s)f + sf˜ , s ∈ [0, 1] . We have:
d Qk (f0s ) = dQk f s , (f˜0 − f0 , f0 − f˜0 ) ≥ 0 , ds
by (2.8). This completes the proof of (ii). ˇ 2.13. Operator Γ
Along with the space Γ(h) we will consider the space Γ(h ⊕ h) ' Γ(h) ⊗ Γ(h). We will use the notation N0 := N ⊗ 1l ,
N∞ := 1l ⊗ N .
Let j0 , j∞ be two operators on h. Set j = (j0 , j∞ ). We identify j with the operator j : h → h⊕ h, jh := (j0 h, j∞ h) . We have j∗ : h ⊕ h → h , ∗ j ∗ (h0 , h∞ ) = j0∗ h0 + j∞ h∞ ,
and ∗ j∞ . j ∗ j = j0∗ j0 + j∞
By second quantization, we obtain the map Γ(j) : Γ(h) → Γ(h ⊕ h) .
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Let U denote the unitary operator identifying Γ(h ⊕ h) with Γ(h) ⊗ Γ(h) introduced in Subsec. 2.7. We define ˇ Γ(j) : Γ(h) → Γ(h) ⊗ Γ(h) , ˇ Γ(j) := U Γ(j) . ˇ Another formula defining Γ(j) is ˇ Γ(j)
n Y
a∗ (hi )Ω :=
i=1
n Y
(a∗ (j0 hi ) ⊗ 1l + 1l ⊗ a∗ (j∞ hi ))Ω ⊗ Ω ,
hi ∈ h .
(2.9)
i=1
Finally, if we denote by Ik the natural isometry between Nk h, then we have:
Nn
h and
Nn−k
h⊗
s Nn = Ik ˇ 1l{k} (N∞ )Γ(j)| h s
n! j0 ⊗ · · · ⊗ j0 ⊗ j∞ ⊗ · · · ⊗ j∞ . {z } | {z } (n − k)!k! | n−k
k
Lemma 2.14. (i) ∗ ˇ ˇ ˜j)∗ 1l{1,...,k} (N∞ )Γ(j) = Qk (˜j0∗ j0 , ˜j∞ j∞ ) , Γ( ∗ ˇ ˜j)∗ 1l{k} (N∞ )Γ(j) ˇ Γ( = Pk (˜j0∗ j0 , ˜j∞ j∞ ) .
(ii) Intertwining properties: ∗ ˇ , ˇ (h) = (a∗ (j0 h) ⊗ 1l + 1l ⊗ a∗ (j∞ h))Γ(j) Γ(j)a ∗ ˇ ˇ Γ(j)a(j 0 h) = a(h) ⊗ 1lΓ(j) , ∗ ˇ ˇ Γ(j)a(j ∞ h) = 1l ⊗ a(h)Γ(j) .
(iii) Commutation properties: ∗ ˇ ˇ ˇ , − Γ(j)a (h) = (a∗ ((1 − j0 )h) ⊗ 1l − 1l ⊗ a∗ (j∞ h))Γ(j) (a∗ (h) ⊗ 1l)Γ(j)
ˇ ˇ ˇ (a(h) ⊗ 1l)Γ(j) − Γ(j)a(h) = −Γ(j)a((1 − j0∗ )h) . ∗ ˇ (iv) Γ(j) is bounded iff kj0∗ j0 + j∞ j∞ k ≤ 1, and then
ˇ kΓ(j)k = 1. Proof. (i) is a direct computation. (ii)–(iv) follow from Subsecs. 2.7 and 2.8. ˇ in the isometric case. Let us note some additional properties of Γ
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Lemma 2.15. Assume ∗ j0∗ j0 + j∞ j∞ = 1 .
(2.10)
(This assumption implies that j is isometric, that is j ∗ j = 1.) Then ˇ ∗ Γ(j) ˇ Γ(j) = 1l . (i) (ii) Intertwining properties: ] ˇ ˇ , Γ(j)a (h) = (a] (j0 h) ⊗ 1l + 1l ⊗ a] (j∞ h))Γ(j)
ˇ ˇ . Γ(j)φ(h) = (φ(j0 h) ⊗ 1l + 1l ⊗ φ(j∞ h))Γ(j) (iii) Let b be an operator on h. Then 1 ˇ ˇ ∗ (dΓ(b) ⊗ 1l + 1l ⊗ dΓ(b))Γ(j) + dΓ(ad2j0 b + ad2j∞ b) . dΓ(b) = Γ(j) 2 Proof. (i) and (ii) are direct consequences of Lemma 2.14. Property (iii) is a kind of IMS localization formula which is shown by direct computation. ˇ 2.14. Operator dΓ(j, k) Let j = (j0 , j∞ ), k = (k0 , k∞ ) be maps from h to h ⊕ h. Let U be the operator constructed in Subsec. 2.7. We set ˇ k) : Γ(h) → Γ(h) ⊗ Γ(h) , dΓ(j, ˇ k) := U dΓ(j, k) . dΓ(j, ˇ k) = U dΓ(k) will be denoted simply by dΓ(k). ˇ The operator dΓ(1, ˇ Lemma 2.16. (i) Heisenberg derivative of Γ(j): dˇ ˇ j, d j , Γ(j) = dΓ dt dt ˇ b (j)) . ˇ ˇ ˇ ad (dΓ(b) ⊗ 1l + 1l ⊗ dΓ(b))Γ(j) − Γ(j)dΓ(b) = dΓ(j, Here b is an operator on h and ˇ b (j) : h → h ⊕ h , ad ˇ b (j)h := ([b, j0 ]h, [b, j∞ ]h) . ad (ii) Intertwining properties: ∗ ˇ k) = dΓ(j, ˇ k)a(j ∗ h) + Γ(j)a(k ˇ a(h) ⊗ 1ldΓ(j, 0 0 h) ,
ˇ k) + (a∗ (k0 h) ⊗ 1l + 1l ⊗ a∗ (k∞ h))Γ(j) ˇ (a∗ (j0 h) ⊗ 1l + 1l ⊗ a∗ (j∞ h))dΓ(j, ˇ k)a∗ (h) . = dΓ(j,
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(iii) Commutation properties: ∗ ˇ ˇ k) − dΓ(j, ˇ k)a(h) = −dΓ(j, ˇ k)a((1 − j ∗ )h) + Γ(j)a(k a(h) ⊗ 1ldΓ(j, 0 0 h) ,
ˇ k) − dΓ(j, ˇ k)a∗ (h) = (a∗ ((1 − j0 )h) ⊗ 1l − 1l ⊗ a∗ (j∞ h))dΓ(j, ˇ k) a∗ (h) ⊗ 1ldΓ(j, ˇ . − (a∗ (k0 h) ⊗ 1l + 1l ⊗ a∗ (k∞ h))Γ(j) ∗ j∞ ≤ 1, k0 , k∞ are self-adjoint, we have the estimate: (iv) If j0∗ j0 + j∞
ˇ k)u1 )| ≤ dΓ(|k0 |) 12 ⊗ 1lu2 dΓ(|k0 |) 12 u1 |(u2 |dΓ(j,
1 1 + 1l ⊗ dΓ(|k∞ |) 2 u2 dΓ(|k∞ |) 2 u1 . ∗ j∞ ≤ 1, then (v) If j0∗ j0 + j∞
ˇ k)u ≤ dΓ(k ∗ k0 + k ∗ k∞ ) 12 u .
(N0 + N∞ )− 12 dΓ(j, 0 ∞ Proof. All statements follow directly from analogous statements in Lemma 2.8 and from the identities in Subsec. 2.7. The only point which deserve some care is (iv). To prove (iv), we write k = k 0 + k ∞ , where k 0 = (k0 , 0), k ∞ = (0, k∞ ), and 1 1 use Lemma 2.8 (v), writing k 0 as r2 r1 with r2 = (|k0 | 2 , 0), r1 = sgn k0 |k0 | 2 , and 1 1 k∞ as r2 r1 with r2 = (0, |k∞ | 2 ), r1 = sgn k∞ |k∞ | 2 . 2.15. Scattering identification operators Let i : h ⊕ h → h, (h0 , h∞ ) 7→ h0 + h∞ . An important role in scattering theory is played by the following identification operator (see [31]): ˇ ∗ )∗ : Γ(h) ⊗ Γ(h) → Γ(h) . I := Γ(i)U ∗ = Γ(i √ Note that since kik = 2, the operator Γ(i) is unbounded. Another formula defining I is: I
n Y i=1
a∗ (hi )Ω ⊗
p Y i=1
a∗ (gi )Ω :=
p Y i=1
a∗ (gi )
n Y
a∗ (hi )Ω ,
hi , g i ∈ h .
(2.11)
i=1
If h = L2 (Rd , dk), then we can write still another formula for I: Z 1 Iu ⊗ ψ = ψ(k1 , . . . , kp )a∗ (k1 ) · · · a∗ (kp )udk , u ∈ Γ(h), ψ ∈ ⊗ps h . 1 (p!) 2 (2.12)
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We deduce from (2.11) that I(N + 1)−k/2 ⊗ 1l restricted to Γ(h) ⊗
k O
h is bounded .
(2.13)
s
Lemma 2.17. Let b be an operator on h. Then dΓ(b)I = I(dΓ(b) ⊗ 1l + 1l ⊗ dΓ(b)) ,
(i) (ii)
1 φ(h)I − I(φ(h) ⊗ 1l) = √ I1l ⊗ a(h) , 2
h ∈ h.
Proof. (i) follows from Lemma 2.16 (i). (ii) follows from Lemma 2.15 (ii).
It is easy to construct a right inverse to the identification operator I. Let j0 , j∞ be two operators on h such that 0 ≤ j0 ≤ 1, 0 ≤ j∞ ≤ 1, and j0 + j∞ = 1. Let j = (j0 , j∞ ) : h → h ⊕ h, as in Subsec. 2.13. Clearly, 0 ≤ j ∗ j ≤ 1, hence kjk ≤ 1, ˇ and therefore Γ(j) is a bounded operator. We have ij = 1, hence ˇ I Γ(j) = 1l . We also have ˇ = Qk (j) , I1l{1,...,k} (N∞ )Γ(j) ˇ = Pk (j) . I1l{k} (N∞ )Γ(j)
(2.14)
2.16. Space of additional degrees of freedom In this subsection we fix some notation which will be used in the next section to define the interaction part of the Hamiltonian. Suppose that K is a Hilbert space. If v ∈ B(K, K ⊗ h), then we can define a∗ (v), a(v), φ(v) as unbounded operators on K ⊗ Γ(h): √ a∗ (v)|K⊗Nn h := n + 1 (1lK ⊗ Sn+1 ) v ⊗ 1lNn h , s
s
∗
∗
a(v) := (a (v)) , 1 φ(v) := √ (a(v) + a∗ (v)) . 2 They satisfy the estimates
]
a (v)(N + 1)− 12 ≤ kvk ,
(2.15)
where kvk is the norm of v in B(K, K ⊗ h). Clearly, the condition v ∈ B(K, K ⊗ h) is equivalent to v ∗ v ∈ B(K) .
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If h = L2 (Rd , dk), then the operator v can be represented as a function k 7→ v(k) ∈ B(K) (defined a.e.), and the condition v ∈ B(K, K ⊗ h) is equivalent to Z (2.16) v ∗ (k)v(k)dk ∈ B(K) . (2.16) is implied in particular by Z kv(k)k2B(K) dk < ∞ .
3. Pauli Fierz Hamiltonians 3.1. Introduction In this section we introduce the class of Hamiltonians that we will study in this paper and we give some examples. We also describe a number of related definitions which will be useful in our study, in particular the “extended Hamiltonian” H ext . At the end of this section we prove some technical estimates concerning the Hamiltonian H. 3.2. Hamiltonian Let K be a Hilbert space representing the degrees of freedom of the atomic system. The Hamiltonian describing the atomic system is denoted by K. We assume that K is selfadjoint on D(K) ⊂ K and bounded below. A condition that will be sometimes imposed is (K + i)−1
is compact .
(H0)
Its physical interpretation is that the atomic system is confined. Let h = L2 (Rd , dk) be the 1-particle Hilbert space in the momentum representation and let Γ(h) be the bosonic Fock space over h, representing the field degrees of freedom. We will denote by k the momentum operator of multiplication by k on L2 (Rd , dk), and by x = i∇k the position operator on L2 (Rd , dk). Let ω ∈ C(Rd , R) be the boson dispersion relation. A general condition which will always be assumed is: ∇ω ∈ L∞ (Rd ) , ∇ω(k) 6= 0 for k 6= 0 , (H1) lim ω(k) = +∞ , |k|→∞ inf ω(k) = ω(0) =: m > 0 . The quantity m = inf ω(k) is called the boson rest mass and plays a very important role in our analysis. The typical example is of course the relativistic dispersion 1 relation ω(k) = (k 2 + m2 ) 2 . We will sometimes need the following smoothness assumption: |∂kα ω(k)| ≤ Cα ,
|α| ≥ 1 .
(H2)
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The Hamiltonian describing the field is equal to dΓ(ω(k)). The Hilbert space of the interacting system is H := K ⊗ Γ(h) . The interaction between the atom and the boson field is described with a coupling operator v satisfying v ∈ B(K, K ⊗ h) . (I1) As we saw in Subsec. 2.16, condition (I1) is implied by the stronger condition Z (I1)0 kv(k)k2B(K) dk < ∞ . The interaction term is equal to: 1 V := φ(v) = √ (a∗ (v) + a(v)) . 2 We consider the Hamiltonian H := H0 + V ,
acting on H ,
where H0 := K ⊗ 1l + 1l ⊗ dΓ(ω(k)) . Proposition 3.1. Assume hypotheses (H1) and (I1). Then V is H0 −bounded with the infinitesimal bound. Consequently H is selfadjoint on D(H0 ) and bounded below. Proof. It follows from (2.15) and from the positivity of the mass that
]
a (v)(H0 + 1)− 12 ≤ Ckvk .
(3.1)
This implies that V is H0 -bounded with the infinitesimal bound.
To study the scattering theory for H, in particular to establish the existence of asymptotic fields, we will need to impose a stronger condition on the interaction v: k1l[R,∞[ (|x|)vkB(K,K⊗h) ≤ CR−1−µ ,
µ > 0, .
(SR)
This assumption is an analog of the short-range condition in non-relativistic scattering. Note that without much additional work, essentially all our results could be proven under a somewhat weaker assumption k1l[R,∞[ (|x|)(i + K)−1 vkB(K,K⊗h) ≤ CR−1−µ ,
µ > 0.
(SR0 )
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We will use the following notations for various Heisenberg derivatives: ∂ + [ω(k), i·] , ∂t ∂ + [dΓ(ω(k)), i·] , D0 = ∂t ∂ + [H, i·] , D= ∂t d0 =
acting on B(h) , acting on B(Γ(h)) , acting on B(H) .
Note that we have D0 dΓ(b) = dΓ(d0 b) . 3.3. Examples Our first example is the spin-boson model, where the small system is simply a two components spin. We have then K = C2 , K = σz , and v(k) = σx ⊗ g(k), for a scalar function g ∈ L2 (Rd , C). Here σx , σy , σz are the Pauli matrices. So the spin-boson Hamiltonian is given by H = σz ⊗ 1l + 1l ⊗ dΓ(ω) + σx ⊗ φ(g) ,
acting on C2 ⊗ Γ(L2 (Rd )) .
Recently there has been a renewed interest in the scattering theory for the spinboson model, in connection with the problem of radiative decay. In [32], Huebner and Spohn proved a Mourre estimate for the massive spin-boson model for a small coupling constant. In [31] the scattering theory for the spin-boson model is connected to the radiative decay problem and various questions are formulated. Our second example is a simplified Hamiltonian of an atom iteracting with a massive relativistic bosonic field. (Note that a similar Hamiltonian was contained in the paper by Pauli and Fierz [41], except that the field was electromagnetic, and hence massless.) In this case K := L2 (R3N ) , K =
N X 1 j=1
2
Dx2 j +
N X j=1
W (xj ) +
X
U (xi − xj ) .
i<j
Here W is the interaction between one electron and the nucleus and U the electronelectron interaction. If we assume that the potential W tends to +∞ at infinity, i.e. that the atom is confined, then condition (H0) is satisfied. 1 The boson dispersion relation is ω(k) = (k 2 + m2 ) 2 . The interaction V is given by N Z X (v(k, xj )a∗ (k) + v(k, xj )a(k))dk , V = j=1
where xj denotes the position of the jth electron and v(k, x) is a function R3 × R3 3 (k, x) 7→ v(k, x) ∈ C .
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In order to satisfy condition (SR), we need to assume that ! 12
Z sup |y|>R
x
≤ CR−1−µ ,
|ˆ v (y, x)| dy 2
(3.2)
where vˆ(y, x) denotes the Fourier transform of v(k, x) with respect to the first variable. (3.2) is satisfied if for instance v(k, x) = ρ(x)eik·x ω(k)− 2 χ(k) , 1
where ρ ∈ S(Rd ) is a spacial cut-off and χ ∈ S(Rd ) is an ultraviolet cutoff. Note that if one uses the condition (SR0 ), then one can treat a somewhat more general, and perhaps even more physical, class of Hamiltonians. (SR0 ) is implied by the following hypothesis: W (x) ≥ C0 hxiσ ,
for some C0 > 0, σ ≥ 0 , ! 12
Z sup x
|y|>R
−2σ
|ˆ v (y, x)| hxi 2
dy
≤ CR−1−µ .
(3.3)
In particular, (3.3) is a consequence of the following conditions: W (x) ≥ C0 hxi1+µ , v(k, x) = e
ikx
for some C0 > 0 , ω(k)− 2 χ(k) , 1
(3.4)
where χ ∈ S(Rd ) is an ultraviolet cutoff. Note that under the condition (3.4) the interaction V is translation invariant. Nevertheless, the atomic Hamiltonian K has to be confining, and hence it is not translation invariant. For more discussion of similar models, their relationship with quantum electrodynamics and their validity the reader should consult [5]. 3.4. Extended Hilbert space Along with the space H = K ⊗ Γ(h) , we will use the “extended space” Hext := H ⊗ Γ(h) = K ⊗ Γ(h) ⊗ Γ(h) . The extended Hilbert space is very convenient to set up the scattering theory for H. We will use the notation N0 := 1l ⊗ N ⊗ 1l ,
N∞ := 1l ⊗ 1l ⊗ N .
We will also need the “extended Hamiltonian” and the “extended free Hamiltonian” H ext := H ⊗ 1l + 1l ⊗ dΓ(ω(k)) , H0ext := H0 ⊗ 1l + 1l ⊗ dΓ(ω(k)) .
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It is useful to introduce the following asymmetric Heisenberg derivatives: ˇ 0 f (t) := ∂ f (t) + (ω(k) ⊕ ω(k))if (t) − if (t)ω(k) , d ∂t
f (t) ∈ B(h, h ⊕ h) ,
ˇ 0 F (t) := ∂ F (t) + (dΓ(ω) ⊗ 1l + 1l ⊗ dΓ(ω))iF (t) − iF (t)dΓ(ω) , D ∂t F (t) ∈ B(Γ(h), Γ(h) ⊗ Γ(h)) , ∂ ˇ DB(t) := B(t) + H ext iB(t) − iB(t)H , ∂t
B(t) ∈ B(H, Hext ) .
Note that we have ˇ 0f ) . ˇ 0 dΓ(f ) = dΓ( ˇ d D For a selfadjoint operator A, we will denote by Hcomp (A) the subspace of vectors u ∈ H such that u = χ(A)u, for some χ ∈ C0∞ (R). In particular, the space Hcomp (N ) is the space Γfin (h) of finite particle vectors. 3.5. Number-energy estimates This subsection is devoted to some rather elementary bounds. Note that these bounds fail if the boson mass m is zero. They imply directly that in the estimates of Sec. 2, the factor (N + 1) can be replaced by (H + i). This observation will be used often in the sequel. Lemma 3.2. Assume the hypotheses (H1), (I1). (i) Then uniformly for z in acompact set of C, we have (N + 1)−m (z − H)−k (N + 1)m+k ∈ O(|Im z|−Cm,k ) . (ii) Let χ ∈ C0∞ (R). Then kN m χ(H)N p k < ∞ ,
n, p ∈ N .
Proof. Clearly, adjN H = φ(ij v). By hypothesis (I1) φ(ij v)(H +i)−1 is bounded. We have (H + z)−1 N k = N (H + z)−1 N k−1 + (H + z)−1 φ(iv)(H + z)−1 N k−1 . Moving repeatedly factors of N to the left, we get (H + z)−1 N k = N k (H + z)−1 +
k X
N k−l (H + z)−1 Bl (z) ,
i=1
with Bl (z) ∈ O(|Im z|−l ). Therefore, using N (H + z)−1 ∈ O(|Im z|−1 )
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we see that (N + 1)−k+1 (H + z)−1 (N + 1)k ∈ O(|Im z|−k ) . This implies (i). (ii) follows directly from (i) by writing N m χ(H)N p =
m Y
N m−k (H + i)−1 N k−m−1 (H + i)m χ(H)(H + i)p
k=1
×
p Y
N k−p−1 (H + i)−1 N p−k .
k=1
We will use the following notation: for an operator B(t) ∈ B(H) depending on some parameter t we will write B(t) ∈ (N + 1)m ON (tp ) if k(N + 1)−m−k B(t)(N + 1)k k ≤ Ck hti , p
k ∈ Z.
Likewise, for an operator C(t) ∈ B(H, Hext ) we will write ˇ N (tp )(N + 1)m C(t) ∈ O if k(N0 + N∞ )−m−k C(t)(N + 1)k k ≤ Ck hti , p
k ∈ Z.
Finally we will frequently use the following functional calculus formula (see [26, 13]) for χ ∈ C0∞ (R): Z i ∂ z χ(z)(z ˜ − A)−1 dz ∧ d z , (3.5) χ(A) = 2π C where χ ˜ ∈ C0∞ (C) is an almost analytic extension of χ satisfying χ ˜|R = χ , ˜ ≤ Cn |Im z|n , |∂ z χ(z)|
n ∈ N.
3.6. Commutator estimates In this subsection we estimate commutators between some operators considered in Sec. 2 and functions of H and H ext . Lemma 3.3. Let f0 ∈ C0∞ (Rd ), f∞ ∈ C ∞ (Rd ), 0 ≤ f0 , 0 ≤ f∞ , f0 + f∞ ≤ 1, f0 = 1 near 0 (and hence f∞ = 0 near 0). Set f := (f0 , f∞ ) and, for R ≥ 1, x x R R ), where f0R (x) = f0 ( R ), f∞ (x) = f∞ ( R ). Assume hypotheses (H1), f R = (f0R , f∞ (H2). Let χ ∈ C0∞ (R). Then for m ∈ N, one has ( under (I1) , o(R0 ) m R N [χ(H), Qk (f )]χ(H) ∈ (3.6) −1 O(R ) under (SR) .
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2 Lemma 3.4. Let j0 ∈ C0∞ (Rd ), j∞ ∈ C ∞ (Rd ), 0 ≤ j0 , 0 ≤ j∞ , j02 + j∞ ≤ 1, j0 = 1 near 0 (and hence j∞ = 0 near 0). Set j := (j0 , j∞ ) and for R ≥ 1 R ). Assume hypotheses (H1), (H2). j R = (j0R , j∞ ( under (I1) , o(R0 ) ext −1 R R −1 ˇ ˇ (i) (H + i) Γ(j ) − Γ(j )(H + i) ∈ O(R−1 ) under (SR) .
(ii) Let χ, χ ˜ ∈ C0∞ (R). Then ( m
(N0 + N∞ ) (χ(H
ext
ˇ R ) − Γ(j ˇ R )χ(H))χ(H) )Γ(j ˜ ∈
o(R0 ) O(R
−1
under (I1) , ) under (SR) .
Proof of Lemma 3.3. Let m(R) := o(R0 ), if hypothesis (I1) holds, and m(R) := O(R−1 ), if hypothesis (SR) holds. Let us show the first estimate. [H0 , Qk (f R )] = dQk (f R , [ω(k), f R ]) ∈ ON (R−1 )(N + 1) ,
(3.7)
using Lemma 2.11 (vi) and pseudodifferential calculus. Using then Lemma 2.9 (iii), and the fact that 0 6∈ supp f∞ , 0 6∈ supp (1 − f0 ) we have 1 R v)Qk−1 (f R ) [V, Qk (f R )] = √ (a∗ ((1 − f0R )v)Qk (f R ) − a∗ (f∞ 2 R − Qk (f R )a((1 − f0R )v) + Qk−1 (f R )a(f∞ v)) 1
∈ n(R)(N + 1) 2 ,
(3.8)
where n(R) = o(R0 ) under hypothesis (I1), and n(R) = O(R−1−µ ) under hypothesis (SR). Hence [H, Qk (f t )] ∈ m(R)(N + 1) . Next we use the functional calculus formula (3.5), which yields Z i m N m [χ(H), Qk (f t )]χ(H) = ∂ z χ(z)N ˜ (z − H)−1 2π C × [H, Qk (f t )](z − H)−1 χ(H)dz ∧ d z .
(3.9)
By Lemma 3.2, N m (z − H)−1 (N + 1)−m+1 ∈ O(|Im z|−Cm ) , k(1 + N )m χ(H)k ˜ ≤C.
(3.10)
This shows that the integrand in (3.9) is bounded by O(|Im z|−p )t−1 for some p, and completes the proof of the lemma. Proof of Lemma 3.4. Using Lemma 2.16 (i), we obtain ˇ ω(k) j R ) ∈ O ˇ R )H0 = dΓ(j ˇ R , ad ˇN (R−1 )(N + 1) . ˇ R ) − Γ(j H0ext Γ(j
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Likewise, using Lemma 2.14 (iii), ˇ R ) − Γ(j ˇ R )V V ⊗ 1lΓ(j 1 R ˇ R ) − Γ(j ˇ R )a((1 − j0R )v)) = √ (a∗ ((1 − j0R )v ⊗ 1l − 1l ⊗ a∗ (j∞ v)Γ(j 2 ˇ N (m(R))(N + 1) 21 . ∈O
(3.11)
Therefore, ˇ R )H ∈ O ˇ N (m(R))(N + 1) . ˇ R ) − Γ(j H ext Γ(j
(3.12)
This implies (i). Using then formula (3.5), we have ˇ R ) − Γ(j ˇ R )χ(H))χ(H) (N0 + N∞ )m (χ(H ext )Γ(j ˜ Z i m ext −1 = ∂ z χ(z)(N ˜ ) 0 + N∞ ) (z − H 2π C ˇ R )H)(z − H)−1 χ(H)dz ˇ R ) − Γ(j × (H ext Γ(j ˜ ∧ dz. By the same argument as in Lemma 3.2, we have (N0 + N∞ )m (z − H ext )−1 (N0 + N∞ )−m+1 ∈ O(|Im z|−Cm ) . Then we argue as in the proof of Lemma 3.3.
4. Spectral Analysis of Pauli Fierz Hamiltonians 4.1. Introduction In this section we study the properties of the spectrum of H. The results of this section are fairly parallel to their analogs in the theory of N -body Schr¨ odinger operators. In Subsec. 4.2 we will show an analog of the HVZ theorem describing the essential spectrum of H. It will obviously imply the existence of a ground state of H. Note that in the massless case under certain additional assumptions, it is also possible to prove the existence of a ground state, but the result is deeper then (see [5, 3, 48]). In Subsec. 4.3 we will prove the finiteness of the imbedded pure point spectrum outside of thresholds — a result that follows from an analog of the Mourre estimate, which we also prove in this subsection. Let us stress that the assumption that the boson mass is positive plays an important role in the results of Subsecs. 4.2 and 4.3. If A is an operator, then σ(A) denotes its spectrum, σpp (A) its pure point spectrum and σess (A) its essential spectrum. For a Borel subset U ⊂ R we use 1lU (A) to denote the spectral projection of A onto U . 4.2. HVZ theorem and existence of a ground state Let us state the main result of this subsection.
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Theorem 4.1. Assume hypotheses (H0), (H1), (I1). Then σess (H) = [inf σ(H) + m, +∞[ . Consequently, inf σ(H) is a discrete eigenvalue of H. We will make use of the partitions of unity of Subsec. 2.12. Recall that to construct this partition we pick functions j0 , j∞ ∈ C ∞ (Rd ) with 0 ≤ j0 ≤ 1, j0 ∈ C0∞ (Rd ), 2 = 1. For R ≥ 1, j R is defined as in Subsec. 3.6. We will j0 = 1 near 0 and j02 + j∞ R R 2 also set q = (j0 ) . Lemma 4.2. Assume hypotheses (H0), (H1) and (I1). Then the operator Γ(q R )(H + i)−1 is compact on H. Proof. Since D(H) = D(H0 ), we see that it is enough to show the compactness of Γ(q R )(H0 + i)−1 . Since 1l[n,+∞[ (N )(H0 + i)−1 tends to 0 in norm when n → ∞, it suffices to prove that Γ(q R )(H0 + i)−1 is compact on every n-particle sector. But R
−1
Γ(q )(H0 + i)
|K⊗Nn h = s
n Y
j02
x
i=1
i
R
i+K +
n X
!−1 ω(Di )
i=1
is compact, using hypotheses (H0) and (H1).
Proof of Theorem 4.1. We prove first the ⊂ part of the theorem. Let χ ∈ − ∞, inf σ(H) + m[). Because of suppχ, we have:
C0∞ (]
χ(H ext ) = χ(H ext )1l{0} (N∞ ) . Hence, using twice Lemma 3.4, we have ˇ R) ˇ R )∗ Γ(j χ(H) = χ(H)Γ(j ˇ R )∗ χ(H ext )Γ(j ˇ R ) + o(R0 ) = Γ(j ˇ R )∗ χ(H ext )1l{0} (N∞ )Γ(j ˇ R ) + o(R0 ) = Γ(j ˇ R )χ(H) + o(R0 ) . ˇ R )∗ 1l{0} (N∞ )Γ(j = Γ(j ˇ R )χ(H) = Γ q R χ(H) is compact by Lemma 4.2. ˇ R )∗ 1l{0} (N∞ )Γ(j The operator Γ(j Hence χ(H) is compact as a limit of compact operators. Let us now prove the ⊃ part of the theorem. Note that it follows from the ⊂ part of the theorem that H admits a ground state. Let R λ > inf σ(H) + m. Let w be a ground state of H. Let h ∈ C0∞ (Rd ) with h(k)dk = 1 and let x0 ∈ Rd , x0 6= 0, k0 ∈ Rd , k0 6= 0, ω(k0 ) = λ − inf σ(H). Choose a sequence (Rj ) such that limj→∞ j −1 Rj = ∞ and define hj ∈ C0∞ (Rd ) by setting hj (k) = j d/2 h(j(k − k0 ))eiRj hk,x0 i .
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Then khj k = 1, w − limj→∞ hj = 0 and limj→∞ (ω(k) − ω(k0 ))hj = 0. Let uj := a∗ (hj )w . We have limj→∞ kuj k = 1 and, by Lemma 2.1 (iii), w − limj→∞ uj = 0. Now (H − λ)uj = a∗ (hj )(H − λ)w + a∗ (ω(k)hj )w + (v|hj )w = a∗ ((ω(k) − ω(k0 ))hj )w + (v|hj )w ∈ o(j 0 ) , when j → ∞. Since uj tends weakly to 0, we have constructed a Weyl sequence for λ. 4.3. The Mourre estimate and local finiteness of point spectrum Let τ := σpp (H) + mN× , where N× is the set of positive integers. Elements of τ will be called thresholds, in analogy with the case of N -particle Schr¨ odinger operators. Let a be the operator on h defined as a = 12 (∇ω(k) · Dk + Dk · ∇ω(k)). Under hypothesis (H2) a is selfadjoint with domain D(a) := {h ∈ h|ah ∈ h}. We assume in this section that kavkB(K,K⊗h) < ∞ .
(I2)
Let A = 1lK ⊗ dΓ(a). Note that [H, iA], defined as a quadratic form on D(A)∩D(H) equals [H, iA] = dΓ(|∇ω(k)|2 ) + φ(iav) . Moreover D(A) ∩ D(H) contains the space Γfin (S(Rd )) which is a core for H. So [H, iA] extends as an operator bounded on D(H), similar to H, with ω(k) replaced by |∇ω(k)|2 and v by iav. Finally using the fact that D(H) = D(H0 ), it is easy to check that eiαA leaves D(H) invariant and that sup|α|<1 kHeiαA ψk < ∞ for ψ ∈ D(H). Consequently Lemma 3.4 applies to [H, iA]. Another consequence of hypothesis (I2) is that [H, iA](H + i)−1 is bounded. Theorem 4.3. Assume the hypotheses (H0), (H1), (H2), (I1), (I2). Then (i) Let λ ∈ R\τ. Then there exists > 0, C0 > 0 and a compact operator K0 such that 1l[λ−,λ+] (H)[H, iA]1l[λ−,λ+] (H) ≥ C0 1l[λ−,λ+] (H) + K0 . (ii) For all [λ1 , λ2 ] such that [λ1 , λ2 ] ∩ τ = ∅, one has dim1lpp [λ1 ,λ2 ] (H) < ∞ . Consequently σpp (H) can accumulate only at τ, which is a closed countable set.
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(iii) Let λ ∈ R\(τ ∪ σpp (H)). Then there exists > 0, C0 > 0 such that 1l[λ−,λ+] (H)[H, iA]1l[λ−,λ+] (H) ≥ C0 1l[λ−,λ+] (H) . Proof. The proof will have the same logical structure as in the case of N -particle Schr¨ odinger operators. Let Ω⊥
d(λ) :=
inf
{σpp (H)+dΓ(ω(k))=λ}
( = inf
n X i=1
˜ := d(λ)
) n X |∇ω(ki )| λ1 + ω(ki ) = λ , n = 1, 2, . . . , λ1 ∈ σpp (H) , 2
i=1
inf
{σpp (H)+dΓ(ω(k))=λ}
( = inf
n X i=1
dΓ(|∇ω(k)|2 )
dΓ(|∇ω(k)|2 )
) n X |∇ω(ki )| λ1 + ω(ki ) = λ , n = 0, 1, 2, . . . , λ1 ∈ σpp (H) . 2
i=1
The superscript Ω⊥ in the definition of d(λ) means that one excludes the vacuum sector to compute the infimum. Let us note that ( d(λ) , λ 6∈ σpp (H) , ˜ := d(λ) 0, λ ∈ σpp (H) , ˜ We introduce also “smeared out” versions of the functions d(λ) and d(λ). We set ∆κλ := [λ − κ, λ + κ] and dκ (λ) := inf κ d(µ) , µ∈∆λ
˜ . d˜κ (λ) := inf κ d(µ) µ∈∆λ
Note that the following inequality holds Ω⊥
inf(d˜κ (λ − dΓ(ω(k))) + dΓ(|∇ω(k)|2 )) ≥ dκ (λ) .
(4.1)
or in other words, if n = 1, 2, . . . , then ! n n X X κ ˜ ω(ki ) + |∇ω(ki )|2 ≥ dκ (λ) . d λ− i=1
i=1
We will use an induction with respect to n ∈ N. Let us first list the statements that we will show. We put E0 := inf σ(H).
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H1 (n) : Let > 0 and λ ∈ [E0 , E0 + nm[. Then there exists a compact operator K0 , an interval ∆ 3 λ such that 1l∆ (H)[H, iA]1l∆ (H) ≥ (d(λ) − )1l∆ (H) + K0 . H2 (n) : Let > 0 and λ ∈ [E0 , E0 + nm[. Then there exists an interval ∆ 3 λ such that ˜ − )1l∆ (H) . 1l∆ (H)[H, iA]1l∆ (H) ≥ (d(λ) H3 (n) : Let κ > 0, 0 > 0 and > 0. Then there exists δ > 0 such that for all λ ∈ [E0 , E0 + nm − 0 ], one has 1l∆δ (H)[H, iA]1l∆δ (H) ≥ (d˜κ (λ) − )1l∆δ (H) . λ
λ
λ
S1 (n) : τ is a closed countable set in [E0 , E0 + nm]. S2 (n) : for all λ1 ≤ λ2 ≤ E0 +nm with [λ1 , λ2 ]∩τ = ∅, we have dim1lpp [λ1 ,λ2 ] (H) < ∞. For all n ∈ N, we will describe the proof the following implications: H1 (n) ⇒ H2 (n) , H2 (n) ⇒ H3 (n) , H1 (n) ⇒ S2 (n) , S2 (n − 1) ⇒ S1 (n) , S1 (n) and H3 (n − 1) ⇒ H1 (n) . Note first that the statements H1 (1) and S1 (1) are immediate since the spectrum of H is discrete in [E0 , E0 + m[. Note also that the implication S2 (n − 1) ⇒ S1 (n) is obvious. The proofs of the implications H1 (n) ⇒ H2 (n), H2 (n) ⇒ H3 (n), H1 (n) ⇒ S2 (n) are standard abstract arguments which adapt directly to the present setting (see e.g. [38] and [20]). It remains to prove that S1 (n) and H3 (n − 1) ⇒ H1 (n). Using Lemma 3.4 for H and [H, iA], we write, for χ ∈ C0∞ ([E0 , E0 + nm[), ˇ R )χ(H)[H, iA]χ(H) ˇ R )∗ 1l{0} (N∞ )Γ(j χ(H)[H, iA]χ(H) = Γ(j ˇ R )∗ 1l[1,∞[ (N∞ )Γ(j ˇ R )χ(H)[H, iA]χ(H) + Γ(j ˇ R )∗ 1l[1,∞[ (N∞ )χ(H ext ) = Γ(q R )χ(H)[H, iA]χ(H) + Γ(j ˇ R ) + o(R0 ) . × [H ext , iA]χ(H ext )Γ(j
(4.2)
The first term of (4.2) is compact by Lemma 4.2. The second term in the r.h.s. of (4.2) we estimate by diagonalizing dΓ(ω(k)) and dΓ(|∇ω(k)|2 ) on the range of 1l[1,+∞[ (N∞ ). Using the closedness of τ in [E0 , E0 + nm], i.e. the induction hypothesis S1 (n), we see that d(λ) = sup dκ (λ) , κ>0
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for λ ∈ [E0 , E0 + nm[. So we may choose κ small enough so that dκ (λ) ≥ d(λ) − /3. Next using H3 (n − 1) we choose δ such that for λ ∈ [E0 , E0 + nm − 0 [, we have 1l∆δ (H +dΓ(ω(k)))([H, iA] ⊗ 1l+1l ⊗ dΓ(|∇ω(k)|2 ))1l∆δ (H +dΓ(ω(k)))1l[1,∞[ (N∞ ) λ λ 1l[1,∞[ (N∞ ) ≥ 1l∆δ (H + dΓ(ω(k))) d˜κ (λ − dΓ(ω(k))) + dΓ(|∇ω(k)|2 ) − λ 3 1l δ (H + dΓ(ω(k)))1l[1,∞[ (N∞ ) ≥ dκ (λ) − 3 ∆λ 2 1l∆δ (H + dΓ(ω(k)))1l[1,∞[ (N∞ ) . ≥ d(λ) − λ 3 Using again Lemmas 3.4 and 4.2, this yields, for supp χ ⊂ [λ − δ, λ + δ], χ(H)[H, iA]χ(H) ≥ (d(λ) − 2/3)χ2 (H) + K1 + o(R0 ) , where K1 is compact. Picking R large enough, this proves H1 (n).
5. Asymptotic Fields and Wave Operators 5.1. Introduction In this section we describe the existence of asymptotic fields. Using these fields we define wave operators. Results of this section follow easily by the Cook method and were well known for a long time (see for example [28, 29]). They are the analog of the existence of the wave operators in non-relativistic scattering theory and serve as the conceptual basis for the scattering theory in QFT. Note that most of the results, after minor modifications, hold even if the mass of the bosons is zero. The most important exception is the unitarity of the wave operator, which implies the Fock property of the asymptotic commutation relations. 5.2. Asymptotic fields In all this section, we will assume the conditions (H1), (H2), (I1) and (SR). For h ∈ h we set ht := e−itω(k) h. We denote by h0 ⊂ h the space C0∞ (Rd \{0}). Theorem 5.1. (i) For all h ∈ h the strong limits W + (h) := s- lim eitH W (ht )e−itH t→+∞
(5.1)
exist. They are called the asymptotic Weyl operators. For h ∈ h0 the limit in (5.1) is a norm limit. For all h ∈ h and > 0 the asymptotic Weyl operators can be also defined using the norm limit: W + (h)(i + H)− = lim eitH W (ht )(i + H)− e−itH . t→+∞
(5.2)
(ii) The map h 3 h 7→ W + (h)
(5.3)
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is strongly continuous and the map h 3 h 7→ W + (h)(i + H)−
(5.4)
is norm continuous. (iii) The operators W + (h) satisfy the Weyl commutation relations: 1
W + (h)W + (g) = ei 2 Im (h|g) W + (h + g) . (iv) The Hamiltonian preserves the asymptotic Weyl operators: eitH W + (h)e−itH = W + (h−t ) .
(5.5)
Proof. It follows from Lemma 2.7 (ii) that W (ht ) = e−itH0 W (h)eitH0 , which implies that, as a quadratic form on D(H0 ), one has ∂t W (ht ) = −[H0 , iW (ht )] .
(5.6)
Using (5.6) and the fact that D(H) = D(H0 ), we have, as quadratic forms on D(H), ∂t eitH W (ht )e−itH = ieitH Im (ht |v)W (ht )e−itH . Integrating this relation we obtain (first as a quadratic form identity on D(H0 ), then by a simple argument, as an operator identity) Z t eisH Im (hs |v)W (hs )e−isH ds . eitH W (ht )e−itH − W (h) = i 0
Using assumption (SR) and stationary phase arguments, we obtain that, for h ∈ h0 , k(ht |v)kB(K) ≤ Ct−1−µ ,
(5.7)
which proves the existence of the norm limit (5.1) for h ∈ h0 . For h ∈ h, let hn ∈ h0 such that h = limn→∞ hn . Using the fact that k(N + 1) (i + H)− k < ∞ and Proposition 2.4 (iii), we have lim sup k(W (hn,t ) − W (ht ))(i + H)− k = 0 .
n→∞ t∈R
This implies the existence of the norm limit (5.2) for all h ∈ h. Now (5.2) implies (5.1). This ends the proof of (i) . We have k(W + (h) − W + (g)k ≤ lim keitH W (ht )(H + i)− e−itH t→+∞
− eitH W (gt )(H + i)− e−itH k = lim k(W (ht ) − W (gt ))(H + i)− k t→+∞
≤ Ckh − gk , by Proposition 2.4 (iii), which implies the norm continuity of (5.4). This implies the strong continuity of (5.3) and completes the proof of (ii). Finally (iii) and (iv) are immediate.
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For two operators A1 , A2 on a Hilbert space H, we make the convention that D(A1 A2 ) := {u ∈ D(A2 )|A2 u ∈ D(A1 )}. Theorem 5.2. (i) For any h ∈ h there exists a selfadjoint operator φ+ (h), called the asymptotic field, such that W + (h) = eiφ
+
(h)
. Qn (ii) For hi ∈ h, 1 ≤ i ≤ n, D((H + i)n/2 ) ⊂ D( 1 φ+ (hi )), and n Y
φ+ (hi )(H + i)−n/2 = lim eitH t→+∞
i=1
We have the bound
n Y
φ(hi,t )e−itH (H + i)−n/2 .
i=1
n n
Y Y
+ −n/2 khi k .
≤ Cn
φ (hi )(H + i)
1
(5.8)
1
(iii) The map h 3 (h1 , . . . , hn ) 7→ n
n Y
φ+ (hi )(H + i)− 2 ∈ B(H) n
1
is norm continuous. (iv) The operators φ+ (h) satisfy in the sense of quadratic forms on D(φ+ (h1 )) ∩ D(φ+ (h2 )) the canonical commutation relations [φ+ (h2 ), φ+ (h1 )] = iIm (h2 |h1 ) .
(5.9)
eitH φ+ (h)e−itH = φ+ (h−t ) .
(v)
(vi) For h ∈ h0 ,φ+ (h) − φ(h) is bounded and lim (eitH φ(ht )e−itH − φ+ (h)) = 0 .
t→∞
Proof. By Theorem 5.1 (ii), s 7→ W + (sh) is a strongly continuous unitary group. Thus the existence of φ+ (h) follows by Stone’s theorem. This proves (i). To prove (ii), let us first establish the existence of the norm limit R(h1 , . . . , hn ) := lim e t→+∞
itH
n Y
φ(hi,t )(H + i)−n/2 e−itH ,
hi ∈ h .
(5.10)
i=1
The Heisenberg derivative Dφ(ht ) (first defined as a quadratic form on D(H), then as a bounded operator on H) equals Im (ht |v). We deduce from this and Lemma 2.3 (i) that similarly D
n Y i=1
φ(hi,t )(H + i)−n/2 =
n X j=1
Im (hj,t |v)
n Y i6=j
φ(hi,t )(H + i)−n/2 .
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For h ∈ h0 , the existence of the limit (5.10) follows then from (5.7) and Lemma 2.3 (i). Moreover, for h ∈ h0 , we have Z +∞ φ+ (h) − φ(h) = eitH Im (ht |v)e−itH dt , 0
φ+ (h) − eitH φ(ht )e−itH =
Z
+∞
eitH Im (ht |v)e−itH dt ,
t
which proves (vi). Next, for hi ∈ h, we approximate hi by sequences hi,n ∈ h0 , and use Lemma 2.3 (ii) to obtain the existence of (5.10). Let us now prove (ii) by induction on n. To prove the induction assumption for Q Q n we have to show that D((H + i)n/2 ) ⊂ D( n1 φ+ (hi )) and that n1 φ+ (hi )(H + i)−n/2 = R(h1 , . . . , hn ). Note that by Lemma 2.3 (i) , we have kR(h1 , . . . , hn )k ≤ Qn Cn i=1 khi k, which will then imply the bound (5.8). This amounts to prove that n Y 1 φ+ (hi )(H + i)−n/2 u , R(h1 , . . . , hn )u = lim (W + (sh1 ) − 1l) s→0 s i=2
u ∈ H.
(5.11)
Qn Note that by the induction assumption, we have D((H + i)n/2 ) ⊂ D( 2 φ+ (hi )) and n Y
φ+ (hi )(H + i)−n/2 = lim eitH t→+∞
i=2
n Y
φ(hi,t )(H + i)−n/2 e−itH .
(5.12)
i=2
Using (5.12) and the fact that eitH W (sh1,t )e−itH is uniformly bounded in t, we have n Y 1 + (W (sh1 ) − 1l) φ+ (hi )(H + i)−n/2 u s i=2 n Y 1 = lim eitH (W (sh1,t ) − 1l) φ(hi,t )(H + i)−n/2 e−itH , t→+∞ s i=2
so to prove (5.11), we have to check that lim lim eitH R(s, t)e−itH u = 0 ,
s→0 t→+∞
for R(s, t) :=
! n n Y Y 1 (W (sh1,t ) − 1l) φ(hi,t ) − i φ(hi,t ) (H + i)−n/2 . s 1 i=2
(5.13)
(5.14)
Using Proposition 2.4 (ii) and Lemma 2.3 (i), we see that R(s, t) is uniformly bounded in s, t. So it suffices to prove (5.13) for u ∈ D((H + i) ), > 0. Again by Proposition 2.4 (ii) and Lemma 2.3 (i), we obtain lim sup kR(s, t)(H + i)− k = 0 ,
s→0 t∈R
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which proves (5.13) for u ∈ D((H + i) ) and hence for all u ∈ H. Property (iii) follows from the existence of the norm limit in (ii) and Lemma 2.3 (ii). Next for ui ∈ D(φ+ (hi )), i = 1, 2 we set f (t2 , t1 ) := (W + (−t2 h2 )u2 |W + (t1 h1 )u1 ) − (W + (−t1 h1 )u2 |W + (t2 h2 )u1 )eit1 t2 Im(h2 |h1 ) . We know that f (t2 , t1 ) = 0. By Stone’s theorem we are allowed to compute the derivative: 0 = ∂t2 ∂t1 f (0, 0) = −(φ+ (h2 )u2 |φ+ (h1 )u1 ) + (φ+ (h1 )u1 |φ+ (h2 )u2 ) − iIm (h2 |h1 )(u2 |u1 ) .
This proves (iv). Finally (v) follows from Theorem 5.1 (iv).
Theorem 5.3. For any h ∈ h, the asymptotic creation and annihilation operators defined on D(a+] (h)) := D(φ+ (h)) ∩ D(φ+ (ih)) by 1 a+∗ (h) := √ (φ+ (h) − iφ+ (ih)) , 2 1 a+ (h) := √ (φ+ (h) + iφ+ (ih)) . 2 are closed. Qn (ii) For hi ∈ h, 1 ≤ i ≤ n, D((H + i)n/2 ) ⊂ D( 1 a+] (hi )) and n Y
a+] (hi )(i + H)− 2 = lim eitH n
t→∞
1
n Y
a] (hi,t )(i + H)− 2 e−itH . n
1
(iii) We have the bound
n n
Y Y
n
+] khj k (H + i) 2 u ,
a (hi )u ≤ C
1
1
and the map hn 3 (h1 , . . . , hn ) 7→
n Y
a+,] (hj )(i + H)− 2 ∈ B(H) n
1
is norm continuous. 1 (iv) The operators a+] satisfy in the sense of forms on D((i+H) 2 ) the canonical commutation relations [a+ (h1 ), a+∗ (h2 )] = (h1 |h2 )1l , [a+ (h2 ), a+ (h1 )] = [a+∗ (h2 ), a+∗ (h1 )] = 0 .
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(v) One has eitH a+] (h)e−itH = a+] (h−t ) .
(5.15)
The following infinitesimal version of (5.15) is true for h ∈ D(ω) in the sense of forms on D(H). It is known as the pullthrough formula: a+∗ (h)H = Ha+∗ (h) − a+∗ (ωh) , a+ (h)H = Ha+ (h) + a+ (ωh) .
(5.16)
(vi) For h ∈ h0 , the operators a+,] (h) − a] (h) are bounded and lim (eitH a] (ht )e−itH − a+,] (h)) = 0 .
t→∞
Proof. The closedness of a] (h) follows from Proposition 2.2 and Theorem 5.2. To prove the pullthrough formula we write in the sense of forms on D(H): +] −isH −isH − 1l − 1l a (h−s ) − a+] (h) e e +] = + a+] (h) . a (h) −is −is −is Letting s tend to 0 and using (iii), we obtain (5.16). The other statements follow from analogous statements in Theorem 5.2. The following result is due to Høegh-Krohn [30]. Corollary 5.4. For h ∈ h, one has: a+ (h)1l]−∞,λ] (H)H ⊂ 1l]−∞,λ−m] (H)H . Proof. It follows from the spectral theorem that u = 1l]−∞,λ] (H)u if and only if the function R 3 t 7→ eitH u ∈ H has an analytic extension to Im z < 0 satisfying keizH uk ≤ Ce|Im z|λ for Im z < 0. Let u = 1l]−∞,λ] (H)u. Using (5.15), we have: eitH a+ (h)u = a+ (eitω(k) h)eitH u
t ∈ R.
Since H and ω(k) are bounded below, the right-hand side is analytic in Im z < 0, with an analytic extension equal to a+ (e−izω(k) h)eizH u for Im z < 0 (remember that a+ (h) is antilinear in h). Moreover using (3.1) one obtains ka+ (e−izω(k) h)eizH uk ≤ ka+ (e−izω(k) h)1l]−∞,λ] (H)k keizH 1l]−∞,λ] (H)uk ≤ Cke−izω(k) hke|Im z|λ ≤ Ckhke|Im z|(λ−m) . This proves that 1l]−∞,λ−m] (H)a+ (h)u = a+ (h)u as claimed.
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5.3. Asymptotic spaces We define the asymptotic matter space to be K+ := {u ∈ H | a+ (h)u = 0, h ∈ h} . The asymptotic space is defined as H+ := K+ ⊗ Γ(h) . Proposition 5.5. (i) K+ is a closed H-invariant space. (ii) K+ is included in the domain of a+∗ (h1 ) · · · a+∗ (hn )
for all h1 , . . . , hn ∈ h .
Hpp (H) ⊂ K+ .
(iii)
Proof. K+ is obviously closed since a+ (h) are closed operators. The fact that K is invariant under e−itH follows from (5.15). Let us prove (ii) by induction on n. Since a+∗ (h) and a+ (h) have the same domain, (ii) is true for n = 1. Assume that (ii) is true for n − 1. By the remark above, it suffices to check that for u ∈ K+ , a+∗ (h2 ) · · · a+∗ (hn )u ∈ D(a+ (h1 )). But this follows from the canonical commutation relations and the fact that u ∈ K+ . Now suppose that Hu = Eu. Then +
eitH a(ht )e−itH u = (E + i)eit(H−E) a(ht )(H + i)−1 u .
(5.17)
But w − limt→∞ ht = 0 and hence by Lemma 2.1 (iii) s- lim a(ht )(H + i)−1 = 0 . t→∞
Therefore, the limit of (5.17) is zero, which means that a+ (h)u = 0. This proves (iii). 5.4. Wave operators The asymptotic matter Hamiltonian and the asymptotic Hamiltonian are defined by the formulas K + := H|K+ ,
H + := K + ⊗ 1l + 1l ⊗ dΓ(ω) .
We also define Ω+ : H + → H , (5.18) Ω+ ψ ⊗ a∗ (h1 ) · · · a∗ (hn )Ω := a+∗ (h1 ) · · · a+∗ (hn )ψ , h1 , . . . , hn ∈ h, ψ ∈ K+ . The map Ω+ is called the wave operator. The following theorem is due to HøeghKrohn [29].
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Theorem 5.6. Ω+ is a unitary map from H+ to H such that : a+] (h)Ω+ = Ω+ 1l ⊗ a] (h) , +
+
h ∈ h,
+
HΩ = Ω H . Proof. Using the canonical commutation relations and the definition of K+ , it is easy to see that Ω+ is isometric. Moreover it follows from Theorem 5.3 (v) that +
eitH Ω+ = Ω+ eitH . Let u ∈ (Ran Ω+ )⊥ . Since Ran Ω+ is H-invariant, we may assume that u = Qn 1l]−∞,λ] (H)u. By Theorem 5.3 (ii), u belongs to the domain of 1 a+ (hi ) for any hi ∈ h, 1 ≤ i ≤ n. By Corollary 5.4, if nm > λ − inf σ(H), then a+ (h1 ) · · · a+ (hn )u = 0 ,
∀ h 1 , . . . , hn ∈ h .
(5.19)
Let n0 be the smallest positive integer with the property (5.19). This implies that v = a+ (h2 ) · · · a+ (hn0 )u ∈ K+ . So we have 0 = u|a+∗ (h2 ) · · · a+∗ (hn0 )v = kvk2 . Thus we have shown that a+ (h2 ) · · · a+ (hn0 )u = 0 ,
∀ h2 , . . . , hn0 ∈ h ,
which is a contradiction. 5.5. Extended wave operator
Recall that in Subsec. 3.4 we introduced the extended Hilbert space and the extended Hamiltonian Hext = H ⊗ Γ(h) ,
H ext = H ⊗ 1l + 1l ⊗ dΓ(ω(k)) .
Clearly, H+ is a subspace of Hext and H + = H ext |H+ . Sometimes we will also need the “extended wave operator”. Its domain can be chosen to be ∞ n M O n ext,+ 2 ) := D (H + i) ⊗ h, D(Ω n=0
s
which is a dense subset of Hext . Now we set Ωext,+ : D(Ωext,+ ) → H , Ωext,+ ψ ⊗ a∗ (h1 ) · · · a∗ (hn )Ω := a+∗ (h1 ) · · · a+∗ (hn )ψ , ψ ∈ D((H + i) 2 ). n
(5.20)
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Note that Ωext,+ is an unbounded operator. Clearly, Ωext,+ |H+ = Ω+ .
(5.21)
We will sometimes treat Ω+ as a partial isometry equal to zero on the orthogonal complement of H+ inside Hext . We can then write the following identity: Ω+ = Ωext,+ 1lH+ .
(5.22)
5.6. Another construction of the wave operators Recall that in Subsec. 2.15, we defined the (unbounded) identification operator I : Γ(h) ⊗ Γ(h) → Γ(h). By the same symbol we will denote the operator 1lK ⊗ I : Hext = K ⊗ Γ(h) ⊗ Γ(h) → H = K ⊗ Γ(h) . Theorem 5.7. (i) Let w ∈ D((H + i)k/2 ) with w = 1l{k} (N∞ )w. Then the limit lim eitH Ie−itH
ext
t→+∞
w
exists and equals Ωext,+ w. (ii) Let w ∈ Hcomp (H ext ). Then the limit lim eitH Ie−itH
ext
t→+∞
w
(5.23)
exists and equals Ωext,+ w. In particular, Ωext,+ χ(H ext ) is bounded for χ ∈ C0∞ (R). (iii) Let w ∈ H+ ∩ Hcomp (H ext ). Then the limit lim eitH Ie−itH w +
(5.24)
t→+∞
exists and equals Ω+ w. Proof. Let us first show (i). Let w ∈ D((H + i)k/2 ) with w = 1l{k} (N∞ )w. Since by (2.13) I(H + i)−k/2 ⊗ 1l{k} (N∞ ) is a bounded operator, it suffices to prove Q (i) for w = ψ ⊗ ki a∗ (hi )Ω, ψ ∈ D((H + i)k/2 ), hi ∈ h. It follows from property (2.11) of I that eitH Ie−itH
ext
ψ⊗
k Y 1
a∗ (hi )Ω = eitH
k Y
a∗ (hi,t )e−itH ψ .
1
(i) follows then from Theorem 5.3 (ii). To prove (ii), we observe that since the boson mass is positive, vectors in Hcomp (H ext ) are also in Hcomp (H) and in Hcomp (N∞ ). So (ii) follows from (i). Finally (iii) follows from (ii) by (5.21).
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6. Propagation estimates In this section, which serves as a technical preparation for Sec. 7, we collect various propagation estimates about the evolution e−itH which will be used in the next section. They closely resemble propagation estimates used in the scattering theory for N -body Schr¨ odinger operators, especially those introduced in [24]. In all this section we will assume the conditions (H1), (H2), (I1), (SR). Finally ext we mention that all the results of this section hold also for the dynamics e−itH with the obvious modifications. 6.1. Large velocity estimate In this subsection we derive a standard large velocity estimate. It means that no boson can asymptotically propagate in the region |x| > vmax t, where the maximal velocity vmax is equal to vmax := sup |∇ω(k)| . k
Proposition 6.1. Let χ ∈ C0∞ (R). For R0 > R > vmax , one has Z 1
∞
2
12
dt
|x|
≤ Ckuk2 . χ(H)e−itH u
dΓ 1l[R,R0 ]
t
t
Proof. Let F ∈ C ∞ (R) be a cutoff function equal to 1 near ∞, to 0 near the origin, with F 0 (s) ≥ 1l[R,R0 ] (s). Let |x| χ(H) , Φ(t) := χ(H)dΓ F t x b(t) := d0 F . t By pseudodifferential calculus, and then taking into account the support of F 0 , we obtain |x| x |x| − + ∇ω(k) + O(t−2 ) b(t) = F 0 t t |x|t C0 |x| + O(t−2 ) , ≤ − F0 t t for some C0 > 0. Hence C0 N |x| |x| = dΓ (b(t)) ≤ − dΓ F 0 +C 2 . D0 dΓ F t t t t Moreover,
1 |x| |x| = φ iF f ∈ ON (t−1−µ )(N + 1) 2 . V, idΓ F t t
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|x| |x| χ(H) + χ(H) V, idΓ χ(H) DΦ(t) = χ(H) D0 dΓ t t |x| ≤ −t−1 C0 χ(H)dΓ F 0 χ(H) + O(t−1−µ ) . t
By Lemma A.1, we obtain the desired result. 6.2. Phase space propagation estimate
This subsection is devoted to a more subtle propagation estimate. Its intuitive meaning is that along the evolution of an asymptotically free boson the instantaneous velocity ∇ω(k) and the average velocity xt converge to each other as time goes to ∞. Proposition 6.2. Let χ ∈ C0∞ (R), 0 < c0 < c1 . Set E x x D x − ∇ω(k), 1l[c0 ,c1 ] − ∇ω(k) . Θ[c0 ,c1 ] (t) := dΓ t t t Then
Z
∞
1
Θ[c ,c ] (t) 12 χ(H)e−itH u 2 dt ≤ Ckuk2 . 0 1 t
Proof. We start by recalling a well-known construction, which can be viewed as a trivial version of the construction of the Graf vector field (see e.g. [24]). It is easy to see that there exists a function R0 (x) such that R0 (x) = 0 , R0 (x) =
for |x| ≤
1 2 x + c, 2
c0 , 2
for |x| ≥ 2c1 ,
∇2x R0 (x) ≥ 1l[c0 ,c1 ] (|x|) . We fix the parameter c1 > vmax + 1, choose a constant c2 > c1 + 1 and consider R(x) := F (|x|)R0 (x) , for F (s) = 1, s ≤ c1 , F (s) = 0, s ≥ c2 . The function R satisfies now: ∇2x R(x) ≥ 1l[c0 ,c1 ] (|x|) − C1l[vmax +2,c2 ] , |∂xα R(x)| ≤ Cα .
(6.1)
It clearly suffices to prove Proposition 6.2 for c1 > vmax + 1, which we will assume in what follows. Let x x E x 1 D − ∇R , − ∇ω(k) + hc . b(t) := R t 2 t t
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We consider the propagation observable Φ(t) = χ(H)dΓ (b(t)) χ(H) . Using first pseudodifferential calculus, and then (6.1), we obtain E x x 1 Dx − ∇ω(k), ∇2 R − ∇ω(k) + O(t−2 ) t t t t 1 x |x| x − ∇ω(k), 1l[c0 ,c1 ] − ∇ω(k) ≥ t t t t C |x| + O(t−2 ) . − 1l[vmax +2,c2 ] t t
d0 b(t) =
This gives D0 Φ(t) = dΓ (d0 b(t)) ≥
C 1 |x| Θ[c1 ,c2 ] (t) − dΓ 1l[vmax +2,c2 ] t t t + ON (t−2 )(N + 1) .
(6.2)
Moreover, using pseudodifferential calculus and hypothesis (SR), we have
|x| −2 −1−µ
≥ c0 v ). kb(t)vk ≤ C F
+ O(t ) ∈ O(t t Hence [V, idΓ (b(t))] = φ(ib(t)v) ∈ ON (t−1−µ )(N + 1) 2 . 1
So we finally obtain DΦ(t) = χ(H)(D0 dΓ(b(t)))χ(H) + χ(H)[iV, dΓ b(t))]χ(H) ≥
1 χ(H)Θ[c1 ,c2 ] (t)χ(H) + O(t−1−µ ) . t
Using Lemma A.1, we obtain the desired result.
6.3. Improved phase-space propagation estimate In this subsection, we will improve Proposition 6.2. Proposition 6.3. Let 0 < c0 < c1 , J ∈ C0∞ ({c0 < |x| < c1 }), χ ∈ C0∞ (R). Then for 1 ≤ i ≤ d Z 1
+∞
2 12
xi −itH dt
dΓ J x − ∂i ω(k) + hc χ(H)e < Ckuk2 . u
t t t
Before starting the proof, we need some technical preparation.
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Lemma 6.4. Let A = ( xt − ∇ω(k))2 + t−δ , δ > 0. Let J, J1 , J2 ∈ C0∞ (Rd ) with J1 = 1 near supp J, J2 = 1 near supp J1 , 0 ≤ J2 ≤ 1. Then 1 (i) J( xt )A 2 = O(1), 1 (ii) [A 2 , J( xt )] = O(tδ/2−1 ), 1 1 (iii) d0 A 2 = − 1t A 2 + O(t−1−δ/2 ). For 1 ≤ i ≤ d and for = inf(δ, 1 − δ/2). 1 (iv) |J( xt )( xti − ∂i ω(k)) + hc| ≤ CJ2 ( xt )A 2 J2 ( xt ) + Ct−/2 , 1 (v) J( xt )( xti −∂i ω(k))A 2 J1 ( xt )+hc ≤ Ch( xt −∇ω(k)), J22 ( xt )( xt −∇ω(k))i+Ct− . 1
1
Proof. (i) is immediate using that kJ( xt )A 2 k = kJ( xt )AJ( xt )k 2 ∈ O(1). To prove (ii) and (iii), we use the identities e
itω(k)
eitω(k) J
1 2
A e
x t
−itω(k)
=
x2 + t−δ t2
12 =: A0 ,
e−itω(k) = J(v) ,
for v := xt + ∇ω(k). It is easy to check using pseudodifferential calculus that [v, A0 ] ∈ O(t−1+δ/2 ). (ii) follows then easily from the following functional calculus formula: Z ihξ,vi ˆ J(v) = (2π)−n J(ξ)e dξ . To prove (iii), we notice that eitω(k) d0 A 2 e−itω(k) = 1
d 1 A0 = − A0 + O(t−δ/2−1 ) , dt t
by a direct computation. Let us now prove (iv). Set B0 := J
x x
i
− ∂i ω(k) + hc ,
t t x 1 x A 2 J1 . B2 := J1 t t
By pseudodifferential calculus and (ii) we have x vi + O(t−1 ) B02 = vi J 2 t x vi + Ct−1 ≤ Cvi J14 t x x vi2 J12 + O(t−1 ) = CJ12 t t x x AJ12 + O(t−δ ) = CJ12 t t x 1 x 1 x A 2 J12 A 2 J1 + O(t− ) = CJ1 t t t = CB22 + O(t− ) .
(6.3)
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Since the function λ 7→ λ 2 is matrix monotone (see [6, Sec. 2.2.2]), we deduce from (6.3) that 1 |B0 | ≤ C(B22 + t− ) 2 ≤ CB2 + Ct−/2 . which proves (iv). To prove (v), we write using pseudodifferential calculus and (ii): 1 x x 1 x x i − ∂i ω(k) A 2 J1 + hc = vi J A 2 + hc + O(t−1+δ/2 ) J t t t t x 1 1 1 A 2 + hc + O(t−1+δ/2 ) = A 2 vi A− 2 J t ≤ CA + Ct−1+δ/2 2 x − ∇ω(k) + Ct− , ≤C t since vi A− 2 is bounded. Next we use that B1 = J2 ( xt )B1 J2 ( xt ) + O(t−∞ ) to obtain (v). 1
Proof of Proposition 6.3. Let b(t) = J where A= and J, J1 ∈
C0∞ ({c0
x t
x t
1
A2J
x t
,
2 − ∇ω(k) + t−δ ,
≤ |x| ≤ c1 }), 0 ≤ J ≤ 1, J = 1 near supp J1 . Let Φ(t) = −χ(H)dΓ(b(t))χ(H) .
Note that by Lemmas 6.4 (i) and 3.2, Φ(t) ∈ O(1). Using Lemmas 6.4 (i) and 3.2 and hypothesis (SR), we have [V, idΓ(b(t))] = φ(ib(t)v) ∈ ON (t−1−µ )(N + 1) 2 . 1
(6.4)
Next, we have D0 dΓ(b(t)) = dΓ(d0 b(t)) , and
x x 1 x x 1 A2 J + hc + J (d0 A 2 )J . d0 b(t) = d0 J t t t t
(6.5)
By Lemma 6.4 (iii) 1
1
d0 A 2 = −
A2 + O(t−1−δ/2 ) , t
and by Lemma 6.4 (iv), we obtain, for some C0 > 0, −J
x t
1
(d0 A 2 )J
x t
≥
C0 x xi − ∂i ω(k) + hc − Ct−1− . J1 t t t
(6.6)
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Next by pseudodifferential calculus, we have x x E x 1D =− ∇J , − ∇ω(k) + hc + O(t−2 ) , d0 J t t t t which, by Lemma 6.4 (v), gives, for J2 ∈ C0∞ ({c0 ≤ |x| ≤ c1 }), J2 = 1 near supp J, E x x x 1 x C Dx A2J + hc ≥ − − ∇ω(k), J2 − ∇ω(k) − d0 J t t t t t t + O(t−1− ) .
(6.7)
Collecting (6.4), (6.6) and (6.7), we obtain finally, for some > 0, −DΦ(t) = χ(H)[V, idΓ(b(t))]χ(H) + χ(H)(d0 dΓ(b(t))χ(H) x x C0 i χ(H)dΓ J1 − ∂i ω(k) + hc χ(H) ≥ t t t D x E x x C − ∇ω(k), J2 − ∇ω(k) χ(H) − χ(H)dΓ t t t t + O(t−1− ) .
(6.8)
Since by Proposition 6.2 the second term in the right-hand side of (6.8) is integrable along the evolution, we obtain the proposition. 6.4. Minimal velocity estimate This subsection is devoted to the proof of the minimal velocity estimate. It will use the Mourre estimate shown in Subsec. 4.3. Proposition 6.5. Assume additionally that (H0) holds. Let χ ∈ C0∞ (R) be supported in R\(τ ∪ σpp (H)). Then there exists > 0 such that Z 1
+∞
2
−itH dt
Γ 1l[0,] |x| χ(H)e ≤ Ckuk2 . u
t t
Proof. Let us first prove the proposition for χ supported near an energy level λ ∈ R\(τ ∪ σpp (H)). By Theorem 4.3, we will find χ ∈ C0∞ (Rd ) equal 1 near λ with supp χ close enough to λ such that for some C0 > 0 χ(H)[H, iA]χ(H) ≥ C0 χ2 (H) .
(6.9)
Let > 0 be a number that will be fixed later on. Let q ∈ C0∞ ({|x| ≤ 2}) such that 0 ≤ q ≤ 1, q = 1 near on {|x| ≤ } and let q t = q( xt ). Let A Φ(t) := χ(H)Γ(q t ) Γ(q t )χ(H) . t Note that A (6.10) ± Γ(q t ) Γ(q t ) ≤ C(N + 1) , t
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which, using Lemma 3.2, shows that Φ(t) is uniformly bounded. We compute its Heisenberg derivative: A A DΦ(t) = χ(H)dΓ(q t , d0 q t ) Γ(q t )χ(H) + hc + χ(H)[V, iΓ(q t )] Γ(q t )χ(H) + hc t t A + t−1 χ(H)Γ(q t )[H, iA]Γ(q t )χ(H) − t−1 χ(H)Γ(q t ) Γ(q t )χ(H) t =: R1 (t) + R2 (t) + R3 (t) + R4 (t) . By Lemma 2.9 (iii) and condition (SR), we have [V, iΓ(q t )] ∈ ON (t−1−µ )(N + 1) 2 , which implies that (6.11) kR2 (t)k ∈ O(t−1−µ ) . 1
Let us now consider R1 (t). We have x E 1 Dx 1 − ∇ω(k), ∇q + hc + rt =: g t + rt , d0 q t = − 2t t t t where rt ∈ O(t−2 ). By Lemma 2.8 (vi), we have
χ(H)dΓ(q t , rt ) A Γ(q t )χ(H) ∈ O(t−2 ) .
t
(6.12)
Next we set B1 := χ(H)dΓ(q t , g t )(N + 1)− 2 , 1
B2∗ := (N + 1) 2 1
A Γ(q t )χ(H) , t
and use the inequality A χ(H)dΓ(q t , g t ) Γ(q t )χ(H) = t−1 B1 B2∗ + t−1 B2 B1∗ t −1 ≥ −−1 B1 B1∗ − 0 t−1 B2 B2∗ . 0 t
(6.13)
We have
A2 (N + 1)Γ(q t )χ(H) . t2 ˜ = χ, q˜q = q, it Introducing cutoff functions χ ˜ ∈ C0∞ (R) and q˜ ∈ C0∞ (Rd ) with χχ is easy to check that B2 B2∗ = χ(H)Γ(q t )
Γ(q t )
A2 Γ(q t ) ≤ C(N + 1)Γ(q t )2 (N + 1) , t2
χ(H)(N + 1) χ(H) ≤ Cχ (H) , p
2
(6.14)
p ∈ N.
This gives using Lemma 3.3 −B2 B2∗ ≥ −Cχ(H)(N + 1)3/2 Γ(q t )2 (N + 1)3/2 χ(H) = −CΓ(q t )χ(H)(N + 1)3 χ(H)Γ(q t ) + O(t−1 ) ≥ −C1 Γ(q t )χ2 (H)Γ(q t ) − Ct−1 ≥ −C1 χ(H)Γ(q t )2 χ(H) − Ct−1 .
(6.15)
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Next we write B1 B1∗ = χ(H)dΓ(q t , g t )(N + 1)−1 dΓ(q t , g t )χ(H) , and use Lemma 2.8 (vi) to obtain
2 1 |(u|B1 B1∗ u)| = (N + 1)− 2 dΓ(q t , g t )χ(H)u
2 1 ≤ dΓ(g t∗ g t ) 2 χ(H)u . Using Proposition 6.2, we obtain Z ∞ dt kB1 e−itH uk2 ≤ Ckuk2 . t 1
(6.16)
Using Lemma 3.3, we have R3 (t) = t−1 Γ(q t )χ(H)[H, iA]χ(H)Γ(q t ) + O(t−2 ) ≥ C0 t−1 Γ(q t )χ2 (H)Γ(q t ) − Ct−2 ≥ C0 t−1 χ(H)Γ2 (q t )χ(H) − Ct−2 .
(6.17)
On the other hand, we have by (6.10) and (6.14) 1 1 −R4 (t) ≤ C χ(H)(N + 1) 2 Γ2 (q t )(N + 1) 2 χ(H) t ≤ C Γ(q t )χ(H)(N + 1)χ(H)Γ(q t ) + Ct−2 t ≤ C2 Γ(q t )χ2 (H)Γ(q t ) + Ct−2 t ≤ C2 χ(H)Γ(q t )2 χ(H) + Ct−2 . t
(6.18)
Collecting (6.15), (6.17) and (6.18) we obtain −0 t−1 B2∗ (t)B2 (t) + R3 (t) + R4 (t) ≥ (−0 C1 + C0 − C2 )t−1 χ(H)Γ(q t )2 χ(H) − Ct−2 .
(6.19)
We pick now and 0 small enough so that C˜0 := −0 C1 + C0 − C2 > 0. Using (6.11), (6.16) and (6.19) we conclude that DΦ(t) ≥
C˜0 χ(H)Γ2 (q t )χ(H) − R(t) − Ct−1−µ , t
where R(t) is integrable along the evolution. By Lemma A.1, this proves the proposition for χ with support close enough to an energy level λ ⊂ R\(τ ∪ σpp (H)). To prove the Proposition for all χ supported in R\(τ ∪ σpp (H)) we use the argument in [13, Proposition 4.4.7].
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7. Asymptotic Completeness 7.1. Introduction In this subsection we describe the main results of this section (and of the whole article). They will be formulated in the following four theorems. Theorem 7.1. Assume hypotheses (H1), (H2), (I1) and (SR). Let q ∈ C0∞ (Rd ), 0 ≤ q ≤ 1, q = 1 near 0. Set q t (x) = q( xt ). Then there exists s- lim eitH Γ(q t )e−itH =: Γ+ (q) , t→∞
We have Γ+ (q q˜) = Γ+ (q)Γ+ (˜ q) , q ) ≤ 1l , 0 ≤ Γ+ (q) ≤ Γ+ (˜
0 ≤ q ≤ q˜ ≤ 1 ,
[H, Γ+ (q)] = 0 . The above theorem, or actually its generalization, will be proven in Theorem 7.5. Using this theorem, we define P0+ := s- lim Γ+ (qn ) , n→∞
where qn ∈ C0∞ is a decreasing sequence of functions such that qn & 1l{0} . Theorem 7.2. P0+ does not depend on the choice of the sequence qn . It satisfies (P0+ )2 = P0+ ,
[H, P0+ ] = 0 .
The first important result of this section is the following theorem, which we call geometric asymptotic completeness. Theorem 7.3. Assume hypotheses (H1), (H2), (I1) and (SR). Then the space of asymptotic matter is equal to the space of states living near the origin: K+ = Ran P0+ . The second main result is the standard asymptotic completeness, which holds under the additional condition (H0). Theorem 7.4. Assume hypotheses (H0), (H1), (H2), (I1) and (SR). Then the space of asymptotic matter is equal to the space of bound states of H: K+ = Hpp (H) .
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7.2. An asymptotic partition of unity Let f0 ∈ C0∞ (Rd ), f∞ ∈ C ∞ (Rd ), 0 ≤ f0 , 0 ≤ f∞ , f0 + f∞ ≤ 1, f0 = 1 near t ), where 0 (and hence f∞ = 0 near 0). Set f := (f0 , f∞ ). Set also f t = (f0t , f∞ x x t t f0 (x) = f0 ( t ), f∞ (x) = f∞ ( t ). Theorem 7.5. (i) The following limits exist: itH Qk (f t )e−itH , Q+ k (f ) := s- lim e t→+∞
+ itH Pk (f t )e−itH . Pk+ (f ) = Q+ k (f ) − Qk−1 (f ) = s- lim e t→∞
(ii) For q ∈ C ∞ (Rd ) such that 0 ≤ q ≤ 1, ∇q ∈ C0∞ (Rd ), q = 1 on a neighborhood of zero and q t := q( xt ), there exists Γ+ (q) := s- lim eitH Γ(q t )e−itH . t→∞
Clearly, + P0+ (f ) = Q+ 0 (f ) = Γ (f0 ) .
[Q+ k (f ), H] = 0 .
(iii)
(iv) If 0 ≤ q ≤ 1, q = 1 near 0, then + + Q+ k (f q) = Qk (f )Γ (q) ,
where qf = (qf0 , qf∞ ). (v)
+ + 0 ≤ Q+ k1 (f ) ≤ Qk2 (f ) ≤ Γ (f0 + f∞ ) ,
k1 ≤ k2 ,
+ s- limk→∞ Q+ k (f ) = Γ (f0 + f∞ ) . + −1 . k(H + i)−1 (Q+ k (f ) − Γ (f0 + f∞ ))k ≤ C(k + 1)
If, moreover, f0 + f∞ = 1, then s- lim Q+ k (f ) = 1l , k→∞
−1 k(H + i)−1 Q+ . k (f ) − 1l k ≤ C(k + 1)
(vi) If f˜ = (f˜0 , f˜∞ ) is another pair of functions satisfying the conditions stated at the beginning of this subsection, and moreover, f˜0 f∞ = 0, then + ˜ + + ˜ ˜ + ˜ Q+ k (f )Pk (f ) = Pk (f )Pk (f ) = Pk (f0 f0 , f∞ f∞ ) .
Proof. Let us first prove (i). Using Lemma 3.3 for m = 0 and a density argument, it suffices to prove the existence of s- lim eitH χ(H)Qk (f t )χ(H)e−itH . t→+∞
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We compute the Heisenberg derivative: χ(H)DQk (f t )χ(H) = χ(H)dQk (f t , d0 f t )χ(H) + χ(H)[V, iQk (f t )]χ(H) , by Lemma 2.11. From (3.8) and Lemma 3.2 we obtain kχ(H)[V, iQk (f t )]χ(H)k ∈ O(t−1−µ ) .
(7.1)
Next we compute: d0 f t =
d X 1 1
t
git + rt ,
where
x 1 xi t t − ∂i ω(k) ∂i f + hc = (gi,0 , gi,∞ ), 2 t t and rt ∈ O(t−2 ). Using Lemma 2.11 (vi) and Lemma 3.2, we obtain that git = −
kχ(H)dQk (f t , rt )χ(H)k ∈ O(t−2 ) .
(7.2)
On the other hand, using Lemma 2.11 (v), we have 1
1
t t |) 2 χ(H)u2 k kdΓ(|gi,0 |) 2 χ(H)u1 k |(u2 |χ(H)dQk (f t , git )χ(H)u1 )| ≤ kdΓ(|gi,0 1
1
t t |) 2 χ(H)u2 k kdΓ(|gi,∞ |) 2 χ(H)u1 k . + kdΓ(|gi,∞
(7.3) Hence the existence of the limits (i) follows from (7.1)–(7.3), Proposition 6.3 and Lemma A.1. (ii) is obvious and (iii) follows by Lemma 3.3. (iv) follows from Qk (f t q t ) = Qk (f t )Γ(q t ) . The first statement of (v) follows from Lemma 2.9 (iv), and the second follows from the third. To see the third statement we first observe that t ))k ≤ (k + 1)−1 , k(N + 1)−1 (Qk (f t ) − Γ(f0t + f∞
which implies t ))k ≤ k(H + i)−1 (N + 1)k(k + 1)−1 . k(H + i)−1 (Qk (f t ) − Γ(f0t + f∞
This completes the proof of (v). (vi) follows from Proposition 2.10.
(7.4)
An analogous theorem is true for the free Hamiltonian, but it is much easier. It follows within each n-particle sector by the stationary phase method. Note that in the free case one does not need to assume that the cutoff is one near zero. Proposition 7.6. Let f0 ∈ C0∞ (Rd ), f∞ ∈ C ∞ (Rd ) with f0 + f∞ ≤ 1. Then s- lim eitdΓ(ω(k)) Qk (f t )e−itdΓ(ω(k)) = Qk (f (∇ω(k))) . t→∞
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7.3. Asymptotic projections In this section, using the monotonicity properties of the observables Qk (f t ) with respect to f∞ , we prove the existence of the limits Pk+ (U ) := s- lim Pk+ (f ) , f∞ →1lU
where U is an open or closed set in Rd \{0}. The range of Pk (U )+ consists of the states with exactly k bosons with an asymptotic velocity in U . Pk+ (U ) are a mutually orthogonal family of projections with sum equal to one. Let U ⊂ Rd be an open set with 0 6∈ U . Let f∞,n ∈ C0∞ (Rd ) be a sequence of cutoff functions. We will say that f∞,n % 1lU if 0 ≤ f∞,n ≤ 1lU , f∞,n ≤ f∞,n+1 , f∞,n+1 f∞,n = f∞,n , f∞,n → 1lU pointwise . (7.5) Similarly if U ⊂ Rd is a closed set with 0 6∈ U , we say that f∞,n & 1lU if 1lU ≤ f∞,n ≤ 1, f∞,n ≤ f∞,n+1 , f∞,n+1 f∞,n = f∞,n+1 , f∞,n → 1lU pointwise . (7.6) For such sequences of cutoff functions and fn = (f0,n , f∞,n ) with f0,n = 1 − f∞,n , + + + , Q+ we denote by Pk,n k,n the operators Pk (fn ), Qk (fn ). Theorem 7.7. (i) Let U ⊂ Rd \{0} be an open (resp. closed) set. Let fn be a sequence of cutoff functions such that f∞,n % 1lU (resp. f∞,n & 1lU ). Then the limits + Q+ k (U ) := s- lim Qk,n n→∞
exist and are independent of the sequence f∞,n . Moreover {Q+ k (U )}k∈N is an increasing family of projections such that [Qk (U )+ , H] = 0 ,
s- lim Qk (U )+ = 1l . k→∞
(ii) The limits + Pk+ (U ) := s- lim Pk,n n→∞
exist and are independent of the sequence fn . Moreover, we have + Pk+ (U ) = Q+ k (U ) − Qk−1 (U ) .
The family {Pk+ (U )}k∈N is a family of mutually orthogonal projections such that [Pk+ (U ), H] = 0 ,
s−
∞ X
Pk+ (U ) = 1l .
k=0
(iii) If f = (f0 , f∞ ) satisfies the hypotheses of Subsec. 7.2 and f0 + f∞ = 1, supp f∞ ⊂ U, then + + (7.7) Q+ k (f )Qk (U ) = Qk (U ) .
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+ d d Of particular importance are the projections Q+ k (R \{0}), Pk (R \{0}), which will + + + be denoted simply by Qk , Pk in what follows. The range of Pk corresponds to the states with exactly k asymptotically free bosons.
Proof of Theorem 7.7. It suffices to consider the case when U is an open set. The case of a closed set is similar. We deduce first from Proposition 2.13 that if f∞,n % 1lU in the sense of (7.5), we have + Q+ k,n+1 ≤ Qk,n ,
l ≤ k, n < m .
+ + Q+ l,m Qk,n = Ql,m ,
(7.8)
Using (7.8) and Lemma A.3, we obtain the existence of + Q+ k (U ) = s- lim Qk,n n→∞
and hence of Pk (U )+ . Let us check that Qk (U )+ is independent of the sequence f∞,n . Let f∞,n , f˜∞,n be two sequences with f∞,n , f˜∞,n % 1lU in the sense of (7.5). Then there exists a sequence mn ∈ N tending to ∞ such that f∞,n ≤ f˜∞,mn . This implies that + ˜ Q+ k (fn ) ≤ Qk (fmn ) ,
which shows that + ˜ + ˜ lim Q+ k (fn ) ≤ lim Qk (fmn ) = lim Qk (fn ) .
n→∞
n→∞
n→∞
Hence Q+ k (U ) is independent of the sequence fn . We deduce from Theorem 7.5 (v) that 0 ≤ Q+ k (U ) ≤ 1 , k(H + i)−1 (Qk (U )+ − 1l)k ≤ C(k + 1)−1 , + which implies the strong convergence of Q+ k (U ) to 1l. The fact that Qk (U ) commutes with H follows also from Theorem 7.5. We deduce from (7.8) that + + Q+ l (U )Qk (U ) = Ql (U ) ,
l ≤ k.
This implies that {Q+ k }k∈N is an increasing family of projections. Finally (iii) follows from Proposition 2.13. 7.4. Asymptotic projections and asymptotic fields In this subsection, we prove that applying an asymptotic annihilation operator a+ (h) amounts to decrease the number of asymptotically free bosons by one. As an immediate consequence we obtain that the range of P0+ is included in the space of asymptotic matter K+ .
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Proposition 7.8. Assume the hypotheses (H1), (H2), (I1), (SR). Then the 1 following identities hold in the sense of quadratic forms on D((H + i) 2 ): + + a+ (h)Q+ k = Qk−1 a (h) ,
h ∈ h.
+ a+ (h) , a+ (h)Pk+ = Pk−1
h ∈ h.
Proof. It is enough to prove the identity involving Q+ k . By the continuity of 1 h 7→ a] (h)(H + i)− 2 it is enough to assume that h ∈ h0 . Let f∞,n % 1lRd \{0} in the sense of (7.5) and f0,n = 1 − f∞,n . We have + + −2 (i + H)− 2 (a+ (h)Q+ k − Qk−1 a (h))(i + H) 1
= lim
1
lim eitH (i + H)− 2 1
n→∞ t→+∞
× (a(ht )Qk (fnt ) − Qk−1 (fnt )a(ht ))(i + H)− 2 e−itH . 1
(7.9)
By Lemma 2.12 (ii), we have t a(ht )Qk (fnt ) − Qk−1 (fnt )a(ht ) = Pk (fnt )a(f0,n ht ) .
(7.10)
Since h ∈ h0 and ht = e−itω(k) h, we see by stationary phase arguments that, for t ht k ∈ o(t0 ). This gives n ≥ n0 , kf0,n
(a(ht )Qk (fnt ) − Qk−1 (fnt )a(ht ))(H + i)− 12 ∈ o(t0 ) ,
which implies that (7.9) is zero.
Corollary 7.9. Assume the hypotheses (H1), (H2), (I1), (SR). Then Ran P0+ ⊂ K+ . Proof. Let u ∈ RanP0+ and let un ∈ D(H) ∩ RanP0+ be a sequence converging to u. By Proposition 7.8 we have a+ (h)un = 0, h ∈ h, and hence un ∈ K+ . Since K+ is closed, u ∈ K+ . 7.5. Geometric inverse wave operators 2 ≤ 1, j0 = 1 near Let j0 ∈ C0∞ (Rd ), j∞ ∈ C ∞ (Rd ), 0 ≤ j0 , 0 ≤ j∞ , j02 + j∞ t ), where 0 (and hence j∞ = 0 near 0). Set j := (j0 , j∞ ). Set also j t = (j0t , j∞ x x t t j0 (x) = j0 ( t ), j∞ (x) = j∞ ( t ). As in Subsec. 2.15, we identify the pair j t with an operator j t : h → h ⊕ h and ˇ t ) : Γ(h) → Γ(h) ⊗ Γ(h). We use the same notation we introduce the operator Γ(j t ˇ t ) : H = K ⊗ Γ(h) → Hext = K ⊗ Γ(h) ⊗ Γ(h). ˇ ) to denote the operator 1lK ⊗ Γ(j Γ(j
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Theorem 7.10. (i) The following limits exist: s- lim eitH
ext
t→+∞
ˇ t )e−itH , Γ(j
ˇ t )∗ e−itH s- lim eitH Γ(j
ext
t→+∞
.
(7.11) (7.12)
If we denote (7.11) by W + (j), then (7.12) equals W + (j)∗ . If we set Wk+ (j) := 1l{k} (N∞ )W + (j) , then Wk+ (j) = s- lim eitH t→+∞
ext
ˇ t )e−itH . 1lk (N∞ )Γ(j
(ii) One has W + (j)(H + i)−1 = (H ext + i)−1 W + (j) , W + (j)χ(H) = χ(H ext )W + (j) ,
χ ∈ C0∞ (R) .
(iii) Let q0 , q∞ ∈ C ∞ (Rd ), ∇q0 , ∇q∞ ∈ C0∞ (Rd ), 0 ≤ q0 , q∞ ≤ 1, q0 = 1 near 0. Set ˜j := (˜j0 , ˜j∞ ) := (q0 j0 , q∞ j∞ ). Then Γ+ (q0 ) ⊗ Γ(q∞ (∇ω(k)))W + (j) = W + (˜j) . (iv) Let q ∈ C ∞ (Rd ), ∇q ∈ C0∞ (Rd ), 0 ≤ q ≤ 1, q = 1 near 0. Then W + (j)Γ+ (q) = W + (qj) , where qj = (qj0 , qj∞ ). (v) Let ˜j = (˜j0 , ˜j∞ ) be another pair satisfying the conditions stated at the beginning of this subsection. (Note that 0 ≤ ˜j0 j0 , 0 ≤ ˜j∞ j∞ , ˜j0 j0 + ˜j∞ j∞ ≤ 1 and ˜j0 j0 = 1 near 0). Then W + (˜j)∗ W + (j) = Γ+ (˜j0 j0 + ˜j∞ j∞ ) , Wk+ (˜j)∗ Wk+ (j) = Pk+ (˜j0 j0 , ˜j∞ j∞ ) . 2 = 1, then W + (j) is isometric. In particular, if j02 + j∞ Nk (vi) Let j0 +j∞ = 1. If u ∈ D((H +i)k/2 ) then Wk+ (j)u ∈ D((H +i)k/2 )⊗ s h ⊂ D(Ωext,+ ) and
Ωext,+ Wk+ (j)u = Pk+ (j)u . If u ∈ Hcomp (H) then W + (j)u ⊂ D(Ωext,+ ) and Ωext,+ W + (j)u = u .
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Proof. Let us first prove the existence of the limit (7.11), the case of (7.12) being similar. Using Lemma 3.4 and a density argument, it suffices to prove the existence of ext ˇ t )χ(H)e−itH , s- lim eitH χ(H ext )Γ(j t→∞
for some χ ∈
C0∞ (R).
We compute the asymmetric Heisenberg derivative
ˇ Γ(j ˇ t )χ(H) = χ(H ext )D ˇ 0 Γ(j ˇ t )χ(H) χ(H ext )D ˇ t ) − Γ(j ˇ t )V )χ(H) . + iχ(H ext )(V ⊗ 1lΓ(j From (3.11), we obtain ˇ t ) − Γ(j ˇ t )V )χ(H)k ∈ O(t−1−µ ) . kχ(H ext )(V ⊗ 1lΓ(j
(7.13)
ˇ 0 j t ), and, by pseuˇ 0 Γ(j ˇ t, d ˇ t ) = dΓ(j On the other hand by Lemma 2.16, we have D dodifferential calculus, d X 1 t ˇt jt = ki + rt , d 0 t 1 where t t , k∞,i ), kit = (k0,i
t k,i =−
x 1 xi − ∂i ω(k) ∂i j + hc , 2 t t
and rt ∈ O(t−2 ). Using Lemma 2.16 (v) and Lemma 3.2, we obtain ˇ t , rt )χ(H)k ∈ O(t−2 ) . kχ(H ext )dΓ(j
(7.14)
Using then Lemma 2.16 (iv), we obtain ˇ t , k t )χ(H)u1 )| |(u2 |χ(H ext )dΓ(j i
1 1 t t |) 2 ⊗ 1l)χ(H ext )u2 dΓ(|k0,i |) 2 χ(H)u1 ≤ (dΓ(|k0,i
1 1 t t + (dΓ(|k∞,i |) 2 ⊗ 1l)χ(H ext )u2 dΓ(|k∞,i |) 2 χ(H)u1 .
(7.15)
Hence the existence of the limit (7.11) follows from (7.13)–(7.15), Proposition 6.3 and Lemma A.1. (ii) follows from Lemma 3.4. (iii) follows from Proposition 7.6 and the fact that t ˇ t ˇ ˜j t ) . )Γ(j ) = Γ( Γ(q0t ) ⊗ Γ(q∞
(iv) follows from t ˇ ˇ t )Γ(j t ) = Γ((jq) ). Γ(j
(v) follows from t t ˇ t ) = Γ(˜j0t j0t + ˜j∞ ˇ ∗ (˜j t )Γ(j j∞ ) , Γ
ˇ t ) = Pk (˜j t j t , ˜j t j t ) . ˇ ∗ (˜j t )1l{k} (N∞ )Γ(j Γ 0 0 ∞ ∞
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Up to technical details due to the unboundedness of I, (vi) can be considered as a special case of (v) with ˜j = (1, 1). To prove (vi) we note that it follows from (ii) that W + (j) is bounded from D((H + i)k ) to D((H ext + i)k ) for k ∈ N. This extends Nk to all k ∈ R+ by interpolation. By Theorem 5.7 (i), we have for w ∈ H ⊗ s h Ωext,+ (H + i)−k/2 w = lim eitH Ie−itH
ext
t→+∞
(H + i)−k/2 w .
Since by (2.13) I(H + i)−k/2 1l{k} (N∞ ) is a bounded operator, we can use the chain rule of the wave operators and write ˇ t )e−itH u = P + (j)u , Ωext,+ Wk+ (j)u = lim eitH I1l{k} (N∞ )Γ(j k t→+∞
by (2.14). Finally the second statement of (vi) is an immediate consequence of the first. 7.6. Asymptotic absolute continuity In this subsection we will prove that if U is a closed set of Lebesgue measure 0 with 0 6∈ U , then P0 (U )+ = 1l, which means that there are no bosons living asymptotically in U . This property will be needed in the next section to effectively decouple bosons close to the origin from bosons close to infinity. Theorem 7.11. Assume the hypotheses (H1), (H2), (I1), (SR). Let U ⊂ Rd \{0} be a compact set of measure zero. Then P0+ (U ) = 1l . 2 Proof. Let j = (j0 , j∞ ) be as in Subsec. 7.5. Assume additionally that j02 +j∞ = ∞ d d 1 and j∞ = 1 near U . Let qn ∈ C (R \{0}), 0 ≤ qn , qn % R \U in the sense of (7.5). For large enough n we have j0 qn = j0 . Hence
Γ+ (qn ) = Γ+ (qn )W + (j)∗ W + (j) = W + (j0 , j∞ qn )∗ W + (j) = W + (j)∗ 1l ⊗ Γ(qn (∇ω(k)))W + (j) , by Theorem 7.10. But s- lim qn (∇ω(k)) = 1l . n→∞
Therefore, s- lim Γ(qn (∇ω(k))) = 1l , n→∞
and P0+ (U ) = s- lim Γ+ (qn ) = 1l . n→∞
Note in parenthesis another result, which follows from exactly the same arguments. (This result will not be used in the proof of asymptotic completeness.)
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Proposition 7.12. Assume hypotheses (H1), (H2), (I1), (SR). Let U ⊂ Rd \ {0} be an open or close set such that U ∩ ∇ω(Rd ) = ∅ . Then P0+ (U ) = 1l . 7.7. Geometric asymptotic completeness In this subsection we will show Theorem 7.3. It will follow from the following theorem, which gives an explicit construction of the inverse wave operator Ω+∗ in terms of the geometric inverse wave operators. Theorem 7.13. Let jn = (j0,n , j∞,n ) satisfy the conditions of Subsec. 7.5. 2 2 + j∞,n = 1 and j∞,n % 1lRd \{0} in the sense of (7.5). Additionally, assume that j0,n Then ∞ X + Wk+ (jn )Pk+ W := s- lim n→∞
k=0
exists. One has W + = Ω+∗ ,
(7.16)
and Ran W + = H+ = Ran P0+ ⊗ Γ(h) . Proof. Set 2 2 fn = (f0,n , f∞,n ) := (j0,n , j∞,n ) 0 0 , f∞,n ) := (j0,n , 1 − j0,n ) , fn0 = (f0,n ∞ ∞ fn∞ = (f0,n , f∞,n ) := (1 − j∞,n , j∞,n ) ,
fn,m = (f0,n,m , f∞,n,m ) := (j0,n j0,m , j∞,n j∞,m ) . Let m ≤ n and u ∈ H such that Pk+ u = u. By density we may assume that u ∈ Hcomp (H). Note that since j∞,n % 1lRd \{0} in the sense of (7.5), we have j0,n j∞,m = 0, which by Theorem 7.5 (vi) gives ∞ Pk+ (fn,m ) = Pk+ (fn0 )Pk+ (fm ).
Now we compute kWk+ (jn )u − Wk+ (jm )uk2 = kWk+ (jn )uk2 + kWk+ (jm )uk2 − 2Re (Wk+ (jn )u|Wk+ (jm )u) = (u|Pk+ (fn )u) + (u|Pk+ (fm )u) − 2Re (u|Pk+ (fn,m )u) ∞ )u|Pk+ (fn0 )u) , = (u|Pk+ (fn )u) + (u|Pk+ (fm )u) − 2Re (Pk+ (fm
(7.17)
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where in the last step we used (7.17). Clearly, 0 ∞ , f∞,n % Rd \{0} f∞,n , f∞,n
in the sense of (7.5). Hence s- lim Pk+ (fn ) = s- lim Pk+ (fn0 ) = s- lim Pk+ (fn∞ ) = Pk+ . n→∞
n→∞
n→∞
Therefore, from Pk+ u = u we see that s- lim kWk+ (jn )u − Wk+ (jm )uk = 0. n,m→∞
In other words, the sequence Wk (jn )u is Cauchy, and hence convergent. Let us check that the limit s- lim Wk+ (jn )u =: Wk+ u n→∞
does not depend on the choice of the sequence jn . In fact if jn , ˜jn are two sequences with jn , ˜jn % 1lRd \{0} , we can find a sequence mn tending to ∞ such that j0,n ˜j∞,mn = 0. Then we argue as above. By Theorem 7.10 (ii) we see that W + χ(H) = χ(H ext )W + . If q ∈ C0∞ (Rd ), q = 1 in a neighborhood of 0, 0 ≤ q ≤ 1 and qj0,n = j0,n , then by Theorem 7.10 (iii) Γ+ (q) ⊗ 1lWk+ (jn ) = Wk+ (jn ) . Therefore, Γ+ (q) ⊗ 1lW + = W + . Hence P0+ ⊗ 1lW + = W + . Thus Ran W + ⊂ Ran P0+ ⊗ Γ(h) ⊂ K+ ⊗ Γ(h) = H+ .
(7.18)
Let us now show (7.16). Let u ∈ Hcomp (H), u = Pk+ u. By Theorem 7.11, we can choose a sequence n & 0 such that 1 1 − n , + n u = u . (7.19) s- lim P0+ n→∞ n n We can demand that the sequence jn = (j0,n , j∞,n ) used to define W + satisfies additionally 1 1 − n , ∞ . supp j0,n ⊂ 0, + n , supp j∞,n ⊂ n n
ASYMPTOTIC COMPLETENESS IN QUANTUM IN FIELD THEORY
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Note that qn := (j0,n + j∞,n )−1 ≤ 1 and qn = 1 outside of [ n1 − n , n1 + n ]. Hence, by (7.19) s- lim Γ+ (qn )u = u . n→∞
Set ˜jn = (˜j0,n , ˜j∞,n ) = (qn j0,n , qn j∞,n ). Then ˜j0,n + ˜j∞,n = 1. Hence, by Theorem 7.10 (vi), W + (˜jn )u ∈ D(Ωext,+ ) and Ωext,+ Wk+ (˜jn ) = Pk+ (˜jn ) . Let χ ∈ C0∞ (R) such that u = χ(H)u. Using the fact that Ωext,+ χ(H ext ) is bounded, we have: u = Pk+ u = lim Pk+ (˜jn )u n→∞
= lim Ωext,+ χ(H ext )Wk+ (˜jn )u n→∞
= Ωext,+ χ(H ext ) lim Wk+ (˜jn )u n→∞
=Ω
ext,+
=Ω
ext,+
χ(H
ext
χ(H
ext
) lim Wk+ (jn )Γ+ (qn )u n→∞
) lim Wk+ (jn )u n→∞
= Ωext,+ χ(H ext )W + u = Ωext,+ W + u . Hence Ωext,+ W + u = u . But by (5.22) Ωext,+ 1lH+ = Ω+ . Therefore, by (7.18) Ω+ W + = 1lH . The fact that Ω+ is unitary from H+ to H implies now W + = Ω+∗ ,
Ran W + = H+ .
Hence by (7.18) we obtain (7.16).
7.8. Asymptotic completeness In this subsection, we will prove that P0+ = 1lpp (H). Combined with Theorem 7.13, this will complete the proof of asymptotic completeness. Theorem 7.15 below will be a consequence of the Mourre estimate of Subsec. 4.3. We first show that condition (SR) implies condition (I2). Lemma 7.14. Hypothesis (SR) implies hypothesis (I2).
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Proof. Assume that (SR) holds. Let us prove that hxiv ∈ B(K, K ⊗ h) which implies (I2). Equivalently we will show that ∞ X
nvn ∈ B(K, K ⊗ h) .
(7.20)
1
for vn := 1l[n,n+1] (hxi)v. If we set wn = 1l[n,+∞[ (hxi)v, use vn = wn+1 − wn and sum by parts in (7.20), we see that the convergence of (7.20) follows from (SR). Theorem 7.15. Assume the hypotheses (H0), (H1), (H2), (I1) and (SR). Then 1lpp (H) = P0+ . Proof. By Proposition 5.5 and geometric asymptotic completeness we already know that Hpp (H) ⊂ K+ = Ran P0+ . Let us now prove that P0+ ≤ 1lpp (H). Let χ ∈ C0∞ (R\(τ ∪ σpp (H))). Using an argument contained eg. in [13, Proposition 4.4.8], we deduce from Proposition 6.5 in Sec. 6 that there exists > 0 such that for q ∈ C0∞ (Rd ) with q(x) = 1 for |x| < we have Γ+ (q)χ(H) = 0. This implies that P0+ ≤ 1lσpp ∪τ (H) . Since τ is a closed countable set and σpp (H) can accumulate only at τ , we see that 1lpp (H) = 1lσpp ∪τ (H). This completes the proof of the theorem. Acknowledgements We would like to thank V. Bach, J. Fr¨ ohlich and H. Spohn for useful discussions. The work of Jan Derezi´ nski is a part of the project nr 2 P03 029 08 financed by Komitet Bada´ n Naukowych in the years 1995–97. Appendix A The following lemma describes an argument commonly used to prove the so called propagation estimates (see [13, Sec. 8.4] and references therein). Lemma A.1. Let H be a self-adjoint operator and D the corresponding Heisenberg derivative d + i[H, ·] . D := dt Suppose that Φ(t) is a uniformly bounded family of self-adjoint operators. Suppose that there exist C0 > 0 and operator valued functions B(t) and Bi (t), i = 1, . . . , n,
ASYMPTOTIC COMPLETENESS IN QUANTUM IN FIELD THEORY
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447
such that DΦ(t) ≥ C0 B ∗ (t)B(t) − Z
n X
Bi∗ (t)Bi (t) ,
i=1 ∞
kBi (t)e−itH φk2 dt ≤ Ckφk2 ,
i = 1, . . . , n .
1
Then there exists C1 such that Z ∞ kB(t)e−itH φk2 dt ≤ C1 kφk2 .
(A.1)
1
Next we describe how one uses propagation estimates to prove the existence of asymptotic observables. Lemma A.2. Let H1 and H2 be two self-adjoint operators. Let 2 D1 be the corresponding asymetric Heisenberg derivative: 2 D1 Φ(t)
:=
d Φ(t) + iH2 Φ(t) − iΦ(t)H1 . dt
Suppose that Φ(t) is a uniformly bounded function with values in self-adjoint operators. Let D1 ⊂ H be a dense subspace. Assume that |(ψ2 |2 D1 Φ(t)ψ1 )| ≤ Z Z
n X
kB2i (t)ψ2 kkB1i (t)ψ1 k ,
i=1 ∞
kB2i (t)e−itH2 φk2 dt ≤ kφk2 ,
φ ∈ H, i = 1, . . . , n ,
1 ∞
kB1i (t)e−itH1 φk2 dt ≤ Ckφk2 ,
φ ∈ D1 , i = 1, . . . , n .
1
Then the limit s- lim eitH2 Φ(t)e−itH1 t→∞
exists. Finally, we describe a simple lemma about the convergence of positive operators. Lemma A.3. Let Qn be a commuting sequence of selfadjoint operators such that : 0 ≤ Qn ≤ 1 , Qn+1 ≤ Qn , Qn+1 Qn = Qn+1 , (i) or (ii)
0 ≤ Qn ≤ 1 ,
Qn ≤ Qn+1 ,
Qn+1 Qn = Qn .
Then the limit Q = s- lim Qn . n→∞
exists and is a projection.
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Proof. Note that case (ii) reduces to case (i) by considering the operators (1 − Qn ), so it suffices to consider case (i). Clearly we have Q = inf Qn = w − lim Qn . n
n→∞
We use the identity Qn Qm = Qm ,
for m > n ,
and let m tend to ∞, which gives Qn Q = Q. Letting then n tend to ∞ we get Q2 = Q. Next we have Q2n ≤ Qn and Qn+1 = Qn+1 Qn ≤ Q2n , which gives Qn+1 ≤ Q2n ≤ Qn . Letting n tend to ∞, we get Q = w − lim Q2n . n→∞
Then we compute lim k(Q − Qn )uk2 = lim (Q2n − Q)u|u = 0 ,
n→∞
which proves that Q = s- limn→∞ Qn .
n→∞
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[42] M. Reed and B. Simon, Methods of Modern Mathematical Physics, vols. I and II, 1976, vol. III, 1979, vol. IV, 1978, Academic Press, London. [43] I. M. Sigal and A. Soffer, “The N -particle scattering problem: Asymptotic completeness for short-range quantum systems”, Ann. Math. 125 (1987) 35–108. [44] B. Simon, The P (φ)2 Euclidean (Quantum) Field Theory, Princeton Univ. Press, Princeton, 1974. [45] E. Skibsted, “Spectral analysis of N -body systems coupled to a bosonic system”, preprint, 1997. [46] H. Spohn, “Ground states of the spin-boson hamiltonian”, Commun. Math. Phys. 123 (1989) 277–304. [47] H. Spohn, “Asymptotic completeness for Raleigh scattering”, preprint, 1996. [48] H. Spohn, “Ground state of a quantum particle coupled to a scalar Bose field”, preprint IHES, 1997. [49] S. Weinberg, The Quantum Theory of Fields, vol. I, Cambridge Univ. Press, 1995. [50] D. R. Yafaev, “Radiation conditions and scattering theory for N -particle hamiltonians”, Commun. Math. Phys. 154 (1993) 523–554.
MEL’NIKOV’S APPROXIMATION DOMINANCE. SOME EXAMPLES G. GALLAVOTTI Dipartimento di Fisica Universit` a di Roma 1 P.le Aldo Moro 2, 00185 Italy
G. GENTILE Dipartimento di Matematica Universit` a di Roma 3 Largo S. Leonardo Murialdo 1 00146, Roma Italy
V. MASTROPIETRO Dipartimento di Matematica Universit` a di Roma 2 V.le della Rierca Scientifica 00133, Roma Italy Received 30 April 1998 We continue a previous paper to show that Mel’nikov’s first order formula for part of the separatrix splitting of a pendulum under fast quasi periodic forcing holds, in special examples, as an asymptotic formula in the forcing rapidity.
0. Introduction Recently there has been renewed interest in a problem treated in the paper [5]. One of the several questions posed in [5] was to find upper and lower bounds on the splitting between the stable and unstable manifolds of the invariant torus with 1 rotation vector ω = η − 2 ω 0 in the Hamiltonian system (“Thirring model”): η− 2 ω 0 · A + 1
1 I2 A·A+ + g 2 J0 (cos ϕ − 1) + µf (α, ϕ) , 2J 2J0
(0.1)
where (I, ϕ) ∈ R×T, (A, α) ∈ Rl−1 ×Tl−1 are canonically conjugated variables, ω 0 ∈ Rl−1 , and J, J0 > 0 (respectively, “rotators’ moments of inertia” and “pendulum’s moment of inertia”), g > 0 (g 2 is the “gravity”); ω 0 , µ are parameters. Here J could be a scalar or a diagonal (l − 1) × (l − 1) matrix. And setting ν = (n, ν) ∈ Zl , Pl−1 |ν| = |n| + |ν| = |n| + i=1 |νi |:
451 Reviews in Mathematical Physics, Vol. 11, No. 4 (1999) 451–461 c World Scientific Publishing Company
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G. GALLAVOTTI, G. GENTILE and V. MASTROPIETRO
X
f (α, ϕ) =
fν,n cos(ν · α + nϕ) ,
(0.2)
ν,n;|ν|≤N0
where fν are fixed constants and N0 > 0 is fixed.a The paper [5] did solve completely the upper bound question for f a trigonometric polynomial: in the sense that it derived generically optimal bounds for the Fourier transform of the 2-dimensional (homoclinic) splitting function ∆↑ (α) (a vector valued function defined in [5, p. 366]). Subsequently it was rightly pointed out, [3], that it would be interesting to just have examples in which some lower bound could be computed as given by the naive first (non trivial) order perturbation theory prediction (i.e. “Mel’nikov integral”). For instance one could study the difference in the free pendulum energy h0 = 2J10 I 2 + J0 g 2 (cos ϕ − 1) evaluated in two “corresponding points” of the stable and unstable manifolds of the invariant torus. Such difference is a very special case of the splitting functions considered in [5]. Of course studying the splitting only via the variations of a single function (“observable”) is very reductive. The stable and unstable manifolds are 2-dimensional surfaces, if observed on a (4-dimensional) section transverse to the motion and at fixed total energy, hence it is clear that studying the difference of the two manifolds at “corresponding points” on the chosen section requires the simultaneous analysis of 2 observables. Thus in [5] two independent observables are considered (more are not necessary: the dimension of the manifolds being 2). One also needs to have good control over the Mel’nikov integral, i.e. on the usually explicitly known first order (in ε) expression of the splitting: not so easy in general. Control is possible in some rather special cases like the 3 degrees of P freedom system (i.e. two rotators) with f (α, ϕ) = ν fν (cos ν · α)(cos ϕ), where fν = e−κ|ν| or more generally F 0 e−κ|ν| ≤ |fν | ≤ F e−κ|ν| for some F 0 , F, κ > 0, and ω 0√= (ω01 , ω02 ) a rotation vector with a golden mean rotation number ω01 /ω02 = 1 2 ( 5 − 1). The problem has been studied, [3], without taking into account [5] but without any saving of work because a theory of the upper bound equivalent to [5] has to be derived one way or other: and in fact the authors of [3] had to restrict considerations to the isochronous cases (J = +∞): taking J < +∞ does not change the first order analysis but changes completely the higher orders treatment. We show here that the analysis of [5] yields an upper bound on the splitting optimal and sufficient to deduce the dominance (hence the asymptoticity) of Mel’nikov’s approximation for the part of the splitting defined by the variation of the function h0 above defined, if the perturbation is: X def f (α, ϕ) = fν cos ν · α cos ϕ , N = η −1 , (0.3) ν,|ν|≤N
a It seems clear that the often sought “first order of perturbation theory” dominance as η → 0 can only hold if N0 < ∞ and fν,n is nonzero and “quite large” for “many” ν’s. Hence N0 < ∞ is a reasonable assumption, together with the assumption that we introduce later on fν .
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453
with fν ≥ F e−κ|ν| for some F, κ > 0 and allb ν. This differs from the function −1 considered in [3] by a quantity O(e−η κ ) which is much smaller than any expected splitting. The latter is expected to be given by the Mel’nikov integral, hence to be −1
of order ≥ O(e−O(η 2 ) ). Extending the sum over ν to all ν’s would require extending the results in [5] to the case in which the perturbation is analytic, while one of the “philosophical assumptions” of [5] was that one should understand first only polynomial cases. However the purpose of [3] was to provide an example of the Mel’nikov integral dominance and the (0.3) is as good for the purpose. Generalizing [5] to non polynomial analytic cases is easy: but to limit the length and the simplicity of the present comment on [5] we do not provide the details. We can indicate that an extension to analytic cases has been performed in the very similar problem studied in [1] (providing again an example of the Mel’nikov integral dominance). The fact that the bounds in [5] are uniform in the matrix J (which is summarized in [5] by calling twistless the invariant torus of (0.1) with rotation vector ω) will imply, below, that the perturbation f in (0.3) also generates a splitting measured by the variation of the function h0 that is asymptotically exactly given by Mel’nikov’s first order perturbation result for all J ≥ J0 . The treatment given below being more general than what would be needed to just obtain the splitting of h0 also solves (in part: see below) the problem posed in [11]: a paper in which the basic strategy is adapted from [5] but which seems to contain incorrect usage of [5] leading to errors in intermediate steps and in the final results, see [9]. The following section assumes full knowledge of [5] and we consider it just as a (conceptually trivial) final comment to it, made interesting by the idea of [3] of studying lower bounds for single observables: therefore we call it Sec. 9 (Sec. 10 below contains a few comments to relate the results to intervened papers), as the paper [5] ends rather abruptly (because of exhaustion of the author, and of the problem) at Sec. 8. We use the notations of [5] and we freely quote the formulae in the eight sections of [5]. The following section has simply to be regarded as a new section of [5]. The above is an introduction and formulae have been labeled by 0 to avoid confusion with the labels used in [5]. The relation between the splitting of the observable h0 and the 2-dimensional splitting ∆↑ (α) introduced in [5] is, on the Poincar´e section considered in [5] (i.e. at ϕ = π, and fixed energy) and if J = +∞, simply −ω · ∆↑ (α). In the anisochronous case (J < +∞) it is slightly more involved. Unless explicitly stated, we shall mean that the “splitting” is the vector ∆↑ (α) of [5, p. 366]. 9. Homoclinic Splitting in the Presence of Several Fast Rotators An interesting consequence of the theory of Sec. 8 is an expression of the splitting 1 in the l = 3 case and when the vector ω has all components fast, i.e. ω = η − 2 ω 0 and η → 0, while ω 0 is fixed and verifies the Diophantine condition (1.3). b This note is made necessary by recent claims that [5] would not be sufficient to get an upper bound which would yield the full result of [3], see [12].
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Examining the tree expansion for ∆↑ (α) one realizes that ∆h↑ν is given by a sum 0
−1 π
of tree values each of which is bounded by ξ ≡ e−|ω·ν |g ( 2 −d) (where d is prefixed 1 and chosen equal to η 2 , as in Sec. 8, to treat the case l = 2) with a ν 0 in general different from ν but nonzero, see p. 379. Each tree will in general contain several bubbles (see Sec. 6, p. 376). Here it will be convenient not to lump together the contributions from the trees with the same free structure: rather it will be convenient to leave them separate: thus a tree value will be described as in Sec. 6 but its fruits will contain their seed inside (see Sec. 6, p. 377). We should call them seeded fruits, for obvious reasons; below we shall simply call them just “fruits”: they can be either ripe or dry, as defined in Sec. 6, p. 376. Therefore we redecompose the resummation trees into single trees by specifying the seed inside each fruit. This is a minor change and it is convenient as it will show that the bound in (8.1) can be trivially and greatly improved in the case of perturbations like (0.3). The small factor ξ arises for some ν 0 with |ν 0 | ≤ N h (N being the cut-off parameter N = η −1 ) in two different ways. (1) In trees with free momentum ν 0 (see Sec. 6, p. 375) and only ripe fruits (see Sec. 6, p. 376) it arises by bounding carefully (i.e. by complex integration) the integrals relative to the free nodes (called v) variables τv and by using the trivial bounds to bound the fruit values (i.e. with no excursion into the complex plane to estimate them, which of course would not be allowed because of lack of analyticity of the integrals that express the values of ripe fruits). (2) In trees with at least one dry fruit it arises from a fruit located inside it, perhaps “very deeply” (i.e. inside many other fruits), which only contains ripe fruits. It exists necessarily (note that no fruit at all is a special case of only ripe fruits) and in this case the free momentum cannot vanish (see Sec. 6): this was the reason why the induction described in Sec. 6 worked. P Therefore there is a natural resummation of the splitting series ν,h µh eiν·α ∆h↑ν : namely we separate the first order contribution and we collect together all contributions from trees whose value is bounded by the same small factor ξ. In this way: ! ∞ X X π 1 h |ω·ν| iν·α − 2g h ˜ ↑ν + ˜ ↑ν (α) , µ∆ e e µ ∆ (9.1) ∆↑ (α) = ν6=0 1
h=2
iν·α ˜ where eiν·α e− 2g |ω·ν| ∆ ↑ν isolates the first order terms, while the quantity e π π h h,ϑ ˜ (α) is defined as the sum of the values eiν·α e− 2g |ω·ν| ∆ ˜ (α) of all e− 2g |ω·ν| ∆ ↑ν ↑ν trees ϑ of order h containing either only ripe fruits (or no fruits at all) or a dry fruit which has inside a seed that is a tree with only ripe fruits (or no fruits at all); see items (1) and (2) above. Since a tree can have several fruits which can be “exponentially bounded” there π is ambiguity in attributing the terms which contain several small factors e− 2g |ω·ν j | with momenta ν 1 , ν 2 , . . . : in the latter cases we just make an arbitrary choice to π
MEL’NIKOV’S APPROXIMATION DOMINANCE. SOME EXAMPLES
455
h
˜ (α) with some ν among ν , ν , . . . (e.g. ν is the first attribute such terms to ∆ 1 2 ↑ν among ν 1 , ν 2 , . . . in some lexicographic order). ˜ h↑ν (α) starts at least at order two because the trees must The series involving µh ∆ be at least of second order, having already separated out the first order contribution. Furthermore each tree contains as a factor the product of the couplings fν v that correspond to its nodes v with order label δv = 1 (see p. 367); and the set of the ν v P must be such that a suitable subset W of the set of nodes verifies v∈W ν v = ν. This means that we can bound the contribution of each tree ϑ with h nodes v of order δv = 1 (and hence total number of nodes m ≤ 2h) by the bounds in Appendix A1, in which the part in the constant c1 due to the factors fν v is explicitly factored out: h,ϑ ˜ (9.2) ∆↑ν (α) ≤ J0 gD0 d−β N 4h (B0 d−β )h−1 h!p |fν 1 | · · · |fν h | , where β, p are the parameters in (8.1),(8.2) with D0 , B0 bounded in Appendix A1, ¯ for some constant B ¯ (which is not the same as see (A1. 2) ÷ (A1. 15), by B, c23 M∗2 N −4 as in (A1. 15) only because we have factored out the product of the factors fν v associated with each of the h nodes or order 1). ˜ h,ϑ (α) is not equal to the total momentum Note that in general the label ν in ∆ ↑ν h ˜ of the tree ϑ (i.e. ∆ (α) is not a Fourier component, as it depends explicitly on ↑ν
α). It is not difficult to see (by working out a special example, see [9] for a trivial example; furthermore the proof of the theorem in Sec. 10 of [2], although incorrect as a proof of the theorem, nevertheless provides a much less trivial example) that the above formula and bounds are optimal, so that a bound like: 0 0 π h (9.3) ∆↑ν < B h η −β e− 2g |ω·ν| for the νth Fourier component of ∆↑ (α) would be incorrect.c Looking at (9.1) we can therefore bound the sum over ν and h of the second addend by first truncating the series to order h ≤ N : the remainder is estimated, as in Sec. 8, via the first of (8.1),d and will be of the order O((η −Q µ)N ) so that −1 if |µ| < η Q+1 it has size not exceeding O(η η ), much smaller than the generically expected contribution from the first order terms. 1 The sum of the first N orders (9.2) can be bounded (as in (8.6) and using d = η 2 ) by: J0 gD0 η −β/2 N 4
N X h=2
µh F h
X
e
−κ
P j
|ν j |
(B0 N 4 η −β/2 )h−1 h!p
ν 1 +...+ν h =ν
2 ≤ 2J0 gD0 η −β/2 cl N 4 µF B0 N l+p+4 l!η −β/2 e−κ|ν| ,
(9.4)
c The above bound, after summation over h, is claimed to hold in [11, Theorem 2.1], for a quantity that the authors appeared to think, in private correspondence, to be quite simply related to the ∆↑ (α) of [5] and of this note. d Which descends, for instance, from the KAM-type theorems in Sec. 5 of [2], hence it holds for analytic perturbations and it is uniform in the perturbation as long as the latter is analytic in |Imαj | < κ and |fν | < F e−κ|ν| , if κ is a prefixed quantity, no matter how small.
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P P −κ |ν j | j where we have used that e ≤ cl e−κ|ν| (lh)lh for some c, ν 1 +···+ν h =ν because the sum has dimension ≤ hl (in fact = h(l − 1)) and we have assumed that µ is so small that (µF B0 N l+p+4 l!η −β/2 )2 < 12 . Hence under the condition: 2 def |µ|µ−1 = |µ| 2D0 cl N 4 J0 gη −β/2 F B0 N l+p+4 l!η −β/2 < 1 (9.5) 0 ˜ 1 is essentiallye e−κ|ν| , we can the sum in (9.4) is bounded by: µe−κ|ν| . Since ∆ ↑ν write 1 X π ˜ + µe−κ|ν| Γ (α) eiν·α e− 2g |ω·ν| ∆ ∆↑ (α) = µ ν ↑ν ν6=0;|ν|≤N
+ O(η η
−1
) + O(µ2 e− 2 η κ
−1
|Γν (α)| < 1 ,
),
−1
(9.6) −1
where O(η η ) is a bound on the terms with h > N and O(µ2 e− 2 η ) is a bound π on the terms with h ≤ N with a bound proportional to e− 2g |ω·ν| with |ν| > N . By using the expression (9.6) for the splitting vector we can compute the determinant of the splitting matrix or, similarly to [3], the variation of the pendulum energy evaluated on the stable and unstable manifolds, both at ϕ = π. In both cases, by conforming to the traditional terminology, we call Mel’nikov integral the first (non trivial) order contribution. If one computes the determinant of the splitting matrix, one has to differentiate once with repect to α the splitting vector ∆↑ (α). Since the derivative of the terms arising from the first order contributions is trivial we only need to say that the derivatives of Γν (α) can be computed graph by graph and in each graph they amount to a multiplication by a component of the total momentum. Therefore under a slightly stronger requirement on µ (since |ν| < η −1 one has to require that µ is smaller than η times the µ0 defined above: |µ| < µ0 η) one finds πµ 2 X ω · ν0 ω·ν i· i hπ hπ det ∂α ∆↑ (α) α=0 = −2 g ω · νg −1 sinh ω · ν 0 g −1 |ν|,|ν 0 |≤N sinh 2 2 0 κ −1 · fν fν 0 (ν ∧ ν 0 )2 +µe−κ(|ν|+|ν |) Γ0ν,ν 0 +O(µ2 e− 2 η ) , (9.7) κ
with |Γ0ν,ν 0 | < 1, under the stated condition on |µ| (i.e. |µ| < ηµ0 ). As to the variation of the pendulum energy, by calling M (α) the Mel’nikov integral (depending on α, contrary to the homoclinic determinant (9.7), which is computed at the homoclinic point ϕ = π, α = 0), one has X π eiν·α e− 2g |ω·ν| ω · ∆1↑ν (9.8) M (α) = µ ν6=0;|ν|≤N
so that, for some constant C, |ω · ∆↑ (α) − M (α)| ≤ C|µ|2
X
e− 2g |ω·ν| |ω|e−κ|ν| π
(9.9)
ν6=0;|ν|≤N e Precisely ∆1 = −iν f 2π ν g ↑ν
4).
ω·ν sinh[ π g −1 ω·ν] 2
π
= −iν fν Φν e− 2 g
−1
|ω·ν|
with 1 ≤ |Φν | < (2|ω · ν|g −1 +
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MEL’NIKOV’S APPROXIMATION DOMINANCE. SOME EXAMPLES
which can be compared with the results in [3], and, unlike [3], it holds also in the anisochronous case. Note that so far no use has been made of the assumption that l = 3 nor that ω has a rotation number equal to the golden mean nor of the contents of the paper [3]. The above (9.8) and (9.7) also seem well suited for extensions to l > 3 (or to more general rotation numbers) of the Mel’nikov integral dominance. Remark. Assuming fν = e−κ|ν| for |ν| < N , by an argument of [3] (exposed in Sec. 6 of [3], see also Sec. 2) we see that, √ if l = 3 and ω 0 has rotation number 1 equal to the golden mean (ω01 /ω02 = 2 ( 5 − 1)), in (9.8) the Mel’nikov integral dominates the asymptotics as η → 0 if µ is small as indicated. In fact, apart for some “special” small intervals of values of η and of α, only two addends matter in (9.8) (see [3]): they correspond to ν equal to ±ν k = ±(fk+1 , −fk ) where fj is Fibonacci’s sequence and k = k(η) is the value that minimizes κ|ν j | +
π −1 η 2 |ω 0 · ν j | 2
(9.10)
(of course |νk | ≤ N ): the corresponding miminum has the form c(η)η − 4 , with c(η) > 0 an explicitly computable quantity bounded above and below by positive constants, uniformly in η → 0. The special values of η are those for which the minimum is taken over two values of k, or of η’s close enough to such values; the special values of α are those for which | sin α · ν k | < b, where b > 0 is an arbitarily prefixed quantity (if b is too small then the size of η for which dominance occurs becomes correspondingly smaller). Excluding the just mentioned values of α and η the asymptotics of the 1
−1
Mel’nikov integral M (α) is proportional to µη 2 e−c(η)η 4 sin ν k · α. Moreover to the r.h.s. of (9.9) the same considerations trivially apply, i.e. only two (or four) addends 1
−1
matter so that it is bounded by a constant times µ2 η − 2 e−c(η)η 4 and the first order dominance for ω ·∆↑ (α) follows, for all α except the special ones. So [5] and the first order analysis (in [3]f ) prove the first order dominance not only in the isochronous case, as done in [3], but also in the anisochronous case. 1
But of course the really interesting quantity is (9.7): i.e. the actual splitting at α = 0. For this quantity we cannot conclude dominance of the first order term, unless some (likely, see [7], but yet undiscovered) cancellations take place: it is manifest from (9.7) that the would be leading term (ν = ν 0 = ν k ) vanishes. Hence no conlusion can be drawn, in the class of models considered here, about the splitting at the homoclinic point (besides the one already derived in Sec. 8 and saying that it is smaller than any power of η). Unless one considers η → 0 along a special sequence in correspondence of which there are two pairs of successive Fibonacci’s vectors with the same, or almost the same, κ|ν k | + π2 |ω · ν k | minimizing (9.10): in this case, obviously, we have dominance of the Mel’nikov term because no two Fibonacci’s vectors are parallel, see (9.7). Note that the expressions “almost the f The rest of the paper [3] provides an interesting partial alternative to the work in [5]; the partiality being mainly due to the isochrony assumption, essential in the theory of [3].
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same” and “close enough”, used above for the variable η, simply mean that the two Fibonacci’s vectors such that the quantity βk = κ|ν k | + π2 |ω · ν k | is closest to its minimum over k correspond to two values of k, say k, k + 1, for which βk − βk+1 is a quantity that is uniformly bounded as η → 0. 10. Comments (1) The above theory is different from that of [3] in a radical way; in fact it works in the isochronous case (J = +∞) as well as in the anisochronous one. This is a new result (see however the acknowledgements below). And the last remark shows that the problem of the homoclinic splitting asymptotics with 2 fast forcing frequences is still open if it is intended as defined by (9.7) (as one should in view of the possible applications to heteroclinic chains and Arnol’d diffusion). This is so in spite of apparent claims to the contrary, [12]. 1 1 (2) In three time scales problems (i.e. when ω = (η − 2 ω01 , η 2 ω02 ) and g = O(1)) the above arguments do not lead to any conclusion because ω·ν is not always 1 1 proportional to η − 2 but it can even be proportional to η 2 also for small |ν|: so that the exponentially small factor is not always present. This means that even the individual components of the splitting ∆↑ (α) in general are not only not dominated by the Mel’nikov integral but have size of the order of a power of η! Regarding the intersection matrix, if (9.3) could be assumed valid three matrix elements would be exponentially small and also the determinant would be (exponentially small and) dominated by the Mel’nikov integral. But, as we said, (9.3) is false. In fact the intersection matrix has, in this case, all matrix elements that have polynomial size as η → 0, starting with second order: this part of the theorem in Sec. 10 of [2] is, of course, valid.g And it is perhaps natural to think that also the splitting (9.7) has size of order of a power of η. This is however an error, as the determinant is indeed exponentially small in η, although the matrix elements have size of the order of a power of η, due to the presence of cancellations in the determinant. This was proved in [7], where such cancellations are exploited. The above error was in fact suffered in [2]: a trivial computational error in one of the about twenty terms contributing to the third order crept in and led the authors of [2] to believe they had checked that even the determinant of the splitting matrix would be polynomially bounded away from zero as η → 0 (i.e. bounded as the individual matrix elements are) under the convergence condition that µ is itself a (large) power of η. In any case these considerations show that there is apparently no hope, without exploiting cancellations in the determinant, to obtain a exponentially small bound (the matrix elements have polynomial size); the derivation of the exponentially small bounds for g The theorem was “proved” by showing that three matrix elements had polynomial size while the fourth was “exponentially small”: the last claim is incorrect because of a trivial computational error which, if corrected, yields that even the fourth matrix elements is of polynomial size.
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MEL’NIKOV’S APPROXIMATION DOMINANCE. SOME EXAMPLES
this case from (9.3), to which many authors seems apparently to believe, is false because (9.3) is false, in general and on the section considered here. (3) But the paper [2] had, nevertheless, laid the foundations of the theory developed in [5]: a very simple and systematic theory, because of its field theoretic viewpoint, and very flexible thanks to the possibility of rapid (graphical) comparison of arbitrarily high order terms of perturbation expansions. Enough to cope with the error and to correct it, as shown in [7] (an achievement scarsely appreciated in [12]). In fact the papers [5], [7] suggest which could be the general solution to the problem of the splitting (9.7) even when the interaction is a trigonometric polynomial: see the conjecture in [6]; a “few” cancellations remain to be checked to prove (or to disprove) the conjecture. In fact the reason the error was not spotted by the authors of [2] (it was pointed out by Gelfreich) was due to the apparent matching of the result with the intuitive idea that the splitting in three time scale problems would be “large”. The splitting, defined as the smallest eigenvalue of the intersection matrix, has the dimension of an action (being the derivative with respect to an angle of the difference in the actions for the stable and for the unstable manifolds) hence it has to be measured with respect to an action. In [2] the relevant action with which to compare the splitting was the size of the gaps, in action space, between the average actions of the invariant tori surviving the perturbation. The latter gap was estimated in [2] by a power, as large as wished, of η times max{J0 g, Jg}: hence the splitting was larger than any power of η −1 with respect to the gaps. The proof was wrong and it was corrected in [7]. And in [8] it has been shown (to the skeptics, see [12])h that the gaps have in this case size of almost −1 O(eO(−η ) ) while the splitting is dominated by the Mel’nikov integral for generic perturbations, even if of polynomial type: the Mel’nikov integral − 1 πω 2
being of order O(e−η 2g ) if the frequency vector is ω = (η − 2 ω1 , η 2 ω). So the splitting is still very large.i (4) Recently we learnt of Eliasson’s remark that the homoclinic splitting Q(α) is a gradient Q(α) = ∂ α Φ(α) of some potential Φ: this is due to the Lagrangian nature of the stable and unstable manifolds. Once this is known one recognizes that the tree expansion in sections 1–8 of [5] provides a power series expansion of the potential Φ, see [6]. (5) One can apply the above theory to the case in which the dimension of the angles α is 1: this is the periodic forcing case discussed in Sec. 8. In this case there is no distinction between the (9.7) and (9.8) and one finds an asymptotic expansion for such splittings (see Sec. 8). This case is closely related to a corresponding problem in the theory of the standard map: the 1
1
h Extending a property and a method that was noted long ago by Neishtadt, [10]. i And even larger than value claimed in [2]: the ratio between the splitting and the gap being now
O(eO(η
−1
) ),
while in [2] it was claimed to be “only” O(η−a ) for arbitrary a.
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basic approach is due to Lazutkin and recently there has been renewed interest in it. The paper [4], which is nice and accurate, also fills some (apparently) missing points in the original work. The approach in [4] is also based on formal power series expansions for the separatrix; but unlike [5] the control of the remainders is not solely made via the analysis of formal series, thus providing an interesting alternative, although (of course) the amount of work is comparable. The drawback, at the moment, is the need of an “integrable normal form” near the invariant tori (circles in this case). Therefore, although adapting the method of [4] to cover the cases of Hamiltonian systems appears straightforward in the case of isochronous systems (which trivially admit an integrable normal form), a possible extension to the anisochronous case seems to require substantial extra work, at least if one proceeds along the lines of [4]. The method of [5] has not (yet) been applied to study the standard map case. Likewise the extension to higher dimension seems to require further extra work (a beginning of the theory is in [3]); and so does the extension to three time scales problems. Once the above extensions will exist they will be be very interesting, and, solving in a substantially different way the problems solved in the latter papers, they will permit a (better) comparison with the methods in [5] and [7]. (6) Finally the reason why one may hope to get large absolutej splitting in three degrees of freedom systems is not because some ω · ν (in the small factors π e−|ω·ν| 2g ) becomes small when ν becomes a good rational approximation of ω⊥ , as sometimes hinted in the literature, see [12]. This is shown by the three time scales problem in (2) above in which ω·ν can be very small even for small ν (i.e. if the fast component of ν vanishes) and nevertheless the splitting does not have a polynomial absolute size, due to remarkable cancellations (see [7]). One has rather to think that the splitting can become large because the perturbation f (α, ϕ) has a small analyticity band in complex ϕ plane: in fact the higher the degree N0 in ϕ the smaller it seems that one has to choose µ with respect to η if one wants to guarantee Mel’nikov’s approximation to dominate.
Acknowledgement The content of this paper was (and is) considered by us a trivial remark on the work [5]: as one can see in the story [12] this view is violently not shared by other specialists. Therefore we wrote it up to clarify our methods and techniques, which in spite of apparently preconceived claims to the contrary (see [12]), are proving more and more well suited for the problems under analysis. We are greatly indebted to G. Benfatto, G. Benettin and A. Carati for clarifying discussions and for support. j I.e. large compared to the natural unit of action J g and not compared to the size of the gaps. 0
MEL’NIKOV’S APPROXIMATION DOMINANCE. SOME EXAMPLES
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References [1] G. Benettin, A. Carati and G. Gallavotti, “A rigorous implementation of the Jeans–Landau–Teller approximation for adiabatic invariants”, Nonlinearity 10 (1997) 479–507. [2] L. Chierchia and G. Gallavotti, “Drift and diffusion in phase space”, Annales de l’Institut Henri Poincar´ e B60 (1994) 1–144. See also the “Erratum”, Annales de l’Institut Henri Poincar´ e B68 (1998), 135. [3] S. Delshams, V. G. Gelfreich, A. Jorba and T. M. Seara, “Exponentially small splitting of separatrices under fast quasiperiodic forcing”, Commun. Math. Phys. 189 (1997) 35–72. [4] V. G. Gelfreich, “A proof of the exponentially small transversality of the separatrices of the standard map”, mp arc@ math. utexas. edu, #98–270, to appear in Commun. Math. Phys. [5] G. Gallavotti, “Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems. A review”, Rev. Math. Phys. 6 (1994) 343–411. [6] G. Gallavotti, “Reminiscences on science at I.H.E.S. A problem on homoclinic theory and a brief review”, preprint, 1998, http://ipparco.roma1.infn.it; chao-dyn@xxx. lanl. gov, #9804044 in “Les relations entre les math´ematiques et la physique th´eorique”, eds. J. P. Bourguignon, L. Michel, Publications Math´ ematiques de l’IHES, special volume, 99–117, Paris (1998). [7] G. Gallavotti, G. Gentile and V. Mastropietro, “Separatrix splitting for systems with three degrees of freedom”, Preprint, in mp [email protected], #97-472 with the title “Pendulum: separatrix splitting” in Commun. Math. Phys. [8] G. Gallavotti, G. Gentile and V. Mastropietro, “Hamilton–Jacobi equation, heteroclinic chains and Arnol’d diffusion in three time scales systems”, archived in chaodyn@xyz. lanl. gov #9801004. [9] G. Gallavotti, G. Gentile and V. Mastropietro, “A comment on the Physica D paper by Rudnev and Wiggins”, mp [email protected], #98–245. See also “Homoclinic splitting, II. A possible counterexample to a claim by Rudnev and Wiggins on Physica D”, chao-dyn 9804017 to appear in Physica D. [10] A. I. Neishtadt, “The separation of motions in systems with rapidly rotating phase”, J. App. Math. and Mechanics 48(2) (1984) 133–139. [11] M. Rudnev and S. Wiggins, “Existence of exponentially small sepratrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near integrable Hamiltonian systems”, Physica D 114 (1998) 3–80. [12] Collection of the comments received by us from colleagues and from referees about the paper [7], and our replies: in http://ipparco.roma1.infn.it.
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS: FROM NUMBER THEORY AND GROUP THEORY TO QUANTUM FIELD AND STRING THEORIES S. C. WOON Department of Applied Mathematics and Theoretical Physics University of Cambridge Cambridge CB3 9EW, UK E-mail : [email protected] Received 29 September 1997 We are used to thinking of an operator acting once, twice, and so on. However, an operator can be analytically continued to the operator raised to a complex power. Applications include (s, r) diagrams and an extension of Fractional Calculus where commutativity of fractional derivatives is preserved, generating integrals and non-standard derivations of theorems in Number Theory, non-integer power series and breaking of Leibniz and Chain rules, pseudo-groups and symmetry deforming models in particle physics and cosmology, non-local effect in analytically continued matrix representations and its connection with noncommutative geometry, particle-physics-like scatterings of zeros of analytically continued Bernoulli polynomials, and analytic continuation of operators in QM, QFT and Strings.
1. Introduction: Analytic Continuation of Operators 1.1. Questions in context R d(1/2+i) • What is dx (dx)(1/2+i) ? (1/2+i) ? Is it meaningful? What about • Given a function f (x), how do we evaluate d(1/2+i) f (x) and dx(1/2+i) • • •
•
•
Z f (x)(dx)(1/2+i) ?
What do they mean? Are these useful? Are there examples of applications? If A is a generic operator, how do we compute A(1/2+i) ? For the creation and annihilation operators in Quantum Mechanics and Field Theories, how do we calculate and interpret a(1/2+i) |ni and the commutator [a(1/2+i) , (a† )(1/2+i) ] ? What about similar generalisation to other operators, e.g. Supersymmetric operators, Vertex operators, Virasoro algebra in String Theories, and Superconformal algebra in Superstring? What are their surprising implications and consequences?
The aim of this paper is to address these issues and questions. 463 Reviews in Mathematical Physics, Vol. 11, No. 4 (1999) 463–501 c World Scientific Publishing Company
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S. C. WOON
1.2. The idea We know that in Complex Analysis [1], functions can be analytically continued from integer points n on the real line to complex plane s, e.g. from factorial n! to Gamma function Γ(s), and from within the area of convergence to beyond via functional equations, e.g. the functional equation of the Riemann zeta function πs Γ(1 − s)ζ(1 − s) (∀ s) . (1) ζ(s) = 2s π s−1 sin 2 We also know that in Euclidean Quantum Gravity [2], spacetime can be analytically continued from Lorentzian
Complex →
spacetime
Euclidean →
spacetime
spacetime
by rotating the signature of spacetime metric from (−1, +1, +1, +1) → (e−2i θ , +1, +1, +1) → (+1, +1, +1, +1) .
(2)
Path integrals ill-defined in Lorentzian spacetime become well-behaved in Euclidean spacetime in which they can be evaluated by methods of infinite-descent and saddlepoint approximation. This gives us a tool to explore non-perturbative, non-linear and topological structures like instantons and wormholes, the Thermodynamics of Hawking radiation [3], the end state of black hole evaporation [4], and Conjectures on the boundary conditions of Complex spacetime [5]. Now, we take analytic continuation a step further. Can operators be analytically continued? Operators and their representations permeate almost every branch of Mathematics and field of Sciences. If the analytic continuation of operators can be consistently defined and computed, then the idea may have broad implications and universal applications. We begin the first step with the Rtwo most familiar operators of all, the differential d and integral operator dx. operator dx 2. Analytic Continuation of Differential and Integral Operators 2.1. Differentiating and integrating in non-integer s-dimensions Analytic continuation of differentiation and integration to non-integer dimensions is straightforward. Differential of an integer n-dimensional function in n-dimensions is ∂ ∂ ∂ ··· f (x1 , x2 , . . . , xn ) ∂x1 ∂x2 ∂xn and the corresponding integral is Z Z xn Z f (x1 , x2 , . . . , xn ) dn x = ··· | {z
x2Z x1
n-times
}
f (x1 , x2 , . . . , xn ) dx1 dx2 · · · dxn .
(3)
(4)
...
465
∂ ∂ n−1 Γ(n/2) ∂ n−1 ∂ ∂ ∂ ··· = n−1 = ∂x1 ∂x2 ∂xn ∂r ∂Ωn−1 2π n/2 ∂rn−1
(5)
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
If f is spherically symmetric, f = f (r), then
can then be analytically continued to differential of a non-integer s-dimensional function in s-dimensions as ∂ s−1 ∂ Γ(s/2) ∂ s−1 ∂rs−1 ∂Ωs−1 2π s/2 ∂rs−1
(6)
and the corresponding integral Z
Z dn x =
∞
Z
0
Z =
Z
2π
rn−1 dr 0
∞
sin θ2 dθ2 · · · 0
Z rn−1 dr
dΩn−1 =
0
as
Z
π
dθ1
Z ds x =
π
sinn−2 θn−1 dθn−1 0
2π n/2 Γ(n/2)
2π s/2 Γ(s/2)
Z
∞
Z
∞
rn−1 dr
(7)
0
rs−1 dr ,
(8)
0
where s can be real or complex. An important physical application example of such analytic continuation is ’t Hooft and Veltman’s Dimensional Regularization where it is used to isolate singularities in divergent integrals in Quantum Field Theory [6]. Another example is in Connes’ noncommutative geometry where the integral in non-integer dimension can be evaluated as a Dixmier trace with Wodzicki residue [19]. However, this is not the only possibility. There is another possible analytic continuation. 2.2. Differentiating and integrating complex s-times in one-dimension Think of the differential and integral as operators. Differentiating or integrating n-times a one-variable function f (x) can be thought of as letting the operator act n-times or n-fold on the function, n dn d : f (x) 7→ n f (x) (9) dx dx Z
x
n Z (dˆ x) : f (ˆ x) 7→ Z xZ
xnZ xn−1
= |
{z
x
f (ˆ x)(dˆ x)n Z
···
x3Z x2
}
f (x1 ) dx1 dx2 · · · dxn−1 dxn .
(10)
n-times
Note that the limits of integration of this analytic continuation are different from those of integrating in n-dimensions.
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S. C. WOON
• Integration in n-dimensions is a product of integrals whereas • Integrating n-times gives a set of n nested integrals with the limits taken at the end of the integration. At this point, it is natural to generalise and combine both the differential and integral operators into one fundamental operator ds (Re(s) > 0) dxs 1 (Re(s) = 0) . (11) Dxs = Z (dx)−s (Re(s) < 0) The analytic continuation of the differential and integral operators to the Ds operator is known as Fractional Calculus [7], a subject of active research and of current interest because of its widespread pure and applied applications in Maths, Physics, Engineering and other Sciences [8]. In this paper, an extension of the conventional Fractional Calculus is introduced. Evaluating Dxs xw is simple and obvious for Re(w) ≥ 0. On the other hand, for the case of Re(w) < 0, it is not so straightforward but will turn out to be simple when mapped to the (s, w) diagrams to be introduced. The surprise is that the consequences of analytic continuation of these operators are not only highly non-trivial but useful. In particular, a number of known and new results in Number Theory are derived in non-standard ways using the idea of analytic continuation of operators in Secs. 4 and 9. These results demonstrate the usefulness and justify the purpose of the idea. In addition, when Ds acts on a standard power series, the result is a non-integer power series. Analysis in Sec. 5 shows that there are interesting relations between non-integer power series and the usual integer power series in limiting forms. In fact, Fractional Calculus can be reinterpreted as differential and integral operators acting non-integer times. Ds is observed to break Leibniz rule and Chain rule when s is non-integer, and thus we are unable to evaluate directly the action of Ds on a function of functions. However, by a trick of series expansion, we can express Ds R as a nested sum of dn /dxn or (dx)n which we can evaluate directly with Leibniz rule and Chain rule. Existing concepts in Group Theory are then extended in Sec. 6 using these results. An extension of Dirac Algebra from the Dirac equation Ris found in Sec. 7. In Finite Difference, the matrix representations of dn /dxn and (dx)n are sparse. When the operator Ds is casted in matrix representation as in Sec. 8, the matrix becomes dense for non-integer s, and so the local finite difference becomes non-local. The cause of this non-local effect can be traced to the appearance of non-integer power series. Its connection with noncommutative geometry is discussed. Towards the end of the paper, problems were raised and challenges were posed on analytic continuation of operators and algebras in Quantum Mechanics, Supersymmetry, and Quantum Field and String Theories.
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
...
467
All in all, the analytic continuation of operators turns out to be quite a general and powerful tool to explore Number Theory, Group Theory, Algebra, Finite Difference and Matrix Representation. Exploring with the idea has motivated the introduction of a few other new ideas and concepts into these fields, each is of very different nature from the other. The intriguing results suggest that the idea of analytic continuation of operators may well find interesting widespread applications in various other fields. 3. Defining the Action of Operator D s The operator Dxn for integer n is well defined since it corresponds to R −n for n ≤ −1. n ≥ 1 and (dx) 0 (0 < m ≤ n) dn m x = m! dxn xm−n (m > n > 0) (m − n)! Z
dn dxn
for
m! xm−n (m > 0, n < 0) (m − n)!
xm (dx)−n =
where m, n ∈ Z. If we tabulate Dxn xm for integers n, m, we observe a pattern emerges as in Table 1. 3.1. Fractional calculus and its extension
R There are several possible ways for analytically continuing d/dx and dx. The first three approaches are well developed, while the last two are new and introduced here. 1. Riemann–Liouville Fractional Calculus In the conventional Riemann–Liouville Fractional Calculus [7], we start with the Riemann–Liouville Integral Z x3Z x2 Z xZ xnZ xn−1 ··· f (x1 ) dx1 dx2 · · · dxn−1 dxn a a | a a a {z } n-times
=
1 Γ(n)
Z
a
x
f (t) dt (x − t)1−n
(12)
as the fundamental defining expression. Fractional integral is analytically continued from this Riemann–Liouville integral (12) as Z x dσ σ f (x) = f (x)(dx)−σ Dx−a f (x) = d(x − a)σ a Z x 1 f (t) = dt (σ < 0, σ, a ∈ R) . (13) Γ(−σ) a (x − t)1+σ
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S. C. WOON
Table 1. Tabulated results of Dxn xm . Dxn xm −1
−3
−2
0
1
2
3
2
2!/5! x5
2!/4! x4
2!/3! x3
x2
2! x
2!
0
1 0
1/4! x4 1/3! x3
1/3! x3 1/2! x2
1/2! x2 x
x 1
1 0
0 0
0 0
m\n
Z
Z
log x(dx)
log x
x−1
−x−2
2! x−3
−3! x−4
−2
log x(dx)2 Z − log x(dx)
− log x
−x−1
x−2
−2! x−3
3! x−4
−4! x−5
−3
1/2! log x
1/2! x−1
−1/2! x−2
x−3
−3!/2! x−4
4!/2! x−5
−5!/2! x−6
−1
Fractional derivative is in turn derived from the fractional integral (13) by ordinary differentiation as (choose m > σ) Dxσ f (x) = Dxm (Dx−(m−σ) f (x))
(σ > 0, m ∈ Z+ )
and it has the property σ Dxσ f (βx) = β σ D(βx) f (βx) .
However, for Dxσ xr , the Riemann–Liouville Integral definition is well defined only for the half plane σ ∈ R, r > −1. 2. Fractional Calculus by Cauchy Integral Cauchy Integral for an analytic function f (z) in the complex plane is Z Γ(1 + n) f (z) dz . f (n) (z0 ) = 1+n 2πi C (z − z0 )
(14)
Generalization of n to non-integer values is however not trivial as the term (z −z0 )1+α may become multi-valued and the result will depend on the choice of branch cut and integration path. 3. Fractional Calculus by Fourier Transform In Fourier Transform, Z +∞ Z +∞ 1 f (x) eikx dx , f (x) = f˜(x) e−ikx dk . f˜(x) = 2π −∞ −∞ Z Dxσ f (x)
+∞
= −∞
Z
+∞
=
σ −ikx ˜ ) dk f(k)D x (e
(15)
(σ ∈ R)
(−ik)σ f˜(k) e−ikx dk
−∞
Z ⇒
Dxσ
≡
+∞
−∞
(−ik)σ e−ikx dk .
(16)
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
...
469
This approach is often known as the pseudo-differential operator approach, and was shown by Z´ avada [10] to be equivalent to the Riemann–Liouville Fractional Calculus and the Fractional Calculus by Cauchy Integral. Now, to compute numerically the fractional derivative or fractional integral of a function f (x) or multi-variable function f (x(1) , x(2) , . . . , x(n) ), we choose to use the Discrete Fast Fourier Transform (DFFT) instead. Take the function values in the interval or region of interest, identify the boundaries so as to make it periodic, feed this into the DFFT algorithm to Fourier transform the function into k-space, multiply the component corresponding to k with (−ik)σ , and feed the result into the Inverse DFFT to Inverse Fourier transform it back to x-space. 4. Extended Fractional Calculus Now, we introduce the Extended Fractional Calculus in which the limit of the ratio of Gamma functions lim
→0
Γ(1 + r + ) Γ(1 + r + − σ)
(17)
is taken as the fundamental defining expression instead. Unlike the Riemann–Liouville Integral, the limit is well defined for the entire plane σ, r ∈ R except along the line intervals r ∈ Z− , σ ∈ R\Z. The analytic continuation to these line intervals will be derived in Subsubsec. 3.2.2, Eq. (26). However, in the wedge-shaped region r < σ, r ≥ 0, there are actually two possible choices, of which one (Type I ) corresponds to the Riemann–Liouville Fractional Calculus in that region where fractional derivatives do not generally commute, and the other (Type II ) an extension of it where commutativity of the fractional derivatives is preserved. This will be described in Subsubsec. 3.2.3. 5. Fractional Calculus by Nested Series Expansion Here, we introduce the method of analytic continuation of operators by nested series expansion with the following observation: Ds can be formally expanded into a nested series as Ds = (w11 − [w11 − D± ])±s ±s 1 ± ±s =w 11 − 11 − D w =w
±s
"n−1 # n ! ∞ X 1 ± (−1)n Y 11 + (±s − k) 11 − D n! w n=1 k=0
=w
±s
"n−1 #! #" m ∞ n X X (−1)n Y −1 n ±m D 11 + (±s − k) 11 + m n! w n=1 m=1 k=0
(18)
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S. C. WOON
where s, w ∈ C, and 11 is the identity operator. In the nested series on r.h.s, all the operators D’s are raised to integer powers ±m, and so Dm corresponds to ordinary m-fold differentiation while D−m corresponds to ordinary m-fold integration. The region of convergence in s and the rate of convergence of the series will be dependent on parameter w, and the function on which D acts. In this way, Fractional Calculus with its analytically continued operator Ds can be reinterpreted as operator D acting non-integer or complex s times on a function. Examples of applications can be found in Subsec. 5.2 and Sec. 10, and in [9]. 3.2. Analytic continuation from integer n to real σ To analytically continue to Dxσ xr for real σ, r, we introduce the (σ, r) diagram in which the coefficient of Dxσ xr is mapped to the point at coordinate (σ, r) of the diagram. The (σ, r) diagram of Dxσ xr can then be characterized into 4 regions as in Fig. 1: r ≥ σ, r ≥ 0 upper r < σ , r < 0 lower . (19) region bounded by r ≥ σ , r < 0 log r < σ, r ≥ 0 zero ( A point lying on the
right
(σ > 0) is a differentiation
left
(σ < 0) is an integration
r
upper
4 3
region
2
zero region
1 -4
-3
-2
log region
-1
1
2
3
-1 -2 -3
lower
region
-4
Fig. 1. (σ, r) diagram of Dxσ xr .
4
σ
.
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
...
471
3.2.1. Upper region The ratio
Γ(1 + r) Γ(1 + r − σ) is finite everywhere in r ≥ σ, r ≥ 0. Thus, Γ(1 + r) xr−σ Dxσ xr = Γ(1 + r − σ) in the upper region.
(20)
3.2.2. Lower and log regions Define Ωr<0 as the union of lower and log regions such that r < 0, σ ∈ R , Ωhorz as the set of the horizontal lines r = −n+ , n+ ∈ Z+ , Ωgrid as the set of integer grid points in Ωr<0 such that σ ∈ Z, r ∈ Z− , Ωdiag as the set of right-sloping diagonals in the lower region r = σ − n+ < 0 ,
n+ ∈ Z+ , σ, r ∈ R .
Figure 2 shows the regions of these sets on the (σ, r) plane. The ratio (20) is finite everywhere in Ωr<0 except in Ωhorz \Ωgrid (along the horizontal lines “mod” the grid points). It is zero in Ωdiag \Ωgrid (along the diagonals in the lower region “mod” the grid points). However, the limit Γ(1 + r + ) (21) lim →0 Γ(1 + r + − σ) evaluated at the grid points in Ωgrid is convergent.
r -4
-3
-2
-1
1
2
3
4
σ
-1
Ωhorz
-2 -3 -4
Ωdiag
grid points in log region grid points in lower region
Fig. 2. Lower and log regions.
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S. C. WOON
Table 2. Some tabulated values of W c(σ) and W s(σ). σ
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
···
W c(σ)
0
1
1
1
1
2
2
2
2
3
···
W s(σ)
0
0
0
1
1
1
1
2
2
2
···
Thus, Dxσ xr = lim
→0
Γ(1 + r + ) xr−σ Γ(1 + r + − σ)
in Ωr<0 \(Ωhorz \Ωgrid ) .
(22)
From Table 2, the natural analytic continuation of Dxσ xr in the part of Ωhorz \ Ωgrid lying in the log region is Γ(1 + r + ) lim →0 Γ()
Z log x (dx)(r−σ) ≡ lim
→0
Γ(1 + r + ) (σ−r) Dx log x . Γ()
It can be shown using (22) and the following expression: Z x 1 x ˆ−1+ dˆ x = lim (x − 1) log x = log x + log 1 = lim →0 1 →0 that
Z
log x (dx)ρ ≡ Dx−ρ log x =
xρ (log x − (ψ(1 + ρ) + γ)) Γ(1 + ρ)
(23)
(24)
(25)
where the digamma function [11] ψ(z) =
Γ0 (z) d log(Γ(z)) = dz Γ(z)
the Euler constant γ = − Γ0 (1) = 0.577215 . . ., and the prime 0 denotes differentiation. This will be proved in Sec. 4, Eq. (35). With (25), expression (23) can then be analytically continued from the part of Ωhorz \Ωgrid lying in the log region into that lying in the lower region. Thus, in Ωhorz \Ωgrid , we have r ∈ Z− , (σ − r) 6∈ Z, and the analytic continuation as Dxσ xr = lim
→0
= lim
→0
Γ(1 + r + ) (σ−r) Dx log x Γ() Γ(1 + r + ) xr−σ (log x − (ψ(1 + r − σ) + γ)) . Γ() Γ(1 + r − σ)
(26)
3.2.3. Zero region As pointed out above, there are two possible choices in this region from the two views or schools of thoughts:
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
...
473
• Type I Fractional Calculus Postulate: Fractional derivative is abstract. Fractional derivative of a constant can be non-zero. So take the Riemann–Liouville Integral as fundamental and derive the fractional derivative in this zero region from it. Then, Dxσ xr =
Γ(1 + r) xr−σ Γ(1 + r − σ)
in the zero region .
(27)
• Type II Fractional Calculus Postulate: Ordinary derivative of a constant is zero and ordinary derivatives commute. Fractional derivative should inherit these property from ordinary derivative as well — fractional derivative of a constant is zero and fractional derivates commute. Dx1 c = 0 ,
c is an arbitrary constant ,
Dxσ c = Dx(σ−1) (Dx1 c) = Dx(σ−1) 0 = 0 for σ > 1 .
(28)
As Dxr xr = Γ(1 + r) for r ≥ 0, by continuity, Dxσ xr = 0 for σ > r, r ≥ 0 ⇒ Dxσ xr = 0 in the zero region and commutativity is preserved. Hence, in both types, there is a mutual trade-off. In Type I, we chose analyticity and lose commutativity in the zero region. In Type II, we chose to preserve and carry over commutativity and lose analyticity at the edges of the zero region. 3.2.4. Entire real (σ, r) plane The combined analytic continuation is then • Type I Γ(1 + r + ) xr−σ lim (log x − (ψ(1 + r − σ) + γ)) →0 Γ() Γ(1 + r − σ) in Ωhorz \Ωgrid . (29) Dxσ xr = Γ(1 + r + ) lim→0 xr−σ elsewhere Γ(1 + r + − σ)
474
S. C. WOON
• Type II (fractional derivatives commute) 0 in the zero region Γ(1 + r + ) xr−σ lim→0 (log x − (ψ(1 + r − σ) + γ)) Γ() Γ(1 + r − σ) . (30) Dxσ xr = in Ωhorz \Ωgrid lim→0 Γ(1 + r + ) xr−σ elsewhere Γ(1 + r + − σ) 3.3. Analytic continuation from real σ to complex s To analytically continue Dσ to Ds where complex s = σ + it, we just generalise σ to s in the combined expressions (29) and (30) for Dxs xw with complex s, w. For Type II Fractional Calculus, commutativity is preserved. The differential R operator d/dx commutes with itself and with its inverse, the integral operator dx, and so does Ds . So in Type II, the commutative operator Ds splits into two, each acting on functions independently. Ds = D(σ+it) = Dσ Dit = Dit Dσ
(31)
as illustrated in the commutative diagram of Fig. 3, with all the limits, if any, taken only at the end after the actions of all the operators have been performed. Consider Dxs xw with complex s = σ + it, w = u + iv. In the zero region of (σ, u) plane, Dxσ xu = 0 ⇒ Dxit Dxσ xu = Dxs xu = 0 . Similarly, in the zero region of (it, iv) plane, Dxit xiv = 0 ⇒ Dxσ Dxit xiv = Dxs xiv = 0 . We can think of the above as Dit expanding the triangular zero region of the (σ, u) plane into a wedge-shaped volume of infinite length along the t-direction, and t
t
Dσ it
s=σ+it
s=σ+it
Ds
it
D
D
σ
r
Dσ
σ
r
Fig. 3. Commutative arrows in (s, r) diagram of Dxs xr in Type II.
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
t
...
475
wedges of zero space t
v= t
σ
u σ
u
v
= σ
Fig. 4. The zero space wedges in (s, w) diagrams where Dxs xw in Type II.
Dσ similarly expanding that in (it, iv) plane into another along the σ-direction as shown in Fig. 4. This defines the zero space in which Dxs xw = 0. For the line intervals in Ωhorz \Ωgrid , Dit extends these intervals in the (σ, u) plane along the t-direction to planar sections, and similarly for Dσ . By now, we have completed the task of defining the action and obtained a complete picture of the operator Ds . In essence, what we have done is analytically continuing an operator to act on a function complex s-times. Let us carry on with applications of the analytically continued Ds to find out more about its properties and consequences. Generating Integrals and 4. Application: Number Theory Analytic Continuation of Finite Series “Nature laughs at the difficulties of integration.” — Laplace 4.1. Generating integral of finite harmonic series It is well known [11] that finite harmonic series h(n) =
n X 1 k
k=1
= 1+
1 1 + ··· + 2 n
= ψ(1 + n) + γ = log n + γ + O(1/n) .
(32)
Now, we observe that finite harmonic series h(n) appears as the coefficient of xn term when we repeatedly integrate log x.
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S. C. WOON
Z
x
log x ˆ(dˆ x) = x(log x − 1) 0
Z
x
x2 log xˆ(dˆ x) = 2 2
0
Z
3 log x − 2
.. . x
xn (log x − h(n)) . n!
log xˆ(dˆ x)n = 0
So by analogy to the concept of generating functions, we take Z
x
log x (dx)n as the generating integral of finite harmonic series h(n) 0
and so the natural analytic continuation of the generating integral takes the form of Z x xρ (log x − h(ρ)) . (33) log x ˆ(dˆ x)ρ = Γ(1 + ρ) 0 From (24), 1 (x − 1) .
log x = lim
→0
The analytic continuation of the integral on l.h.s. of (33) can then be evaluated as Z
x
1 log x ˆ(dˆ x) = lim →0
Z
x
(ˆ x − 1)(dˆ x)ρ
ρ
0
1 = lim →0
0
Z
Z
x
x ˆ (dˆ x) −
0
ρ
x ρ
1(dˆ x) 0
x i 1 h −ρ x Dxˆ x ˆ 0 − Dx−ρ 1 0 ˆ →0 1 xρ Γ(1 + )x − . = lim →0 Γ(1 + + ρ) Γ(1 + ρ) = lim
(34)
Equating this to the expression on r.h.s. of (33) gives the analytic continuation of finite harmonic series 1 Γ(1 + )Γ(1 + ρ) x −1 h(ρ) = log x − lim →0 Γ(1 + + ρ) 1 Γ(1+)Γ(1+ρ) Γ(1+)Γ(1+ρ) Γ(1+)Γ(1+ρ) x − + −1 = log x− lim →0 Γ(1+ + ρ) Γ(1++ρ) Γ(1+ + ρ) Γ(1 + )Γ(1 + ρ) 1 1 Γ(1 + )Γ(1 + ρ) + lim 1− = log x − lim (x − 1) →0 →0 Γ(1 + + ρ) Γ(1 + + ρ)
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
...
477
Γ(1 + )Γ(1 + ρ) 1 1 1− = log x − lim (x − 1) + lim →0 →0 Γ(1 + + ρ) Γ(1 + )Γ(1 + ρ) 1 1− = lim →0 Γ(1 + + ρ) 0 Γ (1 + + ρ)Γ(1 + ) − Γ0 (1 + )Γ(1 + + ρ) = lim Γ(1 + ρ) →0 Γ(1 + + ρ)2
=
Γ0 (1 + ρ) − Γ0 (1) Γ(1 + ρ)
= ψ(1 + ρ) + γ ,
(35)
where the limit → 0 has been taken with L’Hospital rule. The curve h(ρ) = Pn ψ(1 + ρ) + γ passes through points h(n) = k=1 k1 as illustrated in Fig. 5.
3 2.5 2 h(ρ) 1.5
1 0.5 0
0
2
4 ρ
6
8
10
Fig. 5. The curve h(ρ) = ψ(1 + ρ) + γ passes through points h(n) =
Pn
1 k=1 k .
Equation (35) is a result that has been proved in conventional ways in Number Theory. It is somewhat surprising and miraculous that it is also possible to rederive it with the method of the analytically continued operator Ds as above. 4.2. Series of the Riemann zeta function up to finite terms The analytic continuation (35) can be generalised. The Riemann zeta function [12] is defined as ζ(s) =
∞ X 1 ks
(Re(s) > 1) .
(36)
k=1
The series of the Riemann zeta function up to finite terms is defined as the Stegan–Riemann zeta function [11]: ζ(s, n) =
n ∞ X X 1 = ζ(s) − ks k=1
=
k=n+1
sˆ=s dn log(Γ(1 + sˆ)) . n dx sˆ=0
1 ks
(Re(s) > 1)
(37)
(38)
478
S. C. WOON
Its analytic continuation is then given by simply replacing Dw , w ∈ C, sˆ=s log(Γ(1 + s ˆ )) ζ(s, w) = Dsw ˆ
dn dxn
with the operator
(39)
s ˆ=0
which can be evaluated when log(Γ(1 + sˆ)) is expressed in the form of asymptotic series [11]. 4.3. Tables of analytically continued integrals An interesting consequence is that perhaps new editions of Tables of Integrals may have to be compiled, e.g. compute the coefficient of xk term, w(ρ, r, a, k), in the evaluation of Z xr (log x)a (dx)ρ .
Non-integer Power Series, Breaking of 5. Application: Calculus Leibniz Rule and Chain Rule 5.1. Non-integer power series In Type I Fractional Calculus [7], we would write the power series for exp(x) as exp(x) = lim
→0
∞ X n=−∞
X 1 1 xn+ = xn . Γ(1 + + n) Γ(1 + n)
(40)
n∈Z
In Type II, we would write it simply as ∞ X
exp(x) =
n=0
1 xn Γ(1 + n)
(41)
since Dxσ xr = 0 for σ > r, r ≥ 0. Thus, in Type II, operator D acting on a power series real σ-times Dxσ
∞ X k=0
ak xk =
∞ X k=dσe
ak
Γ(1 + k) xk−σ Γ(1 + k − σ)
(σ > 0) ,
(42)
where d e denotes taking the integer ceiling. When σ is not an integer, r.h.s. of (42) is a non-integer power series. Define the notation f (σ, x) ≡ Dxσ f (x) .
(43)
Think of σ in the following way: The one-variable function f (x) is extended to a two-variable function f (σ, x) in which σ has now become a variable of the extended function.
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
...
479
∞ X (−1)k 2k x (2k)!
cos(σ, x) = Dxσ cos(x) = Dxσ
k=0
=
(44)
∞ X
k
k=W c(σ)
(−1) Γ(1 + 2k) x2k−σ (2k)! Γ(1 + 2k − σ)
where W c(σ) =
sin(σ, x) = Dxσ sin(x) = Dxσ
lσ 2
∞ X
m +1 −1.
(45)
(−1)k 2k+1 x (2k + 1)!
k=0
=
(σ > 0)
(46)
∞ X k=W s(σ)
(−1)k Γ(2(k + 1)) x2k+1−σ (2k + 1)! Γ(2(k + 1) − σ)
where W s(σ) =
exp(σ, x) = Dxσ exp(x) = Dxσ
(σ > 0)
1 σ + −1. 2 2
(47)
∞ X (−1)k k x k! k=0
=
∞ X k=dσe
(−1)k Γ(1 + k) xk−σ k! Γ(1 + k − σ)
(σ > 0) .
Table 2 shows some tabulated values of W c(σ) and W s(σ). We then find that ( ( 1 , if σ ∈ 2Z 1 , if σ ∈ 2Z + 1 cos(σ, 0) = , sin(σ, 0) = , 0 , otherwise 0 , otherwise ( 1 , if σ ∈ Z exp(σ, 0) = , 0 , otherwise
(48)
(49)
cos(σ, x) = ± sin(σ ± 1, x) = − cos(σ ± 2, x) , sin(σ, x) = ∓ cos(σ ± 1, x) = − sin(σ ± 2, x) , exp(σ, x) = exp(σ ± 1, x) , in agreement with the definitions of cos(x), sin(x) and exp(x) when σ ∈ Z. In addition, it can be observed from Figs. 6 and 7 that there exist asymptotic limits π cos(σ, x) ∼ cos x + σ 2 π (50) sin(σ, x) ∼ sin x + σ ∀ σ as x → ∞ 2 exp(σ, x) ∼ exp(x) which remain to be proved analytically.
480
S. C. WOON
In fact, any function with a power series definition, e.g. Bessel functions, Fourier series, etc., can similarly have a non-integer power series generalisation. In addition, the generalisation of cos(x) to cos(σ, x) can be further extended to the case of complex s, e.g. cos(s, x) ≡ Dxs cos(x). cos(0.01, x)
cos(0.25, x)
cos(0.5, x)
cos(0.75, x)
1
1
1
1
0.5
0.5
0
0.5
0.5
0
0
0
−0.5
−0.5
−1
−1
−1.5 −2
−0.5
−1.5 0
2π 4π 6π 8π
0
2π 4π 6π 8π
x
1
2π 4π 6π 8π
cos(1.5, x)
cos(2.01, x)
1
0.5
0.5
0
0
0
−0.5
−0.5
−0.5
−1
0
−1
2π 4π 6π 8π
x
2 1.5 1 0.5 0 −0.5 0
−1
2π 4π 6π 8π
x
0
x π σ) 2
as x → ∞.
exp(σ, x) − exp(x)
0
σ=0.01 σ=0.05
−0.2 −0.4
σ=0.25
−0.6 σ=0.5
−0.8 −1
σ=0.75 0
2π 4π 6π 8π
x
Fig. 6. Asymptotic limit: cos(σ, x) ≡ Dxσ cos(x) ∼ cos(x +
2π 4π 6π 8π
x
cos(1.99, x)
0.5
2π 4π 6π 8π
0
x
1
0
−1 0
x
cos(1.25, x)
−1
−0.5
−1
5
10
15
20
x Fig. 7. Asymptotic limit: exp(σ, x) − exp(x) → 0 as x → ∞.
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
...
481
5.2. Breaking of Leibniz rule and Chain rule By definition, Dn with integer n obeys Leibniz rule Dxn {f (x) g(x)}
=
n X n k=0
k
(Dxn−k f (x))(Dxk g(x))
(51)
and D1 obeys Chain rule Dx1 g(f (x)) = (Df1 g)(Dx1 f )
(52)
but Ds with complex s does not in general. However, by observing that Dx1 {f g} = (Dx1 f )Df1 {f g} + (Dx1 g)Dg1 {f g} = (Dx1 f )g + (Dx1 g)f ⇒ Dx1 = (Dx1 f )Df1 + (Dx1 g)Dg1 , we can express Ds where s ∈ C in terms of nested sums of D1 ’s which we can evaluate. Dxs = ((Dx1 f )Df1 + (Dx1 g)Dg1 )s = (1 − (1 − ((Dx1 f )Df1 + (Dx1 g)Dg1 )))s " k # ∞ X (−1)k Y = 1+ (s − m + 1) ((Dx1 f )Df1 + (Dx1 g)Dg1 )k . k! m=1
(53)
k=1
Now, we can evaluate expressions of the form of Dxs {f (x) g(x)} and Dxs g(f (x)) by simply by substituting Ds with the series on the r.h.s. of (53). See Sec. 9, Eq. (71) for the problem of convergence of the series. Analytic Continuation of Groups: 6. Application: Group Theory Mod Groups, Pseudo-Groups, and Symmetry Breaking/Deforming in Groups 6.1. R(mod n) groups The differential operator, and its inverse — integral operator, can act on different functional spaces to generate different discrete groups. These are groups of operators, i.e. groups with operators as elements. dn f (x) = f (x) dxn
482
S. C. WOON
cos( x)
d dx
d dx
sin( x)
sin( x)
d dx Fig. 8. Z4 group flow diagram of
d dx
cos( x) d dx
acting on functional space {± cos(x), ± sin(x)}.
cos( σ, x)
σ
Dx ’ cos( σ’+ σ, x )
4 σ Dx ’
Fig. 9. R(mod
0
4)
group flow diagram of Dxσ acting on functional space {cos(σ, x)|σ ∈ [0, 4)}.
Order
f (x)
n=1
exp(x) (
n=2 ( n=4
Symmetry group d0 d = 11 = dx dx0 2 d d , ≡ 11 Z2 = 11, dx dx2 Z1 = {11} ,
cosh(x) sinh(x)
± cos(x)
Z4 =
± sin(x)
11,
d d2 d3 , , dx dx2 dx3
,
d4 ≡ 11 dx4
Figure 8 shows the Z4 group flow. If elements of the functional space are extended from functions f (x) to their 0 analytic continuations f (σ, x) = Dσ f (x) with real σ, operators Dσ acting on these 0 extended functional spaces will generate continuous groups or Lie groups, e.g., Dxσ acting on the functional space {cos(σ, x)|σ ∈ [0, 4), x ∈ [0, ∞)} generates a natural analytic continuation of the Z4 group, 0
{Dxσ |σ 0 ∈ [0, 4)} ,
Dx4 ≡ 11
as illustrated in Fig. 9. By analogy to the concept of (mod n) congruence in Number Theory, we denote this analytically continued group R(mod 4) . 0
Dxσ f (σ, x) = f (σ, x)
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
, {Dxσ }
d } {dx
...
483
, , { Dxσ+it }
S1 X R Z4
S1
Z4
R (mod 4)
R (mod 4) X R
Fig. 10. Topology change of groups. Order
f (σ, x)
σ0 = 1
exp(σ, x) (
σ0 = 2 ( σ0
=4
Symmetry group
cosh(σ, x) sinh(σ, x)
± cos(σ, x) ± sin(σ, x)
0
Dx1 ≡ 11
0
Dx2 ≡ 11
R(mod
1)
= {Dxσ |σ0 ∈ [0, 1)} ,
R(mod
2)
= {Dxσ |σ0 ∈ [0, 2)} ,
R(mod
4)
= {Dxσ |σ0 ∈ [0, 4)} ,
0
Dx4 ≡ 11
0
For complex s = σ+it, s0 = σ 0 +it0 , Ds acting on functional space {cos(s, x)|σ ∈ 0 [0, 4), t ∈ R, x ∈ [0, ∞)} generates a Lie group R(mod 4) XR since Dit commutes with 0 0 0 Dσ and so Dit acts independently from Dσ . In general, the topology of such analytically continued groups progresses from sets of points on a circle S 1 for discrete groups generated by d/dx, to a circle S 1 for 0 Lie groups generated by Dxσ , and to a 2-dimensional cylinder S 1 XR for Lie group 0 0 generated by Dxσ +it as illustrated in Fig. 10.
6.2. Pseudo-groups in Type II fractional calculus Consider the analytic continuation of the group elements of SO(2) (the group of rotation in a plane) in Type II Fractional Calculus,
R(θ) =
cos θ
sin θ
− sin θ
cos θ
!
where θ ∈ [0, 2π)
7→
R(σ, θ) =
cos(σ, θ)
sin(σ, θ)
− sin(σ, θ)
cos(σ, θ)
! .
(54)
where θ ∈ [0, ∞)
R(σ, θ) forms a set of sets, parametrized at 2 levels. The set of sets is parametrized by σ, and each of these sets is further parametrized by θ. Denote the set of these sets as SO(2; σ, θ).
484
S. C. WOON
Since R(0, θ) ∈ SO(2) ∀ θ
π and R(σ, θ) ∼ R 0, θ + σ 2
as θ → ∞ ,
(55)
we are motivated to introduce the concept of pseudo-groups. A pseudo-group G(ρ1 , ρ2 , . . . , ρk ) of a group G is a set which gradually acquires the group properties or satisfies the group axioms of G as some of the parameters ρ1 , ρ2 , . . . , ρk of the set approach limiting values or tend asymptotically to infinity. SO(2; σ, θ) is a pseudo-group of SO(2) since it is isomorphic to SO(2) when • the parameter σ → n ∈ Z while θ varies freely in the interval [0, ∞), R(σ, θ)R(σ 0 , θ) ∼ R(σ + σ 0 , θ) as σ, σ 0 → n, n0 ∈ Z ⇒ lim
σ→n∈Z
SO(2; σ, θ) ∼ = SO(2) .
(56)
• (alternatively) the parameter θ → ∞ while σ varies freely in the interval (0, 2), π π R(σ, θ)R(σ, θ0 ) ∼ R 0, θ + σ R 0, θ0 + σ 2 2 = R(0, θ + θ0 + πσ)
as θ, θ0 → ∞
⇒ lim SO(2; σ, θ) ∼ = SO(2)
(57)
θ→∞
as shown in Fig. 11. We can define a group property deviation measure W(G(σ, x), G|σ, x) for a pseudo-group G(σ, x), a measure of how much group property the pseudo-group has lost or deviated from the associated “parent” group G from which it is analytically continued. When the pseudo-group becomes isomorphic to the parent group for certain values of the parameter, the measure should be zero.
2
SO(2;σ, 0) σ 1
SO(2) 0
0 Fig. 11. SO(2; σ, θ) plane diagram.
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
...
485
For the case of pseudo-group SO(2; σ, θ), W(SO(2; σ, θ), SO(2)|σ, θ) π = R(σ, θ) − R θ + σ 2 π cos(σ, θ) − cos θ + π σ sin(σ, θ) − sin θ + σ 2 2 = π π − sin(σ, θ) − sin θ + σ cos(σ, θ) − cos θ + σ 2 2 δ cos δ sin = −δ sin δ cos p = (δ cos +δ sin)2 + (−δ sin +δ cos)2 q = 2(δ cos2 +δ sin2 ) s π 2 π 2 = 2 cos(σ, θ) − cos(θ + σ) + sin(σ, θ) − sin(θ + σ) 2 2
(58)
satisfies the requirement. See Fig. 12 for the plot of this measure. Similarly for the simple case of U (1; σ, ix), a pseudo-group of U (1) where x ∈ R, exp(σ, ix1 ) exp(σ 0 , ix2 ) ∼ exp(σ, i(x1 + x2 )) as σ, σ 0 → n, n0 ∈ Z ⇒ lim
σ→n∈Z
W
U (1; σ, x) ∼ = U (1) ,
1
2
0.5
1.5
0
1 2π 0.5
4π
θ
6π 8 π0
Fig. 12. Measure W of SO(2; σ, θ).
σ
486
S. C. WOON
exp(σ, ix1 ) exp(σ, ix2 ) ∼ exp(σ, i(x1 + x2 )) as x1 , x2 → ∞ ⇒ lim U (1; σ, x) ∼ = U (1) , θ→∞
W(U (1; σ, ix), U (1) σ, ix) = exp(σ, ix) − exp(ix) . This measure was plotted in Fig. 7. 6.3. SO(2; σ, θ) rotations and deformations in Type II Figure 13 shows the effect of planar rotations and deformations of SO(2; σ, θ) on a square with vertices {(1, −1), (1, 1), (−1, 1), (−1, −1)} on a sequence of (x, y) planes clipped by square windows of size x ∈ [−2, 2], y ∈ [−2, 2]. The deformation effects seem to be a combination of rotations and contractions/dilations. 6.4. Symmetry breaking/deforming in groups in Type II In the Higgs mechanism of Spontaneous Symmetry Breaking [13], σ \θ
≈ 0 π/16 π/8 π/4 π/2 3π/4 π
3π/2 2π
4π
0 0.001 0.25 0.5 0.75 1 1.25 1.5 1.75 2 3 4
Fig. 13. SO(2; σ, θ) rotations and deformations.
6π
8π
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
...
487
• the symmetry of the effective potential Veff in a Lagrangian density L with respect to a gauge group G is preserved, while • the symmetry of the quantum state ψ satisfying the equations of motion derived from L is broken and reduced to that of a subgroup, H ⊂ G. The profile of Veff changes with energy or temperature. At high energy or temperature, the symmetry of ψ is restored from H → G. In the case here, the symmetry breaking is very different. The symmetry in a group G itself is broken to a subgroup H ⊂ G or “deformed” into an approximate symmetry of G. Take SO(3; (σ1 , x1 ), (σ2 , x2 )), a pseudo-group of SO(3), as an example. When both σ1 , σ2 = 0, SO(3; (σ1 , x1 ), (σ2 , x2 )) is isomorphic to SO(3). Now, choose σ1 6∈ Z and σ2 = 0. The SO(3) symmetry is then small broken approximate for intermediate x , large restored and the SO(3) symmetry in a sphere is “deformed” to an approximate SO(3) symmetry or completely broken to SO(2) in a plane depending on the chosen values of σ1 and x. Now, set both σ1 , σ2 6∈ Z. The SO(2) is further “deformed” to an approximate SO(2) symmetry or broken to Identity. The symmetry breaking/deforming sequence is then σ1 ∈ / Z, σ2 =0
σ1 , σ2 =0
σ1 , σ2 ∈ /Z
SO(3; (σ1 , x1 ), (σ2 , x2 )) →−−−→ SO(3) →−−−−−→ SO(2) →−−−−→ 11 . Similarly for SU (N ; (σ1 , x1 ), (σ2 , x2 ), . . . , (σN , xN )), the symmetry breaking/ deforming sequence is SU (N ; (σ1 , x1 ), (σ2 , x2 ), . . . , (σN , xN )) σ1 ,σ2 ,...,σN = 0
,→−−−−−−−−−−−−−−−−→
SU (N )
σ1 ∈ / Z, σ2 ,...,σN = 0
,→−−−−−−−−−−−−−−−−→ .. .. . .
SU (N − 1)
σ1 ,...,σN −2 ∈ / Z, σN −1 , σN = 0
,→−−−−−−−−−−−−−−−−→
SU (2)
σ1 ,...,σN −1 ∈ / Z, σN = 0
,→−−−−−−−−−−−−−−−−→
U (1)
σ1 ,σ2 ,...,σN ∈ /Z
,→−−−−−−−−−−−−−−−−→
11 .
Perhaps this mode of symmetry breaking/deforming in groups might have some useful applications for models in Particle Physics and Cosmology. In Particle Physics, the symmetry of the flavor of quarks are not exact symmetry but only approximate symmetry of Gell–Mann’s Eightfold way SU (3) [14] or
488
S. C. WOON
GUT SU (5) [15] because different flavors of quarks have different masses. Light quarks do not transform exactly into heavy quarks, and perhaps the effects from the presence of gluons and glueballs in composite particles need to be added into the symmetry. Perhaps SU (3) and SU (5) can be “deformed” in this way to an approximate symmetry that will fit the phenomenological data better. Now, take a pseudo-group to describe the product of residual exact and approximate symmetries of present day Universe. If we set x ∝ T , the temperature of the Universe, as we go back in time, the temperature T goes up, x goes up, and we find that the approximate and other broken symmetries are gradually being restored. The rate the symmetries are being restored will be dependent on the values of σ1 , σ2 , . . . , σN , the parameters of the pseudo-group. The fully restored symmetry will be the symmetry of the parent group of the pseudo-group. Qualitatively, this model resembles the unification of gauge groups in Cosmology [16]. It might be interesting to study and develop this mode of symmetry breaking/deforming for approximate symmetry groups (e.g. iso-spin group, Eightfold way SU (3), and GUT SU (5)) as well as for the gauge groups. Analytic Continuation of 7. Application: Algebra Dirac Equation and Algebra “I think that there is a moral to this story, namely that it is more important to have beauty in one’s equations than to have them fit experiment.” — P.A.M. Dirac in Scientific American, May (1963): Dirac equation ⇒ Klein–Gordon equation ∂ ψ = (−iα . ∇ + β m)ψ ∂t 2 ∂ ψ = (−iα . ∇ + β m)2 ψ ⇒ i ∂t X ∂2 αi αj ∇i ∇j − im(αβ + βα) . ∇ + β 2 m2 ψ ⇒ − 2 ψ = − ∂t i,j i
⇒−
∂2 ψ = (−∇2 + m2 )ψ ∂t2
where
X
≡
i1 ,i2 ,...,ip
XX i1
···
i2
X ip
giving Dirac Algebra [17] ⇒ {αi , αj } = 2 δij 11 ,
{αi , β} = 0 ,
α2i = β 2 = 11 .
(59)
∂ , −i ∇} as a basis. The basis can be analytically continued with Think of {i ∂t s the D operator. Dirac equation can then be analytically continued to
eπ i/p Dt ψ = (e−π i/p α(2/p) . D(2/p) + β (2/p) m2/p )ψ , 2/p
(60)
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
...
489
where p = 1, (60) ⇒ Klein–Gordon equation, (2)
= 11 ,
β (2) = 11 ,
(1)
= αi ,
β (1) = β .
αi p = 2, (60) ⇒ Dirac equation, αi
Introduce the notation for generalised symmetrisation X Aperm1 Aperm2 · · · Apermp , {Ai1 , Ai2 , . . . , Aip } =
(61)
perm(i1 ,i2 ,...,ip )
where the sum is over all permutations of the p indices. The generalised symmetrisation can be re-expressed in terms of a sum of permutations of nested anti-commutators, e.g. {a, b, c} =
1 ({{a, b}, c} + {{b, c}, a} + {{c, a}, b}) . 2(1!)
(62)
{{{a, b}, c}, d} + {{{b, c}, d}, a} + {{{c, d}, a}, b} + {{{d, a}, b}, c} 1 +{{{a, b}, d}, c} + {{{b, c}, a}, d} + {{{c, d}, b}, a} + {{{d, a}, c}, b} . {a, b, c, d} = 2(2!) +{{{a, c}, b}, d} + {{{a, c}, d}, b} + {{{b, d}, c}, a} + {{{b, d}, a}, c}
(63) Now, for p = 3, L.h.s.: 2/3
(eπ i/3 Dt )3 ψ = eπ i Dt2 ψ = −Dt2 ψ = −
∂2 ψ. ∂t2
R.h.s.: (e−π i/3 α(2/3) . D2/3 + β (2/3) m2/3 )3 ψ X (2/3) (2/3) (2/3) 2/3 2/3 2/3 − αi αj αk Di Dj Dk i,j,k (2/3) (2/3) αi αj β X (2/3) (2/3) 2/3 2/3 +e−2 π i/3 m2/3 D D + αi β αj i j i,j (2/3) (2/3) + β αi αj = ψ (2/3) (2/3) 2 α [β ] +e−π i/3 m4/3 + β (2/3) α(2/3) β (2/3) . D2/3 + [β (2/3) ]2 α(2/3) +[β (2/3) ]3 m2
490
S. C. WOON
−
1 X n (2/3) 2/3 (2/3) 2/3 (2/3) 2/3 o αi Di , αj Dj , αk Dk 3!
i,j,k n o X (2/3) 2/3 (2/3) 2/3 −2π i/3 2/3 1 αi m Di , αj Dj , β +e = ψ 3! i,j (2/3) −π i/3 4/3 (2/3) +e m {α , β, β} . D + [β (2/3) ]3 m2 = (−[α(2/3) ]3 . D2 + [β (2/3) ]3 m2 )ψ ≡ (−∇2 + m2 )ψ n (2/3) (2/3) (2/3) o = 3!δijk 11 , , αj , αk αi n o n o (2/3) (2/3) (2/3) αi αi , αj , β (2/3) = 0 , , β (2/3) , β (2/3) = 0 , ⇒ h i α(2/3) 3 = 11 , [β (2/3) ]3 = 11 . i
For general p, Dirac Algebra is analytically continued to o n (2/p) (2/p) (2/p) (2/p) (2/p) = p!δi1 i2 ···ip 11 , αi1 , αi2 , . . . , αip−2 , αip−1 , αip o n (2/p) (2/p) (2/p) (2/p) αi1 , αi2 , . . . , αip−2 , αip−1 , β (2/p) = 0 , o n (2/p) (2/p) (2/p) αi1 , αi2 , . . . , αip−2 , β (2/p) , β (2/p) = 0 , .. . (2/p) αi1 , β (2/p) , . . . , β (2/p) = 0 , | {z }
(64)
(p−1)-times
h
(2/p)
αi
ip = 11 ,
[β (2/p) ]p = 11 .
From here, we may proceed on to find representations of this analytically continued algebra and study the properties of the associated analytically continued spinors. Perhaps they have interesting properties. As a hint, even the matrix representation of the finite difference of Ds itself has surprising properties. To this we turn to the following: 8. Application: Matrix Representation Analytic Continuation of Matrices, and from Local Finite Difference to Non-Local Finite Difference “We [he and Halmos] share a philosophy about linear algebra: We think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury.” — Irving Kaplansky in Paul Halmos: Celebrating 50 Years of Mathematics.
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
...
491
8.1. Analytic continuation of matrices Given a square matrix [M ], we all know how to compute [M ]n , the matrix [M ] raised to an integer power n ∈ Z. It is just trivially multiplying the matrix [M ] by itself n-times. Now we wish to compute [M ]σ , the matrix [M ] raised to a real non-integer power σ ∈ R. A generic matrix may have degenerate eigenvalues and so cannot in general be diagonalized. However, we can obtain [M ]σ as follows: For rational σ = p/q, where p, q ∈ Z, [M ]p/q can be obtained by solving for each element of the matrix [A] in the matrix equation [A]q = [M ]p since formally ([M ]p/q )q = [M ]p/q [M ]p/q · · · [M ]p/q = [M ]p = [A]q | {z } q-times
and so [A] = [M ]p/q . For the case of [M ]u and [M ]s , the matrix [M ] raised to irrational u and complex s respectively, we turn to the series expansion method in Eq. (71) in the next section. 8.2. Local finite difference to non-local finite difference Now, in Finite Difference, if we choose the matrix representations of differentiation D1 to be 1 −1 1 −1 . 1 .. (65) [Dx ] = /(∆x) , 1 −1 1 then integration D−1 is the inverse of D1 , [Dx−1 ] = [Dx1 ]−1
=
1
1 1
1 1 .. .
1 1 1
··· ··· (∆x) , 1 1
m-times
[Dxm ]
=
[Dx1 ]m
z }| { = [Dx1 ][Dx1 ] · · · [Dx1 ] ,
[Dx−m ] = [Dx1 ]−m = [Dx1 ]−1 [Dx1 ]−1 · · · [Dx1 ]−1 , | {z } m-times
(66)
492
S. C. WOON
e.g., [Dx2 ] = [Dx1 ][Dx1 ] = [Dx1 ]2 =
[Dx1 ][f (x)]
=
[Dx−1 ][f (x)]
=
1 −2
1
1
−2 .. .
1 −1 1
−1 .. . 1
1 1
1
1
1 .. .
/(∆x)2 . 1 −2 1 1 −2 1 1
(67)
Dx1 f (xn ) f (xn ) 1 Dx f (xn−1 ) f (xn−1 ) .. .. /(∆x) = , . . 1 D f (x2 ) −1 x f (x2 ) 1 f (x1 ) f (x1 )/(∆x)
−1 Dx f (x)|xxn1 f (xn ) −1 Dx f (x)|xxn−1 1 ··· 1 f (xn−1 ) . .. .. (∆x) = , . −1 x D f (x)| 2 1 1 x1 f (x2 ) x 1 f (x1 ) f (xn )(∆x) 1 ···
where xk = x1 + (k − 1)∆x. [Dx1 ] have degenerate eigenvalues and thus cannot be diagonalized. However, following the above approach, we can compute the matrix representation of Dxσ , σ ∈ R. It can be verified that b(σ, 1) b(σ, 2) b(σ, 3) · · · b(σ, n) b(σ, 1) b(σ, 2) · · · b(σ, n − 1) /(∆x)σ (σ > 0) b(σ, 1) · · · b(σ, n − 2) . .. b(σ, 1) [Dxσ ] = −1 b(−σ, n) b(−σ, 1) b(−σ, 2) b(−σ, 3) · · · b(−σ, 1) b(−σ, 2) · · · b(−σ, n − 1) (∆x)−σ (σ < 0) b(−σ, 1) · · · b(−σ, n − 2) . . . b(−σ, 1) (68) satisfy (69) [Dxσ ] = [Dxσ1 ][Dxσ2 ] · · · [Dxσp ] for σ = σ1 + σ2 + · · · + σp ,
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
...
493
where b(σ, k) =
Γ(1 + σ + ) (−1)(k−1) lim Γ(k) →0 Γ(2 + σ + − k)
which is incidentally the kth term of the binomial expansion (1 + (−x))σ . The matrix representation [Dm ] is sparse while [Dσ ] is in general dense — all the elements in the upper tri-diagonal block become non-zero. In Finite Difference, sparse matrix entails taking the differences between only neighboring sets of points, while dense matrix entails taking the differences among points almost everywhere in the domain — a non-local effect. From (68), Dσ f (xm ) ≡ lim [Dσ ][f (x)] n→∞
= lim
n→∞
x=xm
m X
b(σ, k)f (xm−k+1 )(∆x)−σ ,
(70)
k=1
where 1 m ≤ n. If f (x) is an integer power series, Dσ f (xm ) on l.h.s. is in general a non-integer power series. The corresponding matrix reprensentation on r.h.s. is a sum of ordinary integer power series. The non-local effect can then be seen to arise from approximating the non-integer power series by a sum of ordinary integer power series. This is in parallel with the application of fractional derivative as a pseudodifferential operator in non-local field theory by Barci et al. [18]. 8.3. Evolution of quantum processes in noncommutative geometry There is an interesting connection between Connes’ noncommutative geometry [19] and the analytic continuation of matrices as above. A Markovian matrix is essentially a probability transfer matrix for a quantum physical state in a time step. A bi-graded Markovian matrice is the square root of a Markovian matrix. While solving the distance problem on a 1-dimensional lattice in noncommutative geometry, Atzmon [20] found that bi-graded Markovian matrices can be interpreted as representing non-local Dirac operators in one context and the evolution of a physical quantum state in another. In the latter, the bi-graded Markovian matrix raised to integer power n represents the physical quantum state after n time steps. A straightforward application of the analytic continuation of matrices is that when the matrix raised to integer power n is analytically continued to a matrix raised to arbitray real power σ, the resulting matrix represents the physical quantum state after arbitrary real σ time steps. Hence, the continuous flow of time can be recovered from the discretization of time without sending the separations of lattice points in time to zero. However, when the matrix is analytically continued to a matrix raised to a complex power s, one might extrapolate the interpretation of such a matrix as evolution of a quantum state in complex s time steps. It is not clear if this represents
494
S. C. WOON
a physical situation. Perhaps one may find within the context of noncommutative geometry what physical meaning can be attached to this “quantum evolution” in complex s time steps. 9. Analytic Continuation of Generic Operators Now, let’s go beyond the analytic continuation of differential and integral operators to the analytic continuation of generic operators. 9.1. Nested series expansion approach We are used to thinking of an operator acting once, twice, and so on. However, an operator can be analytically continued to the operator raised to a complex power by making the following observation: A generic operator A : V → V raised to a complex power can be formally expanded into a series as s 1 s s s A = (w11 − [w11 − A]) = w 11 − 11 − A w " # n ! ∞ n−1 X 1 (−1)n Y s = w 11 + (s − k) 11 − A n! w n=1 k=0
= ws
"n−1 #! #" m ∞ n X X (−1)n Y −1 n m A , 11 + (s − k) 11 + m n! w n=1 m=1
(71)
k=0
where s, w ∈ C, 11 is the identity operator. In the nested series on r.h.s, all the operators A’s are raised to integer powers which we can evaluate as usual. The region of convergence in s and the rate of convergence of the series will in general be dependent on operator A, parameter w, and the operand on which A acts. The resulting series then defines As , the analytic continuation of the operator A, in the region of s where it converges. It would be interesting to compare and contrast this new approach to the following well-developed approach: 9.2. Functional analytic approach In the functional analytic approach, Z sin aπ ∞ a−1 λ (λ11 − A)−1 A dλ (−A)a = − π 0
(0 < a < 1) ,
(72)
where evaluation of the integral wrt real variable λ requires various conditions on the spectrum of the operator A. For details of this well-developed functional analysis approach, see [21]. 10. Problems and Challenges “Mathematics is not yet ready for such problems,” — Paul Erd¨ os in The American Mathematical Monthly, Nov. (1992).
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
...
495
10.1. Analytic continuation of Bernoulli numbers and polynomials, a new representation for the Riemann zeta function, and the phenomenon of scattering of zeros Examples of interesting mathematical applications are analytic continuation of Bernoulli numbers and polynomials, the derivation of a new formula for the Riemann zeta function in terms of a nested series of Bernoulli numbers, and the observation of particle-physics-like scattering phenomenon in the zeros of the analytically continued polynomials as described in [9]. For instance, an operator was found in [22] to generate Bernoulli numbers. Applying the series expansion to the operator analytically continues the Bernoulli number to a function " n # ∞ 1 X (−1)n Y s + (s − k) B(s) = w Γ(1 + s) 2 n=1 n! k=1
#! m n Bm+1 1 X −1 n + × m (m + 1)! 2 m=1 w "
(73)
which was verified to converge, for Re(s) > (1/w), real w > 0, to B(s) = −sζ(1 − s)
(74)
as shown in Fig. 14.
1 0.75 0.5
B(s) 0.25 0 -0.25 -0.5 0
2
4
6
8
10
12
14
s Fig. 14. The curve B(s) runs through the points of all Bn except B1 .
Note that B(n) = Bn for n ≥ 2 but B(1) = 1/2 while B1 = −1/2. It was then realised that there is actually an arbitrariness in the sign convention of B1 . The analytic continuation of Bernoulli numbers fixes the arbitrary sign convention, and requires the generating function of Bernoulli numbers to be redefined for consistency
496
S. C. WOON
as
∞ X z Bn n = z (−1)n z e − 1 n=0 n!
or Bn =
(|z| < 2π, n ∈ Z+ )
n−1 (−1)n+1 X n+1 Bk , (−1)k k n+1
B0 = 1
(75)
(76)
k=0
which only changes the sign in the conventional definition [11] of the only non-zero odd Bernoulli number, B1 , from B1 = −1/2 to B1 = B(1) = 1/2. From (73) and (74), by the functional equation of the Riemann zeta function (1), (2πw)s 2 " n #! #" m ∞ n 1 X (−1)n Y Bm+1 1 X −1 n + + (ˆ s − k) 2 m (m + 1)! n! 2 w n=1 m=1 k=1 × lim s ˆ→s πˆ s cos 2
ζ(s) = −
(2πw)s 2 " n #! #" m ∞ n 1 X (−1)n Y 1 X −1 n ζ(−m) + − (ˆ s − k) 2 m n! 2 w m! n=1 m=1 k=1 . × lim s ˆ→s πˆ s cos 2
=−
(77) This is a new representation of the Riemann zeta function in terms of a nested sum of the Riemann zeta function itself evaluated at negative integers. The series converges for Re(s) > (1/w) real w > 0, and the limit only needs to be taken when s = 1, 3, 5, . . . ∈ Z+ odd , the set of positive odd integers, for which the denominator πs cos( 2 ) = 0. 10.2. Analytic continuation of quantum operators “The modern physicist is a quantum theorist on Monday, Wednesday, and Friday and a student of gravitational relativity theory on Tuesday, Thursday, and Saturday. On Sunday he is neither, but is praying to . . . find the reconciliation between the two views.” — Norbert Wiener Quantum Mechanics, Quantum Field Theories and Canonical Quantum Gravity are full of non-commutative operators.
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
...
497
As a start, we begin with the creation and annihilation operators of a simple harmonic oscillator (SHO) √ (78) a† |ni = n + 1 |n + 1i , a|ni =
√ n |n − 1i ,
a|0i = 0 ,
(79)
s a
†m
|ni =
Γ(1 + n + m) |n + mi , Γ(1 + n)
(80)
Γ(1 + n) |n − mi . Γ(1 + n − m)
(81)
s a |ni = m
A possible analytic continuation is s a†s |ri =
Γ(1 + r + s) |r + si , Γ(1 + r)
(82)
Γ(1 + r) |r − si , Γ(1 + r − s)
(83)
s as |ri =
where r, s ∈ R or C, and the Hilbert space of the operators of SHO HSHO = {|ni|n = 0, 1, 2, . . .} is extended to a larger space HR = {|ri|r ∈ R} or HC = {|ri|r ∈ C} . Similar generalization of Hilbert space applies to Fock space |k1 , k2 , . . . , kn i in Quantum Field Theories, and similar analytic continuation applies to spin ladder operators S± , and angular momentum ladder operators J± . For a SHO with position variable q, √ 1 ∂ (84) a† = λ/ 2 q − 2 λ ∂q √ a = λ/ 2
1 ∂ q+ 2 λ ∂q
Analytic continuation of these operators are thus s s √ s √ s 1 ∂ 1 ∂ a†s = λ/ 2 q− 2 1− 2 = λq/ 2 , λ ∂q λ q ∂q s √ s 1 ∂ a = λq/ 2 1+ 2 , λ q ∂q s
(85)
(86)
(87)
498
S. C. WOON
where l.h.s. contain terms of derivatives raised to fractional or complex power s which can be evaluated by various approaches in Fractional Calculus [7]. Now, we turn to the commutators of these analytically continued operators. From the nested series expansion (71), as(w) = (w11 − [w11 − a])s =w
∞ X (−1)p 11 + p! p=1
s
"p−1 Y
#" (s − k)
k=0
m #! p X −1 p am 11 + m w m=1
and similarly for a†s (w) .
(88)
0
A formal nested series expansion of the commutators of as(w) and a†s (w) is s a(w) , a†s (w 0 )
∞ X (−1)p 11 + p! p=1
s
"p−1 Y
m #! p X −1 p , am (s − k) 11 + m w m=1 #"
w k=0 = 0 m0 0 −1 p0 ∞ p0 pY X X 0 (−1) −1 p w0s 11 + 0 0 †m a (s − k ) 1 1 + 0 m p0 ! w0 0 0 0 p =1
m =1
k =0
"p−1 #p0 −1 0 ∞ X Y (−1)p+p Y (s − k) (s0 − k 0 ) 0! p!p p,p0 =1 k=0 k0 =0 0 s = ws (w0 ) . 0 0 0 p p m+m X X 0 (−1) p p m †m [a , a ] 0 0 m 0m m m w w m=1 0
(89)
m =1
From the canonical commutation relations [24], [a, a† ] = 1 ,
[a, a] = 0 = [a† , a† ] ,
(90)
0
the [am , a†m ] in the nested series can be evaluated as usual, and thus the nested series expansion (89) is formally computable. Similar generalization applies to fermionic operators satisfying Grassmann algebra, SUSY operators s 0 Q(w) , Q†s (91) {Q, Q† } = 2H/ω (w 0 ) ,
;
Virasoro generators [25] in String theories [Ln , Lm ] = (n − m)Ln+m +
c n(n2 − 1)δn,−m 12
; L
s s0 n(w) , Lm(w 0 )
,
(92)
Superconformal algebra [26] in Superstring, their respective vertex operators, and Lie algebra in general.
ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS:
...
499
These generalizations seem to have interesting mathematical structures. Further aspects and detailed computations will be presented and explored elsewhere. 11. Conclusion Analytically continued operators have been demonstrated to exhibit intriguing properties. In addition, fractional derivatives in the conventional Riemann–Liouville Fractional Calculus do not generally commute but an extension in which they commute has been found and applied to various fields. These methods of analytic continuation of operators may after all turn out to be a general and powerful exploration tools in Maths, Physics, Sciences, and Engineering. Calculus is never quite the same again. It would be interesting to imagine what Newton and Leibniz would say on this analytic continuation of their discoveries — Calculus, and Dirac of his equation – Dirac equation. Perhaps the most unexpected, and yet “inconsequential”, consequence is that Fig. 14 clearly points out that the commonly adopted definition of the 1st Bernoulli number B1 has the wrong sign. There was actually arbitrariness in its sign convention and the analytic continuation of the operator that generates Bernoulli numbers [9] fixes that arbitrariness, requiring that B1 = −1/2 to be redefined as B1 = 1/2 for consistency. However, the B1 = 1/2 definition has been so widely used — in every Math, Physics and Engineering book or paper where Bernoulli numbers appear, one almost certainly find B1 = −1/2. I can only hope that the readers will be persuaded in the light of this new mathematical fact to change and adopt the consistent definition B1 = 1/2 and the corresponding defining Eqs. (75) and (76). Acknowledgements Special thanks to V. Adamchik, E. Atzmon, D. Bailey, W. Ballman, J. Borwein, P. Borwein, P. D’Eath, U. Dudley, C. Isham, K. Odagiri, Y. L. Loh, B. Lui, A. C. McBride, H. Montgomery, A. Odlyzko, S. Shukla, I. N. Stewart, M. Trott, and B. Wandelt for discussion, all the friends in Cambridge for encouragement, and Trinity College UK Committee of Vice-Chancellors and Principals for financial support. References [1] L. V. Ahlfors, Complex Analysis, 2nd ed. McGraw-Hill, New York, 1966. [2] G. W. Gibbons and S. W. Hawking, eds., Euclidean Quantum Gravity, World Scientific, Singapore, 1993; I. G. Moss, Quantum Theory, Black Holes and Inflation, John Wiley & Sons, New York, 1996. [3] S. W. Hawking, Commun. Math. Phys. 43 (1975) 199. [4] P. D. D’Eath and S. C. Woon, “Black hole evaporation” in Proc. Second Int. A. D. Sakharov Conf. on Physics, eds. I. M. Dremin and A. M. Semikhatov, World Scientific, Singapore, 1997. [5] S. C. Woon, DAMTP 4th Term Report, 1996, http://www.damtp.cam.ac.uk/user/scw21/report/ [6] ‘t Hooft and Veltman, Nucl. Phys. 44B (1972) 189. [7] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974; B. Ross, ed., Fractional Calculus and its Applications, Springer Verlag, Berlin,
500
[8]
[9]
[10] [11]
[12]
[13] [14]
[15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
S. C. WOON
1975; A. C. McBride, Fractional Calculus and Integral Transforms of Generalized Functions, Pitman, London, 1979; A. C. McBride and G. F. Roach, ed, Fractional Calculus, Pitman, London, 1985; H. M. Srivastava and S. Owa, eds., Univalent Functions, Fractional Calculus, and their Applications, Ellis Horwood, Chichester, 1989; K. Nishimoto, Fractional Calculus, Descartes Press, Koriyama, 1991; S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, 1993; K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993; R. N. Kalia, ed., Recent Advances in Fractional Calculus, Global Publishing Co., Sauk Rapids, 1993; V. Kiryakova, Generalized Fractional Calculus and Applications, Longman, Harlow, 1994; K. S. Miller, “Derivatives of non-integer order”, Math. Mag. 68 (1995) 183; M. A. Al-Bassam, Additional Papers on Fractional Calculus and Fractional Equations, Cambridge, 1995; B. Rubin, Fractional Integrals and Potentials, Longman, Harlow, 1996. N. Engheta, J. Electrom. Waves Appl. 9 (1995) 1179; R. S. Rutman, Theor. Math. Fiz. 105 (1995) 191; K. V. Chukbar, JETP 81 (1995) 1025; J. L. Petersen, J. Rasmussen and M. Yu, “Free field realization of SL(2) correlators for admissible representations, and hamiltonian reduction for correlators”, e-print hep-th/9512175; K. M. Kolwankar and A. D. Gangar, “Fractional differentiability of nowhere differentiable functions and dimensions”, e-print chao-dyn/9609016. S. C. Woon, “Analytic continuation of Bernoulli numbers, a new formula for the Riemann zeta function, and the phenomenon of scattering of zeros”, preprint DAMTPR-97/19, e-print physics/9705021. P. Z´ avada, “Operator of fractional derivative in the complex plane”, e-print functan/9608002. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970; H. Bateman, Higher Transcendental Functions, Vol 1., McGraw-Hill, New York, 1953. E. C. Titchmarsh, The Theory of the Riemann zeta-function, Oxford, 1986; S. C. Woon, “Period-harmonic-tupling jumps to chaos in a class of series”, Chaos Solitons & Fractals 5 (1) (1995) 125. P. W. Higgs, Phys. Lett. 12 (1964) 132; Phys. Rev. 145 (1966) 1156. M. Gell-Mann, Phys. Rev. 125 (1962) 1067; Y. Ne’eman, Nucl. Phys. 26 (1961) 222; M. Gell-Mann and Y. Ne’eman, The Eightfold Way, Benjamin, New York, 1964; W. Greiner and B. M¨ uller, Quantum Mechanics: Symmetries, Springer-Verlag, Berlin, 1994. H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32 (1974) 438. G. B¨ orner, The Early Universe, 2nd ed., Springer-Verlag, Berlin, 1992. P. A. M. Dirac, Proc. Roy. Soc. Lon. A117 (1928) 610; A126 (1930) 360. D. G. Barci, C. G. Bollini, L. E. Oxman and M. C. Rocca, “Non-local pseudodifferential operators”, e-print hep-th/9606183 . A. Connes, Noncommutative Geometry, Academic Press, New York, 1994. E. Atzmon, “Bi-graded Markovian matrices as non-local dirac operators and a new quantum evolution”, e-print hep-th/9704200. T. Kato, J. Math. Soc. Japan 13 (1961) 246; T. Kato, J. Math. Soc. Japan 14 (1962) 242. S. C. Woon, “A tree for generating Bernoulli numbers”, Math. Mag. 70 (1) (1997) 51. J. J. Sakurai, Modern Quantum Mechanics, Revised ed., Addison-Wesley, New York, 1994. C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980; L. H. Ryder, Quantum Field Theory, Cambridge, 1985; M. Kaku, Quantum Field Theory, Oxford, 1993.
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[25] A. M. Virasoro, Phys. Rev. Lett. 22 (1969) 37. [26] M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, Vols I and II, Cambridge, 1987; M. Kaku, Strings, Conformal Fields, and Topology, Springer-Verlag, Berlin, 1991.
REVIEWS IN MATHEMATICAL PHYSICS Author Index (1989–1998) ERRATA This correction to the Author Index (1989–1998) which first appeared in Vol. 10 No. 8 is done to distinguish the papers in Vol. 4 from those in the special issue of Vol. 4, as some papers in both issues have the same page numbering. The volume number 4S refers to the special issue of Vol. 4, dedicated to R. Haag on the occasion of his 70th birthday and published in December 1992. Abadie, B. & Exel, R., Hilbert C*-bimodules over commutative C*-algebras and an isomorphism condition for quantum Heisenberg manifolds Abdesselam, A. & Rivasseau, V., An explicit large versus small field multiscale cluster expansion Accardi, L., Noise and dissipation in quantum theory Adler, C., Braid group statistics in twodimensional quantum field theory Affleck, I., The Haldane gap in antiferromagnetic Heisenberg chains Aizenman, M., Localization at weak disorder: some elementary bounds Albeverio, S. & Bogachev, L.V., Brownian survival in clusterized trapping medium Albeverio, S. & Fei, S.-M., Symmetry, integrable chain models and stochastic processes Albeverio, S., Hida, T., Potthoff, J., Rockner, M. & Streit, L., Dirichlet forms in terms of white noise analysis I: Construction and QFT examples Albeverio, S., Hida, T., Potthoff, J., Rockner, M. & Streit, L., Dirichlet forms in terms of white noise analysis II: Closability and diffusion processes Albeverio, S., Gottschalk, H. & Wu, J.-L.,
9(1997)411
Convoluted generalized white noise, Schwinger functions and their analytic continuation to Wightman functions Albeverio, S., Kolokol’tsov, V.N. & Smolyanov, O.G., Continuous quantum measurement: local and global approaches Ali, S.T., Antoine, J.-P., Gazeau, J.P. & Mueller, U.A., Coherent states and their generalizations: A mathematical overview Alicki, R., Andries, J., Fannes, M. & Tuyls, P., An algebraic approach to the Kolmogorov–Sinai entropy Anderson, P.W., The “infrared catastrophe”: When does it trash Fermi liquid theory? Andries, J., see Alicki Angelopoulos, E. & Laoues, M., Masslessness in n-dimensions Aniello, P., Cassinelli, G., de Vito, E. & Levrero, A., Square-integrability of induced representations of semidirect products Antoine, J.-P., see Ali Antoine, J.-P., Inoue, A. & Trapani, C., Partial *algebras of closable operators: A review Araki, H., Symmetries in theory of local observables and the choice of the net of local algebras
9(1997)123
2(1990)127
8(1996)907
6(1994)887
6(1994)1163
10(1998)147
10(1998)723
1(1989)291
1(1989)313
8(1996)763
503
9(1997)907
7(1995)1013
8(1996)167
6(1994)1085
8(1996)167 10(1998)271 10(1998)301
7(1995)1013 8(1996)1
4S(1992)1
504
Ashbaugh, M.S. & Benguria, R.D., The range of values of 2/1 and 3/ 1 for the fixed membrane problem Baadhio, R.A., & Kauffman, L.H., Link manifolds and global gravitational anomalies Bach, A., Emergence of the simultaneous continuous and discrete structure of the electromagnetic field Baker, G.A. Jr., Bessis, D. & Moussa P. Asymptotic behavior of some Hankel-Toeplitz determinants Barata, J.C.A. & Marchetti, D.H.U., The two-point function and the effective fugacity in diluted Ising models on the Cayley tree Barata, J.C.A., S-matrix elements in Euclidean lattice theories Barata, J.C.A., Beliakova, A., Karowski, M., Nill, F., Schmidt, M., Schrader, R. & Wiesbrock, H.-W., On Bert Schroer’s contribution to the development of quantum field theory Baumgärtel, H. & Lledó, F., Superselection structures for C*-algebras with nontrivial center Baumgärtel, H., A modified approach to the Doplicher/Roberts theorem on the construction of the field algebra and the symmetry group in superselection theory Bautista, R., Mucino, J., NahmadAchar, E. & Rosenbaum, M., Classification of gauge-related invariant connections Baxter, R.J., Zero-temperature skewed chiral Potts model Beliakova, A., see Barata Bellissard, J., see Schulz-Baldes Bellissard, J., Bovier, A. & Ghez, J.-M. Gap labelling theorems for one dimensional discrete Schrodinger operators Bellomo, N. & Lachowicz, M., Some mathematical results on the
AUTHOR INDEX
6(1994)999
5(1993)331
7(1995)1
4(1992)65
10(1998)751
6(1994)497
7(1995)523
9(1997)785
9(1997)279
5(1993)69
6(1994)869
7(1995)523 10(1998)1 4(1992)1
1(1989)183
asymptotic behavior of the solutions to the initial value problem for the Enskog equation Bellomo, N. & Gustafsson, T. The discrete Boltzmann equation: A review of the mathematical aspects of the initial and initialboundary value problems Benci, V., Fortunato, D. & Pisani, L., Soliton like solutions of a Lorentz invariant equation in dimension 3 Benguria, R.D., see Ashbaugh Bergmann, O. & Raychowdhury, P.N., Symmetrical treatment of electron and nuclear motions in M. Born’s theory of ideal crystals Bessis, D., see Baker Binnenhei, C., Implementation of endomorphisms of the CAR algebra Blair, A.D., Adelic path space integrals Blanchard, Ph. & Stubbe, J., Bound states for Schrodinger hamiltonians: Phase space methods and applications Bockenhauer, J., An algebraic formulation of level one Wess–Zumino–Witten models Boenkost, W., Vertex operators are not closable Bogachev, L.V., see Albeverio Bolley, C. & Helffer, B., Stability of bifurcating solutions for the Ginzburg–Landau equations Bolley, C. & Helffer, B., Rigorous results for the Ginzburg– Landau equations associated to a superconducting film in the weak k limit Bonneau, P., Topological quantum double Bonora, L., & Toppan, F. Global chiral vertex operators on Riemann surfaces Borchers, H.J. & Yngvason, J., Transitivity of locality and duality in quantum field theory. Some modular aspects
3(1991)137
10(1998)315
6(1994)999 1(1989)497
4(1992)65 7(1995)833
7(1995)21 8(1996)503
8(1996)925
7(1995)51 10(1998)147 10(1998)579
8(1996)43
6(1994)305 4(1992)425
6(1994)597
AUTHOR INDEX
Borchers, H.J., & Yngvason, J. From quantum fields to local von Neumann algebras Borthwick, D., Lesniewski, A. & Rinaldi, M., Notes on the structure of quantized hermitian symmetric spaces Boutet de Monvel-Berthier, A., Georgescu, V. & Soffer, A., N-body Hamiltonians with hardcore interactions Bovier, A. & Kulske, C., A rigorous renormalization group method for interfaces in random media Bovier, A., see Bellissard Bracken, A.J. see Gould Bracken, A.J., see Gould Braga, G.A., Lima, P.C. & O’Carroll, M.L., Exponential decay of truncated correlation functions via the generating function: A direct method Briet, P., General estimates on distorted resolvents and application to stark Hamiltonians Bros, J. & Moschella, U., Two-point functions and quantum fields in de Sitter universe Brunelli, J.C. & Das, A., A nonstandard supersymmetric KP hierarchy Brunelli, J.C., Hamiltonian structures for the generalized dispersionless KdV hierarchy Brunetti, R., Guido, D. & Longo, R., Group cohomology, modular theory and space-time symmetries Bruning, E.A.D., Uniqueness in moment – problems over nuclear spaces and weak convergence of probability measures Buchholz, D. & Verch, R., Scaling algebras and renormalization group in algebraic quantum field theory. II. Instructive examples Buchholz, D. & Schulz Mirbach, H., Haag duality in conformal quantum field theory
4S(1992)15
7(1995)871
6(1994)515
6(1994)413
4(1992)1 3(1991)223 5(1993)533 10(1998)429
8(1996)639
8(1996)327
7(1995)1181
8(1996)1041
7(1995)57
5(1993)631
10(1998)775
2(1990)105
Buchholz, D. & D’Antoni, C., Phase space properties of charged fields in theories of local observables Buchholz, D. & Verch, R., Scaling algebras and renormalization group in algebraic quantum field theory Buchholz, D., Doplicher, S., Longo, R. & Roberts, J.E. A new look at Goldstone’s theorem Busch, P., Cassinelli, G. & Lahti, P.J., Probability structures for quantum state spaces Capps, R.H. & Lyons, M.A., Multiplicity formulas for a class of representations of affine Kac– Moody algebras Carey, A.L., & Wright, J.D., Hilbert space representations of the gauge groups of some two dimensional field theories Carinena, J.F. & Lopez, C. Geometric study of Hamilton’s variational principle Carlen, E.A. & Loss, M., On the minimization of symmetric functionals Carter, B. & Khalatnikov, I.M., Canonically covariant formulation of Landau’s Newtonian superfluid dynamics Casahorran, J., A new supersymmetric version of the Abraham–Moses method for symmetric potentials Cassinelli, G., see Aniello Cassinelli, G., de Vito, E., Lahti, P. & Levrero, A., Symmetries of the quantum state space and group representations Cassinelli, G., see Busch Cassinelli, G., de Vito, E., Lahti, P.J. & Levrero, A., Symmetry groups in quantum mechanics and the theorem of Wigner on the symmetry transformations Chair, N. The (orbifold) Euler characteristic of the moduli space of curves
505
7(1995)527
7(1995)1195
4S(1992)47
7(1995)1105
6(1994)97
5(1993)551
3(1991)379
6(1994)1011
6(1994)277
8(1996)655
10(1998)301 10(1998)893
7(1995)1105 9(1997)921
3(1991)285
506
and the continuum limit of Penner’s connected generating function Chulaevsky, V.A. & Sinai, Ya.G. The exponential localization and structure of the spectrum for 1D quasi-periodic discrete Schrodinger operators Cirelli, R., Mania, A. & Pizzocchero, L., A functional representation for non-commutative C*-algebras Conley, C.H., Geometric realizations of representations of finite length Coquereaux, R. & Jadczyk, A., Conformal theories, curved phase spaces, relativistic wavelets and the geometry of complex domains Coquereaux, R., Jadczyk, A. & Kastler, D. Differential and integral geometry of Grassmann algebras Crehan, P. & Ho, T.G., Geometry of deformed boson algebras ourpevi, M., Geometry of quantum principal bundles II Da Silva, A.R., see Mignaco D’Antoni, C., see Buchholz Das, A., see Brunelli Daubechies, I. & Lagarias, J.C., On the thermodynamic formalism for multifractal functions de Monvel, A.B. & Grinshpun, V., Exponential localization for multidimensional Schrödinger operator with random point potential de Monvel-Berthier, A.B., Georgescu, V. & Mantoiu, M., Locally smooth operators and the limiting absorption principle for N-body hamiltonians de Vito, E., see Aniello de Vito, E., see Cassinelli de Vito, E., see Cassinelli Debievre, S., Hislop, P.D. & Sigal, I.M.
AUTHOR INDEX
3(1991)241
6(1994)675
9(1997)821
2(1990)1
3(1991)63
8(1996)949
9(1997)531
9(1997)689 7(1995)527 7(1995)1181 6(1994)1033
9(1997)425
5(1993)105
10(1998)301 10(1998)893 9(1997)921 4(1992)575
Scattering theory for the wave equation on non-compact manifolds Dell’Antonio, G.F., Variational calculus and stability of periodic solutions of a class of Hamiltonian systems Demuth, M. & van Casteren, J.A., On spectral theory of selfadjoint Feller generators Derezi n´ ski, J., Asymptotic completeness in quantum field theory. A class of Galilei-covariant models Derezinski, J. Algebraic approach to the N-body long range scattering Dick, R., Half-differentials and fermion propagators Digernes, T., Varadarajan, V.S. & Varadhan, S.R.S., Finite approximations to quantum systems Dimock, J., Quantized electromagnetic field on a manifold Dimock, J., Canonical quantization of Yang– Mills on a circle Dittrich, J., Duclos, P. & Gonzalez, N., Stability and instability of the wave equation solutions in a pulsating domain Divakaran, P.P., Symmetries and quantization: structure of the state space Dong, S.-J. & Yang, C.N., Bound states between two particles in a two- or threedimensional infinite lattice with attractive Kronecker-function interaction Doplicher, S., see Buchholz Duclos, P. & Exner, P., Curvature-induced bound states in quantum waveguides in two and three dimensions Duclos, P., see Dittrich Ducomet, B., Hydrodynamical models of gaseous stars Duffield, N.G., & Werner, R.F.
6(1994)1187
1(1989)325
10(1998)191
3(1991)1
7(1995)689 6(1994)621
4(1992)223
8(1996)85
10(1998)925
6(1994)167
1(1989)139
4S(1992)47 7(1995)73
10(1998)925 8(1996)957
4(1992)383
AUTHOR INDEX
Mean-field dynamical semigroups on C*-algebras Durhuus, B., Jakobsen, H.P. & Nest, R., Topological quantum field theories from generalized 6J-symbols Echeverria Enriquez, A., Munoz Lecanda, M.C. & Roman Roy, N. Geometrical setting of time-dependent regular systems. Alternative models Eguchi, T., Kawai, T., Mizoguchi, S. & Yang, S.-K. Character formulas for coset N=2 superconformal theories Eguchi, T., Yamada, Y. & Yang, S.-K., On the genus expansion in the topological string theory Ehrlich P.E., & Emch, G.G. Gravitational waves and causality Ehrlich, P.E., & Emch, G.G. Gravitational waves and causality (Errata) El Gradechi, A.M., On the super-unitarity of discrete series representations of orthosymplectic Lie superalgebras Elizalde, E.& Romeo, A., Regularization of general multidimensional Epstein Zeta-functions Ellis, R.S., Gough, J. & Pule, J.V., The large deviation principle for measures with random weights Emch, G.G., see Ehrlich Emch, G.G., see Ehrlich Enss, V., Geometrical methods in N-body quantum scattering theory Ercolessi, E., Landi, G. & TeotonioSobrinho, P., Noncommutative lattices and the algebras of their continuous functions Esposito, R., Marra, R. & Yau, H.T., Diffusive limit of asymmetric simple exclusion Esposito, R., Marra, R. & Yau, H.T., Erratum: Diffusive limit of asymmetric simple exclusion Exel, R., see Abadie Exner, P.,
5(1993)1
3(1991)301
4(1992)329
7(1995)279
4(1992)163 4(1992)501
10(1998)467
1(1989)113
5(1993)659
4(1992)163 4(1992)501 4S(1992)83
10(1998)439
6(1994)1233
8(1996)905
9(1997)411 7(1995)73
see Duclos Fan, A.H., A proof of the Ruelle operator theorem Fannes, M., Nachtergaele, B. & Slegers L. Functions of Markov processes and algebraic measures Fannes, M., see Alicki Fei, S.-M., see Albeverio Feldman, J., Magnen, J., Rivasseau, V. & Trubowitz, E., Constructive many-body theory Figueroa-O’Farrill, J.M., Ramos, E. & Mas, J. Integrability and bihamiltonian structure of the even order SKdV hierarchies Fioresi, R., Quantizations of flag manifolds and conformal space time Flato, M., Simon, J. & Taflin, E., The Maxwell–Dirac equations: Asymptotic completeness and the infrared problem Fortunato, D., see Benci Fredenhagen, K., Rehren, K.-H. & Schroer, B. Superselection sectors with braid group statistics and exchange algebras II: Geometric aspects and conformal covariance Fredenhagen, K., Gravity induced noncommutative spacetime Froese, R. & Waxler, R., The spectrum of a hydrogen atom in an intense magnetic field Froese, R. & Waxler, R., Ground state resonances of a hydrogen atom in an intense magnetic field Froese, R. & Waxler, R., Errata: Spectrum of a hydrogen atom in an intense magnetic field Frohlich, J. & Gabbiani, F., Braid statistics in local quantum theory Frønsdal, C. & Galindo, A., 8-Vertex correlation functions and twist covariance of q-KZ equation
507
7(1995)1241
4(1992)39
8(1996)167 10(1998)723 6(1994)1095
3(1991)479
9(1997)453
6(1994)1071
10(1998)315 4S(1992)111
7(1995)559
6(1994)699
7(1995)311
8(1996)761
2(1990)251
10(1998)1027
508
Fuchssteiner, B. & Oevel, G., Geometry and action-angle variables of multi soliton systems Furuta, T., Norm inequalities equivalent to Lowner–Heinz theorem Gabbiani, F., see Frohlich Galindo, A., see Frønsdal Gallavotti, G., Twistless KAM Tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems. A review Gannon, T. & Lam, C.S. Gluing and shifting lattice constructions and rational equivalence Gazeau, J.P., see Ali Geerse, C.P.M. & Hof, A. Lattice gas models on self-similar aperiodic tilings Geisler, R., Kostrykin, V. & Schrader, R., Concavity properties of Krein’s spectral shift function Gentile, G. & Mastropietro, V., Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics Georgelin, Y., Wallet, J.-C. & Masson, T., Linear connections on the twoparameter quantum plane Georgescu, V., see de Monvel-Berthier Georgescu, V., see Boutet de Monvel-Berthier Gerard, C., Asymptotic completeness for the spin-boson model with a particle number cutoff Gesztesy, F. & Ratnaseelan, R., An alternative approach to algebro-geometric solutions of the AKNS hierarchy Gesztesy, F. & Holden, H., Trace formulas and conservation laws for nonlinear evolution equations Gesztesy, F. & Holden, H.,
AUTHOR INDEX
1(1989)415
1(1989)135
2(1990)251 10(1998)1027 6(1994)343
3(1991)331
7(1995)1013 3(1991)163
7(1995)161
8(1996)393
8(1996)1055
5(1993)105 6(1994)515 8(1996)549
10(1998)345
6(1994)51
6(1994)673
Errata to “Trace formulas and conservation laws for nonlinear equations” Gesztesy, F., Race, D., Unterkofler, K. & Weikard, R., On Gelfand–Dickey and Drinfeld– Sokolov systems Gesztesy, F., Holden, H., Simon, B. & Zhao, Z., Higher order trace relations for Schrodinger operators Ghez, J.-M., see Bellissard Gonzalez, N., see Dittrich Gootman, E.C., & Lazar, A.J., Quantum groups and duality Gottschalk, H., see Albeverio Gough, J., see Ellis Gould, M.D., Zhang, R.B. & Bracken, A.J. Lie bi-superalgebras and the graded classical Yang–Baxter equation Gould, M.D., see Links Gould, M.D., Tsohantjis, I. & Bracken, A.J., Quantum supergroups and link polynomials Gould, M.D., Tensor product decompositions for affine Kac–Moody algebras Grabowski, J., Z-Graded extensions of poisson brackets Graf, G.M. & Solovej, J.P., A correlation estimate with applications to quantum systems with Coulomb interactions Grinshpun, V., see de Monvel Guido, D., see Brunetti Guille-Biel, C., Sparse Schrödinger operators Guo, B.-L., see Li Gustafsson, T. see Bellomo Haak, G., Schmidt, M. & Schrader, R. Group theoretic formulation of
6(1994)227
7(1995)893
4(1992)1 10(1998)925 5(1993)417 8(1996)763 5(1993)659 3(1991)223
5(1993)345 5(1993)533
6(1994)1269
9(1997)1
6(1994)977
9(1997)425 7(1995)57 9(1997)315 9(1997)675 3(1991)137 4(1992)451
AUTHOR INDEX
the Segal–Wilson approach to integrable systems with applications Haba, Z., Coherent states and quantum dynamics of non-linear systems Hara, T., & Slade G. The lace expansion for selfavoiding walk in five or more dimensions Hayashi, N., Kato, K. & Ozawa, T., Dilation method and smoothing effect of the Schrodinger evolution group Helffer, B., see Bolley Helffer, B., see Bolley Herrin, J. & Howland, J.S., The Born–Oppenheimer approximation: Straight-up and with a twist Hiai, F. & Petz, D., Quantum mechanics in AF C*-systems Hiai, F., & Petz, D., Entropy densities for Gibbs states of quantum spin systems Hida, T., see Albeverio Hida, T., see Albeverio Hillion, P. Plane waves with transverse structure Hillion, P., Boundary value problem for the wave eqautions Hinz, A.M., Regularity of solutions for singular Schrodinger equations Hiroshima, F., Diamagnetic inequalities for systems of nonrelativistic particles with a quantized field Hiroshima, F., A scaling limit of a Hamiltonian of many nonrelativistic particles interacting with a quantized radiation field Hiroshima, F., Functional integral representation of a model in quantum electrodynamics Hislop, P.D. & Nakamura, S.,
8(1996)1061
4(1992)235
7(1995)1123
10(1998)579 8(1996)43 9(1997)467
8(1996)819
5(1993)693
1(1989)291 1(1989)313 3(1991)371
2(1990)177
4(1992)95
8(1996)185
9(1997)201
9(1997)489
2(1990)479
Stark Hamiltonian with unbounded random potentials Hislop, P.D., see Debievre Ho, T.G., Landau, L.J. & Wilkins, A.J., On the weak coupling limit for a fermi gas in a random potential Ho, T.G., see Crehan Hof, A., see Geerse Holden, H., see Gesztesy Holden, H., see Gesztesy Holden, H., see Gesztesy Hoppe, J., Infinite dimensional algebras and 2+1 dimensional field theory Yet another view of gl ( ) Some new algebras Horuzhy, S.S., & Voroni n, A.V., BRST and l(1,1) Howland, J.S., see Herrin Hubner, M.& Spohn, H., Radiative decay: Nonperturbative approaches Hurt, N.E., Three topics on periodic orbit theory: A review Hurt, N.E., Bakers and cats: A review of simple systems in quantum chaology Ichinose, T. & Ichinose, W., On the essential self-adjointness of the relativistic Hamiltonian with a negative scalar potential Ichinose, W., see Ichinose Inoue, A., see Antoine Irac-Astaud, M. & Rideau, G., Bargmann representations for deformed harmonic oscillators Irac-Astaud, M., Differential calculus on a threeparameter oscillator algebra Isozaki, H., Multi-dimensional inverse scattering theory for Schrodinger operators
509
4(1992)575 5(1993)209
8(1996)949 3(1991)163 6(1994)51 6(1994)673 7(1995)893 2(1990)193
5(1993)191 9(1997)467 7(1995)363
5(1993)713
7(1995)103
7(1995)709
7(1995)709 8(1996)1 10(1998)1061
8(1996)1083
8(1996)591
510
Jadczyk, A. see Coquereaux Jadczyk, A., see Coquereaux Jäger, W. & SaitÇ, Y., The uniqueness of the solution of the Schrödinger equation with discontinuous coefficients Jakobsen, H.P., see Durhuus Jaksic, V., & Segert, J. Exponential approach to the adiabatic limit and the Landau–Zener formula Jauslin, H.R., see Monti Jensen, A., & Ozawa, T., Existence and non-existence results for wave operators for perturbations of the Laplacian Junker, W., Hadamard states, adiabatic vacua and the construction of physical states for scalar quantum fields on curved spacetime Karowski, M., see Barata Kastler, D. see Coquereaux Kastler, D. & Schucker, T., A detailed account of Alain Connes’ version of the standard model IV Kastler, D., Algebraic field theory: Recollections and thoughts about the future Kastler, D., A detailed account of Alain Connes’ version of the standard model in non-commutative geometry. I and II. Kastler, D., A detailed account of Alain Connes’ version of the standard model in non-commutative differential geometry III Kato, K., see Hayashi Kato, T. & Yajima, K., Some examples of smooth operators and the associated smoothing effect Kauffman, L.H., see Baadhio
AUTHOR INDEX
3(1991)63 2(1990)1 10(1998)963
5(1993)1 4(1992)529
10(1998)393 5(1993)601
8(1996)1091
7(1995)523 3(1991)63 8(1996)205
4S(1992)155
5(1993)477
8(1996)103
7(1995)1123 1(1989)481
5(1993)331
Kauffman, L.H., Gauss codes, quantum groups and ribbon Hopf algebras Kawai, T., see Eguchi Kay, B.S., The principle of locality and quantum field theory on (non globally hyperbolic) curved spacetimes Kellendonk, J., Noncommutative geometry of tilings and gap labelling Kennedy, T., Some rigorous results on the ground states of the Falicov– Kimball model Kesten, H. & Schonmann, R.H., Behavior in large dimensions of the Potts and Heisenberg models Keyl, M., Causal spaces, causal complements and their relations to quantum field theory Khalatnikov, I.M., see Carter King, C. & Waxler, R., Resonant decay near an accumulation point Kitada, H. Asymptotic completeness of Nbody wave operators I. Short-range quantum systems Klein, M. & Schwarz, E., An elementary approach to formal WKB expansions in Kolokol’tsov, V.N., Localization and analytic properties of the solutions of the simplest quantum filtering equation Kolokol’tsov, V.N., see Albeverio Konopelchenko, B.G., Soliton eigenfunction equations: The IST integrability and some properties Kostrykin, V. & Schrader, R., Cluster properties of one particle Schrödinger operators. II. Kostrykin, V. & Schrader, R., Cluster properties of one particle Schrodinger operators Kostrykin, V., see Geisler Kotani, S., Jacobi matrices with random
5(1993)735
4(1992)329 4S(1992)163
7(1995)1133
6(1994)901
1(1989)147
8(1996)229
6(1994)277 9(1997)227
3(1991)101
2(1990)441
10(1998)801
9(1997)907 2(1990)399
10(1998)627
6(1994)833
7(1995)161 1(1989)129
AUTHOR INDEX
potentials taking finitely many values Koukiou, F., The mean-field theory of directed polymers in random media and spin glass models Krishna, M. & Sunder, V.S., Schrödinger operators with fairly arbitrary spectral features Kulske, C., see Bovier Kupsch, J., A probabilistic formulation of bosonic and fermionic integration Kurasov, P., Energy dependent boundary conditions and the few-body scattering problem Lachowicz, M., see Bellomo Lagarias, J.C., see Daubechies Lahti, P., see Cassinelli Lahti, P.J., see Busch Lahti, P.J., see Cassinelli Lam, C.S. see Gannon Landau, L.J., see Ho Landi, G., see Ercolessi Landsman, N.P. & Wiedemann, U.A., Massless particles, electromagnetism, and Rieffel induction Landsman, N.P., Quantization and superselection sectors I. Transformation group C* algebras Landsman, N.P., Quantization and superselection sectors II. Dirac monopole and Aharonov Bohm effect Landsman, N.P., Induced representations, gauge fields, and quantization on homogeneous space Landsman, N.P., Deformations of algebras of observables and the classical limit of quantum mechanics Landsman, N.P.,
7(1995)183
9(1997)343
6(1994)413 2(1990)457
9(1997)853
1(1989)183 6(1994)1033 10(1998)893 7(1995)1105 9(1997)921 3(1991)331 5(1993)209 10(1998)439 7(1995)923
2(1990)45
2(1990)73
4(1992)503
5(1993)775
9(1997)29
Poisson spaces with a transition probability Laoues, M., see Angelopoulos Laoues, M., Some properties of massless particles in arbitrary dimensions Lazar, A.J., see Gootman Léandre, R., Hilbert space of spinor fields over the free loop space Lebowitz, J.L. & Macris, N., Long range order in the Falicov– Kimball model: Extension of Kennedy–Lieb theorem Lebowitz, J.L., Mazel, A.E. & Suhov, Yu.M., An ising interface between two walls: Competition between two tendencies Lenczewski, R., Addition of independent variables in quantum groups Lesniewski, A., see Borthwick Leukert, P. & Schafer, J., A rigorous construction of abelian Chern–Simons path integrals using white noise analysis Levrero, A., see Aniello Levrero, A., see Cassinelli Levrero, A., see Cassinelli Li, Y.-S. & Guo, B.-L., Attractor for dissipative Zakharov equations in an unbounded domain Lima, P.C., see Braga Links, J.R.,Gould, M.D. & Zhang, R.B., Quantum supergroups, link polynomials and representation of the braid generator Lledó, F., see Baumgärtel Longo, R. & Rehren, K.-H., Nets of subfactors Longo, R., see Buchholz Longo, R., see Brunetti
511
10(1998)271 10(1998)1079
5(1993)417 9(1997)243
6(1994)927
8(1996)669
6(1994)135
7(1995)871 8(1996)445
10(1998)301 10(1998)893 9(1997)921 9(1997)675
10(1998)429 5(1993)345
9(1997)785 7(1995)567 4S(1992)47 7(1995)57
512
Lopez, C. see Carinena Loss, M., see Carlen Lundberg, L.-E., Projective representations of infinite-dimensional orthogonal and symplectic groups Lundberg, L.-E., Quantum mechanics on hyperboloids Lundberg, L.-E., Quantum theory, hyperbolic geometry and relativity Lyons, M.A., see Capps Maciejewski, A. & Rybicki, S., Global bifurcations of periodic solutions of Hénon–Heiles system via degree for S 1-equivariant orthogonal maps Mack, G. & Pordt, A., Convergent weak coupling expansions for lattice field theories that look like perturbation series Mackay, N.J., Quantum affine Toda solitons Macris, N., see Lebowitz Maeda, S., Probability measures on projections in von Neumann algebras Maes, C., Coupling interacting particle systems Magnano, G. & Magri, F. Poisson-Nijenhuis structures and Sato hierarchy Magnen, J., see Feldman Magri, F. see Magnano Majewski, A.W. & Zegarlinski, B., Quantum stochastic dynamics II Mania, A., see Cirelli Mantoiu, M., see de Monvel-Berthier Marchetti, D.H.U., see Barata Marchetti, D.H.U., Upper bound on the truncated connectivity in one-dimensional β |x − y|2 percolation models at β>1
AUTHOR INDEX
3(1991)379 6(1994)1011 6(1994)1
6(1994)19
6(1994)39
6(1994)97 10(1998)1125
1(1989)47
10(1998)1111 6(1994)927 1(1989)235
5(1993)457
3(1991)403
6(1994)1095 3(1991)403 8(1996)689 6(1994)675 5(1993)105 10(1998)751 7(1995)723
Marra, R., see Esposito Marra, R., see Esposito Martin, Ph.A. & Nenciu, G., Semi-classical inelastic S-Matrix for one-dimensional N-states systems Mas, J. see Figueroa-O’Farrill Masson, T., see Georgelin Mastropietro, V., see Gentile Matsui, T., Markov semigroups on UHF algebras Matsutani, S., On time development of a quasiquantum particle in quartic potential (x2 – a2)2 /2g Matsuyama, T., Rapidly decreasing solutions and nonrelativistic limit of semilinear Dirac equations Mazel, A.E., see Lebowitz Messager, A. & Miracle-Sole, S., Low temperature states in the Falicov–Kimball model Mignaco, J.A. Sigaud, C., Vanhecke, F.J. & Da Silva, A.R., The Connes–Lott program on the sphere Miracle-Sole, S., see Messager Mizoguchi, S., see Eguchi Mohri, K., Residues and topological Yang– Mills theory in two dimensions Monti, F. & Jauslin, H.R., Quantum Nekhoroshev theorem for quasi-periodic Floquet Hamiltonians Moriya, H., Entropy density of one-dimensional quantum lattice systems Morosi, C. & Pizzocchero, L., On the continuous limit of integrable lattices II. Volterra systems and SP(N) theories Morosi, C. & Pizzocchero, L., On the bihamiltonian interpretation of the Lax formalism
6(1994)1233 8(1996)905 7(1995)193
3(1991)479 8(1996)1055 8(1996)393 5(1993)587
9(1997)943
7(1995)243
8(1996)669 8(1996)271
9(1997)689
8(1996)271 4(1992)329 9(1997)59
10(1998)393
9(1997)361
10(1998)235
7(1995)389
AUTHOR INDEX
Moschella, U., see Bros Moussa, P., see Baker Mucino, J., see Bautista Mueller, U.A., see Ali Müger, M., Superselection structure of massive quantum field theories in 1+1 dimensions Müller, E., see PodleÑ Munoz Lecanda, M.C. see Echeverria Enriquez Nachtergaele, B., see Fannes Nadaud, F., Generalised deformations, Koszul resolutions, Moyal products Nahmad-Achar, E., see Bautista Nakamura, M. & Ozawa, T., Low energy scattering for nonlinear Schrödinger equations in fractional order Sobolev spaces Nakamura, S., see Hislop Nakamura, S., On Martinez’ method of phase space tunneling Nakanishi, N., On Nambu–Poisson Manifolds Narnhofer, H., Entropy density for relativistic quantum field theory Narnhofer, N., see Thirring Neidhardt, H. & Zagrebnov, V.A., Does each symmetric operator have a stability domain? Neidhardt, H. & Zagrebnov, V., Towards the right hamiltonian for singular perturbations via regularization and extension theory Neidhardt, H. & Zagrebnov, V., On the right Hamiltonian for singular perturbations: General theory Nenciu, G., see Martin Nest, R., see Durhuus Nill, F. & Wiesbrock, H.-W.,
8(1996)327 4(1992)65 5(1993)69 7(1995)1013 10(1998)1147
10(1998)511 3(1991)301 4(1992)39 10(1998)685
5(1993)69 9(1997)397
2(1990)479 7(1995)431
10(1998)499 6(1994)1127
4S(1992)193 10(1998)829
8(1996)715
9(1997)609
7(1995)193 5(1993)1 7(1995)599
A comment on Jones inclusions with infinte index Nill, F., Weyl algebras, Fourier transformations and integrals of finite-dimensional Hopf algebras Nill, F., see Barata Nill, F., On the structure of monodromy algebras and Drinfeld doubles O’Carroll, M.L., see Braga Oevel, G., see Fuchssteiner Oevel, W. & Schief, W., Squared eigenfunctions of the (modified) KP hierarchy and scattering problems of Loewner type Oevel, W., & Rogers, C., Gauge transformations and reciprocal links in 2+1 dimensions Olkiewicz, R., Some mathematical problems related to classical-quantum interactions Orlandi, G., Asymptotic behavior of the Ginzburg–Landau functional on complex line bundles over compact Riemann surfaces Ozawa, T., see Jensen Ozawa, T., see Hayashi Ozawa, T., see Nakamura Packer, J.A., Crossed product C*-algebras and algebraic topology Parthasarathy, K.R., Quantum Ito’s formula Penrose, M.D., Penrose, O. & Stell, G., Sticky spheres in quantum mechanics Penrose, O., see Penrose Petz, D., see Hiai Petz, D., see Hiai Pisani, L., see Benci
513
6(1994)149
7(1995)523 9(1997)371
10(1998)429 1(1989)415 6(1994)1301
5(1993)299
9(1997)719
8(1996)457
5(1993)601 7(1995)1123 9(1997)397 8(1996)623
1(1989)89 6(1994)947
6(1994)947 5(1993)693 8(1996)819 10(1998)315
514
Pizzocchero, L., see Morosi Pizzocchero, L., see Cirelli Pizzocchero, L., see Morosi PodleÑ, P. & Müller, E., Introduction to quantum groups Pordt, A., see Mack Potthoff, J., see Albeverio Potthoff, J., see Albeverio Pron’ko, G.P., Hamiltonian theory of the relativistic string Pule, J.V., see Ellis Pyke, R., Virial relations for nonlinear wave equations and nonexistence of almost periodic solutions Race, D., see Gesztesy Radin, C. Disordered ground states of classical lattice models Raghunathan, M.S., Universal central extensions Ramos, E. see Figueroa-O’Farrill Ratnaseelan, R., see Gesztesy Raychowdhury, P.N., see Bergmann Rehren, K.-H., see Fredenhagen Rehren, K.-H., see Longo Rezende, J., Stationary phase, quantum mechanics and semi-classical limit Rideau, G., see Irac-Astaud Rinaldi, M., see Borthwick Rivasseau, V., see Feldman Rivasseau, V., see Abdesselam Rivera, J.E.M., Global smooth solution and uniform rate of decay in nonlinear viscoelasticity
AUTHOR INDEX
10(1998)235 6(1994)675 7(1995)389 10(1998)511 1(1989)47 1(1989)291 1(1989)313 2(1990)355
5(1993)659 8(1996)1001
6(1994)227 3(1991)125
6(1994)207 3(1991)479 10(1998)345 1(1989)497 4S(1992)111 7(1995)567 8(1996)1161
10(1998)1061 7(1995)871 6(1994)1095 9(1997)123 6(1994)855
Roberts, J.E., see Buchholz Roberts, J.E., The statistical dimension, conjugation and the Jones index Rockner, M., see Albeverio Rockner, M., see Albeverio Rogers, C., see Oevel Roman Roy, N. see Echeverria Enriquez Romeo, A., see Elizalde Rosenbaum, M., see Bautista Ruskai, M.B., Beyond strong subadditivity? Improved bounds on the contraction of generalized relative entropy Rybicki, S., see Maciejewski SaitÇ, Y., see Jäger Salmhofer, M., Improved power counting and Fermi surface renormalization Schafer, J., see Leukert Schief, W., see Oevel Schlingemann, D., Construction of kink sectors for two-dimensional quantum field theory models — An algebraic approach Schlingemann, D., On the algebraic theory of soliton and antisoliton sectors Schlingemann, D., On the existence of kink (soliton) states Schmidt, M., see Haak Schmidt, M., see Barata Schmitt, T., Functionals of classical fields in quantum field theory Schmitt, T., Supergeometry and quantum field theory, or: What is a classical configuration?
4S(1992)47 7(1995)631
1(1989)291 1(1989)313 5(1993)299 3(1991)301 1(1989)113 5(1993)69 6(1994)1147
10(1998)1125 10(1998)963 10(1998)553
8(1996)445 6(1994)1301 10(1998)851
8(1996)301
8(1996)1187
4(1992)451 7(1995)523 7(1995)1249
9(1997)993
AUTHOR INDEX
Schonmann, R.H., see Kesten Schrader, R., see Kostrykin Schrader, R., see Haak Schrader, R., see Kostrykin Schrader, R., see Geisler Schrader, R., see Barata Schroer, B., see Fredenhagen Schroer, B., Reminiscences about many pitfalls and some successes of QFT within the last three decades Schucker, T., see Kastler Schulz Mirbach, H., see Buchholz Schulz-Baldes, H. & Bellissard, J., Anomalous transport: A mathematical framework Schwarz, E., see Klein Segert, J., see Jaksic Sengupta, A., The Moduli space of Yang– Mills connections over a compact surface Shastri, A.R. & Zvengrowski, P. Type of 3-manifolds and addition of relativistic kinks Shulman, V.S., Quasivectors and Tomita– Takesaki theory for operator algebras on II1-spaces Sigal, I.M., see Debievre Sigaud, C., see Mignaco Simon, B., Cyclic vectors in the Anderson model Simon, B., see Gesztesy Simon, J., see Flato Sinai, Ya.G. see Chulaevsky Skibsted, E.,
1(1989)147 10(1998)627 4(1992)451 6(1994)833 7(1995)161 7(1995)523 4S(1992)111 7(1995)645
8(1996)205 2(1990)105 10(1998)1
2(1990)441 4(1992)529 9(1997)77
3(1991)467
9(1997)749
4(1992)575 9(1997)689 6(1994)1183
7(1995)893 6(1994)1071 3(1991)241 10(1998)989
Spectral analysis of N-body systems coupled to a bosonic field Skibsted, E., Smoothness of N-body scattering amplitudes Slade, G., see Hara Slegers, L., see Fannes Smolyanov, O.G., see Albeverio Sobolev, A.V., Discrete spectrum asymptotics for the Schrodinger operator with a singular potential and a magnetic field Soffer, A., see Boutet de Monvel-Berthier Solovej, J.P., see Graf Spera, M. & Wurzbacher, T., Determinants, Pfaffians and quasi-free representations of the CAR algebra Spohn, H., see Hubner Stavracou, T., Theory of connections on graded principal bundles Stell, G., see Penrose Stolz, G., Note to the paper by P.D. Hislop and S. Nakamura: Stark Hamiltonian with unbounded random potentials Streit, L., see Albeverio Streit, L., see Albeverio Stubbe, J., see Blanchard Suhov, Yu.M., see Lebowitz Summers, J. Stephen On the independence of local algebras in quantum field theory Sunder, V.S., see Krishna Suzuki, M., Convergence of exponential product formulas for unbounded operators Szlachanyi, K., Chiral decomposition as a source
515
4(1992)619
4(1992)235 4(1992)39 9(1997)907 8(1996)861
6(1994)515 6(1994)977 10(1998)705
7(1995)363 10(1998)47
6(1994)947 5(1993)453
1(1989)291 1(1989)313 8(1996)503 8(1996)669 2(1990)201
9(1997)343 8(1996)487
6(1994)649
516
of quantum symmetry in the Ising model Taflin, E., see Flato Tahiri, M., Generations and super cohomology Takasaki, K. Geometry of universal Grassmann Manifold from algebraic point of view Takasaki, K. & Takebe, T., Integrable hierarchies and dispersionless limit Takebe, T., see Takasaki Tamura, H., Semi-classical analysis for total cross-sections of magnetic Schrodinger operators in two dimensions Teotonio-Sobrinho, P., see Ercolessi Thirring, W., & Narnhofer, N. Covariant QED without indefinite metric Tomiyama, J., C*-algebras and topological dynamical systems Toppan, F., see Bonora Trapani, C., Quasi*-algebras of operators and their applications Trapani, C., see Antoine Trubowitz, E., see Feldman Truini, P., & Varadarajan, V.S., Quantization of reductive Lie algebras: construction and universality Tsirelson, B.S. & Vershik, A.M., Examples of nonlinear continuous tensor products of measure spaces and non-Fock factorizations Tsohantjis, I., see Gould Tuyls, P., see Alicki Unterkofler, K., see Gesztesy van Casteren, J.A., see Demuth van Elst, A.,
AUTHOR INDEX
6(1994)1071
7(1995)269 1(1989)1
7(1995)743
7(1995)743 7(1995)443
10(1998)439 4S(1992)193
8(1996)741
4(1992)425 7(1995)1303
8(1996)1 6(1994)1095 5(1993)363
10(1998)81
5(1993)533 8(1996)167 6(1994)227 1(1989)325 6(1994)319
Gap-labelling theorems for Schrodinger operators on the square and cubic lattice van Enter, A. & Zegarlinski, B., A remark on differentiability of the pressure functional Vanhecke, F.J., see Mignaco Varadarajan, V.S., see Truini Varadarajan, V.S., see Digernes Varadhan, S.R.S., see Digernes Verch, R., see Buchholz Verch, R., see Buchholz Verch, R., Continuity of symplectically adjoint maps and the algebraic structure of Hadamard vacuum representations for quantum fields on curved spacetime Voronin, A.V., see Horuzhy Wallet, J.-C., see Georgelin Watanabe, K., Spectral concentration and resonances for unitary operators: Applications to self-adjoint problems Waxler, R., see Froese Waxler, R., see Froese Waxler, R., see Froese Waxler, R., see King Wehrhahn, R.F., Scattering with symmetry as the interaction Weikard, R., see Gesztesy Werner, R.F., see Duffield Wiedemann, U.A., see Landsman Wiesbrock, H.-W., Superselection structure and localized Connes’ cocycles Wiesbrock, H.-W., see Barata
7(1995)959
9(1997)689 5(1993)363 6(1994)621 6(1994)621 10(1998)775 7(1995)1195 9(1997)635
5(1993)191 8(1996)1055 7(1995)979
6(1994)699 7(1995)311 8(1996)761 9(1997)227 6(1994)1339
6(1994)227 4(1992)383 7(1995)923 7(1995)133
7(1995)523
AUTHOR INDEX
Wiesbrock, H.-W., see Nill Wilkins, A.J., see Ho Woronowicz, S.L., C*-algebras generated by unbounded elements Wright, J.D., see Carey Wu, J.-L., see Albeverio Yajima, K., see Kato Yamada, Y., see Eguchi Yang, C.N., see Dong Yang, S.-K., see Eguchi Yang, S.-K., see Eguchi Yau, H.T., see Esposito Yau, H.T., see Esposito Yngvason, J., see Borchers Yngvason, J., see Borchers
7(1995)599 5(1993)209 7(1995)481
5(1993)551 8(1996)763 1(1989)481 7(1995)279 1(1989)139 4(1992)329 7(1995)279 6(1994)1233 8(1996)905 4S(1992)15 6(1994)597
Zagrebnov, V., see Neidhardt Zagrebnov, V., see Neidhardt Zagrebnov, V.A., see Neidhardt Zamolodchikov, A.B., Exact solutions of conformal field theory in two dimensions and critical phenomena Zegarlinski, B., Spin systems with long-range interactions Zegarlinski, B., see van Enter Zegarlinski, B., see Majewski Zhang, R.B. see Gould Zhang, R.B., see Links Zhang, R.B., Quantum supergroups and topological invariants of threemanifolds Zhao, Z., see Gesztesy Zvengrowski, P. see Shastri
517
8(1996)715 9(1997)609 10(1998)829 1(1989)197
6(1994)115
7(1995)959 8(1996)689 3(1991)223 5(1993)345 7(1995)809
7(1995)893 3(1991)467
QUANTUM NOETHER’S THEOREM AND CONFORMAL FIELD THEORY: A STUDY OF SOME MODELS SEBASTIANO CARPI Dipartimento di Matematica Universit` a di Roma “La Sapienza” P.le A. Moro 2, 00185 Roma Italy E-mail : [email protected] Received 31 January 1998 Revised 27 April 1998 We study the problem of recovering Wightman conserved currents from the canonical local implementations of symmetries which can be constructed in the algebraic framework of quantum field theory, in the limit in which the region of localization shrinks to a point. We show that, in a class of models of conformal quantum field theory in space-time dimension 1+1, which includes the free massless scalar field and the SU (N ) chiral current algebras, the energy-momentum tensor can be recovered. Moreover we show that the scaling limit of the canonical local implementation of SO(2) in the free complex scalar field is zero, a manifestation of the fact that, in this last case, the associated Wightman current does not exist.
1. Introduction In classical relativistic field theory Noether’s theorem associates a conserved current to every one-parameter group of symmetries of the Lagrangian. Moreover the zero component of this current is a density for the infinitesimal generator of this one-parameter group. Although the presence of conserved currents related to symmetries is a general feature of models of quantum field theory, the understanding of this relation in this context is less satisfactory than in the classical case. In the Lagrangian approach to quantum field theory, for example, a classical symmetry can disappear at the quantum level because of the renormalization procedure. Moreover the classical expression of the currents does not give a well-defined quantum field because it involves multiplications of the basic fields at the same point, so that, to give a precise definition of the current, one need a further renormalization (for a discussion see [14]). On the other hand, if one starts from general assumptions as the Wightman axioms [16], the existence of such conserved currents must be postulated. In the algebraic formulation of quantum field theory (“local quantum physics” [10]) a new approach towards a quantum Nowther’s theorem has been conceived by Doplicher in [5] and developed by Doplicher, Longo and Buchholz in [8] and [2]. In 519 Reviews in Mathematical Physics, Vol. 11, No. 5 (1999) 519–532 c World Scientific Publishing Company
520
S. CARPI
these works it has been proved that, in a theory where the field net satisfies the split property (see [10] and the references quoted there for the meaning and the relevance of the split property in quantum field theory), the global symmetries, including discrete symmetries, space-time symmetries and supersymmetries, can be locally implemented by unitary operators which are canonically construted from the theory in question. If a part of the symmetries considered forms a connected Lie group, then the generators of the corresponding local implementations can be considered as the analogue of the zero component of Wightman conserved currents, smeared with appropriate test functions with support in the region of localization. It has been suggested by Doplicher in [5] (see also [6] and [7]) that the canonical local generators constructed using the split property could be used to construct Wightman current by an appropriate scaling limit in which the region of localization shrinks to a point. The success of this program would give us a complete quantum Noether’s theorem and a general prescription to construct Wightman fields with a definite physical meaning, directly from the algebra of observables. In this paper we study this problem in some models of chiral field theories, which are a special class of conformal field theories in 1 + 1 space time dimensions (see [12] and [4]) that live in the real line, hoping that this will give some enlightenment on the study of a more general situation. The choice of chiral field theories is motivated by their simplicity and by the fact that dilation invariance permits to treat scaling limits in an intrinsic way. We show that, in a class of models of chiral theories, which includes the free massless scalar field and models arising from representation theory of certain loop groups, the energy-momentum tensor can be recovered in the scaling limit of the canonical local generator of translations constructed by the prescriptions given in [2]. Moreover we show that in the case of the free massless complex field, if we consider the canonical local generator of the SO(2) symmetry, the scaling limit is zero. We interpret this result as a consequence of the fact that the conserved current associated to the phase transformation is not a well-defined Wightman field, because of the typical infrared problems of the two-dimensional case. One of the key ingredients of this work is the analysis given in [11] for the construction of pointlike localized fields from conformally invariant Haag–Kastler nets. The present paper is organized in the following way: in Sec. 2 we give assumptions without referring to particular models and show that they are sufficient to recover the energy-momentum tensor (a priori supposed to exist) in the scaling limit of appropriate bounded functions of the canonical local generators. Some of these assumptions are standard but others can be justified only because they work in a non-empty class of models, providing us some non-trivial examples in which the general program can be realized. Their abstraction from the models is then motivated only by the hope of simplifying the exposition and giving a clear idea of the limits of a possible generalization to models not considered in this paper. In Sec. 3 we show that assumptions of Sec. 2 are satisfied in the models cited above, completing our discussion about the energy-momentum tensor. In Sec. 4
QUANTUM NOETHER’S THEOREM AND CONFORMAL FIELD THEORY:
...
521
we consider the case of the complex scalar field and finally in Sec. 5 we make some concluding remarks. 2. General Assumptions, Results and Proofs Let K denote the set of non-empty open bounded intervals of the real line R. We consider a family F = (F (I))I∈K of von Neumann algebras (the field algebra), acting on a separable Hilbert space H and we assume that this family satisfies the following properties: (i) Isotony: F(I1 ) ⊂ F(I2 ) for I1 ⊂ I2 ,
I1 , I2 ∈ K .
(1)
(ii) Locality: F(I1 ) ⊂ F (I2 )0
for I1 ∩ I2 = ∅,
I1 , I2 ∈ K .
(2)
(iii) There exists a strongly continuous unitary representation U of SL(2, R) in H such that U (−1) = 1 (3) and U (α)F (I)U (α)−1 = F (αI)
where SL(2, R) 3 α = acts on R
S {∞} by x → αx =
for I , a c
b d
ax + b cx + d
αI ∈ K ,
(4)
(5)
(note that for every I ∈ K we have αI ∈ K if α is close enough to 1). (iv) The conformal Hamiltonian L0 , which generates the restriction of U to the one-parameter group t t −2 sin cos 2 2 t→ , 1 t t sin cos 2 2 2 has non-negative spectrum. (v) There is a unique (up to a phase) U -invariant unit vector Ω ∈ H (the vacuum vector). (vi) H is the smallest closed subspace containing Ω which is invariant for U and F(I) for every I ∈ K. (vii) Split property: ˜ Let I, I˜ ∈ K such that the closure of I is contained in I˜ (we write I ⊂⊂ I). Then there exists a factor of type I N such that: ˜ . F (I) ⊂ N ⊂ F (I)
(6)
522
S. CARPI
(viii) There exists a strongly continuous unitary representation V of a compact group G leaving the vacuum invariant, commuting with U and such that V (g)F(I)V (g)−1 = F (I) for g ∈ G (ix) Let
T (a) = U
1 0
a 1
and I ∈ K .
(7)
be the group of translations. There exists a Wightman field Θ (the energymomentum tensor) given on an invariant dense domain DΘ containing the vacuum, such that: Θ(f ) is essentially self-adjoint for every function f ∈ SR (SR is the space of real Schwartz test functions); if A(I) := {F ∈ F(I) : V (g)F = F V (g)} for I ∈ K (the observable algebra) then eiΘ(f ) ∈ A(I) , for f ∈ SR with support contained in I; 2 dαx Θ(αx) U (α)Θ(x)U (α)−1 = dx
(8)
for α ∈ SL(2, R);
(9)
and if f ∈ SR is such that f (x) = 1 for x ∈ I then eiaΘ(f ) F e−iaΘ(f ) = T (a)F T (−a) if F ,
T (a)F T (−a) ∈ F(I) .
(10)
(x) Let H0 be the V -invariant subspace of H. Because of the positivity of L0 the representation U splits into a direct sum of irreducible representations τ acting on a subspace Hτ (see [15] and cf. [11]). The equivalence class of each τ is determined by a non-negative integer n(τ ) which is the lower bound of the spectrum (which is discrete and simple) of the restriction of L0 to Hτ . We then assume that, in the decomposition of the restriction of U to H0 (which is U -invariant because U commute with V ), there appear no representations τ with n(τ ) = 1 and only one representation τ with n(τ ) = 2. Note that only here we have a significant restriction for the group G that until now could be taken to be the trivial group. From the first six assumptions several results can be proved. For example PCT theorem, Haag duality, additivity, the Reeh–Schlieder property (see [11, 13, 1]). In particular it has been proved in [11] that these assumptions imply the existence of pointlike localized fields naturally associated to F and the existence of local operator product expansions. The split property, together with the Reeh–Schlieder property and locality, ˜ then the triple Λ = (F (I), F (I), ˜ Ω) is implies that, if for I, I˜ ∈ K we have I ⊂⊂ I, a standard and split inclusion of von Neumann algebras (see [9]). By the results in [2] there exists a local canonical implementation of the translations TΛ (a) = eiaPΛ in the sense that TΛ (a)F TΛ (−a) = T (a)F T (−a) for F ,
T (a)F T (−a) ∈ F (I)
(11)
QUANTUM NOETHER’S THEOREM AND CONFORMAL FIELD THEORY:
...
523
and ˜ . TΛ (a) ∈ F (I)
(12)
Using the transformation properties for the canonical implementations of symmetries (see [2]) we also have ˜ TΛ (a) ∈ A(I)
(13)
T (x)TΛ (a)T (−x) = TΛ+x (a)
(14)
D(λ)TΛ (a)D(λ)−1 = TλΛ (λa)
(15)
where D(λ) = U
1
λ2 0
0 λ− 2 1
is the group of dilations and Λ + x, λΛ are the triples associated to the pairs I + x, I˜ + x and λI, λI˜ respectively. Heuristically one may think of PΛ as an analogue of Θ(fΛ ), where fΛ is a real function with the support in I˜ and equal to 1 in I (cf. [6–8]). In fact they differ by ˜ Then one has the (heuristic) estimate a perturbation in F(I)0 ∩ F(I). Z Z (16) ϕ(x)PλΛ+x dx ∼λ→0 λ fΛ (x) dx Θ(ϕ) for ϕ ∈ S (S is the complex space of Schwartz test functions). To avoid problems with the domain of PΛ (we don’t know if it contains the vacuum) we prefer to consider λi [TλΛ (λa) − (Ω, TλΛ (λa)Ω)] instead of PλΛ (the vacuum mean value subtraction is a necessary renormalization prescription). Then the previous estimate suggests Z lim λ−2 ϕ(x)T (x)D(λ)[TΛ (a) − (Ω, TΛ (a)Ω)]D(λ)−1 T (x)−1 = ηΘ(ϕ) , (17) λ→0
where η is a constant independent of ϕ. Actually a further regularization is needed to avoid possible singularities of the limit, corresponding to the non integrability of fΛ (cf. [11]). Let µ be the Haar measure on SL(2, R) and h ∈ C ∞ (SL(2, R)) have a compact support and 1. For every bounded operator B on H we consider Bh = Rintegral equal to −1 h(α)U (α)BU (α) dµ(α). We can now state the following theorem. Theorem. For every h as described above with support sufficiently close to the identity there is a constant η such that for, every ϕ ∈ S with support contained in an open interval J ∈ K we have Z −2 ϕ(x)T (x)D(λ)[TΛ (a) − (Ω, TΛ (a)Ω)]h D(λ)−1 T (x)−1 dxψ lim λ λ→0
= ηΘ(ϕ)ψ for every ψ ∈ F (J)0 Ω, in the weak topology of H.
(18)
524
S. CARPI
Proof. If the support of h is sufficiently close to the identity then ˆ [TΛ (a) − (Ω, TΛ (a)Ω)]h ∈ A(I) for some Iˆ ∈ K so that Z ϕ(x)T (x)D(λ)[TΛ (a) − (Ω, TΛ (a)Ω)]h D(λ)−1 T (x)−1 dx ∈ A(J) for λ sufficiently small. It is then clear that it is enough to prove our assertion when ψ = Ω. Let HΘ be the closure of the subspace {Θ(f )Ω : f ∈ S}. Then, by the covariance of Θ with respect to SL(2, R), we see that the restriction of U to HΘ is irreducible (see [11]). Let PΘ be the orthogonal projection onto HΘ . Following the arguments given in [11] in the construction of pointlike localized fields, and using the fact that Θ can be identified with (a multiple of) the field ϕΘ associated to HΘ which has been constructed in [11] we find that lim λ−2 PΘ
Z
λ→0
ϕ(x)T (x)D(λ)[TΛ (a) − (Ω, TΛ (a)Ω)]h D(λ)−1 T (x)−1 dxΩ
= ηΘ(ϕ)Ω
(19)
for every ϕ ∈ S and an appropriate constant η. Since, by assumption (x), HΘ is the only (closed) subspace of H0 which is irreducible for U with conformal dimension equal to one or equal to two, for every orthogonal projection Pτ onto a U -irreducible subspace Hτ orthogonal to HΘ we have, following again the arguments given in [11], lim λ−2 Pτ
λ→0
Z
ϕ(x)T (x)D(λ)[TΛ (a) − (Ω, TΛ (a)Ω)]h D(λ)−1 T (x)−1 dxΩ
=0
(20)
for every ϕ ∈ S. Then if ψ is in the linear span of finitely many U -irreducible subspaces of H we have Z lim λ−2 ψ, ϕ(x)T (x)D(λ)[TΛ (a) − (Ω, TΛ (a)Ω)]h D(λ)−1 T (x)−1 dxΩ
λ→0
= (ψ, ηΘ(ϕ)Ω) .
(21)
Since the set of such vectors ψ is dense in H, to prove the weak convergence it is enough to show that the norm of λ−2
Z
ϕ(x)T (x)D(λ)[TΛ (a) − (Ω, TΛ (a)Ω)]h D(λ)−1 T (x)−1 dxΩ
is bounded with respect to λ.
QUANTUM NOETHER’S THEOREM AND CONFORMAL FIELD THEORY:
...
525
We have
2 Z
−2
−1 −1
λ ϕ(x)T (x)D(λ)[TΛ (a) − (Ω, TΛ (a)Ω)]h D(λ) T (x) dxΩ
x−y [TΛ (a) ϕ(y)ϕ(x) ¯ Ω, [TΛ (a) − (Ω, TΛ (a)Ω)]∗h T λ − (Ω, TΛ (a)Ω)]h Ω dy dx . (22)
= λ−4
Z Z
By the conformal cluster theorem [11] the Fourier transform of (Ω, [TΛ (a) − (Ω, TΛ (a)Ω)]∗h T (x)[TΛ (a) − (Ω, TΛ (a)Ω)]h Ω) can be written as ϑ(p)p3 F (p), where ϑ(p) is the Heaviside step function and F (p) is an analytic function of rapid decrease (see [11]) so that we have
2 Z
−2
−1 −1
λ ϕ(x)T (x)D(λ)[TΛ (a) − (Ω, TΛ (a)Ω)]h D(λ) T (x) dxΩ
Z =
∞
Z |ϕ(p)| ˆ F (λp)p dp ≤ max F 2
3
0
∞
2 3 |ϕ(p)| ˆ p dp .
(23)
0
We observe now that this theorem gives a positive answer to our problem if we can show that the constant η is not zero. From the non-vanishing argument given in [11] it follows that if PΘ TΛ (a)Ω 6= 0 (24) an accidental vanishing of η for a given function h can be avoided by an arbitrarily small translation of h on SL(2, R). Unfortunately we are not able to prove the above condition. However we avoid this problem with the aid of the following proposition. Proposition. There exist values of x and a, for arbitrarily large |x|, such that PΘ TΛ (a)TΛ+x (a)Ω 6= 0 .
(25)
Proof. Let’s suppose the contrary, i.e. PΘ TΛ (a)TΛ+x (a)Ω = 0
(26)
for every a and x such that |x| > L. Without loss of generality we can assume L greater than the diameter of the interval I˜ where TΛ is localized. For every x such that |x| > L, we can choose a function fx ∈ SR such that, for small enough eiΘ(fx ) TΛ (a)TΛ+x (a)e−iΘ(fx ) = TΛ (a)TΛ++x (a) .
(27)
526
S. CARPI
It follows that d (Ω, TΛ (a)T (x)TΛ (a)Ω) dx = i(Θ(fx )Ω, TΛ (a)TΛ+x (a)Ω) − i(Ω, TΛ (a)TΛ+x (a)Θ(fx )Ω) =0
(28)
for |x| > L and then (using the conformal cluster theorem) that the function x → (Ω, TΛ (a)T (x)TΛ (a)Ω) − (Ω, TΛ (a)Ω)2 has compact support. Then, using the positivity of the generator of T (positivity of the energy) we have (Ω, TΛ (a)T (x)TΛ (a)Ω) = (Ω, TΛ (a)Ω)2
(29)
for every a, x ∈ R. In particular we have (Ω, TΛ (2a)Ω) = (Ω, TΛ (a)Ω)2 .
(30)
Let E0 be the orthogonal projection onto the subspace of H spanned by Ω. Using the conformal cluster theorem it is not difficult to prove that in the limit a → ∞ T (a) converges to E0 in the weak topology of B(H) (cf. [16]). Thus T (a) ⊗ 1 converges to E0 ⊗ 1 in the weak topology of B(H ⊗ H). Since E0 ⊗ 1 is the orthogonal projection onto the T ⊗ 1-invariant subspace of H ⊗ H and since TΛ is unitarily equivalent to T ⊗ 1 [8], the previous equation leads to (Ω, EΛ Ω) = (Ω, EΛ Ω)2 ,
(31)
where EΛ is the orthogonal projection onto the TΛ -invariant subspace of H. Since Ω cannot be TΛ -invariant because of the Reeh–Schlieder property, the previous equality implies that (32) (Ω, EΛ Ω) = 0 , so that, using the fact that EΛ is local and different from 0, we are led the desired contradiction. It is now clear that if we use TΛ (a)TΛ+x (a), which implements the translations in a disconnected region (here we take x large), instead of TΛ (a), then the result of the previous theorem can be strengthened by the fact that the constant η is different from zero for a suitable choice of h, a and x. Remark. We have proved that the scaling limit converges on the dense domain F(J)0 Ω when the test function ϕ has support in the open interval J. We will show in the appendix that, for every J ∈ K, F (J)0 Ω contains a core for L0 . Thus, in typical models, the energy-bounds proved in [3] imply that F (J)0 Ω is a core for Θ(ϕ).
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3. The Models In this section we consider some models satisfying the assumptions of Sec. 2. Since these models are standard, here we sketch most of their properties referring to the literature for the details (in particular to [12, 4, 13]). 3.1. The free hermitian scalar field This theory is generated by a hermitian Wightman field j on the real line (the U (1)-current) satisfying the canonical commutation relations: [j(x), j(y)] = iδ 0 (x − y) .
(33)
We can define the local field algebras by F (I) = {j(f ) : f ∈ SR and supp f ⊂ I}00
for I ∈ K .
(34)
By the results in [3] and [13] it follows that assumptions from (i) to (vi) of Sec. 2 are satisfied (see also [12] and [4]). The transformation j → −j leaves all the Wightman functions invariant and so they can be unitarily implemented (see [16]) giving a representation of Z2 which satisfies assumption (viii). An energy-momentum tensor satisfying assumption (ix) is given by Θ(x) =
1 2 : j : (x) , 2
(35)
where the double dots indicate the Wick product. Assumption (x) can be proved by showing that in H0 there is only one (up to a multiplicative constant) eigenvector of L0 corresponding to the eigenvalue 2 and no eigenvectors corresponding to the eigenvalue 1 (here we are using the properties of the irreducible representations with positive conformal Hamiltonian stated above). To prove the last statement we use the fact that for every positive integer n the corresponding eigenspace is given by the linear span of vectors of the following form: J−n1 J−n2 . . . J−nk Ω ,
(36)
with n1 ≥ n2 ≥ · · · ≥ nk ≥ 1 ,
(37)
n 1 + n2 + · · · + nk = n ,
(38)
where for every integer m, Jm is the mth Fourier component of j (see [4]). Since for every integer m the projection of Jm Ω onto H0 is zero (because V (−1)Jm V (−1) = −Jm ) then the only eigenvector (up to a multiplicative constant) of L0 in H0 corresponding to the eigenvalue 2 is given by J−1 J−1 Ω .
(39)
Finally the split property (assumption (vii)) follows from the finiteness of the trace of e−βL0 for positive β together with an appropriate estimate for β → 0 [13].
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3.2. Chiral current algebras of simple lie groups Let G a connected, simply connected, simple, simply laced and compact Lie group and let LG be its Lie algebra. We consider theories arising from vacuum representations of the Kac–Moody algebra associated to G (see [12] and [4]). They are generated by a family of hermitian Wightman currents {j u : u ∈ LG} such that the map u → j u is R-linear and, for every u, v ∈ LG, [j u (x), j v (y)] = ij [u,v] (x)δ(x − y) + ikhu, viδ 0 (x − y) ,
(40)
where k is a positive constant and h·, ·i is the unique (up to a normalization) invariant scalar product on LG. With an appropriate normalization for h·, ·i, the possible values of k (levels) are restricted to be positive integers. If we define the field net by (41) F(I) = {j u (f ) : u ∈ LG, f ∈ SR , supp f ⊂ I}00 for I ∈ K then, by the results in [3] and [13], all the assumptions from (i) to (vii) are satisfied and there is a representation V of G with the properties of (viii) and such that, for every u ∈ LG, g ∈ G (42) V (g)j u V (g)−1 = j gu (with gu we denote the adjoint representation of G on LG). An energy-momentum tensor Θ with the properties of (ix) is obtained by the Sugawara construction (see [12, 4, 13]). Finally we show the validity of assumption (x). For every positive integer n the corresponding eigenspace of L0 is given by the linear span of vectors of the following form: u1 uk J u2 . . . J−n Ω, J−n 1 −n2 k
(43)
n1 ≥ n2 · · · ≥ nk ≥ 1 ,
(44)
n1 + n2 · · · + nk = n ,
(45)
with
u is the mth Fourier where u1 , . . . , uk ∈ LG and, for every integer m and u ∈ LG, Jm u component of j (see [4]). If P0 is the orthogonal projection onto H0 then we have, for every integer m and u ∈ LG u Ω = 0. P0 Jm
(46)
Moreover there exists a vector ψ ∈ H such that, for every u, v ∈ LG u v J−1 Ω = hu, viψ . P0 J−1
(47)
By the same argument given above, this implies that assumption (x) holds. Before concluding this section, we shortly describe the case of the chiral current algebras of some semisimple compact Lie groups. We consider a group G which is the direct product of a finite number N of connected, simply connected, simply
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laced, simple compact Lie groups Gi ; i = 1, . . . , N . The corresponding local field algebras are given, for every I ∈ K, by F(I) = F1 (I) ⊗ · · · ⊗ FN (I) ,
(48)
where, for i = 1, . . . , N , Fi is the field net generated by the chiral current algebra of Gi considered above. Moreover the vacuum representation of F is the tensor product of the vacuum representations of Fi ; i = 1, . . . , N . The energy-momentum tensor is given by N X Θi (x) , (49) Θ(x) = i=1
where Θi is the Sugawara energy-momentum tensor corresponding to Gi . Besides it is not difficult to show that, for every triple Λ, the canonical local implementation of the translations is given by TΛ = TΛ1 ⊗ · · · ⊗ TΛN .
(50)
Thus, a similar analysis to that given in Sec. 2, shows that, for every infinitely differentiable function h on SL(2, R) with support close enough to the identity and for every ϕ ∈ S such that supp ϕ ∈ J (J ∈ K open), we have Z −2 ϕ(x)T (x)D(λ)[TΛ (a)TΛ+y (a) − (Ω, TΛ (a)TΛ+y (a)Ω)]h lim λ λ→0
× D(λ)−1 T (x)−1 dxψ =
N X
ηi Θi (ϕ)ψ
(51)
i=1
for every ψ ∈ F(J)0 Ω, in the weak topology of H. Moreover for every i an accidental vanishing of ηi can be avoided by an appropriate choice of y and a and by an arbitrarily small translation of h. Unfortunately we are not able to prove in general that all the constants ηi must be equal. So, in this case the result is less satisfactory than that given for a simple group. However, in the particular case in which all the groups Gi are equal, the equality of the constants ηi follows from the symmetry under the permutation group SN . 4. A “Bad” Example In this section we consider the free complex scalar field. This model is generated by two commuting chiral currents j1 and j2 . For the two point Wightman functions we have (Ω, j1 (x)j2 (y)Ω) = (Ω, j2 (x)j1 (y)Ω) = 0 ,
(52)
(Ω, j1 (x)j1 (y)Ω) = (Ω, j2 (x)j2 (y)Ω) .
(53)
The transformations j1 (x) → cos ϑj1 (x) + sin ϑj2 (x) , j2 (x) → cos ϑj2 (x) − sin ϑj1 (x)
(54)
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S. CARPI
leave then all the Wightman functions invariant and so can be implemented by a unitary representation S of SO(2) leaving the vacuum invariant. Since this model is the tensor product of two free scalar Hermitian field models, the split property is fulfilled. So, by the results in [8], we can consider the canonical local implementations SΛ of S. In this case, as we are considering a gauge symmetry (corresponding to a dimensionless charge), the correct scaling factor in the scaling limit is λ−1 . Let H0 be the S-invariant subspace of H. An argument similar to those given in the previous section shows that, in H0 , there are no irreducible components of the representation U of SL(2, R) with conformal dimension equal to one. Moreover, by the transformation properties of SΛ [2] we have S(ϑ)SΛ (ϑ0 )S(ϑ)−1 = SΛ (ϑ0 ) ,
(55)
so by using the results in [11] we get the following proposition. Proposition. Let h be an infinitely differentiable function on SL(2, R) with compact support. For every ϕ ∈ S with the support contained in some open interval J ∈ K we have Z (56) lim λ−1 ϕ(x)T (x)D(λ) [SΛ (ϑ) − (Ω, SΛ (ϑ)Ω)]h D(λ)−1 T (x)−1 ψ = 0 , λ→0
for every ψ ∈ F (J)0 Ω, in the weak topology of H. This result cannot be avoided using the techniques explained in Sec. 2. Thus we cannot obtain a Wightman current corresponding to the symmetry S in the scaling limit of the canonical local implementations. This fact however is not surprising because in this case this Wightman current does not exist. In fact the classical expression for this current given by the Noether’s theorem does not define a Wightman field because of the infrared divergences typical of the 1+1 dimensional models (by the same reason a proper scalar Wightman field does not exist , cf. [14]). 5. Conclusions We have shown that in some models of conformal field theory the program of recovering Wightman conserved currents from the local canonical implementations of symmetries has a positive issue. We can summarize the ingredients for this success in the following two points: (1) the existence of a local operator product expansion; (2) the transformation properties of the canonical local implementations of symmetries. Although these are general features of models of quantum field theory, we believe that they will not be sufficient, even if necessary, in the study of a more general situation including field theories in a four dimensional space-time. In this study the property of local implementation, which is never used directly in the present work, should play a prominent part. A direct use of this property should also permit to
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give positive results without assuming the existence of the Wightman currents. In this way we should obtain a prescription for their construction. Appendix In this appendix we show that, for every open interval I ∈ K, the domain F(I)Ω contains a core for L0 . This fact is an easy corollary of the following lemma. Lemma. Let B ⊂ R be an inclusion of von Neumann algebras given on a separable Hilbert space H and let ξ ∈ H be a cyclic vector for B. Let U be a strongly continuous one-parameter group of unitary operators acting on H leaving ξ invariant and let H be its self-adjoint generator. If there exists a real number δ > 0 such that U (t)BU (t)−1 ⊂ R f or |t| < δ then Rξ contains a core for H. Proof. If ϕ ∈ S, we denote that if supp ϕ ⊂ (−δ, δ) then
R
ϕ(t)U (t)dt by U (ϕ). Then our assumptions imply
U (ϕ)Bξ ⊂ Rξ .
(57)
We now show that the domain D := {U (ϕ)Bξ : supp ϕ ⊂ (−δ, δ)} is a core for H. If ψ ∈ H, then U (ϕ)ψ is in the domain of H and HU (ϕ)ψ = iU (ϕ0 )ψ .
(58)
Since ξ is cyclic for B, for every ψ ∈ H we can find a sequence Bn ∈ B such that limn→∞ Bn ξ = ψ and thus lim U (ϕ)Bn ξ = U (ϕ)ψ
(59)
lim HU (ϕ)Bn ξ = HU (ϕ)ψ .
(60)
n→∞
n→∞
Finally let ϕ ∈ S be a positive function with integral equal to one and support contained in (−δ, δ) and let ϕn be defined by ϕn (t) = nϕ(nt). Now, for every ψ in the domain of H we have lim U (ϕn )ψ = ψ
(61)
lim HU (ϕn )ψ = Hψ .
(62)
n→∞
n→∞
Thus the closure of the graph of the restriction of H to D contains the graph of H, i.e. D is a core for H. We take now an open non-empty interval J ⊂⊂ I. Our previous assertion follows from the lemma taking B = F (J), R = F (I), ξ = Ω and H = L0 .
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Acknowledgements Special thanks are due to Prof. S. Doplicher for having suggested the problem, enlightening discussions and a constant encouragement. We also wish to thank Prof. K. Fredenhagen for some very useful discussions and hints. References [1] R. Brunetti, D. Guido and R. Longo, “Modular structure and duality in conformal quantum field theory”, Commun. Math. Phys. 156 (1993) 201. [2] D. Buchholz, S. Doplicher and R. Longo, “On Noether’s theorem in quantum field theory”, Ann. Phys. 170 (1986) 1. [3] D. Buchholz and H. Schulz-Mirbach, “Haag duality in conformal quantum field theory”, Rev. Math. Phys. 2 (1990) 105. [4] Ph. Di Francesco, P. Mathieu and D. S´en´echal, Conformal Field Theory, SpringerVerlag, 1996. [5] S. Doplicher, “Local aspects of superselection rules”, Commun. Math. Phys. 85 (1982) 73. [6] S. Doplicher, “Current algebra and the nature of symmetries in local quantum field theory”, in Trend and Developments in the Eighties, eds. S. Albeverio and Ph. Blanchard, World Scientific, 1985. [7] S. Doplicher, “Local observables and the structure of quantum field theory”, in Algebraic Theory of Superselection Sectors, ed. D. Kastler, World Scientific, 1990. [8] S. Doplicher and R. Longo, “Local aspects of superselection rules II”, Commun. Math. Phys. 88 (1983) 399. [9] S. Doplicher and R. Longo, “Standard and split inclusions of von Neumann algebras”, Invent. Math. 75 (1984) 493. [10] R. Haag, Local Quantum Physics, Springer-Verlag, 1992. [11] K. Fredenhagen and M. J¨ orß, “Conformal Haag–Kastler nets, pointlike localized fields and the existence of operator product expansions”, Commun. Math. Phys. 176 (1996) 541. [12] P. Furlan, G. M. Sotkov and I. T. Todorov, “Two-dimensional conformal field theory”, Rivista del Nuovo Cimento 12 (6) (1988) 1. [13] F. Gabbiani and J. Fr¨ ohlich, “Operator algebras and conformal field theory”, Commun. Math. Phys. 155 (1993) 569. [14] C. Itzykson and J. B. Zuber, Quantum Field Theory, MacGraw Hill, 1980. [15] S. Lang, SL2 (R), Springer-Verlag, 1975. [16] R. F. Streater and A. S. Wightman, PCT Spin Statistics and All That, Benjamin, 1964.
GEOMETRY OF QUANTUM HOMOGENEOUS VECTOR BUNDLES AND REPRESENTATION THEORY OF QUANTUM GROUPS I A. R. GOVER Mathematical Sciences Queensland University of Technology Brisbane, Australia
R. B. ZHANG Department of Pure Mathematics University of Adelaide Adelaide, SA 5005 Australia Received 25 June 1997 Revised 5 June 1998 Quantum homogeneous vector bundles are introduced in the context of Woronowicz type compact quantum groups. The bundles carry natural topologies, and their sections furnish finite type projective modules over algebras of functions on quantum homogeneous spaces. Further properties of the quantum homogeneous vector bundles are investigated, and applied to the study of the geometrical structures of induced representations of quantum groups.
1. Introduction There has long been an important interplay between the representation theory of Lie groups and differential geometry. As is well known, induced representations of Lie groups correspond to collections of geometric data on homogeneous vector bundles. This geometric perspective often reveals important properties that may be difficult to establish from the purely representation theoretical point of view. On the other hand, representation theory plays major roles in areas of differential geometry. We mention in particular the interaction between representation theory and the Penrose transforms of twistor theory [1]. Quantum groups play much the same role in noncommutative geometry as that played by Lie groups in classical geometry [2–4]. For this reason, there has been intensive investigations on the underlying geometry of quantum groups in recent years. We refer to the review articles [5, 6] and references therein for details on the subject. In this article we are concerned with the construction and development of some of the structures of quantum geometry and the interplay of these with the representation theory of quantum groups. The geometrical structures of fundamental importance for our purposes are the quantum homogeneous vector bundles. They provide a natural framework for exploring this interplay, and also form the foundations of a geometrical representation theory of quantum groups. Various versions of quantum 533 Reviews in Mathematical Physics, Vol. 11, No. 5 (1999) 533–552 c World Scientific Publishing Company
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deformations of fibre bundles were proposed at the algebraic level (i.e. without any topology) in the literature without the framework of Connes’ noncommutative differential geometry [7]. We mention in particular the work of Brzezinski and Majid and subsequent research by other authors along a similar line, where the primary aim was to develop a version of deformed gauge theory (See [8] for a recent elaboration on their work). Quantum homogeneous vector bundles, in comparison, have been less studied, although they are much more closely related to quantum groups. We introduce quantum homogeneous vector bundles by a direct description of their sections in the context of compact quantum groups of Woronowicz type [9]. The bundles as well as their base spaces, the quantum homogeneous spaces, carry natural topologies. This is a notable difference between the quantum homogeneous spaces studied here and those in the literature [10–13], with the latter being at a purely algebraic level. We should also point out that the quantum homogeneous spaces considered in this paper are all analogues of classical compact manifolds. Recall that in Connes’ theory [7], a noncommutative vector bundle is defined by its space of sections, which is required to be a projective module of finite type over the algebra of functions on the noncommutative base space. We shall prove that the quantum homogeneous vector bundles introduced here possess the required projectivity. This result is of fundamental importance for the development of a differential geometry on quantum homogeneous vector bundles for the reason that projectivity is necessary and sufficient for the existence of connections. We further prove that if the inducing module over the reductive quantum subalgebra Uq (l) is in fact the restriction of a module over the quantized universal enveloping algebra Uq (g) (here Uq (l) is a subalgebra corresponding to the Levi-part l of a parabolic in g), then the associated quantum homogeneous vector bundle is trivial in the sense that its sections form a free module over the algebra of functions on the quantum homogeneous space. Several other classical results are shown to admit quantum analogues. A notion of “quantum holomorphic” sections is established. It is shown, by using representation theoretical techniques, that the space of “holomorphic” sections of a given quantum homogeneous vector bundle is finite dimensional, the corresponding classical result of which is an important fact in elliptic theory. In particular, a “holomorphic” function on a quantum homogeneous space is necessarily a constant. (Recall that we only consider quantum analogues of compact homogeneous spaces in this paper.) As a natural application of quantum homogeneous vector bundles, we investigate the representation theory of quantum groups geometrically. We show that the sections of a quantum homogeneous vector bundle form an induced module of the corresponding quantum group, and there exists a quantum version of Frobenius reciprocity. A quantum analogue of the Borel–Weil theorem is also established. We should point out that algebraic versions of Frobenius reciprocity and the Borel–Weil theorem have been obtained elsewhere (for example [14, 21]) without the framework of quantum homogeneous vector bundles. Our emphasis is on the geometrical interpretation of these results, which in turn enables us to obtain useful results on the geometry of quantum homogeneous vector bundles from representation theory.
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The organization of the paper is as follows. Section 2 introduces the notation and conventions while reviewing the main structural and representation theoretical features of quantum groups. Section 3 introduces quantum homogeneous vector bundles and investigates their basic properties. Section 4 applies quantum homogeneous vector bundles to study a geometrical representation theory of quantum groups on the one hand, and uses representation theory to obtain results for the bundles themselves on the other. Each section begins with a paragraph stating the broad aims and the main new results of the section. 2. Quantum Groups We briefly review here some of the main properties of quantum groups as required for the investigations and developments in later sections. This section also serves the purpose of establishing our notation and conventions. Further details on most of this material can be found in [15] and [16]. Let g be a finite dimensional complex simple Lie algebra of rank r, with the Lr simple roots {αi | i ∈ Nr }, where Nr = {1, 2, . . . , r}. Set E = i=1 Rαi , and let ( , ) : E × E → R be the inner product induced by the Killing form of g. The set of integral elements of E will be denoted by P, and that of the integral dominant elements by P+ . The Jimbo version [17] of the quantized universal enveloping algebra Uq (g) is defined to be the unital associative algebra over C, generated by {ki±1 , ei , fi | i ∈ Nr } subject to the standard relations [16]. We will assume that q is real positive. This is required in order for the Haar functional on the corresponding quantum group to be positive definite. Uq (g) has the structure of a Hopf algebra. We will denote the co-multiplication by ∆, the co-unit by , and the antipode by S. Uq (g) admits a variety of Hopf ∗-algebra structures specified by anti-involutions ∗ satisfying ∗S ∗ S = idUq (g) . We set θ = ∗S, and call θ a quantum Cartan involution. Here we will only consider the ∗-operation defined by e∗i = fi , Let us define
fi∗ = ei ,
ki∗ = ki .
UR q (g0 ) = {x ∈ Uq (g) | θ(x) = x} .
(1)
(2)
It can be readily shown that UR q (g0 ) defines a real associative algebra, which may be regarded as a “real form” of Uq (g). However, the restriction of ∆ does not lead R to a co-multiplication for UR q (g0 ), and so Uq (g0 ) does not possess a natural Hopf algebra structure. Denote by Cq (g0 ) the real vector space spanned by Xi = ei − qi fi , √ Yi = −1(ei + qi fi ) , Zi =
√ ki − ki−1 −1 , qi − qi−1
Si = ki + ki−1 − 2 ,
i ∈ Nr ,
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where qi = q (αi , αi )/2 . Then UR q (g0 ) is generated by Cq (g0 ) ∪ {11Uq (g) }. Note that Cq (g0 ) is a two-sided co-ideal of Uq (g), that is, ∆(Cq (g0 )) ⊂ Cq (g0 ) ⊗R Uq (g) + ∗ Uq (g) ⊗R Cq (g0 ). The complexification of (UR q (g0 )) can be naturally identified with (Uq (g))∗ . Similarly, one can easily establish a one-to-one correspondence between complex representations of UR q (g0 ) and complex representations of Uq (g). A number of Hopf subalgebras of Uq (g) will be of importance later. For any subset Θ of Nr , we introduce the following two sets of elements of Uq (g): Sl = {ki±1 , i ∈ Nr ; ej , fj , j ∈ Θ} ;
Sp = Sl ∪ {ej , j ∈ Nr \Θ} .
Clearly Sl and Sp generate Hopf subalgebras of Uq (g), which we respectively denote by Uq (l) and Uq (p). We call Uq (l) a reductive quantum subalgebra, and Uq (p) a parabolic quantum subalgebra of Uq (g), in analogy to the classical terminology. Note that the image of Uq (p) under the quantum Cartan involution also deserves the name of a parabolic quantum subalgebra. Results presented in the remainder of the paper can also be formulated using such opposite parabolic Hopf subalgebras. Note that Uq (l) is the invariant subalgebra of Uq (p) under the quantum Cartan involution θ. For later use, we also define R UR q (k) = Uq (l) ∩ Uq (g0 ) . R R Then UR q (k) is a real subalgebra of Uq (g0 ), and its complexification is Uq (l). Uq (k) is generated by 11Uq (g) and the set
{Xi , Yi |i ∈ Θ} ∪ {Zi , Si |i ∈ Nr } . We will denote by Cq (k) the linear span of the elements of this set. Then it can be easily shown that Cq (k) is a two-sided co-ideal of Uq (g). The representation theory of Uq (g) is very similar to that of g (see e.g. [16] for details). In particular, all finite dimensional representations are completely reducible. If W (λ) is a finite dimensional irreducible left Uq (g)-module, there exists the unique (up to scalar multiples) highest weight vector v+ , such that ei v+ = 0 ,
ki v+ = q (λ,αi )/2 v+ ,
λ ∈ P+ ,
and the module W (λ) is uniquely determined by the highest weight λ.a We will ¯ The dual module of ¯ and define λ† = −λ. denote the lowest weight of W (λ) by λ, † W (λ) has highest weight λ . We will denote by Modq (g) the set of finite dimensional Uq (g)-modules that are direct sums of a finite number of W (λs ), λs ∈ P+ . Then Modq (g) forms a tensor category. An important fact is that with respect to the ∗-operation given above, every finite dimensional Uq (g)-module W is unitary in the sense that it admits a nondegenerate positive definite sesquilinear form ( , ) satisfying (xv, w) = (v, x∗ w), ∀ v, w ∈ W, x ∈ (λ) Uq (g). This allows us to introduce an orthonormal basis {wi |i = 1, 2, . . . , dλ } a Of course these statements are true only up to the algebra automorphisms of U (g) which multiply q √ the generators by appropriate powers of −1.
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(dλ = dimW (λ)) for each finite dimensional irreducible Uq (g)-module W (λ) such (λ) (λ) (λ) that (wi , wj ) = δij . This basis of W (λ) induces a basis {w ¯i } for (W (λ))∗ defined by (λ)
(λ)
∀ v ∈ W (λ) .
w ¯i (v) = (wi , v) ,
(3) (λ)
Consider an irreducible object W (λ) of Modq (g) with an orthonormal basis {wi }. (λ) Let tij ∈ (Uq (g))∗ be defined by X (λ) (λ) (λ) tji (x)wj = xwi , ∀ x ∈ Uq (g) . (4) j (λ)
By using the quantum PBW theorem, one can easily show that all tij belong to the finite dual (Uq (g))0 of Uq (g). We will denote by t(λ) the irreducible representation (λ) of Uq (g) furnished by W (λ), and call the tji matrix elements of t(λ) . The irreducibility of W (λ) together with the Burnside theorem of matrix algebras implies that t(λ) (Uq (g)) coincides with the entire algebra of dλ × dλ matrices. Furthermore, by considering the left action (9) of the central algebra of Uq (g) on (λ) them, one can show that the entire set {tij | i, j = 1, 2, . . . , dλ , ∀ λ ∈ P+ } is linearly L (λ) d λ Ctij ⊂ (Uq (g))0 , which is clearly independent independent. Define T (λ) = i,j=1 of the choice of bases for W (λ). Let M T (λ) , (5) Tq (g) = λ∈P+
where the direct sum is defined algebraically. Tq (g) is a Hopf subalgebra of (Uq (g))0 , and is essentially the quantum group introduced by Faddeev, Reshetikhin and Takhtajan [18]. As is well known, the multiplication is the pullback of the co-multiplication of Uq (g) (the closure of Tq (g) under this multiplication follows from the complete reducibility of finite dimensional representations of Uq (g)), while the co-multiplication is given by (λ)
∆0 (tij ) =
dλ X
(λ)
(λ)
tik ⊗ tkj .
k=1
The antipode is given by
(λ) S0 (tij )
(λ)
xw ¯i
=
dλ X
=
(λ† ) t˜ji ,
†
(λ with the t˜ji
†
(λ ) (λ) t˜ji (x)w¯j ,
)
∈ Tq (g) defined by
x ∈ Uq (g) ,
j=1
where {w ¯i } is the the basis of W (λ† ) = (W (λ))∗ defined by (3). The unit of Tq (g) is , while the co-unit is 11Uq (g) . From here on we will omit the subscript 0 from ∆0 and S0 . The Cartan involution θ of Uq (g) induces a natural Hopf ∗-algebra structure for Tq (g) with the ∗-operation defined by h∗(a), xi = ha, θ(x)i, ∀ a ∈ Tq (g), x ∈ Uq (g). It follows that (λ)
(λ)
(λ† )
∗(tij ) = t˜ij
.
(6)
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Rl Recall that given a Hopf algebra A, an element ∈ A∗ is called a left integral Rl Rl R r ∗ on 11A i , ∀ x ∈ A∗ . Similarly, called a right integral if R∈ A is R r A if x · =Rhx, r ∗ ·x = hx, 1lA i . A normalised Haar functional ∈ A on A is a left and right integral sending 11A to 1. Such a functionalR is unique if exists. It is an entirely straightforward matter to establish that the ∈ (Tq (g))∗ defined by Z Z (λ) tij = 0 , 0 6= λ ∈ P+ , 1lTq (g) = 1 ; gives rise to a Haar functional on Tq (g). Denote by 2ρ the sum of the positive roots of g. Let K2ρ be the product of −1 = q (2ρ,αi ) ei , ∀ i.R Let Dq (λ) := tr{t(λ) (K2ρ )}. powers of ki±1 ’s such that K2ρ ei K2ρ Then it follows from the left and right invariance of and Schur’s Lemma that Z
(λ) (µ† ) tij t˜rs
(λ)
tsj (K2ρ ) δir δλµ , = Dq (λ)
Z
†
(λ† ) (µ) t˜ij trs
(λ ) t˜ (K2ρ ) δjs δλµ . = ir Dq (λ)
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Observe that, in particular, these formulae imply R that the quantum Haar functional of Tq (g) is positive definite in the sense that (f ∗ f ) > 0, ∀ f 6= 0. We now employ the quantum Haar functional to introduce topologies on Tq (g). Definite a sesquilinear form (·, ·)h for Tq (g) Z (a, b)h = a∗ b , a, b ∈ Tq (g) , and let k · · · kh be the norm on Tq (g) given by p kakh = (a, a)h , a ∈ Tq (g) . This equips Tq (g) with the structure of a pre-Hilbert space. Let us denote by L2q the Hilbert space completion of Tq (g) in this norm. Denote by B(L2q ) the bounded linear operators on L2q . Then the left regular representation of Tq (g) can be extended to the completion L2q , yielding a ∗-representation π : Tq (g) → B(L2q ) in the bounded operators. To prove this claim, note that for any c ∈ Tq (g), (ca, b)h = (a, c∗ b)h , if |(ca, b)h | < ∞. Also observe that kak2h =
P
(λ) ∗ (λ) k (tki ) tki
X
(λ)
tki
∗
X k
(λ)
= 1. Thus, for all a ∈ L2q ,
(λ)
tki a, a
k
=
a, b ∈ L2q ,
(λ)
tki a, tki a
h
h (λ)
≥ ktji ak2h .
Let k · k be the operator norm on B(L2q ). Its pull back under π gives rise to a C ∗ -norm k · kop on Tq (g) such that kakop = sup{kaf kh; f ∈ L2q , kf kh = 1} .
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Finally, it is an elementary exercise to check that the completion in this norm extends Tq (g) to a unital C ∗ -algebra Aq (g). The C ∗ -algebra Aq (g) qualifies as a compact quantum group of the Woronowicz type [9], with Tq (g) a dense subalgebra possessing the structure of a Hopf ∗-algebra. However, we should note that it is not possible to extend the co-unit and antipode of Tq (g) to continuous maps from the entire Aq (g) to appropriate spaces. Furthermore, an extension of the co-multiplication will necessarily map Aq (g) continuously to some completion of Aq (g) ⊗ Aq (g) instead of the algebraic tensor product itself. We should also mention that Tq (g) satisfies all the conditions of a CQG algebra in the sense of [19]. 3. Quantum Homogeneous Vector Bundles In this section, we will introduce quantum homogeneous vector bundles and study some of their basic properties. We believe that the material presented here is largely new. Let us start by introducing two types of actions of Uq (g) on Tq (g). The first action will be denoted by ◦, which corresponds to the right translation in the classical theory of Lie groups. It is defined by X f(1) hf(2) , xi , x ∈ Uq (g) , f ∈ Tq (g) . (9) x◦f = (f )
The other action, which corresponds to the left translation in the classical Lie group theory, will be denoted by ·. It is defined by X hf(1) , S −1 (x)if(2) . (10) x·f = (f )
The two actions commute in the following sense: x ◦ (y · f ) = y · (x ◦ f ) ,
∀ x, y ∈ Uq (g) ,
f ∈ Tq (g) .
These actions can only be extended to certain subspaces of Aq (g). Given any f ∈ Aq (g), we consider the equivalence class of Cauchy sequences, which have the same n → ∞ limit f . If for any two Cauchy sequences {fn } and {gn } in the equivalence class, kx ◦ fn+m − x ◦ fn kop → 0, kx ◦ fn+m − x ◦ gn kop → 0, n → ∞, we define x ◦ f = limn→∞ x ◦ fn . And x · f is defined in an analogous way. Now introduce the following: Definition 1. Eq := {a ∈ Aq (g)|x · a, x ◦ a ∈ Aq (g) ,
|a(x)| < ∞ ,
∀ x ∈ Uq (g)} .
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The Eq clearly forms a subalgebra of Aq (g). We will take it as the quantum analog of the algebra of smooth functions over the group. Let us now turn to the study of quantum homogeneous spaces. Quantum homogeneous spaces have been established and studied in the literature (see, for example, [10–13]). But these treatments are largely at the algebraic level (i.e. without
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introducing any topology). In our current notation this amounts to the algebra of functions, over a quantum homogeneous space, being defined as an appropriate subset of Tq (g). This is similar to a situation in classical analysis where one works with polynomials only. From the point of view of developing a quantum differential geometry this is clearly not satisfactory and thus we now set about lifting the description of quantum homogeneous spaces to a topological setting. As we will see shortly, the well-known fact in classical complex geometry, that any complex analytic function on a compact complex manifold is a constant, also holds in the analogous quantum setting. Of course, in the first instance, we must work in a category of functions that has a richer family of sections. This family should contain enough information to capture the underlying geometrical aspects of the compact quantum homogeneous spaces. On the other hand we want the class of functions (and “bundle sections”) to be closed under operations which generalize classical differentiation. It is natural then to look for the quantum analogs of algebras of smooth functions. As in the classical case this is most easily achieved by working in the “real setting”. Thus we will consider the compact real form of ∗ Uq (g), and regard Tq (g) as a subset of the complexification of (UR q (g0 )) . Let us introduce the following: Definition 2. Eqk := {f ∈ Eq |x ◦ f = (x)f ,
∀ x ∈ UR q (k)} .
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R Note that we may replace UR q (k) by Uq (l) = C ⊗R Uq (k) in the above equation k k without altering Eq . To investigate properties of Eq , we consider the action of Cq (k) R on it. Recall that Cq (k) generates the real subalgebra UR q (k) of Uq (g0 ). Also, it is a two-sided co-ideal of Uq (g) and satisfies (Cq (k)) = 0. For any a, b ∈ Eqk , and x ∈ Cq (k), we have X {x(1) ◦ a}{x(2) ◦ b} = 0 . x ◦ (ab) = (x)
Therefore ab ∈ Eqk , that is, Eqk is a subalgebra of Eq . It will be shown below that this non-commutative algebra is infinite dimensional. We will regard Eq as the quantum analog of the algebra of smooth functions on a quantum homogeneous space, which is the quantum analog of GC /P , where GC is the complexification of a compact Lie group, and P is a parabolic subgroup of GC . Eq contains a dense subalgebra, which coincides with the quantum homogeneous space introduced in [10–13]. Let V be a finite dimensional module over Uq (l), which we will also regard as a R Uq (k)-module by restriction. We extend the actions ◦ and · of Uq (g) on Eq trivially P to actions on Eq ⊗ V : for any ζ = r fr ⊗ vr ∈ Eq ⊗ V X X x ◦ fr ⊗ vr , x · ζ = x · fr ⊗ vr , x ∈ Uq (g) . x◦ζ = r
r
We now introduce another definition, which will be of considerable importance for the remainder of the paper:
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Definition 3. Eqk (V ) := {ζ ∈ Eq ⊗ V |x ◦ ζ = (idAq (g) ⊗ S(x))ζ ,
∀ x ∈ UR q (k)} .
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Note that every ζ ∈ Eqk (V ) satisfies x ◦ ζ = (idAq (g) ⊗ S(x))ζ ,
∀ x ∈ Uq (l) .
Consider the subspace Fq (V ) := {Tq (g) ⊗ V } ∩ Eqk (V ) of Eqk (V ). Since the finite dimensional representations of Uq (l) are completely reducible, the study of its properties reduces to the case when V is irreducible. Let Vµ be a finite dimensional irreducible Uq (l)-module with highest weight µ and lowest weight µ ˜. Any element ζ ∈ Fq (Vµ ) can be expressed in the form X X (λ) (λ) ζ= S(tji ) ⊗ vij , λ∈P+ i,j (λ)
for some vij ∈ Vµ . Fix an arbitrary λ ∈ P+ . For any nonvanishing w ∈ W (λ), the following linear map is clearly surjective: HomC (W (λ), Vµ ) ⊗ w → Vµ , φ ⊗ w 7→ φ(w) . (λ)
(λ)
(λ)
(λ)
(λ)
Thus there exist φi ∈ HomC (W (λ), Vµ ) such that vij = φi (wj ), where {wi } is the basis of W (λ), relative to which the irreducible representation t(λ) of Uq (g) is defined. Now we can rewrite ζ as X X (λ) (λ) (λ) S(tji ) ⊗ φi (wj ) . ζ= λ∈P+ i,j
The defining property of Fq (Vµ ) states that ` ◦ ζ = (idTq (g) ⊗ S(`))ζ ,
∀ ` ∈ Uq (l) .
Thus we have X X X X (λ) (λ) (λ) (λ) (λ) (λ) (λ) S(tki ) ⊗ tjk (S(`))φi (wj ) = S(tji ) ⊗ S(`)φi (wj ) . λ∈P+ i,j
λ∈P+ i,j,k (λ)
(λ)
Recalling that the tki are linearly independent. It follows easily that the S(tki ) also form a linearly independent set. So the above is equivalent to X (λ) (λ) (λ) (λ) (λ) tjk (`)φi (wj ) = `φi (wj ) , ∀ ` ∈ Uq (l) . j (λ)
This equation is precisely the statement that the φi be Uq (l)-module homomorphisms, (λ) φi ∈ HomUq (l) (W (λ), Vµ ) ⊂ HomC (W (λ), Vµ ) , ∀ i .
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Thus finding sections in Fq (Vµ ) is equivalent to finding, for all λ ∈ P+ , the homomorphisms φ(λ) ∈ HomUq (l) (W (λ), Vµ ). Note that each such homomorphism φ(λ) determines dλ linearly independent sections X (λ) (λ) (λ) S(tji ) ⊗ φ(λ) (wj ) . ζi = j
Toward constructing such homomorphisms we consider a couple of useful observations. Note that if W1 → V1 and W2 → V2 are each Uq (l)-homomorphism then these induce a Uq (l)-homomorphism on the tensor product in the obvious manner W1 ⊗ W2 → V1 ⊗ V2 . Now let W (λ1 ) and W (λ2 ) be irreducible Uq (g)-modules of respective highest weights λ1 and λ2 . Let Vµ1 and Vµ2 be irreducible Uq (l)-modules of the highest weights indicated. Then by explicit construction of maximal weights one easily establishes the following: Lemma 1. Suppose there are non-trivial Uq (l)-homomorphisms W (λ1 ) → Vµ1 and W (λ2 ) → Vµ2 . Then there is an induced non-trivial Uq (l)-homomorphism W (λ1 + λ2 ) → Vµ1 +µ2 . Let us consider the case µ = 0, then Fq (Vµ=0 ) = Tq (g) ∩ Eqk . We will show that this has an infinite dimensional vector space of sections. Of course there is a homomorphism from the trivial representation of Uq (g), W (0) = C, onto V0 = C. This gives the constant sections of Tq (g) ∩ Eqk . Let γ be the highest root of g. Recall that l is reductive and there are N = r − |Θ| independent central elements in l. This, transcribed to the quantum setting, implies the existence of this many linearly independent Uq (l)-homomorphisms W (γ) → C. As mentioned above each of these corresponds to d = dim(g) linearly independent sections. So the representation W (γ) determines N d linearly independent sections. Further linearly independent sections may be obtained using Lemma 1. For example there are (m|N ) (partition of m into ≤ N parts) linearly independent homomorphisms W (mγ) → C. It is easily verified that the d(m|N ) sections so obtained are precisely the sections obtained by taking m-fold products of the d sections arising from the homomorphisms W (γ) → Vµ . We have proved the following lemma. Lemma 2. The algebra Eqk is infinite dimensional. Now let us consider the case µ 6= 0. It is an elementary exercise to verify that Vµ is Uq (l)-isomorphic to a Uq (l)-irreducible part of W (λ0 ), where λ0 is the dominant weight in the Weyl group orbit of µ. Thus there is a non-trivial Uq (l)-homomorphism W (λ0 ) → Vµ , and this determines at least dλ0 linearly independent sections in Fq (Vµ ).
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Further linearly independent sections can be constructed explicitly using Lemma 1 which promises a family of homomorphisms W (λ0 + mγ) → Vµ
m ∈ N+ .
Although we have fallen short of a classification of the sections in Eqk (Vµ ) we have established that Eqk (Vµ ) is infinite dimensional. This immediately leads to the following result. Proposition 1. If the weight of any vector of V is Uq (g)-integral, then Eqk (V ) is an infinite dimensional vector space. Ultimately we want to identify Eqk (V ) as the space of sections of a quantum vector bundle over the quantum homogeneous space corresponding to Eqk . However, recall that in classical differential geometry, the space H of sections of a vector bundle over a compact manifold M furnishes a module over the algebra A(M ) of functions. It then follows from the Serre–Swan theorem that this module must be projective and is of finite type. Conversely, any projective module of finite type over A(M ) is isomorphic to the sections of some vector bundle over M . This result is taken as the starting point for studying vector bundles in noncommutative geometry: one defines a vector bundle over a noncommutative space in terms of the space of sections which is required to be a finite type projective module over the noncommutative analog of the algebra of functions on this noncommutative space. Therefore, we need to understand the structure of Eqk (V ) in relation to Eqk . First note the following result. Proposition 2. Eqk (V ) furnishes a two-sided Eqk -module. Explicitly, the left and right actions of Eqk on Eqk (V ) are respectively defined by aζ =
X
afr ⊗ vr ,
r
for all ζ =
P r
ζa =
X
fr a ⊗ vr ,
r
fr ⊗ vr ∈ Eqk (V ). Now, for p ∈ UR q (k), we have p ◦ (aζ) =
X
{p(1) ◦ a}{p(2) ◦ ζ}
(p)
= a{p ◦ ζ} = (idAq (g) ⊗ S(p))aζ ; p ◦ (ζa) =
X
{p(1) ◦ ζ}{p(2) ◦ a}
(p)
= {p ◦ ζ}a = (idAq (g) ⊗ S(p))ζa . This confirms the Proposition. When the inducing module is actually the restriction of a finite dimensional left Uq (g)-module W , Eqk (W ) assumes a particularly simple structure.
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Proposition 3. Let W be a finite dimensional left Uq (g)-module, which we k k regard as a left UR q (k)-module by restriction. Then Eq (W ) is isomorphic to Eq ⊗ W k as both a left and a right Eq -module. Proof. We first construct the right Eqk -module isomorphism. Being a left Uq (g)module, W carries a natural right Aq (g) co-module structure with the co-module action δ : W → W ⊗ Tq (g) ⊂ W ⊗ Aq (g) defined for any element w ∈ W by δ(w)(x) = xw ,
∀ x ∈ Uq (g) .
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Define the map η : Eq ⊗ W → Eq ⊗ W by the composition of the maps P −1
id⊗δ
123 Tq (g) ⊗ Eq ⊗ W → Eq ⊗ W , Eq ⊗ W −→ Eq ⊗ W ⊗ Tq (g) −→
where the last map is the multiplication of Aq (g), and P123 is the permutation map defined by P123 : Aq (g) ⊗ Aq (g) ⊗ V → Aq (g) ⊗ V ⊗ Aq (g) f1 ⊗ f2 ⊗ v 7→ f2 ⊗ v ⊗ f1 . Then η defines a right Eqk -module isomorphism, with the inverse map given by the composition (S⊗id⊗id)P −1
id⊗δ
→ Tq (g) ⊗ Eq ⊗ W → Eq ⊗ W , Eq ⊗ W −→ Eq ⊗ W ⊗ Tq (g) −−−−−−−−123 where the last map is again the multiplication of Aq (g). It is not difficult to show that X (idAq (g) ⊗ x(1) )η(x(2) ◦ ζ) , x ◦ η(ζ) = (x)
x ◦ η −1 (ζ) =
X
(idAq (g) ⊗ S(x(1) ))η −1 (x(2) ◦ ζ) ,
∀ ζ ∈ Eq ⊗ W ,
x ∈ Uq (g) .
(x)
Consider ζ ∈ Eqk (W ). We have p ◦ η(ζ) =
X
(idAq (g) ⊗ p(1) )η(p(2) ◦ ζ)
(p)
=
X
(idAq (g) ⊗ p(1) S(p(2) ))η(ζ)
(p)
= (p)η(ζ) , Hence
⊂
∀ p ∈ Uq (l) .
⊗ W . Conversely, given any ξ ∈ Eqk ⊗ W , we have X (idAq (g) ⊗ S(p(1) ))η −1 (p(2) ◦ ξ) p ◦ η −1 (ξ) =
η(Eqk (W ))
Eqk
(p)
=
X
(idAq (g) ⊗ (p(2) )S(p(1) ))η −1 (ξ)
(p)
= (idAq (g) ⊗ S(p))η −1 (ξ) ,
∀ p ∈ Uq (l) .
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Thus η −1 (Eqk ⊗ W ) ⊂ Eq (W ). Therefore the restriction of η provides the desired right Eqk -module isomorphism. The left module isomorphism is given by the restriction of κ : Eq ⊗ W → Eq ⊗ W defined by the composition of the following maps: id⊗δ
−−−−−−− → Eq ⊗ Tq (g) ⊗ W → Eq ⊗ W , Eq ⊗ W −→Eq ⊗ W ⊗ Tq (g) id⊗(S⊗id)P where P : W ⊗ Aq (g) → Aq (g) ⊗ W , w ⊗ f 7→ f ⊗ w .
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The inverse map κ−1 is given by id⊗δ
−−−−−−−→ Eq ⊗ Tq (g) ⊗ W → Eq ⊗ W . Eq ⊗ W −→ Eq ⊗ W ⊗ Tq (g) id⊗(S⊗id)P
Let Vµ be a finite dimensional irreducible Uq (l)-module with highest weight µ, which is integral with respect to g. Then Vµ can always be embedded into an irreducible Uq (g)-module W (σ(µ)) with a g integral dominant highest weight σ(µ), where σ is some element of the Weyl group W of g. Such a σ always exists, and belongs to the subgroup W l ⊂ W, which leaves invariant the set of the positive roots of l. Since Uq (l) is a reductive subalgebra of Uq (g), all finite dimensional representations of Uq (l) are completely reducible. Hence, W (σ(µ)) can be decomposed into a direct sum of Uq (l)-modules: W (σ(µ)) = Vµ ⊕ Vµ⊥ . Using the complete reducibility of finite dimensional Uq (l)-modules again, we conclude that if the weights of the finite dimensional Uq (l)-module V are all integral with respect to Uq (g), then there exist another Uq (l)-module V ⊥ and a finite dimensional Uq (g)-module W such that V ⊕V⊥ =W . It then immediately follows Proposition 3 that Eqk (V ) ⊕ Eqk (V ⊥ ) = Eqk ⊗ W , that is: Theorem 1. Eqk (V ) is projective and of finite type both as a left and right module over the algebra Eqk of functions on the quantum homogeneous space. We will call Eqk (V ) the space of sections of a quantum vector bundle over the quantum homogeneous space associated with Eqk . As we have already indicated, the existence of connections on a bundle is equivalent to projectivity of the space of sections. Thus Theorem 1 establishes that there is a notion of differential geometry on the quantum homogeneous bundles. This has been the subject of a recent extensive investigation by the authors and the results will be described in a sequel to the present paper.
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Following the classical terminology, we will say that the quantum homogeneous vector bundle is trivial if its sections form a free module over the algebra of functions on the base space. Now Proposition 3 immediately leads to: Proposition 4. The quantum homogeneous vector bundle induced from the restriction of a finite dimensional Uq (g)-module is trivial. 4. Induced Representations In this section, we apply the quantum homogeneous vector bundles to study the geometry of quantum group representations. Two results, the quantum Frobenius reciprocity and a quantum Borel–Weil theorem, will be presented here. Algebraic versions of these results were obtained before by other people [14] without the framework of quantum homogeneous vector bundles. Our main interest is the geometrical significance of these results, part of which will be made clear by the proofs provided for them here. The proofs are also different from and much more elementary than those of [14]. We will also apply representation theoretical techniques to establish some useful results on the geometry of quantum homogeneous vector bundles, which would be difficult to obtain by other means. The importance of quantum homogeneous vector bundles for the representation theory of quantum groups stems from the following results. Theorem 2. Eqk (V ) furnishes (i) a left Uq (g)-module under ·; and (ii) Fq (V ) forms a left Aq (g) co-module under the co-action ω = (∆ ⊗ idV ). Part (i) can be confirmed by the following calculation: for any x ∈ Uq (g), p ∈ Uq (l), ζ ∈ Eqk (V ), p ◦ (x · ζ) = x · (p ◦ ζ) = (idAq (g) ⊗ S(p))(x · ζ) . Part (ii) follows from (idAq (g) ⊗ p◦)ω(ζ) = (idAq (g) ⊗ p◦)(∆ ⊗ idV )ζ = (∆ ⊗ idV )(p ◦ ζ) = ω(idAq (g) ⊗ S(p))ζ . Note that the left Uq (g) action of (i) and left Aq (g) co-action ω on Fq (V ) are closely related: the object ω ˜ = P123 ω defines a right Aq (g) co-action on Fq (V ) which is dual to the left Uq (g) action. We call Eqk (V ) an induced Uq (g)-module, and also call Fq (V ) an induced Aq (g) co-module. Very similar induced modules have been used by Dobrev [20] in the
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study of quantum group invariant difference operators. We expect quantum homogeneous vector bundles to provide a natural framework for further research in this direction. We have the following quantum analog of Frobenius reciprocity. Proposition 5. Let W be a Uq (g)-module, the restriction of which furnishes a (k) module in a natural way. Then there exists a canonical isomorphism UR q HomUq (g) (W, Eqk (V )) ∼ = HomUR (k) (W, V ) , q
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where Uq (g) acts on the left module Eqk (V ) via the · action. Proof. We prove the Proposition by explicitly constructing the isomorphism, which we claim to be the linear map F : HomUq (g) (W, Eqk (V )) → HomUR (k) (W, V ) , q ψ 7→ ψ(1Uq (g) ) , with the inverse map F¯ : HomUR (k) (W, V ) → HomUq (g) (W, Eqk (V )) , q φ 7→ φ¯ = (S ⊗ φ)P δ , where δ : W → W ⊗ Tq (g) ⊂ W ⊗ Aq (g) is the right Aq (g) co-module action defined by (14), and P is the permutation map (15). As for F , we need to show that its image is contained in HomUR (k) (W, V ). q
Consider ψ ∈ HomUq (g) (W, Eqk (V )). For any p ∈ UR q (k) and w ∈ W , we have p(F ψ(w)) = ((idAq (g) ⊗ p)ψ(w))(11Uq (g) ) = (S −1 (p) ◦ ψ(w))(11Uq (g) ) , where we have used the defining property of Eqk (V ). Note that (S −1 (p) ◦ ψ(w))(11Uq (g) ) = (p · ψ(w))(11Uq (g) ) . The Uq (g)-module structure of Eqk (V ) and the given condition that ψ is a Uq (g)module homomorphism immediately leads to p(F ψ(w)) = ψ(pw)(11Uq (g) ) = F ψ(pw) ,
p ∈ UR q (k) ,
w∈W.
In order to show that F¯ is the inverse of F , we first need to demonstrate that the image Im(F¯ ) of F¯ is contained in HomUq (g) (W, Eqk (V )). Note that Im(F¯ ) ⊂ HomC (W, Tq (g) ⊗ V ). Some relatively simple manipulations lead to ¯ ¯ (x · φ(w)) = φ(xw) , ¯ ¯ (p ◦ φ(w)) = (idAq (g) ⊗ S(p))φ(w) ,
∀ x ∈ Uq (g) ,
p ∈ UR q (k) ,
w∈W.
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Therefore, Im(F¯ ) ⊂ HomUq (g) (W, Eqk (V )). Now we show that F and F¯ are inverse to each other. For ψ ∈ HomUq (g) (W, Eqk (V )), and φ ∈ HomUR (k) (W, V ), we have q
(F F¯ φ)(w) = (F¯ φ)(w)(1U q (g) ) = φ(w) , (F¯ F ψ)(w)(x) = (F ψ)(S(x)w) 3 ψ(S(x)w)(1Uq (g) ) = (S(x) · ψ(w))(1Uq (g) ) = ψ(w)(x) ,
x ∈ Uq (g) ,
w∈W.
This completes the proof of the Proposition.
Let Vµ be a finite dimensional irreducible Uq (p)-module with highest weight µ and lowest weight µ ˜. Recall that any two norms on finite dimensional vector spaces determine the same topology. Thus we may speak of convergence of a sequence in such a space without reference to a particular norm. Let us observe here that there is a similar freedom for a certain class of norms on Eq ⊗C Vµ . To each basis {vr } of Vµ we may define a norm on Eq ⊗C Vµ by ζ=
X r
fr ⊗ vr ,
kζk2 =
X
kfr k2op .
r
It is easily verified that convergence in the norm corresponding to one basis for Vµ implies convergence in all other norms defined this way. Thus, given Vµ , we simply fix a basis and define k · k to be the norm relative to that basis. Recall the action of Uq (g) on Eq ⊗ Vµ . Since Vµ is a Uq (p)-module the following is a well defined subspace of Eq ⊗C Vµ , Definition 4. Oq (Vµ ) := {ζ ∈ Eq ⊗ Vµ | p ◦ ζ = (idAq (g) ⊗ S(p))ζ ,
∀ p ∈ Uq (p)} .
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Clearly Oq (Vµ ) ⊂ Eqk (Vµ ). We will regard Oq (Vµ ) as the quantum analog of the space of holomorphic sections. Recall that we use the notation W (λ) to denote the irreducible Uq (g)-module with highest weight λ. We have the following result. Theorem 3. There exists the following Uq (g)-module isomorphism ( W ((−˜ µ)† ), −˜ µ ∈ P+ , ∼ Oq (Vµ ) = 0, otherwise .
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Proof. Let ζ ∈ Oq (Vµ ). Let {ζn } be a sequence in Tq (g) ⊗ Vµ such that ζn → ζ in the norm || · || described above. Each ζn can be expressed in the form X X (λ) (λ),n S(tji ) ⊗ vij , ζn = λ∈P+ i,j (λ),n
for some vij
∈ Vµ (i, j = 1, . . . , dλ ). Arguing as in the proof of Proposition 1 (λ),n
∈ HomC (W (λ), Vµ ) such one concludes, for each λ ∈ P+ , that there exist φi (λ),n (λ),n (λ) (λ) that vij = φi (wj ), where {wi } is the basis of W (λ), relative to which the irreducible representation t(λ) of Uq (g) is defined. Now we can rewrite ζn as X X (λ) (λ),n (λ) ζn = S(tji ) ⊗ φi (wj ) . λ∈P+ i,j
It is clear from this that ζ is determined by the sequences of linear homomorphisms (λ),n . Note that φi kζn+m − ζn k → 0 , n → ∞ . (λ)
Since the S(tij ) are linearly independent, this implies that for each λ ∈ P+ (λ),n
(λ)
and i, j ∈ {1, 2, . . . , dλ }, φi (wj ) is a Cauchy sequence in Vµ . But since Hom(W (λ), Vµ ) is a finite dimensional complex vector space with the basis {vr ⊗ (λ) w ¯j }, it is clear that this further implies that, for each λ ∈ P+ and i ∈ {1, 2, . . . , dλ }, (λ),n
φi
is a Cauchy sequence in Hom(W (λ), Vµ ) and so (λ),n
lim φi
n→∞
(λ)
= φi
∈ HomC (W (λ), Vµ ) .
(λ)
Now we will show that this limit φi must in fact be a Uq (p)-module homomorphism. The defining property of O q (Vµ ) states that p ◦ ζ = (idAq (g) ⊗ S(p))ζ ,
∀ p ∈ Uq (p) .
Thus, for each p, kp ◦ ζn − (idAq (g) ⊗ S(p))ζn k → 0 . (λ)
Again using the linear independence of the S(tij )’s, we see that this implies that, for each i, k ∈ {1, . . . , dλ }, X (λ) (λ),n (λ) (λ),n (λ) tjk (S(p))φi (wj ) − S(p)φi (wk ) j
is a null sequence. Thus in the limit we have (λ)
(λ)
(λ)
(λ)
φi (pwj ) = pφi (wj ) , (λ)
This is precisely the statement that the φi (λ)
φi
∀ p ∈ Uq (p) .
are Uq (p)-module homomorphisms,
∈ HomUq (p) (W (λ), Vµ ) ⊂ HomC (W (λ), Vµ ) ,
∀ i ∈ 1, . . . , dλ .
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It can immediately be shown that (λ)
φi
= ci φ(λ) ,
ci ∈ C ,
and φ(λ) may be nonzero only when ¯=µ λ ˜. Hence, if −˜ µ 6∈ P+ , we have Oq (Vµ ) = 0. When −˜ µ ∈ P+ , we set ν = (−˜ µ)† . Then, we may conclude that Oq (Vµ ) is spanned by X (ν) (ν) S(tji ) ⊗ φ(ν) (wj ) , ζi =
(19)
j
which are obviously linearly independent. Furthermore, X (ν) tji (x)ζj , x ∈ Uq (g) . x · ζi = j
Thus Oq (Vµ ) ∼ = W (ν), and this completes the proof of the theorem.
Note that our proof is constructive. It yields the explicit form of the isomorphism (18): µ ∈ P+ , then the following composition of maps defines Corollary 1. If ν † = −˜ the Uq (g)-module isomorphism W (ν) ∼ = Oq (Vµ ), (ν)
W (ν) (S⊗id)P −−−−−→δ Oq (W (ν)) id⊗φ −−−−→ Oq (Vµ ) ,
(20)
where φ(ν) is the Uq (p)-epimorphism W (ν) → Vµ . We should point out that a quantum Borel–Weil theorem was obtained before for quantum GL(n) in [21, 22], and for arbitrary quantum algebras in [14], in an algebraic setting without the framework of quantum homogeneous vector bundles. Also in [23], a quantum Borel–Weil theorem for the covariant and contravariant tensor representations of quantum GL(m|n) was obtained along a similar line as that adopted here. We should also mention that coherent states of compact quantum groups were investigated in [24, 25] from a representation theoretical viewpoint. The results reported in that reference acquire a natural interpretation within the framework of quantum homogeneous vector bundles. From the quantum Borel–Weil Theorem 3, we can easily deduce useful results on the geometry of quantum homogeneous vector bundles, which would be difficult to establish by other means. Recall that in classical geometry, the space of global holomorphic sections of a vector bundle over a compact complex manifold is finite dimensional. In particular, a complex analytic function on such a manifold must be constant. These results extend to the quantum homogeneous vector bundle setting.
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Corollary 2. (i) dimC Oq (V ) < ∞; (ii) Oq (C) = C. Combining the Corollaries with Proposition 3, we also obtain Corollary 3. Oq (W ) ∼ =⊗W . Acknowledgement This work is supported by the Australian Research Council. References [1] R. J. Baston and M. G. Eastwood, The Penrose Transform; its Interaction with Representation Theory, Oxford Univ. Press, Oxford, 1989. [2] V. G. Drinfeld, Quantum Groups, Proc. Int. Cong. Math., Berkeley, 1 (1986) 789. [3] Yu. I. Manin, Quantum Groups and Noncommutative Geometry, Universite de Montreal, Centre de Recherches Mathematiques, Montreal, PQ, 1988. [4] S. L. Woronowicz, “Differential calculus on compact matrix pseudo groups (quantum groups)”, Commun. Math. Phys. 122 (1989) 125. [5] C. S. Chu, P. M. Ho and B. Zumino, “Some complex quantum manifolds and their geometry”, preprint, 1996. [6] S. Majid, “Advances in quantum and braided geometry”, preprint, 1996. [7] A. Connes, Noncommutative Geometry, Academic Press, 1994. [8] T. Brzezinski and S. Majid, “Quantum differentials and the q-monopole revisited”, preprint, 1997. [9] S. L. Woronowicz, “Compact matrix pseudo groups”, Commun. Math. Phys. 111 (1987) 613. [10] H. J. Schneider, “Principle homogeneous spaces for arbitrary Hopf algebras”, Israel J. Math. 72 (1990) 196. [11] V. Lakshmibai and N. Yu. Reshetikhin, “Quantum deformation of flag and Schubert schemes”, C. R. Acad. Sci. Paris. Ser. I. Math. 313 (3) (1991) 121–126. [12] Y. Soibelman, “On quantum flag manifolds”, Funct. Analy. Appl. 26 (1992) 225–227. [13] M. S. Dijkhuizen and T. H. Koorwinder, “Quantum homogeneous spaces, duality and quantum 2-spheres”, Geom. Dedicata 52 (1994) 291. [14] H. H. Andersen, P. Polo and K. X. Wen, “Representations of quantum algebras”, Invent. Math. 104 (1991) 1–59. [15] Y. Soibelman, “The algebra of functions on compact quantum groups and their representations”, Leningrad Math. J. 2 (1991) 161–178. [16] V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press, Cambridge, 1994. [17] M. Jimbo, “A q-difference analogue of of U (g) and the Yang–Baxter equation”, Lett. Math. Phys. 10 (1985) 63. [18] L. D. Faddeev, N. Yu. Reshetikhin and L. A. Takhtajan, “Quantization of Lie groups and Lie algebras”, Leningrad Math. J. 1 (1990) 193. [19] M. S. Dijkhuizen and T. H. Koorwinder, “CQG algebras: A direct algebraic approach to compact quantum groups”, Lett. Math. Phys. 32 (1994) 315. [20] V. K. Dobrev, J. Phys. A27 (1994) 4841.
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[21] B. Parshall and J. P. Wang, “Quantum Linear Groups”, Memoirs Amer. Math. Soc. 89 (439) (1991) 1–157. [22] M. Noumi, H. Yamada and K. Mimachi, “Finite-dimensional representations of the quantum group GLq (n, C) and zonal spherical functions on Uq (n − 1)\Uq (n)”, Japanese J. Math. 19 (1993) 31. [23] R. B. Zhang, “Structure and representations of the quantum general linear supergroup”, Commun. Math. Phys. 195 (1998) 525–547. [24] Y. Soibelman, “Orbit method for the algebras of functions on quantum groups and coherent states. I.”, Int. Math. Res. Notices 6 (1993) 151. [25] B. Jurco and P. Stovicek, “Coherent states for compact quantum groups”, Commun. Math. Phys. 182 (1996) 221.
DIAGONAL CROSSED PRODUCTS BY DUALS OF QUASI-QUANTUM GROUPS FRANK HAUSSER∗ Freie Universi¨ at Berlin, Institut f¨ ur Theoretische Physik Arnimalle 14, D-14195 Berlin E-mail : [email protected]
FLORIAN NILL∗ Freie Universi¨ at Berlin, Institut f¨ ur Theoretische Physik Arnimalle 14, D-14195 Berlin E-mail : [email protected] A two-sided coaction δ : M → G ⊗ M ⊗ G of a Hopf algebra (G, ∆, , S) on an associative algebra M is an algebra map of the form δ = (λ ⊗ idM ) ◦ ρ = (idM ⊗ ρ) ◦ λ, where (λ, ρ) is a commuting pair of left and right G-coactions on M, respectively. Denoting the associated commuting right and left actions of the dual Hopf algebra Gˆ on M by / and . , respectively, we define the diagonal crossed product M ./ Gˆ to be the algebra generated by M and Gˆ with relations given by ϕ m = (ϕ(1) . m / Sˆ−1 (ϕ(3) )) ϕ(2) , m ∈ M, ϕ ∈ Gˆ . We give a natural generalization of this construction to the case where G is a quasiHopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e. where the coproduct ∆ is non-unital). In these cases our diagonal crossed ˆ even product will still be an associative algebra structure on M ⊗ Gˆ extending M ≡ M ⊗ 1, though the analogue of an ordinary crossed product M o Gˆ in general is not well defined as an associative algebra. Applications of our formalism include the field algebra constructions with quasiquantum group symmetry given by G. Mack and V. Schomerus [31, 47] as well as the formulation of Hopf spin chains or lattice current algebras based on truncated quantum groups at roots of unity. In the case M = G and λ = ρ = ∆ we obtain an explicit definition of the quantum double D(G) for quasi-Hopf algebras G, which before had been described in the form of an implicit Tannaka–Krein reconstruction procedure by S. Majid [35]. We prove that D(G) is itself a (weak) quasi-bialgebra and that any diagonal crossed product M ./ Gˆ naturally admits a two-sided D(G)-coaction. In particular, the above-mentioned lattice models always admit the quantum double D(G) as a localized cosymmetry, generalizing results of Nill and Szlach´ anyi [42]. A complete proof that D(G) is even a (weak) quasi-triangular quasi-Hopf algebra will be given in a separate paper [27].
Contents 1. Introduction and Summary of Results 2. Diagonal Crossed Products by Duals of Hopf Algebras 2.1. Coactions and crossed products 2.2. Two-sided coactions and diagonal crossed products 2.3. Generating matrices 2.4. Hopf spin chains and lattice current algebras 2.5. Double crossed products ∗ Supported
by DF G, SFB 288 “Differentialgeometrie und Quantenphysik”. 553
Reviews in Mathematical Physics, Vol. 11, No. 5 (1999) 553–629 c World Scientific Publishing Company
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3. Diagonal Crossed Products by Duals of Quasi-Hopf Algebras 3.1. Quasi-Hopf algebras 3.2. Coactions of quasi-Hopf algebras 3.3. Two-sided coactions 3.4. The algebras Gˆ ./ M and M ./ Gˆ 3.5. Generating matrices 3.6. Proofs 4. Generalization to Weak Quasi-Hopf Algebras 4.1. Weak quasi-Hopf algebras 4.2. Diagonal crossed products 5. Applications 5.1. The quantum double D(G) 5.2. Two-sided crossed products 5.3. Quasi-Hopf spin chains and lattice current algebras 5.4. Field algebra construction with quasi-Hopf symmetry Appendix A. Representation Theoretic Interpretation References
576 579 581 583 586 589 595 603 604 606 608 609 612 616 618 620 629
1. Introduction and Summary of Results During the last decade quantum groups have become the most fashionable candidates describing the symmetry in low dimensional quantum field theories (QFT)a or lattice modelsb . Here, in an axiomatic approach by a “symmetry algebra” G one means a ∗-algebra acting on the Hilbert space of physical states H, such that: • observables and space-time translations commute with G, • charge creating fields fall into multiplets transforming covariantly under the action of G, • equivalence classes of irreducible representations of G are in one-to-one correspondence with the Doplicher–Haag–Roberts (DHR) superselection sectors of the observable algebra A, such that the fusion rules of RepDHR A and Rep G also coincide. It is well known that the results of Doplicher and Roberts [15, 16] characterizing G as a compact group (or the associated group algebra) break down in low dimensions due to the appearance of braid statistics. It was soon realized that at least for rational theories (i.e. with a finite number of sectors) quantum groups are also ruled out, unless all sectors have integer statistical dimensions (see e.g. [21] for a review or [39] for a specific discussion of q-dimensions in finite quantum groups). Based on the theory of quasi-Hopf algebras introduced by Drinfel’d [18], G. Mack and V. Schomerus [31] have proposed the notion of weak quasi-Hopf algebras G as appropriate symmetry candidates, where “weak” means that the tensor product of two “physical” representations of G may also contain “unphysical” subrepresentations (i.e. of q-dimension ≤ 0), which have to be discarded. Examples are semisimple quotients of q-deformations of classical groups at q = roots of unity. In this way non-integer dimensions could successfully be incorporated. The price to pay was that now commutation relations of G-covariant charged fields involve a See [8, 9, 13, 21–23, 30, 31, 36–38, 43, 46, 49, 51]. b See [2–4, 12, 19, 20, 28, 42, 41, 44, 45, 50].
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operator valued R-matrices and, more drastically, the operator product expansion for G-covariant multiplets of charged fields involves non-scalar coefficients with values in G. Thus, the analogue of the “would-be” DHR-field algebra F is no longer algebraically closed. Instead, Mack and Schomerus have proposed a new “covariant product” for charged fields, which does not lead outside of F , but which is no longer associative. In [47] Schomerus has analyzed this scenario somewhat more systematically in the framework of DHR-theory, showing that a weak quasi-Hopf algebra G and a field “algebra” F may always be constructed such that the combined algebra F ∨ G is associative and satisfies all desired properties, except that F ⊂ F ∨ G is only a linear subspace but not a subalgebra. Technically, the reason for this lies in the fact that the dual Gˆ of a quasi-Hopf algebra is not an associative algebra. One should also remark at this point that the above reconstruction of G from the category of DHR-endomorphisms is not unique. Also, in a more mathematical framework a general Tannaka–Krein like reconstruction theorem for quasi-Hopf algebras has been obtained by S. Majid [33] and for weak quasi-Hopf algebras by [24]. To study quantum symmetries on the lattice in an axiomatic approach, K. Szlach´anyi and P. Vecserny´es [50] have proposed an “amplified” version of the DHR-theory, which also applies to locally finite dimensional lattice models. This setting has been further developed by [41, 42], where based on the example of Hopf spin chains the authors proposed the notion of a universal localized cosymmetry ρ : A → A ⊗ G, incorporating all sectors ρI of A via ρI = (idA ⊗ πI ) ◦ ρ, πI ∈ Rep G. In the specific example studied by [41, 42] G was given by a quantum double and the cosymmetry ρ was given by a coaction of G on A. Related results have later been obtained for lattice current algebras [4], the later actually being a special case of the Hopf spin chains of [42] (see [40] and Sec. 2.4). The analogue of a DHR-field algebra for these models is now given by the standard crossed product F ≡ A o Gˆ [41], where Gˆ is the Hopf algebra dual to G. Now the methods and results of these works were still restricted to ordinary Hopf algebras and therefore to integer dimensions. To formulate lattice current algebras at roots of unity one may of course identify them with the boundary part of lattice Chern–Simons algebras [5–7] defined on a disk. Nevertheless, it remains unclear whether and how for q = root of unity the structural results of [4] survive the truncation to the semi-simple (“physical”) quotients. Similarly, the generalizations of the model, the methods and the results of [42] to weak quasi-quantum groups are by no means obvious. In particular one would like to know whether and in what sense in such models universal localized cosymmetries ρ : A → A ⊗ G still provide coactions and whether G would still be (an analogue of) a quantum double of a quasi-Hopf algebra, possibly in the sense recently described by Majid [35]. In this work we present a theory of left, right and two-sided coactions of a (weak) quasi-Hopf algebra G on an associative algebra M. Based on these structures we then provide a new construction of what we call the diagonal crossed ˆ which we will show to be the appropriate mathematical structure product M ./ G, underlying all constructions discussed above. In particular, M ./ Gˆ will always ˆ On the other hand, the linear be an associative algebra extending M ≡ M ./ 1.
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ˆ unless G is an subspace 1M ./ Gˆ will in general not be a subalgebra of M ./ G, ordinary (i.e. coassociative) Hopf algebra. The basic idea for this construction comes from generalizing the relations defining the quantum double. To this end we start from an algebra M equipped with a (quasi-)commuting pair of right and left G-coactions, ρ : M → M ⊗ G and λ : M → G ⊗ M and denote δl := (λ ⊗ id) ◦ ρ and δr := (id ⊗ ρ) ◦ λ as the associated equivalent two-sided coactions. In the simplest case of G being an ordinary Hopf algebra and (λ, ρ) being strictly commuting (i.e. δl = δr ) this amounts to providing a commuting pair of left and right Hopf module actions . : Gˆ ⊗ M → M (dual to ρ) and / : M ⊗ Gˆ → M (dual to λ) of the dual Hopf algebra Gˆ on M. In this case our diagonal crossed product M ./ Gˆ is defined to be generated by M and Gˆ as unital subalgebras with commutation relations given by (Sˆ : Gˆ → Gˆ being the antipode) (1.1) ϕ m = (ϕ(1) . m / Sˆ−1 (ϕ(3) )) ϕ(2) , m ∈ M, ϕ ∈ Gˆ . Note that for M = G and ρ = λ = ∆ the coproduct on G, these are the defining relations of the quantum double D(G) [17], and therefore G ./ Gˆ = D(G). Introducing the “generating matrix” X ˆ , eµ ⊗ eµ ∈ G ⊗ Gˆ ⊂ G ⊗ (M ./ G) Γ := µ
ˆ (1.1) is equivalent to where eµ ∈ G is a basis with dual basis eµ ∈ G, Γλ(m) = ρop (m)Γ,
∀m ∈ M.
(1.2)
Moreover, in this case Gˆ ⊂ M ./ Gˆ being a unital subalgebra is equivalent to ( ⊗ id)(Γ) = 1 Γ13 Γ23 = (∆ ⊗ id)(Γ) ,
(1.3) (1.4)
ˆ the indices denoting the embeddings where (1.4) is an identity in G ⊗ G ⊗ (M ./ G), of tensor factors. We call Γ the universal normal and coherent λρ-intertwiner in ˆ where normality is the property (1.3) and coherence is the property G ⊗ (M ./ G), (1.4). Again, for M = G and M ./ G = D(G) Eqs. (1.2)–(1.4) are precisely the defining relations for the generating matrix D ≡ Γ D(G) of the quantum double (see e.g. [40, Lemma 5.2]). Inspired by the techniques of [5–7] we show in the main body of this work how to generalize the notion of coherent λρ-intertwiners to the case of (weak) quasi-Hopf algebras G, such that analogues of the Eqs. (1.2)–(1.4) still serve as the defining ˆ We also show that relations of an associative algebra extending M ≡ M ./ 1. diagonal crossed products may equivalently be modeled on the linear spaces M ⊗ Gˆ or Gˆ ⊗ M (or — in the weak case — certain subspaces thereof). The basic model for this generalization is again given by M = G with its natural two-sided G-coactions δl := (∆ ⊗ id) ◦ ∆ and δr := (id ⊗ ∆) ◦ ∆. In this case our construction provides a definition of the quantum double D(G) for (weak) quasiHopf algebras G. In fact, we show that Rep D(G) coincides with what has been called the “double of the category” Rep G in [35]. Hence our definition provides a concrete
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realization of the abstract Tannaka–Krein like reconstruction of the quantum double given by [35]. We also give a proof that D(G) is a (weak) quasi-bialgebra. In [27] we show, that D(G) is in fact a (weak) quasi-triangular quasi-Hopf algebra, and there we also visualize many of our (otherwise almost untraceable) algebraic identities in terms of graphical proofs. The field algebra construction of [30, 47] may also be described as a diagonal crossed product M ./ Gˆ within our formalism by putting M = A⊗G, where A is the observable algebra. In this case the right G-coaction ρ : M → M ⊗ G is a localized cosymmetry acting only on A, whereas the left G-coaction λ : M → G ⊗M only acts on G, where it is given by the coproduct ∆, see Sec. 5.4 for a rough sketch. A more detailed account of this within an appropriate von-Neumann algebraic framework will be given elsewhere. The application of our formalism to G-spin quantum chains is given by putting in the previous example also A = G and ρ = ∆, in which case M ./ Gˆ ∼ = G o Gˆ n G becomes a two-sided crossed product. We take this construction as a building block of a quantum chain living on two neighboring sites (carrying the copies of G) joined ˆ by a link (carrying the copy of G). We show how this construction iterates to provide a local net of associative algebras A(I) for any lattice interval I bounded by sites. Generalizing the methods of [42] we also construct localized coactions of the quantum double D(G) on such (weak) quasi-Hopf spin chains. Periodic boundary conditions for these models are again described as a diagonal crossed product of the open chain by a copy of Gˆ sitting on the link joining the end points. In this way we arrive at a formulation of lattice current algebras at roots of unity by adjusting the transformation rules of [40] to the quasi-coassociative setting. More detailed results on these models are given in [25]. In conclusion we point out that most of our algebraic constructions are based on representation categorical concepts and may therefore also be visualized by graphical proofs. We will put more emphasis on this technique in [26] when providing further results on the quantum double D(G). We also remark that without mentioning explicitly at every instance the (weak, quasi) Hopf algebras G are always supposed to be finite dimensional. Although we believe that many aspects of our formalism would also carry over to an infinite dimensional setting, we don’t consider it worthwhile to discuss this complication at present. More importantly, in applications to quantum physics one should extend our formalism to incorporate C ∗ - or von-Neumann algebraic structures, which we will come back to in the near future when discussing our examples in more detail. Remark. The present paper is a streamlined version of our preprint [26], where the interested reader may also find various technical proofs omitted here for reasons of readability. 2. Diagonal Crossed Products by Duals of Hopf Algebras To strip off all technicalities from the main ideas, in this first section we restrict ourselves to strictly coassociative Hopf algebras G. After reviewing some basic
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notions on coactions and crossed products in Sec. 2.1 we introduce as a new construction the diagonal crossed product M ./ Gˆ of a unital algebra M and the dual Gˆ of a Hopf algebra G in Sec. 2.2. Section 2.3 gives a reformulation of this construction in terms of generating matrices summarized in Theorem 2.13. We will see later on in Sec. 3 that this theorem allows for a generalization to quasi-Hopf algebras given in Theorem 3.1 (and also to weak quasi-Hopf algebras given in Theorem 4.1), which may be viewed as the heart piece of this work. In order to carefully prepare the much more complicated quasi-coassociative scenario we deliberately present this construction in rather elementary steps. In Sec. 2.4 we reformulate the Hopf spin chains of [42] and also the lattice current algebras of [4] — both models being based on a Hopf algebra G — as iterated diagonal crossed products. This opens the way to generalize these models to (weak) quasi-Hopf algebras, thus covering the physically important case of truncated quantum groups at roots of unity, in Sec. 5. Although we do not establish any new results in Sec. 2.4, we think it to be quite illuminating that these two models may be based on the same algebraic construction. In particular the isomorphy of the two models — more exactly the second being obtained by imposing periodic boundary conditions on the first — as already established in [40], becomes rather obvious, as well as the role of the quantum double, describing the representation theory of both models. We conclude this section by relating our diagonal crossed product to the double crossed product construction of [32, 34] in Sec. 2.5. We emphasize that all concepts and constructions given in the subsequent chapters already appear in this first chapter. Thus it may also serve as an overview and the reader is invited to frequently return to Sec. 2 when feeling lost in the much more complicated treatment of the quasi-coassociative case in the following sections. 2.1. Coactions and crossed products To fix our conventions and notations we start with shortly reviewing some basic notions on Hopf module actions, coactions and crossed products. For full textbook treatments see e.g. [1, 34, 48]. We also introduce the “generating” matrix formalism. Throughout by an algebra we will mean an associative unital algebra over C and unless stated differently all algebra morphisms are supposed to be unit preserving. Let G and Gˆ be a dual pair of finite dimensional Hopf algebras. We denote elements of G by Roman letters a, b, c, . . . and elements of Gˆ by Greek letters ˆˆ ˆ Identifying G ˆ ∈ G. ϕ, ψ, χ, . . . . The units are denoted by 1 ∈ G and 1 = G, the dual pairing G ⊗ Gˆ → C is written as ha|ψi ≡ hψ|ai ∈ C,
a ∈ G, ψ ∈ Gˆ .
We denote ∆ : G → G ⊗ G the coproduct, : G → C the counit and S : G → G ˆ ˆ and Sˆ are the structural maps on G. ˆ We will use the antipode. Similarly, ∆, the Sweedler notation ∆(a) = a(1) ⊗ a(2) , (∆ ⊗ id)(∆(a)) ≡ (id ⊗ ∆)(∆(a)) = a(1) ⊗ a(2) ⊗ a(3) , etc., where the summation symbol and the summation indices are cop , where suppressed. Together with G we have the Hopf algebras Gop , G cop and Gop
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“op” refers to opposite multiplication and “cop” to opposite comultiplication. Note cop by S. that the antipode of Gop and G cop is given by S −1 and the antipode of Gop cop cop cop d cop ˆ d ˆ ˆ d = (G)op and Gop = (G)op . Also, Gop = (G) , G The notion of group actions on algebras and the associated crossed products generalize to Hopf algebras as follows: A (left) Hopf module action of G on a unital algebra M is a linear map . : G ⊗ M → M satisfying for all m, n ∈ M and a, b ∈ G a . (b . m) = (ab) . m a . (mn) = (a(1) . m)(a(2) . n) 1 . m = m,
a . 1M = (a) 1M .
Note that for group like elements a ∈ G (i.e. ∆(a) = a ⊗ a and a invertible), a . becomes an algebra automorphism of M, which means that Hopf module actions generalize the notion of group actions. Right Hopf module actions / : m⊗a 7→ m / a are defined analogously. There is also a dual version of Hopf module actions: A right coaction of G on an algebra M is an algebra map ρ : M → M ⊗ G satisfying (ρ ⊗ id) ◦ ρ = (id ⊗ ∆) ◦ ρ
(2.1)
(id ⊗ ) ◦ ρ = id .
(2.2)
Similarly, a left coaction λ is an algebra map λ : M → G ⊗ M satisfying (id ⊗ λ) ◦ λ = (∆ ⊗ id) ◦ λ
(2.3)
( ⊗ id) ◦ λ = id .
(2.4)
Similarly as for coproducts we will use the suggestive notations ρ(m) = m(0) ⊗ m(1) (ρ ⊗ id)(ρ(m)) ≡ (id ⊗ ∆)(ρ(m)) = m(0) ⊗ m(1) ⊗ m(2) (2.5) λ(m) = m(−1) ⊗ m(0) (∆ ⊗ id) ◦ λ ≡ (id ⊗ λ)(λ(m)) = m(−2) ⊗ m(−1) ⊗ m(0) etc., where again summation indices and a summation symbol are suppressed. In this way we will always have m(i) ∈ G for i 6= 0 and m(0) ∈ M. The notions of actions and coactions are dual to each other in the sense that there is a one-toone correspondence between right (left) coactions of G on M and left (right) Hopf module actions, respectively, of Gˆ on M given for ψ ∈ Gˆ and m ∈ M by ψ . m := (id ⊗ ψ)(ρ(m))
(2.6)
m / ψ := (ψ ⊗ id)(λ(m)) ,
(2.7)
where ϕ, ψ ∈ Gˆ and m, n ∈ M. As a particular example we recall the case M = G with ρ = λ = ∆. In this case we denote the associated left and right actions of
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ψ ∈ Gˆ on a ∈ G by ψ * a and a ( ψ, respectively. Analogously, choosing M = G ˆ one arrives at left and right actions of G on G, ˆ denoted by a * ψ with ρ = λ = ∆, and ψ ( a, respectively. Crossed products. Given a right coaction ρ : M → M ⊗ G with dual left ˆ G-action . one defines the (untwisted) crossed product (also called smash product) M o Gˆ to be the vector space M ⊗ Gˆ with associative algebra structure given for m, n ∈ M and ϕ, ψ ∈ Gˆ by (m o ϕ)(n o ψ) = (m(ϕ(1) . n) o ϕ(2) ψ) ,
(2.8)
where we use the notation m o ψ in place of m ⊗ ψ to emphasize the new algebraic ˆ is the unit in M o Gˆ and m 7→ (m o 1), ˆ ϕ 7→ (1M o ϕ) structure. Then 1M o 1 ˆ respectively. Similarly if provide unital inclusions M → M o Gˆ and Gˆ → M o G, λ : M → G ⊗ M is a left coaction with dual right action / then Gˆ n M denotes the associative algebra structure on Gˆ ⊗ M given by (ϕ n m)(ψ n n) = (ϕψ(1) n (m / ψ(2) )n)
(2.9)
containing again M and Gˆ as unital subalgebras. If there are several coactions under consideration we will also write Mρ o Gˆ and Gˆ n λ M, respectively. We note that (2.8) implies that as an algebra ˆ , M o Gˆ = MGˆ = GM ˆ In fact using the antipode ˆ and Gˆ ≡ 1M o G. where we have identified M ≡ M o 1 axioms one easily verifies from (2.8) ˆ . ˆ M o ϕ) = (1M o ϕ(2) )((Sˆ−1 (ϕ(1) ) . m) o 1) m o ϕ = (m o 1)(1
(2.10)
Similar statements hold in Gˆ n M. More generally we have: Lemma 2.1. Let . : Gˆ ⊗ M → M be a left Hopf module action and let A be an algebra containing M and Gˆ as unital subalgebras. Then in A the relations ϕ m = (ϕ(1) . m) ϕ(2) ,
ˆ ∀m ∈ M ∀ ϕ ∈ G,
m ϕ = ϕ(2) (Sˆ−1 (ϕ(1) ) . m) ,
ˆ ∀m ∈ M ∀ ϕ ∈ G,
(2.11) (2.12)
ˆ ⊂ A is a subalgebra and are equivalent and if these hold then MGˆ = GM M o Gˆ 3 (m o ϕ) 7→ m ϕ ∈ MGˆ is an algebra epimorphism. The proof of Lemma 2.1 is obvious from the antipode axioms and therefore omitted. A similar statement of course holds for the crossed product Gˆ n M.
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Generating matrices. We conclude this introductory part by describing crossed products in terms of the “generating matrix” formalism as advocated by the St. Petersburg school. Our presentation will closely follow the review of [40]. First ˆ V)∼ we note that since G is finite dimensional we may identify HomC (G, = G ⊗ V for any C-vector space V . In particular, the relation T (ϕ) = (ϕ ⊗ id)(T) ,
∀ ϕ ∈ Gˆ ,
(2.13)
provides a one-to-one correspondence between algebra maps T : Gˆ → A into some target algebra A and elements T ∈ G ⊗ A satisfying T13 T23 = (∆ ⊗ id)(T) ,
(2.14)
where (2.14) is to be understood as an identity in G ⊗ G ⊗ A, the upper indices indicating the canonical embedding of tensor factors (e.g. T23 = 1G ⊗ T, etc.). Throughout, we will call elements T ∈ G ⊗ A normal, if ( ⊗ id)(T) = 1A , which in (2.13) is equivalent to T : Gˆ → A being unit preserving. In what follows, the target algebra A may always be arbitrary. In the particular case A = End(V ) we would be talking of representations of Gˆ on V , or more generally, as discussed in Lemma 2.3 below, of representations of M o Gˆ or Gˆ n M, respectively, on V . Definition 2.2. Let λ : M → G ⊗ M be a left coaction and let γ : M → A be an algebra map. An implementer of λ in A (with respect to γ) is an element L ∈ G ⊗ A satisfying [1G ⊗ γ(m)] L = L [(idG ⊗ γ)(λ(m))]
(2.15)
for all m ∈ M. Similarly, an implementer in A of a right coaction ρ : M → M⊗G is an element R ∈ G ⊗ A satisfying (denoting ρop = τM⊗G ◦ ρ, τ being the permutation of tensor factors) (2.16) R [1G ⊗ γ(m)] = [(id ⊗ γ)(ρop (m))] R . We now have: Lemma 2.3. Under the conditions of Definition 2.2 the relations γL (ϕ n m) := (ϕ ⊗ id)(L) γ(m) γR (m o ϕ) := γ(m) (ϕ ⊗ id)(R) provide one-to-one correspondences between algebra maps γL : Gˆ n M → A (γR : M o Gˆ → A) extending γ and normal λ-implementers L ∈ G ⊗ A (normal ρimplementers R ∈ G ⊗ A), respectively, satisfying L13 L23 = (∆ ⊗ id)(L) R13 R23 = (∆ ⊗ id)(R) .
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ˆ A) Proof. Writing R(ϕ) := (ϕ ⊗ id)(R) ≡ γR (1M o ϕ) ∈ A and using HomC (G, ∼ = G ⊗ A, the relation R ↔ γR is one-to-one. The implementer property (2.16) is then equivalent to R(ϕ)γ(m) = γ(ϕ(1) . m)R(ϕ(2) ) and R is normal iff γR is unit preserving. Together with the remarks (2.13–2.14) this is further equivalent to γR defining an algebra map, similarly to Lemma 2.1. The argument for γL is analogous. We finally note that the equivalence (2.11) ⇔ (2.12) can be reformulated for implementers as follows: Lemma 2.4. Under the conditions of Definition 2.2 denote λ(m) = m(−1) ⊗ m(0) and ρ(m) = m(0) ⊗ m(1) . Dropping the symbol γ we then have (2.16) ⇔ [1G ⊗ m] R = [S −1 (m(1) ) ⊗ 1A ] R [1G ⊗ m(0) ] ,
∀m ∈ M
(2.17)
(2.15) ⇔ L [1G ⊗ m] = [1G ⊗ m(0) ] L [S −1 (m(−1) ) ⊗ 1A ] ,
∀m ∈ M.
(2.18)
Proof. Suppose R is an implementer of ρ. Then by (2.16) [S −1 (m(1) ) ⊗ 1A ] R [1G ⊗ m(0) ] = [S −1 (m(2) )m(1) ⊗ m(0) ] R = [1G ⊗ m] R by (2.2) and the antipode axioms. Conversely, if R satisfies the right equality in (2.17), then R [1G ⊗ m] = [m(2) S −1 (m(1) ) ⊗ 1A ] R [1G ⊗ m(0) ] = [m(1) ⊗ m(0) ] R proving (2.16). The equivalence (2.18) is proven analogously.
2.2. Two-sided coactions and diagonal crossed products In Sec. 3 we will give a straightforward generalization of the notion of coactions to quasi-Hopf algebras. However, in general an associated notion of a crossed product extension Mo Gˆ will not be well defined as an associative algebra, basically because in the quasi-Hopf case the natural product in Gˆ is not associative. We are now going to provide a new construction of what we call a diagonal crossed product which will allow to escape this obstruction when generalized to the quasi-Hopf case. Our diagonal crossed products are always based on two-sided coactions or, equivalently, on pairs of commuting left and right coactions. These structures are largely motivated by the specific example M = G, where our methods reproduce the quantum double D(G). Definition 2.5. A two-sided coaction of G on an algebra M is an algebra map δ : M → G ⊗ M ⊗ G satisfying (idG ⊗ δ ⊗ idG ) ◦ δ = (∆ ⊗ idM ⊗ ∆) ◦ δ ( ⊗ idM ⊗ ) ◦ δ = idM .
(2.19) (2.20)
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An example of a two-sided coaction is given by M = G and δ := D ≡ (∆⊗id)◦∆. More generally let λ : M → G ⊗ M and ρ : M → M ⊗ G be a left and a right coaction, respectively. We say that λ and ρ commute, if (λ ⊗ id) ◦ ρ = (id ⊗ ρ) ◦ λ .
(2.21)
It is straightforward to check that in this case δ := (λ ⊗ id) ◦ ρ ≡ (id ⊗ ρ) ◦ λ
(2.22)
provides a two-sided coaction. Conversely, given a two-sided coaction δ : M → G ⊗ M ⊗ G then λ := (id ⊗ id ⊗ ) ◦ δ and ρ := ( ⊗ id ⊗ id) ◦ δ provide a pair of commuting left and right coactions, respectively, obeying Eq. (2.22). Thus using the notation (2.5) we may write δ(m) = m(−1) ⊗ m(0) ⊗ m(1) ,
(2.23)
etc. implying again the usual summation conventions. We remark that in the quasicoassociative setting of Sec. 3 the relation between two-sided coactions and pairs (λ, ρ) of left and right coactions becomes more involved, justifying the treatment of two-sided coactions as distinguished objects on their own right also in the present setting. Next, in view of (2.22) we also have a one-to-one correspondence between two-sided coactions δ of G on M and pairs of mutually commuting left and right Hopf module actions, . and / , of Gˆ on M, the relation being given by (ϕ ⊗ id ⊗ ψ)(δ(m)) = ψ . m / ϕ ,
(2.24)
where ϕ, ψ ∈ Gˆ and m ∈ M. This allows to construct as a new algebra the right diagonal crossed product M ./ Gˆ as follows. Proposition 2.6. Let δ = (λ ⊗ idG ) ◦ ρ = (idG ⊗ ρ) ◦ λ be a two-sided coaction of G on M and let . and / be the associated commuting pair of left and right actions of Gˆ on M. Define on M ⊗ Gˆ the product (m ./ ϕ)(n ./ ψ) := (m(ϕ(1) . n / Sˆ−1 (ϕ(3) )) ./ ϕ(2) ψ) ,
(2.25)
where we write (m ./ ϕ) in place of (m⊗ϕ) to distinguish the new algebraic structure. ˆ Then with this product M ⊗ Gˆ becomes an associative algebra with unit (1M ./ 1) ˆ and Gˆ ≡ 1M ./ Gˆ as unital subalgebras. containing M ≡ M ./ 1 Proof. For m, m0 , n ∈ M and ϕ, ψ, ξ ∈ Gˆ we compute [(m ./ ϕ)(m0 ./ ψ)](n ./ ξ) = [m(ϕ(1) . m0 / Sˆ−1 (ϕ(3) )) ./ ϕ(2) ψ](n ./ ξ) = [m(ϕ(1) . m0 / Sˆ−1 (ϕ(5) ))(ϕ(2) ψ(1) . n / Sˆ−1 (ψ(3) )Sˆ−1 (ϕ(4) ))] ./ (ϕ(3) ψ(2) ξ) = m[ϕ(1) . [m0 (ψ(1) . n / Sˆ−1 (ψ(3) ))] / Sˆ−1 (ϕ(3) )] ./ (ϕ(2) ψ(2) ξ) = (m ./ ϕ)[(m0 ./ ψ)(n ./ ξ)] , which proves the associativity. The remaining statements follow trivially from ϕ . 1M = 1M / ϕ = (ϕ)1M and the counit axioms.
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We emphasize that while Proposition 2.6 still is almost trivial as it stands, its true power only appears when generalized to quasi-Hopf algebras G. Definition 2.7. Under the setting of Proposition 2.6 we define the right diagonal crossed product Mδ ./ Gˆ ≡ λ Mρ ./ Gˆ to be the vector space M ⊗ Gˆ with associative multiplication structure (2.25). In cases where the two-sided coaction δ is unambiguously understood from the ˆ We emphasize already at this place that in Sec. 3 context we will also write M ./ G. not every two-sided coaction will be given as δ = (λ⊗idG )◦ρ (or δ = (idG ⊗ρ)◦λ), in which case the notations Mδ ./ Gˆ and λMρ ./ Gˆ will denote different (although still equivalent) extensions of M. Here we freely use either one of them. If δ = idG ⊗ ρ ˆ More generally, λ Mρ ./ Gˆ may for some right coaction ρ then Mδ ./ Gˆ = Mρ o G. ˆ ˆ be identified as a subalgebra of G n (λ Mρ o G) ≡ (Gˆ n λ Mρ ) o Gˆ using the injective algebra map λ Mρ
ˆ n m o 1)(ϕ ˆ (2) n 1M o ϕ(1) ) ./ Gˆ 3 (m ./ ϕ) 7−→ (1 ≡ [ϕ(2) n (m / ϕ(3) ) o ϕ(1) ] ∈ Gˆ n λ Mρ o Gˆ
which we leave to the reader to check. This also motivates our choice of calling the crossed product M ./ Gˆ “diagonal”. The quantum double. In the case M = G and δ := D ≡ (∆ ⊗ id) ◦ ∆, the formula (2.25) coincides with the multiplication rule in the quantum double D(G) [17, 32], i.e. (2.26) D(G) = GD ./ Gˆ . It is well known, that D(G) is itself again a Hopf algebra with coproduct ∆D given by (2.27) ∆D (a ./D ϕ) = (a(1) ./D ϕ(2) ) ⊗ (a(2) ./D ϕ(1) ) , ˆ It turns out that this result generalizes to diagonal crossed where a ∈ G and ϕ ∈ G. products as follows: Proposition 2.8. Let δ : M → G ⊗ M ⊗ G be a two-sided coaction. Then ˆ M ./ Gˆ admits a commuting pair of coactions λD : M ./ Gˆ → D(G) ⊗ (M ./ G) ˆ ⊗ D(G) given by and ρD : M ./ Gˆ → (M ./ G) λD (m ./ ϕ) = (m(−1) ./D ϕ(2) ) ⊗ (m(0) ./ ϕ(1) ) ρD (m ./ ϕ) = (m(0) ./ ϕ(2) ) ⊗ (m(1) ./D ϕ(1) ) , ˆ where elements in D(G) are written as (a ./D ϕ), a ∈ G, ϕ ∈ G. Proof. In view of (2.27) the comodule axioms and the commutativity (2.22) are obvious. That λD and ρD provide algebra maps is shown by direct computation, which we leave to the reader.
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ˆ ∼ Let us also recall the well-known Hopf algebra identity D(G) = D(G)cop , with algebra isomorphism given by ˆ 3 (ϕ ./ ˆ a) 7→ (1 ./D ϕ)(a ./D 1) ˆ ∈ D(G) . D(G) D
(2.28)
This generalizes to diagonal crossed products in the sense that they may equivalently be modeled on the vector space Gˆ ⊗ M. Corollary 2.9. We define the left diagonal crossed product Gˆ ./ Mδ as the vector space Gˆ ⊗ M with multiplication given by (ϕ ./ m)(ψ ./ n) := ϕψ(2) ./ (Sˆ−1 (ψ(1) ) . m / ψ(3) )n .
(2.29)
This defines an associative algebra, and the analog of (2.28) is given by ˆ ≡ ϕ(1) . m / Sˆ−1 (ϕ(3) ) ./ ϕ(2) ∈ M ./ Gˆ , Gˆ ./ M 3 ϕ ./ m 7→ (1M ./ ϕ)(m ./ 1) (2.30) which provides an isomorphism of algebras restricting to the identity on M. The proof of Corollary 2.9 is straightforward from the antipode axioms. The reader is invited to check that in the case M = G and δ = (∆ ⊗ id) ◦ ∆ we recover ˆ Gˆ ./ Mδ = D(G). 2.3. Generating matrices Similarly as in Lemma 2.3 we now describe the defining relations of diagonal crossed products in terms of a generating matrix T. However, whereas in Lemma 2.3 the generating matrices L and R had to fulfill the implementer properties (2.15) or (2.16), respectively, the natural requirement here is that T intertwines the left and right coactions associated with δ. Definition 2.10. Let (λ, ρ) be a commuting pair of left and right G-coactions on M and let γ : M → A be an algebra map into some target algebra A. Then a λρ-intertwiner in A (with respect to γ) is an element T ∈ G ⊗ A satisfying TλA (m) = ρop A (m)T ,
∀m ∈ M,
(2.31)
where λA ≡ (γ ⊗ id) ◦ λ and ρA ≡ (id ⊗ γ) ◦ ρ. A λρ-intertwiner is called coherent if in G ⊗ G ⊗ A it satisfies T13 T23 = (∆ ⊗ id)(T) .
(2.32)
Similarly as in Lemma 2.4 we then have: Lemma 2.11. Let (M, δ) be a two-sided G-comodule algebra with associated commuting left and right G-coactions (λ, ρ), and let γ : M → A be an algebra map. Then for T ∈ G ⊗ A the following properties are equivalent: (i) T is a λρ-intertwiner (ii) T [1G ⊗ γ(m)] = [m(1) ⊗ γ(m(0) )] T [S −1 (m(−1) ) ⊗ 1A ] (iii) [1G ⊗ γ(m)] T = [S −1 (m(1) ) ⊗ 1A ] T [m(−1) ⊗ γ(m(0) )]
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Proof. Suppose T is a λρ-intertwiner. Then [m(1) ⊗ γ(m(0) )] T [S −1 (m(−1) ) ⊗ 1A ] = T [m(−1) S −1 (m(−2) ) ⊗ γ(m(0) )] = T [1G ⊗ γ(m)] by the antipode axiom. Conversely, if T satisfies (ii) then T [m(−1) ⊗ γ(m(0) )] = [m(1) ⊗ γ(m(0) )] T [S −1 (m(−1) )m(−2) ⊗ 1A ] = [m(1) ⊗ γ(m(0) )] T proving (i) ⇔ (ii). The equivalence (i) ⇔ (iii) follows similarly.
We now conclude similarly as in Lemma 2.3. Proposition 2.12. Let (M, δ) be a two-sided G-comodule algebra with associated commuting pair of coactions (λ, ρ), and let γ : M → A be an algebra map. Then the relation (2.33) γT (m ./ ϕ) = γ(m) (ϕ ⊗ id)(T) provides a one-to-one correspondence between normal coherent λρ-intertwiners T and unital algebra maps γT : λ Mρ ./ Gˆ → A extending γ. Proof. Let T (ϕ) := (ϕ ⊗ id)(T). Then (2.32) together with normality is equivalent to Gˆ 3 ϕ 7→ T (ϕ) ≡ γT (1M ./ ϕ) ∈ A being a unital algebra morphism and the correspondence T ↔ γT |1M ./Gˆ is one-to-one. Clearly, γT extends γ and Lemma 2.11 (ii) implies T (ϕ) γ(m) = γ(ϕ(1) . m / S −1 (ϕ(3) )) T (ϕ(2) ) ,
ˆ m∈M ∀ ϕ ∈ G,
ˆ M ./ and therefore γT is an algebra map. Conversely, since (m ./ ϕ) = (m ./ 1)(1 ˆ ϕ), any algebra map γ : λ Mρ ./ G → A is of the form (2.33). We remark that one could equivalently have chosen to work with γTop (ϕ ./ m) := (ϕ ⊗ id)(T) γ(m)
(2.34)
to obtain algebra maps γTop : Gˆ ./ λ Mρ → A. Note that by applying (ϕ ⊗ id) to both sides the equivalence of (ii) and (iii) in Lemma 2.11 ensures that (2.30) is an isomorphism. Applying the above formalism to the case M = G and δ = D ≡ (∆ ⊗ id) ◦ ∆ we realize that (2.31) becomes (suppressing the symbol γ) T∆(a) = ∆op (a)T, ∀ a ∈ A
(2.35)
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In this special case we call T a ∆-flip operator. As already remarked, in this case GD ./ Gˆ ≡ D(G) is the quantum double of G, in which case Proposition 2.12 coincides with [40, Lemma 5.2] describing D(G) as the unique algebra generated by G and the entries of a generating Matrix D ≡ TD(G) ∈ G ⊗ D(G) satisfying (2.32) and (2.35). More generally every diagonal crossed product M ./ Gˆ may be described as the unique algebra generated by M and the entries of a generating matrix Γ ∈ G⊗(M ./ ˆ satisfying (2.31) and (2.32), by choosing A = M ./ Gˆ in Proposition 2.12. G) This construction of diagonal crossed products in terms of generating matrices is summarized in the following theorem, which we state in this explicit form, since it will allow a generalization to (weak) quasi-Hopf algebras in Sec. 3. Theorem 2.13. Let (G, ∆, , S) be a finite dimensional Hopf algebra and let (λ, ρ) be a commuting pair of (left and right) G-coactions on an associative algebra M. 1. Then there exists a unital associative algebra extension M1 ⊃ M together with a linear map Γ : Gˆ → M1 satisfying the following universal property: ˆ and for any algebra map γ : M1 is algebraically generated by M and Γ(G) M → A into some target algebra A the relation γT (Γ(ϕ)) = (ϕ ⊗ id)(T) ,
ϕ ∈ Gˆ
(2.36)
provides a one-to-one correspondence between algebra maps γT : M1 → A extending γ and elements T ∈ G ⊗ A satisfying ( ⊗ idA )(T) = 1A and T λA (m) = ρop A (m) T ,
∀m ∈ M
T13 T23 = (∆ ⊗ idA )(T) ,
(2.37) (2.38)
where λA (m) := (idG ⊗ γ)(λ(m)) and ρA (m) := (γ ⊗ idG )(ρ(m)). ˜ : Gˆ → M ˜ 1 satisfy the same universality property as ˜ 1 and Γ 2. If M ⊂ M ˜1 in Part 1, then there exists a unique algebra isomorphism f : M1 → M ˜ restricting to the identity on M, such that Γ = f ◦ Γ. 3. The linear maps µL : Gˆ ⊗ M 3 (ϕ ⊗ m) 7→ Γ(ϕ) m ∈ M1
(2.39)
µR : M ⊗ Gˆ 3 (m ⊗ ϕ) 7→ m Γ(ϕ) ∈ M1
(2.40)
provide isomorphisms of vector spaces. ˆ µR := id Proof. Putting M1 = M ./ G, M⊗Gˆ and µL the map given in (2.30), Parts 1 and 3 follow from Proposition 2.12 and Corollary 2.9. The uniqueness of M1 up to equivalence follows by standard arguments.
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Putting Γ := eµ ⊗ Γ(eµ ) ∈ G ⊗ M1 Theorem 2.13 implies that Γ itself satisfies the defining relations (2.37) and (2.38). We call Γ the universal λρ-intertwiner in M1 . We again emphasize that once being stated Theorem 2.13 almost appears trivial. Its true power only arises when generalized to the quasi-coassociative setting in Sec. 3. Note that Part 2 of Theorem 2.13 implies that the algebraic structures induced on Gˆ ⊗ M and M⊗ Gˆ via µ−1 L/R from M1 are uniquely fixed. They are given ˆ respectively, by the left- and right diagonal crossed products Gˆ ./ M and M ./ G, defined above in Proposition 2.6 and Corollary 2.9. 2.4. Hopf spin chains and lattice current algebras In this section we reformulate the Hopf spin chains of [42] and the lattice current algebras of [4] as iterated diagonal crossed products, thereby also reviewing the relationship between the two models. It will turn out to be convenient to use the notion of two-sided crossed products, which we will now introduce as a special type of diagonal crossed products. 2.4.1. Two-sided crossed products A simple recipe to produce two-sided G-comodule algebras (M, δ) is by taking a right G-comodule algebra (A, ρ) and a left G-comodule algebra (B, λ) and define M = A ⊗ B and δ(A ⊗ B) := B(−1) ⊗ (A(0) ⊗ B(0) ) ⊗ A(1) ,
(2.41)
where A ∈ A, B ∈ B, ρ(A) = A(0) ⊗ A(1) and λ(B) = B(−1) ⊗ B(0) . In terms of ˆ ˆ the G-actions . on A and / on B dual to ρ and λ, respectively, the G-actions .M and / M dual to (2.41) are given by ϕ .M (A ⊗ B) /M ψ = (ϕ . A ⊗ B / ψ) ,
ϕ, ψ ∈ Gˆ .
(2.42)
Hence, we may construct the diagonal crossed product M ./ G as before. It turns out that this example may be presented differently as a so-called two-sided crossed product. Proposition 2.14. Let . : Gˆ ⊗ A → A and / : B ⊗ Gˆ → B be a left and a right Hopf module action, respectively, with dual G-coactions ρ, λ. Define the “two-sided crossed product” Aρ o Gˆ n λ B to be the vector space A ⊗ Gˆ ⊗ B with multiplication structure (A o ϕ n B) (A0 o ψ n B 0 ) = A(ϕ(1) . A0 ) o ϕ(2) ψ(1) n (B / ψ(2) )B 0 .
(2.43)
ˆ n 1B and Then Aρ o Gˆ n λ B becomes an associative algebra with unit 1A o 1 ˆ ∈ (A ⊗ B) ./ Gˆ f : A o Gˆ n B 3 A o ϕ n B 7→ ((A ⊗ 1B ) ./ ϕ)((1A ⊗ B) ./ 1) (2.44) provides an algebra isomorphism with inverse given by ˆ n B)(A o ϕ n 1B ) . f −1 ((A ⊗ B) ./ ϕ) = (1A o 1
(2.45)
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Instead of giving a direct proof, let us reformulate the above Proposition in P terms of generating matrices. Setting T := µ eµ ⊗ (1A o eµ n 1B ), where as before ˆ the multiplication rule (2.43) is eµ denotes a basis of G with dual basis eµ ∈ G, equivalent to T satisfying (∆ ⊗ id)(T) = T13 T23 [1 ⊗ B] T = T λ(B), T [1 ⊗ A] = ρop T ,
B∈B
(2.46)
A ∈ A,
where we identify A ≡ A⊗1B , λ ≡ λ⊗idB , etc. Thus T is a λρ-intertwiner and since ˆ ˆ it has AoGnB is generated by M = A⊗B and the matrix entries (ϕ⊗id)(T), ϕ ∈ G, ˆ ˆ to be isomorphic to (A ⊗ B) ./ G. Denoting the λρ-intertwiner in G ⊗ ((A ⊗ B) ./ G) by Γ, one verifies that (id ⊗ f )(T) = Γ ,
(id ⊗ f −1 )(Γ) = T ,
which by Proposition 2.12 implies that f is an isomorphism. We leave the details to the reader. As a particular example of the setting of Proposition 2.14 we may choose A = ˆ B = G with its canonical left and right G-action. It turns out that in this case the ˆ two-sided crossed product G o G n G ≡ (G ⊗ G) ./ Gˆ is isomorphic to the iterated ˆ o G. More generally we have: crossed product (G o G) Proposition 2.15. Let A be a right G-comodule algebra and consider the ˆ o G, where G acts on A o Gˆ in the usual way iterated crossed product (A o G) ˆ Then as an algebra by a . (A o ϕ) := A o (a * ϕ), A ∈ A, a ∈ G, ϕ ∈ G. ˆ ˆ (A o G) o G = A o G n G with trivial identification. Proof. The claim follows from A(ϕ(1) . A0 ) o ϕ(2) ψ(1) n (a ( ψ(2) )b = A(ϕ(1) . A0 ) o ϕ(2) (a(1) * ψ)) o a(2) b as an identity in A ⊗ Gˆ ⊗ G, where we have used ψ(1) ⊗ (a ( ψ(2) ) = ψ(1) ha(1) |ψ(2) i ⊗ a(2) = (a(1) * ψ) ⊗ a(2) as an identity in Gˆ ⊗ G.
It will be shown in Sec. 5 that being a particular example of a two-sided (and ˆ o G ≡ A o Gˆ n G therefore of a diagonal) crossed product the analogue of (A o G) may also be constructed for quasi-Hopf algebras G. However, in this case A o Gˆ (if defined to be the linear subspace A ⊗ Gˆ ⊗ 1G ) will no longer be a subalgebra of A o Gˆ n G. We will see in Sec. 5 that this fact is very much analogous to what happens in the field algebra constructions with quasi-Hopf symmetry as given by V. Schomerus [47].
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2.4.2. Hopf spin chains Next, we point out that Propositions 2.14 and 2.15 also apply to the construction of Hopf algebraic quantum chains as introduced in [42]. To see this let us shortly review the model of [42], where one considers even (odd) integers to represent the sites (links) of a one-dimensional lattice and where one places a copy of G ∼ = A2i on each site and a copy of Gˆ ∼ = A2i+1 on each link.
s
G
2i
Gˆ 2i + 1
s
G
2i + 2
Non-vanishing commutation relations are then postulated only on neighboring sitelink pairs, where one requires A2i (a) A2i−1 (ϕ) = A2i−1 (a(1) * ϕ) A2i (a(2) ) (2.47) A2i+1 (ϕ) A2i (a) = A2i (ϕ(1) * a) A2i+1 (ϕ(2) ) . Here G 3 a 7→ A2i (a) ∈ A2i ⊂ A and Gˆ 3 ϕ 7→ A2i+1 (ϕ) ∈ A2i+1 ⊂ A denote the embedding of the single site (link) algebras into the global quantum chain A. Denoting Ai,j ⊂ A as the subalgebra generated by Aν , i ≤ ν ≤ j, we clearly have from (2.47) Ai,j+1 = Ai,j o Aj+1 (2.48) Ai−1,j = Ai−1 n Ai,j . Hence, by Proposition 2.14, we recognize the two-sided crossed products ˆ o G = A2i,2j o Gˆ n G . A2i,2j+2 ≡ (A2i,2j o G)
(2.49)
More generally for all i ≤ ν ≤ j − 1 we have A2i,2j = A2i,2ν o Gˆ n A2ν+2,2j ,
(2.50)
where Gˆ ≡ A2ν+1 . The advantage of looking at it in this way again comes from the fact that the constructions (2.49) and (2.50) generalize to quasi-Hopf algebras G whereas (2.48) do not. This observation will be needed to formulate a theory of Hopf spin models and lattice current algebras at roots of unity, see Sec. 5. Next, we remark that the identifications (2.49), (2.50) may be iterated in the obvious way. This observation also generalizes to the situation where in Proposition 2.14 A and B are both two-sided G-comodules algebras with dual Gˆ actions denoted .A , /A , .B , /B , respectively. Then in the multiplication rule (2.43) only .A and /B appear and one easily checks, that for ϕ, ψ ∈ Gˆ and A ∈ A, B ∈ B the definitions ϕ . (A o ψ n B) := A o ψ n (ϕ .B B) (A o ψ n B) / ϕ := (A /A ϕ) o ψ n B
DIAGONAL CROSSED PRODUCTS BY DUALS OF QUASI-QUANTUM GROUPS
571
again define a two-sided G-comodule structure on A o Gˆ n B. Hence, we have a multiplication law on two-sided G-comodule algebras which is in fact associative, i.e. as a two-sided G-comodule algebra (A o Gˆ n B) o Gˆ n C = A o Gˆ n (B o Gˆ n C)
(2.51)
which the reader will easily check. Obviously, one may also consider mixed cases, e.g. where in (2.51) A is only a right G-comodule algebra, but B and C are two-sided, in which case (2.51) would be an identity between right G-comodule algebras. Let us now formulate the algebraic properties of Hopf Spin chains in terms of generating matrices, using the relations (2.46). Defining the generating “link P operators” L2i+1 := µ eµ ⊗ A2i+1 (eµ ), A2i,2j is the unique algebra generated by ≡ A2i ⊗ A2i+2 ⊗ · · · ⊗ A2j and the entries of generating matrices L2ν+1 ∈ G⊗ G ⊗ A2i,2j , i ≤ ν ≤ j obeying the relations: j−i
23 23 13 L13 2k+1 L2l+1 = L2l+1 L2k+1 ,
∀ k 6= l
[1 ⊗ A2k (a)] L2l+1 = L2l+1 [1 ⊗ A2k (a)] ,
∀ k 6= l, l + 1
(2.52a) (2.52b)
23 L13 2k+1 L2k+1 = (∆ ⊗ id)(L2k+1 )
(2.52c)
[1 ⊗ A2k (a)] L2k−1 = L2k−1 [a(1) ⊗ A2k (a(2) )]
(2.52d)
L2k+1 [1 ⊗ A2k (a)] = [a(2) ⊗ A2k (a(1) )] L2k+1 .
(2.52e)
Let us shortly comment on these relations for the sake of getting a better understanding of the “language” of generating matrices. Equations (2.52a) and (2.52b) express locality in the sense that they give nontrivial commutation relations only on neighboring site link pairs. Equation (2.52c) may be viewed as an operator product expansion. Provided G is quasitriangular with R-matrix R ∈ G ⊗ G, it implies the braiding relations: R12 L13 L23 = L23 L13 R12 . Finally, (2.52d) and (2.52e) express covariance properties of the link operators L. We finish our discussion of Hopf spin chains by noting that the identification (2.50) together with Propositions 2.14 and 2.8 immediately imply that quantum chains of the type (2.47) admit localized commuting left and right coactions of the quantum double D(G), which is precisely the result of Theorem 4.1 of [42]. In fact, applied to the example in Proposition 2.14, Proposition 2.8 gives: Corollary 2.16. Under the setting of Proposition 2.14 we have a commuting pair of left and right coactions ρD : A o Gˆ n B → (A o Gˆ n B) ⊗ D(G) and λD : A o Gˆ n B → D(G) ⊗ (A o Gˆ n B) given by ρD (A o ϕ n B) = (A(0) o ϕ(2) n B) ⊗ (A(1) ./D ϕ(1) ) λD (A o ϕ n B) = (B(−1) ./D ϕ(2) ) ⊗ (A o ϕ(1) n B(0) ) .
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This implies the existence of right coactions ρ2i D of the quantum double D(G) on the quantum chain, which are “localized” (i.e. act nontrivially only) in A2i,2i+1 , where they are given by (using generating matrix notation) ˆ ρ2i D (A2i (a)) = A2i (a(1) ) ⊗ (a(2) ./D 1) (2.53) (id ⊗
ρ2i D )(L2i+1 )
13
=D
L12 2i+1
.
P Here D ∈ G ⊗ D(G) denotes the universal ∆-flip operator D = eµ ⊗ (1 ./D eµ ). Analogously one may define localized left coactions λ2i D which are immediately shown . to commute with ρ2i D 2.4.3. Lattice current algebras
's ps p p p p ps $s
Diagonal crossed products also appear when formulating periodic boundary conditions for the quantum chain (2.47). In this case, starting with the open chain A2,2n localized on [2, 2n] ∩ Z one would like to add another copy of Gˆ sitting on the link 2n + 1 ≡ 1 joining the sites 2n and 2 to form a periodic lattice. Gˆ
Kn :
2
4
2n
Algebraically this means that A1 (≡ A2n+1 ) ∼ = Gˆ should have non-vanishing com∼ G and A G in analogy with (2.47), i.e. mutation relations with A2n ∼ = 2 = A1 (ϕ) A2n (a) = A2n (ϕ(1) * a) A1 (ϕ(2) ) (2.54) A2 (a) A1 (ϕ) = A1 (ϕ(1) ) A2 (a ( ϕ(2) ) . Written in this way Eqs. (2.54) are precisely the relations in Kn := A2,2n ./ Gˆ , where δ : A2,2n → G ⊗ A2,2n ⊗ G is the two-sided coaction given by δ A2 = ∆ ⊗ 1G , δ A2n = 1G ⊗ ∆ and δ A3,2n−1 = 1G ⊗ id ⊗ 1G . Hence, the periodic quantum chain appears as a diagonal crossed product of the open lattice chain by a copy of Gˆ sitting on the link joining the end points. Again we remark that this observation will be needed to give a generalization to (weak) quasi-Hopf algebras. A similar remark applies to the lattice current algebra of [4] defined below. We also conclude from (2.26) that the “periodic chain” K1 consisting of one point and one link is given by the quantum double D(G) K1 = G ./ Gˆ ≡ D(G) :
'$ s&% .
(2.55)
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573
Let us finally review the lattice current algebras of [4]c , which appear as special examples of periodic Hopf spin chains. We follow the review of [40], where the relation with the model of [42] has been clarified. Suppose G to be quasitriangular with R-matrix R ∈ G ⊗ G and define the generating lattice currents J2i+1 := (id ⊗ A2i )(Rop ) L2i+1 .
(2.56)
Using (2.52), these are immediately verified to satisfy the lattice current algebra of [4] [1G ⊗ A2i (a)] J2i−1 = J2i−1 [a(1) ⊗ A2i (a(2) )],
∀a ∈ G
[a(1) ⊗ A2i (a(2) )] J2i+1 = J2i+1 [1G ⊗ A2i (a)] 23 12 J13 2i+1 J2i+1 = R (∆ ⊗ id)(J)2i+1 12 23 23 13 J13 2i−1 R J2i+1 = J2i+1 J2i−1 .
Hence under the additional requirement of G being quasitriangular, the lattice algebras of [42] and [4] are isomorphic. 2.5. Double crossed products We conclude this section by relating our diagonal crossed product with the double crossed product construction of Majid [32, 34]. Here we adopt the version [34, Thm. 7.2.3], according to which a bialgebra B is a double crossed product, written as (2.57) B = M ./ double H , iff M and H are sub-bialgebras of B such that the multiplication map µ : M ⊗ H 3 m ⊗ h 7→ mh ∈ B provides an isomorphism of coalgebras. In this case the bialgebras M and H become a matched pair with mutual actions −>: H ⊗ M → M and <− : H ⊗ M → H given by h −> m := (idM ⊗ H )(µ−1 (hm))
(2.58)
h <− m := (M ⊗ idH )(µ−1 (hm)) ,
(2.59)
see [34, Chap. 7.2] for more details. The guiding example is again given by the quantum double satisfying D(G) = G ./ double Gˆcop .
(2.60)
More generally, any diagonal crossed product λMρ ./ Gˆ becomes a double crossed product ˆ ˆcop (2.61) λMρ ./ G = M ./ double G c For earlier versions of lattice current algebras see also [3, 2, 20].
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provided M is equipped with a bialgebra structure ∆M , M such that the tensor product coalgebra structures ∆B (m ./ ψ) := (m(1) ./ ψ(2) ) ⊗ (m(2) ./ ψ(1) ) B (m ./ ψ) := M (m) ˆ(ψ)
(2.62) (2.63)
give algebra maps ∆B : B → B ⊗ B and B : B → C (here we denote ∆(m) ≡ m(1) ⊗ m(2) ). In this way all generalized quantum doubles M ./ double Gˆcop in the sense of [34, Ex. 7.2.7] are as algebras actually diagonal crossed products in our sense. These may be described in terms of the bialgebra homomorphism κ : M → G induced by the Hopf skew-pairing M ⊗ Gˆcop → C required in [34, Chap. 7.2], which gives rise to the commuting pair of G-coactions λ := (κ ⊗ idM ) ◦ ∆M ,
ρ := (idM ⊗ κ) ◦ ∆M .
(2.64)
For these models the “matched pair” actions (2.58) and (2.59) are given by the coadjoint actions ψ −> m = ψ(1) . m / Sˆ−1 (ψ(2) )
(2.65)
ψ <− m = M (ψ(1) . m / Sˆ−1 (ψ(3) )) ψ(2) = Sˆ−1 (κ(m(1) )) * ψ ( κ(m(2) ) .
(2.66)
Also note that the coactions (2.64) satisfy the compatibility condition (ρ ⊗ idM ) ◦ ∆M = (idM ⊗ λ) ◦ ∆M
(2.67)
and the homomorphism κ : M → G may be recovered as κ = (idG ⊗ M ) ◦ λ = (M ⊗ idG ) ◦ ρ .
(2.68)
Conversely, we have: Proposition 2.17. Let (M, ∆M , M ) be a bialgebra and let (λ, ρ) be a commuting pair of (left and right) G-coactions on M. Then the compatibility condition (2.67) is equivalent to (idG ⊗ M ) ◦ λ = (M ⊗ idG ) ◦ ρ =: κ
(2.69)
being a bialgebra homomorphism κ : M → G which satisfies (2.64). If these conditions are satisfied, then the diagonal crossed product λMρ ./ Gˆ is also a double crossed product with respect to the bialgebra structure (2.62), (2.63), i.e. in this case we have ˆ ˆcop . λMρ ./ G = M ./ double G Proof. Applying (M ⊗ idG ⊗ M ) to (2.67) proves that the two expressions in (2.69) define the same algebra map κ := M → G. Clearly, we have G ◦ κ = M .
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To prove (2.64) we compute (κ ⊗ idM ) ◦ ∆M = (idG ⊗ M ⊗ idM ) ◦ (λ ⊗ idM ) ◦ ∆M (2)
= (M ⊗ idG ⊗ M ⊗ idM ) ◦ (idM ⊗ λ ⊗ idM ) ◦ ∆M
(2)
= (M ⊗ idG ⊗ M ⊗ idM ) ◦ (ρ ⊗ idM ⊗ idM ) ◦ ∆M = (M ⊗ idG ⊗ idM ) ◦ (ρ ⊗ idM ) ◦ ∆M = (M ⊗ λ) ◦ ∆M = λ , (2) ∆M
where = (∆M ⊗ id) ◦ ∆M = (id ⊗ ∆M ) ◦ ∆M and where we have used (2.67) in the third and the fifth line. The identity ρ = (id ⊗ κ) ◦ ∆M is proven similarly. Finally, κ : M → G is a bialgebra map, since (κ ⊗ κ) ◦ ∆M = (idG ⊗ κ) ◦ λ = (idG ⊗ idG ⊗ M ) ◦ (idG ⊗ λ) ◦ λ = (∆G ⊗ M ) ◦ λ = ∆G ◦ κ by the G-coaction property for λ. According to the results of [34, Sec. 7.2] this implies that the “generalized quantum double” M ./ double Gˆcop as an algebra co incides with the diagonal crossed product λMρ ./ Gˆ in our sense. We conclude this section by demonstrating the good use of Proposition 2.12 by giving a short alternative proof of how the compatibility condition (2.67) makes diagonal crossed products into double crossed product bialgebras. Proposition 2.18. Under the conditions (2.64), (2.67), and (2.68) let B := ˆ Then the coalgebra structures ∆B (2.62) and B (2.63) become algebra ./ G. maps. λMρ
Proof. Since ∆B extends ∆M we may use Proposition 2.12 by putting A := B ⊗ B, γ := ∆M : M → M ⊗ M ⊂ A and T := (idG ⊗ ∆B )(Γ) ≡ Γ13 Γ12 ∈ G ⊗ A , where Γ ∈ G ⊗ B is the universal λρ-intertwiner. Thus, ∆B : B → A provides an algebra map if and only if T is a normal coherent λρ-intertwiner with respect to ∆M : M → A. Now normality and coherence of T hold, since the restriction ˆ op is a unital algebra map. To prove the λρ-intertwiner property for T ∆B |Gˆ ≡ ∆ we use the identities (2.64) to compute for all m ∈ M T (idG ⊗ ∆M )(λ(m)) = Γ13 Γ12 [κ(m(1) ) ⊗ m(2) ⊗ m(3) ] = Γ13 Γ12 [λ(m(1) ) ⊗ m(2) ] = Γ13 [κ(m(2) ) ⊗ m(1) ⊗ m(3) ] Γ12 = [κ(m(3) ) ⊗ m(1) ⊗ m(2) ] Γ13 Γ12 = (idG ⊗ ∆M )(ρop (m)) T ,
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(2)
where m(1) ⊗ m(2) ⊗ m(3) ≡ ∆M (m). Hence, T is a λρ-intertwiner and therefore ∆B is an algebra map. Similarly, we prove that B : B → C is multiplicative by putting A = C and T = (idG ⊗ B )(Γ) ≡ 1G . In this case the λρ-intertwiner property reduces to (idG ⊗ M )(λ(m)) = (M ⊗ idG )(ρ(m)) ,
which is precisely the condition (2.69). 3. Diagonal Crossed Products by Duals of Quasi-Hopf Algebras
In Sec. 2 we have reviewed the notions of left and right G-coactions and crossed products and we have introduced as new concepts the notions of two-sided Gcoactions and diagonal crossed products, where throughout G had been supposed to be a standard coassociative Hopf algebra. As an application we have mentioned the Drinfel’d double D(G) and the constructions of quantum chains based on a Hopf algebra G. We now proceed to generalize the above ideas to quasi-Hopf algebras G. In Sec. 3.1 we give a short review of the definitions and properties of quasi-Hopf algebras as introduced by Drinfel’d [18]. In Sec. 3.2 we propose an obvious generalization of the notion of right G-coactions ρ on an algebra M to the case of quasi-Hopf algebras G (and similarly for left coactions λ). As for the coproduct on G, the basic idea here is that (ρ ⊗ id) ◦ ρ and (id ⊗ ∆) ◦ ρ are still related by an inner automorphism, implemented by a reassociator φρ ∈ M ⊗ G ⊗ G. Similarly as for Drinfel’d’s reassociator φ ∈ G ⊗ G ⊗ G, φρ is required to obey a pentagon equation to guarantee McLane’s coherence condition under iterated rebracketings. We also generalize Drinfel’d’s notion of a twist transformation from coproducts to coactions. It is important to realize that φρ has to be non-trivial, if φ is non-trivial. On the other hand, φρ might be non-trivial even if φ = 1G ⊗ 1G ⊗ 1G , in which case the above-mentioned pentagon equation reduce to a cocycle condition for φρ as already considered by [14, 10, 11]. In Sec. 3.3 we pass to two-sided G-coactions (δ, Ψ), which could alternatively be considered as right (G ⊗ G cop )-coactions in the above sense. Correspondingly, Ψ ∈ G ⊗ G ⊗ M ⊗ G ⊗ G is the reassociator for δ, which is again required to obey the appropriate pentagon equation. As in Sec. 2, associated with any two-sided G-coaction (δ, Ψ) we have a pair (λ, φλ ) and (ρ, φρ ) of left and right G-coactions, respectively, which however in this case only quasi-commute. This means that there exists another reassociator φλρ ∈ G ⊗ M ⊗ G such that φλρ (λ ⊗ idG )(ρ(m)) = (idG ⊗ ρ)(λ(m)) φλρ ,
∀m ∈ M.
Also, φλρ obeys in a natural way two pentagon identities involving (λ, φλ ) and (ρ, φρ ), respectively. We show that twist equivalence classes of two-sided coactions are in one-to-one correspondence with twist equivalence classes of quasi-commuting pairs of coactions, i.e. any two-sided coaction δ is twist-equivalent to (λ ⊗ id) ◦ ρ (and also to (id ⊗ ρ) ◦ λ) where λ = (idG ⊗ idM ⊗ ) ◦ δ and ρ = ( ⊗ idM ⊗ idG ) ◦ δ. In Appendix A we give a representation theoretic interpretation of the notions of left, right and two-sided coactions by showing that they give rise to functors
DIAGONAL CROSSED PRODUCTS BY DUALS OF QUASI-QUANTUM GROUPS
577
Rep G×Rep M → Rep M, Rep M×Rep G → Rep M and Rep G×Rep M×Rep G → Rep M, respectively, furnished with natural associativity isomorphisms, obeying the analogue of McLane’s coherence conditions for monoidal categories [29]. In Sec. 3.4 we use our formalism to construct, for any two-sided G-coaction (δ, Ψ) on M, the left and right diagonal crossed products Mδ ./ Gˆ and Gˆ ./ Mδ as associative algebra extensions of M (they are in fact equivalent as will be shown in Sec. 3.5). Up to equivalence, these extensions only depend on the twist-equivalence class of δ’s, and therefore on the twist-equivalence class of quasi-commuting pairs (λ, ρ). The basic strategy for defining the multiplication rules in these diagonal crossed products is to generalize the generating matrix formalism of Sec. 2.3 to the quasi-coassociative setting. In this way one is naturally lead to define λρintertwiners T as in Definition 2.10, where now the coherence condition (2.32) has to be replaced by appropriately injecting the reassociators φλ , φλρ and φρ into the l.h.s., similarly as in Drinfel’d’s definition of a quasitriangular R-matrix for quasiHopf algebras. With these substitutions our main result is given by the following generalization of Theorem 2.13: Theorem 3.1. Let G be a finite dimensional quasi-Hopf algebra and let (λ, φλ , ρ, φρ , φλρ ) be a quasi-commuting pair of (left and right) G-coactions on an associative algebra M. 1. Then their exists a unital associative algebra extension M1 ⊃ M together with a linear map Γ : Gˆ → M1 satisfying the following universal property: ˆ and for any algebra map γ : M1 is algebraically generated by M and Γ(G) M → A into some target algebra A the relation γT (Γ(ϕ)) = (ϕ ⊗ id)(T)
(3.1)
provides a one-to-one correspondence between algebra maps γT : M1 → A extending γ and normal elements T ∈ G ⊗ A satisfying T λA (m) = ρop A (m) T ,
∀m ∈ M
13 132 −1 23 (φ312 ρ )A T (φρλ )A T (φλ )A = (∆ ⊗ idA )(T) ,
(3.2) (3.3)
where λA (m) := (id ⊗ γ)(λ(m)), (φλ )A := (idG ⊗ idG ⊗ γ)(φλ ), etc. ˜ : Gˆ → M ˜ 1 satisfy the same universality property as ˜ 1 and Γ 2. If M ⊂ M ˜1 in Part 1, then there exists a unique algebra isomorphism f : M1 → M ˜ restricting to the identity on M, such that Γ = f ◦ Γ. 3. There exist elements pλ ∈ G ⊗ M and qρ ∈ M ⊗ G such that the linear maps µL : Gˆ ⊗ M 3 (ϕ ⊗ m) 7→ (id ⊗ ϕ(1) )(qρ ) Γ(ϕ(2) ) m ∈ M1
(3.4)
µR : M ⊗ Gˆ 3 (m ⊗ ϕ) 7→ m Γ(ϕ(1) ) (ϕ(2) ⊗ id)(pλ ) ∈ M1
(3.5)
provide isomorphisms of vector spaces.
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Putting Γ := eµ ⊗Γ(eµ ) ∈ G ⊗M1 , Theorem 3.1 implies that Γ itself satisfies the defining relations (3.2) and (3.3). As before, we call Γ the universal λρ-intertwiner in M1 . We remark that it is more or less straightforward to check that the relations (3.2) and (3.3) satisfy all associativity constraints, such that the existence of M1 and its uniqueness up to isomorphisms may not be too much of a surprise to the experts. In this way Parts 1 and 2 of Theorem 3.1 could also be proven without requiring an antipode on G. The main non-trivial content of Theorem 3.1 is stated in Part 3, saying that M1 may still be modeled on the underlying spaces Gˆ ⊗ M or ˆ respectivelyd . However, as a warning against likely misunderstandings we M ⊗ G, emphasize that in general (i.e. for φλρ 6= 1G ⊗ 1M ⊗ 1G ) neither of the maps M ⊗ Gˆ 3 (m ⊗ ϕ) 7→ m Γ(ϕ) ∈ M1 Gˆ ⊗ M 3 (ϕ ⊗ m) 7→ Γ(ϕ) m ∈ M1 need to be injective (nor surjective)e . Also, in general neither of the linear subˆ will be a subalgebra of M1 . Still, the ˆ µL (Gˆ ⊗ 1M ) or µR (1M ⊗ G) spaces Γ(G), invertibility of the maps µL/R guarantees that there exist well defined associative algebra structures induced on Gˆ ⊗ M and M ⊗ Gˆ via µ−1 L/R from M1 . As in Sec. 2 we denote these by Gˆ ./ λMρ ≡ µ−1 L (M1 ) λMρ
./ Gˆ ≡ µ−1 R (M1 ) .
(3.6) (3.7)
They are the analogues of the left and right diagonal crossed products, respectively, constructed in Proposition 2.6 and Corollary 2.9. To actually prove Theorem 3.1 we go the opposite way, i.e. for any two-sided coaction (δ, Ψ) we will first explicitly construct left and right diagonal crossed products Gˆ ./ Mδ and Mδ ./ Gˆ as equivalent algebra extensions of M in Sec. 3.4. As ˆ respecin Sec. 2 these are defined on the underlying spaces Gˆ ⊗ M and M ⊗ G, tively. In Sec. 3.5.1 we describe these constructions in terms of so-called left and right diagonal δ-implementers L and R obeying the relations of Lemma 2.11(iii) and (ii), respectively, together with certain coherence conditions reflecting the mulˆ In Sec. 3.5.2 we generalize Lemma 2.11 tiplication rules in Gˆ ./ M and M ./ G. by showing that coherent (left or right) diagonal δ-implementers are always in oneto-one correspondence with (although not identical to) coherent λρ-intertwiners T, i.e. generating matrices satisfying the relations (3.2) and (3.3) of Theorem 3.1. This will finally lead to a proof of Theorem 3.1 by showing that for δl := (λ ⊗ id) ◦ ρ and ˆ or Mδr ./ Gˆ δr := (id ⊗ ρ) ◦ λ any of the four choices Gˆ ./ Mδl , Gˆ ./ Mδr , Mδl ./ G, explicitly solve all properties claimed in Theorem 3.1. Moreover, in terms of the notations (3.6), (3.7) we will have Gˆ ./ λMρ = Gˆ ./ Mδl λMρ
./ Gˆ = Mδr ./ Gˆ
with trivial identification. d To define the elements p and q one needs an invertible antipode, see (3.71), (3.74). ρ λ e In fact, we don’t even know whether the map Γ : Gˆ → M necessarily has to be injective. 1
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To keep the main part of this section more readable, we have postponed some proofs and technical Lemmata to Sec. 3.6. 3.1. Quasi-Hopf algebras In this section we review the basic definitions and properties of quasi-Hopf algebras as introduced by Drinfeld [18], where the interested reader will find a more detailed discussion. As before algebra morphisms are always supposed to be unital. A quasi-bialgebra (G, ∆, , φ) is an associative algebra G with unit together with algebra morphisms ∆ : G → G ⊗ G (the coproduct) and : G → C (the counit), and an invertible element φ ∈ G ⊗ G ⊗ G, such that (id ⊗ ∆)(∆(a))φ = φ(∆ ⊗ id)(∆(a)) ,
a∈G
(3.8)
(id ⊗ id ⊗ ∆)(φ)(∆ ⊗ id ⊗ id)(φ) = (1 ⊗ φ)(id ⊗ ∆ ⊗ id)(φ)(φ ⊗ 1) ,
(3.9)
( ⊗ id) ◦ ∆ = id = (id ⊗ ) ◦ ∆ ,
(3.10)
(id ⊗ ⊗ id)(φ) = 1 ⊗ 1 .
(3.11)
A coproduct with the above properties is called quasi-coassociative and the element φ will be called the reassociator. The identities (3.8)–(3.11) also imply ( ⊗ id ⊗ id)(φ) = (id ⊗ id ⊗ )(φ) = 1 ⊗ 1 .
(3.12)
As for Hopf algebras we will use the Sweedler notation ∆(a) = a(1) ⊗ a(2) , but since ∆ is only quasi-coassociative we adopt the further convention (∆ ⊗ id) ◦ ∆(a) = a(1,1) ⊗ a(1,2) ⊗ a(2) and (id ⊗ ∆) ◦ ∆(a) = a(1) ⊗ a(2,1) ⊗ a(2,2) , etc. Furthermore, here and throughout we use the notation φ = Xj ⊗ Y j ⊗ Zj ;
¯ j ⊗ Y¯ j ⊗ Z¯ j , φ−1 = X
(3.13)
where we have suppressed the summation symbol. To give an example, Eq. (3.8) written with this notation looks like a(1) X j ⊗ a(2,1) Y j ⊗ a(2,2) Z j = X i a(1,1) ⊗ Y i a(1,2) ⊗ Z i a(2) A quasi-bialgebra G is called quasi-Hopf algebra, if there is a linear antimorphism S : G → G and elements α, β ∈ G satisfying (for all a ∈ G) S(a(1) )αa(2) = α(a) ,
a(1) βS(a(2) ) = β(a)
¯ j )αY¯ j βS(Z¯ j ) , X j βS(Y j )αZ j = 1 = S(X
(3.14) (3.15)
where we have used the notation (3.13). The map S is called an antipode. We will also always suppose that S is invertible. Note that as opposed to ordinary Hopf algebras, an antipode is not uniquely determined, provided it exists. Together with a quasi-Hopf algebra G ≡ (G, ∆, , φ, S, α, β) we also have Gop , G cop cop and Gop as quasi-Hopf algebras, where “op” means opposite multiplication and
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“cop” means opposite comultiplication. The quasi-Hopf structures are obtained by 321 cop −1 , Sop = S cop = (Sop ) := S −1 , putting φop := φ−1 , φcop := (φ−1 )321 , φcop op := φ −1 −1 cop −1 cop −1 cop := S (α), β := S (β), αop := β and αop := S (β), βop := S (α), α cop := α. βop Next we recall that the definition of a quasi-Hopf algebra is “twist covariant” in the following sense: An invertible element F ∈ G ⊗ G, which satisfies ( ⊗ id)(F ) = (id ⊗ )(F ) = 1, induces a so-called twist transformation ∆F (a) := F ∆(a)F −1 ,
(3.16)
φF := (1 ⊗ F ) (id ⊗ ∆)(F ) φ (∆ ⊗ id)(F −1 ) (F −1 ⊗ 1)
(3.17)
It has been noticed by Drinfel’d [18] that (G, ∆F , , φF ) is again a quasi-bialgebra. Setting αF := S(hi )αk i , βF := f i βS(g i ) , where hi ⊗ k i = F −1 and f i ⊗ g i = F , (G, ∆F , , φF , S, αF , βF ) is also a quasi-Hopf algebra. This means that a twist preserves the class of quasi-Hopf algebras [18]. It is well known that the antipode of a Hopf algebra is also an anti coalgebra morphism, i.e. ∆(a) = (S ⊗ S)(∆op (S −1 (a))). For quasi-Hopf algebras this is true only up to a twist: Following Drinfel’d we define the elements γ, δ ∈ G ⊗ G by settingf γ := (S(U i ) ⊗ S(T i )) · (α ⊗ α) · (V i ⊗ W i )
(3.18)
δ := (K j ⊗ Lj ) · (β ⊗ β) · (S(N j ) ⊗ S(M j )) ,
(3.19)
where T i ⊗ U i ⊗ V i ⊗ W i = (1 ⊗ φ−1 ) · (id ⊗ id ⊗ ∆)(φ) , K j ⊗ Lj ⊗ M j ⊗ N j = (∆ ⊗ id ⊗ id)(φ) · (φ−1 ⊗ 1) . With these definitions Drinfel’d has shown in [18], that f ∈ G ⊗ G given by ¯ i )) · γ · ∆(Y¯ i βS(Z¯ i )) . f := (S ⊗ S)(∆op (X
(3.20)
defines a twist with inverse given by ¯ j )αY¯ j ) · δ · (S ⊗ S)(∆op (Z¯ i )) , f −1 = ∆(S(X
(3.21)
such that for all a ∈ G f ∆(a)f −1 = (S ⊗ S)(∆op (S −1 (a))) .
(3.22)
The elements γ, δ and the twist f fulfill the relations f ∆(α) = γ ,
∆(β) f −1 = δ .
(3.23)
Furthermore, the corresponding twisted reassociator (3.17) is given by φf = (S ⊗ S ⊗ S)(φ321 ) . f Suppressing summation symbols.
(3.24)
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Setting h := (S −1 ⊗ S −1 )(f 21 ), the above relations imply h∆(a)h−1 = (S −1 ⊗ S −1 )(∆op (S(a)))
(3.25)
φh = (S −1 ⊗ S −1 ⊗ S −1 )(φ321 )
(3.26)
h∆(S −1 (α)) = (S −1 ⊗ S −1 )(γ 21 )
(3.27)
These identities will be used frequently below as well as the following: Corollary 3.2. For a ∈ G let ∆L (a) := h∆(a) and ∆R (a) := ∆(a)h−1 , where h ∈ G ⊗ G is the twist in (3.25). Then (id ⊗ ∆L )(∆L (a)) φ = (S −1 ⊗ S −1 ⊗ S −1 )(φ321 ) (∆L ⊗ id)(∆L (a)) φ (∆R ⊗ id)(∆R (a)) = (id ⊗ ∆R )(∆R (a)) (S −1 ⊗ S −1 ⊗ S −1 )(φ321 ) ,
∀a ∈ G .
Proof. Writing Eq. (3.26) as (1 ⊗ h) (id ⊗ ∆)(h) φ = (S −1 ⊗ S −1 ⊗ S −1 )(φ321 ) (h ⊗ 1) (∆ ⊗ id)(h) , multiplicating both sides from the right with (∆ ⊗ id)(∆(a)) and using (3.8) yields the first equality. The second equality is proven analogously. We remark that the algebraic properties of a quasi-Hopf algebra G may be translated into corresponding properties of its representation category Rep G. More exactly Rep G is a rigid monoidal category, see Appendix A. Finally we introduce Gˆ as the dual space of G with its natural coassociative ˆ ˆ) given by h∆(ϕ)|a ˆ coalgebra structure (∆, ⊗ bi := hϕ|abi and ˆ(ϕ) := hϕ|1G i, ˆ where ϕ ∈ G, a, b ∈ G and where h·|·i : Gˆ ⊗ G → C denotes the dual pairing. On Gˆ we have the natural left and right G-actions a * ϕ := ϕ(1) hϕ(2) |ai ,
ϕ ( a := ϕ(2) hϕ(1) |ai ,
(3.28)
ˆ By transposing the coproduct on G we also get a multiplication where a ∈ G, ϕ ∈ G. ˆ ˆ ˆ G ⊗ G → G, which however is no longer associative hϕψ|ai := hϕ ⊗ ψ|∆(a)i ,
ˆ h1|ai := (a) .
ˆ ˆ ˆ ˆ = ϕ1 ˆ = ϕ, ∆(ϕψ) Yet, we have the identities 1ϕ = ∆(ϕ) ∆(ψ), a * (ϕψ) = (a(1) * ϕ) (a(2) * ψ) and (ϕψ) ( a = (ϕ ( a(1) )(ψ ( a(2) ) for all ϕ, ψ ∈ Gˆ and a ∈ G. We also introduce Sˆ : Gˆ → Gˆ as the coalgebra anti-morphism dual to S, ˆ i.e. hS(ϕ)|ai := hϕ|S(a)i. 3.2. Coactions of quasi-Hopf algebras The generalization of the definition of coactions as given in (2.1)–(2.4) to the quasi-Hopf case is straightforward:
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Definition 3.3. A left coaction of a quasi-bialgebra (G, 1G , ∆, , φ) on a unital algebra M is an algebra morphism λ : M → G ⊗ M together with an invertible element φλ ∈ G ⊗ G ⊗ M satisfying (id ⊗ λ)(λ(m)) φλ = φλ (∆ ⊗ id)(λ(m)) ,
∀m ∈ M
(3.29a)
(1G ⊗ φλ)(id ⊗ ∆ ⊗ id)(φλ)(φ ⊗ 1M) = (id ⊗ id ⊗ λ)(φλ)(∆ ⊗ id ⊗ id)(φλ) , (3.29b) ( ⊗ id) ◦ λ = id
(3.29c)
(id ⊗ ⊗ id)(φλ ) = ( ⊗ id ⊗ id)(φλ ) = 1G ⊗ 1M .
(3.29d)
Similarly a right coaction of G on M is an algebra morphism ρ : M → M ⊗ G together with φρ ∈ M ⊗ G ⊗ G such that φρ (ρ ⊗ id)(ρ(m)) = (id ⊗ ∆)(ρ(m)) φρ ,
∀m ∈ M
(1M ⊗ φ)(id ⊗ ∆ ⊗ id)(φρ)(φρ ⊗ 1G ) = (id ⊗ id ⊗ ∆)(φρ)(ρ ⊗ id ⊗ id)(φρ) ,
(3.30a) (3.30b)
(id ⊗ ) ◦ ρ = id
(3.30c)
(id ⊗ ⊗ id)(φρ ) = (id ⊗ id ⊗ )(φρ ) = 1M ⊗ 1G
(3.30d)
The triple (M, λ, φλ ) [(M, ρ, φρ )] is called a left [right] comodule algebra over G also denoted λM [Mρ ]. We remark, that of the two counit conditions in (3.29d) and (3.30d), respectively, actually either one of them already implies the other. Clearly, if G is a Hopf algebra, φ = 1G ⊗ 1G ⊗ 1G and φλ = 1G ⊗ 1G ⊗ 1M , one recovers the definitions given in (2.1–2.4). Also, particular examples are given by M = G and λ = ρ = ∆, φλ = φρ = φ. In the general case Eqs. (3.29b), (3.30b) may be understood as a generalized pentagon equation, whereas (3.29a), (3.30a) mean, that λ, ρ respect the quasi-coalgebra structure of G. One should notice, that because of the pentagon Eqs. (3.29b) and (3.30b), φλ and φρ have to be nontrivial if φ is nontrivial (i.e. if G is not a Hopf algebra). On the other hand φλ or φρ may be nontrivial even if φ = 1G ⊗ 1G ⊗ 1G , i.e. if G is a Hopf algebra. In fact, such a restricted setting has been investigated before, see [14, 10, 11]. In [10, 11] Eq. (3.30a) is called a “twisted module condition” and Eq. (3.30b) (for φ = 1 ⊗ 1 ⊗ 1) a “cocycle condition”. We will see in Sec. 3.4, that the twisted crossed products considered in [14, 10, 11] are in fact special types of our diagonal crossed products to be given in Definition 3.9 below. As with Hopf algebras, a left coaction λ (a right coaction ρ) induces a map / : M ⊗ Gˆ → M ( . : Gˆ ⊗ M → M) by m / ϕ := (ϕ ⊗ id)(λ(m)) , ϕ . m := (id ⊗ ϕ)(ρ(m)) ,
(3.31) ˆ m∈M ϕ ∈ G,
(3.32)
which by convenient abuse of notation and terminology we still call a “right action” (“left action”) of Gˆ on M, despite of the fact that Gˆ may not be an associative algebra.
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Similarly as for the coproduct ∆ there is a natural notion of twist equivalence for coactions of quasi-Hopf algebras. Lemma 3.4. Let (ρ, φρ ) be a right coaction of a quasi-bialgebra G on M and let U ∈ M ⊗ G be invertible such that (id ⊗ )(U ) = 1M . Then the pair (ρ0 , φ0ρ ), given by ρ0 (m) := U ρ(m) U −1 φ0ρ := (idM ⊗ ∆)(U ) φρ (ρ ⊗ idG )(U −1 ) (U −1 ⊗ 1G ) again defines a right coaction of G on M (with respect to the same quasi-bialgebra structure on G). The proof of Lemma 3.4 is straightforward and therefore omitted. A similar statement holds for left coactions λ, where one would have to take U ∈ G ⊗ M and φ0λ = (1G ⊗ U ) (idG ⊗ λ)(U ) φλ (∆ ⊗ idM )(U −1 ) . Note that twisting indeed defines an equivalence relation for coactions. Similarly, if ∆F and φF are given by (3.16) and (3.17), then any right (left) G-coaction on M may also be considered as a coaction with respect to the F -twisted structures on G by putting ρF = ρ (λF = λ) and (φρ )F := (1M ⊗ F ) φρ ,
(φλ )F := φλ (F −1 ⊗ 1M ) .
The reader is invited to check that with these definitions (3.29b) and (3.30b) are indeed also twist covariant. Note that in the case λ = ρ = ∆, U = F , one recovers (3.17). 3.3. Two-sided coactions As already mentioned before, the fact that the dual Gˆ fails to be an associative algebra is the reason why there is no generalization of the definitions of ordinary crossed products to the quasi-Hopf algebra case. Nevertheless this will be possible for our diagonal crossed product constructed from two-sided coactions. First we need: Definition 3.5. A two-sided coaction of a quasi-bialgebra (G, ∆, , φ) on an algebra M is an algebra map δ : M → G ⊗ M ⊗ G together with an invertible element Ψ ∈ G ⊗ G ⊗ M ⊗ G ⊗ G satisfying (idG ⊗ δ ⊗ idG )(δ(m)) Ψ = Ψ (∆ ⊗ idM ⊗ ∆)(δ(m)) ,
∀m ∈ M
(3.33a)
(1G ⊗ Ψ ⊗ 1G ) (idG ⊗ ∆ ⊗ idM ⊗ ∆ ⊗ idG )(Ψ) (φ ⊗ 1M ⊗ φ−1 ) = (idG ⊗ idG ⊗ δ ⊗ idG ⊗ idG )(Ψ) (∆ ⊗ idG ⊗ idM ⊗ idG ⊗ ∆)(Ψ)
(3.33b)
( ⊗ idM ⊗ ) ◦ δ = idM
(3.33c)
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(idG ⊗ ⊗ idM ⊗ ⊗ idG )(Ψ) = ( ⊗ idG ⊗ idM ⊗ idG ⊗ )(Ψ) = 1G ⊗ 1M ⊗ 1G .
(3.33d)
The triple (M, δ, Ψ) is called a two-sided comodule algebra, also denoted Mδ . Again we remark, that either one of the two counit axioms in (3.33d) already implies the other. We also note, that two-sided coactions could of course be considered as right coactions of G ⊗ G cop , or left coactions of G cop ⊗ G, respectively. Moreover, if (δ, Ψ) is a two-sided coaction of G on M, then (δ, Ψ−1 ) is a two-sided coaction of Gop on Mop and (δop , Ψop ) is a two-sided coaction of G cop on M, where δop := δ 321 ,
Ψop := Ψ54321 .
(3.34)
An example of a two-sided coaction is given by M = G, δ = (∆ ⊗ id) ◦ ∆ and Ψ := [(id ⊗ ∆ ⊗ id)(φ) ⊗ 1][φ ⊗ 1 ⊗ 1][(δ ⊗ id ⊗ id)(φ−1 )] .
(3.35)
Similarly we could choose δ 0 = (id ⊗ ∆) ◦ ∆ and Ψ0 := [1 ⊗ (id ⊗ ∆ ⊗ id)(φ−1 )][1 ⊗ 1 ⊗ φ−1 ][(id ⊗ id ⊗ δ 0 )(φ)] .
(3.36)
From this example one already realizes that in the present context the relation between two-sided coactions and pairs of commuting left and right coactions gets somewhat more involved as compared to Sec. 2, where we had δ = δ 0 . First, one easily checks that for any two-sided coaction (δ, Ψ) the definitions λ := (idG ⊗ idM ⊗ ) ◦ δ ,
φλ := (idG ⊗ idG ⊗ idM ⊗ ⊗ )(Ψ) ,
(3.37)
ρ := ( ⊗ idM ⊗ idG ) ◦ δ ,
φ−1 ρ := ( ⊗ ⊗ idM ⊗ idG ⊗ idG )(Ψ)
(3.38)
provide us again with a left coaction (λ, φλ ) and a right coaction (ρ, φρ ). Moreover, putting δ (2) := (id ⊗ δ ⊗ id) ◦ δ we have δl := (λ ⊗ idG ) ◦ ρ = ( ⊗ idG ⊗ idM ⊗ ⊗ idG ) ◦ δ (2)
(3.39)
δr := (idG ⊗ ρ) ◦ λ = (idG ⊗ ⊗ idM ⊗ idG ⊗ ) ◦ δ (2) .
(3.40)
However, due to the appearance of the reassociator Ψ in axiom (3.33a), the two expressions (3.39) and (3.40) are in general unequal, and neither one needs to coincide with δ. Indeed, defining Ul := ( ⊗ idG ⊗ idM ⊗ ⊗ idG )(Ψ)
(3.41)
Ur := (idG ⊗ ⊗ idM ⊗ idG ⊗ )(Ψ)
(3.42)
φλρ := Ur Ul−1 ,
(3.43)
(3.37)–(3.40) and (3.33a)–(3.33d) immediately imply δl (m) = Ul δ(m) Ul−1 ,
δr (m) = Ur δ(m) Ur−1 (3.44)
φλρ (λ ⊗ id)(ρ(m)) = (id ⊗ ρ)(λ(m)) φλρ .
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Moreover (λ, ρ, φλ , φρ , φλρ ) provides a quasi-commuting pair of coactions in the following sense. Definition 3.6. Let (G, ∆, , φ) be a quasi-bialgebra. By a quasi-commuting pair of G-coactions on an algebra M we mean a quintuplet (λ, ρ, φλ , φρ , φλρ ), where (λ, φλ ) and (ρ, φρ ) are left and right G-coactions on M, respectively, and where φλρ ∈ G ⊗ M ⊗ G is invertible and satisfies φλρ (λ ⊗ id)(ρ(m)) = (id ⊗ ρ)(λ(m)) φλρ ,
∀m ∈ M
(3.45a)
(1G ⊗ φλρ )(id ⊗ λ ⊗ id)(φλρ )(φλ ⊗ 1G ) = (id ⊗ id ⊗ ρ)(φλ )(∆ ⊗ id ⊗ id)(φλρ ) (3.45b) (1G ⊗ φρ )(id ⊗ ρ ⊗ id)(φλρ )(φλρ ⊗ 1G ) = (id ⊗ id ⊗ ∆)(φλρ )(λ ⊗ id ⊗ id)(φρ ) . (3.45c) Obviously, the conditions (3.45a)–(3.45c) apply to the case M = G, λ = ρ = ∆ and φλ = φρ = φ. Also note, that acting with ( ⊗ ⊗ idM ⊗ idG ) on (3.45b) and with (idG ⊗ idM ⊗ ⊗ ) on (3.45c) and using the invertibility of φλρ one concludes the further identities (idG ⊗ idM ⊗ )(φλρ ) = 1G ⊗ 1M ,
( ⊗ idM ⊗ idG )(φλρ ) = 1M ⊗ 1G .
(3.45d)
We also remark that quasi-commutativity is stable under twisting. The fact that (λ, ρ, φλ , φρ , φλρ ) given by (3.37), (3.38) and (3.43) provides a quasi-commuting pair is shown in detail in [26]. Conversely, one also verifies by direct computation, that every pair of quasi-commuting coactions (λ, ρ, φλ , φρ , φλρ ) provides us with two-sided coactions (Ψl , δl ) and (Ψr , δr ) defined by: δl := (λ ⊗ id) ◦ ρ
(3.46a)
⊗ −1 Ψl := (idG ⊗ λ ⊗ id⊗ G )((φλρ ⊗ 1G )(λ ⊗ idG )(φρ )) [φλ ⊗ 1G ⊗ 1G ]
(3.46b)
δr := (id ⊗ ρ) ◦ λ
(3.46c)
2
2
⊗ −1 −1 Ψr := (id⊗ G ⊗ ρ ⊗ idG )((1G ⊗ φλρ ) (idG ⊗ ρ)(φλ )) [1G ⊗ 1G ⊗ φρ ] . 2
2
(3.46d)
Note that (3.46a)–(3.46d) generalize the examples (3.35) and (3.36). Using this result one is now in the position to show that twist-equivalence classes of quasi-commuting pairs of coactions (λ, ρ, φλ , φρ , φλρ ) are in one-to-one correspondence with twist equivalence classes of two-sided coactions (δ, Ψ), since by (3.44) up to twist equivalence any two-sided coaction is of the type (δl/r , Ψl/r ) given in (3.46a)–(3.46d). Here one uses that two-sided coactions (δ, Ψ) may be twisted in the same fashion as one-sided ones. Definition 3.7. Let (δ, Ψ) and (δ 0 , Ψ0 ) be two-sided coactions of (G, ∆, , φ) on M. Then (δ 0 , Ψ0 ) is called twist equivalent to (δ, Ψ), if there exists U ∈ G ⊗ M ⊗ G
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invertible such that δ 0 (m) = U δ(m) U −1
(3.47a)
Ψ0 = (1G ⊗ U ⊗ 1G ) (idG ⊗ δ ⊗ idG )(U ) Ψ (∆ ⊗ idM ⊗ ∆)(U −1 )
(3.47b)
( ⊗ idM ⊗ )(U ) = 1M .
(3.47c)
The reader is invited to check that for any two-sided coaction (δ, Ψ) and any invertible U satisfying (3.47c) the definitions (3.47a) and (3.47b) indeed produce another two-sided coaction (δ 0 , Ψ0 ). It is also easy to see that twisting does provide an equivalence relation between two-sided coactions. Moreover, similarly as for onesided coactions one readily verifies that if (δ, Ψ) is a two-sided coaction of (G, ∆, , φ) on M, then for any twist F ∈ G ⊗ G the pair (δ, ΨF ) is a two-sided coaction of (G, ∆F , , φF ) on M, where ∆F and φF are the twisted structures on G given by (3.16) and (3.17), and where ΨF := Ψ (F −1 ⊗ 1M ⊗ F −1 ) .
(3.47d)
We summarize the connection between two-sided coactions and quasi-commuting pairs of coactions in the following proposition. Proposition 3.8. Twist-equivalence classes of quasi-commuting pairs of coactions (λ, ρ, φλ , φρ , φλρ ) are in one-to-one correspondence with twist equivalence classes of two-sided coactions (δ, Ψ). In particular the elements Ul/r defined in (3.41)/(3.42) provide a twist equivalence between (δ, Ψ) and (δl/r , Ψl/r ) given by (3.37)–(3.40), (3.43) and (3.46a)–(3.46d). The proof of Proposition 3.8, especially the detailed calculations that all pentagon equations are satisfied, is elementary but quite lengthy and is given in [26]. The importance of Proposition 3.8 stems from the fact that below the diagonal crossed products associated with twist-equivalent two-sided coactions will be shown to be isomorphic. 3.4. The algebras Gˆ ./ M and M ./ Gˆ Having developed our theory of two-sided G-coactions δ for quasi-bialgebras G we are now in the position to generalize the construction of the left and right diagonal crossed products Gˆ ./ Mδ and Mδ ./ Gˆ to the quasi-coassociative setting. Before writing down the concrete multiplication rules we would like to draw the reader’s attention to some important conceptual differences in comparison with the results of Sec. 2. As already remarked, the natural “multiplication” µ ˆ : Gˆ ⊗ Gˆ → Gˆ given as the transpose of the coproduct ∆ : G → G ⊗ G is not associative. Nevertheless, we will ˆ for details see the end of Sec. 3.1. This will still write ϕψ := µ ˆ(ϕ ⊗ ψ), ϕ, ψ ∈ G, imply the fact that although we will have Gˆ ./ Mδ = Gˆ ⊗ M and Mδ ./ Gˆ = M ⊗ Gˆ as linear spaces, the subspaces Gˆ ⊗ 1M and 1M ⊗ Gˆ will not be subalgebras in the
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diagonal crossed product. On the other hand, M will naturally be embedded as ˆ⊗M ∼ ˆ We would also like to stress that the unital subalgebra M ∼ = 1 = M ⊗ 1. Gˆ ./ Mδ ∼ = Mδ ./ Gˆ will still be equivalent algebra extensions of M. However the subspaces Gˆ ./ 1M and 1M ./ Gˆ will not be mapped onto each other under this isomorphism. (Recall that this was the case in (2.30).) We now proceed to the details. Given a two-sided G-coaction (δ, Ψ) on M we still write as before ϕ . m / ψ := (ψ ⊗ idM ⊗ ϕ)(δ(m)) ,
m ∈ M, ϕ, ψ ∈ Gˆ ,
(3.48)
disregarding the fact that δ might be neither of the form (3.46a) nor (3.46c). We also introduce the element ΩL ∈ G ⊗ G ⊗ M ⊗ G ⊗ G built from the reassociator Ψ by ΩL ≡ Ω1L ⊗ Ω2L ⊗ Ω3L ⊗ Ω4L ⊗ Ω5L := (idG ⊗ idG ⊗ idM ⊗ S −1 ⊗ S −1 )(Ψ−1 )·h54 , (3.49) where h ≡ (S −1 ⊗ S −1 )(f 21 ) ∈ G ⊗ G has been introduced in (3.25). As before, we have dropped all summation symbols and summation indices. Definition 3.9. Let (δ, Ψ) be a two-sided coaction of a quasi-Hopf algebra G on an algebra M. We define the left diagonal crossed product Gˆ ./ Mδ to be the vector space Gˆ ⊗ M equipped with the multiplication rule (ϕ ./ m)(ψ ./ n) := [(Ω1L * ϕ ( Ω5L )(Ω2L * ψ(2) ( Ω4L )] ./ [Ω3L (Sˆ−1 (ψ(1) ) . m / ψ(3) ) n] .
(3.50)
In cases where the two-sided coaction is unambiguously understood from the context we also write Gˆ ./ M. Note that (3.50) again implies ˆ ./ m) . (ϕ ./ m) = (ϕ ./ 1M )(1
(3.51)
and for ΩL/R = 1G ⊗ 1G ⊗ 1M ⊗ 1G ⊗ 1G we recover the definition of Sec. 2. Also, in Gˆ ./ M we still have the “commutation relation” m ψ = ψ(2) (Sˆ−1 (ψ(1) ) . m / ψ(3) ) . But for the product (ϕ ./ 1M )(ψ ./ 1M ) to be consistent with this relation one has to inject the reassociator Ψ since δ (2) 6= (∆ ⊗ id ⊗ ∆) ◦ δ and also the twist h since ∆ ◦ S −1 6= (S −1 ⊗ S −1 ) ◦ ∆op in the quasi-coassociative case. We now formulate our first main result. Theorem 3.10. (i) The left diagonal crossed product Gˆ ./ M is an associative algebra with unit ˆ ./ 1M . 1 ˆ ./ M ⊂ Gˆ ./ M is a unital algebra inclusions. (ii) M ≡ 1 We will give a detailed proof of Theorem 3.10 in Sec. 3.6. Let us shortly sketch the idea. Let L ∈ G ⊗ (Gˆ ./ M) be given by L = eµ ⊗ (eµ ./ 1M ), where {eµ } is a ˆ We also abbreviate our notation by identifying basis in G with dual basis {eµ } in G.
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ˆ ./ m), m ∈ M. The multiplication rule (3.50) implies m ≡ (1 [1G ⊗ m] L = [S −1 (m(1) ) ⊗ 1M ] L [m(−1) ⊗ m(0) ] ,
∀m ∈ M
¯ 5 ) ⊗ S −1 (Ψ ¯ 4 ) ⊗ 1M ] [(∆L ⊗ id)(L)] [Ψ ¯1 ⊗ Ψ ¯2 ⊗ Ψ ¯ 3] , L13 L23 = [S −1 (Ψ
(3.52) (3.53)
¯ = where we have introduced the notation δ(m) = m(−1) ⊗ m(0) ⊗ m(1) and Ψ−1 ≡ Ψ 1 2 3 4 5 ¯ ¯ ¯ ¯ ¯ Ψ ⊗ Ψ ⊗ Ψ ⊗ Ψ ⊗ Ψ , and where ∆L (a) := h∆(a), a ∈ G, has been introduced in Corollary 3.2. With these relations the nontrivial associativity constraints to be shown are the following: !
L14 (L24 L34 ) = (L14 L24 )L34 !
[1G ⊗ 1G ⊗ m](L13 L23 ) = ([1G ⊗ 1G ⊗ m]L13 )L23 ,
(3.54) (3.55)
3 where (3.54) is understood as an identity in G ⊗ ⊗ (Gˆ ./ M) and (3.55) as an 2 identity in G ⊗ ⊗ (Gˆ ./ M). Now the identity (3.54) is shown by using the pentagon Eq. (3.33b) for Ψ, whereas (3.55) is implied by the intertwining properties (3.33a) of Ψ and (3.25) of h. The details are given in Sec. 3.6. As for Hopf algebras there is an analogue construction of a right diagonal crossed ˆ which in fact will be proven to be isomorphic to M ./ Gˆ in the product M ./ G, next section.
Definition 3.11. Given a two-sided coaction (δ, Ψ) of G on M, and setting ΩR ≡ Ω1R ⊗Ω2R ⊗Ω3R ⊗Ω4R ⊗Ω5R := (h−1 )21 ·(S −1 ⊗S −1 ⊗idM ⊗idG ⊗idG )(Ψ) , (3.56) we define the right diagonal crossed product Mδ ./ Gˆ to be the vector space M ⊗ Gˆ with the multiplication rule (m ./ ϕ)(n ./ ψ) := [m (ϕ(1) . n / Sˆ−1 (ϕ(3) )) Ω3R ] ./ [(Ω2R * ϕ(2) ( Ω4R )(Ω1R * ψ ( Ω5R )] .
(3.57)
Corollary 3.12. The right diagonal crossed product Mδ ./ Gˆ is an associative ˆ containing M ≡ M ./ 1 ˆ ⊂ M ./ Gˆ as a unital subalgebra. algebra with unit 1M ./ 1, Proof. The proof goes along the same lines as the proof of Theorem 3.10 by noting that under the trivial permutation of tensor factors we have ˆ op = Gˆcop ./ (Mop )δop , (Mδ ./ G) op ˆ op denotes the diagonal crossed product with opposite multipliwhere (Mδ ./ G) cation, and where we recall our remark that with the definition (3.34) the pair cop (δop , Ψ−1 op ) defines a two-sided coaction of Gop on Mop . Before proceeding let us shortly discuss how in the present context one can see that ordinary crossed products Mρ o Gˆ (or Gˆ n λM) in general cannot be defined as associative algebras any more. In the strictly coassociative setting of Sec. 2 these
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could be considered as special types of diagonal crossed products, where δ = 1G ⊗ ρ (or δ = λ ⊗ 1G ). In the present setting it is not clear whether such δ’s give well defined two-sided coactions, since in fact the maps λ0 (m) := 1G ⊗ m ,
ρ0 (m) := m ⊗ 1G
need not even be one-sided coactions. For this one would also need the existence of reassociators φλ0 , φρ0 satisfying the axioms of Definition 3.3 (note that e.g. the choice φρ0 = 1M ⊗ 1G ⊗ 1G will in general not do the job due to the appearance of φ in the pentagon equation (3.30b)). On the other hand, suppose there exists φλ0 such that (λ0 , φλ0 ) is a well-defined left coaction and let (ρ, φρ ) be a right coaction such that (idG ⊗ idG ⊗ ρ)(φλ0 ) = φλ0 ⊗ 1G . Then one immediately checks that (λ0 , ρ, φλ0 , φρ , φλρ := 1G ⊗ 1M ⊗ 1G ) provides a quasi-commuting pair of coactions and therefore in this case Mρ o Gˆ := Mδ ./ Gˆ would indeed be a well defined associative algebra, where δ = 1G ⊗ ρ and Ψ = ˆ (1G ⊗ 1G ⊗ φ−1 ρ )(φλ0 ⊗ 1G ⊗ 1G ). An analogous statement holds for G nλ0 M. Such a scenario may of course be produced trivially by starting with (δ = 1G ⊗ ρ, Ψ = 1G ⊗ 1G ⊗ 1M ⊗ 1G ⊗ 1G ) as a two-sided coaction on a strictly coassociative Hopf algebra (G, ∆) and subsequently passing to a twist equivalent quasi-Hopf algebra structure ∆F on G with ΨF given by (3.47d). As another example one may take a strictly coassociative Hopf algebra (G, ∆) with φ, φλ0 , α, β, h being trivial, but (ρ, φρ ) being a G-coaction on M in the sense of Definition 3.3, with a nontrivial cocycle φρ as considered in [14, 10, 11]. Hence δ := 1G ⊗ ρ, Ψ := 1G ⊗ 1G ⊗ φ−1 ρ would give a well defined two-sided G-coaction in the sense of our Definition 3.5. In this case ΩR in (3.56) would be given by ΩR = 1G ⊗ 1G ⊗ φ−1 ρ , and defining σ : Gˆ ⊗ Gˆ → M by σ(ϕ ⊗ ψ) := (id ⊗ ϕ ⊗ ψ)(φ−1 ρ ), one immediately verifies from Definition 3.11 that in this special case our right diagonal crossed product satisfies Mδ ./ Gˆ = M#σ Gˆ , where the right-hand side is the twisted crossed product considered in [14, 10, 11]. 3.5. Generating matrices We now pass to a formulation of diagonal crossed products in terms of generating matrices similarly as in Sec. 2.3. As discussed in Sec. 3.3 the connection between two-sided coactions δ and (quasi-commuting) pairs of coactions (λ, ρ) becomes more involved in the quasi-coassociative setting. This will make it necessary to distinguish between λρ-intertwiners and what we call left and right δ-implementers, which all
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three coincide in the coassociative setting of Sec. 2 due to Lemma 2.11. The precise relation between these different generating matrices will be clarified in Proposition 3.19, which finally leads to a proof of the main Theorem 3.1. We would like to encourage the reader to frequently glance at Appendix A, where he will find a representation theoretic interpretation of the generating matrices together with their relationships expressed in terms of commuting diagrams. 3.5.1. Left and right diagonal δ-implementers From the associativity proof of Theorem 3.10 in terms of the “generating matrix” L we immediately read off an analogue of Proposition 2.12 describing the conditions under which an algebra map γ : M → A into some target algebra A extends to an algebra map from the diagonal crossed products into A. In view of (3.52) and (3.53) we are lead to the following: Definition 3.13. Let γ : M → A be an algebra map into some target algebra A and let (δ, Ψ) be a two-sided G-coaction on M. A left (right) diagonal δ-implementer in A (with respect to γ) is an element L ∈ G ⊗ A (R ∈ G ⊗ A) satisfying for all m ∈ M, respectively, [1G ⊗ γ(m)] L = [S −1 (m(1) ) ⊗ 1A ] L [m(−1) ⊗ γ(m(0) )]
(3.58)
R [1G ⊗ γ(m)] = [m(1) ⊗ γ(m(0) )] R [S −1 (m(−1) ) ⊗ 1A ] .
(3.59)
A left δ-implementer L (right δ-implementer R) is called coherent if, respectively, L13 L23 = [Ω5L ⊗ Ω4L ⊗ 1A ] (∆ ⊗ id)(L) [Ω1L ⊗ Ω2L ⊗ γ(Ω3L )] R13 R23 = [Ω4R ⊗ Ω5R ⊗ γ(Ω3R )] (∆ ⊗ id)(R) [Ω2R ⊗ Ω1R ⊗ 1A )] ,
(3.60) (3.61)
where ΩL/R have been defined in (3.49)/(3.56). To unburden our terminology from now by a left (right) δ-implementer we will always mean a left (right) diagonal δ-implementer in the sense of the above definition. We trust that the reader will not be confused by this slight inconsistence of terminology (which arises in comparison with Definition 2.2, since two-sided coactions might also be looked upon as one-sided ones). As before, we also call L/R normal, if ( ⊗ id)(L/R) = 1A . Note that in the coassociative setting of Lemma 2.11 left and right δ-implementers always coincide. In the present context we will still have one-to-one correspondences between left and right δ-implementers, however the identifications will not be the trivial ones. Let us note the immediate: Corollary 3.14. Let (M, δ, Ψ) be a two-sided G-comodule algebra and let γ : M → A be an algebra map into some target algebra A. Then the relations γL (ϕ ./ m) = (ϕ ⊗ id)(L)γ(m)
(3.62)
γR (m ./ ϕ) = γ(m)(ϕ ⊗ id)(R)
(3.63)
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provide one-to-one correspondences between algebra maps γL : Gˆ ./ M → A (γR : M ./ Gˆ → A) extending γ and coherent left δ-implementers L (coherent right δ-implementers R), respectively, where γL /γR is unital if and only if L/R is normal. Proof. This follows immediately from (3.52) and (3.53) and the analogue relaP tions in Mδ ./ Gˆ (define L := µ eµ ⊗ γL (eµ ), with {eµ } ⊂ G, {eµ } ⊂ Gˆ being a pair of dual bases). Next, we show that the diagonal crossed products associated with twist equivalent two-sided coactions are equivalent algebra extensions. Proposition 3.15. 1. Let (δ, Ψ) and (δ 0 , Ψ0 ) be twist equivalent two-sided G-coactions on M. Then the diagonal crossed products Mδ ./ Gˆ and Mδ0 ./ Gˆ are equivalent extensions of M. 2. Let (δ, Ψ) be a two-sided G-coaction on M with respect to the coproduct ∆ : G → G ⊗ G, and let (δ, ΨF ) be the two-sided coaction with respect to a twist equivalent coproduct ∆F on G, see (3.47d). Denote the associated diagonal crossed products by M ./ Gˆ and M ./ GˆF , respectively. Then M ./ Gˆ = M ./ GˆF with trivial identification. Proof. 1. Let U ∈ G ⊗ M ⊗ G be a normal twist transformation from (δ, Ψ) ˆ and R0 ∈ G ⊗ (Mδ0 ./ G) ˆ be the generating to (δ 0 , Ψ0 ) and let R ∈ G ⊗ (Mδ ./ G) matrices. By Corollary 3.14, to provide a homomorphism f : Mδ ./ Gˆ → Mδ0 ./ Gˆ restricting to the identity on M we have to find a coherent normal right δ-imple˜ ∈ G ⊗ (Mδ0 ./ G). ˆ We claim that the canonical choice (writing U −1 ≡ menter R 1 2 3 ¯ ⊗U ¯ , where summation symbols are suppressed) ¯ ⊗U U ¯ 2 )]R0 [S −1 (U ¯ 1 ) ⊗ 1A ] ˜ := [U ¯ 3 ⊗ γ(U R
(3.64)
˜ obviously is a normal right δ-implementer and one is left will do the job. Indeed, R with checking the coherence condition with respect to (δ, Ψ). Using (3.47b) this is straightforward and is left to the reader. To prove Part 2 we note that ΨF = Ψ(F −1 ⊗ 1M ⊗ F −1 ) implies by (3.56) (ΩR )F = F 21 ΩR (F −1 )45 since the element h ∈ G ⊗ G transforms under a twist −1 )hF −1 . Hence, by Definition 3.13, R is coherent according to hF = (S −1 ⊗S −1 )(Fop with respect to (δ, Ψ, ∆) if and only if it is coherent with respect to (δ, ΨF , ∆F ). Of course, analogous statements hold for the left diagonal crossed products. Remark 3.16. In view of Proposition 3.8 we may from now on restrict ourselves to two-sided coactions of the form (δ, Ψ) = (δl/r , Ψl/r ) for a quasi-commuting pair (λ, φλ , ρ, φρ , φλρ ), where δl = (λ ⊗ id) ◦ ρ and δr = (id ⊗ ρ) ◦ λ, see (3.46a)– (3.46d). In this light it will also be appropriate to introduce as an alternative
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notation consistent with (3.6), (3.7) Gˆ ./ λMρ := Gˆ ./ Mδl λMρ
./ Gˆ := Mδr ./ Gˆ .
(3.65) (3.66)
By Proposition 3.15(1) we also have Mδr ./ Gˆ ∼ = Mδl ./ Gˆ and Gˆ ./ Mδl ∼ = ˆ G ./ Mδr since δl and δr are twist equivalent. As will be shown below, also Gˆ ./ Mδl and Mδr ./ Gˆ are equivalent extensions of M. Thus we get four equivalent versions of diagonal crossed products associated with any quasi-commuting pair (λ, ρ, φλ , φρ , φλρ ) of G-coactions on M, all of which will be shown to be a realization of the abstract algebra M1 in Theorem 3.1. 3.5.2. Coherent λρ-intertwiners In this subsection we are going to generalize Lemma 2.11 by providing a normality and coherence preserving one-to-one correspondence between right δr -implementers R or left δl -implementers L, respectively, and λρ-intertwiners T, where δr := (id⊗ρ)◦λ and δl := (λ⊗id)◦ρ. This will finally lead to a proof of Theorem 3.1. As a Corollary we get that the left and right diagonal crossed products Gˆ ./ Mδ and Mδ ./ Gˆ are equivalent algebra extensions of M. We start with a generalization of Definition 2.10. Definition 3.17. Let (λ, φλ , ρ, φρ , φλρ ) be a quasi-commuting pair of G-coactions on M and let γ : M ⊗ A be a unital algebra map into some target algebra A. A λρ-intertwiner in A (with respect to γ) is an element T ∈ G ⊗ A satisfying T λA (m) = ρop A (m) T ,
∀m ∈ A.
(3.67)
A λρ-intertwiner is called normal if ( ⊗ id)(T) = 1A and it is called coherent, if 13 −1 132 23 (φ312 ρ )A T (φλρ )A T (φλ )A = (∆ ⊗ id)(T) ,
(3.68)
where the index A refers to the image γ(M) ⊂ A, see also (3.2) and (3.3). We first point out that (3.68) is consistent with (3.67) in the following sense: Lemma 3.18. Under the conditions of Definition 3.17 let T be a λρ-intertwiner in A and define B ∈ G ⊗ G ⊗ A by 13 −1 132 23 B := (φ312 ρ )A T (φλρ ) A T (φλ )A .
(3.69)
Then we have for all m ∈ M B(∆ ⊗ idA )(λ(m)A ) = (∆ ⊗ idA )(ρop (m)A )B .
(3.70)
Proof. This is straightforward from the intertwiner properties of T and of φλ , φλρ and φρ , see (3.67), (3.29a), (3.30a) and (3.45a).
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To provide a bijective map between coherent δ-implementers and coherent λρintertwiners we first need a generalization of formulas like m(0) ⊗ S −1 (m(2) )m(1) = m ⊗ 1G , which are not valid any more due to quasi-coassociativity and the more complicated antipode axioms (3.14)–(3.15). Recall that formulas of this type have been used to prove Lemma 2.11. Associated with any left G-coaction (λ, φλ ) on M we define elements pλ , qλ ∈ G ⊗ M by pλ := φ2λ S −1 (φ1λ β) ⊗ φ3λ , qλ := S(φ¯1λ )αφ¯2λ ⊗ φ¯3λ ,
where φλ = φ1λ ⊗ φ2λ ⊗ φ3λ , ¯1 ¯2 ¯3 where φ−1 λ = φλ ⊗ φλ ⊗ φλ ,
(3.71) (3.72)
and where as before we have dropped summation indices and summation symbols. Here α, β ∈ G are the elements introduced in (3.14). In the case M = G and (λ, φλ ) = (ρ, φρ ) = (∆, φ) analogues of these elements have also been considered by [18, 47]. Denoting λ(m) ≡ m(−1) ⊗ m(0) they satisfy (denoting qλ = qλ1 ⊗ qλ2 , etc.): λ(m(0) )pλ [S −1 (m(−1) ) ⊗ 1M ] = pλ [1G ⊗ m] [S(m(−1) ) ⊗ 1M ]qλ λ(m(0) ) = [1G ⊗ m]qλ λ(qλ2 )pλ [S −1 (qλ1 ) ⊗ 1M ] = 1G ⊗ 1M [S(p1λ ) ⊗ 1M ]qλ λ(p2λ ) = 1G ⊗ 1M . Note that the first two equalities provide a substitute for the non-valid formula m(0) ⊗ S −1 (m(2) )m(1) = m ⊗ 1G , whereas the second pair state some kind of invertibility property of the elements pλ , qλ . Similarly, associated with any right G-coaction (ρ, φρ ) on M we define elements pρ , qρ ∈ M ⊗ G by pρ := φ¯1ρ ⊗ φ¯2ρ βS(φ¯3ρ ) ,
where
qρ := φ1ρ ⊗ S −1 (αφ3ρ )φ2ρ ,
¯1 ¯2 ¯3 φ−1 ρ = φρ ⊗ φρ ⊗ φρ
(3.73)
φρ= φ1ρ ⊗ φ2ρ ⊗ φ3ρ .
(3.74)
where
They obey a similar set of equations. All these equalities together with some kind of “coherence” property are proven below in Lemma 3.21 and Lemma 3.22 in Sec. 3.6. Again the reader may find it helpful to consult the representation theoretic interpretation of the elements qλ , pλ , qρ , pρ given in Appendix A, starting with (A.8). We now state the generalization of Lemma 2.11. Throughout, by a convenient abuse of notation, we are going to omit the symbol γ. Proposition 3.19. Under the conditions of Definition 3.17: 1. Let δr := (id ⊗ ρ) ◦ λ and Ψr ∈ G ⊗ G ⊗ M ⊗ G ⊗ G as in (3.46d) and let pλ , qλ ∈ G ⊗ M be given by (3.71), (3.72). Then the assignments (omitting the symbol γ) T 7→ R := T pλ
(3.75)
R 7→ T := ρop (qλ2 ) R [S −1 (qλ1 ) ⊗ 1A ]
(3.76)
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provide mutually inverse normality and coherence preserving isomorphisms between the space of λρ-intertwiners and the space of right δr -implementers. 2. Similarly let δl := (λ ⊗ id) ◦ ρ and Ψl ∈ G ⊗ G ⊗ M ⊗ G ⊗ G as in (3.46b), and let pρ , qρ ∈ M ⊗ G be given by (3.73), (3.74). Then the assignments T 7→ L := qρop T
(3.77)
L 7→ T := [S −1 (p2ρ ) ⊗ 1A ] L λ(p1ρ )
(3.78)
provide mutually inverse normality and coherence preserving isomorphisms between the space of λρ-intertwiners and the space of left δl -implementers. The proof of Proposition 3.19 is given in Sec. 3.6. The content of the above proposition may also be expressed in terms of commuting diagrams, see (A.17) in Appendix A. We state the immediate: Corollary 3.20. The left and the right diagonal crossed products Gˆ ./ Mδ and Mδ ./ Gˆ defined in Definitions 3.9 and 3.11 are isomorphic algebra extensions of M. Proof. First note that by Remark 3.16 we have Gˆ ./ Mδ ∼ = Gˆ ./ Mδl and µ ˆ Now let Rδr := eµ ⊗ (1M ./ e ) be the coherent δr Mδ ./ Gˆ ∼ = Mδr ./ G. ˆ By (3.76) and (3.77) implementer in Mδr ./ G. ˜ δ := q op ρ(q 2 ) Rδr [S −1 (q 1 ) ⊗ 1A ] L ρ λ λ l is a coherent δl -implementer. Thus, by Corollary 3.14 we get an algebra map f : Gˆ ./ Mδl → Mδr ./ Gˆ by setting ˜ δ ) (m ./ 1) ˆ . f (ϕ ./ m) := (ϕ ⊗ id)(L l Using (3.75) and (3.78) one shows analogously that f is invertible.
(3.79)
We are finally in the position to proof the main Theorem 3.1 stated in the introduction to Sec. 3. Proof of Theorem 3.1. To prove the existence of M1 we choose M1 := Mδr ./ Gˆ and Γ : Gˆ → M1 , Γ(ϕ) = (ϕ ⊗ idM1 )(Γ) ,
Γ = ρop (qλ2 ) Rδr [S −1 (qλ1 ) ⊗ 1A ] ,
(3.80)
where qλ ∈ G ⊗ M is given by (3.72) and where Rδr := eµ ⊗ (1M ./ eµ ) ∈ G ⊗ M1 is the canonical coherent normal δr -implementer. Hence, by Proposition 3.19, Γ is a normal coherent λρ-intertwiner in M1 . Moreover, we obtain for the map µR in (3.5) ˆ Γ(ϕ(1) ) (ϕ(2) ⊗ id)(pλ ) µR (m ⊗ ϕ) := (m ./ 1) ˆ (ϕ ⊗ id)(Γ pλ ) = (m ./ 1)
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ˆ (1M ./ ϕ) = (m ./ 1) ≡ (m ./ ϕ)
(3.81)
by Proposition 3.19(1), and therefore µR : M ⊗ Gˆ → M1 becomes the identity map. ˆ ˆ and Γ(G). This also shows that M1 is algebraically generated by M ≡ (M ./ 1) The universality property follows from Corollary 3.14 — providing a one-to-one correspondence between algebra extensions M1 → A and δr -implementers — and Proposition 3.19 — providing a one-to-one correspondence between δr -implementers R and λρ-intertwiners T. The uniqueness of M1 (up to equivalence) follows by standard arguments from the universality property stated in Part 1 and the fact that M1 is generated by M ˆ and Γ(G). ˆ → We are left with showing that with qρ ∈ M⊗G given by (3.74) also µL : G⊗M M1 given in (3.4) provides a linear isomorphism, which under the identification Gˆ ⊗ M = Gˆ ./ Mδl in fact becomes an algebra map. This is seen by realizing, that µL ≡ f , where f : Gˆ ./ Mδl → Mδr ./ Gˆ is the isomorphism defined in (3.79). 3.6. Proofs In this subsection we have collected the proofs omitted in the previous subsections. Proof of Theorem 3.10. One trivially checks the unit properties in Part (i) and also the identity ˆ ./ n) = (ϕ ./ mn) (ϕ ./ m)(1 (3.82) ˆ thereby proving Part (ii). for all m, n ∈ M and all ϕ ∈ G, We now prove the associativity of the product in Gˆ ./ M. First note that (3.50) and (3.82) immediately imply ˆ ./ m) = X[Y (1 ˆ ./ m)] [XY ](1
(3.83)
for all X, Y ∈ Gˆ ./ M and all m ∈ M. Next we show that ˆ ./ m)]Y = X[(1 ˆ ./ m)Y ] , [X(1
∀ X, Y ∈ Gˆ ./ M, m ∈ M .
(3.84)
To this end we use (id ⊗ )(h) = 1G and therefore ( ⊗ idG ⊗ idM ⊗ idG ⊗ )(ΩL ) = 1G ⊗ 1M ⊗ 1G to conclude for m, m0 , n ∈ M and ψ ∈ Gˆ ˆ ./ m)(ψ ./ n) = ψ(2) ./ (S −1 (ψ(1) ) . m / ψ(3) )n (1
(3.85)
and hence also ˆ ./ m)(ψ ./ n)] = ψ(3) ./ [(S −1 (ψ(2) ) . m0 / ψ(4) )(S −1 (ψ(1) ) . m / ψ(5) )n] ˆ ./ m0 )[(1 (1 = ψ(2) ./ (S −1 (ψ(1) ) . m0 m / ψ(3) ) ˆ ./ m0 m)(ψ ./ n) , = (1
(3.86)
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where we have used that δ is an algebra map. Moreover, (3.85) also implies for all ϕ ∈ Gˆ ˆ ./ m)(ψ ./ n)] (ϕ ./ 1M )[(1 = [(Ω1L * ϕ ( Ω5L )(Ω2L * ψ(2) ( Ω4L )] ./ [Ω3L (S −1 (ψ(1) ) . m / ψ(3) )n] = (ϕ ./ m)(ψ ./ n)
(3.87)
Putting (3.82), (3.83), (3.86) and (3.87) together, we have proven (3.84). In view of (3.82), (3.83) and (3.84), to finish the proof of associativity we are now left with proving the following two identities: (ϕ ./ 1M )[(ψ ./ 1M )(χ ./ 1M )] = [(ϕ ./ 1M )(ψ ./ 1M )](χ ./ 1M )
(3.88)
ˆ ./ m)(ϕ ./ 1M )](ψ ./ 1M ) ˆ ./ m)[(ϕ ./ 1M )(ψ ./ 1M )] = [(1 (1
(3.89)
for all ϕ, ψ, χ ∈ Gˆ and all m ∈ M. To prove these remaining identities we rewrite them using the generating matrix formalism. Let L ∈ G ⊗ (Gˆ ./ M) be given by ˆ We also L = eµ ⊗ (eµ ./ 1M ), where {eµ } is a basis in G with dual basis {eµ } in G. ˆ ./ m), m ∈ M. Then Eqs. (3.88) abbreviate our notation by identifying m ≡ (1 and (3.89) are equivalent, respectively, to L14 (L24 L34 ) = (L14 L24 )L34
(3.90)
[1G ⊗ 1G ⊗ m](L13 L23 ) = ([1G ⊗ 1G ⊗ m]L13 )L23 ,
(3.91)
where (3.90) is understood as an identity in G ⊗ ⊗(Gˆ ./ M) and (3.91) as an identity 2 in G ⊗ ⊗ (Gˆ ./ M). We now use that (3.50) and (3.51) imply 3
[1G ⊗ 1M ] L = L
(3.92)
[1G ⊗ m] L = [S −1 (m(1) ) ⊗ 1M ] L [m(−1) ⊗ m(0) ] ,
∀m ∈ M
(3.93)
¯ 5 ) ⊗ S −1 (Ψ ¯ 4 ) ⊗ 1M ] [(∆L ⊗ id)(L)] [Ψ ¯1 ⊗ Ψ ¯2 ⊗ Ψ ¯ 3 ] (3.94) L13 L23 = [S −1 (Ψ ¯ = where we have introduced the notation δ(m) = m(−1) ⊗ m(0) ⊗ m(1) and Ψ−1 ≡ Ψ 1 2 3 4 5 ¯ ¯ ¯ ¯ ¯ Ψ ⊗ Ψ ⊗ Ψ ⊗ Ψ ⊗ Ψ , and where ∆L (a) := h∆(a), a ∈ G, has been introduced in Corollary 3.2. To prove (3.91) we use (3.93) twice together with (3.84) to get for the r.h.s. of (3.91) ([1G ⊗ 1G ⊗ m] L13 )L23 = [S −1 (m(2) ) ⊗ S −1 (m(1) ) ⊗ 1M ] L13 L23 [m(−2) ⊗ m(−1) ⊗ m(0) ] , where we have used the notation (id ⊗ δ ⊗ id) ◦ δ(m) = m(−2) ⊗ m(−1) ⊗ m(0) ⊗ m(1) ⊗ m(2) . On the other hand, by the intertwiner property (3.25) together with (3.93), (3.94) the l.h.s. of (3.91) gives ¯5 ⊗ Ψ ¯ 4 ) ⊗ 1M ] [1G ⊗ 1G ⊗ m] (L13 L23 ) = [(S −1 ⊗ S −1 ) ∆op (m(1) )(Ψ ¯1 ⊗ Ψ ¯2 ⊗ Ψ ¯ 3] . × [(∆L ⊗ id)(L)] [∆(m(−1) ) ⊗ m(0) ] [Ψ
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597
Using again (3.94) to rewrite the r.h.s. of this formula, Eq. (3.91) follows from the ¯ ≡ Ψ−1 . defining property (3.33a) of Ψ ˆ for another To prove (3.90), we use (3.94) to compute for the l.h.s (writing Ψ copy of Ψ) ¯ 5 ) ⊗ S −1 (Ψ ¯ 4 ) ⊗ 1M ] L14 (L24 L34 ) = [1G ⊗ S −1 (Ψ ¯1 ⊗ Ψ ¯2 ⊗ Ψ ¯ 3] × [(id ⊗ ∆L ⊗ id)(L13 L23 )][1G ⊗ Ψ i h 5 ˆ¯ ⊗ ∆op (Ψ ˆ¯ 4 ))(1 ⊗ Ψ ¯5 ⊗ Ψ ¯ 4 ) ⊗ 1M = (S −1 ⊗ S −1 ⊗ S −1 ) (Ψ G 2 ˆ¯ 1 ⊗ ∆(Ψ ˆ¯ 3 ] ¯ˆ ) ⊗ Ψ × [(id ⊗ ∆L ⊗ id) ◦ (∆L ⊗ id)(L)] [Ψ
¯1 ⊗ Ψ ¯2 ⊗ Ψ ¯ 3] , × [1G ⊗ Ψ
(3.95)
where for the second equality we have used the identity ∆L (S −1 (a)bc) = (S −1 ⊗ S −1 )(∆op (a))∆L (b)∆(c) following from (3.25). For the r.h.s. of (3.90) we get: ¯ 5 ) ⊗ S −1 (Ψ ¯ 4 ) ⊗ S −1 (Ψ ¯ 3 ) ⊗ 1M ] (L14 L24 )L34 = [S −1 (Ψ (1) ¯1 ⊗ Ψ ¯2 ⊗ Ψ ¯3 ¯3 × [∆L ⊗ id ⊗ id)(L13 L23 )][Ψ (−1) ⊗ Ψ(0) ] i h ˆ¯ 5 ) ⊗ Ψ ˆ¯ 4 )(Ψ ¯5 ⊗ Ψ ¯4 ⊗ Ψ ¯ 3 ) ⊗ 1M = (S −1 ⊗ S −1 ⊗ S −1 ) (∆op (Ψ (1) ˆ¯ 1 ) ⊗ Ψ ˆ¯ 2 ⊗ Ψ ˆ¯ 3 ] × [(∆L ⊗ id ⊗ id) ◦ (∆L ⊗ id)(L)] [∆(Ψ ¯2 ⊗ Ψ ¯3 ¯3 ¯1 ⊗ Ψ × [Ψ (−1) ⊗ Ψ(0) ] ,
(3.96)
¯ 3 to the right of L34 and where for the first equality we have used (3.93) to move Ψ in the second equality again (3.25). Now we use that by Corollary 3.2 (id ⊗ ∆L )(∆L (a)) = (S −1 ⊗ S −1 ⊗ S −1 )(φ321 ) ((∆L ⊗ id)(∆L (a)) φ−1 ,
∀a ∈ G .
Hence (3.95) and (3.96) are equal due to the pentagon identity (3.33b) for Ψ, which proves (3.90). This concludes the proof of Parts (i) and (ii) of Theorem 3.10. Properties of the elements pλ , qλ , pρ , qρ Lemma 3.21. 1. Let (λ, φλ ) be a left G-coaction on M and let pλ , qλ be given by (3.71), (3.72). Then the following identities hold for all m ∈ M, where λ(m) ≡ m(−1) ⊗ m(0) : λ(m(0) ) pλ [S −1 (m(−1) ) ⊗ 1M ] = pλ [1G ⊗ m]
(3.97a)
[S(m(−1) ) ⊗ 1M ] qλ λ(m(0) ) = [1G ⊗ m] qλ
(3.97b)
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λ(qλ2 ) pλ [S −1 (qλ1 ) ⊗ 1M ] = 1G ⊗ 1M [S(p1λ ) ⊗ 1M ] qλ λ(p2λ ) = 1G ⊗ 1M .
(3.97c) (3.97d)
Moreover, with f, h ∈ G ⊗ G being the twists given by (3.20), (3.25), the following identities are valid : φ−1 λ (idG ⊗ λ)(pλ ) (1G ⊗ pλ ) = (∆ ⊗ idM )(λ(φ3λ )pλ ) [h−1 ⊗ 1M ] [S −1 (φ2λ ) ⊗ S −1 (φ1λ ) ⊗ 1M ]
(3.97e)
(1G ⊗ qλ ) (idG ⊗ λ)(qλ ) φλ = [S(φ¯2λ ) ⊗ S(φ¯1λ ) ⊗ 1M ] [f ⊗ 1M ] (∆ ⊗ idM )(qλ λ(φ¯3λ )) .
(3.97f)
2. Similarly, let (ρ, φρ ) be a right G-coaction on M and let pρ , qρ be given by (3.73) and (3.74). Then the following identities hold for all m ∈ M, where ρ(m) ≡ m(0) ⊗ m(1) . ρ(m(0) ) pρ [1M ⊗ S(m(1) )] = pρ [m ⊗ 1G ]
(3.98a)
[1M ⊗ S −1 (m(1) )] qρ ρ(m(0) ) = [m ⊗ 1G ] qρ
(3.98b)
ρ(qρ1 ) pρ [1M ⊗ S(qρ2 )] = 1M ⊗ 1G
(3.98c)
[1M ⊗ S −1 (p2ρ )] qρ ρ(p1ρ ) = 1M ⊗ 1G
(3.98d)
φρ (ρ ⊗ idG )(pρ ) (pρ ⊗ 1G ) = (idM ⊗ ∆)(ρ(φ¯1ρ )pρ ) [1M ⊗ f −1 ] [1M ⊗ S(φ¯3ρ ) ⊗ S(φ¯2ρ )]
(3.98e)
(qρ ⊗ 1G ) (ρ ⊗ idG )(qρ ) φ−1 ρ = [1M ⊗ S −1 (φ3ρ ) ⊗ S −1 (φ2ρ )] [1M ⊗ h] (idM ⊗ ∆)(qρ ρ(φ1ρ )) . (3.98f) Proof. Note that Part 2 of Lemma 3.21 is functorially equivalent to Part 1, 321 since (ρ, φρ ) is a right G-coaction if and only if (ρop , (φ−1 ) is a left G cop -coaction. ρ ) −1 Also, considering (ρ, φρ ) as a right Gop -coaction on Mop , the roles of qρ and pρ interchange, which makes it enough to just prove Eqs. (3.98b), (3.98d) and (3.98f) or the corresponding sets of equations in Part 1. Let us begin with (3.98b). Denoting the multiplication in G op by µop one computes [1M ⊗ S −1 (m(1) )] qρ ρ(m(0) ) = [φ1ρ ⊗ S −1 (αφ3ρ m(1) )φ2ρ ] ρ(m(0) )
= (idM ⊗ µop ) ◦ (idM ⊗ idG ⊗ S −1 ) [1M ⊗ 1G ⊗ α] φρ (ρ ⊗ idG )(ρ(m))
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= (idM ⊗ µop ) ◦ (idM ⊗ idG ⊗ S −1 ) [1M ⊗ 1G ⊗ α] (idM ⊗ ∆)(ρ(m)) φρ = [m ⊗ 1G ] qρ , where we have plugged in the definition (3.74) of qρ and used the intertwiner property (3.30a) of φρ and the antipode property (3.14). This proves (3.98b). To prove (3.98d) we introduce for a, b, c ∈ G the notation σ(a ⊗ b ⊗ c) := −1 c S (αbβ) a, to compute for the l.h.s. [1M ⊗ S −1 (p2ρ )] qρ ρ(p1ρ ) ≡ [φ1ρ ⊗ φ¯3ρ S −1 (αφ3ρ φ¯2ρ β)φ2ρ ] ρ(φ¯1ρ ) = (idM ⊗ σ) [φρ ⊗ 1G ] (ρ ⊗ idG ⊗ idG )(φ−1 ρ ) −1 ] (idM ⊗ idG ⊗ ∆)(φρ ) = (idM ⊗ σ) (idM ⊗ ∆ ⊗ idG )(φ−1 ρ ) [1M ⊗ φ = 1M ⊗ φ¯3 S −1 (αφ¯2 β)φ¯1 = 1M ⊗ 1G , where we have used the pentagon identity (3.30b), then the two antipode properties (3.14) together with (id ⊗ id ⊗ )(φρ ) = 1M ⊗ 1G to drop the reassociators φρ and φ−1 ρ and finally (3.15). The proof of (3.98f) is more complicated. First we rewrite l.h.s. (3.98f) = ω(X) , where X = [φ1ρ ⊗ φ2ρ ⊗ 1G ⊗ 1G ⊗ φ3ρ ] [(ρ ⊗ idG ⊗ idG )(φρ ) ⊗ 1G ] [φ−1 ρ ⊗ 1G ⊗ 1G ]
(3.99)
and where the map ω : M ⊗ G ⊗ → M ⊗ G ⊗ is given by 4
2
ω(m ⊗ a ⊗ b ⊗ c ⊗ d) := m ⊗ S −1 (αd) a ⊗ S −1 (αc) b . To rewrite the r.h.s. of (3.98f) in the same fashion we first use the identities (3.25) and (3.27) and the definition (3.74) of qρ to compute i h [1M ⊗ h] (idM ⊗ ∆)(qρ ) = [1M ⊗ h] φ1ρ ⊗ ∆ S −1 (αφ3ρ )φ2ρ = [φ1ρ ⊗ (S −1 ⊗ S −1 )(∆op (φ3ρ ))] × [1M ⊗ h] [1M ⊗ ∆(S −1 (α)φ2ρ )] = [1M ⊗ (S −1 ⊗ S −1 )(γ op ∆op (φ3ρ ))] [φ1ρ ⊗ ∆(φ2ρ )] . Now we use the formula (3.18) for γ implying (S −1 ⊗ S −1 )(γ op ) = S −1 (αφ¯3 φ3(2) )φ1 ⊗ S −1 (αφ¯2 φ3(1) )φ¯1 φ2
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to obtain r.h.s. (3.98f) = ω(Y ) , where Y = [1M ⊗ 1G ⊗ φ−1 ] (idM ⊗ idG ⊗ idG ⊗ ∆)([1M ⊗ φ] (idM ⊗ ∆ ⊗ idG )(φρ )) × ((id ⊗ ∆) ◦ ρ ⊗ idG ⊗ idG )(φρ ) .
(3.100)
Using the pentagon Eq. (3.30b) to replace the second and third reassociator in (3.100) yields 2 2 2 Y = [1M ⊗ 1G ⊗ φ−1 ] (id ⊗ id⊗ ⊗ ∆) (id⊗ ⊗ ∆)(φρ ) (ρ ⊗ id⊗ )(φρ ) [φ−1 ⊗ 1 ] G ρ × ((id ⊗ ∆) ◦ ρ ⊗ id⊗ )(φρ ) 2
= (id⊗ ⊗ (∆ ⊗ id) ◦ ∆)(φρ ) 2 2 × (ρ ⊗ id ⊗ id) [1M ⊗ φ−1 ] (id⊗ ⊗ ∆)(φρ ) (ρ ⊗ id⊗ )(φρ ) [φ−1 ρ ⊗ 1G ⊗ 1G ] 2
= (id⊗ ⊗ (∆ ⊗ id) ◦ ∆)(φρ ) (ρ ⊗ ∆ ⊗ id)(φρ ) 2
× [(ρ ⊗ id⊗ )(φρ ) ⊗ 1G ] [φ−1 ρ ⊗ 1G ⊗ 1G ] , 2
(3.101)
where in the second equation we have used (3.8) and (3.30a) to shift the reassociators φ−1 and φ−1 ρ by one step to the right, and in the third equation again the pentagon identity (3.30b). Hence, when computing ω(Y ), the second factor in (3.101) may be dropped due to the antipode property (3.14) and the two coproducts in the first factor disappear by the same reason. Comparing with (3.99) proves, that ω(X) = ω(Y ) and therefore both sides of (3.98f) are equal. There are also some additional identities in the case where (λ, φλ , ρ, φρ , φλρ ) is a quasi-commuting pair of coactions. Lemma 3.22. Let (λ, φλ , ρ, φρ , φλρ ) be a quasi-commuting pair of G-coactions on M and let pλ/ρ , qλ/ρ be given by Eqs. (3.71)–(3.74). Then putting φ¯λρ ≡ φ−1 λρ , 2 3 −1 1 (φλρ ) ⊗ 1M ⊗ 1G ] φ−1 λρ (idG ⊗ ρ)(pλ ) = [λ(φλρ )pλ ⊗ φλρ ] [S
(3.102a)
(idG ⊗ ρ)(qλ ) φλρ = [S(φ¯1λρ ) ⊗ 1M ⊗ 1G ] [qλ λ(φ¯2λρ ) ⊗ φ¯3λρ ]
(3.102b)
φλρ (λ ⊗ idG )(pρ ) = [φ¯1λρ ⊗ ρ(φ2λρ )pρ ] [1G ⊗ 1M ⊗ S(φ¯3λρ )]
(3.102c)
−1 3 (φλρ )] [φ1λρ ⊗ qρ ρ(φ2λρ )] . (λ ⊗ idG )(qρ ) φ−1 λρ = [1G ⊗ 1M ⊗ S
(3.102d)
Proof of Lemma 3.22. Again we remark that for functorial reasons Eqs. (3.102a)–(3.102d) are all equivalent, see the arguments in the proof of Lemma 3.21. We prove the identity (3.102d). Introducing for a, b ∈ G the map ν(a ⊗ b) := S −1 (αb) a and using the formula (3.74) for qρ we compute
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1 −1 (λ ⊗ id)(qρ ) φ−1 (αφ3ρ )φ2ρ ] φ−1 λρ ≡ [λ(φρ ) ⊗ S λρ = (idG ⊗ idM ⊗ ν) (λ ⊗ id ⊗ id)(φρ ) [φ−1 ⊗ 1 ] G λρ
2 = (idG ⊗ idM ⊗ ν) (id⊗ ⊗ ∆)(φ−1 λρ )[1G ⊗ φρ ] (id ⊗ ρ ⊗ id)(φλρ ) = [1G ⊗ 1G ⊗ S −1 (φ3λρ )] [1G ⊗ qρ ] [φ1λρ ⊗ ρ(φ2λρ )] . Here we have plugged in the pentagon equation (3.45c) and used the fact that φλρ may be dropped due to (3.45d) and the antipode property (3.14). This proves (3.102d) and therefore Lemma 3.22. Proof of Proposition 3.19. We only need to prove Part 1, since Part 2 is functorially equivalent, see the proof of Lemma 3.21. If T is a λρ-intertwiner and R given by (3.75), then ρop (m(0) ) R [S −1 (m(−1) ) ⊗ 1A ] ≡ ρop (m(0) ) T pλ [S −1 (m(−1) ) ⊗ 1A ] = R [1G ⊗ m] by (3.67) and (3.97a), and therefore R is a right δr -implementer. Moreover, (3.97c) implies ρop (qλ2 ) T pλ [S −1 (qλ1 ) ⊗ 1M ] = T λ(qλ2 ) pλ [S −1 (qλ1 ) ⊗ 1M ] = T. Conversely if R is a right δr -implementer and T given by (3.76), then ρop (m) T ≡ ρop (m qλ2 ) R [S −1 (qλ1 ) ⊗ 1A ] = ρop (qλ2 m(0,0) ) R [S −1 (qλ1 m(0,−1) )m(−1) ⊗ 1A ] = ρop (qλ2 ) R [S −1 (qλ1 )m(−1) ⊗ m(0) ] = T λ(m) , where in the second line we have used (3.97b) and in the third line the right δr -implementer property (3.59) of R. Hence T is a λρ-intertwiner. Moreover ρop (qλ2 ) R [S −1 (qλ1 ) ⊗ 1A ] pλ = ρop (qλ2 p2λ(0) ) R [S −1 (qλ1 p2λ(−1) ) p1λ ⊗ 1A ] = R, where we have used the δr -implementer property of R and then (3.97d). Thus the correspondence T ↔ R is one-to-one, and since qλ and pλ are normal it is clearly normality preserving. To prove that it is also coherence preserving assume now that the λρ-intertwiner T satisfies the coherence condition (3.68) and let R = T pλ . Then −1 132 13 A, (∆ ⊗ id)(R) [h−1 ⊗ 1M ] = φ312 ρ T (φλρ )
(3.103)
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where A ∈ G ⊗ G ⊗ M is given by A = T23 φλ (∆ ⊗ idM )(pλ ) [h−1 ⊗ 1M ] = T23 (idG ⊗ λ)(λ(φ¯3λ ) pλ ) [1G ⊗ pλ ] [S −1 (φ¯2λ ) ⊗ S −1 (φ¯1λ ) ⊗ 1M ] = (idG ⊗ ρop )(λ(φ¯3λ ) pλ ) R23 [S −1 (φ¯2λ ) ⊗ S −1 (φ¯1λ ) ⊗ 1M ] .
(3.104)
Here we have used (3.97e) in the second line and the intertwining property (3.67) of T in the third line. Using the intertwiner property (3.45a) of φλρ and (3.102a) we further compute 132 (idG ⊗ ρop )(λ(φ¯3λ ) pλ ) T13 (φ−1 λρ )
i132 h [S −1 (φ1λρ ) ⊗ 1G ⊗ 1M ] . = (ρop ⊗ id)(ρ(φ¯3λ ) (φ2λρ ⊗ φ3λρ )) (R ⊗ 1M ) (3.105) Putting (3.103)–(3.105) together we finally conclude ¯4 ⊗ Ψ ¯5 ⊗ Ψ ¯ 3 ] R13 R23 [S −1 (Ψ ¯ 2 ) ⊗ S −1 (Ψ ¯ 1 ) ⊗ 1M ] , (∆ ⊗ id)(R) [h−1 ⊗ 1M ] = [Ψ r r r r r (3.106) ¯ r ∈ G ⊗ G ⊗ M ⊗ G ⊗ G is given by where Ψ
¯ r := (1G ⊗ 1G ⊗ φρ )(idG ⊗ idG ⊗ ρ ⊗ idG ) (idG ⊗ idG ⊗ ρ)(Φ−1 ) (1G ⊗ φλρ ) . Ψ λ
¯ r = Ψ−1 and therefore (3.106) is equivalent to the coherence By (3.46d) we have Ψ r condition (3.61) for R as a right (δr , Ψr )-implementer. Conversely, assume now that R is a coherent right (δr , Ψr )-implementer and let T be given by (3.76). To prove that T is coherent we have to show that (∆ ⊗ id)(T) = B , where B is given by (3.69). Now writing R = T pλ and going backwards through the derivation (3.103) ← (3.106) we conclude (∆ ⊗ id)(T pλ ) = B (∆ ⊗ id)(pλ ) .
(3.107)
Thus, if pλ were invertible we could immediately conclude that T is coherent. It turns out that we may use (3.97c) as a substitute for the invertibility of pλ , since it implies h i (∆ ⊗ id)(T) ≡ (∆ ⊗ id) ρop (qλ2 ) T pλ S −1 (qλ1 ) ⊗ 1M i h = (∆ ⊗ id)(ρop (qλ2 )) B (∆ ⊗ id) pλ S −1 (qλ1 ) ⊗ 1M h i = B (∆ ⊗ id) λ(qλ2 ) pλ S −1 (qλ1 ) ⊗ 1M = B.
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Here we have used (3.107) in the second line, (3.70) in the third line and again (3.97c) in the last line. Thus T is a coherent λρ-intertwiner, which concludes the proof of Proposition 3.19. 4. Generalization to Weak Quasi-Hopf Algebras In this section we will generalize the definitions and constructions of Sec. 3 to weak quasi-Hopf algebras as introduced in [31]. This contains the physical relevant examples such as truncated quantum groups at roots of unity. As will be shown, it is nearly straightforward to extend all results obtained so far to the case of weak quasi-Hopf algebras. The new feature of weak quasi-Hopf algebras is to allow the coproduct to be non unital, i.e. ∆(1) 6= 1 ⊗ 1. This results in a truncation of tensor products of representations, i.e. the representation (πV ⊗ πW ) ◦ ∆ operates only on the subspace V W := (πV ⊗ πW )(∆(1G ))(V ⊗ W ). Also the invertibility requirement on certain universal elements — such as the reassociator or the R-matrix — is weakened by only postulating the existence of so-called quasi-inverses. Correspondingly coactions and two-sided coactions of a weak quasi-Hopf algebra are no longer supposed to be unital and the associated reassociators are only required to possess quasi-inverses. The diagonal crossed products M1 ≡ Gˆ ./ M may then again be defined by the same relations as before, with the additional requirement that the universal λρ-intertwiner Γ ∈ G ⊗ M1 has to satisfy Γ λ(1M ) ≡ ρop (1M ) Γ = Γ. This will also imply that now as a linear space M1 is only isomorphic to a cerˆ More specifically tain subspace Gˆ ./ λMρ ⊂ Gˆ ⊗ M (or λMρ ./ Gˆ ⊂ M ⊗ G). Theorem 3.1 now reads: Theorem 4.1. Let G be a finite dimensional weak quasi-Hopf algebra and let (λ, φλ , ρ, φρ , φλρ ) be a quasi-commuting pair of (left and right) G-coactions on an associative algebra M. Then Parts 1 and 2 of Theorem 3.1 stay valid with the additional requirement that the normal elements T ∈ G ⊗ A satisfy not only (3.2)/(3.3) but also (4.1) T λA (1M ) ≡ ρop A (1M ) T = T . Part 3 is modified as follows: 3’. There exist elements pλ ∈ G ⊗ M and qρ ∈ M ⊗ G such that the linear maps µL : Gˆ ⊗ M 3 (ϕ ⊗ m) 7→ (id ⊗ ϕ(1) )(qρ ) Γ(ϕ(2) ) m ∈ M1 µR : M ⊗ Gˆ 3 (m ⊗ ϕ) 7→ m Γ(ϕ(1) ) (ϕ(2) ⊗ id)(pλ ) ∈ M1 are surjective. 3”. Let PL : Gˆ ⊗ M → Gˆ ⊗ M and PR : M ⊗ Gˆ → M ⊗ Gˆ be the linear maps given by PL (ϕ ⊗ m) ≡ ϕ ./ m := ϕ(2) ⊗ (ϕ(3) ⊗ id ⊗ Sˆ−1 (ϕ(1) ))(δl (1M ))m PR (m ⊗ ϕ) ≡ m ./ ϕ := m(Sˆ−1 (ϕ(3) ) ⊗ idM ⊗ ϕ(1) )(δr (1M )) ⊗ ϕ(2) , where δl = (λ ⊗ id) ◦ ρ and δr = (id ⊗ ρ) ◦ λ. Then PL and PR are projections with the same kernels as µL and µR , respectively.
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Parts 3’ and 3” of Theorem 4.1 imply that we may put Gˆ ./ λMρ := PL (Gˆ ⊗ M) ˆ to conclude that analogously as in (3.6) and (3.7) the and λMρ ./ Gˆ := PR (M ⊗ G) restrictions µL : Gˆ ./ λMρ → M1 µR : λMρ ./ Gˆ → M1 are linear isomorphisms inducing a concrete realization of the abstract algebra M1 ˆ respectively. on the subspaces Gˆ ./ λMρ ⊂ Gˆ ⊗ M and λMρ ./ Gˆ ⊂ M ⊗ G, As before, we call these concrete realizations the left and right diagonal crossed products, respectively, associated with the quasi-commuting pair of G-coactions (λ, ρ, φλ , φρ , φλρ ) on M. To actually prove Theorem 4.1 we follow the same strategy as in Sec. 3, i.e. we first construct these diagonal crossed products explicitly and then show that they solve the universal properties defining M1 . 4.1. Weak quasi-Hopf algebras Quasi-inverses. We start with a little digression on the notion of quasi-inverses. Let A be an associative algebra and let x, p, q ∈ A satisfy px = x = xq p2 = p ,
q2 = q
(4.2) (4.3)
Then we say that y ∈ A is a quasi-inverse of x with respect to (p, q), if yx = q ,
xy = p ,
yxy = y .
(4.4)
Clearly, given (p, q), a quasi-inverse of x is uniquely determined, provided it exists. This is why we also write y = x−1 , if the idempotents (p, q) are understood. Also note that we have qy = y = yp and xyx = x and therefore x is the quasi-inverse of y with respect to (q, p). All this generalizes in the obvious way to A-module morphisms x ∈ HomA (V, W ), p ∈ EndA (W ) and q ∈ EndA (V ), in which case the quasi-inverse would be an element x−1 ∈ HomA (W, V ). Note that in place of (4.2) we could equivalently add to (4.4) the requirement xyx = x .
(4.5)
In our setting of weak quasi-Hopf algebras the idempotents p, q always appear as images of 1 ∈ G of non-unital algebra maps defined on G, like ∆(1), ∆op (1), (∆ ⊗ id)(∆(1)), . . . , etc., whereas the element x will be an intertwiner between two such maps, like a reassociator φ, and R-matrix R, etc. Hence, throughout we will adopt the convention that if α : G → A and β : G → A are two algebra maps and x ∈ A satisfies x α(g) = β(g) x , ∀ g ∈ G , then the quasi-inverse y = x−1 ∈ A is defined to be the unique (if existing) element satisfying yx = α(1) , xyx = x ,
xy = β(1) yxy = y .
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Clearly this implies conversely α(g) y = y β(g) ,
∀g ∈ G
and therefore x = y −1 . We also note the obvious identities β(g) = x α(g) x−1 , x α(1) = β(1) x = x ,
α(g) = x−1 β(g) x
α(1) x−1 = x−1 β(1) = x−1 .
Weak quasi-Hopf algebras. After this digression we now define, following [31] a weak quasi-bialgebra (G, 1, ∆, , φ) to be an associative algebra G with unit 1, a non-unital algebra map ∆ : G → G ⊗ G, an algebra map : G → C and an element φ ∈ G ⊗ G ⊗ G satisfying (3.8)–(3.10), whereas (3.11) is replaced by (id ⊗ ⊗ id)(φ) = ∆(1)
(4.6)
and where in place of invertibility φ is supposed to have a quasi-inverse ¯ = φ, with respect to the intertwining property (3.8). Hence we have φφφ as well as φ φ¯ = (id ⊗ ∆)(∆(1)) , φ¯ φ = (∆ ⊗ id)(∆(1)) ,
φ¯ ≡ φ−1 ¯ φ¯ = φ¯ φφ (4.7)
implying the further identities (id ⊗ ∆)(∆(a)) = φ (∆ ⊗ id)(∆(a)) φ¯ , φ = φ (∆ ⊗ id)(∆(1)) ,
∀a ∈ G
(4.8)
φ¯ = φ¯ (id ⊗ ∆)(∆(1))
(4.9)
¯ = ∆(1) . (id ⊗ ⊗ id)(φ)
(4.10)
A weak quasi-bialgebra is called weak quasi-Hopf algebra, if there exists a unital algebra antimorphism S and elements α, β ∈ G satisfying (3.14) and (3.15). We will also always suppose that S is invertible. The remarks about the quasi-Hopf cop remain valid as in Sec. 3.1. algebras Gop , G cop and Gop A quasi-invertible element F ∈ G ⊗ G satisfying ( ⊗ id)(F ) = (id ⊗ )(F ) = 1 induces a twist transformation from (G, ∆, , φ) to (G, ∆F , , φF ) as in (3.16) and (3.17), where GF := (G, 1, ∆F , , φF ) is again a weak quasi-bialgebra. The two bialgebras GF and G are called twist-equivalent. Finally the properties of the twists f, h defined as in (3.20) and (3.25) are still valid. In particular (3.21) defines the quasi-inverse of f with respect to the intertwining property (3.22). For implications on the representation theory of G see Appendix A. Weak coactions. The notion of coactions may easily be generalized as well. By a left G-coaction (λ, φλ ) of a weak quasi-bialgebra G on a unital algebra M we mean a (not necessarily unital) algebra map λ : M → G ⊗ M and a quasi-invertible element φλ ∈ G ⊗ G ⊗ M satisfying (3.29a)–(3.29c) as in Definition 3.3 and (id ⊗ ⊗ id)(φλ ) = ( ⊗ id ⊗ id)(φλ ) = λ(1M ) .
(4.11)
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The definition of right coactions is generalized analogously. Lemma 3.4 about twist equivalencies of coactions stays valid, where one has to make the adjustments that a twist U ∈ M ⊗ G only is supposed to be quasi-invertible. By now it should become clear how one has to proceed: Definition 3.5 of twosided coactions is generalized by allowing δ to be non-unital and Ψ to be non¯ ≡ Ψ−1 and by replacing (3.33d) by invertible but with quasi-inverse Ψ (idG ⊗ ⊗ idM ⊗ ⊗ idG )(Ψ) = ( ⊗ idG ⊗ idM ⊗ idG ⊗ )(Ψ) = δ(1M ) .
(4.12)
The definitions of quasi-commuting pairs of coactions, twist equivalencies of twosided coactions etc., are generalized similarly. With these adjustments all results of Sec. 3.3 stay valid for weak quasi-Hopf algebras and are proven analogously. The elements qλ , pλ and qρ , pρ are defined as in (3.71)–(3.74) and obey all the relations stated in Lemma 3.21 with the only modifications that in (3.97c) and (3.97d) the r.h.s. becomes λ(1M ) instead of 1G ⊗ 1M and similarly (3.98c)/(3.98d) where the r.h.s. has to be replaced by ρ(1M ), i.e. λ(qλ2 ) pλ [S −1 (qλ1 ) ⊗ 1M ] = λ(1M )
(4.13)
[S(p1λ ) ⊗ 1M ] qλ λ(p2λ ) = λ(1M )
(4.14)
ρ(qρ1 ) pρ [1M ⊗ S(qρ2 )] = ρ(1M )
(4.15)
[1M ⊗ S −1 (p2ρ )] qρ ρ(p1ρ ) = ρ(1M ) .
(4.16)
Going through the proof of Lemma 3.21, this follows from the fact that one uses (4.11) or the corresponding identity for φρ . 4.2. Diagonal crossed products The definition of diagonal crossed products Gˆ ./ Mδ and Mδ ./ Gˆ as equivalent algebra extensions of M, given in Definitions 3.9/3.11 need some more care in the present context. We will proceed in two steps. First we define an associative algebra ˆ exactly as in Definition 3.9 (or Definition 3.11). structure on Gˆ ⊗ M (or M ⊗ G) Unfortunately in general this algebra is not unital unless the two-sided coaction δ is ˆ is still a left unit) and ˆ ⊗ 1M is still a right unit (1M ⊗ 1 unital. But the element 1 in particular idempotent. The second step then consists in defining the subalgebra ˆ ⊗1M , i.e. Gˆ ./ Mδ := (1⊗1 ˆ Gˆ ./ Mδ ⊂ Gˆ ⊗M as the right ideal generated by 1 M )· ˆ · (1M ⊗ 1)). ˆ i.e. Mδ ./ Gˆ := (M ⊗ G) ˆ (Gˆ ⊗ M) (the left ideal generated by 1M ⊗ 1, ˆ ./ M and of M ≡ These algebras are then unital algebra extensions of M ≡ 1 ˆ respectively. As in Sec. 3.5 one may proceed to a description by left and M ./ 1, right δ-implementers and equivalently by λρ-intertwiners, thus getting a proof of Theorem 4.1. Definition 4.2. Let (δ, Ψ) be a two-sided coaction of a weak quasi-Hopf algebra G on an algebra M. We define Gˆ ⊗δ M to be the vector space Gˆ ⊗ M with multiplication structure given as in (3.50) and the left diagonal crossed product Gˆ ./ Mδ to be the subspace
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ˆ ⊗ 1M ) · (Gˆ ⊗δ M) . Gˆ ./ Mδ := (1
(4.17) ˆ ˆ Analogously M ⊗δ G is defined to be the vector space M ⊗ G with multiplication structure (3.57) and the right diagonal crossed product Mδ ./ Gˆ to be the subspace ˆ · (1M ⊗ 1) ˆ . Mδ ./ Gˆ := (M ⊗δ G)
(4.18)
The elements spanning Gˆ ./ Mδ and Mδ ./ Gˆ will be denoted by, respectively ˆ ⊗ 1M )(ϕ ⊗ m) ≡ (ϕ ⊗ 1M )(1M ⊗ m) ϕ ./ m := (1 ≡ ϕ(2) ⊗ (Sˆ−1 (ϕ(1) ) . 1M / ϕ(3) )m
(4.19)
ˆ ≡ (m ⊗ 1)(1 ˆ M ⊗ ϕ) m ./ ϕ := (m ⊗ ϕ)(1M ⊗ 1) ≡ m(ϕ(1) . 1M / Sˆ−1 (ϕ(3) )) ⊗ ϕ(2) .
(4.20)
Note that Gˆ⊗δ M = Gˆ ./ Mδ , if δ(1M ) = 1G ⊗1M ⊗1G , which means that the above definition generalizes Definition 3.9. We now state the analogue of Theorem 3.10. Theorem 4.3. ˆ ⊗ 1M and (i) Gˆ ⊗δ M and M ⊗δ Gˆ are associative algebras with left unit 1 ˆ right unit 1M ⊗ 1, respectively. Consequently, the diagonal crossed products ˆ respectively, Gˆ ./ Mδ and Mδ ./ Gˆ are subalgebras of Gˆ ⊗δ M and M ⊗δ G, ˆ ⊗ 1M and 1M ./ 1 ˆ ≡ 1M ⊗ 1, ˆ respectively. ˆ ./ 1M ≡ 1 with unit given by 1 ˆ = M ./ 1 ˆ ⊂ Mδ ./ Gˆ ˆ⊗M=1 ˆ ./ M ⊂ Gˆ ./ Mδ and M ≡ M ⊗ 1 (ii) M ≡ 1 are unital algebra inclusions. Proof. We will sketch the proof of Part (i) for Gˆ ⊗ M, the case M ⊗ Gˆ being analogous. From (3.50) one computes that ˆ ⊗ 1M ) = (ϕ ⊗ m) (ϕ ⊗ m)(1 ˆ ⊗ 1M )(ϕ ⊗ m) = ϕ(2) ⊗ (Sˆ−1 (ϕ(1) ) . 1M / ϕ(3) ) m (1 = (1(−1) * ϕ ( S −1 (1(1) )) ⊗ 1(0) m , ˆ 1M is a right unit in Gˆ ⊗δ M where δ(1M ) = 1(−1) ⊗ 1(0) ⊗ 1(1) . This shows that 1⊗ but in general not a left unit. To proof the associativity of the product one proceeds as in the proof of Theorem 3.10. Here one has to take some notational care when translating (3.88)/(3.89) ˜ ≡ into relations of a generating matrix L. First, it is necessary to distinguish L 1 2 µ µ ˜ ˆ ˆ ˜ L ⊗ L := eµ ⊗ (e ⊗ 1M ) ∈ G ⊗ (G ⊗ M) and L := eµ ⊗ (e ./ 1M ) ∈ G ⊗ (G ./ M). Equation (3.92) must then be replaced by ˜ =L [1G ⊗ 1M ] L
(4.21)
˜ since 1M ≡ 1 ˆ ⊗ 1M is and (3.52), (3.53) are at first only valid for L but not for L, ˜ not a left unit in Gˆ ⊗δ M. This can be cured by rewriting for example (3.52) for L more carefully as ˜ 2 ) = S −1 (m(1) ) L ˜ 1 m(−1) ⊗ L ˜ 2 m(0) , ˜ 1 ⊗ mL (L
∀m ∈ M
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in which form it would still be valid. Taking this into account and using that ¯ the proof proceeds as the one of Theorem 3.10(i). The ¯ (1G ⊗ δ(1M ) ⊗ 1G ) = Ψ, Ψ proof of the remaining parts of Theorem 3.10 is straightforwardly adjusted in the same spirit. From now on we will disregard the “unphysical” non-unital algebras M ⊗δ Gˆ and Gˆ ⊗δ M, and stay with Gˆ ./ M and M ./ Gˆ as our objects of interest. With the appropriate notations (4.19), (4.20) all relations of Sec. 3.4 remain valid for these ˆ M (right multiplication algebras. We also remark that the left multiplication by 1⊗1 ˆ by 1M ⊗1) precisely gives the projections PL/R mentioned in Part 3” of Theorem 4.1. Defining left and right δ-implementers as in Definition 3.13 all results in Sec. 3.5.1 also carry over to the present setting. A λρ-intertwiner is then supposed to have the additional property that T λA (1M ) ≡ ρop A (1M ) T = T .
(4.22)
With this the one-to-one correspondence T ↔ R and T ↔ L of Proposition 3.19 is still valid, where (4.22) becomes equivalent to L = [S −1 (1(1) ) ⊗ 1A ] L [1(−1) ⊗ γ(1(0) )] R = [1(1) ⊗ γ(1(0) )] R [S −1 (1(−1) ) ⊗ 1A ] , respectively, which follow from (3.58) and (3.59). One may now prove Theorem 4.1 analogously as Theorem 3.1 where the modifications in Parts 3’ and 3” have their origin in (4.22). 5. Applications As our first important application we will propose in Sec. 5.1 the definition of the quantum double D(G) of a (weak) quasi-Hopf algebra G. We will show that, similarly as for ordinary Hopf algebras, the quantum double D(G) is a weak quasi-bialgebra, where D(G) is weak if and only if G is weak, i.e. iff ∆(1) 6= 1. Analogously as in Proposition 2.8 this will also guarantee that every diagonal crossed product M1 = ˆ λMρ ./ G naturally admits a quasi-commuting pair (λD , ρD , φλD , φρD , φλD ρD ) of coactions of D(G) on M1 . The last observation is of great importance, since it implies that the quantum chains constructed as iterated diagonal crossed products in Sec. 5.3 admit localized D(G)-coactions. In [27] it is shown that D(G) is in fact a quasitriangular quasi-Hopf algebra. In Sec. 5.2 we generalize the notion of two-sided crossed products, defined as special examples of diagonal crossed products in Sec. 2.4.1 to the quasi-coassociative setting. We then define Hopf spin chains based on a weak quasi-Hopf algebra as iterated two-sided crossed products in Sec. 5.3 and arrive at the definition of lattice current algebras by imposing periodic boundary conditions again using our diagonal crossed product construction. We emphasize that this covers the important case of quantum groups at roots of unity.
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Finally we sketch in Sec. 5.4 how the field algebra construction of Mack and Schomerus may be described using two-sided crossed products. 5.1. The quantum double D(G) In view of the identification of the quantum double D(G) of an ordinary Hopf algebra G with the diagonal crossed product G ./ Gˆ in (2.26) we propose the following: Definition 5.1. Let (G, ∆, , φ) be a weak quasi-Hopf algebra. The diagonal crossed product D(G) := Gˆ ./ ∆G∆ ∼ = ∆G∆ ./ Gˆ associated with the quasicommuting pair (λ = ρ = ∆, φλ = φρ = φλρ = φ) of G-coactions on M ≡ G is called the quantum double of G. Following the notations of [40], the universal λρ-intertwiner of the quantum double will be denoted by D ≡ Γ D(G) ∈ G ⊗ D(G). Hence it obeys the relations ( ⊗ id)(D) = 1D(G) and D ∆(1) = ∆op (1) D = D D ∆(a) = ∆op (a) D ,
∀a ∈ G
φ312 D13 (φ−1 )132 D23 φ = (∆ ⊗ id)(D) ,
(5.1) (5.2) (5.3)
where we have suppressed the embedding G ,→ D(G). Property (5.2) motivates to call D the universal flip operator for ∆. Clearly, the relation (5.1) may be omitted if ∆(1) = 1 ⊗ 1. Note that according to Theorem 3.1, the quantum double D(G) may be realized as an algebraic structure on the vector space Gˆ ⊗ G (or, in the weak case, a certain subspace thereof, see Theorem 4.1). We remark that a definition of a quantum double D(G) for quasi-Hopf algebras G has also recently been proposed by S. Majid [35] using a Tannaka–Krein type reconstruction procedure [33]. Unfortunately it is hard to identify this algebra in terms of generators and relations in concrete models. It will be shown in Appendix A that our construction in fact provides a concrete realization of the abstract definition of [35]. We begin with constructing λD : M1 → D(G)⊗ M1 and ρD : M1 → M1 ⊗ D(G) as algebra maps extending the left and right coactions λ : M1 ⊃ M → G ⊗ M ⊂ D(G) ⊗ M1 and ρ : M1 ⊃ M → M ⊗ G ⊂ M1 ⊗ D(G), respectively (see Proposition 2.8). The detailed proof of the next Lemma will also give some flavor of the calculations with generating matrices. (Try to give a proof without using generating matrices!). Lemma 5.2. Let (λ, ρ, φλ , φρ , φλρ ) be a quasi-commuting pair of G-coactions on M and let M1 ≡ λMρ ./ Gˆ be the associated diagonal crossed product with universal λρ-intertwiner Γ ∈ G ⊗ M1 . Then there exist uniquely determined algebra maps λD : M1 → D(G) ⊗ M1 and ρD : M1 → M1 ⊗ D(G) satisfying (we suppress all embeddings M ,→ M1 and G ,→ D(G))
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λD (m) = λ(m) ,
∀ m ∈ M ⊂ M1
231 13 213 Γ φλ D12 φ−1 (id ⊗ λD )(Γ) = (φ−1 λρ ) λ ∈ G ⊗ D(G) ⊗ M1
ρD (m) = ρ(m) ,
∀ m ∈ M ⊂ M1
231 12 −1 D13 φ213 (id ⊗ ρD )(Γ) = (φ−1 ρ ) ρ Γ φλρ ∈ G ⊗ M1 ⊗ D(G) .
(5.4) (5.5) (5.6) (5.7)
Moreover the algebra maps λD , ρD are unital if G is not weak, i.e. if ∆(1) = 1 ⊗ 1. Note that for the case that G is an ordinary Hopf algebra and all reassociators are trivial, we recover the definition of λD , ρD given in Proposition 2.8. Proof. Let us first suppose that ∆(1) = 1 ⊗ 1. Viewing the left G-coaction λ : M → G ⊗ M as a map λ : M → D(G) ⊗ M1 , Theorem 3.1 states that λD is a unital algebra map extending λ if and only if TD := (id ⊗ λD )(Γ) ∈ G ⊗ (D(G) ⊗ M1 ) is a normal coherent λρ-intertwiner. Now normality of TD follows from the normality of Γ. To prove that TD is a λρ-intertwiner we compute for all m ∈ M, 231 13 213 TD (idG ⊗ λD )(λ(m)) = (φ−1 Γ φλ D12 φ−1 λρ ) λ (idG ⊗ λD )(λ(m)) 231 13 213 Γ φλ D12 φ−1 = [(λD ⊗ idG )(ρ(m))]231 (φ−1 λρ ) λ
= (idG ⊗ λD )(ρop (m)) TD , where both sides are viewed as elements in G ⊗ D(G) ⊗ M1 . Here we have used the intertwining properties of Γ and D and of the three reassociators. To show that TD also satisfies the coherence condition, i.e. Eq. (3.3), we compute in G ⊗ G ⊗ D(G) ⊗ M1 — again suppressing all embeddings h i3412 142 24 ) [1 ⊗ φ ] Γ14 (φ−1 Γ (∆ ⊗ id)(TD ) = (id ⊗ id ⊗ ∆)(φ−1 G ρ λρ λρ ) h i3124 × [1G ⊗ φλ ] (id ⊗ ∆ ⊗ id)(φλ )[φ ⊗ 1M ] D13 (φ−1 )132 D23 × [φ ⊗ 1M ](∆ ⊗ id ⊗ id)(φ−1 λ ) i3412 h −1 142 ⊗ 1 ](id ⊗ ρ ⊗ id)(φ ) Γ14 (φ−1 = (λ ⊗ id ⊗ id)(φρ )[φ−1 G λρ λρ λρ ) h i3124 × Γ24 (id ⊗ id ⊗ λ)(φλ )(∆ ⊗ id ⊗ id)(φλ ) D13 (φ−1 )132 D23 −1 × (id ⊗ ∆ ⊗ id)(φ−1 λ )[1G ⊗ φλ ](id ⊗ id ⊗ λ)(φλ )
i3412 h ⊗ 1 ] Γ14 = (λ ⊗ id ⊗ id)(φρ )[φ−1 G λρ h i3142 −1 × (id ⊗ λ ⊗ id)(φ−1 ) [1 ⊗ φ ] (id ⊗ id ⊗ ρ)(φ ) Γ24 D13 G λ λρ λρ
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h i1324 × (∆ ⊗ id ⊗ id)(φλ )[φ−1 ⊗ 1M ] (id ⊗ ∆ ⊗ id)(φ−1 λ ) × D23 [1G ⊗ φ−1 λ ](id ⊗ id ⊗ λ)(φλ ) 3412 14 314 Γ φλ D13 = [(λ ⊗ id ⊗ id)(φρ )[φ−1 λρ ⊗ 1G ]] −1 1324 24 × [(∆ ⊗ id ⊗ id)(φ−1 Γ λρ ) (id ⊗ id ⊗ ρ)(φλ )] −1 23 × φ324 λ D [1G ⊗ φλ ](id ⊗ id ⊗ λ)(φλ )
−1 132 23 13 Γ φλ . = (id ⊗ id ⊗ λD ) φ312 ρ Γ (φλρ ) Here we have used several pentagon identities for the reassociators involved and the intertwining and coherence properties of Γ and D. In the first equality we used (3.3) for Γ and D, and in the second the pentagons (3.45c) and (3.29b). For the third equality we used the intertwining properties of D and Γ to move two more reassociators between D13 and D23 and two more between Γ14 and Γ24 . To arrive at the fourth equality we commuted D13 and Γ24 and used the pentagons (3.45b) and (3.29b) and then again the intertwining properties of D and Γ to bring two reassociators back between D13 and Γ24 . The last equality holds by (5.4), (5.5). Thus we have shown that TD is coherent and therefore the definitions (5.4), (5.5) uniquely define a unital algebra map λD extending λ. Similarly one shows that ρD defines a unital algebra map ρD : M1 → M1 ⊗ D(G) extending ρ. Now let ∆(1) 6= 1 ⊗ 1, then eventually λ is non unital implying that also λD may be non-unital. That λD is an algebra map is proved as above. Choosing in Lemma 5.2 also M = G (i.e. M1 = D(G)) we arrive at the following: Theorem 5.3. Let (G, ∆, , φ) be a weak quasi-Hopf algebra, denote iD : G ,→ D(G) the canonical embedding and let D ∈ G ⊗ D(G) be the universal flip operator. (i) Then (D(G), ∆D , D , φD ) is a weak quasi-bialgebra, where φD := (iD ⊗ iD ⊗ iD )(φ) D (iD (a)) := (a) ,
(5.8)
(id ⊗ D )(D) := 1D(G)
∆D (iD (a)) := (iD ⊗ iD )(∆(a)) ,
(5.9)
∀a ∈ G
(5.10)
231 12 −1 D13 φ213 (iD ⊗ ∆D )(D) := (φ−1 D D φD . D )
(5.11)
Moreover D(G) is weak if and only if G is weak. (ii) Under the setting of Lemma 5.2 denote iM1 : M ,→ M1 the embedding and define φλD := (iD ⊗ iD ⊗ iM1 )(φλ ) ∈ D(G) ⊗ D(G) ⊗ M1 φρD := (iM1 ⊗ iD ⊗ iD )(φρ ) ∈ M1 ⊗ D(G) ⊗ D(G) φλD ρD := (iD ⊗ iM1 ⊗ iD )(φλρ ) ∈ D(G) ⊗ M1 ⊗ D(G) .
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Then (λD , ρD , φλD , φρD , φλD ρD ) provides a quasi-commuting pair of D(G)ˆ coactions on M1 ≡ λMρ ./ G. Proof. Setting M := G and λ = ∆ in Lemma 5.2 implies that ∆D is an algebra morphism, which is unital if and only if ∆ is unital. The property of D being a counit for ∆D follows directly from (5.1) and the fact that (id ⊗ ⊗ id)(φ) = ∆(1). To show that ∆D is quasi-coassociative one computes that [1G ⊗ φD ] · (id ⊗ ∆D ⊗ id)((id ⊗ ∆D )(D)) = (id ⊗ id ⊗ ∆D )((id ⊗ ∆D )(D)) · [1G ⊗ φD ] , where one has to use (5.11), the pentagon equation for φ and the intertwiner property (5.2) of D similarly as in the proof of Lemma 5.2. Thus ∆D is quasicoassociative and this concludes the proof of Part (i). Part (ii) is shown by direct calculation using the intertwiner properties of Γ and D and several pentagon identities for the reassociators involved. The details are left to the reader. Note that viewed in D(G) ⊗ D(G) and D(G)⊗ , respectively, the relations (5.2), (5.3) and (5.11) are precisely the defining properties of a quasitriangular R-matrix [18]. Thus RD := (iD ⊗ id)(D) is an R-matrix for D(G). It is shown in [27] that there also exists an antipode SD for D(G) extending the antipode of G, thus making the quantum double D(G) into a quasitriangular quasi-Hopf algebra. 3
5.2. Two-sided crossed products As in the associative case, a simple recipe to produce two-sided G-comodule algebras (M, δ) is by tensoring a right G-comodule algebra (A, ρ) and a left Gcomodule algebra (B, λ), i.e. by setting M = A ⊗ B and δ(A ⊗ B) := B(−1) ⊗ (A(0) ⊗ B(0) ) ⊗ A(1) as in (2.41). Obviously δ = (λ ⊗ id) ◦ ρ = (id ⊗ ρ) ◦ λ, where we use the same symbols (λ, φλ ) and (ρ, φρ ) for the trivially extended left and right coactions, i.e. λ ≡ λ ⊗ idB , etc. Hence (λ, ρ, φλ , φρ , φλρ = 1G ⊗ 1M ⊗ 1G ) is a strictly commuting pair of G-coactions on M = A ⊗ B. In the terminology of (3.46a)–(3.46d) we have δ = δr = δl , whereas Ψ = Ψr = Ψl is given by Ψ = [1G ⊗ 1G ⊗ φ−1 ρ ] [φλ ⊗ 1G ⊗ 1G ] .
(5.12)
According to Theorem 3.1 the diagonal crossed product M1 = (A ⊗ B)δr ./ Gˆ =: ˆ satisfying the (Aρ ⊗ λ B) ./ Gˆ is generated by {A, B, Γ(ϕ)|A ∈ A, B ∈ B, ϕ ∈ G} defining relations AB = B A [1G ⊗ B] Γ = Γ λ(B) ρop (A) Γ = Γ [1G ⊗ A] 13 23 (∆ ⊗ id)(Γ) = φ312 ρ Γ Γ φλ ,
where Γ = eµ ⊗ Γ(e ) is the universal λρ-intertwiner. µ
(5.13a) (5.13b) (5.13c) (5.13d)
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The next Proposition is an analogue of Proposition 2.14 saying that the diagonal crossed product (Aρ ⊗ λ B) ./ Gˆ may be realized as a two-sided crossed product Aρ o Gˆ n λ B. Note that the isomorphism µ in (5.14) below is different from the isomorphisms µR and µL constructed in Theorem 3.1. Proposition 5.4. Let G be a quasi-Hopf algebra and . : Gˆ ⊗ A → A and / : B ⊗ Gˆ → B be the left and the right action corresponding to a right G-coaction (ρ, φρ ) on A and a left G-coaction (λ, φλ ) on B, respectively. Extend λ and ρ trivially ˆ δr := (id ⊗ ρ) ◦ λ, be the to A ⊗ B and let M1 := (Aρ ⊗ λ B) ./ Gˆ ≡ (A ⊗ B)δr ./ G, diagonal crossed product with universal λρ-intertwiner Γ ∈ G ⊗ M1 . (i) There is a linear bijection µ : A ⊗ Gˆ ⊗ B → M1 given by µ(A ⊗ ϕ ⊗ B) = A Γ(ϕ) B ,
(5.14)
where we have suppressed the embeddings A ,→ M1 and B ,→ M1 . (ii) Denote the induced algebra structure on A ⊗ Gˆ ⊗ B by Aρ o Gˆ n λ B ≡ µ−1 (M1 ). Then we get the following multiplication structure with unit ˆ n 1B on Aρ o Gˆ n λ B 1A o 1 (A o ϕ n B) (A0 o ψ n B 0 ) = A(ϕ(1) . A0 )φ¯1ρ o [φ¯1λ * ϕ(2) ( φ¯2ρ ][φ¯2λ * ψ(1) ( φ¯3ρ ] n φ¯3λ (B / ψ(2) )B 0 . (5.15) Proof. Let pλ ∈ G ⊗ B ≡ G ⊗ (1A ⊗ B) be given by (3.71) and define f : A ⊗ ϕ ⊗ B 7→ A ⊗ ϕ(1) ⊗ (ϕ(2) ⊗ idB )(λ(B)pλ ) .
(5.16)
Using (5.13b) we obtain (µ ◦ f )(A ⊗ ϕ ⊗ B) = A (ϕ ⊗ idM1 )(Γ λ(B) pλ ) = A B (ϕ ⊗ idM1 )(Γ pλ ) = µR (A ⊗ B ⊗ ϕ) , where µR : (A ⊗ B) ⊗ Gˆ → M1 is the linear bijection constructed in Part 3 of Theorem 3.1, see also (3.81). Hence µ is surjective. Conversely, let R := Γ pλ ∈ G ⊗ M1 then by Proposition 3.19 R is a right δr -implementer and Γ = [1G ⊗ qλ2 ] R [S −1 (qλ1 ) ⊗ 1M1 ] , where qλ ∈ G ⊗ B is given by (3.72) and where we have used that ρ is trivial on B. Hence we get for all B ∈ B, using again (5.13b) Γ [1G ⊗ B] = [1G ⊗ qλ2 B(0) ] R [S −1 (qλ1 B(−1) ) ⊗ 1M1 ] , where B(−1) ⊗ B(0) = λ(B). Setting f¯(A ⊗ B ⊗ ϕ) := (Sˆ−1 (ϕ(2) ) ⊗ idB )(qλ λ(B)) ⊗ ϕ(1) ,
(5.17)
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(5.17) implies µ(A ⊗ ϕ ⊗ B) ≡ A Γ(ϕ) B = µR ◦ f¯(A ⊗ B ⊗ ϕ) . But since f¯ is invertible (one directly verifies that f¯ = f −1 , with f given in (5.16)), the injectivity of µR implies the injectivity of µ. This proves Part (i). Part (ii) follows since one straightforward checks that the multiplication rule (5.15) is equivalent to the defining relations (5.13). Next we show, that analogously as in (2.51) the two-sided crossed product construction given in Proposition 5.4 may be iterated if one of the two algebras A and B admits a quasi-commuting pair of coactions. Proposition 5.5. Let (A, ρA , φρA ), (C, λC , φλC ) and (B, ρB , λB , φλB , φρB , φλB ,ρB ) be a right, a left and a two-sided comodule algebra, respectively, and denote the universal λρ-intertwiners Γ AB := eµ ⊗ (1A o eµ n 1B ) ∈ Aρ o Gˆ n λ B Γ BC := eµ ⊗ (1B o eµ n 1C ) ∈ Bρ o Gˆ n λ C ,
(5.18)
where Aρ ≡ AρA , etc., then ρ, φρ˜) given by φρ˜ := φρB (trivially (i) Aρ o Gˆ n λ B admits a right G-coaction (˜ embedded), ρ˜ (A⊗B) := idA ⊗ ρB and (idG ⊗ ρ˜)(Γ AB ) := (Γ AB ⊗ 1G ) φ−1 λB ρB .
(5.19)
˜ φ ˜ ) given by φ˜ := φλ (trivially (ii) Bρ o Gˆ n λ C admits a left G-coaction (λ, B λ λ ˜ embedded), λ (B⊗C) := λB ⊗ idC and −1 231 13 ˜ ΓBC . (idG ⊗ λ)(Γ BC ) := (φλB ρB )
(5.20)
(iii) The trivial identification (Aρ o Gˆ n λ B)ρ˜ o Gˆ n λ C ≡ Aρ o Gˆ nλ˜ (Bρ o Gˆ n λ C)
(5.21)
is an algebra isomorphism. Proof. (i) To show that (5.19) provides a well-defined algebra map ρ˜ : Aρ o ˆ G n λ B → (Aρ o Gˆ n λ B) ⊗ G extending idA ⊗ ρB we have to check that the relations (5.13) are respected. To this end we put TAB := (Γ AB ⊗ 1G ) φ−1 λB ρB and compute for B ∈ B [1G ⊗ ρ˜(B)] TAB = (Γ AB ⊗ 1G ) (λB ⊗ id)(ρB (B)) φ−1 λB ρB = TAB (idG ⊗ ρ˜)(λB (B)) , which is the relation (5.13b). Trivially one also has (since φλB ρB ∈ G ⊗(1A ⊗B)⊗G) TAB [1G ⊗ A ⊗ 1G ] = [ρop (A) ⊗ 1G ] TAB
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verifying (5.13c). Finally, the coherence condition (5.13d) is respected, since in G ⊗ G ⊗ (A o Gˆ n B) ⊗ G we have (∆ ⊗ id ⊗ id)(TAB ) = (∆ ⊗ id)(Γ AB )123 (∆ ⊗ id ⊗ id)(φ−1 λB ρB ) −1 = (∆ ⊗ id)(Γ AB )123 [φ−1 λB ⊗ 1G ] (id ⊗ λB ⊗ id)(φλB ρB )
× [1G ⊗ φ−1 λB ρB ] (id ⊗ id ⊗ ρB )(φλB ) −1 13 23 = φ312 ρA ΓAB ΓAB (id ⊗ λB ⊗ id)(φλB ρB )
× [1G ⊗ φ−1 λB ρB ] (id ⊗ id ⊗ ρB )(φλB ) −1 13 134 23 234 = φ312 ΓAB (φ−1 (id ⊗ id ⊗ ρB )(φλB ) ρA ΓAB (φλB ρB ) λρ ) 13 23 = (id ⊗ id ⊗ ρ˜)(φ312 ˜)(φλB ) . ρA ) TAB TAB (id ⊗ id ⊗ ρ
Here we have used the pentagon identity (3.45b) in the second line, the coherence property (5.13d) of Γ AB in the third line and finally the intertwining property (5.13b). Thus ρ˜ provides a well-defined algebra map, which is also unit preserving ˆ n 1B ) ⊗ 1G . Similarly one shows by a straight since ( ⊗ id ⊗ id)(TAB ) = (1A o 1 forward calculation that the pair (˜ ρ, φρ˜) satisfies (3.30a). Since φρ˜ = φρB , the pentagon Eq. (3.30b) and the counit Eqs. (3.30c) and (3.30d) are clearly satisfied. This proves Part (i). Part (ii) follows analogously. To prove Part (iii) we have to check that we may consistently identify h i h i ! ˆ ˆ ∈ G ⊗ A o G n (B o G n C) G ⊗ (A o Gˆ n B) o Gˆ n C 3 Γ AB ≡ Γ A(BoGnC) ˆ (5.22) h i h i ! G ⊗ (A o Gˆ n B) o Gˆ n C 3 Γ (AoGnB)C ≡ Γ BC ∈ G ⊗ A o Gˆ n (B o Gˆ n C) . ˆ (5.23) For this, the nontrivial commutation relations to be looked at are according to (5.13b), (5.13c) [1G ⊗ X] = ρ˜op (X) Γ (AoGnB)C , Γ (AoGnB)C ˆ ˆ ˜ ), = Γ A(BoGnC) λ(Y [1G ⊗ Y ] Γ A(BoGnC) ˆ ˆ
X ∈ A o Gˆ n B Y ∈ B o Gˆ n C .
ˆ and Y ∈ The remaining cases being trivial, it is enough to consider X ∈ ΓAB (G) ˆ ΓBC (G) for which we get −1 132 23 Γ13 ˜)(Γ AB )132 Γ23 = Γ13 Γ(AoGnB)C Γ23 ˆ ˆ ˆ AB = (idG ⊗ ρ AB (φλB ρB ) (AoGnB)C A(BoGnC) 13 213 132 23 ˜ Γ23 = Γ13 (idG ⊗ λ)(Γ = Γ13 (φ−1 ΓBC BC ) ˆ ˆ ˆ BC ΓA(BoGnC) λB ρB ) A(BoGnC) A(BoGnC)
by the definitions (5.19) and (5.20). This shows that the identifications (5.22) and (5.23) are indeed consistent and therefore proves Part (iii).
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Note that due to Part (iii) of the above proposition the notations Γ AB and Γ BC as in (5.18) are still well defined in iterated two-sided crossed products and the commutation relations of “neighboring” λρ-intertwiners are given by −1 132 23 13 Γ13 ΓBC = Γ23 AB (φλB ρB ) BC ΓAB
(5.24)
Adaption to the weak case. The definition of two-sided crossed products as in Proposition 5.4 has to be slightly modified if G is a weak quasi-Hopf algebra, i.e. if ∆(1) 6= 1 ⊗ 1. The unital algebra Aρ o Gˆ n λ B is now defined on the subspace of A ⊗ Gˆ ⊗ B given by (0) (−1) (1) (0) A o Gˆ n B := span{A o ϕ n B ≡ A1A ⊗ 1B . ϕ / 1A ⊗ 1B B
ˆ , |A ∈ A, B ∈ B, ϕ ∈ G} (0)
(1)
(−1)
where 1A ⊗1A ≡ ρ(1A ) and 1B
(5.25)
(0)
⊗1B ≡ λ(1B ). Again we have a linear bijection
µ : A n Gˆ o B → M1 = (Aρ ⊗ λ B) ./ Gˆ ,
A o ϕ n B 7→ A Γ(ϕ) B
inducing the multiplication rule described by Eq. (5.15). Also Proposition 5.5 stays valid. In particular we note that (5.25) still allows the identification (A o ϕ n B) o ψ n C = A o ϕ n (B o ψ n C) . Moreover, putting (B, λ, φλ ) = (G, ∆, φ) this is also the setting underlying the field algebra constructions with quasi-Hopf symmetry as proposed in Sec. 5.4. 5.3. Quasi-Hopf spin chains and lattice current algebras Due to Proposition 5.5 the definition of Hopf spin chains as reviewed in Sec. 2.4.2 immediately generalizes to the quasi-coassociative case. As in Sec. 2.4.2 we interpret even integers as sites and odd integers as links of a one dimensional lattice and we set ˆ the latter just being a linear space. A local net of associative A2i ∼ = G, A2i+1 ∼ = G, algebras An,m is then constructed inductively for all n, m ∈ 2Z, n ≤ m, by first putting A2i,2i+2 := A2i o A2i+1 n A2i+2 ∼ = G o Gˆ n G , where G is equipped with its canonical two-sided comodule structure (λ = ρ = ∆, φλ = φρ = φλρ = φ). Due to Proposition 5.5 this procedure may be iterated as in (2.49), by setting A2i,2j+2 := A2i,2j o Gˆ n G ≡ G o Gˆ n A2i−2,2j+2 , where the last equality follows from (5.21) by iteration. More generally one has as in (2.50) for all i ≤ µ ≤ j − 1, A2i,2j = A2i,2µ o Gˆ n A2µ+2,2j .
(5.26)
We will now give a description of the finite quasi-Hopf spin chains (i.e. the local algebras A2i,2j ) in terms of generating matrices. Defining the generating “link operators” L2i+1 := Γ A2i A2i+2 as in (5.18), we get:
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Lemma 5.6. Let G be a weak quasi-Hopf algebra. The finite open quasi-Hopf j−i generated by spin chain A2i,2j is the unique unital algebra extension of G ⊗ j−i ≡ A2i ⊗ A2i+1 ⊗ · · · ⊗ A2j and the entries of generating matrices L2ν+1 ∈ G⊗ G ⊗ A2i,2j , i ≤ ν ≤ j, obeying the relations 23 23 13 L13 2k+1 L2l+1 = L2l+1 L2k+1 ,
∀ k 6= l, l − 1, l + 1
23 (φ312 )2k L13 2k+1 L2k+1 φ2k+2 = (∆ ⊗ id)(L2k+1 )
(5.27a) (5.27b)
−1 132 23 13 ) )2k L23 L13 2k−1 ((φ 2k+1 = L2k+1 L2k−1
(5.27c)
[1 ⊗ A2k (a)] L2l+1 = L2l+1 [1 ⊗ A2k (a)] ,
∀ k 6= l, l + 1
[1 ⊗ A2k (a)] L2k−1 = L2k−1 [a(1) ⊗ A2k (a(2) )] L2k+1 [1 ⊗ A2k (a)] = [a(2) ⊗ A2k (a(1) )] L2k+1 ,
(5.27d) (5.27e)
∀a ∈ G ,
(5.27f)
where A2k denotes the identification G ≡ A2k and where φ2k := (id ⊗ id ⊗ A2k )(φ).
Proof. Follows immediately from (5.13) and (5.24). Writing A2i+1 (ϕ) = (ϕ ⊗ id)(L2i+1 ), (5.27c) is equivalent to A2i−1 (φ¯1 * ψ) A2i (φ¯2 ) A2i+1 (ϕ ( φ¯3 ) = A2i+1 (ϕ) A2i−1 (ψ) .
Thus link operators on neighboring links do not commute any more in contrast to the coassociative setting! But the algebras A2i−2,2i and A2i+2,2i+4 still commute, which means that the above construction still yields a local net of algebras now indexed by intervals in 2Z. Next we remark that Theorem 5.3 applied to the special case of two-sided crossed products provides us with localized left and right coactions of the quantum double D(G) on the above quantum chain generalizing (2.53). Indeed, using (5.26) and (5.19), (5.7), we obtain (right) D(G)-coactions ρ2i D : A2k,2j → A2k,2j ⊗ D(G) ,
k
(5.28)
acting trivially on A2k,2i−2 ∪ A2i+2,2j , and given elsewhere by ρ2i D (A2i (a)) := A2i (a(1) ) ⊗ (a(2) ./D 1) ,
a∈G
ˆ (idG ⊗ ρ2i D )(L2i−1 ) := [L2i−1 ⊗ (1 ./D 1)] (idG ⊗ A2i ⊗ iD )(φ) −1 231 −1 ) D13 φ213 L12 , (idG ⊗ ρ2i D )(L2i+1 ) := (φ 2i+1 φ
ˆ ⊂ D(G) and where we have suppressed where iD denotes the embedding G ≡ G ./D 1 the embedding idG ⊗ A2i ⊗ iD of the three reassociators in the last line. As before, D denotes the universal ∆-flip operator in G ⊗ D(G), see Sec. 5. One similarly obtains localized left D(G)-coactions λ2i D using (5.20), (5.5). This generalizes the D(G)-cosymmetry discovered by [42] to weak quasi-Hopf algebras.
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Lattice current algebras. Also the construction of the periodic chain by closing a finite open chain may again be described by a diagonal crossed product. As in Sec. 2.4.3 we define the periodic chain Kn with n sites as Kn := λ(A2,2n )ρ ./ Gˆ ,
(5.29)
where λ and ρ are nontrivial only on A2 o A3 n A4 and A2n−2 o A2n−1 n A2n , respectively, where they are defined as in Proposition 5.5, i.e. λ extends ∆ viewed as a left coaction on A2 ∼ = G to A2 o A3 n A4 and ρ extends the right coaction ∆ on A2n to A2n−2 o A2n−1 n A2n . Let us conclude this subsection by indicating, how the equivalence of the Hopf spin chains of [42] and the lattice current algebras of [4] as shown in [40] generalizes to the quasi-Hopf setting. Following [40] we define the generating current operators by (5.30) J2i+1 := (id ⊗ A2i )(Rop ) L2i+1 , where R ∈ G ⊗ G is supposed to be quasi-triangular. This yields the following commutation relations: [1G ⊗ A2i (a)] J2i−1 = J2i−1 [a(1) ⊗ A2i (a(2) )] ,
∀a ∈ G
[a(1) ⊗ A2i (a(2) )] J2i+1 = J2i+1 [1G ⊗ A2i (a)] −1 23 ˆ J13 2i+1 J2i+1 = R2i φ2i (∆ ⊗ id)(J)2i+1 φ2i+2 23 13 ˆ 23 J13 2i−1 R2i J2i+1 = J2i+1 J2i−1 ,
ˆ 2i := (id ⊗ id ⊗ A2i )(φ213 R12 φ−1 ) and φ2i := (id ⊗ id ⊗ A2i )(φ). These where R relations generalize the defining relations of lattice current algebras as given in [4] to quasi-Hopf algebras. They have also appeared in [5, 6] as lattice Chern– Simons algebras (restricted to the boundary of a disk) in the weak quasi-Hopf algebra setting, where the copies of G sitting at the sites are interpreted as gauge transformations. A more detailed account of these constructions is given in [25]. 5.4. Field algebra construction with quasi-Hopf symmetry In this subsection we sketch how the field algebra construction of Mack and ˜≡ Schomerus [30, 47] may be described by two-sided crossed products. Choosing B ∼ ˆ ˆ G and λB ≡ ∆, one gets a unital algebra M1 := (A⊗ G) ./ G = Ao G n G associated with every right G-comodule algebra (A, ρ, φρ ). Here A ⊂ M1 is to be interpreted as the algebra of observables, the universal λρ-intertwiner Γ is a “master” field operator, and G ⊂ A o Gˆ n G represents the global “quantum symmetry”. The fields are said to transform covariantly, which means that [1G ⊗ a] Γ = Γ λ(a) ≡ Γ ∆(a) ,
a∈G,
whereas the observables A ∈ A are G-invariant aA = Aa,
∀ a ∈ G, A ∈ A .
(5.31)
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The linear subspace F ≡ A o Gˆ := A o Gˆ n 1G is called the field algebra. Note that ˜ But similarly in the quasi-coassociative setting F is not a subalgebra of A o Gˆ n B. as in [30, 31] one may define a new non-associative “product” × on F by setting A × A0 = AA0 for A, A0 ∈ A and Γ13 × Γ23 := (∆ ⊗ id)(Γ) .
(5.32)
This product is quasi-associative in the sense that [φ ⊗ 1] (Γ14 × Γ24 ) × Γ34 = Γ14 × (Γ24 × Γ34 ) [φ ⊗ 1] and it satisfies [1G ⊗ 1G ⊗ a] (Γ13 × Γ23 ) = (Γ13 × Γ23 ) (id ⊗ ∆)(∆(a)) which is the reason why it is called covariant product in [30, 31]. Moreover, if G is quasitriangular, the field operators satisfy the braiding relations Γ13 × Γ23 = R12 (Γ23 × Γ13 ) (R−1 )12 . The difference of this approach with the setting of [30, 31, 47] lies in the fact that here the field operators appear in the form of irreducible matrix-multiplets FIij := (πIij ⊗ id)(Γ) ,
πI ∈ Irrep G .
Correspondingly, the superselection sectors of the observable algebra A are given by the amplimorphisms jk ∗ ij ik ρij I (A) ≡ (id ⊗ π )(ρ(A)) = FI A (FI )
which is equivalent to the defining relation (5.13c). In this sense the above construction fits to the formulation of DHR-sector theory as proposed for lattice theories in [50, 42]. In the terminology of [42] the G-coactions ρ : A → A ⊗ G would then be a “universal cosymmetry”. We will show elsewhere, that such a ρ may indeed be constructed also for continuum theories (more precisely for “rational” theories, where the number of sectors is finite). This will be done by providing suitable multiplets WIi ∈ A satisfying X WIi WIi∗ = 1A , ∀ I i
and putting j i∗ ρij I (A) := WI σI (A) WI ,
where σI is a DHR-endomorphism representing the irreducible sector I. The field operators of [30, 31, 47] are then recovered as ΨiI = WIj FIji . In general the WIi ’s will not generate a Cuntz algebra (i.e. they will not be orthogonal isometries) and therefore the amplimorphisms ρI will be non-unital, implying G to
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be a weak quasi-Hopf algebra. The reassociator φρ ∈ A ⊗ G ⊗ G will be obtained by suitably “blowing up” the quantum field theoretic “6j-symbols” with the help of the WIi ’s. In this way we will finally be able to show, that the inclusion A ⊂ M1 ≡ A o Gˆ n G provides a finite index and depth-2 inclusion of von-Neumann factors satisfying A0 ∩ M1 = G. Appendix A. Representation Theoretic Interpretation As has already been indicated in Sec. 3.1, the algebraic properties of a quasiHopf algebra G may be translated into corresponding properties of its representation category Rep G. We now give a more detailed discussion, where we also provide a representation theoretic interpretation of our notions of left, right and two-sided coactions. We also describe the diagonal crossed products in representation theoretical terms. (In fact this has already be done by proving Theorem 3.1, since algebra maps may be viewed as representations and vice versa.) In particular the complicated formulas given in Lemma 3.21 will be shown to be quite natural by identifying them with certain commuting diagrams. Also the δ-implementers and λρ-intertwiners will give rise to certain morphisms, where the coherence conditions may again be expressed by commuting diagrams. As an application we show that the category Rep D(G) of finite dimensional representations of the quantum double D(G) coincides with what has been called the double category of G-modules (analogue of Drinfeld–Yetter G-modules) by S. Majid [35]. Let G be a quasi-Hopf algebra with invertible antipode S. Let Rep M and Rep G be the category of unital representations of M and G, respectively, where in Rep G we only mean to speak of finite dimensional representations. We denote the objects in Rep G by (U, πU ), (V, πV ), (W, πW ), . . ., where U , V , W, . . . denote the underlying representation spaces and πV : G → EndC (V ) the representation maps. Similarly, we denote the objects in Rep M by (H, γH ), (K, γK ), (L, γL ), . . ., where the Gothic symbols denote the representation spaces and where γH : M → EndC (H), etc. We will also freely use the G-module notation by writing a · v := πV (a)v and V ≡ (V, πV ) (and analogously for M-modules H). The set of morphisms HomG (U, V ) (also called intertwiners) is given by the linear maps f : U → V satisfying f πU (a) = πV (a) f, ∀ a ∈ G. The representation category of a quasi-Hopf algebra. It is well known (see e.g. [18]) that for quasi-Hopf algebras G the category Rep G becomes a rigid monoidal category, where the tensor product (V W, πV πW ) of two representations (V, πV ) and (W, πW ) is given by V W := V ⊗ W ,
πV πW := (πV ⊗ πW ) ◦ ∆ ,
(A.1)
whereas for morphisms (≡ G-module intertwiners) f , g one has f g := f ⊗ g. (The symbol ⊗ always denotes the usual tensor product in the category of vector spaces.) The associativity isomorphisms are given in terms of the reassociator φ by the natural family of G-module isomorphisms φUV W : (U V ) W → U (V W ) ,
φUV W := (πU ⊗ πV ⊗ πW )(φ) .
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The unit object (with respect to ) in Rep G is given by (C, ). Throughout, if C is viewed as a G-module it is always meant to be equipped with the module structure given by the one dimensional representation . The left and right dual of any representation (V, πV ) are defined by ∗ V = V ∗ = Vˆ := HomC (V, C) and πV ∗ := πVt ◦ S −1 ,
π∗ V := πVt ◦ S ,
where t denotes the transposed map. The (left) rigidity structure is given by the family of morphisms (G-module intertwiners) aV : ∗ V V → C ,
vˆ ⊗ v 7→ hˆ v |α · vi
bV : C → V ∗ V ,
1 7→ β · vi ⊗ v i ,
(A.2)
where vi ∈ V and v i ∈ Vˆ are a choice of dual bases and where α, β ∈ G are the elements defined in (3.14), (3.15). Drinfel’d’s antipode axioms for G precisely reflect the fact that aV and bV are morphisms in Rep G fulfilling the rigidity identities (idV aV ) ◦ φV (∗ V )V ◦ (bV idV ) = idV ∗ ∗ (aV id∗ V ) ◦ φ−1 (∗ V )V (∗ V ) ◦ (id V bV ) = id V .
(A.3)
Also note that one has ∗ (V ∗ ) = (∗ V )∗ = V with trivial identification. Next, we recall that in any left-rigid monoidal category one has natural isomorphisms ∗ (U V ) ∼ = ∗ V ∗ U . As already mentioned in our case these isomorphisms are given by fUV : U V → (∗ V ∗ U )∗ ,
u ⊗ v 7→ (πV ⊗ πU )(f ) (v ⊗ u) ,
(A.4)
ˆ )∧ and where the where we trivially identify the vector spaces V ⊗ W ≡ (Vˆ ⊗ W twist f ∈ G ⊗ G is given by (3.20). The fact that fUV is indeed a morphism in Rep G follows from (3.22). Similarly, by (3.25), we have a natural family of isomorphisms hUV : U V → ∗ (V ∗ U ∗ ) ,
u ⊗ v 7→ (πV ⊗ πU )(h) (v ⊗ u) .
Coactions. Now a left G-coaction (λ, φλ ) on M naturally induces a left action of Rep G on Rep M. By this we mean a functor : Rep G × Rep M → Rep M , where for (V, π) ∈ Rep G and (H, γ) ∈ Rep M we define (V H, π γ) ∈ Rep M by V H = V ⊗H,
π γ := (π ⊗ γ) ◦ λ ,
whereas for morphisms we put f g := f ⊗ g. The counit axiom for λ implies γH = γH for all (H, γH ) ∈ Rep M and the axioms for φλ imply the quasiassociativity relations (πV πW ) γH ∼ = πV (πW γH ) ,
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where the isomorphism is given by φV W H := (πV ⊗ πW ⊗ γH )(φλ ) . Finally, the pentagon axiom (3.29b) provides us with the analogue of McLane’s coherence conditions, i.e. the following commuting diagram:
1
φ(U V )W H
((U V ) W ) H HH φ id HHH j UV W
(U V ) (W H)
PPP
PPφPPq
UV (W H)
U (V (W H))
H
(U (V W )) H
-
φU (V W )H
id* φ U
V WH
U ((V W ) H)
(A.5)
With the obvious substitutions analogue statements hold for right coactions (ρ, φρ ), where now these induce a right action : Rep M × Rep G → Rep M given for (γ, H) ∈ Rep M and (π, V ) ∈ Rep G by H V := H ⊗ V ,
γ π := (γ ⊗ π) ◦ ρ .
Finally, a quasi-commuting pair (λ, ρ, φλ , φρ , φλρ ) provides us with both, a left and a right action of Rep G on Rep M, together with a further family of associativity equivalences (πU γH ) πV ∼ = πU (γH πV ), where now the isomorphisms are given by φU HV := (πU ⊗ γH ⊗ πV )(φλρ ) . Again, the conditions (3.45b) and (3.45c) imply further pentagon diagrams of the type (A.5) with objects of the type U V HW or U HV W , respectively, in appropriate bracket positions. Two-sided coactions. In the obvious way the above may also be generalized to arbitrary two-sided coactions (δ, Ψ), in which case we would obtain a functor Rep G × Rep M × Rep G → Rep M U m H l V := U ⊗ H ⊗ V ,
πU m γH l πV := (πU ⊗ γH ⊗ πV ) ◦ δ
(A.6)
with associativity isomorphisms (πU πV ) m γH l (πW πZ ) ∼ = πU m (πV m γH l πW ) l πZ given by ΨUV HW Z := (πU ⊗ πV ⊗ γH ⊗ πW ⊗ πZ )(Ψ)
(A.7)
and obeying analogue “two-sided” pentagon diagrams. Note that the operations m and l are not defined individually, i.e. only the two-sided operation m · l makes sense. According to Proposition 3.8 the relation between two-sided and one-sided Rep G-actions is given by πV γH := πV m γH l ,
γH πV := m γH l πV
(πV γH ) πU ∼ = πV m γH l πU ∼ = πV (γH πU ) ,
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where here the intertwiners are given by (πV ⊗ γH ⊗ πU )(Ul/r ), respectively, see Proposition 3.8. Some families of natural transformations. We now give a representation theoretic interpretation of the elements pλ , qλ ∈ G ⊗ M given in (3.71)–(3.72) by defining for (V, πV ) ∈ Rep G and (H, γH ) ∈ Rep M the natural family of morphismsg PV H : H → V ∗ (V H) , QV H : ∗ V (V H) → H ,
H 7→ v i ⊗ pλ · (vi ⊗ H) vˆ ⊗ v ⊗ H 7→ (ˆ v ⊗ id) qλ · (v ⊗ H)
(A.8)
By (3.97a) and (3.97b) these are indeed morphisms in Rep M, and (3.97c) and (3.97d) imply the “generalized left rigidity” identities QV ∗ (V H) ◦ (idV PV H ) = idV H
(A.9)
(idV QV H ) ◦ P∗ V (V H) = idV H .
Finally, (3.97e) implies the coherence condition given by the following commuting diagram:
-V
P(U V )H
∗
(V H)
?
(U V )∗ [(U V ) H] fV−1 ∗ U ∗ id(U V ) H (V
∗
?
U ∗)
-
idV ∗ PU (V H)
PV H H
V ∗ [U ∗ (U (V H))]
idV ∗ (idU ∗ φ−1 UV H)
-
φV ∗ U ∗ [(U V ) H]
[(U V ) H]
V
∗
[U ∗
(A.10)
?
((U V ) H)]
The reader is invited to draw the analogous diagram implied by (3.97f), now involving the morphisms QV H . Similar statements hold of course for the natural family of morphisms PHV : H → (H V ) ∗ V , ∗
QHV : (H V ) V → H ,
H 7→ pρ · (H ⊗ vi ) ⊗ v i H ⊗ v ⊗ vˆ 7→ (id ⊗ vˆ) qρ · (H ⊗ v)
(A.11)
Adjustments to weak quasi-Hopf algebras. Let us now shortly discuss the adjustments to be made for the case that G is a weak quasi-Hopf algebra. As already discussed in Sec. 4.2, due to the coproduct being non-unital the definition of the tensor product functor in Rep G has to be slightly modified. First note that the element ∆(1) (as well as iterated coproducts of 1) is idempotent and commutes g Again a summation is understood, where {v } is a basis of V with dual basis {vi }. i
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with all elements in ∆(G). Thus, given two representations (V, πV ), (W, πW ), the operator (πV ⊗ πW )(∆(1)) is a projector, whose image is precisely the G-invariant subspace of V ⊗ W on which the tensor product representation operates non trivial. Thus one is led to modify the tensor product of two representations of G as given in (A.1) as follows: V W := (πV ⊗ πW )(∆(1)) V ⊗ W ,
πV πW := (πV ⊗ πW ) ◦ ∆|V W . (A.12)
One readily verifies that with these definition φUV W — restricted to the subspace (U V ) W — furnish a natural family of isomorphisms defining an associativity constraint for the tensor product functor . Moreover, Rep G becomes a rigid monoidal category with rigidity structure defined as before by (A.2)–(A.3). The left action of Rep G on Rep M induced by a left G-coaction (λ, φλ ) on M has to be modified analogously by defining V H := (π ⊗ γ)(λ(1M )) V ⊗ H ,
π γ := (π ⊗ γ) ◦ λ|V H .
The modifications of right actions and of two-sided actions of Rep G on Rep M (induced by (ρ, φρ ) and by (δ, Ψ), respectively) should by now be obvious and are left to the reader. With these adjustments all categorical identities such as the definition of natural families and commuting diagrams given above stay valid. Translating these into algebraic identities one has to take some care with identities in higher tensor products of G. The only equations which have to be modified are (3.97c), (3.97d), where the r.h.s. becomes λ(1M ) instead of 1G ⊗ 1M and similarly (3.98c)/(3.98d), where the r.h.s. has to be replaced by ρ(1M ). This is rather obvious from the categorical point of view, since for example (3.97c) is directly connected with (A.9) where the r.h.s. is given by idV H ≡ (πV ⊗ πH )(λ(1M )). δ-Implementers. We now give a representation theoretic interpretation of our notion of left and right δ-implementers. To this end let (H, γH ) be a fixed representation of M. Putting A := EndC (H), we consider γ ≡ γH : M → A as an algebra map. This leads to: Definition A1. Let (δ, Ψ) be a two-sided G-coaction on M. A representation (H, γ) of M is called δ-coherent if there exists a normal coherent left δ-implementer L ∈ G ⊗ EndC (H) (equivalently right δ-implementer R ∈ G ⊗ EndC (H)). Corollary 3.14 then says that a representation of M is δ-coherent if and only if ˆ it extends to a representation of the diagonal crossed products Gˆ ./ Mδ ∼ = Mδ ./ G. We now provide a category theoretic description of δ-coherent representations (H, γ). Associated with a left δ-implementer L ∈ G ⊗ A, A := EndC (H), we define a family of M-linear morphisms (A.13) v ⊗ id) ◦ LV (v ⊗ H) , lV : V m H l V ∗ → H , v ⊗ H ⊗ vˆ 7→ (ˆ where (V, πV ) ∈ Rep G and LV := (πV ⊗id)(L) and where we have used the notation (A.6). Equation (3.58) guarantees that lV is in fact a morphism in Rep M. The
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normality condition for L implies lC = idH and the coherence condition (3.60) for L translates into the following coherence condition for the lV ’s: lV W ◦ ΩL V W HW ∗ V ∗ = lV ◦ (idV ⊗ lW ⊗ idV ∗ ) , ∗ ∗ where ΩL → (V W ) m H l (V W )∗ V W HW ∗ V ∗ : V m (W m H l W ) l V L is the natural M-linear isomorphism given by ΩV W HW ∗ V ∗ = (idV ⊗ idW ⊗ idH ⊗ fW ∗ V ∗ ) ◦ Ψ−1 V W HW ∗ V ∗ see (A.4), (A.7) and (3.49). Similarly, a right δ-implementer R ∈ G ⊗ A gives rise to a coherent family of M-linear morphisms
rV : H → V ∗ m H l V ,
H 7→ v i ⊗ R21 V (H ⊗ vi ) ,
(A.14)
where RV := (πV ⊗ id)(R). As above, this means ΩR V ∗ U ∗ HUV ◦ rU V = (idV ∗ ⊗ rU ⊗ idV ) ◦ rV , −1 where ΩR V ∗ U ∗ HUV := ΨV ∗ U ∗ HUV ◦ (fV ∗ U ∗ ⊗ idH ⊗ idU ⊗ idV ), see (3.56) and (3.61).
λρ-Intertwiners. We finally give a representation categorical interpretation of the notion of λρ-intertwiners and of their connection with δ-implementers as stated in Proposition 3.19. As before, given a fixed representation (H, γ) of M we consider γ : M → A ≡ EndC (H) as an algebra map. Definition A2. Let (λ, φλ , ρ, φρ , φλρ ) be a quasi-commuting pair of G-coactions on M. A representation (H, γH ) of M is called λρ-coherent, if there exists a normal coherent λρ-intertwiner T ∈ G ⊗ EndC (H). Proposition 3.19 then says, that λρ-coherence is equivalent to δl -coherence for δl = (λ ⊗ id) ◦ ρ (or to δr -coherence for δr = (id ⊗ ρ) ◦ λ). Associated with a λρ-intertwiner T ∈ G ⊗ EndC (H) we now define a family of M-linear morphisms: tV : V H → H V ,
v ⊗ H 7→ T21 V (H ⊗ v) ,
(A.15)
21 where (V, πV ) ∈ Rep G and T21 V := (id ⊗ πV )(T ). Equation (3.67) guarantees that tV is a morphism in Rep M. The normality condition for T implies tC = idH and the coherence condition (3.68) for T translates into the following coherence condition for tV :
tV W = φV W H ◦ (idV ⊗ tW ) ◦ φ−1 V HW ◦ (tV ⊗ idW ) ◦ φHV W .
(A.16)
Note that (A.16) looks precisely like one of the coherence conditions for the braiding in a braided quasi-tensor category with nontrivial associativity isomorphisms. Indeed, as has been shown in [27], for the case M = G, the family of tV ’s may be used to define a braiding in the representation category of the quantum double ˆ D(G) ≡ G ./ G. Using the morphisms PV H , PHV and QV H , QHV given in (A.8) and (A.11), the relation between λρ-intertwiners T, right δr -implementers R and left δl -implementers L may now be described by the following commuting diagrams connecting
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the intertwiner morphisms tV with the maps rV (associated with R) and lV (associated with L) as given in (A.13) and (A.14). r - H V 3 QQQ Q Q P QQ ? Qs lV
(V H) V ∗ tV idV ∗
V
∗
(H V )
6
idV ∗ tV
VH
HV
(H V ) V ∗
V ∗ (V H) (A.17)
((V H) V ∗ ) V
P
QQ QQ
(V H)V ∗
lV idV
QQ Qs
V H tV
- V (V Q
idV rV
? +
∗
(H V ))
V ∗ (H V )
H V
Note that the commutativity of the above diagrams is in fact equivalent to the statements in Proposition 3.19. The representation category of the quantum double. We will now describe the representation category of the quantum double in terms of the representation category of the underlying quasi-Hopf algebra G. In this way we will show that D(G) is a concrete realization of the quantum double as defined by Majid in [35] with the help of a Tannaka–Krein-like reconstruction theorem. We denote the monoidal category of finite dimensional unital representations of D(G) by Rep D(G). As an immediate implication of Theorem 3.1 we state a necessary and sufficient condition, under which a representation of G extends to a representation of D(G), see also Definition A2 for the case λ = ρ = ∆. We only treat the case ∆(1) = 1 ⊗ 1, the generalization to the weak case being obvious. Corollary A3. (1) The objects of Rep D(G) are in one to one correspondence with pairs {(πV , V ), DV }, where (πV , V ) is a finite dimensional representation of G and where DV ∈ G ⊗ EndC (V ) is a normal coherent ∆-flip, i.e. (i) ( ⊗ id)(DV ) = idV (ii) DV · (id ⊗ πV )(∆(a)) = (id ⊗ πV )(∆op (a)) · DV , ∀ a ∈ G −1 132 13 D23 (iii) φ312 V DV (φV ) V φV = (∆ ⊗ id)(DV ), where φV := (id ⊗ id ⊗ πV )(φ). (2) Let {(πV , V ), DV } and {(πW , W ), DW } be as above, then HomD(G) = {t ∈ HomG (V, W )|(id ⊗ t)(DV ) = DW } .
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Proof. Part (1) follows from Theorem 3.1 by choosing λ = ρ = ∆, A = End (V ) and γ = πV , see also (A.15). We shortly repeat the arguments. Define the extended representation πVD on the generators of D(G) by πVD (iD (a)) := πV (a) ,
a∈G
πVD (D(ϕ)) := (ϕ ⊗ idEndV )(DV ) ,
(A.18) ϕ ∈ Gˆ .
(A.19)
Condition (i) implies that πVD is unital whereas conditions (ii), (iii) just reflect the defining relations (5.2) and (5.3) of D(G), which ensures, that πVD is a well-defined algebra morphism. On the other hand, given a representation (πVD , V ) of D(G), we define DV := (idG ⊗ πVD )(D) which clearly satisfies conditions (i)–(iii). This proves Part (1). Part (2) follows trivially. To get the relation with Majid’s formalism [35] we now write a · v := πV (a)v, ¯ ¯ a ∈ G, v ∈ V and define βV : V → G ⊗ V , v 7→ v (1) ⊗ v (2) := DV (1G ⊗ v). With this notation we get Corollary A4. The conditions (i)–(iii) of Corollary A3 are equivalent to the ¯ i ⊗ Y¯ i ⊗ Z¯ i = φ−1 ): following three conditions for βV (as before denoting X (i0 ) ( ⊗ idV ) ◦ βV = idV ¯ ¯ ¯ ¯ (ii0 ) (a(2) · v)(1) a(1) ⊗ (a(2) · v)(2) = a(2) v (1) ⊗ a(1) · v (2) , ∀ v ∈ hV (¯ 1) 0 i (¯ 1) i (¯ 2) (¯ 1) ¯ i i (¯ 2) (¯ 2) −1 321 ¯ ¯ ¯ = (φ ) · (Z¯ i · v)(2) Y¯ i ⊗ (iii ) Z v ⊗ (Y · v ) X ⊗ (Y · v ) i ¯ (1) ¯ i ⊗ (Z¯ i · v)(¯2) , ∀ v ∈ V . (Z¯ i · v)(1) X Proof. The equivalences (i) ⇔ (i’) and (ii) ⇔ (ii’) are obvious. The equivalence 312 from the left and with φ−1 (iii) ⇔ (iii’) follows by multiplying (iii) with (φ−1 V ) V from the right and permuting the first two tensor factors. The conditions stated in the above Corollary agree with those formulated in [35, Proposition 2.2], by taking G cop ≡ (G, ∆op , (φ−1 )321 ) instead of (G, ∆, φ) as the underlying quasi-bialgebra. Thus we have identified the category Rep D(G) with what is called the double category of modules over G in [35]. This proves that our quantum double is a concrete realization of the abstract definition given by S. Majid. References [1] E. Abe, Hopf Algebras, University Press, Cambridge, 1980. [2] A. Alekseev, L. Faddeev and M. Semenov-Tian-Shanski, “Hidden quantum groups inside Kac–Moody algebras”, Commun. Math. Phys. 149 (1992) 335. [3] A. Yu. Alekseev, L. D. Faddeev, M. A. Semenov-Tian-Shansky and A. Yu. Volkov, “The unraveling of the quantum group structure in WZNW theory”, preprint CERNTH-5981/91, 1991.
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[32] S. Majid, “Quasitriangular Hopf algebras and Yang Baxter equations”, Int. J. Mod. Phys. A5 (1990) 1. [33] , “Tannaka–Krein theorem for quasi-Hopf algebras and other results”, Contemp. Math. Phys. 150 (1992) 219–232. , Foundations of Quantum Group Theory, Cambridge Univ. Press, 1995. [34] [35] , “Quantum double for quasi-Hopf algebras”, Lett. Math. Phys. 45 (1998) 1. [36] G. Moore and N. Reshetikhin, “A comment on quantum symmetry in conformal field theory”, Nucl. Phys. B328 (1989) 557. [37] M. M¨ uger, “Disorder operators, quantum doubles and Haag duality in 1 + 1 dimensions”, preprint DESY (1996) 96–237. [38] , “Quantum double actions on operator algebras and orbifold quantum field theories”, Commun. Math. Phys. 191 (1998) 31. [39] F. Nill, “Fusion structures from quantum groups II: Why truncation is necessary”, Lett. Math. Phys. 29 (1993) 83–90. [40] , “On the structure of monodromy algebras and Drinfel’d doubles”, Rev. Math. Phys. 9 (1997) 71–397. [41] F. Nill and K. Szlach´ anyi, “Quantum chains of Hopf algebras and order-disorder filed with quantum double symmetry”, hep-th/9507174, 1995. [42] , “Quantum chains of Hopf algebras with quantum double cosymmetry”, Commun. Math. Phys. 187 (1997) 159. [43] P. Pasquier and H. Saleur, “Common structures between finite systems and conformal field theories through quantum groups”, Nucl. Phys. B330 (1990) 523. [44] V. Pasquier, “Continuum limit of lattice models built on quantum groups”, Nucl. Phys. B295 (1988) 491. [45] V. Pasquier and H. Saleur, “XXZ chain and quantum su(2)”, Fields, Strings, and Critical Phenomena, Les Houches, 1989. [46] N. Reshetikhin and F. Smirnov, “Hidden quantum symmetry and integrable perturbations of conformal field theory”, Commun. Math. Phys. 131 (1990) 157. [47] V. Schomerus, “Constructions of field algebras with quantum symmetry from local observables”, Commun. Math. Phys. 169 (1995) 193. [48] M. E. Sweedler, Hopf Algebras, Benjamin Press, 1969. [49] K. Szlach´ anyi, “Chiral decomposition as a source of quantum symmetry in the Ising model”, Rev. Math. Phys. 6 (1994) 649–671. [50] K. Szlach´ anyi and P. Vescerny´es, “Quantum symmetry and braid group statistics in G-spin models”, Commun. Math. Phys. 156 (1993) 127. [51] P. Vecserny´es, “On the quantum symmetry of the chiral Ising model”, Nucl. Phys. B415 (1994) 557.
DEFORMED HARMONIC OSCILLATOR ALGEBRAS DEFINED BY THEIR BARGMANN REPRESENTATIONS ` MICHELE IRAC-ASTAUD and GUY RIDEAU Laboratoire de Physique Th´ eorique de la mati` ere condens´ ee Universit´ e Paris VII 2 place Jussieu F-75251 Paris Cedex 05, France E-mail : [email protected] Received 9 December 1997 Deformed Harmonic Oscillator Algebras are generated by four operators, two mutually adjoint a and a† , and two self-adjoint N and the unity 1 such as: [a, N ] = a, [a† , N ] = −a† , a† a = ψ(N )
and
aa† = ψ(N + 1) .
The Bargmann Hilbert space is defined as a space of functions, holomorphic in a ring of the complex plane, equipped with a scalar product involving a true integral. In a Bargmann representation, the operators of a Deformed Harmonic Oscillator Algebra act on a Bargmann Hilbert space and the creation (or the annihilation operator) is the multiplication by z. We discuss the conditions of existence of Deformed Harmonic Oscillator Algebras assumed to admit a given Bargmann representation.
1. Introduction In previous papers [1–3], we introduced what we have called Deformed Harmonic Oscillator Algebras (DHOA). Definition 1.1. A Deformed Harmonic Oscillator Algebra is a free algebra generated by four operators: — The annihilation operator a, the creation operator a† that are mutually adjoint, — The self-adjoint energy operator N and the unity 1 satisfying the following commutation relations: [a, N ] = a ,
[a† , N ] = −a† ,
a† a = ψ(N ) ,
aa† = ψ(N + 1) ,
(1)
where ψ is a real analytical function. When ψ(N ) = N + λ, λ being in the field, we recover the commutation relations of the usual harmonic oscillator algebra. Generalizing the pioneer work of Bargmann [12] for the usual harmonic oscillator, we have studied in [1–3] the Bargmann representations of the DHOA, defined by (1). 631 Reviews in Mathematical Physics, Vol. 11, No. 5 (1999) 631–651 c World Scientific Publishing Company
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Definition 1.2. A Bargmann Hilbert space is a space S of functions, holomorphic on a ring D of the complex plane, the scalar product of which is written with a true integral on the form: Z (g, f ) =
F (zz)f (z)g(z) dz d z .
(2)
A Bargmann representation of a Deformed Harmonic Oscillator Algebra is a representation on a Bargmann Hilbert space such as the annihilation or the creation operator admits eigenvectors generating S. Let us stress that in this definition, we discard the occurrence of q-integration in the scalar product, contrarily to many authors, see in particular [8, 9]. In this paper, starting from the reciprocal point of view of that developed in [1– 3], we construct the DHOA defined by a given Bargmann representation. That is, a Bargmann Hilbert space S being given, we look for DHOA that can be represented by operators acting on S. In Sec. 2, we recall briefly the irreducible representations of the DHOA on the basis of the eigenvectors of N and we discuss the existence of coherent states, defined as the eigenstates of the operators a (or a† ). In Sec. 3, we summarize the study of Bargmann representations for the DHOA, done in [1–3] and set up the relations between the weight function F defining the scalar product and the function ψ characterizing the DHOA. Section 4 is devoted to the true subject of this paper. We study the conditions of existence of a function ψ and consequently of a DHOA defined by (1) when we impose that the DHOA has a representation on a given Bargmann Hilbert space. We prove that if the weight function F fulfills sufficient and necessary conditions, the construction can be performed. Furthermore, we obtain a necessary condition on the function ψ in order that the DHOA admits a Bargmann representation when the domain of the coherent states is a true ring. In Sec. 5, we treat several examples illustrating our construction, showing in particular that the sufficient conditions obtained in Sec. 4, are not necessary. 2. Representations 2.1. Eigenvectors of N Let |0i be the eigenvector of N with eigenvalue µ that is assumed to be zero to simplify the notations and is restored when necessary for discussions. We built the normalized vectors |ni ( |ni =
λn a†n |0i ,
n ∈ N+
λn a−n |0i ,
n ∈ N−
(3)
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with
λ−2 n
1, n Y ψ(i) , = ψ(n)! =
i=1 n+1 Y ψ(i) ,
...
633
n=0 n ∈ N + − {0}
(4)
n ∈ N−
i=0
N + and N − are the set of integers ≥ 0 and < 0. The vectors |ni are the eigenvectors of N with eigenvalue n and span the Hilbert space H. The condition that hn|aa† |ni is strictly positive, restricts the spectrum of N . The elements of SpN are the integers in any interval [ν, ν 0 [ in which the function ψ is finite and strictly positive. Eventually one of the edges or both can be infinity. When one edge is finite, it is an integer when µ = 0, and a zero of ψ. We thus get different types of representations [6, 7, 10, 11] accordingly as ψ has no zero, one zero or more, the distance between two consecutive zeros being an integer. The representations are defined by: a† |ni = (ψ(n + 1))1/2 |n + 1i (5) a|ni = (ψ(n))1/2 |n − 1i , n ∈ SpN N |ni = (n)|ni When µ is kept different from zero, all the relations of this section remain valid provided we change ψ(ρ) to ψ(µ + ρ). Proposition 2.1. By construction, the eigenvalue µ of the starting state |0i always belongs to the spectrum of N. The first step to build a Bargmann representation requires the study of coherent vectors that constitute the basis vectors of the representation. 2.2. Coherent states We call the eigenvectors of the operator a or a† coherent states. P The state |zi = p cp |pi is an eigenvector of the annihilation operator a if the coefficients cp verify the recursive relation: zcp = ψ(p + 1)1/2 cp+1 .
(6)
When the spectrum of N is finite, we have proved that a and a† have no eigenvectors, hence no Bargmann representation exists. When the spectrum of N is not upper bounded, let us denote this spectrum SpN = λ + N + , with the convention λ + N + = Z when λ = −∞. It results from Proposition 2.1, that λ ∈ N − + {0}. The eigenvectors |zi of a take the form: |zi =
λ X n=−1
z n (ψ(n)!)1/2 |ni +
∞ X n=0
z n (ψ(n)!)−1/2 |ni .
(7)
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M. IRAC-ASTAUD and G. RIDEAU
The domain D of existence of the coherent states depends on the function ψ. Indeed, |zi belongs to the Hilbert space spanned by the basis |ni only if the series in the right-hand side of (7) are convergent in norm, see the detailed discussion in [1–3]. An analogous reasoning holds when SpN is not lower bounded, for the eigenstates of a† , it results from Proposition 2.1 that, in this case, λ ∈ N + − {0}. We have proved that the eigenstates of a and a† never coexist. Let us summarize the results: Proposition 2.2. The eigenvectors of the annihilation operator of a Deformed Harmonic Oscillator Algebra exist if the function ψ occurring in the relations (1) belongs to two classes : • ψ is a strictly positive function without singularity on the whole real axis, SpN = Z, and limp→−∞ ψ(p)1/2 < limp→∞ ψ(p)1/2 , then the domain of existence of the coherent states is D = z; lim ψ(p)1/2 < |z| < lim ψ(p)1/2 . p→−∞
p→∞
• If λ ∈ N − + {0} such that ψ(λ) = 0 and ψ is strictly positive without singularity when x > λ, SpN = λ + N + then: D = z; |z| < lim ψ(p)1/2 . p→∞
Proposition 2.3. The eigenvectors of the creation operator of a Deformed Harmonic Oscillator Algebra exist if the function ψ occurring in the relations (1) belongs to two classes: • ψ is a strictly positive function without singularity on the whole real axis, SpN = Z, and lim ψ(p)1/2 > lim ψ(p)1/2 , p→−∞
p→∞
then the domain of existence of the coherent states is 1/2 1/2 . < |z| < lim ψ(p) D = z; lim ψ(p) p→∞
p→−∞
• If λ ∈ N + − {0} such that ψ(λ) = 0 and ψ is strictly positive without singularity when x < λ, then SpN = λ + N − and D = z; |z| < lim ψ(p)1/2 . p→−∞
When the eigenvalue of |0i is µ 6= 0, from Proposition 2.1, the point µ, instead of the origin, belongs to SpN and the statements of the propositions must be changed accordingly. Proposition 2.4. The nature of the operators (to be creation or annihilation operators) is inverted in the change: a = a0† ,
N = −N 0 − 1 ,
ψ(ρ) = ψ 0 (−ρ) .
(8)
DEFORMED HARMONIC OSCILLATOR ALGEBRAS
...
635
From this change, we deduce that Proposition 2.3 can be obtained from Proposition 2.2. In the following section, we set up the Bargmann representation when the eigenvectors of the annihilation operator exist and using Proposition 2.4 we obtain the corresponding results when the eigenvectors of the creation operator exist. 3. Bargmann Representation Let |f i be one state of H: |f i =
X
fn |ni ,
n∈SpN
X
|fn |2 < ∞
(9)
n∈SpN
with SpN = λ + N + and λ ∈ N − + {0}. Following the construction [12], in the Bargmann representation any state |f i of H is represented by the function of a complex variable z, f (z) = hz|f i: f (z) =
X n≥0
z n fn ψ(n)!−1/2 +
λ X n<0
z n fn (ψ(n)!)1/2 ,
X
|fn |2 < ∞
(10)
n≥λ
where the variable z belongs to the domain D of definition of the eigenvectors of a. • Let us summarize the results when the eigenvectors of a exist: Proposition 3.1. Let the function ψ characterizing the Deformed Harmonic Oscillator Algebra (1) belong to the first class described in Proposition 2.2. The space S of the Bargmann representation is constituted with holomorphic functions in D = {z; limp→−∞ ψ(p)1/2 < |z| < limp→∞ ψ(p)1/2 }, the Laurent expansions of which read (10) with λ = −∞. In particular, when limp→−∞ ψ(p)1/2 = 0, D is a disk excluding the origin that is a essential singularity point. Proposition 3.2. Let the function ψ characterizing the Deformed Harmonic Oscillator Algebra (1) belong to the second class described in Proposition 2.2. The space S of the Bargmann representation is the subspace of the space of functions holomorphic in D = {z; 0 < |z| < limp→∞ ψ(p)1/2 }, the Laurent expansions of which read (10). The functions of S are of the form z λ g(z) where λ is the lowest bound of the spectrum of N and g(z) is holomorphic in D + {0}. The origin is a pole of multiplicity lower or equal to −λ. In particular, when λ = 0, the functions of S are holomorphic at the origin. In particular, to the basis vectors |ni, n ∈ SpN correspond the monomials: ( n z (ψ(n)!)−1/2 , n ≥ 0 (11) hz|ni = n<0 z n (ψ(n)!)1/2 , • Let us summarize the results when the eigenvectors of a† exist:
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Proposition 3.3. Let the function ψ characterizing the Deformed Harmonic Oscillator Algebra (1) belong to the first class described in Proposition 2.3. The space S of the Bargmann representation is constituted with holomorphic functions in D = {z; limp→+∞ ψ(p)1/2 < |z| < limp→−∞ ψ(p)1/2 }. The functions of S can be expanded in Laurent series of the form: f (z) =
λ−1 X n≥0
z −n fn ψ(n)!1/2 +
X
z −n fn (ψ(n)!)−1/2 ,
n<0
X
|fn |2 < ∞
(12)
n≤λ−1
with λ = +∞. In particular, when limp→∞ ψ(p)1/2 = 0, D is a disk excluding the origin that is an essential singularity point. Proposition 3.4. Let the function ψ characterizing the Deformed Harmonic Oscillator Algebra (1) belong to the second class described in Proposition 2.3. The space S of the Bargmann representation is the subspace of the space of functions holomorphic in D = {z; 0 < |z| < limp→∞ ψ(p)1/2 } that read (12). The functions of S are of the form z −λ+1 g(z) where λ − 1 is the upper bound of the spectrum of N and g(z) is holomorphic in D + {0}. The origin is a pole of multiplicity lower or equal to λ − 1. In particular, when λ = 1, the functions of S are holomorphic at the origin. In particular, to the basis vectors |ni, n ∈ SpN correspond the monomials: ( −n n≥0 z (ψ(n)!)1/2 , . (13) hz|ni = −n −1/2 , n<0 z (ψ(n)!) The function of z, G(ζz) = hz|ζi corresponds to the coherent state |ζi. The holomorphic functions belonging to S have analytical properties strongly depending on ψ. Their growth on the edge of D is controlled by the growth of |G(zz)|1/2 . A Bargmann representation exists if we can obtain a positive real function F (x) such as Z F (zz)|zihz| dz d z = 1 , (14) where the integration is extended to the whole complex plane and where F (|z|2 ) contains the characteristic function of the domain D of existence of the coherent states. The existence of (14) ensures that the scalar product of the representation takes the form (2). We easily prove that: Proposition 3.5. In a Bargmann representation: • either a† is the multiplication by z, a the operator z −1 ψ(zd/dz) and N the operator zd/dz; • or a is the multiplication by z, a† the operator z −1 ψ(−zd/dz) and N the operator −zd/dz − 1.
DEFORMED HARMONIC OSCILLATOR ALGEBRAS
...
Let us introduce the Mellin transform Fˆ (ρ) of F (x): Z ∞ ˆ F (x)xρ−1 dx . F (ρ) =
637
(15)
0
From (11) and (14), we deduce that Fˆ (ρ) exists on all the integers belonging to the spectrum of N and verify the following condition: ( ψ(n)! , n≥0 . (16) Fˆ (n + 1) = −1 (ψ(n)!) , n < 0 , n ∈ SpN Let us remark that Fˆ (ρ) ≤ Fˆ (n) + Fˆ (n + 1) ,
n ≤ Reρ < n + 1
(17)
because F (x) is a positive function. Therefore, the Mellin transform of F exists for any ρ such as Reρ ∈ [λ + 1, +∞[. Formula (16) is equivalent to Fˆ (n + 1) = ψ(n)Fˆ (n) ,
with Fˆ (1) = 1
(18)
which ensures that the operators a† = z, a = z −1 ψ(zd/dz) be adjoint on the basis |ni. In [2] and [3], we have discussed the interpolation of this equation, the simplest one reads: Proposition 3.6. The function ψ characterizing the DHOA defined in (1) and the Mellin transform of the weight function F are related by the following equation: Fˆ (ρ + 1) = ψ(ρ)Fˆ (ρ) ,
with
Fˆ (1) = 1 ,
(19)
when the coherent states are the eigenvectors of the annihilation operator a. We obtain an analogous proposition when the basis vectors of the Bargmann Hilbert space are the eigenvectors of a† . In this case, the annihilation operator is the multiplication by z. Proposition 3.7. The function ψ characterizing the DHOA defined in (1) and the Mellin transform of the weight function F are related by the following equation: Fˆ (−ρ + 1) = ψ(ρ + 1)Fˆ (−ρ) ,
with
Fˆ (1) = 1 ,
(20)
when the coherent states are the eigenvectors of the creation operator a† . When the eigenvalue µ is different from zero, the only change in Eqs. (19) and (20) is ψ(ρ) → ψ(µ + ρ). Proposition 3.6 corresponds to functions ψ that are finite and positive on a nonupper bounded interval ]λ, +∞[ with λ ∈ N − + {0}, as proved in Proposition 2.2.
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The Mellin transform of F given by (19) is finite and positive on the same interval. When λ is finite, Fˆ (λ) is infinite. When λ is infinite, Fˆ (ρ + 1) Fˆ (ρ + 1) < lim . ρ→−∞ ρ→+∞ Fˆ (ρ) Fˆ (ρ) lim
Proposition 3.7 corresponds to functions ψ that are finite and positive on a nonlower bounded interval ]−∞, λ0 +1[ with λ0 ∈ N + as proved in Proposition 2.3. The Mellin transform of F given by (20) is finite and positive on the interval ]− λ0 , +∞[. When λ0 is finite, Fˆ (−λ0 ) is infinite. When λ0 is infinite, Fˆ (−ρ + 1) Fˆ (−ρ + 1) > lim . ρ→−∞ ρ→+∞ Fˆ (−ρ) Fˆ (−ρ) lim
These two cases lead to the same conclusions: Proposition 3.8. The Mellin transform of the weight function of the Bargmann representation is finite and positive on a non-upper bounded interval of the real axis. • When the lower bound of the interval is finite, it is a negative integer or zero and Fˆ is infinite at this point. • When the interval is not lower bounded, Fˆ fulfills the following condition: Fˆ (ρ + 1) Fˆ (ρ + 1) < lim . ρ→−∞ ρ→+∞ Fˆ (ρ) Fˆ (ρ) lim
(21)
Let us remark that, while the spectrum of the energy operator N is a non-lower or non-upper bounded interval in Z, the definition domain of the Mellin transform of the weight function must be a non-upper bounded interval of the real axis containing the origin in order that the construction be possible. 4. Deformed Harmonic Oscillator Constructed from a Weight Function The purpose of the following two sections is to construct a DHOA assumed to admit a given Bargmann representation. More precisely, we start with a Bargmann Hilbert space S of functions, holomorphic on a ring D of the complex plane. The scalar product is written on the form (2) in terms of a given function F positive on the interval ]α, β[. The construction can be summarize as follows: We look for a function ψ, we associate to this function the DHAO defined by (1). Then, remains a consistency condition: we must prove that this algebra admits a Bargmann representation on S, as defined in Sec. 3. Once the DHOA is constructed, the spectrum of N is obtained and, eventually, the representation space must be restricted according to Propositions 3.1–3.4. First, in order to define the function ψ by applying (19) or (20), the Mellin transform Fˆ of the given weight function F must exist. Secondly, in order to define the basis vectors of the representation as the eigenvectors of the annihilation
DEFORMED HARMONIC OSCILLATOR ALGEBRAS
...
639
operator a (or a† ), the function Fˆ must be define on a non-upper bounded interval of the real axis, due to the Proposition 3.8. Let us denote Fˆ the Mellin transform of the weight function F : Z
β
Fˆ (ρ) =
F (x)xρ−1 dx .
(22)
α
From Proposition 3.8, we obtain the following necessary conditions that must satisfy Fˆ in order that the construction of the DHOA be possible: Proposition 4.1. When a Bargmann Hilbert S space is given, a necessary condition in order that a DHOA admits a representation on S is that the Mellin transform of the weight function defining the scalar product (2) exists on a non-upper bounded interval of the real axis. • When the interval is finite, it must be equal to ]ν, +∞[, where ν is a negative integer or zero. The functions of the representation space read: f (z) =
X n≥ν
z n fn Fˆ (n + 1)
1 2
,
X
|fn |2 < ∞ .
(23)
n≥ν
• When the interval is not lower bounded, Fˆ must fulfill the following condition: Fˆ (ρ + 1) Fˆ (ρ + 1) < lim . ρ→−∞ ρ→+∞ Fˆ (ρ) Fˆ (ρ) lim
(24)
The functions of the representation space read (23) with ν = −∞. When the previous conditions are satisfied, we are faced with two possibilities accordingly as we choose that the basis vectors of the Bargmann representation are the eigenstates of the annihilation of the creation operator: Proposition 4.2. When the Mellin transform of the weight function satisfies the necessary conditions of Proposition 4.1, the function: ψ(ρ) =
Fˆ (ρ + 1) Fˆ (ρ)
(25)
put in the relations (1) defines a DHOA that can be represented on the Bargmann Hilbert space spanned with the eigenvectors of the annihilation operator, equipped with the scalar product (2). In this representation, the creation is the multiplication by z. Proposition 4.3. When the Mellin transform of the weight function satisfies the necessary conditions of Proposition 4.1, the function: ψ(ρ) =
Fˆ (−ρ + 2) Fˆ (−ρ + 1)
(26)
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M. IRAC-ASTAUD and G. RIDEAU
put in the relations (1) defines a DHOA that can be represented on the Bargmann Hilbert space spanned with the eigenvectors of the creation operator, equipped with the scalar product (2). In this representation, the annihilation is the multiplication by z. When the eigenvalue µ of the starting state is different from zero, the right-hand side of (25) or (26) is unchanged. The left-hand side is replaced by ψ(µ + ρ), the parameter µ of the representation is involved in the data of the weight function. All the construction is done in terms of the Mellin transform of the weight function. It is worthwhile to note that the reproducing kernel G(x) is always expressed in terms of Fˆ , by the same simple formula: G(x) = Fˆ (1)
X
xn
n≥ν−1
Fˆ (n + 1)
.
(27)
The main point is to prove that the DHOA so constructed admits a representation on the Bargmann Hilbert space S, that is we have to establish the consistency of the construction. 4.1. D = Dαβ ≡ {z; 0 < α < | z |2 < β < +∞} From (22), we obtain the following inequalities: ( Fˆ (ρ) < α−µ Fˆ (ρ + µ) , ∀µ > 0 Fˆ (ρ) < β µ Fˆ (ρ − µ)
(28)
that ensure that if Fˆ (ρ) diverges for one value of ρ, it always diverges and that the Mellin transform of F does not exist on the real axis. In the following we assume that Fˆ exists. Then the Mellin transform (22) exists for any ρ ∈ R and is a strictly positive function. The functions ψ defined by (25) and (26) are also strictly positive functions. Using these functions ψ in (1), we obtain the corresponding DHOAs. Since the function ψ is strictly positive on R, the spectrum of N is Z. The construction is achieved if we prove that the coherent states of the soconstructed DHOAs exist when α < |z|2 < β. We then have to determine the behaviors of the function ψ at ±∞. Let us write (22): Fˆ (ρ) = ρ−1 β ρ
Z
1
1
ρ
F (βx ρ ) dx .
(29)
( ) α β
We easily see that Fˆ (ρ) ' F (β)ρ−1 β ρ when ρ goes to +∞, if F (β) is finite and different from zero. Therefore, the function ψ(ρ) given in (25) (resp. (26)) goes to β when ρ goes to +∞ (resp. −∞). When F (k) (β) = 0, k = 0, . . . , b and when F (b+1) (β) is finite and not equal to zero, Fˆ (ρ) ' (−1)b+1 F (b+1) (β)ρ−(b+2) β b+1+ρ when ρ goes to +∞. Then, the function ψ(ρ) given in (25) (resp. (26)) goes to β when ρ goes to +∞ (resp. −∞).
DEFORMED HARMONIC OSCILLATOR ALGEBRAS
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641
The same reasoning holds for the behavior at −∞, indeed we now write (22): Fˆ (ρ) = ρ−1 αρ
Z ( αβ )ρ
1
F (αx ρ ) dx .
(30)
1
We get now that the limit at −∞: When F (α) is finite and different from zero, Fˆ (ρ) ' F (α)ρ−1 αρ . When F (k) (α) = 0, k = 0, . . . , a and when F (a+1) (α) is finite and different from zero, Fˆ (ρ) ' (−1)a+1 F (a+1) (α)ρ−(a+2) αa+1+ρ . On these conditions, ψ(ρ) given in (25) (resp. (26)) goes to α when ρ goes to −∞ (resp. +∞). As α < β, the necessary condition given in Proposition 4.1 is fulfilled. Using the results of Sec. 3, we prove that the coherent states, eigenstates of a or a† according to the choice of the characteristic function, are defined when α < |z|2 < β. This completes the proof of the consistency of the reconstruction. We obtain the following sufficient conditions in order that the construction be consistent: Proposition 4.4. Let F (x) be a function defined on the interval ]α, β[ such that : • The Mellin transform of F exists; • F (α) is finite and different from zero or F (a+1) (α) is finite and different from zero when F (k) (α) = 0, for k = 0, . . . , a; • F (β) is finite and different from zero or F (b+1) (β) is finite and different from zero when F (l) (β) = 0, for l = 0, . . . , b. One can construct two Deformed Harmonic Oscillator Algebras that admits a representation on a Hilbert space constituted by functions, holomorphic in the 1 1 ring Dαβ = {z; α 2 < |z| < β 2 } and equipped with the scalar product : Z (g, f ) = F (zz)f (z)g(z)θ(zz − α)θ(β − zz) dz d z . (31) • One of this DHOA corresponds to the characteristic function (25) and in the Bargmann representation, its creation operator is the multiplication by z. • The second DHOA corresponds to the characteristic function (26) and in the Bargmann representation, its annihilation operator is the multiplication by z. If the edge conditions are not fulfilled the reconstruction may exist but we have to establish in each specific case that the limits at infinities are the expected ones. Finally, let us remark that Z
Z
β
α
Z
β
F (x)xρ−1 dx ≤
α
β
F (x)xρ x dx ≤ β α
F (x)xρ−1 dx .
(32)
α
Obviously, these inequalities hold when α = 0 or β infinite. We obtain α ≤ ψ(ρ) ≤ β .
(33)
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This constitutes a necessary condition that the function ψ must fulfill in order that the DHOA have a Bargmann representation: Proposition 4.5. Let the function ψ characterizing the deformed harmonic oscillator algebra (1) be strictly positive and such that its limits at infinities are finite and verify ψ(−∞) < ψ(+∞) (resp. ψ(−∞) > ψ(+∞)). The coherent states, eigenstates of the annihilation (resp. creation) operator, exist in the ring of the complex plane D = {z; ψ(−∞)1/2 < |z| < ψ(+∞)1/2 } (resp. D = {z; ψ(+∞)1/2 < |z| < ψ(−∞)1/2 }) .
(34)
The DHOA can be represented on a space of functions holomorphic in D only if ψ takes all its values between the two limiting values: ψ(−∞) < ψ(ρ) < ψ(+∞) (resp. ψ(+∞) < ψ(ρ) < ψ(−∞)) .
(35)
4.2. D = D0β ≡ { z ; | z |2 < β < +∞ } The second inequality of (28) still holds. Let us denote by ν the value such that the integration (22) is divergent when ρ ≤ ν and convergent when ρ > ν. When ν = −∞ the Mellin transform exists for any ρ, while when ν = +∞ it never exists. As Fˆ (ν) is infinite while Fˆ (ν + 1) is finite, ψ(ν) = 0 (resp. ψ(1 − ν) = 0) defined by (25) (resp. (26)) vanishes. The spectrum of N is ν + N + (resp. 2 − ν + N − ) when ν is finite and Z when ν = −∞. From Proposition 4.1, ν must belong to N − + {0}. The expressions of the coherent states contain one or two infinite summations according to whether or not ν is finite. • In the case where ν = −∞, SpN = Z, the limit at −∞ must be done in each specific case and we treat an example in the Subsec. (8.1). This limit is positive or zero since Fˆ (ρ) is positive, it must be zero in order that the domain of existence of the coherent states be consistent with the definition of the given Bargmann Hilbert space. Let us summarize this result: Proposition 4.6. Let F (x) be a function defined on the interval ]0, β[ such that : • The Mellin transform of F exists on the whole real axis. • F (β) is finite and different from zero or F (b+1) (β) is finite and different from zero when F (l) (β) = 0, for l = 0, . . . , b, ˆ • and limρ→−∞ FF(ρ+1) ˆ (ρ) = 0. One can construct two deformed harmonic oscillator algebras, corresponding to characteristic functions given in (25) and (26), that admit a representation
DEFORMED HARMONIC OSCILLATOR ALGEBRAS
...
643
on a Hilbert space constituted by functions, holomorphic in the ring D = 1 {z; 0 < |z| < β 2 } and equipped with the scalar product : Z (36) (g, f ) = F (zz)f (z)g(z)θ(β − zz) dz d z . • In the case where ν is finite, it remains only one infinite summation in (7), this summation is convergent if |z|2 < limρ→+∞ ψ(ρ). As β is finite, this limit is obtained by the same reasoning as in the previous case. In this case, as the spectrum of N is lower or upper bounded according as we choose ψ defined by (25) or by (26), the Laurent expansions of the functions belonging to the representation space contain terms in z n with n ≥ ν as results from Propositions 3.2 and 3.4. We then havex: Proposition 4.7. Let F (x) be a function defined on the interval ]0, β < +∞[ such that : • The Mellin transform of F only exists when ρ > ν, ν finite and belonging to N − + {0}. • F (β) is finite and different from zero or F (b+1) (β) is finite and different from zero when F (l) (β) = 0, for l = 0, . . . , b. One can construct two deformed harmonic oscillator algebras, corresponding to characteristic functions given in (25) and (26), that admit a representation on a Hilbert space equipped with a scalar product (36) and constituted with 1 functions holomorphic in the ring D = {z; 0 < |z| < β 2 } restricted by the condition that the origin be a pole of multiplicity lower or equal to ν. In particular when ν = 0, the functions of the representation space are holomorphic in a disk including the origin. When β is infinite, the consistency of the demonstration must be done in each case, one example is given in Subsec. 8.2. 4.3. D = Dαβ ≡ { z ; 0 < α < | z |2 } In this case, the integration (22) can diverge for ν < ρ and converge for ρ ≤ ν. If ν is finite, ψ(ν) defined by Eq. (25) and ψ(1 − ν) defined by Eq. (26) is infinite. This corresponds to a case where ψ has a singularity at a finite distance, not considered in this paper. If ν = +∞, (22) always converges and the spectrum of N is Z, the consistency must be verified in each case. In this section, we have obtained consistency sufficient conditions to construct of a DHOA from its Bargmann representation when D is a true ring or a true disk in the complex plane, an example will be given in the Subsec 5.3. 5. Examples of Construction In this section, we give four examples of construction of DHOA when its Bargmann representation is given, namely F and the domain D of existence of the coherent states are given. In the first two examples D is the whole complex
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plane, then the sufficient conditions of consistency of the previous section do not apply. In the last two it is a ring, one of them illustrates the results of the Propositions 4.5 and 4.6, while in the last one the sufficient conditions of the previous section are not fulfilled. In the following, the proofs are given for DHOA resulting from characteristic functions given in (25) and for which the coherent states are the eigenvectors of the annihilation operator. Obviously, the same can be developed when ψ is given by (26), leading to DHOA for which the coherent states are the eigenvectors of the creation operator, we just state the results. 2n
5.1. D = C − {0 } and F (x) = exp (− σ (ln x)
)
We assume that σ is a positive real number and that n is a positive integer. As the domain of existence of the coherent states is the whole complex plane, the sufficient condition of the previous section are not fulfilled. This example is an illustration of a case where the existence of the DHOA is established though the characteristic function ψ is not obtained on an explicit form. The Mellin transform of F (x) (15) reads Z +∞ 2n ˆ e−σt +ρt dt . (37) F (ρ) = −∞
As σ is positive and n is a positive integer, we see that Fˆ (ρ) exists and is strictly positive for all ρ. The function ψ given by (25) is then a strictly positive function and the spectrum of N is Z. The reconstruction of the deformed algebra will be achieved if we prove that the coherent states resulting of the function ψ such obtained are defined in the whole complex plane as assumed. We thus have to study the behavior of ψ at infinities. Let us assume that ρ is positive. We write (37): Z
+∞
Fˆ (ρ) =
e−σ(t+v)
2n
+ρ(t+v)
dt ,
(38)
−∞
where we choose
v=
ρ σ(2n − 1)
1 2n−1
,
(39)
in order that the term under the exponential does not contain linear term in t and (38) reads: 1
ρ
2n
2n−1 ( 2n−1 2n−1 ) Fˆ (ρ) = e−2(n−1)σ
Z
+∞
dte
−σ
P p≥2
Cp2n tp v 2n−p
.
(40)
−∞
After a change of variable t = uv 1−n , (40) can be written: 1
ρ
2n
2n−1 ( 2n−1 2n−1 ) Fˆ (ρ) = e−2(n−1)σ
Z
+∞
duv 1−n e−σn(2n−1)u e 2
−σ
P p≥3
Cp2n up v n(2−p)
.
−∞
(41)
DEFORMED HARMONIC OSCILLATOR ALGEBRAS
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645
When ρ → +∞, the integral goes to zero like v 1−n and using (9), we have 4n(n−1)
ψ(ρ) ' e (2n−1)2
ρ σ(2n−1)
2n−1 1 .
(42)
We therefore get lim ψ(ρ) = +∞ .
(43)
Fˆ (−ρ) = −Fˆ (ρ)
(44)
ρ→+∞
From (37), we deduce that so that ψ(−ρ) =
1 Fˆ (−ρ + 1) Fˆ (ρ − 1) . = = ψ(ρ − 1) Fˆ (−ρ) Fˆ (ρ)
(45)
Thus when ρ → −∞, we get that lim ψ(ρ) = 0 .
ρ→−∞
(46)
The domain of existence of the coherent states is the complex plane and the consistency of the reconstruction is established. As SpN = Z, the representation space is the space of the functions holomorphic in the complex plane without the origin that is an essential singularity point. Similarly, to Eq. (26) corresponds another DHOA. Proposition 5.1. One can construct two Deformed Harmonic Oscillator Algebras that can be represented on the Bargmann Hilbert space of functions holomorphic in the complex plane without the origin equipped with the scalar product: Z (g, f ) = exp(−σ(log zz)2n )g(z)f (z) dz d z . • The characteristic function ψ involved in (1) is Z
+∞
exp(−σt2n + (ρ + 1)t) dt ψ(ρ) =
−∞
Z
+∞
exp(−σt2n + ρt) dt −∞
and the spectrum of N is Z and the coherent states are the eigenvectors of a. • The characteristic function ψ involved in (1) is Z ψ(ρ) =
+∞
−∞ Z +∞ −∞
exp(−σt2n + (2 − ρ)t) dt exp(−σt2n + (1 − ρ)t) dt
and the spectrum of N is Z and the coherent states are the eigenvectors of a† .
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In the next subsection, we give another example where the results of the previous section do not apply and in which the explicit calculation of the characteristic function ψ can be performed. The main interest of the next example is to be a generalization of the usual harmonic oscillator algebra. k
5.2. D = C − {0} and F (x) = exp(− x m ) k is put on an irreducible form and that it is positive. When We assume that m = 1, F is the weight function of the Bargmann representation of the usual harmonic oscillator [12]. The Mellin transform of F (15) reads Z +∞ k e−x m xρ−1 dx . (47) Fˆ (ρ) = k m
0 k m
After a change of variable u = x , it reads Z m m m +∞ −u ρ m −1 ˆ e u k du = Γ ρ F (ρ) = k 0 k k and the function ψ characterizing the DHOA and resulting from (25) is m Γ (ρ + 1) k ψ(ρ) = . m ρ Γ k
(48)
(49)
From this explicit expression of ψ, we deduce that this function is strictly positive on the positive axis and vanishes at the origin. The spectrum of N is N + . Using the asymptotic behavior of Γ(z) for large values of |z|, we get m m k ρ . (50) ψ(ρ) ' k Thus limρ→+∞ ψ(ρ) is infinite and the coherent states, as assumed, are defined in the whole C. As SpN = N + , the functions of the representation space are holomorphic in C, including the origin. A similar construction can be performed with the characteristic function (26). Proposition 5.2. One can construct two Deformed Harmonic Oscillator Algebra that can be represented on the Bargmann Hilbert space of functions holomorphic in the whole complex plane equipped with the scalar product: Z k k > 0. (g, f ) = exp(−(zz) m )g(z)f (z) dz d z , m • The characteristic function ψ is Γ ψ(ρ) =
m
(ρ + 1) k m . (ρ) Γ k
The spectrum of N is N + and the coherent states are the eigenvectors of a.
DEFORMED HARMONIC OSCILLATOR ALGEBRAS
• The characteristic function ψ is
...
647
(2 − ρ) k . ψ(ρ) = m (1 − ρ) Γ k Γ
m
The spectrum of N is N − + {0} and the coherent states are the eigenvectors of a† . k = 1, the function ψ resulting of Eq. (49) is the characteristic function When m of the usual harmonic oscillator in a fixed representation, namely a† a = N . We end this subsection by comparing the Bargmann representation considered in this subsection with the Bargmann representation of the usual harmonic oscillator. In this subsection, the scalar product (2) is defined in the space S of holomorphic functions of one complex variable and reads Z k m (51) (g, f ) = dζ d ζe−ζζ f (ζ)g(ζ) , f, g ∈ S .
Denoting ζ = χeiτ , it can be written: Z Z 2πm dτ +∞ 2 −χ 2k −1 (g, f ) = m dχ e m f (χeiτ )g(χeiτ ) . 2π 0 0 The scalar product for the usual Bargmann representation reads Z (gB , fB ) = dz d ze−zz fB (z)gB (z) , fB , gB ∈ SB .
(52)
(53)
It takes the form: (gB , fB ) = k −1
Z 0
2πk
dθ 2π
Z
+∞
dρ2 e−ρ fB (ρeiθ )gB (ρeiθ ) . 2
(54)
0
k
Let us change z = ζ m , we see that 0 ≤ τ < 2πm and that (54) reads Z Z 2πm k k k dτ +∞ 2 −χ 2k k k (gB , fB ) = 2 dχ e m ζ m −1 fB (ζ m )ζ m −1 gB (ζ m ) . m 0 2π 0
(55)
The scalar products written in (55) and in (52) are the same but they are not defined on the same space of functions: P k k Indeed let us write fB (z) = l≤0 fl z l , the functions f (ζ) ≡ ζ m −1 fB (ζ m ) belong to S iff fl = 0 when l 6= nm − 1, n being a strictly positive integer. The functions f such obtained belong to S but do not cover the whole space for they read f (z) =
+∞ X
fnm−1 z kn−1 .
(56)
n=1
Proposition 5.3. Let us consider the two Bargmann Hilbert spaces on which are represented the usual harmonic oscillator algebra and the DHOA considered in
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M. IRAC-ASTAUD and G. RIDEAU
this subsection. When, by a change of variables, their scalar products are written on the same form (52), the functions belonging to the intersection of these two spaces are of the form (56). 5.3. D = Dαβ and F (x) = xσ We start with a Bargmann representation such as the coherent states are defined on a ring of the complex plane 0 ≤ α ≤ ρ ≤ β < +∞. This subsection illustrates the previous section with an example where we obtain an explicit expression for the characteristic function ψ. The Mellin transform of the weight function reads Z
β
Fˆ (ρ) =
xσ+ρ−1 dx .
(57)
α
First, we see that this integration is finite for any ρ and any σ when α 6= 0 and for ρ > −σ when α = 0. The resulting function ψ, defined by (25), takes the form: ψ(ρ) =
σ + ρ β σ+ρ+1 − ασ+ρ+1 . σ + ρ + 1 β σ+ρ − ασ+ρ
(58)
We now must look for the domain of existence of the coherent states in order to verify the consistency of this construction. • When α 6= 0, the function ψ is always positive and the spectrum of N is Z. It is easy to find that the function ψ(ρ) goes to α or β when ρ → −∞ or +∞. This implies that the coherent states are defined for α ≤ ρ2 ≤ β, as expected. As SpN = Z, no restrictions appear on the Laurent expansions of the holomorphic functions of the representation space. The same construction can be done starting with the characteristic function (26) Proposition 5.4. One can construct two Deformed Harmonic Oscillator Algebras that can be represented on the space of functions holomorphic in Dαβ = {z; 0 < α < |z|2 < β} equipped with the scalar product: Z (zz)σ g(z)f (z) dz d z , ∀ σ . (g, f ) = 0<α<|z|2 <β
The characteristic functions are ψ(ρ) or ψ(1 − ρ) expressed in (58), SpN is Z in both cases and the coherent states are the eigenvectors of the annihilation or of the creation operator. • When α = 0, the characteristic function resulting from (25) is ψ(ρ) =
σ+ρ β. σ+ρ+1
(59)
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From Proposition 3.8, we deduce that the construction is only possible when σ ∈ N + . Then SpN = −σ + N + . The coherent states are defined in D = {z; |z|2 < β} as expected and the origin is a pole of multiplicity lower than σ for the functions of the representation space. Let us summarize this result and that obtained starting with Eq. (26): Proposition 5.5. One can construct two Deformed Harmonic Oscillator Algebras that can be represented on the space of functions of the form z −σ f0 (z), where f0 (z) is holomorphic in the whole disk D0β = {z; |z|2 < β} equipped with the scalar product: Z (zz)σ f (z)g(z) dz d z
(g, f ) =
(60)
|z|2 <β
provided that σ be zero or a positive integer. Their characteristic functions are ψ(ρ) or ψ(1 − ρ) written in Eq. (59). The spectrum of N is −σ + N + or σ + N − + {0} and the coherent states are the eigenvectors of the annihilation or of the creation operator. η
5.4. D = D0β and F (x) = xσ (β − x)
This example does not fulfill the general conditions of Sec. 4 for the derivatives of the weight function on the edge β of D is zero or infinite when η is not a positive integer. The Mellin transform of F exists when ρ + σ > 0 and η + 1 > 0 and can be calculated in terms of the B function [13]: Z β ˆ (β − x)η xσ+ρ−1 dx F (ρ) = 0
= β η+σ+ρ B(ρ + σ, η + 1) = β η+σ+ρ
Γ(ρ + σ)Γ(η + 1) Γ(ρ + σ + η + 1)
(61)
The expression of Fˆ (ρ) put in Eq. (25) leads to ψ(ρ) =
ρ+σ β. ρ+σ+η+1
(62)
When η = 0, one recovers the function (59). The reasoning and the results are similar to those of Proposition 5.5: Proposition 5.6. One can construct two Deformed Harmonic Oscillator Algebras that can be represented on the space of functions of the form z −σ f0 (z), where f0 (z) is holomorphic in the whole disk D = {z; |z|2 < β} equipped with the scalar product: Z (g, f ) = |z|2 <β
(β − zz)η (zz)σ f (z)g(z) dz d z
(63)
provided that σ be zero or a positive integer. Their characteristic functions are ψ(ρ) or ψ(1 − ρ) written in Eq. (62). The spectrum of N is −σ + N + or σ + N − + {0}
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M. IRAC-ASTAUD and G. RIDEAU
and the coherent states are the eigenvectors of the annihilation or of the creation operator. −1
5.5. D = Dαβ and F (x) = exp(x − β)
In this example, the general conditions of Sec. 4 are not fulfilled too, for all the derivatives of F (x) vanish on the edge β of D. To simplify, we give the proofs for α = 0. The Mellin transform of F (x) are defined for ρ > 0. Integrating by parts, we get Z xρ 1 β 1 exp dx . (64) Fˆ (ρ) = ρ 0 x − β (β − x)2 Expanding (β − x)−2 , we obtain the relation: ρ = β −2
X n Fˆ (ρ + n + 1) . βn Fˆ (ρ)
(65)
n≥0 ˆ
Now, when ρ → ∞, FF(ρ+1) ˆ (ρ) necessarily goes to the limit β0 ≤ β according Proposition 4.5. This implies Fˆ (ρ + n + 1) lim (66) = β0n . ρ→∞ Fˆ (ρ) This result is used to calculate the right-hand side of Eq. (65) when ρ goes to ∞. As the limit of the left-hand side is infinite, we obtain a contradiction unless β0 6= β. As Fˆ (0) is infinite, ψ(0) = 0 and the coherent states are defined in D0β . The consistency conditions are satisfied. Let us state the result for arbitrary α: Proposition 5.7. One can construct two Deformed Harmonic Oscillator Algebras that can be represented on the space of functions holomorphic in Dαβ equipped with the scalar product : Z exp(zz − β)f (z)g(z) dz d z . (67) (g, f ) = α<|z|2 <β
The spectrum of N is N + or N − + {0} when α = 0 and Z when α 6= 0 and the coherent states are the eigenvectors • of the annihilation when the characteristic function is Z β exp(x − β)−1 xρ dx α , ψ(ρ) = Z β −1 ρ−1 exp(x − β) x dx
(68)
α
• or of the creation operator when the characteristic function is ψ(1 − ρ). From these examples, we conjecture that the propositions of Sec. 4 can be largely extended.
DEFORMED HARMONIC OSCILLATOR ALGEBRAS
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651
6. Conclusion Given a ring D in the complex plane and a positive function F that characterize a functional Bargmann Hilbert space S, we have discussed the conditions under which exists a DHOA that admits a Bargmann representation in S, as defined in 1.2. We have obtained conditions on the weight function in order that solutions exist: — in Proposition 4.1, we give necessary conditions, — in Propositions 4.4–4.7, we give sufficient conditions, when D is not the whole complex plane. When one DHOA solution exists, another DHOA exists, if the annihilation operator of the first one possesses eigenvectors generating the representation space, the same holds for the creation operator of the second one. Finally, we have developed some examples that does not fulfill the sufficient conditions, in particular, when D is the whole complex plane. We have obtained deformations of usual Harmonic Oscillator Algebra through deformations of its Bargmann representation. References [1] M. Irac-Astaud and G. Rideau, Proc. Symposium in honor of Jiri Patera and Pavel Winternitz for their 60th birthday, Algebraic Methods and Theoretical Physics, January 9–11, 1997, Centre de recherches math´ ematiques, Universit´e de Montr´eal. [2] M. Irac-Astaud and G. Rideau, Rev. Math. Phys. 10 (8) (1998) 1061–1078. [3] M. Irac-Astaud and G. Rideau, Czech. J. Phys. 47 (11) (1997) 1147, 6th Colloquium on Quantum Group and Integrable systems, Prague, 19–21 June 1997. [4] L. C. Biedenharn, J. Phys. A: Math. Gene. 22 (1989) L873. [5] A. J. Mac Farlane, J. Phys. A: Math. Gene. 22 (1989) 4581. [6] M. Irac-Astaud and G. Rideau, Proc. Third Int. Wigner Symposium, Oxford, 1993, to appear. [7] M. Irac-Astaud and G. Rideau, “On the existence of quantum bihamiltonian systems: the harmonic oscillator case”, preprint PAR-LPTM 92, Lett. Math. Phys. 29 (1993) 197; Theor. Math. Phys. 99 (1994) 658. [8] R. W. Gray and C. A. Nelson, J. Phys. A: Math. Gen. 23 (1990) L945. [9] M. Chaichian and A. Demichev, Introduction to Quantum Groups, World Scientific, 1996. [10] C. Quesne and N. Vansteenkiste, Helv. Phys. Acta 69 (1996) 141, and many references therein. [11] P. Kosinski, M. Mazewski and P. Maslanka, Czech. J. Phys. 47 (1997) 41, 5th Colloquium on quantum groups and integrable systems, Prague, 20–22 June 1996. [12] V. Bargmann, Commun. on Pure and Applied Math. 14 (1961) 187. [13] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series and Products, Academic Press (1965).
ON NEMATICS STABILIZED BY A LARGE EXTERNAL FIELD ´ NELLY ANDRE D´ epartement de Math´ ematiques, Universit´ e de Tours 37200 Tours, France
ITAI SHAFRIR Department of Mathematics, Technion, Israel Institute of Technology 32000 Haifa, Israel Received 10 May 1998
1. Introduction We study a variational problem motivated by the theory of Liquid Crystals. Let a simply connected smooth bounded domain G of R2 be given, together with a smooth boundary condition g : ∂G → S 2 where S 2 denotes the unit sphere in R3 . We denote by (1.1) Hg1 (G, S 2 ) = {u ∈ H 1 (G, S 2 ); u = g on ∂G} the class of admissible maps. Let (ux , uy , uz ) stand for the component of the map u in the directions x, y and z respectively. For each positive ε let uε denote a minimizer for the energy Z Z 1 |∇u|2 + 2 u2 (1.2) Eε (u) = ε G z G over the set Hg1 (G, S 2 ). We are interested in studying the asymptotic behavior of {uε } as ε goes to zero. In the physical model problem the sample is confined in a long (in the z-direction) cylindrical domain whose cross-section in the xy-plane is G, and is subjected to an high external field, electric or magnetic, of the order 1/ε in the z direction. On the lateral boundary of the domain a z-independent boundary condition g is imposed. If the cylindrical domain is very long we may consider the two dimensional problem as an approximation to the original problem. Eε represents then the deformation energy (under the one constant approximation) corresponding to the macroscopic director field u when the boundary condition g is imposed. For more information on the physical problem we refer to De Gennes [9]. Dynamics of singularities for that problem was studied by Pismen and Rubinstein [15]. Although the motivation for this problem comes from Liquid Crystals, it turns out that from the mathematical point of view the problem is closely related to problems in Ginzburg–Landau theory, and in particular to the work of Brezis, 653 Reviews in Mathematical Physics, Vol. 11, No. 6 (1999) 653–710 c World Scientific Publishing Company
654
´ and I. SHAFRIR N. ANDRE
Bethuel and H´elein [4, 5], see also the papers of Struwe [23] and Bethuel and Rivi`ere [6]. This was first observed by Sandier who in [17] studied the above problem in the case where the boundary condition g is orthogonal to the applied field, i.e. with image in the xy-plane (more generally, Sandier treated also non-constant applied fields, see also below, but the assumption of orthogonality on the boundary is essential for his analysis). In that case, similar analysis to that of [5] applies (as well as alternative approaches such as those of Sandier [16, 17] and Jerrard [11]) and leads to a complete description of the asymptotic behavior of {uε } as ε goes to zero. More precisely, a subsequence {uεn } converges away from a finite number of points a1 , . . . , ad ∈ G (d being the degree of g, see below) to a singular S 1 -valued harmonic map (S 1 is identified with the intersection of S 2 with the xy-plane). Here we are interested in the more general case where g is not necessarily S 1 valued. We are still unable to treat the general case of an arbitrary smooth S 2 valued g and we require that g does not take the values of the two poles, i.e. we assume in the sequel that g(x) ∈ S 2 \ {(0, 0, ±1)} ∀x ∈ ∂G .
(1.3)
The assumption (1.3) enables us to define the degree d of g as d = deg(gh /|gh |) where gh denotes the projection of g on the xy-plane. The z-component of g is denoted by gz , so by (1.3) we have |gz | < 1 on ∂G. We will deal in the sequel with the more interesting case d 6= 0 and without loss of generality d > 0 (see Remark 1.2 below). For topological reasons this assumption implies that the image of each uε covers at least one of the poles (0, 0, ±1). Intuitively we expect the potential term to “force” the image of uε to lie closer and closer to S 1 as ε becomes smaller and smaller. As this is inconsistent with the boundary condition g (which is not assumed to be S 1 -valued), we expect a transition layer near the boundary and an interaction energy term to appear. In fact, we faced a similar problem in our previous work [2] which dealt with minimizers of the Ginzburg–Landau type energy Z Z 1 1 |∇u|2 + 2 (1 − |u|2 )2 2 G 4ε G over the set Hg1 (G, C) = {u ∈ H 1 (G, C) u = g on ∂G} . Indeed, from the qualitative point of view the asymptotic behavior of the minimizers for the two problems is the same. But it turns out that the analysis of the energy (1.2) is more delicate and requires new ideas, although the results of [2] serve as an important tool. Our first main result gives the asymptotic behavior of the energy of the minimizers. Theorem 1. Let g be smooth boundary data of degree d > 0 satisfying (1.3). Then, Z 2 (1 − |gh |) + 2πd| log ε| + O(1), f or all 0 < ε ≤ 1 . (1.4) Eε (uε ) = ε ∂G
ON NEMATICS STABILIZED BY A LARGE EXTERNAL FIELD
655
The first term on the r.h.s. of (1.4) is due to boundary interaction, while the second term results from the degree of g. In the proof of Theorem 1 the most difficult part is the lower bound, while the upper bound is proved by a relatively simple explicit construction. Theorem 1 enables us also to prove the following convergence result. For the definitions of the canonical harmonic map and renormalized energy we refer the reader to [5]. 1 (G \ Theorem 2. For a subsequence εn → 0 we have uεn → u∗ strongly in Hloc 0 {a1 , . . . , ad }) and Cloc (G\{a1 , . . . , ad }) for some d points a1 , . . . , ad ∈ G, where u∗ is the canonical harmonic map associated to the boundary condition gh /|gh | with degree 1 around each of the points a1 , . . . , ad . Moreover, the configuration a = (a1 , . . . , ad ) minimizes the renormalized energy W (gh /|gh |, b) over all configurations of d distinct points b ∈ Gd .
Remark 1.1. It is very probable that with some additional effort the C 0 -convergence can be improved to C k -convergence for all k. Figure 1 gives a schematic description of a minimizer uεn for small εn . Near the boundary there is a transition layer where uεn passes rapidly from the boundary value g(x) to a value close to (gh (x)/|gh (x)|, 0) while its phase remains almost unchanged. In most of the interior remaining region we have uεn ' u∗ , that is, uεn is nearly an S 1 -valued map on this region. What left are d small neighborhoods of the d points a1 , . . . , ad . In each of these neighborhoods, uεn covers almost a complete hemisphere (north or south). A natural question is to determine for a certain ai , which of the two hemispheres is covered by uεn in a small neighborhood around ai . This question is answered by Proposition 2.3 in Sec. 2. As mentioned above, one may consider a more general situation where the applied field varies with x. Indeed this case was treated by Sandier [17] under some restrictive assumptions on the boundary data. We assume thus that we are given a smooth vector field e(x) on G with values in R3 \ {0, 0, 0}. For each ε > 0 we denote by uε a minimizer for the energy Z Z 1 eε (u) = |∇u|2 + 2 (u, e)2 (1.5) E ε G G over the set Hg1 (G, S 2 ). The analogue condition to (1.3) is then that on the boundary e(x) is never parallel to g(x), i.e. g(x) 6 k e(x) ∀x ∈ ∂G .
(1.6)
Next we define e3 (x) = e(x)/|e(x)|. It is easy to see that e3 (x) can be completed to a smooth moving frame (e1 (x), e2 (x), e3 (x)) on G. We may then write g(x) = g1 (x)e1 (x) + g2 (x)e2 (x) + g3 (x)e3 (x),
∀x ∈ ∂G .
(1.7)
In analogy with the case e ≡ (0, 0, 1) we denote gh = (g1 , g2 ) and
gz = g3 .
(1.8)
´ and I. SHAFRIR N. ANDRE
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here the upper hemisphere is covered
.a uz
1
>0
gz > 0 b1
uz < 0
b2 gz < 0
.a
here u(x) lies close to the equator
2
.a
transition layer 3
here the lower hemisphere is covered
Fig. 1.
By our assumption (1.6) gh is never zero, so we may define the degree d of g as . g1 , gb2 ). In order to see why the degree of the S 1 -valued map gb = gh /|gh | = (b 1 -valued map the definition is independent of our choice of e1 , e2 we define the Sp g=b e g1 e1 + gb2 e2 (which depends only on e since ge = (g − (g, e3 )e3 )/ 1 − g32 ) and then claim that Z Z 1 1 ge ∧ geτ · e3 + (e3 )x ∧ (e3 )y · e3 . (1.9) d= 2π ∂G 2π G In order to justify (1.9) we first note that by our definition Z 1 gb ∧ gbτ . d= 2π ∂G
(1.10)
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ON NEMATICS STABILIZED BY A LARGE EXTERNAL FIELD
On the other hand, by a direct calculation we find that g2 ) τ − b g2 (b g1 )τ − (e1 ) · (e2 )τ , ge ∧ (e g )τ · e3 = gb1 (b hence we may rewrite (1.10) as Z Z 1 1 ge ∧ e g τ · e3 + e1 · (e2 )τ . d= 2π ∂G 2π ∂G Note that by Green formula Z Z 1 1 e1 · (e2 )τ = (e1 )x · (e2 )y − (e1 )y · (e2 )x . 2π ∂G 2π G
(1.11)
(1.12)
Finally, we notice the pointwise identity on G: (e1 )x · (e2 )y − (e1 )y · (e2 )x = ((e1 )x · e3 )((e2 )y · e3 ) − ((e1 )y · e3 )((e2 )x · e3 ) = ((e3 )x · e1 )((e3 )y · e2 ) − ((e3 )x · e2 )((e3 )y · e1 ) = (e3 )x ∧ (e3 )y · e3 .
(1.13)
Clearly (1.9) follows from (1.11)–(1.13). We denote a(x) = |e(x)|, so by assumption a(x) is a smooth strictly positive function on G. We are able to prove for this more general situation, a partial analogue to Theorems 1 and 2. We get an energy estimate as in Theorem 1 and a convergence result to a singular limit as in Theorem 2. Yet in contrast with the particular case e(x) ≡ (0, 0, 1), we are not able to give a simple description of the limiting map nor of the location of its singularities. Our result is then Theorem 3. Let g be smooth boundary data of degree d > 0 satisfying (1.6). Then, Z 2 e a(1 − |gh |) + 2πd| log ε| + O(1), for all 0 < ε ≤ 1 . (1.14) Eε (uε ) = ε ∂G 1 (G \ {a1 , . . . , ad }) and For a subsequence εn → 0 we have uεn → u∗ strongly in Hloc 0 Cloc (G\{a1 , . . . , ad }) for some d points a1 , . . . , ad ∈ G, where u∗ : G\{a1 , . . . , ad } → S 2 is a smooth map which satisfies (u∗ (x), e(x)) = 0, ∀ x, u∗ = g on ∂G, and is a solution of (1.15) (∆u∗ ∧ u∗ , e) = 0 in D0 (G) .
Remark 1.2. In the case d = 0 we get by the same methods (which take in fact a much simpler form for that case) analogues to Theorems 1–3. The only difference is that the limiting map u∗ is then smooth on all of G, and in the energy estimate of Theorems 2 and 3 the term of the order | log ε| disappears (i.e. (1.14) continues to hold for d = 0). 2. Preliminaries In this section we shall deal mostly with the case e(x) ≡ (0, 0, 1). The necessary modifications needed for the general case will be given in Sec. 6. We start with some useful notations.
658
´ and I. SHAFRIR N. ANDRE
We denote by δ(x) the distance function to ∂G and for any small r we define . . Γr = {x ∈ G; δ(x) < r} and Gr = {x ∈ G; δ(x) > r}. Since ∂G is smooth, there is µ > 0 such that for every x ∈ Γµ there is a unique point y = y(x) ∈ ∂G satisfying δ(x) = |x − y|, and moreover δ is smooth on Γµ (see Sec. 14.6 in [10]). We can then represent each x ∈ Γµ by the coordinates s = y(x) ∈ ∂G and t = δ(x). Let us fix a smooth vector field V (x) on G such that |V (x)| ≤ 1 on G and for x ∈ Γµ V (x) equals the unit normal to the level curve t = δ(x) directed towards ∂G. We define on ∂G a function α with values in (0, ∞] by α(s) = tanh−1 (|gh (s)|) .
(2.1)
Note that α may take the value +∞; it happens exactly where gz = 0. Recall that a main tool for the analysis of the Ginzburg–Landau energy in [2] was a certain energy decomposition. This was established by writing (following an observation of Lassoued and Mironescu [12]) each admissible map u ∈ H 1 (G, C) as u = ρε v where ρε is the minimizer for the scalar problem i.e. the minimizer for the energy Z Z 1 1 |∇ρ|2 + 2 (1 − ρ2 )2 2 G 4ε G over the set 1 (G) = {ρ ∈ H 1 (G); ρ = |g| on ∂G} . H|g| This leads to the decomposition formula Z Z Z Z 1 1 1 1 2 2 2 2 |∇u| + 2 (1 − |u| ) = |∇ρε | + 2 (1 − ρ2ε )2 2 G 4ε G 2 G 4ε G Z Z 1 1 2 2 + ρ |∇v| + 2 ρ4 (1 − |v|2 )2 . (2.2) 2 G ε 4ε G ε By (2.2) we see that in order to understand the behavior of uε it is enough to study separately the energy of ρε (the solution of the scalar problem) and the weighted energy of uε /ρε . It turns out that the more difficult part is the study of the weighted energy, see [2]. Naturally, we would like to have an analogue decomposition to (2.2) in our situation. It turns out that two such different decompositions are available, and only the combination of the two enables us to establish the estimate (1.4) which is the basis for all further results. We should first find what should be the analogue to the scalar problem in our case. More precisely, it turns out to be an S 1 -valued 1 -valued problem, which can be obviously interpreted as a problem, or even an S+ scalar problem. Recall that we have denoted by gh and gz the planner component and z-component of g respectively. We also denote by gz the component of g in the z-direction. Now let ρε = (ρε,h , ρε,z ) ∈ C 2 (G, S 1 ) denote a minimizer for the energy Z Z 1 |∇ρ|2 + 2 ρ2 (2.3) Eε (ρ) = ε G z G over the set 1 (G, S 1 ) = {ρ = (ρh , ρz ) ∈ H 1 (G, S 1 ); ρ = (|gh |, gz ) on ∂G} . H(|g h |,gz )
ON NEMATICS STABILIZED BY A LARGE EXTERNAL FIELD
659
The uniqueness of ρε will be proved in the next proposition. The boundary condition can be written as π π , ∀ x ∈ ∂G . (2.4) (|gh (x)|, gz (x)) = (cos ϕ(x), sin ϕ(x)) with ϕ(x) ∈ − , 2 2 Since the first component of the boundary data is nonnegative, it follows easily that this property must be preserved by any minimizer. Hence any minimizer ρε can be written too as π π ,∀x ∈ G. (2.5) ρε (x) = (cos ϕε (x), sin ϕε (x)) with ϕε (x) ∈ − , 2 2 Writing the energy in terms of ϕε we find: Z Z 1 2 |∇ϕε | + 2 sin2 ϕε . (2.6) Eε (ρε ) = ε G G The function ϕε associated with a minimizer ρε is then a solution to sin 2ϕε ∆ϕε = 2ε2 in G, ϕε (x) ∈ [−π/2, π/2] in G , ϕε = ϕ on ∂G .
(2.7)
Now we can state: Proposition 2.1. There is a unique solution to (2.7). Moreover, for each integer m ≥ 0 there is an approximate solution ϕ(ε) m =
m X
(ε)
εi ψi
i=0
with
δ(x) (ε) ψi (x) = O e− ε
f or all i ,
such that m+1 ). ϕε − ϕ(ε) m = O(ε
All the estimates above are valid in C k (G) for all k. In particular, the function (ε) ϕ0 = ϕ0 is given on Γµ/2 (using the coordinates s, t as above) by ϕ0 (x) = sgn gz (s) arctan(csch (α(s) + t/ε)) , where
(2.8)
α(s) = tanh−1 (cos ϕ(s)) ∈ (0, ∞] .
The proof of Proposition 2.1 uses results of Angenent [1] and Berger and Fraenkel [3]. A slightly more general version of it is proved in Sec. 7. The main interest of Proposition 2.1 lies in the fact that it provides a uniform approximation on G. It is much easier to show that ϕε , ρz , 1 − ρh and their derivatives decay exponentially away from the boundary as the next proposition shows. The proof is postponed to Sec. 7. Throughout this article we denote by C different constants which do not depend on ε.
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Proposition 2.2. For all x ∈ G we have 1 − ρh (x) ≤ Ce− and C |∇ρh (x)| ≤ δ(x)
"
δ(x) ε
δ(x) 2ε
2
(2.9) #
+ 1 e−
δ(x) 2ε
.
(2.10)
The asymptotic behavior Eε (ρε ) actually follows from Proposition 2.1 but we will give a simple direct proof for it in the next lemma. Lemma 2.1. Eε (ρε ) =
2 ε
Z (1 − |gh |) + O(1),
as ε goes to 0 .
(2.11)
∂G
Proof. It is convenient to work with ϕε rather than ρε . The upper bound can be proved by an explicit construction in the spirit of Modica [14] and Sternberg [22], just as in the appendix of [2], so we omit the details. We give here only the proof of the lower bound which follows the lines of the proof of Proposition 2.2 of [2]. By Cauchy–Schwarz inequality and Green formula, using the vector field V that was defined above, we get: Z Z Z sin2 ϕε 2 2 2 |∇ϕε | + ≥ | sin ϕε ∇ϕε | ≥ ∇(1 − cos ϕε ) · V Eε (ρε ) = ε2 ε G ε G G Z Z 2 2 (1 − |gh |) − (1 − cos ϕε ) div V . (2.12) = ε ∂G ε G Since (1 − cos ϕε )/ sin2 ϕε ≤ C, the upper bound Eε (ρε ) ≤ C/ε yields Z Z C 2 ≤ (1 − cos ϕ ) div V sin2 ϕε ≤ C . ε ε ε G G
Next we describe the two energy decompositions mentioned above. For reasons that will become clear later we write the decompositions on a subdomain of G. In the sequel we write for short ρ instead of ρε . For the first decomposition we define for each u ∈ C 1 (G, S 2 ): uh uz vz = . (2.13) vh = ρh ρz Lemma 2.2. Let u ∈ H 1 (G, S 2 ) with u = g on ∂G be given. Then, for every subdomain Ω ⊆ G with smooth boundary, except (possibly) for a finite number of corners, such that ρz 6= 0 on Ω there holds Z Z u2z ρ2 2 |∇u| + 2 = |∇ρ|2 + 2z + ρ2h |∇vh |2 + ρ2z |∇vz |2 ε ε Ω Ω Z (1 − |vh |2 ) ∂ρz . (2.14) + ρz ∂ν ∂Ω
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ON NEMATICS STABILIZED BY A LARGE EXTERNAL FIELD
Proof. By a standard density argument we may assume that u is smooth. It is easy to see that the Euler–Lagrange equation for ρ reads:
which implies
ρ2 ρh −∆ρh = |∇ρ|2 ρh + z 2 ε 2 −∆ρz = |∇ρ|2 ρz − ρh ρz ε2
(2.15)
2ρ2 ρ2 −∆(ρ2h ) = 2|∇ρ|2 ρ2h + z2 h − 2|∇ρh |2 ε 2 2 −∆(ρ2z ) = 2|∇ρ|2 ρ2z − 2ρz ρh − 2|∇ρz |2 . ε2
(2.16)
Now we get by a direct calculation, using Green formula: Z
Z u2z 1 |∇u| + 2 = ρ2h |∇vh |2 + ∇(ρ2h )∇(|vh |2 ) + |vh |2 |∇ρh |2 Eε (u; Ω) = ε 2 Ω Ω Z 1 1 − ρ2h |vh |2 + ρ2z |∇vz |2 + ∇(ρ2z )∇(vz2 ) + vz2 |∇ρz |2 + 2 ε2 Ω Z 1 − ρ2h |vh |2 = ρ2h |∇vh |2 + |vh |2 |∇ρh |2 + ρ2z |∇vz |2 + vz2 |∇ρz |2 + ε2 Ω Z 1 − ∆(ρ2h )(|vh |2 − 1) + ∆(ρ2z )(vz2 − 1) 2 Ω Z ∂ρ2 ∂ρ2 1 (|vh |2 − 1) h + (vz2 − 1) z . + 2 ∂Ω ∂ν ∂ν 2
Using (2.16) we find Z ρ2h |∇vh |2 + |∇ρh |2 + ρ2z |∇vz |2 + |∇ρz |2
Eε (u; Ω) = Ω
2 2 1 − ρ2h |vh |2 2 2 ρh ρz + (|v | − v ) h z ε2 ε2 Z 1 ∂ρ2 ∂ρ2 + (|vh |2 − 1) h + (vz2 − 1) z . 2 ∂Ω ∂ν ∂ν
+
(2.17)
Now since (|vh |2 − vz2 )ρ2h ρ2z = ρ2h (|vh |2 (1 − |ρh |2 ) − 1 + ρ2h |vh |2 ) = ρ2h (|vh |2 − 1) we find that the integral over Ω in (2.17) coincides with the one in (2.14). To complete the proof of Lemma 2.2, we should prove that the same holds for the boundary term. This follows from the following identity:
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∂ρ
ρh
ρz ∂νz 2 ∂ρh ∂ρz 2 ∂ρh (|vh |2 − 1) + ρz (vz − 1) = ρh (|vh |2 − 1) + (ρh − ρ2h |vh |2 ) ∂ν ∂ν ∂ν ρ2z ρ2h ∂ρz 2 (1 − |vh | ) 1 + 2 = ρz ∂ν ρz =
(1 − |vh |2 ) ∂ρz . ρz ∂ν
The decomposition (2.14) (if we apply it to Ω = G when possible) resembles the decomposition (2.2) in the Ginzburg–Landau case in that the energy of u is decomposed as a sum of the energy of ρε and a kind of weighted energy of v = (vh , vz ). There are however two major difficulties. The first is that the weighted energy of v in our case, namely Z ρ2h |∇vh |2 + ρ2z |∇vz |2 G
does not seem to be related to the weighted Ginzburg–Landau energy, and we can see a priori no reason why should it behave like O(| log ε|) for a minimizer. Secondly, the decomposition (2.14) is only available when ρz 6= 0, so we cannot apply it for Ω = G when gz has zeros. We shall see later how to overcome the first difficulty. In order to overcome the second difficulty we introduce in the next lemma a second decomposition, which is in fact simpler than the first one since it involves only the energy densities (no integration by parts is needed) and moreover, it does not use the minimizing property of ρ. For this decomposition we keep vh as above, but we replace vz of (2.13) by uz . wz = ρh Lemma 2.3. For u ∈ H 1 (G, S 2 ) and any subdomain Ω ⊆ G we have Z Z u2 ρ2 |∇u|2 + 2z = |∇ρ|2 + 2z + ρ2h |∇vh |2 + ρ2h |∇wz |2 ε ε Ω Ω 2 Z 2 ρz ρ + h2 (1 − |vh |2 ) − ρ2h ∇ ε ρh
(2.18)
Ω
Proof. First we write Z ρ2h (|∇vh |2 + |∇wz |2 ) + |∇ρh |2 (|vh |2 + wz2 ) Eε (u; Ω) = Ω
+ 2ρh ∇ρh (vh ∇vh + wz ∇wz ) +
1 − ρ2h |vh |2 . ε2
(2.19)
Next we notice that |∇ρh |2 (|vh |2 + wz2 ) + 2ρh ∇ρh (vh ∇vh + wz ∇wz ) |∇ρh |2 + ρh ∇ρh ∇(|vh |2 + wz2 ) ρ2h 1 1 |∇ρh |2 |∇ρh |2 2 ∇(ρ + )∇ . = − = h ρ2h 2 ρ2h ρ2h
=
(2.20)
ON NEMATICS STABILIZED BY A LARGE EXTERNAL FIELD
Plugging it in (2.19) we are led to Z 1 − ρ2h |vh |2 |∇ρh |2 ρ2h (|∇vh |2 + |∇wz |2 ) + − . Eε (u; Ω) = ε2 ρ2h Ω
663
(2.21)
Now we can do the same calculation with ρ instead of u. Then vh should be replaced by 1 and wz by ρz /ρh in (2.21). We get Z ρ2h
Eε (ρ; Ω) = Ω
2 2 2 ∇ ρz + 1 − ρh − |∇ρh | . 2 ρh ε2 ρh
(2.22)
Combining (2.21) with (2.22) we are led to (2.18). A useful fact about minimizers is proved in the next lemma. Lemma 2.4. When u = uε is a minimizer we have |vh (x)| ≤ 1
on G .
(2.23)
. Proof. Looking for a contradiction, assume that the subdomain Ω = {x ∈ G; |vh (v)| > 1} is nonempty. Then at least one of the domains Ω+ = Ω ∩ {uz > 0} or Ω− = Ω ∩ {uz < 0} is nonempty. Assume without loss of generality that Ω+ is nonempty. Let us consider on Ω+ the S 1 -valued map U = Uε defined by U = (|uh |, uz ) = (ρh |vh |, uz ) . Applying Lemma 2.3 we find that Z
2 uh |uh | ∇ . |uh | 2
Eε (u; Ω+ ) = Eε (U ; Ω+ ) + Ω+
(2.24)
Next let Wε = W = (Wh , Wz ) = (cos ψε , sin ψε ) (with ψε (x) ∈ [0, π/2]) denote a minimizer for Eε on Ω+ for the boundary condition W = U on ∂Ω+ . The uniqueness of W follows from a result of Brezis–Oswald [8] (applied to the positive function π/2 − ψε ). Recall that we also write (ρh , ρz ) = (cos ϕε , sin ϕε ) where ϕε is a solution of sin 2ϕε ∆ϕε = 2ε2 with values in (−π/2, π/2). By Kato’s inequality we have ∆|ϕε | ≥ sgn ϕε ∆ϕε =
sin 2|ϕε | , 2ε2
so |ϕε | is a subsolution for the problem: sin 2ψ ∆ψ = 2ε2 in Ω+ , ψ(x) ∈ [0, π/2] in Ω+ , ψ = ψε on ∂Ω+ .
(2.25)
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664
As mentioned above, the unique solution to (2.25) is ψε . It follows that |ϕε | ≤ ψε on Ω+ which implies that |ρz | ≤ Wz on Ω+ . This is equivalent to Wh ≤ ρh
on Ω+ .
(2.26)
Now let us consider on Ω+ the S 2 -valued map uh , Wz . u b = Wh · |uh | Applying (2.24) to u b we find 2 uh u; Ω+ ) = Eε (W ; Ω+ ) + |Wh | ∇ . Eε (b |uh | Ω+ Z
2
(2.27)
But clearly Eε (W ; Ω+ ) ≤ Eε (U ; Ω+ ) (since W is a minimizer), and by (2.26) Wh ≤ ρh < ρh |vh | = |uh | on Ω+ . u; Ω+ ) < Eε (u; Ω+ ), contradicting our asBy (2.24)–(2.27) we are then led to Eε (b sumption of u being a minimizer. An important corollary of Lemma 2.4 is the next proposition which deals with the sign of uz when u = uε is a minimizer. Notice that in the particular case when 2 2 -valued or S− -valued. Indeed, since gz ≡ 0 on ∂G, any minimizer u is either S+ (uh , |uz |) is a minimizer too and any minimizer is real analytic, we see that either uz ≥ 0 or uz ≤ 0 on G. The degree condition d > 0 excludes the possibility of an S 1 -valued map, hence by the strong maximum principle we have either uz > 0 or uz < 0 in G (the Euler–Lagrange equation for uz takes the form ∆uz = c(x)uz , as can be seen from (4.40) below). Clearly both possibilities can occur since in our case the map obtained by applying a reflection in the xy-plane on the image of u is a minimizer whenever u is. The next proposition deals with the remaining case, when gz is not identically 0. Proposition 2.3. If gz 6≡ 0 on ∂G, then sgn uz = sgn ρz on G. Proof. We first claim that {x ∈ G; uz = 0} ⊆ {x ∈ G; ρz = 0} .
(2.28)
In fact, (2.28) is always true, even if gz ≡ 0. Indeed, consider a point x ∈ G with uz (x) = 0. Then necessarily 1 = |uh (x)| = ρh (x)|vh (x)| . But then Lemma 2.4 clearly implies that ρh (x) = |vh (x)| = 1, hence ρz (x) = 0, as claimed. Next we claim that {x ∈ G; ρz > 0} ⊆ {x ∈ G; uz > 0} .
(2.29)
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665
Consider any connected component D of the set {ρz > 0}. We want to show first that uz > 0 on D. On ∂D ∩ G we clearly have ρz = 0. On ∂D ∩ ∂G we have either ρz = 0 or ρz > 0. We claim that the second possibility occurs on a nonnegligible subset of ∂G. Indeed, otherwise replacing ρz by 0 in D we would get a new minimizer. By analyticity, a minimizer which is zero on an open set must be zero on all of G, contradicting our assumption on the boundary data. By (2.28) uz has no zero in D. Since uz > 0 on part of ∂D ∩ ∂G, we must have uz > 0 in D by the maximum principle. Since D may be any connected component of {ρz > 0} we actually proved (2.29). By symmetry we also have {x ∈ G; ρz < 0} ⊆ {x ∈ G; uz < 0} .
(2.30)
In order to show that {x ∈ G; ρz > 0} = {x ∈ G; uz > 0}
(2.31)
we consider a point x ∈ G with uz (x) > 0. By (2.30) we must have ρz (x) ≥ 0. We claim that ρz (x) = 0 is impossible. Indeed, looking for a contradiction assume that ρz (x) = 0. Then for some small η > 0 we have ρz ≥ 0 on B(x, η) (if not, there exists a sequence xm → x with ρz (xm ) < 0, ∀m, hence uz (xm ) < 0, ∀m and uz (x) ≤ 0, a contradiction). Again by the maximum principle we have either ρz ≡ 0 on B(x, η), which implies ρz ≡ 0 on G as we saw above, impossible, or ρz > 0 on B(x, η) which contradicts our assumption ρz (x) = 0. We thus proved that ρz (x) > 0, completing the proof of (2.31). An identical argument shows also that {x ∈ G; ρz < 0} = {x ∈ G; uz < 0} . The result clearly follows from (2.31)–(2.32).
(2.32)
Remark 2.1. Let us look again at Fig. 1. There it was assumed that gz has z exactly two zeros b1 and b2 (with dg ds (bi ) 6= 0, i = 1, 2) so for each ε the set {x ∈ G; uz (x) = 0} = {x ∈ G; ρz (x) = 0} consists of a smooth curve connecting these two points, separating the two sets {x ∈ G; uz (x) > 0} and {x ∈ G; uz (x) < 0}. In the case of Fig. 1, d = 3 and the singularities are a1 , a2 , a3 . By Theorem 2 they depend only on the phase of the boundary data, i.e. on gh /|gh |. In contrast, the sign of uz near each ai (and in particular which of the two hemispheres is covered by uz in its neighborhood) depends on gz only. Using the second decomposition we can already deduce an upper bound for Eε (uε ). Lemma 2.5. Eε (uε ) ≤ Eε (ρε ) + 2πd| log ε| + C .
(2.33)
Proof. It would be enough to construct for each ε > 0 a map Uε ∈ C 1 (G, S 2 ) which equals g on ∂G for which the upper bound (2.33) holds. We first fix d distinct points a1 , . . . , ad in G, and then r satisfying 0 < r < min{{|ai − aj |/4; 1 ≤ i < j ≤ d}, {δ(ai )/2, 1 ≤ i ≤ d}}
´ and I. SHAFRIR N. ANDRE
666
and a smooth map w : G → S 1 such that w = gh /|gh | on ∂G and w(x) =
x − ai |x − ai |
on B(ai , 2r) \ B(ai , r), i = 1, . . . , d .
We also fix a smooth function η : R2 → [0, 1] such that η(x) = 0 for |x| ≤ 1/2 and η(x) = 1 for |x| ≥ 1. Next we define a map wε on G as follows: w(x) wε (x) =
for x ∈ G \
d [
B(ai , r) , (2.34)
x − a i x − ai η for x ∈ B(ai , r), i = 1, . . . , d . ε |x − ai | i=1
The map Uε is defined then by Uε (x) =
ρh (x)wε (x), sgn ρz (x) 1 − η
2
x − ai ε
1/2 !
ρ2h (x)
,
(2.35)
with i = i(x) such that |ai − x| = min{|aj − x|; j = 1, . . . , d} . Setting Kε =
d [
B(ai , ε)
i=1
we note that by our construction Uε,z = ρz on G \ Kε . Hence applying Lemma 2.3 for u = Uε we find Z Z ρ2h ρ2h |∇wε |2 + (1 − |wε |2 ) Eε (Uε ) = Eε (ρ) + 2 ε G Kε 2 2 Z Z Uε,z ρz 2 ρ2h ∇ − ρ . (2.36) + h ∇ ρh ρh Kε Kε It is easy to see that the last three integrals on the r.h.s. of (2.36) are all O(1) and that Z |∇wε |2 = 2πd| log ε| + O(1) . G
This implies that Eε (Uε ) = Eε (ρ) + 2πd| log ε| + O(1) and (2.33) follows.
3. Sketch of the Proof of the Lower Bound As mentioned above, the more difficult part of Theorem 1 is the lower bound (the upper bound follows from Lemmas 2.1 and 2.5). The detailed proof is given in the next section. For the convenience of the reader we sketch below the proof
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under the additional simplifying assumption gz ≥ γ > 0 on ∂G .
(3.1)
2 -valued. It is easy to see that assumption (3.1) implies that each minimizer is S+ Clearly it would be enough to establish the lower bound for a sequence {uεn } with εn → 0. The main idea is to use the first decomposition (2.14) on a domain of the type Γcεn | log εn | and then the second decomposition (2.18) on Gcεn | log εn | (for a properly chosen constant c > 0). In the sequel we denote by u = uεn a minimizer for Eεn over Hg1 (G, S 2 ) and by ρ = ρεn the minimizer for the scalar problem. Using Proposition 2.1 we may choose a positive c such that
ρz ≥
√ εn on Γcεn | log εn | .
(3.2)
Using Proposition 2.1 again we conclude that for some constant C > 1 we have |∇ρz | C 1 ≤ ≤ Cεn ρz εn
on Γcεn | log εn | .
By (2.14) we have (using the notations (2.13)): Z Eεn (u; Γcεn | log εn | ) = Eεn (ρ; Γcεn | log εn | ) +
(3.3)
ρ2h |∇vh |2 + ρ2z |∇vz |2
Γcεn | log εn |
Z + ∂Gcεn | log εn |
(1 − |vh |2 ) ∂ρz . ρz ∂ν
(3.4)
In order to bound from below the integral on Γcεn | log εn | in (3.4), we use the next lemma that is stated in the general setting (i.e. gz of arbitrary sign) since we are going to use it in the next section. Lemma 3.1. At points x ∈ G where ρz 6= 0 we have ρ2h |∇vh |2 + ρ2z |∇vz |2 ≥
(1 − |vh |2 )2 |∇ρz |2 . 2ρ2z
(3.5)
Proof. Differentiating the identity ρ2h |vh |2 + ρ2z |vz |2 = 1 we find that ρ2h ∇|vh |2 + ρ2z ∇vz2 = −|vh |2 ∇ρ2h − vz2 ∇ρ2z = (vz2 − |vh |2 )∇ρ2h = Using (3.6) and the elementary inequality |v1 |2 + |v2 |2 ≥
1 |v1 + v2 |2 , 2
1 − |vh |2 ∇ρ2h . (3.6) ρ2z
´ and I. SHAFRIR N. ANDRE
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we deduce that ρ2h |∇vh |2 + ρ2z |∇vz |2 ≥ ρ4h |vh |2 |∇|vh ||2 + ρ4z vz2 |∇vz |2 =
1 · (ρ4h |∇|vh |2 |2 + ρ4z |∇vz2 |2 ) 4
≥
1 · |ρ2h ∇|vh |2 + ρ2z ∇vz2 |2 8
=
(1 − |vh |2 )2 2 ρh |∇ρh |2 , 2ρ4z
and the result follows since ρ2h |∇ρh |2 = ρ2z |∇ρz |2 .
From (3.3)–(3.5) it follows that for some positive k we have Z k 1 2 ρh |∇vh |2 + 2 (1 − |vh |2 )2 Eεn (u; Γcεn | log εn | ) = Eεn (ρ; Γcεn | log εn | ) + 4 ε Γcεn | log εn | n Z + ∂Gcεn | log εn |
(1 − |vh |2 ) ∂ρz . ρz ∂ν
(3.7)
Next we claim that a constant c satisfying (3.2) can be chosen so that the boundary term in (3.7) is bounded by some constant. Taking into account (3.3) and Lemma 2.4 we deduce that we only need to show the following: c can be chosen so that Z 1 (1 − |vh |2 ) ≤ C . (3.8) εn ∂Gcεn | log εn | . For that matter we first show that the map Uεn = U = (|uh |, uz ) satisfies Z 2 (1 − |gh |) − C . Eεn (U ; Γcεn | log εn |/2 ) ≥ εn ∂G
(3.9)
The proof of (3.9) is carried out in the next section (in a more general situation). We mention in passing that it relies on a super solution construction for the scalar problem with mixed Dirichlet–Neumann boundary conditions. Next, since clearly Eεn (U ; Γcεn | log εn |/2 ) ≤ Eεn (u; Γcεn | log εn |/2 ) , we deduce from (3.9) and (2.33) that Eεn (u; Gcεn | log εn |/2 ) ≤ 2πd| log εn | + C . Applying the second decomposition (2.18) on Ω = Gcεn | log εn |/2 and using Z Gcεn | log εn |/2
we obtain
2 ∇ ρz ≤ C ρh 1 ε2n
(a consequence of Proposition 2.1)
Z (1 − |vh |2 ) ≤ C| log εn | . Gcεn | log εn |/2
(3.10)
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669
Hence by Fubini theorem we may choose c1 ∈ (c/2, c) (this c1 will replace the previous choice for c) such that Z 1 (1 − |vh |2 ) ≤ C , εn ∂Gc εn | log εn | 1
and (3.8) is satisfied with c = c1 . Note that since c1 < c (3.2) is still valid for c1 . Next, combining (3.7) with the second decomposition (2.18) applied to Ω = Gcεn | log εn | , taking into account (3.8), we get for some k > 0: Z 1 k ρ2h |∇vh |2 + 2 (1 − |vh |2 )2 Eεn (u) − Eεn (ρ) ≥ 4 Γcεn | log εn | εn Z k + ρ2h |∇vh |2 + 2 (1 − |vh |2 )2 − C . (3.11) ε Gcεn | log εn | n Next we are in a position to apply the following result from [2]: Proposition 3.1. Let a bounded, simply connected domain G ⊂ R2 be given, together with a smooth boundary condition g : ∂G → S 1 of degree d > 0. Let λ > 0 be given too. Assume that for each ε > 0 the positive function fε satisfies fε ≥ c0 > 0 on G ,
(3.12.a)
c1 on G , ε
(3.12.b)
|∇fε | ≤ and
|∇fε (x)| + |1 − fε (x)| ≤ c2 e−
c3 δ(x) ε
,
∀ x ∈ Gc4 ε| log ε| ,
(3.12.c)
for some positive constants c0 − c4 . Then for every v ∈ H 1 (G, C) such that v = g on ∂G we have Z Z λ 1 2 2 f |∇v| + 2 (1 − |v|2 )2 ≥ πd| log ε| − C , 2 G ε ε G for some constant C independent of ε. Strictly speaking, Proposition 3.1 was proved in [2] only for some particular weight functions fε , but it is clear that the method of [2] applies to prove Proposition 3.1 as stated. Next let us fix a C ∞ nondecreasing function η satisfying η=
1 on [0, 1] and 2
Setting
fε (x) = η
η = 1 on [2, ∞) .
δ(x) cε| log ε|
(3.13)
ρh (x) ,
(3.14)
it readily follows that the weight functions {fε } satisfy the inequalities (3.12). By (3.11) and Proposition 3.1 we are led to the desired lower bound Eεn (u) − Eεn (ρ) ≥ 2πd| log εn | − C , which completes our sketch of the proof of the lower bound.
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The rigorous proof in the general case, which is carried out in detail in the next section, follows these lines. The main difference is that a more complicated decomposition of the domain G into two subdomains, taking into consideration the zeros of gz , is used. 4. Proof of Theorem 1 In view of Lemmas 2.5 and 2.1 we only need to prove the following lower bound for a minimizer uε : (4.1) Eε (uε ) − Eε (ρε ) ≥ 2πd| log ε| − C . Clearly it is enough to pass to a sequence εn → 0, and prove (4.1) for uεn . We are going to choose a subdomain Ω1 (depending on εn ) on which we will apply . the first decomposition (2.14), while on Ω2 = G \ Ω1 we will apply the second decomposition (2.18). Under the assumption (3.1) in the previous section, Ω1 was simply Γcεn | log εn | for some constant c. In the general case, the construction of Ω1 is more involved. Ω1 will be of the form (using the coordinates s, t defined in Sec. 2) Ω1 = {(s, t); s ∈ ω1 , t < cεn | log εn |} ,
(4.2)
where ω1 is a subset of ∂G consisting of a finite number (depending on εn ) of disjoint boundary segments. We start with: e 1 of G of the form Lemma 4.1. There exists a subdomain Ω 3 e e1 , t < εn | log εn | + θ(εn ) , Ω1 = (s, t); s ∈ ω 4 for some subset ω e1 ⊂ ∂G and some number θ(εn ) of the order εn such that Z e 1) ≥ 2 (1 − |gh |) − C . Eεn (uεn ; Ω εn ∂G
(4.3)
Proof. We start by describing our construction of the set ω e1 . If gz ≡ 0 on ∂G e 1 = ∅. If min∂G |gz | > 0 we set the statement of the lemma is trivial and we set Ω ω e1 = ∂G (this case was treated in the previous section). In the remaining case, we first choose by Sard lemma a constant c0 ∈ (1, 2) such that c20 εn is a regular value of the function ϕ2 for each n (ϕ is defined in (2.4)). Hence, the set √ {x ∈ ∂G; |ϕ(x)| = c0 εn } consists of a finite number of points s1 , . . . , sNn ∈ ∂G labeled in a clockwise manner (we suppress for simplicity the dependence of the points {si } on εn ). For i = 1, . . . , Nn let us denote by hsi , si+1 i the open boundary segment lying between the points si and si+1 in a clockwise manner (here and in the sequel i+1 is taken modulo Nn for i = Nn ). Next we keep only the segments hsi , si+1 i which contain some point √ s with |ϕ(s)| ≥ 4c0 εn . We are left then with Kn ≥ 1 boundary segments s3 , s¯4 i, . . . , h¯ s2Kn −1 , s¯2Kn i . h¯ s1 , s¯2 i, h¯ Denoting the union of these segments by ω b1 , we have √ b1 . |ϕ(s)| > c0 εn , ∀ s ∈ ω
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Note that in particular ϕ has a fixed sign on each of these segments. For each i s2i−1 , s¯2i i after s¯2i−1 with |ϕ(˜ s2i−1 )| = we define s˜2i−1 to be the first point in h¯ √ s2i−1 , s¯2i i with |ϕ(˜ s2i )| = 2c0 εn . Similarly, s˜2i is defined as the last point in h¯ √ 2c0 εn . Next we note that there is a lower bound for the length of each segment √ h˜ s2i−1 , s˜2i i. Indeed, since each such segment contains a point s with |ϕ(s)| = 4c0 εn dϕ and | ds | ≤ C it follows that for some c1 > 0 we have √ (4.4) |˜ s2i − s˜2i−1 | ≥ c1 εn , ∀ i . e 1 (εn ) of G as in the statement of the lemma by e1 = Ω Next we define a subset Ω 3 (s, t); s ∈ h˜ s2i−1 , s˜2i i, 0 < t < εn | log εn | + θ(εn ) , 4 i=1 (4.5) where θ(εn ) is to be determined. We set ζ(εn ) = 34 εn | log εn | + θ(εn ). Clearly Z sin2 wεn e e |∇wεn |2 + , (4.6) Eεn (uεn ; Ω1 ) ≥ Eεn ((|uεn ,h |, uεn ,z ); Ω1 ) ≥ ε2n e1 Ω e1 = Ω
K [n
Di
with Di =
e 1 as the solution on each Di of where wεn is defined on Ω sin 2wεn in Di , ∆wεn = 2ε2 n wεn ∈ (0, π/2) or wεn ∈ (−π/2, 0) in Di , (according to the sign of ϕ on h˜ s2i−1 , s˜2i i) wεn = ϕ on h˜ s2i−1 , s˜2i i , ∂w εn = 0 on ∂Di \ h˜ s2i−1 , s˜2i i . ∂ν
(4.7)
The uniqueness of the solution for (4.7) can be proved, for example, by the method of Brezis–Oswald [8]. Arguing as in the proof of Lemma 2.1 we find that Z Z sin2 wεn 2 2 |∇wεn | + ≥ ∇(1 − cos wεn ) · V (x) ε2n εn Di Di Z 2 (1 − cos wεn )V (x) · ν = εn ∂Di Z 2 (1 − cos wεn )div V (x) − εn Di Z 2 (1 − cos ϕ) ≥ εn h˜s2i−1 ,˜s2i i Z 2 (1 − cos wεn ) − εn ∂Di ∩{t=ζ(εn )} Z C (1 − cos wεn ) . (4.8) − εn Di
´ and I. SHAFRIR N. ANDRE
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Above we used the fact that V is a tangent vector field on the two curves ∂Di ∩{s = s˜2i−1 } and ∂Di ∩ {s = s˜2i }. In order to bound the two last integrals of (4.8), we shall construct a super solution Wεn for the problem (4.7) (we can assume without loss of generality that ϕ > 0 on h˜ s2i−1 , s˜2i i, the other case is treated similarly). First we set 2t 3 on Di ∩ t ≤ εn | log εn | . Wεn = arctan csch 3εn 4 √ Note that since Wεn ≥ c εn for some c > 0, we have ∆Wεn =
2 4 sin 2Wεn 1 sin 2Wεn div V < + · · , 9ε2n 2 3εn cosh 2t 2ε2n 3εn
(4.9)
for small enough εn , where V is the vector field as above. Next we want to complete the definition of Wεn on Di ∩ {t ∈ ( 34 εn | log εn |, 34 εn | log εn | + θ(εn ))} in such a way that the normal derivative on ∂Di ∩ {t = ζ(εn )} is zero (and to decide on the value of θ(εn )). For that matter we define a function j = jεn as the solution of 25 00 j (t) = 36ε2 j(t) for t ∈ (0, ∞) , n j(0) = 1 , (4.10) 3 0 ε | log ε | W n n j 0 (0) = εn 4 . Wεn 34 εn | log εn | Note that by Wε0n we denote the derivative with respect to t as we can view Wεn also as a function of the variable t only. The solution to (4.10) is given by j(t) = αe− 6εn + βe 6εn , 5t
5t
where α and β satisfy = 1, α + β Wε0n 5(β − α) = 6ε Wεn n Since Wε0n
3 4 εn | log εn | 3 4 εn | log εn |
.
(4.11)
3 2 1 εn | log εn | = − , · 4 3εn cosh(| log εn |/2)
the second equation in (4.11) can be written approximately as α − β ' 0.8. So an approximation to the solution of (4.11) is given by α ' 0.9
β ' 0.1 .
Now θ(εn ) is defined as the (unique) zero of j 0 . A simple calculation gives e
5θ(εn ) 3εn
=
α , β
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so θ(εn ) is of the order of εn . Next, on Di ∩ {t ∈ ( 34 εn | log εn |, 34 εn | log εn | + θ(εn ))} we extend Wεn by 3 3 εn | log εn | . (4.12) Wεn (t) = j t − εn | log εn | · Wεn 4 4 The last two equations in (4.10) ensure that Wεn ∈ C 1 (D i ). We claim that it is a super solution for (4.7). Note first that Wεn =
π >ϕ 2
on h˜ s2i−1 , s˜2i i ,
and that the normal derivative of Wεn is zero on ∂Di ∩ G by construction. We need to show that sin 2Wεn on Di . (4.13) ∆Wεn < 2ε2n By (4.9) we already know (4.13) on Di ∩{t < 34 εn | log εn |}. On Di ∩{t > 34 εn | log εn |} we have 3 25 3 0 εn | log εn | divV . Wε − j t − εn | log εn | Wεn (4.14) ∆Wεn = 36ε2n n 4 4 Since Wεn ' sin(2Wεn )/2 and j 0 /j ≤ C/εn (4.13) follows. √ Next, since Wεn (ζ(εn )) = O( εn ) it follows that Z (1 − cos Wεn ) ≤ Cεn . e1 ∩{t=ζ(εn )} ∂Ω By a direct calculation we have also Z (1 − cos Wεn ) ≤ Cεn . e Ω1
(4.15)
(4.16)
Summing (4.8) on i, using (4.15) and (4.16) we are led to Kn Z sin2 wεn 2 X |∇wεn | + ≥ (1 − cos ϕ) − C . ε2n εn i=1 h˜s2i−1 ,˜s2i i e1 Ω
Z
2
But since by our construction √ |ϕ(s)| ≤ 4c0 εn
for s ∈ /
K [n
h˜ s2i−1 , s˜2i i ,
i=1
we have also Z e Ω1 hence the result.
|∇wεn |2 +
sin2 wεn 2 ≥ 2 εn εn
Z (1 − cos ϕ) − C , ∂G
´ and I. SHAFRIR N. ANDRE
674
By the upper bound of Lemma 2.1, Lemma 2.5, and Lemma 4.1 we deduce that e 1 ) ≤ 2πd| log εn | + C . Eεn (uεn ; G \ Ω . e 1 (denoting for short vh = e2 = G\Ω Using the second decomposition (2.18) on Ω uεn ,h /ρh , etc.) yields 2 Z Z 1 ρz 2 2 2 ρ (1 − |v | ) ≤ 2πd| log ε | + ρ +C. h n h h ∇ 2 εn Ω ρh e2 e2 Ω
(4.17)
Noting that the same proof of Lemma 4.1 applied to ρ instead of uεn gives Z 2 e (1 − |gh |) − C , Eεn (ρ; Ω1 ) ≥ εn ∂G and combining it with Lemma 2.1 we are led to e 2) ≤ C . Eεn (ρ; Ω
(4.18)
Alternatively, we could have used Proposition 2.1 to deduce (4.18). Since ρ2h
2 2 ∇ ρz = |∇ϕεn | ≤ C|∇ϕεn |2 , ρh cos2 ϕεn
(4.19)
we deduce from (4.17)–(4.19) that also Z 1 ρ2h (1 − |vh |2 ) ≤ 2πd| log εn | + C . ε2n Ω e2
(4.20)
The estimate (4.20) will enable us to choose, using Fubini theorem, a strictly larger e 1 such that domain Ω1 ⊃ Ω Z (1 − |vh |2 ) ∂ρz (4.21) ∂ν ≤ C , |ρz | ∂Ω1 so that applying the first decomposition (2.14) on Ω1 will produce a bounded boundary integral. We start by noting that the same argument which led to (4.4) shows also that for some constant c1 > 0 √ |¯ s2i−1 − s˜2i−1 | ≥ c1 εn
√ and |¯ s2i − s˜2i | ≥ c1 εn ,
for all i .
(4.22)
. z s, t); t ∈ [0, εn | log εn |]} Next we want to estimate the ratio ρ1z ∂ρ ∂s on the curve γ = {(˜ S s2i−1 , s¯2i i. We claim that for an arbitrary s˜ ∈ i h¯ 1 ∂ρz C (4.23) ρz ∂s ≤ √εn on γ . s, 0) > 0 we use Proposition 2.1 to Assuming without loss of generality that sgn gz (˜ deduce that (ε ) (ε ) (4.24) ϕεn = ϕ0 n + εn ψ1 n + O(ε2n ) in C k (G), ∀ k ,
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and by the same proposition we have (dropping εn for simplicity): t √ ϕ0 ≥ c2 εn e− εn
|ψ1 | + |∇ψ1 | ≤ Ce− εn t
and
Hence (4.23) will follow once we show that 1 ∂ϕ0 C ϕ0 ∂s ≤ √εn Now on γ we have
on γ .
on γ .
α0 (˜ s) ∂ϕ0 (˜ s, t) = − ∂s cosh α(˜ s) +
t εn
(4.25)
(4.26)
,
(4.27)
and in particular for t = 0 α0 (˜ s) ∂ϕ0 (˜ s, 0) = − . ∂s cosh α(˜ s) s) Since | dϕ(˜ ds | ≤ C we get from (4.27)–(4.28) that on γ ∂ϕ0 cosh α(s) C ≤C ≤ √ (˜ s , t) ∂s cosh(α(s) + εtn ) εn cosh α(s) +
(4.28)
t εn
.
Finally, (4.26) follows (hence so does (4.23)) since s, t) ≥ ϕ0 (˜
c
cosh α(˜ s) +
t εn
for some c > 0 .
Now by (4.23), (4.22), (4.20) and Fubini theorem there exists c3 ∈ (0, c1 /2) such √ s2i−1 , s˜2i−1 i with |ˆ s − s¯2i−1 | = c3 εn that, defining sˆ2i−1 to be the point in h¯ √ 2i−1 and sˆ2i to be the point in h˜ s2i , s¯2i i with |ˆ s2i − s¯2i | = c3 εn for all i, we have 2K Xn j=1
Z {s=ˆ sj ,t∈(0,εn | log εn |)}
C (1 − |vh |2 ) ∂ρz ≤ √ ε3/2 | log εn | = Cεn | log εn | ≤ C . |ρz | ∂ν εn n (4.29)
We set then ω1 =
K [n
hˆ s2i−1 , sˆ2i i .
i=1
Using Proposition 2.1 as above it is not difficult to see that 1 ∂ρz C ρz ∂t ≤ ε for s ∈ ω1 and t ∈ (0, εn | log εn |) .
(4.30)
Again by (4.30), (4.20) and Fubini theorem we deduce the existence of c4 ∈ (3/4, 1) such that Z (1 − |vh |2 ) ∂ρz (4.31) ∂t ≤ C . |ρz | {s∈ω1 , t=c4 εn | log εn |}
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Setting Ω1 = {(s, t); s ∈ ω1 , t ∈ (0, c4 εn | log εn |)}, we establish the estimate (4.21) by combining (4.29) with (4.31). Proof of Theorem 1 completed. We apply the same arguments of Sec. 3 with Ω1 replacing Γcεn | log εn | there. Using Proposition 2.1 we find, as in (3.3), that |∇ρz | C 1 ≤ ≤ Cεn ρz εn
on Ω1 .
(4.32)
Now using the first decomposition (2.14) on Ω1 and applying Lemma 3.1 and (4.21) we find that for some k > 0 we have Z k 1 2 ρh |∇vh |2 + 2 (1 − |vh |2 )2 − C . (4.33) Eεn (uεn ; Ω1 ) ≥ Eεn (ρ; Ω1 ) + 4 ε Ω1 Setting Ω2 = G \ Ω1 we get by applying the second decomposition (2.18) on Ω2 : 2 Z Z ρ2 ρz ρ2h |∇vh |2 + h2 (1 − |vh |2 ) − ρ2h ∇ . Eεn (uεn ; Ω2 ) ≥ Eεn (ρ; Ω2 ) + ε ρh Ω2 Ω2 (4.34) But using (4.18) and (4.19) we find that 2 Z Z ρz 2 e 2) ≤ C , ρh ∇ ≤C |∇ϕεn |2 ≤ CEεn (ρ; Ω2 ) ≤ CEεn (ρ; Ω ρh Ω2 Ω2 hence (4.34) leads to
Z
Eεn (uεn ; Ω2 ) ≥ Eεn (ρ; Ω2 ) +
ρ2h |∇vh |2 + Ω2
ρ2h (1 − |vh |2 ) − C . ε2
(4.35)
Combining (4.35) with (4.33) we finally find that for some positive k we have Z 1 k ρ2 |∇vh |2 + 2 (1 − |vh |2 )2 Eεn (uεn ) − Eεn (ρ) ≥ 4 Ω1 h εn Z k + ρ2h |∇vh |2 + 2 (1 − |vh |2 )2 − C ε Ω2 n Z k 1 ρ2 |∇vh |2 + 2 (1 − |vh |2 )2 ≥ 4 Γc4 εn | log εn | h εn Z k + ρ2h |∇vh |2 + 2 (1 − |vh |2 )2 − C . (4.36) ε Gc εn | log εn | n 4
Defining fεn similarly to (3.14), i.e. by δ(x) ρh (x) , fεn (x) = η c4 εn | log εn |
(4.37)
with η defined in (3.13), we may apply Proposition 3.1 to establish the conclusion of Theorem 1. We conclude this section with the following useful estimates:
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677
Proposition 4.1. |∇vh | ≤ and 1 ε2n
C . εn
(4.38)
Z (1 − |vh |2 )2 ≤ C .
(4.39)
G
Proof. To prove (4.38) we use a rescaling argument as in [4, 23]. Note that the Euler–Lagrange equation for uεn reads −∆uεn = |∇uεn |2 uεn +
1 2 (u uε − uεn ,z (0, 0, 1)) . ε2n εn ,z n
(4.40)
Fixing an arbitrary point x0 ∈ G, we define a rescaled sequence by Uεn (x) = uεn (x0 + εn x)
on ε−1 n {(G ∪ B(x0 , εn )) − x0 } .
Uεn is thus a solution of −∆Uεn = |∇Uεn |2 Uεn + Uε2n ,z Uεn − Uεn ,z (0, 0, 1) .
(4.41)
It is well known (see for example Schoen and Uhlenbeck [19, 20]) that a uniform gradient estimate (4.42) |∇Uεn | ≤ C , holds for equations of the type (4.41). Rescaling back, we find that |∇uεn | ≤ hence |∇vh | ≤
C , εn
(4.43)
|∇uεn | |∇ρh | C + ≤ , ρh ρ2h εn
and (4.38) follows. For the proof of (4.39) we first deduce from (4.36), using Lemmas 2.5 and 2.1, and Proposition 3.1, that for some k > 0 there holds: Z Z k 2 2 fεn |∇vh | + 2 (1 − |vh |2 )2 2πd| log εn | − C ≤ εn G G ≤ Eεn (uεn ) − Eεn (ρ) ≤ 2πd| log εn | + C . But again by Proposition 3.1 we have Z Z k fε2n |∇vh |2 + 2 (1 − |vh |2 )2 ≥ 2πd| log εn | − C . 2εn G G Combining (4.44) with (4.45) we are led to (4.39).
(4.44)
(4.45)
´ and I. SHAFRIR N. ANDRE
678
5. The Convergence Result In this section we shall prove our main convergence result Theorem 1. A large part of the proof follows the lines of [2] so some of the details will be omitted. In order to emphasize the dependence of our sequence of minimizers in εn we will not omit εn any more and we will write uεn = (uεn ,h , uεn ,z ) = (ρεn ,h vεn ,h , ρεn ,h wεn ,z ) . Having to our disposal the two basic estimates (4.38) and (4.39), we may apply the argument of Chap. IV of [5] to prove that the set {x ∈ G; |vεn ,h (x)| < 1/2} can be covered by a finite number of discs, called “bad discs”: B(xε1n , λεn ), B(xε2n , λεn ), . . . , B(xεNn , λεn )
(5.1)
with xεi n ’s in G and N, λ which are independent of εn (we may pass to a subsequence if necessary). We denote by a1 , . . . , aN1 ∈ G the distinct limits of the centers of the bad discs. Then we set Λk = {i ∈ {1, 2, . . . , N }; xεi n → ak } . The argument of Lemma V.1 in [5] shows that |deg (vεn ,h , ∂(G ∩ B(xεi n , λεn )))| ≤ C
uniformly in n .
Passing to a further subsequence if necessary we may assume that for all i the degree di = deg(vεn ,h , ∂(G ∩ B(xεi n , λεn ))) is independent of n and then set X di , κj =
∀ j = 1, . . . , N1 .
i∈Λj
We next fix ν > 0 such that ν < min{{|ai − aj |/2; i 6= j} , {δ(ai )/2; ai ∈ G}} . As in Sec. 4 of [2] we get (with fεn defined by (4.37)) Z fε2n |∇vεn ,h |2 ≥ 2π|κj || log(ν/εn )| + |κj |bj,n − C,
for aj ∈ ∂G
(5.2)
G∩B(aj ,ν)
with limn→∞ bj,n = +∞, and Z fε2n |∇vεn ,h |2 ≥ 2π|κj || log(ν/εn )| − C,
for aj ∈ G .
(5.3)
B(aj ,ν)
Combining (5.2) with (5.3) we are led to Z G
fε2n |∇vεn ,h |2
≥ 2π
N1 X j=1
|κj || log εn | +
X aj ∈∂G
|κj |bj,n − C .
(5.4)
ON NEMATICS STABILIZED BY A LARGE EXTERNAL FIELD
Since
PN1 j=1
679
κj = d, we may use (5.4) and the upper bound of (4.44) to infer that κj = 0,
for aj ∈ ∂G
κj ≥ 0,
for aj ∈ G .
and Setting for each γ ≤ ν
[ . B(aj , γ) , Dγ = G \
(5.5)
aj ∈G
we get that
Z |∇vεn ,h |2 ≤ C(γ) .
(5.6)
Dγ
By (5.6) and (4.39) we deduce the weak convergence of a subsequence of vεn ,h , in 1 (G \ {aj ∈ G}) (i.e. in H 1 (K) for every compact K ⊂ G which does not contain Hloc 1 (G \ {aj ∈ G}, S 1 ). Taking into any of the points {aj ∈ G}) to a limit u∗ ∈ Hloc 2 2 2 2 account the terms ρh |∇wz | and ρh (1 − |uh | ) in (2.18) we may now conclude that Z Z |∇wεn ,z |2 + 1 − |uεn ,h |2 ≤ C(K), for every compact K ⊂ G \ {aj ∈ G} . K
K
(5.7) Using (2.18), (5.6) and (5.7) yields Eεn (uεn ; K) ≤ C(K),
for every compact K ⊂ G \ {aj ∈ G} .
(5.8)
1 (G \ {aj ∈ G}) (when we It follows then that uεn converges weakly to u∗ in Hloc identify u∗ with (u∗ , 0)). Actually, as the next lemma shows, we can improve the last weak convergence to a strong one. 1 (G \ {aj ∈ G}) and in Lemma 5.1. uεn converges to u∗ strongly in Hloc \ {aj ∈ G}).
0 (G Cloc
Proof. We apply a modification of an argument due to Lin [13]. Fix any disc B(x0 , R) ⊂ G with min(δ(x0 ), dist(x0 , {aj ∈ G}) > 2R. It would be enough to prove the convergence in both H 1 -norm and C 0 -norm on B(x0 , R). By (5.8) and Fubini theorem we may find R0 ∈ (R, 2R) such that ! Z 2 u ,z ε n |∇uεn |2 + 2 ≤ C for all n . (5.9) εn ∂B(x0 ,R0 ) Clearly (5.9) implies a uniform bound for {uεn } in H 1 (∂B(x0 , R0 )) and C 1/2 (∂B(x0 , R0 )). Since |vεn ,z | ≥ 1/2 on B(x0 , R0 ) (for large n) we may write uεn ,h = |uεn ,h |eiψεn on ∂B(x0 , R0 ) and deduce that uεn → u∗ = eiψ∗
strongly in C 0 (∂B(x0 , R0 )) and H 1/2 (∂B(x0 , R0 )) .
We shall denote by ψεn and ψ∗ also the harmonic extensions of ψεn and ψ∗ respectively inside B(x0 , R0 ) (so that u∗ = eiψ∗ in B(x0 , R0 )).
´ and I. SHAFRIR N. ANDRE
680
Next we define a new sequence of S 2 -valued maps Uεn = (Uεn ,h , Uεn ,z ) on B(x0 , R0 ), using polar coordinates centered at x0 : q r − (R0 − εn ) uεn ,z (R0 , θ) and Uεn ,h (r, θ) = 1 − |Uεn ,z |2 eiψεn Uεn ,z (r, θ) = εn for R0 − εn ≤ r ≤ R0 , and Uεn (r, θ) = (eiψεn , 0) for 0 ≤ r < R0 − εn . Using (5.9) a direct computation shows that Z Eεn (Uεn ; B(x0 , R0 )) ≤
|∇ψεn |2 + Cεn .
B(x0 ,R0 )
Since ψεn → ψ∗ strongly in H 1 (B(x0 , R0 )) it follows that Z Z |∇u∗ |2 ≤ lim |∇uεn |2 ≤ lim Eεn (uεn ; B(x0 , R0 )) B(x0 ,R0 )
B(x0 ,R0 )
Z
≤ lim Eεn (Uεn ; B(x0 , R0 )) ≤
|∇ψ∗ |2 .
(5.10)
B(x0 ,R0 )
From (5.10) we deduce that u∗ = eiψ∗ (hence u∗ is a smooth harmonic map on B(x0 , R0 )) and the strong H 1 -convergence follows as well. From (5.10) we also deduce that Z 1 u2 → 0 . (5.11) ε2n B(x0 ,R0 ) εn ,z By (4.43) and (5.11) we may now conclude as in [4] that uεn ,z → 0
uniformly on B(x0 , R0 ) .
(5.12)
Writing uεn ,h = |uεn ,h |eiφεn , we deduce from (4.40) that φεn satisfies div (|uεn ,h |2 ∇φεn ) = 0
on B(x0 , R0 ) .
(5.13)
Equation (5.13) can be rewritten as −div (|uεn ,h |2 ∇(φεn − ψ∗ )) = div ((|uεn ,h |2 − 1)∇ψ∗ ) .
(5.14)
The uniform convergence of φεn to ψ∗ (and hence also the one of uεn to u∗ ) follows from (5.14) (as in [4, p. 143]) by standard elliptic estimates since |uεn ,h | → 1 in C 0 (B(x0 , R0 )) and φεn → ψ∗ in C 0 (∂B(x0 , R0 )). Applying the argument of [5, Ch. VI] we see that κj = 1,
for aj ∈ G .
(5.15)
We conclude in particular that {aj ∈ G} consists of exactly d points and we may assume then that {aj ∈ G} = {a1 , . . . , ad } ,
ON NEMATICS STABILIZED BY A LARGE EXTERNAL FIELD
681
and denote a = (a1 , . . . , ad ). Next we are going to identify the singular limit u∗ . Recall that the canonical harmonic map associated to the configuration of d distinct points b = (b1 , . . . , bd ) is defined as u0 = eiϕe0
d Y z − bj , |z − bj | j=1
where ϕ e0 is the harmonic function whose trace ϕ0 on ∂G satisfies eiϕ0 = g ·
d Y |z − bj | , z − bj j=1
see [5]. More precisely, the canonical map depends also on a configuration of degrees, but here and in the sequel we shall only consider the configuration of degrees (1, . . . , 1). From the proof of Lemma 5.1 we already know that u∗ ∈ C ∞ (G \ {a1 , . . . , ad }, S 1 )
(5.16)
is an harmonic map. Next we claim: Lemma 5.2. The limit u∗ coincides with u0 — the canonical harmonic map associated to a and the boundary condition gh/|gh|. 1 (G \ {a1 , . . . , ad }). Moreover the argument of Proof. Recall that u∗ ∈ Hloc Appendix IV in [5] shows that
{vεn ,h } is bounded in W 1,p (G),
∀p ∈ [1, 2) .
(5.17)
It follows that vεn ,h * u∗ (weakly) in W 1,p (G)
1,p and uεn ,h * u∗ (weakly) in Wloc (G)∀p ∈ (1, 2) .
We have in particular that u∗ ∈ W 1,p (G), ∀p ∈ (1, 2), and that the trace of u∗ on ∂G is g (as a map in W 1−1/p,p (∂G)). Note that by (4.40) uεn ,h satisfies ∂uεn ,h ∂uεn ,h ∂ ∂ uεn ,h × + uεn ,h × = 0. (5.18) ∂x ∂x ∂y ∂y 1,p (G) Passing to the limit in (5.18), using the weak convergence uεn ,h * u∗ in Wloc for p ∈ (1, 2), we deduce that u∗ satisfies ∂u∗ ∂u∗ ∂ ∂ (5.19) u∗ × + u∗ × = 0 in D0 (G) . ∂x ∂x ∂y ∂y
Since u∗ ∈ H 1 (Dγ , S 1 ) for every small γ (see (5.5) for the definition of Dγ ) it follows from a result of Bethuel and Zheng [7] that we may write locally in Dγ : u∗ = eiϕ∗ with ϕ∗ in H 1 . It follows then from (5.19) that ∆ϕ∗ = 0. Hence u∗ ∈ C ∞ (G \ {a1 , . . . , ad }, S 1 ) is an harmonic map which equals gh /|gh | on ∂G (before we only knew (5.16)). Since also u∗ ∈ W 1,p (G), ∀p ∈ (1, 2), it follows from Remark 1.2 in [5] that u∗ coincides with the canonical harmonic map u0 .
´ and I. SHAFRIR N. ANDRE
682
In order to complete the proof of Theorem 2, we need to show that the configuration a minimizes the renormalized energy W (gh /|gh |, b) over all possible configurations b of d distinct points in G. We refer the reader to [5] for the definition and an explicit formula for W . We just mention here the formula: ( Z ) 1 1 2 |∇ub | − πd log , (5.20) W (gh /|gh |, b) = lim η→0 2 Aη η where ub is the canonical harmonic map associated to the boundary condition gh /|gh | and the configuration b, and where for each η > 0 we denote Aη = G \ d S B(bj , η). Analogously to [5] we define for every ε, δ > 0: j=1
I(ε, δ) = min{Eε (u); u ∈ H 1 (B(0, δ), S 2 ), u = (x/|x|, 0) on ∂B(0, δ)} . As in Ch. III of [5] we see easily by rescaling that if we define I(t) = I(t, 1)∀t > 0 then I(ε, δ) = I(ε/δ). Repeating the proof of Lemma III.1 of [5] we see also that I(t) + 2π log t
is nondecreasing for t > 0 .
(5.21)
Finally we note that from the results of [5] it follows that for some constant C: I(ε) + 2π log ε ≥ −C,
for all ε > 0 .
(5.22)
Indeed, let u be any map in H 1 (B(0, 1), S 2 ) satisfying u = (x/|x|, 0) on ∂B(0, 1). Then by Th. V.3 in [5] we have for all ε > 0: Z Z u2z (1 − |uh |2 )2 1 2 |∇u| + 2 ≥ |∇uh |2 + ≥ 2πd log − C . Eε (u) = 2 ε ε ε B(0,1) B(0,1) From (5.21)–(5.22) we deduce that the finite limit . lim+ I(ε) + 2π log ε = γ ∈ (−∞, +∞) exists .
(5.23)
ε→0
The minimizing property of a will follow from the next two lemmas. Lemma 5.3. Let b be any configuration of d distinct points in G. Then for any small η > 0 we have lim sup (Eεn (uεn ) − Eεn (ρεn ) − dI(εn , η)) ≤ 2W (gh /|gh |, b) + 2πd log (1/η) + oη (1) , n→∞
(5.24) where here and in the sequel oη (1) stands for a quantity which goes to 0 with η. Lemma 5.4. For the configuration a consisting of the distinct limits in G of the centers of bad discs of uεn we have: for every small η > 0, there holds lim inf (Eεn (uεn ) − Eεn (ρεn ) − dI(εn , η)) ≥ 2W (gh /|gh |, a) + 2πd log (1/η) + oη (1) . n→∞
(5.25)
ON NEMATICS STABILIZED BY A LARGE EXTERNAL FIELD
683
Proof of Lemma 5.3. It suffices to construct a sequence of maps Uεn ∈ Hg1 (G, S 2 ) for which (5.24) is valid. We shall use a refinement of the construction . d B(bj , η + εn ) we fix used in the proof of Lemma 2.5. First on Aη+εn = G \ Uj=1 1 an S -valued wεn which minimizes the Dirichlet energy on Aη+εn for the boundary condition gh /|gh | on ∂G and (x − bj )/|x − bj | on each of the circles ∂B(bj , η + εn ). By Remark I.5 in [5] we have Z |∇wεn |2 = 2πd log (1/η) + 2W (gh /|gh |, b) + oη (1) + O(εn ) . (5.26) Aη+εn
On Aη+εn we set Uεn = (ρh wεn , ρz ). Using (2.18) we find Z ρ2h |∇wεn |2 . Eεn (Uεn ; Aη+εn ) = Eεn (ρεn ; Aη+εn ) + Aη+εn
By (5.26) we get lim Eεn (uεn ; Aη+εn ) − Eεn (ρεn ; Aη+εn ) = 2πd log(1/η) + 2W (gh /|gh |, b) + oη (1) .
n→∞
(5.27) . On each annulus Dj = B(bj , η + εn ) \ B(bj , η) we define Uεn by (using polar coordinates around bj ): q r−η ρz (η + εn , θ) and Uεn ,h (r, θ) = 1 − Uε2n ,z eiθ . Uεn ,z (r, θ) = εn Using the exponential decay of ρz to 0 away from ∂G it is easy to see that lim Eεn (Uεn ; Dj ) = 0,
n→∞
∀j .
(5.28)
Finally, we complete the definition of Uεn in each B(bj , η) as a minimizer realizing I(εn , η) (with the obvious change of origin). By (5.27)–(5.28) we deduce that lim (Eεn (Uεn ) − Eεn (ρεn ) − dI(εn , η)) = 2W (gh /|gh |, b) + 2πd log(1/η) + oη (1) ,
n→∞
and (5.24) follows since Eεn (uεn ) ≤ Eεn (Uεn ).
Proof of Lemma 5.4. We denote as above for each small η > 0: Aη = G \
d [
B(aj , η) .
j=1
We are going to use again our energy decompositions (2.14) and (2.18). For that matter, we shall construct a subdomain Ω1 = Ω1 (εn ) on which decomposition (2.14) will be used, while decomposition (2.18) will be used on G \ Ω1 . The construction of Ω1 is similar to the one used in Sec. 4, but slightly simpler. As in Sec. 4, Ω1 will be of the form (4.2). If gz ≡ 0 we simply take Ω1 = ∅. Otherwise we need to choose first the set ω1 . If min∂G |gz | > 0 we take ω1 = ∂G. Otherwise, we first choose by Sard lemma a constant c0 ∈ (1, 2) such that c20 η 2 εn is a regular value of the function ϕ2 for each n. Hence, the set √ {x ∈ ∂G; |ϕ(x)| = c0 η εn }
´ and I. SHAFRIR N. ANDRE
684
consists of a finite number of points s1 , . . . , sNn ∈ ∂G labeled in a clockwise manner. Next we keep only the segments hsi , si+1 i which contain some point s with |ϕ(s)| ≥ √ 4c0 η εn . We are left then with Kn ≥ 1 boundary segments s3 , s¯4 i, . . . , h¯ s2Kn −1 , s¯2Kn i . h¯ s1 , s¯2 i, h¯ As in the proof of Lemma 4.1 we define the points s˜2i−1 , s˜2i in each interval h¯ s2i−1 , s¯2i i which are the first points in this interval, from the left and the right √ respectively, where |ϕ| equals 2c0 η εn . We deduce then that for some constant c1 > 0 we have √ |˜ s2i − s˜2i−1 | ≥ c1 η εn ,
√ |¯ s2i−1 − s˜2i−1 | ≥ c1 η εn
√ and |¯ s2i − s˜2i | ≥ c1 η εn
∀i.
Arguing as in the proof of (4.29) we obtain the existence of some c2 ∈ (0, c1 /2) such s2i−1 , s˜2i−1 i and h¯ s2i , s˜2i i respectively that, defining for all i the points sˆ2i−1 , sˆ2i in h¯ as the point which satisfy √ s2i − s¯2i | = c2 η εn , |ˆ s2i−1 − s¯2i−1 | = |ˆ the following holds: 2K Xn Z j=1
{s=ˆ sj ,t∈(0,εn | log εn |)}
(1 − |vεn ,h |2 ) ∂ρz ∂ν ≤ C(η)εn | log εn | → 0 as n → ∞ . |ρz | (5.29)
We finally set ω1 =
K [n
hˆ s2i−1 , sˆ2i i .
i=1
By Proposition 2.1 we have again (4.30). But now we have better estimates on our hands, so we can do better than (4.31). We claim that we can choose c3 ∈ (3/4, 1) such that Z C (1 − |vεn ,h |2 ) ∂ρz ≤ . (5.30) |ρz | ∂t | log εn | {s∈ω1 , t=c3 εn | log εn |} Indeed, by now we know already (by (5.3) and (5.15)) that Z ρ2h |∇vεn ,h |2 ≥ 2πd| log εn | − C .
(5.31)
Gεn | log εn |/2
Arguing as in the proof of (4.20) we get that Z 1 (1 − |vεn ,h |2 ) ≤ C . ε2n Gεn | log εn |/2
(5.32)
Using (5.32) we may now apply Fubini theorem to choose c3 ∈ (3/4, 1) satisfying (5.30). Setting finally Ω1 = {(s, t); s ∈ ω1 , t ∈ (0, c3 εn | log εn |)} ,
685
ON NEMATICS STABILIZED BY A LARGE EXTERNAL FIELD
we get, applying (2.14): Z lim inf (Eεn (uεn ; Ω1 ) − Eεn (ρεn ; Ω1 )) ≥ lim inf n→∞
n→∞
ρ2h |∇vεn ,h |2 + ρ2z |∇vεn ,z |2 Ω1
Z
+ lim inf n→∞
∂Ω1
1 − |vεn ,h |2 ∂ρz . ρz ∂ν
(5.33)
From (5.29)–(5.30) we know that the boundary term in (5.33) goes to zero. Hence Z ρ2h |∇vεn ,h |2 + ρ2z |∇vεn ,z |2 ≥ 0 . lim inf Eεn (uεn ; Ω1 ) − Eεn (ρεn ; Ω1 ) ≥ lim inf n→∞
n→∞
Ω1
(5.34) Setting Ω2 = D \ Ω1 , it is not difficult to see, using Proposition 2.1, that Z |∇ρεn |2 = oη (1) .
(5.35)
Ω2
Applying the second decomposition (2.18) on Γη ∩ Ω2 and using (5.35) we get Z Eεn (uεn ; Γη ∩ Ω2 ) − Eεn (ρεn ; Γη ∩ Ω2 ) ≥ −
Γη ∩Ω2
ρ2h
2 ∇ ρz ≥ oη (1) . (5.36) ρh
Combining (5.34) with (5.36) yields: lim inf Eεn (uεn ; Γη ) − Eεn (ρεn ; Γη ) ≥ oη (1) . n→∞
(5.37)
eη = Aη ∩ Gη . Applying the second decomposition (2.18) on A˜η , Next we set A eη ), the C k -convergence of using (5.35), the weak convergence of vεn to u0 in H 1 (A ρh to 1 away from the boundary, and (5.20), we get that eη ) − Eεn (ρεn ; A eη )) ≥ 2W (gh /|gh |, a) + 2πd log(1/η) + oη (1) . lim inf (Eεn (uεn ; A n→∞
(5.38) In view (5.37)–(5.38) it is clear that the proof of Lemma 5.4 will be complete once we show that: lim inf (Eεn (uεn ; B(aj , η)) − Eεn (ρεn ; B(aj , η)) − I(εn , η)) ≥ oη (1), ∀ j . n→∞
(5.39)
Since clearly limn→∞ Eεn (ρεn ; B(aj , η)) = 0 by Proposition 2.2, (5.39) is equivalent to (5.40) lim inf (Eεn (uεn ; B(aj , η)) − I(εn , η)) ≥ oη (1), ∀ j . n→∞
For the proof of (5.40), we first construct for each εn an S 2 -valued map Uεn on . B(aj , 2η) which coincides with uεn on B(aj , η). Setting Dη = B(aj , 2η) \ B(aj , η) we need to describe the definition of Uεn on Dη . First, on B(aj , η + εn ) \ B(aj , η) we set, using polar coordinates around aj : q r−η uεn ,h (r, θ) . uεn ,z (r, θ) and Uεn ,h (r, θ) = 1 − Uε2n ,z · Uεn ,z (r, θ) = 1 − εn |uεn ,h |
´ and I. SHAFRIR N. ANDRE
686
Note that 1 ε2n
Z B(aj ,η+εn )\B(aj ,η)
Uε2n ,z ≤
1 ε2n
1 ≤ 2 εn
Z B(aj ,η+εn )\B(aj ,η)
u2εn ,z
Z
Dη
u2εn ,z → 0 ,
as n → ∞ ,
as we saw in (5.11). In addition, the strong convergence of uεn to u0 in H 1 (Dη ) (see Lemma 5.1) implies that Z |∇Uεn |2 → 0 , B(aj ,η+εn )\B(aj ,η)
hence lim Eεn (Uεn ; B(aj , η + εn ) \ B(aj , η)) = 0 .
n→∞
Since |uεn ,h | ≥
1 2
(5.41)
on Dη we may use (5.15) to write uεn ,h = |uεn ,h |ei(θ+ψεn )
on Dη ,
for some smooth ψεn . Note that by our construction we have Uεn = (ei(θ+ψεn ) , 0) on ∂B(aj , η + εn ) .
(5.42)
Finally we extend the definition of Uεn to B(aj , 2η) \ B(aj , η + εn ) by 2η − r ψεn and Uεn ,h (r, θ) = 0 . Uεn ,z (r, θ) = exp i θ + η − εn Note that Uεn = (eiθ , 0) on ∂B(aj , 2η), hence by definition, Eεn (Uεn ; B(aj , 2η)) ≥ I(εn , 2η) .
(5.43)
We may clearly write u0 = ei(θ+ψ0 )
on B(aj , 2η) \ {aj } ,
for some smooth harmonic ψ0 on B(aj , 2η). From the strong H 1 -convergence of uεn to u0 on Dη and (5.41) it follows that Uεn → v0
strongly in H 1 (Dη ) ,
(5.44)
r ψ0 . v0 (r, θ) = exp i θ + 2 − η
where
Note that Z Z |∇v0 |2 = 2π log 2 +
2 r ∇ 2 − ψ0 = 2π log 2 + oη (1) . η Dη
Dη
(5.45)
By (5.43)–(5.45) we are led to lim inf (Eεn (uεn ; B(aj , η)) − I(εn , 2η)) ≥ oη (1) − 2π log 2 . n→∞
(5.46)
ON NEMATICS STABILIZED BY A LARGE EXTERNAL FIELD
But by (5.23)
lim I(εn , 2η) − I(εn , η) = lim I
n→∞
n→∞
εn 2η
−I
εn η
687
= 2π log 2 ,
and the desired (5.40) clearly follows from (5.46)–(5.47).
(5.47)
6. The Case of a Variable Field In this section we are going to prove Theorem 3 which generalizes the results of the previous sections for the case of a variable field e(x). We assume then that we are given a smooth non-zero vector field e(x) with values in R3 \ {(0, 0, 0)} and a smooth boundary condition g : ∂G → S 2 satisfying (1.6). We denote for each eε defined by (1.5) over Hg1 (G, S 2 ). Recall ε > 0 by uε a minimizer for the energy E from the Introduction that we have denoted a(x) = |e(x)| and e3 (x) = e(x)/a(x) and that we have completed e3 to a right orthonormal moving frame (e1 , e2 , e3 ) on G. We also denote b(x) = a(x)2 and a0 = min a(x). We shall use frequently the G
following relations which hold on G: ei ∇ej + ej ∇ei = 0,
∀ i, j .
(6.1)
We assume that the degree d of g (as defined in the introduction) is positive, the case d = 0 is easier, see Remark I.2. Our strategy for proving Theorem 3 will be to try to reduce the case of a general e to the previous case e ≡ (0, 0, 1). For that matter, we shall first introduce a corresponding scalar problem. For each ε > 0 we shall denote by ρε = (ρε,h , ρε,z ) ∈ C 2 (G, S 1 ) a minimizer for the energy Z Z 1 |∇ρ|2 + 2 bρ2 (6.2) Fε (ρ) = ε G z G over the set 1 (G, S 1 ) = {ρ = (ρh , ρz ) ∈ H 1 (G, S 1 ); ρ = (|gh |, gz ) on ∂G} , H(|g h |,gz )
where gh , gz are defined in (1.8). As in Sec. 2 we define ϕ by (2.4) and ϕε by (2.5). The uniqueness of ρε for ε small enough follows from Proposition 7.1. The same proof as that of Lemma 2.1 yields: Z 2 a(1 − |gh |) + 0(1) . Fε (ρε ) = ε ∂G As before, we shall omit for simplicity the subscript ε and write ρ for ρε . Any admissible u ∈ Hg1 (G, S 2 ) can be represented via the frame (e1 , e2 , e3 ) as u = u 1 e1 + u 2 e2 + u 3 e3 . In order to relate the case of a variable field to the previous case it will be convenient to associate to each such u another map defined by u b = (u1 , u2 , u3 ) .
(6.3)
Of course, u b ∈ Hgˆ1 (G, S 2 ) with gˆ = (g1 , g2 , g3 ),
where g = g1 e1 + g2 e2 + g3 e3 .
(6.4)
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We are going to show that the asymptotic behavior (as ε → 0) of the energies Z Z 1 2 e |∇uε | + 2 (uε , e)2 , Eε (uε ) = ε G G for minimizers {uε } is related to the one of the Fε -energies of the corresponding maps {b uε }: Z Z 1 uε ) = |∇b uε |2 + 2 bb u2ε,3 . (6.5) Fε (b ε G G Note that the Fε energy is a slight generalization of the energy we studied in the previous case (where a(x) was constant). Motivated by our analysis of the previous sections, we define for each admissible u as above: uh = (u1 , u2 ), uz = u3 , vh = uh /ρh = (vh,1 , vh,2 ) , wz = uz /ρh
on G
(6.6)
and vz = uz /ρz
on G \ {x; ρz (x) = 0} .
(6.7)
We are going to use the two energy decompositions (2.14), (2.18) for u bε as in the previous sections, the first on a properly chosen Ω1 , and the second on Ω2 = G \ Ω1 . An obvious modification of the proof of Lemma 2.2 gives: b be the Lemma 6.1. Let u ∈ H 1 (G, S 2 ) with u = g on ∂G be given and let u map corresponding to u via (6.3). Then, for every subdomain Ω ⊆ G with smooth boundary, except (possibly) for a finite number of corners, such that ρz 6= 0 on Ω there holds Z (1 − |vh |2 ) ∂ρz . (6.8) u; Ω) = Fε (ρ; Ω) + ρ2h |∇vh |2 + ρ2z |∇vz |2 + Fε (b ρz ∂ν ∂Ω Similarly, a simple modification of the proof of Lemma 2.3 gives: b and any subdomain Lemma 6.2. For u ∈ H 1 (G, S 2 ) with the associated u Ω ⊆ G we have 2 Z Z bρ2h ρz 2 2 2 2 2 2 u; Ω) = Fε (ρ; Ω) + ρh |∇vh | + ρh |∇wz | + 2 (1 − |vh | ) − ρh ∇ . Fε (b ε ρh Ω Ω (6.9) Similarly to the upper bound of Lemma 2.5 we can now prove the following upper bound for the minimizers {uε }: Lemma 6.3.
eε (uε ) ≤ Fε (ρε ) + 2πd| log ε| + C . E
(6.10)
Proof. We let Uε = (Uε,1 , Uε,2 , Uε,3 ) be the same map constructed in the proof of Lemma 2.5 but for the boundary data gˆ defined in (6.4) (and not g). Then we
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eε which coincides with g on ∂G by define for each ε > 0 an S 2 -valued map U eε = Uε,1 e1 + Uε,2 e2 + Uε,3 e3 . U Following the calculation in the proof of Lemma 2.5 we find that Fε (Uε ) ≤ Fε (ρε ) + 2πd| log ε| + C .
(6.11)
It is easy to verify that kUε ||W 1,1 (G) ≤ C hence
∀ε ,
eε ) − Fε (Uε )| ≤ C . eε (U |E
(6.12)
The estimate (6.10) follows from (6.11) and (6.12). In connection with the first decomposition we shall use the next lemma.
Lemma 6.4. Let u ∈ H 1 (G, S 2 ) with u = g on ∂G be given with the corresponding u b via (6.3). Then, for every subdomain Ω ⊆ G with smooth boundary, except (possibly) for a finite number of corners, such that ρz 6= 0 on Ω there holds Z 1 eε (u; Ω) − Fε (b u; Ω)| ≤ C · (|∂Ω| + 1) + ρ2 |∇vh |2 + ρ2z |∇vz |2 , (6.13) |E 2 Ω h for some constant C > 0 independent of ε and u, where |∂Ω| denotes the length of ∂Ω. Proof. We write u = ρh vh,1 e1 + ρh vh,2 e2 + ρz vz e3 and then differentiate to get ∇u = ∇(ρz vz )e3 + ρz vz ∇e3 +
2 X
∇(ρh vh,j )ej + ρh vh,j ∇ej .
j=1
By a direct calculation, using (6.1), we find: u|2 + 2ρ2h (vh,2 ∇vh,1 − vh,1 ∇vh,2 )e1 ∇e2 |∇u|2 = |∇b +2
2 X
[ρz ρh (vz ∇vh,j − vh,j ∇vz ) + vh,j vz (ρz ∇ρh − ρh ∇ρz )]ej ∇e3
j=1
+ 2ρ2h vh,1 vh,2 ∇e1 ∇e2 + 2
2 X
ρh ρz vh,j vz ∇ej ∇e3
j=1 2 2 + ρ2h vh,1 |∇e1 |2 + ρ2h vh,2 |∇e2 |2 + ρ2z vz2 |∇e3 |2
= |∇b u|2 + I1 + I2 + I3 + I4 .
(6.14)
We have clearly |I3 | + |I4 | ≤ C .
(6.15)
´ and I. SHAFRIR N. ANDRE
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Also |I1 | ≤ C|∇vh | , so by Cauchy–Schwarz inequality Z Z 1 |I1 | ≤ C + ρ2h |∇vh |2 . 4 Ω Ω
(6.16)
Similarly, for the first term in brackets in I2 we get Z X Z 2 1 2 ρz ρh (vz ∇vh,j − vh,j ∇vz )ej ∇e3 ≤ C + ρ2 |∇vh |2 + ρ2z |∇vz |2 . (6.17) 8 Ω h Ω j=1 Finally, for the last term of I2 we use the identity ρz ∇ρh − ρh ∇ρz = −∇ϕε ,
(6.18)
the estimate |ϕε |/C ≤ |ρz | ≤ C|ϕε | and integration by parts to conclude that Z Z X 2 ≤ C ·(1+|∂Ω|)+ 1 v v (ρ ∇ρ − ρ ∇ρ ) e ∇e ρ2 |∇vh |2 +ρ2z |∇vz |2 . 2 h,j z z h h z j 3 8 Ω h Ω j=1 (6.19) Combining (6.14) with (6.15)–(6.17) and (6.19) we are led to the result. Next we suppose we are given a sequence of minimizers {uεn } in Hg1 (G, S 2 ) with the associated {b uεn } via (6.3). We are looking for a lower bound eεn (uεn ) ≥ Fεn (ρεn ) + 2πd| log εn | − C . E
(6.20)
We shall need the following generalization of Lemma 4.1: e 1 of G of the form Lemma 6.5. There exists a subdomain Ω 3εn e 1 = (s, t); s ∈ ω e1 , t < | log εn | + θ(εn ) , Ω 4a0 for some subset ω e1 ⊂ ∂G and some number θ(εn ) of the order εn such that Z e 1) ≥ 2 eεn (uεn ; Ω a(1 − |gh |) − C . (6.21) E εn ∂G Proof. The proof is a slight variant of the proof of Lemma 4.1. The necessary modifications are explained below. Using the same construction which led to (4.5) we define ˜1 = Ω
K [n i=1
Di
with Di =
3εn (s, t); s ∈< s˜2i−1 , s˜2i >, 0 < t < | log εn | + θ(εn ) , 4a0 (6.22)
ON NEMATICS STABILIZED BY A LARGE EXTERNAL FIELD
691
(with θ(εn ) to be determined). Next, we define wεn as in (4.7) with the only difference that the first equation in (4.7) is replaced by ∆wεn = b(x)
sin 2wεn . 2ε2n
(6.23)
With this modification (4.8) now reads Z |∇wεn |2 + b Di
sin2 wεn 2 ≥ ε2n εn
Z h˜ s2i−1 ,˜ s2i i
2 − εn C − εn
a(1 − cos ϕ)
Z
n ∂Di ∩ t= 3ε 4a | log εn |
a(1 − cos wεn )
0
Z
a(1 − cos wεn ) .
(6.24)
Di
The super solution Wεn for this modified (4.7) is defined first on Di ∩ n {t ≤ 3ε 4a0 | log εn |} by 2a0 t Wεn = arctan csch . 3εn We define j = jεn as the solution of the following modification of (4.10): 25a20 j 00 (t) = j(t) for t ∈ (0, ∞) , 36ε2n j(0) = 1 , 3εn 0 Wεn | log εn | 4a 0 . 0 (0) = j 3εn Wεn | log εn | 4a0
(6.25)
Then θ(εn ) is defined as the unique zero of j 0 . As in the proof of Lemma 4.1 we 3εn n now extend the definition of Wεn to Di ∩ {t ∈ ( 3ε 4a0 | log εn |, 4a0 | log εn | + θ(εn ))} by the analogue of (4.12) i.e. 3εn 3εn | log εn | · Wεn | log εn | . Wεn (t) = j t − 4a0 4ao As in the proof of Lemma 4.1 we verify that Wεn is indeed a super solution. This enables us to conclude as before that Z Z sin2 wεn 2 |∇wεn |2 + b ≥ a(1 − cos ϕ) − C . (6.26) ε2n εn ∂G e1 Ω e 1 the S 1 -valued map Let us now define on Ω ρh , ρ˜z ) = (cos wεn , sin wεn ) . ρ˜εn = ρ˜ = (˜
´ and I. SHAFRIR N. ANDRE
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e 1 , S 1 ) satisfying the boundClearly ρ˜ is the minimizer for Fεn (h) for maps h ∈ H 1 (Ω ary condition e 1 ∩ ∂G . h = (cos ϕ, sin ϕ) on ∂ Ω Note that in particular ∂ ρ˜ =0 ∂ν
e1 ∩ G . on ∂ Ω
(6.27)
We can now rewrite (6.26) as e 1) ≥ ρ; Ω Fεn (˜
2 εn
Z a(1 − |gh |) − C .
(6.28)
∂G
e 1 with ρ˜ taking the place of ρ. The only Next we apply Lemma 6.4 to uεn on Ω special property of ρ used in the proof of Lemma 6.4 was (6.18), which for ρ˜ reads ρh − ρ˜h ∇˜ ρz = −∇wεn , so the same proof applies here. We set ρ˜z ∇˜ v˜h = uεn ,h /ρ˜h
and v˜z = uεn ,z /ρ˜z
e 1 | ≤ C (since Kn ≤ C/√εn by (4.4)) we get and noting that by construction |∂ Ω Z e 1 ) ≥ Fεn (b e 1) − 1 eεn (uεn ; Ω uεn ; Ω ρ˜2h |∇˜ vh |2 + ρ˜2z |∇˜ vz |2 − C . (6.29) E 2 Ω e1 Note that in the proofs of Lemmas 2.2 and 6.1 we used only the fact that ρ solves the corresponding Euler–Lagrange equation. Since ρ˜ too solves the same equation (the obvious modification of (2.15) for the Fε -energy) we may apply Lemma 6.1 for e 1 but with ρ˜ instead of ρ to get uεn on Ω Z e e uεn ; Ω1 ) = Fε (˜ ρ ; Ω1 ) + ρ˜2h |∇˜ vh |2 + ρ˜2z |∇˜ vz |2 . (6.30) Fεn (b e1 Ω Note that there is no boundary term thanks to (6.27). By (6.28)–(6.30) we are led to (6.21). e 1 we deduce from Lemmas 6.3 and 6.5 that e2 = G \ Ω Setting Ω Z 1 eεn (uεn ; Ω e 2 ) ≤ 2πd| log εn | + C . u2εn ,z ≤ E ε2n Ω e2 Applying the argument of Sec. 4 (see the proof of (4.21) there) we establish the e 1 of the form existence of a domain Ω1 ⊃ Ω Ω1 = {(s, t); s ∈ ω1 , t ∈ (0, c4 εn | log εn |)} , with c4 ∈ ( 4a30 , a10 ) and ω1 ⊆ ∂G such that Z ∂Ω1
u2εn ,z ∂ρz | |≤C. |ρz | ∂t
(6.31)
Next we note that u2εn ,z = 1 − ρ2h |vεn ,h |2 = ρ2h (1 − |vεn ,h |2 ) + 1 − ρ2h .
(6.32)
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We do not know whether |vεn ,h | ≤ 1 everywhere on G (an estimate that was valid in the case e ≡ (0, 0, 1), see Lemma 2.4). But we do have clearly |vεn ,h | ≤ 1/ρh . . On Ω2 = G \ Ω1 we have 1 − ρ2h = ρ2z ≤ Cεn . Hence on Ω2 : 1 − |vεn ,h |2 ≥ 1 −
1 ≥ −Cεn . ρ2h
(6.33)
By (6.31)–(6.33), together with the analogue estimates to (4.30), (4.23), we obtain finally, Z 1 − |vεn ,h |2 ∂ρz ≤C. (6.34) ρz ∂ν ∂Ω1 In the next lemma we prove a lower bound for the energy on Ω1 (an analogue to (4.33)). Lemma 6.6. For some k > 0 we have Z k 1 2 eεn (uεn ; Ω1 ) ≥ Fεn (ρ; Ω1 ) + ρh |∇vεn ,h |2 + 2 (1 − |vεn ,h |2 )2 − C . E εn Ω1 4
(6.35)
Proof. Applying Lemma 6.1 to uεn on Ω1 , taking into account (6.34), and using Lemma 6.4, noting that |∂Ω1 | ≤ C, gives Z 1 e ρ2 |∇vεn ,h |2 + ρ2z |∇vεn ,z |2 − C . (6.36) Eεn (uεn ; Ω1 ) ≥ Fεn (ρ; Ω1 ) + 2 Ω1 h Using an analogue estimate to (4.32) together with Lemma 3.1 yields (as in the proof of (4.33)): Z Z k 1 2 1 2 2 2 2 ρh |∇vεn ,h |2 + 2 (1 − |vεn ,h |2 )2 , ρ |∇vεn ,h | + ρz |∇vεn ,z | ≥ 2 Ω1 h 4 ε Ω1
and (6.35) follows from (6.36). Next we look for an estimate of the energy on Ω2 .
Lemma 6.7. For some c1 > 0 we have Z Z k 2 2 2 2 e ρh |∇vεn ,h | + 2 (1−|vεn ,h | ) −c1 |∇vεn ,h |−C . Eεn (uεn ; Ω2 ) ≥ Fεn (ρ; Ω2 )+ εn Ω2 Ω2 (6.37) Proof. By a direct calculation we find that for some constant c1 > 0, depending only on the frame (e1 , e2 , e3 ), we have uεn |2 | ≤ c1 |∇b uεn | . ||∇uεn |2 − |∇b Applying Lemma 6.2 yields eεn (uεn ; Ω2 ) ≥ Fεn (ρ; Ω2 ) + E
Z ρ2h |∇vεn ,h |2 + ρ2h |∇wεn ,z |2 Ω2
+
bρ2h (1 ε2n
(6.38)
Z
− |vεn ,h | ) − c1
|∇b uεn | − C .
2
Ω2
(6.39)
´ and I. SHAFRIR N. ANDRE
694
Above we used (6.38) and the estimate Z Z ρz ≤ C ρ2h ∇ |∇ϕεn |2 ≤ CFεn (ρ; Ω2 ) ≤ C ρ h Ω2 Ω2
(see (4.18)–(4.19)) . (6.40)
By (6.40) we have also Z Z 1 1 2 2 bρ (1 − |vεn ,h | ) ≥ 2 b(ρ2h − 1) ε2n Ω2 ∩{|vεn ,h |>1} h εn Ω2 ∩{|vεn ,h |>1} Z 1 bρ2 ≥ −C . ≥− 2 εn Ω2 z Since 1 − |vεn ,h |2 ≥ (1 − |vεn ,h |2 )2 when |vεn ,h | ≤ 1 we get Z Z 1 1 2 2 bρ (1 − |v | ) ≥ bρ2 (1 − |vεn ,h |2 )2 − C . ε ,h n ε2n Ω2 h ε2n Ω2 h Replacing k of (6.35) by min(k, a20 min|gh |2 ) we get from (6.39) that ∂G
eεn (uεn ; Ω2 ) ≥ Fεn (ρ; Ω2 ) + E
Z ρ2h |∇vεn ,h |2 + ρ2h |∇wεn ,z |2 Ω2
+
k (1 − |vεn ,h |2 )2 − c1 ε2n
Z |∇b uεn | − C .
(6.41)
Ω2
Since |∇b uεn | ≤ ρh |∇vεn ,h | + ρh |∇wεn ,z | + |vεn ,h ||∇ρh | + |wεn ,z ||∇ρh | , and (6.40) implies in particular that Z |∇ρh | ≤ C , Ω2
while clearly ρ2h |∇wεn ,z |2 − c1 ρh |∇wεn ,z | ≥ −C , we see that (6.41) implies (6.37).
Next we define fεn as in (4.37). Combining Lemma 6.6 with Lemma 6.7 we get Z Z k eεn (uεn )−Fεn (ρ) ≥ fε2n |∇vεn ,h |2 + 2 (1−|vεn ,h |2 )2 −c1 |∇vεn ,h |−C . (6.42) E εn G G Combining (6.42) with the upper bound (6.10) yields: Z Z k fε2n |∇vεn ,h |2 + 2 (1 − |vεn ,h |2 )2 − c1 |∇vεn ,h | ≤ 2πd| log εn | + C . εn G G
(6.43)
We shall next use the following result of Sandier [17] (it also follows from the method of Jerrard [11]):
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Lemma 6.8 (Sandier). Let a smooth boundary data e g : ∂G → S 1 of degree g) d > 0 be given. Then, for every λ, c2 > 0 there exists a constant C0 = C0 (λ, c2 , G, e such that for every u ∈ Hg˜1 (G, C) and every ε > 0 we have Z λ |∇u|2 + 2 (1 − |u|2 )2 − c2 |∇u| ≥ 2πd log(1/ε) − C0 . (6.44) ε G Applying Lemma 6.8 to vεn ,h for the boundary condition ge = gh /|gh |, and some c2 > c1 to be chosen later, we get Z k |∇vεn ,h |2 + 2 (1 − |vεn ,h |2 )2 − c2 |∇vεn ,h | ≥ 2πd| log εn | − C . (6.45) ε G n Subtracting (6.43) from (6.45) yields Z Z (1 − fε2n )|∇vεn ,h |2 − (c2 − c1 ) |∇vεn ,h | ≥ −C , G
or equivalently
G
Z c1
c1 |∇vεn ,h | ≤ c 2 − c1 G
Z G
(1 − fε2n )|∇vεn ,h |2 + C .
Plugging (6.46) into (6.42) gives Z c2 fε2n − c1 k e |∇vεn ,h |2 + 2 (1 − |vεn ,h |2 )2 − C . Eεn (uεn ) − Fεn (ρ) ≥ c − c ε 2 1 G n
(6.46)
(6.47)
Since the sequence {fεn } is bounded away from zero, it is clear that we can choose c2 big enough to ensure that ¯, c2 fε2n − c1 ≥ α > 0 on G and then define
f˜εn =
c2 fε2n − c1 c2 − c1
1/2 .
We may now rewrite (6.47) as Z k eεn (uεn ) − Fεn (ρ) ≥ f˜ε2n |∇vεn ,h |2 + 2 (1 − |vεn ,h |2 )2 − C . E εn G
(6.48)
(6.49)
It is then easy to verify that the weight functions {f˜εn } satisfy the hypotheses of Proposition 3.1 and we obtain from (6.49) that eεn (uεn ) − Fεn (ρ) ≥ 2πd| log εn | − C . E
(6.50)
Combining the lower bound (6.50) with the upper bound (6.10) we are led to the energy estimate (1.14) of Theorem 3. We now turn to the proof of the convergence statement of Theorem 3. Since Proposition 3.1 gives also that Z k f˜ε2n |∇vεn ,h |2 + 2 (1 − |vεn ,h |2 )2 ≥ 2πd| log εn | − C , 2εn G
´ and I. SHAFRIR N. ANDRE
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we may use it with (6.49) and the upper bound (6.10) to deduce the estimate Z 1 (1 − |vεn ,h |2 )2 ≤ C . (6.51) ε2n G It is easy to see that the generalization of the Euler–Lagrange equation (4.40) to our setting reads −∆uεn = |∇uεn |2 uεn +
1 ((uεn , e)2 uεn − (uεn , e)e) . ε2n
(6.52)
Using (6.52) and the argument of the proof of (4.38) leads to |∇vεn ,h | ≤
C . εn
(6.53)
The basic estimates (6.51), (6.53) enable us to repeat the argument of Sec. 5, locating the zeros of vεn ,h in a finite number of bad discs, denoting their distinct limits by {aj } etc, so at that stage we are able to conclude the weak convergence of {uεn } 1 1 (G \ {aj ∈ G}) to a limit u∗ ∈ Hloc (G \ {aj ∈ G}, S 2 ). Moreover, u∗ satisfies in Hloc (u∗ (x), e(x)) = 0 a.e. on G. The following generalization of Lemma 5.1 enables us to get stronger convergences. 1 0 (G\{aj ∈ G}) and in Cloc (G\ Lemma 6.9. uεn converges to u∗ strongly in Hloc {aj ∈ G}).
Proof. For any disc B(x0 , R) ⊂ G with 2R < min(δ(x0 ), dist (x0 , {aj ∈ G}), γ)
(6.54)
(for some positive constant γ later to be fixed) we choose as in the proof of Lemma 5.1 an R0 ∈ (R, 2R) such that (5.9) is satisfied. We then conclude as before that we may write uεn ,h = |uεn ,h |eiψεn on ∂B(x0 , R0 ) and that uεn → u∗ = eiψ∗
strongly in C 0 (∂B(x0 , R0 )) and H 1/2 (∂B(x0 , R0 )) .
We denote by ψεn and ψ∗ also minimizers corresponding to those boundary data (respectively) for the energy Z |∇(cos ψ · e1 + sin ψ · e2 )|2 B(x0 ,R0 )
Z
|∇ψ|2 + |∇e1 |2 cos2 ψ + |∇e2 |2 sin2 ψ
= B(x0 ,R0 )
+ (∇e1 , ∇e2 ) sin 2ψ − 2(e1 ∇e2 , ∇ψ) .
(6.55)
The functions ψεn , ψ∗ are thus both solutions of the corresponding Euler equation: ∆ψ = (|∇e2 |2 − |∇e1 |2 )
sin 2ψ cos 2ψ + 2(∇e1 , ∇e2 ) + div (e1 ∇e2 ) . 2 2
(6.56)
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We do not know whether the minimizers are unique on an arbitrary ball. But the difference h = ψ1 − ψ2 between any pair of solutions satisfies the equation: . Lh = −∆h+(|∇e2 |2 −|∇e1 |2 ) cos 2φb1 (x)·h−2(∇e1 , ∇e2 ) sin 2φb2 (x)·h = 0 , (6.57) where φb1 (x) and φb2 (x) are intermediate points lying between ψ1 (x) and ψ2 (x). The operator L depends on the two solutions ψ1 , ψ2 , but from its form it is clear that it will satisfy λ1 (L) > 0 (for the zero boundary condition problem on B(x0 , R0 )) for R0 ≤ γ with γ depending only on e1 , e2 . It is this value of γ that we take in (6.54). For that choice of γ we also have by elliptic estimates that ψεn → ψ∗ in H 1 (B(x0 , R0 )). Next we define a new sequence of test maps {Uεn } similarly to the proof of Lemma 5.1, but this time we take the x, y, z components of the maps constructed there as coefficients of e1 , e2 , e3 respectively. A direct calculation then gives eεn (Uεn ; B(x0 , R0 )) ≤ E
Z |∇(cos ψεn e1 + sin ψεn e2 )|2 + oεn (1) .
(6.58)
B(x0 ,R0 )
From (6.58) we deduce that u∗ = eiψ∗ and the strong H 1 -convergence of uεn to u∗ . We also obtain the convergences Z
1 ε2n
B(x0 ,R0 )
u2εn ,z → 0 ,
and uεn ,z → 0
uniformly on B(x0 , R0 ) .
Since |uεn ,h | ≥ 1/2 on B(x0 , Ro ) for n large, we may use spheric coordinates and write uεn = cos θεn cos φεn e1 + cos θεn sin φεn e2 + sin θεn e3 .
(6.59)
Note that from the above we already know that θεn → 0 in both C 0 and H 1 norms on B(x0 , R0 ). In the sequel we shall drop the subscript εn for simplicity. When we express the energy of u in terms of φ and θ, we find after a tedious but elementary computation that the Euler–Lagrange equation for φ can be written as sin 2φ div (cos2 θ ∇φ) = cos2 θ (|∇e2 |2 − |∇e1 |2 ) 2 + (∇e1 , ∇e2 ) cos 2φ + div (e1 ∇e2 ) + F (x, θ, ∇θ, φ)
(6.60)
with lim F (x, θεn , ∇θεn , φεn ) = 0
n→∞
in L2 (B(x0 , R0 )) .
(6.61)
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Combining (6.56) (for ψ∗ ) with (6.60) we get div (cos2 θ ∇(φ − ψ∗ )) = div (cos2 θ ∇φ) − ∆ψ∗ + div((1 − cos2 θ)∇ψ∗ ) = (|∇e2 |2 − |∇e1 |2 ) · + 2(∇e1 , ∇e2 ) ·
sin 2φ − sin 2ψ∗ 2
cos 2φ − cos 2ψ∗ 2
− sin θ (|∇e2 |2 − |∇e1 |2 ) 2
sin 2φ 2
+ (∇e1 , ∇e2 ) cos 2φ + div (e1 ∇e2 ) + div (sin2 θ ∇ψ∗ ) + F (x, θ, ∇θ, φ) .
(6.62)
(εn )
similarly to (6.57) by Defining an operator K = K . Kh = −div (cos2 θ ∇h) + (|∇e2 |2 − |∇e1|2 ) cos 2φb1 (x) · h − 2(∇e1 , ∇e2 ) sin 2φb2 (x) · h ,
(6.63)
where φb1 (x) and φb2 (x) are this time intermediate points lying between ψ∗ (x) and φ(x), we see that we can rewrite (6.62) in the following way:
with
K(φ − ψ∗ ) = F1 (x, θ, φ) − F (x, θ, ∇θ, φ) − div (sin2 θ ∇ψ∗ ) ,
(6.64)
lim F1 (x, θεn , φεn ) = 0 in L∞ (B(x0 , R0 )) .
(6.65)
n→∞
Note that by our choice of γ, the operator K satisfies λ1 (K) > 0 (for the zero boundary condition problem on B(x0 , R0 )) for εn small enough (since cos θεn → 1). The r.h.s. of (6.64) consists of three terms: the first goes to 0 in L∞ by (6.65), the second goes to 0 in L2 by (6.61), and the third is the sum of functions which go to 0 in L∞ and derivatives of such functions. Since also φεn → ψ∗ in C 0 (∂B(x0 , R0 )), we conclude by elliptic estimates, similarly to the proof of Lemma 5.1, that φεn → ψ∗ in C 0 (B(x0 , R0 )), hence completing the proof of the lemma. The proof of Lemma 6.9 shows that u∗ is a smooth map in G \ {aj ∈ G} since locally we may write u∗ = eiψ∗ . The same energy considerations of Sec. 5 enable us to conclude that u∗ has exactly d singularities in G, each of degree 1. To conclude the proof of Theorem 3 we need to show that the limit u∗ satisfies (1.15). The same argument of Sec. 5 shows that 1,p (G), ∀ p ∈ (1, 2) . uεn * u∗ in Wloc
From (6.52) we deduce that ∂uεn ∂uεn ∂ ∂ uεn × + uεn × · e = 0. ∂x ∂x ∂y ∂y
(6.66)
(6.67)
Passing to the weak limit in (6.67) we are led to (1.15). This completes the proof of Theorem 3.
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7. The Scalar Problem In this section we prove some properties of the scalar problem which were used in the previous sections. We treat separately the problems of uniqueness and approximation. We begin with a uniqueness results which includes the uniqueness part of Proposition 2.1. Proposition 7.1. Let a positive smooth function a(x) on G be given together with a smooth function ϕ on ∂G which takes its values in (−π/2, π/2). We denote b(x) = a2 (x). Then, for ε small enough there is a unique solution for the problem: sin 2ϕε ∆ϕε = b(x) 2ε2 in G , (7.1) ϕε (x) ∈ [−π/2, π/2] in G , ϕε = ϕ on ∂G . Our proof uses a variant of the method of Angenent [1]. It will be more convenient to work with ψε = π/2 − ϕε and prove uniqueness for ψε which is a solution of sin 2ψε −∆ψε = b(x) 2ε2 in G , (7.2) ψε (x) ∈ [0, π] in G , ψε = π/2 − ϕ on ∂G . We start by establishing estimates for the solutions of (7.2) away from the boundary. Lemma 7.1. For every η > 0 there exists K > 0 such that any solution ψε of (7.2) satisfies π π − η ≤ ψε (x) ≤ + η on GKε . (7.3) 2 2 Proof. We construct a family of subsolutions as in [1]. First we choose ς > 0 such that π sin 2t ≥ ςt, ∀ t ∈ 0, − η , b0 2 2 . where 0 < b0 = minG b(x). Let λ0 be the principal eigenvalue of −∆ for the Dirichlet boundary condition on the unit disk. We choose K > 0 such that K 2 > λ0 /ς. Fix any x0 ∈ GKε . Let φ denote the positive eigenfunction of −∆ on B(x0 , Kε) corresponding to the principal eigenvalue λ = Kλ20ε2 , which satisfies φ(x0 ) = 1. The family {θφ(x); 0 ≤ θ ≤ π/2 − η} forms a family of subsolutions to −∆v = b(x) sin 2v on B(x , Kε) , 0 2ε2 (7.4) v = ψε on ∂B(x0 , Kε) , since
θς sin(2θφ(x)) θλ0 φ(x) ≤ 2 φ(x) ≤ b(x) . 2 2 K ε ε 2ε2 By Serrin’s sweeping principle [18, 21] any positive solution of (7.4) satisfies v(x0 ) ≥ π/2 − η. This yields ψε (x) ≥ π/2 − η on GKε . Applying the same argument to the function π − ψε (x), which is too a solution of (7.2), completes the proof of (7.3). −∆(θφ(x)) =
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Proof of Proposition 7.1. Arguing by contradiction, assume that for a sequence εn → 0 there exists a sequence of pairs of distinct solutions ψεn ,1 , ψεn ,2 to (7.2) with ε = εn . Let us denote for each n by xn a point where maxG |ψεn ,1 (x) − ψεn ,2 (x)| is achieved. We can assume without loss of generality that ψεn ,1 (xn ) > ψεn ,2 (xn ) for all n . Using Lemma 7.1 we choose K corresponding to η = π/6. We claim that xn ∈ ΓKεn for all large n. Indeed, otherwise we would have for some an ∈ (ψεn ,1 (xn ), ψεn ,2 (xn )) −∆(ψεn ,1 (xn ) − ψεn ,2 (xn )) = b(xn )
cos 2an (ψεn ,1 (xn ) − ψεn ,2 (xn )) < 0 , ε2n
impossible for a maximum point. Passing to a subsequence we may assume then that xn → xˆ ∈ ∂G. Using the s, t coordinates introduced in Sec. 2 we write xn = (sn , tn ) ˆ = (ˆ s, 0). Next let us define a sequence of rescaled functions with tn ≤ Kεn and x by Uεn ,i (σ, τ ) = ψεn ,i (sn + εn σ, εn τ ), i = 1, 2, ∀ n . By standard elliptic estimates it is easy to see that Uεn ,i → Ui
2 in Cloc (R2+ ),
i = 1, 2
where R2+ = {(σ, τ ); τ > 0} and Ui is a solution of sin 2Ui inR2+ , x) −∆Ui = b(ˆ 2 Ui (x) ∈ [0, π] in R2+ , Ui = c on ∂R2+ = {τ = 0} ,
(7.5)
with c = π/2 − ϕ(ˆ x). Naturally, the Laplacian above is calculated with respect to the σ, τ coordinates. We claim that the solution to (7.5) is unique, so that U1 = U2 . Indeed, as we shall see below, this follows from a result of Angenent [1]. First notice that we may assume that c ≤ π/2, otherwise we replace Ui by π − Ui , i = 1, 2. Now we claim that (7.6) Ui ≥ c on R2+ i = 1, 2 . Indeed, looking for a contradiction assume that U 1 = c1 < c . inf 2 R+
(7.7)
Since by our assumption c1 ∈ [0, π/2), we may apply an argument very similar to the one of Lemma 7.1 to deduce the existence of L > 0 such that U1 (σ, τ ) > c1
for τ > L .
(7.8)
By (7.7)–(7.8) we deduce that c1 = inf{U1 (σ, τ ); R2+ ∩ {τ ∈ (0, L]} } .
(7.9)
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Now the infimum in (7.9) is not necessarily attained. But we can obtain a new solution of (7.5) (still denoted by U1 ) which is a limit of translations in the σ coordinate of the original U1 , for which the infimum is attained at some point, say U1 (σ0 , τ0 ) = c1 , and clearly τ0 > 0. Since c1 ∈ [θ, π/2) we have 0 ≤ U1 < π/2 in some ball B centered in (σ0 , τ0 ). But then −∆U1 ≥ 0 in B, so the fact that the minimum of U1 is achiered at (σ0 , τo ) contradicts the strong maximum principle. This contradiction establishes (7.6). When c = π/2 it follows immediately that Ui ≡ π/2 since applying the above argument to π − Ui shows that also Ui ≥ c = π/2. Assume then that c < π/2. From the above it follows that Vi = Ui − c is a positive bounded solution of ( −∆Vi = g(Vi ) in R2+ , (7.10) Vi = 0 on ∂R2+ , x) with g(u) = b(ˆ 2 sin(2(u + c)). The function g satisfies all the relevant hypotheses of Theorem B of [1], namely: (i) g(u) > 0 for u ∈ [0, π/2 − c), (ii) g 0 (π/2 − c) < 0.
Hence by [1] it follows that the solution to (7.10) is a function of τ only and is unique. So U1 = U2 = U as claimed. In our case we may even write explicitly the formula for U , but what will be important for us in the sequel is that Uτ > 0 on R2 + and limτ →∞ U = π/2. Going back to our sequences ψεn ,1 , ψεn ,2 we can now conclude from the equality U1 = U2 and Lemma 7.1 that kψεn ,1 − ψεn ,2 kL∞ (G) → 0 .
(7.11)
Let us now consider the sequence {vεn } defined by vεn (x) =
ψεn ,1 (x) − ψεn ,2 (x) . ψεn ,1 (xn ) − ψεn ,2 (xn )
Each vεn is a solution of −∆vεn (x) = b(x)
cos 2hεn (x) vεn (x) , ε2n
with hεn (x) lying between the points ψεn ,1 (x) and ψεn ,2 (x). Moreover |vεn (x)| ≤ 1 for all x and vεn (xn ) = 1. Defining a rescaled sequence by Vεn (σ, τ ) = vεn (sn + εn σ, εn τ ) , we find using standard elliptic estimates and (7.11) that Vεn → V
2 in Cloc (R2+ ) ,
where V (σ, τ ) is a solution of ( −∆V = b(ˆ x) cos(2U )V in R2+ , V = 0 on ∂R2+ , with U as above. Moreover, kV kL∞ = 1 and V (0, τˆ) = 1 for some τˆ.
(7.12)
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Assume first that c < π/2 (recall that c is the boundary value of U ). Then Uτ too solves the equation x) cos(2U )Uτ −∆Uτ = b(ˆ
in R2+ ,
and is positive everywhere on R2 + . Now the argument of [1] (see in particular the discussion after Lemma 2.3 there), shows that for some nontrivial solution of (7.12), which is obtained as a limit of translates of V and which will be still denoted by V , there exists a smallest θ > 0 such that θUτ (x) ≥ V (x)
on R2 +
and θUτ (x0 ) = V (x0 ) for some x0 ∈ R2+ .
(7.13)
But the strong maximum principle implies then that θUτ ≡ V which is impossible since on ∂R2+ we have Uτ > 0 while V = 0. We were led to a contradiction in the case c ∈ (0, π/2). The case c ∈ (π/2, π) is completely analogous. In the remaining case c = π/2, V solves ( ∆V = b(ˆ x)V in R2+ , (7.14) V = 0 on ∂R2+ . In this case we argue as above using the positive solution of (7.14) given by W = e−a(ˆx)τ instead of Uτ . We now turn to the approximation problem, and prove the following generalization of Proposition 2.1. We use the s, t coordinates which were introduced in Sec. 2. Proposition 7.2. Let ϕe denote the unique solution to (7.1). Then, for each integer m ≥ 0 there is an approximate solution ϕ(ε) m =
m X
(ε)
εi ψi
i=0
with (ε) ψi (x)
! i a(s)t t − ε =O e ε
f or all i,
in C k (G), ∀k ≥ 0 ,
such that m+1 ) ϕε − ϕ(ε) m = O(ε
in C k (G), ∀ k ≥ 0 .
(ε)
In particular, the function ϕ0 = ϕ0 is given on Γµ/2 (for µ defined in the beginning of Sec. 2) by ta(s) . (7.15) ϕ0 (x) = sgn gz (s) arctan csch α(s) + ε The proof of Proposition 7.2 follows closely the method of Berger–Fraenkel [3], (ε) which treated in fact a more complicated problem. The construction of the ϕm ’s is done in the next lemma. In the sequel all estimates should be understood as valid in C k (G), ∀ k. We denote by Kε the operator defined by (Kε f )(x) = ε2 ∆f (x) −
b(x) sin 2f (x) . 2
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Lemma 7.2. For every integer m ≥ 0 there exists a function ϕm = ϕm satisfying the boundary condition ϕm = ϕ on ∂G and Kε ϕm = O(εm+1 ) .
(7.16)
Proof. We will look for ϕm of the form ϕm = ζ(w0 + εw1 + · · · + εm wm ) ,
(7.17)
where ζ(x) is a smooth function with values in [0, 1], supported in Γµ such that ζ ≡ 1 on Γµ/2 . The functions w0 , w1 , . . . , wm are determined by equating to zero the coefficients of 1, ε, . . . , εm respectively in the Taylor expansion in ε powers of Kε (w0 + εw1 + · · · + εm wm ) . We write each wi as a function wi (s, τ ) where τ = t/ε, for τ < µ/ε and use these variables also for the functions a and b below when convenient. On Γµ we denote by n a unit vector field in the direction of increment of τ (in fact n = −V of Sec. 2) and by s a unit vector tangential to the curves {τ = c}. Then we have ε2 ∆wi = (wi )τ τ + ε(wi )τ div n + ε2 (wi )ss + ε2 (wi )s div s . Using the Taylor expansion 1 1 1 1 sin(2w0 + x) = sin(2w0 ) + cos(2w0 )x − sin(2w0 )x2 − . . . 2 2 2 4 we get the first two equations (w0 )τ τ −
b(s, 0) sin 2w0 = 0 2
(w1 )τ τ − b(s, 0) cos 2w0 · w1 =
(7.18) τ bt (s, 0) sin 2w0 − (w0 )τ div n , 2
(7.19)
and in general for 1 ≤ k ≤ m : (wk )τ τ − b(s, 0) cos 2w0 · wk = Fk (s, τ, w0 , . . . , wk−1 ) .
(7.20)
The boundary condition for (7.18) and (7.20) are given by: wk (s, τ ) = O(τ k e−a(s,0)τ ) as τ → +∞ (uniformly in s) 0 ≤ k ≤ m ,
(7.21.a)
and w0 (s, 0) = ϕ(s), wk (s, 0) = 0, 1 ≤ k ≤ m,
∀s ∈ ∂G .
(7.21.b)
The solution for w0 is given explicitly by w0 (s, τ ) = (sgn ϕ(s)) arctan(csch (α(s) + a(s, 0)τ )) , where
(7.22)
α(s) = tanh−1 (cos ϕ(s)) ∈ (0, ∞] .
Note that α(s) = +∞ at points where gz (s) = 0, but w0 is smooth everywhere.
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In order to find wk , 1 ≤ k ≤ m we argue as in [3] and we solve the homogeneous equation: (7.23) wτ τ − b(s, 0) cos 2w0 · w = 0 . One solution to (7.23) is obtained by differentiating w0 with respect to τ . This yields the solution A(s, τ ) = (w0 )τ = −
a(s, 0)sgn ϕ(s) . cosh (α(s) + a(s, 0)τ )
(7.24)
Clearly A(s, τ ) = O(e−a(s,0)τ ) as τ → ∞. Next we look for a second solution B(s, τ ) with the asymptotic behavior B(s, τ ) = O(ea(s,0)τ ) as τ → ∞. In order to find B we use the constant Wronskian condition : ABτ − Aτ B = −1 which implies 1 (B A )τ = − A2 . A solution satisfying B(s, 0) = 0 is then given by Z τ d˜ τ B(s, τ ) = −A(s, τ ) 2 (s, τ A ˜) 0 sinh 2α(s) sgn ϕ(s) sinh(α(s) + a(s, 0)τ ) + a(s, 0)τ − . = 2 2a (s, 0) 2 cosh(α(s) + a(s, 0)τ ) (7.25) Next, given any function F (s, τ ) which is O(τ k e−a(s,0)τ ) as τ → ∞ for some k ≥ 0 (uniformly in s), the solution to
satisfying
wτ τ − b(s, 0) cos 2w0 · w = F
(7.26)
w(s, 0) = 0 and ω = O(τ h+1 e−a(s,0)τ ) at infinity
(7.27)
is given by Z
Z
τ
∞
B(s, τ˜)F (s, τ˜) d˜ τ + B(s, τ )
w(s, τ ) = A(s, τ ) 0
A(s, τ˜)F (s, τ˜) d˜ τ.
(7.28)
τ
The uniqueness of the solution to (7.26) and (7.27) is a result of the positiveness . of the operator Lw = −wτ τ + b(s, 0)wz 2w0 · w on H01 (0, ∞), which follows from a similar argument to that of the proof of Lemma 7.3 below. Applying (7.28) to each of the Eq. (7.20), 1 ≤ k ≤ m, we get by induction that wk (s, τ ) = O(τ k e−a(s,0)τ )
as τ → ∞,
1 ≤ k ≤ m.
The result of the lemma follows from (7.29) and (7.17).
(7.29)
Next we want to show that ϕm is close to the exact solution ϕε of (7.1) whose uniqueness (for small ε) is guaranteed by Proposition 7.1. In the sequel we shall omit for simplicity the subscript ε from ϕε . We first write the equation satisfied by . the reminder rm = ϕ − ϕm . We denote fm = Kε ϕm = ε2 ∆ϕm − b
sin 2ϕm . 2
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Then, b (sin 2ϕ − sin 2ϕm ) − fm 2 b = [sin 2ϕm cos 2(ϕ − ϕm ) + cos 2ϕm sin 2(ϕ − ϕm ) − sin 2ϕm ] − fm 2 b = b cos 2ϕm · (ϕ − ϕm ) + cos 2ϕm · [sin 2(ϕ − ϕm ) − 2(ϕ − ϕm )] 2
ε2 ∆rm =
− b sin 2ϕm sin2 (ϕ − ϕm ) − fm . Hence −ε2 ∆rm + b cos 2ϕm rm = Jm (rm ) + fm ,
(7.30)
where
b cos 2ϕm (sin 2y − 2y) 2 (we suppress for simplicity the dependence of Jm in the space variable x). Next we are going to show that the operator on the l.h.s. of (7.30) is positive for small enough ε. Jm (y) = b sin 2ϕm sin2 y −
Lemma 7.3. There exists ν > 0 such that for ε small enough there holds: Z Z Z 2 2 2 2 2 |∇h| + b cos 2ϕm · h ≥ ε ν |∇h|2 , ∀ h ∈ H01 (G) . (7.31) ε G
G
G
Proof. We shall use in the proof 3 positive parameters γ, c and L whose values will be fixed below. First we choose successive points s0 , s1 , . . . , sN ∈ ∂G such that (setting sN +1 = s0 ) we have: |si − si+1 | = γ,
i = 0, . . . , N .
We also denote G(i) = G ∩ {t ≤ Lε, s ∈ hsi , si+1 i},
i = 0, . . . , N .
Applying the arguments of [3] (p. 567) we obtain for any c > 0 the following Poincar´e inequality: Z Z c2 ε 2 h2 ≤ H(cε) · h2t , (7.32) 2 (i) (i) G ∩{t≤cε} G ∩{t≤cε} where H(y) is a continuous function satisfying limy→0 H(y) = 1. By Lemma 7.2 ϕm − w0 = O(ε), hence cos 2ϕm =
sinh2 (α(s) + a(s, 0)τ ) − 1 + O(ε) . cosh2 (α(s) + a(s, 0)τ )
(7.33)
In particular, there exists Lo > o such that: cos 2ϕm ≥
1 2
on GL0 ε .
(7.34)
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By (7.31)–(7.32) we have for some K > 0: Z ε2 |∇h|2 + b(s, t) cos 2ϕm · h2 G(i)
2 − b(s, t) h2 ≥ c2 H(cε) G(i) ∩{t≤cε} Z sinh2 (α(s) + ca(s, 0)) − 1 − Kε h2 . + b(s, t) cosh2 (α(s) + ca(s, 0)) G(i) ∩{cε≤t}
Z
(7.35)
Next we claim that if γ is chosen small enough then c could be found to satisfy √ 2 ∀i. (7.36) max a(s, 0) < min sinh ca(s, 0) > 1 and s∈hsi ,si+1 i s∈hsi ,si+1 i c √ √ Indeed, since log(1 + 2) < 2, we may choose γ > 0 such that √ a(s0 , 0) 0 00 2 0 00 √ . ; s , s ∈ ∂G with |s − s | ≤ γ < max 00 a(s , 0) log(1 + 2) This choice of γ enables us then to choose c satisfying (7.36). Finally we set L = max(L0 , c + 1) (L0 was defined in (7.34)). We have thus fixed the values for the parameters γ, c and L. From (7.35) and (7.30) it follows that for some η1 > 0 there holds: Z Z ε2 |∇h|2 + b cos 2ϕm · h2 ≥ η12 h2 , ∀ ε ≤ ε0 , ∀i . (7.37) G(i)
G(i)
Summing (7.37) on i yields Z Z ε2 |∇h|2 + b cos 2ϕm · h2 ≥ η12 ΓLε
h2 ,
∀ ε ≤ ε0 .
(7.38)
h2 ,
(7.39)
ΓLε
Next note that by (7.34) we have Z Z 2 2 2 2 ε |∇ h| + b cos(2ϕm )h ≥ η2 GLε
GLε
with η2 = 12 minb. Combining (7.38) with (7.39) we are led to ¯ G
Z
Z ε2 |∇h|2 + b cos 2ϕm · h2 ≥ η32 G
h2 ,
(7.40)
G
with η3 = min(η1 , η2 ). But we have also clearly Z Z Z ε2 |∇h|2 + b cos 2ϕm · h2 ≥ ε2 |∇h|2 − b1 h2 , G
G
(7.41)
G
where b1 = maxG b. Combining (7.40) with (7.41) yields Z Z b1 ε2 |∇h|2 + b cos 2ϕm · h2 ≥ ε2 |∇h|2 , 1+ 2 η3 G G and (7.31) follows.
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From Lemma 7.3 and the Lax–Milgram Lemma if follows that the operator Lm defined by Lm h = −ε2 ∆h + cos 2ϕm · h has a bounded inverse which satisfies for ε ≤ ε0 : ||L−1 m h||H01 ≤
||h||H01 ν 2 ε2
,
∀ h ∈ H01 (G) .
(7.42)
In the sequel we let || · || stand for the H01 -norm. Now we can rewrite (7.30) as rm = Rm rm where the operator Rm : H01 (G) → 1 H0 (G) is defined by (7.43) Rm h = L−1 m (Jm (h) + fm ) . Next we claim: Lemma 7.4. For m ≥ 4 there exists a constant cm such that for 0 < ε < ε0 (m) the operator Rm is a contraction self-map of the set Bm = {h ∈ H01 (G); ||h|| ≤ cm εm−1 } .
(7.44)
Proof. We shall first look for an upper bound for kRm h1 −Rm h2 k with h1 , h2 ∈ Bm , assuming cm was already chosen. From the definition of Jm it is clear that there exists a constant C1 > 0 such that 0 (θ(x))| · |h1 (x) − h2 (x)| |Jm (h1 (x)) − Jm (h2 (x))| = |Jm
≤ C1 (|h1 (x)| + |h2 (x)|)|h1 (x) − h2 (x)| . Hence for any ψ ∈ H01 (G) we have by H¨older inequality Z (Jm (h1 ) − Jm (h2 ))ψ ≤ C1 k |h1 | + |h2 | k4 · kh1 − h2 k4 · kψk2 . G
Recall that we denote by k · k the H01 -norm and by k · kp the Lp -norm for p ≥ 1. Applying Sobolev inequalities we get Z kJm (h1 ) − Jm (h2 )k = sup (Jm (h1 ) − Jm (h2 ))ψ ; kψk = 1 G
≤ C2 (kh1 k + kh2 k) · kh1 − h2 k ≤ C2 · 2cm εm−1 kh1 − h2 k .
(7.45)
By (7.42)–(7.45) it follows that kRm h1 − Rm h2 k ≤
1 kJm (h1 ) − Jm (h2 )k ≤ cm C3 εm−3 · kh1 − h2 k . ε2 ν 2
(7.46)
It is clear then that for m ≥ 4 and ε ≤ ε0 (m, cm ) the map Rm is a contraction on Bm .
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Next we want to check that Rm is a self-map of Bm provided cm is chosen large enough. For any h ∈ Bm we apply (7.46) with h1 = h and h2 = 0 to obtain m−3 khk + kRm hk = kRm h − Rm 0 + L−1 m fm k ≤ cm C3 ε
1 ε2 ν 2
kfm k .
Using Lemma 7.2 we find kRm hk ≤ C3 c2m ε2m−4 + C4 εm−1 .
(7.47)
Choosing any cm > C4 we see from (7.47) that Rm is also a self-map of Bm for ε small enough. We are now ready to give the proof of Proposition 7.2. Proof of Proposition 7.2. Assume first that m ≥ 4. By the uniqueness result Proposition 7.1, Lemma 7.4 and the contraction map principle it follows that for ε ≤ ε0 (m), rm = ϕ − ϕm must coincide with the unique fixed point of Rm in Bm . This gives immediately (7.48) krm k ≤ Cεm−1 . Next we rewrite Eq. (7.30) satisfied by rm as −∆r = 1 J (r ) − b cos 2ϕ · r + fm m m m m m ε2 ε2 ε2 rm = 0 on ∂G .
in G ,
(7.49)
Using (7.48) we see that the L2 -norm of the right-hand side of (7.49) is O(εm−3 ). By elliptic estimates we get that the H 2 and C α norms of rm are O(εm−3 ). Plugging it back in (7.49) we get by a standard bootstrap argument that ||rm ||C k (G) ≤ C(k, m)εm−3 ,
∀ m ≥ 4, ∀k ≥ 0, ∀ ε ≤ ε0 (m) .
(7.50)
In order to finish the proof we need to improve the estimate (7.50) by replacing the power εm−3 by the power εm+1 and by removing the restriction m ≥ 4. To this end, we argue as in [3]. For m ≥ 0 we define rem = ϕm+4 − ϕm + rm+4 . Since by construction ϕm+4 −ϕm = O(εm+1 ) in all C k -norms, and by (7.50) rm+4 = O(εm+1 ) it follows that also rem = O(εm+1 ). Since ϕm+4 + rm+4 is the unique solution of (7.1) for small enough ε, it follows that rem = rm and the result follows. We close this section with the proof of Proposition 2.2. Proof of Proposition 2.2. The proof is very similar to the one of Proposition 2.1 in [2]. First notice that by Kato’s inequality ∆|ϕε | ≥ sgn ϕε ∆ϕε =
sin 2|ϕε | , 2ε2
so |ϕε | is a a subsolution for the problem: sin 2ψ ∆ψ = 2ε2 in G , ψ(x) ∈ [0, π/2] in G , ψ = |φ| on ∂G .
(7.51)
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ON NEMATICS STABILIZED BY A LARGE EXTERNAL FIELD
The solution ψε to (7.51) is known to be unique. This follows from a result of Brezis–Oswald [8] (applied to the positive function π/2 − ψε ). Hence we have |ϕε | ≤ ψε
on G .
(7.52)
Since we assume that |ϕ| < π/2 on ∂G we may find a constant c > 0 such that |ϕ| ≤ arctan(csch c) on ∂G . We fix any x ∈ G and denote δ = δ(x). We then define for y ∈ B(x, δ) δ 2 − r2 , where r = r(y) = |y − x| . w(r) = arctan csch c + 3δε
(7.53)
By a direct calculation 4r2 sin 2w 4 sin 2w w0 = 2 2· + · −r 2 r 9δ ε 2 3δε 2 tanh(c + δ23δε )
∆w = w00 + ≤
4 sin 2w 4 sin 2w + · . · 9ε2 2 3δε 2 tanh c
It is easy to see that w is a super solution for (7.51), but on the domain B(x, δ) instead of G for the boundary condition ψε on ∂B(x, δ), provided ε ≤ 15δ 36 · tanh c. Using (7.52) we deduce that |ϕε (y)| ≤ Ce
r2 −δ2 3δε
∀ y ∈ B(x, δ) .
(7.54)
In particular, |ϕε (x)| ≤ Ce−
δ(x) 3ε
which implies (2.9) since 1 − ρh ≤ ρ2z ≤ ϕ2ε . Note that (7.54) implies in particular that |ϕε (y)| ≤ Ce− 4ε δ
∀ y ∈ B(x, δ/2) .
Applying a rescaling argument and standard elliptic estimates as in [2] we find that " # 2 δ(x) δ(x) C + 1 e− 4ε , |∇ϕε (x)| ≤ δ(x) ε which together with (2.9) implies (2.10) since |∇ρh |ρh = |ρz ||∇ρz |.
Acknowledgements The authors are grateful to J. Rubinstein for bringing this problem to their attention, to G. Wolansky for interesting discussions and to H. Brezis for his interest and encouragement. Special thanks to E. Sandier for the references [16] and [17]. Part of this work was done while N. Andr´e was visiting the Technion at Haifa. This visit was supported by the French–Israeli PICS exchange program. She thanks the Mathematics Department for its hospitality.
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References [1] S. B. Angenent, “Uniqueness of the solution of a semilinear boundary value problem,” Math. Ann. 272 (1985), 129–138. [2] N. Andr´e and I. Shafrir, “Minimization of a Ginzburg–Landau type functional with nonvanishing Dirichlet boundary condition,” Calc. Var. Partial Differential Equations 7 (1998), 191–217. [3] M. S. Berger and L. E. Fraenkel, “On the asymptotic solution of a nonlinear Dirichlet problem,” J. Math. Mech. 19 (1970), 553–585. [4] F. Bethuel, H. Brezis and F. H´elein, “Asymptotics for the minimization of a Ginzburg– Landau functional,” Calc. Var. Partial Differential Equations 1 (1993), 123–148. [5] F. Bethuel, H. Brezis and F. H´ elein, Ginzburg–Landau Vortices, Birkh¨ auser, 1994. [6] F. Bethuel and T. Rivi`ere, “Vortices for a variational problem related to superconductivity,” Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire 12 (1995), 243–303. [7] F. Bethuel and X. Zheng, “Density of smooth functions between two manifolds in Sobolev spaces,” J. Func. Anal. 80 (1988), 60–75. [8] H. Brezis and L. Oswald, “Remarks on sublinear elliptic equations,” J. Nonlinear Analysis 10 (1986) 55–64. [9] G. De Gennes, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1974. [10] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin and New York, 1983. [11] R. L. Jerrard, Lower bounds for generalized Ginzburg–Landau functionals, preprint. [12] L. Lassoued and P. Mironescu, Ginzburg–Landau type energy with discontinuous constraint, preprint. [13] F. H. Lin, “Solutions of Ginzburg–Landau equations and critical points of the renormalized energy,” Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire 12 (1995), 599–622. [14] L. Modica, “The gradient theory of phase transitions and the minimal interface criterion,” Arch. Rational Mech. Anal. 98 (1987), 123–142. [15] L. M. Pismen and J. Rubinstein, “Dynamics of disclinations in liquid crystals,” Quarterly of Appl. Math. L. 3 (1992) 535–545. [16] E. Sandier, Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal. 152 (1998), 379–403. [17] E. Sandier, Asymptotics for a nematic in an electric field, preprint. [18] D. H. Sattinger, “Topics in stability and bifurcation theory,” Lect. Notes Math. 309, Springer, 1973. [19] R. Schoen and K. Uhlenbeck, “A regularity theory for harmonic maps,” J. Diff. Geom. 17 (1982), 307–335. [20] R. Schoen and K. Uhlenbeck, “Boundary regularity and the Dirichlet problem for harmonic maps,” J. Diff. Geom. 18 (1983) 253–268. [21] J. Serrin, “Nonlinear elliptic equations of second order,” AMS Symposium in Partial Differential Equations, Berkeley, 1971. [22] P. Strenberg, “The effect of a singular perturbation on nonconvex variational problems,” Arch. Rational Mech. Anal. 101 (1988) 209–260. [23] M. Struwe, “On the asymptotic behavior of minimizers of the Ginzburg–Landau model in 2 dimensions,” Differential Integral Equations 7 (1994) 1613–1624 and Erratum, loc. cit. 8 (1995) 124.
ON GENERALIZED ABELIAN DEFORMATIONS GIUSEPPE DITO∗ Research Institute for Mathematical Sciences Kyoto University, Sakyo-ku Kyoto 606-8502, Japan and Laboratoire Gevrey de Math´ ematique physique Universit´ e de Bourgogne BP 47870 F-21078 Dijon Cedex, France Received 25 February 1998 We study sun-products on Rn , i.e. generalized Abelian deformations associated with star-products for general Poisson structures on Rn . We show that their cochains are given by differential operators. As a consequence, the weak triviality of sun-products is established and we show that strong equivalence classes are quite small. When the Poisson structure is linear (i.e. on the dual of a Lie algebra), we show that the differentiability of sun-products implies that covariant star-products on the dual of any Lie algebra are equivalent each other.
1. Introduction A new kind of deformations was introduced in [4] in connection with the quantization of Nambu–Poisson structures (see also [7]). The main feature of these deformations is that they are not of Gerstenhaber’s type [8] in the sense that one does not have a K[[ν]]-algebra structure on the deformed algebra (K is the ring over which is defined the original algebra A and ν denotes the deformation parameter). More precisely, these deformations are not linear with respect to the deformation parameter; the product operation annihilates the deformation parameter so that one has only a K-algebra structure on the deformed algebra A[[ν]]. The motivation for dealing with these generalized deformations was that they provide non-trivial Abelian deformations of the usual product, and this point was essential for the solution proposed in [4] for the quantization of Nambu–Poisson structures. We recall that in Gerstenhaber’s framework, Abelian deformations of the usual product of smooth functions on some manifold are always trivial (it is a consequence of the fact that a symmetric Hochschild 2-cocycle is a coboundary). Explicit examples of generalized Abelian deformations were constructed in [4, 5]. There are two main classes of generalized Abelian deformations. On the one hand, one has the Zariski products introduced in [4] which involve factorization of polynomials in several variables into irreducible factors. Zariski products are Abelian ∗ Supported
by the Japan Society for the Promotion of Science and the Conseil r´ egional de
Bourgogne. 711 Reviews in Mathematical Physics, Vol. 11, No. 6 (1999) 711–725 c World Scientific Publishing Company
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products on the semi-group algebra generated by irreducible polynomials and can be constructed from any star-product on Rn . Originally the construction of a Zariski product was performed from a Moyal product and it appeared crucial to go over semi-group algebras with a proper notion of derivatives to fulfill algebraic requirements imposed by the Fundamental Identity of Nambu–Poisson structures. This construction is quite sophisticated and little is known about its properties. Actually the Zariski quantization induced by some Zariski product shares many properties with second quantization (appearance of a Fock space generated by irreducible polynomials, etc.). On the other hand, sun-products have been studied in [5]. They have much simpler properties than Zariski products and, roughly speaking, they can be seen as the finite dimensional version of Zariski products. They involve factorization into linear polynomials and can be defined on some algebra of functions over finite dimensional spaces. Still generalized deformations have to find an appropriate algebraic framework and it is the aim of this paper to study sun-products on Rn and to clarify their structure. Our main result is that sun-products are differentiable deformations, i.e. their cochains are differential operators vanishing on constants. This fact allows us to find a complete characterization of the cochains of a sun-product: Any sequence of differential operators vanishing on linear polynomials defines a sun-product and vice-versa. After briefly recalling the most basic facts on star-products and Hochschild cohomology, Sec. 2 provides a study of sun-products associated with star-products on Rn endowed with a general Poisson structure. We show in Theorem 2 the differentiability of sun-products and deduce some consequences of this property. We then specialize our discussion to the important case of the dual of a Lie algebra in Sec. 3. Consider a Lie algebra g. Its dual g∗ is endowed with a canonical Poisson structure. We show that Gutt’s star-product on g∗ is the only covariant star-product on g∗ whose associated sun-product coincides with the usual product on C ∞ (g∗ ). From the differentiability of sun-products one shows that covariant star-products on the dual of any Lie algebra are equivalent each other. In Sec. 4, as another consequence of the differentiable nature of sun-products we show that sun-products are weakly trivial in the sense of [5]. We say that two sun-products are weakly equivalent if there exists an invertible formal series of differential intertwining these sun-products. The sun-product operation kills all of the non-zero powers of the deformation parameter. Weak triviality of a sun-product means weak equivalence with the usual product (on the undeformed algebra). When one allows the deformation parameter coming from the equivalence operator not to be annihilated by the sun-product, one gets the notion of strong equivalence of sun-products. By a simple argument, we remark that strong equivalence classes are rather small. We think that these results might be helpful or give some hints for the definition of a cohomology adapted to generalized deformations.
ON GENERALIZED ABELIAN DEFORMATIONS
713
2. Sun-Products on R n 2.1. Notions on star-products We summarize here basic facts about star-products that we shall need in the present paper. The general reference on star-products theory are the papers [1, 2]. Let M be a Poisson manifold with Poisson bracket P . The space of smooth functions C ∞ (M ) carries two natural algebraic structures: It is an Abelian algebra for the pointwise product of functions and also a Lie algebra for the Poisson bracket P . A star-product on (M, P ) is a formal associative deformation in the Gerstenhaber’s sense [8] of the Abelian algebra structure of C ∞ (M ). More precisely: Definition 1. Let C ∞ (M )[[ν]] be the space of formal series in a parameter ν with coefficients in C ∞ (M ). A star-product on (M, P ) is a bilinear map from P r C ∞ (M ) × C ∞ (M ) to C ∞ (M )[[ν]] denoted by f ∗ν g = r≥0 ν Cr (f, g), f, g ∈ ∞ ∞ ∞ ∞ C (M ), where (the cochains) Cr : C (M ) × C (M ) → C (M ) are bilinear maps satisfying for any f, g, h ∈ C ∞ (M ): (i) C0 (f, g) = f g; 0, for r ≥ 1, c ∈ R; (ii) CX r (c, f ) = Cr (f, c) = X (iii) Cs (Ct (f, g), h) Cs (f, Ct (g, h)), for r ≥ 0; s+t=r s,t≥0
s+t=r s,t≥0
(iv) C1 (f, g) − C1 (g, f ) = 2P (f, g). A star-product ∗ν is naturally extended to a bilinear map on C ∞ (M )[[ν]]. The conditions (i)–(iv) above simply translate, respectively, that a star-product is: (i) a deformation of the pointwise product; (ii) it preserves the original unit (1 ∗ν f = f ∗ν 1 = f ); (iii) it is an associative product; (iv) the associated star-bracket, [f, g]∗ν = (f ∗ν g − g ∗ν f )/2ν, is a Lie algebra deformation of the Lie–Poisson algebra (C ∞ (M ), P ). Usually, one adds one more condition on the cochains Cr of a star-product by requiring that they should be bidifferential operators (necessarily null on constants by condition (ii)). These star-products are called differential star-products. In this paper, star-product will always mean differential star-product. One has a notion of equivalence between star-products given by: Definition 2. Two star-products ∗ν and ∗0ν on (M, P ) are said to be equivalent P if there exists a formal series T = I + r≥1 ν r Tr , where I is the identity map on C ∞ (M ) and the Tr ’s are differential operators on C ∞ (M ) vanishing on constants, such that T (f ∗ν g) = T (f ) ∗0ν T (g) , f, g ∈ C ∞ (M )[[ν]] . For a long time, star-products were known to exist on any symplectic manifold (i.e. when the Poisson bracket P is induced by some symplectic form) [3]. A few months ago, as a consequence of his formality conjecture, Kontsevich showed that in fact star-products exist on any Poisson manifold and gave a complete description of their equivalence classes [10].
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2.2. Hochschild cohomology Hochschild cohomology plays a prevailing role in the deformation theory of associative algebras. It is well known that the obstructions to equivalence of associative deformations are in second Hochschild cohomology space and the obstructions for extending a deformation, given up to certain order in the deformation parameter, to the next order live in the third Hochschild cohomology space. We shall recall here the definition and basic properties of the Hochschild cohomology in the differentiable (null on constants) case. Let A be the Abelian algebra C ∞ (M ) endowed with the pointwise product. Consider the complex C ? (A, A) = {C r (A, A)}r≥0 , where C r (A, A) is the vector space of r-linear differential operators null on constants φ: Ar → A, with coboundary operator δ, defined on an r-cochain C by X (−1)i C(f0 , . . . , fi−1 fi , . . . , fr ) δC(f0 , . . . , fr ) = f0 C(f1 , . . . , fr ) + 1≤i≤r r+1
+(−1)
C(f0 , . . . , fr−1 )fr ,
for any f0 , . . . , fr in A. The Hochschild cohomology (with values in A) is the coho? (A). A mology of the cochain complex (C ? (A, A), δ) and shall be denoted by Hdiff,nc fundamental result is: ? (A) is isomorphic Theorem 1 (Vey [16]). The Hochschild cohomology Hdiff,nc ? to Γ(∧ T M ), the space of skew-symmetric contravariant tensor fields on M . Hence any Hochschild r-cocycle φ can be written as φ = δθ + Λ, where θ is an (r − 1)-cochain and Λ is an r-tensor on M . In particular, a completely symmetric cocycle is a coboundary.
2.3. Notations and definitions We start by making precise our notations. The coordinates of Rn are denoted by (x1 , . . . , xn ). Let N be the R-algebra of smooth functions on Rn . Let Pol be the Rsubalgebra of N consisting of polynomials in R[x1 , . . . , xn ]. For a formal parameter ν, we shall denote by Nν (resp. Polν ) the algebra N[[ν]] (resp. Pol[[ν]]) of formal series in ν with coefficients in N (resp. Pol). We distinguish in Nν a subalgebra N0ν consisting of formal series whose zeroth-order coefficient belongs to Pol. Nν , N0ν and Polν are naturally R[[ν]]-algebras, but we shall often view them as R-algebras. The natural projection π: Nν → N is an R-algebra homomorphism and the same symbol shall be used for the projections of N0ν and Polν on Pol. We now define sun-products. Let S(Pol) denote the symmetric tensor algebra over Pol with symmetric tensor product ⊗, and let λ: Pol → S(Pol) be the R-algebra homomorphism defined by k1
kn
λ(xk11 . . . xknn ) = (x1⊗ ) ⊗ · · · ⊗ (xn⊗ ) ,
∀ k1 , . . . , kn ≥ 0 .
(1)
The map λ sends a polynomial in Pol to an element of S(Pol) by replacing the usual product between linear factors by the symmetric tensor product.
ON GENERALIZED ABELIAN DEFORMATIONS
715
Let P be a Poisson bracket on Rn . Given a star-product ∗ν on (Rn , P ), we define an R-linear map T∗ν : S(Pol) → N0ν by 1 X fσ(1) ∗ν · · · ∗ν fσ(k) , ∀ k ≥ 1 , (2) T∗ν (f1 ⊗ · · · ⊗ fk ) = k! σ∈Sk
where fi ∈ Pol, 1 ≤ i ≤ k, and Sk is the permutation group on k elements. By convention, we set T∗ν (I) = 1, where I the identity of S(Pol). Notice that the zeroth-order coefficient on the right-hand side of (2) is the product of polynomials f1 . . . fk ∈ Pol, but in general the coefficient of ν r for r ≥ 1 is in N. N0ν
Definition 3. To a star-product ∗ν on (Rn , P ), we associate a new product on by the following formula: f ν g = T∗ν (λ(π(f )) ⊗ λ(π(g))) ,
This product is called the
ν -product
f, g ∈ N0ν .
(3)
(or sun-product) associated to ∗ν .
In words, a sun-product on Rn associates to two polynomials f, g ∈ Pol the element f ν g ∈ N0ν obtained by replacing the usual product between linear factors (in some given order) in f g by a star-product ∗ν and then by completely symmetrizing the expression found. The extension of the product to f, g ∈ N0ν is obtained by applying the previous procedure to the zeroth-order coefficient of f g. Hence a sun-product annihilates any non-zero powers of the deformation parameter. Basic properties of sun-products are collected in the following lemma: Lemma 1. A sun-product ν on Rn is an Abelian, associative product on N0ν . It fails to be R[[ν]]-bilinear, but it is R-bilinear. N0ν endowed with a product ν is an Abelian R-algebra. Proof. That the product ν is Abelian is clear from (3). Associativity follows from π(f ν g) = π(f )π(g) for f, g ∈ N0ν , and from the fact that both λ and π are R-algebra homomorphisms: f ν (g ν h) = T∗ν (λ(π(f )) ⊗ λ(π(g ν h))) = T∗ν (λ(π(f gh))) = (f ν g) ν h, for f, g, h ∈ N0ν . Clearly a sun-product does not have a unit on N0ν , nevertheless one has 1 ν f = f when f is a linear polynomial in Pol. From the preceding proof, we see that to every sun-product ν we can associate P a formal series of linear maps ρ = 0≤r ν r ρr , where ρ0 = Id is the identity map on Pol, and ρr : Pol → N for r ≥ 1, such that f ν g = ρ(π(f g)) for f, g ∈ N0ν . We shall (abusively) call the maps ρr the cochains of the sun-product ν . 2.4. Differentiability An example of sun-product has been explicitly computed in [5] for some starproduct on the dual of the Lie algebra su(2) seen as Poisson manifold when endowed with its natural Lie–Poisson bracket. A remarkable feature of this sun-product is that its cochains are differential operators. In the following, we shall show that this fact corresponds to the general situation. As a consequence, any sun-product admits a natural extension from N0ν to Nν .
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Theorem 2. The cochains ρr of a sun-product ν associated to some starproduct ∗ν on (Rn , P ) are given by the restriction to Pol of differential operators on N. Before proving this theorem, we shall derive few lemmas. We consider a sunproduct ν associated with some star-product ∗ν on (Rn , P ). The cochains of the sun-product (resp. star-product) are denoted by ρr (resp. Cr ). P For any map φ: Rk → E, where E is a vector space, (i1 ,...,ik ) φ(xi1 , . . . , xik ) denotes the sum over cyclic permutations of (xi1 , . . . , xik ). Lemma 2. Let ψ: Pol → N be a linear map such that ψ(1) = ψ(xi ) = 0, for 1 ≤ i ≤ n. Let φ: N × N → N be a bidifferential operator null on constants. If the Hochschild coboundary δψ satisfies for any k ≥ 2 and indices (i1 , . . . , ik ) : X X δψ(xi1 , xi2 . . . xik ) = φ(xi1 , xi2 . . . xik ) , (4) (i1 ,...,ik )
(i1 ,...,ik )
then ψ is the restriction to Pol of a differential operator null on constants. Proof. On the right-hand side of Eq. (4), it is clear that is sufficient to consider bidifferential operators of the form (only these are contributing to Eq. (4)): X X φi,J ∂i f ∂J g , φ(f, g) = 1≤i≤n
J |J|≥1
P where J = (j1 , . . . , jn ) is a multi-index, |J| = 1≤s≤n js , ∂i = ∂/∂xi , ∂J = ∂ |J| / ∂xj11 . . . ∂xjnn and, for fixed i and J, φi,J is a smooth function on Rn vanishing if |J| is greater than some integer. Consider the differential operator X X 1 ˜ )=− φi,J ∂iJ f , ψ(f |J| + 1 J 1≤i≤n
|J|+1
f /∂xj11
|J|≥1
. . . ∂xji i +1
. . . ∂xjnn for J = (j1 , . . . , jn ). Notice that where ∂iJ f means ∂ ˜ ˜ i ) = 0, for 1 ≤ i ≤ n. The following property of ψ˜ is established by a ψ(1) = ψ(x straightforward computation: X X ˜ i , xi . . . xi ) = δ ψ(x φ(xi1 , xi2 . . . xik ) (5) 1 2 k (i1 ,...,ik )
(i1 ,...,ik )
for any k ≥ 2 and indices (i1 , . . . , ik ). Then, for ψ: Pol → N satisfying the hypothesis of the lemma, we have X ˜ i , xi . . . xi ) = 0 δ(ψ − ψ)(x (6) 1 2 k (i1 ,...,ik )
˜ Since η(xi ) = 0, 1 ≤ i ≤ n, we for any k ≥ 2 and indices (i1 , . . . , ik ). Let η = ψ − ψ. have δη(xi , f ) = xi η(f ) − η(xi f ) for 1 ≤ i ≤ n and f ∈ Pol. Then Eq. (6) implies that 1 X η(xi1 . . . xik ) = xi1 η(xi2 . . . xik ) , k (i1 ,...,ik )
˜ Pol . and by induction on k, we find that η = 0 on Pol, i.e. ψ = ψ|
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Lemma 3. Let ν be the sun-product associated with some star-product ∗ν on (Rn , P ). The first cochain ρ1 of ν is a differential operator null on constants whose Hochschild coboundary satisfies δρ1 = P − C1 , where C1 is the first cochain of the star-product ∗ν . Proof. From Definition 3, we have for k ≥ 2 and indices (i1 , . . . , ik ): ρ(xi1 . . . xik ) = xi1 ν . . . ν xik =
1 X xiσ (1) ∗ν . . . ∗ν xiσ (k) , k! σ∈Sk
=
1 k
X
xi1 ∗ν ρ(xi2 . . . xik ) .
(7)
(i1 ,...,ik )
The first-order term in ν in the last equation is ρ1 (xi1 . . . xik ) =
1 k
X
C1 (xi1 , xi2 . . . xik ) +
(i1 ,...,ik )
1 k
X
xi1 ρ1 (xi2 . . . xik ) ,
(i1 ,...,ik )
which can be written as X
(δρ1 + C1 )(xi1 , xi2 . . . xik ) = 0 ,
(8)
(i1 ,...,ik )
since ρ1 (xi ) = 0. The associativity condition for a star-product implies that C1 is a Hochschild 2-cocycle and Theorem 1 and condition (iv) in Definition 1 tell us that C1 = P + δθ where θ is a differential operator null on constants. We can always take θ such that θ(xi ) = 0, 1 ≤ i ≤ n, by adding a suitable 1-cocycle to P it (e.g. θ˜ = θ − i θ(xi )∂i ). The Poisson bracket P is a 2-tensor and does not contribute to the left-hand side of Eq. (8). The same argument used in the proof of Lemma 2 (cf. Eq. (6)) leads us to the conclusion that ρ1 = −θ and, consequently, δρ1 = P − C1 . Proof of Theorem 2. Using that the cochains of a sun-product satisfy ρr (xi ) = 0, 1 ≤ r, we can write the equation of the term of order r in Eq. (7) as X X δρr (xi1 , xi2 . . . xik ) = − Cr (xi1 , xi2 . . . xik ) (i1 ,...,ik )
(i1 ,...,ik )
−
X
X
(i1 ,...,ik )
a+b=r a,b≥1
Ca (xi1 , ρb (xi2 . . . xik )) ,
(9)
for k ≥ 2 and r ≥ 1 (for r = 1, the right-hand side has only one sum). Notice that in the right-hand side of Eq. (9) only the first r − 1 cochains of the sun-product ν appear. We already know that ρ1 is a differential operator null on constants from Lemma 3, and with the help of Lemma 2 a simple induction on r proves the theorem.
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Remark 1. A direct consequence of Theorem 2 is that we can extend sunP products, originally defined on N0ν , to Nν by the formula f ν g = π(f g) + r≥1 ν r ρr (π(f g)) for f, g ∈ Nν . Theorem 2 has very simple consequences. We shall end this section by deriving some results about the cochains of a sun-product. In Sec. 3 we shall see that differentiability of sun-products allows one to deduce interesting properties for starproducts on the dual of a Lie algebra. The cochains of a sun-product can be used to construct equivalence operators and this turns out to be a quite powerful tool to establish equivalence relation between certain type of star-products without any cohomological computations. Definition 4. E(P ) is the set of star-products on (Rn , P ) such that their associated sun-products ν coincide with the usual product on Pol, i.e., the cochains ρr = 0 for r ≥ 1. Corollary 1. Any star-product on (Rn , P ) is equivalent to a star-product belonging to E(P ). Proof. Let ∗ν be a star-product and let {ρr }r≥1 be the cochains of its associated sun-product. The maps ρr are defined on N and we shall denote by the same symbol their R[[ν]]-linear extension to Nν . Let us define another star-product ∗0ν P by equivalence from ∗ν with equivalence operator T = I + r≥1 ν r ρr , that is to say: T (f ∗0ν g) = T (f ) ∗ν T (g), f, g ∈ Nν . Since T (xi ) = xi , 1 ≤ i ≤ n, we have for k ≥ 2: T (xi1 ∗0ν . . . ∗0ν xik ) = xi1 ∗ν . . . ∗ν xik , and complete symmetrization gives T (xi1 ν 0 . . . ν 0 xik ) = xi1 ν . . . ν xik . By definition T is invertible and notice that xi1 ν . . . ν xik = T (xi1 . . . xik ), from the equation above we conclude that xi1 ν 0 . . . ν 0 xik = xi1 . . . xik , for any k ≥ 2, i.e. the cochains of ν 0 satisfy ρ0r = 0 for r ≥ 1. Hence ∗0ν belongs to E(P ). In view of the preceding corollary, the problem of classification of equivalence classes of star-products on (Rn , P ) reduces to classifying equivalence classes in E(P ). An order-by-order analysis in ν of star-products in E(P ) makes the second Lichnerowicz–Poisson cohomology [11] space appear explicitly here. It plays the same role in the Poisson case as the one played by the second de Rham cohomology space for the classification of equivalences classes in the symplectic case [14]. Corollary 2. Let {ηi }i≥1 be a sequence of differential operators on N such that ηi (1) = ηi (xk ) = 0, 1 ≤ i, 1 ≤ k ≤ n, and let ∗ν be some star-product on (Rn , P ). There exists a star-product ∗0ν , equivalent to ∗ν , such that the cochains of the sun-product ν 0 associated with ∗0ν are precisely the ηi ’s. Proof. Any star-product ∗ν is equivalent to a star-product ∗00ν in E(P ). For {ηi }i≥1 satisfying the hypothesis of the corollary, we consider a third star-product
ON GENERALIZED ABELIAN DEFORMATIONS
719
P ∗0ν defined by equivalence: T (f ∗00ν g) = T (f ) ∗0ν T (g) where T = I + i≥1 ν i ηi . It is easily verified that the sun-product associated with ∗0ν admits the ηi ’s as cochains. n This shows that the set of possible cochains for a sun-product on R coincides with the set of differential operators on Rn vanishing on polynomial of degree less or equal to one. Also it is sufficient to consider only one equivalence class of starproducts to generate all of the sun-products on Rn . As one could have guessed, there is almost no constraints imposed by the associativity condition on the possible cochains of a sun-product. This fact in our opinion makes the cohomology problem for generalized deformations quite difficult (see the discussion in Sec. 4). 3. Sun-Products on g∗ We shall specialize our discussion to the case of the dual of a Lie algebra. Let g be a real Lie algebra of dimension n. The dual g∗ of g carries a canonical Poisson structure and, by choosing a basis of g, we can identify g∗ as Poisson manifold with Rn endowed with the following Poisson bracket: PC (F, G) =
n X i,j,k=1
k Cij xk
∂f ∂g , ∂xi ∂xj
∀ f, g ∈ N ,
(10)
k ’s are the structure constants of the Lie algebra g expressed in the where the Cij chosen basis. A particular class of star-products which are important for physical applications and in star-representation theory are the covariant star-products:
Definition 5. Let g be a Lie algebra of dimension n. A star-product ∗ν on Rn is said to be g-covariant if X 1 k (xi ∗ν xj − xj ∗ν xi ) = PC (xi , xj ) = Cij xk , 2ν n
∀ 1 ≤ i, j ≤ n ,
(11)
k=1
k ’s are the structure constants of the Lie algebra g in a given basis. where the Cij
Star-products on the dual of a Lie algebra were known from the very beginning of the theory of star-products. The well known Moyal product is such an example, another for so(n)∗ appears in [2] in relation with the quantization of angular momentum. The general case was treated by S. Gutt [9] who defined a star-product on the cotangent bundle of any Lie group T ∗ G. Gutt’s star-product admits a restriction to g∗ that we shall call Gutt’s star-product on g∗ . Gutt’s star-product on g∗ has a simple expression that we briefly recall here (see [9] for further details). Polynomials on g∗ can be considered as elements of the symmetric algebra over g, S(g). Let Sr be the set of homogeneous polynomials of degree r and let U(g) be the universal enveloping algebra of g. The symmetrization map φ: S(g) → U(g) defined by 1 X Xσ(1) ⊗ . . . ⊗ Xσ(k) , φ(Xi1 . . . Xir ) = r! σ∈Sr
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(where ⊗ is the product in U(g)) is a bijection. Let Ur = φ(Sr ), one has U(g) = L L r≥0 Ur and each u ∈ U(g) can be decomposed as u = r≥0 ur , where ur ∈ Ur . Now define a product between P ∈ Sp and Q ∈ Sq , by X (2ν)k φ−1 ((φ(P ) ⊗ φ(Q))p+q−r ) , P ×ν Q = r≥0
and extend it by linearity to all of S(g). It can be shown that the product ×ν is associative and is defined by differential operators. Hence one gets a star-product on S(g) which is naturally extended to C ∞ (g∗ ). This star-product is g-covariant. We shall see that Gutt’s star-product plays a special role in relation with sun-products on g∗ . Lemma 4. Let g be a fixed Lie algebra of dimension n. The set of g-covariant star-products belonging to E(PC ) has only one element. In other words, there is only one g-covariant star-product on g∗ whose sun-product coincides with the usual product on Pol. Proof. Let ∗ν be a g-covariant star-product on (Rn , PC ) with associated sunproduct ν which coincides with usual product on Pol. Let Lin ⊂ Pol be the subspace of linear homogeneous polynomials on Rn . It is easy to verify that the ∗ν -powers, the ν -powers and the usual powers of any X ∈ Lin are identical: X
m ∗
m
= X ν = X m ,
∀ X ∈ Lin, m ≥ 0 .
(12)
m
Obviously, we also have that X ∗ = X m , for any X ∈ Lin[[ν]], m ≥ 0. (As usual, Lin[[ν]] denotes the set of formal series in ν with coefficients in Lin.) For X ∈ Lin[[ν]], consider its ∗ν -exponential defined by exp∗ν (X) =
X 1 r X ∗ν , r!
(13)
r≥0
it is an element of Nν and here exp∗ν (X) is identical to the usual exponential exp(X) for any X ∈ Lin[[ν]]. The fact that ∗ν is a g-covariant star-product allows us to make use of the Campbell–Hausdorff formula in the following form (in the sense of formal series): exp∗ν (sX) ∗ν exp∗ν (tY ) = exp∗ν (Z(sX, tY )) ,
X, Y ∈ Lin, s, t ∈ R ,
(14)
P r where Z(X, Y ) = r≥0 ν Zr (X, Y ) ∈ Lin[[ν]], and the Zr ’s are related to the Campbell–Hausdorff coefficients by Zr (X, Y ) = 2r cr+1 (X, Y ) (where c1 (X, Y ) = X + Y , c2 (X, Y ) = PC (X, Y )/2, etc.). As the ∗ν -exponential of X ∈ Lin[[ν]] is simply the usual exponential, Eq. (14) yields exp(sX) ∗ν exp(tY ) = exp(Z(sX, tY )) ,
X, Y ∈ Lin, s, t ∈ R .
Hence a g-covariant star-product for which the associated product must satisfy the preceding relation.
ν -product
(15)
is the usual
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ON GENERALIZED ABELIAN DEFORMATIONS
Actually Eq. (15) determines the star-product ∗ν completely. Notice that a bidifferential operator B: N × N → N is completely characterized by the functions B(X a , Y b ), a, b ∈ N, X, Y ∈ Lin. The functions Cr (X a , Y b ), 0 ≤ a, b, r, X, Y ∈ Lin, which completely determine the cochains Cr of ∗ν can be easily computed by differentiation with respect to s and t on both sides of Eq. (15) of the coefficient of ν r and by evaluation at s = t = 0. Therefore there is at most one star-product whose associated ν -product is the usual product on Pol. It is easy to show that the star-product defined by Eq. (15) has a the usual product as associated ν -product. By setting Y = X in Eq. (15), we find that a X a ∗ν X b = X a+b , ∀X ∈ Lin, 0 ≤a a, b, which implies by induction that X ∗ν = X a , ∀X, a, b. By Eq. (12) we have X ν = X a , ∀ X, a, and since ν is Abelian, it implies that ν is the usual product on Pol and this proves the lemma. Lemma 5. Let g be a Lie algebra. The g-covariant star-product characterized in Lemma 4 is Gutt’s star-product on g∗ . Proof. We shall use the notations introduced in the proof of Lemma 4. Let g be of dimension n and let ∗ν be the star-product characterized in Lemma 4 by Eq. (15). The identification of the coefficients of ν r in (15) gives Cr (exp(sX), exp(tY )) = Fr (sX, tY ) exp(sX + tY ) ,
∀ X, Y ∈ Lin, s, t ∈ R , (16)
where the Fr ’s are polynomial functions of the (normalized) Campbell–Hausdorff coefficients Zr (sX, tY ) and are defined by the following recursive relation with F0 = 1: k=r−1 1 X (r − k)Zr−k Fk , r ≥ 1 . (17) Fr = r k=0
By induction, one finds the explicit expression for Fr for r ≥ 1 to be Fr =
k=r X
X
k=1
m1 >...>mk ≥1 n1 ,...nk ≥1 m1 n1 +···+mk nk =r
1 (Zm1 )n1 . . . (Zmk )nk . n 1 ! . . . nk !
(18)
Now we shall derive an explicit expression for X ∗ν exp(Y ), X, Y ∈ Lin. Notice that this relation also characterizes ∗ν as any polynomial can be expressed as a ∗ν -polynomial (it is a simple consequence of Eq. (12)). In general, the Campbell– Hausdorff coefficients {ci }i≥1 (c1 (X, Y ) = X + Y , c2 (X, Y ) = 12 [X, Y ], etc.) have the following properties: ci (0, X) = ci (X, 0) = 0 ,
i ≥ 2;
Bi−1 ∂ ci (sX, Y )|s=0 = (adY )i−1 (X) , ∂s (i − 1)!
i ≥ 2;
(19)
where adY : X 7→ [Y, X], and Bn are the Bernoulli numbers. These can be easily derived from the standard recursive formula for the ci ’s, see e.g. [13].
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Also, using Eqs. (18) and (19), along with the definition of Zr (Zr = 2r cr+1 ), one finds that r ≥ 1;
Fr (0, Y ) = 0 ,
∂ 2r Br ∂ Fr (sX, Y )|s=0 = Zr (sX, Y )|s=0 = (adY )r (X) , ∂s ∂s r!
r ≥ 1.
Therefore we can write Cr (X, exp(tY )) = =
∂ (Fr (sX, tY ) exp(sX + tY ))|s=0 ∂s 2r Br (adtY )r (X) exp(tY ) , r!
r ≥ 1.
(20)
For r = 0, we simply have: C0 (X, exp(tY )) = X exp(tY ). Equation (20) is also characterizing Gutt’s star-product on g∗ (compare with Eq. (3.2) in [9]). As a simple consequence of Lemmas 4 and 5 we have the following corollary which tells us that any two covariant star-products on the dual of a Lie algebra are equivalent. Corollary 3. Any covariant star-product on the dual of a Lie algebra g is equivalent to Gutt’s star-product on g∗ . Proof. Let ∗ν be a g-covariant star-product on g∗ , the dual of a Lie algebra of dimension n. From Corollary 1, ∗ν is equivalent to a star-product ∗0ν belonging to E(PC ), where PC is the Lie–Poisson structure on g∗ . The equivalence operator is constructed out from the cochains of the sun-product associated with ∗ν and it leaves invariant linear polynomials, i.e. T (xi ) = xi , 1 ≤ i ≤ n. Consequently, ∗0ν is also a g-covariant star-product. According to Lemmas 4 and 5, ∗0ν must be Gutt’s star-product on g∗ . Remark 2. Though the de Rham cohomology of g∗ is trivial, not all starproducts on g∗ are equivalent. Indeed, in the symplectic case, the second de Rham cohomology space classifies equivalence classes of star-products. In the Poisson case, one has to consider the Lichnerowicz–Poisson cohomology [11] instead, and this cohomology is not in general trivial for the Lie–Poisson structure on g∗ . See [12], for explicit computations of some of the (Chevalley–Eilenberg) cohomology spaces for the dual of a Lie algebra. 4. Weak and Strong Equivalences In the deformation theory of some algebraic structure one has the notion of equivalent deformations. The equivalence of star-products given by Definition 2 is adapted to the associative (differential) case and one has similar notions of equivalence for other algebraic structures (e.g. Lie algebras, Abelian algebras, etc.). Moreover, as mentioned in Sec. 2.2, it is a general result of Gerstenhaber [8] that obstructions for equivalence of deformations reside in the second cohomology space
ON GENERALIZED ABELIAN DEFORMATIONS
723
of an appropriate cohomology. For associative, Lie, Abelian deformations the associated cohomologies are, respectively, Hochschild, Chevalley–Eilenberg, Harrison cohomologies. One may wonder what is the corresponding cohomology for generalized Abelian deformations. Before discussing on that matter, it is important to bear in mind that in Gerstenhaber’s theory of deformations a deformed algebraic structure has a structure of K[[ν]]-algebra, where K is the ground ring of the original structure. This feature, which is crucial to determine the appropriate cohomology, does not hold anymore in the case of generalized deformations. The answer to the cohomology issue raised by generalized deformations might be, as advocated by M. Flato [6], that one has to give a non-commutative ring structure on the space of formal parameters in such a way that R[[ν]]-bilinearity would be restored. This should lead to a non-commutative deformation theory and the first steps toward this program were taken by Pinczon [15] who considered the case where the deformation parameter is acting by different left and right endomorphisms on the algebra (hence the deformation parameter is not required to commute with the undeformed algebra). This point of view produced very interesting results (e.g. deformation of the Weyl algebra yields supersymmetric algebras), but still generalized deformations do not fit in the particular framework considered in [15]. The cohomology problem is still open and in a previous work [5] we have nevertheless considered two notions of equivalence for sun-products. They are mimicking the usual notion of equivalence and take into account that sun-products are not R[[ν]]-bilinear operations, but only R-bilinear. Let us recall their definitions. Definition 6. Two sun-products ν and ν 0 on (Rn , P ) are said to be (a) weakly ((b) strongly) equivalent, if there exists an R[[ν]]-linear map Sν : Nν 7→ Nν P where Sν = r≥0 ν r Sr , with Sr : N → N, r ≥ 1, being differential operators and S0 = I, such that for f, g ∈ N the following holds: (a) Sν (f ν g) = Sν (f ) ν 0 Sν (g), (b) Sν (f ν g) = Sµ (f ) ν 0 Sµ (g)|µ=ν . For weak equivalence, condition (a) above can be equivalently replaced by Sν (f ν g) = f ν 0 g, as sun-products annihilate the deformation parameter ν. In the case of strong equivalence, condition (b), when written in terms of the cochains of the sun-products, simply states that X X Sr (ρs (f g)) = ρ0r (Sa (f )Sb (g)) , f, g ∈ N, t ≥ 0 , (21) r+s=t r,s≥0
r+a+b=t r,a,b≥0
where the ρi ’s (resp. ρ0i ’s) are the cochains of ν (resp. ν 0 ). It can be easily checked that Definition 6 indeed defines equivalence relations on the set of sun-products. Weak or strong triviality has to be understood as weak or strong equivalence with the pointwise product on N. We shall now draw some conclusions for weak and strong equivalences of sunproducts from Theorem 2. It was shown in [5] that a sun-product is weakly trivial if its cochains are differential operators. Hence as a corollary of Theorem 2, we simply have:
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Corollary 7. Let
ν
be a sun-product on (Rn , P ), then
ν
is weakly trivial.
Proof. Let ρi be the cochains of ν . They are differential operators null on P constants by Theorem 2. Then define Sν to be the formal inverse of r≥0 ν r ρr . The map Sν satisfies Sν (f ν g) = f · g for f, g ∈ N, where · denotes the pointwise product, hence ν is weakly equivalent to the pointwise product. On the the hand, we shall see that strong equivalence puts severe conditions on the equivalence operator Sν . By setting g = 1 in Eq. (21), we get with shortened notations that Sν ρ = ρ0 Sν and by substituting this relation in Eq. (21), we find that the equivalence operator should satisfy Sν (f g) = Sν (f )Sν (g). Hence Sν can be nothing else than the exponential of a formal series of derivations of the pointwise product. Actually there are still some supplementary constraints on Sν , but we do not need to be concerned with them. We conclude that strong equivalence classes are very small and can even reduce to a single point in some situations (e.g. the equivalence class of strongly trivial sun-products). Although, we do not know whether weak and strong equivalences are induced by the cohomology of some complexes, these notions provide limiting cases between which a proper notion of equivalence for generalized deformations should lie. Acknowledgements The author would like to thank Mosh´e Flato and Daniel Sternheimer for very useful discussions, and Izumi Ojima for great hospitality at RIMS where this work was finalized. References [1] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, “Deformation theory and quantization. I. Deformations of symplectic structures”, Ann. Physics 111 (1978) 61–110. [2] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, “Deformation theory and quantization. II. Physical applications”, Ann. Physics 111 (1978) 111–151. [3] M. De Wilde and P. B. A. Lecomte, Existence of star-products on and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds, Lett. Math. Phys. 7 (1983) 487–496. [4] G. Dito, M. Flato, D. Sternheimer and L. Takhtajan, “Deformation quantization and Nambu mechanics”, Commun. Math. Phys. 183 (1997) 1–22. [5] G. Dito and M. Flato, “Generalized abelian deformations: Application to Nambu mechanics”, Lett. Math. Phys. 39 (1997) 107–125. [6] M. Flato, private communication (1996). [7] M. Flato, G. Dito and D. Sternheimer, “Nambu mechanics, n-ary operations and their quantization”, in Deformation Theory and Symplectic Geometry (Ascona, 1996), volume 20 of Math. Phys. Stud., pages 43–66. Kluwer Acad. Publ., Dordrecht, 1997. [8] M. Gerstenhaber, “On the deformation of rings and algebras”, Ann. Math. 79 (1964) 59–103. [9] S. Gutt, “An explicit ∗-product on the cotangent bundle of a Lie group”, Lett. Math. Phys. 7 (1983) 249–258. [10] M. Kontsevich, “Deformation quantization of Poisson manifolds I”, preprint I.H.E.S. q-alg/9709040 (1997).
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[11] A. Lichnerowicz, “Les vari´et´es de Poisson et leurs alg` ebres de Lie associ´ ees”, J. Differential Geometry 12 (1977) 253–300. [12] D. M´elotte, “Cohomologie de Chevalley associ´ee aux vari´et´es de Poisson”, Bull. Soc. Roy. Sci. Li` ege 58 (1998) 319–413. ˇ [13] M. A. Na˘ımark and A. I. Stern, “Theory of group representations”, volume 246 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], Springer-Verlag, New York, 1982. [14] R. Nest and B. Tsygan, “Algebraic index theorem”, Commun. Math. Phys. 172 (1995) 223–262. [15] G. Pinczon, “Noncommutative deformation theory”, Lett. Math. Phys. 41 (1997) 101– 117. [16] J. Vey, “D´eformation du crochet de Poisson sur une vari´et´e symplectique”, Comment. Math. Helv. 50 (1975) 421–454.
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEM: LOCAL EQUATIONS OF MOTION AND THEIR HAMILTONIAN PROPERTIES YURI B. SURIS Fachbereich Mathematik, Sekr. MA 8-5, Technische Universit¨ at Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany Received 29 September 1997 Revised 22 July 1998 We develop the approach to the problem of integrable discretization based on the notion of r-matrix hierarchies. One of its basic features is the coincidence of Lax matrices of discretized systems with the Lax matrices of the underlying continuous time systems. A common feature of the discretizations obtained in this approach is non-locality. We demonstrate how to overcome this drawback. Namely, we introduce the notion of localizing changes of variables and construct such changes of variables for a large number of examples, including the Toda and the relativistic Toda lattices, the Volterra and the relativistic Volterra lattices, the second flows of the Toda and of the Volterra hierarchies, the modified Volterra lattice, the Belov–Chaltikian lattice, the Bogoyavlensky lattices, the Bruschi–Ragnisco lattice. We also introduce a novel class of constrained lattice KP systems, discretize all of them, and find the corresponding localizing change of variables. Pulling back the differential equations of motion under the localizing changes of variables, we find also (sometimes novel) integrable one-parameter deformations of integrable lattice systems. Poisson properties of the localizing changes of variables are also studied: they produce interesting one-parameter deformations of the known Poisson algebras.
Contents 1. 2. 3. 4. 5. 6. 7. 8. 9.
Introduction The Problem of Integrable Discretization Lax Representations R-Matrix Poisson Structures Factorization Theorems Recipe for Integrable Discretization Localizing Changes of Variables Basic Algebras and Operators Toda Lattice 9.1. Equations of motion and tri-Hamiltonian structure 9.2. Lax representation 9.3. Discretization 9.4. Local equations of motion for dTL 10. Second Flow of the Toda Hierarchy 10.1. Equations of motion and tri-Hamiltonian structure 10.2. Lax representation 10.3. Discretization 10.4. Local equations of motion for dTL2 11. Volterra Lattice 11.1. Equations of motion and bi-Hamiltonian structure 11.2. Lax representation 727 Reviews in Mathematical Physics, Vol. 11, No. 6 (1999) 727–822 c World Scientific Publishing Company
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Y. B. SURIS
11.3. Discretization 11.4. Local equations of motion for dVL Second Flow of the Volterra Hierarchy 12.1. Equations of motion and bi-Hamiltonian structure 12.2. Lax representation 12.3. Discretization 12.4. Local equations of motion for dVL2 12.5. Local discretization of the KdV Modified Volterra Lattice 13.1. Equations of motion and Hamiltonian structure 13.2. Discretization 13.3. Local equations of motion for dMVL 13.4. Particular case α → ∞ Bogoyavlensky Lattices 14.1. Equations of motion and Hamiltonian structure 14.2. Lax representation 14.3. Discretization of BL1 14.4. Discretization of BL2 14.5. Discretization of BL3 14.6. Particular case p = 2 Alternative Approach to Volterra Lattice 15.1. Equations of motion and bi-Hamiltonian structure 15.2. Lax representation 15.3. Discretization 15.4. Local equations of motion for dVL 15.5. Lax representation for VL2 15.6. Discretization of VL2 15.7. Local equations of motion for dVL2 15.8. Miura relations to the Toda hierarchy Relativistic Toda Lattice 16.1. Equations of motion and tri-Hamiltonian structure 16.2. Lax representation 16.3. Discretization of the relativistic Toda hierarchy 16.4. Discretization of the flow RTL+ 16.5. Local equations of motion for dRTL+ 16.6. Discretization of the flow RTL− 16.7. Local equations of motion for dRTL− 16.8. Third appearance of the Volterra lattice Belov–Chaltikian Lattice 17.1. Equations of motion and bi-Hamiltonian structure 17.2. Lax representation 17.3. Discretization 17.4. Local equations of motion for dBCL Relativistic Volterra Lattice 18.1. Equations of motion and bi-Hamiltonian structure 18.2. Lax representation 18.3. Discretization 18.4. Local equations of motion for dRVL Some Constrained Lattice KP Systems 19.1. Equations of motion and Hamiltonian structure 19.2. Lax representation 19.3. Discretization 19.4. Local equations of motion for dcKPL 19.5. Example 1: = (0, 0, 0) 19.6. Example 2: = (0, 1, 0) 19.7. Example 3: = (0, 1, 1) Bruschi–Ragnisco Lattice
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1. Introduction This paper deals with some aspects of the following general problem: how to discretize one or several of independent variables in a given integrable system, maintaining the integrability property? We call this the problem of integrable discretization. To assure the coincidence of the qualitative properties of the discretized models with that of the continuous ones becomes one of the central ideas of the modern numerical analysis, which therefore comes to a close interplay with different aspects of the theory of dynamical systems. One of the most advanced examples of this approach is the symplectic integration, on which recently the first monograph appeared [57]. The problem of integrable discretization constitutes another aspect of this general line of thinking. It arose in the course of development of the theory of solitons. This theory, born exactly 30 years ago, has grown so tremendously, that it is difficult to keep an overview of the whole variety of different notions and contexts of integrability, not to say about the concrete results. (Recall that recently there appeared a thick book under the title “What is integrability?” [85].) Correspondingly, various approaches to the problem of integrable discretization are currently available. They began to be discussed sporadically in the soliton literature starting from the mid-70s. Following ones should be mentioned: 1. An approach based on the representation of an integrable system as a compatibility condition of two auxiliary linear problems. A natural proposition is to discretize one or both of them [1]. This, however, can be made in a great variety of ways, cf. for instance, different spatial discretizations of the nonlinear Schr¨ odinger equation and of the sine-Gordon equation, found in [1] and in [32]. One attempt of fixing discretization crystallized with the development of the Hamiltonian approach. Namely, Faddeev and Takhtajan, based on the experience of the Leningrad soliton school, formulated in [22] the following rule for a transition from models with one continuous space variable to lattice models: the r-matrix should be preserved, the linear Poisson bracket being replaced by the quadratic one. See [22] for a collection of examples showing productivity of this approach. 2. One of the most intriguing and universal approaches is the Hirota’s one [27], based on the notion of the τ -function and on a bilinear representation of integrable systems. It seems to be able to produce discrete versions of the majority of soliton equations, but still remains somewhat mysterious, and the mechanism behind it is yet to be fully understood. One successful way to do it was proposed in [16], where also a large number of integrable discretizations was derived. Among the most interesting products of this approach is the so called Hirota–Miwa Eq. [28, 38], which is sometimes claimed to contain “everything”, i.e. the majority if not all soliton equations (continuous and discrete) are particular or limiting cases of this single equation, cf. [84].
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3. A fruitful method is based on the “direct linearization” [44, 51, 14, 82, 45, 43]. Its basic idea is to derive integrable nonlinear differential equations which are satisfied by the solutions of certain linear integral equations. A large variety of continuous and discrete soliton equations has been obtained on this way. 4. Approach based on the variational principle (discrete Lagrangian equations), combined with matrix factorizations [79, 40, 18]. Historically, it was the work of Veselov and Moser that consolidated the more or less isolated results to a separate branch of the theory of integrable systems. 5. Considering stationary and restricted flows of soliton hierarchies, and the closely related “nonlinearization” of spectral problems, often leads to interesting discrete equations [52–54]. 6. Differential equations describing various geometric problem (surfaces of the constant mean curvature, motion of the curve in the space, etc.) turn out to be integrable [75, 7, 8]. Correspondingly, a discretization of geometric notions naturally leads to discrete integrable equations [9, 10, 20, 19]. 7. There exist integrable discretizations which belong to the most beautiful examples, but were derived by guess, without any systematical approach [60, 64]. 8. Last but not least we mention an approach to the temporal discretization in which the auxiliary spectral problem is not discretized at all. In other words, the basic feature of this approach is maintaining the Lax matrix of the continuous time system. The first example of this approach is the work by Ablowitz–Ladik [2], further developed towards the practical algorithms in [77]. This feature was also put in the basis of the work by Gibbons and Kupershmidt [25, 35]. The discretizations found in all these papers were somewhat unsatisfactory from the esthetical point of view, namely they suffered from being nonlocal, as opposed to the underlying continuous time systems. Moreover, these authors did not recognize the connection with the factorization problem, which did not allow them to identify their discretizations as certain members of the corresponding hierarchies and to establish the Poisson properties of these discretizations. This led Gibbons and Kupershmidt to call this method “the method of the bizarre ansatz”. Recently the author pushed forward the last mentioned approach to the problem of integrable discretization, putting it in a connection with the r-matrix theory of integrable hierarchies (see [55] for a review of this theory). In this context the method could be understood properly, and became rather natural and simple. It was applied to a number of integrable lattice systems [65–70, 72, 73]. Its clear advantage is universality. The method is in principle applicable to any system admitting an r-matrix interpretation, which is the common feature of the great majority of the known integrable systems. As for the drawback of nonlocality, there exist several ways to repair it. The first one, connected with the notion of discrete time Newtonian equations of motion, was followed in [65, 66, 69, 70]. A splitting of complicated flows into superpositions of simpler ones was used in [72, 73]. The present work is devoted to another way connected with the so-called localizing changes of variables.
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The paper has the following structure. In Sec. 2 we give an accurate formulation of the problem of integrable discretization. Sections 3–5 are devoted to a general framework of integrable r-matrix hierarchies of Lax equations on associative algebras. In Sec. 6 we formulate a general recipe of integrable discretization, and in Sec. 7 we introduce the notion of localizing changes of variables and discuss their general properties. Section 8 contains the description of algebras used for the analysis of all integrable lattice systems in this paper. The rest of the paper is devoted to a detailed elaboration of a number of examples, including the most prominent ones, such as the Toda lattice and the Volterra lattice, and less well known ones, such as the Belov–Chaltikian lattice. 2. The Problem of Integrable Discretization Let us formulate the problem of integrable discretization more precisely. Let X be a Poisson manifold with a Poisson bracket {·, ·}. Let H be a completely integrable Hamilton function on X , i.e. let the system x˙ = {H, x}
(2.1)
possess many enough functionally independent integrals Ik (x) in involution. The problem consists in finding a map X 7→ X described by a formula x e = Φ(x; h) ,
(2.2)
depending on a small parameter h > 0, and satisfying the following requirements: 1. The map (2.2) is a discrete time approximation for the flow (2.1) in the following sense: (2.3) Φ(x; h) = x + h{H, x} + O(h2 ) (of course, one might require also a higher order of approximation). In all our considerations and formulas we pay a special attention to a simple and transparent control of the continuous limit h → 0. 2. The map (2.2) is Poisson with respect to the bracket {·, ·} on X or with respect to some its deformation {·, ·}h such that {·, ·}h = {·, ·} + O(h). 3. The map (2.2) is integrable, i.e. possesses the necessary number of independent integrals in involution Ik (x; h) approximating the integrals of the original system: Ik (x; h) = Ik (x) + O(h).
3. Lax Representations Our approach to the problem of integrable discretization is applicable to any system allowing an r-matrix interpretation, but we formulate the basic recipe in a simplified form, applicable to systems with a Lax representation of one of the following types: (3.1) L˙ = [ L, π+ (f (L))] = −[ L, π− (f (L))] ,
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or L˙ j = Lj · π+ (f (Tj−1 )) − π+ (f (Tj )) · Lj = −Lj · π− (f (Tj−1 )) + π− (f (Tj )) · Lj .
(3.2)
Let us discuss the notations. Let g be an associative algebra. One can introduce in g the structure of Lie algebra in a standard way. Let g + , g − be two subalgebras such that as a vector space g is a direct sum g = g + ⊕ g − . Denote by π± : g 7→ g ± the corresponding projections. Finally, let f : g 7→ g be an Ad-covariant function on g, and let L stand for a generic element of g. Then (3.1) is a certain differential equation on g. Nm Further, let g = j=1 g be a direct product of m copies of the algebra g. A generic element of g is denoted by L = (L1 , . . . , Lm ). We use also the notation Tj = Tj (L) = Lj · . . . · L1 · Lm · . . . · Lj+1 .
(3.3)
Then (3.2) is a certain differential equation on g. Such equations are sometimes called Lax triads. One says that (3.1), resp. (3.2), is a Lax representation of the flow (2.1), if there exists a map L : X 7→ g (resp. L : X 7→ g) such that the former equations of motion are equivalent to the latter ones. Let us stress that when considering Eqs. (3.1), resp. (3.2), in the role of Lax representation, the letter L (resp. L) does not stand for a generic element of the corresponding algebra any more; rather, it represents the elements of the images of the maps L : X 7→ g and L : X 7→ g, correspondingly. The elements L(x), resp. L(x) (and the map L, resp. L, itself) are called Lax matrices. Equations (3.1) and (3.2) have several remarkable features. In particular, they are Hamiltonian under rather general conditions. The corresponding r-matrix Poisson brackets will be discussed in the next section. Further, they may be explicitly solved in terms of a certain factorization problem in the Lie group G corresponding to the Lie algebra g, see Sec. 5. 4. R-Matrix Poisson Structures We give here a brief review of several existing constructions of Poisson brackets on associative algebras implying the Lax form of Hamiltonian equations of motion. Suppose that g carries a non-degenerate scalar product h·, ·i, bi-invariant with respect to the multiplication in g. Let R be a linear operator on g. Definition 4.1 [58]. A linear r-matrix bracket on g corresponding to the operator R is defined by {ϕ, ψ}1 (L) =
1 h[R(∇ϕ(L)), ∇ψ(L)] + [∇ϕ(L), R(∇ψ(L))], Li . 2
If this is indeed a Poisson bracket, it will denoted by PB1 (R).
(4.1)
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Theorem 4.2 [58]. A sufficient condition for (4.1) to define a Poisson bracket is given by the modified Yang–Baxter equation for the operator R: [R(u), R(v)] − R([R(u), v] + [u, R(v)]) = −α [u, v] ∀ u, v ∈ g .
(4.2)
This equation is denoted mYB(R; α). Now let A1 , A2 , S be three linear operators on g, A1 and A2 being skewsymmetric: A∗1 = −A1 , A∗2 = −A2 . (4.3) Definition 4.3 [62]. A quadratic r-matrix bracket on g corresponding to the triple A1 , A2 , S is defined by: {ϕ, ψ}2 (L) =
1 1 hA1 (d0 ϕ(L)), d0 ψ(L)i − hA2 (dϕ(L)), dψ(L)i 2 2 1 1 + hS(dϕ(L)), d0 ψ(L)i − hS ∗ (d0 ϕ(L)), dψ(L)i , 2 2
(4.4)
where we denote for brevity dϕ(L) = L · ∇ϕ(L) ,
d0 ϕ(L) = ∇ϕ(L) · L .
(4.5)
If this expression indeed defines a Poisson bracket, we shall denote it by PB2 (A1 , A2 , S). In what follows we shall usually suppose the following condition to be satisfied: A1 + S = A2 + S ∗ = R .
(4.6)
Then a linearization of PB2 (A1 , A2 , S) in the unit element of g coincides with PB1 (R), and we call the former a quadratization of the latter. Theorem 4.4 [62]. A sufficient condition for (4.4) to be a Poisson bracket is given by the Eq. (4.6) and mYB(R; α) ,
mYB(A1 ; α) ,
mYB(A2 ; α) .
(4.7)
Under these conditions the bracket PB2 (A1 , A2 , S) is compatible with PB1 (R). If the operator R is skew-symmetric and satisfies mYB(R; α), then the Poisson bracket PB2 (R, R, 0) is called Sklyanin bracket [58]. The brackets PB2 (A, A, S) with a skew-symmetric operator A and a symmetric operator S were introduced in [36, 46]. One of the most important properties of the r-matrix brackets is the following one. Theorem 4.5. Ad-invariant functions on g are in involution with respect to the bracket PB1 (R) and with respect to its quadratizations PB2 (A1 , A2 , S). The
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Hamiltonian equations of motion on g corresponding to an Ad-invariant Hamilton function ϕ, have the Lax form 1 L˙ = [L, R(f (L))] , 2
(4.8)
where f (L) = ∇ϕ(L) for the linear r-matrix bracket, and f (L) = dϕ(L) for its quadratizations. Quadratic r-matrix brackets have interesting and important features when conNm sidered on a “big” algebra g = j=1 g. This algebra carries a (non-degenerate, bi-invariant) scalar product hhL, M ii =
m X
hLk , Mk i .
k=1
Working with linear operators on g, we use the following natural notations. Let A : g 7→ g be a linear operator, let (A(L))i be the ith component of A(L); then we set m X (A)ij (Lj ) . (4.9) (A(L))i = j=1
For a smooth function Φ(L) on g we also denote by ∇j Φ, dj Φ, dj0 Φ the jth components of the corresponding objects. Now let A1 , A2 , S be linear operators on g satisfying conditions analogous to (4.3) and to (4.7). One has, obviously ((A1 )ij )∗ = −(A1 )ji ,
((A2 )ij )∗ = −(A2 )ji ,
((S)ij )∗ = (S∗ )ji .
Then one can define the bracket PB2 (A1 , A2 , S) on g. In components it reads {Φ, Ψ}2 (L) =
m m 1 X 1 X h(A1 )ij (d0j Φ), d0i Ψi − h(A2 )ij (dj Φ), di Ψi 2 i,j=1 2 i,j=1
+
m m 1 X 1 X h(S)ij (dj Φ), d0i Ψi − h(S∗ )ij (d0j Φ), di Ψi . (4.10) 2 i,j=1 2 i,j=1
Theorem 4.6 [71]. Let g be equipped with the Poisson bracket PB2 (A1 , A2 , S). Suppose that the following relations hold: (A1 )j+1,j+1 + (S)j+1,j = (A2 )j,j + (S∗ )j,j+1 = R
for all
(A1 )i+1,j+1 = −(S)i+1,j = (S∗ )i,j+1 = −(A2 )i,j
for
1 ≤ j ≤ m;
(4.11)
i 6= j .
(4.12)
Then each map Tj : g 7→ g (3.3) is Poisson, if the target space g is equipped with the Poisson bracket PB2 ((A1 )j+1,j+1 , (A2 )j,j , (S)j+1,j ) .
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Hamilton function of the form Φ(L) = ϕ(Lm · . . . · L1 ), where ϕ is an Ad-invariant function on g, generates Hamiltonian equations of motion on g having the form of Lax triads: 1 (4.13) L˙ j = Lj Bj−1 − Bj Lj , Bj = R(dϕ(Tj )) . 2 (In all formulas the subscripts should be taken (mod m).) This theorem is a far going generalization of the corresponding result for the Sklyanin bracket PB2 (R, R, 0) on g, which arises when S = 0, and
R 0 ... 0 0 R ... 0 A1 = A2 = R = . ... ... ... ... 0 0 ... R In this case each map Tj : g 7→ g (3.3) is Poisson, if the target space g is equipped with the Sklyanin bracket PB2 (R, R, 0) [58]. Certain generalizations of the latter result appeared also, e.g. in [58, 36], but in all previously known formulations the only few non-vanishing “operator entries” for the operators A1 , A2 , S were allowed, namely “diagonal” ones for A1 , A2 , and “subdiagonal” ones for S. In other words, all operators in (4.12) had to vanish. We have discussed above the r-matrix origin of Lax Eq. (4.8). Notice that under some natural conditions one can define also cubic r-matrix brackets on g, compatible with the linear and the quadratic ones and generating (4.8) as Hamiltonian systems. We shall, however, have no opportunity to apply the cubic r-matrix brackets in the present paper. If one is concerned with a Lax representation of a Hamiltonian flow (2.1) on a Poisson manifold (X , {·, ·}), then finding an r-matrix interpretation for it consists of finding an r-matrix bracket on g (or on g) such that the Lax matrix map L : X 7→ g (resp. L : X 7→ g) is a Poisson map. Then the manifold consisting of the Lax matrices is a Poisson submanifold. We close this section by noting that Eqs. (4.8), (4.13) coincide with (3.1), (3.2), respectively, if the operator R is defined as R = π+ − π− .
(4.14)
5. Factorization Theorems As a further remarkable feature of Eqs. (3.1) and (3.2) we consider the possibility to solve them explicitly in terms of a certain factorization problem in the Lie group G corresponding to g [76, 58, 55]. (Actually, this can be done even in a general situation of hierarchies governed by arbitrary R-operators satisfying the modified Yang–Baxter equation, not necessary of the form (4.14), see [55].) The factorization problem is described by the equation U = Π+ (U )Π− (U ) ,
U ∈ G,
Π± (U ) ∈ G± ,
(5.1)
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where G± are two subgroups of G with the Lie algebras g ± , respectively. This problem has a unique solution in a certain neighbourhood of the group unit. In what follows we suppose that G is a matrix group, and write the adjoint action of the group elements on g as a conjugation by the corresponding matrices. [In this −1 .] Correspondingly, we call Ad-covariant context we write Π−1 ± (U ) for (Π± (U ) functions g 7→ g also “conjugation covariant”. This notation has an additional advantage of being applicable also to functions g 7→ G. For the history of the following fundamental theorem and its different proofs the reader is referred to [55]. Theorem 5.1. Let f : g 7→ g be a conjugation covariant function. Then the solution of the differential equation (3.1) with the initial condition L(0) = L0 is given, at least for t small enough, by tf (L0 ) · L · Π etf (L0 ) e L(t) = Π−1 0 + + tf (L0 ) . = Π− etf (L0 ) · L0 · Π−1 − e
(5.2)
Proof. We give a proof based on a direct and simple verification. Denote L(t) = Π+ etf (L0 ) , R(t) = Π− etf (L0 ) , so that etf (L0 ) = L(t) R(t) ,
L(t) ∈ G+ ,
R(t) ∈ G− .
(5.3)
Now we set L(t) = L−1 (t)L0 L(t) = R(t)L0 R−1 (t)
(5.4)
(these two expressions for L(t) are equal due to Ad-covariance of f (L)), and check by direct calculation that this L(t) satisfies the differential equation (3.1). The theorem will follow by the uniqueness of solution. We see immediately that L(t) satisfies the following Lax type equation: ˙ = −[L, R˙ R−1 ] , L˙ = [L, L−1 L] and it remains to show that L−1 L˙ = π+ (f (L)) ,
R˙ R−1 = π− (f (L)) .
Since, obviously, L−1 L˙ ∈ g + , R˙ R−1 ∈ g − , we need to demonstrate only that L−1 L˙ + R˙ R−1 = f (L) .
(5.5)
To do this, we differentiate (5.3) and derive, using Ad-covariance of f and the definition (5.4): L˙ R + L R˙ = etf (L0 ) f (L0 ) = L Rf (L0 ) = L f (L) R . This is equivalent to (5.5).
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For an arbitrary conjugation covariant function F : g 7→ G one can define the map (B¨ acklund transformation) BF : g 7→ g according to the formula e = BF (L) = Π−1 (F (L)) · L · Π+ (F (L)) = Π− (F (L)) · L · Π−1 (F (L)) . L + −
(5.6)
Theorem 5.1 shows that the flows defined by the differential equations (3.1) consist of maps having such a form with F (L) = etf (L) . A very remarkable feature of such maps is their commutativity for different F ’s. Theorem 5.2. For two arbitrary conjugation covariant functions F1 , F2 : g 7→ G (5.7) BF2 ◦ BF1 = BF2 F1 , and therefore the maps BF1 , BF2 commute. Proof. Denote L1 = BF1 (L) ,
L2 = BF2 ◦ BF1 (L) = BF2 (L1 ) .
So, by definition we have −1 L1 = L−1 1 LL1 = R1 LR1 ,
−1 L2 = L−1 2 L1 L2 = R2 L1 R2 ,
(5.8)
where the matrices Li ∈ G+ , Ri ∈ G− (i = 1, 2) come from the following factorizations: F1 (L) = L1 R1 , F2 (L1 ) = L2 R2 . From (5.8) we have L2 = L−1 LL = RLR−1 ,
where L = L1 L2 ∈ G+ ,
R = R2 R1 ∈ G− .
(5.9)
Now the following chain of equalities holds: LR = L1 L2 R2 R1 = L1 F2 (L1 )R1 = L1 F2 (L−1 1 LL1 )R1 = F2 (L)L1 R1 = F2 (L)F1 (L) . In view of (5.9) we get L = Π+ (F2 (L)F1 (L)) ,
R = Π− (F2 (L)F1 (L)) ,
and the theorem is proved.
Theorem 5.2 implies that the flows of two arbitrary differential equations of the form (3.1) commute. Another important consequence of Theorem 5.2 is the following discrete-time counterpart of Theorem 5.1, going back to [76]. Theorem 5.3. Let F : g 7→ G be a conjugation covariant function. Then the solution of the difference equation −1 e = Π−1 L + (F (L)) · L · Π+ (F (L)) = Π− (F (L)) · L · Π− (F (L)) ,
(5.10)
e = L(n + 1), with the initial condition L(0) = L0 , is given by where L = L(n), L −1 n n n n L(n) = Π−1 + (F (L0 )) · L0 · Π+ (F (L0 )) = Π− (F (L0 )) · L0 · Π− (F (L0 )) . (5.11)
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Proof. From Theorem 5.2 there follows by induction that (BF )n = BF n .
Comparing the formulas (5.11), (5.2), we see that the map (5.10) is the time h shift along the trajectories of the flow (3.1) with f (L) = h−1 log(F (L)) . The above results are purely kinematic, in the sense that no additional Hamiltonian structure is necessary neither to formulate nor to prove them. However, as mentioned above, Eq. (3.1) often admit a Hamiltonian or even a multi-Hamiltonian interpretation. If this is the case, then we get some useful additional information. In particular, all maps (5.10) are Poisson with respect to the invariant Poisson bracket of the hierarchy (3.1), being shifts along the trajectories of Hamiltonian flows. Further, if the set of Lax matrices L(X ) for the system at hand forms a Poisson submanifold for one of the r-matrix brackets on g, then this manifold is left invariant by the flows (3.1) and by the maps (5.10). The functions on X of the form I ◦ L, where I are conjugation invariants of g, are integrals of motion of the corresponding systems and in involution with respect to {·, ·}. We close this section by giving analogous results for Lax equations on the direct Nm products g = j=1 g. Theorem 5.4. For a conjugation covariant function f : g 7→ g the solution of Eq. (3.2) with the initial value L(0) is given, at least for t small enough, by the formula etf (Tj (0)) · Lj (0) · Π+ etf (Tj−1 (0)) Lj (t) = Π−1 + tf (Tj−1 (0)) . = Π− etf (Tj (0)) · Lj (0) · Π−1 − e
(5.12)
Theorem 5.5. For a conjugation covariant function F : g 7→ G consider the following system of difference equations on g: −1 e j = Π−1 L + (F (Tj )) · Lj · Π+ (F (Tj−1 )) = Π− (F (Tj )) · Lj · Π− (F (Tj−1 )) .
(5.13)
Its solution with the initial value L(0) is given by the formula n n Lj (n) = Π−1 + (F (Tj (0))) · Lj (0) · Π+ (F (Tj−1 (0))) n = Π− (F n (Tj (0))) · Lj (0) · Π−1 − (F (Tj−1 (0))) .
(5.14)
The proofs are kinematic and absolutely parallel to the case of the “small” algebras g. 6. Recipe for Integrable Discretization The results of the previous section inspire the following recipe for integrable discretization, clearly formulated for the first time in [65, 66].
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Recipe. Suppose you are looking for an integrable discretization of an integrable system (2.1) allowing a Lax representation of the form (3.1). Then as a solution of your task you may take the difference equation (5.10) with the same Lax matrix L and some conjugation covariant function F : g 7→ G such that F (L) = I + hf (L) + O(h2 ) . Analogously, if your system has a Lax representation of the form (3.2) on the algebra g, then you may take as its integrable discretization the difference Lax equation (5.13) with F as above. Of course, this prescription makes sense only if the corresponding factors Π± (F (L)) [resp. Π± (F (Tj ))] admit more or less explicit expressions, allowing to write down the corresponding difference equations in a more or less closed form. The choice of F is a transcendent problem, which however turns out to be solvable for many of (hopefully, for the majority of or even for all) the known integrable systems. The simplest possible choice F (L) = I + hf (L) works perfectly well for a vast set of examples considered below. Let us stress the advantages of this approach to the problem of integrable discretization. • Although we formulate our recipe only for systems with Lax representations of a particular form, it is in fact more universal. Almost without changes it may be applied to any system whose Lax representation is governed by an R-operator satisfying the modified Yang–Baxter equation. • The discretizations obtained in this way share the Lax matrix and therefore the integrals of motion with their underlying continuous time systems. • If the Lax representation (3.1) [resp. (3.2)] allows an r-matrix interpretation, then our discretizations share also the invariant Poisson bracket with the underlying continuous time systems. In particular, if the Lax matrices L [resp. L] form a Poisson submanifold for some r-matrix bracket, then this submanifold is left invariant by the corresponding Poisson map (5.10) [resp. (5.13)]. • The initial value problem for our discrete time equations can be solved in terms of the same factorization in a Lie group as the initial value problem for the continuous time system. • Interpolating Hamiltonian flows also belong to the set of granted by-products of this approach. 7. Localizing Changes of Variables Along with these advantageous properties our recipe has also an important drawback: it produces, as a rule, nonlocal difference equations, when applied to lattice systems with local interactions. Under locality we understand the following property: in some coordinates (x1 , . . . , xN ) on X the equations of motion (2.1) have the form (7.1) x˙ k = φk (x) = φk (xk , xk±1 , . . . , xk±s )
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with a fixed s ∈ N. Non-local difference equations produced by our scheme have the form (7.2) x ek = xk + hΦk (x; h) , Φk (x, 0) = φk (x) , where Φk depends explicitly on all xj , not only on 2s nearest neighbours of xk . The aim of the present paper is to demonstrate on a large number of examples how this drawback can be overcome, i.e. how to bring the latter difference equations into a local form; the price we have to pay is that they become implicit. The general strategy will be to find localizing changes of variables X (x) 7→ X (x) such that in the variables x the map (7.2) may be written as e; h) , ek = xk + hΨk (x, x x
Ψk (x, x; 0) = φk (x) ,
(7.3)
ej ’s with correct indices |j − k| ≤ s. where Ψk depends only on the xj ’s and x Such implicit local equations of motion are much better suited for the purposes of numerical simulation and are much more satisfactory from the esthetical point of view. Moreover, in all our examples the functions Ψk actually depend only on xj ’s ej ’s with k − s ≤ j ≤ k, which makes the practical with k ≤ j ≤ k + s and on x implementation of the corresponding difference equations even more effective (if, e, then for instance, one uses the Newton’s iterative method to solve (7.3) for x one has to solve only linear systems whose matrices are triangular and have a band structure, i.e. only s non-zero diagonals). Further, it has to be remarked that, when considered as equations on the lattice (t, k), Eqs. (7.3) often allow transformations of independent variables (mixing t and k) bringing these equations into local explicit form (cf. [48]). This last remark will be the subject of a separate publication [74]. It is by no means evident that such localizing changes of variables exist, but we give in this paper a large number of examples which, hopefully, will convince the reader that this is indeed the case. The first examples probably appeared in the context of the Bogoyavlensky lattices in [67], and will be reproduced here in a clarified form for the sake of completeness. The localizing changes of variables turn out to have many additional remarkable properties. They are always given by the formulas xk = xk + hΞk (x; h)
(7.4)
with local functions Ξk . However, the inverse change of variables is always non-local. Therefore nothing guarantees a priori that the pull-back of the differential equations of motion (7.1) under the change of variables (7.4) will be given by local formulas. Nevertheless, this turns out to be the case. This gives a way of producing (sometimes novel) one-parameter families of integrable local deformations of lattice systems (see [34] for a general concept and some examples of integrable deformations). The system (7.1) often admits one or several invariant local Poisson brackets. Nothing guarantees a priori that the pull-backs of these brackets under the change of variables (7.4) are also given by local formulas. Indeed, as a rule these pull-backs are non-local. However, in the multi-Hamiltonian case it often turns out that pullbacks of certain linear combinations of invariant Poisson brackets are local again!
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These facts still wait to be completely understood. It seems that the remarkable properties of the localizing maps have the same nature as that of the Miura maps (“miraculous cancellations”). Moreover, actually our localizing changes of variables are Miura maps, and in this image some of them already appeared in [34]. However, the observation that they bring integrable discretizations into a local form, seems to be completely new. The scope of the present work is restricted to elaborating a large number of examples of localizing changes of variables, along with their Poisson properties, in a hope of attracting the attention of the soliton community to these fascinating and beautiful objects. 8. Basic Algebras and Operators Two concrete algebras play the basic role in our presentation. They are well suited to describe various lattice systems with the so called open-end and periodic boundary conditions, respectively. Here are the relevant definitions. For the open-end case we always set g = gl(N ), the algebra of N × N matrices with the usual matrix product, the Lie bracket [L, M ] = LM − M L, and the nondegenerate bi-invariant scalar product hL, M i = tr(L · M ). As a linear space, g may be presented as a direct sum g = g+ ⊕ g− , where the subalgebras g + and g − consist of lower triangular and of strictly upper triangular matrices, respectively. The Lie group G corresponding to the Lie algebra g is GL(N ), the group of N × N non-degenerate matrices. The subgroups G+ , G− corresponding to the Lie algebras g + , g − consist of non-degenerate lower triangular matrices and of upper triangular matrices with unit diagonal, respectively. The Π+ Π− factorization is well known in the linear algebra under the name of the LU factorization. In the periodic case we always choose g as a certain twisted loop algebra over gl(N ). A loop algebra over gl(N ) is an algebra of Laurent polynomials with coefficients from gl(N ) and a natural commutator [Lλj , M λk ] = [L, M ]λj+k . Our twisted algebra g is a subalgebra singled out by the additional condition g = {L(λ) ∈ gl(N )[λ, λ−1 ] : ΩL(λ)Ω−1 = L(ωλ)} , where Ω = diag(1, ω, . . . , ω N −1 ), ω = exp(2πi/N ). In other words, elements of g satisfy X X (p) λp `jk Ejk . (8.1) L(λ) = p
j−k≡p (mod N )
(Here and below Ejk stands for the matrix whose only non-zero entry is on the intersection of the jth row and the kth column and is equal to 1.) The nondegenerate bi-invariant scalar product is chosen as hL(λ), M (λ)i = tr(L(λ) · M (λ))0 , the subscript 0 denoting the free term of the formal Laurent series.
(8.2)
742
Y. B. SURIS
As a linear space, g is again a direct sum g = g+ ⊕ g− , with the subalgebras g+ =
M
λk g k ,
k≥0
g− =
M
λk g k .
(8.3)
k<0
The group G corresponding to the Lie algebra g is a twisted loop group, consisting of GL(N )-valued functions U (λ) of the complex parameter λ, regular in CP 1 \{0, ∞} and satisfying ΩU (λ)Ω−1 = U (ωλ). Its subgroups G+ and G− corresponding to the Lie algebras g + and g − , are singled out by the following conditions: • U (λ) ∈ G+ are regular in the neighbourhood of λ = 0; • U (λ) ∈ G− are regular in the neighbourhood of λ = ∞ and U (∞) = I. We call the corresponding Π+ Π− factorization the generalized LU factorization. It is uniquely defined in a certain neighbourhood of the unit element of G. As opposed to the open-end case, finding the generalized LU factorization is a problem of the Riemann–Hilbert type which is solved in terms of algebraic geometry rather than in terms of linear algebra. The basic operator governing the hierarchies of Lax equations, is, as already mentioned, (8.4) R = π+ − π− . Denote by R0 , P0 its skew-symmetric and its symmetric parts, respectively: R0 = (R − R∗ )/2 ,
P0 = (R + R∗ )/2 .
Obviously, in the open-end case P0 assigns to each matrix L its diagonal part, and in the periodic case P0 assigns to each Laurent series L(λ) its free term. Denote by g 0 the range of the operator P0 . Let the skew-symmetric operator W act on g 0 according to X X W (Ekk ) = Ejj − Ejj , j
j>k
and on the rest of g according to W = W ◦ P0 . Finally, define A1 = R0 + W ,
A2 = R0 − W ,
S = P0 − W ,
S ∗ = P0 + W .
(8.5)
These operators will be basic building blocks in almost all quadratic r-matrix brackets appearing in this paper (except those for Bogoyavlensky lattices, see Theorem 14.1). 9. Toda Lattice 9.1. Equations of motion and tri-Hamiltonian structure The equations of motion of the Toda lattice (hereafter TL) read a˙ k = ak (bk+1 − bk ) ,
b˙ k = ak − ak−1 ,
1≤k≤N,
(9.1)
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with one of the two types of boundary conditions: open-end (a0 = aN = 0), or periodic (all subscripts are taken (mod N ), so that a0 ≡ aN , bN +1 ≡ b1 ). The phase space of the TL in the case of the periodic boundary conditions is T = R2N (b1 , a1 , . . . , bN , aN ) .
(9.2)
There exist three compatible local Poisson brackets on T such that the system TL is Hamiltonian with respect to each one of them [3, 34], see also [15]. We adopt once and forever the following conventions: the Poisson brackets will be defined by writing down all nonvanishing brackets between the coordinate functions; the indices in the corresponding formulas are taken (mod N ). The “linear” Poisson structure on T is defined by the brackets {bk , ak }1 = −ak ,
{ak , bk+1 }1 = −ak ,
(9.3)
the corresponding Hamilton function for the flow TL is given by H2 (a, b) =
1X 2 X bk + ak . 2 N
N
k=1
k=1
(9.4)
The “quadratic” Poisson structure has the following definition: {bk , ak }2 = −ak bk ,
{ak , bk+1 }2 = −ak bk+1 ,
{ak , ak+1 }2 = −ak+1 ak ,
{bk , bk+1 }2 = −ak .
(9.5)
The Hamilton function generating TL in this bracket is H1 (a, b) =
N X
bk .
(9.6)
k=1
Finally, the “cubic” bracket on T is given by the relations {bk , ak }3 = −ak (b2k + ak ) ,
{ak , bk+1 }3 = −ak (b2k+1 + ak ) ,
{ak , ak+1 }3 = −2ak ak+1 bk+1 ,
{bk , bk+1 }3 = −ak (bk + bk+1 ) ,
{ak , bk+2 }3 = −ak ak+1 ,
{bk , ak+1 }3 = −ak ak+1 .
(9.7)
The expression for the corresponding Hamilton function, suitable in both the periodic and the open-end case, is non-local in the coordinates (a, b). However, in the periodic case one has an alternative Hamilton function: 1X log(ak ) . 2 N
H0 (a, b) =
(9.8)
k=1
9.2. Lax representation The Lax representation of the Toda lattice [24, 37] lives in the algebra g introduced in Sec. 8. Actually, there exist different versions of the Lax representation
744
Y. B. SURIS
connected with different ways to represent the algebra g as a direct sum of its two subalgebras [17]. We discuss here only the so-called LU version which is the one working also for all other examples concerned in this paper. The Lax matrix L : T 7→ g of the TL corresponding to the generalized LU decomposition is L(a, b, λ) = λ−1
N X
ak Ek,k+1 +
k=1
N X
bk Ek,k + λ
k=1
N X
Ek+1,k .
(9.9)
k=1
We use here and below a convention according to which in the periodic case EN +1,N = E1,N , EN,N +1 = EN,1 ; in the open-end case EN +1,N = EN,N +1 = 0 and we may set λ = 1. The equations of motion (9.1) are equivalent to the Lax equations L˙ = [L, B] = −[L, C] ,
(9.10)
with B(a, b, λ) = π+ (L) =
N X
bk Ek,k + λ
k=1
C(a, b, λ) = π− (L) = λ−1
N X
Ek+1,k ,
(9.11)
k=1 N X
ak Ek,k+1 ,
(9.12)
k=1
where π± : g 7→ g ± are the projections to the subalgebras g ± defined as in Sec. 8 (the generalized LU decomposition). Spectral invariants of the Lax matrix L(a, b, λ) serve as integrals of motion of this system. Note that all Hamilton functions in different Hamiltonian formulations belong to these spectral invariants. For instance, H2 (a, b) =
1 (tr L2 (a, b, λ))0 , 2
H1 (a, b) = (tr L(a, b, λ))0 ,
where the subscript “0” is used to denote the free term of the corresponding Laurent series. All spectral invariants turn out to be in involution with respect to each of the Poisson brackets (9.3), (9.5), (9.7). Most directly it follows from the r-matrix interpretation of the Lax equation (9.10), which can be given for all three brackets. We cite here the results for the linear and the quadratic brackets. Theorem 9.1. The Lax matrix L(a, b, λ) : T 7→ g is a Poisson map, if T carries the linear bracket {·, ·}1 and g is equipped with PB1 (R), and also if T carries the quadratic bracket {·, ·}2 and g is equipped with PB2 (A1 , A2 , S). The first statement is due to [4], see also [17] for similar results connected with other matrix decompositions. The second statement is due to [62], while a more simple version connected with the QR decomposition appeared earlier in [46]. The
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latter paper contained also an r-matrix interpretation of the cubic bracket {·, ·}3 , but only in the QR setting. See also the review [39]. 9.3. Discretization In order to find an integrable time discretization for the flow TL, we apply the recipe of Sec. 6 with F (L) = I + hL, i.e. we take as a solution of this problem the map described by the discrete time Lax equation e = B −1 LB = CLC −1 L
with
B = Π+ (I + hL) ,
C = Π− (I + hL) .
(9.13)
Theorem 9.2 [65] (see also [25]). The discrete time Lax equation (9.13) is equivalent to the map (a, b) 7→ (e a, eb) described by the following equations: ak−1 βk+1 ak e , (9.14) , bk = bk + h − e a k = ak βk βk βk−1 where the functions βk = βk (a, b) = 1 + O(h) are uniquely defined by the recurrent relation h2 ak−1 , (9.15) βk = 1 + hbk − βk−1 and have the asymptotics βk = 1 + hbk + O(h2 ) .
(9.16)
Remark. The matrices B, C have the following expressions: B(a, b, λ) = Π+ (I + hL) =
N X
βk Ek,k + hλ
k=1
C(a, b, λ) = Π− (I + hL) = I + hλ−1
N X
Ek+1,k ,
(9.17)
k=1 N X ak Ek,k+1 . βk
(9.18)
k=1
Proof. The bi-diagonal structure of the factors B, C as well as the expressions for the entries of B − , follow from the tri-diagonal structure of the matrix L. The recurrent relation (9.15) for the entries of B is equivalent to BC = I + hL. The equations of motion (9.14) are now nothing but the componentwise form of the e = LB. matrix equation B L The map (9.14) will be denoted dTL. Due to the asymptotics (9.16) it is easy to see that the equations of motion (9.14) of the dTL serve as a difference approximation to the Toda flow TL (9.1). The construction assures numerous positive properties of this discretization: the map dTL is Poisson with respect to each one of the Poisson brackets (9.3), (9.5), (9.7), it has the same integrals of motion as the flow TL, the Lax representation for the dTL lies in its very definition, etc. The only unpleasant property of the equations of motion (9.14), when compared with their continuous time counterparts (9.1), is the nonlocality. The source of nonlocality are
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Y. B. SURIS
the functions βk . In the open-end case they have explicit expressions in terms of finite continued fractions: βk = 1 + hbk −
h2 ak−1 1 + hbk−1 − . . .
. h 2 a1 − 1 + hb1
In the periodic case for small h the βk ’s may be expressed as analogous infinite (periodic) continued fractions. 9.4. Local equations of motion for dTL Fortunately, there exist different ways to bring these equations of motion into a local form connected with a localizing change of variables in the sense of Sec. 7. Consider another copy of the phase space T . The coordinates in this another copy will be denoted by ak , bk . The localizing change of variables for dTL is the map T (a, b) 7→ T (a, b) defined by the following formulas: ak = ak (1 + hbk ) ,
bk = bk + hak−1 .
(9.19)
The implicit functions theorem assures that this is a local diffeomorphism between the two copies of T by small enough values of h. This change of variables appeared in [34] as a Miura map in connection with the problem of deformation of integrable systems, but without any relation to integrable discretizations. Theorem 9.3. The change of variables (9.19) conjugates the map dTL with e described by the following equations of motion: the map (a, b) 7→ (e a, b) e k ) = ak (1 + hbk+1 ) , e ak (1 + hb
e k = bk + h(ak − e b ak−1 ) .
(9.20)
Proof. The key point is the following observation: the auxiliary functions βk aquire in the new coordinates local expressions, namely, βk = 1 + hbk .
(9.21)
To demonstrate this, it is enough to notice that from (9.19) there follows that the quantities 1 + hbk satisfy the same recurrent relation as the quantities βk , namely 1 + hbk = 1 + hbk −
h2 ak−1 . 1 + hbk−1
Due to the uniqueness of the solution with the asymptotics 1 + O(h) the formula (9.21) is proved. The statement of the theorem follows now immediately from (9.14) and (9.21). Of course, the map (9.20) is Poisson with respect to pull-backs of the three invariant Poisson structures of the Toda lattice. These pull-backs are described by highly non-local and non-polynomial formulas. However, there exist certain linear combinations of the basic Poisson structures in coordinates (a, b) whose pull-backs to the coordinates (a, b) are local.
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Theorem 9.4. (a) The pull-back of the bracket {·, ·}1 + h{·, ·}2
(9.22)
on T (a, b) under the change of variables (9.19) is the following bracket on T (a, b): {bk , ak } = −ak (1 + hbk ) ,
{ak , bk+1 } = −ak (1 + hbk+1 ) .
(9.23)
(b) The pull-back of the bracket {·, ·}2 + h{·, ·}3
(9.24)
on T (a, b) under the change of variables (9.19) is the following bracket on T (a, b): {bk , ak } = −ak (bk + hak )(1 + hbk ) , {ak , bk+1 } = −ak (bk+1 + hak )(1 + hbk+1 ) , {ak , ak+1 } = −ak ak+1 (1 + hbk+1 ) , {bk , bk+1 } = −ak (1 + hbk )(1 + hbk+1 ) .
(9.25)
(c) The brackets (9.23) and (9.25) are compatible. The map (9.20) is Poisson with respect to both of them. Proof. To prove the theorem, one has, for example, in the (less laborious) case (a) to verify the following statement: the formulas (9.23) imply that the nonvanishing pairwise Poisson brackets of the functions (9.19) are {bk , ak } − ak (1 + hbk ) , {ak , ak+1 } = −hak ak+1 ,
{ak , bk+1 } = −ak (1 + hbk+1 ) , {bk , bk+1 } = −hak .
This verification consists of straightforward calculations. In what follows we do not repeat analogous arguments in similar situations. The map (9.20) was first found in [30], along with the Lax representation. In [59] it was stressed that this map is nothing other than the so-called qd algorithm well known in the numerical analysis. Its Poisson structure and its place in the continuous time Toda hierarchy were not discussed in [30]. The previous theorem provides a bi-Hamiltonian structure of the qd algorithm. This result in a slightly different form was found in [65]. Theorem 9.5 [34]. The pull-back of the flow TL under the change of variables (9.19) is described by the following differential equations: a˙ k = ak (bk+1 − bk ) ,
b˙ k = (ak − ak−1 )(1 + hbk ) .
(9.26)
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Y. B. SURIS
Proof. To determine the pull-back of the flow TL, we can use the Hamiltonian formalism. An opportunity to apply it is given by the Theorem 9.4. We shall use PN the statement (a) only. Consider the function h−1 H1 (a, b) = h−1 k=1 bk . It is a Casimir of the bracket {·, ·}1 , and generates exactly the flow TL in the bracket h{·, ·}2 . Hence it also generates the flow TL in the bracket (9.22). The pull-back PN of this Hamilton function is equal to h−1 k=1 (bk + hak−1 ). It remains only to calculate the flow generated by this function in the Poisson brackets (9.23). This results in the equations of motion (9.26). 10. Second Flow of the Toda Hierarchy Now we want to demonstrate that our method for finding integrable discretizations and local equations of motion for them works not only for the flow TL, but equally well for the higher flows of the Toda hierarchy. We consider here the second flow (called hereafter TL2.) 10.1. Equations of motion and tri-Hamiltonian structure This is the flow on T governed by the differential equations a˙ k = ak (b2k+1 − b2k + ak+1 − ak−1 ) ,
b˙ k = ak (bk+1 + bk ) − ak−1 (bk + bk−1 ) . (10.1)
This flow is Hamiltonian with respect to all three brackets (9.3), (9.5) and (9.7). The corresponding Hamilton functions are: H3 (a, b) =
1X 3 X 1 bk + bk (ak + ak−1 ) = (tr L3 (a, b, λ))0 3 3 N
N
k=1
k=1
(10.2)
for the bracket {·, ·}1 , H2 (a, b) for the bracket {·, ·}2 , and H1 (a, b) for the bracket {·, ·}3 . 10.2. Lax representation Naturally, the Lax matrix for the flow TL2 is the same as for the flow TL. The difference lies in the auxiliary matrices B, C taking part in the Lax representation (since in this section we are dealing only with the flow TL2, using the same notations B, C as in Sec. 9 will not lead to confusion; the same also holds for some other notations in this section). The equations of motion (10.1) are equivalent to the Lax equations in g: L˙ = [L, B] = −[L, C] with B = π+ (L2 ) ,
C = π− (L2 ) ,
(10.3)
so that B(a, b, λ) =
N X
(b2k + ak + ak−1 )Ek,k + λ
k=1
N X
(bk+1 + bk )Ek+1,k + λ2
k=1
N X
Ek+2,k ,
k=1
(10.4) C(a, b, λ) = λ−1
N X
(bk+1 + bk )ak Ek,k+1 + λ−2
k=1
N X k=1
ak+1 ak Ek,k+2 .
(10.5)
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10.3. Discretization In order to obtain an integrable discretization of the flow TL2 we can apply the recipe of Sec. 6 with F (L) = I + hL2 , i.e. consider the map described by the discrete time Lax equation e = B −1 LB = CLC −1 L
with
B = Π+ (I + hL2 ) ,
C = Π− (I + hL2 ) . (10.6)
Theorem 10.1. The discrete time Lax equation (10.6) is equivalent to the map (a, b) 7→ (e a, eb) described by the following equations: ak−1 δk−1 βk+1 ak δ k e , (10.7) , bk = bk + h − e a k = ak βk βk βk−1 where the auxiliary functions δk = δk (a, b) = O(1) and βk = βk (a, b) = 1 + O(h) are uniquely defined for h small enough by the recurrent relations δk = bk+1 + bk − βk = 1 + h(b2k + ak + ak−1 ) −
hak−1 δk−1 , βk−1
(10.8)
2 h2 ak−1 δk−1 h2 ak−1 ak−2 − , βk−1 βk−2
(10.9)
and have the asymptotics δk = bk+1 + bk + O(h) ,
(10.10)
βk = 1 + h(b2k + ak + ak−1 ) + O(h2 ) .
(10.11)
Remark. The matrices B, C from this theorem have the following expressions: B(a, b, λ) =
N X
βk Ek,k + hλ
k=1
C(a, b, λ) = I + hλ−1
N X
δk Ek+1,k + hλ2
k=1
N X
Ek+2,k ,
(10.12)
k=1
N N X X ak δ k ak+1 ak Ek,k+1 + hλ−2 Ek,k+2 . (10.13) βk βk k=1
k=1
Proof. The tri-diagonal structure of the matrix L assures that the factors B, C have the structure as in (10.12) and (10.13). The expressions for the entries of C and the recurrent relations for the entries of B follow easily from the equality BC = I + hL2 . After that the equations of motion follow from the equation e = LB. BL 10.4. Local equations of motion for dTL2 For the map (10.7), called hereafter dTL2, there exists a localizing change of variables, however different from the one used for the map dTL. It is given by the formulas ak = ak (1 + hak−1 )(1 + hb2k ) ,
bk = bk (1 + hak−1 ) + hbk−1 ak−1 .
(10.14)
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Y. B. SURIS
(Actually, discretizations of all higher flows of the Toda hierarchy should possess their own localizing changes of variables). The name “localizing change of variables” is justified by the following theorems. Theorem 10.2. The change of variables (10.14) conjugates the map dTL2 with e governed by the following local equations of motion: the map (a, b) 7→ (e a, b) e 2 ) = ak (1 + hak+1 )(1 + hb2 ) , e ak−1 )(1 + hb ak (1 + he k k+1 e k−1 + b ek ) . e k − bk = hak (bk + bk+1 ) − he ak−1 (b b
(10.15)
Proof. This time the key point of the proof is obtaining the following local expressions for the coefficients of the factor Π+ (I + hL2 ): βk = (1 + hak )(1 + hak−1 )(1 + hb2k ) ,
(10.16)
δk = (1 + hak )(bk + bk+1 ) .
(10.17)
Indeed, the equations of motion (10.15) follow directly from (10.7) and the latter formulas. To prove the latter formulas, define the quantities βk , δk by Eqs. (10.16) and (10.17). A straightforward calculation convinces that then the recurrent relations (10.9) and (10.8) are satisfied. The uniqueness of solution to these equations proves the desired expressions for βk , δk . There still exists a linear combination of invariant Poisson brackets of the Toda hierarchy, whose pull-back with respect to the above change of variables is local. Theorem 10.3. The pull-back of the bracket {·, ·}1 + h{·, ·}3
(10.18)
on T (a, b) under the change of variables (10.14) is the following bracket on T (a, b): {bk , ak } = −ak (1 + hak )(1 + hb2k ) , (10.19) {ak , bk+1 } = −ak (1 + hak )(1 + hb2k+1 ) . The map (10.15) is Poisson with respect to (10.19). Proof. By straightforward verification.
Theorem 10.4. The pull-back of the flow TL under the map (10.14) is described by the following differential equations: a˙ k = ak (bk+1 − bk )(1 + hak ) ,
b˙ k = (ak − ak−1 )(1 + hb2k ) .
(10.20)
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Proof. The equations of motion we are looking for describe the flow TL as a Hamiltonian system in the Poisson bracket (10.18). The corresponding Hamilton function is, obviously, given by (I + hL2 )∇H(L) = L . Hence we have H(L) = (2h)−1 (log det(I + hL2 ))0 . From the expressions for the factors of the generalized LU factorization of the matrix I + hL2 given in the PN formulas (10.12) and (10.13) we conclude that H(L) = (2h)−1 k=1 log(βk ). Taking into account the expressions (10.16), we find finally H=
1X 1 X log(1 + hb2k ) + log(1 + hak ) . 2h h N
N
k=1
k=1
(10.21)
The Hamiltonian equations of motion generated by this Hamilton function and the Poisson brackets (10.19), coincide with (10.20). The differential equations (10.20) are a particular case of a two-parameter deformation of TL found in [34]. 11. Volterra Lattice 11.1. Equations of motion and bi-Hamiltonian structure The second flow (10.1) of the Toda hierarchy, unlike the first one, allows an important reduction bk = 0, and it results in the famous Volterra lattice (hereafter VL), known also under the names of Lotka–Volterra system, discrete KdV equation, Langmuir lattice, Kac–van Moerbecke lattice, etc. [37, 33]: a˙ k = ak (ak+1 − ak−1 ) .
(11.1)
The phase space of VL in the case of periodic boundary conditions is V = RN (a1 , . . . , aN ) .
(11.2)
Clearly, it is the subspace bk = 0 of T (a, b). Unfortunately, neither of three invariant Poison brackets {·, ·}1 , {·, ·}2 , and {·, ·}3 of the Toda hierarchy can be properly restricted to the set V. However, it is easy to see that the quadratic bracket {·, ·}2 allows a Dirac reduction to this set, the reduced bracket being defined by the relations (11.3) {ak , ak+1 }2 = −ak ak+1 . The system VL is Hamiltonian with respect to this bracket with the Hamilton function N X ak . H1 (a) = k=1
There exists one more local Poisson bracket on V invariant with respect to VL and compatible with (11.3) [34, 21]. It is given by the relations {ak , ak+1 }3 = −ak ak+1 (ak + ak+1 ) ,
{ak , ak+2 }3 = −ak ak+1 ak+2 ,
(11.4)
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Y. B. SURIS
the corresponding Hamilton function being equal to 1X log(ak ) . 2 N
H0 (a) =
k=1
11.2. Lax representation The Lax representation of the previous section survives in the process of restriction to V. So, we obtain the Lax representation of the VL with the matrices from g [37, 33]: L˙ = [L, B] = −[L, C] , (11.5) where L(a, λ) = λ−1
N X
ak Ek,k+1 + λ
k=1
B(a, λ) = π+ (L2 ) =
N X
Ek+1,k ,
(11.6)
k=1 N X
(ak + ak−1 )Ek,k + λ2
k=1
C(a, λ) = π− (L2 ) = λ−2
N X
Ek+2,k ,
(11.7)
k=1 N X
ak+1 ak Ek,k+2 .
(11.8)
k=1
The Volterra hierarchy consists of the flows allowing Lax representations of the form (11.5) with B = π+ (f (L2 )), C = π− (f (L2 )), where f : g 7→ g is Ad-covariant. In particular, the second flow corresponds to f (L) = L4 . All these equations allow an r-matrix interpretation in case of the quadratic bracket {·, ·}2 . Theorem 11.1 [63]. The Poisson bracket PB2 (A1 , A2 , S) admits a Dirac reduction to the submanifold V ⊂ g consisting of matrices (11.6). The Lax matrix L(a, λ) : V 7→ V defines a Poisson map, if V carries the quadratic bracket {·, ·}2 , and V is equipped with the Dirac reduction of PB2 (A1 , A2 , S). If ψ is an Adinvariant function on g, then the vector fields generated by the Hamilton function ϕ(L) = ψ(L2 ) on V in the Dirac reduced and in the unreduced Poisson brackets, coincide and have the Lax form. Each two such functions ψ1 (L2 ) and ψ2 (L2 ) are in involution on V with respect to the Dirac reduced bracket. 11.3. Discretization The discretization of the second Toda flow (10.7) may also be restricted to the set V. It is easy to see from (10.8) and (10.7) that in this situation we have with necessity δk = 0, and the recurrent relations (10.9) turn into βk = 1 + h(ak + ak−1 ) −
h2 ak−1 ak−2 . βk−2
(11.9)
Theorem 11.2 [67] (see also [35]). The discrete time Lax equation e = B −1 LB = CLC −1 L
with
B = Π+ (I + hL2 ) ,
C = Π− (I + hL2 ) (11.10)
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is equivalent to the following map a 7→ e a: e ak = a k
βk+1 , βk
(11.11)
where the quantities βk = βk (a) = 1 + O(h) are uniquely defined by the recurrent relations βk−1 hak−1 βk − hak = =1+ (11.12) βk−1 − hak−1 βk−1 − hak−1 and have the asymptotics βk = 1 + h(ak + ak−1 ) + O(h2 ) .
(11.13)
Remark. The matrices B, C have the following expressions: B(a, λ) = Π+ (I + hL2 ) =
N X
βk Ek,k + hλ2
k=1
N X
Ek+2,k ,
(11.14)
Ek,k+2 .
(11.15)
k=1
C(a, λ) = Π− (I + hL2 ) = I + hλ−2
N X ak ak+1 k=1
βk
Proof. Referring to Theorem 10.1, we only need to prove that the auxiliary quantities βk defined by the recurrent relations (11.9), may be alternatively charace = LB. terized as solutions to (11.12). To do this, consider the matrix equation B L In coordinates it is equivalent to the set of two scalar equations: (11.11) and βk+1 − hak+1 =
βk (βk−1 − hak−1 ) . βk−1
The last equation means that the following quantity does not depend on k: (βk − hak )(βk−1 − hak−1 ) = c = const , βk−1
(11.16)
and we need only to prove that c = 1. But applying (11.16) twice, we have βk − hak = c + = c+
hcak−1 βk−1 − hak−1 hak−1 (βk−2 − hak−2 ) βk−2
= c + hak−1 −
h2 ak−1 ak−2 . βk−2
Comparing this with (11.9), we conclude that c = 1.
The map (11.11) and (11.12) will be denoted dVL. It is Poisson with respect to the brackets (11.3) and (11.4), but non-local, as opposed to the flow VL. In the
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Y. B. SURIS
open-end case we have terminating continued fractions for the nonlocal quantities βk (in the periodic case for small h the βk may be expressed as analogous infinite periodic continued fractions). From (11.9) we derive, for example, for βk with odd indices: β2k+1 = 1 + h(a2k+1 + a2k ) −
h2 a2k a2k−1 1 + h(a2k−1 + a2k−2 ) − . .
(11.17) 2 . h a2 a1 − 1 + ha1
(with obvious changes for β2k ). Analogously, from (11.12) we derive alternative continued fractions: βk − hak = 1 + 1+
hak−1 hak−2 1 + .. .
+
.
(11.18)
ha2 1 + ha1
One says that the continued fractions (11.17) are obtained from (11.18) with the help of compression. 11.4. Local equations of motion for dVL The localizing change of variables V(a) 7→ V(a) for dVL follows from (10.14) upon setting bk = 0: (11.19) ak = ak (1 + hak−1 ) . Again, this change of variables was found in [34] in the context of integrable deformations, but with no relation to integrable disacretizations. As it follows from the proof of Theorem 10.2, in the variables ak the auxiliary quantities βk become local: βk = (1 + hak )(1 + hak−1 ) ,
(11.20)
and we obtain the following result. Theorem 11.3. The change of variables (11.19) conjugates the map dVL with the map a 7→ e a governed by the following local equations of motion: ek (1 + he ak−1 ) = ak (1 + hak+1 ) . a
(11.21)
The only known local invariant Poisson bracket (10.19) for the local form of dTL2 does not admit a proper restriction or a reduction to the subset V(a) of T (a, b) characterized by bk = 0. Nevertheless, there appears to exist a local bracket on V(a) invariant under the local form of dVL. Theorem 11.4. The pull-back of the bracket {·, ·}2 + h{·, ·}3
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on V(a) under the change of variables (11.19) is the following bracket on V(a): {ak , ak+1 } = −ak ak+1 (1 + hak )(1 + hak+1 ) .
(11.22)
This bracket is invariant under the map (11.21).
Proof. By direct calculation.
The local equations of motion (11.21) together with Lax representation were found by different methods in [78, 49, 67]. The invariant Poisson structure and the relation to factorization problem were pointed out only in the latter of these references. Theorem 11.5 [34]. The pull-back of the flow VL under the map (11.19) is described by the following equations of motion: a˙ k = ak (1 + hak )(ak+1 − ak−1 ) .
(11.23)
PN Proof. Since the flow VL has a Hamilton function (2h)−1 k=1 log(ak ) in the Poisson bracket h{·, ·}3 , and this function is a Casimir of the bracket {·, ·}2 , we conclude that this function generates the flow VL also in the bracket {·, ·}2 +h{·, ·}3 . That means that the pull-back of the flow VL is a Hamiltonian flow in the bracket (11.22) with the Hamilton function (2h)−1
N X
log(ak (1 + hak−1 )) .
k=1
Calculating the corresponding equations of motion, we arrive at (11.23). The system (11.23) is known under the name of the modified Volterra lattice. Remark. An additional map ak 7→ ak =
ak 1 + hak
delivers an alternative version of the localizing change of variables ak =
ak . (1 − hak )(1 − hak−1 )
(11.24)
The corresponding local version of the dVL reads ek ak a = . (1 − he ak )(1 − he ak−1 ) (1 − hak )(1 − hak+1 )
(11.25)
This version was introduced in [29] under the name “discrete Lotka–Volterra equation of type II” (while the type I was assigned to (11.21)). The modified Volterra lattice in the variables ak takes the following form: ak−1 ak+1 . (11.26) − a˙ k = ak 1 − hak+1 1 − hak−1
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A remarkable feature of the variables ak (not mentioned in [29]) is the formal coincidence of the invariant Poisson bracket (11.22) with the quadratic invariant Poisson bracket (11.3) of the continuous time flow VL. Indeed, in the variables ak the bracket (11.22) takes the form {ak , ak+1 } = −ak ak+1 .
(11.27)
12. Second Flow of the Volterra Hierarchy Now we apply the general procedure of integrable discretization to the second flow of the Volterra hierarchy. We use the notations βk , B± etc. for objects analogous to those of the previous section without danger of confusion. 12.1. Equations of motion and bi-Hamiltonian structure The second flow of the Volterra hierarchy (hereafter VL2) is described by the following differential equations on V: a˙ k = ak (ak+1 (ak+2 + ak+1 + ak ) − ak−1 (ak + ak−1 + ak−2 )) .
(12.1)
This flow is Hamiltonian with respect to both Poisson brackets (11.3) and (11.4). The corresponding Hamilton functions are H2 (a) =
1X 2 X ak + ak+1 ak 2 N
N
k=1
k=1
for the quadratic bracket {·, ·}2 , and H1 (a) for the cubic bracket {·, ·}3 . 12.2. Lax representation The Lax representation for the flow VL2 is of the type (3.1) with f (L) = L4 . Theorem 12.1. The flow (12.1) admits the following Lax representation in g: L˙ = [L, B] = −[L, C]
(12.2)
with the matrices B(a, λ) = π+ (L4 ) =
N X
(ak+1 ak + (ak + ak−1 )2 + ak−1 ak−2 )Ekk
k=1
+ λ2
N X
(ak+2 + ak+1 + ak + ak−1 )Ek+2,k + λ4
k=1
C(a, λ) = π− (L4 ) = λ−2
N X
k=1
(ak+2 + ak+1 + ak + ak−1 )ak+1 ak Ek,k+2
k=1
+ λ−4
N X
N X k=1
ak+3 ak+2 ak+1 ak Ek,k+4 .
Ek+4,k ,
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12.3. Discretization Applying the recipe of Sec. 6 with F (L) = I + hL4 , we take as a discretization of the flow VL2 the map described by the discrete time Lax equation e = B −1 LB = CLC −1 L
with
B = Π+ (I + hL4 ) ,
C = Π− (I + hL4 ) . (12.3)
Theorem 12.2. The discrete time Lax equations (12.3) are equivalent to the map a 7→ e a described by the following equations: e a k = ak
βk+1 , βk
(12.4)
where the functions βk = βk (a) = 1 + O(h) are uniquely defined for h small enough simultaneously with the functions δk = O(1) by the system of recurrent relations βk−1 , βk−1 − h(δk−1 − ak+1 )ak−1
βk − h(δk − ak+2 )ak =
δk = ak+2 + ak+1 + ak + ak−1 −
(12.5)
hak−1 ak−2 δk−2 . (12.6) βk−2
The auxiliary functions βk have the asymptotics βk = 1 + h(ak+1 ak + (ak + ak−1 )2 + ak−1 ak−2 ) + O(h2 ) .
(12.7)
Remark. The matrices B, C have the following expressions: B(a, λ) =
N X
βk Ek,k + hλ2
k=1
N X
δk Ek+2,k + hλ4
k=1
C(a, λ) = I + hλ−2
N X ak+1 ak δk k=1
βk
N X
Ek+4,k ,
k=1
Ek,k+2 + hλ−4
N X ak+3 ak+2 ak+1 ak k=1
βk
Ek,k+4 .
Proof. The scheme of the proof is standard. First of all, the general structure of the factors B, C is clear from the bi-diagonal structure of the matrix L. The expressions for the entries of C and the recurrent relations for the entries of B follow from the equality BC = I + hL4 . However, in this way we come to the recurrent relation (12.6) for δk and the following recurrent relation for βk : βk = 1 + h(ak+1 ak + (ak + ak−1 )2 + ak−1 ak−2 ) −
2 h2 ak−1 ak−2 δk−2 h2 ak−1 ak−2 ak−3 ak−4 − , βk−2 βk−4
(12.8)
different from (12.5). In order to “decompress” this recurrent relation for βk and to e = LB bring it into the form (12.5), we proceed as follows. The matrix equation B L
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is equivalent to the set of three scalar ones: the equation of motion (12.4) and βk+1 − hδk ak+1 =
βk (βk−1 − hδk−1 ak−1 ) , βk−1
(12.9)
βk . βk−1
(12.10)
δk+1 − ak+3 = δk − ak−1
From the last two equations it is easy to derive βk+1 − h(δk+1 − ak+3 )ak+1 =
βk (βk−1 − h(δk−1 − ak+1 )ak−1 ) , βk−1
which means that the following quantity does not depend on k: (βk − h(δk − ak+2 )ak )(βk−1 − h(δk−1 − ak+1 )ak−1 ) = c = const . βk−1
(12.11)
It remains to prove that this constant is equal to 1. To this end we apply (12.11) twice (as in the proof of Theorem 11.2) to derive βk = c + h(δk − ak+2 )ak + h(δk−1 − ak+1 )ak−1 −
h2 (δk−1 − ak+1 )(δk−2 − ak )ak−1 ak−2 . βk−2
Transforming the second and the third terms on right-hand side with the help of (12.6), and the fourth one with the help of (12.10), we see that the O(h)-terms of the last formula exactly coincide with the O(h)-terms on the right-hand side of (12.8). Equating the free terms, we get c = 1. 12.4. Local equations of motion for dVL2 The localizing change of variables for the map dVL2 is given by the formulas ak = ak
(1 + ha2k−1 ) . (1 − hak ak−1 )(1 − hak−1 ak−2 )
(12.12)
Theorem 12.3. The change of variables (12.12) conjugates the map dVL2 with the map a 7→ e a described by the following local equations of motion: e ak
(1 + he a2k−1 ) (1 + ha2k+1 ) = ak . (12.13) ak−1 )(1 − he ak−2 ) (1 − he ak e ak−1 e (1 − hak+2 ak+1 )(1 − hak+1 ak )
Proof. The statement will follow immediately, if we prove the following expressions for the auxiliary quantities βk : βk =
(1 + ha2k )(1 + ha2k−1 ) . (1 − hak+1 ak )(1 − hak ak−1 )2 (1 − hak−1 ak−2 )
(12.14)
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To do this, it is sufficient to verify that the recurrent relations (12.5) and (12.6) are satisfied by the quantities (12.14) and δk =
ak + ak−1 ak+2 + ak+1 + . (1 − hak+2 ak+1 )(1 − hak+1 ak ) (1 − hak+1 ak )(1 − hak ak−1 )
(12.15)
Such a verification is a matter of straightforward, though somewhat tedious algebra. We omit it, giving only an important intermediate formula: βk − h(δk − ak+2 )ak =
(1 + ha2k−1 ) (1 − hak ak−1 )(1 − hak−1 ak−2 )
(12.16)
(immediately it implies (12.5)).
Unlike the situation with the previously encountered localizing changes of variables, we did not find a linear combination of invariant Poisson brackets of the Volterra hierarchy, which would be pulled back into a local bracket on V(a) (presumably, such a linear combination does not exist). Therefore, to find a pull back of the flow VL under the change of variables (12.12), we cannot use the Hamiltonian formalism and are forced to turn to a direct analysis of equations of motion. Theorem 12.4. The pull-back of the flow VL under the change of variables (12.12) is described by the following equations of motion: ak−1 ak+1 . (12.17) − a˙ k = ak (1 + ha2k ) 1 − hak+1 ak 1 − hak ak−1 Proof. It is a matter of straightforward calculations to verify that the Eq. (12.17) are sent into (11.1) by the change of variables (12.12). 12.5. Local discretization of the KdV Probably the most interesting feature of the flow VL2 is that a certain linear combination of this flow with the original VL gives a direct spatial discretization of the famous KdV [1]. Indeed, the equations of motion of the linear combination α · VL + VL2 read a˙ k = ak (ak+1 (α + ak+2 + ak+1 + ak ) − ak−1 (α + ak + ak−1 + ak−2 )) .
(12.18)
Setting α = −6 and ak = 1 + ε 2 pk , we get the equations of motion p˙ k = (1 + ε2 pk )(pk+2 − 2pk+1 + 2pk−1 − pk−2 + ε2 pk+1 (pk+2 + pk+1 + pk ) − ε2 pk−1 (pk + pk−1 + pk−2 )) . Assuming that pk (t) ≈ p(t, kε) with a smooth p(t, x) and rescaling the time t 7→ t/(2ε3 ), we see that the previous equation approximates pt = pxxx + 6px p ,
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Y. B. SURIS
which is the KdV. Therefore a local discretization of the linear combinations of the flows VL and VL2 will result (for α = −6) in a local spatio-temporal discretization of the KdV (see [77] for a highly non-local integrable discretization). Theorem 12.5. A localizing change of variables for the discretization of the above flow corresponding to F (T ) = I + h(αL2 + L4 ) is given by the formula: ak = ak
(1 + hαak−1 + ha2k−1 ) . (1 − hak ak−1 )(1 − hak−1 ak−2 )
(12.19)
This change of variables conjugates the above discretization with the map a 7→ e a described by the local equations of motion e ak
(1 + hαe ak−1 + he a2k−1 ) (1 + hαak+1 + ha2k+1 ) = ak . (12.20) ak−1 )(1 − he ak−2 ) (1 − he ak e ak−1 e (1 − hak+2 ak+1 )(1 − hak+1 ak )
Proof. Completely analogous to that of Theorem 12.3, and will not be repeated here. 13. Modified Volterra Lattice 13.1. Equations of motion and Hamiltonian structure As we have seen many times in the preceding exposition, the localizing changes of variables are interesting not only in their connection with the problem of integrable discretization, but already on the level of continuous time systems. Namely, they may be viewed as Miura transformations and used to find the so-called modified equations of motion. These modified systems are often interesting in their own rights — and one can also wish to discretize them as well. We consider in the present section one example of modified systems, namely, the modified Volterra lattice (MVL). It appeared for the first time in a slightly different form (see (13.18) below) in [26] under the name of the “self-dual network equations”. We use the change of variables (11.19) in slightly different notations. Namely, we define a Miura change of variables M : MV(q) 7→ V(a) by the formula ak = qk (1 + αqk−1 ) .
(13.1)
We denote the parameter α instead of h (α is not related to the time step of discretizations and is not supposed to be small). The MVL is the pull-back of the VL (11.1) under the change of variables (13.1). Its equations of motion were already determined in Sec. 11 to be q˙k = qk (1 + αqk )(qk+1 − qk−1 ) .
(13.2)
As usual, this system may be considered under periodic or open-end boundary conditions, the phase space in case of the periodic ones being MV = RN (q1 , . . . , qN ) .
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As pointed out in Sec. 11, this system is Hamiltonian with respect to the following Poisson bracket on MV: {qk , qk+1 } = −qk qk+1 (1 + αqk )(1 + αqk+1 ) ,
(13.3)
and with one of the Hamilton functions H0 (q) =
N X
log(qk ) or H0 (q) = α−1
k=1
N X
log(1 + αqk )
(13.4)
k=1
(their difference is a Casimir of the bracket (13.3)). The bracket (13.3) is the pullback under the Miura change of variables (13.1) of the invariant Poisson bracket {·, ·}2 + α{·, ·}3 of the VL (see (11.3), (11.4) for the relevant definitions). 13.2. Discretization We define the discretization dMVL of MVL (13.2) as a pull-back of the dVL under the Miura change of variables (13.1). Theorem 13.1. The equations of motion of the map dMVL read: γk+1 qek = qk , γk
(13.5)
where the quantities γk = γk (q) = 1 + O(h) are uniquely defined by h small enough by the system of recurrent relations γk − hqk =
γk−1 + hαqk qk−1 γk−1 − hqk−1
⇔
γk − hqk + αqk = (1 + αqk )
γk−1 , γk−1 − hqk−1 (13.6)
and have the asymptotics γk = 1 + h(qk + qk−1 + αqk qk−1 ) + O(h2 ) .
(13.7)
Proof. From the equations of motion (13.5) and the definition (13.6) we derive 1 αqk (γk + hαqk+1 qk ) 1 + 1 + αe qk = γk γk − hqk = (1 + αqk )
γk−1 (γk+1 − hqk+1 ) . γk (γk−1 − hqk−1 )
(13.8)
Hence the variables ak defined as in (13.1) satisfy the equations of motion (11.11) with γk (γk−1 − hqk−1 )(γk−2 − hqk−2 ) . (13.9) βk = γk−2 It remains to prove that these quantities βk satisfy the recurrent relations (11.12). But this follows immediately from the formula βk − hak =
γk−1 (γk−2 − hqk−2 ) , γk−2
which is an easy consequence of (13.9), (13.1) and (13.6).
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Clearly, by definition the map dMVL is Poisson with respect to the bracket (13.3), but the above theorem renders dMVL non-local. In particular, in the openend case we have terminating continued fractions: γk − hqk = 1 + 1+
hqk−1 (1 + αqk ) hqk−2 (1 + αqk−1 ) 1+ . ..
+
.
(13.10)
hq2 (1 + αq3 ) 1 + hq1 (1 + αq2 )
In the periodic case these continued fractions are also periodic. 13.3. Local equations of motion for dMVL It is possible to find a localizing change of variables MV(q) 7→ MV(q) for the map dMVL: qk (1 + hqk−1 ) . (13.11) qk = 1 − hαqk qk−1 As usual, this map is a local diffeomorphism for h small enough. Theorem 13.2. The change of variables (13.11) conjugates the map dMVL e described by the following local equations of motion: with the map q 7→ q ek qk q (1 + he qk−1 ) = (1 + hqk+1 ) , 1 + αe qk 1 + αqk
(13.12)
1 + αqk 1 + αe qk = . ek−1 1 − hαe qk q 1 − hαqk+1 qk
(13.13)
or, equivalently,
Proof. The first step is to prove the following local expressions for the quantities γk in the variables qk : (1 + hqk )(1 + hqk−1 ) γk = . (13.14) 1 − hαqk qk−1 Indeed, from (13.5), (13.11) and (13.14) we derive immediately the equations of motion in the form ek (1 + he qk (1 + hqk+1 ) qk−1 ) q = , ek−1 1 − hαe qk q 1 − hαqk+1 qk which is equivalent to either of (13.12) and (13.13). To prove (13.14), it is as usual enough to verify that the quantities defined by this formula satisfy the recurrent relations (13.6). This verification is the matter of a simple algebra. The equations of motion (13.12) were introduced in [29]. In general, the only expression for a Poisson bracket invariant with respect to the map (13.12) is non-local and non-polynomial (it may be characterized as the pull-back of the bracket (13.3) under the localizing change of variables (13.11)). However, a direct analysis of equations of motion allows to prove the following statement.
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Theorem 13.3. The pull-back of the flow (13.2) under the change of variables (13.11) is described by the following equations of motion: q˙ k = qk (1 + αqk )(1 + hqk )
qk−1 qk+1 − 1 − hαqk+1 qk 1 − hαqk qk−1
.
(13.15)
This system may be called the second modification of the Volterra lattice. (In a somewhat different form this system appeared for the first time in [83].) Notice its remarkable symmetry with respect to the interchange α ↔ h. Also the expressions for ak in terms of qk enjoy this symmetry. Indeed, the composition of two changes of variables (13.1) and (13.11) results in ak =
qk (1 + hqk−1 )(1 + αqk−1 ) . (1 − hαqk qk−1 )(1 − hαqk−1 qk−2 )
(13.16)
However, in the discrete equations of motion (13.12) this symmetry gets lost. The formula (13.16) also allows to translate the Miura map M : MV(q) 7→ V(a) into the language of localizing variables. Namely, if we define the map M : MV(q) 7→ V(a) by the formula ak =
qk (1 + αqk−1 ) , 1 − hαqk qk−1
(13.17)
an easy calculation shows the commutativity of the following diagram: M MV(q)
-
V(a)
(13.11)
(11.19)
?
? -
MV(q)
V(a)
M Remark. Sometimes another form of MVL is more convenient: c˙k = (c2k − ε2 )(ck+1 − ck−1 ) .
(13.18)
It is related to (13.2) with α = (2ε)−1 by means of a linear change of variables: qk = ck − ε ,
(13.19)
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and time rescaling t 7→ t/α. The correct way to discretize MVL in the form (13.18) turns out to be the following. First of all, rescale the time step h: √ 2hε 1 − 1 − 2hε √ = hε2 ⇐⇒ h = . (1 + hε2 )2 1 + 1 − 2hε Then perform the following change of localizing variables, approximating (13.19): qk = (1 + hε2 )
ck − ε . 1 − hεck
(13.20)
It is easy to see that this implies 1 + αqk =
ck + ε 1 − hε2 · , 2ε 1 − hεck
1 + hqk =
1 − hε2 1 + hεck · . 1 + hε2 1 − hεck
and
Henceforth we end up with the following local integrable discretization of the flow (13.18): e ck−1 ck − ε 1 + hεck+1 ck − ε 1 + hεe · · = . (13.21) e ck + ε 1 − hεe ck−1 ck + ε 1 − hεck+1 This discretization (derived for the first time in [27] in a completely different way) share with (13.18) the following important property: both systems have real solutions in the case ε = iβ. To close this section, we mention the relation between the variables ck and localizing variables ck . This relation follows from (13.19), (13.20) and (13.11), and reads ck − ε =
(ck − ε)(1 + hεck−1 ) . 1 − hck ck−1
(13.22)
13.4. Particular case α→∞ There exists a particular case of MVL, for which a more detailed information is available. It corresponds to the α → ∞ limit of the previous constructions, which has to be accompanied by the time rescaling t 7→ t/α. (Alternatively, one could consider the ε → 0 limit of the system (13.18)). Let us list the corresponding results. The equations of motion read q˙k = qk2 (qk+1 − qk−1 ) ,
(13.23)
and the Miura map M : MV(q) 7→ V(a) relating this system to the VL, takes the form (13.24) ak = qk qk−1 . An invariant Poisson bracket (13.3) in the limit α → ∞ turns (being suitably rescaled) into 2 (13.25) {qk , qk+1 }3 = −qk2 qk+1
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(see below for the reason for assuming the index “3” to this bracket). The Hamilton function of the flow (13.23) in this bracket is equal to H0 (q) =
N X
log(qk ) .
k=1
The localizing change of variables (13.11) after rescaling h 7→ h/α and going to the limit α → ∞ becomes qk qk = , (13.26) 1 − hqk qk−1 and the discrete time equations of motion (13.13) become ek qk q = , ek−1 1 − he qk q 1 − hqk qk+1
(13.27)
ek−1 ) . ek − qk = he qk qk (qk+1 − q q
(13.28)
or, equivalently,
Finally, the pull-back of the system (13.23) under the localizing change of variables (13.26) reads qk−1 qk+1 2 . (13.29) − q˙ k = qk 1 − hqk+1 qk 1 − hqk qk−1 Now notice that the bracket (13.25) is the pull-back with respect to the Miura map M (13.24) of the cubic invariant bracket {·, ·}3 of the Volterra hierarchy. It turns out that in this particular case the pull-back of the quadratic bracket {·, ·}2 is given by more elegant formulas than in the general case. Namely, the corresponding bracket on MV(q) is quadratic albeit still non-local: {qk , qj }2 = πkj qk qj , with the coefficients 0 πkj = −1 1
(13.30)
(in the case of odd N )
k−j =0
k − j > 0 and odd or k − j < 0 and even . k − j > 0 and even or k − j < 0 and odd
(13.31)
The corresponding Hamilton function of the flow (13.23) is given by H(q) =
N X
qk qk+1 .
k=1
So, in this case we know a simple formula for an additional invariant Poisson bracket. This allows to find a nice formula for an invariant Poisson structure of the map (13.28) and of the flow (13.29) — the result lacking in the general case.
766
Y. B. SURIS
Theorem 13.4. The pull-back of the bracket {·, ·}2 + h{·, ·}3
(13.32)
on MV(q) under the map (13.26) is the following bracket on MV(q): {qk , qj } = πkj qk qj ,
(13.33)
with the coefficients (13.31). The map (13.28) and the flow (13.29) are Poisson with respect to this bracket. Proof. A direct calculation shows that the brackets (13.33) for the variables qk are sent by the map (13.26) into the brackets (13.32) for the variables qk . We close this section by noticing that the translation of the Miura map M (13.24) into the localizing variables is the map M : MV(q) 7→ V(a) given by the formula qk qk−1 . (13.34) ak = 1 − hqk qk−1 For the localizing variables ak of VL introduced by (11.24), we have ak = qk qk−1
(13.35)
(which formally coincides with (13.24)). 14. Bogoyavlensky Lattices 14.1. Equations of motion and Hamiltonian structure There are three basic families of integrable lattice systems carrying the name of Bogoyavlensky [11] (although some of them were found earlier in [42, 34, 31]). These systems are enumerated by integer parameters m, p ≥ 1 (p > 1 for the third one) and read: m m X X ak+j − ak−j , a˙ k = ak (14.1) a˙ k = ak
j=1
j=1
p Y
p Y
ak+j −
j=1
a˙ k =
p−1 Y
a−1 k+j −
j=1
ak−j ,
(14.2)
j=1 p−1 Y
a−1 k−j .
(14.3)
j=1
We shall call these systems BL1(m), BL2(p), and BL3(p), respectively. The lattices BL1(m) and BL2(p) serve as generalizations of the Volterra lattice, which arises from them by m = 1 and p = 1, respectively. The lattice BL3(p) after the change of variables ak 7→ qk = a−1 k and t 7→ −t turns into p−1 p−1 Y Y qk+j − qk−j . (14.4) q˙k = qk2 j=1
j=1
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
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We call the latter system modified BL2(p). It serves as a generalization of the modified Volterra lattice (13.23), which is the p = 2 particular case of (14.4). These systems may be considered on an infinite lattice (all the subscripts belong to Z), and also admit also periodic finite-dimensional reductions (all indices belong to Z/N Z, where N is the number of particles). The lattices BL1 and BL2 admit also finite-dimensional versions with the open-end boundary conditions. The phase space of the periodic BL’s is B = RN (a1 , . . . , aN ) .
(14.5)
The Hamiltonian structure of the BL1 was determined in [34] and later in [11]; for the BL2 and BL3 in the infinite setting this was done in [86]; in the open-end and the periodic setting (where some subtleties come out) the Hamiltonian structures were determined in [63]. We reproduce here the corresponding result for the periodic boundary conditions. The invariant quadratic Poisson brackets for Bogoyavlensky lattices are given by the formula (we set p = 1 for BL1(m), m = 1 for BL2(p), and m = −1 for BL3(p)): {ak , aj }2 = πkj ak aj ,
(14.6)
with the coefficients 1 (p) (p) (p) (p) δk,j+m − δk+m,j + wk+m,j+m − wk,j+m − wk+m,j + wk,j . πkj = 2
(14.7)
(p)
Here, in turn, the coefficients wkj are given in the N -periodic case with g.c.d.(N, p) = 1 by the formula ( sgn(k − j) k ≡ j (mod N ) , (p) wkj = (14.8) 2n/p − 1 k − j ≡ nN (mod p), 1 ≤ n ≤ p − 1 . The Poisson structure (14.6) is non-local unless p = 1 (the case of BL1(m)), when it is given by the following brackets: {ak , ak+1 } = −ak ak+1 . . . ,
{ak , ak+m } = −ak ak+m .
(14.9)
In particular, for m = 1 we obtain the quadratic Poisson bracket for the Volterra lattice. The corresponding Hamiltonians are H(a) =
N X
ak
for BL1(m) ,
(14.10)
k=1
H(a) =
N X
ak ak+1 . . . ak+p−1
for BL2(p) ,
(14.11)
k=1
H(a) =
N X k=1
−1 −1 a−1 k ak+1 . . . ak+p−1
for BL3(p) .
(14.12)
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Y. B. SURIS
14.2. Lax representations The Lax representations of the Bogoyavlensky lattices fall into the class considered in [34], and were also specified in [11]. They have the form L˙ = [L, B] or L˙ = −[L, C] ,
(14.13)
with the spectral-dependent (in the periodic case) matrices L = L(a, λ) ∈ g, B = B(a, λ), C = C(a, λ). For BL1(m) the matrices T , B are given by L(a, λ) = λ
N X
Ek+1,k + λ−m
k=1
N X
ak Ek,k+m ,
(14.14)
k=1
B(a, λ) = π+ (Lm+1 ) =
N X
m+1
(ak + ak−1 + . . . + ak−m )Ek,k + λ
k=1
N X
Ek+m+1,k ; (14.15)
k=1
for BL2(p) the matrices T , C are given by p
L(a, λ) = λ
N X
−1
Ek+p,k + λ
k=1
C(a, λ) = π− (L
N X
ak Ek,k+1 ,
(14.16)
k=1
p+1
−p−1
)=λ
N X
ak ak+1 . . . ak+p Ek,k+p+1 ,
(14.17)
k=1
and for BL3(p) they are given by L(a, λ) = λp
N X
Ek+p,k + λ
k=1
N X
ak+1 Ek+1,k ,
(14.18)
k=1
C(a, λ) = π− (L−p+1 ) = λ−p+1
N X
−1 −1 a−1 k+1 ak+2 . . . ak+p−1 Ek,k+p−1 . (14.19)
k=1
All three Lax representations above may be seen as having the form L˙ = [L, ±π± (dϕ(L))] with appropriate Ad-invariant functions ϕ on g, namely, ϕ(L) =
1 (tr(Lm+1 ))0 m+1
ϕ(L) =
1 (tr(Lp+1 ))0 p+1
ϕ(L) = −
for BL1(m) , for BL2(p) ,
1 (tr(L−p+1 ))0 p−1
for BL3(p) .
(14.20) (14.21) (14.22)
(It is easy to see that the values of these functions in coordinates ak coincide with (14.10)–(14.12), respectively.)
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
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These Lax equations allow an r-matrix interpretation. Theorem 14.1 [63]. Let the skew-symmetric operator W (p) act on g 0 according to W (p) (Ekk ) =
N X
(p)
wkj Ejj ,
k=1
and on the rest of g according to W (p) = W (p) ◦ P0 . Define (p)
(p)
A1 = R0 + W (p) , A2 = R0 − W (p) , S (p) = P0 − W (p) , S (p)∗ = P0 + W (p) . (14.23) (p)
(p)
Then the Poisson bracket PB2 (A1 , A2 , S (p) ) admits a Dirac reduction to the submanifold B ⊂ g consisting of Lax matrices of the corresponding Bogoyavlensky lattice: (14.14), (14.16), or (14.18), respectively. The Lax matrices L(a, λ) : B 7→ B define Poisson maps, if B carries the bracket {·, ·}2 , and B is equipped with the (p) (p) Dirac reduction of PB2 (A1 , A2 , S (p) ). Let ψ be an Ad-invariant function on g, and let n = (p1 + m1 )n1 with some n1 ∈ Z and m = m1 d, p = p1 d, d = g.c.d.(m, p). Then the vector fields generated by the Hamilton function ϕ(L) = ψ(Ln ) on B in the Dirac reduced and in the unreduced Poisson brackets, coincide and have the Lax form. Each two such functions ψ1 (Ln ) and ψ2 (Ln ) are in involution on B with respect to the Dirac reduced bracket. It is easy to see that Theorem 11.1 for VL is a p = 1 particular case of the latter result. 14.3. Discretization of BL1 We are now in a position to apply the recipe of Sec. 6 to the problem of discretizing the Bogoyavlensky lattices. This was done for the first time in [67]. Closely related results were obtained in [78, 49] by different methods. We reproduce here the results of [67] without proofs. As usual, the construction gives automatically for all discrete time systems (called hereafter dBL1, dBL2, dBL3, respectively) the invariant Poisson structure, the Lax representation, the integrals of motion, the interpolating Hamiltonian flows, etc. The maps of all three families are non-local, but we demonstrate how to bring them to a local form by means of a suitable change of variables. Of course, the local forms of the maps dBL1–dBL3 are Poisson with respect to the Poisson brackets on B(a) which are the pull-backs of the corresponding brackets on B(a). However, we did not succeed in finding more or less nice formulas for such pull-backs (the only exceptions — the Volterra and the modified Volterra lattices). We define dBL1(m) as the map described by the discrete time Lax equation e = B −1 LB , L
B = Π+ (I + hLm+1 ) .
(14.24)
Theorem 14.2 [67]. The discrete time Lax equation (14.24) is equivalent to the map a 7→ e a described by the equations βk+m e ak = ak , (14.25) βk
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Y. B. SURIS
where the functions βk = βk (a) = 1 + O(h) are uniquely defined for h small enough by the recurrent relations m Y hak−j 1+ , (14.26) βk − hak = βk−j − hak−j j=1 and have the following asymptotics: βk = 1 + h
m X
ak−j + O(h2 ) .
(14.27)
j=0
Remark. The factor B is of the form B(a, λ) = Π+ (I + hLm+1 ) =
N X
βk Ek,k + hλm+1
k=1
N X
Ek+m+1,k .
(14.28)
k=1
The map (14.25) is non-local due to the functions βk . In the simplest case m = 1 one has the continued fractions, terminating for the open-end boundary conditions: hak−1 , 1 + .. ha2 . + 1 + ha1
βk − hak = 1 +
(14.29)
and periodic for the periodic boundary conditions. For m > 1 there lacks even such an expressive mean as continued fractions to represent these non-local functions. The localizing change of variables B(a) 7→ B(a) for dBL1(m) is ak = ak
m Y
(1 + hak−j ) .
(14.30)
j=1
Obviously, the map (14.30) for h small enough is a local diffeomorphism. Theorem 14.3. The change of variables (14.30) conjugates the map dBL1(m) with the following one: e ak
m Y
(1 + he ak−j ) = ak
j=1
m Y
(1 + hak+j ) .
(14.31)
j=1
Proof. Let us define the quantities βk by the relation βk =
m Y
(1 + hak−j ) ,
(14.32)
j=0
and prove that they satisfy the recurrent relations (14.26). Indeed, from (14.30) and (14.32) it follows that βk − hak =
m Y j=1
(1 + hak−j ) =
ak . ak
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
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Hence ak = ak /(βk −hak ), which, being substituted in the previous formula, implies (14.26). Now the uniqueness of solution of this latter recurrent system yields (14.32). Plugging (14.32) and (14.30) into the equations of motion (14.25) results in (14.31). Theorem 14.4. The pull-back of equations of motion (14.1) under the change of variables (14.30) is given by the following formula: m Y
a˙ k = ak (1 + hak )
(1 + hak+j ) −
j=1
m Y
(1 + hak−j )
j=1
.
h
(14.33)
Proof. By a direct calculation one checks that Eq. (14.33) under the change of variables (14.30) are mapped to the equations of motion (14.1). 14.4. Discretization of BL2 We define dBL2(p) as the map described by the discrete time Lax equation e = CLC −1 , L
C = Π− (I + hLp+1 ) .
(14.34)
Theorem 14.5 [67]. The discrete time Lax equation (14.34) is equivalent to the map a 7→ e a described by the equations e ak =
ak − hγk−p ak+p+1 , ak+p+1 − hγk+1
(14.35)
where the functions γk = γk (a) = O(1) are uniquely defined for h small enough by the recurrent relations ak , (14.36) ak − hγk−p = p Y 1+h (ak−j − hγk−p−j ) j=1
and have the asymptotics γk =
p Y
ak+j (1 + O(h)) .
(14.37)
j=0
Remark. The factor C is of the form C(a, λ) = Π− (I + hLp+1 ) = I + hλ−(p+1)
N X
γk Ek,k+p+1 .
(14.38)
k=1
The quantities γk render the equations of motion (14.35) non-local. The localizing change of variables for dBL2(p) reads p Y ak−j . ak = ak 1 + h (14.39) j=1
As usual, its bijectivity follows from the implicit functions theorem.
772
Y. B. SURIS
Theorem 14.6. The change of variables (14.39) conjugates the map dBL2(p) with the following one: p p Y Y e e ak−j = ak 1 + h ak+j . ak 1 + h (14.40) j=1
j=1
Proof. We proceed according to the, by now, usual scheme. Define the quantities γk by the formula p Y γk = ak+j . (14.41) j=0
Then we immediately derive ak − hγk−p = ak ,
(14.42)
and plugging this expression for ak into (14.39) shows that the recurrent relations (14.36) are satisfied. The uniqueness of the system of functions ak −hγk−p satisfying these relations justifies the expressions (14.41). Now putting (14.39) and (14.42) into the equations of motion (14.35) allows to rewrite them as (14.40). Theorem 14.7. The pull-back of the equations of motion (14.2) under the change of variables (14.39) is given by the following formula: ! p p p p Y Y Y Y 1+h (14.43) ak+j − ak−j ak−n+i . a˙ k = ak j=1
n=1
j=1
i=1
Proof. A direct, though tedious calculation shows that the equations of motion (14.43) are mapped on (14.2) by means of the change of variables (14.39). 14.5. Discretization of BL3 We define dBL3(p) as the map with the discrete time Lax representation e = CLC −1 , L
C = Π− (I + hL−p+1 ) .
(14.44)
Theorem 14.8 [67]. The discrete time Lax equation (14.44) is equivalent to the map a 7→ e a described by the equations e ak =
ak − hαk−p ak+p−1 , ak+p−1 − hαk−1
(14.45)
where the functions αk = αk (a) = O(1) are uniquely defined for h small enough by the recurrent relations p−1 Y 1 αk = , (14.46) a − hαk+j−p k+j j=1 and have the asymptotics αk =
p−1 Y j=1
a−1 k+j (1 + O(h)) .
(14.47)
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
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Remark. The factor C is of the form C(a, λ) = Π− (I + hL−p+1 ) = I + hλ−p+1
N X
αk Ek,k+p−1 .
(14.48)
k=1
The non-locality of the equations of motion (14.45) is due to the functions αk (a). For example, for p = 2 they can be expressed as periodic continued fractions of the following structure: h . hαk = h ak+1 − h ak − ak−1 − . . . The localizing change of variables for dBL3(p) is given by p−1 Y . ak = ak 1 + h a−1 (14.49) k−j j=0
The bijectivity of this map is assured by the implicit functions theorem. Theorem 14.9. The change of variables (14.49) conjugates the map dBL3(p) with the following one: p−1 p−1 Y Y = ak 1 + h . e ek 1 + h a−1 a−1 (14.50) a k−j k+j j=0
j=0
Proof. Defining the quantities αk by the formula αk =
p−1 Y
a−1 k+j ,
(14.51)
j=1
we obtain with the help of (14.49): ak − hαk−p = ak .
(14.52)
Substituting this expression into (14.51), we see that the recurrent relations (14.46) are satisfied, which proves (14.51). Substituting (14.49) and (14.52) into (14.45), we immediately arrive at the equations of motion (14.50). Theorem 14.10. The pull-back of the equations of motion (14.3) under the change of variables (14.49) is given by the following formula: !−1 p−1 p−1 p−1 p−1 Y Y Y Y −1 −1 −1 1+h ak+j − ak−j ak−n+i . (14.53) a˙ k = j=1
j=1
n=0
Proof. By direct (but tiresome) calculations.
i=0
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Y. B. SURIS
We close the discussion of the local equations of motion for the dBL3(p) by noticing that under the change of variables ak 7→ qk = a−1 k , h 7→ −h the map (14.50) turns into −1 −1 p−1 p−1 Y Y ek−j = qk 1 − h e k 1 − h q q qk+j , (14.54) j=0
j=0
which is a local integrable discretization of the system (14.4), while the differential equations (14.53) under the change of variables ak 7→ qk = q−1 k , h 7→ −h, t 7→ −t go into !−1 p−1 p−1 p−1 p−1 Y Y Y Y 2 1−h q˙ k = qk qk+j − qk−j qk−n+i , (14.55) j=1
n=0
j=1
i=0
which is an integrable one-parameter deformation of (14.4). 14.6. Particular case p = 2 The case p = 2 of the Bogoyavlensky lattice BL3 is remarkable in several respects. The equations of motion for this case read a˙ k =
1 1 − . ak+1 ak−1
(14.56)
The localizing change of variables for the discretization of the system (14.56) obtained in the previous subsection, is h h = ak + . (14.57) ak = ak 1 + ak ak−1 ak−1 In the variables ak we have the following discretization: 1 1 e . ak − ak = h − e ak−1 ak+1
(14.58)
(When considered on the lattice (t, k), Eq. (14.58) is equivalent to the so-called lattice KdV, which is a very popular object nowadays, cf. [43, 41], and references therein.) The localizing change of variables (14.57) brings the system (14.56) itself into the form −1 −1 h h 1 1 1+ 1+ − a˙ k = ak+1 ak−1 ak−1 ak ak ak+1 1 1 − . (14.59) = ak ak+1 ak + h ak ak−1 + h The special properties of this case begin with the following observation. The subset of g consisting of the Lax matrices L(a, λ) = λ2
N X k=1
Ek+2,k + λ
N X k=1
ak+1 Ek+1,k
(14.60)
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
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775 (2)
(2)
is a Poisson submanifold for the quadratic r-matrix bracket PB2 (A1 , A2 , S (2) ), so that the Dirac reduction is not needed in giving an r-matrix interpretation to the bracket (14.61) {ak , aj }2 = πkj ak aj . The coefficients πkj are given (in the case of odd N ) by the formula (13.31). Moreover, not only the quadratic r-matrix bracket, but also the linear one PB1 (R) may be properly restricted to the set of the matrices (14.60). The coordinate representation of the induced bracket on this set is given by {ak , ak+1 }1 = −1 .
(14.62)
The two Poisson brackets (14.61) and (14.62) are compatible, hence the system (14.56) and its discretization given by Theorem 14.8 with p = 2 are bi-Hamiltonian. Obviously, the Hamilton function of the system (14.56) in the Poisson bracket (14.62) may be taken as N X log(ak ) . (14.63) H0 (a) = k=1
It is easy to check that this function is a Casimir of the quadratic bracket. Being bi-Hamiltonian, the system (14.56) and its discretization admit also an arbitrary linear combination of the brackets (14.61) and (14.62) as an invariant Poisson structure. A further remarkable feature is the following: there exists a linear combination of these two brackets whose pull-back under the map (14.57) allows a nice representation in the variables ak . Theorem 14.11. The pull-back of the bracket {·, ·}2 + h{·, ·}1
(14.64)
on B(a) under the map (14.57) is the following bracket on B(a): {ak , aj } = πkj ak aj ,
(14.65)
with the coefficients (13.31). The map (14.58) and the flow (14.59) are Poisson with respect to this bracket. Proof. A direct calculation shows that the brackets (14.65) for the variables ak are sent by the map (14.57) into the brackets (14.64) for the variables ak . This theorem also allows to derive the equations of motion (14.59) in a Hamiltonian manner. Indeed, these equations describe the Hamiltonian flow with the Hamilton function N X −1 −1 log(ak (1 + ha−1 h k ak−1 )) k=1
PN in the Poisson brackets (14.65). Indeed, this function is a pull-back of h−1 k=1 log(ak ), which generates (14.56) in the bracket (14.64). A direct calculation shows that this Hamiltonian system is governed by the differential equations (14.59).
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Y. B. SURIS
Remark. The results of this subsection agree with the results of the Subsec. 13.4 −1 after the change of variables ak 7→ qk = a−1 k , ak 7→ qk = ak . 15. Alternative Approach to Volterra Lattice Starting from this point, we consider the systems with Lax representations in the direct products g = g ⊗ m rather than in g itself. We start with an alternative approach to the Volterra lattice which delivers the simplest example of a Lax representation in g ⊗ g. 15.1. Equations of motion and bi-Hamiltonian structure The version of VL we consider here is u˙ k = uk (vk − vk−1 ) ,
v˙ k = vk (uk+1 − uk ) .
(15.1)
Usually we let the subscript k change in the interval 1 ≤ k ≤ N and consider either open-end boundary conditions (v0 = uN +1 = 0) or periodic ones (all indices are taken (mod N ). We consider mainly the periodic case, because the open-end one is similar and more simple. The relation to the form (11.1) is achieved by renaming the variables according to uk 7→ a2k−1 ,
vk 7→ a2k .
(15.2)
So, in the present setting the N -periodic VL consists of 2N particles. The phase space of VL in the case of periodic boundary conditions is the space W = R2N (u1 , v1 , . . . , uN , vN ) .
(15.3)
Two compatible local Poisson brackets on W invariant under the flow VL are given by the relations {uk , vk }2 = −uk vk ,
{vk , uk+1 }2 = −vk uk+1 ,
(15.4)
and {uk , vk }3 = −uk vk (uk + vk ) ,
{vk , uk+1 }3 = −vk uk+1 (vk + uk+1 ) , (15.5)
{uk , uk+1 }3 = −uk vk uk+1 ,
{vk , vk+1 }3 = −vk uk+1 vk+1 ,
respectively. The corresponding Hamilton functions for the flow VL are equal to H1 (u, v) =
N X k=1
and H0 (u, v) =
N X k=1
uk +
N X
vk
(15.6)
k=1
log(uk ) or H0 (u, v) =
N X
log(vk )
(15.7)
k=1
(the second function makes sense only in the periodic case; the difference of these two functions is a Casimir of {·, ·}3 whose value is fixed to ∞ in the open-end case).
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
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15.2. Lax representation Consider the following two matrices: U (u, v, λ) =
N X
uk Ek,k +λ
k=1
N X
Ek+1,k ,
−1
V (u, v, λ) = I +λ
k=1
N X
vk Ek,k+1 . (15.8)
k=1
These formulas define the “Lax matrix” (U, V ) : W 7→ g = g ⊗ g. Theorem 15.1 [34] (see also [71]). The flow (15.1) admits the following Lax representation in g ⊗ g: U˙ = U B2 − B1 U = C1 U − U C2 , (15.9) V˙ = V B1 − B2 V = C2 V − V C1 , with the matrices B1 (u, v, λ) =
N X
(uk + vk−1 )Ekk + λ
k=1
B2 (u, v, λ) =
N X
N X
Ek+1,k ,
(15.10)
k=1
(uk + vk )Ekk + λ
k=1
C1 (u, v, λ) = λ−1
N X
Ek+1,k ,
(15.11)
k=1 N X
uk vk Ek,k+1 ,
(15.12)
uk+1 vk Ek,k+1 .
(15.13)
k=1
C2 (u, v, λ) = λ−1
N X k=1
Corollary. The matrices T1 (u, v, λ) = U (u, v, λ)V (u, v, λ) ,
T2 (u, v, λ) = V (u, v, λ)U (u, v, λ)
(15.14)
satisfy the usual Lax equations in g: T˙i = [Ti , Bi ] = −[Ti , Ci ] ,
i = 1, 2 .
(15.15)
The matrices T1,2 are easy to calculate explicitly. From the corresponding formulas one sees that the matrices B1,2 , C1,2 allow the following representations: Bi = π+ (Ti ) ,
Ci = π− (Ti ) ,
i = 1, 2 .
The Lax equations (15.9) may be given an r-matrix interpretation in the case of quadraic Poisson brackets.
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Y. B. SURIS
Theorem 15.2 [71]. (a) Supply the algebra g ⊗ g with the bracket PB2 (A1 , A2 , S) defined by the operators A1 =
A1 −S S ∗ A1
S=
S S S −S ∗
! , ! ,
! A2 −S ∗ A2 = , S A2 ! S∗ S∗ ∗ S = , S ∗ −S
(15.16)
(15.17)
where operators A1 , A2 , S, S ∗ are as in (8.5). Then the Lax matrix (U (u, v, λ), V (u, v, λ)) : W 7→ g ⊗ g defines a Poisson map, if W carries the quadratic bracket {·, ·}2 . (b) The monodromy maps M1,2 : g ⊗ g 7→ g, M1 : (U, V ) 7→ U V = T1 ,
M2 : (U, V ) 7→ V U = T2
(15.18)
are Poisson, if the target space g is equipped with the Poisson bracket PB2 (A1 , A2 , S). (c) Let ϕ be an Ad-invariant function on g. Then the Hamiltonian equations of motion on g ⊗ g with the Hamilton function ϕ ◦ M1,2 may be presented in the form of the “Lax triads”: 1 1 U˙ = U · R(dϕ(T2 )) − R(dϕ(T1 )) · U , 2 2
(15.19)
1 1 V˙ = V · R(dϕ(T1 )) − R(dϕ(T2 )) · V . 2 2
(15.20)
15.3. Discretization To find an integrable time discretization for the flow VL, we apply the recipe of Sec. 6 with F (T ) = I + hT , i.e. we consider the map described by the discrete time “Lax triads”: −1 Ve = B −1 2 U B1 = C 2V C 1 ,
e = B −1 U B 2 = C 1 U C −1 , U 1 2
(15.21)
with B i = Π+ (I + hTi ) ,
C i = Π− (I + hTi ) ,
i = 1, 2 .
Theorem 15.3. The discrete time Lax equations (15.21) are equivalent to the map (u, v) 7→ (e u, ve) described by the following equations: u ek = uk
γk , βk
vek = vk
βk+1 , γk
(15.22)
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
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779
where the functions βk = βk (u, v) = 1 + O(h), γk = γk (u, v) = 1 + O(h) are uniquely defined for h small enough by the recurrent relations βk − huk =
γk−1 hvk−1 =1+ , γk−1 − hvk−1 γk−1 − hvk−1
(15.23)
γk − hvk =
βk huk =1+ , βk − huk βk − huk
(15.24)
and have the asymptotics βk = 1 + h(uk + vk−1 ) + O(h2 ) ,
(15.25)
γk = 1 + h(uk + vk ) + O(h2 ) .
(15.26)
Remark. The matrices B 1,2 , C 1,2 have the following expressions: B 1 (u, v, λ) = Π+ (I + hU V ) =
N X
βk Ek,k + hλ
k=1
B 2 (u, v, λ) = Π+ (I + hV U ) =
N X
N X
Ek+1,k ,
(15.27)
Ek+1,k ,
(15.28)
k=1
γk Ek,k + hλ
k=1
C 1 (u, v, λ) = Π− (I + hU V ) = I + hλ−1
k=1 N X uk vk k=1
C 2 (u, v, λ) = Π− (I + hV U ) = I + hλ−1
N X
βk
Ek,k+1 ,
N X uk+1 vk k=1
γk
Ek,k+1 .
(15.29)
(15.30)
Proof. The general structure of the factors B 1,2 , C 1,2 , as given in (15.27)– (15.30), as well as the expressions for the entries of C 1,2 , follow directly from the defining equalities B i C i = I + hTi . For the entries βk , γk of B 1,2 one obtains the following recurrent relations: βk = 1 + h(uk + vk−1 ) −
h2 uk−1 vk−1 , βk−1
γk = 1 + h(uk + vk ) −
h2 uk vk−1 . (15.31) γk−1
Now notice that these relations coincide with (11.9) after renaming (15.2) and βk 7→ β2k−1 ,
γk 7→ β2k .
Hence we may use the proof of Theorem 11.2 to establish the alternative recurrent relations (15.23) and (15.24). The equations of motion (15.22) follow directly from e = U B 2 , B 2 Ve = V B 1 . It is important to notice that (15.22) become identical B1U with (11.11) after the above-mentioned renamings of variables. This allows to denote consistently the map constructed in this theorem by dVL.
780
Y. B. SURIS
15.4. Local equations of motion for dVL Now we can simply reformulate the results of Sec. 11 in our new notations. The localizing change of variables W(u, v) 7→ W(u, v) for dVL is given by the following formulas: uk = uk (1 + hvk−1 ) ,
vk = vk (1 + huk ) .
(15.32)
Due to the implicit function theorem, these formulas define a local diffeomorphism for h small enough. Theorem 15.4. The change of variables (15.32) conjugates the map dVL with e ) governed by the following local equations of motion: the map (u, v) 7→ (e u, v e k (1 + he vk−1 ) = uk (1 + hvk ) , u
ek (1 + he v uk ) = vk (1 + huk+1 ) .
(15.33)
Let us mention that the functions βk (u, v), γk (u, v) in the localizing variables are given by the formulas βk = (1 + huk )(1 + hvk−1 ) ,
γk = (1 + hvk )(1 + huk ) ,
(15.34)
so that βk − huk = 1 + hvk−1 ,
γk − hvk = 1 + huk .
(15.35)
Remark. Cosidering Eqs. (15.33) as lattice equations on the lattice (t, k), and performing a linear change of independent variables, one arrives at the explicit version of dVL [74], cf. also [80, 23]. Theorem 15.5. The pull-back of the bracket {·, ·}2 + h{·, ·}3
(15.36)
on W(u, v) under the change of variables (15.32) is the following bracket on W(u, v): {uk , vk } = −uk vk (1+huk )(1+hvk ) ,
{vk , uk+1 } = −vk uk+1 (1+hvk )(1+huk+1 ) . (15.37) The map (15.33) is Poisson with respect to the bracket (15.37).
Theorem 15.6. The pull-back of the flow VL under the map (15.32) is described by the following equations of motion: u˙ k = uk (1 + huk )(vk − vk−1 ) ,
v˙ k = vk (1 + hvk )(uk+1 − vk ) .
(15.38)
15.5. Lax representation for VL2 We consider now the g ⊗ g formulation of the second flow VL2 of the Volterra hierarchy. We use the notations βk , γk , B1,2 , C1,2 , etc. for objects analogous to those relevant for VL without danger of confusion.
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
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781
The flow VL2 is described by the following differential equations on W: u˙ k = uk (vk (uk+1 + vk + uk ) − vk−1 (uk + vk−1 + uk−1 )) , (15.39) v˙ k = vk (uk+1 (vk+1 + uk+1 + vk ) − uk (vk + uk + vk1 )) . The Hamilton functions of this flow are H2 (u, v) =
X 1X 2 (uk + vk2 ) + (uk+1 vk + vk uk ) 2 N
N
k=1
k=1
in the quadratic bracket {·, ·}2 , and H1 (u, v) in the cubic bracket {·, ·}3 . The Lax representation for the flow VL2 is of the type (3.2) with m = 2 and f (T ) = T 2 . Theorem 15.7. [34]. The flow (15.39) admits the following Lax representation in g ⊗ g: U˙ = U B2 − B1 U = C1 U − U C2 , (15.40) V˙ = V B1 − B2 V = C2 V − V C1 , with the matrices B1 (u, v, λ) =
N X
((uk + vk−1 )2 + uk vk + uk−1 vk−1 )Ekk
k=1
+λ
N X
(uk+1 + vk + uk + vk−1 )Ek+1,k + λ2
k=1
B2 (u, v, λ) =
N X
N X
Ek+2,k ,
k=1
((uk + vk )2 + uk+1 vk + uk vk−1 )Ekk
k=1
+λ
N X
(uk+1 + vk+1 + uk + vk )Ek+1,k + λ2
k=1
C1 (u, v, λ) = λ−1
N X
N X k=1
(uk+1 + vk + uk + vk−1 )uk vk Ek,k+1
k=1
+ λ−2
N X
uk+1 vk+1 uk vk Ek,k+2 ,
k=1
C2 (u, v, λ) = λ−1
N X
(uk+1 + vk+1 + uk + vk )uk+1 vk Ek,k+1
k=1
+ λ−2
N X k=1
uk+2 vk+1 uk+1 vk Ek,k+2 .
Ek+2,k ,
782
Y. B. SURIS
Obviously, the expressions for B1,2 may be obtained from (10.4) with the help of the substitutions (15.52) and (15.53), respectively, and C1,2 follow analogously from (10.5). 15.6. Discretization of VL2 Applying the recipe of Sec. 6 with F (T ) = I + hT 2 , we take as a discretization of the flow VL2 the map described by the discrete time Lax triads −1 Ve = B −1 2 U B1 = C 2V C 1 ,
e = B −1 U B 2 = C 1 U C −1 , U 1 2
(15.41)
with B i = Π+ (I + hTi2 ) ,
C i = Π− (I + hTi2 ) ,
i = 1, 2 .
Theorem 15.8. The discrete time Lax equations (15.41) are equivalent to the map (u, v) 7→ (e u, ve) described by the following equations: u ek = uk
γk , βk
vek = vk
βk+1 , γk
(15.42)
where the functions βk = βk (u, v) = 1 + O(h), γk = γk (u, v) = 1 + O(h) are uniquely defined for h small enough simultaneously with the functions δk = O(1), k = O(1) by the system of recurrent relations βk − h(δk − uk+1 )uk =
γk−1 , γk−1 − h(k−1 − vk )vk−1
(15.43)
γk − h(k − vk+1 )vk =
βk , βk − h(δk − uk+1 )uk
(15.44)
δk = uk+1 + vk + uk + vk−1 −
huk−1 vk−1 δk−1 , βk−1
(15.45)
k = uk+1 + vk+1 + uk + vk −
huk vk−1 k−1 . γk−1
(15.46)
The auxiliary functions βk , γk have the asymptotics βk = 1 + h((uk + vk−1 )2 + uk vk + uk−1 vk−1 ) + O(h2 ) ,
(15.47)
γk = 1 + h((uk + vk )2 + uk+1 vk + uk vk−1 ) + O(h2 ) .
(15.48)
Remark. The matrices B 1,2 have the following expressions: B 1 (u, v, λ) = Π+ (I + h(U V )2 ) =
N X
βk Ek,k + hλ
k=1
B 2 (u, v, λ) = Π+ (I + h(V U )2 ) =
N X k=1
N X
δk Ek+1,k + hλ2
k=1
γk Ek,k + hλ
N X k=1
N X
Ek+2,k ,
k=1
k Ek+1,k + hλ2
N X k=1
Ek+2,k .
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
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783
Proof. The scheme of the proof is standard. First of all, the general structure of the factors B 1,2 , C 1,2 is clear from the bi-diagonal structure of the matrices U , V . The recurrent relations for the entries of the matrices B 1,2 follow, in principle, from (10.8), (10.9) with the help of substitutions (15.52), (15.53). It is easy to see that these relations coincide with (12.8), (12.6) after re-naming (15.2) and βk 7→ β2k−1 ,
γk 7→ β2k ,
δk 7→ δ2k−1 ,
k 7→ δ2k .
e = U B 2 , B 2 Ve = V B 1 , also The equations of motion (15.42), following from B 1 U coincide with (12.4) after the above re-namings. Thus the discretization introduced in this theorem agrees with the one from Sec. 12 and may be consistently denoted by dVL2. 15.7. Local equations of motion for dVL2 Here we translate the corresponding results from Sec. 12 into our present notations. The localizing change of variables for the map dVL2 is given by the formulas uk = uk
2 (1 + hvk−1 ) , (1 − huk vk−1 )(1 − hvk−1 uk−1 )
vk = vk
(1 + hu2k ) . (1 − hvk uk )(1 − huk vk−1 ) (15.49)
Theorem 15.9. The change of variables (15.49) conjugates the map dVL2 e ) described by the following local equations of motion: with the map (u, v) 7→ (e u, v ek u
2 (1 + he vk−1 ) (1 + hvk2 ) = uk , ek−1 )(1 − he e k−1 ) (1 − he uk v vk−1 u (1 − huk+1 vk )(1 − hvk uk )
(15.50) (1 + hu2k+1 ) (1 + he u2k ) ek = vk v e k )(1 − he ek−1 ) (1 − he vk u uk v (1 − hvk+1 uk+1 )(1 − huk+1 vk ) . Theorem 15.10. The pull-back of the flow VL under the change of variables (15.49) is described by the following equations of motion: vk−1 vk , − u˙ k = uk (1 + hu2k ) 1 − hvk uk 1 − huk vk−1 (15.51) uk uk+1 2 . − v˙ k = vk (1 + hvk ) 1 − huk+1 vk 1 − hvk uk 15.8. Miura relations to the Toda hierarchy We have seen in Sec. 11 that the flow VL may be considered as a restriction of the second flow TL2 of the Toda hierarchy. There exists a relation of a completely different nature with the Toda hierarchy. Namely, the flow VL is Miura related to the first flow TL of the Toda hierarchy, while the flow VL2 is Miura related to
784
Y. B. SURIS
TL2. To see this notice that the matrices T1,2 (u, v, λ) from (15.14) have the same tri-diagonal structure as the Toda lattice Lax matrix (9.9) with ak = uk vk ,
bk = uk + vk−1 ,
(15.52)
bk = uk + vk ,
(15.53)
or ak = uk+1 vk ,
respectively. These two pair of formulas may be considered as two Miura maps M1,2 : W(u, v) 7→ T (a, b). The following holds (see [34]): (1) Both the maps M1,2 are Poisson, if W is equipped with the bracket (15.4), and T is equipped with the bracket (9.5). (2) Both the maps M1,2 are Poisson, if W is equipped with the bracket (15.5), and T is equipped with the bracket (9.7). (3) The flow VL (15.1) is conjugated with the flow TL (9.1) and the flow VL2 (15.39) is conjugated with the flow TL2 (10.1) by either one of the maps M1,2 . We now translate these statements into the language of localizing variables. Since we have two different localizing changes of variables (for dVL and dVL2), two different translations are necessary. We start with the case of the localizing change of variables (15.32). Theorem 15.11. [34]. (a) Define two maps M1,2 : W(u, v) 7→ T (a, b) by the formulas (15.54) ak = uk vk , 1 + hbk = (1 + huk )(1 + hvk−1 ) , and ak = uk+1 vk ,
1 + hbk = (1 + huk )(1 + hvk ) ,
(15.55)
respectively. Then the following diagram is commutative: M1,2 W(u, v)
-
T (a, b)
(15.32)
(9.19)
?
? -
W(u, v)
T (a, b)
M1,2 (b) Both the maps M1,2 are Poisson, if W(u, v) is equipped with the bracket (15.37), and T (a, b) is equipped with the bracket (9.25).
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
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785
(c) The local form of the dVL (15.33) is conjugated with the local form of the dTL (9.20) by either of the maps M1,2 . Proof. The first statement is verified by a direct check, the second and the third ones are its consequences. In [34] this theorem was formulated without any relation to the problem of integrable discretization. In the context of discrete time systems the formulas (15.54) and (15.55) were found also in [29], however, without discussing Poisson properties of these maps. Concerning the localizing change of variables (15.49) for dVL2, we get the following results. Theorem 15.12. Define the maps N1,2 : W(u, v) 7→ T (a, b) by the formulas ak =
uk vk , 1 − huk vk
ak =
uk+1 vk , 1 − huk+1 vk
and
bk =
uk + vk−1 , 1 − huk vk−1
(15.56)
uk + vk , 1 − huk vk
(15.57)
bk =
respectively. Then the following diagram is commutative: N1,2 W(u, v)
-
T (a, b)
(15.49)
(10.14)
?
? -
W(u, v)
T (a, b)
M1,2 and the local form of the dVL2 (15.50) is conjugated with the local form of the dTL2 (10.15) by either of the maps N1,2 . Proof. By a direct check.
16. Relativistic Toda Lattice 16.1. Equations of motion and tri-Hamiltonian structure The relativistic Toda lattice was invented by Ruijsenaars [56], and further studied in [12, 47] and numerous other papers. In particular, the tri-Hamiltonian structure was elaborated in the latter reference.
786
Y. B. SURIS
We consider here two flows of the relativistic Toda hierarchy: d˙k = dk (ck − ck−1 ) ,
and d˙k = dk
c˙k = ck (dk+1 + ck+1 − dk − ck−1 ) ,
ck−1 ck − dk dk+1 dk−1 dk
,
c˙k = ck
1 1 − dk dk+1
(16.1)
.
(16.2)
Both the systems may be considered either under open-end boundary conditions (c0 = cN = 0), or under periodic ones (all the subscripts are taken (mod N ), so that dN +1 ≡ d1 , d0 ≡ dN , cN +1 ≡ c1 , c0 ≡ cN ). We shall denote the first flow by RTL+, and the second one by RTL−. The phase space of the flows RTL± in the case of the periodic boundary conditions may be defined as R = R2N (d1 , c1 , . . . , dN , cN ) .
(16.3)
This space carries three compatible local Poisson bracket, with respect to which the flows RTL± are Hamiltonian. The linear Poisson structure on R is defined as {dk , ck }1 = −ck ,
{ck , dk+1 }1 = −ck ,
{dk , dk+1 }1 = ck .
(16.4)
The quadratic invariant Poisson structure on R is given by the brackets {dk , ck }2 = −dk ck ,
{ck , dk+1 }2 = −ck dk+1 ,
{ck , ck+1 }2 = −ck ck+1 .
(16.5)
Finally, the cubic Poisson bracket on R is given by the relations {dk , ck }3 = −dk ck (dk + ck ) ,
{ck , dk+1 }3 = −ck dk+1 (ck + dk+1 ) ,
{dk , dk+1 }3 = −dk ck dk+1 ,
{ck , ck+1 }3 = −ck ck+1 (ck + 2dk+1 + ck+1 ) ,
{dk , ck+1 }3 = −dk ck ck+1 ,
{ck , dk+2 }3 = −ck ck+1 dk+2 ,
{ck , ck+2 }3 = −ck ck+1 ck+2 . (16.6) The Hamilton functions for the flow RTL+ in the brackets (16.4), (16.5) and (16.6) are equal to H2 (c, d), H1 (c, d), and H0 (c, d), respectively, where H2 (c, d) =
H1 (c, d) =
X 1X (dk + ck )2 + (dk + ck )ck−1 , 2 N
N
k=1
k=1
N X
(16.7)
(dk + ck ) ,
(16.8)
log(dk ) .
(16.9)
k=1
H0 (c, d) =
N X k=1
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
...
787
Similarly, the Hamilton functions for the flow RTL− in these three brackets are equal to −H0 (c, d), H−1 (c, d), H−2 (c, d), respectively, where H−1 (c, d) =
N X dk + ck k=1
dk dk+1
,
(16.10)
1 X (dk + ck )2 X (dk−1 + ck−1 )ck + . H−2 (c, d) = 2 d2k+1 d2k dk+1 d2k dk−1 N
N
k=1
k=1
(16.11)
16.2. Lax representation The most natural Lax representation for the relativistic Toda hierarchy is the one living in g = g ⊗ g [62], which is in many respects analogous to the Lax representation for the Volterra hierarchy considered in the previous section. Introduce the matrices U (c, d, λ) =
N X k=1
dk Ekk + λ
N X
V (c, d, λ) = I − λ−1
Ek+1,k ,
k=1
N X
ck Ek,k+1 . (16.12)
k=1
Theorem 16.1. [62]. The equations of motion (16.1) are equivalent to the following Lax equations in g ⊗ g (Lax triads): U˙ = U B − AU ,
V˙ = V B − AV ,
(16.13)
where A(c, d, λ) =
N X
(dk + ck−1 )Ekk + λ
k=1
B(c, d, λ) =
N X
N X
Ek+1,k ,
(16.14)
k=1
(dk + ck )Ekk + λ
k=1
N X
Ek+1,k .
(16.15)
k=1
The equations of motion (16.2) are equivalent to the following Lax triads: U˙ = U D − CU ,
V˙ = V D − CV ,
(16.16)
N X ck Ek,k+1 , dk+1
(16.17)
N X ck Ek,k+1 . dk
(16.18)
where C(c, d, λ) = −λ−1
k=1
D(c, d, λ) = −λ−1
k=1
Corollary. The matrices T1 (c, d, λ) = U (c, d, λ)V −1 (c, d, λ) ,
T2 (c, d, λ) = V −1 (c, d, λ)U (c, d, λ)
(16.19)
788
Y. B. SURIS
satisfy the usual Lax equations in g. Namely, for the flow (16.1): T˙1 = [T1 , A] ,
T˙2 = [T2 , B] ,
(16.20)
T˙1 = [T1 , C] ,
T˙2 = [T2 , D] .
(16.21)
and for the flow (16.2):
The following formulas are easy to establish by a direct calculation: A = π+ (T1 ) ,
B = π+ (T2 ) ,
C = π− (T1−1 ) ,
D = π− (T2−1 ) .
(16.22) (16.23)
Hence the Lax equations (16.20) and (16.21) may be presented as T˙i = [Ti , π+ (Ti )] and T˙i = [Ti , π− (Ti−1 )] . respectively (i = 1, 2). These Lax equations in g may be given an r-matrix interpretation in the cases of the linear and of the quadratic Poisson brackets; for the Lax triads in g ⊗ g an r-matrix interpretation is known in the case of the quadratic bracket only. Theorem 16.2. [62]. (a) The Lax matrices T1,2 (c, d, λ) : R 7→ g define Poisson maps, if R carries the linear bracket {·, ·}1 and g is equipped with the bracket PB1 (R), and also if R carries the quadratic bracket {·, ·}2 and g is equipped with the bracket PB2 (A1 , A2 , S). (b) The Lax matrix (U (c, d, λ), V −1 (c, d, λ)) : R 7→ g ⊗ g defines a Poisson map, if R carries the quadratic bracket {·, ·}2 and g ⊗ g is equipped with the Poisson bracket PB2 (A1 , A2 , S) defined by the operators (15.16) and (15.17). (c) The both monodromy maps M1,2 : g ⊗ g 7→ g, M1 : (U, V ) 7→ U V −1 = T1 ,
M2 : (U, V ) 7→ V −1 U = T2
(16.24)
are Poisson, if g ⊗ g is equipped with PB2 (A1 , A2 , S), and the target space g is equipped with PB2 (A1 , A2 , S). (d) Let ϕ be an Ad-invariant function on g, and consider the Hamiltonian flow on R with either of two brackets {·, ·}1,2 with the Hamilton function ϕ(T1 ) = ϕ(T2 ). Then the evolution of the matrices U, V may be presented in the form of the “Lax triads” 1 1 U˙ = U · R(f (T2 )) − R(f (T1 )) · U , 2 2
(16.25)
1 1 V˙ = V · R(f (T2 )) − R(f (T1 )) · V , 2 2
(16.26)
where f = ∇ϕ for the linear bracket, and f = dϕ for the quadratic one.
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
...
789
16.3. Discretization of the relativistic Toda hierarchy We see that actually the pairs (U, V −1 ) satisfy the Lax equations of the form (3.2) (with m = 2). This allows to apply our general recipe to find integrable discretizations of the flows RTL±. This results in considering the following discrete time Lax triads: e = Π−1 (F (T1 )) · U · Π+ (F (T2 )) = Π− (F (T1 )) · U · Π−1 (F (T2 )) , U + − −1 Ve = Π−1 + (F (T1 )) · V · Π+ (F (T2 )) = Π− (F (T1 )) · V · Π− (F (T2 )) ,
with F (T ) = I + hT
and F (T ) = I − hT −1 ,
respectively. It turns out that for the flow RTL+ the version with the Π+ factors is more suitable, while for the RTL− flow the version with the Π− factors is preferable. 16.4. Discretization of the flow RTL+ Consider the discrete time Lax triad e = A−1 U B , U
Ve = A−1 V B ,
(16.27)
with A = Π+ (I + hT1 ) ,
B = Π+ (I + hT2 ) ,
implying also either of the two equivalent forms of a convenient Lax equation: Te1 = A−1 T1 A ,
Te2 = B −1 T2 B .
(16.28)
e Theorem 16.3. [66]. Equation (16.27) is equivalent to the map (c, d) 7→ (e c, d) described by the following equations: bk , dek = dk ak
c k = ck e
bk+1 , ak
(16.29)
where the functions ak = ak (c, d) = 1 + O(h) are uniquely defined by the recurrent relation hck−1 , (16.30) ak = 1 + hdk + ak−1 and the coefficients bk = bk (c, d) = 1 + O(h) are given by bk = ak−1
ak + hck ak+1 − hdk+1 = ak . ak−1 + hck−1 ak − hdk
(16.31)
The following asymptotics hold: ak = 1 + h(dk + ck−1 ) + O(h2 ) ,
(16.32)
bk = 1 + h(dk + ck ) + O(h2 ) .
(16.33)
790
Y. B. SURIS
Remark. The auxiliary matrices A, B are bi-diagonal: A(c, d, λ) = Π+ (I + hT1 ) =
B(c, d, λ) = Π+ (I + hT2 ) =
N X
ak Ekk + hλ
N X
k=1
k=1
N X
N X
bk Ekk + hλ
k=1
Ek+1,k ,
(16.34)
Ek+1,k .
(16.35)
k=1
Proof. The general bi-diagonal structure of the factors A, B follows from their definition. The simplest way to derive the recurrent relations (16.30) for the entries of A is to notice that A = Π+ (I + hU V −1 ) = Π+ (V + hU ) , because V ∈ G− . e = The equations of motion (16.29) and the relations (16.31) follow now from AU U B, AVe = V B. The map (16.29) will be denoted dRTL+. Due to the asymptotic relations (16.32), (16.33) it clearly approximates the flow RTL+. As usual, it is tri-Poisson, allows the same integrals and the same Lax matrix as the flow RTL, but has a drawback of non-locality. Its source are the functions ak , which in the open-end case may be expressed as terminating continued fractions: ak = 1 + hdk +
hck−1 1 + hdk−1 + . . .
. hc1 + 1 + hd1
In the periodic case the ak ’s may be expressed as infinite periodic continued fractions of an analogous structure. 16.5. Local equations of motion for dRTL+ Introduce another copy of the phase space R parametrized by the variables ck , dk and consider the change of variables R(c, d) 7→ R(c, d) defined by the following formulas: (16.36) dk = dk (1 + hck−1 ) , ck = ck (1 + hdk )(1 + hck−1 ) . Obviously, due to the implicit function theorem, for h small enough this map is locally a diffeomorphism. Theorem 16.4. The change of variables (16.36) conjugates dRTL+ with the map described by the following local equations of motion: e k (1 + he ck−1 ) = dk (1 + hck ) , d e k )(1 + he e ck−1 ) = ck (1 + hdk+1 )(1 + hck+1 ) . ck (1 + hd
(16.37)
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
...
791
Proof. The crucial point in the proof of this theorem is the following observation: the parametrization of the variables (c, d) according to (16.36) allows to find the coefficients ak (defined by the recurrent relations (16.30)) in the closed form, namely: (16.38) ak = (1 + hdk )(1 + hck−1 ) . Indeed, if we accept the last formula as the definition of the quantities ak , then we obtain successively from (16.38) and (16.36): ak = 1 + hdk (1 + hck−1 ) + hck−1 = 1 + hdk +
hck−1 . ak−1
So, the quantities defined by (16.38) satisfy the recurrent relation (16.30), and due to the uniqueness of solution our assertion is demonstrated. Now (16.31) and (16.38) yield (16.39) bk = (1 + hdk )(1 + hck ) , and the equations of motion (16.37) follow directly from (16.29), (16.38) and (16.39). It turns out that the pull-back of either of the brackets (16.4), (16.5) and (16.6) is highly non-local. Nevertheless, there exist certain linear combinations thereof, whose pull-backs are local. Theorem 16.5. (a) The pull-back of the bracket {·, ·}1 + h{·, ·}2
(16.40)
on R(c, d) under the change of variables (16.36) is the following Poisson bracket on R(c, d): {dk , ck } = −ck (1 + hdk ) , {ck , dk+1 } = −ck (1 + hdk+1 ) , {dk , dk+1 } = ck
(16.41)
(1 + hdk )(1 + hdk+1 ) . (1 + hck )
(b) The pull-back of the bracket {·, ·}2 + h{·, ·}3
(16.42)
on R(c, d) under the change of variables (16.36) is the following Poisson bracket on R(c, d): {dk , ck } = −dk ck (1 + hdk )(1 + hck ) , {ck , dk+1 } = −ck dk+1 (1 + hck )(1 + hdk+1 ) ,
(16.43)
{ck , ck+1 } = −ck ck+1 (1 + hck )(1 + hdk+1 )(1 + hck+1 ) . (c) The brackets (16.41) and (16.43) are compatible. The map (16.37) is Poisson with respect to both of them.
792
Y. B. SURIS
Proof. By a direct check. Notice a curious feature of the bracket (16.41): it is non-polynomial in coordinates (though still local). Theorem 16.6. The pull-back of the flow RTL+ under the map (16.36) is described by the following equations of motion: d˙ k = dk (1 + hdk )(ck − ck−1 ) , (16.44) c˙ k = ck (1 + hck )(dk+1 + ck+1 + hdk+1 ck+1 − dk − ck−1 − hdk ck−1 ) . Proof. We will use by the proof only the bracket of the part (b) of the previous theorem. Obviously, the pull-back we are looking for is a Hamiltonian system with the Hamilton function, which is a pull-back of h−1 H0 (c, d) (indeed, this function is a Casimir function for {·, ·}2 and is the Hamilton function of the flow RTL+ in the bracket h{·, ·}3 ). Calculating the equations of motion generated by the Hamilton function N X log(dk (1 + hck−1 )) h−1 k=1
in the Poisson brackets (16.43), we obtain (16.44). 16.6. Discretization of the flow RTL− Consider the discrete time Lax triad e = C U D−1 , U
Ve = C V D−1 ,
(16.45)
with C = Π− (I − hT1−1 ) ,
D = Π− (I − hT2−1 ) ,
implying also the convenient Lax equations: Te1 = C T1 C −1 ,
Te2 = D T2 D−1 .
(16.46)
e Theorem 16.7. [66]. Equation (16.45) is equivalent to the map (c, d) 7→ (e c, d) described by the following equations: ck ck , e ck = ck+1 , (16.47) dek = dk+1 dk dk+1 where the functions dk = dk (c, d) = O(1) are uniquely defined by the recurrent relation ck , (16.48) dk = dk − h − hdk−1 and the coefficients ck = ck (c, d) = O(1) are given by ck = dk
dk − hdk−1 ck + hdk = dk+1 . dk+1 − hdk ck+1 + hdk+1
The following asymptotics hold: ck + O(h) , dk = dk
ck =
ck dk+1
+ O(h) .
(16.49)
(16.50)
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
...
793
Remark. The auxiliary matrices C, D in the discrete time Lax equations admit the following expressions: C(c, d, λ) = I + hλ−1
N X
ck Ek,k+1 ,
(16.51)
dk Ek,k+1 .
(16.52)
k=1
D(c, d, λ) = I + hλ−1
N X k=1
Proof. Analogous to that of Theorem 16.3.
The map (16.47) will be called hereafter dRTL−. Like dRTL+, it is tri-Poisson, etc. It is non-local because of the presence of the functions dk , which in the open-end case have the following finite continued fractions expressions: dk =
ck hck−1 dk − h − dk−1 − h − . . .
. −
hc1 d1 − h
16.7. Local equations of motion for dRTL− To bring the map dRTL− to the local form, another change of variables R(c, d) 7→ R(c, d) is necessary: hck−1 h dk = dk 1 + , ck = ck 1 − . (16.53) dk−1 dk dk Again, for h small enough this map is locally a diffeomorphism, due to the implicit function theorem. Theorem 16.8. The change of variables (16.53) conjugates dRTL− with the map described by the following local equations of motion: ! he ck−1 hck e = dk 1 + dk 1 + , dk dk+1 e k−1 d ek d (16.54) ! h h e = ck 1 − . ck 1 − dk+1 e d k
Proof. This time the crucial component of the proof is the following remarkably simple local formula for the coefficients dk (defined by the recurrent relations (16.48)) in the coordinates (ck , dk ): dk =
ck . dk
(16.55)
794
Y. B. SURIS
Indeed, if we use both (16.53) and (16.55) as definitions, then we obtain ck = dk − h = dk − h − hdk−1 . dk Hence, the quantities defined by (16.55) satisfy the recurrent relation (16.48), and due to the uniqueness of solution our assertion is proved. From (16.49) and (16.55) we also obtain ck . (16.56) ck = dk+1 Now the equations of motion (16.54) follow directly from (16.47), (16.55) and (16.56). The Poisson properties of the change of variables (16.53) are similar to that of (16.36). Namely, the pull-backs of either of the brackets (16.4), (16.5) and (16.6) are non-local, but there exist linear combinations thereof, whose pull-backs are local. Theorem 16.9. (a) The pull-back of the Poisson bracket {·, ·}2 − h{·, ·}1
(16.57)
on R(c, d) under the change of variables (16.53) is the following bracket on R(c, d): h h , {ck , dk+1 } = −ck dk+1 1 − , {dk , ck } = −dk ck 1 − dk dk+1 h {ck , ck+1 } = −ck ck+1 1 − . (16.58) dk+1 (b) The pull-back of the Poisson bracket {·, ·}3 − h{·, ·}2
(16.59)
on R(c, d) under the change of variables (16.53) is the following bracket on R(c, d): h , {dk , ck } = −dk ck (dk + ck ) 1 − dk h {ck , dk+1 } = −ck dk+1 (ck + dk+1 ) 1 − , dk+1 h h {dk , dk+1 } = −dk ck dk+1 1 − 1− , dk dk+1 h {ck , ck+1 } = −ck ck+1 (ck + 2dk+1 + ck+1 ) 1 − , (16.60) dk+1 h h {dk , ck+1 } = −dk ck ck+1 1 − 1− , dk dk+1 h h {ck , dk+2 } = −ck ck+1 dk+2 1 − 1− , dk+1 dk+2 h h {ck , ck+2 } = −ck ck+1 ck+2 1 − 1− . dk+1 dk+2
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
...
795
(c) The brackets (16.58) and (16.60) are compatible. The map (16.54) is Poisson with respect to both of them.
Proof. Consists of straightforward calculations.
Theorem 16.10. The pull-back of the flow RTL− under the map (16.53) is described by the following equations of motion: −1 −1 ! hck hck−1 ck−1 ck ˙dk = (dk − h) , 1+ 1+ − dk dk+1 dk dk+1 dk−1 dk dk−1 dk (16.61) −1 hck 1 1 c˙ k = ck 1+ − . dk dk+1 dk dk+1 Proof. We use the part a) of the previous theorem. The flow under consideration is a Hamiltonian system with the Hamilton function, which is a pull-back of h−1 H0 (c, d) (indeed, this function is a Casimir function for {·, ·}2 and is a Hamilton function of the flow RTL− in the bracket −h{·, ·}1 ). Calculating the equations of motion generated by the Hamilton function h
−1
N X
log(dk ) + h
k=1
−1
N X
log 1 +
k=1
hck dk dk+1
in the Poisson brackets (16.58), we arrive at (16.61).
16.8. Third appearance of the Volterra lattice It is interesting to remark that the flow RTL+ allows the reduction dk = 0, in which it turns into the Volterra lattice c˙k = ck (ck+1 − ck−1 ) . The Lax representation of the RTL+ flow survives this reduction, delivering a new (third) Lax representation for the VL. Also the quadratic and the cubic Poisson brackets (16.5) and (16.6) allow this reduction and turn into the corresponding objects for the VL. It may be verified that the map dRTL+ in the reduction dk = 0 turns into dVL, although this discretization is based on a completely different Lax representation. Naturally, the same holds for the local forms of these maps. 17. Belov Chaltikian lattice 17.1. Equations of motion and bi-Hamiltonian structure In their studies of lattice analogs of W -algebras, Belov and Chaltikian [6, 5] found an interesting integrable lattice (hereafter BCL): b˙ k = bk (bk+1 − bk−1 ) − ck + ck−1 ,
c˙k = ck (bk+2 − bk−1 ) .
(17.1)
796
Y. B. SURIS
This system may be viewed as an extension of the Volterra lattice (which appears as a ck = 0 reduction of the above system). Belov and Chaltikian also established the bi-Hamiltonian structure of this system. Namely, its phase space, which in the periodic case is (17.2) BC = R2N (b1 , c1 , . . . , bN , cN ) carries two compatible local Poisson brackets, with respect to which the system BCL is Hamiltonian. The first (“quadratic”) Poisson bracket is given by {bk , bk+1 }2 = −bk bk+1 + ck , {bk , ck+1 }2 = −bk ck+1 ,
{ck , ck+1 }2 = −ck ck+1 , {ck , bk+2 }2 = −ck bk+2 ,
(17.3)
{ck , ck+2 }2 = −ck ck+2 , the corresponding Hamilton function being H1 (b, c) =
N X
bk .
(17.4)
k=1
The second (“cubic”) Poisson bracket on BC is given by {bk , ck }3 = −ck (bk bk+1 − ck ) , {ck , bk+1 }3 = −ck (bk bk+1 − ck ) , {bk , bk+1 }3 = −(bk + bk+1 )(bk bk+1 − ck ) , {ck , ck+1 }3 = −ck ck+1 (bk + bk+2 ) , {bk , ck+1 }3 = −bk ck+1 (bk + bk+1 ) + ck ck+1 , {ck , bk+2 }3 = −ck bk+2 (bk+1 + bk+2 ) + ck ck+1 ,
(17.5)
{bk , bk+2 }3 = −bk bk+1 bk+2 + bk ck+1 + ck bk+2 , {ck , ck+2 }3 = −ck ck+2 (bk+1 + bk+2 ) , {bk , ck+2 }3 = −ck+2 (bk bk+1 − ck ) , {ck , bk+3 }3 = −ck (bk+2 bk+3 − ck+2 ) , {ck , ck+3 }3 = −ck bk+2 ck+3 , and the corresponding Hamilton function is equal to 1X H0 (b, c) = log(ck ) . 3 N
(17.6)
k=1
17.2. Lax representation The Lax matrix of the BCL found in [6, 5], is given in terms of the matrices U (λ) = λ
N X k=1
Ek+1,k ,
V (b, c, λ) = I − λ−1
N X k=1
bk Ek,k+1 + λ−2
N X
ck Ek,k+2 .
k=1
(17.7)
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
...
797
Theorem 17.1. The equations of motion (17.1) are equivalent to the following matrix differential equation: V˙ = V B − AV , (17.8) where A(b, c, λ) = π+ (U V
−1
)=
N X
bk−1 Ekk + λ
k=1
B(b, c, λ) = π+ (V
−1
U) =
N X
N X
Ek+1,k ,
k=1
bk Ekk + λ
k=1
N X
Ek+1,k ,
(17.9)
k=1
so that also the following equation holds identically: U B − AU = 0 .
(17.10)
T (b, c, λ) = U · V −1 (b, c, λ)
(17.11)
Corollary [6, 5]. The matrix
satisfies the usual Lax equation in g: T˙ = [T, A] .
(17.12)
This Lax equation can be given an r-matrix interpretation in the case of quadratic Poisson bracket. Theorem 17.2. (a) The Lax matrix (U (b, c, λ), V −1 (b, c, λ)) : BC 7→ g ⊗ g defines a Poisson map, if BC carries the quadratic bracket {·, ·}2 and g ⊗ g is equipped with the Poisson bracket PB2 (A1 , A2 , S) defined by the operators (15.16) and (15.17). (b) The both monodromy maps M1,2 : g ⊗ g 7→ g, M1 : (U, V ) 7→ U V −1 = T1 ,
M2 : (U, V ) 7→ V −1 U = T2
(17.13)
are Poisson, if g ⊗ g is equipped with PB2 (A1 , A2 , S), and the target space g is equipped with PB2 (A1 , A2 , S). (c) Let ϕ be an Ad-invariant function on g, and consider the Hamiltonian flow on BC with the bracket {·, ·}2 with the Hamilton function ϕ(T1 ) = ϕ(T2 ). Then the evolution of the matrix V may be presented in the form of the “Lax triad” 1 1 V˙ = V · R(dϕ(T2 )) − R(dϕ(T1 )) · V , 2 2
(17.14)
while there holds also the following identity: 1 1 U · R(dϕ(T2 )) − R(dϕ(T1 )) · U = 0 . 2 2
(17.15)
798
Y. B. SURIS
17.3. Discretization Since the Lax equation (17.12) has the form (3.1), and moreover, the pairs (U, V −1 ) satisfy the Lax triads of the form (3.2), we can apply the recipe of Sec. 6. Taking, as usual, F (T ) = I + hT , we come to the discrete time matrix equation Ve = A−1 V B
(17.16)
with A = Π+ (I + hU V −1 ) ,
B = Π+ (I + hV −1 U ) .
Moreover, since the equation U = A−1 U B
(17.17)
holds, we have also the usual Lax equation Te = A−1 T A .
(17.18)
Theorem 17.3. The equations (17.16) and (17.17) are equivalent to the following equations: αk+3 ebk = bk αk+2 − h ck 1 − ck−1 αk+2 , e c k = ck , (17.19) αk αk αk αk−1 αk where the coefficients αk = αk (b, c) = 1 + O(h) are uniquely defined by the recurrent relation hbk−1 h2 ck−2 αk = 1 + + . (17.20) αk−1 αk−1 αk−2 The following asymptotics hold: αk = 1 + hbk−1 + O(h2 ) .
(17.21)
Remark. The auxiliary matrices A, B are bi-diagonal: A(b, c, λ) =
N X k=1
B(b, c, λ) =
N X k=1
αk Ekk + hλ
N X
Ek+1,k ,
(17.22)
k=1
αk+1 Ekk + hλ
N X
Ek+1,k .
(17.23)
k=1
Proof. Standard. The general bi-diagonal structure of the factors A, B follows from A = Π+ (I + hU V −1 ) = Π+ (V + hU ), the latter representation implies also the recurrent relation for the entries αk of the matrix A. From (17.17) one derives immediately βk = αk+1 . The equations of motion (17.19) follow then easily from AVe = V B.
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
...
799
Hereafter we call the map (17.19) dBCL. By construction, it is bi-Poisson with respect to the brackets (17.3) and (17.5), approximates the flow BCL due to the asymptotics (17.21), but is non-local due to the nature of the auxiliary quantities αk . 17.4. Local equations of motion for the dBCL The localizing change of variables for dBCL is the map BC(b, c) 7→ BC(b, c) given by the formulas: bk = bk (1 + hbk−1 ) − hck−1 ,
ck = ck (1 + hbk−1 ) .
(17.24)
As usual this is a local diffeomorphism for h small enough. Theorem 17.4. The change of variables (17.24) conjugates the map dBCL with the following one: e k−1 ) − he e k (1 + hb ck−1 = bk (1 + hbk+1 ) − hck , b e k−1 ) = ck (1 + hbk+2 ) . e ck (1 + hb
(17.25)
Proof. Introducing the quantities αk = 1 + hbk−1 ,
(17.26)
we immediately see via simple check that they satisfy the recurrent relations (17.20). Hence they represent the unique solution of these recurrencies with the asymptotics αk = 1 + O(h). Now the equations of motion follow directly from (17.19), (17.24) and (17.26). Theorem 17.5. The pull-back of the bracket {·, ·}2 + h{·, ·}3
(17.27)
on BC(b, c) under the change of variables (17.24) is the following local Poisson bracket on BC(b, c): {bk , ck } = −hck (bk bk+1 − ck )(1 + hbk ) , {ck , bk+1 } = −hck (bk bk+1 − ck )(1 + hbk+1 ) , {bk , bk+1 } = −(bk bk+1 − ck )(1 + hbk )(1 + hbk+1 ) ,
(17.28)
{ck , ck+1 } = −ck ck+1 (1 + hbk + hbk+2 + h2 (bk bk+1 − ck ) + h2 (bk+1 bk+2 − ck+1 )) , {bk , ck+1 } = −ck+1 (bk + h(bk bk+1 − ck ))(1 + hbk ) {ck , bk+2 } = −ck (bk+2 + h(bk+1 bk+2 − ck+1 ))(1 + hbk+2 ) , {ck , ck+2 } = −ck ck+2 (1 + hbk+1 + hbk+2 + h2 (bk+1 bk+2 − ck+1 )) . The map (17.25) is Poisson with respect to this bracket.
800
Y. B. SURIS
Proof. By a straightforward but tiresome calculation.
Theorem 17.6. The pull-back of the flow BCL under the change of variables (17.24) is described by the following equations of motion: b˙ k = (1 + hbk )(bk (bk+1 − bk−1 ) − ck + ck−1 ) , (17.29) c˙ k = ck (bk+2 (1 + hbk+1 ) − bk−1 (1 + hbk ) − hck+1 + hck−1 ) . Proof. We can use the Hamiltonian formalism. The pull-back we are looking for, is a Hamiltonian system on BC(b, c) with the Poisson bracket (17.28) and the PN Hamilton function which is a pull-back of (3h)−1 k=1 log(ck ). Indeed, this function is a Casimir function for {·, ·}2 and is the Hamilton function for BCL in the bracket h{·, ·}3 . Calculating the equations of motion generated by the Hamilton function (3h)−1
N X
log(ck ) + (3h)−1
k=1
N X
log(1 + hbk )
k=1
in the bracket (17.28), we arrive at (17.29).
18. Relativistic Volterra Lattice 18.1. Equations of motion and bi-Hamiltonian structure Consider the following lattice system: u˙ k = uk (wk − wk−1 + αuk wk − αuk−1 wk−1 ) , (18.1) w˙ k = wk (uk+1 − uk + αuk+1 wk+1 − αuk wk ) . It is Miura related to the relativistic Toda lattice in the same way as the Volterra lattice is related to the Toda lattice, therefore we adopt the name “relativistic Volterra lattice” (RVL) for it. Indeed, the Miura map ck = uk wk ,
dk = uk + wk−1
brings (18.1) into d˙k = (1 + αdk )(ck − ck−1 ) ,
c˙k = ck (dk+1 − dk + αck+1 − αck−1 ) .
But this is the RTL+ flow, written in the form which is a perturbation of the usual Toda lattice TL (perform in (16.1) the change of variables dk 7→ 1 + αdk , ck 7→ α2 ck in order to get the latter system). The phase space of RVL in the periodic case is RW = R2N (u1 , w1 , . . . , uN , wN ) .
(18.2)
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
...
801
The system RVL is bi-Hamiltonian. First of all, it is Hamiltonian with respect to a quadratic Poisson bracket on RW which is identical with the invariant quadratic Poisson bracket of the Volterra lattice: {uk , wk }2 = −uk wk ,
{wk , uk+1 }2 = −wk uk+1 .
(18.3)
The corresponding Hamilton function is equal to H1 (u, w) =
N X
(uk + wk + αuk wk ) .
(18.4)
k=1
The second (“cubic”) invariant Poisson bracket, compatible with the previous one, is different from the cubic bracket of the VL and is given by {uk , wk }3 = −uk wk (uk + wk + αuk wk ) , {uk , uk+1 }3 = −uk uk+1 (wk + αuk wk ) ,
{wk , uk+1 }3 = −wk uk+1 (wk + uk+1 ) ,
{wk , wk+1 }3 = −wk wk+1 (uk+1 + αuk+1 wk+1 ) ,
{wk , uk+2 }3 = −αwk uk+1 wk+1 uk+2 .
(18.5)
The corresponding Hamilton function may be taken as H0 (u, w) =
N X
log(uk ) or H0 (u, w) =
k=1
N X
log(wk )
(18.6)
k−1
(the difference between these two functions is a Casimir of the bracket (18.5)). 18.2. Lax representation The Lax representation for RVL is given in terms of three matrices from g: U (u, w, λ) =
N X
uk Ek,k + λ
k=1
N X
Ek+1,k ,
k=1
V (u, w, λ) = I − λ−1 α
N X
uk wk Ek,k+1 ,
(18.7)
k=1
W (u, w, λ) = I + λ−1
N X
wk Ek,k+1 .
k=1
Theorem 18.1. The equations of motion (18.1) are equivalent to the following matrix differential equations: U˙ = U C − AU,
˙ = W B − CW , W
(18.8)
and imply also the matrix differential equation V˙ = V B − AV ,
(18.9)
802
Y. B. SURIS
with the auxiliary matrices A(u, w, λ) =
N X
(uk + αuk−1 wk−1 + wk−1 )Ekk + λ
k=1
B(u, w, λ) =
N X
N X
Ek+1,k ,
k=1
(uk + αuk wk + wk−1 )Ekk + λ
k=1
C(u, w, λ) =
N X
N X
Ek+1,k ,
(18.10)
k=1
(uk + αuk wk + wk )Ekk + λ
k=1
N X
Ek+1,k .
k=1
Proof. An elementary check. It is easy to establish the following fact: A = π+ (U W V −1 ) ,
B = π+ (V −1 U W ) ,
C = π+ (W V −1 U ) ,
so that the triples (U, V −1 , W ) ∈ g ⊗ g ⊗ g satisfy the Lax equations of the type (3.2) with m = 3 and f (T ) = T . These equations may be given an r-matrix interpretation, at least in the case of the quadratic bracket {·, ·}2 . The corresponding quadratic bracket on g ⊗ g ⊗ g turns out to be identical with the one introduced in [71] for the Bogoyavlensky lattice BL(2). Theorem 18.2. (a) The Lax matrix 7 g⊗g⊗g (U (u, w, λ), V −1 (u, w, λ), W (u, w, λ)) : RW → defines a Poisson map, if RW carries the quadratic bracket {·, ·}2 and g ⊗ g ⊗ g is equipped with the Poisson bracket PB2 (A1 , A2 , S) defined by the operators
A1 −S −S A2 −S ∗ −S ∗ A1 = S ∗ A1 S ∗ , A2 = S A2 −S ∗ , S ∗ −S A1 S S A2 ∗ ∗ ∗ S S S S S S S = S −S ∗ −S ∗ , S∗ = S ∗ −S S ∗ . S S −S ∗ S ∗ −S −S
(18.11)
(18.12)
(b) The monodromy maps M1,2,3 : g ⊗ g ⊗ g 7→ g, M1 : (U, V −1 , W ) 7→ U W V −1 = T1 ,
M2 : (U, V −1 , W ) 7→ V −1 U W = T2 ,
M3 : (U, V −1 , W ) 7→ W V −1 U = T3
(18.13)
are Poisson, if g ⊗ g ⊗ g is equipped with PB2 (A1 , A2 , S), and the target space g is equipped with PB2 (A1 , A2 , S).
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
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803
(c) Let ϕ be an Ad-invariant function on g, and consider the Hamiltonian flow on RW with the bracket {·, ·}2 with the Hamilton function ϕ(T1,2,3 ). Then the evolution of the matrices U, V, W may be presented in the form of the “Lax triads”: 1 1 U˙ = U · R(dϕ(T3 )) − R(dϕ(T1 )) · U , 2 2
(18.14)
1 1 V˙ = V · R(dϕ(T2 )) − R(dϕ(T1 )) · V , 2 2
(18.15)
˙ = W · 1 R(dϕ(T2 )) − 1 R(dϕ(T3 )) · W . W 2 2
(18.16)
18.3. Discretization To discretize RVL, we can apply the recipe of Sec. 6 with F (T ) = I + hT . So, we have to consider the following discrete time Lax representation: e = A−1 U C , U
Ve = A−1 V B ,
f = C −1 W B , W
(18.17)
where A = Π+ (I + hU W V −1 ) ,
B = Π+ (I + hV −1 U W ) ,
C = Π+ (I + hW V −1 U ) .
Theorem 18.3. The discrete time Lax equations (18.17) are equivalent to the following equations of motion: u ek = uk
ck , ak
w ek = wk
bk+1 , ck
(18.18)
where the functions ak = ak (u, w) = 1 + O(h) are uniquely defined by the system of recurrent relations ak+1 = 1 + h(uk+1 + wk ) +
h(α − h)uk wk , ak
(18.19)
and the coefficients bk = bk (u, w) = 1 + O(h), ck = ck (u, w) = 1 + O(h) are given by ak + hαuk wk ak+1 − huk+1 bk = ak−1 , ck = ak . (18.20) ak−1 + hαuk−1 wk−1 ak − huk The following asymptotics hold: ak = 1 + h(uk + wk−1 + αuk−1 wk−1 ) + O(h2 ) ,
(18.21)
bk = 1 + h(uk + wk−1 + αuk wk ) + O(h2 ) ,
(18.22)
ck = 1 + h(uk + wk + αuk wk ) + O(h2 ) .
(18.23)
804
Y. B. SURIS
Remark. The auxiliary matrices A, B, C are bi-diagonal: A=
N X
ak Ek,k + hλ
k=1
B=
N X
N X
Ek+1,k ,
(18.24)
Ek+1,k ,
(18.25)
Ek+1,k .
(18.26)
k=1
bk Ek,k + hλ
k=1
C=
N X
N X k=1
ck Ek,k + hλ
k=1
N X k=1
Proof. We have: A = Π+ (I + hU W V −1 ) = Π+ (V + hU W ), since V ∈ G− . Here V +hU W =
N X
(1+huk +hwk−1 )Ek,k −λ−1
k=1
N X
(α−h)uk wk Ek,k+1 +hλ
k=1
N X
Ek+1,k ,
k=1
and the recurrent relations (18.19) for ak , the entries of the Π+ factor of this tridiagonal matrix, follow immediately. The expressions for bk , ck through ak , as well e = U C, AVe = V B, as the equations of motion (18.18), follow directly from AU f C W = W B. We denote the map defined in this theorem by dRVL. As usual, it shares with the system RVL the bi-Hamiltonian structure, the integrals of motion, and so on, but is highly non-local. 18.4. Local equations of motion for dRVL The localizing change of variables for the map dRLV is given by the formulas uk = uk
1 + hwk−1 , 1 − hαuk−1 wk−1
wk = wk
1 + huk . 1 − hαuk wk
(18.27)
Indeed, the following statement holds. Theorem 18.4. The change of variables (18.27) conjugates the map dRVL with the following one: ek u
e k−1 1 + hw 1 + hwk = uk , e k−1 1 − hαe uk−1 w 1 − hαuk wk (18.28) 1 + he uk 1 + huk+1 ek = wk . w ek 1 − hαe uk w 1 − hαuk+1 wk+1
Proof. The statement will follow immediately, if we establish the local expressions for the quantities ak : ak =
(1 + huk )(1 + hwk−1 ) . (1 − hαuk−1 wk−1 )
(18.29)
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
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805
Indeed, from (18.27), (18.29) and the formulas (18.20) we derive immediately: bk =
(1 + huk )(1 + hwk−1 ) , (1 − hαuk wk )
(18.30)
ck =
(1 + huk )(1 + hwk ) , (1 − hαuk wk )
(18.31)
and then (18.18) imply (18.28). To prove (18.29), we take this formula as a definition of the quantities ak and by means of a simple algebra verify that the recurrent relations (18.19) hold. The reference to the uniqueness of solution to these recurrent relations finishes the proof. Theorem 18.5. The pull-back of the bracket {·, ·}2 + h{·, ·}3
(18.32)
on RW(u, w) under the change of variables (18.27) is the following bracket on RW(u, w): {uk , wk } = −uk wk (1 + huk )(1 + hwk ) , {wk , uk+1 } = −wk uk+1 (1 + hwk )(1 + huk+1 ) .
(18.33)
The map (18.28) is Poisson with respect to the bracket (18.33). Proof. By a straightforward verification.
It is very interesting that the bracket (18.33) again turns out to be identical with the invariant local bracket (15.37) of the local version of dVL (so that the contributions of different cubic brackets for VL and RVL are somehow compensated by different localizing changes of variables). Theorem 18.6. The pull-back of the flow RVL under the change of variables (18.27) is described by the following differential equations: wk−1 + αuk−1 wk−1 wk + αuk wk , − u˙ k = uk (1 + huk ) 1 − hαuk wk 1 − hαuk−1 wk−1 (18.34) uk + αuk wk uk+1 + αuk+1 wk+1 ˙ k = wk (1 + hwk ) . − w 1 − hαuk+1 wk+1 1 − hαuk wk Proof. To obtain these differential equations, one has to calculate the Hamiltonian equations of motion generated by the Hamilton function h−1
N X
(log(wk ) + log(1 + huk ) − log(1 − hαuk wk ))
k=1
with respect to the Poisson brackets (18.33).
806
Y. B. SURIS
19. Some Constrained Lattice KP Systems In this section we introduce a large family of systems generalizing simultaneously the Volterra lattice, the relativistic Toda lattice, the Belov–Chaltikian lattice, and the relativistic Volterra lattice. For some reason it is convenient to call these systems constrained lattice KP systems. 19.1. Equations of motion and Hamiltonian structure Each system of this family may be treated as consisting of m sorts of particles. The phase space of such systems in the periodic case is described as Km = RmN (v (1) , . . . , v (m) ) ,
(19.1)
where each vector (j)
(j)
v (j) = (v1 , . . . , vN ) ∈ RN
(19.2)
represents the set of particles of the jth sort. We introduce the notion of the signature of the constrained lattice KP as an ordered m-tuple of numbers: j ∈ {0, 1} ,
= (1 , . . . , m ) ,
1 = 0 .
(19.3)
The constrained KP lattice (hereafter cKPL(m)) with the signature is the following system of differential equations: j−1 m X X (j) (j) (i) (i) (j) (j) (i) (i) v˙ k = vk (vk+1 − vk ) + j (vk+1 − vk−1 ) + (vk − vk−1 ) . (19.4) i=1
i=j+1
Obviously, the Volterra lattice VL belongs to this class and is characterized by the signature = (0, 0). The RTL+ flow of the relativistic Toda hierarchy also belongs to this class and has the signature = (0, 1). We will see later on that RVL is a reduction of cKPL(3) with = (0, 1, 0), while BCL is Miura related to a simple reduction of cKPL(3) with = (0, 1, 1). The system cKPL(m) with the signature consisting only of zeros, = (0, . . . , 0) is nothing but the Bogoyavlensky (1) lattice BL1(m − 1). Also the vk = 0 reduction of cKPL(m) with the signature = (0, 1, 1, . . . , 1) coincides with BL1(m − 1). Theorem 19.1. The system (19.4) is Hamiltonian with respect to the following Poisson bracket on Km : o n (j) (j) (j) (j) (19.5) vk , vk+1 = −j vk vk+1 , o n (i) (j) (i) (j) = −vk vk , vk , vk 2
2
o n (j) (i) (j) (i) vk , vk+1 = −vk vk+1
for
2
1 ≤ i < j ≤ m, (19.6)
with the Hamilton function H1 (v) = H1 (v (1) , . . . , v (m) ) =
N m X X j=1 k=1
(independent of the signature).
(j)
vk
(19.7)
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
...
807
This statement can be easily checked and generalizes the quadratic brackets (15.4) and (16.5) for the VL and the RTL, respectively. It would be important to find out, when do the analogs of the linear bracket (for the RTL) and of the cubic bracket (for both the VL and the RTL) hold, and to find the corresponding expressions. 19.2. Lax representation The natural Lax representation of the cKPL(m) (19.4) lives in g ⊗ m and is given in terms of the following matrices from g: U1 (v
(1)
N X
, λ) = λ
Ek+1,k +
k=1
Vj (v
(j)
N X
(1)
vk Ek,k ,
(19.8)
k=1 −1
, λ) = I + σj λ
N X
(j)
vk Ek,k+1 ,
j = 1, 2, . . . , m ,
(19.9)
k=1
where σj =
1 −1
j = 0 j = 1
= 1 − 2j .
(19.10)
Theorem 19.2. The equations of motion (19.4) are equivalent to the following matrix differential equations: U˙ 1 = U1 Bm − B1 U1 , ( Vj Bj−1 − Bj Vj ˙ Vj = Vj Bj − Bj−1 Vj
(19.11) j = 0 j = 1
) ,
2 ≤ j ≤ m,
(19.12)
where Bj (v, λ) =
j N X m N X X X (i) (i) ( vk + vk−1 )Ek,k + λ Ek+1,k . k=1 i=1
i=j+1
(19.13)
k=1
The evolution of the monodromy matrices σ
σ
j+1 (λ) (19.14) Tj (v (1) , . . . , v (m) , λ) = Vj j (λ) · . . . · V2σ2 (λ) · U1 (λ) · Vmσm (λ) · . . . · Vj+1
is governed by usual Lax equations: T˙j = [Tj , Bj ] .
(19.15)
The matrices Bj allow the representation Bj = π+ (Tj ) . Proof. An easy check.
(19.16)
808
Y. B. SURIS
It is easy to see that this Lax representation is exactly of the form (3.2), if one considers the m-tuple of matrices (U1 , V2σ2 , . . . , Vmσm ) ∈ g ⊗ m as the Lax matrix. This Lax representation allows an r-matrix interpretation for the quadratic bracket of the previous theorem. As a matter of fact, the corresponding quadratic bracket in g ⊗ m literally coincides with the bracket introduced for the BL1 in [71]. Theorem 19.3. (a) The Lax matrix map (U1 , V2σ2 , . . . , Vmσm ) : Km 7→ g ⊗ m is Poisson, if Km carries the bracket {·, ·}2 , and g ⊗ m is equipped with the bracket PB2 (A1 , A2 , S), where operators A1 , A2 , S are defined according to the formulas: A1 if i = j , (A1 )ij = −S if i = 1, j > 1 or i > j > 1 , ∗ if i > 1, j = 1 or j > i > 1 , S A2 if i = j , S if i > j , (A2 )ij = ∗ −S if j > i , ( S if i = 1 or i > j , (S)ij = −S ∗ if j ≥ i > 1 , ( ∗
(S )ij =
−S if i ≥ j > 1 , S∗
if j > i or j = 1 .
(b) The monodromy maps Mj : g ⊗ m 7→ g σj
Mj : (U1 , V2σ2 , . . . , Vmσm ) 7→ Tj = Vj
σ
j+1 · . . . · V2σ2 · U1 · Vmσm · . . . · Vj+1
(19.17)
are Poisson, if the target space is equipped with the bracket PB2 (A1 , A2 , S). (c) If ϕ is an Ad-invariant function on g, then the Hamiltonian equations of motion on g ⊗ m with the Hamilton function ϕ ◦ Mj may be presented in the form 1 1 U˙ 1 = U1 · R(dϕ(Tm )) − R(dϕ(T1 )) · U1 , 2 2 1 1 Vj · R(dϕ(Tj−1 )) − R(dϕ(Tj )) · Vj , 2 2 V˙ j = V · 1 R(dϕ(T )) − 1 R(dϕ(T )) · V , j j j−1 j 2 2
σj = +1 σj = −1
,
2 ≤ j ≤ m.
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
...
809
19.3. Discretization The Lax representation (19.11) and (19.12) serves as a starting point for applying the recipe of Sec. 6. Taking in this recipe F (T ) = I + hT , we come to the discrete time Lax equations e1 = B −1 U1 B m , U 1 ( −1 B j Vj B j−1 Vej = B −1 j−1 Vj B j
j = 0
)
j = 1
,
2 ≤ j ≤ m,
(19.18)
with B j = Π+ (I + hTj ). Theorem 19.4. The discrete time Lax equations (19.18) are equivalent to the map v 7→ e v described by the equations (j−1) (j) bk+1 vk j = 0 (j) (m) bk (1) (1) bk (j) (2 ≤ j ≤ m) , (19.19) , v e = vek = vk k (1) (j) bk bk+1 (j) j = 1 vk (j−1) bk (j)
(j)
where the functions bk = bk (v) = 1 + O(h) satisfy the following equations: (1)
(m)
bk
=
(1)
bk
(j−1)
bk+1 bk
(j)
(1)
(j−1)
=
,
(19.20)
(j)
bk+1 − hvk+1 (j)
(j)
bk − hvk (j−1)
bk+1 bk
(1)
bk − hvk (j)
=
(j)
(1)
bk+1 − hvk+1
,
(j−1)
2 ≤ j ≤ m,
(19.21)
(j)
bk+1 + hvk+1 bk
j = 0 ,
(j)
+ hvk
,
j = 1 ,
2 ≤ j ≤ m.
(19.22)
Proof. First of all notice that the matrices B j must have the following structure: B j (λ) =
N X
(j)
bk Ek,k + hλ
k=1
N X
Ek+1,k .
k=1
Now the equations of motion are derived straightforwardly. For example, the j = 1 variant of the last m − 1 equations in (19.18), i.e. the matrix equation B j−1 Vej = Vj B j , is equivalent to the following system of scalar equations: ( (j−1) (j) (j) (j) bk vek = vk bk+1 , (j−1)
bk
(j)
(j)
(j)
− he vk−1 = bk − hvk .
This is equivalent to the corresponding variant of equations of motion (19.19) together with the relation (19.22).
810
Y. B. SURIS
Remark. It is important to notice that the statement of the last theorem deviates from the usual scheme in that it does not contain a system of equations (j) which determine bk uniquely. In fact, in order to find such a system one has to deduce from the equations (19.20)–(19.22) m formulas of the type (j)
bk+q (j)
(j)
=
bk
Ψk
.
(j)
Ψk+1 (j)
Here the number q does not depend on j; Ψk are certain expressions of the type Qm (i) (i) (i) (i) (i) (i−1) (i) + hvk )−1 . This, in turn, i=1 ψk+ni , and all ψk = bk − hvk or ψk = (bk implies that there hold certain equations of the type (j)
(j)
(j)
bk · . . . · bk+q−1 Ψk = const
(19.23)
(here we assumed for definiteness that q > 0). The value of the constant on the right-hand side is uniquely defined by the conditions B j = Π+ (I + hTj ). (As a matter of fact, it is easy to see that this constant does not depend on j.) The value of the constant being determined, Eq. (19.23) give the desired system which defines (j) bk uniquely. However, the outfit of this system depends heavily on the signature of the cKPL, and the general formulas would contain too many indices to be instructive enough. It is simpler to derive such formulas for each concrete signature separately. Nevertheless, the formulas (19.20)–(19.22) completely characterize the coefficients of the matrices which serve as the factors Π+ (αI + hTj ) with some α, so that every solution of this system leads to a discretization based on the factorization of I + h0 T with h0 = h/α, which enjoys all the positive properties of our general construction. We call the maps introduced in the previous theorem dcKPL(m). 19.4. Local equations of motion for dcKPL The dcKPL(m) can be brought into the local form for an arbitrary signature . Theorem 19.5. The change of variables Km (v) 7→ Km (v), (j)
(j)
vk = vk
j−1 Y
(i)
m Y
(j)
(1 + hvk ) · (1 + j hvk−1 ) ·
i=1
(i)
(1 + hvk−1 )
(19.24)
i=j+1
conjugates dcKPL(m) with the following map: (j)
ek v
j−1 Y
(i)
(j)
(1 + he vk ) · (1 + j he vk−1 ) ·
i=1
m Y
(i)
(1 + he vk−1 )
i=j+1
(j)
= vk
j−1 Y i=1
(i)
(j)
(1 + hvk+1 ) · (1 + j hvk+1 ) ·
m Y i=j+1
(i)
(1 + hvk ) .
(19.25)
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
...
811 (j)
Proof. It is easy to calculate that if (19.24) holds, and if the quantities bk are defined by the formula j m Y Y (i) (i) = (1 + hvk ) (1 + hvk−1 ) ,
(j) bk
i=1
(19.26)
i=j+1
then (j)
(j)
bk − hvk =
j−1 Y i=1
(j−1) bk
+
(j) hvk
=
m Y
(i)
(1 + hvk )
j Y
(i)
(1 + hvk−1 ) ,
j = 0 ,
i=j+1
(1 +
(i) hvk )
i=1
m Y (i) (1 + hvk−1 ) ,
j = 1 ,
i=j
and it is easy to check now that the equations (19.20)–(19.22) are satisfied. Indeed, for j = 0 we find: (j−1)
(j)
bk (j)
=
(j)
bk − hvk
bk+1 (j)
(j)
(j)
bk+1 − hvk+1
= 1 + hvk ,
which proves (19.21), while for j = 1 we find (j−1)
bk
(j−1)
(j)
+ hvk
(j−1)
bk
=
(j)
bk+1 + hvk+1 (j)
bk+1
(j)
= 1 + hvk ,
which proves (19.22). The verification of (19.20) is completely analogous. The pull-back of the equations of motion is now calculated straightforwardly. Unfortunately, we do not know a general formula for the second invariant Poisson structure for cKPL’s. This prevents us from applying our general scheme for finding the local invariant Poisson brackets for the localized maps. However, by a direct analysis of equations of motion the following statement can be proved. Theorem 19.6. The pull-back of the system (19.4) under the change of variables (19.24) is given by the formula v˙ k = vk (1 + hvk ) (j)
(j)
(j)
j−1 Y
(i)
(j)
(1 + hvk+1 ) · (1 + j hvk+1 ) ·
i=1
−
j−1 Y i=1
(i)
m Y
(i)
(1 + hvk )
i=j+1
(j)
(1 + hvk ) · (1 + j hvk−1 ) ·
m Y i=j+1
(1 + hvk−1 ) /h . (i)
(19.27)
812
Y. B. SURIS
19.5. Example 1: =(0,0,0) As illustrations we consider the systems with m = 3 — the simplest possible ones after VL and RTL+. We denote for simplicity (1)
vk = uk ,
(2)
(3)
vk = vk ,
vk = wk .
The Hamilton function is in all cases equal to H1 (u, v, w) =
N X
(uk + vk + wk ) .
k=1
The non-vanishing Poisson brackets consist of the signature independent part, {uk , vk }2 = −uk vk , {uk , wk }2 = −uk wk , {vk , wk }2 = −vk wk ,
{vk , uk+1 }2 = −uk+1 vk , {wk , uk+1 }2 = −uk+1 wk ,
(19.28)
{wk , vk+1 }2 = −vk+1 wk ,
supplemented by (j)
(j)
(j) (j)
{vk , vk+1 }2 = −vk vk+1 for those j where j = 1. In particular, if all j = 0, then all non-vanishing Poisson brackets of the coordinate functions are exhausted by (19.28), and we arrive at the system u˙ k = uk (vk + wk − vk−1 − wk−1 ) , v˙ k = vk (uk+1 + wk − uk − wk−1 ) ,
(19.29)
w˙ k = wk (uk+1 + vk+1 − uk − vk ) , which becomes the usual Bogoyavlensky lattice BL1(2) after the renaming uk 7→ a3k−2 ,
vk 7→ a3k−1 ,
wk 7→ a3k .
The localizing change of variables for its discretization: uk = uk (1 + hvk−1 )(1 + hwk−1 ) , vk = vk (1 + huk )(1 + hwk−1 ) ,
(19.30)
wk = wk (1 + huk )(1 + hvk ) . The local discretization of the system (19.29): e k−1 ) = uk (1 + hvk )(1 + hwk ) , e k (1 + he vk−1 )(1 + hw u e k−1 ) = vk (1 + huk+1 )(1 + hwk ) , ek (1 + he uk )(1 + hw v e k (1 + he uk )(1 + he vk ) = wk (1 + huk+1 )(1 + hvk+1 ) . w
(19.31)
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
...
813
19.6. Example 2: =(0,1,0) In this example the signature dependent part of the Poisson brackets reads {vk , vk+1 }2 = −vk vk+1 , and the equations of motion take the following form: u˙ k = uk (vk + wk − vk−1 − wk−1 ) , v˙ k = vk (uk+1 + vk+1 + wk − uk − vk−1 − wk−1 ) ,
(19.32)
w˙ k = wk (uk+1 + vk+1 − uk − vk ) . The localizing change of variables for the discretization of this system: uk = uk (1 + hvk−1 )(1 + hwk−1 ) , vk = vk (1 + huk )(1 + hvk−1 )(1 + hwk−1 ) ,
(19.33)
wk = wk (1 + huk )(1 + hvk ) .
(19.34)
The local form of equations of motion for the discretization of (19.32): e k (1 + he e k−1 ) = uk (1 + hvk )(1 + hwk ) , u vk−1 )(1 + hw e k−1 ) = vk (1 + huk+1 )(1 + hvk+1 )(1 + hwk ) , ek (1 + he uk )(1 + he vk−1 )(1 + hw v e k (1 + he uk )(1 + he vk ) = wk (1 + huk+1 )(1 + hvk+1 ) . w
(19.35)
Let us mention that the system (19.32) allows an interesting reduction vk = αuk wk ,
(19.36)
which is, moreover, compatible with the quadratic Poisson brackets. In this reduction we arrive at the system RVL. It is easy to check that in the variables uk , vk , wk the reduction (19.36) takes the form vk = αuk wk (1 + hvk ), so that vk =
αuk wk , 1 − hαuk wk
1 + hvk =
1 . 1 − hαuk wk
This makes a link with the results of Sec. 18. 19.7. Example 3: =(0,1,1) In this case the Poisson brackets (19.28) have to be supplemented by {vk , vk+1 }2 = −vk vk+1 ,
{wk , wk+1 }2 = −wk wk+1 ,
814
Y. B. SURIS
and the equations of motion take the form: u˙ k = uk (vk + wk − vk−1 − wk−1 ) , v˙ k = vk (uk+1 + vk+1 + wk − uk − vk−1 − wk−1 ) ,
(19.37)
w˙ k = wk (uk+1 + vk+1 + wk+1 − uk − vk − wk−1 ) . The localizing change of variables for the discretization of this system: uk = uk (1 + hvk−1 )(1 + hwk−1 ) , vk = vk (1 + huk )(1 + hvk−1 )(1 + hwk−1 ) ,
(19.38)
wk = wk (1 + huk )(1 + hvk )(1 + hwk−1 ) . The local form of equations of motion for the discretization of (19.37): e k (1 + he e k−1 ) = uk (1 + hvk )(1 + hwk ) , u vk−1 )(1 + hw e k−1 ) = vk (1 + huk+1 )(1 + hvk+1 )(1 + hwk ) , ek (1 + he uk )(1 + he vk−1 )(1 + hw v e k−1 ) = wk (1 + huk+1 )(1 + hvk+1 )(1 + hwk+1 ) . e k (1 + he uk )(1 + he vk )(1 + hw w (19.39) Let us discuss the following reduction of the system (19.37): uk = 0 .
(19.40)
It is compatible with the quadratic Poisson brackets, so that we arrive at the following reduced system: v˙ k = vk (vk+1 + wk − vk−1 − wk−1 ) , (19.41) w˙ k = wk (vk+1 + wk+1 − vk − wk−1 ) . Interestingly enough, this system is again nothing but the usual Bogoyavlensky lattice BL1(2), which becomes obvious after the renaming vk 7→ a2k−1 ,
wk 7→ a2k .
So, we have found the third Lax representation for BL1(2). It is easy to see that the maps M1,2 : K3 (0, v, w) 7→ BC(b, c) defined as M1 :
bk = vk + wk ,
ck = vk wk+1 ,
(19.42)
and M2 :
bk = vk+1 + wk ,
ck = vk+2 wk ,
(19.43)
conjugate the flow (19.41) with the Belov–Chaltikian flow BCL, and are Poisson, if both spaces are equipped with the brackets {·, ·}2 . So, the system BCL is Miura
INTEGRABLE DISCRETIZATIONS FOR LATTICE SYSTEMS:
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related to BL1(2) (this fact is similar to the Miura relation between the Toda and the Volterra hierarchy). The localizing change of variables for the discretization of the reduced system (19.41) is given by vk = vk (1 + hvk−1 )(1 + hwk−1 ) ,
wk = wk (1 + hvk )(1 + hwk−1 ) ,
(19.44)
and the corresponding local equations of motion read: ek (1 + he e k−1 ) = vk (1 + hvk+1 )(1 + hwk ) , v vk−1 )(1 + hw (19.45) e k−1 ) = wk (1 + hvk+1 )(1 + hwk+1 ) . e k (1 + he vk )(1 + hw w So, the discretizations of BL1(2) based on different Lax representations, agree with one another. It turns out that the Miura maps M1,2 are still given by nice local formulas, when translated to the localizing variables. Namely, the following diagram is commutative: M1,2
-
K3 (0, v, w)
BC(b, c)
(19.44)
(17.24)
?
? -
K3 (0, v, w)
BC(b, c)
M1,2 if the maps M1,2 are defined by the formulas M1 :
1+hbk = (1+hvk )(1+hwk ) ,
ck = vk wk+1 (1+hwk )(1+hvk+1 ) , (19.46)
and M2 :
1 + hbk = (1 + hvk+1 )(1 + hwk ) ,
ck = vk+2 wk (1 + hvk+1 )(1 + hwk+1 ) . (19.47) This statement may be verified by a simple calculation. 20. Bruschi Ragnisco Lattice The Bruschi–Ragnisco lattice (hereafter BRL) was introduced in [13]: b˙ k = bk+1 ck − bk ck−1 ,
c˙k = ck (ck − ck−1 ) .
(20.1)
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Y. B. SURIS
It may be considered either under open-end boundary conditions (bN +1 = c0 = cN = 0), or under periodic ones (all the subscripts are taken (mod N ), so that c0 ≡ cN , bN +1 ≡ b1 ). The phase space of the Bruschi–Ragnisco lattice: BR = R2N (b1 , c1 , . . . , bN , cN ) . Two compatible brackets may be defined on BR such that the system (20.1) is Hamiltonian with respect to each one of them. The linear Poisson bracket is given by (20.2) {bk , ck }0 = −{bk+1 , ck }0 = −ck , while the quadratic one — by {bk , bk+1 }1 = −bk+1 ck ,
{bk , ck }1 = c2k ,
{bk , ck+1 }1 = −ck ck+1 .
(20.3)
The corresponding Hamilton functions are H1 (b, c) =
N X
bk+1 ck
and H0 (b, c) =
k=1
N X
bk
(20.4)
k=1
for the brackets {·, ·}0 and {·, ·}1 , respectively. It has been pointed out in [13] that this system allows a very complete study by different methods of the soliton theory. As was demonstrated in [61], this is due to its extreme simplicity. Namely, in a certain gauge the Lax representation of this system is a linear matrix equation. Namely, if the entries of the Lax matrix L = L(b, c) ∈ g = gl(N ) are defined as j−1 Y b ci k ≤j, j i=k −1 (20.5) Lkj = k−1 Y ci k >j, bj i=j
then the system (20.1) is equivalent to the equation L˙ = [L, M ]
(20.6)
with the constant matrix M M=
N −1 X k=1
Ek,k+1
or
N −1 X
Ek,k+1 + CEN,1
(20.7)
k=1
for the open-end and periodic case, respectively (in the latter case it is supposed that the dynamics of the BRL is restricted to the set c1 . . . cN = C). The r-matrix interpretation of the brackets (20.2) and (20.3) was given in [61].
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Theorem 20.1. The Lax matrix map L(b, c) : BR 7→ g = gl(N ) is Poisson, if BR carries the bracket {·, ·}0 and gl(N ) is equipped with the standard Lie–Poisson bracket, and also if BR carries the bracket {·, ·}1 and gl(N ) is equipped with the non-standard Lie–Poisson bracket corresponding to the non-standard commutator in gl(N ) : [L1 , L2 ]1 = L1 M L2 − L2 M L1 . For an arbitary Hamilton function ϕ(L) the corresponding Hamiltonian equations of motion in the brackets {·, ·}0 and {·, ·}1 read: L˙ = [L, ∇ϕ(L)] ,
resp.
L˙ = L∇ϕ(L)M − M ∇ϕ(L)L .
For ϕ(L) = tr(LM m ) these equations reduce to L˙ = [L, M m ] ,
resp.
L˙ = [L, M m+1 ] .
The Hamilton functions (20.4) are equal to H1 (b, c) = tr(LM ) ,
H0 (b, c) = tr(L) .
So, the whole hierarchy of the BRL consists of equations L˙ = [L, M m ] , which are linear and may be immediately integrated: L(t) = exp(−tM m ) · L(0) · exp(tM m )
(20.8)
Obviously, the recipe of Sec. 6 cannot be literally applied to BRL. However, the philosophy behind this recipe is, of course, applicable, and requires to seek for the discrete time Bruschi–Ragnisco lattice in the same hierarchy. It should share the Lax matrix with the continuous time system, and its explicit solution should be given by (20.9) L(nh) = (I + hM )−n L(0)(I + hM )n (cf. (20.8)). Hence the corresponding discrete Lax equation should have the form e = (I + hM )−1 L(I + hM ) . L
(20.10)
Theorem 20.2. The discrete time Lax equation (20.10) is equivalent to the following map on the space BR: ebk (1 + he ck−1 ) = bk + hbk+1 ck ,
Proof. An easy calculation.
e c k = ck
1 + he ck . 1 + he ck−1
(20.11)
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Y. B. SURIS
By construction, this map is Poisson with respect to both brackets (20.2) and (20.3). We see that the extreme simplicity of the BRL allows to find its local discretization in the original variables. The localizing change of variables is not necessary for this system. 21. Conclusion This paper contains a rich collection of examples illustrating the procedure of constructing local integrable discretizations for integrable lattice systems. The construction is based on the notion of the r-matrix hierarchy and consists of three steps of a rather different nature. The first step is to find a Lax representation for a given lattice system, living in an associative algebra g. This Lax representation has to be a member of a hierarchy governed by an R-operator on g satisfying the modified Yang–Baxter equation. In all examples treated here this operator is simply a difference of projections to two complementary subalgebras. The second step is an application of a general recipe for integrable discretization. This step is almost algorithmic, the only non-formalized (and, probably, nonformalizable) point being the choice of the function F (L) : g 7→ G approximating exp(hL) for L ∈ g (cf. Sec. 6). In all examples treated here the simplest possible choice F (L) = I + hL works perfectly. The difference equations obtained on this step share the invariant Poisson structures, the integrals of motion, the Lax matrices, etc. with the underlying continuous time systems. However, as a rule, they are non-local. This feature is unpleasant from both the esthetical and the practical point of view, because it makes the equations ugly and not well suited for practical realization. The third step is finding the localizing change of variables. This step is again absolutely non-algorithmic (at least, at our present level of knowledge). These changes of variables have remarkable properties: they often produce one-parameter local deformations of Poisson brackets algebras, and always produce one-parameter integrable deformations of the lattice systems themselves. At the moment we cannot provide a rational explanation neither for these properties nor for the mere existence of localizing changes of variables. However, our collection seems to be representative enough to convince that these phenomena are very general. We feel that they are connected with the Poisson geometry of certain r-matrix brackets on associative algebras and of monodromy maps, but we prefer to stop at this point. References [1] M. Ablowitz and J. Ladik, “Nonlinear differential-difference equations.” J. Math. Phys. 16 (1975) 598–603; “Nonlinear differential-difference equations and Fourier analysis,” J. Math. Phys. 17 (1976) 1011–1018. [2] M. Ablowitz and J. Ladik, “A nonlinear difference scheme and inverse scattering.” Stud. Appl. Math. 55 (1976) 213–229; “On solution of a class of nonlinear partial difference equations.” Stud. Appl. Math. 57 (1977) 1–12. [3] M. Adler, “On a trace functional for formal pseudo-differential operators and the symplectic structure for Korteweg–de Vries type equations,” Invent. Math. 50 (1979) 219–248.
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ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY R. DICKSON Department of Mathematics University of Missouri, Columbia, MO 65211, USA E-mail : [email protected]
F. GESZTESY Department of Mathematics University of Missouri, Columbia, MO 65211, USA E-mail : [email protected]
K. UNTERKOFLER Institute for Theoretical Physics Technical University of Graz A–8010 Graz, Austria E-mail : [email protected] Received 2 June 1998 We continue a recently developed systematic approach to the Bousinesq (Bsq) hierarchy and its algebro-geometric solutions. Our formalism includes a recursive construction of Lax pairs and establishes associated Burchnall–Chaundy curves, Baker–Akhiezer functions and Dubrovin-type equations for analogs of Dirichlet and Neumann divisors. The principal aim of this paper is a detailed theta function representation of all algebro-geometric quasiperiodic solutions and related quantities of the Bsq hierarchy.
1. Introduction The Boussinesq (Bsq) equation, utt = uxx + 3(u2 )xx − uxxxx ,
(1.1)
was originally introduced in 1871 as a model for one-dimensional weakly nonlinear dispersive water waves propagating in both directions (cf. the recent discussion in [48]). It is customary to cast the equation in yet another form and instead write it as the system of equations 1 2 q0,t + q1,xxx + q1 q1,x = 0 , 6 3
q1,t − 2q0,x = 0 .
(1.2)
Introducing q1 (x, t) = −(6u(x, 3−1/2 t) + 1)/4 ,
(1.3)
Eq. (1.1) results upon eliminating q0 (cf. also [24]). The principal subject of this paper concerns algebro-geometric quasi-periodic solutions of the completely integrable hierarchy of Boussinesq equations, of which 823 Reviews in Mathematical Physics, Vol. 11, No. 7 (1999) 823–879 c World Scientific Publishing Company
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(1.2) is just the first of infinitely many members. In order to be able to give a more precise description of the concepts involved, we briefly recall some basic notation in connection with the Boussinesq hierarchy. The Boussinesq hierarchy is defined in terms of Lax pairs (L3 , Pm ) of differential expressions, where L3 is a fixed one-dimensional third-order linear differential expression, 1 d3 d + q1,x + q0 , (1.4) L3 = 3 + q1 dx dx 2 and Pm is a differential expression of order m 6= 0(mod 3), such that the commutator of L3 and Pm becomes a differential expression of order one. For the Boussinesq Eq. (1.2) itself, we have m = 2, that is, P2 =
d2 2 + q1 , 2 dx 3
(1.5)
and the resulting Lax commutator representation of the Boussinesq equation then reads ( q0,t + 16 q1,xxx + 23 q1 q1,x = 0 , d Bsq2 (q0 , q1 ) = L3 − [P2 , L3 ] = 0 , that is, dt q1,t − 2q0,x = 0 . (1.6) A systematic, in fact, recursive approach to all differential expressions Pm will be reviewed in Sec. 2. However, before turning to the contents of each section, it seems appropriate to review the existing literature on the subject and its relation to our approach. Despite a fair number of papers on the Boussinesq system, the current status of research has not yet reached the high level of the KdV hierarchy, or more generally, that of the AKNS hierarchy. From the perspective of completely integrable systems, the reasons for this discrepancy are easily traced back to the enormously increased complexity when making the step from the second-order operator L2 associated with the KdV hierarchy to the third-order operator L3 in connection with the Bsq hierarchy. On an algebro-geometrical level this difference amounts to hyperelliptic curves in the KdV (and AKNS) context as opposed to non-hyperelliptic ones that typically arise in the Bsq case. The classical paper on the Bsq equation, or perhaps more appropriately, the nonlinear string equation, is due to Zakharov [57]. In particular, he introduced the basic Lax pair (L3 , P2 ) and discussed the infinite set of polynomial integrals of motion. In many ways closest in spirit to our approach is the seminal paper by McKean [43] (see also [42]) describing spatially periodic solutions of the Bsq equation. In contrast to [43] though, we concentrate here on the algebro-geometric (i.e. finite-genus) case and make no assumptions of periodicity in order to describe all algebro-geometric quasi-periodic solutions. The application of inverse scattering techniques for the third-order differential expression L3 to the initial value problem of the Bsq equation is discussed in great detail by Deift, Tomei, and Trubowitz [13] and Beals, Deift, and Tomei [4]. General existence theorems (local and global in time) for solutions of the Bsq equation can also be found, for instance, in Craig
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
825
[12], Bona and Sachs [6], and Fang and Grillakis [18], and the references therein. In particular, [4, 6, 12, 13, 37, 43, 44] further discuss and contrast the blow-up mechanism for solutions of the nonlinear string equation obtained by Kalantarov and Ladyzhenskaya [31]. Other special classes of solutions have been considered by a variety of authors. For instance, certain classes of rational Bsq solutions are treated by Airault [2], Airault, McKean, and Moser [3], Chudnovsky [11], and Latham and Previato [36]. In addition, the classical dressing method of Zhakarov and Shabat to construct particular classes of solutions for very general systems of integrable equations, as described, for instance, in [58, 59, 60, 61], should be mentioned in this context. Moreover, certain algebro-geometric Bsq solutions, obtained as special solutions of the Kadomtsev–Petviashvili (KP) equation or by the reduction theory of Riemann theta functions, are briefly discussed by Dubrovin [16], Matveev and Smirnov [38, 39, 40], Previato [49, 50], Previato and Verdier [52], and Smirnov [54, 55]. The latter solutions appear as special cases of a general scheme of constructing algebro-geometric solutions of completely integrable systems developed by Krichever [33, 34, 35] and Dubrovin [15, 17] (see also [5, 22, 47, 53]). Our principal contribution to this subject is a unified framework that yields all algebro-geometric quasi-periodic solutions of the entire Boussines hierarchy at once. In Sec. 2 we briefly recall a recursive construction of the stationary Bsq hierarchy following the approach first outlined in our paper [14]. The stationary Boussinesq hierarchy is then obtained by imposing the t-independent Lax commutator relations [Pm , L3 ] = 0,
m 6= 0 (mod 3) ,
(1.7)
assuming q0 and q1 to be t-independent. From the differential expression Pm we construct two polynomials Sm (z) and Tm (z) in z, which are both x-independent. This leads immediately to the classical Burchnall–Chaundy polynomial (cf. [9, 10]), and hence to a (generally, non-hyperelliptic) curve Km−1 of arithmetic genus m − 1, the central object in the analysis to follow. In Sec. 3, the stationary formalism, and in particular, the curve Km−1 are briefly reviewed. Rather than studying the Baker–Akhiezer function ψ (i.e. the common eigenfunction ψ of the commuting operators L3 and Pm ) directly, our main object is a meromorphic function φ equal to the logarithmic x-derivative of ψ, such that φ satisfies a nonlinear second-order differential equation. Moreover, we describe Dubrovin-type equations for the analogs of Dirichlet and Neumann eigenvalues when compared to the KdV hierarchy. Section 4 then presents our first set of new results, the explicit theta function representations of the Baker–Akhiezer function, the meromorphic function φ, and in particular, that of the potentials q1 and q0 for the entire Boussinesq hierarchy (the latter being the analog of the celebrated Its–Matveev formula [29] in the KdV context). Sections 5 and 6 then extend the analyses of Secs. 3 and 4, respectively, to the time-dependent case. Each equation in the hierarchy is permitted to evolve in terms of an independent deformation (time) parameter tr . As initial data we use a stationary solution of the mth equation of the Boussinesq hierarchy and then construct
826
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
a time-dependent solution of the rth equation of the Boussinesq hierarchy. The Baker–Akhiezer function, the meromorphic function φ, the analogs of the Dubrovin equations, and the theta function representations of Sec. 4 are all extended to the time-dependent case. In Appendix A we provide an introduction to the theory of Riemann surfaces and their theta functions. Appendix B is a collection of results on trigonal Riemann surfaces associated with Bsq-type curves. It should perhaps be noted at this point that our elementary algebraic approach to the Bsq hierarchy and its algebro-geometric solutions is in fact universally applicable to 1+1-dimensional hierarchies of soliton equations such as the KdV hierarchy [25], the AKNS hierarchy [23], the combined sine-Gordon and mKdV hierarchy [21], and the Toda and Kac–van Moerbeke hierarchies [8] (see also [22]). 2. The Recursive Approach to the Boussinesq Hierarchy In this section we briefly recall the necessary material from our previous paper [14] without proofs. Suppose q0 , q1 are meromorphic on C and introduce the third-order differential expression 1 d3 d + q1,x + q0 , x ∈ C . (2.1) L3 = 3 + q1 dx dx 2 For each fixed m ∈ N0 (= N ∪ {0}) with m 6= 0(mod 3) we write m = 3n + ε ,
ε ∈ {1, 2} ,
(2.2)
and then construct two distinct differential expressions of order 3n + 1 and 3n + 2, respectively, denoted by Pm , where m = 3n + 1 or m = 3n + 2. In order for these differential expressions Pm to commute with L3 , one proceeds as follows (cf. [14] for more details). (ε) Pick n ∈ N0 , ε ∈ {1, 2}, and define the sequences {f` (x)}`=0,...,n+1 and (ε) {g` (x)}`=0,...,n+1 recursively by ( (0, 1) for ε = 1 , (ε) (ε) (ε) (ε) (2) d0 ∈ C , (f0 , g0 ) = (c0 , d0 ) = (2) (1, d0 ) for ε = 2 , (ε)
(ε)
(ε)
(ε)
(ε)
(ε)
3f`,x = 2g`−1,xxx + 2q1 g`−1,x + q1,x g`−1 + 3q0 f`−1,x + 2q0,x f`−1 , (ε)
1 (ε) 5 5 (ε) (ε) (2.3) f − q1 f`−1,xxx − q1,x f`−1,xx 6 `−1,xxxxx 6 4 2 2 2 3 1 (ε) (ε) q1,xx + q1 f`−1,x − q1,xxx + q1 q1,x f`−1 , − 4 3 6 3 (ε)
(ε)
3g`,x = 3q0 g`−1,x + q0,x g`−1 −
` = 1, . . . , n + 1 . However, as most of the ensuing discussion can be made for both cases simultaneously, we write (ε) (ε) (2.4) f ` = f` , g ` = g` , and only make the distinction explicit when necessary.
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
827
Explicitly, one computes (i) Let m = 1 (mod 3) (i.e. ε = 1): (1)
= 0,
(1)
= q1 + 3c1 ,
(1)
=
f0 3f1 3f2
(1)
g0 = 1 , (1)
(1)
(1)
3g1 = q0 + 3d1 ,
2 4 (1) (1) (1) q0,xx + q0 q1 + c1 2q0 + d1 q1 + 3c2 , 3 3 1 1 2 4 1 2 q1,xxxx − q1,x − q13 − q1 q1,xx + q02 18 6 27 3 3 1 1 (1) (1) (1) − q1,xx − q12 + d1 q0 + 3d2 , + c1 6 3
(1)
3g2 = −
(2.5)
etc. (ii) Let m = 2 (mod 3) (i.e. ε = 2): (2)
= 1,
(2)
= 2q0 + d0 q1 + 3c1 ,
f0
3f1
(2)
3f2
(2)
3g2
(2)
(2)
g0 = d0 ∈ C ,
1 1 (2) (2) (2) 3g1 = − q1,xx − q12 + d0 q0 + 3d1 , 6 3 1 5 5 3 5 2 5 q1 − q1,x + q02 = − q1,xxxx − q1 q1,xx − 9 9 27 12 3 4 2 (2) (2) (2) (2) q0,xx + q0 q1 + c1 2q0 + d1 q1 + 3c2 , + d0 3 3 1 5 2 5 5 5 q0 q1,xx − q1 q0,xx − q0,x q1,x = − q0,xxxx − q1 q0 − 9 9 18 9 18 1 1 2 4 3 1 2 2 (2) q − q1 q1,xx + q0 − q1,xxxx − q1,x − + d0 18 6 27 1 3 3 1 1 (2) (2) (2) − q1,xx − q12 + d1 q0 + 3d2 , (2.6) + c1 6 3 (2)
(2)
etc., (ε)
(2)
(ε)
where {c` }`≥1 , d0 , {d` }`≥1 are integration constants, which arise when solving (2.3). It is convenient to introduce the homogeneous case where all free integration constants vanish. We denote (ε) (ε) fˆ` = f` |c(ε) =d(ε) =0, p=1,...,` , p
(ε) gˆ`
=
p
(ε) g` |c(ε) =d(ε) =0, p=1,...,` p p
(2.7)
.
and use (cf. (2.3)) (1)
c0 = 0,
(2)
c0 = 1,
(1)
d0 = 1,
(2)
d0 = 0.
(2.8)
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R. DICKSON, F. GESZTESY and K. UNTERKOFLER
We do not list these functions explicitly, however, this notation allows us to write (ε)
f`
=
` X
(1) (2) dp(ε) fˆ`−p + cp(ε) fˆ`−p ,
(ε)
g`
p=0
=
` X
(1) (2) dp(ε) gˆ`−p + cp(ε) gˆ`−p .
(2.9)
p=0
Given (2.3) one defines the differential expression Pm of order m by n 2 X 1 (ε) d (ε) d (ε) Pm = fn−` 2 + gn−` − fn−`,x dx 2 dx `=0
+
2 1 (ε) (ε) (ε) fn−`,xx − gn−`,x + q1 fn−` 6 3
km,` ∈ C ,
` = 0, . . . , n,
L`3 +
n X
km,` L`3 ,
(2.10)
`=0
m = 3n + ε, ε ∈ {1, 2}, n ∈ N0 ,
and verifies that (ε)
[Pm , L3 ] = 3 fn+1,x
3 (ε) d (ε) + fn+1,xx + 3 gn+1,x , dx 2 m = 3n + ε, ε ∈ {1, 2}, n ∈ N0
(2.11)
(where [ · , · ] denotes the commutator symbol). The pair (L3 , Pm ) represents the Lax pair for the Bsq hierarchy. Varying n ∈ N0 and ε ∈ {1, 2}, the stationary Bsq hierarchy is then defined by the vanishing of the commutator of Pm and L3 in (2.11), that is, by [Pm , L3 ] = 0,
m = 3n + ε, ε ∈ {1, 2}, n ∈ N0 ,
(2.12)
(ε)
(2.13)
or equivalently, by (ε)
fn+1,x = 0 ,
gn+1,x = 0 ,
ε ∈ {1, 2}, n ∈ N0 .
Explicitly, one obtains for the first few equations of the stationary Boussinesq hierarchy, m = 1 (i.e. n = 0 and ε = 1) : q0,x = 0 ,
q1,x = 0.
m = 2 (i.e. n = 0 and ε = 2) : −
2 1 (2) q1,xxx − q1 q1,x + d0 q0,x = 0 , 6 3
(2)
2 q0,x + d0 q1,x = 0 .
m = 4 (i.e. n = 1 and ε = 1) : 1 2 4 4 1 q1,xxxxx − q1 q1,xxx − q1,x q1,xx − q12 q1,x + q0 q0,x 18 3 3 9 3 1 2 (1) (1) − q1,xxx − q1 q1,x + d1 q0,x = 0 , + c1 6 3
−
4 4 2 (1) (1) q0,xxx + q1 q0,x + q1,x q0 + c1 2q0,x + d1 q1,x = 0 , 3 3 3 etc.
(2.14)
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
829
By definition, solutions (q0 , q1 ) of any of the stationary Bsq Eqs. (2.14) are called stationary algebro-geometric Bsq solutions or simply algebro-geometric Bsq potentials. Next, we introduce two polynomials Fm and Gm , both of degree at most n with respect to the variable z ∈ C, Fm (z, x) =
n X
(ε)
fn−` (x)z ` ,
(2.15)
`=0
Gm (z, x) =
n X
(ε)
gn−` (x)z ` ,
m = 3n + ε, ε ∈ {1, 2}, n ∈ N0 .
(2.16)
`=0
In terms of homogeneous quantities we define (cf. (2.7) and (2.8)) Fb` = F` |c(ε) =d(ε) =0, p=1,...,n , p
p
b ` = G` | (ε) (ε) G . c =d =0, p=1,...,n p
p
(2.17)
We may then write Fm =
n X
(cn−j Fb3j+2 + dn−j Fb3j+1 ) , (ε)
(ε)
Gm =
j=0
n X
b3j+2 + d b (cn−j G n−j G3j+1 ) . (2.18) (ε)
(ε)
j=0
Explicitly, the first few polynomials Fm , Gm read F1 = 0 ,
G1 = 1 ,
F2 = 1 ,
G2 = d0 ,
F4 =
(2)
1 (1) q1 + c1 , 3
F5 = z +
G4 = z +
1 (1) q0 + d1 , 3
2 (2) 1 (2) q0 + d0 q1 + c1 , 3 3
(2.19) (2)
G5 = d0 z −
1 1 (2) 1 (2) q1,xx − q12 + d0 q0 + d1 , 18 9 3
etc. Given (2.15) and (2.16), (2.12) (or equivalently, (2.13)) becomes 2 Gm,xxx + 2 q1 Gm,x + q1,x Gm − 3 (z − q0 )Fm,x + 2 q0,x Fm = 0 , (2.20) 5 5 2 1 3 Fm,xxxxx + q1 Fm,xxx + q1,x Fm,xx + q1,xx + q12 Fm,x 6 6 4 4 3 2 1 q1,xxx + q1 q1,x Fm + 3(z − q0 )Gm,x − q0,x Gm = 0 . + (2.21) 6 3 Both equations can be integrated (cf. [14]) to get 1 1 1 2 5 Fm,xx − q1 Fm,xx Fm Sm (z) = − Fm,xxxx Fm + Fm,xxx Fm,x − 6 6 12 6 5 5 1 1 2 2 q1,x Fm,x Fm + q1 Fm,x q1,xx + q12 Fm − − + 2 Gm,xx Gm 12 12 3 2 − G2m,x + q1 G2m − 3(z − q0 )Fm Gm ,
(2.22)
830
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
where the integration constant Sm (z) is a polynomial in z of degree at most 2n − 1 + ε, m = 3n + ε, ε ∈ {1, 2}, n ∈ N0 , Sm (z) =
2 n−1+ε X
sm,p z p ,
m = 3n + ε, ε ∈ {1, 2}, n ∈ N0 ,
(2.23)
p=0
and Tm (z) =
1 1 2 Fm,xxxx Fm,xx Fm − Fm,xxxx Fm,x 18 24 +
1 1 1 1 3 2 2 Fm,xxx Fm,xx Fm,x − Fm,xx q1 Fm,xxxx Fm − Fm Fm,xxx + 36 108 36 18
−
1 1 1 2 2 q1,x Fm,xxx Fm q1,xx Fm,xx Fm − q1 Fm,xxx Fm,x Fm + 18 9 18
+
2 7 7 2 2 q1,x Fm,xx Fm,x Fm − q1 Fm,xx Fm,x q1 Fm,xx + Fm 9 72 36
5 2 1 7 1 2 2 3 2 q1 Fm,xx Fm q1,xx Fm,x q1,x Fm,x q1,x q1 Fm,x Fm − Fm − + 18 24 48 12 1 1 2 1 2 3 2 3 q1 − q1,x + q1,xx q1 + (z − q0 )2 Fm − q12 Fm,x Fm + 6 27 36 18 1 1 + (z − q0 )G3m + Fm,xxxx G2m − Fm,xxx Gm,x Gm + Fm G2m,xx 6 3 1 2 + Fm,xx G2m,x + Gm,xx Gm − Fm,x Gm,xx Gm,x − q1 (z − q0 )Fm Gm 3 +
+
2 2 5 4 7 q Fm G2m + q1 Fm,xx G2m − q1 Fm,x Gm,x Gm + q1,x Fm,x G2m 3 1 6 3 12
+
1 4 1 1 q1 Fm G2m,x + q1 Fm Gm,xx Gm + q1,xx Fm G2m − q1,x Fm Gm,x Gm 3 3 6 3
+ (z − q0 )Fm,x Fm Gm,x −
1 2 2 (z − q0 )Fm,x Gm − 2(z − q0 )Fm Gm,xx , 4 (2.24)
where the integration constant Tm (z) is a monic polynomial of degree m, Tm (z) = z m +
m−1 X
tm,q z q ,
m = 3n + ε, ε ∈ {1, 2}, n ∈ N0 .
(2.25)
q=0
Next, we consider the algebraic kernel of (L3 − z), z ∈ C (i.e. the formal nullspace in a purely algebraic sense), ker(L3 − z) = {ψ : C → C ∪ {∞} meromorphic | (L3 − z)ψ = 0} ,
z ∈ C . (2.26)
Taking into account (2.12), that is, [Pm , L3 ] = 0, computing the restriction of Pm to ker(L3 -z), and using (2.27) ψxxx = −q1 ψx + z − 2−1 q1,x − q0 ψ , etc.
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
831
to eliminate higher-order derivatives of ψ, one obtains from (2.3), (2.10), (2.13), (2.15), (2.16), (2.20), and (2.21) 1 d2 d + Hm = Fm 2 + Gm − Fm,x . (2.28) Pm dx 2 dx ker(L3 −z) ker(L3 −z) Here Hm (z, x) =
1 2 Fm,xx (z, x) + q1 (x)Fm (z, x) − Gm,x (z, x) + km (z) 6 3
(2.29)
and (cf. (2.10)) km (z) =
n X
km,` z `
(2.30)
`=0
is an integration constant. The presence of this constant km (z) in (2.29), and hence in (2.28), corresponds to adding an arbitrary polynomial in L3 to the non-trivial part of the differential expression Pm (cf. (2.10)). This polynomial in L3 obviously commutes with L3 , and without loss of generality we henceforth choose to suppress its presence by setting km (z) = 0. (ε) (ε) Still assuming fn+1,x = gn+1,x = 0 as in (2.13), [Pm , L3 ] = 0 in (2.10) yields an algebraic relationship between Pm and L3 by appealing to a result of Burchnall and Chaundy [9, 10] (see also [20, 27, 51, 56]). In fact, one can prove (ε)
(ε)
Theorem 2.1 [14]. Assume fn+1,x = gn+1,x = 0, that is, [Pm , L3 ] = 0, m = 3n + ε, ε ∈ {1, 2}, n ∈ N0 . Then the Burchnall–Chaundy polynomial Fm−1 (L3 , Pm ) of the pair (L3 , Pm ) explicitly reads (cf. (2.23) and (2.25)): 3 + Pm Sm (L3 ) − Tm (L3 ) = 0 , Fm−1 (L3 , Pm ) = Pm
Sm (z) =
2 n−1+ε X
sm,p z p ,
Tm (z) = z m +
p=0
m−1 X
tm,q z q ,
(2.31)
q=0
m = 3n + ε, ε ∈ {1, 2}, n ∈ N0 . Remark 2.2. Fm−1 (L3 , Pm ) = 0 naturally leads to the plane algebraic curve Km−1 , (2.32) Km−1 : Fm−1 (z, y) = y 3 + y Sm (z) − Tm (z) = 0 of (arithmetic) genus m − 1. For m ≥ 4 these curves are non-hyperelliptic. Finally, introducing a deformation parameter tm ∈ C into the pair (q0 , q1 ) (i.e., q` (x) → q` (x, tm ), ` = 0, 1), the time-dependent Bsq hierarchy is defined as a collection of evolution equations (varying m = 3n + ε, ε ∈ {1, 2}, n ∈ N0 ) d L3 (tm ) − [Pm (tm ), L3 (tm )] = 0 , d tm (x, tm ) ∈ C2 , m = 3n + ε, ε ∈ {1, 2}, n ∈ N0 ,
(2.33)
832
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
or equivalently, by Bsqm (q0 , q1 ) =
q0,tm − 3 g (ε) = 0 , n+1,x q
(ε)
1,tm
− 3 fn+1,x = 0 ,
(x, tm ) ∈ C2 , m = 3n + ε, ε ∈ {1, 2}, n ∈ N0 ,
(2.34)
that is, by
Bsqm (q0 , q1 ) =
1 5 5 q0,tm + Fm,xxxxx + q1 Fm,xxx + q1,x Fm,xx 6 6 4 2 2 3 1 2 + 4 q1,xx + 3 q1 Fm,x + 6 q1,xxx + 3 q1 q1,x Fm + 3(z − q0 )Gm,x − q0,x Gm = 0 , q1,tm − 2Gm,xxx − 2q1 Gm,x − q1,x Gm + 3(z − q0 )Fm,x − 2q0,x Fm = 0 , (x, tm ) ∈ C2 , m = 3n + ε, ε ∈ {1, 2}, n ∈ N0 .
(2.35)
Explicitly, one obtains for the first few equations in (2.34), ( q0,t1 − q0,x = 0 , Bsq1 (q0 , q1 ) = q1,t1 − q1,x = 0 , q0,t + 1 q1,xxx + 2 q1 q1,x − d(2) q0,x = 0 , 2 0 6 3 Bsq2 (q0 , q1 ) = (2) q1,t2 − 2 q0,x − d0 q1,x = 0 , (2.36) 1 1 2 4 2 q0,t4 + q1,xxxxx + q1 q1,xxx + q1,x q1,xx + q1 q1,x 18 3 3 9 4 2 1 (1) (1) q1,xxx + q1 q1,x − d1 q0,x = 0 , − q0 q0,x + c1 Bsq4 (q0 , q1 ) = 3 6 3 q1,t − 2 q0,xxx − 4 q1 q0,x − 4 q1,x q0 − c(1) 2q0,x − d(1) q1,x = 0 , 4 1 1 3 3 3 etc. 3. The Stationary Boussinesq Formalism In this section we continue our review of the Bsq hierarchy as discussed in [14] and focus our attention on the stationary case. Following [25] we outline the connections between the polynomial approach described in Sec. 2 and a fundamental meromorphic function φ(P, x) defined on the Boussinesq curve Km−1 in (2.32). Moreover, we discuss in some detail the associated stationary Baker–Akhiezer function ψ(P, x, x0 ), the common eigenfunction of L3 and Pm , and associated
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
833
positive divisors of degree m − 1 on Km−1 . The latter topic was originally developed by Jacobi [30] in the case of hyperelliptic curves and applied to the KdV case by Mumford [46, Sec. III.a.1] and McKean [45]. Before we enter any further details we should perhaps stress one important point. In spite of the considerable complexity of the formulas displayed at various places in Secs. 2–3, the basic underlying formalism is a recursive one as described in depth in [14]. Consequently, the majority of our formalism can be generated using symbolic calculation programs (such as Mathematica or Maple). We recall the Bsq curve Km−1 in (2.32) Km−1 : Fm−1 (z, y) = y 3 + y Sm (z) − Tm (z) = 0 , Sm (z) =
2 n−1+ε X p=0
sm,p z p ,
Tm (z) = z m +
m−1 X
tm,q z q ,
(3.1)
q=0
m = 3n + ε, ε ∈ {1, 2}, n ∈ N0 , (where m = 3n + ε, ε ∈ {1, 2}, n ∈ N0 will be fixed throughout this section) and denote its compactification (adding the branch point P∞ ) by the same symbol Km−1 . (In the following Km−1 will always denote the compactified curve.) Thus Km−1 becomes a (possibly singular) three-sheeted Riemann surface of arithmetic genus m− 1 in a standard manner. We will need a bit more notation in this context. Points P on Km−1 are represented as pairs P = (z, y) satisfying (3.1) together with P∞ , the point at infinity. The complex structure on Km−1 is defined in the usual way by introducing local coordinates ζP0 : P → (z − z0 ) near points P0 ∈ Km−1 which are neither branch nor singular points of Km−1 , ζP∞ : P → z −1/3 near the branch point P∞ ∈ Km−1 (with an appropriate determination of the branch of z 1/3 ) and similarly at branch and/or singular points of Km−1 . The holomorphic map ∗, changing sheets, is defined by ( Km−1 → Km−1 , ∗: P = (z, yj (z)) → P ∗ = (z, yj+1(mod 3) )(z)) , j = 1, 2, 3, P ∗∗ := (P ∗ )∗ , etc. ,
(3.2)
where yj (z), j = 1, 2, 3 denote the three branches of y(P ) satisfying Fm−1 (z, y) = 0. Finally, positive divisors on Km−1 of degree m − 1 are denoted by
DP1 ,...,Pm−1
K → N0 , m−1
k if P occurs k : (3.3) times in {P1 , . . . , Pm−1 } , P → DP1 ,...,Pm−1 (P ) = 0 if P 6∈ {P1 , . . . , Pm−1 } .
Specific details on curves of Bsq-type (i.e. trigonal curves with a triple point at P∞ ) as defined in (3.1) can be found in Appendix B.
834
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
Given these preliminaries, let ψ(P, x, x0 ) denote the common normalized eigenfunction of L3 and Pm , whose existence is guaranteed by the commutativity of L3 and Pm (cf., e.g., [9, 10]), that is, by [Pm , L3 ] = 0 ,
m = 3n + ε
(3.4)
for a given ε ∈ {1, 2}, and n ∈ N0 , or equivalently, by the requirement (ε)
fn+1,x = 0 ,
(ε)
gn+1,x = 0 .
(3.5)
Explicitly, this yields L3 ψ(P, x, x0 ) = z(P ) ψ(P, x, x0 ) ,
Pm ψ(P, x, x0 ) = y(P ) ψ(P, x, x0 ) , (3.6) P = (z, y) ∈ Km−1 \{P∞ }, x ∈ C .
Assuming the normalization, ψ(P, x0 , x0 ) = 1 ,
P ∈ Km−1 \{P∞ }
(3.7)
for some fixed x0 ∈ C, ψ(P, x, x0 ) is called the stationary Baker–Akhiezer function for the Bsq hierarchy. Closely related to ψ(P, x, x0 ) is the following meromorphic function φ(P, x) on Km−1 defined by φ(P, x) =
ψx (P, x, x0 ) , ψ(P, x, x0 )
such that
Z
x
ψ(P, x, x0 ) = exp
P ∈ Km−1 , x ∈ C ,
d x φ(P, x ) , 0
0
P ∈ Km−1 \{P∞ } .
(3.8)
(3.9)
x0
Since φ(P, x) is a fundamental object for the stationary Bsq hierarchy, we next intend to establish its connection with the recursion formalism of Sec. 2. In pursuit of this connection, it is necessary to define a variety of further polynomials Am , Bm , Cm , Dm−1 , Em , Jm , and Nm with respect to z ∈ C, Am (z, x) = −Gm (z, x)2 −
1 1 q1 (x) Fm (z, x)2 + Fm,x (z, x)2 3 4
1 Fm (z, x) Fm,xx (z, x) , (3.10) 3 1 2 2 Bm (z, x) = (z − q0 (x)) −2 Fm (z, x) Gm (z, x) + Fm (z, x) Fm,x (z, x) 2 −
− Gm (z, x)2 Gm,x (z, x) +
1 Fm,x (z, x)2 Gm,x (z, x) 4
−
1 1 q1,x (x) Fm (z, x)2 Gm (z, x) − q1,x (x) Fm (z, x)2 Fm,x (z, x) 6 2
+
11 1 Gm (z, x)2 Fm,xx (z, x) − q1 (x) Fm (z, x)2 Fm,xx (z, x) 6 18
−
1 1 Fm,x (z, x)2 Fm,xx (z, x) + Fm (z, x) Fm,xx (z, x)2 24 36
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
835
+
2 2 q1 (x) Fm (z, x) Gm (z, x)2 − q1 (x)2 Fm (z, x)3 3 9
−
2 1 q1 (x)Fm (z, x) Gm (z, x) Fm,x (z, x) + q1 (x)Fm (z, x) Fm,x (z, x)2 3 6
+ Fm (z, x) Gm (z, x) Gxx (z, x) −
1 Fm (z, x) Fm,x (z, x) Gm,xx (z, x) 2
−
1 1 q1,xx (x) Fm (z, x)3 − Fm (z, x) Gm (z, x) Fm,xxx (z, x) 6 6
+
1 1 Fm (z, x) Fm,x (z, x) Fm,xxx (z, x) − Fm (z, x)2 Fm,xxxx (z, x) 12 6
− Fm (z, x) Gm,x (z, x)2 ,
(3.11)
1 Cm (z, x) = Fm (z, x) Jm (z, x) − Gm (z, x) + Fm,x (z, x) Hm (z, x) , 2 2 Dm−1 (z, x) = (Fm (z, x) Bm (z, x) − A2m (z, x) − Sm (z) Fm (z, x)) −1 1 × ε(m) Gm (z, x) + Fm,x (z, x) , 2
1 Em (z, x) = −(Am (z, x) Cm (z, x) − Bm (z, x) Gm (z, x) + Fm,x (z, x) 2 1 + Sm (z) Fm (z, x) Gm (z, x) + Fm,x (z, x)) Fm (z, x)−1 , 2 1 Jm (z, x) = Hm,x (z, x) + z − q0 (x) − q1,x (x) Fm (z, x) , 2 1 2 Nm (z, x) = (Cm (z, x) + Em (z, x) Gm (z, x) + Fm,x (z, x) 2 + Sm (z)(Gm (z, x) + where
( ε(m) =
1 Fm,x (z, x))2 )ε(m) Fm (z, x)−1 , 2 1
for m = 2 (mod 3) ,
−1
for m = 1 (mod 3) .
(3.12)
(3.13)
(3.14) (3.15)
(3.16)
(3.17)
Explicit (though rather lengthy) formulas for Cm , Dm−1 , Em , and Nm , directly in terms of Fm and Gm and their x-derivatives, which prove their polynomial character with respect to z, can be found in [14]. Moreover we recall the relations (cf. [14]), 1 Bm Cm + Am Em + Sm Am Gm + Fm,x − Fm Cm 2 1 −Tm Fm Gm + Fm,x = 0 , 2
(3.18)
836
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
Bm =
2 1 Sm Fm + ε(m) Dm−1,x , 3 3
2 − Am Bm , ε(m) Cm Dm−1 = Tm Fm 1 Dm−1 Nm = Bm Em − Tm Am Gm + Fm,x − Fm Cm , 2 2 1 ε(m) Am Nm = Tm Gm + Fm,x − Cm Em , 2 1 Nm,x Gm + Fm,x = Nm (q1 Fm + Fm,xx ) 2 1 − ε(m) Jm 2 Gm + Fm,x Sm + 3 Em . 2
(3.19) (3.20) (3.21) (3.22)
(3.23)
Next we recall explicit expressions for φ(P, x). Lemma 3.1 [14]. Let P = (z, y) ∈ Km−1 and (z, x) ∈ C2 . Then (Gm (z, x) + 2−1 Fm,x (z, x))y(P ) + Cm (z, x) Fm (z, x)y(P ) − Am (z, x)
(3.24)
=
Fm (z, x)y(P )2 + Am (z, x)y(P ) + Bm (z, x) ε(m)Dm−1 (z, x)
(3.25)
=
−ε(m)Nm (z, x) . (3.26) (Gm (z, x) + 2−1 Fm,x (z, x))y(P )2 − Cm (z, x)y(P ) − Em (z, x)
φ(P, x) =
By inspection of (2.15) and (2.16) one infers that Dm−1 and Nm are monic polynomials with respect to z of degree m − 1 and m, respectively. Hence we may write Dm−1 (z, x) =
m−1 Y j=1
(z − µj (x)) ,
Nm (z, x) =
m−1 Y
(z − ν` (x)) .
(3.27)
`=0
Defining
Am (µj (x), x) ∈ Km−1 , j = 1, . . . , m − 1 , x ∈ C , (3.28) µj (x), Fm (µj (x), x) Cm (ν` (x), x) ∈ Km−1 , νˆ` (x) = ν` (x), − Gm (ν` (x), x) + 12 Fm,x (ν` (x), x)
µ ˆj (x) =
` = 0, . . . , m − 1 , x ∈ C , (3.29) one infers from (3.24) that the divisor (φ(P, x)) of φ(P, x) is given by (cf. (3.3)) (φ(P, x)) = Dνˆ0 (x),...,ˆνm−1 (x) (P ) − DP∞ ,ˆµ1 (x),...,ˆµm−1 (x) (P ) .
(3.30)
ˆ1 (x), . . . , µ ˆ m−1 (x) That is, νˆ0 (x), . . . , νˆm−1 (x) are the m zeros of φ(P, x) and P∞ , µ its m poles.
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
837
Further properties of φ(P, x) and ψ(P, x, x0 ) are summarized in: Theorem 3.2 [14]. Assume (3.4)–(3.8), P = (z, y) ∈ Km−1 \{P∞ }, and let (z, x, x0 ) ∈ C3 . Then (i) φ(P, x) satisfies the second-order -equation φxx (P, x) + 3 φx (P, x)φ(P, x) + φ(P, x)3 + q1 (x) φ(P, x) = z − q0 (x) −
1 q1,x (x) . 2
(ii) φ(P, x) φ(P ∗ , x) φ(P ∗∗ , x) =
(3.31)
Nm (z, x) . Dm−1 (z, x)
(iii) φ(P, x) + φ(P ∗ , x) + φ(P ∗∗ , x) =
(3.32)
Dm−1,x (z, x) . Dm−1 (z, x)
(3.33)
(iv) y(P ) φ(P, x) + y(P ∗ ) φ(P ∗ , x) + y(P ∗∗ ) φ(P ∗∗ , x) =
3 Tm (z) Fm (z, x) − 2 Sm (z) Am (z, x) . ε(m)Dm−1 (z, x)
(v) ψ(P, x, x0 ) ψ(P ∗ , x, x0 ) ψ(P ∗∗ , x, x0 ) =
Dm−1 (z, x) . Dm−1 (z, x0 )
(vi) ψx (P, x, x0 ) ψx (P ∗ , x, x0 ) ψx (P ∗∗ , x, x0 ) = (vii) ψ(P, x, x0 ) =
Dm−1 (z, x) Dm−1 (z, x0 )
1/3
(3.34)
Z
x
exp
Nm (z, x) . Dm−1 (z, x0 )
(3.35) (3.36)
d x0 ε(m)Dm−1 (z, x0 )−1
x0
× Fm (z, x0 ) y(P )2 + Am (z, x0 ) y(P ) +
2 Fm (z, x0 ) Sm (z) . 3
(3.37)
Thus, up to normalizations, Dm−1 represents the product of the three branches of ψ and Nm the product of the three branches of ψx , their zeros represent the analogs of Dirichlet and Neumann eigenvalues of L3 with the corresponding boundary conditions imposed at the point x ∈ C when compared to the KdV Lax expression L2 . Returning to Dm−1 (z, x) and Nm (z, x) for a moment, we note that (2.3), (2.15), (2.16), (3.13), and (3.16) yield D0 = 1 , D1 = z − q0 (x) − 6−1 q1,x (x) − d0 q1 (x) − (d0 )3 , (2)
etc. ,
(2)
(3.38)
838
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
and N1 = z − q0 (x) , 2 (2) N2 = z − q0 (x) + 6−1 q1,x (x) − d0 (z − q0 (x))q1 (x) − 6−1 q1 (x)q1,x (x) − 6−1 (d0 )2 q1,xx (x) − (d0 )3 (z − q0 (x)) , (2)
(2)
(3.39)
etc. Concerning the dynamics of the zeros µj (x) and ν` (x) of Dm−1 (z, x) and Nm (z, x) one obtains the following Dubrovin-type equations. Lemma 3.3 [14]. Assume (3.5) to hold. (i) Suppose the zeros {µj (x)}j=1,...,m−1 of Dm−1 ( · , x) remain distinct in Ωµ , where Ωµ ⊆ C is open and connected. Then {µj (x)}j=1,...,m−1 satisfy the system of differential equations −ε(m) Fm (µj (x), x) 3y(ˆ µj (x))2 + Sm (µj (x)) µj,x (x) = , j = 1, . . . , m − 1 , m−1 Y (µj (x) − µk (x)) k=1 k6=j
(3.40) with initial conditions {ˆ µj (x0 )}j=1,...,m−1 ⊂ Km−1 ,
(3.41)
for some fixed x0 ∈ Ωµ . The initial value problem (3.40), (3.41) has a unique solution {ˆ µj (x)}j=1,...,m−1 ⊂ Km−1 satisfying µ ˆj ∈ C ∞ (Ωµ , Km−1 ) ,
j = 1, . . . , m − 1 .
(3.42)
(ii) Suppose the zeros {ν` (x)}`=0,...,m−1 of Nm ( · , x) remain distinct in Ων , where Ων ⊆ C is open and connected. Then {ν` (x)}`=0,...,m−1 satisfy the system of differential equations −ε(m) Jm (ν` (x), x) 3y(ˆ ν` (x))2 + Sm (ν` (x)) ν`,x (x) = , ` = 0, . . . , m − 1 , (3.43) m−1 Y (ν` (x) − νk (x)) k=0 k6=j
with initial conditions {ˆ ν` (x0 )}`=0,...,m−1 ⊂ Km−1 ,
(3.44)
for some fixed x0 ∈ Ων . The initial value problem (3.43), (3.44) has a unique solution {ˆ ν` (x)}`=0,...,m−1 ⊂ Km−1 satisfying νˆ` ∈ C ∞ (Ων , Km−1 ) ,
` = 0, . . . , m − 1 .
(3.45)
For trace formulas expressing certain combinations of q0 , q1 and their x-derivatives in terms of µj (x) and ν` (x) we refer to [14].
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
839
The following example illustrates our recursion formalism for the simplest genus g = 1 case. Further examples can be found in [14]. Example 3.4. m = 2 (genus g = 1): q1 (x) = −3℘(x) ,
q0 (x) = 0 ,
(3.46)
3 0 d3 d d2 − ℘ − 3 ℘(x) (x) , P = − 2 ℘(x) , 2 dx3 dx 2 d x2 g2 g3 y − z2 − = 0, F1 (z, y) = y 3 − 4 4
L3 =
F2 (z, x) = 1 ,
G2 (z, x) = 0 ,
D1 (z, x) = z +
1 0 ℘ (x) , 2
φj (z, x) =
N2 (z, x) =
2 1 z − ℘0 (x) , 2
=
(3.50)
(3.51)
yj2 + yj ℘(x) + ℘(x)2 − z+
(3.48) (3.49)
z − 12 ℘0 (x) yj − ℘(x)
=
(3.47)
1 2
g2 4
(3.52)
℘0 (x)
(z − 12 ℘0 (x))2 (z − 12 ℘0 (x))yj − ℘(x)(z −
1 2
℘0 (x))
,
1 ≤ j ≤ 3,
(3.53)
where yj , 1 ≤ j ≤ 3 denote the roots of (3.48) and ℘(x) denotes the elliptic Weierstrass function (cf., e.g., [1], Ch. 18). 4. Stationary Algebro-Geometric Solutions of the Boussinesq Hierarchy In this section we continue our study of the stationary Bsq hierarchy, but now direct our efforts towards obtaining explicit Riemann theta function representations for the fundamental quantities φ and ψ, introduced in Sec. 3, and especially, for each of the potentials q0 and q1 associated with the differential expression L3 . As a result of our preparatory material in Secs. 2 and 3, we are now able to simultaneously treat the class of algebro-geometric quasi-periodic solutions of the entire Bsq hierarchy, one of our principal aims in this paper. In the following we freely employ the notation established in Appendices A and B and refer to this material whenever appropriate. Lemma 4.1. Let x ∈ C. Near P∞ ∈ Km−1 , in terms of the local coordinate ζ = z −1/3 , one has ∞
φ(P, x) =
ζ→0
1X βj (x)ζ j as P → P∞ , ζ j=0
(4.1)
840
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
where β0 = 1, 1 βj = − 3 +
1 β2 = − q1 , 3
β1 = 0,
βj−2,xx + q1 βj−2 +
1 1 β3 = − q0 + q1,x , 3 6
j−1 X
(3βk,x βj−k−1 + βk βj−k )
k=2 j−1 X ` X
! βk β`−k βj−`
,
j ≥ 4.
(4.2)
`=1 k=0
Proof. In terms of the local coordinate ζ = z −1/3 , (3.31) reads φxx + 3φφx + φ3 + q1 φ = ζ −3 − q0 − 2−1 q1,x .
(4.3)
A power series ansatz in (4.3) then yields the indicated Laurent series.
Let θ(z) denote the Riemann theta function (cf. (A.59)) associated with Km−1 and an appropriately fixed homology basis. We assume Km−1 to be nonsingular for the remainder of this section. Next, choosing a convenient base point P0 ∈ Km−1 \{P∞ }, the vector of Riemann constants ΞP0 is given by (A.66), and the Abel maps AP0 ( · ) and αP0 ( · ) are defined by (A.56) and (A.57), respectively. For brevity, define the function z : Km−1 × σ m−1 Km−1 → Cm−1 by z(P, Q) = ΞP0 − AP0 (P ) + αP0 (DQ ) , P ∈ Km−1 , Q = (Q1 , . . . , Qm−1 ) ∈ σ m−1 Km−1 .
(4.4)
We note that by (A.81) and (A.82), z( · , Q) is independent of the choice of base point P0 . (3) The normalized differential ωP∞ ,ˆν0 (x) of the third kind (dtk) is the unique differential holomorphic on Km−1 \{P∞ , ν0 (x)} with simple poles at P∞ and νˆ0 (x) with residues ±1, respectively, that is, (3) (4.5) ωP∞ ,ˆν0 (x) (P ) = ζ −1 + O(1) dζ as P → P∞ . ζ→0
Then
Z
P
P0
(3)
ωP∞ ,ˆν0 (x) = ln(ζ) + e(3) (P0 ) + O(ζ) as P → P∞ , ζ→0
(4.6)
(2)
where e(3) (P0 ) is an appropriate constant. Furthermore, let ωP∞ ,2 denote the normalized differential defined by ( 2n m−1 m = 3n + 1 , z dz , X 1 (2) (4.7) λj ηj (P ) − ωP∞ ,2 (P ) = − 3y(P )2 + Sm (z) y(P )z n dz , m = 3n + 2 , j=1 where the constants {λj }j=1,...,m−1 are determined by the normalization condition Z (2) ωP∞ ,2 = 0 , j = 1, . . . , m − 1 , (4.8) aj
841
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
and the differentials {ηj (P )}j=1,...,m−1 (defined in (B.7)) form a basis for the space (2) of holomorphic differentials. The b-periods of the differential ωP∞ ,2 are denoted by Z 1 (2) (2) (2) (2) (2) ω , j = 1, . . . , m − 1 . (4.9) U 2 = (U2,1 , . . . , U2,m−1 ) , U2,j = 2πi bj P∞ ,2 A straightforward Laurent expansion of 4.7 near P∞ yields the following result. Lemma 4.2. Assume the curve Km−1 to be nonsingular. Then the differential (2) ωP∞ ,2 defined in (4.7) is a differential of the second kind (dsk), holomorphic on Km−1 \{P∞ } with a pole of order 2 at P∞ . In particular, near P∞ in the local (2) coordinate ζ, the differential ωP∞ ,2 has the Laurent series (2)
ωP∞ ,2 (P ) =
ζ→0
where u= and w=
ζ −2 + u + wζ + O(ζ 2 ) dζ as P → P∞ ,
λm−1 − c(1) 1
for m = 1 (mod3) , (2)
λm−n−1 − (d0 )2
(4.11) for m = 2 (mod 3) ,
λm−n−1 − 2d(1) 1
(2)
(2)
(4.10)
for m = 1 (mod 3) , (4.12)
(2)
(d0 )3 − c1 − d0 λm−n−1 + λm−1
for m = 2 (mod 3) .
From Lemma 4.2 one infers Z P (2) (2) ωP∞ ,2 = −ζ −1 + e2 (P0 ) + uζ + 2−1 wζ 2 + O(ζ 3 ) as P → P∞ ,
(4.13)
ζ→0
P0
(2)
where e2 (P0 ) is an appropriate constant. The theta function representation of φ(P, x) then reads as follows. Theorem 4.3. Assume that the curve Km−1 is nonsingular. Let P = (z, y) ∈ Km−1 \{P∞ } and let x, x0 ∈ Ωµ , where Ωµ ⊆ C is open and connected. Suppose that Dµˆ (x) , or equivalently, Dνˆ(x) is nonspecial for x ∈ Ωµ . Then θ(z(P∞ , µ ˆ(x))) θ(z(P, νˆ(x))) exp e(3) (P0 ) − φ(P, x) = θ(z(P∞ , νˆ(x))) θ(z(P, µ ˆ(x)))
Z
P
P0
! (3) ωP∞ ,ˆν0 (x)
.
(4.14)
Proof. Let Φ be defined by the right-hand side of (4.14) with the aim to prove that φ = Φ. From (4.6) it follows that ! Z P
exp e(3) (P0 ) − P0
(3)
ωP∞ ,ˆν0 (x)
= ζ −1 + O(1) .
ζ→0
(4.15)
842
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
Using (3.30) we immediately see that φ has simple poles at µ ˆ(x) and P∞ , and simple zeros at νˆ0 (x) and νˆ(x). By (4.15) and a special case of Riemann’s vanishing theorem (Theorem A.22), we see that Φ has the same properties. Using the Riemann–Roch theorem (Theorem A.12), we conclude that the holomorphic function Φ/φ = c, a constant with respect to P . Using (4.15) and Lemma 4.1, one computes (1 + O(ζ))(ζ −1 + O(1)) Φ = = 1 + O(ζ) as P → P∞ , ζ→0 φ ζ→0 ζ −1 + O(ζ)
(4.16)
from which one concludes c = 1.
Similarly, the theta function representation of the Baker–Akhiezer function ψ(P, x, x0 ) is summarized in the following theorem. Theorem 4.4. Assume that the curve Km−1 is nonsingular. Let P = (z, y) ∈ Km−1 \{P∞ } and let x, x0 ∈ Ωµ , where Ωµ ⊆ C is open and connected. Suppose that Dµˆ (x) , or equivalently, Dνˆ(x) is nonspecial for x ∈ Ωµ . Then ψ(P, x, x0 ) =
θ(z(P, µ ˆ(x))) θ(z(P∞ , µ ˆ(x0 ))) θ(z(P∞ , µ ˆ(x))) θ(z(P, µ ˆ(x0 ))) Z × exp (x − x0 )
(2) e2 (P0 )
P
− P0
!! (2) ωP∞ ,2
.
(4.17)
Proof. Assume temporarily that e µ ⊆ Ωµ , µj (x) 6= µj 0 (x) for j 6= j 0 and x ∈ Ω
(4.18)
e µ is open and connected. For the Baker–Akhiezer function ψ we will use where Ω the same strategy as was used in the previous proof. However, the situation is slightly more involved in that ψ has an essential singularity at P∞ . Let Ψ denote the right-hand side of (4.17). In order to prove that ψ = Ψ, one first observes that since Z x dx0 φ(P, x0 )
ψ(P, x, x0 ) = exp
,
(4.19)
x0
the zeros and poles of ψ can come only from simple poles in the integrand (with positive and negative residues respectively). Using (3.28) and (3.40), one computes φ= =
Fm y 2 + Am y + 23 Fm Sm + 13 ε(m)Dm,x ε(m)Dm 1 3Am y + Fm Sm 1 Fm 1 Dm,x + 3y 2 + Sm + 3 ε(m)Dm 3 ε(m)Dm 3 Dm
X µk,x 1 m−1 2 Fm 2 3y + Sm − = + O(1) 3 ε(m)Dm 3 z − µk k=1
µj,x =− + O(1), as P → µ ˆj (x) . z − µj
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
843
More concisely, φ(P, x0 ) = Hence
Z
x
exp x0
∂ ln(z − µj (x0 )) + O(1) for P near µ ˆj (x0 ) . ∂x0
dx0
(4.20)
∂ 0 ln(z − µ (x )) + O(1) j ∂x0
(z − µj (x))O(1) = O(1) (z − µj (x0 ))−1 O(1)
for P near µ ˆj (x) 6= µ ˆ j (x0 ) , for P near µ ˆj (x) = µ ˆ j (x0 ) ,
(4.21)
for P near µ ˆj (x0 ) 6= µ ˆj (x) ,
where O(1) 6= 0 in (4.21). Consequently, all zeros of ψ and Ψ on Km−1 \{P∞ } are simple and coincide. It remains to identify the essential singularity of ψ and Ψ at P∞ . From (4.1), we infer Z x dx0 φ(P, x0 ) = (x − x0 )(ζ −1 + O(ζ)) as P → P∞ . (4.22) x0
ζ→0
Looking at (4.13) we see that this coincides with the singularity in the exponent of Ψ near P∞ . The uniqueness result in Lemma A.26 for Baker–Akhiezer functions then completes the proof that Ψ = ψ as both functions share the same singularities e µ to x ∈ Ωµ then simply and zeros. The extension of this result from x ∈ Ω follows from the continuity of αP0 and the hypothesis of Dµˆ (x) being nonspecial for x ∈ Ωµ . Next it is necessary to introduce two further polynomials Km and Lm with respect to the variable z ∈ C, Km (z, x) = (ε(m)Nm (z, x) − Jm (z, x)Cm (z, x))(Gm (z, x) + 2−1 Fm,x (z, x))−1 ,
(4.23)
Lm (z, x) = (ε(m)Dm−1 (z, x) − (Gm (z, x) − 2−1 Fm,x (z, x))Am (z, x))Fm (z, x)−1 .
(4.24)
In analogy to our polynomials Am –Nm introduced in (3.10)–(3.16), it is possible to derive explicit expressions of Km and Lm directly in terms of Fm and Gm and their x-derivatives. These expressions then prove, in particular, the polynomial character of Km and Lm with respect to z, but we here omit the rather lengthy formulas since they can be generated with the help of symbolic calculation programs such as Maple or Mathematica. Lemma 4.5. Let x ∈ C. Then µj (x)) , Lm (µj (x), x) = − Gm (µj (x), x) − 2−1 Fm,x (µj (x), x) y(ˆ
(4.25)
844
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
for j = 1, . . . , m − 1 and ν` (x)) , Km (ν` (x), x) = Jm (ν` (x), x)y(ˆ
(4.26)
for ` = 0, . . . , m − 1. The well-known linearization property of the Abel map for completely integrable systems of soliton-type, is next verified in the context of the Bsq hierarchy. Theorem 4.6. Assume that the curve Km−1 is nonsingular and let x, x0 ∈ C. Then (2)
αP0 (Dµˆ (x) ) = αP0 (Dµˆ (x0 ) ) + U 2 (x − x0 ) ,
(4.27) (2)
AP0 (ˆ ν0 (x)) + αP0 (Dνˆ(x) ) = AP0 (ˆ ν0 (x0 )) + αP0 (Dνˆ(x0 ) ) + U 2 (x − x0 ) . (4.28) Proof. We prove only (4.27) as (4.28) follows mutatis mutandis (or from (4.27) and Abel’s theorem, Theorem A.14). Assume temporarily that eµ ⊆ C , µj (x) 6= µj 0 (x) for j 6= j 0 and x ∈ Ω
(4.29)
e µ is open and connected. Then using (3.40), (B.7), and (B.9), one computes where Ω m−1 X d αP0 ,` (Dµˆ (x) ) = µj,x (x)ω` (ˆ µj (x)) dx j=1
= −ε(m)
m−n−1 X k=1
×
m−1 Y
e` (k)
m−1 X
µj (x)k−1 Fm (µj (x), x)
j=1
(µj (x) − µp (x))−1
p=1 p6=j
− ε(m)
n X
e` (k + m − n − 1)
k=1
×
m−1 Y
(µj (x) − µp (x))−1 .
m−1 X
µj (x)k−1 Am (µj (x), x)
j=1
(4.30)
p=1 p6=j
Next we consider the two cases m = 3n + 1 and m = 3n + 2 separately and substitute the polynomials Fm (µj (x), x) and Am (µj (x), x) in the variable µj (x) into (4.30). Using a standard Lagrange interpolation argument then yields ( m = 3n + 1 , e` (m − 1) , d αP0 ,` (Dµˆ (x) ) = − (4.31) dx e` (m − n − 1) , m = 3n + 2 . e µ , using (4.9), (4.31), (B.11), and (B.16). By The result now follows for x ∈ Ω e µ to x ∈ C. continuity of αP0 , this result extends from x ∈ Ω
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
845
We conclude this section with the theta function representations for the stationary Bsq solutions q0 , q1 (the analog of the Its–Matveev formula in the KdV context). Theorem 4.7. Assume that the curve Km−1 is nonsingular and let x ∈ Ωµ , where Ωµ ⊆ C is open and connected. Suppose that Dµˆ (x) , or equivalently, Dνˆ(x) is nonspecial for x ∈ Ωµ . Then ˆ(x)))) + (3/2)w , q0 (x) = 3 ∂U (2) ∂x ln(θ(z(P∞ , µ
(4.32)
ˆ(x)))) + 3u , q1 (x) = 3 ∂x2 ln(θ(z(P∞ , µ
(4.33)
3
with u and w defined in (4.11) and (4.12), that is, λm−1 − c(1) for m = 1 (mod 3) , 1 u= (2) λm−n−1 − (d0 )2 for m = 2 (mod 3) , and w=
λm−n−1 − 2d(1) 1
(2)
(2)
(4.34)
for m = 1 (mod 3) , (4.35)
(2)
(d0 )3 − c1 − d0 λm−n−1 + λm−1
for m = 2 (mod 3) .
Proof. Using Lemma 4.2 and Theorem 4.4, one can write ψ near P∞ in the coordinate ζ, as ψ(P, x, x0 ) = 1 + α1 (x)ζ + α2 (x)ζ 2 + O(ζ 3 ) ζ→0
× exp (x − x0 )(ζ −1 − uζ − 2−1 wζ 2 + O(ζ 3 )) as P → P∞ ,
(4.36)
where the terms α1 (x) and α2 (x) in (4.36) come from the Taylor expansion about P∞ of the ratios of the theta functions in (4.17), and the exponential term stems from substituting (4.13) into (4.17). Using (4.36) and its x-derivatives one can show that ψxxx + 3(u − α1,x)ψx + 3(2−1 w − α1,xx + α1 α1,x − α2,x )ψ − ζ −3 ψ = O(ζ)ψ . (4.37) Since O(ζ)ψ is another Baker–Akhiezer function with the same essential singularity at P∞ and the same divisor on Km−1 \{P∞ }, the uniqueness theorem for Baker– Akhiezer functions (cf. Lemma A.26) then yields O(ζ) = 0. Hence (4.38) q0 (x) = 3 2−1 w − 2−1 α1,xx (x) + α1 (x)α1,x (x) − α2,x (x) , q1 (x) = 3(u − α1,x (x)) ,
(4.39)
where ˆ(x))) , α1,x (x) = −∂x2 ln θ(z(P∞ , µ
(4.40)
ˆ(x))) . (4.41) −2−1 α1,xx (x) + α1 (x)α1,x (x) − α2,x (x) = ∂U (2) ∂x ln θ(z(P∞ , µ 3
846
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
Here ∂U (2) = 3
m−1 X
(2)
U3,j
j=1
∂ ∂zj
(4.42) (2)
denotes the directional derivative in the direction of the vector of b-periods U 3 , defined by Z 1 (2) (2) (2) (2) (2) ω , j = 1, . . . , m − 1 , (4.43) U 3 = (U3,1 , . . . , U3,m−1 ) , U3,j = 2πi bj P∞ ,3 (2)
with ωP∞ ,3 the dsk holomorphic on Km−1 \{P∞ } with a pole of order 3 at P∞ , (2)
ωP∞ ,3 (P ) =
ζ→0
ζ −3 + O(1) dζ as P → P∞ .
(4.44)
Combining (4.38)–(4.41) then proves (4.32) and (4.33).
For interesting spectral characterizations of third-order (in fact, odd-order) selfadjoint differential operators with quasi-periodic coefficients we refer to [26]. 5. The Time-Dependent Boussinesq Formalism In this section we return to the recursive approach outlined in Sec. 2 and briefly recall our treatment of the time-dependent Bsq hierarchy in [14]. (0) (0) We start with a stationary algebro-geometric solution (q0 (x), q1 (x)) associated with Km−1 satisfying (ε) −3 fn+1,x = 0, (0) (0) x ∈ C, m = 3n + ε (5.1) Bsqm (q0 , q1 ) = (ε) −3 gn+1,x = 0 , for some fixed ε ∈ {1, 2}, n ∈ N0 , and a given set of integration constants (ε) (ε) {c` }`=1,...,n , {d` }`=0,...,n . Our aim is to construct the rth Bsq flow Bsqr (q0 , q1 ) = 0 ,
(0)
(0)
(q0 (x, t0,r ), q1 (x, t0,r )) = (q0 (x), q1 (x)) ,
x ∈ C,
r = 3s + ε0 (5.2) for some fixed ε0 ∈ {1, 2}, s ∈ N0 , and t0,r ∈ C. In terms of Lax pairs this amounts to solving d L3 (tr ) − [Per (tr ), L3 (tr )] = 0 , d tr
tr ∈ C ,
(5.3)
[Pm (t0,r ), L3 (t0,r )] = 0 .
(5.4)
As a consequence one obtains [Pm (tr ), L3 (tr )] = 0 ,
tr ∈ C ,
Pm (tr )3 + Pm (tr ) Sm (L3 (tr )) − Tm (L3 (tr )) = 0 , since the Bsq flows are isospectral deformations of L3 (t0,r ).
(5.5) tr ∈ C ,
(5.6)
847
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY 0
0
(ε ) (ε ) (ε) We emphasize that the integration constants {˜ c` } and {d˜` } in Per , and {c` } (ε) and {d` } in Pm , are independent of each other (even for r = m). Hence we er , H e r , etc., in order to distinguish them from shall employ the notation Per , Fer , G Pm , Fm , Gm , Hm , etc. In addition we follow a more elaborate approach inspired by Hirota’s τ -function approach and indicate the individual rth Bsq flow by a separate time variable tr ∈ C. (The latter notation suggests considering all Bsq flows simultaneously by introducing t = (t1 , t2 , t4 , t5 , . . .).) Instead of working directly with (5.3) and (5.5) we find it preferable to take the following two equations as our point of departure (never mind their somewhat intimidating size), 1 e 5 5 2 2 e 3 e e q1,xx + q1 Fr,x q0,tr = − Fr,xxxxx − q1 Fr,xxx − q1,x Fr,xx − 6 6 4 4 3 2 1 e r,x + q0,x G er , q1,xxx + q1 q1,x Fer − 3(z − q0 ) G − (5.7) 6 3
e r,xxx + 2 q1 G er,x + q1,x G er − 3 (z − q0 ) Fer,x + 2 q0,x Fer , q1,tr = 2 G −
(x, tr ) ∈ C2 ,
1 1 1 2 5 Fm,xxxxFm + Fm,xxx Fm,x − Fm,xx − q1 Fm,xx Fm 6 6 12 6
5 5 2 q1,x Fm,x Fm + q1 Fm,x 12 12 1 1 2 2 q1,xx + q1 Fm − + 2 Gm,xxGm − G2m,x + q1 G2m 3 2 −
− 3 (z − q0 )Fm Gm = Sm (z) ,
(x, tr ) ∈ C2 ,
(5.8)
1 1 2 Fm,xxxxFm,xx Fm − Fm,xxxxFm,x 18 24 +
1 1 2 q1 Fm,xxxx Fm Fm,xxx Fm,xx Fm,x + 18 36
−
1 1 1 1 2 2 Fm Fm,xxx q1,x Fm,xxx Fm F3 − − q1 Fm,xxx Fm,x Fm − 36 18 9 108 m,xx
+
2 1 7 2 2 q1,x Fm,xx Fm,x Fm + q1,xx Fm,xx Fm q1 Fm,xx Fm,x − 9 18 72
+
5 2 7 1 2 2 2 q Fm,xx Fm q1 Fm,xx q1,xx Fm,x + Fm − Fm 18 1 36 24
7 1 1 3 2 2 q1,x Fm,x q1,x q1 Fm,x Fm − q12 Fm,x Fm + 48 6 12 1 2 1 2 3 3 q1 − q1,x + q1,xx q1 + (z − q0 )2 Fm + + (z − q0 )G3m 27 36 18 −
+
1 1 Fm,xxxxG2m − Fm,xxx Gm,x Gm + Fm G2m,xx 6 3
848
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
+
1 Fm,xx G2m,x + Gm,xx Gm − Fm,x Gm,xx Gm,x 3
2 − q1 (z − q0 )Fm Gm +
2 2 5 q Fm G2m + q1 Fm,xx G2m 3 1 6
−
4 1 7 q1 Fm,x Gm,x Gm + q1 Fm G2m,x + q1,x Fm,x G2m 3 3 12
+
4 1 1 q1 Fm Gm,xx Gm + q1,xx Fm G2m − q1,x Fm Gm,x Gm 3 6 3
+ (z − q0 )Fm,x Fm Gm,x −
1 2 (z − q0 )Fm,x Gm 4
2 − 2 (z − q0 )Fm Gm,xx = Tm (z) ,
(x, tr ) ∈ C2 ,
(5.9)
where (cf. (2.15), (2.16)) Fm (z, x, tr ) =
n X
(ε)
fn−` (x, tr )z ` ,
Fm (z, x, t0,r ) =
`=0
Gm (z, x, tr ) =
n X
n X
(ε),(0)
fn−` (x)z ` , (5.10)
`=0 (ε)
gn−` (x, tr )z ` ,
Gm (z, x, t0,r ) =
`=0
n X
(ε),(0)
gn−` (x)z `
(5.11)
`=0
for fixed t0,r ∈ C, m = 3n + ε, r = 3s + ε0 , n, s ∈ N0 , ε, ε0 ∈ {1, 2}. Here (ε) (ε) (ε),(0) (ε),(0) f` (x, tr ), g` (x, tr ) and f` (x), g` (x) are defined as in (2.3) with (q0 (x), (0) (0) q1 (x)) replaced by (q0 (x, tr ), q1 (x, tr )), and (q0 (x), q1 (x)), respectively. In analogy to (3.27) one introduces Dm−1 (z, x, tr ) =
m−1 Y
(z − µj (x, tr )) ,
Nm (z, x, tr ) =
j=1
m−1 Y
(z − ν` (x, tr )) , (5.12)
`=0
where Dm−1 and Nm are defined as in (3.13) and (3.16). This implies in particular (cf. (3.21)), Dm−1 (z, x, tr )Nm (z, x, tr ) = Bm (z, x, tr ) Em (z, x, tr ) − Tm (z)(Am (z, x, tr ) , × (Gm (z, x, tr ) + 2−1 Fm,x (z, x, tr )) − Fm (z, x, tr ) Cm (z, x, tr )) ,
(5.13)
and Am , Bm , Cm , Dm−1 , Em , Jm , and Nm are defined as in (3.10)–(3.16). Hence (3.18)–(3.23) also hold in the present context. Moreover, we recall Lemma 5.1 [14]. Assume (5.7)–(5.11) and let (z, x, tr ) ∈ C3 . Then er (z, x, tr ) − 1 Fer,x (z, x, tr ) (i) Dm−1,tr (z, x, tr ) = Dm−1,x (z, x, tr ) G 2 Fer (z, x, tr ) − Fm (z, x, tr )
!! 1 Gm (z, x, tr ) − Fm,x (z, x, tr ) 2
849
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
!
e e r (z, x, tr ) − Fr (z, x, tr ) Hm (z, x, tr ) . + Dm−1 (z, x, tr ) 3 H Fm (z, x, tr )
(5.14)
(ii) Nm,tr (z, x, tr ) e er (z, x, tr ) + 1 Fer,x (z, x, tr ) − Jr (z, x, tr ) = Nm,x (z, x, tr ) G 2 Jm (z, x, tr ) ×
!! 1 Gm (z, x, tr ) + Fm,x (z, x, tr ) 2
− Nm (z, x, tr ) q1 (x, tr ) Fer (z, x, tr ) + Fer,xx (z, x, tr ) ! Jer (z, x, tr ) q1 (x, tr ) Fm (z, x, tr ) + Fm,xx (z, x, tr ) . − Jm (z, x, tr )
(5.15)
Similarly, Lemma 3.1 remains valid and one obtains φ(P, x, tr ) =
(Gm (z, x, tr ) + 12 Fm,x (z, x, tr ))y(P ) + Cm (z, x, tr ) Fm (z, x, tr )y(P ) − Am (z, x, tr )
(5.16)
=
Fm (z, x, tr )y(P )2 + Am (z, x, tr )y(P ) + Bm (z, x, tr ) ε(m)Dm−1 (z, x, tr )
(5.17)
=
(Gm (z, x, tr ) +
1 2
−ε(m)Nm (z, x, tr ) , (5.18) Fm,x (z, x, tr ))y(P )2 − Cm (z, x, tr )y(P ) − Em (z, x, tr ) P = (z, y) ∈ Km−1 .
In analogy to (3.28) and (3.29) one then introduces (the analogs of) Dirichlet and Neumann data by Am (µj (x, tr ), x, tr ) ∈ Km−1 , µ ˆ j (x, tr ) = µj (x, tr ), Fm (µj (x, tr ), x, tr ) j = 1, . . . , m − 1 , (x, tr ) ∈ C2 , (5.19) Cm (ν` (x, tr ), x, tr ) ∈ Km−1 , νˆ` (x, tr ) = ν` (x, tr ), − Gm (ν` (x, tr ), x, tr ) + 12 Fm,x (ν` (x, tr ), x, tr ) ` = 0, . . . , m − 1 , (x, tr ) ∈ C2
(5.20)
and hence infers that the divisor (φ(P, x, tr )) of φ(P, x, tr ) is given by (φ(P, x, tr )) = Dνˆ0 (x,tr ),...,ˆνm−1 (x,tr ) (P ) − DP∞ ,ˆµ1 (x,tr ),...,ˆµm−1 (x,tr ) (P ) . (5.21)
850
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
Next we define the time-dependent BA-function ψ(P, x, x0 , tr , t0,r ) Z
x
ψ(P, x, x0 , tr , t0,r ) = exp
d x0 φ(P, x0 , tr ) +
Z
tr
d s Fer (z, x0 , s)
t0,r
x0
er (z, x0 , s) × φx (P, x0 , s) + φ(P, x0 , s)2 + (G 1 1 e Fr,xx (z, x0 , s) − Fer,x (z, x0 , s))φ(P, x0 , s) + 2 6 ! 2 e e + q1 (x0 , s)Fr (z, x0 , s) − Gr,x (z, x0 , s) , (5.22) 3 P ∈ Km−1 \{P∞ } ,
(x, tr ) ∈ C2 ,
with fixed (x0 , t0,r ) ∈ C2 . The following theorem recalls the basic properties of φ(P, x, tr ) and ψ(P, x, x0 , tr , t0,r ). Theorem 5.2 [14]. Assume (5.7)–(5.11), P = (z, y) ∈ Km−1 \{P∞ } and let (z, x, x0 , tr , t0,r ) ∈ C5 . Then (i) φ(P, x, tr ) satisfies φxx (P, x, tr ) + 3 φx (P, x, tr ) φ(P, x, tr ) + φ(P, x, tr )3 + q1 (x, tr ) φ(P, x, tr ) = z − q0 (x, tr ) − 2−1 q1,x (x, tr ),
(5.23)
er (z, x, tr ) φtr (P, x, tr ) = ∂x (Fer (z, x, tr )(φ(P, x, tr )2 + φx (P, x, tr )) + (G e r (z, x, tr )) . −2−1 Fer,x (z, x, tr ))φ(P, x, tr ) + H
(5.24)
(ii) ψ(P, x, x0 , tr , t0,r ) satisfies ψxxx (P, x, x0 , tr , t0,r ) + q1 (x, tr )ψx (P, x, x0 , tr , t0,r ) + (q0 (x, tr ) + 2−1 q1,x (x, tr ) − z)ψ(P, x, x0 , tr , t0,r ) = 0 ,
(5.25)
ψtr (P, x, x0 , tr , t0,r ) = (Fer (z, x, tr )(φ(P, x, tr )2 + φx (P, x, tr )) e r (z, x, tr ) − 2−1 Fer,x (z, x, tr ))φ(P, x, tr ) +(G e r (z, x, tr ))ψ(P, x, x0 , tr , t0,r ) +H
(5.26)
(i.e., (L3 − z)ψ = 0, (Pm − y)ψ = 0, ψtr = Per ψ) . (iii) φ(P, x, tr ) φ(P ∗ , x, tr ) φ(P ∗∗ , x, tr ) =
Nm (z, x, tr ) . Dm−1 (z, x, tr )
(iv) φ(P, x, tr ) + φ(P ∗ , x, tr ) + φ(P ∗∗ , x, tr ) =
Dm−1,x (z, x, tr ) . Dm−1 (z, x, tr )
(5.27) (5.28)
(v) y(P ) φ(P, x, tr ) + y(P ∗ ) φ(P ∗ , x, tr ) + y(P ∗∗ ) φ(P ∗∗ , x, tr ) =
3 Tm (z) Fm (z, x, tr ) − 2 Sm (z) Am (z, x, tr ) . ε(m)Dm−1 (z, x, tr )
(5.29)
851
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
(vi) ψ(P, x, x0 , tr , t0,r )ψ(P ∗ , x, x0 , tr , t0,r )ψ(P ∗∗ , x, x0 , tr , t0,r ) =
Dm−1 (z, x, tr ) . Dm−1 (z, x0 , t0,r )
(5.30)
(vii) ψx (P, x, x0 , tr , t0,r )ψx (P ∗ , x, x0 , tr , t0,r )ψx (P ∗∗ , x, x0 , tr , t0,r ) =
Nm (z, x, tr ) . Dm−1 (z, x0 , t0,r )
(viii) ψ(P, x, x0 , tr , t0,r ) Dm−1 (z, x, tr ) Dm−1 (z, x0 , t0,r )
=
(5.31)
!1/3
Z
x
exp
d x0 ε(m)Dm−1 (z, x0 , tr )−1
x0
" × Fm (z, x0 , tr ) y(P )2 + Am (z, x0 , tr ) y(P ) # 2 0 + Fm (z, x , tr ) Sm (z) 3 Z −
"
tr
ds
−1
ε(m)Dm−1 (z, x0 , s)
Fm (z, x0 , s)y(P )2
t0,r
# " 2 er (z, x0 , s) + Am (z, x0 , s) y(P ) + Fm (z, x0 , s) Sm (z) × G 3 1 − Fer,x (z, x0 , s) − 2
! 1 Gm (z, x0 , s) − Fm,x (z, x0 , s) 2
# !! Fer (z, x0 , s) Fer (z, x0 , s) + y(P ) . × Fm (z, x0 , s) Fm (z, x0 , s)
(5.32)
The dynamics of the zeros µj (x, tr ) and ν` (x, tr ) of Dm−1 (z, x, tr ) and Nm (z, x, tr ), in analogy to Lemma 3.3, are then described in terms of Dubrovin-type equations as follows. Lemma 5.3 [14]. Assume (5.7)–(5.11). (i) Suppose the zeros {µj (x, tr )}j=1,...,m−1 of Dm−1 ( · , x, tr ) remain distinct for (x, tr ) ∈ Ωµ , where Ωµ ⊆ C2 is open and connected. Then {µj (x, tr )}j=1,...,m−1 satisfy the system of differential equations, 3y(ˆ µj (x, tr ))2 + Sm (µj (x, tr )) µj,x (x, tr ) = −ε(m) Fm (µj (x, tr ), x, tr ) , m−1 Y (µj (x, tr ) − µk (x, tr )) k=1 k6=j
j = 1, . . . , m − 1 ,
(5.33)
852
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
er (µj (x, tr ), x, tr ) µj,tr (x, tr ) = −ε(m) Fm (µj (x, tr ), x, tr ) G − 2−1 Fer,x (µj (x, tr ), x, tr ) + Fer (µj (x, tr ), x, tr ) Gm (µj (x, tr ), x, tr ) − 2−1 Fm,x (µj (x, tr ), x, tr ) 3y(ˆ µj (x, tr ))2 + Sm (µj (x, tr )) , j = 1, . . . , m − 1 , (5.34) × m−1 Y µj (x, tr ) − µk (x, tr ) k=1 k6=j
with initial conditions {ˆ µj (x0 , t0,r )}j=1,...,m−1 ∈ Km−1 ,
(5.35)
for some fixed (x0 , t0,r ) ∈ Ωµ . The initial value problem (5.34), (5.35) has a unique solution satisfying µ ˆj ∈ C ∞ (Ωµ , Km−1 ) ,
j = 1, . . . , m − 1 .
(5.36)
(ii) Suppose the zeros {ν` (x, tr )}`=0,...,m−1 of Nm ( · , x, tr ) remain distinct for (x, tr ) ∈ Ων , where Ων ⊆ C2 is open and connected. Then {ν` (x, tr )}`=0,...,m−1 satisfy the system of differential equations, 3y(ˆ ν` (x, tr ))2 + Sm (ν` (x, tr )) ν`,x (x, tr ) = −ε(m) Jm (ν` (x), x, tr ) , m−1 Y (ν` (x, tr ) − νk (x, tr )) k=0 k6=`
` = 0, . . . , m − 1 ,
(5.37)
e r (ν` (x, tr ), x, tr ) ν`,tr (x, tr ) = −ε(m) Jm (ν` (x, tr ), x, tr ) G + 2−1 Fer,x (ν` (x, tr ), x, tr ) − Jer (ν` (x, tr ), x, tr ) Gm (ν` (x, tr ), x, tr ) + 2−1 Fm,x (ν` (x, tr ), x, tr ) 3y(ˆ ν` (x, tr ))2 + Sm (ν` (x, tr )) , ` = 0, . . . , m − 1 , × m−1 Y (ν` (x, tr ) − νk (x, tr )) k=0 k6=`
(5.38) with initial conditions {ˆ ν` (x0 , t0,r )}`=0,...,m−1 ∈ Km−1 ,
(5.39)
for some fixed (x0 , t0,r ) ∈ Ων . The initial value problem (5.38), (5.39) has a unique solution satisfying νˆ` ∈ C ∞ (Ων , Km−1 ) ,
` = 0, . . . , m − 1 .
(5.40)
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
853
(iii) The initial condition (0)
(0)
x∈C
(5.41)
j = 1, . . . , m − 1 ,
x ∈ C,
(5.42)
` = 0, . . . , m − 1 ,
x∈C
(5.43)
(q0 (x, t0,r ), q1 (x, t0,r )) = (q0 (x), q1 (x)) , effects (0)
ˆj (x) , µ ˆj (x, t0,r ) = µ (0)
νˆ` (x, t0,r ) = νˆ` (x) , (cf. (5.10)–(5.12)).
6. Time-Dependent Algebro-Geometric Solutions of the Boussinesq Hierarchy In our final and principal section we extend the results of Sec. 4 from the stationary Bsq hierarchy, to the time-dependent case. In particular, we obtain Riemann theta function representations for the time-dependent Baker-Akhiezer function and the time-dependent meromorphic function φ. We finish this section with the corresponding theta function representation for general time-dependent algebro-geometric quasi-periodic Bsq solutions q0 , q1 . We start with the theta function representation of our fundamental object φ(P, x, tr ). Theorem 6.1. Assume that the curve Km−1 is nonsingular. Furthermore, let P = (z, y) ∈ Km−1 \{P∞ }, and let (x, tr ), (x0 , t0,r ) ∈ Ωµ , where Ωµ ⊆ C2 is open and connected. Suppose also that Dµˆ(x,tr ) , or equivalently, Dνˆ(x,tr ) is nonspecial for (x, tr ) ∈ Ωµ . Then φ(P, x, tr ) =
θ(z(P∞ , µ ˆ(x, tr ))) θ(z(P, νˆ(x, tr ))) θ(z(P∞ , νˆ(x, tr ))) θ(z(P, µ ˆ (x, tr ))) Z
P
× exp e(3) (P0 ) − P0
(3) ωP∞ ,ˆν0 (x,tr )
! .
(6.1)
Proof. The proof carries over ad verbatim from the stationary case, Theorem 4.3. Let ωP∞ ,r , r = 3s + ε0 , ε0 ∈ {1, 2}, s ∈ N0 , be the normalized dsk holomorphic on Km−1 \{P∞ }, with a pole of order r at P∞ , (2)
ωP∞ ,r (P ) = (ζ −r + O(1))dζ as P → P∞ , (2)
ζ→0
r = 3s + ε0 , ε0 ∈ {1, 2}, s ∈ N0 .
(6.2)
Furthermore, define the normalized dsk e (2) Ω P∞ ,r+1 =
s X `=0
(ε0 )
(2)
c˜s−` (3` + 2) ωP∞ ,3`+3 +
s X
0
(ε ) (2) d˜s−` (3` + 1) ωP∞ ,3`+2 ,
`=0
r = 3s + ε0 , ε0 ∈ {1, 2}, s ∈ N0 ,
(6.3)
854
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
where (cf. 2.3))
( (ε0 ) (ε0 ) (˜ c0 , d˜0 )
=
for ε0 = 1 ,
(0, 1) (2) (1, d˜0 )
0
for ε = 2 ,
(2) d˜0 ∈ C .
(6.4)
e (2) In addition, we define the vector of b-periods of the dsk Ω P∞ ,r+1 Z e (2) , . . . , U e (2) e (2) = 1 e (2) = (U e (2) U , j = 1, . . . , m − 1 U Ω r+1 r+1,1 r+1,m−1 ) , r+1,j 2πi bj P∞ ,r+1 r = 3s + ε0 , ε0 ∈ {1, 2}, s ∈ N0 .
(6.5)
Motivated by the second integrand in (5.22) one defines the function Ir (P, x, tr ), meromorphic on Km−1 × C2 by Ir (P, x, tr ) = Fer (z, x, tr )(φx (P, x, tr ) + φ(P, x, tr )2 ) e r (z, x, tr ) , er (z, x, tr ) − 2−1 Fer,x (z, x, tr ))φ(P, x, tr ) + H + (G
(6.6)
for r = 3s + ε0 , ε0 ∈ {1, 2}, s ∈ N0 . Denote by Ibr (P, x, tr ) the associated homogeer , H e r by the corresponding homogeneous polynomials neous quantity replacing Fer , G be be be F r , Gr , H r . Theorem 6.2. Let r = 3s + ε0 , ε0 ∈ {1, 2}, ζ = z −1/3 be the local coordinate near P∞ . Then
s ∈ N0 , (x, tr ) ∈ C2 , and
Ibr (P, x, tr ) = ζ −r + O(ζ) as P → P∞ . ζ→0
(6.7)
Proof. One easily verifies (6.7) by direct computation for r = 1 and r = 2. Assume (6.7) is true with r = 3s + ε0 , ε0 ∈ {1, 2}, s ∈ N0 . Then one may rewrite (6.7) as ∞ X δj (x, tr ) ζ j as P → P∞ , (6.8) Ibr (P, x, tr ) = ζ −r + ζ→0
j=1
for some coefficients {δj (x, tr )}j∈N . Compare coefficients of ζ in (4.1) and (6.8) by means of (5.24) and (6.6) to obtain 1 δ1,x (x, tr ) = − q1,tr (x, tr ) , 3
(6.9)
δ2,x (x, tr ) =
1 1 q1,tr x (x, tr ) − q0,tr (x, tr ) , 6 3
(6.10)
δ3,x (x, tr ) =
1 1 q0,tr x (x, tr ) − q1,tr xx (x, tr ) . 3 18
(6.11)
From (2.34) one infers (ε0 )
δ1 (x, tr ) = γ1 (tr ) − fˆs+1 (x, tr ) , (ε0 )
(6.12) (ε0 )
δ2 (x, tr ) = γ2 (tr ) + 2−1 fˆs+1,x (x, tr ) − gˆs+1 (x, tr ) , (ε0 )
(ε0 )
δ3 (x, tr ) = γ3 (tr ) − 6−1 fˆs+1,xx (x, tr ) + gˆs+1,x (x, tr ) ,
(6.13) (6.14)
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ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
where γ1 (tr ), γ2 (tr ), and γ3 (tr ) are integration constants. Next we note that the coefficients of the power series for φ(P, x, tr ) in the coordinate ζ near P∞ (cf. b Lemma 4.1), and the coefficients of the homogeneous polynomials Fe r (ζ, x, tr ) and b be e G r (ζ, x, tr ), (and hence those of H r (ζ, x, tr )) are differential polynomials in q0 and q1 , with no arbitrary integration constants in their construction. From the definition of Ibr in (6.6) it follows that it also can have no arbitrary integration constants, and must consist purely of differential polynomials in q0 and q1 . From these considerations it follows that γ1 (tr ) = γ2 (tr ) = γ3 (tr ) = 0. Hence one concludes (ε0 ) (ε0 ) (ε0 ) Ibr (P, x, tr ) = ζ −r − fˆs+1 ζ + 2−1 fˆs+1,x (x, tr ) − gˆs+1 (x, tr ) ζ 2 ζ→0
0 (ε ) (ε0 ) + gˆs+1,x (x, tr ) − 6−1 fˆs+1,xx (x, tr ) ζ 3 + O(ζ 4 ) as P → P∞ , (ε0 )
(6.15)
(ε0 )
where the functions fs (x, tr ) and gs (x, tr ) are defined as in (2.3) with (q0 (x), q1 (x)) replaced by (q0 (x, tr ), q1 (x, tr )). We note that one may write b b (ε0 ) Fe r+3 (ζ, x, tr ) = ζ −3 Fe r (ζ, x, tr ) + fˆs+1 (x, tr ),
(6.16)
be be with analogous expressions for G r and H r . It follows that 0
(ε ) Ibr+3 (P, x, tr ) = ζ −3 Ibr (P, x, tr ) + fˆs+1 (x, tr ) φx (P, x, tr ) + φ(P, x, tr )2 1 (ε0 ) (ε0 ) + gˆs+1 (x, tr ) − fˆs+1,x (x, tr ) φ(P, x, tr ) 2
+
2 1 ˆ(ε0 ) (ε0 ) (ε0 ) fs+1,xx (x, tr ) + q1 (x, tr )fˆs+1 (x, tr ) − gˆs+1,x (x, tr ) . (6.17) 6 3
Using Lemma 4.1 and (6.15), (6.17) yields Ibr+3 (P, x, tr ) = ζ −r−3 + O(ζ) as P → P∞ ,
(6.18)
ζ→0
and the result follows by induction. By (2.18) one infers Ir =
s X
0
(ε ) c˜s−` Ib3`+2 +
`=0
s X
0
(ε ) d˜s−` Ib3`+1 ,
r = 3s + ε0 , ε0 ∈ {1, 2}, s ∈ N0 .
(6.19)
`=0
Thus, Z
tr
t0,r
s X (ε0 ) c˜s−` Ir (P, x, τ )dτ = (tr − t0,r ) ζ→0
`=0
1 ζ 3`+2
+ O(ζ) as P → P∞ .
0
(ε ) + d˜s−`
1
ζ 3`+1 (6.20)
856
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
Furthermore, integrating (6.3) yields Z
P
P0
e (2) Ω P∞ ,r+1 =
ζ→0
s X
Z
(ε0 )
c˜s−` (3` + 2)
`=0
= −
s X
(ε0 )
c˜s−`
`=0
1 ζ 3`+2
ζ
ζ0
−
dξ ξ 3`+3
s X
0
+
Z
0
(ε ) d˜s−` (3` + 1)
ζ
ζ0
`=0
(ε ) d˜s−`
`=0
s X
dξ ξ 3`+2
1 ζ 3`+1
(2)
+ er+1 (P0 ) + O(ζ) as P → P∞ ,
(6.21)
(2)
where er+1 (P0 ) is a constant that arises from evaluating all the integrals at their lowers limits P0 , and summing accordingly. Combining (6.20) and (6.21) yields ! Z P Z tr (2) (2) e + O(ζ) as P → P∞ . Ir (P, x, s)ds = (tr − t0,r ) e (P0 ) − Ω t0,r
ζ→0
r+1
P0
P∞ ,r+1
(6.22) Given these preparations, the theta function representation of ψ(P, x, x0 , tr , t0,r ) reads as follows. Theorem 6.3. Assume that the curve Km−1 is nonsingular. Furthermore, let P = (z, y) ∈ Km−1 \{P∞ }, and let (x, tr ), (x0 , t0,r ) ∈ Ωµ , where Ωµ ⊆ C2 is open and connected. Suppose also that Dµˆ(x,tr ) , or equivalently, Dνˆ(x,tr ) is nonspecial for (x, tr ) ∈ Ωµ . Then ψ(P, x, x0 , tr , t0,r ) =
θ(z(P, µ ˆ (x, tr ))) θ(z(P∞ , µ ˆ(x0 , t0,r ))) θ(z(P∞ , µ ˆ(x, tr ))) θ(z(P, µ ˆ (x0 , t0,r ))) Z × exp (x − x0 )
(2) e2 (P0 )
P
− P0
Z + (tr − tr,0 )
(2) er+1 (P0 )
P
− P0
!
(2) ωP∞ ,2
!! e (2) Ω P∞ ,r+1
.
(6.23)
Proof. We present only a proof of the time variation here, and refer the reader to Theorem 4.4 for the argument concerning the space variation. Let ψ(P, x, x0 , tr , t0,r ) be defined as in (5.22) and denote the right-hand side of (6.23) by Ψ(P, x, x0 , tr , t0,r ). Temporarily assume that e µ ⊆ Ωµ , µj (x, tr ) 6= µ0j (x, tr ) for j 6= j 0 and (x, tr ) ∈ Ω
(6.24)
e µ is open and connected. In order to prove that ψ = Ψ one uses (5.17), where Ω (5.14), the time-dependent analog of (3.19), and Fm (φx + φ2 ) + (Gm − 2−1 Fm,x )φ + Hm = y ,
(6.25)
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ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
to compute er er − 1 Fer,x φ + H Ir = Fer (φx + φ2 ) + G 2 1 e r − 1 Fer,x − Fer Gm − 1 Fm,x e r − Fer Hm ) + Fm G y Fer + (Fm H φ = Fm 2 2 1 1e 1 1 Dm,tr e e y Fr + Fm (Gr − Fr,x ) − Fr Gm − Fm,x + = 3 Dm Fm 2 2 2 −1 × Fm y 2 + Am y + Fm Sm ε(m)Dm 3 =
er − 1 Fer,x ) − Fer (Gm − 1 Fm,x ) X µj,t 1 m−1 y Fer 2 Fm (G r 2 2 + 3y 2 + Sm − 3 ε(m)Dm 3 z − µk Fm k=1
=−
µj,tr y Fer µj,tr + + O(1) = − + O(1) z − µj Fm z − µj
(6.26)
as P → µ ˆj (x, tr ). More concisely, Ir (P, x0 , s) =
∂ ln(z − µj (x0 , s)) + O(1) for P near µ ˆj (x0 , tr ) . ∂s
Hence Z
tr
ds
exp t0,r
(6.27)
! ∂ ln(z − µj (x0 , s)) + O(1) ∂s
(z − µj (x0 , tr ))O(1) = O(1) (z − µj (x0 , t0,r ))−1 O(1)
for P near µ ˆj (x0 , tr ) 6= µ ˆ j (x0 , t0,r ) , ˆj (x0 , t0,r ) , for P near µ ˆj (x0 , tr ) = µ
(6.28)
for P near µ ˆj (x0 , t0,r ) 6= µ ˆ j (x0 , tr ) ,
where O(1) 6= 0 in (6.28). Consequently, all zeros and poles of ψ and Ψ on Km−1 \ {P∞ } are simple and coincide. It remains to identify the essential singularity of ψ and Ψ at P∞ . By (6.22) we see that the singularities in the exponential terms of ψ and Ψ coincide. The uniqueness result in Lemma A.26 for Baker–Akhiezer e µ . The extension of the result from functions completes the proof that ψ = Ψ on Ω e µ to (x, tr ) ∈ Ωµ follows from the continuity of αP and the hypothesis (x, tr ) ∈ Ω 0 that Dµˆ (x,tr ) is nonspecial for (x, tr ) ∈ Ωµ . The straightening out of the Bsq flows by the Abel map is contained in our next result. Theorem 6.4. Assume that the curve Km−1 is nonsingular, and let (x, tr ), (x0 , t0,r ) ∈ C2 . Then (2) e (2) (tr − t0,r ) , αP0 (Dµˆ(x,tr ) ) = αP0 (Dµˆ (x0 ,t0,r ) ) + U 2 (x − x0 ) + U r+1
(6.29)
858
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
and ν0 (x, tr )) + αP0 (Dνˆ(x,tr ) ) = AP0 (ˆ ν0 (x0 , t0,r )) + αP0 (Dνˆ(x0 ,t0,r ) ) AP0 (ˆ (2)
e + U 2 (x − x0 ) + U r+1 (tr − t0,r ) . (2)
(6.30)
Proof. As in the context of Theorem 4.6, it suffices to prove (6.29). Temporarily assume that Dµˆ (x,tr ) is nonspecial for (x, tr ) ∈ Ωµ ⊆ C2 , where Ωµ is open and connected. Introduce the meromorphic differential Ω(x, x0 , tr , t0,r ) =
∂ ln(ψ( · , x, x0 , tr , t0,r )) dz . ∂z
(6.31)
From the representation (6.23) one infers (2) e (2) Ω(x, x0 , tr , t0,r ) = −(x − x0 )ωP∞ ,2 − (tr − t0,r )Ω P∞ ,r+1
−
m−1 X
(3)
ωµˆj (x0 ,t0,r ),ˆµj (x,tr ) + ω ,
(6.32)
j=1
P where ω denotes a holomorphic differential on Km−1 , that is, ω = m−1 j=1 ej ωj for some ej ∈ C, j = 1, . . . , m − 1. Since ψ( · , x, x0 , tr , t0,r ) is single-valued on Km−1 , all a- and b-periods of Ω are integer multiples of 2πi and hence Z Z Ω(x, x0 , tr , t0,r ) = ω = ek , j = 1, . . . , m − 1 (6.33) 2πimk = ak
ak
for some mk ∈ Z. Similarly, for some nk ∈ Z, Z Z Z (2) e (2) 2πink = Ω(x, x0 , tr , t0,r ) = −(x − x0 ) ωP∞ ,2 − (tr − t0,r ) Ω P∞ ,r+1 bk
−
bk
m−1 XZ j=1
(3)
bk
ωµˆj (x0 ,t0,r ),ˆµj (x,tr ) + 2πi Z
= −(x − x0 ) bk
(2) ωP∞ ,2
− (tr − t0,r )
m−1 X Z µˆj (x0 ,t0,r ) j=1
m−1 X
Z mj
bk
ωk + 2πi
µ ˆ j (x,tr )
m−1 X
ωj bk
j=1
Z
− 2πi
bk
e (2) Ω P∞ ,r+1 Z mj
ωj bk
j=1
e = −2πi(x − x0 ) U2,k − 2πi(tr − t0,r ) U ˆ (x,tr ) ) r+1,k + 2πiαP0 ,k (Dµ (2)
− 2πiαP0 ,k (Dµˆ (x0 ,t0,r ) ) + 2πi
(2)
m−1 X
mj τj,k ,
(6.34)
j=1
where we used (A.36). By symmetry of τ (see Theorem A.4) this is equivalent to (2) e (2) (tr − t0,r ) , αP0 (Dµˆ(x,tr ) ) = αP0 (Dµˆ (x0 ,t0,r ) ) + U 2 (x − x0 ) + U r+1
(6.35)
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
859
for (x, tr ) ∈ Ωµ . This result extends from (x, tr ) ∈ Ωµ to (x, tr ) ∈ C2 using the continuity of αP0 and the fact that positive nonspecial divisors are dense in the space of positive divisors (cf. [19], p. 95). Our principal result, the theta function representation of the class of timedependent algebro-geometric quasi-periodic Bsq solutions now quickly follows from the material prepared thus far. Theorem 6.5. Assume that the curve Km−1 is nonsingular and let (x, tr ) ∈ Ωµ , where Ωµ ⊆ C2 is open and connected. Suppose also that Dµˆ (x,tr ) , or equivalently, Dνˆ(x,tr ) is nonspecial for (x, tr ) ∈ Ωµ . Then q0 (x, tr ) = 3 ∂U (2) ∂x ln(θ(z(P∞ , µ ˆ(x, tr )))) + (3/2)w ,
(6.36)
ˆ(x, tr )))) + 3u , q1 (x, tr ) = 3 ∂x2 ln(θ(z(P∞ , µ
(6.37)
3
where u and w are defined by (4.34) and (4.35), respectively, and ∂U (2) denotes the 3 directional derivative introduced in (4.42). Proof. The proof carries over ad verbatim from the stationary case, Theorem 4.7. Appendix A. Algebraic Curves and Their Theta Functions in a Nutshell This appendix treats some of the basic aspects of complex algebraic curves and their theta functions as used at numerous places in this paper. The material below is standard (see, e.g. [7, 19, 28, 32, 41]), and we include it for two major reasons: On the one hand it allows us to introduce a large part of the notation used in Secs. 4 and 6 (which otherwise would take up considerable space and disrupt the flow of arguments in these sections) and on the other hand, it permits a fairly self-contained presentation of the Bsq hierarchy and its algebro-geometric solutions in this paper. Definition A.1. An affine plane (complex) algebraic curve K is the locus of zeros in C2 of a (nonconstant) polynomial F (z, y) in two variables. The polynomial F is called nonsingular at a root (z0 , y0 ) if ∇F(z0 , y0 ) = (Fz (z0 , y0 ), Fy (z0 , y0 )) 6= 0 .
(A.1)
The affine plane curve K of roots of F is called nonsingular at P0 = (z0 , y0 ) if F is nonsingular at P0 . The curve K is called nonsingular, or smooth, if it is nonsingular at each of its points. The Implicit Function Theorem allows one to conclude that a smooth affine curve K is locally a graph and to introduce complex charts on K as follows. If F(P0 ) = 0 with Fy (P0 ) 6= 0, there is a holomorphic function gP0 (z) such that in
860
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
a neighborhood UP0 of P0 , the curve K is characterized by the graph y = gP0 (z). Hence the projection ˜z (UP0 ) ⊂ C , π ˜z : UP0 → π
(z, y) 7→ z ,
(A.2)
yields a complex chart on K. If, on the other hand, F (P0 ) = 0 with Fz (P0 ) 6= 0, then the projection ˜y (UP0 ) ⊂ C , π ˜y : UP0 → π
(z, y) 7→ y ,
(A.3)
defines a chart on K. In this way, as long as K is nonsingular, one arrives at a complex atlas on K. The space K ⊂ C2 is second countable and Hausdorff. In order to obtain a Riemann surface one needs connectedness of K which is implied by adding the assumption of irreducibility of the polynomial F . Thus, K equipped with charts (A.2) and (A.3) is a Riemann surface if F is nonsingular and irreducible. Affine plane curves K are unbounded as subsets of C2 , and hence noncompact. The compactification of K is conveniently described in terms of the projective plane CP2 , the set of all one-dimensional (complex) subspaces of C3 . In order to simplify notations, we temporarily abbreviate x0 = x, x1 = y, and x2 = z. Moreover, we denote the linear span of (x2 , x1 , x0 ) ∈ C3 \{0} by [x2 : x1 : x0 ]. In particular, [x2 : x1 : x0 ] ∈ CP2 with L∞ = {[x2 : x1 : x0 ] ∈ CP2 | x0 = 0} representing the line at infinity. Since the homogeneous coordinates [x2 : x1 : x0 ] satisfy (A.4) [x2 : x1 : x0 ] = [cx2 : cx1 : cx0 ] , c ∈ C\{0} , the space CP2 can be viewed as the quotient space of C3 \{0} by the multiplicative action of C\{0}, that is, CP2 = (C3 \{0})/(C\{0}), and hence CP2 inherits a Hausdorff topology which is the quotient topology induced by the natural map ι: C3 \{0} → CP2 ,
(x2 , x1 , x0 ) 7→ [x2 : x1 : x0 ] .
(A.5)
Next, define the open sets U m = {[x2 : x1 : x0 ] ∈ CP2 | xm 6= 0} ,
Then f :U → C , 0
0
2
[x2 : x1 : x0 ] 7→
m = 0, 1, 2 . x2 x1 , x0 x0
(A.6)
(A.7)
with inverse (f 0 )−1 : C2 → U 0 ,
(x2 , x1 ) 7→ [x2 : x1 : 1] ,
(A.8)
and analogously for functions f 1 and f 2 (relative to sets U 1 and U 2 , respectively), are homeomorphisms. In particular, U 0 , U 1 , and U 2 together cover CP2 . Moreover, CP2 is compact since it is covered by the closed unit (poly)disks in U 0 , U 1 , and U 2 . Let P be a (nonconstant) homogeneous polynomial of degree d in (x2 , x1 , x0 ), that is, (A.9) P(cx2 , cx1 , cx0 ) = cd P(x2 , x1 , x0 ) ,
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
861
and introduce K = {[x2 : x1 : x0 ] ∈ CP2 | P(x2 , x1 , x0 ) = 0} .
(A.10)
The set K is well defined (even though P(u, v, w) is not for [u : v : w] ∈ CP2 ) and closed in CP2 . The intersections, Km = K ∩ U m ,
m = 0, 1, 2
(A.11)
are affine plane curves when transported to C2 , that is, K0 ∼ = {(x2 , x1 ) ∈ C2 | P(x2 , x1 , 1) = 0}
(A.12)
represents the affine curve F (z, y) = 0, where F (x2 , x1 ) = P(x2 , x1 , 1), and analogously for K1 and K2 . We recall that F (x2 , x1 ) is irreducible if and only if P(x2 , x1 , x0 ) is irreducible. Given the affine curve defined by F (x2 , x1 ) = 0, the associated homogeneous polynomial P(x2 , x1 , x0 ) can be obtained from x2 x1 d , (A.13) P(x2 , x1 , x0 ) = x0 F , x0 x0 where d denotes the degree of F (and P). The element [x2 : x1 : 0] ∈ CP2 represents the point at infinity along the direction x2 : x1 in C2 (identifying [x2 : x1 : 0] ∈ CP2 and [x2 : x1 ] ∈ CP1 ). The set of all such elements then represents the line at infinity, L∞ , and yields the compactification CP2 of C2 . In other words, CP2 ∼ = C2 ∪ L∞ , CP1 ∼ = C ∪ {∞}, and 1 ∼ L∞ = CP . The projective plane curve K then intersects L∞ in a finite number of points (the points at infinity). Definition A.2. A projective plane (complex) algebraic curve K is the locus of zeros in CP2 of a homogeneous polynomial P in three variables. A homogeneous (nonconstant) polynomial P(x2 , x1 , x0 ) is called nonsingular if there are no common solutions (x2,0 , x1,0 , x0,0 ) ∈ C3 \{0} of P(x2,0 , x1,0 , x0,0 ) = 0 ,
(A.14)
∇P(x2,0 , x1,0 , x0,0 ) = (Px2 , Px1 , Px0 )(x2,0 , x1,0 , x0,0 ) = 0 .
(A.15)
The set K is called a smooth projective plane curve (of degree d ∈ N) if P is nonsingular (and of degree d ∈ N). One verifies that the homogeneous polynomial P(x2 , x1 , x0 ) is nonsingular if and only if each Km is a smooth affine plane curve in C2 . Moreover, any nonsingular homogeneous polynomial P(x2 , x1 , x0 ) is irreducible and consequently each Km is a Riemann surface for m = 0, 1, 2. The coordinate charts on each Km are simply the projections, that is, x2 /x0 and x1 /x0 for K0 , x2 /x1 and x0 /x1 for K1 , and finally, x1 /x2 and x0 /x2 for K2 . These separate complex structures on Km are compatible on K and hence induce a complex structure on K.
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R. DICKSON, F. GESZTESY and K. UNTERKOFLER
The zero locus in CP2 of a nonsingular homogeneous polynomial P(x2 , x1 , x0 ) defines a smooth projective plane curve K which is a compact Riemann surface. Topologically, this Riemann surface is a sphere with g handles where g = (d − 1)(d − 2)/2 ,
(A.16)
with d the degree of P(x2 , x1 , x0 ). In particular, K has topological genus g and we indicate this by writing Kg in our main text, or simply Kg if no confusion can arise. In general, the projective curve Kg can be singular even though the associated affine curve Kg0 is nonsingular. In this case one has to account for the singularities at infinity and properly amend the genus formula (A.16) according to results of Clebsch, Noether, and Pl¨ ucker. If Kg is a nonsingular projective curve, associated with the homogeneous polynomial P(z, y, x) of degree d, the set of finite branch points of Kg is given by {[z : y : 1] ∈ CP2 | P(z, y, 1) = Py (z, y, 1) = 0} .
(A.17)
Similarly, branch points at infinity are defined by {[z : y : 0] ∈ CP2 | P(z, y, 0) = Py (z, y, 0) = 0} .
(A.18)
The set of branch points B of Kg then being the union of points in (A.17) and (A.18). Given B = {P1 , . . . , Pr } one can cut the complex plane along smooth nonintersecting curves Cq (e.g., straight lines if P1 , . . . , Pr are arranged suitably) connecting Pq and Pq+1 for q = 1, . . . , r − 1, and defines holomorphic functions f1 , . . . , fd on the cut plane Π = C\ ∪r−1 q=1 Cq such that P(z, y, 1) = 0 for y ∈ Π if and only if y = fj (z) for some j ∈ {1, . . . , d} .
(A.19)
This yields a topological construction of Kg by appropriately gluing together d copies of the cut plane Π, the result being a sphere with g handles (g depending on the order of the branch points in B). If Kg is singular, this procedure requires appropriate modifications. Next, choose a homology basis {aj , bj }gj=1 on Kg for some g ∈ N in such a way that the intersection matrix of the cycles satisfies aj ◦ bk = δj,k ,
j, k = 1, . . . , g
(A.20)
(with aj and bk intersecting to form a right-handed coordinate system). Turning briefly to meromorphic differentials (1-forms) on Kg , we state the following result. Theorem A.3 (Riemann’s period relations). Let g ∈ N and suppose ω and ν to be closed differentials (1-forms) on Kg . Then (i) ZZ ω∧ν = Kg
g X j=1
! Z
Z ω aj
! ν
bj
! Z
Z −
ω bj
!! ν
aj
.
(A.21)
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
If, in addition ω and ν are holomorphic 1-forms on Kg , then ! Z ! ! Z !! Z Z g X ω ν − ω ν = 0. j=1
aj
bj
bj
aj
(A.22)
aj
(ii) If ω is a nonzero holomorphic 1-form on Kg , then ! Z ! Z g X ω ω > 0. Im j=1
863
(A.23)
bj
The proof of Theorem A.3 is usually based on Stokes’ theorem and a canonical b g of the dissection of Kg along its cycles yielding the simply connected interior K b fundamental polygon ∂ Kg given by b g = a1 b1 a−1 b−1 a2 b2 a−1 b−1 . . . a−1 b−1 . ∂K g g 1 1 2 2
(A.24)
Given the cycles {aj , bj }gj=1 , we denote by {ωj }gj=1 a normalized basis of the space of holomorphic differentials (also called Abelian differentials of the first kind, denoted dfk) on Kg , that is, Z ωk = δj,k , j, k = 1, . . . , g . (A.25) aj
The b-periods of ωk are then defined by Z ωk , τj,k =
j, k = 1, . . . , g .
(A.26)
bj
Theorem A.3 then implies the following result. Theorem A.4. The matrix τ is symmetric, that is, τj,k = τk,j ,
j, k = 1, . . . , g ,
(A.27)
with a positive definite imaginary part, Im(τ ) = (τ − τ ∗ )/(2i) > 0 .
(A.28)
Abelian differentials of the second kind (abbreviated dsk), say ω (2) , are characterized by the property that all their residues vanish. They are normalized by the vanishing of all their a-periods (achieved by adding a suitable linear combination of dfk’s) Z ω (2) = 0 ,
j = 1, . . . , g ,
(A.29)
aj
which determines them uniquely. (We will always assume that the poles of dsk’s b g , that is, do not lie on ∂ K b g . This can always be achieved by an on Kg lie in K appropriate choice of the cycles aj and bj .) We may add in this context that the
864
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
sum of the residues of any meromorphic differential ν on Kg vanishes, the residue at a pole Q0 ∈ Kg of ν being defined by Z 1 ν, (A.30) resQ0 (ν) = 2πi γQ0 where γQ0 is a smooth, simple, closed contour, oriented counter-clockwise, encircling Q0 , but no other pole of ν. (2)
Theorem A.5. Let g ∈ N. Assume ωQ1 ,n to be a dsk on Kg , whose only pole b g with principal part ζ −n dζQ1 for some n ∈ N0 and ω (1) a dfk on Kg of is Q1 ∈ K Q1 P∞ m the type ω (1) = m=0 cm (Q1 )ζQ dζQ1 near Q1 . Then 1 ! Z ! ! Z !! Z Z g X 2πi (2) (2) (1) (1) cn−2 (Q1 ) , = ω ωQ1 ,n − ω ωQ1 ,n (n − 1) aj bj bj aj j=1 n ≥ 2.
(A.31)
P∞ (2) m In particular, if ωQ1 ,n is normalized and ω (1) = ωj = m=0 cj,m (Q1 )ζQ dζQ1 , then 1 Z 2πi (2) cj,n−2 (Q1 ) , n ≥ 2, j = 1, . . . , g . ωQ1 ,n = (A.32) (n − 1) bj Any meromorphic differential ω (3) on Kg not of the first or second kind is said to be of the third kind, written dtk. It is common to normalize a dtk ω (3) , by the vanishing of its a-periods, that is, by Z ω (3) = 0 , j = 1, . . . , g . (A.33) aj (3) b g by A normal dtk, denoted ωQ1 ,Q2 , associated with two distinct points Q1 , Q2 ∈ K `+1 for ` = 1 and 2, vanishing definition has simple poles at Q` with residues (−1) a-periods, and is holomorphic anywhere else.
Theorem A.6. Let g ∈ N. Suppose ω (3) to be a dtk on Kg whose only singub g with residues cn for n = 1, . . . , N. Denote by larities are simple poles at Qn ∈ K (1) ω a dfk on Kg . Then ! Z ! ! Z !! Z Z Z Qn g N X X (1) (3) (1) (3) − = 2πi ω ω ω ω cn ω (1) , j=1
aj
bj
bj
aj
n=1
Q0
(A.34) b g is any fixed base point. In particular, if ω (3) is normalized and where Q0 ∈ K ω (1) = ωj , then Z ω (3) = 2πi bj
N X n=1
Z
Qn
cn
ωj , Q0
j = 1, . . . , g .
(A.35)
865
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
(3)
Moreover, if ωQ1 ,Q2 is a normal dtk on Kg holomorphic on Kg \{Q1 , Q2 }, then Z bj
Z
(3)
ωQ1 ,Q2 = 2πi
Q1
ωj ,
j = 1, . . . , g .
(A.36)
Q2
We shall always assume (without loss of generality) that all poles of dsk’s and b g (i.e. not on ∂ K b g ) and that integration paths on the right hand dtk’s on Kg lie on K side of (A.34)–(A.36) do not touch any cycles aj or bk . Next, we turn to divisors on Kg and the Jacobi variety J(Kg ) of Kg . Let H(Kg ) (M(Kg )) and H1 (Kg ) (M1 (Kg )) denote the set of holomorphic (meromorphic) functions (i.e. 0-forms) and holomorphic (meromorphic) 1-forms on Kg for some g ∈ N0 . Definition A.7. Let g ∈ N0 . Suppose f ∈ M(Kg ), ω = h(ζQ0 )dζQ0 ∈ M (Kg ), and (UQ0 , ζQ0 ) a chart near Q0 ∈ Kg . P∞ −1 (i) If (f ◦ ζQ )(ζ) = n=m0 cn (Q0 )ζ n for some m0 ∈ Z (which turns out to be 0 independent of the chosen chart), the order νf (Q0 ) of f at Q0 is defined by 1
νf (Q0 ) = m0 .
(A.37)
One defines νf (P ) = ∞ for all P ∈ Kg if f is identically zero on Kg . P∞ n for some m0 ∈ Z (which again is inde(ii) If hQ0 (ζQ0 ) = n=m0 dn (Q0 )ζQ 0 pendent of the chart chosen), the order νω (Q0 ) of ω at Q0 is defined by νω (Q0 ) = m0 .
(A.38)
Definition A.8. Let g ∈ N0 . (i) A divisor D on Kg is a map D: Kg → Z, where D(P ) 6= 0 for only finitely many P ∈ Kg . On the set of all divisors Div(Kg ) on Kg one introduces the partial ordering D ≥ E if D(P ) ≥ E(P ) ,
P ∈ Kg .
(ii) The degree deg(D) of D ∈ Div(Kg ) is defined by X deg(D) = D(P ) .
(A.39)
(A.40)
P ∈Kg
(iii) D ∈ Div(Kg ) is called nonnegative (or effective) if D ≥ 0,
(A.41)
where 0 denotes the zero divisor 0(P ) = 0 for all P ∈ Kg . (iv) Let D, E ∈ Div(Kg ). Then D is called a multiple of E if D≥E.
(A.42)
D and E are called relatively prime if D(P )E(P ) = 0 ,
P ∈ Kg .
(A.43)
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R. DICKSON, F. GESZTESY and K. UNTERKOFLER
(v) If f ∈ M(Kg )\{0} and ω ∈ M1 (Kg )\{0}, then the divisor (f ) of f is defined by (f ): Kg → Z , P 7→ νf (P ) (A.44) (thus f is holomorphic, f ∈ H(Kg ), if and only if (f ) ≥ 0), and the divisor of ω is defined by (A.45) (ω): Kg → Z , P 7→ νω (P ) (thus ω is a dfk, ω ∈ H1 (Kg ), if and only if (ω) ≥ 0). The divisor (f ) is called a principal divisor, and (ω) a canonical divisor. (vi) The divisors D, E ∈ Div(Kg ) are called equivalent, written D ∼ E, if D − E = (f )
(A.46)
for some f ∈ M(Kg )\{0}. The divisor class [D] of D is defined by [D] = {E ∈ Div(Kg ) | E ∼ D} .
(A.47)
Clearly, Div(Kg ) forms an Abelian group with respect to addition of divisors. The principal divisors form a subgroup DivP (Kg ) of Div(Kg ). The quotient group Div(Kg )/DivP (Kg ) consists of the cosets of divisors, the divisor classes defined in (A.47). Also the set of divisors of degree zero, Div0 (Kg ), forms a subgroup of Div(Kg ). Since DivP (Kg ) ⊂ Div0 (Kg ), one can introduce the quotient group Pic(Kg ) = Div0 (Kg )/DivP (Kg ) called the Picard group of Kg . Theorem A.9. Let g ∈ N0 . Suppose f ∈ M(Kg ) and ω ∈ M1 (Kg ). Then deg((f )) = 0 and deg((ω)) = 2(g − 1) .
(A.48)
Definition A.10. Let g ∈ N0 , and define L(D) = {f ∈ M(Kg ) | (f ) ≥ D} ,
L1 (D) = {ω ∈ M1 (Kg ) | (ω) ≥ D} .
(A.49)
Both L(D) and L1 (D) are linear spaces over C. We denote their (complex) dimensions by r(D) = dim L(D) , i(D) = dim L1 (D) . (A.50) i(D) is also called the index of specialty of D. Lemma A.11. Let g ∈ N0 and D ∈ Div(Kg ). Then deg(D), r(D), and i(D) only depend on the divisor class [D] of D (and not on the particular representative D). Moreover, for ω ∈ M1 (Kg )\{0} one infers i(D) = r(D − (ω)),
D ∈ Div(Kg ) .
(A.51)
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
867
Theorem A.12 (Riemann Roch). Let g ∈ N0 and D ∈ Div(Kg ). Then r(−D) and i(D) are finite and r(−D) = deg(D) + i(D) − g + 1 .
(A.52)
In particular, Riemann’s inequality r(−D) ≥ deg(D) − g + 1
(A.53)
holds. Next we turn to the Jacobi variety and the Abel map. Definition A.13. Let g ∈ N and define the period lattice Lg in Cg by Lg = {z ∈ Cg | z = N + τ M , N , M ∈ Zg } .
(A.54)
Then the Jacobi variety J(Kg ) of Kg is defined by J(Kg ) = Cg /Lg ,
(A.55)
and the Abel maps are defined by AP0 : Kg → J(Kg ),
P 7→ AP0 (P ) = (AP0 ,1 (P ), . . . , AP0 ,g (P )) ! Z Z P
=
P
ω1 , . . . , P0
ωg
(mod Lg ) ,(A.56)
P0
and αP0 : Div(Kg ) → J(Kg ),
D 7→ αP0 (D) =
X
D(P )AP0 (P ),
(A.57)
P ∈Kg
where P0 ∈ Kg is a fixed base point and (for convenience only) the same path is chosen from P0 to P for all j = 1, . . . , g in (A.56) and (A.57).a Clearly, AP0 is well-defined since changing the path from P0 to P amounts to adding a closed cycle whose contribution in the integral (A.56) consists in adding a vector in Lg . Moreover, αP0 is a group homomorphism and J(Kg ) is a complex torus of (complex) dimension g that depends on the choice of the homology basis {aj , bj }gj=1 . However, different homology bases yield isomorphic Jacobians, see [19, p. 137] and [28, Sec. 8(b)]. Theorem A.14 (Abel’s theorem). Let g ∈ N. Then D ∈ Div(Kg ) is principal if and only if (A.58) deg(D) = 0 and αP0 (D) = 0 . a This convention allows one to avoid the multiplicative version of the Riemann–Roch Theorem at various places in this paper.
868
R. DICKSON, F. GESZTESY and K. UNTERKOFLER
Next, we turn to Riemann theta functions and a constructive approach to the Jacobi inversion problem. We assume g ∈ N for the remainder of this appendix. Given the curve Kg , the homology basis {aj , bj }gj=1 , and the matrix τ of b-periods of the dfk’s {ωj }gj=1 , the Riemann theta function associated with Kg and the homology basis is defined as X exp (2πi(n, z) + πi(n, τ n)) , z ∈ Cg , (A.59) θ(z) = n∈Zg
Pg g where (u, v) = j=1 uj vj denotes the scalar product in C . Because of (A.28), g θ is well-defined and represents an entire function on C . Elementary properties of θ are, for instance, θ(z1 , . . . , zj−1 , −zj , zj+1 , . . . , zn ) = θ(z),
z = (z1 , . . . , zg ) ∈ Cg ,
θ(z + m + τ n) = θ(z) exp (−2πi(n, z) − πi(n, τ n)) ,
(A.60)
m, n ∈ Zn , z ∈ Cg . (A.61)
Lemma A.15. Let ξ ∈ Cg and define bg → C , F:K where b g → Cg , b :K A P0
P 7→ θ(AbP0 (P ) − ξ) ,
(A.62)
b (P ) = A bP0 ,1 (P ), . . . , A bP0 ,g (P ) P 7→ A P0 Z
Z
P
=
ω1 , . . . , P0
!
P
ωg
.
(A.63)
P0
b g , that is, F 6≡ 0. Then F has precisely g Suppose F is not identically zero on K b zeros on Kg counting multiplicities. bg . Lemma A.15 is traditionally proven by integrating d ln(F ) along ∂ K Theorem A.16. Let ξ ∈ Cg and define F as in (A.62). Assume that F is not b g , and let Q1 , . . . , Qg ∈ Kg be the zeros of F (multiplicities identically zero on K included) given by Lemma A.15. Define the corresponding positive divisor DQ of degree g on Kg by ( m if P occurs m times in {Q1 , . . . , Qg } , DQ : Kg → N0 , P 7→ DQ (P ) = 0 if P 6∈ {Q1 , . . . , Qg } ,
Q = (Q and recall the Abel map αP0 in (A.57). Then there exists a vector ΞP0 ∈ Cg , the vector of Riemann constants, such that αP0 (DQ ) = (ξ − ΞP0 )(mod Lg ) .
(A.65)
869
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
The vector ΞP0 = (ΞP0,1 , . . . , ΞP0,g ) is given by X 1 (1 + τj,j ) − 2 g
ΞP0,j =
`=1 `6=j
Z
Z
P
ω` (P ) a`
ωj ,
j = 1, . . . , g .
(A.66)
P0
bP0,j (P )d ln(F (P )) along ∂ K bg . For the proof of Theorem A.16 one integrates A Clearly, ΞP0 depends on the base point P0 and on the choice of the homology basis {aj , bj }gj=1 . Remark A.17. Theorem A.14 yields a partial solution of Jacobi’s inversion problem which can be stated as follows: Given ξ ∈ Cg , find a divisor DQ ∈ Div(Kg ) such that (A.67) αP0 (DQ ) = ξ(mod Lg ). b (P ) + ξ) 6≡ 0 on K b g , the zeros Q1 , . . . , Qg ∈ K b g of Fe Indeed, if Fe (P ) = θ(ΞP0 − A P0 (guaranteed by Lemma A.15) satisfy Jacobi’s inversion problem by (A.65). Thus it bg . remains to specify conditions such that Fe 6≡ 0 on K Remark A.18. While θ(z) is well-defined (in fact, entire) for z ∈ Cg , it is not well-defined on J(Kg ) = Cg /Lg because of (A.61). Nevertheless, θ is a “multiplicative function” on J(Kg ) since the multipliers in (A.61) cannot vanish. In particular, if z 1 = z 2 (mod Lg ), then θ(z 1 ) = 0 if and only if θ(z 2 ) = 0. Hence it is meaningful to state that θ vanishes at points of J(Kg ). Since the Abel map AP0 maps Kg into J(Kg ), the function θ(AP0 (P ) − ξ) for ξ ∈ Cg , becomes a multiplicative function on Kg . Again it makes sense to say that θ(AP0 ( · ) − ξ) vanishes at points of Kg . In the following we use the obvious notation X + Y = {(x + y) ∈ J(Kg ) | x ∈ X, y ∈ Y } , −X = {−x ∈ J(Kg ) | x ∈ X},
(A.68)
X + z = {(x + z) ∈ J(Kg ) | x ∈ X} , for X, Y ⊂ J(Kg ) and z ∈ J(Kg ). Furthermore, we may identify the nth symmetric power of Kg , denoted σ n Kg , with the set of nonnegative divisors of degree n ∈ N on Kg . Moreover, we introduce the convenient notation (N ∈ N) DP0 Q = DP0 + DQ ,
DQ = DQ1 + · · · + DQN ,
Q = (Q1 , . . . , QN ) ∈ σ N Kg , (A.69)
where for any Q ∈ Kg , DQ : Kg → N0 ,
( P 7→ DQ (P ) =
1 for P = Q , 0 for P ∈ Kg \{Q} .
(A.70)
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R. DICKSON, F. GESZTESY and K. UNTERKOFLER
Definition A.19. (i) Define W 0 = {0} ⊂ J(Kg ) ,
W n = αP0 (σ n Kg ) ,
n ∈ N.
(A.71)
(ii) A positive divisor D ∈ Div(Kg ) is called special if i(D) ≥ 1, otherwise D is called nonspecial. (iii) Q ∈ Kg is called a Weierstrass point of Kg if i(gDQ ) ≥ 1, where gDQ = Pg j=1 DQ . Remark A.20. (i) Since i(DP ) = 0 for all P ∈ K1 , the curve K1 has no Weierstrass points. For g ≥ 2, and Kg hyperelliptic, the Weierstrass points of Kg are given precisely by the 2g + 2 branch points of Kg . (ii) The special divisors of the type DQ with Q = (Q1 , . . . , QN ) ∈ σ N Kg and deg(Q) = N ≥ g are precisely the critical points of the Abel map αP0 : σ N Kg → J(Kg ), that is, the set of points D at which the rank of the differential dαP0 is less than g. (iii) While σ m Kg 6⊂ σ n Kg for m < n, one has W m ⊂ W n for m < n. Thus W n = J(Kg ) for n ≥ g by Theorem A.12. Theorem A.21. The set W g−1 + ΞP0 ⊂ J(Kg ) is the complete set of zeros of θ on J(Kg ), that is, θ(X) = 0 if and only if X ∈ W g−1 + ΞP0 (A.72) (i.e., if and only if X = αP0 (D) + ΞP0 (modLg ) for some D ∈ σ g−1 Kg ). The set W g−1 + ΞP0 has complex dimension g − 1. Theorem A.22 (Riemann’s vanishing theorem). Let ξ ∈ Cg . (i) If θ(ξ) 6= 0, then there exists a unique D ∈ σ g Kg such that ξ = αP0 (D) + ΞP0 (modLg )
(A.73)
and i(D) = 0 .
(A.74)
(ii) If θ(ξ) = 0 and g = 1, then ξ = ΞP0 (modL1 ) = 2−1 (1 + τ )(modL1 ) ,
L1 = Z + τ Z ,
−iτ > 0 .
(A.75)
(iii) Assume θ(ξ) = 0 and g ≥ 2. Let s ∈ N with s ≤ g − 1 be the smallest integer 6 F such that such that θ(W s − W s − ξ) 6= 0 (i.e., there exist E, F ∈ σs Kg with E = θ(αP0 (E) − αP0 (F) − ξ) 6= 0). Then there exists a D ∈ σ g−1 Kg such that ξ = αP0 (D) + ΞP0 (modLg ) (A.76) and i(D) = s .
(A.77)
All partial derivatives of θ with respect to AP0 ,j for j = 1, . . . , g of order strictly less than s vanish at ξ, whereas at least one partial derivative of θ of order s is nonzero at ξ. Moreover, s ≤ (g + 1)/2 and the integer s is the same for ξ and −ξ.
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
871
Note that there is no explicit reference to the base point P0 in the formulation of Theorem A.22 since the set W s − W s ⊂ J(Kg ) is independent of the base point while W s alone is not. Theorem A.23 (Jacobi’s inversion theorem). The map αP0 is surjective. More precisely, given ˜ξ = (ξ + ΞP0 ) ∈ Cg , the divisors D in (A.73) and (A.76) (resp. D = DP0 if g = 1) solve the Jacobi inversion problem for ξ ∈ Cg . We summarize some of this analysis in the following remark. Remark A.24. Consider the function g X b (P ) + b (Qj ) , G(P ) = θ ΞP0 − A A P0 P0
P, Qj ∈ Kg ,
j = 1, . . . , g (A.78)
= θ ΞP0 + αP0 (D(Q1 ,...,Qk−1 ,Qk+1 ,...,Qg ) ) = 0 ,
k = 1, . . . , g (A.79)
j=1
on Kg . Then G(Qk ) = θ ΞP0 +
g X j=1 j6=k
b (Qj ) A P0
by Theorem A.21. Moreover, by Lemma A.15 and Theorem A.22, the points Q1 , . . . , Qg are the only zeros of G on Kg if and only if DQ is nonspecial, that is, if and only if (A.80) i(DQ ) = 0 , Q = (Q1 , . . . , Qg ) ∈ σ g Kg . Conversely, G ≡ 0 on Kg if and only if DQ is special, that is, if and only if i(DQ ) ≥ 1. We also mention the elementary change in the Abel map and in Riemann’s vector if one changes the base point, AP1 = AP0 − AP0 (P1 ) (mod Lg ) , ΞP1 = ΞP0 + (g − 1)AP0 (P1 ) (mod Lg ) ,
(A.81) P0 , P1 ∈ Kg .
(A.82)
Remark A.25. Let ξ ∈ J(Kg ) be given, assume that θ(ΞP0 − AP0 ( · ) + ξ) 6≡ 0 g on Kg and suppose that α−1 P0 (ξ) = (Q1 , . . . , Qg ) ∈ σ Kg is the unique solution of Jacobi’s inversion problem. Let f ∈ M(Kg )\{0} and suppose f (Qj ) 6= ∞ for j = 1, . . . , g. Then ξ uniquely determines the values f (Q1 ), . . . , f (Qg ). Moreover, any symmetric function of these values is a single-valued meromorphic function of ξ ∈ J(Kg ), that is, an Abelian function on J(Kg ). Any such meromorphic function on J(Kg ) can be expressed in terms of the Riemann theta function on Kg . For instance,
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R. DICKSON, F. GESZTESY and K. UNTERKOFLER
for the elementary symmetric functions of the second kind (Newton polynomials) one obtains from the residue theorem in analogy to the proof of Lemma A.15 that g g Z X X X n f (Qj ) = f (P )n ωj (P ) resP =Pr f (P )n d ln(θ(ΞP0 − AP0 (P ) + ξ)) , j=1
j=1
aj
Pr ∈K f (Pr )∞
(A.83) −1 b g = a1 b1 a−1 b−1 . . . a−1 where an appropriate homology basis {aj , bj }gj=1 with ∂ K g bg 1 1 avoiding {Q1 , . . . , Qg } and the poles {Pr } of f has been chosen. (We also note that Lemma A.15 just corresponds to the case n = 0 in (A.83).)
Finally, we formulate the following auxiliary result (cf., e.g., Lemma 3.4 in [23]). Lemma A.26. Let ψ( · , x), x ∈ U, U ⊆ R open, be meromorphic on Kg \{P∞ } e (2) with an essential singularity at P∞ (and Ω P∞ ,r+1 defined as in (6.3)) such that e · , x) defined by ψ( ! Z P (2) e e (A.84) ΩP∞ ,r+1 ψ( · , x) = ψ( · , x) exp −i(x − x0 ) P0
is multi-valued meromorphic on Kn and its divisor satisfies e · , x)) ≥ −Dµˆ (x) . (ψ(
(A.85)
e · , x)) = D0 (x) − Dµˆ (x) . (ψ(
(A.86)
D0 (x) ∈ σ g Kg , D0 (x) ≥ 0, deg(D0 (x)) = g .
(A.87)
Define a divisor D0 (x) by
Then Moreover, if D0 (x) is nonspecial for all x ∈ U, that is, if i(D0 (x)) = 0, then ψ( · , x) is unique up to a constant multiple (which may depend on x ∈ U). Appendix B. Trigonal Curves of Boussinesq-Type We give a brief summary of some of the fundamental properties and notations needed from the theory of trigonal curves of Boussinesq-type (i.e., those with a triple point at infinity). First we investigate what happens at the point (or possibly points) at infinity on our Bsq-type curves. Fix g ∈ N. The Bsq-type curve Kg of arithmetic genus g = m − 1 is defined by Fm−1 (z, y) = y 3 + y Sm (z) − Tm (z) = 0 , Sm (z) =
2 n−1+ε X p=0
sm,p z p ,
Tm (z) = z m +
m−1 X
tm,q z q ,
q=0
m = 3n + ε, ε ∈ {1, 2}, n ∈ N0 .
(B.1)
ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
873
Following the treatment in [46] one substitutes the variable u = z −1 into (B.1) to obtain u3n+ε y 3 + sm,0 u2n−1+ε + · · · + sm,2n−1+ε un+1 y (B.2) − tm,0 u3n+ε + · · · + tm,m−1 u + 1 = 0 . Let v = un+1 y in (B.2) to obtain v 3 + (sm,0 u2n−1+ε + · · · + sm,2n−1+ε )u3−ε v −(tm,0 u3n+ε + · · · + tm,3n−1+ε u + 1)u3−ε = 0 .
(B.3)
Let u → 0 (corresponding to z → ∞) in (B.3) to obtain v 3 = 0. This corresponds to one point of multiplicity three at infinity (in both cases ε = 1 and ε = 2), given by (u, v) = (0, 0). We therefore use the coordinate ζ = z −1/3 at the branch point at infinity, denoted by P∞ . The curve (B.1) is compactified by adding the point P∞ at infinity. In homogeneous coordinates, the point at infinity we add is [1 : 0 : 0] ∈ CP2 if g = 0 or g = 1, otherwise the point at infinity we add is [0 : 1 : 0] ∈ CP2 . The point P∞ is singular in all cases except when g = 1, or when g = 2 and sm,0 = −1/3. Although not directly associated with the Bsq hierarchy, we note that the case ε = 0 in (B.1) is analogous to AKNS, Toda, and Thirring-type hyperelliptic curves, which are not branched at infinity. In fact, a similar argument to that above, with the coordinate v = un y in (B.2), yields the equation v 3 = 1 as u → 0. This corresponds to three distinct points P∞,j , j = 1, 2, 3 at infinity (each with multiplicity one), given by the three points (u, v) = (0, ωj ) for j = 1, 2, 3, where ω1 , ω2 , ω3 are the third roots of unity. As each point at infinity has multiplicity one, none are branch points, and consequently each admits the local coordinate u = 1/z for |z| sufficiently large. In [10, p. 561], Burchnall and Chaundy define the g-number of an algebraic curve as the maximum number of double points possible in the finite plane. For Bsq-type curves the g-number is g = m − 1. For a curve that is smooth in the finite plane, the g-number coincides with the arithmetic genus of the curve, but in the presence of double points, the g-number remains the same, while the genus is diminished (according to results of Clebsch, Noether, and Pl¨ ucker, see, e.g., [7, 41]). We now prove that the g-number of Kg , and hence the arithmetic genus of Kg if Kg is smooth in the finite plane, is m − 1 using a special case of the Riemann–Hurwitz theorem. Theorem B.1. Let π ˜z : Kg → CP1 be the projection map with respect to the z coordinate. Then X πz ) − 1 = 2g + 4 , (B.4) νP (˜ P ∈Kg
πz ) denotes the multiplicity of π ˜z at P ∈ Kg , and g is the arithmetic genus where νP (˜ of the curve Kg .
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If Eq. (B.1) has only double points, this implies that the discriminant ∆(z) of the curve (B.1), defined by ∆(z) = 27Tm (z)2 + 4 Sm (z)3
(B.5)
(modulo constants), is non-zero. ∆(z) is easily seen to be a polynomial of degree 2m. Hence in the finite complex plane, the Riemann surface defined by the compactification of (B.1) can have at most 2m double points, corresponding to the possible 2m zeros of ∆(z). If all finite branch points are distinct double points (taking into P πz ) − 1 = 2m + 2, and account the triple point at infinity) one obtains P ∈Kg νP (˜ so by (B.4), one infers g = m − 1. Let B denote the set of branch points and let |B| denote the number of branch points counted according to multiplicity. In the case of Bsq-type curves, deg(˜ πz ) = πz ) = 1 for all P ∈ K \B. Moreover, ν (˜ π ) ∈ {2, 3} for all P ∈ B. 3, and νP (˜ g P z P πz ) − 1 ≤ 2|B|, and (B.4) reduces to Hence |B| ≤ P ∈Kg νP (˜ g + 2 ≤ |B| ≤ 2g + 4 .
(B.6)
Thus one arrives at an upper and lower bound on the number of branch points on Kg . When m = 1, corresponding to g = 0, there are no non-zero holomorphic differentials on Kg . When m = 2, corresponding to g = 1, the only holomorphic differential on Kg is dz/(3y(P )2 + Sm (z)). Recall also that m 6= 0(mod 3), so we need not consider holomorphic differentials for the case m = 3. One verifies that dz/(3y(P )2 + Sm (z)) and y(P )dz/(3y(P )2 + Sm (z)) are holomorphic differentials Kg with zeros at P∞ of order 2(m − 2) and (m − 4), respectively, for m ≥ 4. It follows that the differentials (m = 3n + ε, ε ∈ {1, 2}) ( `−1 for 1 ≤ ` ≤ g − n , z dz 1 (B.7) η` (P ) = 3y(P )2 + Sm (z) y(P )z `+n−g−1 dz for g − n + 1 ≤ ` ≤ g , form a basis in the space of holomorphic differentials H1 (Kg ). Introducing the invertible matrix Υ ∈ GL(g, C), R Υ = (Υj,k )j,k=1,...,g , Υj,k = ak ηj , (B.8) e(k) = (e1 (k), . . . , eg (k)) , ej (k) = Υ−1 j,k , the normalized differentials ωj for j = 1, . . . , g, ωj =
g X
Z ej (`)η` ,
ωj = δj,k ,
j, k = 1, . . . , g
(B.9)
ak
`=1
form a canonical basis for H1 (Kg ). Near P∞ one infers ω = (ω1 , . . . , ωg ) =
ζ→0
(ε) (ε) (ε) α0 + α1 ζ + α3 ζ 3 + O(ζ 4 ) dζ ,
(B.10)
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ALGEBRO-GEOMETRIC SOLUTIONS OF THE BOUSSINESQ HIERARCHY
where (ε) α0
( = −
e(g) ,
ε = 1, (B.11)
e(g − n) , ε = 2 ,
(ε)
ε = 1, −e(g − n) , = d(2) e(g − n) − e(g) , ε = 2 , 0
(ε)
(1) (1) ε = 1, d1 e(g) + c1 e(g − n) − e(g − 1) , = (2c(2) − (d(2) )3 )e(g − n) − e(g − n − 1) + (d(2) )2 e(g) , ε = 2 , 1 0 0
α1
α3
(B.12)
(B.13) etc., and
y(P ) =
ζ→0
(ε) (ε) (ε) (ε) c0 + d0 ζ + c1 ζ 3 + d1 ζ 4 + O(ζ 6 ) ζ −3n−2 as P → P∞ , (
with (ε) (ε) (c0 , d0 )
=
(0, 1) ,
ε = 1,
(2) (1, d0 ) ,
ε = 2,
(2)
d0 ∈ C .
In particular, using (A.32), (B.10), and (B.11), one obtains Z Z 1 1 1 (ε) (2) (ε) (2) ωP∞ ,2 = α0,j and ω = α1,j . 2πi bj 2πi bj P∞,3 2
(B.14)
(B.15)
(B.16)
Finally, we turn our attention to special divisors. From the theory of elementary symmetric polynomials one infers the following lemma. Lemma B.2. Pick z ∈ C, and denote by y1 (z), y2 (z), and y3 (z), the three solutions of (B.1). These solutions are distinct if and only if the discriminant ∆(z) 6= 0. Moreover, introduce Qj = (z, yj ) ∈ Kg for j = 1, 2, 3. Then P3 yj (z) = 0. (i) Pj=1 3 yj (z)yk (z) = Sm (z). (ii) Q3j
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Pr Lemma B.3. Let m1 , . . . , mr ∈ N with j=1 mj = g and Qj = (z, yj ), j = 1, 2, 3 as in Lemma B.2. Suppose P1 , . . . , Pr ∈ Kg . If {Q1 , Q2 , Q3 } ⊆ {P1 , . . . , Pr } ,
(B.17)
then the divisor Dm1 P1 +···+mr Pr ∈ σ g Kg is special. In particular, if one of the points Pj ∈ {P1 , . . . , Pr } is a triple point, then the divisor Dm1 P1 +...+mr Pr ∈ σ g Kg is special. Proof. Using the identities in Lemma B.2, one readily computes 3 X j=1
1 = 0, 3yj (z)2 + Sm (z)
3 X j=1
yj (z) = 0. 3yj (z)2 + Sm (z)
(B.18)
Thus, choosing for simplicity the base point P0 = P∞ , a comparison of (A.56), (B.7), and (B.18) yields 3 X
AP∞ (Qj ) = 0 (mod Lg ) .
(B.19)
j=1
Thus Dm1 P1 +···+mr Pr ∈ σ g Kg is special by Theorem A.21.
Acknowledgments K. U. would like to thank G. Teschl for numerous helpful discussions. Moreover, he is indebted to the Department of Mathematics at the University of Missouri, Columbia for the extraordinary hospitality extended to him during a stay in the Spring of 1998. References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. [2] H. Airault, “Solutions of the Boussinesq equation”, Physica D 21 (1986) 171–176. [3] H. Airault, H. P. McKean and J. Moser, “Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem”, Commun. Pure Appl. Math. 30 (1977) 95–148. [4] R. Beals, P. Deift and C. Tomei, Direct and Inverse Scattering on the Line, Amer. Math. Soc. Providence, R.I., 1988. [5] E. D. Belokolos, A. I. Bobenko, V. Z. Enol’skii, A. R. Its and V. B. Matveev, AlgebroGeometric Approach to Nonlinear Integrable Equations, Springer, Berlin, 1994. [6] J. L. Bona and R. L. Sachs, “Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation”, Commun. Math. Phys. 118 (1988) 15–29. [7] E. Brieskorn and H. Kn¨ orrer, Plane Algebraic Curves, Birkh¨ auser, Basel, 1981. [8] W. Bulla, F. Gesztesy, H. Holden and G. Teschl, “Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac–van-Moerbeke hierarchies”, Memoirs Amer. Math. Soc., 135 (641) (1998), 1–79. [9] J. L. Burchnall and T. W. Chaundy, “Commutative ordinary differential operators”, Proc. London Math. Soc. 21(2) (1923) 420–440.
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[56] G. Wilson, “Algebraic curves and soliton equations”, in Geometry Today, eds. E. Arbarello, C. Procesi, and E. Strickland, Birkh¨ auser, Boston, 1985, pp. 303–329. [57] V. E. Zakharov, “On stochastization of one-dimensional chains of nonlinear oscillators”, Sov. Phys. JETP 38 (1974) 108–110. [58] V. E. Zakharov and S. V. Manakov, “Multidimensional nonlinear integrable systems and methods for constructing their solutions”, J. Sov. Math. 31 (1985) 3307–3316. [59] V. E. Zakharov and S. V. Manakov, “Construction of higher-dimensional nonlinear integrable systems and of their solutions”, Funct. Anal. Appl. 19 (1985) 89–101. [60] V. E. Zakharov and A. B. Shabat, “A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I”, Funct. Anal. Appl. 8 (1974) 226–235. [61] V. E. Zakharov and A. B. Shabat, “Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II”, Funct. Anal. Appl. 13 (1979) 166–174.
INTERACTING QUANTUM FIELDS GLENN ERIC JOHNSON Litton-TASC, 4801 Stonecroft Blvd. Chantilly, VA 20151-3822, USA E-mail : [email protected] Received 29 January 1998 A wide class of quantum field theories (QFTs) describing interacting neutral scalar bosons are constructed. It is shown that a redefinition of the vacuum expectation values (VEV) of quantum fields as generalized functions leads to constructions of interacting fields that are operators in a Hilbert space and satisfy the physical requirements for a field theory. The field operators are constructed directly, without perturbation expansion, and constructed in spacetime of any number of dimensions. The corresponding scattering theory is developed. The limited function class underlying the QFT imposes a direction to time.
1. Introduction Since the inception of quantum field theory (QFT), many efforts have been directed at discovering a consistent definition for quantized fields. While QFT provides widely accepted models and precise predictions for the physics of elementary particles, inconsistencies in the descriptions of fields as operators on Hilbert spaces have plagued its development. This paper documents constructions of interacting quantum field models. These constructions demonstrate that a wide variety of interacting field models can be defined as operators in Hilbert spaces and that the fields satisfy requirements for causality, nonnegative energy, wave-particle duality, and Lorentz covariance. The models are not derived from perturbation expansions using model Lagrangians but are explicitly constructed to satisfy the physical requirements for quantum fields. Several properties unanticipated in established field theory developments [1, 2] are exhibited by these quantum field theories (QFT), including an imposed direction on time, and non-self-adjointness of the field even though the field is real. These QFT are in spacetimes of any number of dimensions. The constructed interacting fields model neutral scalar bosons. The constructions include models with the (exact) differential cross-section for two-particle elastic scattering |p|d−4 dσ = co , dΩ (ωm )2 where ωm is the energy and |p| the magnitude of the momentum for a scattered particle in the center-of-mass frame in d dimensional spacetime. The constant is determined in an expansion in moments of the characteristic function for underlying random processes identified below. At nonrelativistic momenta, this corresponds to scattering by a central potential that in first Born approximation for four dimensions is given by the modified Bessel function, V (r) = −b K1(mr)/r. The equivalent 881 Reviews in Mathematical Physics, Vol. 11, No. 7 (1999) 881–928 c World Scientific Publishing Company
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potential for this reaction is attractive and short range as anticipated from one particle exchange of a neutral meson [2]. The number of spacetime dimensions, d, decisive in perturbation expansions and constructive quantum field theory, plays little role. This paper, however, does not address the representation of the Hamiltonian operator in terms of field operators. This correspondence, like the equivalent potential, is anticipated to be dependent upon the number of dimensions, d. The constructed VEV for the interacting fields are functions of the energy and momenta except for an overall energy-momentum conserving and mass shell Dirac delta measures. The VEV in the spacetime representation are Fourier transforms of these functions defined as generalized functions of the spatial arguments, and differentiable functions of time. The significant deviation from prior attempts to construct interacting quantum field models is in the definition of field operators. The field operators are not indexed by test functions, but by “sampling functions” that are generalized functions of time, and test functions of space. The fields are defined only for sampling functions of particular forms, that for the example of the simplest constructed models include Z dp e−ip·x −iωm tk p e g(p)(ωm δ(x(o) − tk ) + iδ 0 (x(o) − tk )) (1) ϕ(x; ˆ tk ) = (2π)d−2 with the parameter tk and g(p) a test function. For this example, the two-point function is of the form Z −ip(x1 −x2 ) 2 d δ(p2 − m2 ) . W2 (x1 , x2 ) = dp e θ(p(o) ) 1 + a d(p2 ) This defines a nonnegative bilinear form for the selected sampling functions, Z hΦ(ϕ(t))0, ˆ Φ(ϕ(t))0i ˆ = (2π)d dp (2ωm )|g(p)|2 ≥ 0 . These sampling functions can be recognized as defining the asymptotic particle creation operators, a∗m (g, t). That is, Z Z dp p e−iωm tk −ip·x g(p) Φ(ϕ(t ˆ k )) = dx (2π)d−2 × (ωm δ(xk(o) − tk ) + iδ 0 (xk(o) − tk )) Φ(x) ! Z Z dp −iωm tk −ip·x p = −i dx e g(p) ∂o Φ(tk , x) (2π)d−2 Z dx p = u(tk , x)∂o Φ(tk , x) (2π)d−2 √ = 2π a∗m (g, tk ) .
(2)
There is a particular choice for sampling functions, ϕ(x; ˆ tk ), for each class of QFT models.
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The class of sampling functions is adequate to define the scattering matrix since the Hilbert space of the quantum fields contains incoming and outgoing particle-like states, yet, the definition of the VEV is sufficiently limited that technical obstacles to construction of QFTs have been removed. However, established results, for example the necessity of the K¨ all´en–Lehmann representation [6] for the two-point function, do not survive the technical changes to QFT descriptions. The VEV are now only conditionally positive generalized functions. Formally, the reconstructed fields, reconstructed from the VEV, are Φ(t, x) ≈ e−itH Ψ(0, x)eitH with Ψ(τ, x) a generalized random process on Rd and H the Hamiltonian. The Wightman functions, with boundary values that are the VEVs, are defined as the family of generalized functions parametrized by complex time, * n + Y e−(τk +itk )H Ψ(0, xk )e(τk +itk )H 0 . Wn (t − iτ, x)n = 0, k=1
These Wightman functions are analytic extensions in the time differences of the moment functions of the random process, and are analytic in Euclidean time-ordered domains that include a region with each τk > τk+1 . Sharp time fields, however, are not defined, and only the fields in the particular linear combinations with their temporal derivatives define operators. The VEV define a linear functional for sampling function sequences ϕ = (ϕo , ϕ1 (x), ϕ2 (x)2 , . . .) with ϕo a constant and ϕk (x)k a sampling function of k arguments in d dimensional spacetime, W(ϕ) = ϕo + W1 (ϕ1 ) + W2 (ϕ2 ) + · · · . With definition of an involution of sequences, ϕ∗ = (ϕ∗o , . . . , ϕn (xn , . . . , x1 )∗ , . . .), and the product of sequences, ! n X ϕk (x1 , . . . , xk )ψn−k (xk+1 , . . . , xn ), . . . , ϕ × ψ = ϕo ψo , . . . , k=0
the VEV define a bilinear form that equals the squared lengths of vectors. That is, the magnitudes of vectors are,
2 Z Z
ϕo + dx1 Φ(x1 )ϕ1 (x1 ) + dx1 dx2 Φ(x2 )Φ(x1 )ϕ2 (x1 , x2 ) + · · · 0
= W(ϕ∗ × ϕ) and are nonnegative if W(ϕ∗ × ϕ) ≥ 0. There is a Hilbert space realization of the fields if the VEV define this seminorm on sequences of sampling functions. Nonnegativity of this form holds only for a limited class of sampling functions,
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denoted E+ , that include (1) for the example of the simplest constructed models. In the Wightman axioms, the VEV are sampled by arbitrary tempered test functions in both space and time, while in this development the VEV are sampled in the spatial dimensions by tempered test functions and sampled in time using particular linear combinations of the function and its time derivatives. Achievement of this seminorm is the key step in development of these QFTs. This step strongly restricts the form for the VEV as generalized functions and this restriction turns out to allow just the class of sampling functions required to achieve asymptotically free particle states. The restriction permits only definition of asymptotically free creation operators, and annihilation operators cannot be constructed as adjoints of these creation operators. The established expression for an annihilation operator in terms of the field operator is not defined within the physical Hilbert space even though the particle creation operator is defined. Due to the restriction on sampling functions limiting the domain and range of field operators, the field operators cannot be self-adjoint operators even though the field is real. This reemphasizes that there is no unitary equivalence of interacting and free QFTs. The restriction on sampling functions is required to even define field operators. The definition of the field, Φ(ϕ(t)), ˆ holds only for ϕ(t) ˆ ∈ E+ and even though formally ˆ ∗ ∈ / E+ . The Φ(x)∗ = Φ(x), the implied adjoint of the sampling functions, ϕ(t) modified definition for the VEV serves as an alternative to discovery of random processes that exhibit the Markoff property [3]. Since only the VEV of the fields plus time derivatives in one sense define positive forms, there is an imposed, preferred direction to time in these models. The direction to time is not established by a lack of symmetry of the Wightman functions, but the restricted extent of the class of sampling functions for which the VEV define positive forms does not support time reflection. And, although the quantum fields are Poincar´e covariant, this development emphasizes the “3 + 1” structure [4] to physics. The constructions begin with generating functionals described in Euclidean space. The selected method of constructing generating functionals uses the characteristic functions for generalized random processes realized as linear combinations of independent random processes, Z Ψ(ξ) = dξ 0 R(ξ 0 − ξ)Ψo (ξ 0 ) . The underlying random process, Ψo (ξ), has independent, identically distributed values on Rd . The Fourier transform of the expansion coefficients is denoted the a ˜ The root functions are restricted to forms: root function, R(p). 1 ˜ . R(p) = Y (|p|2 + b2k ) k a Note added in proof. This approach to construction of Schwinger functions was applied with
conventional test functions and root functions of a form 1/(|p|2 + b2 )α with 0 < α < 1, achieving analytic extension to Wightman functions and two-point functions of K¨ all´ en–Lehmann form but not positivity or scattering, in [5].
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Normal (Gaussian) processes underlie free field models, and nonnormally distributed processes underlie interacting fields. For these constructions, nonnormal statistics of the random process is a sufficient condition for the constructed QFT to exhibit scattering. QFT derived from these root functions do not include all QFT models. Exceptions include p the free−1quantum fields. Free fields can be constructed using root functions (( |p|2 + m2 ) ), but then the demonstration of positivity uses the normal statistics of the free field. To achieve interaction, structures that exclude the established free field model are considered. However, to achieve asymptotic equality with free particle states, a free field theory normal random process is added to each construction. The Wightman axioms [7, 8] describe essential physics of quantum fields as conditions on their vacuum expectation values (VEV). Vacuum expectation values are described as generalized functions (functionals) dual to linear topological spaces of test functions. The Wightman axioms may be summarized as six conditions: temperedness, Poincar´e covariance, positivity, locality, nonnegative energy, and cluster decomposition. Temperedness is a technical assertion, since, although it has physical ramifications, the definition of the VEV as generalized functions is not directly motivated by the observable properties of fields. The initial selections for linear topological spaces were K{Mp } spaces, predominately the (Schwartz) space of tempered test functions, S. It has been recognized that modification to the temperedness axiom could be required [8] and various alternatives have been suggested [9]. The models presented within this paper represent fruition of a program suggested in [10]. With the substitution for the temperedness axiom, the constructed field theories satisfy the remaining Wightman axioms. Once the structure of the QFTs, inherited from an underlying random process, has been set out, three significant issues are resolved. First, it is shown that the constructed generalized functions exhibit physical (Osterwalder–Schrader [11]) positivity on a domain E+ (Rd ) of sampling functions suitable for construction of a QFT. Next, it is shown that the constructed generalized functions analytically continue to boundary values that define VEV of quantum fields. Finally, within Sec. 5, scattering modeled by the constructed QFTs is established. The development concentrates on one explicit, example class of model QFT. In the course of the construction, several generalizations to the example constructions are identified. The construction is fully developed only for one class of models that establish the existence of interacting QFT models. The many models within this class include a QFT model for every generalized random process with independent identically distributed values. Nevertheless, additional root functions and random processes defined using derivatives of test functions suggest constructions for a much wider class of fields, including many particle species models. The constructions also generalize to models that lack Symanzik–Nelson positivity, models that do not derive from an underlying random process. The goal of these constructions is to discern a mathematical form consistent with the physics of interacting quantum fields. The method of construction, deriving the
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G. E. JOHNSON
vacuum expectation values (VEV) from a generating functional, does not readily permit imposition of requirements that the QFTs model familiar interactions. Rather, these explicit interacting quantum field models are intended as convenient prototypes, analogous to the role provided by the linear harmonic oscillator models in quantum mechanics. Although model QFTs in one and two spatial dimensions have been constructed by Glimm–Jaffe–Spencer and others [12], these results have not extended to physical spacetimes. And, in spite of a characterization of required generalized functions [13, Chap. II, Sec. 4.1], before these constructions, no explicit formulation of quantum field theory had been fully realized in physical spacetimes. 2. A Random Process Construction Random processes are associated with quantum fields through analytic extension of their vacuum expectation values (VEV) to imaginary times [3, 8, 14, 15]. When an additional condition is satisfied, the VEV of quantum fields are the analytic extensions of the moment functions of a generalized random process. The method adopted within this paper is to adapt elementary constructions of random processes to arrive at a QFT for scalar bosons. This approach led to the conjecture that either there were no interacting QFTs associated with very wide classes of random processes, or the technical aspect of the Wightman axioms, the temperedness axiom, must be relaxed. The result is construction of quantum fields using sampling functions peculiar to each model. The first task is to define a class of generalized random processes compatible with the structure of a QFT. The construction begins with the assumption that moment functions of a generalized random process [16] are Schwinger functions [11, 17] for a QFT. Moment functions are constructed using one of the variety of methods for constructing random processes [18]. The appropriate question becomes whether there are any nonnormal (nonGaussian) random processes with statistical moments that analytically continue to physical spacetime and define positive generalized functions that exhibit scattering. There are many known examples of random processes that lead to QFTs but exhibit trivial physics, for example, the normal random processes associated with generalized free QFTs [15, 19]. The characteristic functions of random processes are constructed. In contrast to defining only VEV of products of the fields, the characteristic function fully defines the random process. A random process is not always fully defined by its statistical moment functions [20]. The characteristic function of the random process will be utilized as the generating functional for the equal time quantum fields. Two elementary methods, designated the distribution distortion and correlation distortion methods, construct random processes with selected marginal statistics and covariances [18]. The correlation distortion method uses functions of normal random variables and consequently produces monomials [21] of (generalized) free fields. Assuming that theories generated from functions of normally distributed random fields are all physically trivial, the interesting QFT candidates derived by these elementary methods must be generated by the distribution distortion method.
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The distribution distortion method develops correlated generalized random processes from linear combinations of random processes with independent, identically distributed values. At each Euclidean domain spacetime point, ξ = (τ, x) ∈ Rd , a random process, Ψ(ξ), is represented in terms of a random process with independent, identically distributed values, Ψo (ξ), as Z Ψ(ξ) = dξ 0 R(ξ 0 − ξ)Ψo (ξ 0 ) . (3) A wide class of generalized random processes with independent, identically distributed values have been constructed [16]. Convolutions of the expansion coefficients, R(ξ), with the generalized random fields must be defined. It is sufficient to require R(ξ) to be the Fourier transform of a rational function, 1 ˜ . (4) R(p) = N Y 2 2 (|p| + bk ) k=1
˜ With this definition, R(p) will be a multiplier of the generalized functions since it is infinitely differentiable (b2k > 0) and slow growth. (This selection could be generalized with a numerator that is a function with appropriate holomorphy and growth in the upper half p(o) plane.) This result rests on the yet to be defined ˜ selection of test functions. R(p) is designated the root function since its squared magnitude is proportional to the Fourier transform of the covariance function of the random process. ˜ to depend only on the Euclidean Rotational invariance is ensured by choosing R d length of the argument, p ∈ R . Selecting the expansion to depend only on spacetime differences implements translation invariance. Rotational and translational symmetries of the random process are necessary [11] for Poincar´e covariance of the constructed QFT. The definition (3) provides that the characteristic function for the constructed random process, Ψ(ξ), equals the characteristic function of the process with independent values evaluated for functions of the form: Z (5) R ∗ ϕ = dξ 0 R(ξ − ξ 0 )ϕ(ξ 0 ) . That is, E[e−iΨ(ϕ) ] = L(R ∗ ϕ)
(6)
with L(f ) the characteristic function of the generalized random process with independent values at each point. The argument of the process, ϕ ∈ E+ (Rd ), is an element of the as yet undefined class of sampling functions. A variety of the required characteristic functions have been identified in [16] for random processes defined upon S. These results are summarized in Appendix A. For the moment, assume that L(R ∗ ϕ) can be defined given L(f ) with f ∈ S(Rd ). A sufficient condition for L(f ) to be the characteristic function of a generalized
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G. E. JOHNSON
random processes with independent, identically distributed values is for L(f ) to be of the form: Z log(L(f )) = dξ G(f (ξ)) . (7) The function may include derivatives (i.e. G(f (ξ), f 0 (ξ), . . . , f (n) (ξ))), but this additional generality will not be carried along. The derivatives of the real sampling functions may be mixed among the components of ξ in any manner and be of any finite order, but for the constructions, only the rotationally invariant forms are of interest. The models defined by (7) without derivatives of the sampling functions will be designated the base models. The next issue is to determine whether L(R ∗ ϕ) is defined given the definition of L(f ) for f ∈ S(Rd ). The elements ϕ ∈ E+ (Rd ) derive from test functions h ∈ S(Rd−1 ). Define these generalized functions as the union of positive time translates of the class of generalized functions: ϕ(ξ) ∈ E+,0 if Z Y dp e−ipξ ϕ(ξ) = h(p) (ωbk + ip(o) ) d/2 (2π) k
ϕ(τ − τ 0 , x) ∈ E+,τ 0 if [ E+ = E+,τ 0
ϕ(ξ) ∈ E+,0
and τ 0 ≥ 0
(8)
τ0
with the indicated product associated with the root function definition, 1 ˜ . R(p) = Y 2 (bk + |p|2 ) k
p The energies, ωb = ωk = b2 + pk · pk , will be labeled either by mass or momenta (and c = 1). The indicated summations Z dp(o) e−ip(o) τ (p(o) )n = 2πin δ (n) (τ ) are understood as generalized functions, (T, f ) = (T˜, f˜), with the convention, (δ (n) , f ) = (−1)n
dn f (0) . dτ n
These sampling functions are particular generalized functions of the Euclidean time with the VEV of the fields serving as the test functions. For one example of a theory
INTERACTING QUANTUM FIELDS
889
dp e−ip·x h(p)(ωb δ(τ − τ 0 ) − δ 0 (τ − τ 0 )) . (2π)d/2−1
(9)
with a single mass, Z ϕ(ξ) =
The structure of E+ at this point is unmotivated, but as will be developed in Secs. 3.1 and 5.1, the selection is natural. The elements of E+ are fixed generalized functions (the Dirac delta and derivatives) of time and test functions in space. The Fourier transforms of E+ are test functions of the momenta and polynomials in the index of characters for the Euclidean time, p(o) . Even with this change from temperedness, the supports of the Wightman functions remain defined. Support is defined as the support of a function in the temporal domain, and as the support of a generalized function in space, energy, and momenta. The arguments at which the functions vanish are in the complement of the support, and the space of tempered functions is sufficiently rich in functions of compact support to define the support of generalized functions. The test functions, h(p), are taken to be from a nuclear, countably normed function space such as the Schwartz space of tempered functions, S(Rn(d−1) ), or other K{Mp } spaces. Since each sampling function is uniquely associated with an element of S(Rn(d−1) ), nuclearity is sufficient to define the Wightman functions. The definition of L(f ) can be extended from f ∈ S to functions of the form (5) given (8). In the spatial arguments, the functions R ∗ ϕ are summable since the root functions are multipliers. The product of the root function and the Fourier transform of the sampling function is square summable in p(o) , so the Plancherel Fourier inversion theorem provides the convolution for the temporal dimension. Explicitly, Z R ∗ ϕ(ξ) =
0
e−ipξ eip(o) τ h(p) dp Y (ωbk − ip(o) ) k
= 2π θ(τ − τ 0 )
XZ k
0
e−ip·x e−ωb` (τ −τ ) h(p) dp Y (ωb` − ωbk )
(10)
`6=k
which is absolutely summable in the Euclidean time. These functions have support limited to τ ≥ τ 0 ≥ 0. These functions are bounded and exhibit at least tempered decay in each of their arguments. Consequently, the summation (7) is defined, and L(R ∗ ϕ) is the characteristic function of a correlated random process indexed by ϕ. The definition of L(f ), (10), requires only the summability implied by boundedness and the tempered decay of the functions at large arguments. 2.1. Schwinger functions The characteristic function of a generalized random process is a generating functional for the statistical moment functions. The statistical moment functions are
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the candidate Schwinger functions: n d i E[e−iβΨ(ϕ) ] dβ β=0 n d = i L(βR ∗ ϕ) dβ β=0 n Z d = i exp dξ G(βR ∗ ϕ(ξ)) dβ β=0
Sn (ϕ · · · × ϕ) =
(11)
for the base models. Then, "
Z Sn (ξ1 , ξ2 , . . . , ξn ) = Sn (ξ)n =
E
n Y
# Ψo (ξk0 )
R(ξk0 − ξk ) dξk0 .
k=1
The link-cluster identity [22, Sec. 4.4.3] defines the connected, truncated functions, n d T Sn (ϕ × · · · ϕ) = i log(L(βR ∗ ϕ)) (12) dβ β=0 and then, formally, L(R ∗ ϕ) =
X (−i)n n!
Sn (ϕ × · · · ϕ) = exp
X (−i)n n!
Sn (ϕ × · · · ϕ) .
T
(13)
The truncated Schwinger functions for base models (models defined without derivatives of the sampling functions) are of the form: Z dξ(R ∗ ϕ(ξ))n
T
Sn (ϕ)n = cn
Z d
= (2π) cn
˜ 2 )ϕ(p ˜ n−1 )ϕ(p ˜ 1 )ϕ(p dp1 . . . dpn−1 R(p ˜ 1 )R(p ˜ 2 ) . . . R(p ˜ n−1 )
˜ ˜ × R(−p 1 − p2 . . . − pn−1 )ϕ(−p 1 − p2 . . . − pn−1 )
(14)
with cn = dn G(λ)/dλn at λ = 0. In the example of an underlying Poisson random process, the constants that determine the scattering and production amplitudes are cn = (−λ)n with λ a real constant. 2.2. Root functions Every function of the form (4) can be represented as Z ˜ R(p) = 0
with
R
∞
dη(m) (|p|2 + m2 )N
(15)
dη(m) a postive measure with support that excludes the origin. Using
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891
methods presented in [23], write, Z
1 N Y
Z
1
= (N − 1)! 0
(|p|2 + b2k )
Z
1
du1
du2 u2 . . . 0
1
duN −1 (uN −1 )N −2
0
k=1
×
1 . (|p|2 + b21 u1 u2 . . . uN −1 + b22 (1 − u1 )u2 . . . uN −1 + · · · + b2N (1 − uN −1 ))N (16)
Inspection reveals that the support of the measure is bounded, m2 ∈ (min b2k , max b2k ). That is, min b2k ≤ b21 u1 u2 . . . uN −1 + b22 (1 − u1 )u2 . . . uN −1 + · · · + b2N (1 − uN −1 ) ≤ max b2k . This explicitly defines bound,
R
dη(m) as a convergent summation. Inspection provides a
Z βo = sup p
0
∞
N dη(m) 1 ≤ . (p2 + m2 )N mink b2k
(17)
Analytic continuations based on properties of 1/(|p|2 + m2 ) will remain valid for multiple factor root functions since the additional factors only improve the convergence of required summations. The constructions assert that each constituent particle has finite mass, although the masses may be arbitrarily small. Refinement of the bounds for analytic extension is required to establish theories that include massless particles. 2.3. Cluster decomposition axiom The candidate QFTs naturally satisfy the cluster decomposition axiom. The cluster decomposition property follows from the definition of the root function and the structure of the construction. The cluster decomposition property implies uniqueness of the vacuum state [9]. Cluster decomposition requires that Wn+m (f · g) → Wn (f )Wm (g) as the supports of f and g become arbitrarily greatly spacelike separated. The link-cluster identity together with the equivalence of Euclidean and physical spacetime conditions demonstrated in [11] imply that the condition is satisfied if the truncated Schwinger functions vanish for large spatial differences. From the representation (14), find that the truncated functions vanish with large spacelike displacement of proper subsets of the arguments if the root function decays with large spatial displacements. That is, if R(ξ + λ(0, a)) → 0 as λ → ∞, then Z T
Sn (ξ)n = cn
dξ
n Y
R(ξ − ξk ) → 0
k=1
as a proper subset of the xk grow without bound.
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G. E. JOHNSON
Using the representation (15) for the root function, apply rotational invariance to align a with the Euclidean time axis. Then a contour integration provides Z
Z
R(ξ + λ(0, a)) =
dη(m)
2πi = (N − 1)!
eipξ+ipo λa dp d/2 (|p|2 + m2 )N (2π)
Z
Z dη(m)
dp eip·x dN −1 eip(o) (λa+τ ) −1 (p N (2π)d/2 dpN (o) + iω) (o)
p(o) =iω
√ with ω > 0, and a = a · a > 0. This vanishes in the indicated limit since the sum and coefficients can be bounded and there is a finite upper bound on multinomials in p(k) times e−λwa . Z |R(ξ + λ(0, a)| ≤
Z dη(m)
N −1 ip(o) (λa+τ ) d e dq sup (1 + |p|M ) N −1 N 1 + |q|M p (p + iω) dp(o) (o)
. p(o) =iω
This upper bound vanishes as λ → ∞ resulting in limλ→∞ R(ξ + λ(0, a)) = 0 and satisfaction of the cluster decomposition condition. 3. Physical Positivity At this point, Euclidean covariance and symmetry conditions have been incorporated into the constructions. The next goal is construction of an appropriate seminorm that enables construction of the Hilbert space realization for the quantum field. The appropriate condition is physical (Osterwalder–Schrader) positivity and this follows from the structure of the generating functional (6) for the sampling functions within E+ . The constructed Schwinger functions have both the positivity of a QFT as well as the natural (Symanzik–Nelson) positivity of a random process. Construction of the physical Hilbert space consists of completion of a linear vector space using a seminorm expressed as a mean value of the random process. To achieve nonnegativity of a suitable seminorm requires a significant deviation from the Euclidean domain (Osterwalder–Schrader) statement of the Wightman positivity axiom. The Hilbert space spanned by the field operators is obtained by completion of a vector space indexed by elements of E+ . The required seminorm will be positive on the restricted space of sampling functions, E+ , but not on S. The natural positivity of a random process is denoted Symanzik–Nelson positivity and is manifest in the nonnegativity of matrices formed from the characteristic function: 2 X X E αk e−iΨ(ϕk ) = α` α∗k M`,k ≥ 0 k
`,k
M`,k = E[e−iΨ(ϕ` ) (e−iΨ(ϕk ) )∗ ] = L(R ∗ (ϕ` − ϕ∗k )) .
(18)
INTERACTING QUANTUM FIELDS
893
The natural seminorm derived from this inner product is, kuk2o = (u, u)o with (u, v)o = E[u∗ v] and with elements of a linear vector space defined X αk e−iΨ(ϕk ) . (19) u= k
For functions of random variables in this form, the Osterwalder–Schrader positivity condition [11, 24] is nonnegativity of all matrices Q`,k = E[e−iΨ(ϕ` ) (e−iΨ(ϑϕk ) )∗ ] = L(R ∗ (ϕ` − ϑϕ∗k )) .
(20)
A Euclidean domain time reflection operator, ϑ, is defined by ϑϕ(τ, x) = ϕ(−τ, x) .
(21)
Nonnegativity of Q`,k does not follow from the construction of the random process and is an additional condition. No interesting models can exhibit both Symanzik–Nelson and Osterwalder– Schrader positivity for all sampling functions. Since the Symanzik–Nelson positivity is unrestricted, the Osterwalder–Schrader positivity condition can only be valid on a limited domain of the sampling functions. Indeed, since the Cauchy inequality for the Symanzik–Nelson positive inner product provides p (ϑu, u)o ≤ (u, u)o (ϑu, ϑu)o = (u, u)o , if nonnegativity of the Osterwalder–Schrader form were also unrestricted, then the Cauchy inequality for the reflection positive form would provide p (u, u)o = hϑu 0, u 0i ≤ hu 0, u 0ihϑu 0, ϑu 0i = (ϑu, u)o . Together, the unrestricted nonnegativity of the two forms would imply that (ϑu, u)o = (u, u)o . Limiting the domain of the Osterwalder–Schrader positive form evades triviality of the models. Independence of the underlying random process at each Euclidean point together with reflection symmetries of the root function leads to a factorization of the matrix Q`k when the sampling functions are restricted to E+ . From the representation (15), it follows that R is a real function with reflection symmetry, R(τ, x) = R(−τ, x). Because R is invariant under time reversal, R ∗ (ϑϕ) = ϑ(R ∗ ϕ) . For the functions R ∗ ϕk with support limited to τ ≥ 0, Q`,k = L(R ∗ ϕ` − ϑ(R ∗ ϕ∗k )) = L(R ∗ ϕ` )L(−ϑ(R ∗ ϕ∗k )) = L(R ∗ ϕ` )L(−R ∗ ϕ∗k ) = L(R ∗ ϕ` )L(R ∗ ϕk )∗ .
(22)
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G. E. JOHNSON
Since the supports of R ∗ ϕ` and ϑR ∗ ϕk are disjoint, the characteristic function factors. It is a property of generalized random processes with independent values at each point that L(ϕ1 + ϕ2 ) = L(ϕ1 )L(ϕ2 ) if ϕ1 ϕ2 = 0 [16]. It was demonstrated in (10) that R∗ϕ has support limited to τ ≥ 0 for every ϕ ∈ E+ . For these characteristic functions, L(ϑϕ) = L(ϕ), and L(−ϕ∗ ) = L(ϕ)∗ . This factored matrix is evidently nonnegative. This demonstration of Osterwalder–Schrader positivity is the key development in construction of a QFT from the random processes. The random process provides the form and organization of the Schwinger functions, but the selection of root and sampling functions implements positivity. To achieve nonnegativity of (20), it is not necessary that L(ϕ) be the characteristic function of a random process. Nonnegativity requires only the properties L(ϕ1 + ϕ2 ) = L(ϕ1 )L(ϕ2 ) if ϕ1 ϕ2 = 0, L(ϑϕ) = L(ϕ), and L(−ϕ∗ ) = L(ϕ)∗ . This is another source for generalizations of the constructed QFT. In these more general cases, the models would lack the Symanzik–Nelson positivity of a random process and require a regularity condition [24] for L(ϕ). The required positivity condition holds for functions ϕ such that R ∗ ϕ has support only for τ > 0. Such functions evidently include the Fourier transforms of the product of any test function with support limited to τ ≥ 0 with the reciprocal ˜ −1 is justified since polynomials are of the root function. (Multiplication by R(p) infinitely differentiable and slow growth.) As developed in Sec. 5.1, the selection (8) denoted E+ for the sampling functions achieves positivity and results in a definition for particle states. 3.1. Model augmentations and fields The relation (20) immediately implies a method to generate new quantum field models from old. Schur’s theorem [25] states that the Hadamard product (Mij = Aij Bij ) of two nonnegative matrices is nonnegative. Consequently, the QFT derived from the sum of two random processes that each produce a QFT is also a QFT model. The characteristic function of the sum of the two independent random processes is the product of the original characteristic functions. Thus, the Hadamard product of the nonnegative matrices from two QFT models produce a new QFT. Together with the result (13) that the truncated functions of the two theories will add, it is evident that the new model will satisfy all the Wightman axioms (including the modified temperedness requirement) as long as both theories are defined on common sampling functions E+ . To be developed in Sec. 5.2, to achieve a sensible scattering theory, a free quantum field must be added to the field constructed from the random process. This augmentation to the model requires additional notation. The Schwinger and Wightman functions constructed from the random process will remain designated by Sn and Wn , the Schwinger and Wightman functions for the free field theory will be designated Snf and Wnf , and the augmented Schwinger and Wightman functions for the composition will be designated Sne and Wne . The relations between Sn , Snf and
895
INTERACTING QUANTUM FIELDS
Sne are given by the link-cluster identity (13), T e Sn
= TSn + T Snf .
(23)
The relation for identifying VEV of quantum fields, Φ(ϕ), evaluated at equal times is Z ∞ X (−i)n d(ξ)n Sen (ξ)n (ϕ` − ϑϕ∗k )n he−iΦ(ϕk ) 0, e−iΦ(ϕ` ) 0i = n! n=0 Z ∞ X (−i)n im e d(ξ)n+m Sn+m = (ξ)n+m (ϕ` )n (ϑϕ∗k )m . n! m! n,m=0 (24) (ϕ` )n (ϕk )m denotes ϕ` (ξ1 ) . . . ϕ` (ξn )ϕk (ξn+1 ) . . . ϕk (ξn+m ), and ϕk ∈ E+,0 . A field Φx (ϕ) is associated with the functions Sn , and a free field Φf (ϕ) is associated with the functions Snf . 3.2. Special properties of the constructions The factoring of the nonnegative matrices produces identities among the Schwinger functions. These identities are developed from the characteristic function, ∗ i h L(R ∗ (ϕ` − ϑϕ∗k )) = E e−iΨ(ϕ` ) e−iΨ(ϑϕk ) Z ∞ X (−i)n im d(ξ)n+m Sn+m (ξ)n+m (ϕ` )n (ϑϕ∗k )m = n! m! n,m=0 = L(R ∗ ϕ` )L(−R ∗ ϑϕ∗k ) Z ∞ X (−i)n im d(ξ)n+m Sn (ξ)n Sm (ξ)m (ϕ` )n (ϑϕ∗k )m . = n! m! n,m=0 (25) Then, (22) is equivalent to the series of identities, Z d(ξ)n+m (Sn+m (ξ)n+m − Sn (ξ1 , . . . , ξn )Sm (ξn+1 , . . . , ξn+m ))(ϕ` )n (ϑϕ∗k )m = 0 (26) for ϕ` , ϕk ∈ E+ . These relations follow from the limited support of R ∗ ϕ. However, these relations do not persist among the analytic extensions of the Schwinger functions. Analytic extension to physical times occupies the following section. The limitation on the domain of sampling functions, E+ , implies that the field cannot be self-adjoint. The field as constructed is real, Φ(x)∗ = Φ(x), (the Schwinger functions are real and the Wightman functions will have the Hermiticity property
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G. E. JOHNSON
[8]), but the sampling function of the adjoint is not in E+ . The adjoint of the field operator would be defined if the index set included functions in E− , the sampling functions defined so that the support of R ∗ ϕ is limited to τk < 0. However, the validity of the positivity condition is restricted to either E+ or E− .b Extension of the function space to include E− so that the field could be selfadjoint is inconsistent with nonnegativity of the form (20). Indeed, the field cannot be a self-adjoint operator in this construction since hΦ(ϕ)0, Φ(ϕ)0i = 6 h0, Φ(ϕ)Φ(ϕ)0i even for real sampling functions. The two-point function equals Z h0, Φ(ϕ)Φ(ϕ)0i = dξ1 dξ2 S2e (ξ1 , ξ2 )ϕ(ξ1 )ϕ(ξ2 )
(27)
(28)
while that component of the two-point function due to the random process, hΦx (ϕ)0, Φx (ϕ)0i , involves only the vacuum polarization, TS1 , and not TS2 . As 1 , 2 → 0, the property, (22), of the characteristic function provides that −i1 Φx (ϕ) − 1 e−i2 Φx (ϕ) − 1 e 0, 0 1 2 " ∗ −iΨ(2 ϕ) # −1 e eiΨ(1 ϑϕ) − 1 = hΦx (ϕ)0, Φx (ϕ)0i = E 1 2 exp =
Z
Z ∗ dξ G(−1 ϑ(R ∗ ϕ)) − 1 exp dξ G(2 R ∗ ϕ) − 1 1 2
Z 2 = c1 dξ R ∗ ϕ . The constant c1 can be freely selected (see Appendix) independently from c2 , but (27) follows since equality will not hold for all ϕ. The field is real but the field operator is not even Hermitian. Indeed, hΦx (ϕ)u 0, v 0i = hu 0, Φ∗x(ϕ)v 0i Z = dξ ϑϕ(ξ)∗ E[(ϑu∗ )Ψ(ξ)v] = hu 0, Φx(ϑϕ∗ )v 0i b With D denoting the domain of an operator A: an operator A is formally self-adjoint if A Ajk = (ej , Aek ) = A∗kj in a separable Hilbert space with basis {ek }, or for L2 , the differential expression of A equals that for its adjoint; an operator is Hermitian if (Au, v) = (u, Av) for every u, v ∈ DA ; an Hermitian operator is symmetric if it has a dense domain; an operator is self-adjoint if DA = DA∗ and Au = A∗ u for every u ∈ DA . In Hilbert spaces, each property is stronger than its predecessor, although in finite dimensional spaces, the four properties are equivalent.
INTERACTING QUANTUM FIELDS
and of course,
Z hu 0, Φx(ϕ)v 0i =
897
dξ ϕ(ξ) E[(ϑu∗ )Ψ(ξ)v] .
There can be no ϕ ∈ E+ such that ϕ = ϑϕ∗ . 4. Analytic Extension of the Moment Functions 4.1. Representations of truncated Schwinger functions Within this section, it is shown that no additional conditions are required to analytically continue the moments of the constructed random process from Euclidean to physical spacetime. A technical digression to isolate time dependence and achieve bounds on the statistical moment functions results in the analytic extensions. With this analytic extension, the Hamiltonian operator is defined. As a consequence of translation invariance, the Schwinger functions can be considered generalized functions of difference arguments, yk = ξk − ξk+1 . Within a Euclidean domain with time-ordered arguments, the Schwinger functions extend to complex time differences. Considered as a function of time differences and as a generalized function of the spatial difference variables, (14) is readily manipulated into the following convenient representation for the truncated Schwinger functions of the base models, T
Sn (y)n−1 = cˆn
n−1 YZ k=1
Z dpk
dη(mk ) (|pk |2 + m2k )N
Z
dη(mn ) eipk (ξk −ξn ) . (|p1 + p2 + · · · + pn−1 |2 + m2n )N (29)
Normalizations for the Fourier transforms were absorbed into the constant defined by (2π)nd/2 cˆn = (2π)d cn and cn = dn G(λ)/dλn at λ = 0. This generalized function can be many times differentiable in time depending upon the value of N (controlling the decay of 1/(|p|2 +m2 )N with large p(o) ). The goal is now analytic extension of these truncated functions in the time differences. The method is to observe that the Fourier transforms of the Schwinger functions in the difference variables are slow growth functions of the momenta and consequently define generalized functions for S(Rn(d−1) ). These Fourier transforms remain slow growth for all complex times in Euclidean time-ordered domains. 4.2. Recursive evaluation of the Schwinger functions A recursive construction for the Schwinger functions enables verification of holomorphy properties by explicitly isolating the time dependence. An absolutely convergent summation over a coordinate pk(o) can be segregated from the representation (29) for TSn . The summation attenuates as p−4 k(o) for large
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G. E. JOHNSON
arguments, so a pk(o) summation in T
Sn (y)n−1 = cˆn
n−1 YZ
Z dpk
×
∞
dη(mk ) 0
k=1
Z
Z
∞
dη(mn ) eipk ·(xk −xn )
0
eipk(o) (τk −τn ) dpk(o) 2 (pk(o) + ωk2 )N
1 ((p1(o) + p2(o) + · · · + pn−1(o) )2 + ωn2 )N
converges. The ωn is defined subject to the momentum conservation (translation invariance) condition n X pk = 0 . k=1
The choice for Euclidean domain time-ordering determines the analytic extension as one of the n! possible argument orderings for the Wightman functions associated with each Schwinger function. An ordering of the noncommutative quantum fields derives from this selection for analytic continuation. The n! permutations of the arguments of a Wightman function coincide in Euclidean space. With the temporal arguments satisfying the ordering τk > τk+1
k = 1, 2, . . . , n − 1
the Euclidean spacetime difference variables designated as yk = (τk − τk+1 , xk − xk+1 ) have positive time components. This choice for ordering corresponds to the Wightman function, Wn (x1 − x2 , x2 − x3 , . . . , xn−1 − xn ) = (0, Φ(x1 )Φ(x2 ) . . . Φ(xn )0) .
(30)
The correspondence of time-ordered Euclidean domain and argument ordering in the Wightman function is fixed by the result [8] that Wn (ζ1 , ζ2 , . . . , ζn−1 ) is analytic if each ζk ∈ Rd − iV+ . The forward cone, V+ , is defined as the set of vectors, q ∈ V+ if q(o) > 0, and q 2 > 0. In particular, a Wn (ζ)n−1 satisfying the original Wightman axioms is analytic if each ζk = (tk −tk+1 −i(τk −τk+1 ), xk −xk+1 ) and τk > τk+1 . It is convenient to define the functions: n−1 YZ eiuk (τk −τn ) 1 N duk 2 Υn (τ1 , . . . , τn ; γ) = (31) N !2 (uk + ωk2 )N n−1 k=1 X u` + iγ + ωn2 `=1
and then, Z T
Sn (y)n−1 = cˆn
dη(mn )
n−1 YZ
Z dη(mk )
dpk eipk ·(xk −xn ) N Υn ((τ )n ; 0) .
(32)
k=1
The utility of the definition (31) is that a recursion can be set up to evaluate the functions N Υn .
899
INTERACTING QUANTUM FIELDS
The pk(o) coordinate summation is evaluated using a closed contour integration. Depending upon the selection of time ordering, the relevant contour is closed in the upper or lower complex pk(o) half plane. The summand defining N Υn includes four poles, uk = ±iωk
n−1 X
uk = ±iωn − iγ −
u` .
`6=k
For τk > τk+1 , the required summations equal the integrations over contours closed in the upper half planes. There are one, two, or three poles enclosed within the integration contour depending upon the value of γ: iωk is always enclosed by the contour ; iωn − iγ −
n−1 X
u` is enclosed by the contour if, ωn − γ > 0 ;
`6=k
−iωn − iγ −
n−1 X
u` is enclosed by the contour if, −ωn − γ > 0 .
`6=k
The development is now limited to base models with the simplest root function, 1/(|p|2 + m2 ). This is sufficient to establish existence of interacting quantum field models yet simplifies notation. Choosing to perform the un−1 summation, the Cauchy–Goursat residue theorem provides 1
Υn ((τ )n ; γ) =
n−2 YZ k=1
eiuk (τk −τn ) duk 2 uk + ωk2
2π 2ωn−1
n−2 X
e−ωn−1 (τn−1 −τn ) !2 u` + iγ + iωn−1
+ ωn2
`=1
− ωn − γ + i 2π e + θ(ωn − γ) 2ωn
n−2 X
!
n−2 X
(τn−1 − τn )
u`
`=1
!2
u` + iγ − iωn
2 + ωn−1
`=1
− −ωn − γ + i 2π e − θ(−ωn − γ) 2ωn
n−2 X
!
(τn−1 − τn ) . (33) ! 2 n−2 X 2 u` + iγ + iωn +ω u`
`=1
n−1
`=1
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G. E. JOHNSON
This is recognized as an order recursion among the functions 1 Υn , −ωn−1 (τn−1 −τn ) e 1 1 Υn ((τ )n ; γ) = 2π Υn−1 ((τ )n−2 , τn ; γ + ωn−1 ) 2ωn−1 + θ(ωn − γ)
e−(ωn −γ)(τn−1 −τn ) 1 Υn−1 ((τ )n−1 ; γ − ωn ) 2ωn
− θ(−ωn − γ)
e−(−ωn −γ)(τn−1 −τn ) 1 Υn−1 ((τ )n−1 ; γ + ωn ) . (34) 2ωn
A redundant θ(ωn ) could be included with the first term to emphasize that the magnitude of each factor of the exponential function is less than unity in the time-ordered domain. Analytic continuation of 1 Υn to complex times τn−1 − τn + i(tn−1 − tn ) is now manifest given holomorphy of lower order functions since the exponential functions are entire and all time dependence has been isolated in the exponentials. Since only 1 Υn ((τ )n ; 0) is associated with TSn , this order recursion does not extend to the Schwinger functions. The recursion simplifies significantly for γ = 0. n n n −ki ωi (τi −τk ) X Y X e 1 1 Υn ((τ )n ; 0) = π ` Υn−` (τ )n−`−1 , τk ; kj ωj |ki | ω k=n−` i=n−` j=n−` i (35) with an antisymmetric matrix denoted 1 j
1
4.3. Bounds on 1 Υ2 for ordered complex times Given the recursion (35), 1 Υn will be analytic for complex times if 1 Υ2 is analytic (with a different extension from each Euclidean time-ordered region). The recursions for the functions 1 Υn are initiated with 1
Υ2 (τ1 , τ2 ; γ) = 2π
e−ω1 (τ1 −τ2 ) 1 2 2ω1 ω2 − (ω1 + γ)2
+ 2πθ(ω2 − γ)
e−(ω2 −γ)(τ1 −τ2 ) 1 2ω2 ω12 − (ω2 − γ)2
− 2πθ(−ω2 − γ)
e−(−ω2 −γ)(τ1 −τ2 ) 1 . 2 2ω2 ω1 − (ω2 + γ)2
(36)
INTERACTING QUANTUM FIELDS
901
This essentially defines, 1 Υ1 (τk ; γ) as 1/(ωk2 − γ 2 ). Together with (35), (32) and the definition for the Fourier transform of a generalized function [13], this provides the analytic extension of the Schwinger functions in time provided 1 Υ2 remains a slow growth function of momenta for all Euclidean time-ordered, complex time differences and all values of γ. To develop a bound that ensures analytic extension, (36) is bounded separately in the three domains: ω1 ω22 e−ω1 z γ > ω2 (ω1 + γ)2 − ω 2 2 ω1 ω22 (e−ω1 z − e−(ω2 −γ)z ) ω22 − (ω1 + γ)2 ω1 ω2 (ω1 + ω2 ) e−(ω2 −γ)z ω2 > γ > −ω2 + (ω1 + ω2 )2 − γ 2 2 2 1 ω1 ω2 | Υ2 (z1 , z2 ; γ)| ≤ π ω12 ω2 (e−(ω2 −γ)z − e−ω1 z ) (ω2 − γ)2 − ω12 2 ω1 ω2 (e(ω2 +γ)z − e−ω1 z ) + ω12 − (ω2 + γ)2 ω1 ω22 e−ω1 z + −γ > ω2 . (ω1 − γ)2 − ω 2 2
(37) The notation was condensed by denoting the time difference, now extended to complex times, as z = τ + it = z1 − z2 = τ1 − τ2 + i(t1 − t2 ) with τ1 > τ2 . Each term either has a factor e−ωk z with a magnitude bounded by e−ωk τ ≤ 1, or the term can be put in the form (e−βk z −e−βj z )/(βj −βk ) times a finite, real constant. In every case, β` ≥ 0. Since the real component of z = τ + it is nonnegative, −β z e k − e−βj z |z| ≤ p . βj − βk 1 + (βj − βk )2 |z|2 /12 This inequality follows from |1 − e−z |2 ≤
|z|2 1 + |z|2 /12
which is proved by elementary means. Putting this inequality into polar form with z = reiθ , the τ > 0 constraint becomes −π/2 < θ < π/2 and the demonstration reduces to the discovery that for each r, the minimum values over θ of f (r, θ) =
r2 − 2e−r cos θ (cosh(r cos θ) − cos(r sin θ)) 1 + r2 /12
occur on the boundary, θ = ±π/2. The inequality then follows by inspection of the function on these boundaries (r2 ≥ (2 + r2 /6)(1 − cos r)).
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G. E. JOHNSON
Individually bounding each term in (37), the bounds x ≤1 x+y−v for x, y > 0 and v ≤ y, 1 1 1 ≤ < 2 2 2 2 (x + y) − v (x + y) − y x(x + y) for x, y > 0 and v 2 ≤ y 2 together with the bounds on the time dependence produces a bound on 1 Υ2 . For complex time differences with a nonnegative real component, −ω τ γe 1 γ > ω2 2 ω22 |z| p 2 2 1 + (ω1 + γ − ω2 ) |z| /12 ω2 > γ > −ω2 + ω2 e−(ω2 −γ)τ 2 2 1 ω1 ω2 | Υ2 (z1 , z2 ; γ)| ≤ π ω1 ω2 |z| (38) p 2 |z|2 /12 1 + (ω + γ − ω ) 1 2 ω1 ω2 |z| +p 1 + (ω1 + γ + ω2 )2 |z|2 /12 |γ|e−ω1 τ −γ > ω2 . + 2 This is bounded for all time differences, even as |z| → ∞ if z has a nonnegative real component, unless γ = ±ω2 − ω1 . In cases with γ = ±ω2 − ω1 , the growth is no worse than linear with the time difference. 4.4. Analytic extension of the truncated Schwinger functions Equipped with an upper bound on the modulus of 1 Υ2 , analytic continuation in time of TSn follows. The L1 equivalance class of every function that lacks singularties and has slow growth defines a generalized function, and in particular, the constructed Schwinger functions define generalized functions dual to S(Rn(d−1) ). These generalized functions are parametrized by complex times that satisfy the Euclidean time-ordering. Substituting the recursion (35) into the expression for the truncated Schwinger functions, (32), and denoting τk + itk as zk , T
Sn (z, x)n = π n−2 cˆn
n−1 YZ
dpj eipj ·(xj −xn )
j=1
×
n n X Y e−ki ωi (zi −zk ) |ki |
k=2
i=2
ωi
1
Υ2 (z1 , zk ;
n X
! k` ω` )
.
(39)
`=2
For all complex time differences, ki (zi − zk ) = ki (τi − τk + i(ti − tk )) with the Euclidean times ordered ki (τi −τk ) > 0, the exponential functions and 1 Υ2 are slow
INTERACTING QUANTUM FIELDS
903
growth functions of the momenta, pj , since ωj2 = m2 + pj · pj . Consequently (29) defines generalized functions. These generalized functions are twice differentiable functions of each time. This representation, (39), holds for the particular Euclidean time-ordered domain, τk > τk+1 , (and in the other domains, the time order puts the exponential, time dependent factors in a form such that | exp(±ωi (zi − zk ))| ≤ 1). The demonstration of analytic continuation together with Euclidean covariance and physical positivity implies the spectrum property [11, 17]. 4.5. The Hamiltonian Given the seminorm (20) and the analytic extension of the moments of the random process, the effort now returns to construction of the physical Hilbert space. The construction began with vectors defined by (19) and the characteristic function of the underlying random process defined by (6) and (7). The Hilbert space is defined as the completion of the equivalence classes of the linear span of exponentials of the random field e−iΨ(ϕ) with ϕ ∈ E+ and with the seminorm derived from (20). This construction results in a rigged Hilbert space constructed from a seminorm defined ultimately on a union of nuclear countably normed spaces, since each sampling function is uniquely associated with a test function in S(Rn(d−1) ). Construction of the Hamiltonian closely follows [11, 17, 24] except that the Schwinger functions have already been shown to be analytic functions of time (within time-ordered domains) and the nonnegativity of the Osterwalder–Schrader seminorm is limited to E+ . The Hamiltonian is defined as the generator of complex time translations. h0, Φ(ϕ)k−1 e−Hz Φ(ϕ)eHz Φ(ϕ)n−k 0i Z = d(ξ)n (ϕ(ξ))n Sne (ξ1 , . . . , ξk−1 , (τk + z, xk ), ξk+1 , . . . , ξn ) .
(40)
The analytic extension, (39), displays the Schwinger functions as holomorphic for complex times z within the Euclidean time-ordered domains. Then, Z d(ξ)n (ϕ(ξ))n Sne (ξ1 , . . . , ξk−1 , (τk + z, xk ), ξk+1 , . . . , ξn ) Z X z` S e(`) (ξ1 , . . . , ξk−1 , (τk , xk ), ξk+1 , . . . , ξn ) = d(ξ)n (ϕ(ξ))n `! n `
of the vector space, (19), where d` f (τ )/(dτ )` = f (`) (τ ). Expansions in elements R −i dξ Φ(τ +,x)ϕ(ξ) ˆ −iΦ(ϕ) ˆ ˆ − 1)/ → −iΦ(ϕ) ˆ and (e − e−iΦ(ϕ) )/2 → such R as (e −i dξ(dΦ(ξ)/dτ )ϕ(ξ), ˆ and the definition of equal time fields, (24), display Cauchy sequences within the Hilbert space that converge to |e−Hz e−iΦ(ϕ) 0i. The densely defined operation of e−Hz generates the analytic extension of the Schwinger functions, and its range is within the constructed Hilbert space. Time translation invariance of the Schwinger functions provides that e e ((τ − z ∗ , x)n , (τ + z, x)m ) = Sn+m ((τ, x)n , (τ + z + z ∗ , x)m ) Sn+m
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G. E. JOHNSON
so the bilinear form defined by (19), (20) and (40) has the property that ∗
∗
he−Hz ueHz 0, e−Hz ueHz 0i = hu 0, e−H(z+z ) ueH(z+z ) 0i .
(41)
The Cauchy inequality implies ∗
∗
|he−Hz ueHz 0, e−Hz ueHz 0i| ≤ kuk ke−H(z+z ) ueH(z+z ) k .
(42)
It was shown in Sec. 3 that for positive Euclidean (z + z ∗ is real) time translations (correponding to ϕ(ξ 0 ) ∈ E+,τo → ϕ(τ 0 − τ, x0 ) ∈ E+,τo +τ with τo , τ > 0) the bilinear form is nonnegative. Then, equivalence classes are preserved under complex time translation. Repeated application of the Cauchy–Schwarz inequality [11] shows that time translation is a contraction in the Hilbert space, ke−Hz k ≤ 1 for z with nonnegative real components. Time translation is then a strongly continuous, contractive semigroup with a self-adjoint generator of nonnegative spectrum [26]. The self-adjoint generator of this holomorphic semigroup is denoted the Hamiltonian, H. Self-adjointness implies that H is Hermitian, hΦ(ϕ)k−1 e−Hz Φ(ϕ)eHz Φ(ϕ)n−k 0, 0i Z ∗ ∗ = d(ξ)n (ϑϕ(ξ)∗ )n h0, Φ(0, xn ) . . . eHz Φ(0, xk )e−Hz . . . Φ(0, x1 ) 0i Z =
d(ξ)n (ϑϕ(ξ)∗ )n Sen (ξn , . . . , ξk+1 , (τk − z ∗ , xk ), ξk−1 , . . . , ξ1 ) .
This can be verified also by employing the representation of TSn in (36) and (39), the association of Euclidean time-ordering with VEV (30), and reflecting summation variables. (This is the Hermiticity property [8] of Wightman functions.) The VEV of quantum field operators, Φ(ϕ(t)) = e−iHt Φ(ϕ)eiHt are now defined by the analytic continuation of the Schwinger functions, * 1 + n Y Y Φ(ϕk (tk ))0, Φ(ϕk (tk )) 0 k=m
k=m+1
Z =
d(ξ)n
m Y k=1
∗
ϑϕk (ξk )
n Y k=m+1
ϕk (ξk ) Sen (τ, x)n
.
(43)
τk →τk +itk
This derives from the expression for the fields evaluated at equal times, (24). Recall that ϕ ∈ E+,0 , and now the product must be ordered, hΦm . . . Φ2 Φ1 0, Φm+1 . . . Φn 0i. These operators Φ(ϕ(t)) are defined by the boundary value, τk → 0, of the analytic extensions of the Schwinger functions into the complex time, Euclidean time-ordered domain with τk > τk+1 . This together with the definitions for the Schwinger functions, (6), (11) and (23), are the defining relations for the subsequent analysis.
905
INTERACTING QUANTUM FIELDS
4.6. Two-point function The two-point function is the analytic continuation of the covariance function of the random process. The two-point truncated Schwinger function for the base ˜ model example with R(p) = 1/(|p|2 + m2 ) is most readily evaluated beginning with (29). Z eip(ξ1 −ξ2 ) T S2 (ξ1 , ξ2 ) = cˆ2 dp . (44) (|p|2 + m2 )2 Isolating upon the p(o) summation, for τ1 > τ2 Laurent expansion provides Z
eip(o) (τ1 −τ2 ) dp(o) 2 = 2 )2 (p(o) + ωm
Z dp(o)
eip(o) (τ1 −τ2 ) (p(o) − iωm )2 (p(o) + iωm )2
d eip(o) (τ1 −τ2 ) = 2πi dp(o) (p(o) + iωm )2 p
(o) =iωm
=π
e−ωm (τ1 −τ2 ) 2 2ωm
1 (τ1 − τ2 ) + . ωm
Repetition for τ1 < τ2 yields Z T
S2 (ξ1 , ξ2 ) = πˆ c2
−ωm |τ1 −τ2 | d e dp ip·(x1 −x2 ) − e . 2ωm dωm ωm
(45)
This expression can be manipulated into a covariant representation of the twopoint Wightman function. Z e−iωm (t1 −t2 )+ip·(x1 −x2 ) 1 T i(t W2 (x1 , x2 ) = πˆ c2 dp − t ) + 1 2 2 2ωm ωm −ip(x1 −x2 ) Z d e 2 )θ(p(o) ) − = πˆ c2 dp δ(p2(o) − ωm dp(o) p(o) Z d 2 δ(p2(o) − ωm = πˆ c2 dp e−ip(x1 −x2 ) θ(p(o) ) ). p(o) dp(o) This last step used the definition of the derivative of a generalized function. This expression can be placed into the manifestly Poincar´e covariant form Z d T δ(p2 − m2 ) . W2 (x1 , x2 ) = 2πˆ c2 dp e−ip(x1 −x2 ) θ(p(o) ) (46) d(p2 ) The spectral support property is also evident. As permitted by the weakened temperedness axiom, this two-point function is not a K¨ all´en-Lehmann form since the Fourier transform with respect to the difference variable is not a positive measure. The development of the two-point functions will be simplified by asserting that the first moment of the random process vanishes. As seen in the appendix, the vacuum polarization ( TS1 ) may be specified independently of the remaining moments.
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G. E. JOHNSON
The truncated two-point function for sampling functions from E+ is defined by (14). The contribution from the constructed random process to the two-point VEV at unequal times is the boundary value at physical times of the analytic extension of the Schwinger function. Sampling functions supported at τ1 = ε > 0 and τ2 = 0 with a limit ε → 0 approach the physical time boundary. (8) together with (4) and (14) provide that hΦx (ϕ1 (t1 )) 0, Φx (ϕ2 (t2 )) 0i = TS2 (ϑϕ∗1 , ϕ2 ) Z d ˜ R(−p) ˜ (ϑϕ∗1 )˜(p) ϕ˜2 (−p) = (2π) cˆ2 dp R(p) Z (ωp − ip(o) )2 d = (2π) cˆ2 dp e−ip(o) ε h1 (−p )∗ h2 (−p ) (|p|2 + m2 )2 Z 1 = (2π)d cˆ2 dp e−ip(o) ε h1 (−p )∗ h2 (−p ) (ωp + ip(o) )2 =0 (47)
since the summation equals a closed contour integration that encloses no poles. The Euclidean domain time reflection operator, ϑ, was defined in (21). The analytic extension also vanishes. A conventional notation, TS2 (ϕ1 , ϕ2 ), was retained although a clearer notation for the evaluation in the temporal arguments would be, (ϕ1 , ϕ2 ∗ TS2 ), with ∗ indicating convolution, since the sampling function is the generalized function and the Schwinger function is the test function in the temporal arguments. The vanishing of this truncated two-point function at equal times follows from the identity (26) implied by the implementation of positivity. Since it vanishes at equal times, it must vanish at all times as a consequence of the Cauchy inequality in the Hilbert space that includes quantum field operators. The contribution from the constructed random process to the VEV of the square of the field also derives from (14), and in this case, the analytic extension results from setting ε = i(t1 − t2 ) since there is no Euclidean time reflection. h0, Φx (ϕ1 (t1 ))Φx (ϕ2 (t2 )) 0i = TS2 (ϕ1 , ϕ2 ) Z ˜ R(−p) ˜ ϕ˜1 (p) ϕ˜2 (−p) = (2π)d cˆ2 dp R(p) ip(o) ε e = (2π)d cˆ2 dp 2 h (p )h (−p ) 1 2 2 ωm + p(o) ε=i(t1 −t2 ) Z −iωm (t1−t2 ) dp e = (2π)d+1 cˆ2 h1 (p )h2 (−p ). (48) 2ωm Z
For t1 = t2 and h2 (−p ) = h1 (p )∗ , h0, |Φx |2 0i ≥ 0.
INTERACTING QUANTUM FIELDS
907
The definition (14) for (ϕ1 , ϕ2 ∗TS2 ) requires that the Schwinger function be twice differentiable in time (for this simplest example model). Normalize the covariance for the free field, a Pauli–Jordan function, as Z S2f (ξ) =
eipξ dp = 2 2π |p| + m2
Z dp
e−ωm |τ |+ip·x . 2ωm
Since the Pauli–Jordan function is not differentiable at the origin in Euclidean time, the covariance for the free field normal random process indexed by elements from E+ requires a definition distinct from (14). For the free field component, the two-point VEV defines the Euclidean region function as the analytic (and hence differentiable) extension. The appropriate analytic continuation for the implied field ordering is, Z W2f (ζ1
− ζ2 ) =
dp
e−ωm (τ1−τ2+it1−it2 )+ip·(x1−x2 ) . 2ωm
Derivatives at τ1 = τ2 refer to the boundary values of the appropriate analytic function. This representation exhibits nontempered growth in the momenta for τ1 < τ2 , and hence a generalized function for E+ is defined by this extension only for τ1 ≥ τ2 . The VEV of the free field defines the covariance of the random process for E+ . In contrast, the contributions to the VEV from the constructed random process are defined by analytic extensions of (14) to physical times. The two-point VEV for a free field indexed by sampling functions from E+ can be derived by employing sampling functions supported at τ1 = τ2 = 0. hΦf (ϕ1 (t1 )) 0, Φf (ϕ2 (t2 )) 0i = S2f (ϑϕ∗1 , ϕ2 ) −ωm (τ1−τ2+i(t1−t2 )) Z Z e eip·(x1 −x2 ) = dξ1 dξ2 dp 2ωm Z dq1 eiq1 x1 h1 (q1 )∗ (ωq1 δ(−τ1 ) − δ 0 (−τ1 )) (2π)d/2−1 Z dq2 e−iq2 x2 h2 (q2 ) (ωq2 δ(τ2 ) − δ 0 (τ2 )) (2π)d/2−1 Z = (2π)d dp (2ωm ) e−iωm (t1−t2 ) h1 (−p )∗ h2 (−p ) Z = (2π)d
(49)
dp (2ωm ) e−iωm (t1−t2 ) h1 (p )∗ h2 (p ).
This does not vanish at equal times. To be developed in Sec. 5.1, this result agrees with the squared norm of the state resulting from a free field creation operator acting on the vacuum. That is, this result provides the squared norm of the one particle states.
908
G. E. JOHNSON
The contribution from the augmentation by a free field to the VEV of the square of the field vanishes.
h0, Φf (ϕ1 (t1 ))Φf (ϕ2 (t2 )) 0i = S2f (ϕ1 , ϕ2 ) −ωm (τ1−τ2+i(t1−t2 )) Z Z e eip·(x1 −x2 ) = dξ1 dξ2 dp 2ωm Z dq1 e−iq1 x1 h1 (q1 ) (ωq1 δ(τ1 ) − δ 0 (τ1 )) (50) (2π)d/2−1 Z dq2 e−iq2 x2 h2 (q2 ) (ωq2 δ(τ2 ) − δ 0 (τ2 )) (2π)d/2−1 = 0.
4.7. Four-point function The four-point function describes the elastic scattering amplitudes. Scattering can be established by examining this four-point function (d > 2). Only the four˜ point truncated functions for the base model example with R(p) = 1/(|p|2 + m2 ) is developed. The four-point truncated functions for free quantum fields vanish. The scattering amplitudes for any-in, any-out are evaluated from the multiplepoint functions evaluated using (35) substituted into the definition for the Schwinger functions, (32). In the case of the four-point function, this substitution produces,
T
S4 (ξ)4 = (2π)3 cˆ4 ×
3 Z Y
dpk eipk ·(xk −x4 )
k=1
e−ω1 (τ1 −τ4 ) e−ω3 (τ3 −τ4 ) e−ω2 (τ2 −τ4 ) 2 2ω3 2ω2 2ω1 (ω4 − (ω1 + ω2 + ω3 )2 )
+
e−ω4 (τ3 −τ4 ) e−ω2 (τ2 −τ3 ) e−ω1 (τ1 −τ3 ) 2 2ω4 2ω2 2ω1 (ω3 − (ω1 + ω2 − ω4 )2 )
+
e−ω4 (τ2 −τ4 ) e−ω3 (τ2 −τ3 ) e−ω1 (τ1 −τ2 ) 2 2ω4 2ω3 2ω1 (ω2 − (ω1 − ω3 − ω4 )2 )
+
e−ω4 (τ1 −τ4 ) e−ω3 (τ1 −τ3 ) e−ω2 (τ1 −τ2 ) 2 2ω4 2ω3 2ω2 (ω1 − (ω2 + ω3 + ω4 )2 )
.
(51)
Applying the analytic extension, the Wightman function is obtained as a boundary value. With δ ± (p2(o) − ω 2 ) = θ(±p(o) )δ(p2(o) − ω 2 ), a procedure similar to that
909
INTERACTING QUANTUM FIELDS
followed in the evaluation of the two-point function results in 4 Z Y T 3 dpk e−ipk xk δ(p1 + p2 + p3 + p4 ) W4 (x)4 = (2π) cˆ4 k=1
×
δ + (p21(o) − ω12 )δ + (p22(o) − ω22 )δ + (p23(o) − ω32 )
+ δ + (p21(o) − ω12 )δ + (p22(o) − ω22 ) + δ + (p21(o) − ω12 )
ω32
ω42
1 − p24(o)
1 δ − (p24(o) − ω42 ) − p23(o)
1 δ − (p23(o) − ω32 )δ − (p24(o) − ω42 ) ω22 − p22(o)
! 1 δ − (p22(o) − ω22 )δ − (p23(o) − ω32 )δ − (p24(o) − ω42 ) . (52) + 2 ω1 − p21(o) The Poincar´e covariance of the boundary value of the analytic continuation is manifest, and the spectral support property of this component of the Wightman function is also apparent after a short analysis. The spectral support property is a consequence of the properties of the constructed Schwinger functions [11]. Nevertheless, it is reassuring to explicitly evaluate the consequences of the spectral support property for T W4 . In the first three terms of (52), p1 is in the closed forward cone V + explicitly. In the fourth term, p1 = −(p2 + p3 + p4 ) with p2 , p3 , p4 ∈ V − and consequently p1 ∈ V + in this term also. In the first two terms, p1 + p2 is in the forward cone explicitly. In the third and fourth terms, p1 + p2 = −(p3 + p4 ) with p3 , p4 ∈ V − so p1 + p2 is in the closed forward cone. In the first term, p1 + p2 + p3 is in the forward cone explicitly. In the latter three terms, p1 + p2 + p3 = −p4 with p4 ∈ V − so p1 + p2 + p3 is in the closed forward cone. Translation invariance provides that p1 + p2 + p3 + p4 = 0 which is in the closed forward cone. The apparent principal value singularities in the four-point function at p2k = m2 are precursors of the scattering exhibited by these constructed field models. These singularities cancel in pairs in (51), consistent with the previous bounds on the Schwinger functions. 5. Scattering Amplitudes The remaining task is to establish the physics modeled by the constructed QFTs. The constructed QFTs of self-coupled bosons exhibit scattering when the underlying random process has nonnormal statistics. Scattering is described given the decomposition of the fields into particle states. Within a Wightman axiom-based QFT model, the Haag–Ruelle scattering theory establishes definitions of free particle states, and the Lehmann–Symanzik–Zimmerman (LSZ) reduction evaluates scattering amplitudes in terms of VEV of the fields [9]. The structure of E+ does not allow the established definition of an asymptotically free annihilation operator yet an LSZ reduction can be developed for these constructed QFTs using only the particle creation operator. This section develops asymptotic free particle states and their scattering theory for the constructed QFTs.
910
G. E. JOHNSON
5.1. Asymptotic conditions For brevity and conciseness of the notation, the development will again be ˜ limited to the augmented base models with root functions given by R(p) = 2 2 1/(|p| + m ). The concept of scattering is that as incoming and scattered states separate and become noninteracting, the quantized fields decompose into free fields of particular masses. Free particle descriptions are tied to the interacting field descriptions through asymptotic equality of operators defined in the interacting QFT with free field theory particle creation operators. In established developments, asymptotically free creation (a∗m ) operators are related to the field by particular sampling functions, that with the adopted sign conventions are given by [9] Z dp ˜ ∗. √ g(p)(ωm + p(o) ) e−i(ωm −p(o) )t Φ(p) (53) a∗m (g, t) = 2π Asymptotically, the values of the test function g(p) are the coefficients in an expansion of a packet state into plane waves of particle mass m. (The Haag–Ruelle development includes a mollifier function that is unnecessary in this construction.) The association of the creation operator with the constructed field begins with manipulation of the definitions for derivative and Fourier transform of generalized functions. The field is not self-adjoint in the constructed model, but with the require˜ ˜ ∗ . Assuming that the definition ment that the field is real, identify Φ(−p) = Φ(p) (53) is sensible, find that, Z Z dp(o) ˜ √ dp e−iωm t g(p)(ωm + p(o) ) eip(o) t Φ(−p) a∗m (g, t) = 2π Z Z dp(o) ∂ ˜ √ eip(o) t Φ(−p) = dp e−iωm t g(p) ωm − i ∂t 2π Z Z ∂ dx −iωm t p e−ip·x Φ(x) . dp e = g(p) ωm − i (54) d−1 ∂t (2π) Applying the definition of the field operators (43), vacuum expectation values can be associated with Schwinger functions. Z Z dx ∗ −i(ωm t+p·x) p dp e g(p) h0, . . . , am (g, t) . . . 0i = (2π)d−1 ∂ Sne (. . . , (τ + it, x) . . .) . × ωm − i ∂t τ =0 In these expressions, the appropriate analytic extension must be selected. The appropriate analytic continuation corresponds to the physical domain field ordering (a Euclidean domain time-ordering). Given that the field operators defining the VEV of the creation operators are in the order Φ(x1 )Φ(x2 ) . . . Φ(xn ), the Euclidean times must be in the order τk > τk+1 . The analytic extension is in the difference variables, τk − τk+1 + itk − itk+1 .
INTERACTING QUANTUM FIELDS
911
Within these regions of holomorphy, the Schwinger functions have derivatives. Consequently, there is an identity relating Euclidean and physical domain temporal derivatives for each indicated field ordering. ∂ ∂ e Sn (. . . , (τ + it, x) . . .) = ωm + Sne (. . . , (τ + it, x) . . .) . ωm − i ∂t ∂τ
(55)
The expression for the VEV of the creation operators becomes Z dx p dp e−i(ωm t+p·x) g(p) (2π)d−1 ∂ Sne (. . . , (τ + it, x) . . .) × ωm + ∂τ τ =0 ! Z Z dp e−i(ωm t+p·x) 0 p = dξ g(p)(ωm δ(τ ) − δ (τ )) (2π)d−1
h0, . . . , a∗m (g, t) . . . 0i =
Z
× Sne (. . . , (τ + it, x) . . .) . Recalling the definition of the quantum fields (43), the creation operator is identified as, √ 2π a∗m (g, t) = Φ(ϕ(t)) ˆ (56) with Z ϕ(ξ; ˆ t) = Z =
dp e−iωm t−ip·x p g(p)(ωm δ(τ ) − δ 0 (τ )) (2π)d−2 dp e−ipξ p g(p) e−iωm t (ωm + ip(o) ) . (2π)d
(57)
These ϕˆ are complex valued. These ϕ(ξ; ˆ t) are solutions of the Klein–Gordon equation when considered as functions of (t, x). It is remarkable that the same set of sampling functions that leads to a positive seminorm enabling construction of the field operators contains the functions required to formulate the asymptotic condition. The creation operators are associated with selected elements within E+,0 , elements that are parametrized by a physical time. These elements have a test function h(p) equal to e−iωm t g(p) and g(p) is independent of time. Following the preceding steps backwards provides a definition within the constructed Hilbert space of the creation operator given by (53). In established developments, the particle annihilation operator is the adjoint of the creation operator. In these models constructed upon E+ , the established definition for an annihilation operator in terms of the adjoint of the field is not in the constructed Hilbert space. The constructed function space, E+ , lacks the functions required to define the established form for an annihilation operator. Identifying sin(ωt)/(πω) as a delta sequence for t → ∞, the Lorentz covariance
912
G. E. JOHNSON
of the asymptotic limit of (53) for g(p) → δ(p − pk ) is evident in the form Z dp ˜ a∗m (g, t) − a∗m (g, −t) = −2i √ g(p)(ωm + p(o) ) sin((ωm − p(o) )t) Φ(−p) 2π √ Z sin((ωm − p(o) )t) ˜ Φ(−p) . = i 2π dp g(p)(p2 − m2 ) π(ωm − p(o) )
(58)
5.2. Convergence to free fields To establish the interpretation of the asymptotic states as freely propagating particles that approach, scatter and produce particles, it is sufficient to show that the VEV of the constructed models coincide with free field theories as t → ±∞. If the truncated functions defined by (12) vanish as all tk = t → ±∞, then the augmentation by a free quantum field (Sec. 3.1) achieves asymptotic convergence of the interacting field theory to a free quantum field. Asymptotically, the free field VEV will be all that remains. Since the constructed Wightman functions are slow growth functions of time and elements of the duals of S n(d−1) , they are elements of the duals of S nd and Haag– Ruelle results [9] can be cited to prove convergence of the truncated functions to zero for n ≥ 3. Also, the identity (26) implies that the truncated functions vanish on timelike planes except possibly for VEV of powers of the field (h0, Φn 0i). For coinciding times, (26) implies that TSn+m ((ϑϕ∗k )n (ϕ` )m ) = 0 which is equivalent to T
hΦn 0, Φm 0i = 0
at equal times, as long as n ≥ 1 and m ≥ 1. These relations hold for all equal times, finite or infinite, and all elements of E+ . With the Haag–Ruelle results and the random process contribution to hΦ0, Φ 0i vanishing at equal times, () and (26), asymptotic convergence to a free field will be demonstrated if (59) lim T h0, a∗m (g1 , t)a∗m (g2 , t) 0i → 0 . t→∞
Using (48) and the relation of g to h, lim
t→∞
T
h 0, a∗m (g1 , t)a∗m (g2 , t) 0i = lim (2π)d cˆ2 t→∞
Z
dp g1 (p) g2 (−p) e−i2ωm t . 2ωm
Bounding, Z t
dp g1 (p) g2 (−p) e−iωm t = i 2ωm
Z
de−iωm t dp g1 (p) g2 (−p) 2ωm dωm
using the chain rule, ωm d d|p| d d = = dωm dωm d|p| |p| d|p| integration by parts, and that the remaining factors are absolutely summable (gk (p) are test functions), it follows that T h0, a∗m (g1 , t)a∗m (g2 , t) 0i < C/|t| for a constant
913
INTERACTING QUANTUM FIELDS
C and d > 2. These results imply that all the truncated VEV of products of a∗m vanish as |t| → ∞ for n > 2, and for n = 2, the VEV converge to those of a free field theory. The vanishing of the truncated Schwinger functions at equal times as time grows without bound leaves only the contributions of the free field to the VEV in the asymptotic regions. This implies that the VEV of a∗m converge to the VEV of a free field theory creation operator and ensures a particle interpretation of the interacting field in the asymptotic regions. Individual particle states will be labeled gj and describe (in the asymptotic regime) freely propagating wave packets of rest mass m. The states of definite momentum (simultaneous eigenstates of each of the generators of spacetime translations) are not elements of the Hilbert space, but plane wave states can be approximated by packet states that are strongly supported near one momentum. There are delta sequences within S(Rd−1 ). One class of delta sequences will be distinguished as G(p) and used to construct a subclass of E+ that models plane wave states. These Gk ∈ S(Rd−1 ) are particular choices for the test functions g defined by (57), nd−1 e−n (p−pk ) √ π d−1 2
Gk (p) =
2
(60)
and lim G(p − pk ) → δ(p − pk ) .
n→∞
An additional time dependent factor, ei(p−pk )·uk tk with a velocity selected to be compatible with the momentum, ωk uk = pk , provides the more intuitive picture of a packet that propagates once through the interaction region. However, the scattering results in the plane wave limit are independent of uk , so to be consistent with the earlier assumption that g is independent of time, this factor is neglected. The plane wave limit for the two-point VEV is derived from (), (49), and the summation, Z ∞ p 2 2 ds e−αs +βs = π/α eβ /(4α) . −∞
The result is ha∗m (G1 , t)0, a∗m (G2 , t)0i
Z
2ωm n2(d−1) e−n (p−p1 ) e−n = dp 2π π d−1 d−1 2 2 2ωm n √ = e−n (p1 −p2 ) /2 2π 2π ≈
2ωm δ(p1 − p2 ) 2π
2
2
2
(p−p2 )2
(61)
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G. E. JOHNSON
as n → ∞. The free field truncated function was included with a coefficient of unity, and any change in the relative scale of the free field component can be absorbed into the constants cˆn or equivalently, cn . In the center-of-mass (barycentric) coordinates for the collision of two particles, one of the momenta is (|pk |, 0, . . . , 0) and the other is (−|pk |, 0, . . . , 0). The dimensions of the wave packets are (nearly) independent of time since the measure of the delta sequences are strongly concentrated near p = pk . ˆ with ϕˆ ∈ E+ defined by The creation operators, a∗m (G, t), derived from Φ(ϕ(t)) (57) and (60) are asymptotically free creation operators for particles near a definite momentum. 5.3. Scattering amplitudes Interaction is exhibited if the elementary particles are deflected from their initial states. Define asymptotically free multiple particles states as
lim
t→±∞
n Y
a∗m (gk , t)|0i = |g1 . . . gn(out/in) i .
k=1
The (out) label corresponds to the t → ∞ limit, and the (in) label corresponds with the t → −∞ limit. The Schwinger functions are symmetric so the equal time fields commute, (24). The states of bosons must be symmetric in the particle labels so for the in or out states the state label of the first particle may be taken as g1 and (out) (out) so forth since |g1 g2 i = |g2 g1 i from [a∗m (g1 , t), a∗m (g2 , t)] = [Φ(ϕˆ1 (t)), Φ(ϕˆ2 (t))]/2π = 0 .
The two-particle elastic scattering amplitude for nonforward directions is given by hg1 g2 (out) |g3 g4 i = lim ha∗m (g1 , t)a∗m (g2 , t) 0, a∗m (g3 , −t)a∗m (g4 , −t) 0i (in)
t→∞
= lim
t→∞
Z
dξ1 dξ2 dξ3 dξ4 (ϑϕˆ∗1 (ξ1 ; t))(ϑϕˆ∗2 (ξ2 ; t))
× ϕˆ3 (ξ3 ; −t)ϕˆ4 (ξ4 ; −t)
1 2π
2
T S4 (t − iτ, x)4
(62) τk >τk+1
evaluated with t3 = t4 = −t and t1 = t2 = t as t → ∞. Substitution of the truncated four-point Wightman function (52) gives
915
INTERACTING QUANTUM FIELDS
(out) (in) hg1 g2 |g3 g4 i
= (2π)ˆ c4
4 Z Y
dξ`
`=1
×
dpk eipk ·(xk −x4 )
k=1
(ϑϕˆ∗1 (ξ1 ; t))(ϑϕˆ∗2 (ξ2 ; t))ϕˆ3 (ξ3 ; −t)ϕˆ4 (ξ4 ; −t)
×
3 Z Y
e−ω3 (τ3 −τ4 ) e−ω2 (τ2 −τ4 +2it) e−ω1 (τ1 −τ4 +2it) (2ω1 )(2ω2 )(2ω3 )(ω42 − (ω1 + ω2 + ω3 )2 )
+
e−ω4 (τ3 −τ4 ) e−ω2 (τ2 −τ3 +2it) e−ω1 (τ1 −τ3 +2it) (2ω1 )(2ω2 )(2ω4 )(ω32 − (ω1 + ω2 − ω4 )2 )
+
e−ω4 (τ2 −τ4 +2it) e−ω3 (τ2 −τ3 +2it) e−ω1 (τ1 −τ2 ) (2ω1 )(2ω3 )(2ω4 )(ω22 − (ω1 − ω3 − ω4 )2 )
+
e−ω4 (τ1 −τ4 +2it) e−ω3 (τ1 −τ3 +2it) e−ω2 (τ1 −τ2 ) (2ω2 )(2ω3 )(2ω4 )(ω12 − (ω2 + ω3 + ω4 )2 )
(63)
defined with (ω4 )2 = m2 + (p1 + p2 + p3 )2 . Substitution for the ϕˆ and the Fourier Inversion theorem provides that, Z 2d+1
= (2π)
cˆ4
dτ1 dτ2 dτ3 dτ4
3 Z Y
dpk eiω1 t g1 (−p1 )∗ (ω1 δ(τ1 ) + δ 0 (τ1 ))
k=1
×e
iω2 t
∗
g2 (−p2 ) (ω2 δ(τ2 ) + δ 0 (τ2 )) eiω3 t g3 (p3 )(ω3 δ(τ3 ) − δ 0 (τ3 ))
× eiω4 t g4 (−p1 − p2 − p3 )(ω4 δ(τ4 ) − δ 0 (τ4 )) −ω3 (τ3 −τ4 ) −ω2 (τ2 −τ4 +2it) −ω1 (τ1 −τ4 +2it) e e e × (2ω1 )(2ω2 )(2ω3 )(ω42 − (ω1 + ω2 + ω3 )2 ) +
e−ω4 (τ3 −τ4 ) e−ω2 (τ2 −τ3 +2it) e−ω1 (τ1 −τ3 +2it) (2ω1 )(2ω2 )(2ω4 )(ω32 − (ω1 + ω2 − ω4 )2 )
+
e−ω4 (τ2 −τ4 +2it) e−ω3 (τ2 −τ3 +2it) e−ω1 (τ1 −τ2 ) (2ω1 )(2ω3 )(2ω4 )(ω22 − (ω1 − ω3 − ω4 )2 )
e−ω4 (τ1 −τ4 +2it) e−ω3 (τ1 −τ3 +2it) e−ω2 (τ1 −τ2 ) + (2ω2 )(2ω3 )(2ω4 )(ω12 − (ω2 + ω3 + ω4 )2 )
.
(64)
The Euclidean time integrations of two of the terms vanish, but the remaining terms become 2d+1
= (2π)
cˆ4
3 Z Y
dpk eiω1 t g1 (−p1 )∗ eiω2 t g2 (−p2 )∗ eiω3 t g3 (p3 ) eiω4 t
k=1
× g4 (−p1 − p2 − p3 )
e−2iω3 t e−2iω4 t e−2iω1 t e−2iω2 t + ω3 + ω4 − ω1 − ω2 ω1 + ω2 − ω3 − ω4
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G. E. JOHNSON
2d+1
= (2π)
× 2i
cˆ4
3 Z Y
dpk g1 (−p1 )∗ g2 (−p2 )∗ g3 (p3 )g4 (−p1 − p2 − p3 )
k=1
sin(ω1 + ω2 − ω3 − ω4 )t ω1 + ω2 − ω3 − ω4
.
(65)
For large times, the support of the scattering amplitude becomes very peaked near a surface that corresponds to energy conservation, sin(ωt) → δ(ω) . t→∞ πω lim
The delta sequence test functions are used to evaluate the scattering amplitudes. Neglecting forward scattering, the amplitude is given by d−1 4 Z n (out) (in) dq1 dq2 dq3 |G3 G4 i = i(2π)2d+2 cˆ4 √ hG1 G2 π d−1 ×
sin(ω1 + ω2 − ω3 − ω4 )t −n2 (q1 +q2 −q3 −p4 )2 e π(ω1 + ω2 − ω3 − ω4 )
× e−n
2
(q1 −p1 )2
e−n
2
(q2 −p2 )2 −n2 (q3 −p3 )2
e
(66)
with ωk2 = m2 + (qj )2 for j = 1, 2, 3 and ω42 = m2 + (q1 + q2 − q3 )2 . Noting that the supports of the delta sequences become highly concentrated, factors with relatively slowly varying momentum dependence may be factored from the summations. Then, d−1 4 n (out) (in) hG1 G2 |G3 G4 i ≈ i(2π)2d+2 cˆ4 √ π d−1 3 Z 2 2 2 2 sin(ωp1 + ωp2 − ωp3 − ωp4 )t Y dq` e−n (q` −p` ) e−n (q1 +q2 −q3 −p4 ) × π(ωp1 + ωp2 − ωp3 − ωp4 ) `=1
with corrections on the order of n−2 . These summations can be viewed as the expected values of independent normal random variates with zero mean and variance 1/(2n2 ). Summation produces (out)
hG1 G2
(in)
|G3 G4
i ≈ i(2π)2d+2 cˆ4 s
×
n2 4π
sin((ω1 + ω2 − ω3 − ω4 )t) π(ω1 + ω2 − ω3 − ω4 )
d−1
e−n
2
(p1 +p2 −p3 −p4 )2 /4
.
(67)
As the packets approach plane waves, (out)
hG1 G2
(in)
|G3 G4
i = i(2π)2d+2 cˆ4 δ(p1 + p2 − p3 − p4 )
(68)
evaluated at pk(o) = ωk . The constant cˆ4 is determined in an expansion in moments of the characteristic function for the underlying random process. In the example
917
INTERACTING QUANTUM FIELDS
of an underlying Poisson random process, the constants that determine scattering and production amplitudes are cˆn = (2π)d−nd/2 (−λ)n with λ a real constant. This is the complete two-particle elastic scattering amplitude for this augmented ˜ base model with R(p) = 1/(|p|2 + m2 ). This describes a QFT of self-coupled bosons that must exhibit scattering since the elastic scattering amplitude is nonvanishing. A similar procedure would evaluate the five point function to find the amplitude for scattering with production of a single particle, and the six point for the amplitude for scattering with production of a pair. In first Born approximation, the nonforward amplitude is −i times real functions of the momentum transfer [2], including a factor, Z V˜ (q) =
dr V (r) e−iq·r
(69)
with V (r) the interaction potential. The overall sign change with respect to (68) indicates that the equivalent potential for the exact reaction is attractive (negative) as anticipated from one particle exchange of a neutral meson [2]. 5.4. LSZ reduction This scattering amplitude can also be evaluated in a more established manner, from LSZ reduction. The relation of the field to the creation operator results in the LSZ expression for the scattering amplitudes [9]. In the adopted sign conventions, (out)
hG1 . . . Gk
n Z Y √ dpj Gj (pj ) |Gk+1 . . . Gn(in) i = (−i 2π)n j=1
× (p2j − m2 )h0|T˜ (p1 , . . . , pk , −pk+1 , . . . , −pn )|0i
.
(70)
pj(o) =ωj
T˜(p)n is the Fourier transform of the time-ordered Wightman function, and for the constructed fields, the time-ordered Wightman distribution is defined without mollification since the Wightman functions are bounded functions of time. Multiplication of bounded functions by the Heaviside function is unambiguous. The time-ordered function is the sum over all distinct permutations of the arguments in a product of a Wightman function with Heaviside functions that ensure time-ordered arguments, T
T4 (x)4 =
X
θ(tπ1 − tπ2 )θ(tπ2 − tπ3 )θ(tπ3 − tπ4 ) T W4 (xπ1 , xπ2 , xπ3 , xπ4 ) .
π{1,...,4}
The Fourier transform of the Heaviside function is provided by θ(t) =
i 2π
Z du
e−itu . u + i0+
In a notation with p + u = (po + u, p), the Fourier transform of the time-ordered
918
G. E. JOHNSON
four-point function is T
T˜4 (p)4 =
X
4 Z Y
π{1,...,4} k=1
dxk eipk xk (2π)d/2
× θ(tπ1 − tπ2 )θ(tπ2 − tπ3 )θ(tπ3 − tπ4 ) T W4 (xπ1 , xπ2 , xπ3 , xπ4 ) i3 = (2π)3
X
3 Z Y
π{1,...,4} `=1
du` u` + i0+
˜ 4 (pπ1 − u1 , pπ2 + u1 − u2 , pπ3 + u2 − u3 , pπ4 + u3 ) . × TW
(71)
Dropping the infinitesimal contour deformations defining the Heaviside functions, substitution of the Fourier transform derived from (52) provides T
T˜4 (p)4 = (−iˆ c4 )(2π)2d
X
4 Y δ(p1 + p2 + p3 + p4 ) 16ω1 ω2 ω3 ω4
π{1,...,4} k=1
×
1 (pπ1 (o) − ωπ1 )(pπ2 (o) − ωπ2 + pπ1 (o) − ωπ1 )
×
2ωπ4 (pπ3 (o) − ωπ3 + pπ2 (o) − ωπ2 + pπ1 (o) − ωπ1 )(ωπ2 4 − (ωπ1 (o) + ωπ2 (o) + ωπ3 (o) )2 )
+
1 (pπ1 (o) − ωπ1 )(pπ2 (o) − ωπ2 + pπ1 (o) − ωπ1 )
×
2ωπ3 (ωπ2 3 − (ωπ4 − ωπ1 − ωπ2 )2 )(−pπ4 (0) − ωπ4 )
+
2ωπ2 (pπ1 (o) − ωπ1 )(ωπ2 2 − (ωπ3 + ωπ4 − ωπ1 )2 )
×
1 (−pπ3 (o) − ωπ3 − pπ4 (o) − ωπ4 )(−pπ4 (o) − ωπ4 )
2ωπ1 (ωπ2 1 − (ωπ2 + ωπ3 + ωπ4 )2 )(−pπ2 (o) − ωπ2 − pπ3 (o) − ωπ3 − pπ4 (o) − ωπ4 ) 1 . × (−pπ3 (o) − ωπ3 − pπ4 (o) − ωπ4 )(−pπ4 (o) − ωπ4 )
+
(72)
In (70), the time-ordered product of fields is a sum over all 4! distinct permutations for four arguments times the four terms of T W4 . The result is a sum of 96 terms in the expression for the time-ordered function. To evaluate the exQ4 pression k=1 (p2k − m2 ) T˜4 (p1 , p2 , −p3 , −p4 ) at pk(o) = ωk , it is convenient to set pk(o) − ωk = k ≈ 0 with the translation invariance constraint p1 + p2 = p3 + p4 . Q4 The summation over permutations has a limit, k=1 (p2k −m2 ) T˜4 (p1 , p2 , −p3 , −p4 ) = iˆ c4 (2π)2d δ(p1 + p2 − p3 − p4 ). This limit of the sum is not equal to the sum of the limits. Substitution into (70) agrees with (68).
919
INTERACTING QUANTUM FIELDS
5.5. Cross sections and equivalent potentials The joint likelihood for the result of a measurement corresponding to the (out) scattered state described by |G1 G2 i is the trace of the initial state density matrix projected onto the subspace spanned by the final states [27, 28]. For a system pre(in) pared in the pure state |G3 G4 i, projection onto the subspace of final states near (out) |G1 G2 i gives the likelihood (out)
Trace(P ρ) = µ(dp1 )µ(dp2 )
|hG1 G2
(in)
|G3 G4
(out) kG1 G2 k2
i|2
(in) kG3 G4 k2
with the evident definitions for the projection P and the state density matrix ρ. (in) The states |G1 G2 i have momenta near p1 , p2 . The measure on subsets of state labels, µ(dpk ), is fixed by the idempotence property (P 2 = P ) of the projection P . The squared magnitude of the scattering amplitude (67) is used to evaluate the transition probability for nonforward elastic scattering. The cross section depends only on the likelihood of a transition to the final state conditioned upon preparation of the initial state. Plane wave states are not within the Hilbert space of states, but a cross section is defined in the plane wave limit. This well-known result is briefly reproduced here in the adopted notation with definitions appropriate for these constructions of scattering amplitudes. The established correspondence of QFT to measurements of the number of particles scattered into an angular segment by a target of known composition from a beam of known flux is the differential cross section [1, 2, 29]. The differential cross section for two-particle scattering without production is defined by summation of the differential cross section for scattering into a momentum increment dp1 dp2 , dσ = lim A Trace(P ρ) n→∞
(73)
with a flux corrected interaction area defined as A=
V ω3 ω4 p . 2t (p3 · p4 )2 − m23 m24
(74)
In the barycentric frame this is A=
V 2t|v3 − v4 |
with the volume V fixed by the overlapping support of the incoming wave packets. To evaluate the cross section, the required four-point functions result from substitution of the two- and four-point truncated functions, (61) and (67), into a linkcluster identity. W4 (x)4 = T W2 (x1 , x2 ) T W2 (x3 , x4 ) + T W2 (x1 , x3 ) T W2 (x2 , x4 ) + T W2 (x1 , x4 ) T W2 (x2 , x3 ) + T W4 (x)4 .
(75)
920
G. E. JOHNSON
The vacuum polarization, T W1 , vanishes for the Poisson distributed models, although vacuum polarization could readily be included into this model by selection of a finite a1 . Substitution of (51), (59), and (61) provides the result for norming the states. (in)
hG1 G2
(in)
|G3 G4
i = lim ha∗m (G1 , t)a∗m (G2 , t) 0|0ih0|a∗m (G3 , t)a∗m (G4 , t) 0i t→∞
+ ha∗m (G1 , t) 0|a∗m (G3 , t) 0iha∗m (G2 , t) 0|a∗m (G4 , t) 0i + ha∗m (G1 , t) 0|a∗m (G4 , t) 0iha∗m (G2 , t) 0|a∗m (G3 , t) 0i !2 2 2 2 2 nd−1 e−n (p1 −p3 ) /2 e−n (p2 −p4 ) /2 = 2ω1 2ω2 p (2π)d+1 + 2ω1 2ω2
nd−1
!2 e−n
2
p (2π)d+1
(p1 −p4 )2 /2 −n2 (p2 −p3 )2 /2
e
(in)
.
(in)
The truncated components of four-point functions T hGj Gk |Gj Gk i vanish as a result of the convergence of a∗m to a free field. This holds for either in or out states. For nonforward scattering, p1 6= p2 and p3 6= p4 , and (61) provides (in)
(out) 2 kj6=k
kGj Gk k2j6=k = kGj Gk (in) kGk k2
=
(out) kGk k2
(in) 2
≈ kGj
2ωk = 2π
(in)
k kGk k2
n √ 2π
d−1 .
The measure on state labels is evaluated using Trace(P 2 ) = Trace(P ). For large n, Z µ(dp1 )µ(dp2 ) Z
Z µ(dp1 )µ(dp2 )
= Z ≈
µ(dp1 )µ(dp2 )µ(dp0 1 )µ(dp0 2 ) Z
+
µ(dp0 1 )µ(dp0 2 )
(out)
|hG1 G2
(out) 2 k
kG1 G2
(out)
|G10 G20
i|2
(out) 2 k
kG10 G20
(2π)d−1 δ(p1 − p0 1 )δ(p2 − p0 2 ) n2d−2
µ(dp1 )µ(dp2 )µ(dp0 1 )µ(dp0 2 )
(2π)d−1 δ(p1 − p0 2 )δ(p2 − p0 1 ) n2d−2
solved by 1 nd−1 dpk . µ(dpk ) = √ p 2 (2π)d−1 Substitution of (67) and the normalization of states results in the evaluation for
921
INTERACTING QUANTUM FIELDS
nonforward scattering. For p1 , p2 6= p3 and p1 , p2 6= p4 , 2 sin((ω1 + ω2 − ω3 − ω4 )t) (out) (in) 2 4d+4 2 |hG1 G2 |G3 G4 i| ≈ (2π) (ˆ c4 ) π(ω1 + ω2 − ω3 − ω4 ) 2d−2 2 2 n × √ e−n (p1 +p2 −p3 −p4 ) /2 4π (out) (in) |hG1 G2 |G3 G4 i|2 2t sin((ω1 + ω2 − ω3 − ω4 )t) 6d+5 2 Y = (2π) (ˆ c ) 4 (out) 2 (in) 2 π(ω + ω − ω − ω ) 1 2 3 4 kG1 G2 k kG3 G4 k (2ωk ) k
×
n3
1 √ 8π
d−1
n √ 2π
d−1
e−n
2
(p1 +p2 −p3 −p4 )2 /2
.
In the limit of plane waves, box normalization [1, Sec. 3.4] identifies V δ(p) (δ(p))2 = (2π)d−1 and this establishes a relationship between the volume defined for beam flux, (74), and the parameter of the delta sequences, n. The square of the delta sequence in (67) results in the identification !d−1 √ d−1 d−1 2π n n 1 d−1 √ √ = . V = (2π) 2 4π 2 These results, collected together in (73), produce the exact differential cross section for elastic nonforward two-particle scattering in the base model QFT. 2d−2 (out) (in) |hG1 G2 |G3 G4 i|2 n dσ = lim lim dp1 dp2 A √ (out) (in) t→∞ V →∞ 2π kG1 G2 k2 kG3 G4 k2 =
dp1 dp2 (2π)5d+6 (ˆ c4 )2 p δ(p1 + p2 − p3 − p4 ) . 2ω1 2ω2 22d+1 (p3 · p4 )2 − m4
(76)
In the center-of-mass (barycentric) frame, p4 = (p3(o) , −p3 ), and on the mass shell p (p3 · p4 )2 − m4 = 2ω3 |p3 | . Integration over p2 and all magnitudes for p1 defines the elastic scattering cross section as Z ∞ δ(2ω1 − 2ω3 ) (2π)5d+6 (ˆ dσ c4 )2 = |p1 |d−2 d|p1 | 2 2d+1 dΩ 4(ω1 ) 2|p3 |ω3 2 0 =
|p3 |d−4 (2π)5d+6 (ˆ c4 )2 . 16(ω3 )2 22d+1
(77)
An equivalent potential is defined by setting the center-of-mass frame elastic scattering cross section equal to the first Born approximation for scattering by a
922
G. E. JOHNSON
˜ potential. In the case of the base model with root function R(p) = 1/(|p|2 + m2 ), this potential is attractive since the signs of (68) and (69) are opposite [2]. Using the result for scattering from a potential in four dimensions (d = 4), identify m 2 dσ ≈ |V˜ (q)|2 dΩ 4π 2 (2π)13 cˆ4 √ = (78) 64 2 ω3 with a momentum transfer defined as q = p3 − p1 . Extracting a factor of e−ip1 ·x from the absolute value, Z dq iq·x ˜ e V (x) = V (q) (2π)3 Z cˆ4 dp ip·x e . = (2π)11 √ ωm 32 2 m If the relativistic momentum dependence (1/ωm ) is discounted, this produces a point potential, δ(x), but including this momentum dependence produces Z eikr cos θ cˆ4 k 2 dk sin θ dθdφ √ V (x) = (2π)11 √ 32 2 m m2 + k 2 Z ∞ d cos(kr) cˆ4 = −(2π)12 √ dk √ dr 16 2 mr m2 + k 2 0 = (2π)12 cˆ4
K1 (mr) √ . 16 2 r
At small r, K1 (mr)/r ∝ 1/r2 , and at large r, K1 (mr)/r ∝ e−mr /r3/2 [30]. In four dimensions and in the first Born approximation, this central potential has the same elastic differential cross section as the constructed QFT. The form of this effective potential depends on the dimensionality of spacetime d. Appendix. Characteristic Functions The form of the function appearing in the representation for the characteristic function of a generalizedR random process with independent, identically distributed values (7) (log(L(f )) = dξ G(f (ξ), f 0 (ξ), . . . , f (k) (ξ))) is given in [16], Z dσ(λ)(e
G(ρ) =
i(λ,ρ)
kλk>0
− α(λ)(1 + i(λ, ρ))) +
2 X |k|=0
with σ a positive tempered measure such that the summations Z dσ(λ)kλk2 < ∞ 0
Z
dσ(λ) < ∞ 1
ak
(iρ)k (k!)n
(79)
923
INTERACTING QUANTUM FIELDS
converge. The function α is a Fourier transform of a test function of compact support such that (α(λ) − 1) has a third order zero at λ = 0. Also, for |k| = 2, X ar+s βr βs ≥ 0 |r|=|s|=1
Z
and a0 +
kλk>0
dσ(λ)(1 − α(λ)) = 0 .
The notation includes ρ = (ρ1 , ρ2 , . . . , ρn ) kλk2 = λ21 + λ22 . . . + λ2n |k| = k1 + k2 . . . + kn ρk = ρk11 ρk22 . . . ρknn (k!)n = k1 ! k2 ! . . . kn ! (λ, ρ) =
n X
λk ρk .
k=1
This structure can be understood by noting several results intermediate to this final representation. Every positive-definite continuous distribution such that L(0) = 1 defines a generalized random process with a characteristic function L(ϕ) (Borcher’s theorem). For L(ϕ) to be positive-definite, it is sufficient that esG(ρ) be positivedefinite for any s > 0. Finally, it is necessary and sufficient for esG(ρ) to be positiveP P definite that G(ρj −ρk )cj c∗k ≥ 0 for all complex coefficients constrained by ck = 0. The form (79) then follows from the Bochner–Schwartz result that the positive generalized functions are Fourier transforms of positive measures. The freedom P permitted by the constraint ck = 0 and the condition G(0) = 0 then determine the form. The base models include only one component, ρ = (ρ1 ). Examples include free and generalized free fields which can be derived from a stationary, normally distributed process with independent values at each point, R 2 1 L(ϕ) = e− 2 dξ ϕ(ξ) Z
1
Pϕ (x) = q R 2π dξ ϕ(ξ)2
−
x
ds e
2
R
s2 dξ ϕ(ξ)2
.
−∞
These normal processes are the archetypal examples of generalized random processes. Nonnormally distributed base models include the generalized Poisson processes. The Poisson process models have a measure σ(λ) concentrated on one point λ0 , and a0 = a1 = a2 = 0, α(λ0 ) = 1. In this case, 0
G(ρ) = eiλ ρ − 1 − iλ0 ρ
924
G. E. JOHNSON
and the characteristic function of the random process is given by (7) Z 0 log(L(ϕ)) = dξ(eiλ ϕ(ξ) − 1 − iλ0 ϕ(ξ)) . For this Poisson distributed case, (12) provides that n Z d T iλ0 βR∗ϕ(ξ) 0 dξ(e Sn (ϕ)n = i − 1 − iλ βR ∗ ϕ(ξ)) dβ β=0 Z = (−λ0 )n dξ(R ∗ ϕ(ξ))n for n ≥ 2 and the constants defining the nonforward scattering and boson production amplitudes are given below (29) cˆn = (2π)d−nd/2 cn = (2π)d−nd/2 (−λ0 )n . Appendix.
1
Υn ((τ )n ; 0)
A combinatorial identity for the functions 1 Υn ((τ )n ; 0), (35), is proved by induction. Recall that an antisymmetric matrix is denoted 1 j
the indicated substitution produces, 1
Υn ((τ )n ; 0) = π
`+1
n X
n Y e−ki ωi (τi −τk ) e−ωn−`−1 (τn−`−1 −τk ) | | ωn−`−1 ω ki
k=n−` i=n−`
i
× 1Υn−`−1 ((τ )n−`−2 , τk ; χ`,k + ωn−`−1 ) + θ(ωk − χ`,k ) −(ωk −χ`,k )(τn−`−1 −τk ) e 1 × Υn−`−1 ((τ )n−`−1 ; χ`,k − ωk ) ωn−`−1 −(−ωk −χ`,k )(τn−`−1 −τk ) e − θ(−ωk − χ`,k ) ωn−`−1 ! × 1Υn−`−1 ((τ )n−`−1 ; χ`,k + ωk )
.
INTERACTING QUANTUM FIELDS
925
The definition of the antisymmetric matrix and a combination of factors simplifies this to 1
Υn ((τ )n ; 0) = π
`+1
n X
n Y
k=n−`
i=n−`−1
e−ki ωi (τi −τk )
! 1
|ki |
+ π `+1
n X
ωi
n Y e−ki ωi (τi −τn−`−1 ) |ki |
ωi
k=n−` i=n−`
× θ(ωk − χ`,k ) − π `+1
Υn−`−1 ((τ )n−`−2 , τk ; χ`+1,k )
e−ωk (τn−`−1 −τk ) 1 Υn−`−1 ((τ )n−`−1 ; χ`,k − ωk ) ωn−`−1
n X
n Y e−ki ωi (τi −τn−`−1 ) |ki |
ωi
k=n−` i=n−`
× θ(−ωk − χ`,k )
eωk (τn−`−1 −τk ) 1 Υn−`−1 ((τ )n−`−1 ; χ`,k + ωk ) . ωn−`−1
Relabeling the dummy summation variable in the second set of terms to k 0 = k − 1, and regrouping terms gives
1
n X
Υn ((τ )n ; 0) = π `+1
n Y
k=n−`−1
e−ki ωi (τi −τk ) |ki |
i=n−`−1
ωi
n X
× 1 Υn−`−1 (τ )n−`−2 , τk ;
kj ωj
j=n−`−1
+ π `+1
n−1 X
n Y e−k0 +1,i ωi (τi −τn−`−1 )
k0 =n−` i=n−`
× θ(ωk0 +1 − χ`,k0 +1 )
|k0 +1,i |
ωi
e−ωk0 +1 (τn−`−1 −τk0 +1 ) ωn−`−1
× 1Υn−`−1 ((τ )n−`−1 ; χ`,k0 +1 − ωk0 +1 ) − π `+1
n−1 X
n Y e−ki ωi (τi −τn−`−1 ) |ki |
k=n−` i=n−`
× θ(−ωk − χ`,k )
ωi
eωk (τn−`−1 −τk ) 1 Υn−`−1 ((τ )n−`−1 ; χ`,k + ωk ) . ωn−`−1
P The k = n term in the third set of terms vanishes since θ(− ωj ) = 0, and the k 0 = n − ` − 1 term in the second set of terms was added to the first set of terms P after manipulation and setting θ( ωj ) = 1. Finally, the definition of χ`,k together
926
G. E. JOHNSON
with the identities ak +
n X
n X
kj aj = −ak+1 +
j=n−`
k+1,j aj
j=n−`
=
k X
aj −
j=n−`
n X
aj
j=k+1
provides that sum of the second and third sets of terms vanish. The terms of the second and third sets cancel in pairs (with k = k 0 terms paired). This completes the demonstration of the identity (35).
Table 1. Summary of notation. Notation xk(j)
Description jth component of kth coordinate vector
xk = (tk , xk )
spacetime coordinates, tk = xk(o)
ξk = (τk , xk )
Euclidean coordinates
px = p(o) t − p · x pξ =
d−1 X
p(k) ξ(k)
k=0 d−1 X
|p|2 =
(p(k) )2
Minkowski inner product Euclidean inner product
Euclidean norm squared
k=0
(x)n = x1 , x2 , . . . , xn Sn (ξ)n = Sn (ξ1 − ξ2 , . . . , ξn−1 − ξn ) * + n Y = 0, Ψ(ξ)0
multiple arguments n-point Schwinger functions expected value of random process Ψ
k=1
Wn (x)n = Wn (x1 − x2 , . . . , xn−1 − xn ) boundary values of Wightman functions, * + n Y −(τk +itk )H (τk +itk )H Wn (t − iτ, x)n = 0, e Ψ(0, xk )e 0 k=1
Wn (x)n = h0, Φ(x1 )Φ(x2 ) . . . Φ(xn )0i Z d/2 ˜ ϕ(ξ) = dp e−ipξ ϕ(p)/(2π)
dx eipx W (x)/(2π)d/2
sign convention set by spatial components Z ˜ (p)/(2π)d/2 W (x) = dp e−ipx W
dξ e−ipx S(ξ)/(2π)d/2
S(ξ) =
Z ˜ (p) = W Z ˜ S(p) =
for quantum field operators, Φ(x) Z d/2 ϕ(x) = dp eipx ϕ(p)/(2π) ˜
Z d/2 ˜ dp eipξ S(p)/(2π)
INTERACTING QUANTUM FIELDS
927
References [1] S. Weinberg, The Quantum Theory of Fields, Volume I, Foundations, New York, NY, Cambridge Univ. Press, 1995. [2] F. Gross, Relativistic Quantum Mechanics and Field Theory, New York, NY, John Wiley and Sons, 1993. [3] E. Nelson, “Construction of quantum fields from Markoff fields”, J. Funct. Anal. 12 (1973) 97–112. [4] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, San Francisco, CA, W. H. Freeman and Co., 1973. [5] S. Albeverio, H. Gottschalk and J.-L. Wu, “Convoluted generalized white noise, Schwinger functions and their analytic continuation to Wightman functions”, Rev. Math. Phys. 8 (1996) 763–817. [6] O. Steinman, “Structure of Two-Point Functions”, J. Math. Phys. 4 (1963) 583–588. [7] A. S. Wightman, “Quantum Field Theory in Terms of Vacuum Expectation Values”, Phys. Rev. 101 (1956) 860. [8] R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That, Reading, MA, W. A. Benjamin, 1964. [9] N. N. Bogolubov, A. A. Logunov and I. T. Todorov, Introduction to Axiomatic Quantum Field Theory, trans. by Stephen Fulling and Ludmilla Popova, Reading, MA, W. A. Benjamin, 1975. [10] G. E. Johnson and D. I. Fivel, “Consequences of weakening the positivity property of Wightman quantum field theories”, J. Math. Phys. 21 April (1980) 891–895. [11] K. Osterwalder and R. Schrader, “Axioms for Euclidean Green’s functions”, Commun. Math. Phys. 31 (1973) 83–112. [12] G. Velo and A. Wightman, eds., Constructive Quantum Field Theory (Lecture Notes in Physics, No. 25), Springer-Verlag, Berlin and New York, 1973. [13] I. M. Gel’fand, and G. E. Shilov, Generalized Functions, Vol. 2, trans. M. D. Friedman, A. Feinstein and C. P. Peltzer, New York, NY, Academic Press, 1968. [14] K. Symanzik, “Euclidean Quantum Field Theory, I. Equations for a Scalar Model”, J. Math. Phys. 7 (1966) 510–525. [15] B. Simon, The P (φ)2 Euclidean (Quantum) Field Theory, Princeton, NJ, Princeton Univ. Press, 1974. [16] I. M. Gel’fand, and N. Ya. Vilenkin, Generalized Functions, Vol. 4, trans. A. Feinstein, New York, NY, Academic Press, 1964. [17] K. Osterwalder and R. Schrader, “Axioms for Euclidean Green’s functions II”, Commun. Math. Phys. 42 (1975) 281–305. [18] G. E. Johnson, “Constructions of particular random processes”, Proc. of the IEEE 82 Feb. (1994) 270–285. [19] O. W. Greenberg, “Generalized free fields and models of local field theory”, Ann. Phys. 16 (1961) 158. [20] J. A. Shohat and J. D. Tamarkin, The Problem of Moments, Providence, RI: Amer. Math. Soc., 1963. [21] L. G˚ arding and A. S. Wightman, “Fields as operator-valued distributions in relativistic quantum field theory”, Arkiv Fysik 28 (1965) 129. [22] D. Ruelle, Statistical Mechanics: Rigorous Results, Reading, MA, W. A. Benjamin, 1974. [23] R. P. Feynman, “Space-time approach in quantum electrodynamics”, Phys. Rev. 76 (1949) 769. [24] J. Glimm and A. Jaffe, “Functional integral methods in quantum field theory”, pp. 35–66 in New Developments in Quantum Field Theory and Statistical Mechanics, ed. M. L´evy and P. Mitter, New York, Plenum Press, 1976. [25] R. G. Horn and C. R. Johnson, Topics in Matrix Analysis, New York, NY, Cambridge
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Univ. Press, 1991. [26] E. Hille and S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. colloq. pub. 31 1957. [27] J. von Neumann, Mathematical Foundations of Quantum Mechanics, trans. R. T. Beyer, Princeton, NJ, Princeton Univ. Press, 1974. [28] P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford, The Oxford Univ. Press, reprinted 1991. [29] M. E. Goldberger and K. M. Watson, Collision Theory, New York, John Wiley and Sons, 1964. [30] G. Arfken, Mathematical Methods for Physicists, New York, Academic Press, 1970.
A NEW ALGEBRA FROM THE REPRESENTATION THEORY OF FINITE GROUPS J. JUYUMAYA∗ International Centre for Theoretical Physics P.O. Box 586, 34100 Trieste, Italy Received 7 November 1997 1991 Mathematics Subject Classification: 20C33, 20F36 In this work we define a new algebra. The definition of our algebra arises naturally in the study of certain generators (non standard) for Yokonuma–Hecke algebra [8]. This algebra is linked to the Knot theory via the Vassiliev algebra defined by J. Baez [2].
1. Introduction Let n be a natural number. The purpose of this work is to define and make a preliminary study of a new algebra Jn (u) over C, at parameter u. Our algebra is defined by generators 1, T1 , . . . , Tn−1 , Q1 , . . . , Qn−1 and certain relations, see Definition 2.1. As in the Iwahori–Hecke algebras, we have the generators Ti ’s satisfying braid relations. Further, the generators Ti ’s and Qi ’s satisfy a relation of the Vassiliev algebra [2]: Ti − Ti−1 = (u − u−1 )Qi . Also, we demand Ti Qi = Qi Ti = −u−1 Qi . These relations imply that our generators satisfy all the relations of a generalized braid monoid defined by L. Smolin [1], see also [4]. Then, we obtain that the generators Ti ’s and Qi ’s also satisfy all relations of the Vassiliev algebra, hence the Vassiliev algebra surjects onto our algebra, see (4.3). Now, with the usual picture for the braids Ti ’s and singular braids Qi ’s, we have by the rule of concatenation.
∗ Ti
≡ −u−1
=
Qi
Qi
Na¨ıvely, in the picture the second “equality”, is obtained by using the u-generalized Reidemeister moves (c.f. [11, 13, 14]). In the other words we have a resolution ∗ Regular
Associate at ICTP. This research was partially supported from Chile by DIPUV 01-99. Address after 15 December: Gran Breta¯ na 1041, Valparai¯so, Chile. 929 Reviews in Mathematical Physics, Vol. 11, No. 7 (1999) 929–945 c World Scientific Publishing Company
930
J. JUYUMAYA
in Jn (u) for the concatenation of braids with singular braids, which do not occur in the Vassiliev algebra. The other relation, of skein type, that defines our algebra is: Ti−1 Tj Ti−1 + uTj Ti−1 Tj = Tj−1 Ti Tj−1 + uTi Tj−1 Ti , (u + 1)Qi Qj Qi + uTi Qj Qi = uQj Qi . The original motivation for the definition of the algebra Jn (u), arose from our work on the new generators for the commuting algebra (Yokonuma–Hecke algebra) Yn (q) associated with the permutation representation of the general linear group over Fq , with respect to the maximal upper unipotent subgroup, see [8–10]. More precisely, the Ti ’s and Qi ’s can be realized as operators of intertwining in Yn (q). In fact, the operators Ti ’s correspond to certain operators of the Fourier transform type, and the Qi ’s are simple average operators, see [8]. Thus, we can realize Jn (q) as a subalgebra of Yn (q), see Theorem 3.1. Taking advantage of this realization of Jn (u), we obtain a partial result about their algebraic structure. See Theorem 4.2 and Eq. (4.3). In Sec. 5, we pose several questions that will be studied in a subsequent work. 2. The Algebra Jn (u) Definition 2.1. For n a natural number and u ∈ C\{−1, 0, 1}, let Jn (u) be the associative algebra over C, with generators: −1 , Q1 , . . . , Qn−1 , 1, T1 , . . . , Tn−1 , T1−1 , . . . , Tn−1
subject to the following relations: Ti Ti−1 = Ti−1 Ti = 1
(2.1.1)
Ti − Ti−1 = (u − u−1 )Qi
(2.1.2)
Ti Qi = Qi Ti = −u−1 Qi
(2.1.3)
[Ti , Tj ] = [Qi , Qj ] = [Ti , Qj ] = 0
if |i − j| > 1
(2.1.4)
when |i − j| = 1: Ti Tj Ti = Tj Ti Tj
(2.1.5)
Ti−1 Tj Ti−1 + uTj Ti−1 Tj = Tj−1 Ti Tj−1 + uTi Tj−1 Ti
(2.1.6)
(u + 1)Qi Qj Qi + uTi Qj Qi = uQj Qi .
(2.1.7)
Note that we have a natural inclusion of Jn−1 (u) as a subalgebra in Jn (u). Remark. We can define Jn (u) for u ∈ {−1, 1}. In this case one obtains that the group algebra of the symmetric group is a subalgebra of our algebra, via the map (i, i + 1) 7→ Ti .
A NEW ALGEBRA FROM THE REPRESENTATION THEORY OF FINITE GROUPS
931
By multiplying the relation (2.1.2) by Qi and applying (2.1.3), one has Q2i = Qi .
(2.2)
The Ti ’s satisfy the cubic equation p(Ti ) = 0, where p(x) = x3 +u−1 x2 −x−u−1 . In fact, multiplying the relation (2.1.2) by Ti , we have Ti2 − 1 = (u − u−1 )Qi Ti . Now, from (2.1.3) and (2.1.2): Qi Ti = −u−1 Qi = −u−1 (u − u−1 )−1 {Ti − Ti−1}. So, Ti2 − 1 = −u−1 {Ti − Ti−1 } = (u−2 − 1)Qi .
(2.3)
Finally, multiplying this last equation by Ti , we obtain the cubic relation: Ti3 + u−1 Ti2 − Ti − u−1 = {Ti + u−1 }{Ti + 1}{Ti − 1} = 0 .
(2.4)
Therefore, our algebra is a quotient of the cubic Hecke algebra Hn (p(x)), Hn (p(x)) := C[Bn ]/(p(bj ) ; 0 < j < n) , where Bn denotes the Braid group with the standard Artin presentation, via the generators b1 , . . . , bn−1 . The cubic Hecke algebra was studied by L. Funar [7]. We use from [7] the following result: Funar Lemma. Let wn,j = bn−1 bn−2 · · · bj+1 b2j bj+1 · · · bn−2 bn−1 and rn,j = bj bj+1 · · · bn−2 . In Hn (p(x)) we have bi wn,j = wn,j bi
i 6= j − 1, 1 < i < n − 1
if
−1 , bn−1 wn,j = −u−1 wn,j + wn−1 bn−1 + u−1 rn,j+1 wn,j+1 rn,j −1 wn,j+1 rn,j . wn,j bn−1 = −u−1 wn,j + bn−1 wn−1 + u−1 rn,j+1
Other relations that follow easily from the previous relations are: Qi Tj Ti = Tj Ti Qj , if |i − j| = 1 (from the braid relation and (2.3))
(2.5)
Ti2 Tj Ti = Tj Ti Tj2 , if |i − j| = 1 (from (2.3) and (2.5) )
(2.6)
Ti−2 = 1 + (u2 − 1)Qi = 1 + u−1 {Ti − Ti−1 }
(2.7)
Ti−1 = uTi2 + Ti − u .
(2.8)
We will need the following relations Proposition 2.9. For all i, j such that |i − j| = 1, we have Ti2 Tj Ti2 −Tj2 Ti Tj2 −Ti Tj2 Ti +Tj Ti2 Tj = Tj Ti2 −Ti Tj2 +Ti2 Tj −Tj2 Ti +Ti −Tj +Tj2 −Ti2 . Proof. Using (2.8) and (2.6), we have Ti−1 Tj Ti−1 − Tj−1 Ti Tj−1 = u2 {Ti2 Tj Ti2 − Tj2 Ti Tj2 + Tj − Ti − Tj Ti2 − Ti2 Tj + Ti Tj2 + Tj2 Ti }
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J. JUYUMAYA
On the other hand, from (2.8) Ti Tj−1 Ti = uTi Tj2 Ti + Ti Tj Ti − uTi2. Thus, the claim follows applying (2.1.6). Lemma 2.10. For all i, j such that |i − j| = 1, we have: (u − 1){Ti Qj Ti − Tj Qi Tj } = −(u − u−1 )2 {Qi Qj Qi − Qj Qi Qj } (2.10.1) (u − 1){Qi Tj Qi − Qj Ti Qj } = u(u − u−1 ){Qi Qj Qi − Qj Qi Qj }
(2.10.2)
(u − 1){Qi Qj Ti − Tj Qi Qj } = −(u − u−1 ){Qi Qj Qi − Qj Qi Qj }
(2.10.3)
Qi Qj Qi Qj − Qj Qi Qj Qi = u(u + 1)−2 {Qi Qj − Qj Qi } .
(2.10.4)
Proof. These claims follow directly from (2.1.2), (2.1.5)–(2.1.7). For instance, we shall prove (2.10.1) and (2.10.4). Using (2.1.2) we have: (u − u−1 )3 Qi Qj Qi = {Ti − Ti−1 }{Tj − Tj−1 }{Ti − Ti−1 } = Ti Tj Ti − Ti Tj−1 Ti − Ti−1 Tj Ti + Ti−1 Tj−1 Ti − Ti Tj Ti−1 + Ti Tj−1 Ti−1 + Ti−1 Tj Ti−1 − Ti−1 Tj−1 Ti−1 . Then, from the braid relation we have: (u − u−1 )3 {Qi Qj Qi − Qj Qi Qj } = {−Ti Tj−1 Ti + Ti−1 Tj Ti−1 } − {−Tj Ti−1 Tj + Tj−1 Ti Tj−1 } = (u − 1){Ti Tj−1 Ti − Tj Ti−1 Tj }
(from (2.1.6))
= (u − 1){Ti {Tj − (u − u−1 )Qj }Ti − Tj {Ti − (u − u−1 )Qi }Tj }
(from (2.1.2))
= −(u − 1)(u − u−1 ){Ti Qj Ti − Tj Qi Tj } , from which we obtain (2.10.1). In order to prove (2.10.4) we multiply (2.1.7) by Ti , and we obtain (u + 1) Ti Qj Qi Qj + uTi Tj Qi Qj = uTi Qi Qj . From (2.5) and (2.1.3) one has (Ti Tj Qi )Qj = Qj Ti Tj Qj = −u−1 Qj Ti Qj , and Ti Qi Qj = −u−1 Qi Qj . Solving for Ti Qj Qi Qj we get Ti Qj Qi Qj = (u + 1)−1 {Qj Ti Qj − Qi Qj } . (2.11) On the other hand, multiplying (2.1.7) by Qj we get (u+1)Qi Qj Qi Qj +uTi Qj Qi Qj = uQj Qi Qj . Then, from (2.11) we obtain Qi Qj Qi Qj = u(u + 1)−1 {Qj Qi Qj − (u + 1)−1 {Qj Ti Qj − Qi Qj }} . Thus, the claim (2.10.4) follows from this last relation and (2.10.2).
A NEW ALGEBRA FROM THE REPRESENTATION THEORY OF FINITE GROUPS
933
Corollary 2.12. For all i, j such that |i − j| = 1, we have Qi Tj Qi Tj − Tj Qi Tj Qi = −u−1 {Qi Qj − Qj Qi } . Proof. Multiplying the relation (2.10.1) by Qi from the left, and applying (2.1.3) yields (u − 1)Qi Tj Qi Tj = (u − u−1 )2 {Qi Qj Qj − Qi Qj Qi Qj } − u−1 (u − 1)Qi Qj Ti . Similarly, we get (u − 1)Tj Qi Tj Qi = (u − u−1 )2 {Qi Qj Qj − Qj Qi Qj Qi } − u−1 (u − 1)Ti Qj Qi . Thus, (u − 1){Qi Tj Qi Tj − Tj Qi Tj Qi } = (u − u−1 )2 {Qj Qi Qj Qj − Qi Qj Qi Qj } − u−1 (u − 1){Qi Qj Ti − Ti Qj Qi } = −u(u − u−1 )2 (u + 1)−2 {Qi Qj − Qj Qi } − u−1 (u − 1){Qi Qj − Qj Qi } from which the Corollary follows.
(from (2.10.4)) ,
In order to study the structure of Jn (u), we note that the map ϕn defined by Ti 7→ (i, i + 1), Ti−1 7→ (i, i + 1) and Qi 7→ 0, gives a homomorphism of Jn (u) onto the group algebra of the symmetric group Sn . Hence, the dimension of Jn (u) is at least n!, and we have (2.13) Jn (u) ' Kerϕn ⊕ C[Sn ] . 3. A Realization of Jn (u) Generalities. Let G be GLn (k), where k is the finite field Fq . Let U be the subgroup of G consisting of upper unipotent matrices. We denote by B the Borel subgroup corresponding to U , and by T the subgroup of G of all diagonal matrices; of course B = U T = T U . We consider the induced representation IndG U 1. 1 affords the principal series of G. More precisely, Recall that the spectrum of IndG U G B G one has IndG U 1 ≡ IndB (IndU 1) ≡ ⊕λ IndB λ, where the sum is over all characters Tˆ of T . We denote by Yn (q) the commuting algebra of the representation IndG U1 (Yokonuma–Hecke algebra.) It is known that the elements of the standard basis of Yn (q) are parametrized by the elements of the group NG (T ). Actually, NG (T ) is the semidirect product Sn n T , where Sn is the symmetric group. Let λs denote the action (by permutation) of s ∈ Sn on λ ∈ Tˆ , one has that λs (t) = λ(sts−1 ), for all t ∈ T . We denote by Xλ the orbit of λ. Let {Rγ ; γ ∈ NG (T )} be the standard basis of Yn (q). We note that each operator of homothethy Rt (t ∈ T ) is diagonalizable, moreover they constitute a commutative group. Therefore we have that its simultaneous diagonalization yields
934
J. JUYUMAYA
G an eigenspace decomposition: IndG U 1 = ⊕λ∈Tˆ Vλ . Notice that Vλ ≡ IndB λ. We know that either HomG (Vλ , Vβ ) = 0 or Vλ ≡ Vβ ; moreover, this together occurs if and only if β ∈ Xλ . L Set Wλ := γ∈Xλ Vγ . We denote by Hn (λ, q) the commuting algebra of the representation Wλ . Recall that one has, by restriction, a natural epimorphism πλ of Yn (q) onto Hn (λ, q). In the case λ = 1, the algebra Hn (λ, q) is so-called the Iwahori–Hecke algebra, which we will denote simply by Hn (q). The Iwahori–Hecke algebra has a presentation with the standard generators L1 , . . . , Ln−1 and relations
L2i = q + (q − 1)Li , Li Lj = Lj Li
if |i − j| > 1 ,
Li Lj Li = Lj Li Lj
if |i − j| = 1 .
Notice, that π1 (Ri ) is just the generator Li of the standard basis of Hn (q) associated to simple transposition (i, i + 1) ∈ Sn , where Ri := R(i,i+1) . The Ji ’s and Pi ’s operators. In [8] we define in a “geometric way”certain operators Pi ’s (average operators) and Ji ’s (Fourier transform) in the algebra Yn (q). We shall use the definition of these operators here via the standard basis of Yn (q), cf. [10]. Henceforth, ψ will be a fixed non-trivial character of the additive group (k, +). We denote by Hti the element of the standard basis of Yn (q) associated to the double coset U t(i)U , where t(i) is the diagonal matrix with t in the (i, i)-position, t−1 in the (i + 1, i + 1)-position, and 1 otherwise. We define the following (Gauss) sum of operators Ψi :=
X
ψ(t)Hti ,
and Ii :=
t∈k×
X
Hti .
t∈k×
Now, for all 0 < i < n, we define the operators Ji ’s by Ji :=
1 {Ii + Ψi Ri } , q
and the operators Pi ’s as Pi :=
q {Ii + Ii Ri } . q − q −1
We can prove that Pi = (q − q −1 )−1 {Ii − Ii Ji }, see [10]. The following theorem realizes the algebra Jn (q) as a subalgebra of the Yokonuma–Hecke algebra Yn (q). Theorem 3.1. The operators Ji ’s and Pi ’s satisfy the relations (2.1.1)–(2.1.7), putting Ji in the place of Ti , Pi in the place of Qi , and q in the place of u.
A NEW ALGEBRA FROM THE REPRESENTATION THEORY OF FINITE GROUPS
935
For the proof of this theorem we will need some facts from Proposition 2.25 [9]. More precisely: Lemma 3.2. For all 1 ≤ i, j ≤ n, we have:
where Iij =
P r∈k×
Ii2 = (q − 1)Ii
(3.2.1)
Ji Ii = Ii Ji
(3.2.2)
Ji Ij = Iij Ji , if |i − j| = 1
(3.2.3)
Ii Iij = Ii Ij , if |i − j| = 1 ,
(3.2.4)
Hri Hrj .
From (3.2.1) and (3.2.2) it is easy to check that Pi Ii = Ii Pi = (q − 1)Pi .
(3.3)
3.4. Proof of Theorem 3.1. The relations (2.1.1)–(2.1.4) are easy to verify, see Proposition 2.20 [9]. The braid relation (2.1.5) was proved in Proposition 2.2 [9], see also Theorem 2.12. [10]. From the proof of Theorem 2.26 [9] we have the skein relation (2.1.6). We will now prove the relation (2.1.7). Let i, j be such that |i − j| = 1. From the definition of Pi , (3.2.3) and (3.2.4), we obtain Pj Pi = (q − q −1 )−2 Ii Ij {1 − Jj − Ji + Jj Ji } .
(3.4.1)
On the other hand, we have: Ji Pj Pi = (q − q −1 )−2 Ji {Ij − Ij Jj }{Ii − Ii Ji } = (q − q −1 )−2 {Ji Ij Ii − Ji Ij Jj Ii − Ji Ij Ii Ji + Ji Ij Jj Ii Ji } = (q − q −1 )−2 {Ii Iij Ji − Ij Iij Ji Jj − Iij Ii Ji2 + Ij Iij Ji Jj Ji }
(from (3.2.3))
= (q − q −1 )−2 {Ii Iij Ji − Ij Iji Ji Jj − Iji Ii − (q −2 − 1)Iji Ii Pi + Ij Iji Ji Jj Ji } = (q − q −1 )−2 {Ii Iji Ji − Ij Iji Ji Jj − Iji Ii − (q −2 − 1)(q − q −1 )−1 {Iji Ii2 − Iji Ii2 Ji } + Ij Iji Ji Jj Ji } . Using (3.2.1) and (3.2.4), we obtain Ji Pj Pi = (q − q −1 )−2 Ii Ij {−q −1 + q −1 Ji − Ji Jj + Ji Jj Ji } .
(3.4.2)
936
J. JUYUMAYA
We have Pi Pj Pi = (q − q −1 )−2 {Pi Ij Ii − Pi Ij Jj Ii − Pi Ij Ii Ji + Pi Ij Jj Ii Ji } = (q − q −1 )−2 {(q − 1)Pi Ij − Pi Ij Iij Jj − (q − 1)Pi Ij Ji + Pi Ij Iij Jj Ji } = (q − q −1 )−3 {(q − 1){Ii Ij − Ii Ji Ij } − Ii Ij Iij Jj + Ii Ji Ij Iij Jj − (q − 1)Ii Ij Ji + (q − 1)Ii Ji Ij Ji + Ii Ij Iij Jj Ji − Ii Ji Ij Iij Jj Ji } = (q − q −1 )−3 {(q − 1)Ii Ij − (q − 1)Ii Iji Ji − Ii Ij Iij Jj + Ii Iji Ij Ji Jj − (q − 1)Ii Ij Ji + (q − 1)Ii Iji − q −1 (q − 1)Ii Iji Ii + q −1 (q − 1)Ii Iji Ii Ji + Ii Ij Iij Jj Ji − Ii Ij Iji Ji Jj Ji } . Thus, from Lemma 3.2 one has Pi Pj Pi = (q − q −1 )−3 (q − 1)Ii Ij {(q −1 + 1) − (q −1 + 1)Ji − Jj + Ji Jj + Jj Ji − Ji Jj Ji } . This together with (3.4.1) and (3.4.2) implies the relation (2.1.7) for Ji and Pi . P Now, we have that πλ (Rt ) (λ ∈ Tˆ , t ∈ T ) is γ∈Xλ γ(t)pλ,γ , where pλ,γ is the projector of Wλ onto Vγ . Then, πλ (Ii ) =
X X
γ(t(i))pλ,γ ,
t∈k× γ∈Xλ
πλ (Ψi ) =
X X
ψ(t)γ(t(i))pλ,γ .
t∈k× γ∈Xλ
Thus, we can calculate πλ (Pi ) and πλ (Ji ). For example, let λ be such that |Xλ | = n!. P Then πλ (Pi ) = 0, and πλ (Ji ) = γ∈Xλ gγ pλ,γ , where gλ denote the Gauss sum X
ψ(t)λ(t(i)) .
t∈k×
In the case λ = 1 we have π1 (Ji ) = q −1 {q − 1 − Li } = −L−1 i , and π1 (Pi ) = (q + 1)−1 {1 + Li } =: Ei . These operators are just the projections used in the work of V. Jones to give another presentation of the Hecke algebra Hn (q) (a presentation that helps to understand the Temperley–Lieb algebras as a quotient of Hecke algebras, see [5]). Setting η = 2 + q + q −1 , the algebra Hn (q) has a presentation with generators E1 , . . . , En−1 and with relations Ei2 = Ei , Ei Ej = Ej Ei
if |i − j| > 1 ,
Ei Ej Ei − η −1 Ei = Ej Ei Ej − η −1 Ej
if |i − j| = 1 .
(3.5)
937
A NEW ALGEBRA FROM THE REPRESENTATION THEORY OF FINITE GROUPS
From this we obtain, in particular, that in Yn (q) the following family is C-free {1, P1 , P2 , P1 P2 , P2 P1 , P1 P2 P1 } .
(3.6)
Given a reduced word w = si1 · · · sim ∈ Sn , we define Jw := Ji1 · · · Jim . Jw is well defined, that is, Jw it is independent of the choice of reduced word for w. See [10]. Another property of the operators Ji ’s that we will use here, is that they permute the representations IndG B λ as follows (see (3.5.2) [9]): ( G w (w ∈ Sn ) Jw (IndG B λ) = IndB λ (3.7) G G (0 < i < n) Pi (IndB λ) ⊆ IndB λ The Generalized Steinberg Representations. It is well known that IndG Bλ has only one irreducible component Stλ , whose dimension is a polynomial in q of degree n(n − 1)/2. These series are called the generalized Steinberg representations of G, see [6, 12] (in the case λ = 1, we obtain the classical Steinberg representation.) For instance, if G = GL3 (k) we have the following families of generalized Steinberg representations of G: St(α,α,α)
of dimension q 3 ,
St(α,β,β) ≡ St(β,β,α) ≡ St(β,α,β)
of dimension q(q 2 + q + 1) ,
St(α,β,γ) = IndG B (α, β, γ)
of dimension (q + 1)(q 2 + q + 1) ,
where α, β and γ are the distinct characters of k × . ˇλ (λ ∈ Tˆ ) the G-subrepresentation of Wλ such that HomG We denote by St L ˇλ ≡ Wλ / L (Wλ , γ∈Xλ Stγ ) = 0 (clearly St γ∈Xλ Stγ ). It is easy to check that EndG (Stˇλ ) = {φ ∈ Hn (λ, q) ; φ(Stγ ) = 0,
∀γ ∈ Xλ } .
Let Pn (q) be the ideal of Jn (q) generated by P1 , . . . , Pn−1 . It is obvious that each element in Pn (q) is a linear combination of monomials of type M = Jw1 Pi1 Jw2 Pi2 · · · Jwm Pim Jwm+1 . From (3.7) and the fact that Pi (Stλ ) = 0 we deduce M (Stλ ) = 0, for all λ ∈ Tˆ (see [12, 9, 10]). Hence: Proposition 3.8. For all characters λ of T, we have a morphism φλ : Pn (q) −→ ˇ λ ), where φλ := πλ |P (q) . EndG (St n The case n = 3. In [9] we constructed (geometrical) models for the series of generalized Steinberg representations. For example, if G = GL3 (k), we have: St(α,α,α) ≡ {f ∈ V(α,α,α) ; P1 f = P2 f = 0} , St(α,β,β) ≡ {f ∈ V(α,β,β) ; P2 f = 0} . The spectrum of IndG U 1 is described by the following direct sum: 2
1 q IndG U 1 = ⊕α [Uα ⊕ 2Uα
+q
⊕ St(α,α,α) ] ⊕α,β [St⊥ (α,β,β) ⊕ St(α,β,β) ]
⊥ ⊕α,β [St⊥ (β,β,α) ⊕ St(β,β,α) ] ⊕α,β [St(β,α,β) ⊕ St(β,α,β) ]
⊕α,β,γ V(α,β,γ) ,
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J. JUYUMAYA
i where St⊥ is the irreducible λ is the orthogonal complement of Stλ in Vλ , and Uα 1 q2 +q ⊕ St(α,α,α) . subrepresentation of dimension i such that V(α,α,α) = Uα ⊕ 2Uα
Proposition 3.9. If q > 2, then the family B 0 = {P1 , P2 , P1 P2 , P2 P1 , J1 P2 , J2 P1 , P2 J1 , P1 J2 , J1 J2 P1 , J2 J1 P2 , J2 P1 J2 , J1 P2 J1 , P1 P2 P1 , P1 J2 P1 } is C-free in P3 (q). Proof. Suppose that X
Zb b = 0 ,
for some Zb ∈ C .
(3.9.1)
b∈B0
Let Vi be the kernel of Pi . From (2.10.1) we have J2 P1 J2 = J1 P2 J1 , in the level V1 ∩ V2 . Therefore, Eq. (3.9.1) at the level V1 ∩ V2 is given by ZP2 J1 P2 J1 + ZP1 J2 P1 J2 + (ZJ2 P1 J2 + ZJ1 P2 J1 )J1 P2 J1 = 0 . Let α, β be two different characters of k × . For all f ∈ St⊥ (α,β,α) ⊂ V(α,β,α) ⊂ V1 ∩V2 , ⊥ ⊥ we have (P2 J1 )f ∈ St(β,α,α) , (P1 J2 )f ∈ St(α,α,β) , and (J2 P1 J2 )f ∈ St⊥ (α,β,α) . Thus, from the last equation we deduce ZP2 J1 = ZP1 J2 = ZJ2 P1 J2 + ZJ1 P2 J1 = 0 . Now, in the level St⊥ (β,α,α) ⊂ V1 , it is obvious that P1 = P2 P1 = J2 P1 = J1 J2 P1 = P1 P2 P1 = 0. Also we have P1 P2 = J2 P1 P2 = J2 P1 J2 − J1 P2 J1 = 0, in the level St⊥ (β,α,α) , because ⊥ P2 (St⊥ (β,α,α) ) ⊆ St(β,α,α) .
Thus, Eq. (3.9.1) yields ZP2 (P2 )f + ZJ1 P2 (J1 P2 )f + ZJ2 J1 P2 (J2 J1 P2 )f = 0 ,
for all f ∈ St⊥ (β,α,α) .
From this last equation we obtain ZP2 = ZJ1 P2 = ZJ2 J1 P2 = 0, because (P2 )f ∈ ⊥ ⊥ St⊥ (β,α,α) , (J1 P2 )f ∈ St(α,β,α) , and (J2 J1 P2 )f ∈ St(α,α,β) . By a similar argument, if considering Eq. (3.9.1) in the level St⊥ (α,α,β) , we obtain ZP1 = ZJ2 P1 = ZJ1 J2 P1 = 0. Thus, Eq. (3.9.1) is reduced to ZP1 P2 P1 P2 + ZP2 P1 P2 P1 + ZJ1 P2 J1 (J1 P2 J1 − J2 P1 J2 ) + ZP1 P2 P1 P1 P2 P1 + ZP1 J2 P1 P1 J2 P1 = 0 . Now, π1 (P1 J2 P1 ) = P1 − q −1 (q + 1)P1 P2 P1 , and by (2.10.1) and (3.6) we deduce ZP1 P2 =ZP2 P1 =ZJ1 P2 J1 =ZP1 P2 P1 =ZP1 J2 P1 =0. This completes the proof. Corollary 3.10. Let q > 2. The family B = {1, J1 , J2 , J1 J2 , J2 J1 , J1 J2 J1 , Rj ; Rj ∈ B 0 } , is C-free in J3 (q).
A NEW ALGEBRA FROM THE REPRESENTATION THEORY OF FINITE GROUPS
Proof. Consider the equation X Zb b = 0 ,
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for some Zb ∈ C .
b∈B
We use this equation in the level
L α,β,γ
V(α,β,γ) to write
Z1 1 + ZJ1 J1 + ZJ2 J2 + ZJ1 J2 J1 J2 + ZJ2 J1 J2 J1 + ZJ1 J2 J1 J1 J2 J1 = 0 . We then deduce that Z1 = ZJ1 = ZJ2 = ZJ1 J2 = ZJ2 J1 = ZJ1 J2 J1 = 0. The proof follows from Proposition 3.9. Theorem 3.11. Let γ = (α, α, α) and δ = (α0 , α0 , β) be two distinct characters ˇ γ ) ⊕ EndG (St ˇ δ ) is of T. The morphism Φ : x 7→ φγ (x) + φδ (x) of P3 (q) in EndG (St an isomorphism. Hence P3 (q) is semisimple, and P3 (q) ' M1 (C) ⊕ M2 (C) ⊕ M3 (C) . Proof. We can assume that γ = 1. According to (3.5) a basis of EndG (W1 ) = ˇ 1 ), Hn (q) is {1, E1 , E2 , E1 E2 , E2 E1 , E1 E2 E1 }. Take the following basis of EndG (St C1 := {φ1 (b) ; b ∈ C} , where C := {P1 P2 , P2 P1 , P1 P2 P1 , P2 P1 P2 , P2 J1 P2 }. Note that φ1 (P2 J1 P2 ) = E2 − q −1 (q + 1)E2 E1 E2 , because φ1 (Ji ) = −L−1 = 1 − q −1 (q + 1)Ei , i
(0 < i < n) .
Using this relation we get φ1 (M ) = 0, where M := P1 − q −1 (q + 1)P1 P2 P1 − P1 J2 P1 . Also, we have that φδ (M ) = φδ (P1 ) 6= 0. Thus, we can consider the following basis ˇ δ ): of EndG (St D1 := {φδ (Js1 M Js−1 ) ; s1 , s2 ∈ {1, w, w0 }} , 2
0
0
where w, w are such that Xδ = {δ, δ , δ w }. ˇ 1 ) ⊕ EndG (St ˇ δ ), and Now, C1 ∪ D1 is a basis of EndG (St w
Φ(x) = φ1 (x) Φ(Js−1 M Js ) = φδ (Js1 P1 Js−1 ) 2
for all x ∈ C , for all s1 , s2 ∈ {1, w, w0 } .
So, Φ is surjective. As the algebras in question have the same dimension, Φ is an isomorphism. The decomposition of P3 (q) in matrices results from the spectrum of IndG U 1. 4. Partial Results In this section we will see partial results about the structure of Jn (u). In this sense we will prove the following claims:
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Proposition 4.1. For n = 2, 3, 4, in the algebra Jn (u) any word in Ti , Ti2 (0 < i < n) is a linear combination of words having degree in Tn−1 at most 2. Theorem 4.2. The dimension of J2 (u) is 3, and the dimension of J3 (u) is 20. We believe that for generical u the algebra Jn (u) is semisimple. For instance, for n = 2 J2 (u) = (R1 ) ⊕ (R2 ) ⊕ (R3 ) , where the Ri are the idempotents: R1 = Q1 , R2 = R3 = − 21 [T1 + (u−1 + 1)Q1 − 1]. For n = 3, from (2.13) and Theorem 3.11 we get
1 2 [T1
+ (u−1 − 1)Q1 + 1],
J3 (u) ' 3M1 (C) ⊕ 2M2 (C) ⊕ M3 (C)
(4.3)
In the proof of Proposition 4.1, we need the following lemmas: Lemma 4.4. For all δ ∈ {1, 2}, the word T32 T2 T1δ T22 T3 is a linear combination of words whose degree in T3 is at most 2. Proof. For δ = 1, we have from (2.6) that T32 T2 T1δ T22 T3 = T32 T12 T2 T1 T3 , hence T32 T2 T1δ T22 T3 = T12 T32 T2 T3 T1 = T12 T2 T3 T22 T1 . For δ = 2, we have from Funar’s Lemma T32 {T2 T1δ T22 }T3 = T32 {T12 T22 T1 − u−1 {T2 T12 T2 − T1 T22 T1 } + T2 T12 − T22 T1 }T3 = T12 {T32 T22 T3 }T1 − u−1 {T32 T2 T12 T2 T3 − T1 T32 T22 T3 T1 } + T32 T2 T3 T12 − T32 T22 T3 T1 . Using Funar’s Lemma on the words: T32 T22 T3 , T32 T2 T12 T2 T3 , and as T32 T2 T3 = T2 T3 T22 , the lemma is proved. Lemma 4.5. Let L be a word in 1, Ti , Ti2 (i = 1, 2). Then the word T3 T22 T3 LT3 is a linear combination of words whose degree in T3 is at most 2. Proof. If L does not contain T2δ (δ = 1, 2), the claim is true from Funar’s Lemma. If L contains T2 , we have that L is a linear combination of monomials of type T1δ T2 T1 , T1δ T22 T1 with δ, ∈ {0, 1, 2}. Therefore, it is sufficient to prove that the degree in T3 of the following monomials can be reduced to 0, 1, 2: T3 T22 T3 T2 T3
(4.5.1)
T3 T22 T3 T1 T2 T3
(4.5.2)
T3 T22 T3 T1δ T22 T3 ,
(δ = 1, 2)
(4.5.3)
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In (4.5.1) and (4.5.2) the degree is reduced by the braid relation. For (4.5.3), from Proposition 2.9 we have T3 T22 T3 T1δ T22 T3 = {T32T2 T32 − T22 T3 T22 + T2 T32 T2 + T3 T22 − T2 T32 + T22T3 − T32 T2 + T2 − T3 − T22 + T32 }T1δ T22 T3 = T32 T2 T32 T1δ T22 T3 − T22 T3 T22 T1δ T22 T3 + T2 T32 T2 T1δ T22 T3 + T3 T22 T1δ T22 T3 − T2 T32 T1δ T22 T3 + T22 T3 T1δ T22 T3 − T32T2 T1δ T22 T3 + T2 T1δ T22 T3 − T3 T1δ T22 T3 − T22T1δ T22 T3 + T32 T1δ T22 T3 . From Lemma 4.4 the degree in T3 of the monomial T32 T2 T1δ T22 T3 can be reduced. For the monomial T2 T32 T1δ T22 T3 , T32 T1δ T22 T3 the degree is reduced from Funar’s Lemma. Finally, we have T32 T2 T32 T1δ T22 T3 = T32 T2 T1δ T32 T22 T3 . Thus, from Funar’s Lemma we have T32 T2 T32 T1δ T22 T3 = T32 T2 T1δ {T2 T32 T22 − u−1 {T3 T22 T3 − T2 T32 T2 } + T22 T3 − T2 T32 } = T32 T2 T1δ T2 T32 T22 − u−1 {T32 T2 T1δ T3 T22 T3 − T32 T2 T1δ T2 T32 T2 } + T32 T2 T1δ T22 T3 − T32 T2 T1δ T2 T32 . The degree in the monomial T32 T2 T1δ T22 T3 can be reduced by Lemma 4.4. For the monomial T32 T2 T1δ T3 T22 T3 one has T32 T2 T1δ T3 T22 T3 = T32 T2 T3 T1δ T22 T3 = T2 T3 T22 T1δ T22 T3 . For the other monomials the degree can be reduced by Funar’s Lemma.
4.6. Proof of Proposition (4.1). For n = 2 it is trivial. For n = 3 it is a consequence of Lemma 2.4 [7], because the algebra Jn (u) is a quotient of the cubic Hecke algebra. We will prove the proposition for the case n = 4. Let M be one word in T1 , T2 , T3 , T12 , T22 , T32 such that the degree in T3 is at least 3. Then M can be written as: M = AT33 Z
(4.6.1)
M = AT3 BT32 Z
(4.6.2)
M = AT32 BT3 Z
(4.6.3)
M = AT3 BT3 CT3 Z ,
(4.6.4)
where A, B, C ∈ J3 (u) and Z ∈ J4 (u). We will prove that in all these cases the degree of T3 can be reduced. From the fact that T3 satisfies the cubic relation, the degree of T3 can be reduced in (4.6.1).
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The proof that in (4.6.2) and (4.6.3) the degree of T3 can be reduced is the same. Now, in the situation (4.6.2), we have that B is a linear combination of monomials in which the degree in T2 is at most 2. Therefore, M is a linear combination of the following monomials: (a) (b) (c) (d)
AT3 U T32 Z AT3 U T2 V T32 Z AT3 U T22 V T32 Z AT3 U T2 V T2 W T32 Z,
where U , V , W ∈ J2 (u), A, B, C ∈ J3 (u) and Z ∈ J4 (u). In (a), as T3 commutes with U , it is trivially true from the cubic relation that the degree of T3 can be reduced. In (b) we have AT3 U T2 V T32 Z = AU T3 T2 T32 V Z = AU T22 T3 T2 V Z, so the degree T3 is reduced. For the case (c), we have AT3 U T22 V T32 Z = AU T3 T22 T32 V Z. Thus, from Funar’s Lemma the degree of T3 can be reduced. For (d) we have AT3 U T2 V T2 W T32 Z = AU T3 T2 V T2 T32 W Z, where V ∈ {1, T1 , 2 T1 }. For V = 1, T12 , from Funar’s Lemma the degree can be reduced. If V = T1 , we have: AT3 U T2 V T2 W T32 Z = AU T3 T2 T1 T2 T32 W Z = AU T3 T1 T2 T1 T32 W Z = AU T1 T3 T2 T32 T1 W Z = AU T1 T22 T3 T2 T1 W Z . Thus the degree of T3 can be reduced in the situation (4.6.2). Suppose that we are in (4.6.4). The monomial B is a sum of monomials whose degree in T2 is at most 2. Thus, the monomial in (4.6.4) is a linear combination of the following monomials: (e) AT3 R1 T2 R10 T3 CT3 Z (f) AT3 R1 T22 R10 T3 CT3 Z (g) AT3 R10 T2 R1 T2 R100 T3 CT3 Z, where R1 , R10 ∈ J2 (u), A, C ∈ J3 (u) and Z ∈ J4 (u). In the case (e), one has AT3 R1 T2 R10 T3 CT3 Z = AR1 T3 T2 T3 R10 CT3 Z = AR1 T2 T3 T2 R10 CT3 Z , i.e. the degree in T3 is reduced. In (f) one has AT3 R1 T22 R10 T3 CT3 Z = AR1 T3 T22 T3 R10 CT3 Z. Lemma 4.2 the degree in T3 is reduced. In (g) we have AT3 R10 T2 R1 T2 R100 T3 CT3 Z = AR10 T3 T2 R1 T2 T3 R100 CT3 Z , where R1 , R10 , R100 ∈ {1, T1, T12 }.
Hence, from
A NEW ALGEBRA FROM THE REPRESENTATION THEORY OF FINITE GROUPS
943
If R1 = T1 , we have AT3 R10 T2 R1 T2 R100 T3 CT3 Z = AR10 T3 T2 T1 T2 T3 R100 CT3 Z = AR10 T3 T1 T2 T1 T3 R100 CT3 Z = AR10 T1 T3 T2 T3 T1 R100 CT3 Z = AR10 T1 T2 T3 T2 T1 R100 CT3 Z. If R1 = T12 , we get AT3 R10 T2 R1 T2 R100 T3 CT3 Z = AR10 T3 T2 T12 T2 T3 R100 CT3 Z. Now, from Funar’s Lemma the monomial R100 C ∈ J3 (u) commutes with T3 T2 T12 T2 T3 . So AT3 R10 T2 R1 T2 R100 T3 CT3 Z = AR100 CR10 T3 T2 T12 T2 T3 T3 Z. But, from Funar’s Lemma again, the degree in T3 of the monomial T3 T2 T12 T2 T3 T3 can be reduced. If R1 = 1, we have AT3 R10 T2 R1 T2 R100 T3 CT3 Z = AR10 T3 T22 T3 R100 CT3 Z. From Lemma 4.5 the degree in T3 can be reduced.
4.7. Proof of Theorem 4.2. It is easy to prove that the dimension of J2 (u) is 3 (a basis is {1, T1 , P1 }). From Proposition 4.1 we obtain that J3 (u) is generated by the following family: D000 = {1, T1, T2 , T1 T2 , T2 T1 , T1 T2 T1 , Rj ; Rj ∈ D00 } , where D00 := {T12 , T22 , T1 T22 , T22 T1 , T12 T2 , T2 T12 , T12 T22 , T22 T12 , T1 T22 T1 , T12 T22 T1 , T1 T22 T12 , T1 T2 T12 , T12 T2 T1 , T12 T22 T12 }. Now, from (2.3) and (2.1.7), it is easy to check that each element of D000 is a linear combination of element of D D = {1, T1 , T2 , T1 T2 , T2 T1 , T1 T2 T1 , Rj ; Rj ∈ D0 } , where D0 := {1, Q1 , Q2 , Q1 Q2 , Q2 Q1 , T1 Q2 , T2 Q1 , Q2 T1 , Q1 T2 , T 1 T 2 Q1 , T 2 T 1 Q2 , T 2 Q1 T 2 , T 1 Q2 T 1 , Q1 Q2 Q1 , Q1 T 2 Q1 } . Hence, the C-vector space J3 (u) is spanned by D. Now, we consider the algebra homomorphism φ of J3 (u) to H3 (q) defined by Ti 7→ Ji , Qi 7→ Pi . From the fact that φ is C-linear and Proposition 3.9, we obtain that D is a basis for J3 (u).
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5. Remarks 5.1. A conjecture. We use the notations of Sec. 3. In the generic case, the algebra Jn (u) is semisimple. More precisely, Jn (u) ' C[Sn ] ⊕ EndG (W ), M W = Wλ , λ`n
where Wλ ’s are certain G-subrepresentations of IndG U 1. 5.2. Via our algebra it is possible to construct linear representations of Braid groups. For instance, for n = 3 the ideal (Q2 Q1 , Q1 Q2 Q1 , Q1 T2 Q1 ) of Jn (u) provides the representation 1 1 1 0 0 − − −1 u u+1 1 1 u+1 +1 − 0 − b1 7→ . , b2 7→ 0 0 u u u 1 1 u−1 0 0 − 0 u u+1 u 5.3. An observation that links the algebra Jn (u) with the Knot theory, is the fact that the Vassiliev algebra Vn (u − u−1 ) surject onto the algebra Jn (u). Recall that the Vassiliev algebra Vn () defined by J. Baez [2] is C[Mn ]/hgi − gi−1 − ai i ,
( ∈ C) ,
where Mn is the generalized braid monoid [1] generated by elements invertibles 1, g1 , . . . , gn−1 and elements a1 , . . . , an−1 , satisfying the following relations g i ai = ai g i . For all i, j such that |i − j| > 1, gi gj = gj gi
ai g j = g j ai
ai aj = aj ai
And when |i − j| = 1, gi gj gi = gj gi gj ,
g i g j ai = aj g i g j .
Thus, the map gi 7→ Ti , ai 7→ Qi defines a surjective homomorphism from Vn (u − u−1 ) onto Jn (u). −1 Now, it is easy to check that the assignment Ti 7→ −L−1 i , Qi 7→ (u+1) {1+Li} defines a surjective homomorphism π from Jn (u) onto the Iwahori–Hecke algebra Hn (u). On the other hand, recall that A. Ocneanu has defined a (Markov) linear trace tr on the inductive limit H∞ (u) of the algebras of Iwahori–Hecke Hn (u). The trace tr : H∞ (u) −→ C is defined by tr(1) = 1 ,
tr(xy) = tr(yx) ,
tr(Ln x) = tr(Ln )tr(x)
(x ∈ Hn (u)) .
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Thus, we can consider the “linear trace” Tr = tr◦ π on the inductive limit J∞ (u) of the algebras Jn (u), Tr : J∞ (u) −→ C It is well known that with tr we can define the famous Jones polynomial for knots. Roughly speaking, we believe that is possible to define, via Tr, a polynomial for knots with singularities. Acknowledgements I would like to express my deepest gratitude to M. S. Narasimhan for believing in my research at ICTP. I would like to thank P. Cartier for carefully reading the preliminary version of this paper. The author would also like to thank R. Gambini for the numerous discussions and for his interest in this work. References [1] A. Ashtekar and L. Bombelli, New Perspectives in Canonical Gravity, Monographs and textbooks in physical science, Lecture notes, V. 5, 1988. [2] J. Baez, “Link invariants of finite type and perturbation theory”, Lett. in Math. Phys. 26 (1992) 43–51. [3] R. Baeza, Knots, Lecture at CECS, Santiago, 1996. [4] B. Br¨ ugmann, R. Gambini and J. Pullin, “Jones polynomials for intersecting knots as physical states of quantum gravity”, Nuclear Physics B385 (1992) 587–603. [5] A. Connes, Indice des sous-facteurs, alg`ebres de Hecke et th´eorie des noeuds, S´eminaire Bourbaki, 647, Juin 1985. [6] S. W. Dagger, “A class of irreducible characters for certain classical groups”, London Math. Soc. 22 (1970) 513–520. [7] L. Funar, “On the Quotients of cubic Hecke algebra”, Commun. Math. Phys. 173 (1995) 513–558. [8] J. Juyumaya, “R´epresentation du groupe sym´etrique par des op´erateurs de FourierGrassmann”, C.R. Acad. Sc. Paris 315 (1992) 755–758. GL (F ) [9] J. Juyumaya, “Op´erateurs de type Fourier-Grassmann sur 11U n q ”, J. Algebra 185 (1996) 796–818. [10] J. Juyumaya, “Sur les nouveaux g´en´erateurs de l’alg` ebre de Hecke H(G, U, 11)”, J. Algebra 204 (1998) 49–68. [11] L. Kauffman, “Invariants of graphs in three-space”, Transactions of the AMS 311 (1989) 697–710. [12] R. Kilmoyer, “Principal series representations of finite Chevalley groups”, J. Algebra 51 (1978) 300–319. [13] D. Ugon, R. Gambini and P. Mora, “Link invariant for intersecting loops”, Physics Lett. B305 (1993) 214–222. [14] D. Ugon, R. Gambini and P. Mora, “Intersecting braid and intersecting knots theory”, J. Knots and Its Ramifications 4 (1995) 1–12.
COVARIANT TECHNIQUES FOR COMPUTATION OF THE HEAT KERNEL I. G. AVRAMIDI∗ Department of Mathematics, The University of Iowa 14 MacLean Hall, Iowa City IA 52242-1419, USA E-mail : [email protected] Received 13 September 1997 Revised 22 July 1998 The heat kernel associated with an elliptic second-order partial differential operator of Laplace type acting on smooth sections of a vector bundle over a Riemannian manifold, is studied. A general manifestly covariant method for computation of the coefficients of the heat kernel asymptotic expansion is developed. The technique enables one to compute explicitly the diagonal values of the heat kernel coefficients, so called Hadamard–Minakshisundaram– De Witt–Seeley coefficients, as well as their derivatives. The elaborated technique is applicable for a manifold of arbitrary dimension and for a generic Riemannian metric of arbitrary signature. It is very algorithmic, and well suited to automated computation. The fourth heat kernel coefficient is computed explicitly for the first time. The general structure of the heat kernel coefficients is investigated in detail. On the one hand, the leading derivative terms in all heat kernel coefficients are computed. On the other hand, the generating functions in closed covariant form for the covariantly constant terms and some low-derivative terms in the heat kernel coefficients are constructed by means of purely algebraic methods. This gives, in particular, the whole sequence of heat kernel coefficients for an arbitrary locally symmetric space.
1. Introduction In this paper we report on recent progress on developing some computational methods for the heat kernel that turned out to be very powerful for carrying out explicit computations [1–4, 7, 5, 6]. We will start with the definition of the heat kernel and, then, will try to explain the main ideas of our approach and present the main results without going much into details. The heat kernel proved to be a very powerful tool in mathematical physics as well as in quantum field theory. It has been the subject of much investigation in recent years in mathematical as well as in physical literature. Since it is almost impossible to provide an exhaustive bibliography on this subject, we give only some key references (see, for example, [?, ?, ?, ?, ?, ?, ?] and references therein). The study of the heat kernel is motivated, in particular, by the fact that it gives a general framework of covariant methods for investigating the quantum field theories with local gauge symmetries, such as quantum gravity and gauge theories [?]. ∗ On
leave of absence from Research Institute for Physics, Rostov State University, Stachki 194, 344104 Rostov-on-Don, Russia. Present address: Department of Mathematics, New Mexico Tech, Socorro, NM 87801, USA. E-mail : [email protected] 947 Reviews in Mathematical Physics, Vol. 11, No. 8 (1999) 947–980 c World Scientific Publishing Company
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1.1. Preliminaries To define the heat kernel one has to remember some preliminary facts from the differential geometry [?]. Let (M, g) be a smooth Riemannian manifold of dimension d with a positive definite Riemannian metric g. To simplify the exposition we assume additionally that it is compact and complete, i.e. without boundary, ∂M = ∅. Let T M and T ∗ M be the tangent and cotangent bundles of the manifold M . On the tangent bundle T M of a Riemannian manifold there is always a unique canonical connection, so called Levi–Civita connection, ∇T M , which is torsion-free and compatible with the metric g. Let V be a smooth vector bundle over the manifold M , End (V ) be the bundle of all smooth endomorphisms of the vector bundle V , and C ∞ (M, V ) and C ∞ (M, End (V )) be the spaces of all smooth sections of the vector bundles V and End (V ). Further, we will also assume that V is a Hermitian vector bundle, i.e. there is a Hermitian pointwise fibre scalar product hϕ, ψi for any two sections of the vector bundle ϕ, ψ ∈ C ∞ (M, V ). The dual vector bundle V ∗ is naturally identified with V , so that (1) hϕ, ψi = tr V (ϕ¯ ⊗ ψ) , where ψ ∈ C ∞ (M, V ), and ϕ¯ ∈ C ∞ (M, V ∗ ) and tr V is the fibre trace. Using the invariant Riemannian volume element d vol (x) on the manifold M we define a natural L2 inner product Z Z (ϕ, ψ) = Tr L2 (ϕ¯ ⊗ ψ) = d vol (x)hϕ, ψi = d vol (x) tr V (ϕ¯ ⊗ ψ) . (2) M
M
The Hilbert space L (M, V ) is defined to be the completion of C ∞ (M, V ) in this norm. Let ∇V be a connection, or covariant derivative, on the vector bundle V 2
∇V : C ∞ (M, V ) → C ∞ (M, T ∗ M ⊗ V ) .
(3)
The connection defines a parallel transport, which can be always used to extend some geometric objects, known locally, to the whole manifold. In particular, the Hermitian metric on the vector bundle V can be always defined in such a way that the connection ∇V is compatible, i.e. ∇hϕ, ψi = h∇V ϕ, ψi + hϕ, ∇V ψi .
(4)
On the tensor product bundle T ∗ M ⊗ V we define the tensor product connection by means of the Levi–Civita connection ∇T
∗
M⊗V
= ∇T
∗
M
⊗ 1 + 1 ⊗ ∇V .
(5)
Similarly, we extend the connection ∇V with the help of the Levi–Civita connection to C ∞ (M, V )-valued tensors of all orders and denote it just by ∇. Usually there is no ambiguity and the precise meaning of the covariant derivative is always clear from the nature of the object it is acting on.
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The composition of two covariant derivatives is a mapping ∇T
∗
M⊗V
∇V : C ∞ (M, V ) → C ∞ (M, T ∗ M ⊗V ) → C ∞ (M, T ∗ M ⊗T ∗ M ⊗V ) . (6)
Let, further, trg denote the contraction of sections of the bundle T ∗ M ⊗ T ∗ M ⊗ V with the metric on the cotangent bundle trg = g ⊗ 1 : C ∞ (M, T ∗ M ⊗ T ∗ M ⊗ V ) → C ∞ (M, V ) .
(7)
Then we can define a second-order differential operator, called the generalized Laplacian, by = trg ∇T
∗
M⊗V
∇V
(8)
: C ∞ (M, V ) → C ∞ (M, T ∗ M ⊗ V ) → C ∞ (M, T ∗ M ⊗ T ∗ M ⊗ V ) → C ∞ (M, V ) . (9) Further, let Q be a a smooth Hermitian section of the endomorphism bundle, End (V ), i.e. hϕ, Qψi = hQϕ, ψi . (10) Finally, we define a Laplace type differential operator F as the sum of the generalized Laplacian and the endomorphism Q F = − + Q .
(11)
1.2. Laplace type operator in local coordinates The generalized Laplacian can be easily expressed in local coordinates. Let x , (µ = 1, 2, . . . , d), be a system of local coordinates and ∂µ and dxµ be the local coordinate frames for the tangent and the cotangent bundles. We adopt the notation that the Greek indices label the tensor components with respect to local coordinate frame and range from 1 through d = dim M . Besides, a summation is always carried out over repeated indices. Let g = gµν dxµ ⊗ dxν be the metric on the tangent bundle, g ∗ = g µν ∂µ ⊗ ∂ν be the metric on the cotangent bundle, |g| = det gµν , and A = Aµ dxµ be the connection 1-form of ∇V . Then it is not difficult to obtain for the generalized Laplacian µ
= g µν ∇µ ∇ν = |g|−1/2 (∂µ + Aµ )|g|1/2 g µν (∂ν + Aν ) .
(12)
Therefore, a Laplace type operator is a second-order partial differential operator of the form (13) F = −g µν ∂µ ∂ν − 2aµ ∂µ + q , where aµ is a End (V )-valued vector 1 aµ = g µν Aν + |g|−1/2 ∂ν (|g|1/2 g νµ ) 2
(14)
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and q is a section of the endomorphism bundle End (V ) q = Q − g µν Aµ Aν − |g|−1/2 ∂µ (|g|1/2 g µν Aν ) .
(15)
Thus, a Laplace type operator is constructed from the following three pieces of geometric data: • a metric g on M , which determines the second-order part; • a connection 1-form A on the vector bundle V , which determines the firstorder part; • an endomorphism Q of the vector bundle V , which determines the zeroth order part. It is worth noting that every second-order differential operator with a scalar leading symbol given by the metric tensor is of Laplace type and can be put in this form by choosing the appropriate connection ∇V and the endomorphism Q. 1.3. Self-adjoint operators Using the L2 inner product we define the adjoint F ∗ of a differential operator F by (16) (F ∗ ϕ, ψ) = (ϕ, F ψ) . It is not difficult to prove that if the connection ∇ is compatible with the Hermitian metric on the vector bundle V and the boundary of the manifold M is empty, then the generalized Laplacian , and, obviously, any Laplace type operator F , is an elliptic symmetric differential operator (ϕ, ψ) = (ϕ, ψ),
(F ϕ, ψ) = (ϕ, F ψ) ,
(17)
with a positive principal symbol. Moreover, the operator F is essentially selfadjoint, i.e. there is a unique self-adjoint extension F¯ of the operator F . We will not be very careful about distinguishing between the operator F and its closure F¯ , and will simply say that the operator F is elliptic and self-adjoint. Spectral theorem. There is a well-known theorem about the spectrum of any elliptic self-adjoint differential operator F acting on smooth sections of a vector bundle V over a compact manifold M , F : C ∞ (M, V ) → C ∞ (M, V ), with a positive definite principal symbol [?]. Namely, • the operator F has a discrete real spectrum, λn , (n = 1, 2, . . . ,), bounded from below (18) λn ≥ λ1 > −C , with some real positive constant C, • all eigenspaces of the operator F are finite-dimensional and the eigenvectors, ϕn , of the operator F , (19) F ϕn = λn ϕn , are smooth sections of the vector bundle V , which form a complete orthonormal basis in L2 (M, V ). (20) (ϕn , ϕm ) = δmn .
COVARIANT TECHNIQUES FOR COMPUTATION OF THE HEAT KERNEL
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It is convenient to assume for simplicity that the endomorphism Q is bounded from below by a sufficiently large constant, so that the Laplace type operator F is strictly positive. This is equivalent to replacing F → F − λ with a sufficiently large negative constant λ, i.e. λ < λ1 . This does not influence all the conclusions but simplifies significantly the technical details needed to treat the negative and zero modes of the operator F . This can always be done as long as we study only asymptotic properties of the spectrum for large eigenvalues but not the structure and the dimension of the null space and related cohomolgical and topological questions. We will also point out which formulas or arguments are harder when there are zero or negative modes. 1.4. Heat kernel Thus all eigenfunctions of the Laplace type operator F are smooth sections of the vector bundle V and, if the manifold M is compact, F has a unique self-adjoint extension, which we denote by the same symbol F . Then the operator U (t) = exp(−tF ) for t > 0 is well defined as a bounded operator on the Hilbert space of square integrable sections of the vector bundle V . These operators form a one-parameter semi-group. The kernel U (t|x, x0 ) of this operator is defined by X e−tλn ϕn (x) ⊗ ϕ¯n (x0 ) , (21) U (t|x, x0 ) = exp(−tF )δ(x, x0 ) = n
where δ(x, x0 ) is the covariant Dirac distribution along the diagonal of M × M , and each eigenvalue is counted with multiplicities. It can be regarded as an endomorphism from the fiber of V over x0 to the fiber of V over x. The kernel U (t|x, x0 ) of the operator exp(−tF ) satisfies the heat equation (∂t + F )U (t|x, x0 ) = 0
(22)
U (0+ |x, x0 ) = δ(x, x0 ) .
(23)
with the initial condition
That is why, it is called the heat kernel. It can be proved that there is a unique smooth solution, called the fundamental solution, of the heat equation satisfying that initial condition. Thus, the heat kernel is the fundamental solution of the heat equation. For t > 0 the heat kernel is a smooth section of the external tensor product of the vector bundles V V ∗ over the tensor product manifold M × M : U (t|x, x0 ) ∈ C ∞ (R+ × M × M, V V ∗ ). It is not difficult to prove that for Re λ > −λ1 the resolvent, G(λ) = (F − λ)−1 , of the operator F is a bounded operator with the kernel given by Z ∞ dt etλ U (t|x, x0 ) . (24) G(λ|x, x0 ) = 0
Note that this formula is not valid if the operator F has negative or zero modes.
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1.5. Trace of the heat kernel and the spectral functions As we already said above, for any t > 0 the heat semi-group U (t) = exp(−tF ) of a Laplace type operator F on a compact manifold M is a bounded operator on the Hilbert space L2 (M, V ) and is trace-class, with a well-defined trace given by the formula: Z X d vol (x) tr V U (t|x, x) = e−tλn . (25) Tr L2 exp(−tF ) = M
n
The trace of the heat kernel is obviously a spectral invariant of the operator F . It determines other spectral functions by integral transforms. 1. The distribution function (also called counting function), N (λ), defined as the number of eigenvalues below the level λ, is given by Z c0 +i∞ dt etλ Tr L2 exp(−tF ) , (26) N (λ) = #{λn ≤ λ} = c0 −i∞ 2πi t where c0 is a positive constant. 2. The density function, ρ(λ), is defined by derivative of the distribution function and is obviously d N (λ) = ρ(λ) = dλ
Z
c0 +i∞
c0 −i∞
dt tλ e Tr L2 exp(−tF ) . 2πi
(27)
3. The generalized zeta-function, ζ(s, λ), defined as the trace of the complex power of the positive operator (F − λ), is given by Z ∞ 1 −s dtts−1 etλ Tr L2 exp(−tF ) , (28) ζ(s, λ) = Tr L2 (F − λ) = Γ(s) 0 where s and λ are complex variables with Re λ < λ1 and Re s > d/2. These spectral functions are very useful tools in studying the spectrum of the operator F . In principle, if known exactly, they determine the spectrum. Of course, this is not valid for asymptotic expansions of the spectral functions. There are examples of operators that have the same asymptotic series of the spectral functions but different spectrum. The zeta function enables one to define, in particular, the regularized determinant of a positive operator (F − λ), ∂ ζ(s, λ) = − log Det (F − λ) , (29) ζ 0 (0, λ) ≡ ∂s s=0 which determines the one-loop effective action in quantum field theory. All these functions are, in principle, equivalent to each other. However, the heat kernel is a smooth function whereas the distribution and especially the density function are extremely singular. That is why the heat kernel seems to be more convenient for practical purposes.
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2. Asymptotic Expansion of the Heat Kernel In the following we are going to study the heat kernel only locally, i.e. in the neighbourhood of the diagonal of M ×M , when the points x and x0 are close to each other. The exposition will follow mainly our papers [?, ?]. We will keep a point x0 of the manifold fixed and consider a small geodesic ball, i.e. a small neighbourhood of the point x0 : Bx0 = {x ∈ M |r(x, x0 ) < ε}, r(x, x0 ) being the geodesic distance between the points x and x0 . We will take the radius of the ball sufficiently small, so that each point x of the ball of this neighbourhood can be connected by a unique geodesic with the point x0 . This can be always done if the size of the ball is smaller than the injectivity radius of the manifold, ε < rinj . Let σ(x, x0 ) be the geodetic interval, also called world function, defined as one half the square of the length of the geodesic connecting the points x and x0 σ(x, x0 ) =
1 2 r (x, x0 ) . 2
(30)
The first derivatives of this function with respect to x and x0 define tangent vector fields to the geodesic at the points x and x0 uµ = g µν ∇ν σ,
0
0 0
uµ = g µ ν ∇0ν 0 σ
(31)
and the determinant of the mixed second derivatives defines a so-called Van Vleck–Morette determinant ∆(x, x0 ) = |g|−1/2 (x) det [−∇µ ∇0ν 0 σ(x, x0 )]|g|−1/2 (x0 ) .
(32)
Let, finally, P(x, x0 ) denote the parallel transport operator along the geodesic from the point x0 to the point x. It is a section of the external tensor product of the vector bundle V V ∗ over M × M , or, in other words, it is an endomorphism from the fiber of V over x0 to the fiber of V over x. Near the diagonal of M × M all these two-point functions are smooth singlevalued functions of the coordinates of the points x and x0 . Let us note from the beginning that we will construct the heat kernel in form of covariant Taylor series in coordinates. In C ∞ case these series do not necessarily converge. However, if one assumes additionally that the two-point funtions are analytic, then the Taylor series converge in a sufficiently small neighbourhood of the diagonal. Further, one can easily prove that the function 1 0 −d/2 0 0 ∆(x, x ) exp − σ(x, x ) P(x, x0 ) (33) U0 (t|x, x ) = (4πt) 2t satisfies the initial condition U0 (0+ |x, x0 ) = δ(x, x0 ) .
(34)
Moreover, locally it also satisfies the heat equation in the free case, when the Riemannian curvature of the manifold, Riem, the curvature of the bundle connection, R, and the endomorphism Q vanish: Riem = R = Q = 0. Therefore, U0 (t|x, x0 ) is
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the exact heat kernel for a pure generalized Laplacian in flat Euclidean space with a flat trivial bundle connection and without the endomorphism Q. This function gives a good framework for the approximate solution in the general case. Namely, by factorizing out this free factor we get an ansatz 1 (35) U (t|x, x0 ) = (4πt)−d/2 ∆(x, x0 ) exp − σ(x, x0 ) P(x, x0 )Ω(t|x, x0 ) . 2t The function Ω(t|x, x0 ), called the transport function, is a section of the endomorphism vector bundle End (V ) over the point x0 . Using the definition of the functions σ(x, x0 ), ∆(x, x0 ) and P(x, x0 ) it is not difficult to find that the transport function satisfies a transport equation 1 (36) ∂t + D + L Ω(t) = 0 , t where D is the radial vector field, i.e. operator of differentiation along the geodesic, defined by D = ∇u = uµ ∇µ , (37) and L is a second-order differential operator defined by L = P −1 ∆−1/2 F ∆1/2 P .
(38)
The initial condition for the transport function is obviously Ω(t|x, x0 ) = IV ,
(39)
where IV is the identity endomorphism of the vector bundle V over x0 . It is obvious that if we replace the operator F by (F − λ), with Re λ < λ1 , then the heat kernel and the transport function are simply multiplied by etλ , i.e. ΩF −λ (t) = etλ ΩF (t). Further, for λ < λ1 the operator (F − λ) becomes a positive operator. Therefore, the function etλ Ω(t) satisfies the following asymptotic conditions: lim tα ∂tN [etλ Ω(t)] = 0 for λ < λ1 , α > 0, N ≥ 0 .
t→∞,0
(40)
In other words, as t → ∞ the function etλ Ω(t) and all its derivatives decreases faster than any power of t, it actually decreases exponentially, and as t → 0 the product of etλ Ω(t) with any positive power of t vanishes. Hereafter we fix λ < λ1 , so that (F − λ) is a positive operator. Now, let us consider a slightly modified version of the Mellin transform of the function etλ Ω(t) introduced in [?] Z ∞ 1 dtt−q−1 etλ Ω(t) . (41) bq (λ) = Γ(−q) 0 Note that for fixed λ this is a Mellin transform of etλ Ω(t) and for a fixed q this is a Laplace transform of the function t−q−1 Ω(t). The integral (??) converges for
COVARIANT TECHNIQUES FOR COMPUTATION OF THE HEAT KERNEL
955
Re q < 0. By integrating by parts N times and using the asymptotic conditions (??) we also get Z ∞ 1 dtt−q−1+N (−∂t )N [etλ Ω(t)] . (42) bq (λ) = Γ(−q + N ) 0 This integral converges for Re q < N − 1. Using this representation one can prove that [?] • the function bq (λ) is analytic function of q everywhere, i.e. it is an entire function, • the values of the function bq (λ) at the integer positive points q = k are given by X bk (λ) = (−∂t )k [etλ Ω(t)] t=0 = 0≤n≤k
where
Γ(k + 1) (−λ)k−n an , n!Γ(k − n + 1)
ak = (−∂t )k Ω(t) t=0 ,
(43)
(44)
• bq (λ) satisfies an asymptotic condition lim
|q|→∞, Re q
Γ(−q + N )bq (λ) = 0,
for any N > 0 .
(45)
By inverting the Mellin transform we obtain a new ansatz for the transport function and, hence, for the heat kernel Z c+i∞ 1 dq tq Γ(−q)bq (λ) , (46) Ω(t) = e−tλ 2πi c−i∞ where c < 0 and Re λ < λ1 . Clearly, since the left-hand side of this equation does not depend on λ, neither does the right-hand side. Thus, λ serves as an auxiliary parameter that regularizes the behavior at t → ∞. Instead if we invert the Laplace transform, we obtain another representation Z tq+1 γ+i∞ dλ e−tλ bq (λ) , (47) Ω(t) = Γ(−q) 2πi γ−i∞ where γ < λ1 and Re q < 0. Substituting this ansatz into the transport equation we get a functional equation for the function bq 1 1 + D bq (λ) = (L − λ) bq−1 (λ) . (48) q The initial condition for the transport function is translated into b0 (λ) = IV .
(49)
Thus, we have reduced the problem of solving the heat equation to the following problem: one has to find an entire function bq (λ|x, x0 ) that satisfies the functional equation (??) with the initial condition (??) and the asymptotic condition (??).
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Although the variables q and λ seem to be independent they are very closely related to each other. In particular, by differentiating with respect to λ we obtain an important result ∂ bq (λ) = −qbq−1 (λ) . (50) ∂λ Moreover, one can actually manifest the dependence of bq on λ. It is not difficult to prove that [?] Z c1 +i∞ 1 Γ(−p)Γ(p − q) (−λ)q−p ap , bq (λ) = dp (51) 2πi c1 −i∞ Γ(−q) with Re q < c1 < 0, satisfies Eq. (??) if ap satisfies this equation for λ = 0, i.e. 1 (52) 1 + D aq = L aq−1 . q a0 = IV .
(53)
For integer q = k = 1, 2, . . . the functional equation (??) becomes a recursion system that, together with the initial condition (??), determines all the HMDS-coefficients ak . From here, we also obtain the asymptotic expansion of bq (λ) as λ → −∞ bq (λ) ∼
X n≥0
Γ(q + 1) (−λ)q−n an . n!Γ(q − n + 1)
(54)
For integer q this coincides with (??). The function bq (λ) turns out to be extremely useful in computing the heat kernel, the resolvent kernel, the zeta-function and the determinant of the operator F . It contains the same information about the manifold as the heat kernel. In some cases the function bq can be constructed just by analytical continuation from the integer positive values bk [?]. Now we are going to do the usual trick, namely, to move the contour of integration over q to the right. Due to the presence of the gamma function Γ(−q) the integrand has simple poles at the non-negative integer points q = 0, 1, 2, . . ., which contribute to the integral while moving the contour. So, we get (N −1 ) X (−t)k −tλ bk (λ) + RN (t) , (55) Ω(t) = e k! k=0
where 1 RN (t) = 2πi
Z
cN +i∞
cN −i∞
dq tq Γ(−q)bq (λ)
(56)
with cN is a constant satisfying the condition N − 1 < cN < N . As t → 0 the rest term RN (t) behaves like O(tN ), so we obtain an asymptotic expansion as t → 0 Ω(t|x, x0 ) ∼ e−tλ
X (−t)k k≥0
k!
bk (λ|x, x0 ) =
X (−t)k ak (x, x0 ) . k!
k≥0
(57)
COVARIANT TECHNIQUES FOR COMPUTATION OF THE HEAT KERNEL
Using our ansatz (??) we find immediately the trace of the heat kernel Z c+i∞ −d/2 −tλ 1 Tr L2 exp(−tF ) = (4πt) e dq tq Γ(−q)Bq (λ) , 2πi c−i∞
957
(58)
Z
where Bq (λ) = Tr L2 bq (λ) =
d vol (x) tr V bq (λ|x, x) .
(59)
M
The trace of the heat kernel has an analogous asymptotic expansion as t → 0 Tr L2 exp(−tF ) ∼ (4πt)−d/2 e−tλ
X (−t)k X (−t)k Bk (λ) = Ak . k! k!
k≥0
(60)
k≥0
This is the famous Minakshisundaram–Pleijel asymptotic expansion. Physicists call it the Schwinger–De Witt expansion [?]. Its coefficients Ak are also sometimes called Hadamard–Minakshisundaram-De Witt–Seeley (HMDS) coefficients. This expansion is of great importance in differential geometry, spectral geometry, quantum field theory and other areas of mathematical physics, such as theory of Huygens’ principle, heat kernel proofs of the index theorems, Korteweg-De Vries hierarchy, Brownian motion, etc. (see, for example, [?]). One should stress, however, that this series does not converge, in general. In that sense our ansatz (??) or (??) in the form of a Mellin transform of an entire function is much better since it is exact and gives an explicit formula for the rest of the term. Let us apply our ansatz for computation of the complex power of a positive operator (F − λ) defined by Z ∞ 1 p −p dt tp−1 etλ U (t) . (61) G (λ) = (F − λ) = Γ(p) 0 Using our ansatz for the heat kernel we obtain Z c+i∞ 1 Γ(−q)Γ(−q − p + d/2) σ q+p−d/2 dq bq (λ) , Gp (λ) = (4π)−d/2 ∆1/2 P 2πi c−i∞ Γ(p) 2 (62) where c < −Re p + d/2. Outside the diagonal, i.e. for σ 6= 0, this integral converges for any p and defines an entire function of p. The integrand in this formula is a meromorphic function of p with some simple and maybe double poles. If we move the contour of integration to the right, we get contributions from the simple poles in the form of powers of σ and a logarithmic part due to the double poles (if any). This gives the complete structure of diagonal singularities of the complex power of the operator (F − λ), Gp (x, x0 ). Thus the function bq (λ) turns out to be very useful to study the diagonal singularities. In the particular case p = 1 we recover in this way the singularity structure of the resolvent σ + Greg (λ) , (63) G(λ) = (4π)−d/2 ∆1/2 P Φ(λ) + Ψ(λ) log 2
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I. G. AVRAMIDI
where d/2−1−k X (−1)k 2 Γ(d/2 − 1 − k) bk (λ) Φ(λ) = k! σ d/2−1
(64)
k=0
for odd d 0, d/2 Ψ(λ) = (−1) bd/2−1 (λ) for even d Γ(d/2) Greg = (4π)−d/2 ∆1/2 P
1 2πi
Z
α+i∞
dq Γ(−q)Γ(−q − 1 + d/2) α−i∞
(65)
σ q+1−d/2 2
bq (λ) ,
(66) where [d/2] − 1 < α < [d/2] − 1/2. We see that due to the absence of the double poles in the integrand there is no logarithmic singularity in odd dimensions. Thus, the singular part of the resolvent is determined by the HMDS-coefficients bk (λ) (and is, therefore, polynomial in λ) and the regular part is determined by the function bq (λ). Now, let us consider the diagonal limit of Gp . By taking the limit σ → 0 we obtain a very simple formula in terms of the function bq Gpdiag (λ) = (4π)−d/2
Γ(p − d/2) bd/2−p (λ|x, x) . Γ(p)
(67)
This gives automatically the zeta-function of the operator F [?] ζ(p, λ) = (4π)−d/2
Γ(p − d/2) Bd/2−p (λ) . Γ(p)
(68)
Thus, we see that both Gpdiag (λ) and ζ(p, λ) are meromorphic functions of p with simple poles at the points p = [d/2]+1/2−k, (k = 0, 1, 2, . . .) and p = 1, 2, . . . , [d/2]. In particular, the zeta-function is analytic at the origin. Its value at the origin is given by 0 for odd d d/2 (69) ζ(0, λ) = (−1) −d/2 Bd/2 (λ) for even d . (4π) Γ(d/2 + 1) This gives the regularized number of all modes of the operator F , since formally X 1. (70) ζ(0, λ) = Tr L2 I = n
Moreover, the derivative of the zeta-function at the origin is also well defined. As mentioned above it determines the regularized determinant of the operator (F − λ) since formally X log (λn − λ) = −ζ 0 (0, λ) . (71) log Det (F − λ) = Tr L2 log (F − λ) = n
COVARIANT TECHNIQUES FOR COMPUTATION OF THE HEAT KERNEL
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Thus we obtain for the determinant log Det (F − λ) = −(4π)−d/2
π(−1)(d+1)/2 Bd/2 (λ) Γ(d/2 + 1)
for odd d
(72)
and log Det (F − λ) = (4π)−d/2
(−1)d/2 0 Bd/2 (λ) − [Ψ(d/2 + 1) + C]Bd/2 (λ) (73) Γ(d/2 + 1)
for even d. Here Ψ(z) = (d/dz) log Γ(z) is the psi-function, C = −Ψ(1) is the Euler constant, and ∂ 0 Bd/2 Bq (λ) (λ) = . (74) ∂q q=d/2 3. Non-Recursive Solution of the Recursion System The main problem we study is to compute the HMDS-coefficients, not only R the integrated ones Ak = M d vol (x) tr V ak (x, x), which are determined by the diagonal values of ak (x, x), but also the off-diagonal coefficients ak (x, x0 ). They are determined by a recursion system which is obtained simply by restricting the complex variable q in Eq. (??) to positive integer values q = 1, 2, . . .. This problem was solved in [?, ?] where a systematic technique for calculation of ak was developed. The formal solution of this recursion system is −1 −1 −1 1 1 1 D L 1+ L··· 1 + D L·I. ak = 1 + D k k−1 1
(75)
So, the problem is to give a precise practical meaning to this formal operator solution. To do this one has, first of all, to define the inverse operator (1 + D/k)−1 . This can be done by constructing the complete set of eigenvectors of the operator D. However, first we introduce some auxiliary notions from the theory of symmetric tensors. 3.1. Algebra of symmetric tensors Let ω a and ea be the bases in the cotangent T ∗ M and tangent T M bundles, S (M ) be the bundle of symmetric contravariant tensors of rank n, Sn (M ) be the n (M ) = Sm (M ) ⊗ S n (M ) be the bundle of bundle of symmetric n-forms and Sm symmetric tensors of type (m, n) with the basis n
...am = ω (a1 ⊗ · · · ⊗ ω am ) ⊗ e(b1 ⊗ · · · ⊗ ebn ) , sab11...b n
(76)
where the parenthesis mean the symmetrization over all indices included. In the space Snn there is a natural unity symmetric tensor I(n) = s1...n 1...n , which is an identical endomorphism of the vector bundles S n and Sn .
(77)
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I. G. AVRAMIDI
We define the following binary operations on symmetric tensors: (a) the exterior symmetric tensor product ∨ n+i n × Sji → Sm+j ∨ : Sm
(78)
by (b ...b
b
...b
)
a ...a
n+i s 1 m+j , A ∨ B = A(a11 ...anm Ban+1 m+1 ...am+j ) b1 ...bn+i
(79)
(b) and an inner product ? n i × Sni → Sm , ? : Sm
(80)
b1 ...bi a1 ...am n A ? B = Aca11...c ...am Bc1 ...cn sb1 ...bi .
(81)
by Further, we also define an exterior symmetric covariant derivative ∇S on symmetric tensors m (82) ∇S : Snm → Sn+1 by a ...a
1 n+1 m . ∇S A = ∇(a1 Aba12...b ...an+1 ) sb1 ...bm
(83)
Everything said above remains true if we consider End (V )-valued symmetric n ⊗ End (V ), for some vector bundle tensors, i.e. sections of the vector bundle Sm V over M . The product operations then include the usual endomorphism (matrix) inner product as well. 3.2. Covariant taylor basis Let us consider the space C ∞ (Bx0 ) = {|f i ≡ f (x, x0 )| x ∈ Bx0 } of smooth two-point functions in a small neighbourhood Bx0 of the diagonal x = x0 . Here we denote the elements of this space by |f i. Let us define a special set of such functions |ni ∈ C ∞ (Bx0 ), labeled by a natural number n ∈ N, by |0i = 1 |ni =
(−1)n n 0 ∨ u, n!
(n = 1, 2, . . .) ,
(84)
where u0 is the tangent vector field to the geodesic connecting the points x and x0 at the point x0 given by the first derivative of the geodetic interval σ u0 = (g 0ab ∇0b σ)e0a ,
(85)
where prime 0 denotes the objects and operations at the point x0 . The functions |ni are two-point geometric objects, which are scalars at the point x and symmetric contravariant tensors at the point x0 , more precisely, they are sections of the vector bundle S n over the point x0 . Let us also define the dual space of linear functionals C ∞∗ (Bx0 ) = {hf | : C ∞ (Bx0 ) → C} ,
(86)
COVARIANT TECHNIQUES FOR COMPUTATION OF THE HEAT KERNEL
961
with the basis hn| dual to the basis |ni. The values of the dual basis functionals on the two-point functions are sections of the vector bundle of symmetric forms Sn defined to be the diagonal values of the symmetric exterior covariant derivative ∇S hn|f i = [(∇S )n f ] ,
(87)
where the square brackets mean restriction to the diagonal x = x0 . The basis hn| is dual to |mi in the sense that hn|mi = δmn I(n) .
(88)
Using this notation the covariant Taylor series for an analytic function |f i can be written in the form X |ni ? hn|f i . (89) |f i = n≥0
Remember that for smooth functions the Taylor series is only an asymptotic expansion, which does not necessarily converge. For analytic functions, however, the Taylor series converges in a sufficiently small neighbourhood of the diagonal. Therefore, our set of functions |ni forms a complete basis in the subspace of analytic functions L(Bx0 ) ⊂ C ∞ (Bx0 ) due to the fact that there is no nontrivial analytic function which is “orthogonal” to all of the eigenfunctions |ni. In other words, an analytic function that is equal to zero together with all symmetrized derivatives at the point x = x0 is, in fact, identically equal to zero in Bx0 . It is easy to show that these functions satisfy the equation D|ni = n|ni
(90)
and, hence, are the eigenfunctions of the operator D with positive integer eigenvalues. Note, however, that the space of analytical functions L(Bx0 ) is not a Hilbert space with a scalar product hf |gi defined above since there are a lot of analytic functions for which the norm hf |f i diverges. If we restrict ourselves to polynomial functions of some order then this problem does not appear. Thus the space of polynomials is a Hilbert space with the inner product defined above. 3.3. Covariant Taylor series for HMDS-coefficients ak The complete set of eigenfunctions |ni can be employed to present an arbitrary linear differential operator L in the form X |mi ? hm|L|ni ? hn| , (91) L= m,n≥0
where hm|L|ni are the “matrix elements” of the operator L that are just End (V )n valued symmetric tensors, i.e. sections of the vector bundle Sm (M ) ⊗ End (V ). We will not study the question of convergency of the expansion (??). It can be regarded as just a formal series. When acting on an analytic function, this series is nothing
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but the Taylor series and converges in a sufficiently small region Bx0 ; for a smooth function it gives the asymptotic expansion. Now it should be clear that the inverse operator (1 + k1 D)−1 can be defined by −1 X 1 k |ni ? hn| . (92) = 1+ D k k+n n≥0
Using this representation together with the analogous one for the operator L, (??), we obtain a covariant Taylor series for the coefficients bk X ak = |ni ? hn|ak i (93) n≥0
with the covariant Taylor coefficients hn|ak i given by [?, ?] X
hn|ak i =
n1 ,...,nk−1 ≥0
k−1 1 k · ··· k + n k − 1 + nk−1 1 + n1
×hn|L|nk−1 i ? hnk−1 |L|nk−2 i ? · · · ? hn1 |L|0i ,
(94)
where hm|L|ni are the matrix elements of the operator L (??). It is not difficult to show that for a differential operator L of second order, the matrix elements hm|L|ni do not vanish only for n ≤ m + 2. Therefore, the sum (??) always contains only a finite number of terms, i.e. the summation over ni is limited from above n1 ≥ 0,
ni ≤ ni+1 + 2,
(i = 1, . . . , k − 1; nk ≡ n) .
(95)
3.4. Matrix elements hm|L|ni Thus we reduced the problem of computation of the HMDS-coefficients ak to the computation of the matrix elements of the operator L. The matrix elements hm|L|ni are symmetric tensors of the type (m, n), i.e. sections of the vector bundle n (M ). Sm The matrix elements hn|L|mi of a Laplace type operator have been computed in our papers [?, ?]. They have the following general form: hm|L|m + 2i = −g ∗ ∨ I(m) hm|L|m + 1i = 0
(97) m
hm|L|ni =
n +
(96)
! I(n) ∨ Z(m−n) + m n−2
m
!
n−1
I(n−1) ∨ Y(m−n+1)
! I(n−2) ∨ X(m−n+2) ,
(98)
where g ∗ is the metric on the cotangent bundle, Z(n) is a section of the vector bundle Sn (M ) ⊗ End (V ), Y(n) is a section of the vector bundle Sn1 (M ) ⊗ End (V )
963
COVARIANT TECHNIQUES FOR COMPUTATION OF THE HEAT KERNEL
2 and X(n) is a section of the vector bundle Sn (M ). Here it is also meant that the n binomial coefficient k is equal to zero if k < 0 or n < k. We will not present here explicit formulas for the objects Z(n) , Y(n) , and X(n) , (they have been computed for arbitrary n in our papers [?, ?]), but note that all these quantities are expressed polynomially in terms of three sorts of geometric data:
• symmetric tensors of type (2, n), i.e. sections of the vector bundle Sn2 (M ) K(n) = (∇S )n−2 Riem ,
(99)
where Riem is the symmetrized Riemann tensor Riem = R(c (a d) b) sab cd ;
(100)
• sections of the vector bundle End (V ) ⊗ Sn1 (M ) R(n) = (∇S )n−1 R ,
(101)
where R is the curvature of the connection on the vector bundle V in the form R = Ra b sab ;
(102)
• End (V )-valued symmetric forms, i.e. sections of the vector bundle End (V ) ⊗ Sn (M ), constructed from the symmetrized covariant derivatives of the endomorphism Q of the vector bundle V Q(n) = (∇S )n Q .
(103)
From the dimensional arguments it is obvious that the matrix elements hn|L|ni are expressed in terms of the Riemann curvature tensor, Riem, the bundle curvature, R, and the endomorphism Q; the matrix elements hn + 1|L|ni — in terms of the quantities ∇Riem, ∇R and ∇Q; the elements hn + 2|L|ni — in terms of the quantities of the form ∇∇Riem, Riem · Riem, etc. 3.5. Diagramatic technique In the computation of the HMDS-coefficients by means of the matrix algorithm a “diagrammatic” technique, i.e. a graphic method for enumerating the different terms of the sum (??), turns out to be very convenient and pictorial [?]. The matrix elements hm|L|ni are presented by some blocks with m lines coming in from the left and n lines going out to the right (Fig. 1), ( m
.. .
.. .
Fig. 1.
) n
964
I. G. AVRAMIDI
and the product of the matrix elements hm|L|ki?hk|L|ni — by two blocks connected by k intermediate lines (Fig. 2), ( m
(
.. .
k
.. .
.. .
) n
Fig. 2.
that represents the contractions of the corresponding tensor indices (the inner product). To obtain the coefficient hn|ak i one should draw, first, all possible diagrams which have n lines incoming from the left and which are constructed from k blocks connected in all possible ways by any number of intermediate lines. When doing this, one should keep in mind that the number of the lines, going out of any block, cannot be greater than the number of the lines, coming in, by more than two and by exactly one. Then one should sum up all diagrams with the weight determined for each diagram by the number of intermediate lines from the analytical formula (??). Drawing of such diagrams is of no difficulties. This helps to keep under control the whole variety of different terms. Therefore, the main problem is reduced to the computation of some standard blocks, which can be computed once and for all. For example, the diagrams for the diagonal values of the HMDS-coefficients [ak ] = h0|ak i have the form:
j jj jj jjj jjj jjj jjj jjj jjj j j j jj [a1 ] =
[a2 ] =
[a3 ] =
+
+
2 1 · 4 2
+
1 3
1 3
+
+
(104)
(105)
2 4
2 1 · 4 3
+
2 1 · 4 5
.(106)
As an illustration let us compute the coefficients [a1 ] and [a2 ]. We have [?, ?] 1 = h0|L|0i = Z(0) = Q − R IV 6
(107)
= h0|L|2i = −g ab
(108)
= h2|L|0i = Z(2)ab
(109)
= h0|L|2ih2|L|0i = − tr g Z(2) = −g ab Z(2)ab ,
(110)
COVARIANT TECHNIQUES FOR COMPUTATION OF THE HEAT KERNEL
965
where 1 1 Z(2)ab = ∇(a ∇b) Q − Rc(a Rc b) + ∇(a ∇|c| Rc b) 2 2 3 1 1 + IV − ∇a ∇b R − Rab + Rac Rc b 20 20 15 1 1 − Racde Rb cde − Rcd Rc a d b . 30 30
(111)
Here Rabcd and Rab = g cd Racbd are the components of the Riemann and Ricci tensors respectively, R = g ab Rab is the scalar curvature and Rab are the components of the curvature of the bundle connection ∇V . Hence, we immediately get 1 [a1 ] = Q − R IV , 6
(112)
and, by taking the trace of Z(2) and using the identity ∇a ∇b Rab = 0, we obtain the well-known result [?] 2 1 1 1 − Q + Rab Rab [a2 ] = Q − R IV 6 3 6 1 1 1 R − Rab Rab + Rabcd Rabcd . (113) + IV 15 90 90 3.6. Remarks Let us make some remarks about the elaborated technique. • This technique is applicable for a generic Riemannian manifold M and for a generic vector bundle V of arbitrary dimensions. • This technique is manifestly covariant, which is an inestimable advantage in quantum field theory, especially in quantum gravity and gauge theories. A manifestly covariant calculus is such that every step is invariant, or covariant; i.e. it is not something which proceeds through non-invariant steps to an invariant “bottom line”. Roughly speaking, it deals with the curvatures and its covariant derivatives instead of partial derivatives of the metric, that is why a covariant technique is much more effective. Besides, when doing nonlocal two-point calculations, i.e. off-diagonal heat kernel or Green functions in quantum field theory, the result is also expressed in terms of non-local covariant geometric objects, e.g. the geodetic interval and its covariant derivatives, operators of parallel transport, etc. • Since it is purely local, it is also valid for manifolds with boundary and noncompact manifolds, provided one considers the local HMDS-coefficients bk (x, x0 ) in a small neighbourhood B of the diagonal of M × M that does not intersect with the boundary, B ∩ ∂M = ∅. • Moreover, this technique also works in the case of pseudo-Riemannian manifolds and hyperbolic differential operators.
966
I. G. AVRAMIDI
• This method is direct, or straightforward — it works without using any additional properties of the heat kernel coefficients Ak (for an overview of different approaches for calculating the heat kernel coefficients see [?]). • It gives not only the diagonal values of the HMDS-coefficients [ak ] but also the diagonal values of all their derivatives; thus it gives immediately the asymptotics of the trace of derivatives of the heat kernel Tr L2 P exp(−tF ) ,
(114)
where P is a differential operator. • Due to the use of symmetric forms and symmetric covariant derivatives the famous “combinatorial explosion” in the complexity of the HMDS-coefficients is avoided. • The developed technique is very algorithmic and well suited to automated computation — there are a number of usual algebraic operations on symmetric tensors that seems to be easily programmed, the needed input, i.e. the matrix elements hn|L|mi, is computed in advance analytically and is already known. Recently, considerable attention has been focused on this problem [?, ?, ?, ?]. • The developed method is very powerful; it enables us to compute for the first time the diagonal value of the fourth HMDS-coefficient [a4 ] [?, ?]. (The third coefficient [a3 ] was first computed by Gilkey [?].) • Lastly, this technique enables one not only to carry out explicit computations, but also to analyse the general structure of the HMDS-coefficients ak for all orders k.
4. Covariant Approximation Schemes for the Heat Kernel 4.1. General structure of HMDS-coefficients Now we are going to investigate the general structure of the HMDS-coefficients. We will follow mainly our papers [1–6] (see also our review papers [7–9]). Our analysis will again be purely local. Since locally one can always expand the metric, the connection and the endomorphism Q in the covariant Taylor series, they are completely characterized by their Taylor coefficients, i.e. the covariant derivatives of the curvatures, more precisely by the objects K(n) , R(n) and Q(n) introduced above. We introduce the following notation for all of them: <(n) = {K(n+2) , R(n+1) , Q(n) },
(n = 0, 1, 2, . . .) ,
(115)
and call these objects covariant jets. n will be called the order of a jet <(n) . It is worth noting that the jets are defined by symmetrized covariant derivatives. This makes them well defined as the order of the derivatives becomes unimportant. It is only the number of derivatives that plays a role. Further we introduce an infinite set of covariant jets of all orders J = {<(n) ;
(n = 0, 1, 2, . . .)} .
(116)
967
COVARIANT TECHNIQUES FOR COMPUTATION OF THE HEAT KERNEL
As we already know, the first two HMDS-coefficients have a very well-known form [?, ?]: Z d vol (x) tr V IV , 1 A1 = d vol (x) tr V Q − R IV . 6 M
A0 =
(117)
ZM
(118)
As far as the higher order coefficients Ak , (k ≥ 2), are concerned they are integrals of local invariants which are polynomial in the jets [?]. One can classify all the terms in them according to the number of the jets and their order. The terms linear in the jets in higher order R coefficients Ak , (k ≥ 2), are given by integrals of total derivatives, symbolically M d vol (x) tr V k−1 <. They are calculated explicitly in [?]. Since the total derivative do not contribute to an integral over a complete compact manifold, it is clear that the linear terms vanish. Thus Ak , (k = 2, 3, . . .), begin with the terms quadratic in the jets. These terms contain the jets of highest orderR(or the leading derivatives of the curvatures) and can be shown to be of the form M d vol (x) tr V <k−2 <. Then it follows a class of terms cubic in the jets etc. The last class of terms does not contain any covariant derivatives at all but only the powers of the curvatures. In other words, the higher order HMDS-coefficients have a general structure, which can be presented symbolically in the form:
Z Ak =
d vol (x) trV M
X
+···+
<k−2 < +
X
< ∇i < ∇2k−6−i <
0≤i≤2k−6
0≤i≤k−3
.
(119)
4.2. Leading derivatives in heat kernel asymptotics More precisely, all quadratic terms can be reduced to a finite number of invariant structures, viz. [?, ?]
Ak,2
k!(k − 2)! = 2(2k − 3)! (3)
Z
n d vol (x) trV
M (4)
(1)
(2)
fk Qk−2 Q + 2fk Rbc ∇b k−3 ∇a Ra c (5)
+ fk Qk−2 R + fk Rab k−2 Rab + fk Rk−2 R (i)
o ,
(120)
where fk are some numerical coefficients. These numerical coefficients can be computed by the technique developed in the previous section. From the formula
968
I. G. AVRAMIDI
(??) we have for the diagonal coefficients [ak ] up to cubic terms in the jets X X (−1)k−1 h0; k − 1|L|0i + (−1)k [ak ] = h0|ak i = 2k − 1 ni =0 i=1 k 2k − 1 i h0; k − i − 1|L|ni ihni ; i − 1|L|0i + O(<3 ) , (121) × 2k − 1 2i + ni − 1 i k k−1 2(k−i−1)
where hn; k|L|mi = (∨k g ∗ ) ? hn|L|mi
(122)
3
and O(< ) denote terms of third order in the jets. By computing the matrix elements in the second order in the jets and integrating over M one obtains [?, ?] (1)
=1
(2)
=
1 2(2k − 1)
(124)
(3)
=
k−1 2(2k − 1)
(125)
(4)
=
fk fk fk fk
(5)
fk
=
(123)
1 2(4k 2
− 1)
k2 − k − 1 . 4(4k 2 − 1)
(126) (127)
One should note that the same results were obtained by a completely different method by Branson, Gilkey and Ørsted [?]. 4.3. “Summation” of asymptotic expansion Let us consider the situation when the curvatures are small but rapidly varying, i.e. the derivatives of the curvatures are more important than the powers of them. Then the leading derivative terms in the heat kernel are the largest ones. Thus the trace of the heat kernel has the form t2 (128) Tr L2 exp(−tF ) ∼ (4πt)−d/2 A0 − tA1 + H2 (t) + O(<3 ) , 2 where H2 (t) is some complicated nonlocal functional that has the following asymptotic expansion as t → 0: H2 (t) ∼ 2
X (−t)k−2 Ak,2 + O(<3 ) . k! k≥2
(129)
COVARIANT TECHNIQUES FOR COMPUTATION OF THE HEAT KERNEL
969
Using the results for Ak,2 one can easily construct such a functional H2 just by a formal summing of the leading derivatives Z 1 d vol (x) tr V Qγ (1) (−t)Q + 2Rac ∇a γ (2) (−t)∇b Rbc H2 = M − 2Qγ (3)(−t)R + Rab γ (4) (−t)Rab + Rγ (5) (−t)R , (130) where γ (i) (z) are entire functions defined by [?, ?] Z 1 X 1 − ξ2 k! (i) fk z k = z dξ f (i) (ξ) exp − γ (i) (z) = (2k + 1)! 4 0
(131)
k≥0
where f (1) (ξ) = 1
(132)
f (2) (ξ) =
1 2 ξ 2
(133)
f (3) (ξ) =
1 (1 − ξ 2 ) 4
(134)
f (4) (ξ) =
1 4 ξ 6
(135)
f (5) (ξ) =
1 (3 − 6ξ 2 − ξ 4 ) . 48
(136)
Therefore, H2 (t) can be regarded as generating functional for quadratic terms Ak,2 (leading derivative terms) in all HMDS-coefficients Ak . It also plays a very important role in investigating the nonlocal structure of the effective action in quantum field theory in high-energy approximation [?]. Let us also note that the function Bq (λ) introduced in the Sec. 1 can be obtained just by analytical continuation of the formula for Bk (λ) from integer points k to a complex plane q. 4.4. Covariantly constant background Let us consider now the opposite case, when the curvatures are strong but slowly varying, i.e. the powers of the curvatures are more important than the derivatives of them. The main terms in this approximation are the terms without any covariant derivatives of the curvatures, i.e. the lowest order jets. We will consider mostly the zeroth order of this approximation which corresponds simply to covariantly constant background curvatures: ∇Riem = 0,
∇R = 0,
∇Q = 0 .
(137)
The asymptotic expansion of the trace of the heat kernel Tr L2 exp(−tF ) ∼ (4πt)−d/2
X (−t)k k≥0
k!
Ak
(138)
970
I. G. AVRAMIDI
determines then all the terms without covariant derivatives (highest order terms in the jets), Ak,k , in all HMDS-coefficients Ak . These terms do not contain any covariant derivatives and are just polynomials in the curvatures and the endomorphism Q. Thus the trace of the heat kernel is a generating functional for all HMDScoefficients for a covariantly constant background, in particular, for all symmetric spaces. Thus the problem is to calculate the trace of the heat kernel for covariantly constant background. 4.4.1. Algebraic approach There exist a very elegant indirect way to construct the heat kernel without solving the heat equation but using only the commutation relations of some covariant first order differential operators [3–6]. The main idea is in a generalization of the usual Fourier transform to the case of operators and consists in the following. Let us consider for a moment a trivial case, where the curvatures vanish but the potential term does not: Riem = 0,
R = 0,
∇Q = 0 .
(139)
In this case the operators of covariant derivatives obviously commute and form together with the potential term an Abelian Lie algebra [∇µ , ∇ν ] = 0,
[∇µ , Q] = 0 .
(140)
It is easy to show that the heat semigroup operator can be presented in the form Z 1 −d/2 exp(−tF ) = (4πt) exp(−tQ) d vol (k) exp − hk, gki + k · ∇ , (141) 4t Rd where hk, gki = k µ gµν k ν , k · ∇ = k µ ∇µ . Here, of course, it is assumed that the covariant derivatives also commute with the metric [∇, g] = 0 .
(142)
Acting with this operator on the Dirac distribution and using the obvious relation exp(k · ∇)δ(x, x0 ) = δ(k) , (143) x=x0
one integrates easily over k and obtains the trace of the heat kernel Z d vol (x) tr V exp(−tQ) . Tr L2 exp(−tF ) = (4πt)−d/2
(144)
M
In fact, the covariant differential operators ∇ do not commute, their commutators being proportional to the curvatures <. The commutators of covariant derivatives ∇ with the curvatures < give the first derivatives of the curvatures, i.e. the jets <(1) , the commutators of covariant derivatives with <(1) give the second jets <(2) , etc. Thus the operators ∇ together with the whole set of the jets J form an infinite dimensional Lie algebra G = {∇, <(i) ; (i = 1, 2, . . .)}.
COVARIANT TECHNIQUES FOR COMPUTATION OF THE HEAT KERNEL
971
Now, let us remember that the heat kernel diagonal is a functional of the jets, with the jets being defined by symmetrized covariant derivatives. This makes the order of a jet be well defined. For example, the structures involving commutators of covariant derivatives, like [∇a , ∇b ]Re c f d , which involve 2-jets of the Riemann tensor on the left but (after using the Ricci identity) only 0-jets on the right, are not allowed. After symmetrizing over abcd this jet vanish. So, if we express the final answer for the heat kernel diagonal or for the heat kernel coefficients in terms of the symmetrized jets, then there is a natural filtration with respect to order of the jets involved. In other words, one can always say, what is the maximal order of symmetrized covariant derivative of the curvature involved in the result. This is especially true for the heat kernel coefficients Ak since they are polynomial in the jets. If we identify a small parameter ε with each derivative (this can be always done by a local deformation of the operator, i.e. the metric, connection, etc.) then a jet of order n is, actually, of order εn . Thus, we get a perturbation theory in this small parameter (see [?, ?] for details). Since the derivatives are naturally identified with the momentum (or energy), the physicists call a situation when the derivatives are small the low-energy approximation. To evaluate the heat kernel in the considered (low-energy) approximation one can take into account a finite number of low-order jets, i.e. the low-order covariant derivatives of the background fields, {<(i) ; (i ≤ N )}, and neglect all the higher order jets, i.e. the covariant derivatives of higher orders, i.e. put <(i) = 0 for i > N . Then one can show that there exist a set of covariant differential operators that together with the background fields and their low-order derivatives generate a finite dimensional Lie algebra GN = {∇, <(i) ; (i = 1, 2, . . . , N )} [7–9]. One should stress here what problem one can solve this way. We try to answer the following concrete question: How do the heat coefficients look if we throw away all the (symmetrized) jets of order higher than N ? Thus one can try to generalize the above idea in such a way that (??) would be the zeroth approximation in the commutators of the covariant derivatives, i.e. in the curvatures. Roughly speaking, we are going to find a representation of the heat semigroup operator in the form: Z 1 (145) dk Φ(t, k) exp − hk, Ψ(t)ki + k · T , exp(−tF ) = 4t RD µ where hk, Ψ(t)ki = k A ΨAB (t)k B , k · T = k A TA , (A = 1, 2, . . . , D), TA = XA ∇µ + YA are some first order differential operators and the functions Ψ(t) and Φ(t, k) are expressed in terms of commutators of these operators — i.e. in terms of the curvatures. In general, the operators TA do not form a closed finite dimensional algebra because at each step taking more commutators there appear more and more derivatives of the curvatures. It is the low-energy reduction G → GN , i.e. the restriction to the low-order jets, that actually closes the algebra G of the operators TA and the background jets, i.e. makes it finite dimensional.
972
I. G. AVRAMIDI
Using this representation one can, as above, act with exp(k · T ) on the Dirac distribution to get the heat kernel. The main point of this idea is that it is much easier to calculate the action of the exponential of the first order operator k · T on the Dirac distribution than that of the exponential of the second order operator . 4.4.2. Covariantly constant bundle curvature and covariantly constant endomorphsim Q in flat space Let us now consider the more complicated case of nontrivial covariantly constant curvature of the connection on the vector bundle V in flat space: Riem = 0,
∇R = 0,
∇Q = 0 .
(146)
Using the condition of covariant constancy of the curvatures one can show that in this case the covariant derivatives form a nilpotent Lie algebra [?] [∇µ , ∇ν ] = Rµν ,
(147)
[∇µ , Rαβ ] = [∇µ , Q] = 0 ,
(148)
[Rµν , Rαβ ] = [Rµν , Q] = 0 .
(149)
For this algebra one can prove a theorem expressing the heat semigroup operator in terms of an average over the corresponding Lie group [?] tR 1/2 −d/2 (150) exp(−tQ) det End (T M) exp(−tF ) = (4πt) sinh (tR) Z 1 × d vol (k) exp − hk, gtR coth (tR)ki + k · ∇ , (151) 4t Rd where k ·∇ = k µ ∇µ . Here functions of the curvatures R are understood as functions of sections of the bundle End (T M ) ⊗ End (V ), and the determinant det End (T M) is taken with respect to End (T M ) indices, End (V ) indices being intact. It is not difficult to show that in this case we also have (152) exp(k · ∇)δ(x, x0 ) x=x0 = δ(k) . Subsequently, the integral over k µ becomes trivial and we obtain immediately the trace of the heat kernel [?] Z tR 1/2 −d/2 2 . d vol (x) tr V exp(−tQ) det End (T M) Tr L exp(−tF ) = (4πt) sinh (tR) M (153) Expanding it in a power series in t one can find all covariantly constant terms in all HMDS-coefficients Ak . As we have seen the contribution of the bundle curvature Rµν is not as trivial as that of the potential term. However, the algebraic approach does work in this case too. It is a good example how one can get the heat kernel without solving
COVARIANT TECHNIQUES FOR COMPUTATION OF THE HEAT KERNEL
973
any differential equations but using only the algebraic properties of the covariant derivatives. 4.4.3. Contribution of two first derivatives of the endomorphism Q In fact, in flat space it is possible to do a bit more, i.e. to calculate the contribution of the first and the second derivatives of the potential term Q [?]. That is we consider the case when the derivatives of the endomorphism Q vanish only starting from the third order, i.e.: ∇R = 0,
Riem = 0,
∇∇∇Q = 0 .
(154)
Besides we assume the background to be Abelian, i.e. all the nonvanishing background quantities, Rαβ , Q, Q;µ ≡ ∇µ Q and Q;νµ ≡ ∇µ ∇ν Q, commute with each other. Thus we have again a nilpotent Lie algebra [∇µ , ∇ν ] = Rµν
(155)
[∇µ , Q] = Q;µ
(156)
[∇µ , Q;ν ] = Q;νµ
(157)
all other commutators being zero. Now, let us represent the endomorphism Q in the form: Q = Ω − αik Ni Nk ,
(158)
where (i = 1, . . . , q; q ≤ d), αik is some constant symmetric nondegenerate q × q matrix, Ω is a covariantly constant endomorphism and Ni are some endomorphisms with vanishing second covariant derivative: ∇Ω = 0,
∇∇Ni = 0 .
(159)
Next, let us introduce the operators XA = (∇µ , Ni ), (A = 1, . . . , d + q) and the matrix ! Rµν Ni;µ , (160) (FAB ) = −Nk;ν 0 with Ni;µ ≡ ∇µ Ni . The operator F can now be written in the form: F = −GAB XA XB + Ω , where AB
(G
)=
g µν
0
(161)
!
0 αik
(162)
and the commutation relations (??) take a more compact form [XA , XB ] = FAB all other commutators being zero.
(163)
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I. G. AVRAMIDI
This algebra is again a nilpotent Lie algebra. Thus one can also apply the previous theorem in this case to get [?] tF exp(−tF ) = (4πt)−(d+q)/2 exp(−tΩ) det 1/2 sinh (tF ) Z 1 dkG1/2 exp − hk, GtF coth (tF )ki + k · X , (164) × 4t Rd+q where G = det GAB and k · X = k A XA . Therefore we have expressed the heat semigroup operator in terms of the operator exp(k · X). The integration over k is Gaussian except for the noncommutative part. Splitting the integration variables (k A ) = (q µ , ω i ) and using the Campbell– Hausdorf formula we obtain [?] exp(k · X)δ(x, x0 )|x=x0 = exp(ω · N )δ(q) ,
(165)
where ω · N = ω i Ni . Further, after taking off the trivial integration over q and a Gaussian integral over ω, we obtain the trace of the heat kernel in a very simple form [?] Z d vol (x) tr V Φ(t) Tr L2 exp(−tF ) = (4πt)−d/2 M
1 × exp −tQ + t3 h∇Q, Ψ(t)g ∗ ∇Qi , 4
(166)
where h∇Q, Ψ(t)g ∗ ∇Qi = ∇µ QΨµν (t)g νλ ∇λ Q, −1/2 −1/2 Φ(t) = det End (T M) K(t) det End (T M) 1 + t2 [E(t) − S(t)K −1 (t)S(t)]P −1/2
× det End (T M) [1 + t2 C(t)P ] ,
(167)
Ψ(t) = {Ψµν (t)} = [1 + t2 C(t)P ]−1 C(t) ,
(168)
P is the matrix determined by second derivatives of the potential term, P = {P µ ν },
P µν =
1 µλ g ∇ν ∇λ Q , 2
(169)
and the matrices C(t) = {C µ ν (t)}, K(t) = {K µ ν (t)} S(t) = {S µ ν (t)} and E(t) = {E µ ν (t)} are defined by I dz t coth (tz −1 )(1 − zR − z 2 P )−1 , (170) C(t) = C 2πi I dz t sinh (tz −1 )(1 − zR − z 2 P )−1 , (171) K(t) = 2 C 2πi z I dz t sinh (tz −1 )(1 − zR − z 2 P )−1 , (172) S(t) = C 2πi z I dz t sinh (tz −1 )(1 − zR − z 2 P )−1 , (173) E(t) = C 2πi
COVARIANT TECHNIQUES FOR COMPUTATION OF THE HEAT KERNEL
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where the integrals are taken along a sufficiently small closed contour C that encircles the origin counter-clockwise, so that F (z) = (1 − zR − z 2 P )−1 is analytic inside this contour. The formula (??) exhibits the general structure of the trace of the heat kernel. Namely, one sees immediately how the endomorphism Q and its first derivatives ∇Q enter the result. The nontrivial information is contained only in a scalar, Φ(t), and a tensor, Ψµν (t), functions which are constructed purely from the curvature Rµν and the second derivatives of the endomorphism Q, ∇∇Q. So, we conclude that the HMDS-coefficients Ak are constructed from three different types of scalar (connected) blocks, Q, Φ(n) (R, ∇∇Q) and ∇µ QΨµν (n) (R, ∇∇Q) ∇ν Q. They are listed explicitly up to A8 in [?]. 4.4.4. Symmetric spaces Let us now generalize the algebraic approach to the case of the curved manifolds with covariantly constant Riemann curvature and the trivial bundle connection [?, ?]: ∇Riem = 0, R = 0, ∇Q = 0 . (174) First of all, we give some definitions. The condition (??) defines, as we already said above, the geometry of locally symmetric spaces. A Riemannian locally symmetric space which is simply connected and complete is globally symmetric space (or, simply, symmetric space). A symmetric space is said to be of compact, noncompact or Euclidean type if all sectional curvatures K(u, v) = Rabcd ua v b uc v d are positive, negative or zero. A direct product of symmetric spaces of compact and noncompact types is called semisimple symmetric space. A generic complete simply connected Riemannian symmetric space is a direct product of a flat space and a semisimple symmetric space. It is well known that the HMDS-coefficients have a universal structure, i.e. they are polynomials in the curvatures, with the numerical coefficients that do not depend on the global properties of the manifold, on the dimension, on the signature of the metric, etc. It is this universal structure we would like to study. To solve this problem one has to compute the heat kernel diagonal in the case when all symmetrized jets of higher order (those involving derivatives) vanish but there are no algebraical restrictions on the curvatures. However, in this setting the problem is too complicated. By considering the symmetric spaces we simplify the problem in that we also throw away some contribution in this universal structure that is also constructed from the zero-jets, but vanishes in symmetric spaces due to special algebraical identities, which follow from the covariant constancy of the curvature. It should be noted that our analysis in this paper is purely local. We are looking for a universal (in the category of locally symmetric spaces) local generating function of the curvature invariants, that reproduces adequately the asymptotic expansion of the trace of the heat kernel. This function should give all the terms without covariant derivatives of the curvature in the asymptotic expansion of the heat kernel, i.e. in other words all HMDS-coefficients Ak for any locally symmetric space. It turns out to be much more convenient to obtain a universal generating function of t
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whose Taylor coefficients reproduce the heat kernel coefficients Ak than to compute them directly. It is obvious that any flat subspaces do not contribute to the HMDS-coefficients Ak . Therefore, to find this universal structure it is sufficient to consider only semisimple symmetric spaces. Moreover, since HMDS-coefficients are analytic in the curvatures, one can restrict oneself to only symmetric spaces of compact type. Using the factorization property of the heat kernel and the duality between compact and noncompact symmetric spaces one then can obtain the results for the general case by analytical continuation. That is why we consider only the case of compact symmetric spaces when the sectional curvatures and the metric are positive definite. Let ea be a basis in the tangent bundle which is covariantly constant (parallel) along the geodesic. The frame components of the curvature tensor of a symmetric space are, obviously, constant and can be presented in the form Rabcd = βik E i ab E k cd ,
(175)
where E i ab , (i = 1, . . . , p; p ≤ d(d − 1)/2), is some set of antisymmetric matrices and βik is some symmetric nondegenerate p × p matrix. The traceless matrices Di = {Da ib } defined by Da ib = −βik E k cb g ca = −Da bi
(176)
are known to be the generators of the holonomy algebra H [Di , Dk ] = F j ik Dj ,
(177)
where F j ik are the structure constants. In symmetric spaces a much richer algebraic structure exists. Indeed, let us define the quantities C A BC = −C A CB , (A = 1, . . . , D; D = d + p): C i ab = E i ab ,
C a ib = Da ib ,
C i kl = F i kl ,
(178)
C a bc = C i ka = C a ik = 0 , and the matrices CA = {C B AC } = (Ca , Ci ): ! 0 Db ai , Ci = Ca = E j ac 0
(179)
Db ia
0
0
F j ik
! .
(180)
One can show that they satisfy the Jacobi identities [?, ?] [CA , CB ] = C C AB CC
(181)
and, hence, define a Lie algebra G of dimension D with the structure constants C A BC , the matrices CA being the generators of adjoint representation. In symmetric spaces one can find explicitly the generators of the infinitesimal isometries, i.e. the Killing vector fields ξA , and show that they form a Lie algebra
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of isometries that is (in case of semisimple symmetric space) isomorphic to the Lie algebra G, viz. (182) [ξA , ξB ] = C C AB ξC . Moreover, introducing a symmetric nondegenerate D × D matrix ! gab 0 , γAB = 0 βik
(183)
that plays the role of the metric on the algebra G, one can express the operator F in semisimple symmetric spaces in terms of the generators of isometries F = −γ AB ξA ξB + Q ,
(184)
where γ AB = (γAB )−1 . Using this representation one can prove a theorem that presents the heat semigroup operator in terms of some average over the group of isometries G [?, ?] 1 exp(−tF ) = (4πt)−D/2 exp −t Q − RG 6 Z 1 sinh (k · C/2) 1/2 exp − hk, γki + k · ξ , dk|g|1/2 det Ad(G) × k · C/2 4t RD (185) where γ = det γAB , k · C = k A CA , k · ξ = k A ξA , and RG is the scalar curvature of the group of isometries G 1 (186) RG = − γ AB C C AD C D BC . 4 Acting with this operator on the Dirac distribution δ(x, x0 ) one can, in principle, evaluate the off-diagonal heat kernel exp(−tF )δ(x, x0 ), i.e. for non-coinciding points x 6= x0 (see [?]). To calculate the trace of the heat kernel, it is sufficient to compute only the coincidence limit x = x0 . Splitting the integration variables k A = (q a , ω i ) and solving the equations of characteristics one can obtain the action of the isometries on the Dirac distribution [?, ?] sinh (ω · D/2) −1 δ(q) , (187) = det exp (k · ξ) δ(x, x0 ) End (T M) ω · D/2 x=x0 where ω · D = ω i Di . Using this result one can easily integrate over q in √ (??) to get the heat kernel diagonal. After changing the integration variables ω → tω it takes the form [?, ?] 1 1 −d/2 exp −t Q − R − RH [U (t)] = (4πt) 8 6 Z 1 dω β 1/2 exp − hω, βωi × (4π)−p/2 4 Rp √ √ sinh ( tω · F/2) sinh ( tω · D/2) 1/2 −1/2 √ √ × det Ad(H) det End (T M) , tω · F/2 tω · D/2 (188)
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where ω · F = ω i Fi , Fi = {F j ik } are the generators of the holonomy algebra H in adjoint representation and 1 RH = − β ik F m il F l km 4
(189)
is the scalar curvature of the holonomy group. Here β = det βik and β ik = (βik )−1 . The remaining integration over ω in (??) can be done in a rather formal way [?, ?]. Let a∗i and ak be operators acting on a Hilbert space, that form the following Lie algebra: [aj , a∗k ] = δkj ,
(190)
[ai , ak ] = [a∗i , a∗k ] = 0 .
(191)
Let |0i be the “vacuum vector” in the Hilbert space, i.e. h0|0i = 1 , ai |0i = 0,
h0|a∗k = 0 .
(192) (193)
Then the heat kernel (??) can be presented in an formal algebraic form without any integration 1 1 −d/2 exp −t Q − R − RH [U (t)] = (4πt) 8 6 √ √ ta · F/2) ta · D/2) sinh ( sinh ( 1/2 −1/2 √ √ det End (T M) × 0 det Ad(H) ta · F/2 ta · D/2 (194) × exp(ha∗ , β −1 a∗ i) 0 , where a · F = ak Fk and a · D = ak Dk . This formal solution should be understood as a power series in the operators ak and a∗k and determines a well-defined asymptotic expansion in t → 0. Let us stress that these formulas are manifestly covariant because they are expressed in terms of the invariants of the holonomy group H, i.e. the invariants of the Riemann curvature tensor. They can be used now to generate all HMDS-coefficients [bk ] for any locally symmetric space, i.e. for any manifold with covariantly constant curvature, simply by expanding it in an asymptotic power series as t → 0. Thereby one finds all covariantly constant terms in all HMDS-coefficients in a manifestly covariant way. This gives a very nontrivial example how the heat kernel can be constructed using only the Lie algebra of isometries of the symmetric space. 5. Conclusion In present paper we have presented recent results in studying the heat kernel obtained in our papers [1–6]. We discussed some ideas connected with the problem of developing consistent covariant approximation schemes for calculating the heat kernel. Special attention is paid to the low-energy approximation. It is shown
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that in the local analysis there exists an algebraic structure (the Lie algebra of background jets) that turns out to be extremely useful for the study of the lowenergy approximation. Based on the background jets algebra we have proposed a new promising approach for calculating the low-energy heat kernel. Within this framework we have obtained closed formulas for the heat kernel diagonal in the case of covariantly constant background. Besides, we were able to take into account the first and second derivatives of the endomorphism Q in flat space. The obtained formulas are manifestly covariant and applicable for a generic covariantly constant background. This enables the results to be treated as the generating functions for the whole set of the Hadamard–Minakshisundaram–De Witt–Seeley-coefficients. In other words, we have calculated all covariantly constant terms in all HMDS-coefficients. Needless to say that the investigation of the low-energy effective action is of great importance in quantum gravity and gauge theories because it describes the dynamics of the vacuum state of the theory. Acknowledgments I would like to thank Peter Bor-Luh-Lin and Thomas Branson for hospitality expressed to me at the University of Iowa, where this work was completed. It was supported by the Deutsche Forschungsgemeinschaft. I am also grateful to the anonymous referee whose report contributed significantly to the improvement of the manuscript. References [1] I. G. Avramidi, “Covariant methods for the calculation of the effective action in quantum field theory and investigation of higher-derivative quantum gravity, PhD thesis”, Moscow State Univ. (1986), UDK 530.12:531.51, 178 pp. [in Russian]; Transl.: hep-th/9510140, 159 pp. [2] I. G. Avramidi, “A covariant technique for the calculation of the one-loop effective action”, Nucl. Phys. B355 (1991) 712–754. Erratum: Nucl. Phys. B509 (1998) 557– 558. [3] I. G. Avramidi, “A new algebraic approach for calculating the heat kernel in gauge theories”, Phys. Lett. B 305 (1993) 27–34. [4] I. G. Avramidi, “Covariant algebraic method for calculation of the low-energy heat kernel”, J. Math. Phys. 36 (1995) 5055–5070. [5] I. G. Avramidi, “The heat kernel on symmetric spaces via integrating over the group of isometries”, Phys. Lett. B 336 (1994) 171–177. [6] I. G. Avramidi, “A new algebraic approach for calculating the heat kernel in quantum gravity”, J. Math. Phys. 37 (1996) 374–394. [7] I. G. Avramidi, “New algebraic methods for calculating the heat kernel and the effective action in quantum gravity and gauge theories”, in Heat Kernel Techniques and Quantum Gravity, ed. S. A. Fulling, Discourses in Mathematics and Its Applications, College Station, Texas: Department of Mathematics, Texas A&M Univ. 1995, pp. 115–140. [8] I. G. Avramidi, “Nonperturbative methods for calculating the heat kernel”, Proc. Int. Workshop “Global Analysis, Differential Geometry and Lie Algebras”, Thessaloniki, Greece, Dec. 15–17, 1994; ed. G. Tsagas, Balcan Press, 1998, pp. 7–21.
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[9] I. G. Avramidi, “Covariant approximation schemes for calculation of the heat kernel in quantum field theory”, Proc. VIth Moscow Int. Seminar “Quantum Gravity”, Moscow, June 12–19, Singapore, World Scientific, 1997, pp. 61–78. [10] I. G. Avramidi and R. Schimming, “Algorithms for the calculation of the heat kernel coefficients”, in Quantum Field Theory under the Influence of External Conditions, ed. M. Bordag, Teubner-Texte zur Physik, Band 30, Stuttgart: Teubner, 1996, pp. 150–162. [11] A. O. Barvinsky and G. A. Vilkovisky, “The generalized Schwinger–De Witt technique in gauge theories and quantum gravity” Phys. Rep. C 119 (1985), (1) 1–74. [12] N. Berline, E. Getzler and M. Vergne, Heat Kernels and Dirac Operators, Springer, Berlin, 1992. [13] M. Booth, “HeatK: A Mathematica program for computing heat kernel coefficients”, JHU-TIPAC-98005, hep-th/9803113. [14] T. Branson, P. B. Gilkey and B. Ørsted, “Leading terms in the heat invariants”, Proc. Amer. Math. Soc. 109 (1990) 437. [15] S. A. Fulling, Ed., Heat Kernel Techniques and Quantum Gravity, Discourses in Mathematics and Its Applications, No. 4, Dept. Mathematics, Texas A&M Univ., College Station, Texas, 1995. [16] S. A. Fulling, Private communication 1997. [17] P. B. Gilkey, “The spectral geometry of Riemannian manifold”, J. Diff. Geom. 10 (1975) 601–618. [18] P. B. Gilkey, Invariance Theory, the Heat Equation and the Atiyah–Singer Index Theorem, FL: Chemical Rubber Company, Boca Raton, 1995. [19] N. E. Hurt, Geometric Quantization in Action: Applications of Harmonic Analysis in Quantum Statistical Mechanics and Quantum Field Theory, Reidel, Dordrecht, Holland, 1983 [20] A. E. M. van de Ven, “Index free heat kernel coefficients”, Class. Quant. Grav. 15 (1998) 2311–2344. [21] S. Yajima, “Evaluation of the heat kernel in Riemann–Cartan space using the covariant Taylor expansion method”, Class. Quant. Grav. 14 (1997) 2853–2868.
THE τ FUNCTIONS OF THE AKS HIERARCHY AND TWISTOR CORRESPONDENCE PARTHA GUHA S.N. Bose National Centre for Basic Sciences JD Block, Sector-3 Salt Lake, Calcutta- 700091, India and Institut des Hautes Etudes Scientifiques 35, Route de Chartres 91440- Bures-sur-Yvette, France Received 9 June 1998 Revised 25 August 1998 Adler–Kostant–Symes scheme provides a geometrical method for constructing different integrable systems. We construct an AKS hierarchy and obtain the τ function solutions of this hierarchy. We also show that this AKS hierarchy is a reduction of self dual Yang–Mills (SDYM) equation hierarchy and discuss its twistor construction. Hence we re-establish once again that SDYM hierarchy is a universal integrable hierarchy, so that by appropriate reduction and suitable choice of gauge group this hierarchy produces all the well-known hierarchies of the soliton equations 1.
Contents 1. Introduction 2. Adler–Kostant–Symes Scheme 2.1. AKS Theorem 2.2. Examples of AKS scheme 3. AKS Hierarchy and Commuting Flows 3.1. Example: AKNS hierarchy 4. The τ -function of the AKS Hierarchy 5. Reduced SDYM Hierarchy and Twistor Correspondence 5.1. AKS Hierarchy as a reduction of SDYM hierarchy 5.2. Twistor correspondences 6. What Next? Acknowledgement References
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1. Introduction The theory of τ function in the integrable system was discovered by the Kyoto School. It first appeared in the KP hierarchy and provides solution to the equations of the KP hierarchy. The τ -function of the KP hierarchy is a function τ (t1 , t2 , . . .) ˆ 1 , t2 , . . . , λ) of the KP of some “time” variables such that the Baker function Ψ(t hierarchy is expressed as 2 −1 ˆ = τ (t1 − 1/λ, t2 − 1/2λ , . . .) = τ (t − [λ ]) . Ψ τ (t1 , t2 , . . .) τ (t)
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(1)
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The KP hierarchy is generated by a pseudo-differential operator M = ∂ + u1 ∂ −1 + u2 ∂ −2 + · · · = S∂S −1 , where S = 1 + operator. Let
P∞ 1
(2)
si ∂ −i is a monic pseudo-differential operator, known as dressing ∞ X
Ψ(t, λ) = S exp
! i
ti λ
i=1
=
∞ X
−i
si λ
∞ X
exp
0
! ti λ
1
ˆ λ) exp = Ψ(t,
∞ X
i
!
ti λi
1
is the formal Baker function and it satisfies KP hierarchy: M Ψ = λΨ
(3)
∂M l = [M+ , M] . ∂tl
(4)
It is well known that a systematic procedure of obtaining most finite dimensional completely integrable systems is obtained from Adler, Kostant and Symes (AKS) theorem [3, 4, 17, 18] applying to some Lie algebra G equipped with an ad-invariant non-degenerate bilinear form. We assume G be a vector space, presented as the linear sum of two subalgebras G = K+L. The bilinear form induces an isomorphism G ' G ∗ . Hence with the help of bilinear form h , i we can identify K∗ ∼ L⊥ and L∗ ∼ K⊥ where (5) hK⊥ , Ki = hL⊥ , Li = 0 . So K⊥ acquires a Poisson structure from that of L∗ . The co-adjoint action of L on K⊥ ∼ L∗ is given by g ◦ p = πk⊥ (gpg −1 ) for g ∈ L and p ∈ K⊥ . Then the infinitesimal action is ⊥ [η, p] η(p) = πK
for η ∈ l. The symplectic manifold here is some co-adjoint L-orbit M ⊂ K⊥ ' L∗ . We associate to it a Hamiltonian equation of suitable ad-invariant function f : G −→ R for all f |M . In our case G is a loop algebra. As mentioned many important equations can be derived from this approach, e.g. Adler and van Moerbeke [4] obtained Euler–Arnold equation as a geodesic flow on ellipsoid, Ratiu [12] obtained C. Neumann equation and so on. In fact we [6, 9] have also obtained coupled KdV and nonlinear Schr¨ odinger equation by applying this AKS theorem. Hence AKS proves
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to be a very general systematic procedure of obtaining many completely integrable Hamiltonian system. Recently Ablowitz et al. (ACT) [1] have developed a hierarchy of self dual Yang– Mills (SDYM) equations via the introduction of “dressing operator”. This approach to the SDYM hierarchy is based on the general concept of intertwining operators which was introduced by Schur. The concept of intertwinning operator has been applied by Mikio Sato in his theory of KP system [16]. These intertwiners transform “bare unperturbed” auxiliary linear systems into “perturbed” linear systems which have been treated as a formal power series in λ−1 . In fact Ablowitz et al. have shown that these intertwiners are the non-local functionals of the Yang–Mills potentials. As an application to two dimensional systems, they have shown that upon appropriate reduction and suitable choice of gauge group this hierarchy produces all the well-known hierarchies of the soliton equation in 1 + 1 dimensions. Hence they call it universal integrable hierarchy. Ward and Mason et al. [19, 20, 11] have demonstrated how most well known or not so well-known integrable systems arise as reductions of self dual Yang–Mills equation. Thus all such integrable systems fall under the twistor framework. Certainly this give us a clue that we can reformulate the theory of integrable systems in terms of symmetric reductions of the SDYM equations. By reduction we mean (a) one can reduce the number of independent variables to fewer than four by factoring out by a subgroup of the Poinca´re group and (b) one can reduce the number of dependent variables by imposing algebraic constraints on the connection. We organize this paper in the following way: The following section is devoted to the Adler–Kostant–Symes scheme, we also present some well-known example and application of this theorem. We derive nonlinear Schr¨ odinger and KdV equation from this scheme. We describe the commuting flows of this AKS hierarchy, we also show how AKNS hierarchy is a special case of AKS hierarchy. We derive the τ function of the AKS hierarchy in Sec. 4. In Sec. 5 we show that AKS hierarchy is a reduction of SDYM hierarchy. We give a twistor correspondence of this hierarchy. In the final section we discuss various open problems of twistor theory. The author would like to confess that the article makes very little claim to originality. All the results are either standard or appeared in his previous publications [8, 9]. In this paper I give a much better exposition and also compute the τ -function of the AKS hierarchy following G. Wilson [21]. Of course the result of Wilson is not completely original, either it has been floating or deduced by others [5, 7]. Finally the result of the paper has a cultural interaction between two different schools, Oxford and Kyoto. The main thrust of this survey is to give a concrete realization of AKS hierarchy, using the dressing operator approach we introduce infinitely many commuting flows of this hierarchy. We show that this hierarchy is the reduction of self dual Yang– Mills hierarchy, so this provides us a twistor description of the hierarchy. Finally we discuss the τ functions of the hierarchy. So far integrable theorists always put emphasis on AKS scheme, and they hardly discuss anything about the hierarchy associated with the scheme. In this survey we shed some light on it.
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2. Adler Kostant Symes Scheme We lay out this section into two parts, in the first part we shall discuss Adler– Kostant–Symes theorem of a group with a two cocycle [6, 13–15]. In the second part we shall discuss the higher flows of AKS scheme and show that all the higher flows are commuting. 2.1. AKS Theorem A two form σ on M is called weak-symplectic form if σ is closed and the induced map from T M to T ∗ M defined by u 7−→ σ(◦, u) , if it is injective then (M, σ) is called symplectic manifold. Let M ⊂ K⊥ ' L∗ be the co-adjoint L orbit. Then the weak symplectic form is the Kostant–Kirillov two form on M is defined by (6) σT (X, Y ) = hT, [ξ, η]i for all ξ, η ∈ g and T ∈ M. Let ΩG = gl(n, C)⊗C[λ, λ−1 ] be the loop algebra of semi-infinite formal Laurent series in λ with coefficients in gl(n, C), for example, an element X(λ) ∈ ΩG can be expressed as a formal series of the form X(λ) =
m X
xi λi for all xi ∈ gl(n, C) ,
i=∞
and the Lie bracket with Y (λ) =
Pl j=−∞
[X(λ), Y (λ)] =
yj λj given by
m+l X
X
[xi , yj ]λk .
(7)
k=−∞ i+j=k
We define a Poisson bracket of two smooth functions g1 and g2 on ΩG ∗ by {g1 , g2 } = hα, [∇g1 , ∇g2 ]i ,
(8)
where α ∈ ΩG ∗ and ∇ is the usual gradient defined by hη, ∇gi (ζ)i =
d gi (ζ + ηt)|t=0 for all ζ, η ∈ g . dt
(9)
We define a non-degenerate ad-invariant bilinear form on ΩG hX(λ), Y (λ)i := Resλ=0 (λ−1 X(λ)Y (λ)) = tr (X(λ)Y (λ))0 .
(10)
There is a natural splitting in the loop algebra ΩG = ΩG + ⊕ ΩG − , where ΩG + denote the subalgebra of ΩG given by the polynomial in λ and ΩG − is the subalgebra of strictly negative series. These mean by ΩG + consisting of all loops with a Fourier P − i consisting of all loops expansion of the form i Xi λ with i ≥ 0 and by ΩG P i X λ with i < 0. i i
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The above decomposition of ΩG do not correspond to global decomposition of the loop group ΩG, but we have dense open subset ΩG− ΩG+ ⊂ ΩG
(11)
consisting of all loops φ that can be factorized in the form φ = φ− φ+
(12)
with φ− ∈ Ω− G, φ+ ∈ Ω+ G. We refer to this subset of ΩG as the big cell . Let us consider the Grassmannian like homogeneous space ΩG/ΩG+ . The image in ΩG/ΩG+ of the complement of the big cell in ΩG is a divisor in ΩG/ΩG+ , it therefore corresponds to a holomorphic line bundle L over ΩG/ΩG+ . We denote by ˜ the automorphism group of L. The pullback of L to ΩG ˆ is canonically trivial. ΩG ˜ Hence ΩG turns out to be the central extension of ΩG by C× . ˜ −→ ΩG −→ 1 . 1 −→ C× −→ ΩG
(13)
Hence we obtain the central extension corresponding to the Lie algebra ˜ −→ ΩG −→ 0 . 0 −→ C −→ ΩG Remark. At this moment we want to point to our reader that in this survey we discuss two types of loop algebras. One is ΩG = gl(n, C) ⊗ C[λ, λ−1 ] and other one is C ∞ (S 1 , gl(n, C)), where S 1 is parametrized by x, which plays the role of the physical space variable. For example, Eq. (14) below is defined on the second loop algebra, not on ΩG. Abusing the notation we will continue to denote the second loop algebra by ΩG. The one dimensional central extension of ΩG defined by two cocycle Z X 0 Y dx . (14) ω(X, Y ) = S1
˜ = ΩG ⊕ C. The Lie bracket of the loop The corresponding loop algebra ΩG ˜ algebra ΩG satisfies Z tr (XY 0 ) , (15) [(X(λ), a), (Y (λ), b)] = [X, Y ], S1
where X ∈ ΩG and a ∈ R. In general the map ˜ ≡ ΩG ⊕ C κ : ΩG −→ ΩG
(16)
is not a Lie algebra homomorphism, only its restriction to ΩG + is a Lie algebra homomorphism, since the central extension term vanishes identically. Then the corresponding induced map ˜ (17) κ : ΩG+ −→ ΩG yields a canonical holomorphic trivialization of the part of the fibration lying over ΩG+ .
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Let φ = φ− φ+ be an element of the big cell then that induces a factorization of κφ κ(φ) = κ(φ− )κ(φ+ ) .
(18)
˜ that lies over the big cell of ΩG. κφ dense open subset of ΩG ˜ by The bilinear form on ΩG can be extended to define the bilinear form on ΩG Z h(X, a), (Y, b)i = ab + tr (XY ) . (19) S1
˜ − , where as ˜ = ΩG ˜ + ⊕ ΩG There is a natural splitting in the loop algebra ΩG + ˜ denote the subalgebra of ΩG ˜ given by the polynomial in λ and ΩG ˜ − is usual ΩG the subalgebra of strictly negative series. Then via ad-invariant bilinear form h , i we identify ˜ − )⊥ and (ΩG ˜ − )∗ ∼ ΩG ˜ + )⊥ . ˜ + )∗ ∼ (ΩG (ΩG
(20)
Under the above identification, we can define the infinitesimal action of the ˜ ⊥ and this is given by ˜ − on ΩG coadjoint action of ΩG (π+ (ad∗ X)µ + cπ+ X 0 , 0) .
(21)
˜ ⊥ stratifies into Poisson submanifolds corresponding to different The dual space ΩG values of the parameter; each of them is endowed with a Poisson bracket. Let us ˜ ∗ . By abuse of notation we fix c = 1, so we confine ourselves to a hyperplane in ΩG ∗ ˜ shall continue to call it ΩG . ˜ ∗ for the two smooth Proposition 1. The Poisson bracket in the space of ΩG functions has the form Z d∇f2 , (22) ∇f1 {f1 , f2 }(Y ) = h[∇f1 , ∇f2 ], Y i + dx S1 where Y ∈ ΩG. Let I(g ∗ ) denote the ring of infinitesimally Ad∗ invariant function on G ∗ ⊕ C. So ∇F ∈ I(G ∗ ) will be ad-invariant function if and only if ∂∇F (µ, 0), [X, ∇F ] + =0 (23) ∂x for all X ∈ G and µ ∈ G ∗ , where ∇F is thought of as an element of G ∼ G ∗∗ . In the absence of any central extension term Ad∗ invariant function satisfies hµ, [∇F, X]i = 0 . Let fˆ1 and fˆ2 be the ad-invariant function and when they are restricted to L ∼ K⊥ these satisfy {fˆ1 , fˆ2 }l∗ = 0. ∗
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˜ = ΩG ˜ + ⊕ ΩG ˜ − and M ⊂ ΩG ˜ + a coadjoint orbit equipped Theorem 2. Let ΩG ˜ −→ R be the set with a natural weak orbit symplectic structure ω. Let Hi : ΩG ∗ + ⊥ ˜ ) is an involutive system on the ad-invariant functions in I(g ) restricted to (ΩG ˜ ∗ generated by the coadjoint orbit. The Hamiltonian equations of motion on ΩG Hamiltonian (ad-invariant function) have the form ∂P ∂γ = + [P, γ] , ∂t ∂x
(24)
where P = π+ [grad H]. So this define a flat connection γdx+P dt on a cylinder S 1 ×R associated with the above zero curvature equation. In order to apply Adler–Kostant–Symes scheme we have to know about ad-invariant function. But Adler and van Moerbeke or Reiman and Semenov–Tian–Shansky gave a nice formalism to construct these functions. We will skip this discussions here. Let us define the Hamiltonians by H(γ) =
1 tr λ−1 γ 2 , 2
where γ is the orbit. Let us assume our orbit to be γ = λ2 A + λQ2 + Q1 , where A is the constant diagonal matrix and Q2 and Q1 are the off diagonal and diagonal matrices. Although we take a very special orbit, this can be generalized to higher orbits. Then the Hamiltonian equation would be (λ2 A + λQ1 + λQ2 )t = [λ2 A + λQ1 + Q2 , λA + Q1 ] + (λA + Q1 )x .
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Remark. When we apply Fordy–Kulish method to AKS scheme we obtain different Hamiltonian systems associated to different Hermitian symmetric spaces. In this case Q1 and Q2 will take special values which depend on the decomposition of the Lie algebra. 2.2. Examples of AKS scheme We apply Fordy–Kulish scheme for the derivation of orbit. Let us evaluate an orbit through a point λ2 A, γ = (B −1 [λ2 A]B) , where
−1
−2
−3
B = (b1 λ−1 , eβ1 λ )(b2 λ−2 , eβ2 λ )(b3 λ−3 , eβ3 λ ) . Then the orbit γ is given by γ = λ2 A + λQ + (P − i(n + 1)[Q− , Q+ ]) , where Q = [A, β1 ] and P = [A, β2 ] + 12 [Q, β1 ].
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Let us fix the algebra to be sl(2, C), hence the subspaces of Q corresponding to eigenvalues 2i and −2i of ad A are Q+ and Q− respectively. Let us fix ! 1 i 0 A= , 2 0 −i ! 1 0 q . Q= 2 r 0 Then the AKS equation reduces to AKNS system iqx = qtt − 2q 2 r −irx = rtt − 2r2 q , when we constructed the orbit through [λ3 A] by B=
4 Y
(bi λ−i , eβi λ−i ) .
i=1
Then the orbit γ is γ = λ3 A + λ2 Q + λ(P − 2i[Q− , Q+ ]) + T + [S, Q] , where
1 1 1 T = [A, β3 ] + [[A, β1 ], β2 ] + [[A, β2 ], β1 ] + [[Q, β1 ], β1 ] 2 2 6
and S = 2i[P+ − P− ] + cQ . We can evaluate the following higher order AKNS systems from the zero curvature AKS equation −qx = qttt − 6qrqt , −rx = rttt − 6rqrt . The τ function solution of this AKNS system has been done by Wilson and Dickey [21, 7]. 3. AKS Hierarchy and Commuting Flows This part has already appeared in [8]. P<<∞ Let ΩG((λ−1 )) be an element of the loop algebra and ΩG|+ = 0≤k ⊕ΩGλk P be its polynomial part and ΩG|− = k≤−1 ⊕ΩGλk be it pure Laurent part. We assume ΩG|− be the group corresponding to subalgebra ΩG|− . Let γ be the orbit and it is defined by γ := Q1 + λQ2 + λ2 A .
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Hence the gradient of the Hamiltonian would be ∇H = Q2 + λA .
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We will denote ∇H by γ2 . Lemma 3. There exists S(x, λ) ∈ ΩG|− such that it satisfies ∂x − Q1 − λQ2 − λ2 A = S(∂x − λ2 A)S −1 .
(27)
Proof. We shall follow the proof of ACT. This S is a Sato type operator introduced by ACT in this context. It is easy to see the above expression reduces to ∂SS −1 = Γ − λ2 SAS −1 . Assume, ∂SS −1 =
∞ X Vn λn n=0
and SAS −1 = A +
(28)
∞ X Sn . λn n=1
Here Vn , Sn ∈ ΩG, substituting this expression to the equation we obtain S2 = Q1 − V0 ; S1 = Q2 and Sn = −Vn−2 n > 2 . Consider the differential equation ∂(SAS −1 ) = [∂SS −1 , SAS −1 ] = [γ, SAS −1 ] , this produces a recurssion relation among Sn ’s (ad A)Sn+2 + (ad Q2 )Sn+1 = (∂ − ad Q1 )Sn . Since the coefficient Sn and Vn can be determined recursively so the existence ˆ −. of S(x, λ) ∈ G| Definition 4. (1) We define φ := SAS
−1
=
∞ X φn 0
λn
,
where φ0 = A and φ1 = Q2 . (2) ∂k := ∂tk and γk = (λk−1 φ)+ for all n ≥ 0, where γ2 := Q2 + λA.
(29)
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Lemma 5. Coefficients φn ∈ ΩG, n ≥ 1, are uniquely determined by the initial conditions φ0 = A and φ1 = Q2 . Proof. We know φ = SAS −1 and ∂x − γ = S(∂x − λ2 A)S −1 , hence we obtain [∂ − γ, φ] = S[∂x − λ2 A, A]S −1 = 0 . Thus we get ∂x φ = [γ, φ] . P∞ Now we consider the expansion φ = n=0 λφnn and γ = λ2 A + λQ2 + Q1 . So using this expression we obtain the recursion relations (∂x − ad Q1 )φn + (ad A)φn+2 − (ad Q2 )φn+1 = 0 . Hence once we specify the initial conditions φ0 = A and φ1 = Q2 we can uniquely obtain all other φn recursively. Definition 6. An AKS hierarchy is defined by the infinite sequence of flows with respect to higher times tk and the kth member of the family is given by [∂x − γ, ∂k − γk ] = 0
k ≥ 2,
(30)
i.e. ∂k γ − ∂x γk + [γ, γk ] = 0 .
(31)
This zero curvature equation arises as the compatibility condition for the following linear systems: ∂x ξ = γξ, and ∂k ξ = γk ξ . Definition 7. The kth flow in the AKS hierarchy can be considered as a “dressing” of a “bare” solution by the intertwinner ∂x − γ = S(∂ − λ2 A)S −1 , ∂k − γk = S(∂k − λk−1 A)S −1 . Then by simple rearrangement and manipulation we obtain γk = ∂k SS −1 + λk−1 SAS −1 = ∂k SS −1 + λk−1 φ .
(32)
Since by definition we know γk = (λk−1 φ)+ and ∂k SS −1 ∈ ΩG|− hence we obtain (33) ∂k SS −1 = −(λk−1 φ)− . Lemma 8. The evolution equation for φ is given by ∂k φ = [γk , φ] .
(34)
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Proof. We use the following equalities: ∂k − γk = S(∂k − λk−1 A)S −1 and φ = SAS −1 . Hence, it is not hard to see [∂k − γk , φ] = S[∂k − λk−1 A, A]S −1 = 0 .
Theorem 9. The higher flows of AKS are commuting. Proof. Let us rewrite Eq. (33) by ∂k S = −(λk−1 φ)− S for k ≥ 2 . The compatibility condition gives us ∂k (λl−1 φ)− − ∂l (λk−1 φ)− + [(λk−1 φ)− , (λl−1 φ)] = 0 ,
(35)
and this is valid only for k, l ≥ 2. Now we shall use Lemma 8, we multiply λl−1 to that equation for kth time and λk−1 to that of lth time and then substracting these two, we obtain the following equation: ∂l (λk−1 φ) − ∂k (λl−1 φ) = [γl , (λk−1 φ)] − [γk , (λl−1 φ)] .
(36)
When we project this equation to “negative” loops or ΩG|− we get ∂l (λk−1 φ)− − ∂k (λk−1 φ)− = [γl , (λk−1 φ)]− − [γk , (λl−1 φ)]− .
(37)
Hence using (35) and (37) we obtain −[(λl−1 φ)+ , (λk−1 φ)]− + [(λk−1 φ)+ , (λl−1 φ)]− + [(λk−1 φ)− , (λl−1 φ)− ]− = [(λk−1 φ), (λl−1 φ)]− = 0 , where we have used γk := (λk−1 φ)+ .
3.1. Example: AKNS hierarchy Definition 10. This is a commuting flows of ξx = [λA + Q(x, tn )]ξ , # " n X i Bn−i (x, tn )λ ξ . ξtn = i=0
The coefficient of λn+1 , λn are required to vanish [A, B0 ] = 0 , [A, B1 ] + [Q, B0 ] − ∂x B0 = 0 .
(38) (39)
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For a special choice B0 = A, B1 = Q leads to
ξtn
ξx = [λA + Q]ξ , # "n−2 X = λn−1 ξx + Bn−i (x, tn )λi ξ . i=0
Hence AKNS hierarchy follows from AKS hierarchy. Remark. This AKNS hierarchy is a special case of a more general hierarchy
ξtn
ξx = [λA + Q(x, y, tn )]ξ , # "n−2 X n−1 i =λ ξy + Bn−i (x, y, tn )λ ξ
n≥2
i=0
when the symmetry ∂x − ∂y = 0 is reimposed. If the symmetry ∂tn = 0 is reimposed we obtain “negative flows” ξx = [λA + Q]ξ , "n−2 # X 1 Bn−i λi ξ . ξy = − n−1 λ i=0 This “negative flows” are also explainable in terms of our AKS formalism. 4. The τ -Function of the AKS Hierarchy Our construction is inherently close to George Wilson’s review [21], so we accept it as a guiding profile for this section. Let us recall that AKS hierarchy is given by [∂x − γ, ∂k − γk ] = 0 , P∞ and γk = (λk−1 φ)+ = ( n=0 λk−n−1 φn ) for all n ≥ 0, k ≥ 2 and
where ∂k := ∂tk γ2 := Q2 + λA. We have seen from Lemma 3 of Sec. 3 that the above pair of flat connections can be obtained by dressing the bare connections such that the dressing operator S satisfies (28) and it is given by ! ∞ X −n . Vn (x, t)λ S = exp 1
˜ that lies over the big cell of ΩG. As Let us consider a dense open subset of ΩG ˜ can be written as we have seen from Sec. 2 that any element φˆ in the big cell ΩG φˆ = µκ(φ) , ˜ where µ ∈ C× belongs to the centre of ΩG.
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Wilson defined a unique holomorphic function υ that assigns to an element of ˜ to its central component. the big cell ΩG The following proposition records some properties of υ: ˜ then υ satisfies Proposition 11. (1) If µ ∈ C× belongs to the centre of ΩG ˆ = µυ(φ) ˆ f or any φˆ ∈ ΩG ˜ . υ(µφ) (2) Let S = S − S + be an element of the big cell of ΩG such that S − ∈ ΩG− and ˜ lying over S and µ ∈ C× belongs to the S ∈ ΩG+ , let Sˆ be any element of ΩG ˜ centre of ΩG, then υ satisfies +
ˆ ˆ f or any χ ˜ . υ(χκ(S)) ˆ = υ(χ ˆS)/υ( S) ˆ ∈ ΩG (3) The function υ is invariant with respect to left multiplication by elements of ˜ and with respect to right multiplication by elements of the subgroup κ(ΩG− ) of ΩG, + κ(ΩG ).
Proof. See Wilson.
Let G is a simply laced algebra. Let φ− ∈ ΩG− be an arbitrary element, given by −
φ (λ) = exp
∞ X
! −i
φi λ
.
(40)
1
The φ1 can be decomposed (Cartan decomposition) by X φ1 = X + φα 1 eα ,
(41)
α
where X ∈ K (Cartan Subalgebra) and eα ∈ L is a basis element for root space of α. We have a map Aα : ΩG− −→ C, defined by Aα (φ− ) = φα 1 . This homomorphism follows from ΩG to G. × Let e∨ α ∈ K be the coroot corresponding to α. Let us define a map χ : C −→ K, such that (42) χα (λ) = exp((log λ)e∨ α) . Restricting to the unit circle S 1 ⊂ C× χα yields a loop in K, hence essentially it is an element of ΩG. Suppose χ ˆα is an element over χα . We define a map Bα : ΩG− −→ C such that ˆα κ(φ− )) , Bα (φ− ) = ω(χ ˜ lies over the big cell of ΩG. where κ(φ− ) is the element in the big cell of ΩG − − Since each φ ∈ ΩG has a unique decomposition − φ− = φ− 0 φα ,
(43)
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that induces the following factorization − Bα (φ− ) = ν(χ ˆα κ(φ− )) = ν(χ ˆα κ(φ− ˆα κ(φ− ˆ−1 ˆα κ(φ− α χ α )) . 0 )κ(φα )) = ν(χ 0 )χ
The final step follows from the third property of Proposition 11. The next result is the fundamental lemma for our construction. Lemma 12. For a suitable choice of χ ˆα we have an identity Aα (φ− ) = Bα (φ− ) = φα 1
(44)
for all φ− ∈ ΩG− . The proof has been given in [21, Sec. 5]. ˜ lying over it. We fix an element g in the big cell of ΩG and an element gˆ of ΩG Definition 13. τ (x, t) = ν(κ[exp(xAλ2 + tAλ)]ˆ g) ,
(45)
where exp(xAλ2 + tAλ) belongs to the subgroup ΩG+ of ΩG, thus to the big cell, hence κ is perfectly well defined. Definition 14. ˆα κ[exp(xAλ2 + tAλ)]ˆ g) . τα (x, t) = ν(χ
(46)
Theorem 15. For each root α, we set Qα (x, t) = α(A)τα (x, t)/τ (x, t) , then the g-valued function Q(x, t) =
X
Qα (x, t)eα
(47)
(48)
α
coincides with the solution the AKS equation [∂x − γ, ∂t − γ2 ] = 0 . 5. Reduced SDYM Hierarchy and Twistor Correspondence 5.1. AKS Hierarchy as a reduction of SDYM Hierarchy In this section we will establish a link between self dual Yang–Mills (SDYM) equation (hierarchy) and AKS scheme (hierarchy). We shall begin this part with some preliminary definitions [2]. Let V be a trivial vector bundle over R4 with fibres isomorphic as linear vector spaces to the Lie algebra g. We define the connection and curvature are g-valued P3 1 and 2 forms respectively and these are given by A := µ=0 Aµ (x)dxµ and F := P3 µ ν µ 4 µν=0 Fµν (x)dx ∧ dx , where x = (x ) are the usual co-ordinates of R and Fµν = ∂µ Aν − ∂ν Aµ − [Aµ , Aν ] .
(49)
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Definition 16. (1) The SDYM equations are system of first order partial differential equations given by F01 = F23 ,
F02 = F31 ,
F03 = F12 ,
(50)
connections are defined upto gauge transformation. (2) In terms of complex coordinates on R4 SDYM equations can be rewritten Fyz = Fy¯z¯ = 0,
Fyy¯ + Fzz¯ = 0 ,
(51)
where we introduce y = x1 + ix2 , z = x0 − ix3 , y¯ and z¯ are the complex conjugates. (3) The SDYM equations can be realized as the compatibility condition for the following linear system: D1 ψ = A1 ψ,
D2 ψ = A2 ψ ,
(52)
D1 = ∂y + λ∂z¯,
D2 = ∂z − λ∂y¯ ,
(53)
A1 = Ay + λAz¯,
A2 = Az − λAy¯ ,
(54)
where
and λ ∈ CP 1 is the spectral parameter. These linear pair of SDYM was introduced by Belavin et al. [11]. Under a suitable gauge transformation ACT have shown SDYM equation can be reduced to Fyz = 0,
Fyy¯ + Fzz¯ = 0 .
(55)
In this representation Ay¯ and Az¯ are the diagonal matrices. SDYM equations can be dimensionally reduced and when we reduce to 1 + 1 dimensions, these equations depend only on the coordinate y and z. Here we shall denote y and z by x and t. We choose Az¯ = A1 , Ay¯ = −A2 and Ay = U1 , Az = U2 such that Ai s and U i s are diagonal and off diagonal matrices respectively. Definition 17. The 1 + 1 dimensional reduction of SDYM can be obtained from the compatibility condition of ∂x ψ = (U 1 + λA1 )ψ,
∂t ψ = (U 2 + λA2 )ψ ,
(56)
i.e. these look like ∂t U 1 − ∂X U 2 + [U 1 , U 2 ] = 0,
[A1 , U 2 ] = [A2 , U 1 ] .
(57)
Theorem 18 (ACT). (1) The reduced SDYM hierarchy satisfies the following evolution equation at kth time ∂k U 1 = ∂x χk−1 − [U 1 , χk−1 ],
t2 = t,
k ≥ 2,
(58)
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where χ is defined by χ = A2 +
∞ X χk k=1
λk
.
(59)
(2) These hierarchy can be realized as the compatibility condition of following linear system: (60) ∂k ψ = Wk ψ, k ≥ 1 , where t1 = x, W1 = U 1 + λA1 and Wk = (λk−1 χ)+ . (3) Higher flows commute: ∂k Wl − ∂l Wk + [Wl , Wk ] = 0 .
(61)
Next we shall show that our AKS equation is a special case of 1 + 1 dimensional reduced SDYM equation. Lemma 19. AKS system is 1 + 1 reduction of SDYM equation. Proof. It is easy to see that when we replace A1 = λA, U 1 = Q1 + λQ2 , A2 = A and U 2 = Q2 we obtain AKS equation. Moreover, if we put these values into the constraint equation (62) [A1 , U 2 ] = [A2 , U 1 ] we obtain [A, Q1 ]. Since A is a constant diagonal matrix and Q1 is also a diagonal matrix they commute, i.e. [A, Q1 ] = 0. By a similar calculation it can be proved easily the following: Theorem 20. AKS hierarchy is the reduction of SDYM hierarchy. 5.2. Twistor correspondences Since AKS equation are reductions of the SDYM equations on R4 the standard twistor correspondence for the full self dual Yang–Mills equation can be reduced to give a correspondence for solutions of AKS scheme. Let us recall that Bogomoln´ yi equation is obtained when a single non-null translational symmetry is imposed on the self dual Yang–Mills equation. The solutions of the Bogomoln´ yi equation correspond to bundles invariant under the corresponding symmetry on CP 3 (twistor space). Since there is no fixed point on CP 3 , bundles here are the pull back bundles on the quotient of CP 3 by that symmetry. This quotient is called mini twistor space and is denoted by O(2); it is a holomorphic line bundle of Chern class 2 over CP 1 . This idea can be generalized to any member of the Bogomoln´ yi hierarchy. Let O(n) denote the twistor space, the complex line bundle of Chern class n ≥ 2 over Riemann sphere CP 1 . By stereographic projection CP 1 = C ∪ {∞}
(63)
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with coordinates λ on C with λ0 = λ1 a coordinate on C 0 = {C ∪ ∞} − {0}. The affine coordinate of the line bundles O(n) can then be given (µ, λ) over C and (µ0 , λ0 ) = (µ/λn , 1/λn ) on C 0 with λ an affine coordinates on CP 1 . Let Γ(n) be the space of global holomorphic section of O(n). It can be shown Γ(n) ∼ = C n+1 , the points of C n+1 can be holomorphically embedded CP 1 via the incidence relation n X ti λi (64) µ= i=0 n+1
. Like all other dynamical systems, one is of course where ti are coordinates on C interested in real solutions to the Bogomoln´ yi equation or AKS systems. We put in the real structure as given by the fixed points of an involution corresponding to complex conjugation. Hence our bundle O(n) has a real structure. Let OR be the real points of O. Ward correspondence provides a rank n vector bundle E on O(n) from a solution of the SL(n, C) Bogomoln´ yi equation on Cn+1 . Theorem 21 (MS). There exists a 1 : 1 correspondence between the solution of the nth Bogomoln´ yi hierarchy on a domain U ⊂ C n+1 and the holomorphic vector bundle with structure group on the open region in O(n) swept out by the sections of O(n) corresponding to the points of U . This bundle is trivial when it is restricted to the holomorphic sections of O(n) corresponding to points of U . In the case AKS scheme we impose one additional symmetry. It is not possible to factor out this extra symmetry. Hence the solutions of the AKS equation has one to one correspondence with the holomorphic vector bundles satisfying certain additional symmetry corresponding to this extra symmetry and reality condition on O(2). It is known that the AKS hierarchy is the reduction of Bogomolny hierarchy. Hence the solutions of the AKS hierarchy correspond to holomorphic vector bundles on O(n) satisfying the appropriate symmetry and reality condition. 6. What Next? AKS scheme works elegantly in 1 + 1 dimensional integrable systems. But there is no unified approach to 2 + 1 dimensional systems. Formally one can apply pseudo differential operators in the AKS scheme to construct 2 + 1 dimensional integrable systems. However, that is a brute force method and one sacrifices the beauty of the geometry when one applies pseudo differential operators. So far KP system has not been expressed as a reduction of the self duality equation. Unless the KP equation has a completely different Lax pair, it will be impossible to express it as a reduction of self duality equation. Lax pair (or linear systems) plays a key role to link between the twistor theory and integrable systems. π Let E −→ M 4 be the Yang–Mills vector bundle over Minkowski space M 4 . Then linear system can be reformulated as a distribution on a space Ex × CP 1 , where Ex is a fibre of π at x ∈ M 4 and CP 1 is the spectral parameter space. If we
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introduce homogeneous coordinates πA (A is the Spinor index) then the Lax pair π A DAA0 span a subbundle of the horizontal distribution on E pulled back to the spin bundle. There are some open problems in the twistor programme, for example, can we introduce twistor programme to those systems which in fact do have linear pairs but not Lax pairs? The linear system for a set of nonlinear equations is an overdetermined set of linear equations whose coefficients contain the dependent variables of the nonlinear equation and whose consistency conditions are the original partial differential equations. As for example Einstein vacuum equation has a linear pair, Rarita–Schwinger equation for the potential of a helicity-3/2 field. Most of the equations of this class are non integrable. Of course we know that Lax pair guarantees integrability, i.e. Hamiltonian flow is realized as a linear flow as the real part of the Jacobian of the spectral curve. Can we establish a twistor correspondence of the nonintegrable systems using these linear systems? Acknowledgements The Author is grateful to the papers of Ablowitz et al. and Wilson. He thanks Professors K. Takasaki, M. Mulase, M. Jimbo, L. J. Mason, N. Woodhouse and N. T. Zung for discussions at various stages of this work. Finally he is grateful to Professor Maxim Kontsevich for his kind invitation where the final part of the work was done in an extremely cordial atmosphere. References [1] M. J. Ablowitz, S. Chakravarty and L. J. Takhtajan, “A self dual Yang–Mills hierarchy and its reductions to integrable systems in 1 + 1 and 2 + 1 dimensions,” Commun. Math. Phys. 158 (1993) 289–314. [2] M. F. Atiyah, Classical Geometry of Yang–Mills fields, Fermi Lectures, Scuola Normale Pisa, 1980. [3] M. Adler, “On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg de Vries type equations,” Invent. Math. 50 (1979) 219–248. [4] M. Adler and P. van Moerbeke, “Completely integrable systems, Euclidean Lie algebras and curves,” Adv. Math. 38 (1980) 267. [5] M. J. Bergveld and A. P. E. ten Kroode, “τ -functions and zero curvature equations of Toda-AKNS type,” J. Math. Phys. 29 (1988) 1308–1320. [6] A. R. Chowdhury and P. Guha, “Current algebra, AKS theorem and new super evolution equation,” J. Phys. A: Math. Gen. 23 (1990) L639. [7] L. A. Dickey, “On Segal–Wilson’s definition of the τ -function and hierarchies AKNS-D and mcKP,” in Integrable systems: the Verdier memorial conference, eds. O. Babelon, et al. Boston, MA: Birkh¨ auser, Prog. Math 115 (1993) 147–161. [8] P. Guha, “On Commuting flows of AKS hierarchy and twistor correspondence,” J. Geom. Phys. 20 (1996) 207–217. [9] P. Guha, “AKS construction, Bihamiltonian manifolds and KdV equations,” J. Math. Phys. 38 (1997) 5167–5182. [10] I. Marshall, “Some integrable systems related to affine Lie algebras and homogeneous spaces,” Phys. Lett. 127A (1988) 19. [11] L. J. Mason and G. Sparling, “Nonlinear Schr¨ odinger and Korteweg de Vries are reductions of self dual Yang–Mills,” Phys. Lett. 137A (1989) 29–33.
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[12] T. Ratiu, “The C. Neumann problem as a complete integrable system on an adjoint orbit,” Trans. Amer. Math. Soc. 264 (1981) 321–329. [13] A. G. Reiman and M. A. Semenov-Tian-Sanskii, “Reduction of Hamiltonian systems, affine Lie algebras and Lax equations I,” Invent. Math. 54 (1979) 81–100. [14] A. G. Reiman and M. A. Semenov-Tian-Sanskii, “Current algebras and nonlinear partial differential equations,” Soviet Math. Dokl. 21 (1980) 630–634. [15] M. A. Semenov-Tian-Sanskii, “What is classical r matrix?” Funct. Anal. Appl. 17 (1983) 259–272. [16] M. Sato and Y. Sato, “Soliton equations as dynamical systems on infinite dimensional Grassmann manifold,” Lect. Notes in Num. Appl. Anal. 5 (1982) 259–271. [17] W. Symes, “Systems of Toda type, inverse spectral problems and representation theory,” Invent. Math. 59 (1980) 13–51. [18] W. Symes, “Hamiltonian group actions and integrable systems,” Physica D1 (1980) 339–374. [19] R. S. Ward, “On self dual gauge fields,” Phys. Lett. 61A (1977) 81. [20] R. S. Ward, “Integrable systems in twistor theory,” in Twistors in Mathematics and Physics, eds. by T. N. Bailey and R. J. Baston, Cambridge Univ. Press, 1990. [21] G. Wilson, “The τ - functions of the gAKNS equations,” in Integrable systems: the Verdier memorial conference, eds. by O. Babelon et al. Boston, MA: Birkh¨ auser, Prog. Math. 115 (1993) 131–145.
ON CONVERGENCE OF THE FEYNMAN PATH INTEGRAL FORMULATED THROUGH BROKEN LINE PATHS Dedicated to the Memory of Professor Kenzo Shinkai WATARU ICHINOSE∗ Department of Mathematical Science Shinshu University, Matsumoto 390 Japan Received 2 February 1998 We study the convergence of the Feynman path integral formulated through broken line paths in nonrelativistic quantum mechanics. The rigorous proof of its convergence had been given little except for special cases for a long time. In the preceding paper the author showed for a class of potentials that this path integral converges and gives the probability amplitude, i.e. the solution of the Schr¨ odinger equation. In the present paper we generalize this result, showing the boundedness theorem on the weighted Sobolev spaces for some integral operators and joining this boundedness theorem to the method in the preceding paper. We note that the result obtained is gauge invariant.
1. Introduction In the present paper we consider some charged particles in an electromagnetic field. For the sake of simplicity suppose charge = one and mass = m > 0. Let x ∈ Rn and t ∈ [0, T ]. We denote by E(t, x) = (E1 , . . . , En ) ∈ Rn and (Bjk (t, x))1≤j
n X
Aj dxj =
j=1
∂V ∂Aj − (j = 1, . . . , n) , ∂t ∂xj X
Bjk dxj ∧ dxk
on Rn .
(1.1)
1≤j
It is well known that the electromagnetic potentials are not unique, but gauge dependent. That is, the gauge transformation V0 =V −
∂ψ , ∂t
A0j = Aj +
∂ψ ∂xj
(j = 1, 2, . . . , n)
(1.2)
leaves E and (Bjk (t, x))1≤j
partially supported by Grant-in-Aid for Scientific No. 10640176, Ministry of Education, Science, and Culture, Japanese Government. 1001
Reviews in Mathematical Physics, Vol. 11, No. 8 (1999) 1001–1025 c World Scientific Publishing Company
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paper we always assume that V, ∂V /∂xj , ∂Aj /∂t, and ∂Aj /∂xk (j, k = 1, 2, . . . , n) are continuous in [0, T ] × Rn . In classical mechanics the Lagrangian function is given by L(t, x, x) ˙ =
m 2 |x| ˙ + x˙ · A − V . 2
(1.3)
Let (Rn )[s,t] be the space of all paths γ : [s, t] → Rn and S(γ) the classical action Z t dγ (θ) (1.4) L(θ, γ(θ), γ(θ))dθ, ˙ γ(θ) ˙ = S(γ) = dt s for γ ∈ (Rn )[s,t] . Then all classical paths γ, ∂L d ∂L (t, γ(t), γ(t)) ˙ = (t, γ(t), γ(t)) ˙ , dt ∂ x˙ ∂x are defined by the condition that S(γ) is an extremum in the variational problem. This is called the principle of least action (cf. [3]). So S(γ) is fundamental in classical mechanics. The Hamiltonian function H(t, x, p) =
1 |p − A|2 + V 2m
(1.5)
is defined by the Legendre transformation H = x·p−L, ˙ p = ∂L/∂ x˙ of L(t, x, x) ˙ in x. ˙ In quantum mechanics the Hamiltonian operator H(t) is defined after replacing pj by ~i−1 ∂/∂xj in H(t, x, p). It should be noted that H(t) has the ordering ambiguity (cf. [16]). Then the temporal evolution U (t, s)f of the probability amplitude can be described as the solution of the Schr¨ odinger equation i~
∂ u(t) = H(t)u(t), ∂t
u(s) = f .
(1.6)
On the other hand Feynman in [4] and [5] proposed another method of the quantization. He claimed that the value of the probability amplitude U (t, s)f at x can be described as the sum, in a sense, of N −1 (exp i~−1 S(γ))f (γ(s)) over all paths γ ∈ (Rn )[s,t] such that γ(t) = x. Here N is a normalization factor independent of x and γ. This sum is called the Feynman path integral or simply the path integral. Since then, much work has been devoted by physicists and mathematicians to give the rigorous meaning of the Feynman path integral. Let A be a constant magnetic potential and V a sum of |x|2 and the Fourier transform of a complex measure of bounded variation on Rn . The path integral for such potentials was studied in [1, 2, 12, 13, 15, 19] and so on in various ways. The path integral formulated through piecewise classical paths was studied by Fujiwara and Yajima in [7, 8, 20] for a wider class of potentials than the above. They showed that these path integrals are well defined and satisfy the Schr¨ odinger Eq. (1.6) with H(t) determined by 1 X (~Dxj − Aj )2 + V, 2m j=1 n
H(t) =
Dx j =
1 ∂ . i ∂xj
(1.7)
ON CONVERGENCE OF THE FEYNMAN PATH INTEGRAL FORMULATED
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1003
In the present paper we study the path integral formulated through broken line paths, which is very familiar with us and well known to be useful (cf. [17] and its references). Let ∆ : 0 = t0 < t1 < . . . < tν = t be a subdivision of the interval [0, t] and put |∆| = max1≤j≤ν (tj − tj−1 ). Let x(j) ∈ Rn (j = 0, 1, . . . , ν − 1) and denote by γ∆ = γ∆ (x(0) , x(1) , . . . , x(ν−1) , x) ∈ (Rn )[0,t] the broken line path joining (tj , x(j) )(j = 0, 1, . . . , ν, x(ν) = x). Let S be the space of all rapidly decreasing functions on Rn with the usual topology. Let L2 = L2 (Rn ) be the space of all square integrable functions on Rn with inner product (·, ·) and norm k · k. We define the operator C(∆) on S by the oscillatory integral (cf. [14]) C(∆)f =
ν r Y j=1
≡
ν r Y j=1
n
m Os− 2πi~(tj − tj−1 )
Z
Z ··· Rn
ei~
−1
S(γ∆ )
f (x(0) )dx(0) dx(1) . . .dx(ν−1)
Rn
Z Z n −1 m lim ··· ei~ S(γ∆ ) χ(x(1) ) . . . χ(x(ν−1) ) →0 Rn 2πi~(tj − tj−1 ) n R
×f (x(0) )dx(0) dx(1) . . . dx(ν−1)
in L2 ,
(1.8)
√ where χ(x) ∈ S such that χ(0) = 1 and i = eiπ/4 . For the sake of simplicity we assume in the present paper that χ is real-valued. The path integral formulated through broken line paths is defined by lim|∆|→0 C(∆)f . For a multi-index α = (α1 , . . . , αn ) we write ∂xα = (∂/∂x1 )α1 · · · (∂/∂xn )αn and Pn |α| = j=1 αj . Suppose the following. There exist constants δ > 0, υ ≥ 0, and Cα such that |∂xα V (t, x)| ≤ Cα , |α| ≥ 2,
|∂xα ∂t V (t, x)| ≤ Cα hxiυ , |α| ≥ 1
|∂xα Aj (t, x)| ≤ Cα hxi−(1+δ) , |α| ≥ 2,
|∂xα ∂t Aj (t, x)| ≤ Cα , |α| ≥ 1
p in [0, T ] × Rn for j = 1, 2, . . . , n, where hxi = 1 + |x|2 . Then the author proved in [11] that the path integral formulated through broken line paths is well defined and satisfies the Schr¨ odinger Eq. (1.6) where s = 0 and H(t) is determined by (1.7). In the present paper we generalize this result. Our aim in the present paper is to prove the following. Theorem. Let ∂xα Ej (t, x) (j = 1, 2, . . . , n), ∂xα Bjk (t, x), and ∂t Bjk (t, x) (1 ≤ j < k ≤ n) be continuous in [0, T ] × Rn for all α and suppose |∂xα Ej (t, x)| ≤ Cα , |α| ≥ 1, |∂xα Bjk (t, x)| ≤ Cα hxi−(1+δ) , |α| ≥ 1
(1.9)
in [0, T ]× Rn for some δ > 0 and Cα . Let V and A be arbitrary potentials. Then we have: (1) If |∆| is small, C(∆) on S is well defined and can be extended to a bounded operator on L2 . (2) As |∆| → 0, C(∆)f for f ∈ L2 converges in L2 uniformly in t ∈ [0, T ]. So the path integral formulated through broken line paths is well defined.
1004
W. ICHINOSE
(3) The path integral defined by (2) satisfies the Schr¨ odinger Eq. (1.6) where s = 0 and H(t) is determined by (1.7). We note that the assumptions put in Theorem is a little more general than those in [7, 8, 20]. The of the proof of Theorem is as follows. Let fˆ be the Fourier transR outline −ix·ξ f (x)dx. We denote by B a (a ≥ 0) the weighted Sobolev space {f ∈ form e L2 ; kf kB a ≡ kh·ia f k + kh·ia fˆk < ∞} and by B −a its dual space with norm kf kB −a . t,s ∈ (Rn )[s,t] by Let us define γx,y t,s (θ) = y + γx,y
θ−s (x − y) (s ≤ θ ≤ t) t−s
and set for f ∈ S Z p m/(2πi~(t − s)) n (exp i~−1 S(γ t,s ))f (y)dy, x,y C(t, s)f = f,
(1.10)
s < t,
(1.11)
s = t.
Then we can easily write from (1.8) for f ∈ S C(∆)f = lim C(t, tν−1 )χ(·)C(tν−1 , tν−2 )χ(·) · · · C(t2 , t1 )χ(·)C(t1 , 0)f . →0
(1.12)
It is shown from the assumptions in Theorem that there exist the potentials V and A satisfying V (t, x) = 0 and |∂xα Aj (t, x)| + |∂t ∂xα Aj (t, x)| ≤ Cα ,
|α| ≥ 1, (t, x) ∈ [0, T ] × Rn
for j = 1, 2, . . . , n (Lemma 6.1 in the present paper). Consider C(t, s) for these V and A. Then we can prove under the assumptions in Theorem: (i) There exist constants ρ∗ > 0 and K ≥ 0 such that if 0 ≤ t − s ≤ ρ∗ , C(t, s) can be extended to a bounded operator on L2 and satisfies kC(t, s)f k ≤ eK(t−s) kf k,
f ∈ L2 .
(1.13)
(ii) Let |∆| ≤ ρ∗ . Then C(∆) on S is well defined and can be extended to a bounded operator on L2 . We also have C(∆)f = C(t, tν−1 )C(tν−1 , tν−2 ) · · · C(t2 , t1 )C(t1 , 0)f, f ∈ L2 .
(1.14)
(iii) There exist constants a ≥ 0 and C such that
C(t, s)f − f √
i~ − H(s)f ≤ C t − skf kB a , 0 < t − s ≤ ρ∗ , f ∈ B a .
t−s (1.15) The properties (i) and (ii) above can be proved by the analogous argument used in [11]. On the other hand to prove (iii) we need a different argument from that in [11]. In [11] the proof of (iii) was rather easy. It followed from the direct study of C(t, s)f for f ∈ S (Theorem 2.4 in [11]). It should be emphasized that our
ON CONVERGENCE OF THE FEYNMAN PATH INTEGRAL FORMULATED
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1005
operator C(t, s) is much different from that in [7, 8, 20, 11] and so even the continuity of C(t, s)f for f ∈ S in 0 ≤ s ≤ t ≤ T and x ∈ Rn is not so clear (Lemma 2.1 and Remark 4.1 in the present paper). In the present paper we prove (iii) by showing the boundedness theorem on B a for the operator C(t, s) and the like (Theorem 4.4). We will prove (i)–(iii) for a little more general potentials V and A than the above |∂xα V (t, x)| + |∂t ∂xα V (t, x)|hxi−υ ≤ Cα hxi, |∂xα Aj (t, x)| + |∂t ∂xα Aj (t, x)|hxi−υ ≤ Cα ,
|α| ≥ 1 , |α| ≥ 1, (t, x) ∈ [0, T ] × Rn (1.16)
for j = 1, 2, . . . , n, where υ ≥ 0 and Cα are constants. From (i)–(iii) the same assertions (1)–(3) as in Theorem can be proved. Using this result and the argument on the gauge transformation, we can complete the proof of Theorem. The plan of the present paper is as follows. Section 2 is devoted to preliminaries. We prove (i) and (ii) in Sec. 3. In Sec. 4 we show the boundedness theorem on B a for C(t, s) and the like. Property (iii) is proved in Sec. 5. In Sec. 6 we complete the proof of Theorem. 2. Preliminaries As in [11] we write x = (t, x) ∈ Rn+1 and set A = (−V, A) ∈ Rn+1
(2.1)
and t,s t,s t,s : γx,y (θ) = (θ, γx,y (θ)) ∈ Rn+1 γx,y
(s ≤ θ ≤ t) .
(2.2)
Then we have from (1.3), (1.4), and (1.10) t,s ) S(γx,y
m|x − y|2 + = 2(t − s)
Z t,s γx,y
A · dx
Z 1 m|x − y|2 + (x − y) · A(s + θ(t − s), y + θ(x − y))dθ = 2(t − s) 0 Z 1 V (s + θ(t − s), y + θ(x − y))dθ −(t − s) 0
Z 1 m|x − y|2 + (x − y) · A(t − θ(t − s), x − θ(x − y))dθ = 2(t − s) 0 Z 1 V (t − θ(t − s), x − θ(x − y))dθ . −(t − s)
(2.3)
0
Let M ≥ 0 and p(x, w) an infinitely differentiable function in R2d satisfying α β ∂x p(x, w)| ≤ Cα,β hx; wiM , |∂w
x, w ∈ Rn
for all α and β with constants Cα,β , where hx; wi =
(2.4)
p 1 + |x|2 + |w|2 . For f ∈ S
1006
W. ICHINOSE
we define p n m/(2πi~(t − s)) Z s < t, √ t,s × (exp i~−1 S(γx,y ))p(x, (x − y)/ t − s)f (y)dy , (2.5) P (t, s)f = p n m/(2πi~) Os Z s = t. × − (exp i~−1 m|w|2 /2)p(x, w)dwf (x) , We can easily see that the formal adjoint operator P (t, s)∗ of P (t, s) on S, (P (t, s)f, g) = (f, P (t, s)∗ g) for f and g ∈ S, is given by p n im/(2π~(t − s)) Z s < t, √ t,s × (exp −i~−1 S(γy,x ))p(y, (y − x)/ t − s)f (y)dy , P (t, s)∗ f = p n im/(2π~) Os Z s = t, − (exp −i~−1 m|w|2 /2)p(x, w)dwf (x) , (2.6) where p(x, w) is the complex conjugate of p(x, w). Remark 2.1.R Set p(x, w) = 1 p in (2.5). Then P (t, s) = C(t, s) follows from n i~−1 m|w|2 /2 dw = 2πi~/m . (1.11) and Os − e The following is fundamental. Lemma 2.1. Assume that there exists an M 0 ≥ 0 satisfying |∂xα V | +
n X
0
|∂xα Aj | ≤ Cα hxiM , (t, x) ∈ [0, T ] × Rn
(2.7)
j=1
for all α with constants Cα . Suppose p(x, w) satisfies (2.4). Let f ∈ S. Then ∂xα (P (t, s)f ) are continuous in 0 ≤ s ≤ t ≤ T and x ∈ Rn for all α. √ Proof. Make the change of variables: y → w = (x− y)/ t − s in (2.5) for s < t. Then from (2.3) we can write (2.5) as r Z n −1 √ m Os − ei~ φ(t,s;x,w) p(x, w)f (x − ρw)dw, s ≤ t , (2.8) P (t, s)f = 2πi~ Z 1 √ m 2 √ A(t − θρ, x − θ ρw)dθ φ(t, s; x, w) = |w| + ρw · 2 0 Z 1 √ V (t − θρ, x − θ ρw)dθ −ρ 0
√ m = |w|2 + ψ(t, s; x, ρw), 2
ρ = t− s,
(2.9)
ON CONVERGENCE OF THE FEYNMAN PATH INTEGRAL FORMULATED
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1007
where Z
Z
1
A(t − θρ, x − θξ)dθ − ρ
ψ(t, s; x, ξ) = ξ · 0
1
V (t − θρ, x − θξ)dθ . 0
We have from the assumption (2.7) |∂ξα ∂xβ ψ| ≤ Cα,β hx; ξiM
0
+1
, 0 ≤ s ≤ t ≤ T, x, ξ ∈ Rn
(2.10)
for all α and β. Pn Let L = hwi−2 (1 − i~m−1 j=1 wj ∂wj ) and t L its transposed operator. Then Z ei~
−1
φ(t,s;x,w)
χ(w)p(x, w)f (x −
√ ρw)dw =
Z ei~
−1
m|w|2 /2 t
( L)l
n −1 o √ √ × ei~ ψ(t,s;x, ρw) χ(w)p(x, w)f (x − ρw) dw, 0 < ≤ 1
(2.11)
for l = 0, 1, 2, . . .. We have from (2.4) and (2.10) √ ρw)}| 0 √ √ ≤ Cl,N hwi−l hx; ρwil(M +1) hx; wiM hx − ρwi−N
|(t L)l {ei~
−1
√ ψ(t,s;x, ρw)
χ(w)p(x, w)f (x −
for any N = 0, 1, 2, . . ., √ where Cl,N is independent of 0 < ≤ 1. So using hx; yi ≤ hxihyi and hx + yi−1 ≤ 2hxihyi−1 , √ χ(w)p(x, w)f (x − ρw)}| 0 0 √ 0 hxil(M +1)+M+N h ρwil(M +1)−N hwiM−l . ≤ Cl,N
|(t L)l {ei~
−1
ψ
Taking l and N so that l ≥ M + n + 1 and N ≥ l(M 0 + 1), we have |(t L)l {ei~
−1
ψ
χ(w)p(x, w)f (x −
≤ Chxil(M
0
+1)+M+N
√ ρw)}|
hwi−(n+1) , 0 ≤ s ≤ t ≤ T, x ∈ Rn
(2.12)
with some constant C independent of 0 < ≤ 1. Hence applying the Lebesgue dominated convergence theorem, we can see from (2.8), (2.11), and (2.12) that P (t, s)f is continuous in 0 ≤ s ≤ t ≤ T and x ∈ Rn . Using (2.10), in the same way we can prove for all α that ∂xα (P (t, s)f ) are also continuous. We write for 0 ≤ σ2 ≤ σ1 ≤ 1 τ (σ) = τ (σ1 , σ2 ) = t − σ1 (t − s) , ζ(σ) = ζ(σ1 , σ2 ; x, y, z) = z + σ1 (x − z) + σ2 (y − x)
(2.13)
and set Bjk = −Bkj , 1 ≤ k < j ≤ n,
Bjj = 0, j = 1, 2, . . . , n .
(2.14)
1008
W. ICHINOSE
Lemma 2.2. We assume that V and Aj (j = 1, 2, . . . , n) are continuously differentiable in [0, T ] × Rn . Suppose p(x, w) satisfies (2.4). Let f ∈ S. Then for any 0 < ≤ 1 and 0 ≤ s < t ≤ T we have n Z m f (y)dy P (t, s)∗ χ(·)2 P (t, s)f = 2π~(t − s) Z √ √ mΦ 2 p(z, (z − x)/ t − s)p(z, (z − y)/ t − s)dz , × χ(z) exp i(x − y) · ~(t − s) Φ = Φ(t, s; x, y, z) = (Φ1 , . . . , Φn ) , Z t−s 1 xj + yj + Aj (s, y + θ(x − y))dθ Φj = z j − 2 m 0 Z 1 Z σ1 n t−s X (zk − xk ) Bjk (τ (σ), ζ(σ))dσ2 dσ1 − m 0 0
(2.15)
k=1
−
(t − s)2 m
Z
1
Z
σ1
Ej (τ (σ), ζ(σ))dσ2 dσ1 . 0
(2.16)
0
Proof. The proof of this lemma is analogous to that of Proposition 3.3 in [11]. We can write from (2.5) and (2.6)
n Z m f (y)dy P (t, s) χ(·) P (t, s)f = 2π~(t − s) Z t,s t,s ) − S(γz,y ))} × χ(z)2 {exp −i~−1 (S(γz,x ∗
2
√ √ × p(z, (z − x)/ t − s)p(z, (z − y)/ t − s)dz . (2.17) s,s s,s : γx,y (θ) = (s, y + θ(x − y)) ∈ Rn+1 (0 ≤ θ ≤ 1). Then applying the Stokes Set γx,y theorem, we have by (2.3) t,s t,s ) − S(γz,y ) S(γz,x
m = − (x − y) · t−s
Z ZZ x+y A · dx − d(A · dx) z− − s,s 2 γx,y ∆
Z 1 x+y m A(s, y + θ(x − y))dθ z− − (x − y) · = − (x − y) · t−s 2 0 ZZ d(A · dx) , (2.18) − ∆
where ∆ = ∆(t, s, x, y, z) is the 2-dimensional plane with oriented boundary cons,s t,s t,s , γz,y , and −γz,x . sisting of −γx,y
ON CONVERGENCE OF THE FEYNMAN PATH INTEGRAL FORMULATED
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Let x 6= y. Then the mapping : {(σ1 , σ2 ); 0 ≤ σ2 ≤ σ1 ≤ 1} 3 (σ1 , σ2 ) → (τ (σ), ζ(σ)) ∈ ∆(t, s, x, y, z) is homeomorphic and σ give the positive orientation of ∆. The equation n X
d(A · dx) = −
X
Ej (t, x)dt ∧ dxj +
j=1
Bjk dxj ∧ dxk on [0, T ] × Rn (2.19)
1≤j
follows from (1.1). So we have from (2.13) ZZ d(A · dx) ∆
=−
n ZZ X ∆
j=1
Z
1
Z
=− +
σ1
0
Z
X
1
Z
σ1
Bjk det 0
1≤j
= − (t − s)
n X
0
Z
X
1
∂(τ, ζj ) dσ2 dσ1 ∂(σ1 , σ2 )
∂(ζj , ζk ) dσ2 dσ1 ∂(σ1 , σ2 )
Z
σ1
(xj − yj )
Ej dσ2 dσ1 0
j=1
−
Bjk dxj ∧ dxk ∆
1≤j
Ej (τ (σ), ζ(σ)) det 0
ZZ
X
Ej (t, x)dt ∧ dxj +
0
Z
1
Z
{(xk − yk )(xj − zj ) − (xj − yj )(xk − zk )}
Bjk dσ2 dσ1 , 0
1≤j
σ1
0
(2.20) where ∂(τ, ζj )/∂(σ1 , σ2 ) and ∂(ζj , ζk )/∂(σ1 , σ2 ) are the Jacobian matrices. Hence we can prove Lemma 2.2 together with (2.14), (2.17), and (2.18). We write Φ defined by (2.16) as x+y t−s + Φ(t, s; x, y, z) = z − 2 m −
Z
1
A(s, y + θ(x − y))dθ 0
(t − s)2 0 t−s 0 B (t, s; x, y, z) − E (t, s; x, y, z) , m m
(2.21)
where E 0 = (E10 , . . . , En0 ) and B 0 = (B10 , . . . , Bn0 ). 3. Proof of (i) and (ii) in Introduction Lemma 3.1. Suppose the same assumptions as in Theorem. Then we have for E and B 0 in (2.21) 0
|∂xα ∂yβ ∂zγ Ej0 | ≤ Cα,β,γ ,
|α + β + γ| ≥ 1 ,
|∂xα ∂yβ ∂zγ Bj0 | ≤ Cα,β,γ ,
|α + β + γ| ≥ 1, 0 ≤ s ≤ t ≤ T, x, y, z ∈ Rn . (3.2)
(3.1)
1010
W. ICHINOSE
Proof. The proof of this lemma is analogous to that of (3.14) and (3.15) in [11]. It follows from (2.13), (2.16), and (2.21) that Z 1 Z σ1 Ej (t − σ1 (t − s), z + σ1 (x − z) + σ2 (y − x))dσ2 dσ1 , Ej0 = 0
Bj0
=
n X
0
Z
1
Z
σ1
(zk − xk )
Bjk (t − σ1 (t − s), z + σ1 (x − z) + σ2 (y − x))dσ2 dσ1 .
0
0
k=1
(3.3) So we have (3.1) from the assumption (1.9). It is easilyRproved from (1.9) that R1 ∞ |Bjk (t, x) − Bjk (t, 0)| ≤ Const. 0 hθxi−(1+δ) |x|dθ ≤ Const. 0 hθi−(1+δ) dθ < ∞. So Bjk is bounded in [0, T ] × Rn . Let x 6= z. Then setting ξ = x − z, η = y − x, and Ω = ξ/|ξ|, we have Z 1 Z σ1 ∂ Bjk (τ (σ), z + σ1 (x − z) + σ2 (y − x))dσ2 dσ1 |x − z| ∂zl 0 0 Z 1 Z σ1 hz + σ1 ξ + σ2 ηi−(1+δ) dσ2 dσ1 ≤ C|ξ| 0
0
Z
Z
1
= C|ξ| 0
Z
Z dσ2
0
dσ2 0
Z ≤C
dσ2 0
|ξ|
σ2 |ξ|
Z
1
|ξ|
σ2 |ξ|
Z
1
≤C
hz + σ1 ξ + σ2 ηi−(1+δ) dσ1
σ2
1
=C Z
1
dσ2
∞
−∞
hz + σ1 Ω + σ2 ηi−(1+δ) dσ1 hσ1 + Ω · (z + σ2 η)i−(1+δ) dσ1
hσ1 i−(1+δ) dσ1 < ∞ .
(3.4)
In the same way we can show (3.2) generally.
Lemma 3.2. Suppose the same assumptions as in Theorem. Then we have for Φ = Φ(t, s; x, y, z) defined by (2.16): (1) There exists a constant ρ∗ > 0 such that the mapping: Rn 3 z → ξ = Φ ∈ Rn is homeomorphic and det ∂Φ/∂z > 0 for each fixed 0 ≤ t − s ≤ ρ∗ , x, and y. We write its inverse mapping as Rn 3 ξ → z = z(t, s; x, y, ξ) ∈ Rn . (2) We add the assumption |∂xα Aj (t, x)| ≤ Cα , |α| ≥ 1, (t, x) ∈ [0, T ] × Rn , j = 1, 2, . . . , n
(3.5)
to the above. Then we have |∂xα ∂yβ ∂ξγ zj (t, s; x, y, ξ)| ≤ Cα,β,γ ,
|α + β + γ| ≥ 1 ,
0 ≤ t − s ≤ ρ∗ , x, y, ξ ∈ Rn , j = 1, 2, . . . , n .
(3.6)
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1011
Proof. We have from (2.21) (t − s)2 ∂E 0 t − s ∂B 0 ∂Φ = In − − , ∂z m ∂z m ∂z
(3.7)
where In is the identity matrix. So applying Theorem 1.22 in [18], we can prove (1) from Lemma 3.1. It follows from Lemma 3.1 and (3.5) that |∂xα ∂yβ ∂zγ Φ| ≤ Cα,β,γ , |α + β + γ| ≥ 1, 0 ≤ s ≤ t ≤ T, x, y, z ∈ Rn .
Hence we can easily prove (2) from Lemma 3.1 and (3.7). Hereafter we fix ρ∗ > 0 determined in Lemma 3.2.
Theorem 3.3. We assume that V and Aj (j = 1, 2, . . . , n) are continuously differentiable in [0, T ] × Rn . Besides the assumptions in Theorem suppose (3.5). Then (i) and (ii) in introduction hold. In addition, if |∆| ≤ ρ∗ , we have kC(∆)f k ≤ eKt kf k, 0 ≤ t ≤ T, f ∈ L2 .
(3.8)
Proof. First we prove (i). Let t = s. Then it is clear that kC(t, s)f k = kf k. Let 0 < t − s ≤ ρ∗ . We can write from Remark 2.1 and Lemma 2.2 for f ∈ S C(t, s)∗ χ(·)2 C(t, s)f n Z Z m mΦ dz . = f (y)dy χ(z)2 exp i(x − y) · 2π~(t − s) ~(t − s) Make the change of variables: z → ξ = Φ(t, s; x, y, z), noting Lemma 3.2. Then n Z m f (y)dy C(t, s)∗ χ(·)2 C(t, s)f = 2π~(t − s) Z ∂z mξ dξ . det × χ(z(t, s; x, y, ξ))2 exp i(x − y) · ~(t − s) ∂ξ It follows from Lemmas 3.1, 3.2, and (3.7) that det
∂z = 1 + (t − s)h(t, s; x, y, ξ) , ∂ξ
|∂xα ∂yβ ∂ξγ h(t, s; x, y, ξ)| ≤ Cα,β,γ
for all α, β, γ .
(3.9) (3.10)
Consequently C(t, s)∗ χ(·)2 C(t, s)f =
1 2π
n Z
Z f (y)dy
χ(z(t, s; x, y, ξ))2
× ei(x−y)·η (1 + (t − s)h(t, s; x, y, ξ))dη, ξ = ~(t − s)η/m . (3.11)
1012
W. ICHINOSE
Hence we can easily show together with (3.6) and (3.10) n 1 ∗ 2 Os lim C(t, s) χ(·) C(t, s)f = f + (t − s) →+0 2π ZZ − ei(x−y)·η h(t, s; x, y, ~(t − s)η/m)f (y)dydη (3.12) in the topology of S. It should be noted that the second term on the right-hand side above is a pseudo-differential operator with double symbol (cf. [14]). It follows from (3.10) that we can apply the Calder´ on–Vaillancourt theorem (cf. [14]) to the second term on the right-hand side above. Then it is proved by Fatou’s lemma that (1.13) holds for f ∈ S. Hence we can easily complete the proof of (i). We prove (ii). Let f ∈ L2 . We write C(t, tν−1 )χ(·)C(tν−1 , tν−2 )χ(·) · · · C(t2 , t1 )χ(·)C(t1 , 0)f − C(t, tν−1 )C(tν−1 , tν−2 ) · · · C(t2 , t1 )C(t1 , 0)f =
ν−1 X
C(t, tν−1 ) · · · C(tj+1 , tj )(χ(·) − 1)
j=1
× C(tj , tj−1 )χ(·) · · · C(t2 , t1 )χ(·)C(t1 , 0)f . It follows from (1.13) and the Lebesgue dominated convergence theorem that the L2 norm of each term on the right-hand side above converges to zero as → 0. Thus we can prove (ii) by (1.12). The inequality (3.8) can be easily shown from (i) and (ii). 4. Boundedness Theorem on B a Lemma 4.1. Besides the assumptions in Theorem suppose (3.5). Let zj (t, s; x, y, ξ) (j = 1, 2, . . . , n) be the function defined in Lemma 3.2. Then √ √ √ (zj (t, s; x, x + ρy, ~ρη/m + ~ ρζ/m) − xj )/ ρ can be extended to be continuous in 0 ≤ t − s ≤ ρ∗ and x, y, η, ζ ∈ Rn , where ρ = t − s. We also have √ √ |zj (t, s; x, x + ρy, ~ρη/m + ~ ρζ/m) − xj | √ (4.1) ≤ C0,0,0,0 ρ(1 + |x| + |y| + |η| + |ζ|) , 0
0
|∂ηα ∂ζα ∂xβ ∂yβ (zj (t, s; x, x +
√ √ √ ρy, ~ρη/m + ~ ρζ/m) − xj )| ≤ Cα,α0 ,β,β 0 ρ ,
|α + α0 + β + β 0 | ≥ 1, 0 ≤ t − s ≤ ρ∗ , x, y, η, ζ ∈ Rn .
(4.2)
√ √ Proof. Let z = z(t, s; x, x + ρy, ~ρη/m + ~ ρζ/m). Then we have from (2.21) Z √ √ ~ ρζ 2x + ρy ρ 1 √ ~ρη + =z− + A(s, x + (1 − θ) ρy)dθ m m 2 m 0 −
√ ρ2 √ ρ 0 B (t, s; x, x + ρy, z) − E 0 (t, s; x, x + ρy, z) m m
ON CONVERGENCE OF THE FEYNMAN PATH INTEGRAL FORMULATED
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and so
√ √ Z 1 √ ρ ρ 0 ρ3/2 0 y ~ ρη ~ζ z−x + − B + E . Adθ + (4.3) √ = + ρ 2 m m m 0 m m √ Hence (zj − xj )/ ρ can be extended to be continuous in 0 ≤ t − s ≤ ρ∗ and x, y, η, ζ ∈ Rn . We also get (4.1) and (4.2) from Lemma 3.1, (3.5), and (3.6). Lemma 4.2. Suppose the same assumptions as in Theorem 3.3. Let p(x, w) be a function satisfying (2.4). We set n ZZ √ √ √ 1 Os − e−iy·ζ p(z, (z − x)/ ρ)p(z, (z − x − ρy)/ ρ) q(t, s; x, η) = 2π × det
∂z √ √ (t, s; x, x + ρy, ~ρη/m + ~ ρζ/m)dydζ , ∂ξ
0 ≤ t − s ≤ ρ∗ , x, η ∈ Rn , where ρ = t − s and z = z(t, s; x, x +
(4.4)
√ √ ρy, ~ρη/m + ~ ρζ/m). Then we have:
(1) For any α and β there exists a constant Cα,β such that |∂ηα ∂xβ q(t, s; x, η)| ≤ Cα,β hx; ηi2M ,
0 ≤ t − s ≤ ρ∗ , x, η ∈ Rn .
(4.5)
(2) It holds that for f ∈ S lim P (t, s)∗ χ(·)2 P (t, s)f = Q(t, s; x, Dx )f in S, 0 ≤ t − s ≤ ρ∗ ,
(4.6)
→+0
where Q(t, s; x, Dx )f denotes the pseudo-differential operator (2π)−n × q(t, s; x, η)fˆ(η)dη.
R
eix·η
Proof. Set for 0 ≤ ≤ 1 n ZZ 1 √ Os − e−iy·ζ χ(z)2 p(z, (z − x)/ ρ) q (t, s; x, η) = 2π × p(z, (z − x −
√ √ ∂z √ √ (t, s; x, x + ρy, ~ρη/m + ~ ρζ/m)dydζ , ρy)/ ρ) det ∂ξ
(4.7) 0 ≤ t − s ≤ ρ∗ , x, η ∈ Rn , √ √ where z = z(t, s; x, x + ρy, ~ρη/m + ~ ρζ/m). We note q0 (t, s; x, η) = q(t, s; x, η). Let l and l0 be large integers. We have from (3.6), (4.1), and (4.2) n Z Z 0 0 1 e−iy·ζ hyi−2l (1 − ∆ζ )l hζi−2l (1 − ∆y )l q (t, s; x, η) = 2π √ √ √ ∂z 2 × χ(z) p(z, (z − x)/ ρ)p(z, (z − x − ρy)/ ρ) det dydζ ∂ξ n Z Z 1 (4.8) e−iy·ζ q,l,l0 dydζ ≡ 2π
1014
W. ICHINOSE
and 0 √ √ |q,l,l0 | ≤ C1 hyi−2l hζi−2l hz; (z − x)/ ρiM hz; (z − x)/ ρ − yiM 0
≤ C2 hyi−2l hζi−2l (1 + |x| + |y| + |η| + |ζ|)2M 0
≤ C3 hyi2M−2l hζi2M−2l hx; ηi2M for 0 ≤ ≤ 1, 0 ≤ t − s ≤ ρ∗ , and x, y, η, ζ ∈ Rn . Hence we have |q (t, s; x, η)| ≤ C4 hx; ηi2M by (4.8). In the same way we get |∂ηα ∂xβ q (t, s; x, η)| ≤ Cα,β hx; ηi2M , 0 ≤ ≤ 1, 0 ≤ t − s ≤ ρ∗ , x, η ∈ Rn
(4.9)
for all α and β. So (1) holds. Let 0 < ≤ 1 and f ∈ S. Let 0 < t − s ≤ ρ∗ . As in the proof of (3.11) we get from Lemmas 2.2 and 3.2 P (t, s)∗ χ(·)2 P (t, s)f =
1 2π
n Z
Z f (y)dy
χ(z)2 ei(x−y)·η
√ √ × p(z, (z − x)/ ρ)p(z, (z − y)/ ρ) ∂z (t, s; x, y, ~ρη/m)dη ∂ξ n Z Z 1 ≡ f (y)dy ei(x−y)·η q˜ (t, s; x, y, η)dη , (4.10) 2π × det
where z = z(t, s; x, y, ~ρη/m). It is clear from (4.7) and (4.10) that
1 2π
n
ZZ Os −
=
1 2π
e−iy·ζ q˜ (t, s; x, x + y, η + ζ)dydζ
n
ZZ Os −
e−iy·ζ q˜ (t, s; x, x +
√ √ ρy, η + ζ/ ρ)dydζ
= q (t, s; x, η) .
(4.11)
Let us use the result on pseudo-differential operators with double symbol. Applying Theorem 2.5 in Chap. 2 of [14] to (4.10) an (4.11), we have P (t, s)∗ χ(·)2 P (t, s)f = Q (t, s; x, Dx )f .
(4.12)
Let t = s in (4.7). It follows from (2.21) and (4.3) that z(s, s; x, x, 0) = x, det(∂z/∂ξ) √ √ √ (s, s; x, y, ξ) = 1, and limt→s (z(t, s; x, x + ρy, ~ρη/m + ~ ρζ/m) − x)/ ρ = y/2 +
ON CONVERGENCE OF THE FEYNMAN PATH INTEGRAL FORMULATED
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~ζ/m. So we can easily prove q (s, s; x, η) n ZZ 1 Os − e−iy·ζ p(x, y/2 + ~ζ/m)p(x, −y/2 + ~ζ/m)dydζχ(x)2 = 2π ZZ m n −1 = Os − e−i~ my·ζ p(x, y/2 + ζ)p(x, −y/2 + ζ)dydζχ(x)2 2π~ ZZ m n −1 2 = |Os − ei~ m|w| /2 p(x, w)dw|2 χ(x)2 2π~ and so Q (s, s; x, Dx )f =
2 ZZ m n i~−1 m|w|2 /2 χ(x)2 f (x) . (4.13) Os − p(x, w)dw e 2π~
Consequently we can show from (2.5) and (2.6) that (4.12) also holds for t = s. Hence we get (4.6) from (4.9). Proposition 4.3. Suppose the same assumptions as in Theorem 3.3. p(x, w) be a function satisfying (2.4). Then we have kP (t, s)f k ≤ Ckf kB M , 0 ≤ t − s ≤ ρ∗ , f ∈ B M
Let
(4.14)
for some constant C. Proof. Let a ≥ 0. Then there exist a constant µa ≥ 0 and a wa (x, η) such that |∂ηα ∂xβ wa (x, η)| ≤ Cα,β hx; ηi−a
(4.15)
for all α and β and Wa (x, Dx ) = (µa + hxia + hDx ia )−1
on S
(4.16)
(Lemma 2.3 in [9]). Let 0 ≤ t − s ≤ ρ∗ and f ∈ S. It follows from (2) in Lemma 4.2 that lim kχ(·)P (t, s)f k2 = (Q(t, s; x, Dx )f, f )
→+0
= (WM (x, Dx )Qf, (µM + hxiM + hDx iM )f ) . (4.17) Applying Lemmas 2.1 and 2.5 in [9], we have from (4.5) and (4.15) kWM (x, Dx )Q(t, s; x, Dx )f k ≤ Const. kf kB M . Hence we obtain (4.14) by (4.17) and Fatou’s lemma.
(4.18)
1016
W. ICHINOSE
Theorem 4.4. Besides the assumptions in Theorem 3.3 we suppose 0
|∂xα V (t, x)| ≤ Cα hxiM , |α| ≥ 1, (t, x) ∈ [0, T ] × Rn
(4.19)
for some constant M 0 ≥ 0. Set M ∗ = max(M 0 , 1). Let p(x, w) be a function satisfying (2.4). Then we have for a = 0, 1, 2, . . . ∗
kP (t, s)f kB a ≤ Ca kf kB M +aM ∗ , 0 ≤ t − s ≤ ρ∗ , f ∈ B M+aM .
(4.20)
Proof. We have for all a ≥ 0 and α k∂xα f kB a ≤ Ca,α kf kB a+|α| .
(4.21)
For applying the Calder´ on–Vaillancourt theorem to h·ia ∂xα Wa+|α| , we have from (4.15) and (4.16) k∂xα f kB a = kh·ia ∂xα f k + kh·ia ξ α fˆk ≤ kh·ia ∂xα Wa+|α| (µa+|α| + hxia+|α| + hDx ia+|α| )f k + kh·ia+|α| fˆk ≤ Const.k(µa+|α| + hxia+|α| + hDx ia+|α| )f k + kh·ia+|α| fˆk ≤ Const.kf kB a+|α| . Let f ∈ S. We know in Lemma 2.1 that ∂xα (P (t, s)f ) exists in 0 ≤ s ≤ t ≤ T and x ∈ Rn for all α. Let 0 ≤ t − s ≤ ρ∗ . Using (2.8) and (2.9), we can write X Pβ (t, s)(∂xα−β f ) , ∂xα (P (t, s)f ) = β≤α
where β ≤ α implies βj ≤ αj (j = 1, 2, . . . , n). We see from the assumptions on V, Aj , and p(x, w) 0
0
α β ∂x pβ (t, s; x, w)| ≤ Cα0 ,β 0 hx; wiM+|β|M |∂w
∗
for all α0 and β 0 . Hence we have from Proposition 4.3 and (4.21) X k∂xα−β f kB M +|β|M ∗+k kh·ik ∂xα (P (t, s)f )k ≤ Const. β≤α
≤ Const.kf kB M +|α|M ∗+k
(4.22)
and so for a = 0, 1, 2, . . . kh·ia P (t, s)f k +
X
k∂xα (P (t, s)f )k ≤ Const.kf kB M +aM ∗ .
|α|≤a
This shows (4.20).
Remark 4.1. It follows from Theorem 4.4 as in [7, 8, 20, 11] that P (t, s)f for f ∈ S belongs to S and the mapping: S 3 f → P (t, s)f ∈ S is bounded uniformly in 0 ≤ s ≤ t ≤ T for t − s ≤ ρ∗ .
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5. Proof of (iii) in Introduction Lemma 5.1. Suppose (2.7) and that there exists a constant M 00 ≥ 0 satisfying |∂t ∂xα V
n X
|+
00
|∂t ∂xα Aj | ≤ Cα hxiM , (t, x) ∈ [0, T ] × Rn
(5.1)
j=1
for all α. Let H(t) be the Hamiltonian operator determined by (1.7). Then there exists a continuous function r(t, s; x, w) in 0 ≤ s ≤ t ≤ T and x, w ∈ Rn satisfying (2.4) for an M ≥ 0 such that for f ∈ S r n √ √ m ∂ i~ − H(t) C(t, s)f = t − sR(t, s)f ≡ t − s ∂t 2πi~(t − s) Z −1 t,s x−y f (y)dy , 0 ≤ s < t ≤ T . (5.2) × ei~ S(γx,y ) r t, s; x, √ t−s Proof. This lemma was proved essentially in Proposition 2.3 in [11]. In this proof we give a more precise formula than that in [11], which will be applied in Remarks 5.1, 6.1, and 6.2 to the case where Bjk (1 ≤ j < k ≤ n) is constant. Let 0 ≤ s < t ≤ T and f ∈ S. Using the assumptions, we get by direct calculations r nZ −1 t,s m ∂ ei~ S(γx,y ) i~ − H(t) C(t, s)f = − ∂t 2πi~(t − s) i~ r2 (t, s; x, y) f (y)dy , (5.3) × r1 (t, s; x, y) + 2m
t,s )+ r1 = ∂t S(γx,y
r2 =
n 2 1 X t,s ) − Aj (t, x) + V (t, x) , ∂xj S(γx,y 2m j=1
X nm t,s − ∆x S(γx,y )+ ∂xj Aj (t, x) . t−s j
Set ρ = t − s. It follows from (2.3) that t,s ) − Aj (t, x) ∂xj S(γx,y
=
m(xj − yj ) + ρ +
X
Z
1
Aj (t − θρ, x − θ(x − y)) − Aj (t, x)dθ 0
Z
1
(xk − yk )
(1 − θ) 0
k
Z
1
(1 − θ)
−ρ 0
∂Ak (t − θρ, x − θ(x − y))dθ ∂xj
∂V (t − θρ, x − θ(x − y))dθ . ∂xj
(5.4)
(5.5)
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W. ICHINOSE
The potentials V and Aj are continuously differentiable in [0, T ] × Rn and infinitely differentiable in Rn . So we have by the Taylor expansion m(xj − yj ) t,s + ) − Aj (t, x) = ∂xj S(γx,y ρ +
( −
1X ∂Aj (xk − yk ) (t, x) 2 ∂xk k
∂ 2 Aj 1X (xk − yk )(xl − yl ) (t, x) 6 ∂xk ∂xl k,l
Z
1
Z
−ρ 0
( +
0
1
∂Aj (t − θθ0 ρ, x)dθdθ0 θ ∂t
)
∂Ak 1X (xk − yk ) (t, x) 2 ∂xj k
) ∂ 2 Ak 1X (xk − yk )(xl − yl ) (t, x) − 6 ∂xl ∂xj k,l
x−y 1 ∂V (t, x) + ρ3/2 q1 t, s; x, √ , − ρ 2 ∂xj ρ where q1 (t, s; x, w) is a continuous function in 0 ≤ s ≤ t ≤ T and x, w ∈ Rn satisfying (2.4) for an M . From this we can prove m|x − y|2 1 X t,s (∂xj S(γx,y ) − Aj (t, x))2 = − (x − y) 2m j=1 2ρ2 n
Z
1
Z
·
1
θ 0
0
∂A (t − θθ0 ρ, x)dθdθ0 ∂t
1 − (x − y) · ∇x V (t, x) 2 x−y , + ρq2 t, s; x, √ ρ
(5.6)
where ∇x V = (∂x1 V, . . . , ∂xn V ). The same argument shows that ∂A 1 m|x − y|2 1 (t, x) + (x − y) · ∇x V (t, x) − V (t, x) + (x − y) · 2 2ρ 2 ∂t 2 Z 1 ∂A ∂A (t − θρ, x) − (t, x) dθ (1 − θ) + (x − y) · ∂t ∂t 0 x−y + ρq3 t, s; x, √ , (5.7) ρ
t,s )=− ∂t S(γx,y
ON CONVERGENCE OF THE FEYNMAN PATH INTEGRAL FORMULATED
t,s ∆x S(γx,y )−
X
...
1019
nm 1 X ∂ 2 Aj − (xk − yk ) (t, x) ρ 3 ∂xj ∂xk
∂xj Aj (t, x) =
j
j,k
+
∂ 2 Ak 1X (xk − yk ) (t, x) 3 ∂x2j j,k
x−y . + ρq4 t, s; x, √ ρ
(5.8)
Substituting (5.6)–(5.8) into (5.3)–(5.5), we have ∂A ∂A 0 (t − θθ ρ, x) − (t, x) dθdθ0 θ r1 = −(x − y) · ∂t ∂t 0 0 Z 1 ∂A ∂A (t − θρ, x) − (t, x) dθ + (x − y) · (1 − θ) ∂t ∂t 0 x−y x−y + ρ q2 t, s; x, √ + q3 t, s; x, √ , ρ ρ Z
r2 =
1
Z
1
(5.9)
1X ∂ 2 Aj (xk − yk ) (t, x) 3 ∂xj ∂xk j,k
∂ 2 Ak x−y 1X (xk − yk ) (t, x) − ρq4 t, s; x, √ . − 3 ∂x2j ρ
(5.10)
j,k
Thus we could complete the proof.
Remark 5.1. We assume that Bjk (1 ≤ j < k ≤ n) is constant and add the following to the assumptions in Theorem. There exists a constant υ 0 ≥ 0 such that 0
|∂t ∂xα Ej (t, x)| ≤ Cα hxiυ , |α| ≥ 1, (t, x) ∈ [0, T ] × Rn P for j = 1, 2, . . . , n. Then since d( nj=1 Ej dxj ) = 0 (cf. (6.1)), V (t, x) = R1 Pn − 0 E(t, θx) · xdθ and Aj (t, x) = 1/2 l=1 Blj xl (j = 1, 2, . . . , n) give the potentials. We can easily see that these V and A satisfy (1.16). Take these V and A in Lemma 5.1. Then it follows from (5.9) and (5.10) that the right-hand side of (5.2) can be replaced by (t − s)R(t, s). t,s ) = Examples. Set ρ = t − s. (1) Let V = |x|2 and A = 0. Then S(γx,y 2 2 2 m|x− y| /(2ρ)− ρ(|x| + x·y + |y| )/3. So it follows from (5.3)–(5.5) that r(t, s; x, w) √ in (5.2) is given by −ρ2 |3x − ρw|2 /(18m) − i~nρ/(3m). t,s ) = (2) Let V = −E · x and A = 0, where E ∈ Rn is constant. Then S(γx,y 2 2 2 m|x − y| /(2ρ) + ρE · (x + y)/2 and so r(t, s; x, w) = −ρ |E| /(8m).
Theorem 5.2. We assume that V and Aj (j = 1, 2, . . . , n) are continuously differentiable in [0, T ] × Rn . Besides the assumptions in Theorem we suppose (3.5),
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W. ICHINOSE
(4.19), and (5.1). Let M and M ∗ be the constants determined in Lemma 5.1 and Theorem 4.4, respectively. Then we have for a = 0, 1, 2, . . .
√
i~ ∂ − H(t) C(t, s)f ≤ Ca t − skf kB M +aM ∗ ,
∂t Ba 0 < t − s ≤ ρ∗ , f ∈ B M+aM
∗
(5.11)
with some constant Ca . Proof. proof.
Applying Theorem 4.4 to R(t, s)f in (5.2), we can complete the
Corollary 5.3. Suppose the same assumptions as in Theorem 5.2. Then there exist constants a ≥ 0 and C such that (1.15) holds. So we obtain (iii) in introduction. Proof. Let ρ = t − s > 0 and f ∈ S. We can easily prove from (2.8) and (2.9) that ∂C(t, s)f /∂t is continuous in 0 ≤ s < t ≤ T and x ∈ Rn . We use Lemma 2.1. Then we have ∂C C(s + ρ, s)f − f = i~ (s + θρ, s)f i~ ρ ∂t for some 0 < θ < 1 and so by Theorem 5.2
C(s + ρ, s)f − f
≤ C √ρkf kB M
i~ − H(s)f
ρ + kH(s + θρ)(C(s + θρ, s)f − f )k + kH(s + θρ)f − H(s)f k .
(5.12)
We also have by Lemma 2.1 ∂C (s + θ0 θρ, s)f ∂t ∂C −1 0 0 0 = (i~) θρ i~ (s + θ θρ, s)f − H(s + θ θρ)C(s + θ θρ, s)f ∂t
C(s + θρ, s)f − f = θρ
+ (i~)−1 θρH(s + θ0 θρ)C(s + θ0 θρ, s)f for some 0 < θ0 < 1 and so as in the proof of (4.21) and (5.12) from Theorems 4.4, 5.2, the assumptions (3.5), and (4.19) kH(s + θρ)(C(s + θρ, s)f − f )k ≤ C 0 ρkf kB a0
(5.13)
for some a0 ≥ 0. It is easy to see from the assumption (5.1) that kH(s + θρ)f − H(s)f k ≤ Const. ρkf kB a00 for some a00 ≥ 0. Hence we can complete the proof together with (5.12) and (5.13).
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6. Proof of Theorem Lemma 6.1. Assume the same assumptions as in Theorem. Then there exist potentials V and A such that V (t, x) = 0 and |∂xα Aj (t, x)| + |∂t ∂xα Aj (t, x)| ≤ Cα ,
|α| ≥ 1, (t, x) ∈ [0, T ] × Rn
for j = 1, 2, . . . , n. Proof. The proof below is analogous to that of Lemma 2.2 of [10]. First we note that n X X X Ej dxj = − ∂t Bjk dxj ∧ dxk , d Bjk dxj ∧ dxk = 0 d j=1
1≤j
1≤j
(6.1) on Rn follows from the Maxwell equation. Consequently V and A can be defined by (1.1). The first equation of (1.1) is equivalent to dV = −
n X
on Rn .
(6.2)
Using (2.14), we define functions A0j (j = 1, 2, . . . , n) by n Z 1 X Bjk (t, θx)θxk dθ . A0j (t, x) = −
(6.3)
(Ej + ∂t Aj )dxj
j=1
k=1
0
It follows from the assumption on Bjk that A0j are continuously differentiable in [0, T ] × Rn and infinitely differentiable in Rn . We also see from the second equation of (6.1) and the Poincar´e lemma (cf. [6]) that A0 = (A01 , . . . , A0n ) satisfies the second equation of (1.1). It was proved in the proof of Lemma 3.1 that Bjk (t, x) are bounded in [0, T ] × Rn . In the same way we can prove that ∂xα A0j (t, x) are bounded R1 in [0, T ] × Rn for all |α| ≥ 1. Define V 0 (t, x) = − 0 (E(t, θx) + ∂t A0 (t, θx)) · xdθ, which is continuous in [0, T ] × Rn and infinitely differentiable in Rn . It is clear that V 0 satisfies the first equation of (1.1). Let us define V and A by the gauge transformation Z t 0 V 0 (θ, x)dθ = 0 , V (t, x) = V (t, x) − ∂t 0
Aj (t, x) = A0j (t, x) + ∂xj
Z
t
V 0 (θ, x)dθ .
(6.4)
0
Since V = 0, we have E = −∂t A from (1.1) and so ∂xα E = −∂t ∂xα A. Hence it follows from the assumption on E that ∂t ∂xα Aj are bounded in [0, T ] × Rn for all |α| ≥ 1. We write Z 1 (∂t Aj )(θt, x)dθ Aj (t, x) = Aj (0, x) + t 0
= A0j (0, x) − t
Z
1
Ej (θt, x)dθ . 0
(6.5)
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W. ICHINOSE
So we can see that ∂xα Aj are bounded in [0, T ] × Rn for all |α| ≥ 1. Thus we could complete the proof. We denote by Ctj ([s, T ]; B a ) the space of all B a -valued j times continuously differentiable functions in t ∈ [s, T ]. The following is proved in [9]. Lemma 6.2. Assume |∂xα V (t, x)| ≤ Cα hxi, |α| ≥ 1 , |∂xα Aj (t, x)| ≤ Cα , |α| ≥ 1, (t, x) ∈ [0, T ] × Rn for j = 1, 2, . . . , n. Then for any f ∈ B a (−∞ < a < ∞) there exists a unique solution U (t, s)f ∈ Ct0 ([s, T ]; B a ) ∩ Ct1 ([s, T ]; B a−2 ) of (1.6) with H(t) determined by (1.7). In addition, there exists a constant Ca (T ) such that kU (t, s)f kB a ≤ Ca (T )kf kB a , 0 ≤ s ≤ t ≤ T .
(6.6)
In particular, when a = 0, we have kU (t, s)f k = kf k, 0 ≤ s ≤ t ≤ T .
(6.7)
Proposition 6.3. Assume the same assumptions as in Theorem. Let V and A be the potentials satisfying (1.16). Then (i)–(iii) in introduction hold. In addition, the same assertions (1)–(3) as in Theorem hold. Proof. The properties (i)–(iii) in introduction follow from Theorem 3.3 and Corollary 5.3. Let ρ = t − s > 0 and f ∈ S. Then we have from Lemma 6.2 i~
U (t, s)f − f − H(s)f t−s = i~
∂U (s + θρ, s)f − H(s)f ∂t
= H(s + θρ) (U (s + θρ, s)f − f ) + (H(s + θρ) − H(s))f = θρH(s + θρ)
∂U (s + θ0 θρ, s)f + (H(s + θρ) − H(s))f ∂t
= (i~)−1 θρH(s + θρ)H(s + θ0 θρ)U (s + θ0 θρ, s)f + (H(s + θρ) − H(s))f for some 0 < θ, θ0 < 1. As in the proof of (4.21) we can prove from (1.16) and (6.6) kH(s + θρ)H(s + θ0 θρ)U (s + θ0 θρ, s)f k ≤ Const. kf kB a0 , k(H(s + θρ) − H(s))f k ≤ Const. ρkf kB a0 for some a0 ≥ 0 and so
U (t, s)f − f 0
i~ ≤ Const. (t − s)kf kB a0 , 0 ≤ s < t ≤ T, f ∈ B a . (6.8) − H(s)f
t−s
ON CONVERGENCE OF THE FEYNMAN PATH INTEGRAL FORMULATED
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Hence we get together with (1.15) kC(t, s)f − U (t, s)f k ≤ C(t − s)3/2 kf kB b , 0 ≤ t − s ≤ ρ∗
(6.9)
for all f ∈ B b , where b = max(a, a0 ). Let f ∈ B b and |∆| ≤ ρ∗ . We can write by (1.14) C(∆)f − U (t, 0)f = C(t, tν−1 ) · · · C(t1 , 0)f − U (t, tν−1 ) · · · U (t1 , 0)f =
ν X
C(t, tν−1 ) · · · C(tj+1 , tj )(C(tj , tj−1 ) − U (tj , tj−1 ))U (tj−1 , 0)f . (6.10)
j=1
Applying (1.13), (6.6), and (6.9) to the above, we have kC(∆)f − U (t, 0)f k ≤ Const.|∆|1/2
ν X
eK(t−tj ) (tj − tj−1 )kf kB b
j=1
≤ Const. eKT T |∆|1/2 kf kB b .
(6.11)
Consequently C(∆)f for f ∈ B b converges to U (t, 0)f in L2 uniformly in t ∈ [0, T ] as |∆| → 0. Hence using (3.8) and (6.7), we can easily complete the proof. Remark 6.1. Suppose the same assumptions as in Remark 5.1 and take V (t, x) R1 Pn = − 0 E(t, θx) · xdθ and Aj (t, x) = 1/2 l=1 Blj xl (j = 1, 2, . . . , n) as the potentials. These V and A satisfy (1.16) as was noted in Remark 5.1. Following the proof of Theorem 5.2 and Corollary 5.3, we can easily show from Remark 5.1 that √ t − s on the right-hand side of both (5.11) and (1.15) can be replaced by t − s. Using this, as in the proof of Proposition 6.3 we can prove (6.9) where (t − s)3/2 is replaced by (t − s)2 and so kC(∆)f − U (t, 0)f k ≤ Const.eKT T |∆|kf kB b , |∆| ≤ ρ∗
(6.12)
for all f ∈ B b . Let us prove Theorem. We can choose from Lemma 6.1 the potentials V 0 and A satisfying (1.16). We fix them. Define H(t)0 by (1.7) for these V 0 and A0 . Then it follows from Lemma 6.2 that there exists a unique solution U (t, s)0 f of (1.6) for H(t)0 . In the same way we define C(t, s)0 f and C(∆)0 f . Let V and A be arbitrary potentials, where V, ∂V /∂xj , ∂Aj /∂t, and ∂Aj /∂xk (j, k = 1, 2, . . . , n) are assumed to be continuous in [0, T ] × Rn . We have by (1.1) 0
∂ ∂ 0 (A − Aj ) + (V 0 − V ) = 0 (j = 1, . . . , n), ∂t j ∂xj n X d (A0j − Aj )dxj = 0 on Rn . j=1
(6.13)
1024
W. ICHINOSE
R1 Set ψ1 (t, x) = 0 (A0 (t, θx) − A(t, θx)) · xdθ, which is continuously differentiable in P [0, T ] × Rn . Then we have dψ1 = j (A0j − Aj )dxj on Rn and so A0j (t, x) = Aj (t, x) +
∂ψ1 (t, x) ∂xj
(j = 1, 2, . . . , n) .
It follows from this that ∂ 2 ψ1 /∂xj ∂xk and ∂ 2 ψ1 /∂t∂xj are continuous in [0, T ]×Rn. Consequently we get together with the first equation of (6.13) ∂ ∂ 0 ψ1 + V − V = 0 (j = 1, 2, . . . , n) ∂xj ∂t and hence V 0 (t, x) − V (t, x) R= −∂ψ1 /∂t + ψ2(t), where ψ2 (t) is continuous in [0, T ]. t Setting ψ(t, x) = ψ1 (t, x) − 0 ψ2 (θ)dθ, we have (1.2) and so A0 · dx − A · dx = dψ
on [0, T ] × Rn .
(6.14)
We note that ∂ 2 ψ/∂xj ∂xk and ∂ 2 ψ/∂t∂xk are continuous in [0, T ] × Rn . We can easily see from (6.14) Z Z A · dx = A0 · dx − ψ(t, x) + ψ(s, y) t,s γx,y
t,s γx,y
and so from (1.11) and (2.3) C(t, s)f = e−i~
−1
ψ(t,·)
C(t, s)0 (ei~
−1
ψ(s,·)
f ), 0 ≤ t − s ≤ ρ∗ .
(6.15)
Since C(t, s)0 satisfies (1.13), so does C(t, s). Hence as in the proof of Theorem 3.3 we can easily prove that (1.14) holds too. So we have from (6.15) C(∆)f = e−i~
−1
ψ(t,·)
C(∆)0 (ei~
−1
ψ(0,·)
f ), |∆| ≤ ρ∗ , f ∈ L2 .
(6.16)
This shows that C(∆) is well defined and can be extended to a bounded operator on L2 . In addition, we see from Proposition 6.3 that C(∆)f for f ∈ L2 converges −1 −1 to e−i~ ψ(t,·) U (t, 0)0 (ei~ ψ(0,·) f ) in L2 uniformly in t ∈ [0, T ]. Since U (t, 0)f = e−i~
−1
ψ(t,·)
U (t, 0)0 (ei~
−1
ψ(0,·)
f) ,
(6.17)
we could prove Theorem. Remark 6.2. We suppose the same assumptions as in Theorem. In addition we assume that the potentials V and A satisfy (2.7). Then using (1.2), (6.11), (6.16), and (6.17), we can easily prove kC(∆)f − U (t, 0)f k ≤ Const.|∆|1/2 kf kB c ,
0 ≤ t ≤ T, |∆| ≤ ρ∗
for some c ≥ 0. In the same way we can prove from (6.12) kC(∆)f − U (t, 0)f k ≤ Const. |∆|kf kB c ,
0 ≤ t ≤ T, |∆| ≤ ρ∗
if we suppose the same assumptions as in Remark 5.1.
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References [1] S. Albeverio and Z. Brze´zniak, “Oscillatory integrals on Hilbert spaces and Schr¨ odinger equation with magnetic fields”, J. Math. Phys. 36 (1995) 2135–2156. [2] S. Albeverio and R. J. Høegh-Krohn, Mathematical Theory of Feynman Path Integrals, Lecture Notes in Math. 523, Springer, 1976. [3] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, Berlin, 1978. [4] R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics”, Rev. Mod. Phys. 20 (1948) 367–387. [5] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGrawHill, New York, 1965. [6] H. Flanders, Differential Forms with Applications to the Physical Sciences, Academic Press, New York, 1963. [7] D. Fujiwara, “A construction of the fundamental solution for the Schr¨ odinger equation”, J. D’Analyse Math. 35 (1979) 41–96. [8] D. Fujiwara, “Remarks on convergence of the Feynman path integrals”, Duke Math. J. 47 (1980) 559–600. [9] W. Ichinose, “A note on the existence and ~-dependency of the solution of equations in quantum mechanics”, Osaka J. Math. 32 (1995) 327–345. [10] W. Ichinose, “On the semi-classical approximation of the solution of the Heisenberg equation with spin”, Ann. Inst. Henri Poincar´e, Phys. Th´eor. 67 (1997) 59–76. [11] W. Ichinose, “On the formulation of the Feynman path integral through broken line paths”, Commun. Math. Phys. 189 (1997) 17–33. [12] K. Itˆ o, “Wiener integral and Feynman integral”, pp. 227–238 in Proc. 4th Berkeley Symposium on Mathematical Statistics and Probability 2, Univ. of California Press, Berkeley, 1961. [13] K. Itˆ o, “Generalized uniform complex measures in the Hilbertian metric space with their application to the Feynman integral”, pp. 145–161 in Proc. 5th Berkeley Symposium on Mathematical Statistics and Probability 2, Univ. of California Press, Berkeley, 1967. [14] H. Kumano-go, Pseudo-Differential Operators, MIT Press, Cambridge, 1981. [15] V. P. Maslov and A. M. Chebotarev, “Jump-type process and their applications in quantum mechanics”, Transl. Journal of Soviet Math. 13 (1980) 315-358. [16] A. Messiah, M´ecanique Quantique, Dunod, Paris, 1959. [17] J. M. Rabin, “Introduction to quantum field theory for mathematicians”, pp. 183– 269 in Geometry and Quantum Field Theory, eds. D. S. Freed and K. K. Uhlenbeck, Amer. Math. Soc., 1995. [18] J. T. Schwartz, Nonlinear Functional Analysis, Gordon and Breach Science Publishers, New York, London, Paris, Montreux, Tokyo, 1969. [19] A. Truman, “The polygonal path formulation of the Feynman path integral”, pp. 73– 102 in Feynman Path Integrals, Lecture Notes in Phys. 106, Springer, 1979. [20] K. Yajima, “Schr¨ odinger evolution equations with magnetic fields”, J. D’Analyse Math. 56 (1991) 29–76.
NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS ANDREAS KNAUF Mathematisches Institut der Universitaet Erlangen-Nuernberg Bismarckstr. 1 12 D-91054 Erlangen, Germany E-mail : [email protected] Received 28 May 1998 In these lecture notes connections between the Riemann zeta function, motion in the modular domain and systems of statistical mechanics are presented.
1. Introduction Counting was the earliest mathematical activity. Number theory was thus among the first subjects of mathematics. It was shown by Euclid that every integer n ∈ N has a unique factorization Y pαp (1) n= p∈P
in terms of the primes P ⊂ N, and a class of rings like Z sharing this property thus bears his name. To some extent the beauty of number theory seems to be related to the contradiction between the simplicity of the integers and the complicated structure of the primes, their building blocks. This has always attracted people. A nonrepresentative example is the following, from a book by Sacks: He describes a dialogue between twins: “John would say a number — a six-figure number. Michael would catch the number, nod, smile and seem to savour it. Then he, in turn, would say another six-figure number, and now it was John who received, and appreciated it richly. They looked, at first, like two connoisseurs wine-tasting, sharing rare tastes, rare appreciations.” After having recognized with the help of a table that these numbers are prime, Sacks returns the favour, when they meet next time, by presenting an eight-digit prime number. Some minutes later John answers with a nine-digit integer . . . The twins described in the book The man who mistook his wive for a hat by the neuro-psychologist Oliver Sacks were inhabitants of a mental hospital. They were unable to perform even simple arithmetic operations. According to their own description, they saw the landscape of integers, which is indeed very bizarre. 1027 Reviews in Mathematical Physics, Vol. 11, No. 8 (1999) 1027–1060 c World Scientific Publishing Company
1028
A. KNAUF
Maybe I should quote a source which is nearer to the way mathematics is done at a Max-Planck-Institute. In his inaugural lecture at Bonn University, Don Zagier argues: “There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers belong to the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these laws with almost military precision.”[55] This series of lecture notes is aimed at presenting a current attempt to use ideas from mathematical physics, more specifically from the ergodic theory of dynamical systems and equilibrium statistical mechanics, to solve problems in number theory. The group of people having contributed to this attempt includes M. Berry, J.-B. Bost, A. Connes, P. Contucci, P. Cvitanovi´c, M. Feigenbaum, F. Guerra, D. Harari, B. Julia, J. Keating, P. Kleban, O. E. Lanford, E. Leichtnam, A. Lopes, D. Mayer, ¨ uk, M. Pollicott, I. Procaccia, L. Ruedin, D. Spector, T. T´el and myself. A. Ozl¨ One basic observation in this attempt, and the one on which I will concentrate, is that scattering in the modular domain of PSL(2, Z) may be analyzed by using the thermodynamic formalism. The quantum scattering amplitude (which is basically a quotient of Riemann zeta functions) can be written in terms of the relative time delay of the geodesics coming from and going to the cusp. Finally, that time delay on the set of scattering geodesics may be considered as the energy function of a spin chain, and the scattering amplitude turns out to be the partition function of that system. The spin chain shows a phase transition, and its interaction function (the negative Fourier transform of the energy function) turns out to be positive. This is of some interest because of the analogy with partition functions of ferromagnetic Ising systems, whose zeroes all lie on a line (Lee–Yang Theorem). At this moment the above attempt has not been successful in deriving new number-theoretical results, so that its importance for number theory naturally remains controversial. On the other hand, there are now numerous applications of results from number theory to physical problems. To mention just a few: • p-adic analysis plays a role in string theory [7]; • a key element in understanding the growth of plants like sunflowers is the modular domain [31];
NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS
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• some properties of information networks are optimal if they have the form of so-called Ramanujan graphs [49]; • finally, the Riemann zeta function is the mother of all zeta functions, including the ones considered in the theory of dynamical systems [4, 43, 48]. As I shall deal with p-adic analysis, the modular domain and Ramanujan graphs, these lecture notes might be of some use even if I cannot convince you that there is a relevant leftward arrow in Number Theory ⇐⇒ Physics . I begin by reviewing some elementary properties of the Riemann zeta function. Instead of giving detailed proofs, I shall indicate the main ideas and refer for details to the introductory literature. 2. The Riemann Zeta Function A central fact supporting Zagier’s second observation is the: Prime Number Theorem. The number π(x) := |P ∩ [0, x]| of primes not exceeding x is asymptotic to Z dy . π(x) ∼ Li(x) := ln(y) 2 See [41] for a short proof based on explicit formulae. In turn Li(x) ∼
x , ln(x)
but with slower convergence. Loosely speaking, one may thus say that near x, the primes have density 1/ ln(x).
2000
1500
1000
500
5000
10000
15000
20000
Fig. 1. The number π(x) of primes not exceeding x. For comparison: logarithmic integral (above) and function x/ ln(x) (below).
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A. KNAUF
Gauss discovered this law in 1792, at the age of 15, by studying prime number tables. Later on he found further evidence by listing the prime numbers up to three million. In fact it is not hard to see that if the limit lim
x→∞
π(x) Li(x)
(2)
exists, it must equal one. But it took over 100 years until in 1896 Hadamard and de la Val´ee Poussin proved independently convergence of (2) and thus the Prime Number Theorem [12]. Their proof was based on the Riemann zeta function, defined by the Dirichlet series ∞ X n−s (<(s) > 1) , (3) ζ(s) := n=1
which is absolutely convergent in that half-plane. One obtains the Euler product form Y 1 (<(s) > 1) , ζ(s) = 1 − p−s
(4)
p∈P
which exhibits more clearly its connection with the primes, by expanding the geometric series in (4), and using uniqueness of the prime factorization (1). (Ordinary) Dirichlet series have the form ∞ X
a(n)n−s
n=1
for some arbitrary arithmetic function a : N → C, see, e.g. [2] for an introduction. They converge, resp. converge absolutely, in open half-planes of the form <(s) > σc
resp. <(s) > σa ,
and diverge, resp. diverge absolutely, in the complement of their closures. As can be read off from (3), ζ has a pole at one. So both its abscissa of convergence σc and its abscissa of absolute convergence σc equal one. However, s = 1 is the only pole of the Riemann zeta function, analytically continued to C. This will follow from the functional equation and the Theorem. s = 1 is the only pole of ζ in the half-plane <(s) > 0. Proof. For s > 0, (1 − 21−s ) · ζ(s) =
∞ X
n−s − 2
n=1
=
∞ X
∞ X
(2n)−s
n=1
(2n − 1)−s − (2n)−s
n=1
=
∞ X n=1
O(n−1−s ) 6= ∞
NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS
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and similarly (1 − 31−s ) · ζ(s) 6= ∞
(s > 0) ,
so that these Dirichlet series have σc = 0. The only common zero of 1 − 21−s and 1 − 31−s is s = 1, which proves the claim. The functional equation for the Riemann zeta function ζ takes a simple form if one introduces the complete zeta function ζA (s) := π −s/2 · Γ(s/2) · ζ(s) .
(5)
ζA (1 − s) ≡ ζA (s)
(6)
Theorem.
Proof. The Mellin transform of a function f is given by Z ∞ (f (t) − f (∞))ts−1 dt . Mf (s) :=
(7)
0
Setting f (t) := e−nt ,
Mf (s) = Γ(s) · n−s
(n > 0) ,
since for n = 1 this is the definition of the Gamma function, whereas for general n this follows by substitution of nt. P So f (t) := g(e−t ) for a power series g(z) := n a(n)z n is transformed into X a(n)n−s . Mf (s) = Γ(s) · n
In particular we obtain the identity ζA (2s) = Mf (s) with
f (t) :=
1 θ(it) , 2
where θ denotes the Jacobi theta series, defined on the upper half plane H ≡ H∞ := {z ∈ C | =(z) > 0} by θ(z) :=
X
exp(iπn2 z) .
n∈Z
Now applying the Poisson summation formula X X F h(y) = h(x) y∈Z
x∈Z
for the Fourier transform F h of a Schwarz function h to ht (x) := exp(−πtx2 ), one sees that √ (8) θ(−1/z) = −iz θ(z) (z ∈ H) , so that the functional equation follows.
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Equation (6) is indeed a remarkable identity, since it relates half-planes of divergence to half-planes of convergence. Remark. In addition we note several things here. • The Mellin transform is important in number theory since it relates power series and Dirichlet series. • Fourier transformation, through Poisson summation, leads to the kind of duality encoded in the functional equation. • In addition to (8), the Jacobi theta function meets θ(z + 2) = θ(z). So it is a so-called modular function (see [3, 39, 49]) for the subgroup of SL(2, Z) and 10 21 . generated by the elements 01 −1 0 Here M := ac db ∈ SL(2, R) acts on H by M¨ obius transformations: ˆ (z) := az + b . z 7→ M cz + d
(9)
Thus we see how the modular group Γ ≡ PSL(2, Z) := SL(2, Z)/{±1} and the modular domain Γ \ H come into play. Theorem. The Riemann zeta function is a meromorphic function on C with a single, simple pole at s = 1, the trivial zeroes s = −2n, n ∈ N, all other zeroes being contained in the critical strip 0 ≤ <(s) ≤ 1, and ζ(¯ s) = ζ(1 − s) = 0 if ζ(s) = 0 .
Proof. We first show that there are no zeroes ζ(s) = 0 with <(s) > 1. Zeroes of ζ become poles of its inverse ∞ Y X 1 = (1 − p−s ) = µ(n)n−s ζ(s) n=1
(10)
p∈P
with the M¨obius functiona µ
Y
p
αp
(
=
P (−1) αp , 0,
αp ≤ 1 . otherwise
But the Dirichlet series (10) converges for <(s) > 1, since |µ| ≤ 1. The remaining statements follow immediately by inspection of the functional equation (6), see [2, 12]. Remark. The analytical proofs of the Prime Number Theorem basically reduce to a proof that the non-trivial zeroes of ζ are located in the interior of the critical strip. Riemann Hypothesis (RH) The non-trivial zeroes are on the critical line <(s) = 12 . a The M¨ obius function has found important generalizations in combinatorics, see [1].
NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS
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20
1000
2000
3000
4000
5000
-20
-40
√ Fig. 2. The sum function M (x) of the M¨ obius µ function. For comparison: parabolae ± 6π −2 x.
Despite constant efforts this remains an open question. If true, it implies a very smooth distribution of the primes [27], that is, small fluctuations of π away from Li: √ (RH) =⇒ π(x) = Li + O( x · ln(x)) . So much for the regular properties of the primes mentioned by Zagier. The Dirichlet series (10) of 1/ζ converges in the half-plane <(s) > 12 if and only if the Riemann Hypothesis holds true. P Thus setting M (x) := n≤x µ(n), we obtain from partial summation 1 = ζ(s)
Z 1
∞
x−s dM (x) =
Z
∞
M (x) · s · x−s−1 dx
1
which converges for <(s) > σ if M (x) = O(xσ ). In fact the (modified) Mertens conjecture M (x) = O x1/2+ε (ε > 0) (11) is equivalent to the Riemann Hypothesis, see [52]. Now the graph of M looks very much like a random walk. This lead Denjoy (see [12]) and Good and Churchhouse in [17] to a probabilistic motivation for RH. Indeed (11) would follow with probability one if the µ(n) were i.i.d. random variables with the above distribution, since then n 7→ M (n) would correspond to a symmetric random walk. However, arithmetical functions like µ are of course deterministic, and thus Good and Churchhouse remark that “all our probability arguments are put forward in a purely heuristic spirit” [17]. One goal of these lectures is to show how to convert this analogy into a mathematical question concerning certain Markov processes. To speak metaphorically, it is clear that the primes are generated by a deterministic random number generator, but it is unclear how random this generator is. Coming back to the functional equation (6), one may argue that ζA is a function even more fundamental than ζ itself. But why so? What is the significance of the Γ function appearing in its definition? This leads us to our next point.
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3. Some p-adic Analysis The Riemann zeta function is the only most prominent member of the family of so-called Dedekind zeta functions [56] which are associated to certain fields (algebraic extensions of Q). The relevant field for ζ is the quotient field Q of the ring Z. We recall that the field R of real numbers is defined by forming Cauchy sequences of rationals, that is, by means which are not purely algebraic. Abstractly one starts from an absolute value k · k : Q → [0, ∞) with the properties kxk = 0 iff x = 0 , kx1 · x2 k = kx1 k · kx2 k
and kx1 + x2 k = kx1 k + kx2 k ,
takes the ring of Cauchy sequences w.r.t. to that absolute value and observes that their quotient w.r.t. the zero sequences forms an extension field of Q. In the case of the reals Q∞ := R one simply takes the absolute value | · |∞ := | · |. But this is not the only choice. Namely the p-adic field extensions Qp , for p ∈ P, are constructed using the absolute value | · |p with |0|p := 0 and Y q αp ∈ Q∗ := Q \ {0} . |x|p := p−αp for x = ± q∈P
We immediately notice the closure relation Y |x|v = 1 (x ∈ Q∗ ) , v∈P∪{∞}
from which one gets the right impression that these are essentially the only absolute values (Theorem of Ostrowski, [54]). Considering for a moment the ring Zp ⊂ Qp of p-adic integers a, arising as limit points of Cauchy sequences ai ∈ Z of ordinary integers, we notice that we can represent them as formal sequences ∞ X
an p n
with
an ∈ {0, . . . , p − 1}
(12)
n=m
and m = 0, since the coefficients ain eventually stabilizes. Elements of Qp can still represented in the form (12), but now with arbitrary negative m. From an algebraic point of view all these field extensions Qv ⊃ Q, v ∈ P ∪ {∞}, are equally important. They are all locally compact so that they have Haar measures dxv , which are unique up to normalization. Thus dx∗∞ :=
dx∞ |x|∞
resp. dx∗p :=
p dxp p − 1 |x|p
(p ∈ P)
are Haar measures on the multiplicative groups Q∗p . Similar to the additive characters χu,∞ : Q∞ → C∗
χu,∞ (x) := exp(2πiux) (u ∈ Q∞ )
NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS
1035
on Q∞ = R, one has additive characters χu,p on Qp , given by χu,p (x) := exp(2πi[ux]p ) (u ∈ Qp ) with the fractional part [x]p :=
−1 X
an p n
for
∞ X
x=
n=m
an p n ∈ Q p .
n=m
So we have formally Fourier Transformation Z Fv f (u) := f (x)χu,v (x)dxv Qv
on this field and may ask ourselves about its eigenfunctions γv for eigenvalue 1. In the case of R we know that γv (x) = exp(−πx2 ) is such an eigenfunction. For p ∈ P ( γp (x) :=
1,
|x|p ≤ 1
0,
|x|p > 1
solve the problem for the p-adic case, since for |u|p ≤ 1 the exponential equals 1, whereas for |u|p > 1 we have effectively a sum over unit roots. Similarly, for s ∈ C x 7→ |x|sv is a multiplicative character on Qv , v ∈ P ∪ {∞}. Setting now Z ζv (s) :=
Qv
γv (x)|x|sv dx∗v
(v ∈ P ∪ {∞}) ,
one calculates ζ∞ (s) = π −s/2 Γ(s/2) and ζp (s) =
1 1 − p−s
(p ∈ P) ,
so that the complete zeta function has the form ζA =
Y
ζv .
v∈P∪{∞}
The index A indicates that ζA is an adelic object, i.e. one in which all valuations of Q contribute, see [7, 29]. 4. Scattering in the Modular Domain Since the times of Minkowski [38], the geometrical aspects of number theory have turned out to be more and more important. We have already seen the appearance of the group Γ, and will now consider motion on the surface Γ \ H.
1036
A. KNAUF
F
i
e2iπ/3 + 1
2iπ/3
e
H
−1
0
1
Fig. 3. The modular domain F.
The modular domain (see Fig. 3) F :=
z∈H|−
1 1 < <(z) < , |z| > 1 2 2
is a fundamental domain of Γ, that is, every z ∈ H can be carried by M ∈ Γ to a ˆ (z) ∈ F, ¯ and no point of F can be carried to another point of F in such a point M way. The upper half plane is endowed with the Riemannian metric ds2 =
dx2 + dy 2 , y2
(z ≡ x + iy ∈ H)
(13)
of curvature −1, which is invariant under arbitrary M¨ obius transformations (9). As can be seen from (13), the Laplace operator ∆ on H has the form ∆ = y 2 (∂x2 + ∂y2 ). The wave equation ∂2u = ∂t2
1 u ∆+ 4
(14)
has the x-independent special solutions hω (z) · eiωt
with
1
hω (z) := y 2 +iω
(z ∈ H) .
Starting from hω , we construct (Maass) automorphic functions f : H → C, which ˆ = f for M ∈ Γ). are solutions of −(∆ + 14 )f = ω 2 f and are Γ-invariant (f ◦ M hω is already invariant under the subgroup Γ∞ ⊂ Γ of integer translations z 7→ z + n. So we only sum over the right cosets Γ∞ \ Γ and obtain the Eisenstein series X ˆ (z)) e(z, ω) := hω (M (15) M∈Γ∞ \Γ
1037
NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS
which is known to converge for frequencies ω with =(ω) < − 21 . By construction (15) is automorphic w.r.t. z ∈ H. The scattering matrix b S(ω) is defined with the help of the zeroth x-Fourier coefficient Z 12 (0) e(x + iy, ω)dx, y > 0 , e (y, ω) := − 12
namely e(0) (y, ω) = y 2 +iω + S(ω)y 2 −iω . 1
1
(16)
Faddeev and Pavlov showed (see also [30]) that: Theorem [13]. S(ω) =
ζA (2iω) , ζA (1 + 2iω)
(17)
ζA being the complete zeta function (5). Proof. If M = ac db ∈ SL(2, Z) is not in the coset space of the identity, then c 6= 0 and we can assume c > 0 because we are dealing with Γ = SL(2, Z)/{±1}. b−nd , Then the other elements of the right coset of M are of the form a−nc c d n ∈ Z, so that there is a unique representative with 0 < a ≤ c. The greatest common divisor (c, d) = 1, since ad − bc = 1. If, on the other hand, we are given c > 0 and d with (c, d) = 1, then there is a unique M = ( a bc d ) ∈ SL(2, Z) with 0 < a ≤ c. Furthermore, 12 +iω 1 y 2 +iω az + b ˆ hω (M (z)) = = = , 1 cz + d [(cx + d)2 + c2 y 2 ] 2 +iω so that one need not determine the integers a and b. Thus one gets X X 1 1 e(z, ω) = y 2 +iω 1 + [(cx + d)2 + c2 y 2 ]− 2 −iω c∈N d∈Z, (c,d)=1
and
e(0) (y, ω) = y 2 +iω 1 + 1
X
Z
X
c∈N d∈Z,(c,d)=1
1 = y 2 +iω 1 +
X X Z c∈N
1≤d≤c
1 2
1 [(cx + d)2 + c2 y 2 ]− 2 −iω dx
− 12
∞
−∞
1 [(cx + d)2 + c2 y 2 ]− 2 −iω dx
(c,d)=1
= y 2 +iω + y 2 −iω 1
1
Γ( 12 )Γ(iω) X ϕ(c) Γ( 12 + iω) c∈N c1+2iω
(18)
with the Euler totient ϕ(c) := |{d ∈ {1, . . . , c} | (c, d) = 1}|. b One should rather call it scattering amplitude. However, a similar construction for surfaces with n cusps leads to an n × n matrix [39].
1038
A. KNAUF
By (16) this implies the Faddeev–Pavlov formula (17) for the scattering matrix S(ω), since the quotient ζ(s − 1)/ζ(s) has the Dirichlet series X ϕ(n) ζ(s − 1) = . (19) ζ(s) ns n∈N
Remark. Berry, Keating and others noted similarities between the spectra of operators whose underlying dynamics is ergodic, and the non-trivial zeroes of the Riemann zeta function [5]. Here the poles of the scattering amplitude of ∆ are related to these zeroes, and the geodesic flow on Γ \ H is mixing. This explains why such a connection exists. The appearance of the adelic zeta function in (17) may come as a surprise, since the modular surface Γ \ H is defined in terms of the upper half plane H ≡ H∞ which is the homogeneous space H = SL(2, R)/SO(2, R)
√ (since SO(2, R) ⊂ SL(2, R) is the isotropy group of −1 ∈ H). So H∞ is a real object. However Γ \ H may be defined in purely adelic terms. The p-adic analogue Hp = PGL(2, Qp )/PGL(2, Zp ) of H∞ is naturally identified with the vertex set of the (p + 1)-regular tree, and the scattering matrix on the modular domain has the form Y Sv , S= v∈P∪{∞}
the Sv being the scattering matrices on Hv , see [7]. Remark. Motivated by [20] and [51], in [6] Bost and Connes considered the abstract statistical mechanical system whose partition function is the Riemann zeta function. A relation with the Riemann Hypothesis was stated in [8], see also [19]. It is unknown to me whether their (adelic) constructions are directly related to scattering theory and the approach presented here. Now we will give a semiclassical geometrical interpretation of the Faddeev– Pavlov formula in terms of geodesics [25]. 5. The Farey Tesselation ¯ we obtain the When we identify corresponding points on the boundary of F, Riemannian surface Γ \ H which is of finite volume. But the action of the modular group Γ on H is not free: the point i ∈ H is a fixed point of the involutive transfor0 1 2πi/3 2πi/3 ˆ +1) = e +1 for S = −1 1 ∈ SL(2, Z), mation z 7→ −1/z, and similarly S(e ±S ∈ Γ generating an order three subgroup. Instead we will consider the Farey tesselation [50] of H, whose fundamental domain 1 1 , G := z ∈ H | 0 < <(z) < 1, z − > 2 2 is the interior of a geodesic triangle, see Fig. 4.
NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS
R−1
G
1039
R
L LL
0 1
LR
1 3
1 2
2 3
1 1
Fig. 4. The fundamental domain G of the Farey tesselation.
G has three times the volume of F and the M¨ obius transformation Sˆ leads to a cyclic permutation of the three cusps at 0, 1 and ∞. The images of G w.r.t. the matrices in ab ∈ SL(2, Z) | 0 ≤ a ≤ c, 0 ≤ b ≤ d U := {1} ∪ cd form a tesselation of the strip {z ∈ H | 0 ≤ <(z) ≤ 1}, since they map the cusps (0, 1, ∞) of G onto themselves resp. onto b a+b a b a+b a , , with 0 ≤ < < ≤ 1. d c+d c d c+d c On the other hand the sum in the second line of (18) equals X X X cˆ−1−2iω = (c + d)−1−2iω a b cˆ∈N 1≤d≤ˆ ˆ c, (ˆ ˆ c,d)=1 (c d)∈U
(20)
ˆ since there is a unique ac db ∈ U with c = cˆ − dˆ and d = d. ab To every M = c d ∈ U we associate the scattering geodesic cp/q in H coming ˆ (G). This geodesic is from ∞ and going to the middle of the three cusps of M vertical, with real part p/q, where p = a + b and q = c + d, see Fig. 5(a). The first geodesic c1 obtained in this way has real part one. We now compare the length of a geodesic cp/q with the the length of that geodesic. Although both lengths are, of course, infinite, the length differences turn out to be well-defined finite quantities.
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A. KNAUF
0 1
1 1 2 1 3 2 3 4 3 5 2 5 3 4
0 1
1 1
1 2
1 1
Fig. 5. (a) Geodesics related to the Farey tesselation. (b) The Ford circles.
So let in a general setting Φt : X → X be the geodesic flow on the unit tangent bundle X of a Riemannian surface, with distance d. The (un)stable manifold of x ∈ X is given by W ± (x) := y ∈ X | lim d(Φt (y), Φt (x)) = 0 . t→±∞
In the case of the geodesic flow for Γ \ H and x a point on the geodesic flow line corresponding to c1 any scattering geodesics flow line in X (corresponding to a line in H with real part p/q) has unique points y ± ∈ W ± (x). Definition. The time delay T (p/q) of the scattering geodesic cp/q is defined by ΦT (p/q) (y − ) = y + . This definition does not depend on the choice of the point x on the reference geodesic c1 . Theorem. If gcd(p, q) = 1, then the time delay equals T (p/q) = 2 ln(q). Proof. The Ford circle (see, e.g. [3]) in H with center at pq + 2qi 2 and radius 1 −2 touches the cusp at pq . The geodesics that cross that circle perpendicularly 2q converge at the cusp. Thus it is the projection of the stable manifold of a point y + on the geodesic cp/q , and y + projects to p/q + i/q 2 ∈ H. The Ford circle is the image of the horizontal line =(z) = 1 under the M¨ obius ˆ Sˆ−1 (z), M = a b . That horizontal line is the projection of transformation z 7→ M cd the unstable manifold W − (y − ), where the point y − lies on the geodesic cp/q and projects to p/q + i. Equivalently, it is the image of the Ford circle |z − (1 + i/2)| = 12 under the ˆ (z). transformation z 7→ M Its vertical metric distance to that line equals 2 ln(q), see Fig. 5(b). Thus the term −2( 12 +iω)
(c + d)
1 + iω = exp −T (p/q) · 2
NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS
1041
in (20) may be interpreted as a phase shift of a partial wave due to the time delay of T (p/q) = 2 ln(c + d) of the (unit speed) geodesic cp/q relative to the vertical geodesic c1 in H. We thus have related the scattering amplitude S to a classical dynamical zeta function. Unlike conventional dynamical zeta functions here one does not sum over closed orbits but over scattering orbits. Now comes a crucial point: These scattering geodesics connected to the Farey tesselation naturally come in families of 2k members. One may see this by observing that every matrix M ∈ U \ {1} can be uniquely presented as Y 10 11 Xi M = LX with L := , R := and X = 11 01 i being a finite product of matrices Xi = L resp. R (including the case X = 1). ˆ and R ˆ from the right means going to the Geometrically an application of L neighbouring left resp. right Farey triangle which lies between the given one and the real axis, see Fig. 4. ˆL ˆR ˆ L(G) ˆ Example. L is the triangle with the cusps ( 13 , 38 , 25 ). ˆ (G), M ∈ U, are in bijection with the rationals Q ∩ (0, 1]. The middle cusps of M In order to recover the matrix X for a given x ∈ Q ∩ (0, 1], we apply the map f , see Fig. 6, 1 x 0≤x≤ 1 − x 2 f : [0, 1] → [0, 1], x 7→ 1 2 − 1 <x≤1 x 2 until f k (x) = 12 , setting Xi := L if f i−1 (x) < 12 and Xi := R if f i−1 (x) > 12 .c Example. For x := X = LRL.
3 8
< 12 , f (x) =
3 5
> 12 , f 2 (x) =
1 3
< 12 , f 3 (x) = 12 , so that
1 0.8 0.6 0.4 0.2 0 0
0.2
0.4
0.6
0.8
1
Fig. 6. The map f . c One may relate these preimages of 0 and the fixed points of f , using the Minkowski ?-function [38], whose inverse appears in the Appendix of [22].
1042
A. KNAUF
The definition f (x) = x/(1 − x) of f for 0 ≤ x ≤ 12 coincides with the appliˆ −1 to that interval, whereas for 1 < x ≤ 1 we have f (x) = 2 − 1/x = cation of L 2 −1 −1 ˆ S(x) ˆ (the conjugation by S is necessary in order to transform everything Sˆ R to the unit interval). Iterated maps of the interval have been studied in depth. We may consider f as a continuously differentiable map on R/Z, and for 0 < x < 1 it has the expanding property f 0 (x) > 1. However at x = 0 the map is tangent to the diagonal, a fact which leads to particular properties, see [11, 14, 33]. Remark. f is also related to the Gauss map g : [0, 1] → [0, 1] x 7→ 1/x − [1/x] studied by Mayer [35, 36], Series [50], Lanford and Ruedin [28], and others in the context of closed geodesics in the modular domain. One such connection is given by the Lewis functional equation [32] λ · ψ(x) = ψ(x + 1) + x−β ψ(1 + 1/x) , since the free energy (28) of the number-theoretical spin chain (see below) equals F (β) = − ln(λ(β))/β [26]. At this point I would like to mention a relation between Farey fractions and the Riemann zeta function. The set Fn of Farey fractions of order n (see, e.g. [3]) is the Pn set of (reduced) fractions p/q ∈ (0, 1], with q ≤ n (so that |Fn | = k=1 ϕ(k) which is asymptotic to 3n2 /π 2 , see [2]). For example 1 1 1 2 1 3 2 3 4 1 , , , , , , , , , . F5 = 5 4 3 5 2 5 3 4 5 1 For q ∈ N prime the relation X
exp 2πi pq = µ(q)
(q ∈ N)
(21)
0
gcd(p,q)=1
(µ being the M¨ obius function) follows, since the qth roots of unity have sum 0, and only the root 1 is missing. For arbitrary q one shows (21) by the inclusion-exclusion principle. From (21) we conclude X x∈Fn
exp(2πix) = M (n) ≡
X
µ(q) .
q≤n
Thus setting N := |F (n)| and denoting by xl ∈ Fn the lth element in ascending order, by (11) the Riemann Hypothesis follows if N X l = O(n1/2+ε ) , exp(2πixl ) − exp 2πi M (n) = N l=1
NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS
1043
thus in particular if the Farey fractions xl of order n have a small mean distance N X xl − l = O n1/2+ε N l=1
to the points l/N and are thus quite equidistributed in (0, 1] (Franel and Landau, see [12]). The M ∈ U with M = 1 or of the form M =L
l Y
Xi
i=1
with 0 ≤ l < k lead to cusps 0 < x1 < · · · < x2k = 1 which make up the set Fk of so-called modified Farey fractions of order k, with Fk ⊃ Fk . For example 1 1 2 1 3 2 3 1 , , , , , , , . F3 = 4 3 5 2 5 3 4 1 We now concentrate on the denominators of Fk . This leads us to classical statistical mechanics. 6. The Number-Theoretical Spin Chain Since |Fk | = 2k , it is natural to enumerate the denominators by elements of the additive group Gk := (Z/2Z)k ,
with
Z/2Z = ({0, 1}, +) .
We then inductively set h0 := 1,
hk+1 (σ, 0) := hk (σ)
and hk+1 (σ, 1) := hk (σ) + hk (1 − σ) ,
(22)
where σ = (σ1 , . . . , σk ) ∈ Gk and 1 − σ := (1 − σ1 , . . . , 1 − σk ) is the inverted configuration. Writing the hk (σ) in the row number k using the lexicographic order of the σ ∈ {0, 1}k , we obtain what could be called Pascal’s triangle with memory, see Fig. 7. Like in the usual Pascal triangle one writes the sum of neighbouring integers in row no. k into the next row. But in addition one also copies the integers from row no. k to the (k + 1)st row. Notice that these sequences hk (σ) of integers coincide with the denominators of the modified Farey sequence Fk , except that 1 now has the zeroth instead the 2kth position.
k k k k k
=0 =1 =2 =3 =4
1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1547385727583745
Fig. 7. Pascal’s triangle with memory.
1
1
1
1
1
1044
A. KNAUF
8 7 6 5 4 3 2 1 0
5000
10000 15000 20000 25000 30000
Fig. 8. Graph of the energy function Hk , k = 15, in lexicographic ordering of σ ∈ Gk .
We now formally interpret σ ∈ Gk as a configuration of a spin chain with k spins and energy function Hk := ln(hk ) , see Fig. 8. Thus we may interpret X
Zk (s) :=
exp(−s · Hk (σ))
σ∈Gk
as the partition function of that finite spin chain for inverse temperature s. The quotient ∞ ζ(s − 1) X ≡ ϕ(n)n−s (23) Z(s) := ζ(s) n=1 appearing in the scattering matrix (17) is simply the thermodynamic limit lim Zk (s) = Z(s) (<(s) > 2)
k→∞
(24)
P∞ −s , since 0 ≤ ϕk (n) ≤ ϕ(n) and of partition functions Zk (s) = n=1 ϕk (n)n ϕk (n) = ϕ(n) if n ≤ k + 1. That number-theoretical spin chain was introduced in [22]. The Gibbs measure for inverse temperature β ∈ R assigns probabilities σ 7→
exp(−βHk (σ)) Zk (β)
(σ ∈ Gk )
(25)
to the configurations of the spin chain. We denote the expectation of a random variable by hf ik (β) :=
X σ∈Gk
f (σ)
exp(−βHk (σ)) Zk (β)
(f : Gk → R) .
To show that the analogy with statistical mechanics is not only formal, the Fourier coefficients of Hk were estimated in [22]. One notes that the dual group
NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS
1045
G∗k of Gk is naturally isomorphic to Gk , since the characters on Gk can be written in the form Pk χt : Gk → {−1, 1}, χt (σ) := (−1) l=1 σi ti (t ∈ G∗k ) . The Fourier coefficients jk (t) := −2−k
X
Hk (σ) · χt (σ)
(t ∈ G∗k )
σ∈Gk
of −Hk are called interaction coefficients in the statistical mechanics terminology, and X jk (t) · χσ (t) . Hk (σ) = − t∈G∗ k
The negative mean jk (0) of Hk has special properties. In the thermodynamic limit it is asymptotic to jk (0) ∼ −c · k for some c > 0 [23], but it is the only coefficient whose value does not affect the Gibbs probability measure (25). When we write t ≡ (t1 , . . . , tk ) ∈ G∗k \ {0} in the form t = (0, . . . , 0, 1, tl+1 , . . . , tr−1 , 1, 0, . . . , 0) , s := r − l will be called the size of t, and d := min(l, k + 1 − r) its distance from Pk the ends of the chain at 1 and k. Finally we say that t is even (odd ) if i=1 ti is even (odd ). With these notations the following estimates were shown. Theorem [22]. 1. The even interactions decay exponentially in the size: jk (t) < 2−s
(t ∈ G∗k \ {0} even) ,
(26)
whereas in the odd case one even has jk (t) < 2−(k−l)
(t ∈ G∗k odd) .
(27)
So odd interactions are small in comparison to the even ones except near the right end of the chain. 2. The interaction is asymptotically translation invariant in the sense that, up to a relative error which is exponentially small in the distance from the ends, the interactions only depend on the relative positions of the spins involved:d 0 ≤ (jk+1 (0, t) − jk+1 (t, 0)) · 2s < C · 2−d
(t ∈ G∗k ) .
3. The interaction has a thermodynamic limit in the sense 0 ≤ (jk+1 (0, t) − jk (t)) · 2s < C · 2−d
(t ∈ G∗k \ {0}) .e
d For a related spin chain with an exactly translation invariant interaction see [21] and [8a]. e C = 5 covers both cases.
1046
A. KNAUF
For β > 0 the thermodynamic limit F (β) := lim Fk (β) k→∞
with Fk (β) := −
1 ln (Zk (β)) β·k
(28)
of the free energy per spin exists [24]. 4. The interaction is ferromagnetic, that is, jk (t) ≥ 0 5. The effective interaction X Ak (l, r) :=
(t ∈ G∗k \ {0}) .
(29)
jk (0, . . . , 0, 1, tl+1 , . . . , tr−1 , 1, 0, . . . , 0)
t0 ∈G∗ r−l−1
between spins at positions l and r decays quadratically with their distance s = r − l in the bulk : 1 2−(k−r) . (30) Ak (l, r) ≤ 2 + s s Remarks. • Properties (1)–(3) are typical for many systems of statistical mechanics, and it should not be astonishing to find them in an application of the thermodynamic formalism. Since for <(s) > 2 the partition function Z of the infinite chain is finite, it is clear that F (β) = 0 in the low temperature range β > 2. The free energy is important in statistical mechanics, since many physical quantities like the internal energy per spin U (β) := lim Uk (β) k→∞
with
Uk (β) :=
1 d hHk ik (β) = (βFk (β)) k dβ
can be calculated from it. Points of non-analyticity of F are called phase transitions. • The ferromagnetic property (4), however, comes as a surprise. There is no a priori reason why the Fourier transform of −Hk should be positive (except for the zeroth coefficient). I consider the ferromagnetic property as the main motivation for studying the ferromagnetic spin chain and return to that point in the next section. • By (26) the individual coefficients decay exponentially w.r.t. the distance s of the spins involved in the interaction, but the number of such components increases exponentially, with the same rate. So (5) sharpens (26) and (27) in the mean, and since that estimate is somewhat sharp, many-body interactions play an important role. In statistical mechanics it is known [44] that spin chains with a decay rate s−α of the effective interaction show no phase transition if α > 2. Since the number-theoretical spin chain has a phase transition [23], we are in the borderline situation.
NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS
1047
Switching to a multiplicative representation si (σ) := (−1)σi of the ith spin, an important variable is the mean magnetization per spin k 1X Mk (β) := hsi ik (β) k i=1
and its thermodynamic limit M (β). By analyzing a Perron–Frobenius operator with PF eigenvalue exp(−β · F (β)), the following statements were proved. Theorem [9]. The only phase transition of the number-theoretical spin chain occurs for inverse temperature βcr := 2. For lower temperatures F (β) = U (β) = 0 and M (β) = 1
(β > βcr ) ,
whereas for high temperatures U (β) ≥
1 (β − βcr ) (1 < β < βcr ) and M (β) = 0 4
(0 ≤ β < βcr ) .
Remarks. (1) Thus for low temperatures β > βcr , due to the long range of the interaction, the chain is in a frozen state, and with probability one only finitely many spins equal one, being located near the (left) end of the infinite chain. The ferromagnetic property together with the GKS inequality (31) below abstractly implies that M (β) ≥ 0 (if the thermodynamic limit exists). So the jump from M (β) = 1 to M (β) = 0 is the sharpest possible, see Fig. 9 The phase transition is at most of second order, since the derivative of U is discontinuous at βcr . (2) Amongst other things in [6] a, transition from a type I∞ state for β > 1 to a type III factor for β ≤ 1 was shown for the KM Sβ states related to the Riemann zeta functions. In the language of C ∗ algebras, this corresponds to a (a)
(b)
111111111101100010001111101110110000100001111111110111000000 000000000000000000000000000000000011000000100101100000111111 001111101110000110100011111111111111111111111111111111111111 111111111111111111011111111111110111101110011111010000000110 111000000000000000000000000010010011000000000000000000000000 000000000011100011111010000111110111111111111111111000110111 111100000011100011111111111100000100100101111110010111111110 101110011111111110101101100000010101101111111111111101111111 111111111111111111111111111000010111100010000000000000000111 111110001111111101000000011111110111001111110000000000000000 110100010000011100000010001001111010000101110100010001111110 101101001101111111111000111111111111101000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000
Fig. 9. Typical configurations of k = 600 spins, for inverse temperature (a) β = 1.95 and (b) β = 2.05 near phase transition at βcr := 2.
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A. KNAUF
similar phase transition as the one described above. Observe that in our case the partition function equals ζ(β − 1)/ζ(β) so that the location of the phase transition is shifted by one. 7. Polymer Models and Ferromagnetism Although there is no a priori reason for the positivity of the interaction, in a sense it did not come as a surprise. Instead, Kac (see his Comments in P´olya [43], pp. 424–426), Newman [40], Ruelle [46] and others had conjectured the existence of a ferromagnetic spin system related to the Riemann zeta function. One motivation for that conjecture has been the Lee–Yang circle theorem of statistical mechanics. In its basic form it states that all zeroes of the partition function Y Y X exp(h|X|) axy Z(h) := X⊂Λ
x∈X y∈Λ−X
of a ferromagnetic (axy = ayx ∈ [−1, 1]) Ising model occur at imaginary values of the external magnetic field h (see [47] for a proof). Here we are interested in the zeroes of a partition function Z(s) in the complex s plane, s being the inverse temperature. There exist variants of the Lee–Yang circle theorem which predict zero-free half-planes of the inverse temperature for certain ferromagnets, see, e.g. Ruelle [45]. Unfortunately, these theorems do not apply to our situation since our spin chain includes many-body interactions. It is known that the Lee–Yang theorem does not hold for general ferromagnetic systems with many-body interactions. Here, however, the interaction coefficients are of a rather special nature, as the proof of ferromagnetism (see below) will elucidate. The proof of a first version of the circle theorem by Kac had been based on a technique applied by P´olya in [43]. There he took the asymptotics of the Fourier transform of ξ (with ξ(iz) = 12 (z 2 − 1/4)ζA ( 12 + z)), and proved the Riemann Hypothesis for the inverse Fourier transform (which he called “verf¨ alschte ξ-Funktion”) of that function. A second (admittedly vague) motivation to believe in the importance of the ferromagnetic property might be that positivity of a certain involution plays a central role in A. Weil’s proof of the Riemann Hypothesis for curves over finite fields [41]. Thirdly there are correlation inequalities valid for expectations of ferromagnetic spin systems. As an example, in our additive notation the GKS inequalities [16] state that for β ≥ 0 the expectations hχt ik (β) ≥ 0
(t ∈ G∗k )
(31)
and hχt1 χt2 ik (β) ≥ hχt1 ik (β) · hχt2 ik (β)
(t1 , t2 ∈ G∗k ) .
Thus much stronger information is available for ferromagnetic than for general spin systems.
1049
NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS
After a first proof in [22] of the ferromagnetic property, a second, more conceptional proof in [18] was based on the polymer model technique. I shortly introduce polymer models and sketch the proof of ferromagnetism. In an abstract setting one starts with a finite (or denumerable) set P whose elements are called polymers. Two given polymers γ1 , γ2 ∈ P may or may not overlap (be incompatible). Incompatibility is assumed to be a reflexive and symmetric relation on P . Thus one may associate to a k-polymer X := (γ1 , . . . , γk ) ∈ P k an undirected graph G(X) = (V (X), E(X)) with vertex set V (X) := {1, . . . , k}, vertices i 6= j being connected by the edge {γi , γj } ∈ E(X) if γi and γj are incompatible. Accordingly the k-polymer X is called connected if G(X) is path-connected and disconnected if it has no edges (E(X) = ∅). The corresponding subsets of P k are called C k resp. Dk , with D0 := P 0 := {∅} S∞ consisting of a single element. Moreover P ∞ := k=0 P k with the subsets D∞ := S S∞ k ∞ k k := ∞ k=0 D and C k=1 C . We write |X| := k if X ∈ P . Statistical weights or activities z : P → C of the polymers are multiplied to give Qk the activities z X := i=1 z(γi ) of k-polymers. Definition. A system of statistical mechanics is called polymer model if its partition function Z has the form X
Z=
X∈D ∞
zX . |X|!
(32)
Then, up to the factor −1/(β|Λ|), the free energy is given by Theorem [15]. ln(Z) =
X n(X) zX , |X|! ∞
(33)
X∈C
with n(X) := n+ (X) − n− (X), n± (X) being the number of subgraphs of G(X) connecting the vertices of G(X) with an even resp. odd number of edges. For example, n+ = 1 and n− = 4 if the graph G is a quadrangle. Example. We consider the 2-dim. nearest neighbour Ising model with energy function HΛ : {±1}Λ → R over a finite region Λ ⊂ Z2 (say, Λ := {0, . . . , L} × {0, . . . , L} ⊂ Z × Z) 1 X HΛ (s) := − si sj . (34) 2 i,j∈Λ |i−j|1 =1
Then (neglecting boundary effects) in a low-temperature expansion 1 1 exp(−2β|Λ|) · ZΛ (β) = 2 2
X s∈{±1}Λ
exp −
β 2
X
(si − sj )2 /2
i,j∈Λ,|i−j|1 =1
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A. KNAUF
P with partition function ZΛ (β) = s exp(−βHΛ (s)) can be written in the form (32) where the polymers are the loops encircling regions of positive spin.f The activity equals z(γ) := exp(−2β|γ|) , (35) where |γ| is the length (number of edges) of γ. In fact the low-temperature expansion is the first step in one method of solving (finding an explicit expression for the free energy) the 2-dim. Ising model. There are several ways in which Dirichlet series can be related to polymer models [10]. However, a specific relation is useful to analyze the interaction of the ferromagnetic spin chain [18]: Proof of the ferromagnetic property (29). In our case the energy function Hk = log(hk ) is the logarithm of a function (defined in (22)), whose Fourier coefficients X Jk (t) := 2−k hk (σ) · χt (σ) σ∈Gk
can be easily calculated: The set Pk := {pl,r , pl ∈ G∗k | 1 ≤ l, r ≤ k; l < r} of polymers in G∗k is given by pl,r (i) := δl (i) + δr (i),
pl (i) := δl (i) (i = 1, . . . , k) .
(36)
We define their support by supp(pl,r ) := {i | l ≤ i ≤ r} for the even polymers and supp(pl ) := {i | l ≤ i ≤ k} for the odd ones. Two polymers γ1 , γ2 ∈ Pk are called overlapping or incompatible if supp(γ1 ) ∩ supp(γ2 ) 6= ∅. So in particular all the odd polymers pl , 1 ≤ l ≤ k, are mutually incompatible. Now an arbitrary group element t = (t1 , . . . , tk ) ∈ G∗k can be uniquely decomposed into the sum of n := [ 12 (|t| + 1)] compatible polymers: n X pli ,ri t=
i=1 n−1 X
, |t| even (37)
pli ,ri + pln , |t| odd .
i=1
whose indices li , ri may be assumed to be increasing in i and are determined by tli = tri = 1. As an example (0, 1, 0, 1, 1, 0) = p2,4 + p5 ∈ G∗6 . Pn We write this decomposition in the short form t = i=1 γi and set the activity of a polymer γ ∈ G∗k equal to z(γ) := −3−|supp(γ)| . f Technically, one takes all loops which have no self-crossing and when meeting a vertex v ∈ Λ from all sides, do not cross the main diagonal through v. Then two such polymers γ1 , γ2 ∈ P are incompatible if the loops contain a common edge, they cross, or meet at a vertex crossing the main diagonal.
NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS
Then Jk (t) = (3/2)k
n Y
1051
z(γi ) ,
i=1
so that the interaction coefficients jk (t) can be written as X Hk (σ)χt (σ) jk (t) = −2−k σ∈Gk
= −2
−k
X
ln
σ∈Gk
X
Jk (s) · χσ (s) χt (σ)
s∈G∗ k
= −δt,0 · k ln(3/2) − 2−k
X σ∈Gk
ln
X
∞ X∈Dk
z˜σX · χt (σ) , |X|!
(38)
where the redefined single-polymer activities z˜σ (γ), γ ∈ Pk are given by z˜σ (γ) := z(γ) · χσ (γ) = −|z(γ)| · (−1)σ·γ . For all σ ∈ Gk the terms in the sum (38) are logarithms of expressions of the form (32), and we can use formula (33). One notes in addition that the relevant graphs G are interval graphs,g so that [18] ( 0 G not connected . sign(n(G)) = |V | G connected −(−1) Thus for t 6= 0 we only have positive contributions in (38), proving the ferromagnetic property. Note that not the free energy of the chain but its interaction coefficients were considered as abstract polymer models. One may try to prove the Lee–Yang theorem for the number-theoretical spin chain, since its many-body interactions have that special combinatorial structure. Most systems of statistical mechanics cannot be non-trivially presented as polymer models. Sometimes, however, a system can be presented in more than one way. Example. In the high temperature expansion one uses the identity e±β = cosh(β) · (1 ± tanh(β)) in order to write the partition function ZΛ of the 2-dim. nearest neighbour Ising model (34) in the form ZΛ (β) = (cosh2 (β))|Λ|
X
L−1 Y
(1 + xsi,k · si,k+1 ) · (1 + xsi,k · si+1,k )
s∈{±1}Λ i,k=0 g A graph G = (V, E) is called interval graph if there exists an isomorphism I : V → V ¯ between the set V of vertices and a set V¯ of intervals in R such that vi , vj ∈ V are connected by the edge {vi , vj } ∈ E precisely if the corresponding intervals I(vi ) and I(vj ) have non-empty intersection.
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A. KNAUF
with x := tanh(β). When the summation over s is performed, one obtains X x|G| , ZΛ (β) = (2 cosh2 (β))|Λ| · G∈Γ
where Γ consists of the graphs G with vertex set Λ which have an even number of edges at every vertex, and |G| denotes the number of edges. Similar to the low-temperature expansion above, we can define the polymers to be the loop graphs γ, with activities z(γ) := (tanh β)|γ| ,
(39)
and represent the graphs G ∈ Γ uniquely as disconnected multipolymers. The Kramers–Wannier duality of the Ising model relates the partition function at temperatures β and 1 β ∗ := − ln(tanh β) . 2 ∗ The map β 7→ β is involutive, interchanges the activities (35) and (39), and thus leads to the functional equation Z(β) =
1 (2 cosh(β) sinh(β))|Λ| Z(β ∗ ) . 2
Its fixed point is the point of phase transition. In our language, it corresponds to an isomorphism of the polymer models for the low- and high temperature expansions. It is know since long [37] that, similar to (6), it can be understood on the basis of Poisson summation. One open problem consists in an interpretation of the functional equation (6) of the Riemann zeta function in terms of a statistical mechanics duality. 8. Markov Chains and Ramanujan Graphs The pole of ζ at s = 2 gives rise to a phase transition and leads to a divergence of the Dirichlet series (19) for smaller real parts. If we seek for a convergent Dirichlet series providing us information about ζ inside the critical strip, we must “add weight with signs”, instead of just “adding weights”, much as in the case of the Dirichlet series (10) for 1/ζ. Instead of considering 1/Z, let us change signs in the Euler product representation Y 1 − p−s (<(s) > 2) . Z(s) = 1 − p1−s p∈P
Then the “twisted” partition function ˜ Z(s) :=
Y 1 + p−s ζ(s) · ζ(2(s − 1)) = 1 + p1−s ζ(s − 1) · ζ(2s)
(40)
p∈P
has the Dirichlet series ˜ Z(s) =
∞ X n=1
λ(n) · ϕ(n) · n−s
(<(s) > 2) ,
(41)
NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS
1053
Q and the Liouville function λ( p∈P pαp ) := (−1)Σp αp resembles the µ function in the Dirichlet series of 1/ζ. By (40) the pole of Z at s = 2 is converted into a zero ˜ which in turn has a pole at 3/2. The non-trivial zeroes s of ζ generate poles of Z, of Z˜ at s + 1. In [25] it was observed numerically that X λ(hk (σ)) · exp(−s · Hk (σ)) Z˜k (s) := σ∈Gk
˜ approximates Z(s) very well in the half-plane <(s) > 3/2, in fact much better than the truncations of the Dirichlet series (41). In this half plane only the term ζ(s − 1) in the product (40) has no convergent Dirichlet series. Thus a proof of convergence Z˜k → Z˜ for that half-plane would imply the Riemann Hypothesis. Applying the heuristic random walk analogy of Sec. 2, we expect that in the thermodynamic limit the expectation hλik = O(2k/2+ε )
(ε > 0) ,
(42)
since the cardinality of our ensemble equals |Gk | = 2k . Here we use the notation X hf ik := hf ◦ hk ik (0) = 2−k f ◦ hk (σ) σ∈Gk
for expectations of arithmetic functions f : N → R on the ensemble of values hk (σ), σ ∈ Gk . The conceptional advantage of the ensemble hk (σ), σ ∈ Gk , of integers over the ensemble {1, . . . , k} used in Sec. 2 lies in the relation with the ensemble hk+1 (τ ), τ ∈ Gk+1 , imposed by the definition (22) of hk . Setting for m ∈ N the function χm : N → {0, 1} equal to χm (n) := 1 if m divides n and zero otherwise, we have λ=
∞ ∞ YY YY (−1)χpl = (1 − 2χpl ) . p∈P l=1
p∈P l=1
Furthermore for m1 , m2 ∈ N one has χm1 · χm2 = χm
with
m :=
m 1 m2 , gcd(m1 , m2 )
(43)
so that instead of estimating expectations (42), we may formally consider sums of expectations hχm ik . Now this leads us directly to the consideration of a Markov chain with state space Ωm := (Z/mZ) × (Z/mZ) . ˜ ∈ Vm in the Hilbert space Namely χm descends to the function χ X f¯(ω)g(ω) , Vm := {f : Ωm → C} with inner product (f, g) := ω∈Ωm
1054
A. KNAUF
χ(a, ˜ b) := δa,0 ,
((a, b) ∈ Ωm ) .
Setting the initial probability vector vm,0 ∈ Vm of the Markov chain equal to vm,0 (a, b) := δa,1 · δb,1
((a, b) ∈ Ωm ) ,
and defining its transition matrix Tm : Vm → Vm by 1 f ◦ L−1 + f ◦ R−1 2
TM f := with L =
10 11
and R =
11 01
(f ∈ Vm ) ,
, we have ˜m , vm,k ) hχm ik = (χ
(44)
for the vector vm,k := Tm vm,k−1
(k ∈ N)
(45)
of probabilities at time k. vm,k (a, b) can be interpreted as the normalized frequency of occurrence of the neighbouring pair (a,b) in the Pascal triangle with memory modulo m. For example, v3,4 (0, 2) = 1/16, since the pair (0, 2) appears only once in the fourth line of Fig. 10. The thermodynamic limit k → ∞ leads to the unique equilibrium state of the Markov chain, and the limit expectations are calculated as −1 Y (1 + 1/p) . (46) hχm i∞ := lim hχm ik = m k→∞
p∈P,p|m
We are interested in the deviations hχm ik − hχm i∞ from the thermodynamic limit. If the χm ◦ hk (σ), σ ∈ Gk were i.i.d. random variables, we would obtain hχm ik − hχm i∞ = O(2−k/2+ε ). Instead, by using the relation k ˜m , Tm (vm,0 ) − vm,∞ ) , hχm ik − hχm i∞ = (χ
(47)
we see that the exponential decay rate of that deviation equals the spectral radius of the transition matrix Tm (omitting √ the trivial PF eigenvalue 1). So one guesses that this spectral radius equals 1/ 2. The actual spectral radius turns out to be larger, but the Markov chain (Ωm , Tm , vm,0 ) carries information about the frequency of pairs (a, b) ∈ Ωm of integers (mod m), whereas we are only interested in divisibility of a. So we do not use
k k k k k
=0 =1 =2 =3 =4
1 1 2 1 0 2 0 1 1 0 2 2 2 0 1 1211022121220112
1
1
1
1
1
Fig. 10. Pascal’s triangle with memory, modulo m = 3.
NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS
1055
the full information encoded in the transition matrix Tn and may therefore reduce it. Assuming for simplicity of the presentation that m is a prime, we observe that ˜m is the Ωm is a two-dimensional vector space over the field Z/mZ, and that χ characteristic function of a line in that vector space. So we consider the reduced transition matrix T¯m on the Hilbert space over the projective space P 1 (Z/mZ) on which L and R act by M¨ obius transformations z 7→ z/(z + 1) resp. z 7→ z + 1. Example. In the basis corresponding to the representation (1, . . . , m, ∞) of P 1 (Z/mZ) we have for m = 3, 01 0 0 L= 0 0 10
00 0 1 , 1 0 00
so that
0 0 R= 1 0
02 1 00 ¯ T3 = 2 10 10
10 01 00 00
0 0 , 0 1
00 1 1 . 1 0 01
(48)
¯ m (for the projector Π ¯ m onto We could not show that the spectral radius of T¯m − Π √ the Perron–Frobenius eigenvector vm,∞ of T¯m ) equals 1/ 2. Instead, we proved the following: Theorem [26]. There is a B < 1 so that for all m ∈ N, • All real eigenvalues e ∈ R \ {1} of T¯m have modulus |e| ≤ B. √ • All non-real eigenvalues e ∈ C \ R of T¯m have modulus |e| = 1/ 2. In particular the spectral radius of T¯m on the ortho-complement of the Perron– Frobenius eigenspace is bounded from above by B. Remark. The important point here is the independence of the bound B from m. From (47) we see that the deviation of hχm ik from its thermodynamic limit (46) is of the form hχm ik − hχm i∞ = O(B k ) .
(49)
This not only implies small fluctuations for the proportion of those σ ∈ Gk for which hk (σ) is divisible by a given integer m. In addition, by (43) and (46) these divisibility properties are only weakly correlated in the sense h(χm1 − hχm1 ik ) · (χm2 − hχm2 ik )ik = O(B k ) for relatively prime integers m1 , m2 .
1056
A. KNAUF
Eigenvalues, m 229 1 0.75 0.5 0.25 -1 -0.75-0.5-0.25
0.25 0.5 0.75 1
-0.25 -0.5 -0.75 -1
Fig. 11. The spectrum of the reduced Markov transition matrix T¯m for the 50th prime m = 229.
The proof of the theorem is based on the identity −1 2 2 ) = 9Bm , (2T¯m + T¯m
(50)
where Bm is a self-adjoint operator. √ (50) directly implies that the non-real spectrum is located on a circle of radius 1/ 2. Further, if e is an eigenvalue of Bm , then 3(1−e) is an eigenvalue of the Laplacian of a certain three-regular graph Gm , which is constructed as follows. The matrices M± ∈ SL(2, Z/mZ) M+ :=
−1 −1 1
0
! = −L
−1
R
and M− :=
−1 1 −1 0
! = −LR−1
(51)
act by left transformations on the group SL(2, Z/mZ). For m > 2 the orbits of these actions are of size three, and the M+ -orbit and the M− -orbit through g ∈ SL(2, Z/mZ) have only g in common. Definition. We denote by V+ (V− ) the set of M+ (M− )-orbits and consider V := V+ ∪ V− as the vertex set of an undirected graph Gm = (V, E). A pair {v+ , v− }, v± ∈ V of vertices belongs to the set E of edges iff v+ ∈ V+ , v− ∈ V− and the orbits v+ and v− contain a common group element g ∈ SL(2, Z/mZ). Figure 12 shows such a graph. A distinctive feature is its large girth (length of the shortest closed circuit).
NUMBER THEORY, DYNAMICAL SYSTEMS AND STATISTICAL MECHANICS
1057
Fig. 12. The graph G3 for the group SL(2, Z/3Z).
The proof of the second statement in the above theorem is based on consideration of these graphs, using known properties of the so-called Fell topology on the set of (equivalence classes of irreducible unitary) representations of SL(2, Z), see [34]. Definition. A r-regular graph, i.e. one with r edges at every vertex, is called Ramanujan if the non-trivial spectrum σ(∆)\{0, 2r} of its Laplacian ∆ is contained in the spectrum √ √ r − 2 r − 1, r + 2 r − 1 of the r-regular tree. It may be shown that there is no family of finite r-regular graphs with diverging size meeting a better bound, so that Ramanujan graphs are in a sense optimal, see [34, 49, 53]. Theorem [26]. If the graph Gm has the Ramanujan property, then √ the spectral radius of T¯m on the complement of the PF eigenspace equals B = 1/ 2. This would then indeed imply a decay of correlations (49) like in the case of an ensemble of i.i.d. random variables. 9. Final Remarks We have seen how questions of number theory are related to geometrical and dynamical questions, and how these questions are reformulated using equilibrium statistical mechanics. The general impression is that many different-looking concepts are in fact closely related. While this may be beautiful mathematics, one may ask whether it only leads to a vicious circle or it helps to answer old questions.
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The most prominent such question is of course the one about the truth of the Riemann Hypothesis. Here one should be pessimistic, seen the long history of unsuccessful attempts. On the other hand, as a general rule, these attempts, although missing their direct aim, led to stimulating developments in different fields. Applied to the present context, one is motivated to consider anew the Lee–Yang theorem, to ask about the significance of positive interaction function of a dynamical systems, cohomological questions of geodesic motion, to mention just a few aspects. Moreover there are of course many simpler number-theoretical questions which one may try to answer using methods from dynamical systems and statistical mechanics. Acknowledgements I thank G. Rudolph and E. Zeidler for the opportunity to give this series of lectures in the “Oberseminar Mathematische Physik”. References [1] M. Aigner, Combinatorial Theory, New York, Springer, 1979. [2] T. M. Apostol, Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, New York, Springer, 1976. [3] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Graduate Texts in Mathematics 41, New York, Springer, 1990. [4] V. Baladi, “A brief introduction to dynamical zeta functions”, in Classical Nonintegrability, Quantum Chaos. eds. A. Knauf, Ya. Sinai, DMV Seminar, Vol. 27. Basel, Birkh¨ auser, 1997. [5] M. Berry and J. P. Keating, “H = xp and the Riemann Zeros”, to appear in Supersymmetry and Trace Formulae: Chaos and Disorder, eds. J. P. Keating and I. V. Lerner, Plenum, 1998. [6] J.-B. Bost and A. Connes, “Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory”, Selecta Mathematica, New Series, 1 (3) (1995), 411–457. [7] L. Brekke and P. Freund, “p-adic numbers in physics”, Physics Reports 233 (1993) 1–66. [8] A. Connes, “Formule de trace en g´eom´etrie non-commutative et hypoth`ese de Riemann”, C. R. Acad. Sci. Paris, Ser. I 323 (1996) 1231–1236. [8a] P. Contucci, P. Kleban and A. Knauf, “A fully magnetizing phase transition,” to appear in J. Statistical Phys. (1999). [9] P. Contucci and A. Knauf, “The phase transition of the number-theoretical spin chain”, Forum Math. 9 (1997) 547–567. [10] P. Contucci and A. Knauf, “The low activity phase of some dirichlet series”, J. Math. Phys. 37 (1996) 5458–5475. [11] P. Cvitanovi´c, “Circle maps: irrationally winding”, in From Number Theory to Physics, eds. M. Waldschmidt, P. Moussa, J. M. Luck, C. Itzykson, Berlin, Heidelberg, New York, Springer, 1992. [12] H. M. Edwards, Riemann’s Zeta Function, New York, Academic Press, 1974. [13] L. D. Faddeev and B. S. Pavlov, “Scattering theory and automorphic functions”, Proc. Steklov Inst. Math. 27 (1972) 161–193; English translation: J. Sov. Math. 3 (1975) 522–548. [14] M. Feigenbaum, I. Procaccia and T. T´el, “Scaling properties of multifractals as an eigenvalue problem”, Phys. Rev. A39 (1989) 5359–5372.
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[39] W. M¨ uller, “Spektraltheorie automorpher Formen”, in Forschungsbericht 8, Akademie der Wissenschaften zu Berlin, Berlin, New York, de Gruyter 1993. [40] Ch. Newman, “Gaussian correlation inequalities for ferromagnets”, Z. Wahrscheinlichkeitstheorie verw. Gebiete 33 (1975) 75–93. [41] S. J. Patterson, The Theory of the Riemann Zeta-function, Cambridge, Cambridge Univ. Press, 1995. [42] M. Pollicott, “Closed geodesics and zeta functions”, in Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, eds. T. Bedford, M. Keane and C. Series, Oxford, Oxford Univ, Press 1992. [43] G. P´ olya, Collected Papers, Vol. II: Locations of Zeros, ed. R. P. Boas, Cambridge, M.I.T. Press, 1974. [44] D. Ruelle, “Statistical mechanics of a one-dimensional lattice gas”, Commun. Math. Phys. 9 (1968) 267–278. [45] D. Ruelle, “Some remarks on the location of zeroes of the partition function for lattice systems”, Commun. Math. Phys. 31 (1973) 265–277. [46] D. Ruelle, “Is our Mathematics natural? The case of equilibrium statistical mechanics”, Bull. AMS 19 (1988) 259–268. [47] D. Ruelle, Statistical Mechanics: Rigorous Results, Addison-Wesley, Redwood City, 1989. [48] D. Ruelle, “Dynamical Zeta functions: Where do they come from and what are they good for?” in Mathematical Physics X, ed. K. Schm¨ udgen, Berlin, Springer, 1991. [49] P. Sarnak, Some Applications of Modular Forms, Cambridge, New York, Cambridge Univ. Press, 1990. [50] C. Series, “The modular surface and continued fractions”, J. London Math. Soc. 31 (1985) 69–80. [51] G. Spector, “Supersymmetry and the M¨ obius inversion function”, Commun. Math. Phys. 127 (1990) 239–252 (1990). [52] E. C. Titchmarsh, The Theory of the Riemann Zeta Function, London, Oxford Univ. Press, 1967. [53] A. Venkov and A. Nikitin, “Selberg trace formula, Ramanujan graphs and some problems of mathematical physics”, Algebra Anal. 5 (1993) 1–76. [54] B. van der Waerden, Algebra, Berlin, Springer, 1993. [55] D. Zagier, “The first 50 million prime numbers”, Math. Intell. 0 (1977) 7–19. [56] D. Zagier, Zetafunktionen und quadratische K¨ orper, Berlin, Springer, 1981.
ON THE SPECTRAL PROPERTIES ¨ OF DISCRETE SCHRODINGER OPERATORS: THE MULTI-DIMENSIONAL CASE ANNE BOUTET DE MONVEL and JAOUAD SAHBANI Institut de Math´ ematiques de Jussieu, CNRS UMR 7586 Physique math´ ematique et G´ eom´ etrie Universit´ e Paris 7-Denis Diderot U.F.R. de Math´ ematiques, case 7012 Tour 45–55, 5-` eme ´ etage 2, place Jussieu, 75251 Paris Cedex 05, France E-mail : [email protected] E-mail : [email protected] Received 23 February 1998 We use the method of the conjugate operator to prove the limiting absorption principle and the absence of the singular continuous spectrum for the discrete Schr¨ odinger operator. We also obtain local decay estimates. Our results apply to a large class of perturbating potentials V tending arbitrarily slowly to zero at infinity.
1. Introduction Our purpose in this work is to study the absolute continuity of the spectrum of the discrete Schr¨ odinger operators. We also investigate the propagation properties of these operators. Our method works for a large class of arbitrarily slowly decaying potentials V that will be explicitly described in our theorems. Our study is based on the method of the conjugate operator. This theory shows, in an abstract frame, that a Hamiltonian H has nice spectral and propagation properties if it has a conjugate operator A, i.e. a self-adjoint operator A such that the commutator [H, iA] is strictly positive in a convenient sense (see [1–3, 9] and references therein). This theory was used efficiently for the spectral and scattering theory of (pseudo) differential operators (see [1, 8] and references therein). It is natural to apply it to the study of discrete operators. The configuration space is the multidimensional lattice Zν for some integer ν > 0. For a multi-index α = (α1 , . . . , αν ) ∈ Zν we set |α|2 = α21 + · · · + α2ν . Let us consider the Hilbert space H = `2 (Zν ) of square integrable sequences ψ = (ψ(α))α∈Zν . We are interested here in the discrete Schr¨ odinger operators H = H0 +V acting in H, where H0 is the finite difference
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operator defined by (H0 ψ)(α) = −
X
ψ(β) ,
∀ ψ ∈ H,
β∈Zν |β−α|=1
and V is the multiplication operator by a real valued sequence (V (α))α∈Zν : (V ψ)(α) = V (α)ψ(α) . Using a Fourier transform, one sees that H0 is a bounded self-adjoint operator in H, and that its spectrum is purely absolutely continuous: σac (H0 ) = [−2ν, 2ν] ; (1.1) σs (H0 ) = σsc (H0 ) ∪ σpp (H0 ) = ∅ . Our purpose here is to study the essential stability of this spectral structure under perturbation by a potential V that decays at infinity. By essential stability we mean that the absolutely continuous spectrum of the perturbed operator H does not change, but the singular spectrum that can occur is a countable set with only ±2ν as possible accumulation points, and in particular H has no singular continuous spectrum. Such questions have been studied by many authors and there is a large literature about it. In particular, in the one-dimensional case (ν = 1) which we shall discuss briefly in the rest of this introduction (see also [4]). It is not difficult to see that if V is in `1 (Zν ), e.g. if for ε > 0, |V (α)| ≤
C , |α|1+ε
then the spectrum of the associated Schr¨ odinger operator H is absolutely continuous. Indeed, V is trace class in this case. On the other hand, if V decreases slower than 1/|α| then the absolute continuity of the spectrum of H can be partially or completely destroyed. Indeed, in the one dimensional case Simon has given in [10] an example of a potential V such that |V (α)| ≤
C 1 , |α| 2
and that the associated Schr¨odinger operator has only point spectrum, so σac (H) = ∅. Nevertheless, the different components of the spectrum can coexist if |V (α)| ≤
C , |α|δ
δ>
1 . 2
Indeed, again in the one-dimensional case, Kiselev [5] recently proved that the absolutely continuous component of the spectrum fills the whole essential spectrum
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[−2, +2], if δ > 34 . On the other hand, Naboko and Yakovlev [6] have constructed a potential V such that C(α) , α with C(α) → ∞ for α → ∞ (arbitrarily slowly) |V (α)| ≤
for which the set of eigenvalues of the corresponding discrete Schr¨odinger operator H is dense in [−2, +2]. Here we will describe a class of perturbations V decaying arbitrarily slowly to zero at infinity which leave essentially stable the spectral structure (1.1). We give a compromise between regularity and decay at infinity by allowing several components in the perturbations, behaving each differently at infinity. Another goal of this work is to study the propagation properties of H. More precisely, we establish estimates of local decay type, i.e. estimates of the form ke−iHt ϕkX ≤ CkϕkY for some ϕ ∈ Hac in adequate Banach spaces X, Y . These estimates play a fundamental role in scattering theory. The paper is organized as follows. Section 2 contains a detailed description of our main results. In Sec. 3 we recall what we need of the method of the conjugate operator. Section 4 contains the main step of our proofs, namely the construction of the conjugate operator for the unperturbed hamiltonian H0 . In Sec. 5 we prove our results. In the appendix we give a criterion to verify the regularity requirements of the abstract theory of the conjugate operator method. 2. Main Results Let us consider N = (N1 , . . . , Nν ), where Nj is the diagonal operator in H given by (Nj ψ)(α) = αj ψ(α) . Nj is a self-adjoint operator with domain ( D(Nj ) =
ψ ∈ H such that
X
) |αj ψ(α)| < ∞ . 2
α∈Zν
For each real s we denote by Hs the Sobolev √ space associated to N and defined by the norm kf ks = khN is f k, with hxi = 1 + x2 . By interpolation we obtain the Besov space Hs,p : Hs,p = (Hs1 , Hs2 )θ,p for s1 < s2 , 0 < θ < 1, s = θs1 + (1 − θ)s2 , 1 ≤ p ≤ ∞ . We are specially interested in the space K := H 12 ,1 and its topological adjoint K∗ = H− 12 ,∞ . We also consider P = (P1 , . . . , Pν ), where Pj is the finite difference operator given in H by (Pj ψ)(α) = ψ(α + ej ) − ψ(α) .
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We have denoted by ej the element of Zν whose components all vanish except the jth component which is equal to 1. For a multi-index β ∈ Nν we set P β = P1β1 . . . Pνβν and |β| = β1 + · · · + βν . Let us set C± = {z ∈ C| ± Im z > 0}. Theorem 2.1. Assume that the potential V can be decomposed as V = Vs + Vl + VM , where Vs , Vl and VM are real-valued sequences such that for each j = 1, . . . , ν, and each β ∈ Zν , |β| = 1, 2 we have Z ∞ sup |Vs (α)|dr < ∞, (2.1) 1
r<|α|<2r
Z Vl (α) → 0 as |α| → ∞ and
∞
sup 1
|(Pj Vl )(α)| dr < ∞ ,
(2.2)
r<|α|<2r
VM (α) → 0 and |(P β VM )(α)| = O(|α|−|β| ) as |α| → ∞ .
(2.3)
Then: (i) the set σp (H) of eigenvalues of H has no accumulation points except (probably) ±2ν, and each eigenvalue is finitely degenerate; (ii) the singular continuous spectrum of H is empty; (iii) the holomorphic maps C± 3 z 7→ (H − z)−1 ∈ B(K, K∗ ) extend to a weak∗ continuous function on C± ∪ [(−2ν, 2ν) \ σp (H)]. Let us indicate what kind of behavior at infinity is allowed by our assumptions. Example 2.1. For simplicity assume that V is radial, i.e. V (α) = V (|α|). The assumption (2.1) is fulfilled if for some ε > 0 we have |Vs (α)| ≤ C|α|−1 · (ln |α|)−1−ε . Assumption (2.2) is satisfied for example if Vl (α) → 0 as |α| → ∞ and |(Pj Vl )(α)| ≤ C|α|−1 · (ln |α|)−1−ε . Finally it is easy to see that VM (α) = (ln ln |α|)−1 for |α| ≥ n0 satisfies the condition (2.3). We see then our conditions allow potentials tending arbitrarily slowly to zero at infinity. Example 2.2. In order to compare with the results cited above we shall give some examples in the one-dimensional case. Let C be a bounded real-valued sequence such that |C(n) − c| = O(n−β ), for some β > 0 and some constant c. Let α be a positive number such that α + β > 1, and set Vl (n) = n−α · C(n) , ∀ |n| > n0 , for some n0 > 0 . Clearly Vl satisfies assumption (2.2). Then Theorem 2.1 works for V = Vl (compare with Simon’s result [10]).
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From this example one can deduce a class of real-valued sequences C tending to infinity at infinity such that the potentials V of the form V (n) = n−1 · C(n) , satisfy the condition (2.2) of Theorem 2.1 (compare with [6]). The assertion (iii) of Theorem 2.1 is usually called the “limiting absorption principle” and has important consequences in scattering theory. For example, it allows to establish that for each ϕ ∈ C0∞ (µ(H)), for each ψ ∈ H, and each s > 12 we have Z k(1 + |N |)−s e−iHt ϕ(H)ψk2 dt ≤ Ckψk2 . R
In fact sharper propagation properties of the unitary group e−iHt generated by H can be obtained, for example: Theorem 2.2. Let s > 0 be a positive number and denote by [s] the largest P integer in s. Assume that V = 0≤k≤[s] Vk , where Vk is a real-valued sequence tending to zero at infinity and such that (P β Vk )(α) = O(|α|−s+k−|β| )
∀ 0 ≤ |β| ≤ k .
(2.4)
Let us set κ = 1 − (2s − 1)−1. Then for each σ ∈ [0, κ − 1/2] and ϕ ∈ C0∞ ((−2ν, 2ν) \ σp (H)) we have khN i−σ e−iHt ϕ(H)hN i−σ k ≤ Const.hti−κσ .
(2.5)
3. The Conjugate Operator Method As we have explained above the proofs of our main theorems are based on the method of the conjugate operator described in [1–3] (see also [9]). In this section we give a brief review on these considerations. 3.1. H¨ older Zygmund space Let (E, k · k) be a Banach space and f : R → E a bounded continuous function. Let ε > 0 and m ∈ N an integer. We define the modulus of continuity of order m of f as
X
m
m j
f (x + jε) (3.1) wm (f, ε) = sup (−1)
. j x∈R j=1
One says that f belongs to the H¨ older–Zygmund space Λα,p , α > 0, p ∈ [1, +∞) if and only if there is an integer l > α such that the function ε 7→ ε−α wl (ε) belongs to Lp ((0, 1), ε−1 dε) . For p = ∞ we set Λα = Λα,∞ .
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3.2. Besov space associated to A Let A be a self-adjoint operator in a Hilbert space H. 3.2.1. For each real s we denote by HsA the√Sobolev space associated to A, defined by the norm kf ks = khAis f k, with hxi = 1 + x2 . For all real numbers t ≤ s we have a continuous dense embedding HsA ⊂ HtA . A By interpolation we obtain the Besov space Hs,p associated to A, namely A Hs,p = (HsA1 , HsA2 )θ,p
for s1 < s2 , 0 < θ < 1, s = θs1 + (1 − θ)s2 , 1 ≤ p ≤ ∞ . 3.2.2. Let S be a bounded operator in H. For each integer k, we denote adkA (S) the sesquilinear form on D(Ak ) defined by induction as follows: ad0A (S) = S , ad1A (S) = [S, A] = SA − AS and X k! k−1 adkA (S) = ad1A (adA (S)) = (−1)i Ai SAj . i!j! i,j≥0 i+j=k
We say that S is of class C k (A) if the sesquilinear form adkA (S) has a continuous extension to H, which we identify with the its associated bounded operator in H (from the Riesz Lemma) and we denote it by the same symbol. In this case one can prove easily that the function τ 7→ S(τ ) := e−iτ A Se−iτ A ∈ B(H) is strongly of class C k . Moreover ik adkA (S) =
dk S(τ ) . dτ k τ =0
Using the continuity properties of the function S(τ ) one can define another class of regularity of operators. For s > 0 and 1 ≤ p ≤ ∞, we say that S is of class C s,p (A) if the function S(τ ) is of class Λs,p . In the appendix we give an abstract tool which will enable us to verify this regularity. 3.3. Mourre estimate From now on let us assume that S is at least of class C 1 (A). In particular, the commutator [S, iA] is a bounded operator in H. Then one may consider the real open set µA (S) of points λ such that E(J)[S, iA]E(J) ≥ aE(J)
(3.2)
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for some number a > 0, and some neighborhood J of λ. Then we will say that A is locally strictly conjugate to S on µA (S), and the estimate (3.2) is called the strict Mourre estimate. Similarly, we define the real open set µ ˜A (S) of points λ such that E(J)[S, iA]E(J) ≥ aE(J) + K
(3.3)
for some number a > 0, some compact operator K in H and a suitable neighborhood J of λ. Then we say that A is locally conjugate to S on µ ˜A (S), and the estimate (3.3) is called the Mourre estimate. Remark 3.1. In general one cannot found explicitly the set µA (S). For this reason we have introduced the set µ ˜A (S), which one can describe rather explicitly in many interesting situations. Another advantage of µ ˜A (S) is its stability under weakly relatively compact perturbations. More precisely: If S, T are two bounded and symmetric operators in H, which are of class Cu1 (A) (e.g. C 1,1 (A)) and such that (S + i)−1 − (T + i)−1 is a compact operator in H, then ˜A (T ) . µ ˜A (S) = µ In the following proposition we describe the difference between the strict and large Mourre estimates. Proposition 3.1. Assume that S is of class C 1 (A). Then µA (S) and µ ˜A (S) are open real sets, µA (S) ⊂ µ ˜A (S), and the set µ ˜A (S) \ µA (S) does not have accuA A A mulation points in µ ˜ (S). Moreover, µ ˜ (S) \ µ (S) consists of eigenvalues of S of finite multiplicities and the spectrum of H in µA (S) is purely continuous. We do not know whether the C 1 (A) regularity property is sufficient for the absence of singularly continuous spectrum of S in µA (S). But it is proved in Chap. 7 of [1] that the limiting absorption principle (L.A.P.) breaks down if S is not more regular. The next theorem singles out a sufficient condition ensuring the L.A.P. Theorem 3.1. Assume S is of class C 1,1 (A). Then the boundary values of the resolvent R(λ ± i0) = w∗ - limµ→±0 R(λ ± iµ) exist in B(H 12 ,1 , H− 12 ,∞ ) locally uniformly for λ ∈ µA (S). In particular, the spectrum of H is purely absolutely continuous on µA (S). Theorem 3.1 remains valid if S is only locally of class C 1+0 (A), i.e. for each ϕ ∈ C0∞ (R), the operator ϕ(H) is of class C 1+0 (A) (see [9]). This fact allows us to study quite singular Hamiltonians without any gap in their spectrum (see [8]). Another important consequence of the limiting absorption principle is the socalled estimate of local decay type, namely that for each ϕ ∈ C0∞ (µA (S)) and ε > 0 we have Z 1 khAi− 2 −ε e−iSt ϕ(S)ψk2 dt ≤ Ckψk2 , R
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which plays an important role in scattering theory. The point is that if S is more regular, then we have more precise propagation properties. 1
Theorem 3.2. Let s > 0, and S be a self-adjoint operator of class C s+ 2 (A). Let us set κ = 1−(2s−1)−1. Then for each σ ∈ [0, κ−1/2] and for each ϕ ∈ C0∞ (µA (S)) there exists a constant C < ∞ such that khAi−σ e−iSt ϕ(S)hAi−σ k ≤ Chti−κσ ,
t ∈ R.
(3.4)
We finish this section with some technical result which allows us to check the Mourre estimate for some particular operators. Proposition 3.2. Let S1 and S2 be two self-adjoint bounded operators in Hilbert spaces H1 , H2 respectively. Assume that there exist two self-adjoint operators A1 , A2 in H1 , H2 such that Si is of class C 1 (Ai ), and that Ai is strictly conjugate to Si on Ji . Then the operator S = S1 ⊗ 1 + 1 ⊗ S2 is of class C 1 (A), with A = A1 ⊗ 1 + 1 ⊗ A2 . Moreover, A is conjugate to S on J = {λ1 + λ2 | λi ∈ Ji }. 4. The Conjugate Operator As is explained in the appendix, a key point of our proofs is the construction of a suitable conjugate operator A for the free hamiltonian H0 . This section is entirely devoted to this fact. 4.1. The one-dimensional case It is instructive to study the one-dimensional case (see also [4]). More precisely, the Hilbert space is H = `2 (Z), and H0 is given by (H0 ψ)(n) = −ψ(n + 1) − ψ(n − 1) ,
∀ ψ = (ψ(n))n∈Z ∈ H .
In this case N is given by (N ψ)(n) = nψ(n) and the finite difference operator P has only one component which we denote by the same symbol P given in H by (P ψ)(n) = ψ(n + 1) − ψ(n) . By straightforward computations one can see that H0 = P ∗ P − 2 . Let us consider the self-adjoint operator A in H such that iA = N P − P ∗ N . Lemma 4.1. H0 is of class C ∞ (A) and µA (H0 ) = R \ {±2}.
(4.1)
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Proof. Let us calculate the first commutator [H0 , iA] = [P ∗ P, N P − P ∗ N ] = [P ∗ P, N ]P − P ∗ [P ∗ P, N ] = [H0 , N ]P − P ∗ [H0 , N ] .
(4.2)
On the other hand, for any ψ ∈ H with compact support we have hψ, [H0 , N ]ψi = hH0 ψ, N ψi − hN ψ, H0 ψi X (ψ(n + 1) + ψ(n − 1))nψ(n) =− n∈Z
+
X
nψ(n)(ψ(n + 1) + ψ(n − 1))
n∈Z
=
X
ψ(n)(ψ(n − 1) − ψ(n + 1)) .
n∈Z
Then it is clear that this quadratic form can be extended to a continuous quadratic form in H. Moreover, we can also obtain that [H0 , iA] = 4 − H02 = (2 − H0 )(2 + H0 ) .
(4.3)
So, H0 is of class C ∞ (A). On the other hand, since −2 ≤ H0 ≤ 2, A is strictly conjugate to H0 on R \ {±2}. 4.2. The multidimensional case In this section we shall show how one passes to the multidimensional case in Lemma 4.1. We start by recalling some obvious, but useful, commutation relations between the operators Pi and Nj . We have [Ni , Nj ] = [Pi , Pj ] = 0 if i 6= j
[Ni , Pj ] = 0 [Pi , Ni ] = τei ,
[Ni , Pi∗ ] = τ−ei
with (τα ψ)(β) = ψ(α + β). By a straightforward calculation we obtain ∗
P P =
ν X
Pi∗ Pi = H0 + 2ν .
i=1
This operator is clearly a bounded self-adjoint operator in H, and it is purely absolutely continuous with spectrum [0, 4ν]. Let us consider the self-adjoint operator A defined by iA =
ν X
Ni Pi − Pi∗ Ni ≡ N P − P ∗ N ,
i=1
with its natural domain D(A) = D(N ).
(4.4)
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Lemma 4.2. H0 is of class C ∞ (A) and µA (H0 ) = R \ {±2ν}. Proof. For ν = 1, Lemma 4.2 is Lemma 4.1. In the two-dimensional case, we have H ≈ H1 ⊗ H2 , with H1 = H2 = `2 (Z) , and H0 = H0,1 ⊗ 1 + 2 ⊗ H0,2 , A = A1 ⊗ 1 + 1 ⊗ A2 ,
(4.5) (4.6)
with for j = 1, 2 H0,j = Pj∗ Pj − 2 , iAj = Nj Pj − Pj∗ Nj ,
(4.7) (4.8)
acting in the Hilbert space Hj . On the other hand, H0,j is of class C ∞ (Aj ) and Aj is locally strictly conjugate to H0,j on R \ {±2}. It follows then, from Proposition 3.1, that H0 is of class C ∞ (A) and that A is locally strictly conjugate to H0 on R\{±4}. Now, we are done by an obvious induction. 5. Proofs Applying the theorems of Sec. 3, in order to prove our main results we must check that H is sufficiently regular with respect to A and that the Mourre estimate between H and A holds. More precisely, we shall prove the following proposition. Proposition 5.1. Assume that the assumptions of Theorem 2.1 hold. Then (i) the operator V (hence H) is of class C 1,1 (A); (ii) A is locally conjugate to H on R \ {±2ν}, i.e. µ ˜A (H) = µA (H0 ) = R \ {±2ν}; (iii) if V satisfies hypothesis (2.4) of Theorem 2.2 then V (hence H) is of class C s (A). Proof of Theorems 2.1 and 2.2. Combining Theorem 3.1 and the two first assertions of the preceding proposition one can easily conclude the proof of Theorem 2.1. Similarly, Theorem 2.2 follows from the second and third assertions of the preceding proposition combined with Theorem 3.2. Proof of Proposition 5.1. (i) First we note that the second assertion of Proposition 5.1 follows from the first. Indeed, V tends to zero at infinity. Then the difference of the resolvents (H + i)−1 − (H0 + i)−1 is a compact operator in H. It
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follows from Remark 3.1 that if H is of class C 1,1 (A) (in fact we only need Cu1 (A)); thus µ ˜A (H) = µA (H0 ) = R \ {±2ν} . (ii) Now we prove the first assertion of Proposition 5.1 For this we shall treat each component separately. More precisely, we shall use Theorem 6.1 of the Appendix by taking for A the operator defined by (4.4), Λ = hN i, G = G ∗ = H, and for T one of the component of V . This is possible because of the obvious lemma: Lemma 5.1. For each positive number s > 0, hN i−s As is a bounded operator in H. (iii) Taking T = Vs in (ii), it is not difficult to see that assumption (2.2) is equivalent to hypothesis (6.1) for s = p = 1. Then Vs is of class C 1,1 (A). Now we shall deal with Vl . For this we have to calculate the first commutator between Vl and A. We have [Vl , iA] = [Vl , N P − P ∗ N ] =
ν X [Vl , Ni Pi − Pi∗ Ni ] i=1
=
ν X
Ni [Vl , Pi ] − [Vl , Pi∗ ]Ni
i=1
=
ν X
Ni [Vl , Pi ] + (Ni [Vl , Pi ])∗ .
i=1
On the other hand, it is easy to see that [Vl , Pi ] is a bounded operator in H, for each i = 1, . . . , ν, and that ([Vl , Pi ]ψ)(α) = (Vl (α) − Vl (α + ei ))ψ(α + ei ) = (Pi Vl )(α)ψ(α + ei ) ≡ V˜l (α)ψ(α + ei ) or equivalently Ti := [Vl , Pi ] = V˜l · τei . Then we have [Vl , iA] =
ν X
Ni Ti + Ti∗ Ni .
(5.1)
i=1
It follows that the commutator [Vl , iA] is a bounded operator in H if Ni Ti also is a bounded operator for each i = 1, . . . , ν. But (Ni Ti ψ)(α) = αi (Vl (α) − Vl (α + ei ))ψ(α + ei )
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defines a bounded operator in H if and only if (Pi Vl )(α) = O(|α|−1 ) at infinity. But this property is contained in hypothesis (2.3), and so Vl is of class C 1 (A). Moreover, hypothesis (2.3) implies condition (6.5) of Theorem 6.1 for k = s = p = 1, so Vl is of class C 1,1 (A). (iv) To establish the first assertion of Proposition 5.1, it remains to show that VM is also regular. In fact, we shall prove that VM is of class C 2 (A) which is more than we need. Now replacing Vl by VM in the preceding computations, one concludes that [VM , iA] is a bounded operator in H, and so VM is of class C 1 (A). We have to show that [[VM , iA], iA] is a bounded operator in H. For this it suffices to compute the commutator [Ni Ti , iA], with (Ni Ti ψ)(α) = αi (Pi VM )(α)ψ(α + ei ). We have [Ni Ti , iA] =
ν X [Ni Ti , Nj Pj − Pj∗ Nj ] j=1
=
ν X
Ni [Ti , Nj Pj − Pj∗ Nj ] + [Ni , Nj Pj − Pj∗ Nj ]Ti .
j=1
But it is not difficult to see that ν X [Ni , Nj Pj − Pj∗ Nj ]Ti = −Ni τei Ti − τ−ei Ni Ti j=1
which, as we saw before, is a bounded operator in H. On the other hand, a simple computation shows that X (ψ, Ni [Ti , Nj Pj − Pj∗ Nj ]ψ) = ψ(α) · αi [(αj + δij )(Pi VM )(α) α
− αj (Pi VM )(α + ej )]ψ(α + ei + ej ) X + ψ(α) · αi [(αj − 1)(Pi VM )(α − ej ) α
− (αj + δij − 1)(Pi VM )(α)]ψ(α + ei − ej ) where we have denoted by δij the Kronecker symbol. It follows that this expression defines a bounded operator in H if αi [(αj + δij )(Pi VM )(α) − αj (Pi VM )(α + ej )] = αi αj (Pj Pi VM )(α) + δij αi (Pi VM )(α) is bounded. But this holds if (Pj Vl )(α) = O(|α|−1 ) and (Pj Pi Vl )(α) = O(|α|−2 ) as |α| → ∞ . Consequently, VM is of class C 2 (A) if it satisfies assumption (2.3) of Theorem 2.1. This finishes the proof of the first assertion of Proposition 5.1.
¨ ON THE SPECTRAL PROPERTIES OF DISCRETE SCHRODINER OPERATORS
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(v) Similarly, one can prove that if V satisfies assumption (2.3) then V is of class C s (A). Indeed, by an induction argument one can show that if (P β W )(α) = O(|α|−|β| ) at infinity for each multi-index |β| ≤ k, then W is of class C k (A). After that, the second part of Theorem 6.1 allows us to finish the proof easily. 6. Appendix The efficiency of the method of the conjugate operator in applications closely depends on our ability to verify the regularity hypothesis of the studied operator with respect to its conjugate operator. The goal of this appendix is to develop abstract tools which will enable us to check this. Since such result can be applied to other situations (see for example [8]) we describe them in general form. Let G, H be two Hilbert spaces such that G ⊂ H. Then via the Riesz identification we have G ⊂ H ≈ H∗ ⊂ G ∗ . Let A be a self-adjoint operator in H such that its associated group eiAt leaves invariant G and G ∗ . Hence it induces two strongly continuous groups of bounded operators in G and G ∗ , which we still denote by the same symbol eiAt . We then get an automorphism group on X := B(G, G ∗ ) denoted Wt and defined by Wt [T ] = Wt T = e−iAt T eiAt ,
∀T ∈X.
In this context one can introduce new regularity classes of operators. In what follows the numbers s, p, k are such that: s ≥ 0, 1 ≤ p ≤ ∞ and k is a non-negative integer. Definition 6.1. (a) Let s > 0. An operator T ∈ X is of class C s,p (A; G, G ∗ ) (resp. C k (A; G, G ∗ )) if the function t 7→ Wt T ∈ X is of class Λs,p on R (resp. strongly C k ). (b) For s = 0 and p = 1, we say that T ∈ X is of class C 0,1 (A; G, G ∗ ) ≡ C +0 (A; G, G ∗ ) if the function t 7→ Wt T ∈ X is Dini continuous. It is not difficult to see that T is of class C s,p (A; G, G ∗ ) if and only if there exists an integer l > s such that (with the usual convention if p = ∞): Z 0
1
kε−s (Wε − 1)l T kpX
dε ε
p1 < ∞.
(6.1)
Similarly we define for operators the regularity classes C s,p (A; G ∗ , G), C k (A; G ∗ , G) and C +0 (A; G ∗ , G). Theorem 6.1. Let Λ be a self-adjoint operator in H bounded from below by a strictly positive constant such that (i) eiΛτ G ⊂ G and keiΛτ kB(G) ≤ Chτ iN with N < ∞; (ii) the operator Al Λ−l is continuous in G ∗ for some integer l ≥ 1. Let 0 ≤ σ < l. Then a bounded symmetric operator T ∈ X is of class C s,p (A; G, G ∗ ) if there exists a function θ ∈ C0∞ (R) with θ(x) > 0 for 0 < a < |x| < b < ∞ such that (with the usual convention if p = ∞):
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A. BOUTET DE MONVEL and J. SAHBANI
Z
∞
kr
1
s
θ(Λ/r)T kpX
dr r
p1
< ∞.
(6.2)
In particular, if p = 1 or ∞ and if the operator T is of class C k (A; G, G ∗ ) for an integer 0 ≤ k ≤ s and Z
∞
kr
s−k
k
θ(Λ/r)A
1
[T ]kpX
dr r
p1
< ∞,
(6.3)
then T is of class C s,p (A; G, G ∗ ). Proof. (i) Let us denote Wt0 , resp. Wt00 the operators defined on X by Wt0 [T ] = eiAt T, resp. Wt00 [T ] = T eiAt . 0 Then Wt = Wt00 W−t . Hence we get 0 0 0 Wε − 1 = Wε00 W−ε − 1 = (Wε00 − 1)W−ε + (W−ε − 1) .
Now let us calculate the powers (Wε − 1)l as follows. By applying Newton’s formula we obtain l X l l 0k 0 (Wε − 1) = (W−ε − 1)l−k . (Wε00 − 1)k W−ε k k=0
More explicitly, using the definition of Wε0 and Wε00 , we get l X l −kiAε −iAε (Wε − 1) [T ] = e (e − 1)l−k · T · (eiAε − 1)k . k l
k=0
On the other hand, h ε i ε ε m ε Aε = (2i)m eiAm 2 sinm (eiAε − 1)m = eiAm 2 eiA 2 − e−iA 2 . 2 Consequently, (6.1) follows from Z
1
kε 0
−s
sin (Aε) · T · sin m
n
(Aε)kpX
dε ε
p1 <∞
(6.4)
for any integers m, n such that m + n = l. But if we set ϕ(x) = sin x + i sinx x , it is easy to see that there exists a finite constant C which depends only on ϕ such that k(sin Aε)T kX ≤ kεA(εA + i)−1 T kX kϕ(εA)kB(G ∗ ) ≤ CkεA(εA + i)−1 T kX .
¨ ON THE SPECTRAL PROPERTIES OF DISCRETE SCHRODINER OPERATORS
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Then (6.4) is a consequence of Z
∞
−1 m
kr (A(A + ir) s
) T (A(A +
1
ir)−1 )n kpX
dr r
p1
< ∞,
(6.5)
for any integers m, n such that m + n = l. (ii) Let us set Ar = A(A + ir)−1 and Λr = Λ(Λ + r)−1 . It is clear that I = Λr + r(Λ + r)−1 . It follows that for each integer m we have −1 m
I = (Λr + r(Λ + r)
)
=
m X i=0
m! ri (Λ + r)−i Λrm−i . i!(m − i)!
Consequently we have in B(G ∗ ) the identity Am r =
m X i=0
=
m X i=0
m! −i m−i ri Am r (Λ + r) Λr i!(m − i)! m! Am−i (r(A + ir)−1 )i Ai Λ−i Λm r i!(m − i)! r
≡ Br Λm r . m iAt Similarly Am defines a strongly continuous group of r = Λr Cr in B(G). Since e ∗ −1 bounded operators in G , kr(A + ir) kB(G ∗ ) is bounded by a finite constant independent of r, this is also the case for Ar = I −r(A+ir)−1 in B(G ∗ ). Using condition (ii) and an interpolation argument, we deduce that kBr kB(G ∗ ) is dominated by a finite constant independent of r. Similarly kCr kB(G) ≤ C, independently of r. Then (6.5) is a consequence of
Z 1
∞
krs (Λ(Λ + r)−1 )m T (Λ(Λ + r)−1 )n kpX
dr r
p1
< ∞,
(6.6)
for any integers m, n such that m + n = l. (iii) In (6.6) the terms given by m = 0 and n = l dominate all other terms: Lemma 6.1. For a given T ∈ X , there exists a finite constant C independent of r such that m+n kΛnr T Λm T kX . r kX ≤ CkΛr Proof of lemma. Lemma 6.1 is obtained by complex interpolation. For this we have to define the powers Λzr for a complex number z and to estimate them conveniently. From assumption (i) of Theorem 6.1, eiΛτ induces a continuous group G with polynomially growth at infinity. Using Theorem 3.7.10 of [1], we see that for each function ϕ ∈ BC ∞ (R) (i.e. ϕ is a bounded function of class C ∞ (R) with
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A. BOUTET DE MONVEL and J. SAHBANI
bounded derivatives) the bounded operator ϕ(Λ) in H leaves G invariant, so its associated bounded operator in G, which we denote by the same symbol, satisfies kϕ(Λ)kB(G) ≤ CkϕkBC k for some constant C and some integer k. We know by hypothesis that there exists a number a > 0 such that Λ ≥ a, then σ(Λ) ⊂ [a, ∞). Let 0 < a0 < a and let η ∈ C ∞ (R) such that ( a0 a0 if x < 2 η(x) = x if x > a then η(Λ) = Λ in H. Let us consider the function ϕ(x) = log(η(x)(η(x) + r)−1 ), then ϕ(Λ) = log(Λr ) in H. But ϕ belongs to BC ∞ (and its norm is independent of r) then ϕ(Λ) = log(Λr ) in G also. It follows that Λr = exp(ϕ(Λ)) in G and consequently Λzr = exp(zϕ(Λ)) for each z ∈ C. Clearly the function z 7→ Λzr ∈ B(G) is holomorphic and kΛzr kB(G) ≤ exp(cr |z|) for cr = kϕ(Λ)kB(G) . Moreover when z = iy ∈ iR this estimate is uniform with respect to r. Indeed, for −1 ≤ y ≤ 1 we have iy iy −iy Λiy . r = (ϕ(Λ)) = η(Λ) (η(Λ) + r)
Since ψ(x) = (η(x) + r)iy is of class BC ∞ and all its derivatives have a supremum independent of r, the norm of ψ(Λ) in B(G) is bounded by a constant independent of r. Then (for more details see [1, p. 329]) there exists a constant c < ∞ independent of r such that kΛiy r kB(G) ≤ c ∀ y ∈ [−1, 1] . It follows that there exists a similar constant c such that c|y| kΛiy r kB(G) ≤ ce
∀ y ∈ R.
Let us set M = Λr . For g ∈ G let us consider ∗
2
z 7→ F (z) = hM z g, T M l−z giez , which is holomorphic in the strip {x + iy | y ∈ R, x ∈ (0, l)}, and is continuous 2 on the closure of this strip; |F (z)| ≤ Ce−y /2 with a constant C independent of r. Then ( ) 2
hM n g, T M l−ngien = |F (n)| ≤ max
sup |F (iy)|, sup |F (l + iy)| y∈R
y∈R
≤ sup e−y kM iy k2B(G) kM l T kX kgk2G 2
y∈R
≤ CkM l T kX kgk2G .
¨ ON THE SPECTRAL PROPERTIES OF DISCRETE SCHRODINER OPERATORS
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This implies that there exists a constant C independent of r such that for n = 1, . . . , l − 1, we have kM n T M l−n kX ≤ CkM l T kX .
This ends the proof of the lemma. (iv) Consequently (6.6) follows from Z
∞
−1 l
kr (Λ(Λ + r) s
1
)
T kpX
dr r
p1
< ∞.
(6.7)
But Theorem 3.5.11(b) of [1], p. 144 shows that (6.7) follows from (6.2): Z 1
∞
kr
s
θ(Λ/r)T kpX
dr r
p1
< ∞.
This finishes the proof of the first part of our theorem. (v) Now let us consider the case p = 1. Let us recall that for 0 < k < s, 1 ≤ p ≤ ∞, we have ( T ∈ C k (A; G, G ∗ ) and T ∈ C s,p (A; G, G ∗ ) ⇔ (6.8) T ∈ C s−k,p (A; G, G ∗ ) If condition (6.3) of the theorem is satisfied for an integer 0 < k < s then the first part implies that Ak [T ] (G → G ∗ ) is of class C s−k,1 (A; G, G ∗ ) and (6.8) finishes the proof in this case. It remains to prove our assertion in the case where s is an integer and k = s. In this case, T is of class C k (A; G, G ∗ ), so Ak [T ] ∈ X . Moreover, condition (6.3) coincides with (6.2) for s = 0 and with Ak [T ] instead of T . We deduce from the first part that Ak T is of class C +0 (A; G, G ∗ ), i.e. T is of class C k+0 (A; G, G ∗ ). It suffices to note that C k+0 (A; G, G ∗ ) ⊂ C k,1 (A; G, G ∗ ). The case p = ∞ is similar. Note that in this case, the result is trivial for s = k because C k (A; G, G ∗ ) ⊂ C k (A; G, G ∗ ). This theorem has been proved in [7] (see also [1] where this theorem is proved in the case σ = 1). References [1] W. Amrein, A. Boutet de Monvel and V. Georgescu, C0 -Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians, Birkh¨ auser, Progress in Math. Ser. 135, Basel 1996. [2] A. Boutet de Monvel, V. Georgescu and J. Sahbani, “Boundary values of resolvent families and propagation properties”, C. R. Acad. Sci. Paris S´er. I Math. 322 (1996) 289–294. [3] A. Boutet de Monvel, V. Georgescu and J. Sahbani, “Higher order estimates in the conjugate operator theory”, Helv. Phys. Acta 71 (1998) 518–553 & preprint Institut de Math´ematiques de Jussieu, no. 59, 1996. [4] A. Boutet de Monvel and J. Sahbani, “On the spectral properties of discrete Schr¨ odinger operators”, C. R. Acad. Sci. Paris S´er. I Math. 326 (1998) 1145–1150.
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[5] A. Kiselev, “Absolutely continuous spectrum of one-dimensional Schr¨ odinger operators and Jacobi matrices with slowly decreasing potentials”, Commun. Math. Phys. 179 (1996) 377–400. [6] S. N. Naboko and S. I. Yakovlev, “On the point spectrum of discrete Schr¨ odinger operator”, Func. Analys. Appl. 26 (1992) 145–147. [7] J. Sahbani, “Th´eor`emes de propagation, Hamiltoniens localement r´eguliers et applications”, PhD thesis, Univ. Paris 7, July 1996. [8] J. Sahbani, “Propagation theorems for some classes of pseudo-differential operators”, J. Math. Anal. Appl. 211 (1997) 481–497. [9] J. Sahbani, “The conjugate operator method for locally regular Hamiltonians”, J. Operator Theory 38 (1997) 297–322. [10] B. Simon, “Some Jacobi matrices with decaying potential and dense point spectrum”, Commun. Math. Phys. 87 (1982) 253–258.
3D SINGLETONS AND THEIR BOUNDARY 2D CONFORMAL FIELD THEORY ´ ´ SEBASTIEN MICHEA Laboratoire Gevrey de Math´ ematique Physique Universit´ e de Bourgogne B.P. 400, F-21011 Dijon Cedex France E-mail : [email protected] Received 31 August 1998 This paper is a continuation of recent work of Flato and Frønsdal on singletons in 1 + 2 anti De Sitter universe and their link with 2D conformal field theories on the boundary. More specifically we show that in this framework we can construct a 3D-singleton model in the bulk, the limit of which on the boundary of De Sitter space is a Gupta–Bleuler triplet for two commuting copies of the Witt algebra. We also generalize this result to the case of WZNW models.
1. Introduction Singletons in 4D anti De Sitter space (AdS4 ) were discovered 35 years ago by Dirac [7] as very degenerate representations of SO(3, 2). In the past 20 years there have been recurrent bouts of serious interest in physical applications of that notion, starting (also in four dimensions) 20 years ago [10] with fundamental works by Flato and Frønsdal (see e.g. [12] for a partial account) and continuing with a series of works on (super)membranes by Salam and coworkers (see e.g. [5]). In the past months there has been a flurry of e-prints (many by famous authors) on hep-th dealing with a variety of aspects of physics (and cosmology) in anti De Sitter universes, typically in at least (usually more than) four dimensions; see e.g. the list of references in [9] (and references of references). As developed in e.g. [10, 11], field theory on AdS has a number of very interesting properties, such as invariant infrared regularization, natural discrete spectra and a role similar to quarks of singletons as building blocks of particles, massless particles being dynamically treated as states of two (geometrically confined) singletons. A spectacular achievement is a construction of QED [14], where massless photon fields are built as composite of two singleton fields (the latter obeying nonconventional statistics [15]). A distinguished feature of singleton field models is that they are topological gauge theories. The physical modes are so sparse that all the physics is in fact determined by what happens at the spatial infinity boundary of AdS. There we have a conformal field theory (in one dimension less). Strictly speaking, all this has been extensively studied mainly for AdS4 . In higher dimensions, we have to define properly what we mean by singletons and 1079 Reviews in Mathematical Physics, Vol. 11, No. 9 (1999) 1079–1090 c World Scientific Publishing Company
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massless particles in AdS universe. For a discussion of these points, see [1, 20]; a short study of singleton representations for AdS3 was also done in [20]. Much of the latest interest in AdS physics comes from the conjecture [21] of a (not yet completely proved) duality between supergravity and string theories in AdSd (d being the dimension) on the one hand, and conformal invariant field theories in d − 1 dimensions on the other hand. For instance, a connection was made [8] between conformal Maxwell theory and singletons in AdS5 ; then type IIB supergravity on AdS5 × S 5 was linked with super Yang–Mills theories in four dimensions (see e.g. [24, 19]). These are only a few of the recent results involving singletons in high dimensional AdS (see also [9] and references quoted therein for a few more references on singletons and AdS), among which we can stress [17] an extension to this framework of previous works dealing with composite electrodynamics [14]. Furthermore this intensive study triggered a huge interest in all possible aspects of physics in various AdS universes of dimension at least 4. Nevertheless there is a significant interest (both mathematically and physically) to focus attention on three dimensional singletons, since the boundary physics is then a conformal field theory on a two-dimensional torus. Indeed it is well known that whereas the conformal group of n-dimensional Minkowskian space is SO(n, 2) for n > 2, the situation changes drastically in two dimensions. The reason is that the 2D Minkowskian metric can be factorized: ds2 = dx2 − dt2 = (dx − dt)(dx + dt) = dzd¯ z so that the whole conformal group is the direct product of diffeomorphisms in the z variable by those in z¯. The Lie algebra of this infinite dimensional Lie group is a direct sum of two copies of the well-known Witt algebra with generators ln , ¯ln , n ∈ Z (realized as the vector fields −z n+1 ∂z and −¯ z n+1 ∂z¯ respectively) satisfying the commutation relations [ln , lm ] = (n − m)ln+m ; [¯ln , ¯lm ] = (n − m)¯ln+m ; [ln , ¯lm ] = 0; n, m ∈ Z .
(1)
When dealing with quantized theories and therefore projective pseudogroup representations we obtain two copies of the Virasoro algebra (the unique non trivial central extension of the Witt algebra) with the same values of the central charge c: c 3 (n − n)δn,−m ; 12 c [¯ln , ¯lm ] = (n − m)¯ln+m + (n3 − n)δn,−m ; 12 [ln , ¯lm ] = 0; n, m ∈ Z .
[ln , lm ] = (n − m)ln+m +
This algebra imposes a infinite set of constraints on the conformal invariant theory. Since the seminal work of B.P.Z. [4] that framework has been used with great success to solve non pertubatively several two dimensional conformal invariant models. Using the Sugawara construction [23], the Virasoro algebra can be found in a completion of the enveloping algebra of an affine Kac–Moody algebra. Furthermore
3D SINGLETONS AND THEIR BOUNDARY 2D
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the conformal symmetry has been extended to an affine Kac–Moody algebra in the context of the WZNW model [25]. It has been shown in [16] that singletons on AdS3 , where the invariance group is the linear conformal group SO(2, 2), describe on the AdS 3 boundary a conformal field theory. There the authors restricted their analysis to the finite dimensional conformal group SO(2, 2). Here we generalize this analysis and show that, by suitably modifying the 3D singleton model, we get on the boundary a gauge theory invariant under two commuting copies of the Witt algebra. In Sec. 2 we develop the theory of 3D singletons on the De Sitter space AdS3 and its associated 2D conformal Gupta–Bleuler triplet on the boundary. In Sec. 3 we discuss the implementation of the Witt symmetry in three dimensions. Finally in Sec. 4 we apply this result to the case of the WZNW model and extend the symmetry to the loop algebra. 2. Three Dimensional Singletons and the Witt Algebra The fields we are interested in live in the anti De Sitter space realized as the one sheet hyperboloid M3 (with curvature ρ = 1) embeded in R4 y 2 ≡ y0 2 − y1 2 − y2 2 + y5 2 = 1 . We identify the space (noted F) of differentiable functions φ on M3 with homogeneous functions on {y 2 > 0, y ∈ R4 } of degree zero, i.e. functions such that y · ∂φ ≡ (y0 ∂0 + y1 ∂1 + y2 ∂2 + y5 ∂5 )φ = 0 . It is useful to consider the following system of coordinates: 1 1 x+ = √ (y0 − iy5 ); x− = √ (y0 + iy5 ) ; 2 2 1 1 y+ = √ (y1 + iy2 ); y− = √ (y1 − iy2 ) ; 2 2 where y 2 = 2(x+ x− − y+ y− ); y · ∂ = x+ ∂x+ + x− ∂x− + y+ ∂y+ + y− ∂y− . On M3 we define a representation of so(2, 2) = sl(2, R)⊕sl(2, R) with generators L−1 = −(y− ∂x+ + x− ∂y+ ); L1 = y+ ∂x− + x+ ∂y− ; 1 (−x+ ∂x+ + x− ∂x− − y+ ∂y+ + y− ∂y− ) ; 2 ¯ 1 = y− ∂x− + x+ ∂y+ ; = −(y+ ∂x+ + x− ∂y− ); L
L0 = ¯ −1 L
¯ 0 = 1 (−x+ ∂x+ + x− ∂x− + y+ ∂y+ − y− ∂y− ) . L 2 They verify the commutation relations (1) (for n, m = −1, 0, 1).
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We consider “fundamental particles” on De Sitter space as unitary, irreducible, highest weight projective representations of SO(2, 2), labelled by their energy E and spin s. Since so(2, 2) = so(2, 1) ⊕ so(2, 1) we can write these representations as D(E, s) = D(s1 ) ⊗ D(s2 ), E = s1 + s2 , s = s1 − s2 ; 2s1 , 2s2 ∈ N. The spectrum of ¯ 0 ) on each factor of the tensor product is given by {s1 + n, n ∈ N} −L0 (resp. −L (resp. {s2 + n, n ∈ N}). We denote by D(−s) the unitary irreducible representation of the discrete series of so(2, 1) contragredient to D(s). Our study will be based on the covariant dipole equation 2 φ = 0
(3)
introduced in 4 dimensions in [13]. Here the d’Alembertian operator is the representative of the so(2, 2) Casimir ¯ 20 − L ¯ 1L ¯ −1 + L ¯ 0 ) = −y 2 ∂ 2 + (y · ∂)(y · ∂ + 2) . = 2(L20 − L1 L−1 + L0 ) + 2(L We want to construct a quantizable Gupta–Bleuler triplet of singleton solutions of (3), i.e. fields that transform trivially (modulo gauge transformations) under one of the two factors of SO(2, 2). Following [16], we start with the 3D massless Klein–Gordon equation: φ = 0 .
(4)
We see, studying the radial wave equation at infinity, that it describes a gauge theory. Gauge modes, except for constants, are characterized by a fall-off at infinity as R−2 (where R2 = y02 + y52 ) while physical modes have a non zero limit. Consider now the following gauge modes: f0 = 1, E = 0; f3 =
y2 y2 0 , E = 2; f = , E = −2 , 3 2x2− 2x2+
(5)
on which the action of so(2, 2), that generate an infinite dimensional space denoted by H3 , is L−1 f3 = 0 ,
L0 f3 = −f3 ,
L1 f3 = −2f1 f3 ,
L−1 f30 = 2f10 f30 ,
L0 f30 = f30 ,
L1 f30 = 0 ,
¯ −1 f3 = 0 , L
¯ 0 f3 = −f3 , L
¯ 1 f3 = −2f2 f3 , L
¯ −1 f30 = 2f20 f30 , L
¯ 0 f30 = f30 , L
¯ 1 f30 = 0 ; L
where f1 , f10 , f2 , f20 are the physical modes y+ , E = 1; x− y− f10 = , E = −1 ; x+
f1 =
y− , E = 1; x− y+ f20 = , E = −1 , x+ f2 =
(6)
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with corresponding action of so(2, 2) (generating a space denoted by H2 ) L−1 f1 = −1 ,
L−1 f2 = 0 ,
L−1 f10 = f 0 21 ,
L−1 f20 = −f30 ,
L0 f1 = −f1 ,
L 0 f2 = 0 ,
L0 f10 = f10 ,
L0 f20 = 0 ,
L1 f1 = −f12 ,
L 1 f2 = f3 ,
L1 f10 = 1 ,
L1 f20 = 0 ,
¯ −1 f1 = 0 , L
¯ −1 f2 = −1 , L
¯ −1 f 0 = −f 0 , L 1 3
¯ −1 f 0 = f 0 2 , L 2 2
¯ 0 f1 = 0 , L
¯ 0 f2 = −f2 , L
¯ 0f 0 = 0 , L 1
¯ 0f 0 = f 0 , L 2 2
¯ 1 f1 = f3 , L
¯ 1 f2 = −f 2 , L 2
¯ 1f 0 = 0 , L 1
¯ 1f 0 = 1 . L 2
(7)
To obtain some indications about the representation content of the Klein–Gordon Eq. (4), it is useful to look at its reproducing kernel (two-point function). Indeed when we consider the invariant two point function K(y, y 0 ) associated with (4), expressed in terms of z = y · y 0 , it satisfies the differential equation (1 − z 2 )
d2 K dK −3 = 0. 2 d z dz
In the Fourier development of the general solution, a hypergeometrical function, only the three gauge modes (5) are present. This indicates that the space of solutions of (4) carries an indecomposable representation of SO(2, 2). From the actions (6) and (7) we can describe it as follows. The gauge space H3 generated by the action of so(2, 2) over the modes f0 , f3 , f30 carries the representation Id ⊕ (D(1) ⊗ D(1)) ⊕ (D(−1) ⊗ D(−1)). The space H2 generated by the action on f1 , f2 , f10 , f20 contains H3 . The physical space H2 /H3 carries the representation (D(1) ⊗ Id) ⊕ (Id ⊗ D(1)) ⊕ (D(−1) ⊗ Id) ⊕ (Id ⊗ D(−1)). Following the notations of Frønsdal, we denote the “biplet” indecomposable representation on H2 (“H2 leaking into H3 ”) by [(D(1) ⊗ Id) ⊕ (Id ⊗ D(1)) ⊕ (D(−1) ⊗ Id) ⊕ (Id ⊗ D(−1))] → [Id ⊕ (D(1) ⊗ D(1)) ⊕ (D(−1) ⊗ D(−1))] . In mathematical terms, this means [18] that we have an exact sequence of so(2, 2) modules: 0 → H2 /H3 → H2 → H3 → 0 . We know that to get a valid quantized theory, such a representation is to be supplemented by scalar modes canonically associated with gauge modes. This is precisely the role of the dipole equation; the Klein–Gordon equation plays then the role of gauge fixing. Moreover, as shown by Araki [3], when all irreducible representations involved are unitary, the gauge and scalar representations are equivalent. Therefore we add the scalar modes of (3): x− x+ x− , E = 0; f5 = , E = 2; f50 = , E = −2 . f4 = log x+ x− x+
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These modes correspond (respectively) to the gauge modes f0 , f3 , f30 . The action of so(2, 2) on them is L−1 f4 = f10 ,
L−1 f5 = −f2 ,
L−1 f50 = f50 f10 ;
L 0 f4 = 1 ,
L0 f5 = −f5 ,
L0 f50 = f50 ;
L 1 f4 = f1 , ¯ −1 f4 = f 0 , L 2
L1 f5 = −f1 f5 , ¯ −1 f5 = −f1 , L
L1 f50 = f20 ; ¯ −1 f 0 = f 0 f 0 ; L
¯ 0 f4 = 1 , L ¯ 1 f4 = f2 , L
¯ 0 f5 = −f5 , L ¯ 1 f5 = −f2 f5 , L
¯ 0f 0 = f 0 ; L 5 5 0 ¯ L1 f5 = f10 .
5
5 2
In this way we get H1 , the scalar space of (2), which contains H2 . The quotient H1 /H2 carries the representation Id ⊕ (D(1) ⊗ D(1)) ⊕ (D(−1) ⊗ D(−1)). Note that f5 and f50 do not satisfy the Klein–Gordon equation. To kill f4 we impose in addition (for gauge fixing) a condition of univaluedness. Finally we get a quantizable Gupta–Bleuler triplet [Id ⊕ (D(1) ⊗ D(1)) ⊕ (D(−1) ⊗ D(−1))] → [(D(1) ⊗ Id) ⊕ (Id ⊗ D(1)) ⊕ (D(−1) ⊗ Id) ⊕ (Id ⊗ D(−1))] → [Id ⊕ (D(1) ⊗ D(1)) ⊕ (D(−1) ⊗ D(−1))] .
(8)
Remark. The representations of negative energy that appear above have to be considered as describing anti-particles. It is only when quantized that the unitarity is restored, in complete analogy with the case of Dirac equation. As said in the Introduction, an important feature of this 3D model is its behavior on the boundary of De Sitter space. AdS3 being homeomorphic to S 1 × S 1 × R+ , its boundary is the torus S 1 × S 1 . Namely, if we set 1 1 1 p 2 1 p 2 x+ = √ Re−it ; x− = √ Reit ; y+ = √ R − 1eiθ ; y− = √ R − 1e−iθ 2 2 2 2 the boundary values are obtained by taking the limit R → ∞. The boundary values of the physical modes f1 , f2 , f10 , f20 are solutions of the 2D massless Klein–Gordon equation: (∂θ2 − ∂t2 )φ = 0 , a conformally covariant equation which has a long story. It has been studied by many people (see e.g. [22] for references) but the complete indecomposable representation of SO(2, 2) on its one particle space has been clarified only recently [6]. In this model, the gauge modes are only the constant functions. As we shall see, these are exactly the boundary values of the functions on which the indecomposable representation (8) acts. Indeed, consider the limit of the gauge modes f0 → 1, f3 → 0, f30 → 0 .
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so that the boundary values of the functions in H3 reduce to a constant. Note that for the physical modes f1 → ei(θ−t) , f2 → e−i(θ+t) , f10 → e−i(θ−t) , f20 → ei(θ+t) . The limit of H2 is then the vector space generated by ein(θ−t) , ein(θ+t) n ∈ Z .
(9)
Finally the lone limiting scalar mode (associated with the gauge mode f0 ) is f4 → 2it. Let us act now by two commuting copies {ln }, {¯ln }, n ∈ Z of the Witt algebra on these spaces by the differential operators π(ln ) = iein(θ−t) ∂θ−t , π(¯ln ) = iein(θ+t) ∂θ+t .
(10)
We then have: Proposition 1. The SO(2, 2) Gupta–Bleuler triplet (8) leads at spatial infinity to the following Gupta–Bleuler triplet for two commuting copies of the Witt algebra Id → Π(1, 1) → Id . Here we denote by Π(1, 1) the representation defined by the action of the operators (10) on the space (9), quotiented by the constant mode. 3. Difficulties in the Implementation of the Witt Algebra on AdS3 There are many difficulties in finding an infinite dimensional extension of the so(2, 2) symmetry in the bulk. The major one comes from the fact that the generators {L−1 , L0 , L1 } form a basis of the F-module of vector fields on the De Sitter space M3 . For instance, 2 2 ¯ −1 = − 2y+ L−1 + 4x− y+ L0 − 2x− L1 , L y2 y2 y2 2 2 ¯ 1 = − 2x+ L−1 + 4x+ y− L0 − 2y− L1 L y2 y2 y2 ¯ 0 = − 2x+ y+ L−1 + 2 x+ x− + y+ y− L0 − 2x− y− L1 L y2 y2 y2
(11)
and the d’Alembertian is simply = 4(L20 − L1 L−1 + L0 ) .
(12)
Indeed the whole space of solutions of the dipole equation cannot carry an infinite dimensional representation of the Witt algebra. Since this Lie algebra is Liegenerated by {l−2 , l−1 , l0 , l1 , l2 } and the representation is to be infinite dimensional, the representatives L2 , L−2 of l2 , l−2 cannot be zero.
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Now suppose that L2 is a non zero vector field on M3 which keeps invariant the space of solutions of the dipole Eq. (3), i.e. there exists a function λ such that [2 , L2 ] = λ2 . So 2[, L2 ] + [, [, L2 ]] − λ2 = 0 . Writing C = [, L2 ] − λ2 we get 2C + [, C] = 0 .
(13)
We now write (13) as relation in the representation of the enveloping algebra U(sl(2)) with generators {L−1 , L0 , L1 } and see (looking at the coefficients, starting with the highest order) that in fact C = 0. We conclude that L2 keeps invariant the space of solutions of the Klein–Gordon equation, i.e. λ [, L2 ] = . (14) 2 Expressing L2 in the preceding basis (over F) L2 = h−1 L−1 + h0 L0 + h1 L1 ; (h−1 , h0 , h1 ) ∈ F ,
(15)
we get [L1 , L2 ] = [L1 , h−1 ]L−1 + ([L1 , b] + 2h−1 )L0 + ([L1 , h1 ] + h0 )L1 . Using [L−1 , L2 ] = −3L1 we have [L−1 , h−1 ] = h0 , [L−1 , h0 ] = 2h1 , [L−1 , h1 ] = −3 ,
(16)
and [L0 , L2 ] = −2L2 gives [L0 , h−1 ] = −3h−1 , [L0 , h0 ] = −2h0 , [L0 , h1 ] = −h1 .
(17)
Thus 1 [, L2 ] = − [L1 , h−1 ]L2−1 − 4h0 L20 + 3L21 − 4h1 L1 L0 − ([L1 , h0 ] + 6h−1 )L0 L−1 4 − ([L1 , h1 ] + h0 )L1 L−1 + 2h0 L0 + 6h−1 L−1 + 2h1 L1 . The presence of the term 3L21 shows that this equation is incompatible with (12) and (14). Therefore L2 = 0, and the representation is finite dimensional. Another difficulty comes from the fact that the representation of so(2, 2) cannot be extended to any representation of two commuting copies of the Witt algebra. ¯ −1 , L2 ] = 0 (commutativity of Indeed, let us write L2 as in (15). Assuming then [L the two copies of the Witt algebra) we get with (11) [L1 , z] = 2
y2 y+ [L0 , z] − + [L−1 , z], z ∈ {h−1 , h0 , h1 } x− x2−
(18)
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¯ 1 , L2 ] = 0 gives while [L [L1 , z] = 2
x2 x+ [L0 , z] − + 2 [L−1 , z], z ∈ {h−1 , h0 , h1 } . y− y−
From these two equations we obtain 1 y+ x+ [L−1 , z], z ∈ {h−1 , h0 , h1 } [L0 , z] = + 2 x− y− which (on the basis of the previous calculations (16) and (17)) implies 3 2 1 y+ x+ 3 y+ x+ 3 y+ x+ + , h0 = − + , h1 = + h−1 = . 8 x− y− 4 x− y− 2 x− y− A contradiction comes from the fact that [L1 , h1 ] =
−3 2
2 x2 y+ + + 2 2 x− y−
does not satisfy (18). We must conclude that in three dimensional space-time, we cannot extend the so(2, 2) symmetry to a direct sum of two copies of the Witt algebra. Preliminary calculations show that even if we restrict to subspaces such as H1 , this remains true. 4. Wess Zumino Novikov Witten Model We now apply results of the last section to the WZNW model. With the obvious exception of the massless Klein–Gordon equation, this is the only known 2D conformal field theory to be described in terms of an action. Let φ(x) be a field, valued in some simple Lie algebra g satisfying the dipole equation: 2g φ(x) = 0 , (19) where g is the covariant Laplace operator, defined in terms of a fixed metric g on a three dimensional space. φ(x) has to be seen as a linear approximation of a field U (x) = exp(φ(x)) valued in a compact Lie group with Lie algebra g. So (19) is the linearized version of the equation 1 1 ρ 1 ρ √ µν µ ∂ gg ∂ ∂ A + A , ∂ ∂ A = 0, (20) √ µ √ √ ν ρ µ ρ g g g where Aµ is the flat connection associated with U (x): Aµ = U −1 (∂µ U ) . In the linearized version one has Aµ = ∂µ φ. It is shown in [2] that under suitable boundary conditions on A we can find a bilinear form g and a Lagrangian whose variation gives (20) and such that it reduces on the 2D boundary to the WZNW model (using the notations of the preceding section): ∂θ+t Aθ−t = 0
(21)
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so that in the limit R → ∞ the field φ satisfies the conformal covariant equation (∂θ2 − ∂t2 )φ = 0 . To give a representation of the Witt algebra (and of the loop algebra over g) we specify the metric g to be of the De Sitter type and again restrict the space of solutions of (20) to some quantizable Gupta–Bleuler triplet. Let ta , a ∈ I, be a basis of g and fi , fi0 the modes introduced in Sec. 2. We a define the modes fia = fi ta , i = {0, 1, 2, 3, 4, 5}, a ∈ I and f 0 i = fi0 ta , i = {1, 2, 3, 5}, a ∈ I. Then a straightforward generalization of Proposition 1 shows that, taking again the SO(2, 2) generators as (2), their action on the scalar modes f4a , f5a , f 0 a5 , a ∈ I generates a vector space carrying a triplet similar to (8) where each irreducible representation appears N times (N being the dimension of g). We start the vectorial formulation with the gradient of these solutions. Since the constant gauge mode is mapped to zero, we may forget its associated logarithmic scalar modes. The (linearized) flat connection defined by a 0 0 Ai,a µ = ∂µ fi , A µ = ∂µ f i ; µ = 0, 1, 2, 5, i = 1, 2, 3, 5, a ∈ I i,a
a
generate the subspace H10 of solutions of the non-linear Eq. (20) which satisfy i,a Fµν = ∂µ Ai,a ν − ∂ν Aµ = 0 ,
F 0 µν = ∂µ A0 ν − ∂ν A0 µ = 0 . i,a
i,a
We can define a representation of so(2, 2) on the vector space generated by the gradients of all the solutions of (19) (with De Sitter metric) by ¯ i φ; i = −1, 0, 1 . ¯ 0 Aµ ≡ ∂µ L ∂µ φ = Aµ 7→ L0 i Aµ ≡ ∂µ Li φ; Aµ 7→ L i It is well defined since Aµ = ∂µ φ = ∂µ ψ, ∀ µ, implies φ − ψ = constant, on which so(2, 2) acts trivially. From there follows immediately that the space H10 has the Gupta–Bleuler structure over so(2, 2) N [(D(1) ⊗ D(1)) ⊕ (D(−1) ⊗ D(−1))] → N [(D(1) ⊗ Id) ⊕ (Id ⊗ D(1)) ⊕ (D(−1) ⊗ Id) ⊕ (Id ⊗ D(−1))] → N [(D(1) ⊗ D(1)) ⊕ (D(−1) ⊗ D(−1))] . Each irreducible representation appears N times, where N is the dimension of g. Let us finally study the boundary values of this modes. At spatial infinity the gauge modes are mapped to zero, and it is not necessary to consider their associated scalar modes. Nevertheless the physical modes √ √ R2 − 1 i(θ−t) R2 − 1 −i(θ+t) 1,a 2,a Aθ−t = i e ta , Aθ+t = e ta , R iR √ √ R2 − 1 −i(θ−t) R2 − 1 i(θ+t) 0 1,a 0 2,a A θ−t = e ta , A θ+t = i e ta , a ∈ I iR R have a non trivial limit satisfying the WZNW Eq. (21) and ∂θ−t Aθ+t = 0 .
(22)
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2,a 0 0 The other physical modes A1,a θ+t , Aθ−t , A θ+t and A θ−t are zero, while 1,a
2,a
A1,a R =
ei(θ−t) ta √ , R2 R2 − 1
A2,a R =
e−i(θ+t) ta √ , R2 R2 − 1
A0 R =
e−i(θ−t) ta √ , R2 R2 − 1
A0 R =
ei(θ+t) ta √ R2 R2 − 1
1,a
2,a
have zero for limit. We obtain then at spatial infinity the vector space W generated by ein(θ−t) ta , ein(θ+t) ta ; n ∈ Z − {0}, a ∈ I ,
(23)
which again carries a representation of two commuting copies of the Witt algebra given by (10). Furthermore W carries a representation of the loop algebra ˆg = g ⊗ C[z, z −1 ]. A basis of ˆg is given by Jna , n ∈ Z, a ∈ I with commutation relations b c c [Jna , Jm ] = kab Jn+m , c where kab are the structure constants of g. Then (23) carries two representations ρ1 and ρ2 of gˆ given by
ρ1 (Jna ) = ein(θ−t) ta , ρ2 (Jna ) = ein(θ+t) ta . Note that these generators are nothing but the Fourier components of the general solutions Aθ−t (resp. Aθ+t ) of (21) (resp. (22)). In conclusion, when we restrict the metric g to be of De Sitter type, we can construct in the bulk a so(2, 2) Gupta–Bleuler triplet of solutions of the non-linear Eq. (20). Furthemore we obtain at infinity solutions of the WZNW model that carry a representation of the loop algebra gˆ and of a direct sum of two Witt algebras. Acknowledgements The author is very grateful to Mosh´e Flato for suggesting the problem and discussing it with patience, to Daniel Sternheimer for helpful criticism and style corrections of the manuscript and to Christiane Martin and Mourad Laoues for useful discussions. References [1] E. Angelopoulos and M. Laoues, “Masslessness in n-dimensions”, Rev. Math. Phys. 10(3) (1998) 271–299 (hep-th/9806100). [2] A. M. Harun Ar-Rashid, C. Fronsdal and M. Flato, “Three-D Singletons and 2-D C.F.T.”, Int. J. Mod. Phys. A7 (1992) 2193–2206. [3] H. Araki, “Indecomposable representations with invariant inner product. A theory of Gupta–Bleuler triplet”, Commun. Math. Phys. 97 (1985) 149–159. [4] A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, “Infinite conformal symmetry in two dimensional quantum field theory”, Nucl. Phys. B241 (1984) 333–380.
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[5] E. Bergshoeff, A. Salam, E. Sezgin and Y. Tanii, “Singletons, higher spin massless states and the supermembrane”, Phys. Lett. B205 (1988) 237–244. [6] S. De Bi`evre and J. Renaud, “The indecomposable representation of SOo (2, 2) on the one-particle space of the massless field in 1 + 1 dimension”, Lett. Math. Phys. 34 (1995) 385–393. [7] P. A. M. Dirac, “A remarkable representation of the 3 + 2 De Sitter group”, J. Math. Phys. 4 (1963) 901–909. [8] S. Ferrara and C. Frønsdal, “Conformal Maxwell theory as a singleton field theory on AdS5 , IIB three branes and duality”, hep-th/9712239. [9] M. Flato, “Two disjoint aspects of the deformation programme: quantizing Nambu mechanics; singleton physics” hep-th/9809073, to be published in Particles, Fields and Gravitation, ed. J. Rembieli´ nski, AIP Press, 1998. [10] M. Flato and C. Frønsdal, “One massless particle equals two Dirac Singletons”, Lett. Math. Phys. 2 (1978) 421–426. [11] M. Flato and C. Fronsdal, “Singletons: fundamental gauge theory”, in Topological and Geometrical Methods in Field Theory, Espoo, 1986, World Sci. Publ., Teaneck NJ, 1986, pp. 273–290. [12] M. Flato and C. Fronsdal, “Quarks or singletons”, Phys. Lett. B172 (1986) 412–416. [13] M. Flato and C. Fronsdal, “The singleton dipole”, Commun. Math. Phys. 108 (1987) 469–482. [14] M. Flato and C. Fronsdal, “Composite electrodynamics”, J. Geom. Phys. 5 (1988) 37–61. [15] M. Flato and C. Fronsdal, “Parastatistics, highest weight osp(n, ∞) modules, singleton statistics and confinement”, J. Geom. Phys. 6 (1989) 293–309. [16] M. Flato and C. Fronsdal, “Three-dimensional singletons”, Lett. Math. Phys. 20 (1990) 65–74. [17] M. Flato and C. Frønsdal, “Interacting singletons”, Lett. Math. Phys. 44 (1998) 249– 259 (hep-th/9803013). [18] M. Flato, C. Fronsdal and D. Sternheimer, “Singletons as a basis for composite conformal quantum electrodynamic”, in Math. Phys. Studies 10, Quantum Theories and Geometry, Kluwer Academic Publ., 1988, pp. 65–76. [19] D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, “Correlation functions in the CF Td /AdSd+1 correspondence”, hep-th/9804058. [20] M. Laoues, “Massless particles in arbitrary dimensions”, hep-th/9806101, Rev. Math. Phys. 10 (1998). [21] J. M. Maldacena, “The large N limit of superconformal field theories and supergravity”, Adv. Theor. Math. Phys. 2 (1998) 231–252 (hep-th/9711200). [22] G. Morchio, D. Pierotti and F. Strocchi, “Infrared and vacuum structure in two dimensional quantum fields theory models. The massless scalar field”, J. Math. Phys. 31 (1990) 1467–1477. [23] H. Sugawara, “A field theory of currents”, Phys. Rev., Second Series 170(5) (1968) 1659–1662. [24] E. Witten, “Anti de Sitter space and holography”, Adv. Theor. Math. Phys. 2 (1998) 253–291 (hep-th/9802150). [25] J. Wess and B. Zumino, “Consequences of anomalous Ward identities”, Phys. Lett. 37B (1971) 95–97; S. P. Novikov, “The Hamiltonian formalism and a many-valued analogue of Morse theory”, Russ. Math. Surv. 37(5) (1982) 1–56 (Uspekhi Mat. Nauk 37(5) (1982) 3–49, 248); “Multivalued functions and functionals. An analogue of the Morse theory” Sov. Math., Dokl. 24 (1981) 222–226 (Dokl. Akad. Nauk SSSR 260 (1981) 31–35); E. Witten, “Nonabelian bosonization in two dimensions”, Commun. Math. Phys. 92 (1984) 455–472.
SELF-DUAL YANG MILLS: SYMMETRIES AND MODULI SPACE A. D. POPOV Bogoliubov Laboratory of Theoretical Physics JINR, 141980 Dubna, Moscow Region, Russia E-mail : [email protected] Received 16 June 1998 Geometry of the solution space of the self-dual Yang–Mills (SDYM) equations in Euclidean four-dimensional space is studied. Combining the twistor and group-theoretic approaches, we describe the full infinite-dimensional symmetry group of the SDYM equations and its action on the space of local solutions to the field equations. It is argued that owing to the relation to a holomorphic analogue of the Chern–Simons theory, the SDYM theory may be as solvable as 2D rational conformal field theories, and successful nonperturbative quantization may be developed. An algebra acting on the space of self-dual conformal structures on a 4-space (an analogue of the Virasoro algebra) and an algebra acting on the space of self-dual connections (an analogue of affine Lie algebras) are described. Relations to problems of topological and N = 2 strings are briefly discussed.
Contents 1. Introduction 2. Self-Duality and Manifest Symmetries 2.1. Definitions and notation 2.2. Gauge symmetries 2.3. Conformal symmetries 3. Complex Geometry of Twistor Spaces 3.1. Complex structure on R4 3.2. Riemann sphere CP1 3.3. Twistor space 3.4. Real structure on twistor space 4. The Penrose–Ward Correspondence 4.1. Complex vector bundles over U and P 4.2. Self-duality ⇒ holomorphy 4.3. Gauge transformations and holomorphic equivalence 4.4. Unitarity conditions 4.5. Riemann–Hilbert problems 4.6. Holomorphy ⇒ self-duality ˇ 5. Holomorphic Bundles in the Cech Approach 5.1. Moduli space of holomorphic bundles over the twistor space P 5.2. Action of the group C 0 (U, H) on the space Z 1 (U, H) 5.3. Action of the group C 1 (U, H) on the space Z 1 (U, H) 5.4. The group H(P) of automorphisms of the complex manifold P 5.5. Action of the group H(P) on the space Z 1 (U, H) 6. Symmetries in Holomorphic Setting 6.1. Germs of sets and groups 6.2. Holomorphic triviality of bundles E 0 on CP1x ,→ P 6.3. Jumping points and jumping lines 6.4. Representatives M0 and N0 of the germs M and N 1091 Reviews in Mathematical Physics, Vol. 11, No. 9 (1999) 1091–1149 c World Scientific Publishing Company
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ˇ 6.5. Symmetries of local solutions in the Cech approach 6.6. Unitarity conditions 7. Holomorphic Bundles: the Dolbeault Description 7.1. Some definitions ˆ Bˆ0,q and Bˆ 7.2. The sheaves S, 7.3. The sheaves S, B0,q and B 7.4. Exact sequences of sheaves 7.5. The group H 0 (P, S) and the cohomology set H 1 (P, S) 7.6. Exact sequences of cohomology sets 7.7. Unitarity conditions 8. Symmetries in Terms of Smooth Sheaves 8.1. Riemann–Hilbert problems from the cohomological point of view 8.2. Action of the symmetry group Gτ on real solutions of the SDYM equations 8.3. Gauge fixing and some formulae 8.4. Generalization to self-dual manifolds 9. Discussion 9.1. What is integrability? 9.2. Holomorphic Chern–Simons–Witten theory 9.3. N = 2 and N = 4 topological strings 9.4. Integrable 4D conformal field theories ˇ 9.5. The Cech description of the Virasoro algebra 9.6. Infinitesimal deformations of self-dual conformal structures 9.7. Quantization 10. Conclusion Acknowledgements Appendix A. Actions of Groups on Sets Appendix B. Sheaves of (non-Abelian) Groups Appendix C. Cohomology Sets and Vector Bundles References
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1. Introduction In the past two decades significant progress in understanding integrable, conformal and topological quantum field theories in two dimensions has been achieved. In many respects this progress was related to the existence of an infinite number of symmetries making it possible not only to describe the space of classical solutions to 2D field equations, but also to advance essentially the nonperturbative quantization of 2D theories. Among symmetry algebras of 2D models, the most important role is played by the Virasoro and affine Lie algebras (see e.g. [1–5]). The use of transformation groups, their orbits and representations forms the basis of the dressing transformation method [6–10] and of the Kyoto’s school approach [1, 11, 12] to solvable equations of 2D and 3D field theories. In four dimensions there also exist integrable, conformal and topological field theories, and naturally the following question arises: Can the methods and results of 2D theories be transferred to 4D theories? On the whole, the answer is positive for 4D integrable and topological field theories. At the same time, the knowledge of 4D conformal field theories (CFT) beyond the trivial case of free field theories is much less explicit, and not so many exact results are obtained (see e.g. [13–16] and references therein). Usually, one connects this with the fact that, unlike the 2D case, the conformal group in four dimensions is finite-dimensional, and constraints arising from conformal invariance are not sufficient for a detailed description of 4D
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CFT’s. One of our aims is to demonstrate that for a special subclass of CFT’s — integrable 4D CFT’s — this is a wrong impression based on the consideration of only local (manifest) symmetries. There actually exists only one nonlinear integrable model in 4D described by the self-dual Yang–Mills (SDYM) equations defined on a 4-manifold with the selfdual Weyl tensor [17–19]. This unique theory is conformally invariant, and it is usually considered as a 4D analogue of the 2D WZNW theory. It is expected that many results of 2D rational CFT’s can be extended to the SDYM theory. This was discussed, for instance, in [20, 21], where the quantization of the SDYM theory on K¨ ahler manifolds was considered. We shall give additional arguments in favour of the conjecture that the SDYM model is a good starting point for the development of 4D quantum CFT’s. The main purpose of our paper is to describe all symmetries of the SDYM equations and, in particular, algebras generalizing the Virasoro and affine Lie algebras to the 4D case. In contrast with the WZNW model, most symmetries of the SDYM model are nonlocal. These symmetries are local symmetries of a holomorphic analogue of the Chern–Simons theory on a 6D twistor space, and the SDYM theory is connected with this model via the nonlocal Penrose–Ward transform. The use of this correspondence makes it possible to simplify considerably the investigation of symmetries of the SDYM equations. The lift from a 4D self-dual space to its 6D twistor space is useful for understanding correct degrees of freedom and correct symmetries of the SDYM theory. We show that just as in the case of the 2D WZNW theory these symmetries completely define the space of local solutions to the field equations and therefore the quantization of the SDYM theory is connected with the construction of representations of a symmetry algebra. Roughly speaking, the symmetry group of a system of differential equations is the group that maps solutions of this system into one another. From this point of view the transformation groups of type Map(X3 ; G) (maps: space X3 → group G, dimR X3 = 3) considered in [20, 21] are not symmetry groups, since in general their action does not preserve the solution space. The above-mentioned groups Map(X3 ; G) can be considered as “off-shell” symmetry groups reflecting only the field content of the theory and acting on free fields. These groups are not connected with the integrability and can be introduced in a space-time of an arbitrary dimension (see e.g. [22]). Study of “on-shell” infinitesimal symmetries of the SDYM equations (in 4D Euclidean space) began from the papers [23] and was continued in [24–33]. In [24, 25] it has been shown that the obtained infinitesimal symmetries form the affine Lie algebra g ⊗ C [λ, λ−1 ] when the gauge potential A = Aµ dxµ takes values in the Lie algebra g of a group G. Ueno and Nakamura [26] have shown that on the solution space of the SDYM equations it is possible to define an infinitesimal action of a larger Lie algebra of holomorphic maps from a domain on the twistor space Z of R4 into the algebra g. Takasaki [27] described this algebra in terms of Sato’s approach to soliton equations. But Crane [28] showed that in general the group
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corresponding to the Ueno–Nakamura algebra does not preserve the solution space of the SDYM equations and indicated a vagueness of geometrical meaning of these transformations. The above-mentioned infinitesimal symmetries do not exhaust all symmetries of the SDYM equations, which has been shown in the papers [30–32] where Virasoro-type symmetries were described. In this paper, we describe the full symmetry group of the SDYM equations. 2. Self-Duality and Manifest Symmetries 2.1. Definitions and notation We consider the Euclidean space R4 with the metric δµν , a gauge potential A = Aµ dxµ and the Yang–Mills field F = dA + A ∧ A with components Fµν = ∂µ Aν − ∂ν Aµ + [Aµ , Aν ], where µ, ν, . . . = 1, . . . , 4, ∂µ := ∂/∂xµ . The fields Aµ and Fµν take values in the Lie algebra g of an arbitrary semisimple compact Lie group G. We suppose that G is a matrix group G ⊂ GL(n, C). The SDYM equations have the form: 1 εµνρσ Fρσ = Fµν , 2
(2.1)
where εµνρσ is the completely antisymmetric tensor in R4 and ε1234 = 1. Here, and throughout the paper, we use the Einstein summation convention, unless otherwise stated. In this paper, we study the space of smooth local solutions to the SDYM Eqs. (2.1). More precisely, we suppose Aµ to be smooth on an arbitrary open ball U in R4 , and we do not fix boundary conditions for Aµ . As U we shall also consider open subsets in R4 which can be dense subsets in R4 and can even coincide with R4 . We shall consider smooth solutions of the SDYM equations on U lying in an open neighbourhood of some fixed solution A0µ , for instance, in a neighbourhood of the vacuum A0µ = 0 (local solutions). The set of local solutions is an infinitedimensional space and contains finite-dimensional moduli spaces of global solutions, (instantons, monopoles, etc.) as subspaces. 2.2. Gauge symmetries Equation (2.1) are manifestly invariant under the group of gauge transformations Aµ 7→ Agµ = g −1 Aµ g + g −1 ∂µ g ,
g Fµν 7→ Fµν = g −1 Fµν g ,
(2.2)
where g = g(x) ∈ G, x ∈ U ⊂ R4 . For infinitesimal gauge transformations we have δϕ Aµ = Dµ ϕ ≡ ∂µ ϕ + [Aµ , ϕ] ,
(2.3)
where ϕ(x) ∈ g, x ∈ U . The fields Aµ and Agµ differing by the gauge transformation (2.2) are considered to be equivalent. That is why gauge transformations are “trivial” symmetries.
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2.3. Conformal symmetries It is well known that the SDYM Eqs. (2.1) are invariant with respect to (w.r.t.) the group of conformal transformations of the space R4 . This group is locally isomorphic to the group SO(5, 1). On the coordinates xµ and components Aµ of the gauge potential A the group of conformal transformations acts in the following way: translations: xµ 7→ x ˜µ = xµ + aµ , ˜µ = aµν xν , rotations: xµ 7→ x dilatations: xµ 7→ x ˜µ = eα xµ , special conformal transformations:
˜µ = xµ 7→ x =
Aµ (xν ) 7→ A˜µ = Aµ (xν + aν ) , (2.4a)
Aµ (xν ) 7→ A˜µ = (a−1 )σµ Aσ (aνρ xρ ) ,
(2.4b)
Aµ (xν ) 7→ A˜µ = e−α Aµ (eα xν ) ,
(2.4c)
µ
µ 2
x +α x , 1 + 2αν xν + α2 x2
Aµ (xν ) 7→ A˜µ
∂xσ Aσ (˜ xν ) , ∂x ˜µ
(2.4d)
where a = (aµν ) ∈ SO(4), aµ , α, αµ ∈ R, α2 := αν αν , x2 := xν xν . For infinitesimal conformal transformations we have δN Aµ = LN Aµ ≡ N ν ∂ν Aµ + Aν ∂µ N ν ,
(2.5)
where LN is the Lie derivative along a vector field N and N = N ν ∂ν is any generator of the 15-parameter conformal group, b Xa = δab ηµν xµ ∂ν ,
Kµ =
b Ya = δab η¯µν xµ ∂ν ,
1 2 x ∂µ − xµ xν ∂ν , 2
D = xν ∂ν .
Pµ = ∂µ , (2.6)
Here {Xa } and {Ya }, a, b, . . . = 1, 2, 3, generate two commuting SO(3) subgroups in SO(4), Pµ are the translation generators, Kµ are the generators of special confora mal transformations, D is the dilatation generator, ηµν = {abc , µ = b, ν = c; δµa , ν = a 4; −δνa , µ = 4} are the self-dual ’t Hooft tensors and η¯µν = {abc , µ = b, ν = a a c; −δµ , ν = 4; δν , µ = 4} are the anti-self-dual ’t Hooft tensors. Remark. It is well known that for semisimple structure groups G there are no local symmetries of the SDYM equations differing from the gauge and conformal symmetries described above. 3. Complex Geometry of Twistor Spaces 3.1. Complex structure on R 4 To write down a linear system for Eqs. (2.1) and to clarify its geometrical meaning, it is necessary to introduce a complex structure J on R4 (and thus on any open subset U ⊂ R4 ). This means that we must introduce on R4 a tensor Jµν such that Jµσ Jσν = −δµν . It is well known that all constant complex structures on R4 are parametrized by the two-sphere S 2 ' SO(4)/U (2), and the most general
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form of Jµν is a σν Jµν = sa η¯µσ δ , a
2
(3.1)
a
where real numbers s parametrize S : sa s = 1. Using the identities for the ’t Hooft tensors a b c η¯νσ = δ ab δµν + abc η¯µν , (3.2) η¯µσ it can be shown that J 2 = −1. The other admissible choice of the complex struca σν ture J˜µν = sa ηµσ δ corresponds to choosing the opposite orientation on R4 and transition from self-duality to anti-self-duality equations. Eigenvalues of the operator J = (Jµν ) (applied to vectors) are ±i, and we can introduce two subspaces in C4 = R4 ⊗ C, V 1,0 = {V ∈ C4 : Jσµ V σ = iV µ } ,
V 0,1 = {V ∈ C4 : Jσµ V σ = −iV µ } .
(3.3a)
As a basis in V 1,0 and V 0,1 one may take vectors with the components 1 i 1¯ i ¯ 1¯ i ¯ 1 i (1)µ (1)µ {V1 } = , − , − λ, λ , {V2 } = λ, λ, , , (3.3b) 2 2 2 2 2 2 2 2 1 i i 1 i 1 i 1 (1)µ (1)µ , , − λ, − λ , {V¯2 } = λ, − λ, , − , (3.3c) {V¯1 } = 2 2 2 2 2 2 2 2 ¯ are local holomorphic and antiholomorphic coordinates on the sphere where λ and λ S 2 , λ = (s1 + is2 )/(1 + s3 ). (1) (1)µ Using J, one can introduce vector fields VA = VA ∂µ of the type (1, 0) and (1) (1)µ vector fields V¯A = V¯A ∂µ of the type (0, 1) w.r.t. J, where A, B, . . . = 1, 2. We have 1 λ (1) (1)µ V¯1 = V¯1 ∂µ = (∂1 + i∂2 ) − (∂3 + i∂4 ) = ∂y¯1 − λ∂y2 , (3.4a) 2 2 1 λ (1) (1)µ V¯2 = V¯2 ∂µ = (∂3 − i∂4 ) + (∂1 − i∂2 ) = ∂y¯2 + λ∂y1 , 2 2
(3.4b)
where y 1 = x1 + ix2 ,
y 2 = x3 − ix4 ,
y¯1 = x1 − ix2 ,
y¯2 = x3 + ix4
(3.5)
are the canonical complex coordinates on R4 ' C2 . 3.2. Riemann sphere C P 1 In (3.3) we have introduced the complex coordinate λ on S 2 ' CP1 , parametrizing complex structures on R4 . Using the stereographic projection S 2 → R2 , one can introduce two coordinate patches Ω1 ' R2 and Ω2 ' R2 of the sphere with the coordinates s1 s2 2 and ω = on Ω1 , ω11 = 1 1 + s3 1 + s3 ω21 =
s1 1 − s3
and ω22 =
in which the metric on S 2 is conformally flat.
s2 1 − s3
on Ω2 ,
(3.6)
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We introduce the standard complex structure j on S 2 with the components j = (jA B) ,
C A jA C jB = −δB ,
j12 = −j21 = −1
(3.7)
in the coordinates {ω1A }. Now we can introduce vector fields, holomorphic and antiholomorphic w.r.t. j, on Ω1 as (1)
V3
=
1 (∂ 1 − i∂ω12 ) = ∂λ , 2 ω1
(1)B
jA B V3
(1)A
= iV3
,
(3.8)
1 (1) ¯ (1)B = −iV¯ (1)A , V¯3 = (∂ω11 + i∂ω12 ) = ∂λ¯ , jA (3.4c) B V3 3 2 where λ = ω11 +iω12 is the complex coordinate on Ω1 ' C. Analogously, we introduce (2) the complex coordinate ζ = ω21 − iω22 on Ω2 ' C and vector fields V3 = ∂ζ , (2) V¯3 = ∂ζ¯ on Ω2 . So the sphere S 2 can be covered by two coordinate patches Ω1 , Ω2 , with Ω1 , the neighbourhood of λ = 0, and Ω2 , the neighbourhood of λ = ∞. Let us fix α1 , α2 : 0 ≤ α1 < 1 < α2 ≤ ∞ and put Ω1 = {λ ∈ C : |λ| < α2 } ,
Ω2 = {λ ∈ C ∪ ∞ : |λ| > α1 } .
(3.9)
The sphere S 2 , considered as a complex projective line CP1 = Ω1 ∪ Ω2 , is the complex manifold obtained by patching together Ω1 and Ω2 with the coordinates λ and ζ related by ζ = λ−1 on Ω1 ∩ Ω2 . For example, if Ω1 = {λ ∈ C : |λ| < ∞} and Ω2 = {λ ∈ C ∪ ∞ : |λ| > 0}, Ω1 ∩ Ω2 is the multiplicative group C∗ of complex numbers λ 6= {0, ∞}. 3.3. Twistor space We consider an open subset U in R4 . As a smooth manifold the twistor space P ≡ P(U ) of U is a direct product of the spaces U and CP1 : P = U × CP1 and is the bundle of complex structures on U [19]. This space can be covered by two coordinate patches: P = U1 ∪ U2 ,
U1 = U × Ω1 ,
U2 = U × Ω2 ,
(3.10a)
¯ on U1 and {xµ , ζ, ζ} ¯ on U2 . The two-set open cover with the coordinates {xµ , λ, λ} 1 O = {Ω1 , Ω2 } of the Riemann sphere CP was described in Sec. 3.2. We shall consider the intersection U12 of U1 and U2 U12 := U1 ∩ U2 = U × (Ω1 ∩ Ω2 )
(3.10b)
¯ ∈ Ω12 := Ω1 ∩ Ω2 . Thus, the twistor space P is with the coordinates xµ ∈ U , λ, λ ¯ → {xµ } a trivial bundle π : P → U over U with the fibre CP1 , where π : {xµ , λ, λ} is the canonical projection. We shall also consider the twistor space Z ≡ Z(R4 ) of R4 which as a smooth manifold is a direct product Z = R4 × CP1 . The twistor space P is an open subset
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of Z. In its turn, Z ' CP3 − CP1 is an open subset in the space CP3 which is the twistor space of the sphere S 4 . Formally, P coincides with Z if we take U = R4 ; that is why we denote the cover of Z by the same letters U1 = R4 ×Ω1 , U2 = R4 ×Ω2 . Since P is an open subset of Z, a complex structure will be discussed for Z. Having the complex structure J on R4 and the complex structure j on S 2 , we (1) can introduce a complex structure J = (J, j) on Z. The vector fields {V¯a } on U1 , introduced in (3.4), are vector fields of the type (0,1) w.r.t. the complex structure (2) J . Vector fields {V¯a } of the type (0,1) on U2 have the form: (2) V¯1 = ζ∂y¯1 − ∂y2 ,
(2) V¯2 = ζ∂y¯2 + ∂y1 ,
(2) V¯3 = ∂ζ¯ ,
(3.11a,b,c)
and we have (1) (2) V¯1 = λV¯1 ,
(1) (2) V¯2 = λV¯2 ,
(1) ¯ 2 V¯ (2) V¯3 = −λ 3
(3.12a,b,c)
on U12 = U1 ∩ U2 . Now we can introduce complex coordinates {z1a } on U1 and {z2a } on U2 as solu(1) (2) tions of the equations V¯a (z1b ) = 0 and V¯a (z2b ) = 0. We have z11 = y 1 − λ¯ y2 ,
z12 = y 2 + λ¯ y1 ,
z13 = λ ,
(3.13a)
z21 = ζy 1 − y¯2 ,
z22 = ζy 2 + y¯1 ,
z23 = ζ
(3.13b)
and on the intersection U12 these coordinates are connected by the holomorphic transition function f12 a z1a = f12 (z2b )
⇔
2 z12 = f12 (z2b ) =
z22 , z23
1 z11 = f12 (z2b ) =
z21 , z23
3 z13 = f12 (z2b ) =
1 . z23
(3.13c)
From (3.13) it is not difficult to derive the formulae ∂ (1) = γ1 V¯1 , ∂ z¯11
∂ (1) = γ1 V¯2 , ∂ z¯12
∂ (1) (1) (1) = V¯3 + y¯2 γ1 V¯1 − y¯1 γ1 V¯2 , ∂ z¯13
(3.14a)
¯ Analogously, on U2 where γ1 = 1/(1 + λλ). ∂ (2) = γ2 V¯1 , ∂ z¯21
∂ (2) = γ2 V¯2 , ∂ z¯22
∂ (2) (2) (2) = V¯3 − y¯1 γ2 V¯1 − y¯2 γ2 V¯2 , ∂ z¯23
(3.14b)
¯ where γ2 = 1/(1 + ζ ζ). It is easy to check that the local basis (0,1)-forms w.r.t. J are 1 ¯ 2 ), θ¯2 = γ1 (d¯ ¯ 1 ), θ¯3 = dλ ¯ on U1 , θ¯(1) = γ1 (d¯ y 1 − λdy y 2 + λdy (1) (1)
(3.15a)
1 ¯ y 1 − dy 2 ), θ¯2 = γ2 (ζd¯ ¯ y 2 + dy 1 ), θ¯3 = dζ¯ on U2 . θ¯(2) = γ2 (ζd¯ (2) (2)
(3.15b)
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¯ d = ∂ + ∂, ¯ where The exterior derivative d on Z splits into ∂ and ∂: ∂ a ¯ (1) ∂¯ = d¯ z1a a = θ¯(1) Va ∂ z¯1
on U1 ,
(3.16a)
∂ a ¯ (2) ∂¯ = d¯ z2a a = θ¯(2) Va ∂ z¯2
on U2 ,
(3.16b)
and the operator ∂ is connected with ∂¯ by means of complex conjugation. As usual ¯ = 0. d2 = ∂ 2 = ∂¯2 = ∂ ∂¯ + ∂∂ It follows from (3.12), (3.13) and (3.15) that as a complex manifold Z is not a direct product C2 ×CP1 , but is a nontrivial holomorphic vector bundle p : Z → CP1 . Moreover, from (3.12), (3.13) and (3.15) it follows that Z coincides with a total space of the rank 2 holomorphic vector bundle L−1 ⊕ L−1 over CP1 , p : Z = L−1 ⊕ L−1 −→ CP1 ,
(3.17)
where L is the tautological complex line bundle over CP1 with the transition function λ−1 , and the first Chern class c1 (L) equals −1: c1 (L) = −1. Its dual L−1 is isomorphic to the hyperplane bundle (Chern class c1 (L−1 ) = 1) over CP1 . The twistor space P of U ⊂ R4 is an open subset of Z and Z = L−1 ⊕ L−1 ' CP3 − CP1 is an open subset of CP3 . Holomorphic sections of the bundle (3.17) are projective lines y 2 , z12 = y 2 + λ˜ y1 λ ∈ Ω1 : z11 = y 1 + λ˜ 1 CPy = (3.18) ζ ∈ Ω2 : z21 = ζy 1 + y˜2 , z22 = ζy 2 + y˜1 parametrized by the points y = {y 1 , y 2 , y˜1 , y˜2 } ∈ C4 . 3.4. Real structure on twistor space A real structure on the complex twistor space Z is defined as an antiholomorphic ¯ on the CP1 factor, involution τ : Z → Z, defined by the antipodal map λ 7→ −1/λ ¯ , τ (xµ , λ) = (xµ , −1/λ)
τ2 = 1 .
(3.19)
This involution takes the complex structure J on Z to its conjugate −J , i.e., it is antiholomorphic. It is obvious from the definition (3.19) that τ has no fixed points on P ⊂ Z but does leave the fibres CP1x , x ∈ U , of the bundle P → U invariant. The same is true for the fibres CP1x of the bundle Z → R4 . Fibres CP1x of the bundle P → U are also real holomorphic sections of the bundle (3.17) for which we have y˜1 = y¯1 , y˜2 = −¯ y 2 in (3.18), i.e., they are parametrized by {xµ } = {y A , y¯A } ∈ U . An extension of the involution τ to complex functions f (xµ , λ) has the form [34]: ¯ −1 ) . τ : f (x, λ) 7→ τ (f (x, λ)) ≡ fτ (x, λ) := f (τ (x, λ)) = f (x, −λ
(3.20)
In particular, for the complex coordinates {z1a } and {z2a } on Z we have τ (z11 ) = z22 , τ (z1a ) = Bba z2b ,
τ (z12 ) = −z21 , B21 = 1 ,
τ (z13 ) = −z23 ,
B12 = −1 ,
⇔
B33 = −1 .
All the rest components of the constant matrix B = (Bba ) are equal to zero.
(3.21)
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A. D. POPOV
Using (3.21), it is not difficult to verify that for the transition function (3.13c) compatible with the real structure τ , we have a b τ (f12 ) = Bba f˜12 ,
(3.22)
where f˜12 is the transition function inverse to f12 z1 1 z21 = f˜12 (z1b ) = 13 , z1
z2 2 z22 = f˜12 (z1b ) = 13 , z1
1 3 z23 = f˜12 (z1b ) = 3 . z1
(3.23)
So all the holomorphic data are compatible with τ . 4. The Penrose Ward Correspondence 4.1. Complex vector bundles over U and P Let us consider a principal G-bundle P = P (U, G) = U ×G over U ⊂ R4 . Then, a gauge potential A = Aµ dxµ (a connection 1-form) defines a connection D := d+A = dxµ (∂µ + Aµ ) on the bundle P , and the 2-form F = dA + A ∧ A = 12 Fµν dxµ ∧ dxν is the curvature of the connection form A. We shall consider irreducible connections. Suppose a representation of G in the complex vector space Cn is given. In the standard manner we associate with P the complex vector bundle E = P ×G Cn ' U × Cn , which is topologically trivial. Using the projection π : P → U of the twistor space P on U , we can pull back E to a bundle E 0 := π ∗ E over P, and the pulled back bundle E 0 is trivial on the fibres CP1x of the bundle P → U . We can set components of π ∗ A along the fibres equal to zero and then the pulled back connection D0 will have the form ¯ ¯ (on U1 ) = D + dζ∂ζ + dζ∂ ¯ ¯ (on U2 ). D0 = D + dλ∂λ + dλ∂ λ
ζ
4.2. Self-duality ⇒ holomorphy The twistor space P of the space U ⊂ R4 is a complex three-dimensional manifold with the coordinates {z1a } on U1 ⊂ P and {z2a } on U2 ⊂ P, P = U1 ∪ U2 . Using the (0,1)-forms (3.15), we introduce the (0,1) components Ba of the connection ¯ + B by the formulae 1-form π ∗ A = Aµ dxµ = B 1,0 + B 0,1 ≡ B (1)
:= Ay¯1 − λAy2 ,
B2 := Ay¯2 + λAy1 ,
(2)
:= ζAy¯1 − Ay2 ,
B2 := ζAy¯2 + Ay1 ,
{B1 {B1
(1)
(1)
B3 := 0} on U1 ,
(1)
(4.1a)
(2)
B3 := 0} on U2 .
(2)
(4.1b)
(2)
a Notice that Ba = λBa on U12 . One can also introduce the components Bz¯1,2 of a B along the antiholomorphic vector fields ∂z¯1,2 from (3.14),
(1)
(1)
(2)
(2)
(1)
(1)
{Bz¯11 := γ1 B1 , Bz¯12 := γ1 B2 , Bz¯13 := y 2 γ1 B1 − y 1 γ1 B2 } (2)
(2)
on U1 ,
(4.2a)
{Bz¯21 := γ2 B1 , Bz¯22 := γ2 B2 , Bz¯23 := −¯ y 1 γ2 B1 − y¯2 γ2 B2 } on U2 . (4.2b)
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¯ + B and Then we have π ∗ A = B a B ≡ B 0,1 = Bz¯1a d¯ z1a = Ba(1) θ¯(1)
on U1 ,
(4.3a)
a z2a = Ba(2) θ¯(2) B ≡ B 0,1 = Bz¯2a d¯
on U2 .
(4.3b)
Now we can introduce components of the connection D0 on the complex vector bundle E 0 which are (0,1) components w.r.t. the complex structure J on P, D0 := ∂B¯ + ∂¯B ,
∂¯B = ∂¯ + B ,
(4.4)
where the operator ∂¯ was introduced in (3.16), the (0,1)-form B was introduced in ¯ is the (1,0) component of the operator D0 . (4.3) and the operator ∂B¯ = ∂ + B Remark. In most cases we shall further write down formulae and equations in the trivialization over U1 ⊂ P. Let us consider the equations ∂¯B ξ = 0
(4.5)
on a smooth local section ξ of the bundle E 0 . Local solutions of these equations are by definition local holomorphic sections of the complex vector bundle E 0 . The 2 bundle E 0 → P is said to be holomorphic if Eqs. (4.5) are compatible, i.e., ∂¯B =0⇒ 0 the (0,2) components of the curvature of D are equal to zero. In the trivialization over U1 , Eqs. (4.5) are equivalent to the equations ¯ = 0, [(D1 + iD2 ) − λ(D3 + iD4 )]ξ1 (x, λ, λ)
(4.6a)
¯ = 0, [(D3 − iD4 ) + λ(D1 − iD2 )]ξ1 (x, λ, λ)
(4.6b)
¯ = 0, ∂λ¯ ξ1 (x, λ, λ)
(4.6c)
and analogously in the trivialization over U2 . Equation (4.6c) simply means that ¯ ξ1 is a function of xµ and λ (does not depend on λ). If Eq. (4.6c) is solved, the remaining two Eqs. (4.6a,b) for ξ1 (x, λ) are usually called the linear system for the SDYM equations [35]. It is readily seen that the compatibility conditions 2 ∂¯B = 0 of Eqs. (4.6) are identical to the SDYM Eqs. (2.1), which in the coordinates {y 1 , y 2 , y¯1 , y¯2 } have the form: Fy1 y2 = 0 ,
Fy¯1 y¯2 = 0 ,
Fy1 y¯1 + Fy2 y¯2 = 0 ,
(4.7)
2 = 0. Therefore, if a gauge potential i.e., Eqs. (4.7) follow from the equations ∂¯B µ A = Aµ dx is a smooth solution of Eqs. (4.7) on a domain U in R4 , there exist solutions of Eqs. (4.5), and the bundle E 0 → P is holomorphic. For the cover U = {U1 , U2 } of P = U1 ∪ U2 , Eqs. (4.5) have a local solution ξ1 over U1 , a local solution ξ2 over U2 and ξ1 = ξ2 on the overlap U12 = U1 ∩ U2 (i.e., it is a section over P). We can always represent ξ1 , ξ2 in the form ξ1 = ψ1 χ1 , ξ2 = ψ2 χ2 , where GC -valued functions ψ1 and ψ2 nonsingular on U1 and U2 satisfy the equations ∂¯B ψ1 = 0 , ∂¯B ψ2 = 0 (4.8)
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on U1 and U2 , respectively. The vector-functions χ1,2 ∈ Cn are holomorphic on U1,2 , (4.9) V¯a(1) χ1 = 0 , V¯a(2) χ2 = 0 . It follows from (4.8) that (∂y¯1 ψ1 − λ∂y2 ψ1 )ψ1−1 = (∂y¯1 ψ2 − λ∂y2 ψ2 )ψ2−1 = −(Ay¯1 − λAy2 ) ,
(4.10a)
(∂y¯2 ψ1 + λ∂y1 ψ1 )ψ1−1 = (∂y¯2 ψ2 + λ∂y1 ψ2 )ψ2−1 = −(Ay¯2 + λAy1 ) ,
(4.10b)
∂λ¯ ψ1 = ∂λ¯ ψ2 = 0 .
(4.10c)
Moreover, the vector-functions χ1 and χ2 are related by
on U12 , i.e.,
χ1 = F12 χ2
(4.11)
F12 := ψ1−1 ψ2
(4.12)
−1 . From Eqs. (4.8), is the transition matrix in the bundle E 0 and F21 := ψ2−1 ψ1 = F12 C (4.10) it follows that F12 is the holomorphic G -valued function on U12 with nonvanishing determinant.
Remark. (1) The matrices ψ1 and ψ2 are matrix fundamental solutions, i.e., the columns of ψ1 , ψ2 form frame fields for E 0 over U1 , U2 . In other words, matrix-valued functions ψ1 , ψ2 define a trivialization of the bundle E 0 over U1 , U2 . At the same ˇ time, χ1 = χ1 (z1a ) and χ2 = χ2 (z2a ) are Cech fibre coordinates of the bundle E 0 over U1 and U2 . The representation of ξ1,2 in the form ξ1 = ψ1 χ1 , ξ2 = ψ2 χ2 is simply an expansion of the sections ξ1,2 in the basis sections ψ1,2 with the components χ1,2 (see e.g. [36]). (2) The matrix-valued functions ψ1,2 are C∞ -functions on U1,2 , and any transition matrix of the form (4.12) defines a bundle E 0 , which is topologically trivial, but holomorphically nontrivial, since ψ1,2 are not holomorphic functions on U1,2 . On the other hand, Eqs. (4.10c) mean that the restriction of E 0 to any real projective line CP1x (x ∈ U ) is holomorphically trivial: E 0 |CP1x ' CP1x × Cn . 4.3. Gauge transformations and holomorphic equivalence It is easy to see that the local gauge transformations (2.2) of the gauge potential A are induced by the transformations ψ1 7→ ψ1g := g −1 (x)ψ1 ,
ψ2 7→ ψ2g := g −1 (x)ψ2 ,
(4.13)
and the transition matrix F12 = ψ1−1 ψ2 is invariant under these transformations because (ψ1g )−1 ψ2g = ψ1−1 ψ2 . On the other hand, the components {Aµ } of the gauge potential A in (4.10) will not change after transformations ψ1 7→ ψ1h1 := ψ1 h−1 1 ,
ψ2 7→ ψ2h2 := ψ2 h−1 2 ,
(4.14)
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where h1 is any regular holomorphic GC -valued function on U1 and h2 is any regular holomorphic GC -valued function on U2 . This means that a class of holomorphically equivalent bundles over the twistor space P corresponds to a self-dual connection on U . Recall that holomorphic bundles with the transition matrices Fˆ12 and F12 are called holomorphically equivalent if Fˆ12 = h1 F12 h−1 2
(4.15)
for some regular matrices h1 , h2 such that h1 is holomorphic on U1 and h2 is holomorphic on U2 . 4.4. Unitarity conditions It follows from Eqs. (4.10) that in the general case the components {Aµ } of the gauge potential A will take values in the Lie algebra gC , because ψ1,2 are GC valued. This is equivalent to the consideration of Aµ with values in the Lie algebra g, but with complex components Akµ in the expansion Aµ = Akµ Tk in the generators {Tk } of the Lie group G. If we want to consider real gauge fields, we have to impose additional reality conditions on the bundle E 0 induced by the real structure τ on P (see Sec. 3.4) and by an automorphism σ ˜ of the Lie algebra gC such that g = {a ∈ gC : σ ˜ (a) = a, σ ˜ 2 = id}. Such a reality structure in the bundle E 0 exists for any compact Lie group G [19], and we shall describe it for the case G = SU (n), g = su(n). Namely, in the case g = su(n) we have A†µ = −Aµ and therefore A†y1 = −Ay¯1 ,
A†y2 = −Ay¯2 ,
(4.16a)
where † denotes Hermitian conjugation. Then the matrices F12 ∈ SL(n, C) and ψ1 , ψ2 ∈ SL(n, C) have to satisfy on U12 the following unitarity conditions (see e.g. [28]): † F12 (τ (¯ z1a )) = F12 (z1a ) ,
(4.16b)
ψ1† (τ (x, λ)) = ψ2−1 (x, λ) ,
(4.16c)
where the action of τ on the coordinates of the space P was described in Sec. 3.4. Remark. For simplicity, we shall always consider the case G = SU (n) when discussing real gauge fields. Thus, starting from a bundle E over U ⊂ R4 with a self-dual connection, we have constructed a topologically trivial holomorphic vector bundle E 0 over P satisfying the conditions: (1) E 0 is holomorphically trivial on each real projective line CP1x , x ∈ U , in P; (2) E 0 has a real structure. 4.5. Riemann Hilbert problems Suppose we have a nonsingular matrix-valued function F(x, λ) ∈ SL(n, C) on Ω1 ∩ Ω2 ⊂ CP1 (see Sec. 3.2) depending holomorphically on λ and smoothly on
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some parameters {xµ }. Then a parametric Riemann–Hilbert problem is to find matrix-valued functions ψ1 , ψ2 ∈ SL(n, C) on Ω1 ∩ Ω2 such that ψ1 can be extended continuously to a regular (i.e., holomorphic with a non-vanishing determinant) matrix-valued function on Ω1 , ψ2 can be extended to a regular matrix-valued function on Ω2 and F(x, λ) = ψ1−1 (x, λ)ψ2 (x, λ) (4.17) on Ω1 ∩ Ω2 . It follows from the Birkhoff decomposition theorem (see e.g. [37]) that for a fixed x any holomorphic on Ω1 ∩ Ω2 nonsingular matrix-valued function F admits a decomposition F = ψ1 Λψ2 , (4.18) where ψ1 , ψ2 are defined above and Λ is a diagonal matrix whose entries are integral powers ki ∈ Z of λ, k1 + · · · + kn = 0. The ki ’s are unique up to permutation and are Chern classes of the holomorphic line bundles over CP1 which occur in the decomposition of the holomorphic vector bundle over CP1 with F as a transition matrix (Grothendieck’s theorem). If Λ is the identity matrix, the decomposition (4.18) is called a solution to the Riemann–Hilbert problem. For these matrices F, the factorization is unique up to a transformation ψ1 (x, λ) 7→ ψ1g = g −1 (x)ψ1 (x, λ) ,
ψ2 (x, λ) 7→ ψ2g = g −1 (x)ψ2 (x, λ) ,
(4.19)
for some matrix g(x) ∈ SL(n, C). So the Riemann–Hilbert problem can only be solved “generically” and (4.17) may not have a solution for all values of the parameters xµ . But if a factorization (4.17) exists at some xµ0 , then it exists in an open neighbourhood U of xµ0 . Usually, Λ 6= 1 on a submanifold of codimension 1 (or more) of the parameter space. The points xµ for which Λ 6= 1 are called jumping points, and projective lines CP1x corresponding to these points x are called jumping lines. In the twistor construction the jumping points x ∈ R4 give rise to singularities in the SDYM potential A. For details see e.g. [39, 40]. 4.6. Holomorphy ⇒ self-duality Suppose we have a topologically trivial holomorphic vector bundle E 0 over P with the cover U = {U1 , U2 } and a transition matrix F12 satisfying the unitarity condition (4.16b). Considering F12 for fixed xµ ∈ U , we obtain a parametric Riemann–Hilbert problem on CP1 . Then in a set of all possible transition matrices we choose those for which a solution of the Riemann–Hilbert problem exists. (1) After finding a Birkhoff decomposition (4.17) for F12 we consider (V¯a ψ1 )ψ1−1 (1) and (V¯a ψ2 )ψ2−1 as functions on U1 and U2 with values in the Lie algebra sl(n, C). (1) (2) For definitions of the (0,1) vector fields V¯a , V¯a see Sec. 3. From the holomorphy of F12 it follows that (V¯a(1) ψ1 )ψ1−1 = (V¯a(1) ψ2 )ψ2−1 (4.20)
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on U12 . Notice that as functions on CP1 the matrices ψ1 and ψ2 are regular on Ω1 and Ω2 , respectively. Hence, ψ1,2 can be expanded on Ω1 ∩ Ω2 in powers of λ: ∞ X
ψ1 (x, λ) =
λn ψ1n (x) ,
ψ2 (x, λ) =
n=0
∞ X
λ−n ψ2n (x) .
(4.21)
n=0
If we substitute the expansion of ψ1,2 in powers of λ into (4.20), both the sides of (4.20) must be linear in λ, and we have (∂y¯1 ψ1 − λ∂y2 ψ1 )ψ1−1 = (∂y¯1 ψ2 − λ∂y2 ψ2 )ψ2−1 = −(Ay¯1 (x) − λAy2 (x)) ,
(4.22a)
(∂y¯2 ψ1 + λ∂y1 ψ1 )ψ1−1 = (∂y¯2 ψ2 + λ∂y1 ψ2 )ψ2−1 = −(Ay¯2 (x) + λAy1 (x)),
(4.22b)
where (1) Ay1 := − Res λ−2 (V¯2 ψ2 )ψ2−1 λ=0
I ≡ −
S1
dλ ¯ (1) (V ψ2 )ψ2−1 = −(∂y1 ψ20 )(ψ20 )−1 , 2πiλ2 2
(4.23a)
Ay2 := Res λ−2 (V¯1 ψ2 )ψ2−1 (1)
λ=0
I
dλ ¯ (1) (V ψ2 )ψ2−1 = −(∂y2 ψ20 )(ψ20 )−1 , 2πiλ2 1
≡ S1
(4.23b)
(1) Ay¯1 := − Res λ−1 (V¯1 ψ1 )ψ1−1 λ=0
I ≡ −
S1
dλ ¯ (1) (V ψ1 )ψ1−1 = −(∂y¯1 ψ10 )(ψ10 )−1 , 2πiλ 1
(4.23c)
Ay¯2 := − Res λ−1 (V¯2 ψ1 )ψ1−1 (1)
λ=0
I ≡ −
S1
dλ ¯ (1) (V ψ1 )ψ1−1 = −(∂y¯2 ψ10 )(ψ10 )−1 . 2πiλ 2
(4.23d)
Here, the contour S 1 = {λ ∈ CP1 : |λ| = 1} circles once around λ = 0 and the contour integral determines residue Res at the point λ = 0. The components {Aµ } of the gauge potential defined by (4.23) satisfy the SDYM equations on U which are the compatibility conditions of Eqs. (4.22). Thus, starting from a holomorphic matrix-valued function F12 which is a transition matrix of a holomorphic vector bundle E 0 over the twistor space P, we have completed the procedure of reconstructing a gauge potential A which defines a self-dual connection on a complex vector bundle E over U ⊂ R4 . As it was explained in Sec. 4.3, the transformations (4.14), (4.15) of F12 into a holomorphically equivalent transition matrix h1 F12 h−1 do not change Aµ , and gauge transformations Aµ 7→ Agµ inducing the 2
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transformations (4.13) do not change F12 . It follows from the twistor construction that a self-dual gauge potential A is real-analytic. To sum up, we have described a one-to-one correspondence between gauge equivalence classes of solutions to the SDYM equations on an open subset U of the Euclidean 4-space and equivalence classes of holomorphic vector bundles E 0 over the twistor space P satisfying the conditions: (i) bundles E 0 are holomorphically trivial on each real projective line CP1x , x ∈ U , in P, (ii) each E 0 has a real structure. This is the Euclidean version of Ward’s theorem [39, 41]. Remark. A twistor correspondence between self-dual gauge fields and holomorphic bundles also exists in a more general situation [19]. Let us consider a real oriented four-manifold M with a metric g of signature (++++). The 4-manifold M is called self-dual if its Weyl tensor is self-dual. In [19] it was proved that the twistor space Z ≡ Z(M ) for a self-dual manifold M is a complex analytic 3-manifold. There is a natural one-to-one correspondence between self-dual bundles E over M (in particular, over R4 , S 4 , T 4 , . . .) and holomorphic vector bundles E 0 over the twistor space Z. In the general case, bundles E and E 0 are not topologically trivial, as it takes place in the case of Euclidean space R4 , when P ⊂ Z(R4 ) = R4 × CP1 . ˇ 5. Holomorphic Bundles in the Cech Approach We are going to analyse the twistor correspondence between self-dual complex vector bundles E over U ⊂ R4 and holomorphic vector bundles E 0 over P from the group-theoretic point of view, i.e., we want to describe groups acting on the space of transition matrices F12 of the bundles E 0 , on the space of self-dual gauge potentials A and on the moduli space of self-dual gauge fields. In our discussion, we shall use the notion of local groups, (local) actions of (local) groups on sets, germs, sheaves ˇ and Cech cohomology, definitions of which are recalled in Appendices A, B and C. In this section, we shall describe symmetries and the moduli space of all holomorphic vector bundles over P. This means that we shall consider holomorphic bundles over P which are not necessarily holomorphically trivial over CP1x ,→ P, x ∈ U , and do not satisfy the unitarity condition (4.16b). As recalled in Appendices B and C, there is a one-to-one correspondence between the set of isomorphism classes of holoˇ morphic bundles over a complex space X and the Cech 1-cohomology set H 1 (X, H) GC of the space X with values in the sheaf H = O of germs of holomorphic maps from X into the complex Lie group GC . We shall consider this correspondence for our case of the complex twistor space P and the group GC = SL(n, C) and describe it from the group-theoretic point of view. 5.1. Moduli space of holomorphic bundles over the twistor space P We consider the two-set open cover U = {U1 , U2 } of P (see Sec. 3.3), where U1 , U2 are Stein manifolds. For this cover we have the following q-simplexes hUα0 , . . . , Uαq i: hU1 i, hU2 i, hU1 , U2 i, hU2 , U1 i, supports U1 , U2 , U12 := U1 ∩ U2 of which are nonempty sets. Further, a q-cochain of the cover U with the coefficients in the sheaf H = OSL(n,C) is a map f , which associates with any q-simplex hUα0 , . . . , Uαq i a section
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of the sheaf H over Uα0 ∩. . .∩ Uαq : fα0 ...αq ≡ f (Uα0 ∩. . .∩ Uαq ) ∈ H(Uα0 ∩. . .∩ Uαq ). In other words, a q-cochain of the cover U with values in H is a collection f = {fα0 ...αq } of sections of the sheaf H over nonempty intersections Uα0 ∩ . . . ∩ Uαq . The set of q-cochains is denoted by C q (U, H) (see Appendix C). In the considered case we have the sets of 0-cochains C 0 (U, H) and 1-cochains C 1 (U, H). The set C 0 (U, H) is a group under a pointwise multiplication. For h = {h1 , h2 }, f = {f1 , f2 } ∈ C 0 (U, H) we have hf = {(hf )1 , (hf )2 } := {h1 f1 , h2 f2 } ,
(5.1)
where hα , fα ∈ H(Uα ) ≡ Γ(Uα , H), α = 1, 2. The set C 1 (U, H) of all 1-cochains forms a group under the following operation: if h = {h12 , h21 }, f = {f12 , f21 } ∈ C 1 (U, H), then (5.2) hf = {(hf )12 , (hf )21 } := {h12 f12 , h21 f21 } , where h12 , h21 , f12 , f21 ∈ H(U12 ) ≡ Γ(U12 , H). Notice that h12 and h21 (f12 and f21 ) are elements of two different groups H(U12 ) : {h12 , h21 } ∈ H(U12 ) × H(U12 ). For the two-set open cover U, sets of 0- and 1-cocycles are defined by the formulae: Z 0 (U, H) = {{h1 , h2 } ∈ C 0 (U, H) : h1 = h2
on U12 } ,
Z 1 (U, H) = {{h12 , h21 } ∈ C 1 (U, H) : h12 = h−1 21 } ,
(5.3) (5.4)
and the space Z 0 (U, H) coincides with the group H 0 (P, H) ≡ Γ(P, H) of global sections of the sheaf H. The set Z 1 (U, H) is not a group for the non-Abelian sheaf H. Finally, two cocycles F, Fˆ ∈ Z 1 (U, H) are said to be equivalent, Fˆ ∼ F, if Fˆ12 = h1 F12 h−1 2 ,
(5.5)
for some element h = {h1 , h2 } ∈ C 0 (U, H) restricted to U12 . A set of equivalence ˇ classes of 1-cocycles F with respect to the equivalence relation (5.5) is called a Cech 1 1-cohomology set and denoted by H (U, H). In the general case we should take the direct limit of these sets H 1 (U, H) over successive refinement of cover U of P ˇ to obtain H 1 (P, H), the Cech 1-cohomology set of P with coefficients in H. But in our case U1 , U2 are Stein manifolds and therefore H 1 (U, H) = H 1 (P, H). The cohomology set H 1 (P, H) is identified with the set of all holomorphic vector bundles over P with the group SL(n, C) which are considered up to equivalence (5.5), i.e., with the moduli space of holomorphic vector bundles E 0 . 5.2. Action of the group C 0 (U, H) on the space Z 1 (U, H) Suppose that we are given a cover {Uγ } of the space P, γ = 1, 2, . . ., and the groups C 0 ({Uγ }, H) and C 1 ({Uγ }, H) of 0-cochains and 1-cochains. Let us define the following action of the group C 0 on the group C 1 (automorphism σ0 (h, .)): σ0 (h, f )αβ = hβ fαβ h−1 β (no summation) ,
(5.6)
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where h = {hα } ∈ C 0 ({Uγ }, H), f = {fαβ } ∈ C 1 ({Uγ }, H). Now we can define a twisted homomorphism δ 0 : C 0 → C 1 of the group C 0 into the group C 1 by the formula [42] δ 0 (h)αβ = hα h−1 (5.7a) β , where δ 0 (h) = {δ 0 (h)αβ } ∈ C 1 ({Uγ }, H). It is not difficult to see that δ 0 (hg) = δ 0 (h)σ0 (h, δ 0 (g)) ,
(5.7b)
i.e., the homomorphism δ 0 is “twisted” by σ0 . The twisted homomorphism δ 0 permits one to define an action ρ0 of the group C 0 on C 1 as on a set. The corresponding transformations act on C 1 by the formula [42] ρ0 (h, f ) = δ 0 (h)σ0 (h, f ) ⇔ ρ0 (h, f )αβ = hα fαβ h−1 β (no summation) , ρ0 (gh, f ) = ρ0 (g, ρ0 (h, f )) ,
(5.8a) (5.8b)
where h, g ∈ C 0 ({Uγ }, H), f ∈ C 1 ({Uγ }, H). Of course, in (5.6)–(5.8) it is implied that the components hα of the element h ∈ C 0 are restricted to Uαβ . It is not difficult to verify that the action (5.8) preserves the space of 1-cocycles Z 1 ({Uγ }, H) ⊂ C 1 ({Uγ }, H). For a two-set open cover U = {U1 , U2 } of P, the action ρ0 of the group C 0 on the space Z 1 (U, H) of 1-cocycles has the form: ρ0 (h, F)12 = h1 F12 h−1 2 ,
(5.9)
where h ∈ C 0 (U, H), F ∈ Z 1 (U, H). As already said, the action (5.9) (the special case of (5.8)) preserves the space Z 1 , and the quotient space ρ0 (C 0 )\Z 1 (ρ0 (C 0 ) acts on Z 1 on the left), i.e., the space of orbits of the group C 0 in Z 1 , H 1 (P, H) = H 1 (U, H) := ρ0 (C 0 (U, H))\Z 1 (U, H) ,
(5.10)
ˇ is the Cech 1-cohomology set. 5.3. Action of the group C 1 (U, P) on the space Z 1 (U, P) For a two-set open cover U of P one may define an automorphism σ(h, .) : C 1 → C , h ∈ C 1 , of the group of 1-cochains by the formula 1
σ(h, f )12 = h21 f12 h−1 21 ,
(5.11)
where h, f ∈ C 1 (U, H), and a twisted homomorphism δ : C 1 → C 1 by the formula −1 δ(h) = {δ(h)12 , δ(h)21 } = {h12 h−1 21 , h21 h12 } ,
(5.12)
where h, δ(h) ∈ C 1 (U, H). With the help of the homomorphisms σ and δ one can define the action of the group C 1 on itself as follows: ρ(h, f ) = δ(h)σ(h, f ) ⇔ ρ(h, f )12 = h12 f12 h−1 21 ,
(5.13a)
ρ(gh, f ) = ρ(g, ρ(h, f )) ,
(5.13b)
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where g, h, f ∈ C 1 (U, H). This action preserves the set Z 1 (U, H) of 1-cocycles, and for a cocycle F ∈ Z 1 (U, H) we have F˜12 := ρ(h, F)12 = h12 F12 h−1 21 .
(5.14)
−1 −1 −1 It is easy to see that F˜21 := h21 F21 h−1 = F˜12 , i.e., F˜ is a 12 = (h12 F12 h21 ) 1-cocycle. For h = {h12 , h21 } ∈ C 1 (U, H), the matrices h12 , h21 ∈ SL(n, C) are arbitrary holomorphic matrix-valued functions on U12 and therefore with the help of the action (5.14) one can obtain any cocycle from Z 1 (U, H). In other words, the action of C 1 (U, H) on Z 1 (U, H) is transitive, and the set Z 1 can be identified with a 1 homogeneous space C 1 /C4 , 1 Z 1 (U, H) = C 1 (U, H)/C4 (U, H) ,
(5.15a)
1 C4 (U, H) = {{h12 , h21 } ∈ C 1 (U, H) : h21 = h12 }
(5.15b)
where 0 1 is the stability subgroup of the trivial cocycle F12 = 1. The group C4 (U, H) is the 1 kernel of the homomorphism (5.12). Thus, the group C (U, H) acts transitively on the space Z 1 (U, H) of holomorphic bundles E 0 over P.
Remark. The description of the group C 1 and of its action on the space Z 1 of cocycles in terms of matrix-valued functions depends on a cover of the space P. For a general system of local trivializations with an open cover {Uγ }, γ ∈ I, the elements F of Z 1 ({Uγ }, H) must satisfy the conditions −1 Fαα = 1 (no summation) on Uα , Fβα = Fαβ
on Uαβ := Uα ∩ Uβ ,
(5.16a)
Fαβ Fβγ Fγα = 1 (no summation) on Uαβγ := Uα ∩ Uβ ∩ Uγ 6= ∅ .
(5.16b)
Then C 1 ({Uγ }, H) acts on F ∈ Z 1 ({Uγ }, H) as follows: Fαβ 7→ F˜αβ := ρ(h, F)αβ = hαβ Fαβ h−1 βα (no summation) .
(5.17)
It is easily checked that the conditions (5.16a) for F˜ are satisfied, and from the conditions (5.16b) imposed on F˜αβ it follows that hαβ |Uαβγ = hαγ |Uαβγ .
(5.18)
It simply means that hαβ are defined on ∪ Uαβ ,
α,β∈I
(5.19)
and we denote by C¯ 1 ({Uγ }, H) the subgroup of all elements h = {hαβ } ∈ C 1 ({Uγ }, H) satisfying (5.18). Thus, we obtain 1 Z 1 ({Uγ }, H) = C¯ 1 ({Uγ }, H)/C¯4 ({Uγ }, H) ,
(5.20a)
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where 1 C¯4 ({Uγ }, H) = {{hαβ } ∈ C¯ 1 ({Uγ }, H) : hβα = hαβ }
(5.20b)
0 is the stability subgroup of the trivial cocycle Fαβ = 1. For a two-set open cover 1 1 ¯ U = {U1 , U2 } we have C (U, H) = C (U, H). It follows from the definitions that the groups C 0 (U, H) and C 1 (U, H) are direct products
C 0 (U, H) = H(U1 ) × H(U2 ) ≡ Γ(U1 , H) × Γ(U2 , H) 3 {h1 , h2 } ,
(5.21a)
C 1 (U, H) = H(U12 ) × H(U12 ) ≡ Γ(U12 , H) × Γ(U12 , H) 3 {h12 , h21 } ,
(5.21b)
of the groups H(U1 ), H(U2 ) and H(U12 ) of sections over U1 , U2 and U12 of the sheaf 1 (U, H) coincides with the diagonal subgroup in the group H . Respectively, C4 H(U12 )×H(U12 ), and Z 1 (U, H) coincides with the subset of elements h = {h12 , h−1 12 } from the group C 1 (U, H). Collating formulae (5.10) and (5.15), we obtain that 1 , H 1 (P, H) = ρ0 (C 0 ) \ C 1 /C4
(5.22)
i.e., the moduli space of holomorphic bundles E 0 over P is parametrized by the double coset space (5.22). It is not difficult to see that the 1-cohomology set (5.22) is isomorphic to 1 -orbits in Y 1 := ρ0 (C 0 ) \ C 1 , (i) the set of C4 1 (ii) the set of C 0 -orbits in Z 1 = C 1 /C4 , 1 1 1 (iii) the set of C -orbits in Y × Z , where an action of h ∈ C 1 on (y, z) ∈ Y 1 × Z 1 is defined by the formula C 1 × (Y 1 × Z 1 ) 3 (h, (y, z)) : (y, z) 7→ (yh, ρ(h−1 , z)) ∈ Y 1 × Z 1 . To sum up, for the space Z 1 (U, H) of holomorphic bundles E 0 over P, the group C (U, H) of 1-cochains for the cover U with values in the sheaf H of non-Abelian groups acts on the transition matrices F12 of bundles E 0 by the left multiplication on matrices h12 ∈ H(U12 ) and by the right multiplication on matrices h−1 21 ∈ H(U12 ). This group acts on Z 1 transitively, and the space Z 1 is the coset space (5.15a) (or (5.20a) for an arbitrary cover of P). So C 1 (U, H) is the symmetry group of the space ˇ of holomorphic bundles E 0 in the Cech approach. The moduli space H 1 (P, H) of 0 bundles E is the double coset space (5.22). 1
5.4. The group H(P) of automorphisms of the complex manifold P Let X be a compact smooth manifold, G a compact simple connected Lie group and Aut G a group of automorphisms of the group G. Consider the group Map(X; G) of smooth maps from X into G and the connected component of the unity Map0 (X; G) of the group Map(X; G). It is well known that the group of automorphisms of the group Map0 (X; G) is a semidirect product Diff (X) n Map(X; Aut G)
(5.23)
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of the diffeomorphism group Diff (X) of the manifold X and the group of automorphisms Map(X; Aut G) (for proof see Sec. 3.4 in [37]). As a set the space Z 1 (U, H) considered above coincides with the group Map(U12 ; SL(n, C)) of holomorphic maps from U12 into SL(n, C) and it is an analogue of the group Map0 (X; G). The group C 1 (U, H) acting on the space Z 1 (U, H) is respectively an analogue of the group of automorphisms Map(X; Aut G). It is clear that there should be an analogue of the diffeomorphism group from (5.23), i.e., some group of transformations of the coordinates of the space P acting on the set Z 1 (U, H). Remember that as a smooth manifold the twistor space is P = U × S 2 . At the same time, P is a complex 3-manifold, and in Sec. 3.3 we have introduced the complex coordinates z1 : U1 → C3 , z2 : U2 → C3 on P and the holomorphic transition function f12 connecting z1 and z2 on U12 . Let η : P → P be an arbitrary transformation from the group Diff (P) of diffeomorphisms of the twistor space P. Let us denote by U˜1 := η(U1 ), U˜2 := η(U2 ) the images of the open sets U1 , U2 in P. We have η(P) = η(U1 ∪ U2 ) = η(U1 ) ∪ η(U2 ) = U˜1 ∪ U˜2 ,
(5.24a)
η(U12 ) = η(U1 ∩ U2 ) = η(U1 ) ∩ η(U2 ) = U˜1 ∩ U˜2 ,
(5.24b)
since the map η is a bijection. Let us consider the restriction of the map η to U12 , i.e., the local diffeomorphism η |U12 : U12 → P. On U˜12 = η(U12 ) one can always introduce complex coordinates zˆ1 : U˜12 → C3 , zˆ2 : U˜12 → C3 related by a holomorphic transition function fˆ12 such that the map η |U12 : U12 → U˜12 will be holomorphic in the chosen coordinates. In other words, domains U12 and U˜12 are biholomorphic and there exist holomorphic functions η1 , η2 such that [43] zˆ1a ◦ η = η1a (z1b ) ,
zˆ2a ◦ η = η2a (z2b ) ,
a zˆ1a = fˆ12 (ˆ z2b ) .
(5.25)
These maps form the (local) group H(U12 ). Having the group H(U12 ) of local holomorphic maps η |U12 : U12 → P, one can define its action on transition matrices F of holomorphic bundles over P. But in this connection the following questions arise: 1. Is it possible to introduce on U˜1 ∪ U˜2 complex coordinates z˜1 : U˜1 → C3 , z˜2 : U˜2 → C3 related by a holomorphic transition function f˜12 ? 2. Can the coordinates zˆ1 , zˆ2 on U˜12 be extended to U˜1 , U˜2 and will they be equivalent to the coordinates z˜1 , z˜2 ? The diffeomorphism group Diff (P) acts not only on transition matrices of bundles E 0 over P, but also on the complex structure of the space P. But a change of the complex structure of the space P leads to a change of the conformal structure and a metric on U ⊂ R4 by virtue of the twistor correspondence [17, 19]. If we are interested in symmetries of the SDYM equations on the space U with a conformally flat metric, then we have to consider only those diffeomorphisms η ∈ Diff (P) which preserve the complex structure of P. These maps η : P → P form the group of
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biholomorphic transformations of the space P which we shall denote by H(P). It is a subgroup of the diffeomorphism group: H(P) ⊂ Diff (P). In the coordinates z1 , z2 , z˜1 , z˜2 transformations η ∈ H(P) are defined by the holomorphic functions z˜1a ◦ η = η1a (z1 ) ,
z˜2a ◦ η = η2a (z2 ) ,
a z˜1a = f˜12 (˜ z2b ) .
(5.26)
Formulae (5.26) are not always convenient because there the coordinates zα are calculated at points p ∈ P, and the coordinates z˜α are calculated at points q = η(p) ∈ P. It is often more convenient to define η by transition functions ηαβ from zα to z˜β in the domains Uα ∩ U˜β (if Uα ∩ U˜β 6= ∅), then zα and z˜β are calculated at the same points p ∈ Uα ∩ U˜β . For example, the conformal transformations (2.4) of z1a } the space R4 induce such holomorphic transformations of coordinates {z1a } 7→ {˜ of the twistor space Z = Z(R4 ) that on U1 ∩ U˜1 we have: translations: z˜11 = z11 + a1 − a ¯2 z13 , 1 z˜1 c d rotations 2 ¯ z˜ = −d c¯ induced by{Xa } : 1 3 z˜1 0 0 rotations induced by{Ya } :
z˜11 =
z11 , a − ¯bz13 a −¯b
dilatations : special conformal transformations :
b a ¯
z˜12 = z12 + a2 + a ¯1 z13 , z˜13 = z13 , 1 ! 0 z1 c d 2 0 z1 , ∈ SUL (2) , −d¯ c¯ 3 1 z1 z˜12 = !
z12 , a − ¯bz13
z˜13 =
b+a ¯z13 , a − ¯bz13
∈ SUR (2) ,
z˜11 = eα z11 ,
z˜12 = eα z12 ,
z˜11 =
z11 , 1 + α1 z11 + α2 z12
z˜13 =
z13 − α ¯ 1 z11 + α ¯ 2 z12 . 1 1 + α1 z1 + α2 z12
z˜12 =
z˜13 = z13 , z12 , 1 + α1 z11 + α2 z12
Here a1 , a2 , α1 , α2 ∈ C, α ∈ R. 5.5. Action of the group H(P) on the space Z 1 (U, P) Action of the group H(P) of complex-analytic diffeomorphisms of the space P on transition matrices of holomorphic bundles E 0 over P is defined in the following way. We consider a two-set open cover U = {U1 , U2 } of P and a transition matrix F ∈ Z 1 (U, H) of a bundle E 0 . After a transformation H(P) 3 η : P → P we have ˜ = {U˜1 , U˜2 }, U˜1 = η(U1 ), U˜2 = η(U2 ). Let us consider the common a new cover U refinement both of the covers. Denote Uˆ1 := U1 ∩ U˜1 ,
Uˆ2 := U1 ∩ U˜2 ,
Uˆ3 := U2 ∩ U˜1 ,
ˆ = {Uˆ1 , Uˆ2 , Uˆ3 , Uˆ4 }. to give the refined cover U
Uˆ4 := U2 ∩ U˜2 ,
(5.27a)
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ˆ H): The cocycle F ∈ Z 1 (U, H) induces the following 1-cocycle Fˆ ∈ Z 1 (U, Fˆ = {Fˆ12 , Fˆ13 , Fˆ14 , Fˆ23 , Fˆ24 , Fˆ34 } := {1, F12 (z1a ), F12 (z1a ), F12 (z1a ), F12 (z1a ), 1} , (5.27b) where Fˆαβ is defined in Uˆαβ := Uˆα ∩ Uˆβ and Uˆ12 := Uˆ1 ∩ Uˆ2 = U1 ∩ U˜12 ,
Uˆ13 := Uˆ1 ∩ Uˆ3 = U12 ∩ U˜1 ,
Uˆ14 := Uˆ1 ∩ Uˆ4 = U12 ∩ U˜12 ,
Uˆ23 := Uˆ2 ∩ Uˆ3 = U12 ∩ U˜12 ,
Uˆ24 := Uˆ2 ∩ Uˆ4 = U12 ∩ U˜2 ,
Uˆ34 := Uˆ3 ∩ Uˆ4 = U2 ∩ U˜12 .
(5.28)
The cocycle Fˆ is equivalent to the cocycle F, and the group H(P) acts on F ∈ Z 1 (U, H) as follows: η ˆη ˆ η ˆ η ˆ η ˆ η H(P) 3 η : F 7→ Fˆ 7→ ρ(η, F) ≡ Fˆ η = {Fˆ12 , F13 , F14 , F23 , F24 , F34 } , η Fˆ12 := 1 ,
η Fˆ13 := F12 (η1a (z1 )) ,
η := F12 (η1a (z1 )) , Fˆ23
η Fˆ14 := F12 (η1a (z1 )) ,
η a Fˆ24 := F12 (f12 (η2b (z2 ))) ,
(5.29)
η Fˆ34 := 1 .
In the general case cocycles Fˆ and Fˆ η are not equivalent and therefore the group H(P) of biholomorphic transformations of the twistor space P acts nontrivially on the space Z 1 (U, H). This action includes refining of the cover and a transition to an equivalent cocycle. It is usually considered that elements η ∈ H(P) which are close to the identity do not move the covering sets. That is, if η is close to the identity, it is possible to define the action of such η ∈ H(P) as follows: η ρ(η, .) : F12 7→ ρ(η, F)12 = F12 = F12 (η1 (z1 )) ,
(5.30)
ˆ In other words, the action of a neighbourhood i.e., without using the refined cover U. of unity of the group H(P) maps Z 1 (U, H) into itself. In what follows we shall study just this case. Returning to Sec. 5.3 and to the beginning of Sec. 5.4, we come to the conclusion that the full group of continuous symmetries acting on the space Z 1 (U, H) of holomorphic bundles E 0 over P is a semidirect product H(P) n C 1 (U, H)
(5.31)
of the group H(P) of holomorphic automorphisms of the space P and of the group C 1 (U, H) of 1-cochains for the cover U with values in the sheaf H of holomorphic maps of the space P into the Lie group SL(n, C). 6. Symmetries in Holomorphic Setting 6.1. Germs of sets and groups Let B be a set with a marked point e ∈ B. The element e is called the unity. If B and C are sets with the marked points which we denote by the same letter e, then a homomorphism of the set B into the set C is such a map ϕ : B → C that
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A. D. POPOV
ϕ(e) = e. The homomorphism B → C is said to be the isomorphism if it maps B onto C bijectively. The set Ker ϕ = ϕ−1 (e) with the marked point e is called the kernel of the homomorphism ϕ. Let X be a set with a marked point e, and let Y1 , Y2 be two subsets of the set X also containing the point e. The sets Y1 , Y2 are called equivalent at the point e if there exists such a neighbourhood Y3 of this point that Y1 ∩ Y3 = Y2 ∩ Y3 . The class of all sets equivalent to the set Y1 is called the germ of this set at the point e and denoted by Y [44]. The sets Y1 , Y2 , Y3 are representatives of the germ Y of sets. In Appendix A, a notion of group germs G [45] based on the definition of germs of sets is introduced. Representatives of the group germ G are local groups, i.e., open neighbourhoods G of the identity e ≡ 1, which are closed under all group operations (multiplication, operation of inverse, etc). In particular, we shall consider the germs C and H of the groups C 1 (U, H) and H(P) described in Sec. 5. ,→ P 6.2. Holomorphic triviality of bundles E 0 on C P 1x ,→ Let us consider the twistor space Z ≡ Z(R4 ) of R4 and the moduli space H (Z, H) of holomorphic bundles E 0 over the space Z. With the sheaf H = OSL(n,C) (of germs) of holomorphic maps from Z into the group SL(n, C) one associates the sheaf Osl(n,C) (of germs) of holomorphic maps from Z into the Lie algebra sl(n, C). The Abelian group (by addition) of cohomologies H 1 (Z, Osl(n,C) ) of the space Z with values in the sheaf Osl(n,C) parametrizes infinitesimal deformations of the trivial bundle E00 = Z × Cn and dim H 1 (Z, Osl(n,C) ) = ∞, i.e., in an arbitrarily small neighbourhood of the trivial bundle E00 there exists an infinite number of holomorphically nontrivial bundles E 0 . Let us fix an arbitrary point x0 ∈ R4 and consider a real projective line CP1x0 sl(n,C) embedded into Z. Now we consider the restriction Ox0 := Osl(n,C) |CP1x of the 1
0
sl(n,C)
sheaf Osl(n,C) to CP1x0 and the cohomology group H 1 (CP1x0 , Ox0 ) parametrizing 0 := CP1x0 × Cn . It infinitesimal deformations of the trivial holomorphic bundle E0x 0 is easily seen that H 1 (CP1x0 , Oxsl(n,C) ) = 0, (6.1) 0 because H 1 (CP1 , O) = 0, where O is the sheaf of germs of holomorphic functions on CP1 . The equality (6.1) means that there exists a sufficiently small open neighbourhood U ⊂ R4 of the point x0 and an open subset M 3 e of the space H 1 (P, H), where P ⊂ Z is the twistor space of U , such that for the bundles E 0 , representing points [E 0 ] of the space M ⊂ H 1 (P, H), their restriction Ex0 to CP1x ,→ P will be holomorphically trivial for any x ∈ U (version of the Kodaira theorem). In other words, small enough deformations do not change the trivializability of the bundle E 0 over real projective lines in a neighbourhood of a given projective line CP1x0 (for discussion see e.g. [39, 40]). Projective lines {CP1x }x∈U form a family of complex 1-manifolds parametrized by x ∈ U , and CP1x coincides with CP1 × {x} in the direct product CP1 × U ' P. We consider holomorphic bundles E 0 over P with transition matrices F from Z 1 (U, H) and their restriction to CP1x ,→ P, x ∈ U . Then a family of holomorphic
SELF-DUAL YANG–MILLS: SYMMETRIES AND MODULI SPACE
1115
maps F12 (x, λ) from U × Ω12 into SL(n, C) determines a family of vector bundles Ex0 := E 0 |CP1x over CP1x , labelled by the parameters x ∈ U ⊂ R4 . In this family 0 = CP1x × Cn . there exists a marked family of holomorphically trivial bundles E0x Finally, we have introduced an open subset M of the set H 1 (P, H) (being an open neighbourhood of the marked point e ∈ H 1 (P, H)) of moduli of those bundles E 0 from H 1 (P, H), which are holomorphically trivial on CP1x ,→ P for all x ∈ U . With each point m ≡ [E 0 ] ∈ M one can associate a bundle Em := E 0 (m) over P. Then we have a family {Em }m∈M of holomorphic bundles over P, parametrized by m ∈ M. The marked point in this family is the trivial bundle E0 := E 0 (e) (the isomorphism class of the bundle E00 ). Let X be a complex space. Consider a family of holomorphic vector bundles of rank n with the base X and a family of complex parameters T , i.e., a holomorphic vector bundle E of rank n over X ×T . The space T is called the base of deformation. For t ∈ T , we denote by Et a bundle over X which is induced by restriction of E to X × {t} with a natural identification X ↔ X × {t} [46]. In our case, we have a holomorphic vector bundle E of rank n over P × M, M ⊂ H 1 (P, H). Using the definitions of Sec. 6.1, one can consider sets equivalent to the set M, and a class of all open subsets in H 1 (P, H), equivalent to the set M, defines the germ M of this set at the point e. Of course, the notion of equivalence is supplemented here by the demand that all representatives M, M0 , . . . of the germ M should be moduli spaces of those bundles from Z 1 (U, H) which are holomorphically trivial on CP1x , x ∈ U . Let us stress that a choice of a concrete representative M, M0 , . . . of the germ M is not essential since a different choice gives equivalent deformations of the bundle E00 . That is why in the modern deformation theory of complex spaces and holomorphic bundles as a base of deformation one takes not a set with a marked point e but the germ of this set at the marked point (see e.g. [46]). Now we take a point m = [E 0 ] ∈ M and the transition matrix F(m) ∈ Z 1 (U, H) in the bundle E 0 representing this point. Acting on F(m) by all possible elements of C 0 (U, H) by formulae (5.8), (5.9), we obtain an orbit ρ0 (C 0 )(F(m)) of the point F(m) ∈ Z 1 (U, H) under the action ρ0 of the group C 0 (U, H). This orbit coincides with the space C(U, H) := C 0 (U, H)/H 0 (P, H), and we denote it by Cm (U, H). Consider the union of orbits N =
∪ Cm (U, H) .
m∈M
(6.2)
The space N ⊂ Z 1 (U, H) is a bundle over M associated with the principal fibre bundle P (M, C 0 ), N = P (M, C 0 (U, H)) ×C 0 (U,H) C(U, H) ,
(6.3)
and the group C 0 acts on N on the left. The space N is a neighbourhood of the unity F 0 = 1 in the space Z 1 (U, H). We consider an open subset N 0 ⊂ Z 1 (U, H) equivalent to N and such that for all transition matrices F from N 0 there exists a solution of the Riemann–Hilbert problem (4.17) on CP1x and F 0 ∈ N 0 . Then we can introduce the germ N of the set N at the point F 0 as a class of sets equivalent to N .
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A. D. POPOV
The group C 0 (U, H) acts on any representative N of the germ N, and we have M = ρ0 (C 0 )\N ,
(6.4)
i.e., M is a set of orbits of the group C 0 in the space N (cf. (5.10)). By virtue of the Penrose–Ward correspondence described in Sec. 4, there is a bijection between the space M and the moduli space of real-analytic solutions to the SDYM equations on an open set U ⊂ R4 which are sufficiently close to the trivial solution A0 = 0. A set of all such solutions is called the space of local solutions (a small open neighbourhood of the point A0 = 0). So, M is bijective to the moduli space of local solutions to the SDYM equations with the marked point A0 = 0. However, as a marked ˆ 0 over P, point in Z 1 (U, H) one can choose a transition matrix Fˆ of a bundle E 1 ˆ holomorphically trivial on CPx0 , which corresponds to a solution A of the SDYM equations. Then one can consider bundles (trivial on CP1x , x ∈ U ) with transition ˆ ⊂ Z 1 (U, H) of the point Fˆ and the moduli matrices from an open neighbourhood N 0 ˆ of these bundles. This space M ˆ will be bijective to the space ˆ = ρ0 (C )\N space M ˆ of local solutions to the SDYM equations that are near the solution A. 6.3. Jumping points and jumping lines Let us consider a holomorphic bundle E 0 over the twistor space Z = L−1 ⊕L−1 ' R × CP1 such that its restriction to P ⊂ Z belongs to the space N ⊂ Z 1 (U, H) introduced in Sec. 6.2. In general the bundle E 0 will be holomorphically trivial on real projective lines CP1x parametrized not by x from U but by x from a “wider” open set U 0 ⊃ U . Those points x from R4 , for which E 0 |CP1x are not holomorphically trivial, are called the jumping points, and projective lines CP1x corresponding to them are called the jumping lines. In the Ward construction the jumping points give rise to singularities in the gauge potential A. The set R4 −U 0 of jumping points has codimension 1 (hypersurface) or more, i.e., the set U 0 is an open dense subset in R4 . Lines CP1x with x ∈ U 0 are called generic lines, and semi-stability of the bundle E 0 is equivalent to being trivial on the generic line. For more details see e.g. [39, 40]. Now we consider a holomorphic bundle E 00 over Z such that its restriction E 00 |P to P belongs to N , and E 00 is nonequivalent to the bundle E 0 considered above. So, E 0 |P and E 00 |P correspond to different points from the moduli space M. The bundle E 00 will be holomorphically trivial on CP1x with x from an open set U 00 ⊃ U and in the general case U 0 6= U 00 . In other words, subsets of jumping points for the 0 bundles E 0 and E 00 do not coincide. At last, one can consider bundles Einst over Z 4 4 0 which have no jumping points in R ⊂ S . The restriction of Einst to P belongs to N , and instantons are parametrized by a subset Ninst in the set N . It is clear that Ninst ⊂ N is a “small” subset of N , and for a fixed topological charge the dimension of the moduli space Minst is finite. 4
6.4. Representatives M0 and N0 of the germs M and N In Sec. 6.2 the germ M at the point e of the set M and the germ N at the point F 0 of the set N have been introduced. As an example, we shall describe some representatives M0 and N0 of these germs using the standard ε-δ language.
SELF-DUAL YANG–MILLS: SYMMETRIES AND MODULI SPACE
1117
Consider the twistor space P for an open ball U = {x ∈ R4 : (x − x0 )2 < r02 } of the radius r0 with a center at the point x0 ∈ R4 , the cover U = {U1 , U2 } of P and the space Z 1 (U, H) of holomorphic vector bundles over P. For the cover O = {Ω1 , Ω2 } of CP1 from Sec. 3.2, we consider the closure Ω12 := {λ ∈ C : α1 ≤ |λ| ≤ α2 } of the open set Ω12 = Ω1 ∩ Ω2 . Let U be the closure of the open set U : U = {x ∈ R4 : (x − x0 )2 ≤ r02 }. Then the closure of the open set U12 = U × Ω12 is U 12 = U × Ω12 ,
(6.5)
and U 12 is a compact subset of the set P. We assume that matrix-valued transition functions F12 of bundles E 0 are not only holomorphic on U12 , but also smooth on U 12 . This mild assumption can be replaced by the condition of holomorphy of F12 in an open δ-neighbourhood of the set U 12 with sufficiently small δ > 0 [44]. Length |ξ| of a vector ξ = (ξ1 , . . . , ξn ) ∈ Cn P P 2 is given by the formula |ξ|2 = i |ξi | = i ξi ξ i . We consider complex n × n matrices A = (aij ) defining a linear transformation A : ξ 7→ Aξ. For the matrices A we define a norm |A| by setting (see e.g. [44]): |A| := max ξ6=0
|Aξ| = max |Aξ| . |ξ| |ξ|=1
(6.6a)
Now let us introduce a norm k · k on the space Z 1 (U, H) setting kFk = max |F12 (z1 )| z1 ∈U 12
(6.6b)
for F ∈ Z 1 (U, H). Then Z 1 (U, H) turns into a topological space. It follows from the equality (6.1) discussed in Sec. 6.2, that there exists such a positive number r1 (x) depending on x ∈ U that the bundle Ex0 = E 0 |CP1x will be holomorphically trivial if its transition matrix satisfies the condition max |F12 (x, λ) − 1| < r1 (x).
(6.7a)
λ∈Ω12
The function r1 (x) : U → R can always be chosen smooth. It maps the compact space U into R and therefore r1 (x) ≥ r1 := min r1 (x) ,
(6.7b)
x∈U
i.e., it is bounded from below. Moreover, one can always choose such a radius r0 of an open ball U that r1 will be positive: r1 > 0. We fix the radius r0 of an open ball U and consider all F ∈ Z 1 (U, H) such that kF − 1k ≡ max |F12 (z1 ) − 1| < r1 , z1 ∈U 12
(6.8)
i.e., we consider the transition matrices F ∈ Z 1 (U, H) close to the identity in the norm (6.6b). By virtue of (6.7b), all such transition matrices will satisfy the condition (6.7a) for any x ∈ U and therefore holomorphically nontrivial bundles E 0
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A. D. POPOV
over P, associated with them, will be holomorphically trivial on CP1x ,→ P for all x ∈ U. Notice that in the general case the action (5.9) of the group C 0 (U, H) does not preserve the condition (6.8) on F ∈ Z 1 (U, H), but it preserves the condition of holomorphic triviality of bundles E 0 on CP1x . As such, we can act by the group C 0 (U, H) on the space of all F’s satisfying inequality (6.8) and “spread” this space over the space Z 1 (U, H). As usual, two matrices F and Fˆ satisfying the condition (6.8) are considered to be equivalent if they are connected by formula (5.9). Factorizing the space of all transition matrices satisfying (6.8) by this equivalence relation, we get a moduli space M0 . The space M0 is one of representatives of the germ M at the point e = [E00 ] of the moduli space of holomorphic bundles introduced in Sec. 6.2. Now, following Sec. 6.2, we introduce the space N0 =
∪
m∈M0
Cm (U, H) ,
(6.9)
obtained by the “spread” of F(m) over the space Z 1 (U, H) with the help of the action of the group C 0 (U, H). We have (cf. (6.4)) M0 = ρ0 (C 0 )\N0 ,
(6.10)
i.e., M0 is the space of orbits of the group C 0 (U, H) in the space N0 . The space N0 is an open neighbourhood of F 0 = 1 in the set Z 1 (U, H) and is one of representatives of the germ N at the point F 0 = 1 of the space of holomorphic bundles described in Sec. 6.2. So, for transition matrices F12 from N0 the Birkhoff decomposition (4.17) exists for all x ∈ U . ˇ 6.5. Symmetries of local solutions in the Cech approach We consider the space Z 1 (U, H) of holomorphic bundles E 0 over P and the open subset N in Z 1 (U, H) introduced in Sec. 6.2. In Secs. 5.3–5.5 we have defined the group G(P, H) := H(P)nC 1 (U, H) and described its action ρ on the space Z 1 (U, H). This action, of course, does not map N into itself (or into another representative of the germ N), and one should consider a local action of the group G(P, H). Let us consider an open neighbourhood H of the identity of the group H(P), an open neighbourhood C of the identity of the group C 1 (U, H) and an open neighbourhood G := H n C of the identity of the group G(P, H). As explained in Appendix A and Sec. 6.1, the local groups H, C and G are representatives of the germs H, C, G at the identity of the groups H(P), C 1 (U, H) and G(P, H), respectively. As local groups, H, C and G are isomorphic to the groups H(P), C 1 (U, H) and G(P, H). The above-mentioned representatives of the germs H and C can always be chosen so that the local group G will map the set N into itself. In more detail, there exists a subset N 0 of the set N (N 0 is another representative of the germ N) such that we have a map ρ : G × N 0 → N . The map ρ : N 00 → N , where N 00 = {(a, F) ∈ G×N : ρ(a, F) ∈ N } is an open subset in G×N containing {e}×N , is also defined. In this case, the properties ρ(e, F) = F, ρ(a, ρ(b, F)) = ρ(ab, F) etc. are fulfilled for all (a, F) ∈ N 00 . In particular, the local group H of biholomorphisms acts on the space N by formula (5.30) from Sec. 5.5.
SELF-DUAL YANG–MILLS: SYMMETRIES AND MODULI SPACE
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For the matrix local group C we introduce the diagonal subgroup 1 C4 := C ∩ C4 (U, H) ,
(6.11)
which is the local stability subgroup of the marked cocycle F 0 ∈ N . For the 1 definition of the group C4 (U, H) see (5.15b). Then, by repeating all the arguments of Sec. 5.3 in terms of the local groups, we have N ' C/C4 ,
(6.12a)
i.e., N is a coset space. In other words, for each representative N of the germ N of the space of bundles, holomorphically trivial on CP1x ,→ P, one can always choose a representative C of the germ C of the group of 1-cochains such that (6.12a) will take place. In fact, (6.12a) is a consequence of an isomorphism of germs N ' C/C4 .
(6.12b)
Combining (6.12a) and (6.4), we obtain M ' ρ0 (C 0 )\C/C4 ,
(6.13a)
i.e., the moduli space of local solutions to the SDYM equations is a double coset space. Again, (6.13a) is a consequence of the isomorphism of germs M ' ρ0 (C0 )\C/C4 .
(6.13b)
Thus, the full group of continuous symmetries acting on the space N is a semidirect product G=HnC (6.14) of the local group H of holomorphic automorphisms of the space P and of the local group C of 1-cochains of the cover U with values in the sheaf H = OSL(n,C) of holomorphic maps of the space P into the group SL(n, C). 6.6. Unitarity conditions As discussed in Sec. 4.4, the transition matrices F12 in holomorphic bundles E 0 → P which are compatible with the real structure τ on P have to satisfy the additional condition (4.16b). Denote by n o † Zτ1 (U, H) := F ∈ Z 1 (U, H) : F12 (τ (¯ z1 )) = F12 (z1 ) (6.15) a subset of transition matrices satisfying this unitarity conditions. We should next define subgroups Cτ0 in C 0 and Cτ1 in C 1 such that their action, described by formulae (5.9) and (5.14), will preserve Zτ1 (U, H). It is not hard to see that n o Cτ0 (U, H) = {h1 , h2 } ∈ C 0 (U, H) : h†1 (τ (¯ z1 )) = h−1 (z ) , (6.16) 1 2 n o Cτ1 (U, H) = {h12 , h21 } ∈ C 1 (U, H) : h†12 (τ (¯ z1 )) = h−1 21 (z1 ) .
(6.17)
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Actions of these groups on Zτ1 have the form: z1 )) , F12 7→ Fˆ12 (z1 ) := ρ0 (h, F)12 = h1 (z1 )F12 (z1 )h†1 (τ (¯ F12 7→ F˜12 (z1 ) := ρ(h, F)12 = h12 (z1 )F12 (z1 )h†12 (τ (¯ z1 )) ,
h ∈ Cτ0 , h ∈ Cτ1 .
(6.18) (6.19)
By the definitions (6.16) and (6.17), Cτ0 (U, H) and Cτ1 (U, H) are real subgroups in C 0 (U, H) and C 1 (U, H), respectively. ˆ 0 , F12 ∼ The cocycles F12 and Fˆ12 from (6.18) define equivalent bundles E 0 ∼ E 1 ˆ F12 , and one can introduce a 1-cohomology set Hτ (U, H) as a set of orbits of the group ρ0 (Cτ0 ) in the space Zτ1 (U, H) of transition matrices compatible with the real structure τ on P, Hτ1 (U, H) := ρ0 (Cτ0 (U, H))\Zτ1 (U, H) ⊂ H 1 (U, H) .
(6.20)
For the cover U = {U1 , U2 } we have Hτ1 (P, H) = Hτ1 (U, H). So, the real structure τ on P induces a real structure on H 1 (P, H), and Hτ1 (P, H) is a set of real “points” of the space H 1 (P, H) corresponding to the bundles E 0 with the unitary structure (6.15). Consider the action of the group Cτ1 on Zτ1 . As a stability subgroup of the element F 0 = 1 compatible with the real structure we have the group 1 Cτ14 := Cτ1 ∩ C4 = {{h12 , h21 } ∈ Cτ1 (U, H) : h12 = h21 } ,
(6.21)
and the space Zτ1 (U, H) can be identified with the quotient space Zτ1 (U, H) = Cτ1 (U, H)/Cτ14 (U, H) .
(6.22)
The moduli space Hτ1 (P, H) of holomorphic bundles E 0 with the unitary structure coincides with the double coset space Hτ1 := ρ0 (Cτ0 )\Cτ1 /Cτ14 ,
(6.23)
and this set is isomorphic to (i) the set of Cτ14 -orbits in Yτ1 := ρ0 (Cτ0 )\Cτ1 , (ii) the set of Cτ0 -orbits in Zτ1 , (iii) the set of Cτ1 -orbits in Yτ1 × Zτ1 . As to the group H(P), the action of which on Z 1 (U, H) was described in Sec. 5.5, one should choose in it a subgroup Hτ (P) of those transformations η ∈ H(P) which are compatible with the real structure τ on P. In terms of the functions η1 and η2 from (5.26) representing η in the chosen coordinates it means that η1a (τ (¯ z1 )) = Bba η2b (z2 ) ,
(6.24)
where the coefficients Bba are written down in (3.21). Thus, the symmetry group acting on the space Zτ1 (U, H) of holomorphic bundles E 0 satisfying the unitarity conditions is the group Gτ (P, H) := Hτ (P) n Cτ1 (U, H) . This group is a real subgroup in the group (5.31).
(6.25)
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Further, going over to local solutions, we introduce a subset Nτ of those transition matrices from N which satisfy the condition (4.16b), i.e., Nτ := N ∩ Zτ1 (U, H). One analogously introduces the moduli space Mτ := M ∩ Hτ1 (P, H), the real local groups Hτ := H ∩ Hτ (P), Cτ := C ∩ Cτ1 (U, H) and the germs Hτ , Cτ corresponding to them. Then one obtains isomorphisms Nτ ' Cτ /Cτ 4 ,
Mτ ' ρ0 (Cτ0 )\Cτ /Cτ 4 ,
(6.26)
corresponding to the isomorphisms (6.12), (6.13). At last, as the symmetry group ˇ of the space of real local solutions in the Cech approach one gets the local group Gτ = Hτ n Cτ ,
(6.27)
which is a semidirect product of the local groups Hτ and Cτ . 7. Holomorphic Bundles: the Dolbeault Description 7.1. Some definitions The well-known Dolbeault theorem reduces a computation of cohomology spaces of a manifold X with the coefficients in a sheaf of germs of holomorphic maps from X into a complex Abelian group T to problems of calculus of T-valued differential forms ˇ of the type (0,q) on the manifold X (isomorphism between Cech and Dolbeault cohomology groups) [36]. We want to describe an analogue of the Dolbeault theorem for the sheaf H of germs of holomorphic maps of the space P into the non-Abelian group SL(n, C), following mainly the papers [42]. This will permit us to describe symmetries of the space of local solutions to the SDYM equations on U ⊂ R4 . But first, let us recall some definitions for objects which will be considered below. Let K be a sheaf of groups and A a sheaf of sets on X. We shall say that K acts on A if for any x ∈ X the group Kx acts on Ax , and also this action is continuous in the topology of the sheaves K and A. It is said that K transitively acts on A, if Kx transitively acts on Ax for each x ∈ X. In this case A can be identified with a quotient sheaf K/K 0 , where K 0 is a sheaf of stability subgroups Kx0 and stalks of the sheaf K/K 0 are quotient spaces Kx /Kx0 . Conversely, if K 0 is a subsheaf of subgroups in K, the sheaf K/K 0 can be considered as a sheaf of sets with a marked section x 7→ Kx0 , x ∈ X, on which K transitively acts on the left. ˆ Bˆ0,q and Bˆ 7.2. The sheaves S, Consider the sheaf Sˆ of germs of smooth maps from P into the group SL(n, C). The sheaf H of germs of holomorphic maps P → SL(n, C) is a subsheaf of the ˆ and there exists a canonical embedding i : H → S. ˆ Consider also the sheaf sheaf S, Bˆ0,q (q = 1, 2, . . .) of germs of smooth (0,q)-forms on P with values in the Lie algebra sl(n, C). Let us define a map δ¯0 : Sˆ → Bˆ0,1 given for any open set U of the space P by the formula ˆ ψˆ−1 , δ¯0 ψˆ = −(∂¯ψ) (7.1)
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¯ Let us also introduce an operator ˆ where ψˆ ∈ S(U), δ¯0 ψˆ ∈ Bˆ0,1 (U), d = ∂ + ∂. 1 0,1 0,2 ¯ ˆ ˆ δ : B → B , defined for any open set U ⊂ P by the formula ˆ = ∂¯B ˆ +B ˆ ∧B ˆ, δ¯1 B
(7.2)
ˆ ∈ Bˆ0,2 (U). In other words, the maps of sheaves δ¯0 : Sˆ → ˆ ∈ Bˆ0,1 (U), δ¯1 B where B Bˆ0,1 and δ¯1 : Bˆ0,1 → Bˆ0,2 are defined by means of localizations. In particular, on U1 ⊂ P we have (1) (δ¯0 ψˆ1 )a = −(V¯a ψˆ1 )ψˆ1−1
(7.10 )
ˆ (1) )ab = V¯a(1) B ˆ (1) − V¯ (1) B ˆa(1) + [B ˆa(1) , B ˆ (1) ] . (δ¯1 B b b b
(7.20 )
The sheaf Sˆ acts on the sheaves Bˆ0,q (q = 1, 2, . . .) with the help of the adjoint representation. In particular, for any open set U ⊂ P we have ˆ B) ˆ 7→ Ad(ψ, ˆ = ψˆ−1 B ˆ ψˆ + ψˆ−1 ∂¯ψˆ , B
(7.3a)
ˆ Fˆ ) = ψˆ−1 Fˆ ψˆ , Fˆ 7→ Ad(ψ,
(7.3b)
ˆ ˆ ∈ Bˆ0,1 (U), Fˆ ∈ Bˆ0,2 (U). where ψˆ ∈ S(U), B ˆ ˆ with Denote by B the subsheaf in Bˆ0,1 consisting of germs of (0,1)-forms B 1 ˆ ¯ ˆ values in sl(n, C) such that δ B = 0, i.e., sections B over any open set U of the sheaf Bˆ = Ker δ¯1 satisfy the equations ˆ +B ˆ ∧B ˆ = 0, ∂¯B ˆ ∈ where B connections
(7.4)
Bˆ0,1 (U). So the sheaf Bˆ can be identified with the sheaf of (0,1)¯ ˆ in the holomorphic bundle E 0 over P. ∂Bˆ = ∂¯ + B
7.3. The sheaves S, B0,q and B Recall that P is the fibre bundle with fibres CP1x over the points x from U ⊂ R4 , and the canonical projection π : P → U is defined. The typical fibre CP1 has the SU (2)-invariant complex structure j (see Sec. 3.2), and the vertical distribution V = Ker π∗ inherits this complex structure. A restriction of V to each fibre CP1x , x ∈ U , is the tangent bundle to that fibre. The (flat) Levi–Civita connection on U generates the splitting of the tangent bundle T (P) into a direct sum T (P) = V ⊕ H
(7.5)
of the vertical distribution V and the horizontal distribution H. Using the complex structures j, J and J on CP1 , U and P respectively, one can split the complexified tangent bundle of P into a direct sum T C (P) = (V 1,0 ⊕ H 1,0 ) ⊕ (V 0,1 ⊕ H 0,1 )
(7.6)
of vectors of type (1,0) and (0,1). So we have the integrable distribution V 0,1 of (1) (2) antiholomorphic vector fields with the basis V¯3 = ∂λ¯ on U1 ⊂ P and V¯3 = ∂ζ¯
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on U2 ⊂ P. The vector fields (3.4a), (3.4b) and (3.11a), (3.11b) form a basis in the normal bundle H 0,1 of a line CP1x ,→ P. Having the canonical distribution V 0,1 on the space P, we introduce the sheaf S of germs of partially holomorphic maps ψ : P → SL(n, C), which are annihilated by vector fields from V 0,1 . In other words, sections of the sheaf S over open subsets U ⊂ P are SL(n, C)-valued functions ψ on U, which satisfy the equations ∂λ¯ ψ = 0
on U ∩ U1 ,
∂ζ¯ψ = 0 on U ∩ U2 ,
(7.7)
i.e., they are holomorphic along CP1x ,→ P, x ∈ U . It is obvious that the sheaf H of holomorphic maps from P into SL(n, C), i.e., smooth maps which are annihilated ˆ by vector fields from V 0,1 ⊕ H 0,1 , is a subsheaf of S and S is a subsheaf of S. 0,q 0,1 ˆ Consider now the sheaves B , introduced in Sec. 7.2. Let B be the subsheaf of (0,1)-forms from Bˆ0,1 vanishing on the distribution V 0,1 . In components this means that for any open set U ⊂ P (1)
B3 = 0 on U ∩ U1 ,
(2)
B3 = 0
on U ∩ U2 ,
(7.8)
where B (1) belongs to the section of the sheaf B 0,1 over U1 , and B (2) belongs to the section of the sheaf B 0,1 over U2 . So B 0,1 is the subsheaf of Bˆ0,1 . The map δ¯0 , introduced in Sec. 7.2, induces a map δ¯0 : S → B 0,1 , defined for any open set U of the space P by the formula −1 ¯ , δ¯0 ψ = −(∂ψ)ψ
(7.9a)
where ψ ∈ S(U), δ¯0 ψ ∈ B 0,1 (U). Analogously, the operator δ¯1 induces a map δ¯1 : B 0,1 → Bˆ0,2 , given for any open set U ⊂ P by the formula ¯ +B∧B, δ¯1 B = ∂B
(7.10a)
where B ∈ B 0,1 (U), δ¯1 B ∈ Bˆ0,2 (U). In particular, on U1 ⊂ P we have a δ¯0 ψ1 = −{(V¯a(1) ψ1 )ψ1−1 }θ¯(1) ,
(7.9b)
1 (1) (1) (1) a b δ¯1 B (1) = {V¯a(1) Bb − V¯b Ba(1) + [Ba(1) , Bb ]}θ¯(1) ∧ θ¯(1) , (7.10b) 2 a where the (0,1)-forms {θ¯1,2 } were introduced in Sec. 3.3, ψ1 ∈ S(U1 ), B (1) ∈ B 0,1 (U1 ). The sheaf S acts on the sheaves B 0,1 and Bˆ0,q by means of the adjoint representation. In particular, for B 0,1 and Bˆ0,2 we have the same formulae (7.3) with replacement ψˆ by ψ ∈ S(U), ¯ , B 7→ Ad(ψ, B) = ψ −1 Bψ + ψ −1 ∂ψ
(7.11a)
Fˆ 7→ Ad(ψ, Fˆ ) = ψ −1 Fˆ ψ ,
(7.11b)
where B ∈ B 0,1 (U), Fˆ ∈ Bˆ0,2 (U).
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At last, let us denote by B the subsheaf of B 0,1 consisting of germs of sl(n, C)valued (0,1)-forms B such that δ¯1 B = 0, i.e., sections B of the sheaf B = Ker δ¯1 satisfy the equations ¯ +B ∧B = 0. ∂B (7.12a) In components for B ∈ B 0,1 (U1 ) on the open set U1 Eqs. (7.12a) have the form: (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) V¯1 B2 − V¯2 B1 + [B1 , B2 ] = 0 , V¯3 B1 = 0 , V¯3 B2 = 0 ,
(7.12b)
(1)
since B3 = 0. We have analogous equations on U2 ⊂ P. 7.4. Exact sequences of sheaves ˆ Bˆ0,1 , Bˆ0,2 } with the ˆ Bˆ0,1 and Bˆ0,2 . The triple {S, Let us consider the sheaves S, maps δ¯0 and δ¯1 is a resolution of the sheaf H, i.e., the sequence of sheaves δ¯ δ¯ i 1 −→ H −→ Sˆ −→ Bˆ0,1 −→ Bˆ0,2 , 0
1
(7.13)
where i is an embedding, is exact. For proof see [42]. Restricting δ¯0 to S ⊂ Sˆ and δ¯1 to B 0,1 ⊂ Bˆ0,1 , we obtain the exact sequence of sheaves δ¯ δ¯ i 1 −→ H −→ S −→ B 0,1 −→ Bˆ0,2 , 0
1
(7.14)
where 1 is the identity of the sheaf H. By virtue of the exactness of the sequence (7.13), we have δ¯0 Sˆ = Ker δ¯1 = Bˆ .
(7.15a)
Since δ¯0 is the projection, connected with the action (7.3a) of the sheaf Sˆ on Bˆ0,1 , ˆ the sheaf Sˆ acts transitively with the help of Ad on Bˆ and Bˆ ' S/H. Thus, we obtain the exact sequence of sheaves δ¯ δ¯ i 1 −→ H −→ Sˆ −→ Bˆ −→ 0 . 0
1
(7.15b)
For more details see [42]. Restricting the map δ¯0 to S and δ¯1 to B, we obtain the exact sequence of sheaves i
δ¯0
δ¯1
1 −→ H −→ S −→ B −→ 0 ,
(7.16)
since δ¯0 S = Ker δ¯1 (the exactness of the sequence (7.14)), and S acts on B transitively (B ' S/H). For sections of the sheaf B over U1 and U2 we have (1) (1) (1) (1) (1) (1) B1 = −(V¯1 ψ1 )ψ1−1 , B2 = −(V¯2 ψ1 )ψ1−1 , B3 = −(V¯3 ψ1 )ψ1−1 ≡ 0 ,
(7.17a) (2) B1
=
(2) −(V¯1 ψ2 )ψ2−1
,
(2) B2
=
(2) −(V¯2 ψ2 )ψ2−1
,
(2) B3
=
(2) −(V¯3 ψ2 )ψ2−1
≡ 0, (7.17b)
where ψ1,2 ∈ S(U1,2 ), B (1,2) ∈ B(U1,2 ).
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7.5. The group H 0 (P, S) and the cohomology set H 1 (P, S) Having the sheaf S of partially holomorphic smooth maps from P into SL(n, C) and the two-set open cover U = {U1 , U2 }, we consider the groups of cochains C 0 (U, S) = {maps ψ1 : U1 → S(U1 ), ψ2 : U2 → S(U2 )} = S(U1 ) × S(U2 ) , (7.18a) C 1 (U, S) = {maps f12 : U12 → S(U12 ), f21 : U12 → S(U12 )} = S(U12 ) × S(U12 ) , (7.18b) where S(U) is a space of sections of the sheaf S over an open set U ⊂ P. For 0- and 1-cocycles we have Z 0 (P, S) = {ψ = {ψ1 , ψ2 } ∈ C 0 (U, S) : ψ1 = ψ2 on U12 } ,
(7.19a)
−1 Z 1 (U, S) = {f = {f12 , f21 } ∈ C 1 (U, S) : f21 = f12 }.
(7.19b)
By definition, H 0 (P, S) := Z 0 (P, S) = Γ(P, S). As usual, two cocycles F, Fˆ ∈ Z 1 (U, S) are called equivalent if Fˆ12 = ψ1 F12 ψ2−1 for some ψ = {ψ1 , ψ2 } ∈ C 0 (U, S). ˇ A set of equivalence classes of 1-cocycles F is the Cech 1-cohomology set H 1 (U, S). For the considered cover U we have H 1 (P, S) = H 1 (U, S). By replacing the sheaf H by the sheaf S in the formulae of Sec. 5.2, one can define the action of the group C 0 (U, S) on C 1 (U, S) by automorphisms σ0 , σ0 (ψ, f )12 = ψ2 f12 ψ2−1 , ψ = {ψ1 , ψ2 } ∈ C 0 (U, S) ,
σ0 (ψ, f )21 = ψ1 f21 ψ1−1 , f = {f12 , f21 } ∈ C 1 (U, S) ,
(7.20)
and define a twisted homomorphism δ 0 : C 0 (U, S) → C 1 (U, S) by the formulae δ 0 (φ)12 = φ1 φ−1 2 ,
δ 0 (φ)21 = φ2 φ−1 1 ,
δ 0 (hφ) = δ 0 (h)σ0 (h, δ 0 (φ)) ,
(7.21)
where φ = {φ1 , φ2 } ∈ C 0 (U, S), δ 0 (φ) ∈ Z 1 (U, S) ⊂ C 1 (U, S). Then we have H 0 (P, S) = Ker δ 0 ,
(7.22)
and the image Im δ 0 = δ 0 (C 0 (U, S)) ⊂ Z 1 (U, S)
(7.23)
of the map δ corresponds to the marked element e ∈ H (P, S), i.e., to the class of smoothly trivial bundles over P which are holomorphically trivial over CP1x ,→ P, x ∈ U . Transition matrices F ∈ Im δ 0 have the form (4.17): F12 = ψ1−1 (x, λ)ψ2 (x, λ). Finally, for ψ ∈ C 0 (U, S), F ∈ Z 1 (U, S), the formula 0
1
ρ0 (ψ, F) := δ 0 (ψ)σ0 (ψ, F) ⇔ ρ0 (ψ, F)12 = ψ1 F12 ψ2−1
(7.24)
defines the action of the group C 0 (U, S) on the set Z 1 (U, S), and we obtain H 1 (U, S) = ρ0 (C 0 (U, S))\Z 1 (U, S) . For the chosen cover U we have H 1 (P, S) = H 1 (U, S).
(7.25)
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7.6. Exact sequences of cohomology sets From (7.15b) we obtain the exact sequence of cohomology sets [42] i
δ¯0
δ¯1
ϕ ˆ
∗ ∗ ∗ ˆ −→ ˆ −→ ˆ , e −→ H 0 (P, H) −→ H 0 (P, S) H 0 (P, B) H 1 (P, H) −→ H 1 (P, S)
(7.26) where e is a marked element (identity) of the considered sets, and a homomorphism ϕˆ coincides with the canonical embedding, induced by the embedding of sheaves ˆ The kernel Ker ϕˆ = ϕˆ−1 (e) of the map ϕˆ coincides with a subset i : H → S. ˆ of those elements from H 1 (P, H), which are mapped into the class e ∈ H 1 (P, S) of topologically (and smoothly) trivial bundles. This means that representatives of the subset Ker ϕˆ are those transition matrices F ∈ Z 1 (U, H) for which there exists a splitting ¯ 2 (x, λ, λ) ¯ F12 = ψ1−1 (x, λ, λ)ψ (7.27) with smooth matrix-valued functions ψ1 , ψ2 ∈ SL(n, C). Similarly, from (7.16) we obtain the exact cohomology sequence i
δ¯0
δ¯1
ϕ
∗ ∗ ∗ e −→ H 0 (P, H) −→ H 0 (P, S) −→ H 0 (P, B) −→ H 1 (P, H) −→ H 1 (P, S) ,
(7.28) where a homomorphism ϕ is an embedding, induced by the embedding of sheaves i : H → S. The kernel Ker ϕ = ϕ−1 (e) of the map ϕ coincides with a subset of those elements from H 1 (P, H), which are mapped into the class e ∈ H 1 (P, S) of smoothly trivial bundles over P, which are holomorphically trivial on any projective line CP1x ,→ P, x ∈ U . This means that representatives of the subset Ker ϕ of the 1-cohomology set H 1 (P, H) are those transition matrices F ∈ Z 1 (U, H) for which there exists a Birkhoff decomposition (cf. (4.17)) F12 = ψ1−1 (x, λ)ψ2 (x, λ)
(7.29)
with smooth matrix-valued functions ψ1 , ψ2 ∈ SL(n, C) that are holomorphic in λ. The map δ¯0 corresponds a global section ¯ 1 )ψ −1 on U1 , B (2) = −(∂ψ ¯ 2 )ψ −1 on U2 , B (1) = B (2) on U12 } , B = {B (1) = −(∂ψ 1 2 (7.30) of the sheaf B over P to {ψ1 , ψ2 } ∈ C 0 (U, S). The equality B (1) = B (2) on U12 , which means that the (0,1)-form B ∈ H 0 (P, B) is defined globally, follows from the identity: ¯ 2 = ψ −1 {−(∂ψ ¯ 12 = ∂(ψ ¯ −1 ψ2 ) = (∂ψ ¯ −1 )ψ2 + ψ −1 ∂ψ ¯ 1 )ψ −1 + (∂ψ ¯ 2 )ψ −1 }ψ2 = 0 . ∂F 1 1 1 1 1 2 (7.31) The group S(P) := H 0 (P, S) = Z 0 (P, S) = Γ(P, S) of global sections of the sheaf S acts on the set H 0 (P, B) with the help of Ad(g, ·) transformations ¯ , Ad(g, B) = g −1 Bg + g −1 ∂g
(7.32)
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where g ∈ H 0 (P, S), B ∈ H 0 (P, B). Notice that from the definition (7.19a) of the group H 0 (P, S) and from the Liouville theorem for CP1x ,→ P it follows that the elements g ∈ H 0 (P, S) do not depend on λ. Comparing (7.12) and (7.30) with (4.20)–(4.23), we conclude that the 0-cohomology set H 0 (P, B) coincides with the space of (complex) local solutions to the SDYM equations on U ⊂ R4 , the group H 0 (P, S) coincides with the group of (complex) gauge transformations, and the quotient space H 0 (P, B)/H 0 (P, S) coincides with the moduli space of (complex) local solutions to the SDYM equations on U . The space Ker ϕ is a representative of the germ M at the point e ∈ H 1 (P, H) of the moduli space of bundles E 0 over P, holomorphically trivial on CP1x ,→ P, x ∈ U . We will denote it by M := Ker ϕ; this set was described in detail in Sec. 6. From the exactness of the sequence (7.28) it follows that the set M = Ker ϕ ⊂ H 1 (P, H) is bijective to the moduli space H 0 (P, B)/H 0 (P, S) of (complex) solutions to the SDYM equations, M ' H 0 (P, B)/H 0(P, S) . (7.33) This correspondence is a non-Abelian analogue of the Dolbeault theorem about the ˇ isomorphism of (Abelian) Cech and Dolbeault 1-cohomology groups. ˆ considered in Secs. 7.2, 7.4 and 7.6, one Remark. Using the sheaves Sˆ and B, can introduce a Dolbeault 1-cohomology set H∂0,1 ¯ ˆ (P) as a set of orbits of the group B 0 0 ˆ ˆ H (P, S) in the set H (P, B), i.e., 0 0 ˆ ˆ H∂0,1 ¯ (P) := H (P, B)/H (P, S) . ˆ B
(7.34)
The set H 0 (P, B)/H 0 (P, S) considered above is an open subset in the Dolbeault 1-cohomology set H∂0,1 ¯ (P). It follows from the exactness of the sequence (7.26) that ˆ B
H∂0,1 ˆ i.e., the moduli space H∂0,1 ¯ ˆ (P) ' Ker ϕ, ¯ ˆ (P) of global solutions of Eq. (7.4) on B B P is bijective to the moduli space of holomorphic bundles over P which are trivial as smooth bundles. Transition matrices of such bundles have the form (7.27). Using the bijection (7.33), we will identify the spaces M and H 0 (P, B)/H 0(P, S) and denote them by the same letter M. It also follows from (7.33) that H 0 (P, B) is a principal fibre bundle H 0 (P, B) = P (M, H 0 (P, S))
(7.35)
with the base space M and the structure group H 0 (P, S). 7.7. Unitarity conditions In Sec. 6.6 we discussed the imposition of a unitarity condition on transition matrices F ∈ Z 1 (U, H) and defined various subsets of transition matrices and their moduli satisfying the unitarity condition. As discussed in Sec. 4.4, the matrices ψ1 , ψ2 ∈ SL(n, C) corresponding to gauge fields with values in the algebra su(n) have to satisfy the condition (4.16c). The conditions (4.16a) for components of the gauge potential follow from (4.16c), (4.22) and
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(4.23). To satisfy these conditions, consider the following real subgroup Cτ0 (U, S) (a real form) of the group C 0 (U, S): Cτ0 (U, S) := {ψ = {ψ1 , ψ2 } ∈ C 0 (U, S) : ψ1† (τ (x, λ)) = ψ2−1 (x, λ)} ,
(7.36)
compatible with the real form τ on P. Of course, one can also define other real forms of the complex group C 0 (U, S) assuming ψ1† (τ (x, λ)) = Πψ2−1 (x, λ) ,
(7.37)
where Π is a diagonal matrix with m copies of +1 and n − m copies of −1. For all these subgroups the matrices δ 0 (ψ −1 ) = ψ1−1 ψ2 ∈ Z 1 (U, H) will satisfy the unitarity condition (4.16a) and therefore δ 0 (ψ −1 ) ∈ Zτ1 (U, H). The map δ 0 : Cτ0 (U, S) → Zτ1 (U, S) defines in Zτ1 (U, S) a subset of matrices −1 ψ1 ψ2 with {ψ1 , ψ2 } ∈ Cτ0 (U, S) which corresponds to the element e ∈ Hτ1 (P, S). The set Hτ1 (P, S) is defined analogously with the set H 1 (P, S) (see Sec. 7.5). The kernel Ker ϕτ = ϕ−1 τ (e) of the map ϕτ := ϕ |Hτ1 (P,H) : Hτ1 (P, H) → Hτ1 (P, S)
(7.38)
coincides with the moduli space Mτ of transition matrices F ∈ Zτ1 (U, H), for which there exists a Birkhoff decomposition (7.29) with ψ1 , ψ2 satisfying the unitarity conditions (4.16c). The map δ¯0 associates with ψ1 , ψ2 the global section (7.17), (7.30) of the sheaf B satisfying the unitarity condition (4.16a). We denote the space of all these solutions by Hτ0 (P, B). The matrices g ∈ SL(n, C) from the group H 0 (P, S) do not depend on λ, and the subgroup Hτ0 (P, S) = {g ∈ H 0 (P, S) : g † = g −1 }
(7.39)
of unitary matrices g(x) ∈ SU (n) preserves the space Hτ0 (P, B). So we have a one-to-one correspondence between Mτ and the moduli space Hτ0 (P, B)/Hτ0(P, S) of real local solutions to the SDYM equations, Mτ ' Hτ0 (P, B)/Hτ0(P, S) .
(7.40)
8. Symmetries in Terms of Smooth Sheaves 8.1. Riemann Hilbert problems from the cohomological point of view In Sec. 7.5 we described the twisted homomorphism δ 0 : C 0 (U, S) → C 1 (U, S), the image of which Im δ 0 = δ 0 (C 0 (U, S)) belongs to the set Z 1 (U, S) ⊂ C 1 (U, S). More precisely, we have Im δ 0 ' C 0 (U, S)/H 0 (P, S), where the group H 0 (P, S) = Ker δ 0 is a kernel of the map δ 0 . Hence δ 0 (C 0 (U, S)) can be identified with C 0 (U, S)/ H 0 (P, S), and δ 0 : C 0 (U, S) → C 0 (U, S)/H 0 (P, S) (8.1) is a projection of the group C 0 (U, S) onto the homogeneous space Q := C 0 (U, S)/ H 0 (P, S). So, the group C 0 (U, S) can be considered as a principal fibre bundle C 0 (U, S) = P (Q, H 0 (P, S))
(8.2)
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with the structure group H 0 (P, S) and the base space Q ⊂ Z 1 (U, S), points of which correspond to smoothly trivial bundles. As described in detail in Secs. 5–7, the space Q contains as a subset the set N of those holomorphic bundles which are not only trivial as smooth bundles, but also holomorphically trivial on CP1x ,→ P, x ∈ U . The group C 0 (U, S) acts on Q transitively by formula (7.24) and therefore for any cocycle F ∈ N ⊂ Q there exists an element ψ = {ψ1 , ψ2 } ∈ C 0 (U, S) such that the action ρ0 (ψ, ·) transforms F into F 0 = 1, ρ0 (ψ, F)12 = ψ1 F12 ψ2−1 = 1 ⇒ F12 = ψ1−1 ψ2 , (8.3) and to solve the Riemann–Hilbert problem means to find such an element ψ from the group C 0 (U, S). Of course, this element ψ ∈ C 0 (U, S) is not unique; it is defined up to an element g from the stability subgroup H 0 (P, S) of the point F 0 = 1. Indeed, if ψ1 F12 ψ2−1 = 1, then (g −1 ψ1 )F12 (g −1 ψ2 )−1 = 1 for any g ∈ H 0 (P, S). In other words, to solve the Riemann–Hilbert problem means to define a section s : N → C 0 (U, S) (8.4) over N ⊂ Q of the bundle (8.2). The section s is not uniquely defined, and the group H 0 (P, S) defines a transformation g of the section s into an equivalent section sg . Remark. It should be stressed that the cohomological description of the construction of solutions is applicable not only to the SDYM equations, but also to all equations integrable with the help of a Birkhoff decomposition of matrices on CP1 (the dressing method [6–10]. For such equations, one can write an exact sequence of sheaves like (7.16) and an exact sequence of cohomology sets like (7.28). In many cases this can be done by reductions of the sheaves H, S and B, which explains the known fact that (almost) all integrable equations in 2D can be obtained by reductions of the SDYM equations (see e.g. [40, 47, 48, 49] and references therein). Consider the restriction P (N , H 0 (P, S)) := P (Q, H 0 (P, S)) |N = (δ 0 )−1 (N )
(8.5)
of the principal fibre bundle P (Q, H 0 (P, S)) to the subset N ⊂ Q. As described in Sec. 6.2, the group C 0 (U, H) acts on the space N on the left, and this action can be lifted up to the action on P (N , H 0 (P, S)), since this (left) action commutes with the (right) action of the group H 0 (P, S) on the space P (N , H 0 (P, S)). Thus, we have the space P (M, H 0 (P, S)) as a space of orbits of the group C 0 (U, H) in the space P (N , H 0 (P, S)), P (M, H 0 (P, S)) = P (ρ0 (C 0 (U, H))\N , H 0 (P, S)) = ρ0 (C 0 (U, H))\P (N , H 0 (P, S)) .
(8.6a)
At the same time, it follows from (7.35) that this space coincides with the space H 0 (P, B) = P (M, H 0 (P, S)) of (complex) local solutions to the SDYM equations.
(8.6b)
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Finally, it follows from (8.6) that the moduli space of (complex) local solutions to the SDYM equations is M ' ρ0 (C 0 (U, H))\P (N , H 0 (P, S))/H 0 (P, S) ,
(8.7)
i.e., M is the biquotient space of the space P (N , H 0 (P, S)) under the action of the groups C 0 (U, H) and H 0 (P, S). Using Sec. 7.7, where we discussed the unitarity conditions in terms of F12 , ψ ∈ C 0 (U, S), etc. one can rewrite all formulae of Sec. 8.1 in a way compatible with the real structure τ on P. In particular, for the moduli space Mτ of (real) local solutions to the SDYM equations we have Mτ ' ρ0 (Cτ0 (U, H))\P (Nτ , Hτ0 (P, S))/Hτ0 (P, S) .
(8.8)
Then gauge fields take values in the Lie algebra su(n). 8.2. Action of the symmetry group Gτ on real solutions of the SDYM equations We consider the cover U = {U1 , U2 } of the twistor space P and holomorphic bundles E 0 ∈ Nτ ⊂ Zτ1 (U, H). In Secs. 6.5 and 6.6, the (local) action of the local group Gτ = Hτ n Cτ on the space Nτ ' Cτ /Cτ 4 was described. Let us choose an arbitrary transition matrix F12 = ψ1−1 ψ2 ∈ Nτ and an element h = {η, a} ∈ Hτ n Cτ . Consider the action ρ(h, ·) of the element h ∈ Gτ given by formulae h (6.19), (5.30) and (6.24). Then we have ρ(h, ·) : F12 7→ F12 = ρ(h, F)12 . Since h the local action preserves Nτ , then F ∈ Nτ and therefore there exists an element ψ h = {ψ1h , ψ2h } ∈ Cτ0 (U, S) such that h F12 = (ψ1h )−1 ψ2h .
(8.9)
Let us introduce φ(h) = {φ1 (h), φ2 (h)} ∈ Cτ0 (U, S) by the formulae φ1 (h) := ψ1h ψ1−1 ,
φ2 (h) := ψ2h ψ2−1 .
(8.10)
Then we have a map φ : Gτ → Cτ0 (U, S)
(8.11)
Cτ0 (U, S).
of the group Gτ into the group The elements φ(h) = {φ1 (h), φ2 (h)} of the group Cτ0 (U, S) act by definition on ψ = {ψ1 , ψ2 } ∈ P (Nτ , Hτ0 (P, S)) as follows: h : ψ = {ψ1 , ψ2 } 7→ ρ(h, ψ) := ψ h = {ψ1h , ψ2h } = {φ1 (h)ψ1 , φ2 (h)ψ2 } .
(8.12)
From (7.11) it follows that B = {B (1) , B (2) } is transformed by the formulae ¯ −1 (h) ⇒ h : B 7→ ρ(h, B) ≡ B h := φ(h)Bφ−1 (h) + φ(h)∂φ
(8.13a)
¯ −1 B (1) 7→ φ1 (h)B (1) φ−1 1 (h) + φ1 (h)∂φ1 (h) , ¯ −1 B (2) 7→ φ2 (h)B (2) φ−1 2 (h) + φ2 (h)∂φ2 (h) .
(8.13b)
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With the help of formulae (8.13), (4.22) and (4.23) it is not difficult to write down explicit formulae for transformations of components Aµ of the gauge potential A. We shall not do this. Consider now a transformation fh † h F12 7→ F12 7→ F12 = f12 h12 F12 h†12 f12 . f
h
(8.14)
It is easy to see that ¯ −1 (f h) B f h = φ(f h)Bφ−1 (f h) + φ(f h)∂φ ¯ −1 (f ) = φ(f )B h φ−1 (f ) + φ(f )∂φ ¯ )φ(h))−1 . = φ(f )φ(h)B(φ(f )φ(h))−1 + φ(f )φ(h)∂(φ(f
(8.15)
It follows from (8.14), (8.15) that φ(f h) = φ(f )φ(h) ,
(8.16)
i.e., the map (8.11) is a homomorphism of the local Lie group Gτ into the group Cτ0 (U, S). 8.3. Gauge fixing and some formulae The SDYM Eqs. (4.7) for Aµ ∈ sl(n, C) imply that the components of the gauge potential can be written in the form: Ay1 = Θ−1 ∂y1 Θ ,
Ay2 = Θ−1 ∂y2 Θ ,
˜ −1 ∂y¯1 Θ ˜, Ay¯1 = Θ
˜ −1 ∂y¯2 Θ ˜, Ay¯2 = Θ (8.17)
˜ are some SL(n, C)-valued functions on U ⊂ R4 . One may perform where Θ and Θ the following gauge transformation: ˜ ˜ −1 + Θ∂ ˜ −1 = 0 , ˜ ˜ y¯1 Θ Ay¯1 7→ AΘ y¯1 = ΘAy¯1 Θ ˜ ˜ ˜ −1 + Θ∂ ˜ y¯2 Θ ˜ −1 = 0 , Ay¯2 7→ AΘ y¯2 = ΘAy¯2 Θ
(8.18a)
˜
˜ ˜ −1 + Θ∂ ˜ y1 Θ ˜ −1 = Φ−1 ∂y1 Φ , Ay1 7→ AΘ y 1 = ΘAy 1 Θ ˜
˜ ˜ −1 + Θ∂ ˜ y2 Θ ˜ −1 = Φ−1 ∂y2 Φ , Ay2 7→ AΘ y 2 = ΘAy 2 Θ
(8.18b)
˜ ˜ Θ ˜ −1 ∈ SL(n, C), and thus fix the gauge AΘ where Φ := ΘΘ y¯1 = Ay¯2 = 0 [23–27]. Then Eqs. (4.7) are replaced by the matrix equations
∂y¯1 (Φ−1 ∂y1 Φ) + ∂y¯2 (Φ−1 ∂y2 Φ) = 0 ,
(8.19)
which are the SDYM equations in the Yang gauge. Equations (8.19) are a 4D analogue of the 2D WZNW equations.
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It is also possible to perform the gauge transformation Ay¯1 7→ ΘAy¯1 Θ−1 + Θ∂y¯1 Θ−1 = Φ∂y¯1 Φ−1 , Ay¯2 7→ ΘAy¯2 Θ−1 + Θ∂y¯2 Θ−1 = Φ∂y¯2 Φ−1 , Ay1 7→ ΘAy1 Θ−1 + Θ∂y1 Θ−1 = 0 ,
(8.20)
Ay2 7→ ΘAy2 Θ−1 + Θ∂y2 Θ−1 = 0 , then Eqs. (4.7) get converted into the equations ∂y1 (Φ∂y¯1 Φ−1 ) + ∂y2 (Φ∂y¯2 Φ−1 ) = 0 .
(8.21)
From the linear system (4.10) it is easy to see that Θ = ψ2−1 (ζ = 0) ,
˜ = ψ −1 (λ = 0) , Θ 1
(8.22)
where the SL(n, C)-valued function ψ1 is defined on U1 , and the SL(n, C)-valued function ψ2 is defined on U2 . Eqs. (8.19) are the compatibility conditions of the linear system ∂y¯1 ψ˜1 − λ(∂y2 + Φ−1 ∂y2 Φ)ψ˜1 = 0 ,
∂y¯2 ψ˜1 + λ(∂y1 + Φ−1 ∂y1 Φ)ψ˜1 = 0 , (8.23)
obtained from (4.10) for ψ1 by performing the gauge transformation ψ1 (x, λ) 7→ ˜ ψ˜1 (x, λ) = ψ1−1 (x, 0)ψ1 (x, λ) = Θ(x)ψ 1 (x, λ), λ ∈ Ω1 . Analogously, Eqs. (8.21) are the compatibility conditions for the linear system ζ(∂y¯1 + Φ∂y¯1 Φ−1 )ψ˜2 − ∂y2 ψ˜2 = 0 ,
ζ(∂y¯2 + Φ∂y¯2 Φ−1 )ψ˜2 + ∂y1 ψ˜2 = 0 , (8.24)
where ψ˜2 (x, ζ) = ψ2−1 (x, 0)ψ2 (x, ζ) = Θ(x)ψ2 (x, ζ) is well defined for ζ ∈ Ω2 . We have ψ˜1 (x, λ = 0) = 1 and therefore ψ˜1 = 1 + λΨ + O(λ2 )
(8.25)
for some Lie algebra valued function Ψ ∈ sl(n, C). By substituting (8.25) into (8.23), we find that Φ−1 ∂y2 Φ = ∂y¯1 Ψ ,
Φ−1 ∂y1 Φ = −∂y¯2 Ψ .
(8.26)
Then after substitution (8.26) into (8.23), the compatibility conditions of the linear system (8.23) will be ∂y1 ∂y¯1 Ψ + ∂y2 ∂y¯2 Ψ + [∂y¯1 Ψ, ∂y¯2 Ψ] = 0 .
(8.27)
Equations (8.27) are the SDYM equations in the so-called Leznov–Parkes form. Notice that the condition ψ1 (x, λ = 0) = 1 ,
(8.28)
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leading to the gauge fixing Ay¯1 = Ay¯2 = 0, can be imposed from the very beginning. Then the Birkhoff factorization (8.3) is unique, which corresponds to the choice of the fixed section (8.4) of the bundle (8.5). Nevertheless, the gauge (8.28) does not remove all degrees of freedom related to holomorphic transformations of the group C 0 (U, H), and if we want to obtain the moduli space M, we have to factorize s(N ) ' N w.r.t. the action of the subgroup in C 0 (U, H) preserving the gauge (8.28). The same gauge may be used in the description of the moduli space Mτ discussed in Sec. 8.1. 8.4. Generalization to self-dual manifolds As has been mentioned in Sec. 4.6, the twistor correspondence between selfdual gauge fields and holomorphic bundles exists not only for the Euclidean space R4 , but also for 4-manifolds M , the Weyl tensor of which is self-dual. Twistor spaces Z ≡ Z(M ) for such manifolds M are three-dimensional complex spaces. The description of symmetries of local solutions to the SDYM equations can be easily generalized to this general case. It can be done as follows. Fix an open set U ⊂ M such that Z|U ' U × CP1 and choose coordinates xµ on U . Consider the restriction of the twistor bundle π : Z → M to U and put P := Z|U . The space P is an open subset of Z, and, as a real manifold, P is diffeomorphic to the direct product U ×CP1 . Now a metric on U is not flat, and a conformal structure on U is coded into a complex structure J on P [17, 19]. In this “curved” case we again have a natural one-to-one correspondence between solutions of the SDYM equations on U and holomorphic bundles E 0 over P, holomorphically trivial on (real) projective lines CP1x ,→ P, ∀ x ∈ U . In our group-theoretic analysis of the twistor correspondence we did not use the explicit form of the complex structure J on P and therefore did not use the explicit form of the metric on U . This explicit form was used only in some illustrating formulae, which can easily be generalized. That is why, all statements about local solutions and symmetry groups are also true for the SDYM equations on self-dual manifolds M . Thus, as the local symmetry group we again obtain the group Gτ = Hτ nCτ from Secs. 6–8 acting on the space of local solutions to the SDYM equations defined on a self-dual 4-manifold M . 9. Discussion 9.1. What is integrability? In books and papers on soliton equations one often poses the question: What is integrability? There is no general answer to this question, and usually one connects the integrability with the existence of Lax or zero curvature representations. Then non-Abelian cohomology, local groups and deformation theory of bundles with holomorphic or flat connections form the basis of integrability. In other words, there are always exact sequences of sheaves and cohomology sets of type (7.16), (7.28) hiding behind the integrability. This explains, in particular, why almost all integrable equations in two dimensions can be obtained by reductions of the SDYM equations (see e.g. [40, 47, 48, 49] and references therein).
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In [50–55] generalized SDYM equations in dimension D>4 and their solutions have been considered. Some of these equations in dimension D = 4n [51, 52] are integrable, since with the help of the twistor approach these quaternionic-type SDYM equations can be rewritten as holomorphy conditions of the Yang–Mills bundle over an auxiliary (twistor) (4n + 2)-space. The situation with the integrability of other generalized SDYM equations in D>4 is much less clear. Solutions of these equations, e.g. octonionic-type SDYM equations in D = 8 [50, 53], were used in constructing solitonic solutions of string theories [56]. The modification of these generalized SDYM equations arising after replacement of commutators by Poisson brackets are considered in supermembrane theory (see e.g. [57]). At the moment it is not clear whether all these equations can be interpreted as an existence condition of flat or holomorphic connections in bundles over some auxiliary spaces. This interesting problem deserves further study. 9.2. Holomorphic Chern Simons Witten theory Let us consider a smooth six-dimensional manifold Z with an integrable almost complex structure J . Then Z is a complex 3-manifold, and one can introduce a cover {Uα } of Z and coordinates zα : Uα → C3 . Let E 0 be a smooth complex vector ˆ be the (0, 1)-component of a connection 1-form bundle of rank n over Z and let B 0 ˆ satisfies the equations on the bundle E . Suppose that B ˆ +B ˆ ∧B ˆ = 0, ∂¯B
(9.1)
¯ The special case of where ∂¯ is the (0, 1) part of the exterior derivative d = ∂ + ∂. Eqs. (9.1) on the twistor space P of U ⊂ R4 was considered in Sec. 7. Equations (9.1) mean that the (0, 2) part of the curvature of the bundle E 0 is equal to zero: F 0,2 := 2 0 ¯ ˆ 2 ∂¯B ˆ = (∂ + B) = 0 and, therefore, the bundle E is holomorphic. We shall call Eqs. (9.1) defined on a complex 3-manifold Z the field equations of holomorphic Chern–Simons–Witten (CSW) theory. Equation (9.1) were suggested by Witten [58] for a special case of bundles over Calabi–Yau (CY) 3-folds Z as equations of a holomorphic analogue of the ordinary Chern–Simons theory. Witten obtained Eqs. (9.1) from open N = 2 topological strings with a central charge cˆ = 3 (6D target space) and the CY restriction c1 (Z) = 0 arose from N = 2 superconformal invariance of a sigma model used in constructing the topological string theory. The connection of Eqs. (9.1) with topological strings was also considered in [59]. Equations (9.1) on CY 3-folds were considered by Donaldson and Thomas [54] in the frames of program on extending the results of Casson, Floer, Jones and Donaldson to manifolds of dimension D>4. Donaldson and Thomas [54] pointed out that one may try to consider a more general situation with Eqs. (9.1) on complex manifolds Z which are not Calabi–Yau (c1 (Z) 6= 0). This is important since the CY restriction cannot be imposed if one uses the twistor correspondence between 4D and 6D theories. In Sec. 5 we considered the special case of the holomorphic CSW theory when field equations are defined not on an arbitrary complex 3-manifold, but on the twistor space P of U ⊂ R4 . The manifold P can be covered by two charts, and
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in Sec. 5 we described the moduli space and symmetries of the holomorphic CSW ˇ theory in the Cech approach. In Sec. 7 (see formulae (7.13), (7.26) and (7.34)) we gave the Dolbeault description of this moduli space. This analysis of the moduli space and symmetries of the holomorphic CSW theory can be generalized without difficulties to an arbitrary complex 3-manifold Z. 9.3. N = 2 and N = 4 topological strings The coupling of topological sigma models and topological gravity gives the above-mentioned N = 2 topological strings [60] which were further studied in [58, 59, 61]. They have critical dimension D = 6 and are related to topological sigma models with the 6D target space. There are two classes of such models, called A- and B-models. In the open string sector of the critical topological string theories there are A and B versions of these theories. The A-model is related to the ordinary Chern–Simons theory in 3 real dimensions and the B-model is related to the holomorphic Chern–Simons–Witten theory in 3 complex dimensions. We discuss only the B-model, the field equations for which concide with Eqs. (9.1) on a CY 3-fold. Besides N = 2 topological strings with cˆ = 3 (6D target space) there are N = 4 topological strings with a central charge cˆ = 2 (4D target space) [62] and nontopological N = 2 strings (see e.g. [63–65] and references therein). In [62] it was shown that N = 2 strings are a special case of N = 4 topological strings. The N = 2 string theories describe quantum SDYM fields on a self-dual gravitational background [62–65]. For heterotic N = 2 strings [66] besides SDYM fields there are also matter fields depending on the details of the construction. Comparing the above-mentioned string theories and field theories corresponding to them, one obtains the following “commutative” diagram ? y
−−−−→ N = 4 topological strings −−−−→ N = 2 strings y y
Holomorphic CSW Holomorphic CSW theory SDYM theory theory on complex −−−−→ on twistor spaces of self- −−−−→ on self-dual 43-manifolds dual 4-manifolds manifolds (9.2) The arrows mean that one theory can be derived from another one. The difference between the holomorphic CSW theory on a general complex 3-manifold and the one defined on a twistor space Z is stipulated by the existence in Z of a bundle structure π : Z → M with a self-dual 4D manifold M as a base space and CP1 as a typical fibre. In the general case, complex 3-spaces are arbitrary. Into the box with the question-mark from (9.2) one cannot substitute “N = 2 topological strings”, since they are obtained from sigma models on CY 3-folds. One should substitute there some generalized N = 2 topological strings on a complex 3manifold without the CY restriction. The possibility of introducing such strings was
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pointed out in the papers [59, 64]. Ooguri and Vafa [64] gave reasons for possible equivalence of N = 4 topological strings and generalized N = 2 topological strings on the twistor space with a holomorphic (2, 0)-form turned on. It would be very interesting to study this possibility. 9.4. Integrable 4D conformal field theories It is well known that the ordinary 3D Chern–Simons theory is connected with 2D conformal field theories if one supposes that a 3-manifold has the form Σ × R, where Σ is a 2-manifold with or without a boundary [67, 68]. In particular, if Σ has a boundary, the quantum Hilbert space HΣ is infinite-dimensional and is a representation space of the chiral algebra of CFT on Σ. Analogously, the holomorphic Chern–Simons–Witten theory on a complex 3-manifold Z is connected with integrable 4D CFT’s on a self-dual 4-manifold M if one supposes that Z is the twistor space of M . This means that Z is the bundle π : Z → M over M with CP1 as a typical fibre. On M it is possible to consider a CFT of fields of an arbitrary spin. Most of these CFT’s will describe free fields in a fixed background. By considering local solutions of field equations on M we take an open set U ⊂ M and consider the twistor space P = Z|U of U which is an open subset in Z. In this paper, we actually discuss how the concrete nonlinear 4D CFT — the SDYM theory — is connected with the holomorphic CSW theory on the twistor space P of U . The SDYM model on an open ball U ⊂ R4 is a generalization of the WZNW model on the complex plane C, and we mainly consider sets U with the flat metric. We described symmetries of the SDYM model and the moduli space of self-dual gauge fields on U . Naturally, the following questions arise: 1. What is an analogue of affine Lie algebras of 2D CFT’s? 2. What is an analogue of the Virasoro algebra? In this paper we have not discussed symmetry algebras yet. But knowing the symmetry groups of the SDYM equations, described in Secs. 7 and 8, it is not difficult to write down the algebras corresponding to them. A symmetry algebra of integrable 4D CFT’s is connected with the algebra Gh of functions that are holomorphic on U12 = U1 ∩ U2 ⊂ P and take values in the Lie algebra g of a complex Lie group G. The algebra Gh with pointwise commutators generalizes affine Lie algebras. The symmetry algebra is the algebra g C 1 (U, OP ) ' Gh ⊕ Gh
(9.3)
g of 1-cochains of the cover U = {U1 , U2 } of the space P with values in the sheaf OP of holomorphic maps from P into the Lie algebra g. We mainly considered the case g = sl(n, C). The algebra (9.3) was also considered by Ivanova [33]. Notice that the affine Lie algebra g ⊗ C[λ, λ−1 ] (without a central term) is the algebra of g-valued meromorphic functions on CP1 ' C∗ ∪ {0} ∪ {∞} with the poles at λ = 0, λ = ∞ and holomorphic on Ω12 = Ω1 ∩ Ω2 ' C∗ . Hence, it is a subalgebra in the algebra g −1 C 1 (O, OCP ] ⊕ g ⊗ C[λ, λ−1 ] (9.4) 1 ) ' g ⊗ C[λ, λ
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of 1-cochains of the cover O = {Ω1 , Ω2 } of CP1 with values in the sheaf of holomorphic maps from CP1 into the Lie algebra g. Thus, the algebra (9.3) is an analogue of the 2D affine Lie algebra (9.4). Notice that (central) extensions of the algebras (9.3) and (9.4) will appear after the transition to quantum theory. ˇ 9.5. The Cech description of the Virasoro algebra Elements of the Virasoro algebra Vir0 (with zero central charge) are meromorphic vector fields on CP1 having poles at the points λ = 0, λ = ∞ and holomorphic on the overlap Ω12 = Ω1 ∩ Ω2 ' C∗ = CP1 − {0} − {∞}. This algebra has the ˇ following Cech description. Let us consider the sheaf VCP1 of holomorphic vector ˇ fields on CP1 . Then for the space of Cech 1-cochains with values in VCP1 we have C 1 (O, VCP1 ) ' Vir0 ⊕ Vir0 .
(9.5)
Notice that for {v12 , v21 } ∈ C 1 (O, VCP1 ) the antisymmetry condition cannot be imposed on cohomology indices of the holomorphic vector fields v12 , v21 , since it is not preserved under commutation. So we have v21 6= −v12 in the general case. The space Z 1 (O, VCP1 ) of 1-cocycles of the cover O = {Ω1 , Ω2 } of CP1 with values in the sheaf VCP1 coincides with the algebra Vir0 as a vector space, since Z 1 (O, VCP1 ) ' (Vir0 ⊕ Vir0 )/diag (Vir0 ⊕ Vir0 ) .
(9.6)
Further, by virtue of the equality H 1 (CP1 , VCP1 ) = 0 ,
(9.7)
which means the rigidity of the complex structure of CP1 , any element v from Vir0 ' Z 1 can be represented in the form v = v1 − v2 .
(9.8)
Here, v1 can be extended to a holomorphic vector field on Ω1 , and v2 can be extended to a holomorphic vector field on Ω2 . It follows from (9.6)–(9.8) that the algebra Vir0 is connected with the algebra C 0 (O, VCP1 )
(9.9)
of 0-cochains of the cover O with values in the sheaf VCP1 by the (twisted) homomorphism δ˙ 0 : C 0 (O, VCP1 ) −→ C 1 (O, VCP1 ) ⇔
(9.10a)
δ˙ 0 : {v1 , v2 } 7→ {v1 − v2 , v2 − v1 } .
(9.10b)
Just the cohomological nature of the algebra Vir0 permits one to define its local action on Riemann surfaces of arbitrary genus and on the space of conformal structures of Riemann surfaces [69]. A central extension arises under an action of the Virasoro algebra on holomorphic sections of line bundles over moduli spaces (quantization).
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9.6. Infinitesimal deformations of self-dual conformal structures Here we briefly answer the question of Sec. 9.4 about an analogue of the Virasoro algebra (without a central term). In Secs. 5.4, 5.5 and 8.2 we described the local group H of biholomorphisms of the twistor space P and its action on the space of local solutions to the SDYM equations. To this group there corresponds the algebra (cf. (9.9), (9.10)) C 0 (U, VP )
(9.11)
of 0-cochains of the cover U = {U1 , U2 } of P with values in the sheaf VP (of germs) of holomorphic vector fields on P = U1 ∪ U2 . However, this algebra is not a correct generalization of the Virasoro algebra. An analogue of the Virasoro algebra is the algebra VP (U12 ) of holomorphic vector fields on U12 = U1 ∩ U2 ⊂ P. It is a subalgebra of the algebra C 1 (U, VP ) ' VP (U12 ) ⊕ VP (U12 )
(9.12)
of 1-cochains of the cover U with values in the sheaf VP . Elements of the algebra C 1 (U, VP ) are the collections of vector fields ∂ ∂ χ = {χ12 , χ21 } = χa12 a , χa21 a (9.13) ∂z1 ∂z2 with ordered “cohomology indices”. From the Kodaira–Spencer deformation theory [70] it follows that the algebra (9.12) acts on the transition function f12 of the space P (see Sec. 3.3) by the formula a δf12 = χa12 −
a ∂f12 a ∂ χb21 ⇔ δf12 := δf12 = χ12 − χ21 . b ∂z1a ∂z2
(9.14)
Accordingly, one may define the following action of the algebra C 1 (U, VP ) on the transition matrices F12 of holomorphic bundles E 0 over the twistor space P: δχ F12 = χ12 (F12 ) .
(9.15)
The algebra C 0 (U, VP ) acts on the transition function f12 of the space P and on the transition matrices F12 of bundles E 0 over P by formulae (9.14),(9.15) via the twisted homomorphism δ˙ 0 : C 0 (U, VP ) 3 {χ1 , χ2 } 7→ {χ1 − χ2 , χ2 − χ1 } ∈ C 1 (U, VP )
(9.16)
of the algebra C 0 (U, VP ) into the algebra C 1 (U, VP ). Notice that δf := {δf12 , δf21 } ∈ Z 1 (U, VP ), and the quotient space H 1 (U, VP ) := Z 1 (U, VP )/δ˙ 0 (C 0 (U, VP ))
(9.17)
describes nontrivial infinitesimal deformations of the complex structure of P. For a cover U = {U1 , U2 }, where U1 , U2 are Stein manifolds, we have H 1 (P, VP ) = H 1 (U, VP ). In contrast with the 2D case (9.7) now we have H 1 (P, VP ) 6= 0. Hence,
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the transformations (9.14) of the transition function in general change the complex structure of P and therefore change the conformal structure on U . Recall that a conformal structure [g] is called self-dual if the Weyl tensor for any metric g in the conformal equivalence class [g] is self-dual [19]. In virtue of the twistor correspondence [17, 19] the moduli space of self-dual conformal structures on a 4manifold M is bijective to the moduli space of complex structures on the twistor space of M . All algebras of infinitesimal symmetries of the self-dual gravity equations known by now (see e.g. [71] and references therein) are subalgebras in the algebra C 1 (U, VP ). The action of the algebra C 0 (U, VP ) (and the group H(P) corresponding to it) transforms f12 into an equivalent transition function and therefore preserves the conformal structure on U . At the same time, the action of the algebra C 0 (U, VP ) on transition matrices of holomorphic bundles E 0 → P is not trivial. If we want to define an action of the algebra C 1 (U, VP ) on the coordinates {z1a }, a {z2 }, q-forms, etc. we should define: (1) a sheaf T 1,0 of (1, 0) vector fields on P, holomorphic along fibres CP1x of the bundle P → U ; (2) a sheaf W of (0, 1)-forms W on P with values in T 1,0 , vanishing on the distribution V 0,1 (see Sec. 7.3) and satisfying the equations ¯ =0 ∂W (9.18) on any open set U ⊂ P, where W ∈ W(U). Then we have the exact sequence of sheaves 0 −→ VP −→ T 1,0 −→ W −→ 0 (9.19) and the corresponding exact sequence of cohomology spaces 0 −→ H 0 (P, VP ) −→ H 0 (P, T 1,0 ) −→ H 0 (P, W) −→ H 1 (P, VP ) −→ 0 , (9.20) describing infinitesimal deformations of the complex structure of the twistor space P. From (9.20) it follows that for any element δf ∈ Z 1 (U, VP ) ⊂ Z 1 (U, T 1,0 ) there exists an element {ϕ1 , ϕ2 } ∈ C 0 (U, T 1,0 ) such that δf = {χ12 − χ21 , χ21 − χ12 } = {ϕ1 − ϕ2 , ϕ2 − ϕ1 } ∈ δ˙ 0 (C 0 (U, T 1,0 )) .
(9.21)
Then for infinitesimal transformations of coordinates on P = U1 ∪ U2 we have δz1a := ϕa1 (z1 , z¯1 ) ,
δz2a := ϕa2 (z2 , z¯2 ) .
(9.22)
To preserve the reality of the conformal structure on U , one should define real subalgebras of the algebras C 1 (U, VP ) and C 0 (U, T 1,0 ) by analogy with Secs. 6.6 and 7.7. We shall not write down transformations of the metric and conformal structure on U , since this will require a lot of additional explanations. Details will be published elsewhere. 9.7. Quantization Some problems related to the quantization of the SDYM model were discussed in [20, 21, 72]. The quantization was carried out in four dimensions in terms of
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g-valued fields Aµ or in terms of a G-valued scalar field by using the Yang gauge. But the obtained results are fragmentary; the picture is not complete and far from what we have in 2D CFT’s. Remembering the connection between 2D CFT’s and the ordinary 3D CS theory, one may come to the reasonable conclusion that the quantization of integrable 4D CFT’s may be much more successful if we use the 6D holomorphic CSW theory. When quantizing the holomorphic CSW theory on the twistor space P one may use the results on the quantization of the ordinary CS theory (see e.g. [67, 68] and references therein) after a proper generalization. We are mainly interested in ˆ3 = 0 in Eqs. (9.1), which quantizing the SDYM model. As such, we have to put B leads to the equations (cf. (7.12)) ¯ +B ∧B =0 ∂B
(9.23)
equivalent to the SDYM equations, as has been discussed in this paper. The com¯ may parison with the ordinary CS theory in the Hamiltonian approach shows that λ be considered as (complex) time of the holomorphic CSW theory. Further, one can use two standard approaches to the quantization of constrained systems: (1) one first solves the constraints and then performs the quantization of the moduli space; (2) one first quantizes the free theory and then imposes (quantum) constraints. The first approach will mainly be discussed. We shall write down the list of questions and open problems whose solutions are necessary to give the holomorphic CSW and the SDYM theories a status of quantum field theories. 1. One should rewrite a symplectic structure ω ˜ on the space of gauge potentials or their relatives [20, 21, 40, 72] in terms of fields on the twistor space P. This 2-form ω ˜ induces a symplectic structure ω on the moduli space M of solutions to Eqs. (9.23), and the cohomology class [ω] ∈ H 2 (M, R) has to be integral. 2. Over the moduli space M one should define a complex line bundle L with the Chern class c1 (L) = [ω]. Then L admits a connection with the curvature 2-form equal to ω. 3. A choice of a complex structure J on the twistor space P endows the moduli space M with a complex structure which we shall denote by the same letter J . Then the bundle L over (M, J ) has a holomorphic structure, and a quantum Hilbert space of the SDYM theory can be introduced as the space HJ of (global) holomorphic sections of L. 4. Is it possible to introduce the bundle L → M as the holomorphic determinant line bundle Det∂¯B of the operator ∂¯B = ∂¯ + B on P? 5. The action functional of the holomorphic CSW theory on a Calabi–Yau 3-fold has a simple form [58, 59] analogous to the action of the standard CS theory. How should one modify this action if we go over to the case of an arbitrary complex 3-manifold? 6. One should lift the action of the symmetry groups and algebras described in this paper up to an action on the space HJ of holomorphic sections of the bundle L over M. What is an extension (central or not) of these groups and algebras? Finding
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g of an extension of the algebra C 1 (U, OP ) is equivalent to finding a curvature of the bundle L since this curvature represents a local anomaly. 7. What can be said about representations of the algebras C 1 (U, VP ) and g C 1 (U, OP )? Which of these representations are connected with the Hilbert space HJ ? 8. In the quantum holomorphic CSW and SDYM theories there exist Sugawaratype formulae, i.e., generators of the algebra C 1 (U, VP ) can be quadratically exg ). This follows from the fact pressed in terms of generators of the algebra C 1 (U, OP that any transformation of transition matrices of a holomorphic bundle E 0 → P under the action of the algebra C 1 (U, VP ) can be compensated by an action of the g algebra C 1 (U, OP ). What are the explicit formulae connecting the generators of these algebras? 9. One should write down Ward identities resulting from the symmetry algeg bra C 1 (U, VP ) u C 1 (U, OP ). To what extent do these identities define correlation functions?
Clearly, to carry out this quantization program, it will be necessary to overcome a number of technical difficulties. The general picture arising as a result of quantization of the SDYM model on a self-dual 4-manifold M and the holomorphic CSW theory on the twistor space Z of M resembles the one that arises in the quantization of the ordinary CS theory and is as follows: Let [g] be a self-dual conformal structure on a 4-manifold M and let J be a complex structure on the twistor space Z of M . As has already been noted, there exists a bijection [17, 19] between the moduli space of self-dual conformal structures on M and the moduli space X of complex structures on Z. Let M be a moduli space of solutions to the SDYM equations on M and let HJ be the quantum Hilbert space of holomorphic sections of the line bundle L over (M, J ). The space HJ depends on J ∈ X and one can introduce a holomorphic vector bundle ˜ −→ X p:H
(9.24)
with fibres HJ at the points J ∈ X. Then one may put a question about the existence of a (projectively) flat connection in the bundle (9.24). If such a connection exists, then as a quantum Hilbert space one may take a space of covariantly constant ˜ sections of the vector bundle H. 10. Conclusion In this paper, the group-theoretic analysis of the Penrose–Ward correspondence was undertaken. Having used sheaves of non-Abelian groups and cohomology sets we have described the symmetry group acting on the space of local solutions to the SDYM equations and the moduli space M of local solutions. It has been shown that M is a double coset space. The full algebra of infinitesimal deformations of self-dual conformal structures on a 4-space M has also been described. We have discussed the program of quantization of the SDYM model on M based on the equivalence of this model to a subsector of the holomorphic CSW model on the twistor space Z of M . There are a lot of open problems, which deserve further study.
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Acknowledgements The author is grateful to Yu. I. Manin and I. T. Todorov for helpful discussions. He also thanks for its hospitality the Max-Planck-Institut f¨ ur Mathematik in Bonn, where part of this work was done, and the Alexander von Humboldt Foundation for support. This work is supported in part by the grant RFBR-98-01-00173. Appendix A. Actions of Groups on Sets The left action of a group G on a set Υ is a map ρ : G ×Υ → Υ with the following properties: ρ(e, x) = x ,
(A.1a)
ρ(a, ρ(b, x)) = ρ(ab, x) ,
(A.1b)
for any x ∈ Υ, a, b, e ∈ G. Here e is the identity in the group G. If we are given an action ρ on a set Υ, to any a ∈ G we can correspond a bijective transformation ρa : x 7→ ρ(a, x) of the set Υ such that a map γ : a 7→ ρa is a homomorphism of the group G into the group SΥ of all permutations (bijective transformations) of the set Υ. Conversely, any homomorphism γ : G → SΥ defines the action of the group G on Υ by the formula ρ(a, x) := γ(a)(x) (A.2) for any a ∈ G, x ∈ Υ. If Υ is a smooth manifold, then to define an action of G on Υ is equivalent to assigning a homomorphism γ : G → Diff (Υ) of the group G into the group of diffeomorphisms of the manifold Υ. Usually the left action of the group G is represented as a multiplication of elements from Υ by elements of the group G and written as ρ(a, x) = ax, a ∈ G, x ∈ Υ. One also considers the right action of the group G on Υ in the definition of which the condition (A.1b) is replaced by the condition ρ(a, ρ(b, x)) = ρ(ba, x) .
(A.1c)
Then the notation ρ(a, x) = xa is used. Recall that a space G is called a local group, if for elements a, b sufficiently close to the identity e (marked element) the multiplication ab is defined, the inverse elements a−1 , b−1 exist and all group axioms are fulfilled every time the objects participating in these axioms are defined. More precisely, a space G is called a local group if: (1) some element e (identity) of G is chosen; (2) a neighbourhood V ⊂ G of the element e is chosen; (3) there is a map V × V → G, (a, b) 7→ ab (multiplication) satisfying the conditions ea = ae = a and (ab)c = a(bc) for a, b, c, ab, bc ∈ V. From these conditions it follows that there exists a neighbourhood W ⊂ G of the identity and a map ı : W → W, a 7→ a−1 (inversion) such that aa−1 = a−1 a = e. Choosing V = W = G, one can consider any group G as a local group; this is why we use the same letter G for groups and for local groups. If one replaces G and V by open subsets G 0 ⊂ G, V 0 ⊂ V ∩ G 0 satisfying the condition V 0 V 0 ⊂ G 0 , one obtains a local group G 0 , called a restriction or a part
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of the initial one. Two local groups are called equivalent, if some of their parts coincide. The equivalence class of the local group G is called the germ of the group G at the point e ∈ G and denoted by G. An action of a group G on a set Υ can be localized if one considers G as a local group. Namely, let ρ be an action of the group G on the set Υ and let N be an open subset in Υ. The action ρ, generally speaking, does not map N into itself and therefore does not define an action of the whole group G on N . However, an action of G as a local group is defined, i.e., a map ρ : W → N is defined, where W = {(a, x) ∈ G × N : ρ(a, x) ∈ N } is an open subset in G × N containing {e} × N . Moreover, for any fixed point x ∈ N there exists a neighbourhood V of the identity in G and a neighbourhood N 0 of the point x in N such that ρ(V × N 0 ) ⊂ N . In a more general situation, a local action of a local group G on a set N is a map ρ : W → N , where W is an open set in G × N containing {e} × N , and the properties (A.1) are satisfied for all a, b ∈ G, x ∈ N for which both parts of the equality (A.1b) are defined. A local action ρ of the local group G on the set N generates a local action of G on any open subset N 0 ⊂ N . This action is called a restriction of the action ρ to the subset N 0 . A local action of the group G is called globalizable if it is a localization of some global action of the group. Appendix B. Sheaves of (non-Abelian) Groups Let us consider a topological space X and recall the definitions of a presheaf and a sheaf of groups over X (see e.g. [44, 45]). One has a presheaf {S(U ), rVU } of groups over a topological space X if with any nonempty open set U of the space X one associates a group S(U ) and with any two open sets U and V with V ⊂ U one associates a homomorphism rVU : S(U ) → S(V ) U satisfying the following conditions: (i) the homomorphism rU : S(U ) → S(U ) is U V U the identity map idU ; (ii) if W ⊂ V ⊂ U , then rW = rW ◦ rV . A sheaf of groups over a topological space X is a topological space S with a local homeomorphism π : S → X. This means that any point s ∈ S has an open neighbourhood V in S such that π(V ) is open in X and π : V → π(V ) is a homeomorphism. A set Sx = π−1 (x) is called a stalk of the sheaf S over x ∈ X, and the map π is called the projection. For any point x ∈ X the stalk Sx is a group, and the group operations are continuous. A section of a sheaf S over an open set U of the space X is a continuous map s : U → S such that π ◦ s =idU . A set S(U ) := Γ(U, S) of all sections of the sheaf S of groups over U is a group. Corresponding to any open set U of the space X the group S(U ) of sections of the sheaf S over U and to any two open sets U, V with V ⊂ U the restriction homomorphism rVU : S(U ) → S(V ), we obtain the presheaf {S(U ), rVU } over X. This presheaf is called the canonical presheaf. On the other hand, one can associate a sheaf with any presheaf {S(U ), rVU }. Let Sx = lim S(U ) −→ x∈U
be a direct limit of sets S(U ). There exists a natural map rxU : S(U ) → Sx , x ∈ U ,
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sending elements from S(U ) into their equivalence classes in the direct limit. If s ∈ S(U ), then sx := rxU (s) is called a germ of the section s at the point x, and s is called a representative of the germ sx . In other terms, two sections s, s0 ∈ S(U ) are called equivalent at the point x ∈ U if there exists an open neighbourhood V ⊂ U such that s|V = s0 |V ; the equivalence class of such sections is called the germ sx of section s at the point x. Put S = ∪ Sx x∈X
and let π : S → X be a projection mapping points from Sx into x. The set S is equipped with a topology, the basis of open sets of which consists of sets {sx , x ∈ U } for all possible s ∈ S(U ), U ⊂ X. In this topology π is a local homeomorphism, and we obtain the sheaf S. Let X be a smooth manifold. Consider a complex (non-Abelian) Lie group ˆ ), rU } of groups by putting G = GC and define a presheaf {S(U V ˆ ) := {C ∞ -maps f : U → G} , S(U
(B.1)
ˆ ) its image and using the canonical restriction homomorphisms rVU when for f ∈ S(U U ˆ ), V ⊂ U . To each elements αx and βx from Sˆx := rxU (S(U ˆ )) rV (f ) equals f |V ∈ S(V ˆ ), rU } one can correspond their pointwise multiplication αx βx . To this presheaf {S(U V there corresponds the sheaf Sˆ of germs of smooth maps of the space X into the group G. Suppose now that X is a complex manifold. Then one can define a presheaf {H(U ), rVU } of groups assuming that H(U ) ≡ OG (U ) := {holomorphic maps h : U → G} ,
(B.2)
and associate with it the sheaf H ≡ OG of germs of holomorphic maps of the space X into the complex Lie group G. Appendix C. Cohomology Sets and Vector Bundles We shall consider a complex manifold X and a sheaf S coinciding with either the sheaf Sˆ or the sheaf H introduced in Appendix B. So S is the sheaf of germs of smooth or holomorphic maps of the space X into the complex Lie group G. ˇ Cech cohomology sets H 0 (X, S) and H 1 (X, S) of the space X with values in the sheaf S of groups are defined as follows [34, 44, 45]. Let there be given an open cover U = {Uα }, α ∈ I, of the manifold X. The family hU0 , . . . , Uq i of elements of the cover such that U0 ∩ . . . ∩ Uq 6= ∅ is called a q-simplex. The support of this simplex is U0 ∩ . . . ∩ Uq . Define a 0-cochain with coefficients in S as a map f associating with α ∈ I a section fα of the sheaf S over Uα : fα ∈ S(Uα ) := Γ(Uα , S) . (C.1) A set of 0-cochains is denoted by C 0 (U, S) and is a group under the pointwise multiplication.
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Consider now the ordered set of two indices hα, βi such that α, β ∈ I and Uα ∩ Uβ 6= ∅. Define a 1-cochain with coefficients in S as a map f associating with hα, βi a section of the sheaf S over Uα ∩ Uβ : fαβ ∈ S(Uα ∩ Uβ ) := Γ(Uα ∩ Uβ , S) .
(C.2)
A set of 1-cochains is denoted by C 1 (U, S) and is a group under the pointwise multiplication. Subsets of cocycles Z q (U, S) ⊂ C q (U, S) for q = 0, 1 are defined by the formulae Z 0 (U, S) = {f ∈ C 0 (U, S) : fα fβ−1 = 1 on Uα ∩ Uβ 6= ∅} ,
(C.3)
−1 Z 1 (U, S) = {f ∈ C 1 (U, S) : fβα = fαβ on Uα ∩ Uβ 6= ∅ ,
fαβ fβγ fγα = 1 on Uα ∩ Uβ ∩ Uγ 6= ∅} .
(C.4)
It follows from (C.3) that Z 0 (U, S) coincides with the group H 0 (X, S) := S(X) ≡ Γ(X, S) of global sections of the sheaf S. The set Z 1 (U, S) is not in general a subgroup of the group C 1 (U, S). It contains the marked element 1, represented by the 1-cocycle fαβ = 1 for any α, β such that Uα ∩ Uβ 6= ∅. For h ∈ C 0 (U, S), f ∈ Z 1 (U, S) let us define an action ρ0 of the group C 0 (U, S) on the set Z 1 (U, S) by the formula ρ0 (h, f )αβ = hα fαβ h−1 β .
(C.5)
So we have a map ρ0 : C 0 × Z 1 3 (h, f ) 7→ ρ0 (h, f ) ∈ Z 1 . A set of orbits of the group C 0 in Z 1 is called a 1-cohomology set and denoted by H 1 (U, S). In other ˜ if words, two cocycles f, f˜ ∈ Z 1 are called equivalent, f ∼ f, f˜ = ρ0 (h, f )
(C.6)
for some h ∈ C 0 , and by the 1-cohomology set H 1 = ρ0 (C 0 )\Z 1 one calls a set of equivalence classes of 1-cocycles. Finally, we should take the direct limit of these sets H 1 (U, S) over successive refinement of the cover U of X to obtain H 1 (X, S), the 1-cohomology set of X with coefficients in S. In fact, one can always choose a cover U = {Uα } such that it will be H 1 (U, S) = H 1 (X, S) and therefore it will not be necessary to take the direct limit of sets. This is realized, for instance, when the coordinate charts Uα are Stein manifolds (see e.g. [44]). Recall that S is the sheaf of germs of (smooth or holomorphic) functions with values in the complex Lie group G. Suppose we are given a representation of G in Cn . It is well known that any 1-cocycle {fαβ } from Z 1 (U, S) defines a unique complex vector bundle E 0 over X, obtained from the direct products Uα × Cn by glueing with the help of fαβ ∈ G. Moreover, two 1-cocycles define isomorphic complex vector bundles over X if and only if the same element from H 1 (X, S) corresponds to them. Thus, we have a one-to-one correspondence between the set H 1 (X, S) and the set of equivalence classes of complex vector bundles of the rank ˆ and holomorphic n over X. Smooth bundles are parametrized by the set H 1 (X, S)
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bundles are parametrized by the set H 1 (X, H), where the sheaves Sˆ and H were described in Appendix B. For more details see e.g. [44, 45]. References [1] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, “Transformation groups for soliton equations”, in Nonlinear Integrable Systems-Classical Theory and Quantum Theory, World Scientific, Singapore, 1983, pp. 39–120; M. Jimbo and T. Miwa, “Solitons and infinite-dimensional Lie algebras”, Publ. RIMS, Kyoto Univ. 19 (1983) 943–1001. [2] E. Witten, “Non-Abelian bosonization in two dimensions”, Commun. Math. Phys. 92 (1984) 455–472. [3] A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, “Infinite conformal symmetry in two-dimensional quantum field theory”, Nucl. Phys. B241 (1984) 333–380. [4] V. G. Knizhnik and A. B. Zamolodchikov, “Current algebra and Wess–Zumino model in two dimensions”, Nucl. Phys. B247 (1984) 83–103. [5] A. C. Newell, Solitons in Mathematics and Physics, SIAM, Philadelphia, 1985. [6] V. E. Zakharov and A. B. Shabat, “Integration of nonlinear equations by the inverse scattering method. II”, Funct. Anal. Appl. 13 (1979) 166–174. [7] M. Mulase, “Complete integrability of the Kadomtsev–Petviashvili equation”, Adv. Math. 54 (1984) 57–66. [8] G. Segal and G. Wilson, “Loop groups and equations of KdV type”, Publ. Math. IHES 61 (1985) 5–65. [9] M. A. Semenov-Tian-Shansky, “Dressing transformations and Poisson group actions”, Publ. RIMS, Kyoto Univ. 21 (1985) 1237–1260. [10] L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, Berlin, 1987. [11] M. Sato, “Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds”, RIMS Kokyuroky 439 (1981) 30–46. [12] K. Ueno and K. Takasaki, “Toda lattice hierarchy”, Adv. Stud. Pure Math. 4 (1984) 1–95; K. Takasaki, “Geometry of universal Grassmannian manifold from algebraic point of view”, Rev. Math. Phys. 1 (1989) 1–46. [13] A. P. Hodges, R. Penrose and M. A. Singer, “A twistor conformal field theory for four space-time dimensions”, Phys. Lett. B216 (1989) 48–52. [14] J. L. Cardy, “Operator content and modular properties of higher-dimensional conformal field theories”, Nucl. Phys. B366 (1991) 403–419. [15] H. Osborn and A. Petkos, “Implications of conformal invariance in field theories for general dimensions”, Ann. Phys. 231 (1994) 311–362; J. Erdmenger and H. Osborn, “Conserved currents and the energy momentum tensor in conformally invariant theories for general dimensions”, Nucl. Phys. B483 (1997) 431–474. [16] P. Howe and P. West, “Non-perturbative Green’s functions in theories with extended superconformal symmetry”, hep-th/9509140; P. C. Argyres, M. R. Plesser, N. Seiberg and E. Witten, “New N = 2 superconformal field theories in four dimensions”, Nucl. Phys. B461 (1996) 71–84. [17] R. Penrose, “Nonlinear gravitons and curved twistor theory”, Gen. Rel. Grav. 7 (1976) 31–52. [18] R. S. Ward, “On self-dual gauge fields”, Phys. Lett. A61 (1977) 81–82. [19] M. F. Atiyah, N. J. Hitchin and I. M. Singer, “Self-duality in four-dimensional Riemannian geometry”, Proc. R. Soc. Lond. A362 (1978) 425–461. [20] V. P. Nair and J. Schiff, “K¨ ahler–Chern–Simons theory and symmetries of anti-selfdual gauge fields”, Nucl. Phys. B371 (1992) 329–352.
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FROM EUCLIDEAN FIELD THEORY TO QUANTUM FIELD THEORY DIRK SCHLINGEMANN II. Institut f¨ ur Theoretische Physik Universit¨ at Hamburg and The Erwin Schr¨ odinger International Institute for Mathematical Physics (ESI) Vienna Received 26 February 1998 Revised 19 May 1998 In order to construct examples for interacting quantum field theory models, the methods of Euclidean field theory turned out to be powerful tools since they make use of the techniques of classical statistical mechanics. Starting from an appropriate set of Euclidean n-point functions (Schwinger distributions), a Wightman theory can be reconstructed by an application of the famous Osterwalder–Schrader reconstruction theorem. This procedure (Wick rotation), which relates classical statistical mechanics and quantum field theory, is, however, somewhat subtle. It relies on the analytic properties of the Euclidean n-point functions. We shall present here a C ∗ -algebraic version of the Osterwalder–Schrader reconstruction theorem. We shall see that, via our reconstruction scheme, a Haag–Kastler net of bounded operators can directly be reconstructed. Our considerations also include objects, like Wilson loop variables, which are not pointlike localized objects like distributions. This point of view may also be helpful for constructing gauge theories.
1. Introduction Why Euclidean field theory? During the last two decades it turned out that the techniques of Euclidean field theory are powerful tools in order to construct quantum field theory models. Compared with the method of canonical quantization in Minkowski space, which, for example, has been used for the construction of P (φ)2 and Yukawa2 models [10, 11, 12, 14, 21, 22], the functional integral methods of Euclidean field theory simplify the construction of interactive quantum field theory models. In particular, the existence of the φ43 model as a Wightman theory has been established by using Euclidean methods [5, 24, 18] combined with the famous Osterwalder–Schrader reconstruction theorem [19]. For this model the methods of canonical quantization are much more difficult to handle and lead by no means as far as Euclidean techniques do. Only the proof of the positivity of the energy has been carried out within the Hamiltonian framework [10, 13]. One reason why the functional integral point of view simplifies a lot is that the theory of classical statistical mechanics can be used. For example, renormalization 1151 Reviews in Mathematical Physics, Vol. 11, No. 9 (1999) 1151–1178 c World Scientific Publishing Company
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group analysis [9] and cluster expansions [2] can be applied in order to perform the continuum and the infinite volume limit of a lattice regularized model. Instead of working with non-commutative objects, one considers the moments Z Sn (x1 , . . . , xn ) = dµ(φ) φ(x1 ) . . . φ(xn ) of reflexion positive measures µ, usually called Schwinger distributions or Euclidean correlation functions, on the space of tempered distributions. Heuristically, the functional integral point of view leads to conceptionally simple construction scheme for a quantum field theory. Starting from a given lagrangian density L, the measure µ under consideration is simply given by Z O −1 dφ(x) exp − dx L(φ(x), dφ(x)) , dµ(φ) = Z x∈Rd
where the factor Z −1 is for normalization. Therefore, the lagrangian L can be interpreted as a germ of a quantum field theory. Moreover, this also leads to a nice explanation of the minimal action principle. However, to give the expression above a rigorous mathematical meaning is always accompanied with serious technical difficulties. Some comments on the Osterwalder Schrader reconstruction theorem. In order to motivate the main purpose of our paper, we shall make some brief remarks on the Osterwalder–Schrader reconstruction theorem [19] which relates Schwinger and Wightman distributions. Let T (S) be the tensor algebra over the space of test functions S (in Rd ) and let us denote by JE (E stands for Euclidean) the two-sided ideal in T (S), which is generated by elements f1 ⊗ f2 − f2 ⊗ f1 ∈ T (S) where f1 and f2 have disjoint supports. We build the algebra TE (S) := T (S)/JE and take the closure TET (S) of it in an appropriate locally convex topology. We claim that the Euclidean group E(d) acts naturally by automorphisms (αg , g ∈ E(d)) on TET (S). A linear functional η ∈ TET (S)∗ fulfills the Osterwalder–Schrader axioms if the following conditions hold: (E0) η is continuous and unit preserving: hη, 1i = 1. (E1) η is invariant under Euclidean transformations: ω ◦ αg = ω. (E2) η is reflexion positive: The sesqui-linear form a ⊗ b 7→ hη, ιe (a∗ )bi is a positive semi-definite on those elements which are localized at positive times with respect to the direction e ∈ S d−1 where ιe is the automorphism which corresponds to the reflexion e 7→ −e. Given a linear functional η which satisfies the conditions (E0) to (E2), the analytic properties of the distributions Sn (f1 , . . . , fn ) := hη, f1 ⊗ · · · ⊗ fn i
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and Sn (ξ1 , . . . , ξn ) = Sn+1 (x0 , . . . , xn ); ξj = xj+1 − xj lead to the result: ˜ n ∈ S 0 (Rnd ) supported in the Theorem 1.1. There exists a distribution W n ¯ n-fold closed forward light cone (V+ ) which is related to Sn by the Fourier–Laplace transform: Z ˜ n (q) . Sn (ξ) = dnd q exp(−ξ 0 q 0 − iξq) W The proof of this theorem [19] relies essentially on the choice of the topology T . It does not apply for the ordinary S-topology, i.e. it is not enough to require that the Sn ’s are tempered distributions. This was stated wrongly in the first paper of [19] and was later corrected in the second one. We claim that, nevertheless, the Theorem might be true for the ordinary S-topology, but, at the moment, there is no correct proof for it. These problems show that the relation between Euclidean field theory and quantum field theory is indeed subtle. In order to formulate the famous Osterwalder–Schrader reconstruction theorem from a more algebraic point of view, we shall briefly introduce the notion of a local net and a vacuum state. P↑+ -covariant local nets: A P↑+ -covariant local net of ∗ -algebras is an isotonousa prescription A : O 7→ A(O), which assigns to each double cone O = V+ + x ∩ V− + y a unital ∗ -algebra A(O), on which the the Poincar´e group P↑+ acts covariantly on A, i.e. there is a group homomorphism α ∈ Hom(P↑+ , Aut A), such that αg A(O) = A(gO). Here A denotes the ∗ -inductive limit of the net A. Furthermore, the net fulfills locality, i.e. if O, O1 are two space-like separated regions O ⊂ O10 then [A(O), A(O1 )] = {0}. A P↑+ -covariant local net of C ∗ -algebras is called a Haag– Kastler net. Vacuum states: A state ω on A is called a vacuum state iff ω is P↑+ -invariant (or translationally invariant), i.e. ω ◦ αg = ω for each g ∈ P↑+ , and for each a, b ∈ A Z dxhω, aα(1,x) (b)if (x) = 0 for each test function f ∈ S with supp(f˜) ∩ V¯+ = ∅. This implies that there exists a strongly continuous representation U of P↑+ on the GNS Hilbert space of ω such that U (g)π(a)U (g)∗ = π(αg a) and the spectrum of U (1, x) is contained in the closed forward light cone. Here π is the GNS representation of ω. Usually it is required that a vacuum state ω is a pure state. This aspect is not so important for our purpose and we do not assume this here. a Isotony: O ⊂ O implies A(O ) ⊂ A(O ). 1 2 1 2
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An example for a P↑+ -covariant local net of ∗ -algebras is given by the prescription T M (S) : O 7−→ TM (S(O)) , where TM (S) := T (S)/JM b is the well-known Borchers–Uhlmann algebra. We should mention here that now the test functions in S are test functions in Minkowski space-time. Let τ ∈ Hom(P↑+ , GL(S)) be the action of the Poincar´e group on the test functions which is given by τg f = f ◦ g −1 then αg (f1 ⊗ · · · ⊗ fn ) := τg f1 ⊗ · · · ⊗ τg fn defines a covariant action of P↑+ on T M (S). Now, the theorem above leads to the famous Osterwalder–Schrader reconstruction theorem: Theorem 1.2. Given a linear functional η which satisfies the conditions (E0) to (E2), then there exists a vacuum state ωη on the Borchers algebra TM (S) such that hωη , f1 ⊗ · · · ⊗ fn i = Wn (f1 , . . . , fn ) , where Wn is defined by Wn (x) =
Z ˜ n (q); ξj = xj+1 − xj . dnd q exp(−iξq) W
The fact that ωη is a vacuum state on the Borchers algebra is completely equivalent to the statement that the distributions Wn fulfill the Wightman axioms in its usual form (except the clustering)(see [25]). A heuristic proposal for the treatment of gauge theories. As mentioned above, the main reason for using Euclidean field theory is for constructing quantum field theory models with interaction. In four space time dimensions, the most promising candidates for interactive quantum field theory models are gauge theories. Scalar or multi-component scalar field theories of P (φ)4 type are less promising to describe interaction, since their construction either run into difficulties with renormalizability or, as conjectured for the φ44 -model, they seem to be trivial [7]. The description of gauge theories within the Wightman framework leads to some conceptional problems. For example, in order to study gauge invariant objects in quantum electrodynamics one may think of vacuum expectation values of products of the field strength Fµν Wµ1 ν1 ,...,µn νn (x1 , . . . , xn ) = hΩ, Fµ1 ν1 (x1 ) . . . Fµn νn (xn )Ωi which satisfy the Wightman axioms. Here, the problem arises when one wish to include fermions. In this case it is natural to consider correlation functions of b The ideal J M (M stands for Minkowski) is the two-sided ideal in T (S), which is generated by elements f1 ⊗ f2 − f2 ⊗ f1 ∈ T (S) where f1 and f2 have space-like separated supports.
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products of gauge fields and fermion fields. Since then one deals with non gauge invariant objects one is faced with such well-known problems as indefinite metric, solving constraints and so forth. Moreover, there is another problem which we would like to mention here. Within the Wightman framework the quantized version of the gauge field uµ is an operator valued distribution. On the other hand, the classical concept of a gauge field leads to the notion of a connection in a vector or principal bundle over some manifold which suggests to consider as gauge invariant objects Wilson loop variables Z u wγ [u] = tr Pexp γ
and string-like objects Z ¯ sγ [u, ψ] = ψ(r(γ))Pexp
u ψ(s(γ)) ,
γ
where ψ is a smooth section in an appropriate vector bundle and γ is an oriented path which starts at s(γ) and ends at r(γ). Unfortunately, to express wγ (u) in terms of Wightman fields leads to difficulties. From a perturbation theoretical point of view one expects that the distribution u is too singular in order to be restricted to a one-dimensional sub-manifold. To motivate our considerations, we shall discuss here, heuristically, an alternative proposal which might be related to a quantized version of a gauge theory. It is concerned with the direct quantization of regularized Wilson loops Z wγ (f )[u] = dx wγ+x [u] f (x) . Here we allow f ∈ E 0 (Rd ) to be a distribution with compact support which has the form f (x) = fΣ (x)δΣ (x) , where Σ is a d − 1-dimensional hyper-plane and fΣ ∈ C0∞ (Σ) and δΣ is the natural measure on Σ. We claim that such a type of regularization is necessary since in d-dimensional quantum field theories there are no bounded operators which are localized within d − 2-dimensional hyper-planes [4]. Such a point of view has been discussed by J. Fr¨ ohlich [6], E. Seiler [23] or more recently by A. Ashtekar and J. Lewandowski [1]. In order to describe a quantum gauge theory in terms of regularized Wilson loop variables one wishes to construct a function γ 7→ wγ which assigns to each path γ an operator valued distribution wγ : f 7→ wγ (f ), where the operators wγ (f ) are represented by operators on some Hilbert space H. Heuristically, one expects that the operators wγ (f ) are unbounded [20]. (1) The operators wγ (f ) are self-adjoint for real-valued test functions with a joint core D ⊂ H.
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(2) w should transform covariantly under the action of the Poincar´e group, i.e. wgγ (f ◦ g −1 ) = U (g)wγ (f )U (g)∗ ; g ∈ P↑+ , where U is a unitary strongly continuous representation of the Poincar´e group on H and the spectrum of the translations is contained in the closed forward light cone V¯+ . (3) Moreover, the operators wγ (f ) should satisfy the locality requirement, i.e. [E(γ,f ) (∆1 ), E(γ1 ,f1 ) (∆)] = 0 if the (convex hulls) of the regions γ + supp(f ) and γ1 + supp(f1 ) are spacelike separated. Here Z wγ (f ) = dE(γ,f ) (λ) λ is the spectral resolution of wγ (f ). According to [6, 23], it has been suggested to reconstruct Wilson loop operators wγ from Euclidean correlation functions of loops γ1 , . . . , γn 7−→ Sn (γ1 , . . . , γn ) , which satisfy the analogous axioms as the usual Schwinger distributions do, namely the reflexion positivity and the symmetry. However, within the analysis of J. Fr¨ ohlich, K. Osterwalder and E. Seiler [8, 23], the correlation function may have singularities in those points where two loops intersect and there are some additional technical conditions assumed which are related to the behavior of these singularities. They have proven (compare also [6]) that one can reconstruct from the Euclidean correlation functions Sn an operator valued function γ 7→ wγ together with a unitary strongly continuous representation of P ↑+ on H [8]. Here wγ is only defined for loops which are contained in some space-like plane and it fulfills the covariance condition (2). E. Seiler [23] has also discussed an idea how to proof locality (3). We shall come back to this point later. For our purpose, we look from an algebraic point of view at the problem of reconstructing a quantum field theory from Euclidean data. Let us consider functions Z ◦ a : AE 3 u 7−→ a dx wγj +x (u) fj (x); j = 1, . . . , n on the space of smooth connections AE in a vector bundle E over the Euclidean space Rd where a◦ is a bounded function on Rn . These functions are bounded and thus they generate an abelian C ∗ -algebra A with C ∗ -norm kak = sup |a(u)| . u∈AE
We assign to a given bounded region U ⊂ Rd the C ∗ -sub-algebra A(U) ⊂ A which is generated by all functions of Wilson loop variables wγ (f ) with γ + supp(f ) ⊂ U.
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The Euclidean group E(d) acts naturally by automorphisms on A, namely the prescription αg : a 7−→ a ◦ g −1 : u 7−→ a(u ◦ g) defines for each g ∈ E(d) an appropriate automorphism of A, which, of course, acts covariantly on the isotonous net A : U 7−→ A(U) , namely we have: αg A(U) = A(gU). Motivated by the work of E. Seiler, J. Fr¨ohlich and K. Osterwalder [23, 6, 8] as well as that of A. Ashtekar and J. Lewandowski [1], we propose to consider reflexion positive functionals on A, i.e. linear functionals η ∈ A∗ which fulfill conditions, corresponding to the axioms (E0)–(E2) above. These functionals can be interpreted as the analogue of the functional integral. Note, if η is a state, then η is nothing else but a measure on the spectrum X of the C ∗ -algebra A. The advantage of this point of view is based on the fact that abelian C ∗ -algebras are rather simple objects namely algebras of continuous functions on a (locally)-compact Hausdorff space. Overview. In order to make the comprehension of the subsequent sections easier, we shall give an overview of the content of our paper by stating the main ideas and results. This paragraph is also addressed to quick readers who are not so much interested in technical details. Motivated by the considerations above, in Sec. 2 we make a suggestion for axioms which a Euclidean field theory should satisfy. We start from an isotonous net A : U 7−→ A(U) ⊂ A of C ∗ -algebra on which the Euclidean group E(d) acts covariantly by automorphisms of α : E(d) → Aut A, like in the example of Wilson loop variables given in the previous paragraph. However, we assume a somewhat weaker condition than commutativity for A. For our considerations we only have to assume that two operators commute if they are localized in disjoint regions. In addition to that, we consider a reflexion positive functional η on A. We shall call the triple (A, α, η), consisting of the net A of C ∗ -algebra, the action of the Euclidean group α, and the reflexion positive functional, a Euclidean field. We show in Sec. 3 how to construct from a given Euclidean field a quantum field theory in a particular vacuum representation. In order to point out the relation between the Euclidean field (A, α, η) and the Minkowskian world, we briefly describe the construction of a Hilbert space H on which the reconstructed physical observables are represented. According to our axioms, the map a ⊗ b 7−→ hη, ιe (a∗ )bi is a positive semidefinite sesqui-linear form on the algebra A(e) of operators which are localized in eR+ + Σe where Σe is the hyper plane orthogonal to the Euclidean
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time direction e ∈ S d−1 . Here ιe is the automorphism on A which corresponds to the reflexion e 7→ −e. By dividing the null-space and taking the closure we obtain a Hilbert space H. The construction of the observables, which turn out to be bounded operators on H, is based on two main steps. Step 1 : In Sec. 3.1, we reconstruct a unitary strongly continuous representation of the Poincar´e group U on H. To carry through this analysis, it is not necessary to impose new ideas. The construction is essentially analogous to that which has been presented in [8] (compare also [23]). In order to keep the present paper self contained, we feel obliged to discuss this point within our context in more detail. Step 2 : We discuss in Sec. 3.2 the construction of the physical observables. At the moment this can only be done, if we assume that the algebra A contains operators which are localized at sharp times, i.e. we require that the algebra A(e) ∩ A(−e) is large enough. A precise formulation of this condition is given in Sec. 3.2. We shall abbreviate this condition by (TZ) which stands for time-zero. For the fix-point algebra B(e) of ιe in A(e) ∩ A(−e) we obtain a ∗ -representation π on H, where an operator π(b), b ∈ B(e), is given by the prescription π(b)p(a) 7−→ p(ba) . Here p is the canonical projection onto the quotient, identifying an operator a ∈ A(e) with its equivalence class p(a) in H. Now, we consider for a given Poincar´e transform g ∈ P↑+ and a given time-zero operator b ∈ B(e) the following bounded operator: Φ(g, b) := U (g)π(b)U (g)∗ . We shall say that Φ(g, b) is localized in a double cone O in Minkowski space if b is localized in U ⊂ Σe and the transformed region gU is contained in O. Let us denote the C ∗ -algebra which is generated by all operators Φ(g, b), which are localized in O, by A(O). Hence we get an isotonous net of C ∗ -algebras A : O 7−→ A(O) indexed by double cones in Minkowski space on which the Poincar´e group acts covariantly by the automorphisms αg := Ad(U (g)), g ∈ P↑+ . The main result: (1) The reconstructed isotonous net A is a Haag–Kastler net: locality holds, i.e. if O, O1 are two double cones such that O ⊂ O10 then [A(O), A(O1 )] = {0}. (2) Furthermore, the P↑+ -invariant vector Ω = p(1) induces a vacuum state ω : a 7−→ hω, ai := hΩ, aΩi . The non-trivial aspect of this statement is the proof of locality. As mentioned above, E. Seiler has discussed an idea on how to prove locality for a net of Wilson loops wγ . This idea does not rely on the fact that one considers loops and it can
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also be used for general Euclidean fields. In order to close the gaps within the sketch which is given in [23], we present here (Sec. 3.2) a detailed proof of locality. Our strategy is based on the analytic properties of the functions ˆ F (z1 , z2 ) := hψ, ΦX1 (z1 , b1 )ΦX2 (z2 , b2 )ψi ˆ , Fˆ (z1 , z2 ) := hψ, ΦX2 (z2 , b2 )ΦX1 (z1 , b1 )ψi where we have introduced the operators ΦX (z, b) := U (exp(zX))π(b)U (exp(−zX)) . Here b ∈ B(e) is a time-zero operator and iX is a boost generator or the Hamiltonian H with respect to the time direction e. Roughly, the argument for the proof of locality goes as follows: Suppose bj is ˆ in which F (Fˆ ) are localized in Uj ⊂ Σe . We shall show that the regions G (G) holomorphic are (a) connected and they contain pure imaginary points (is1 , is2 ) and ˆ contains all those points (t1 , t2 ) for which O1 = (b) the intersection G ∩ G exp(t1 X)U1 and O2 = exp(t2 X2 )U2 are space-like separated. But F and Fˆ coincide in the pure imaginary points since operators which are localized in disjoint regions commute. This implies F |G∩Gˆ = Fˆ |G∩Gˆ and thus by (b) we conclude ˆ =0 hψ, [ΦX1 (t1 , b1 ), ΦX2 (t2 , b2 )]ψi if ΦX1 (t1 , b1 ) and ΦX2 (t2 , b2 ) are localized in space-like separated regions. We claim ˆ However, one can ˆ depend on the choice of the vector ψ. that the regions G and G ˆ ˆ for all ψˆ ∈ D. find a dense subspace D such that F (F ) are holomorphic in G (G) Thus the commutator [ΦX1 (t1 , b1 ), ΦX2 (t2 , b2 )] vanishes on a dense subspace and, since ΦX (t, b) is bounded for real points t ∈ R, the commutator vanishes on H. In order to get analyticity of F within a region G which is large enough, we prove in the appendix an statement which is the analogue of the famous Bargmann–Hall– Wightman theorem [15, 16, 25]. In Sec. 4, we discuss some miscellaneous consequences of our result. Note that for the application of our reconstruction scheme it was crucial to assume that the there are enough non-trivial Euclidean operators which can be localized at sharp times. We shall give some remarks on the condition (TZ) in Sec. 4.1. Our considerations can easily be generalized to the case in which there are also fermionic operators present or even to super-symmetric theories. Here one starts with an isotonous net F : U → 7 F (U) of Z2 -graded C ∗ -algebras which fulfills the time-zero condition (TZ), i.e. the fix-point algebra B(e) of ιe in F (e) ∩ F (−e) is large enough. The Euclidean group acts covariantly by automorphisms on F and
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we require that the graded commutator [a, b]g = 0 vanishes if a and b are localized in disjoint regions. Let η be a reflexion positive functional, then, by replacing the commutator by the graded commutator, we conclude that the operators Φ(g, b) = U (g)π(b)U (g)∗ ; b ∈ B(e) and g ∈ P↑+ generate a fermionic net F of C ∗ -algebras. This can really be done analogously to the construction of the Haag–Kastler net A, described above. Finally, we close our paper by the Sec. 5 conclusion and outlook. 2. Axioms for Euclidean Field Theories In the present section we make a suggestion for axioms which a Euclidean field theory should satisfy. In the first step, we introduce the notion of a Euclidean net of C ∗ -algebras. Within our interpretation this notion is related to physical observations. Definition 2.1. A d-dimensional Euclidean net of C ∗ -algebras is given by a pair (A, α) which consists of an isotonous net: A : Rd ⊃ U 7−→ A(U) of C ∗ -algebras, indexed by bounded subsets in Rd and a group homomorphism α ∈ Hom(E(d), Aut(A)).c We require that the pair fulfills the conditions: (1) Locality: U1 ∩ U2 = ∅ implies [A(U1 ), A(U2 )] = {0}. (2) Euclidean covariance: αg A(U) = A(gU) for each U. For a Euclidean direction e ∈ S d−1 we consider the reflection θe : e 7→ −e. and the sub-group Ee (d − 1) which commutes with θe . Moreover, we set ιe := αθe . As in the introduction, we denote by A(e) the C ∗ -algebra A(eR+ + Σe ) where Σe is the hyper-plane orthogonal to e. Now we formulate a selection criterion for linear functionals on A which corresponds to the selection criterion for physical states. We shall see that class of functional, which is introduced below, is the Euclidean analogue of the set of vacuum states. Definition 2.2. We define S(A, α) to be the set of all continuous linear functionals η on A which fulfill the following conditions: (1) e-reflexion positivity: There exists a Euclidean direction e ∈ S d−1 such that ∀a ∈ A(e) : hη, ιe (a∗ )ai ≥ 0 . (2) Unit preserving: hη, 1i = 1. (3) Invariance: ∀g ∈ E(d) : η ◦ αg = η. c We denote the the C ∗ -inductive limit of A by A. For an unbounded region Σ the algebra A(Σ) denotes the C ∗ -sub-algebra which is generated by the algebras A(U ), U ⊂ Σ.
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Remark. We easily observe that the definition of S(A, α) is independent of the chosen direction e. In the subsequent section, we call the functionals in S(A, α) reflexion positive. For our purpose it is necessary to require a further condition for the functionals under consideration. Definition 2.3. We denote by SR (A, α) the set of all reflexion positive functionals η of A for which the map E(d) 3 g 7−→ hη, a(αg b)ci is a continuous function for each a, b, c ∈ A. These functionals are called regular reflexion positive. We shall call a triple (A, α, η) which consists of a Euclidean net and a regular reflexion positive functional η a Euclidean field. As mentioned in the introduction, we have to assume that the operators of the Euclidean net can be localized at a sharp d − 1-dimensional hyper plane. For a Euclidean time direction e and for a region U ⊂ Σe , we denote by B(e, U) the fix-point algebra B(e, U) := [A(R+ e + U) ∩ A(−R+ e + U)]ιe under the reflexion ιe . We call the algebras B(e, U) the time-zero algebras of the net (A, α). For a given region Uˆ ⊂ Rd we introduce the algebra ˆ := ATZ (U)
[
αg (B(e, U))
k·k
.
ˆ (g,U ):g∈E(d),U ⊂Σe ,gU ⊂U
Remark. Then the net
(1) Let (A, α) be a d-dimensional Euclidean net of C ∗ -algebras. B(e) : Σe ⊃ U 7−→ B(e, U)
together with the group homomorphism β e := α|Ee (d−1) is, of course, a d − 1dimensional Euclidean net of C ∗ -algebras. We denote by B(e) its C ∗ -inductive limit. (2) The pair (ATZ , α) which consists of the net ˆ ATZ : Rd ⊃ Uˆ 7−→ ATZ (U) and the the group homomorphism α is a d-dimensional Euclidean net of C ∗ -algebras. In particular, it is a subnet of (A, α). Condition (TZ). A d-dimensional Euclidean net of C ∗ -algebras (A, α) fulfills the time-zero condition (TZ) if (ATZ , α) = (A, α) .
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Remark. The condition (TZ) states that the structure of (A, α) is determined by the time-zero algebras (B(e), β e ) and the action of the full Euclidean group α restricted to the time-zero algebras. In Sec. 4.1 we shall see that the quantum field theory model, which can be reconstructed from (A, α) via our reconstruction scheme, only depends on the subnet (ATZ , α). Therefore, if the condition (TZ) is fulfilled, then our reconstruction scheme does not miss relevant information contained in (A, α) (Proposition 4.4). On the other hand, if B(e) is too small in the sense that for some region Uˆ the algebra ATZ (Uˆ) is a proper subalgebra of A(Uˆ), then our reconstruction scheme ˆ As misses all (physical) aspects which are described by operators in A(Uˆ)\ATZ (U). an example, we consider two nets (A1 , α1 ) and (A2 , α2 ) with (A1,TZ , α1 ) = (A1 , α1 ) (A2,TZ , α2 ) = C1 . We obtain for the tensor product (A1 ⊗ A2 , α1 ⊗ α2 ) ((A1 ⊗ A2 )TZ , α1 ⊗ α2 ) = (A1 , α1 ) and an application of our reconstruction scheme to the net ((A1 ⊗ A2 )TZ , α1 ⊗ α2 ) misses the subtheory (A2 , α2 ). 3. From Euclidean Field Theory to Quantum Field Theory In the present section, we discuss how to pass from a Euclidean field (A, α, η) to a quantum field theory in a particular vacuum representation. In the first step we construct from a given Euclidean field (A, α, η) a unitary strongly continuous representation of the Poincar´e group (Sec. 3.1). In the second step we have to require that condition (TZ) is satisfied in order to show that a concrete Haag–Kastler net can be reconstructed from the elements of the time-zero algebras and the representation of the Poincar´e group (Sec. 3.2). 3.1. Reconstruction of the Poincar´ e group For e ∈ S d−1 we introduce a positive semidefinite sesqui-linear form on A(e) as follows: a ⊗ b 7−→ hη, ιe (a∗ )bi . Its null space is given by N (e, η) := {a ∈ A(e)|∀b ∈ A(e) : hη, ιe (a∗ )bi = 0} and we obtain a pre-Hilbert space D(e, η) := A(e)/N (e, η) The corresponding quotient map is denoted by p(e,η) : A(e) −→ D(e, η) and its closure H(e, η) is a Hilbert space with scalar product
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hp(e,η) (a), p(e,η) (b)i := hη, ιe (a∗ )bi . Lemma 3.1. The map T(e,η) : s ∈ R+ 7−→ T(e,η) (s) : p(e,η) (a) 7−→ p(e,η) (α(1,se) a) is a strongly continuous semi-group of contractions with a positive generator H(e,η) ≥ 0. Proof. Since hη, ιe (b∗ )ai = 0 for each b ∈ A(e) implies hη, ιe (b∗ )αse ai = hη, ιe (αse b∗ )ai = 0 for each b ∈ A(e), we conclude that T(e,η) (s)p(e,η) (a) = 0 for a ∈ N (e, η). Hence T(e,η) is well-defined. The fact that T(e,η) is a semi-group of contractions follows by standard arguments, i.e. a multiple application of the Cauchy–Schwartz inequality. Finally, the strong continuity follows from the regularity of η. We consider the set Con(e) of all cones Γ (in Euclidean space) of the form Γ = R+ (Bd (r) + e) + e where Bd (r) denotes the ball in Rd with center x = 0 and radius 0 < r ≤ 1. In addition, we define the following subspace of H(e, η): D(Γ; η) := p(e,η) A(Γ) . Lemma 3.2. For each cone Γ ∈ Con(e), the vector space D(Γ, η) is a dense subspace of H(e, η). Proof. Lemma 3.1 states that T(e,η) is a semi-group of contractions with a positive generator. Furthermore, D(Γ, η) is mapped into itself by T(e,η) (s). Since for each operator a ∈ A(e) there exists an s > 0 such that T(e,η) (s)p(e,η) (a) ∈ D(Γ, η) , we can apply a Reeh–Schlieder argument in order to prove that D(Γ, η) is a dense subspace of H(e, η). Lemma 3.3. Let V ⊂ E(d) be a small neighborhood of the unit element 1 ∈ E(d) and let Γ ∈ Con(e) be a cone such that VΓ ⊂ eR+ +Σe . Then a ∈ A(Γ)∩N (e, η) implies αg a ∈ N (e, η) for each g ∈ V. Proof. We have hη, ιe (b∗ )αse ai = 0 for each b ∈ A(Γ) and hence hη, ιe (b∗ )αg ai = hη, ιe (αθe g b∗ )ai = 0. Since we may choose V to be θe -invariant, we have αθe g b∗ ∈ A(e) and the result follows by Lemma 3.2.
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Theorem 3.4. Let η ∈ SR (A, α) be a regular reflexion positive functional. Then for each e ∈ S d−1 there exists a unitary strongly continuous representation U(e,η) of the d-dimensional Poincar´e group P↑+ U(e,η) ∈ Hom[P↑+ , U (H(e, η))] such that the spectrum of the translations x → U(e,η) (1, x) is contained in the closed forward light cone V¯+ . Proof. The theorem can be proven using the techniques presented in [8] (compare also [23, Theorem 8.10]). We briefly illustrate the construction of the representation U(e,η) . Let V ⊂ E(d) be a small neighborhood of the unit element 1 ∈ E(d). Then there exists a cone Γ ∈ Con(e) such that VΓ ⊂ eR+ + Σe . According to Lemma 3.3 we may define for each g ∈ V the operator V(e,η) (g)p(e,η) (a) := p(e,η) (αg a) with domain D(Γ, η). If g belongs to the group Ee (d − 1) then we conclude that V(e,η) (g) = U(e,η) (g) is a unitary operator. Let e(d) be the Lie algebra of E(d) and let ee (d−1) ⊂ e(d) be the sub-Lie algebra of Ee (d−1) ⊂ E(d). We decompose e(d) into a direct sum of linear spaces as follows: e(d) = ee (d − 1) ⊕ me (d − 1) and we obtain another real Lie algebra: p(d) := ee (d − 1) ⊕ ime (d − 1) which is the Lie algebra of the Poincar´e group P↑+ . For each X ∈ me (d − 1) there exists a self adjoint operator L(e,η) (X) where D(Γ, η) consists of analytic vectors for L(e,η) (X) and for each s ∈ R with exp(sX) ∈ V we have V(e,η) (exp(sX)) = exp(sL(e,η) (X)) . According to [23, Theorem 8.10] we conclude that the unitary operators U(e,η) (exp(isX)) := exp(isL(e,η) (X)); X ∈ me (d − 1) U(e,η) (g) := V(e,η) (g); g ∈ Ee (d − 1) induce a unitary strongly continuous representation of the Poincar´e group P↑+ . The positivity of the Energy follows from the fact that the transfer matrix T(e,η) (1) is a contraction. Remark. The vector Ω(e,η) := p(e,η) (1) is invariant under the action of the Poincar´e group. 3.2. Reconstruction of the net of local observables In the subsequent, we consider a Euclidean net of C ∗ -algebras (A, α) which fulfills the condition (TZ).
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Proposition 3.5. Let η be a regular reflexion positive functional on A. Then the map π(e,η) : B(e) 3 b 7−→ π(e,η) (b) : p(e,η) (a) 7−→ p(e,η) (ba) is a well-defined ∗ -representation of B(e). Proof. For each a ∈ N (e, η) and for each c ∈ A(e) we have hη, ιe (c∗ )bai = hη, ιe (c∗ b)ai = 0 and hence π(e,η) (b) is a well-defined linear and bounded operator. By construction it is clear that π(e,η) is a ∗ -homomorphism. Remark. The restriction of η|B(e) is a state of B(e). Of course, the GNSrepresentation of η|B(e) is a sub-representation of π(e,η) . Definition 3.6. (1) Let O be a double cone in Rd . Then we define A(e,η) (O) to be the C ∗ -algebra on H(e, η) which is generated by operators Φ(e,η) (g, b) := U(e,η) (g)π(e,η) (b)U(e,η) (g)∗ with b ∈ B(e, U), g ∈ P↑+ and gU ⊂ O. (2) We denote by A(e,η) the net of C ∗ -algebras which is given by the prescription A(e,η) : O 7−→ A(e,η) (O) . Theorem 3.7. The pair (A(e,η) , Ad(U(e,η) )) is a P↑+ -covariant Haag–Kastler net which is represented on H(e, η). Remark. (1) Note that ω(e,η) : A(e,η) 3 a 7−→ hΩ(e,η) , aΩ(e,η) i is a vacuum state since U(e,η) is a positive energy representation of the Poincar´e group. However, in general ω(e,η) is not a pure state. (2) In general, Haag duality for the reconstructed net A(e,η) can be violated. We do not take the dual net of von Neumann algebras O 7−→ Ad(e,η) (O) := A(e,η) (O0 )0 since this might lead to problems with locality. Preparation of the proof of Theorem 3.7. For a Lie algebra element X ∈ ime (d−1) and a complex number z ∈ C we define a linear (unbounded) operator on H(e, η) by Φ(e,η,X) (z, b) := U(e,η) (exp(zX))π(e,η) (b)U(e,η) (exp(−zX)) on a dense domain D(Γ, η) where Γ ∈ Con(e) an appropriate cone.
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In order to formulate our next result, we define for two generators X1 , X2 ∈ ime (d − 1), for an interval I, for a neighborhood V of the unit element in P+ (C), with L↑+ ⊂ V, and for two subsets Uj ⊂ Σe , j = 1, 2, the region [n G(V; X1 , X2 ; U1 , U2 ; I) := (z1 , z2 ) ∈ (R × iI)2 ∀xj ∈ Uj : g∈V
o e Im[g(exp(z1 X1 )x1 − exp(z2 X2 )x2 )] ∈ R+ . We shall prove in the appendix the lemma given below which is the analogue of the famous BHW theorem (compare also [16, 25] and references given there): Lemma 3.8. For a given interval I, there exists a dense subspace D ⊂ H(e, η), such that the function F(X1 ,X2 ,b1 ,b2 ) : (z1 , z2 ) 7−→ hψ1 , Φ(e,η,X1 ) (z1 , b1 )Φ(e,η,X2 ) (z2 , b2 )ψ2 i is holomorphic in G(V; X1 , X2 ; U1 , U2 , I) for each ψ1 , ψ2 ∈ D. We claim that the E(d) invariance of η yields that the dense subspace D ⊂ H(e, η) can be chosen in such a way that I(V; X1 , X2 ; U1 , U2 ; I) := G(V; X2 , X1 ; U2 , U1 ; I) ∩ G(V; X1 , X2 ; U1 , U2 ; I) ∩ iR2 6= ∅ . Lemma 3.9. If U1 ∩ U2 = ∅ and (s1 , s2 ) ∈ I(V; X1 , X2 ; U1 , U2 ; I), then F(X1 ,X2 ,b1 ,b2 ) (is1 , is2 ) = F(X2 ,X1 ,b2 ,b1 ) (is2 , is1 ) . Proof. The lemma is a direct consequence of the Euclidean covariance and the locality of the net A. Proof of Theorem 3.7. We conclude from Theorem 3.4 and the construction of the algebras A(e,η) (O) that A(e,η) is a Poincar´e covariant net of C ∗ -algebras, represented on H(e, η). It remains to be proven that A(e,η) is a local net. For this purpose it is sufficient to show that for each pair (t1 , t2 ) ∈ R(X1 , X2 ; U1 , U2 ) := {(t1 , t2 ) ∈ R2 |exp(t1 X1 )U1 ⊂ (exp(t2 X2 )U2 )0 } the commutator [Φ(e,η,X1 ) (t1 ), Φ(e,η,X2 ) (t2 )]|D = 0 vanishes on an appropriate dense domain D ⊂ H(e, η). Since the points in R(X1 , X2 ; U1 , U2 ) are space-like points, we conclude that there exist complex Lorenz boosts g± ∈ V such that Im g± R(X1 , X2 ; U1 , U2 ) ⊂ V± .
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Hence we have R(X1 , X2 ; U1 , U2 ) ⊂ G(V; X1 , X2 ; U1 , U2 ; I) ∩ G(V; X2 , X1 ; U2 , U1 ; I) . Using Lemma 3.9, we conclude that F(X1 ,X2 ,b1 ,b2 ) (z1 , z2 ) = F(X2 ,X1 ,b2 ,b1 ) (z2 , z1 ) for (z1 , z2 ) 3 G(V; X1 , X2 ; U1 , U2 ; I) ∩ G(V; X2 , X1 ; U2 , U1 ; I) which finally yields F(X1 ,X2 ,b1 ,b2 ) (t1 , t2 ) = F(X2 ,X1 ,b2 ,b1 ) (t2 , t1 ) for each (t1 , t2 ) ∈ R(X1 , X2 ; U1 , U2 ). This proves the locality of A(e,η) .
4. Discussion of Miscellaneous Consequences Due to Theorem 3.7 we are able to pass from a Euclidean field (A, α, η) to a quantum field theory in a particular vacuum representation. One crucial condition to apply our method is the existence of the time-zero algebras. We shall see that the discussion of Sec. 4.1 covers all possible situations for Euclidean fields which fulfill the condition (TZ). Afterwards, we discuss in Sec. 4.2 how the reconstruction scheme has to be generalized in order to include fermionic operators. 4.1. Some remarks on Euclidean fields which satisfy the time-zero condition Let us consider a d − 1-dimensional Euclidean net (B, β) of abelian C ∗ -algebras. Definition 4.1. Let G be a group which contains E(d − 1) as a sub-group and let U (B) be the group of unitary operators in B. We define G(G; B, β) to be the group which is generated by pairs (g, v) ∈ G × U (B) modulo the relations: (1) For each g ∈ G, the map v 7−→ (g, v) is a group-homomorphism. (2) For each g ∈ G, for each h ∈ E(d − 1), and for each v ∈ U (B): (gh, v) = (g, βh v) We equip G(G; B, β) with the discrete topology. Definition 4.2. We denote by A0 (G; B, β) the ∗ -algebra of functions on G(G; B, β) with compact support, where the product of two functions a1 , a2 is defined by the convolution X a1 · a2 : v 7−→ (a1 · a2 )(v) = a1 (v1 )a2 (v1−1 v) v1 ∈G(G;B,β)
and the ∗ -involution is given by a∗ : v 7−→ a ¯(v−1 ) .
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The algebra A0 (G; B, β) possesses a faithful state which is given by ω1 (a) := a(1) , where 1 is the unit element in G(G; B, β). The algebra A0 (G; B, β) has a natural C ∗ -norm which is given by kak := kπω1 (a)k , where πω1 is the GNS representation with respect to ω1 . Note that πω1 is faithful and that the inequality X kπω1 (a)k2 ≤ |a1 (v)|2 v∈G(G;B,β)
holds. The closure of A0 (G; B, β) with respect to the norm, given above, is denoted by A(G; B, β). Remark. (1) There is a natural group homomorphism α ∈ Hom(G, Aut A(G; B, β)) and a natural faithful embedding φ ∈ Hom(U (B), U A(G; B, β)) given by: αg (g1 , v) := (gg1 , v) φ(v) := (1, v) , where U A(G; B, β) is the group of unitary operators in A(G; B, β). Of course, we have for each h ∈ E(d − 1): φ ◦ βh = αh ◦ φ . ˆ := A(E(d − 1); B, β) which contains a canonical (2) We obtain a C ∗ -algebra B ˆ such that closed ideal JB ⊂ B ˆ B. B = B/J ˆ as Furthermore, for each group G ⊃ E(d − 1) the C ∗ -algebra A(G; B, β) contains B a sub-C ∗ -algebra. (3) We are mostly interested in two cases for G, namely G = P↑+ and G = E(d). For both groups A(G; B, β) has a natural local structure since P↑+ and E(d) act as groups on Rd . Definition 4.3. For a region O ∈ Rd we define A(G; B, β|O) to be the C ∗ sub-algebra in A(G; B, β) which is generated by elements (g, v) with v ∈ B(U) and gU ⊂ O and we obtain nets A(G; B, β) : O 7−→ A(G; B, β|O) . In order to get a Haag–Kastler net for G = P↑+ and a Euclidean net for G = E(d), we consider the following ideals: (1) Jc (P↑+ ; B, β) is the two-sided ideal which is generated by JB and elements [(g, v), (g1 , v1 )] where (g, v) and (g1 , v1 ) are localized in space like separated regions.
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(2) Jc (E(d); B, β) is the two-sided ideal which is generated by JB and elements [(g, v), (g1 , v1 )] where (g, b) and (g1 , v1 ) are localized in disjoint regions. Thus the prescription AG : O 7−→ AG (O) := A(G; B, β|O)/Jc (G; B, β) is a P↑+ -covariant Haag–Kastler net for G = P↑+ , and a Euclidean net of C ∗ -algebras for G = E(d). Proposition 4.4. Let (A, α) be a d-dimensional Euclidean net which fulfills the condition (TZ) and let (B, β) be the d − 1-dimensional Euclidean net, corresponding to the hyper plane Σe . Then the map χ : AE(d) 3 [(g, v)] −→ αg (v) ∈ A is a surjective ∗ -homomorphism which preserves the net structure, where [(g, v)] denotes the equivalence class of (g, v) in AE(d) . Proof. By using the relations in Definition 4.2 and the definition of the ideal JB we conclude, by some straight forward computations, that χ is a well-defined a ∗ -homomorphism which preserves the net structure. According to the definition of the subnet (ATZ , α) and according to the condition (TZ) (Sec. 2), we conclude χ(AE(d) ) = ATZ = A which proves that χ is surjective. An application of Theorem 3.7 gives: Corollary 4.5. For each regular reflexion positive functional η on AE(d) there exists a vacuum state ωη on AP↑ such that +
ωη |B = η|B . Remark. (1) Note that we may view B as a common subalgebra of AE(d) and AP↑ since B ∩ Jc (G; B, β) = {0}. +
(2) Given a Euclidean field (A, α, η), for which the time zero algebra B := B(e) is non trivial. By Proposition 4.4, we conclude that there is a positive energy representation π(e,η) of AP↑ on the Hilbert space H(e, η) whose image is precisely + the net A(e,η) . In particular the GNS-representation of ωη is a sub-representation of π(e,η) . (3) Both, the algebra AP↑ of observables in Minkowski space and the Euclidean + algebra AE(d) can be considered as sub-algebras of AP+ (C) where the algebra AP+ (C) is defined by AP+ (C) := A(P+ (C); B, β)/[Jc (P↑+ ; B, β) ∪ Jc (E(d); B, β)] . We close this section by illustrating the situation by the commutative diagram, given below.
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D. SCHLINGEMANN
AP+(C )
[[ ^
u
# A+ u [ P
"
{
[[! ] [
[
y
B
[ e w
AE(d)
C
u
u
A
Here (A, α) is a Euclidean net of C ∗ -algebras and B is the time-zero algebra which corresponds to the hyper-plane Σe . 4.2. The treatment of fermionic operators In order to discuss the treatment of fermionic operators we introduce the notion of a fermionic Euclidean net. The axioms for such a net coincide with those of a Euclidean net, except the locality requirement. Definition 4.6. An isotonous and E(d)-covariant net (F , α) F : Rd ⊃ U 7−→ F (U) = F+ (U) ⊕ F− (U) of Z2 -graded C ∗ -algebras is called a fermionic Euclidean net iff U1 ∩ U2 = ∅ implies [F (U1 ), F (U2 )]g = {0}, where [· , ·]g denotes the graded commutator. For a given d − 1-dimensional fermionic net (F , β), we build the C ∗ -algebras A(E(d); F, β) and A(P↑+ ; F, β) as introduced in the previous section. Note, that the algebra A(P↑+ ; F, β) possesses a Z2 -grading, namely we have A(P↑+ ; F, β) = A+ (P↑+ ; F, β) ⊕ A− (P↑+ ; F, β) , where the algebra A+ (P↑+ ; F, β) is spanned by products of elements (g, v) containing an even number of generators in G × U (F− ): (g1 , v1 ) . . . (g2n , v2n ) . Therefore the sub-space A− (P↑+ ; F, β) is spanned by elements which are products of elements (g, v) containing an odd number of generators in G × F− : (g1 , v1 ) . . . (g2n−1 , v2n−1 ) . Analogously to the purely bosonic case, we consider the two-sided ideals: (1) Jg (P↑+ ; F, β) which is generated by JF and graded commutators [(g, b), (g1 , b1 )]g , where (g, b) and (g1 , b1 ) are localized in space like separated regions and (2) Jg (E(d); B, β) which is generated by JF and graded commutators [(g, b), (g1 , b1 )]g , where (g, b) and (g1 , b1 ) are localized in disjoint regions.
FROM EUCLIDEAN FIELD THEORY TO QUANTUM FIELD THEORY
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Thus the prescription FG : O 7−→ FG (O) := A(G; F, β|O)/Jg (G; F, β) is a fermionic P↑+ -covariant Haag–Kastler net for G = P↑+ , and a fermionic Euclidean net for G = E(d). By following the arguments in the proof of Theorem 3.7 and keeping in mind that the ordinary commutator has to be substituted by the graded commutator, we get the result: Corollary 4.7. For each regular reflexion positive functional η on the fermionic Euclidean net FE(d) there exists a vacuum state ωη on FP↑ such that +
ωη |F = η|F . Remark. As described in Sec. 3.2 the state is defined by * + * + n n Y Y ωη , (gj , vj ) = Ω(e,η) , Φ(e,η) (gj , vj )Ω(e,η) . j=1
j=1
5. Conclusion and Outlook 5.1. Concluding remarks and comparison We have shown, how a quantum field theory can be reconstructed form a given Euclidean field (A, α, η) which fulfills the condition (TZ). We think, that in comparison to the usual Osterwalder–Schrader reconstruction theorem the reconstruction of a quantum field theory from Euclidean fields (in our sense) has the following advantages: ⊕ The Osterwalder–Schrader reconstruction theorem relates Schwinger distributions to a Wightman theory. One obtains an operator valued distribution Φ which satisfies the Wightman axioms. The reconstructed field operators Φ(f ) are, in general, unbounded operators and in order to get a Haag–Kastler net of bounded operators one has to prove that not only the field operators Φ(f ), Φ(f1 ) commute if f and f1 have space-like separated supports, but also its corresponding spectral projections. Furthermore, as mentioned in the introduction, in order to apply the results of [19] one has to prove that the Schwinger distributions are continuous with respect to an appropriate topology. Since our considerations are based on C ∗ -algebras, we directly obtain, via our reconstruction scheme, a Haag–Kastler net of bounded operators. In our case, the technical conditions which a reflexion positive functional has to satisfy are more natural. It has to be continuous and regular where the continuity is automatically fulfilled if one considers reflexion positive states. Our reconstruction scheme does also include objects, like Wilson loop variables, which are not point-like localized objects in a distributional sense. This point of view may also be helpful for constructing gauge theories.
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D. SCHLINGEMANN
Furthermore, one also may start with an abelian C ∗ -algebra like the example of Wilson loop variables, given in the introduction. Abelian C ∗ -algebras are rather simple objects, namely nothing else but continuous functions on a compact Hausdorff space. In comparison to the construction of reflexion positive functional on the tensor algebra TET (S), one may hope that it is easier to construct reflexion positive functionals for abelian C ∗ -algebras. This might simplify the construction of quantum field theory models. Nevertheless, we also have to mention some drawbacks: Unfortunately, our reconstruction scheme is not a complete generalization of the Osterwalder–Schrader reconstruction. This is due to that fact, that we have assumed the existence of enough operators in A which can be localized on a sharp d− 1-dimensional hyper plane (condition (TZ)). Such a condition is not needed within the Osterwalder–Schrader framework and there are indeed examples of quantum field theories which do not fulfill this condition, for instance the generalized free field for which the mass distribution is not L1 . On the other hand, the known interacting models like the P (φ)2 , the Yukawa2 as well as the φ43 model fulfill the condition (TZ). Thus we think that the existence of the time-zero algebras is not such a harmful requirement. 5.2. Work in progress The main aim of our work in progress is concerned with the construction of examples for Euclidean fields which go beyond the free fields. It would also be desirable to develop a generalization of our reconstruction scheme which also lead directly to a Haag–Kastler net but which do not rely on the condition (TZ). A further open question is concerned with a reconstruction scheme for Euclidean fields with cutoffs. The main motivation for such a considerations is based on the work of J. Magnen, V. Rivasseau, and R. S´en´eor [17] where it is claimed that the Yang–Mills exists within a finite Euclidean volume. A. Analytic Properties Within this appendix, we give a complete proof of Lemma 3.8. We shall use a simplified version of the notation introduced in the previous sections by dropping the indices (e, η). Let (A, α, η) be a Euclidean field and let U be the corresponding strongly continuous representation of the Poincar´e group on H = H(e, η) which has been constructed by Theorem 3.4. Furthermore, let π be the ∗ -representation of the time-zero algebra B on H. For a given tuple (X, b) ∈ im(d−1)n ×B n , we like to study the analytic properties of the function Ψn [X, b] : C2n 3 (z, z 0 ) 7−→
n Y j=1
UXj (zj )π(bj )UXj (zj0 )ψ ,
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where ψ ∈ D(Γ, η) and Γ is a cone which is contained in Con(e) and we write: UX (ζ) := U (exp(−iζX)) . For this purpose, we introduce some technical definitions. Definition A.1. For a generator X ∈ im(d − 1), for an operator b ∈ B(U) and for a cone Γ ∈ Con(e), we define the regions I(Γ, X) := {s0 | exp(−is0 X)Γ ⊂ eR+ + Σe } J(Γ, X, b, s0 ) := {s| exp(−isX)[exp(−is0 X)Γ ∪ U] ⊂ eR+ + Σe } [ [R + iJ(Γ, X, b, s0 ) × R + i{s0 }] , G(Γ|X, b) := s0 ∈I(Γ,X)
where the region J(Γ, X, b, s0 ) depends on b only via its localizing region U. Definition A.2. (1) Consider a region U which is contained in Σe + eτ , τ ≥ 0. We define the corresponding time-zero algebra by B(U) := αeτ B(U − eτ ). (2) For a given tuple (X, b, s, s0 ) ∈ im(d − 1)n × B(U1 ) × · · · × B(Un ) × R2n , we define recursively the regions Γ0 := Γ Γ1 (s1 , s01 ) := conv(exp(−is1 X1 )[exp(−is01 X1 )Γ ∪ U1 ]) Γn (s1 . . . sn , s01 . . . s0n ) := conv(exp(−isn Xn )[exp(−is0n Xn ) × Γn−1 (s1 . . . sn−1 , s01 . . . s0n−1 ) ∪ Un ]) . Definition A.3. For each n ∈ N we introduce the region: Gn (Γ; X, b) := {(s1 . . . sn , s01 . . . s0n )|∀k ≤ n : Γk (s1 . . . sk , s01 . . . s0k ) ⊂ eR+ + Σe } . See also Fig. 1 for illustration. Lemma A.4. For a given tuple (X, b) ∈ im(d − 1)n × B(U1 ) × · · · × B(Un ) the function Ψn [X, b] is holomorphic in R2n + iGn (Γ; X, b). Proof. We prove the statement by induction. The vector ψ ∈ D(Γ, η) is contained in the domain of UX1 (is01 ) as long as s01 ∈ I(Γ, X1 ). For a fixed value s01 ∈ I(Γ, X1 ) the vector π(b1 )UX1 (is01 )ψ is contained in the domain of UX1 (is1 ) for
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D. SCHLINGEMANN
s1
1.
2.
s1
s1
s1
s1
3.
s1
Fig. 1. The figure above shows, according to the Malgrange–Zerner theorem, regions of analyticity, which are contained in G1 (Γ; X, b), for the following cases: (1) b1 is localized in Σe . (2) b1 is localized in Σe + e but is is not localized in Γ. (3) b1 is localized in Γ. If we translate b1 in positive e-direction, then we increase the region of analyticity and the maximal region is given by case 3.
Γ Γ( s1 )
U1
Fig. 2. The figure illustrates the fact that the vector π(b1 )UX1 (is01 )ψ is contained in the domain of UX1 (is1 ) for an sufficient small s1 . Here Γ(s01 ) is the cone rotated by s01 and U1 is the localizing region of b1 .
s1 ∈ J(Γ, X1 , b1 , s01 ). This implies that Ψ1 [X1 , b1 ] is holomorphic in G(Γ|X1 , b1 ) ⊃ R + iG1 (Γ; X, b) (see Fig. 2 for illustration). Suppose Ψn−1 [X1 . . . Xn−1 , b1 . . . bn−1 ] is holomorphic in R2(n−1) + iGn−1 (Γ; X, b). By the same argument as above we conclude that for a fixed values (s, s0 ) ∈ Gn−1 (Γ; X, b) the function (zn , zn0 ) 7−→ Ψn [X, b](is, zn , is0 , zn0 ) is holomorphic in G(Γn−1 (s, s0 )|Xn , bn ) and hence it is holomorphic in
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FROM EUCLIDEAN FIELD THEORY TO QUANTUM FIELD THEORY
[ (s,s0 )∈G
R2(n−1) + i{(s, s0 )} × G(Γn−1 (s, s0 )|Xn , bn )
n−1 (Γ;X,b)
which is a region containing Gn (Γ; X, b). B. Proof of Lemma 3.8 For a given Euclidean field (A, α, η) we introduce the following notions: Definition B.1. (1) We define the subspace ˆ D(Γ; η) := p(e,η) A(Γ) and D(Γ; η) := U(e,η) (L↑+ )D(Γ; η) . Definition B.1. (1) We define the subspace ˆ D(Γ; η) := p(e,η) A(Γ) and D(Γ; η) := U(e,η) (L↑+ )D(Γ; η) .
(2) Let X ∈ im(d − 1). For two regions Γ1 ⊂ Γ we define I(Γ1 , Γ; X) := {s ∈ R+ | exp(−isX)Γ1 ⊂ Γ} . (3) For a generator X ∈ im(d − 1) we define the region U(s, X) := exp(−isX)U for each s ∈ R. (4) Given two regions U1 , U2 in Rd , we define n Ge (X1 , X2 ; U1 , U2 ; I) := (z1 , z2 ) ∈ (R × iI)2 ∀xj ∈ Uj : e Im(exp(z1 X1 )x1 − exp(z2 X2 )x2 ) ∈ R+
o
n Gge (X1 , X2 ; U1 , U2 ; I) := (z1 , z2 ) ∈ (R × iI)2 ∀xj ∈ Uj : o e Im[g(exp(z1 X1 )x1 − exp(z2 X2 )x2 )] ∈ R+ , where g ∈ P+ (C) is a complex Poincar´e transformation. Lemma B.2. Let Γ1 , Γ ∈ Con(e) be two conic regions such that gΓ1 ⊂ Γ is a proper inclusion. Then there exists an interval I such that for each b1 ∈ B(U1 ), ˆ 1 ; η) the function b2 ∈ B(U2 ) and for each ψ1 , ψ2 ∈ D(Γ (ψ ,ψ )
F(X11 ,X22 ,b1 ,b2 ) : (z1 , z2 ) 7−→ hψ1 , ΦX1 (z1 , b1 )ΦX2 (z2 , b2 )ψ2 i is holomorphic in Ge (X1 , X2 ; U1 , U2 ; I).
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Proof. First we obtain by an application of Lemma A.4, that for each ψ1 ∈ H(e, η) and for each ψ ∈ D(Γ, η), the function (z, ζ) 7−→ hψ1 , ΦX2 (z, b2 )UX (ζ)ψ2 i is holomorphic for Im ζ ∈ I(Γ1 , Γ; X) and Im z ∈ I(Γ; X2 ) for X ∈ im(d − 1). The holomorphy is due to the fact that U is a strongly continuous representation of the Poincar´e group and that D(Γ; η) consists of analytic vectors for the boost generators. For a fixed values s0 ∈ I(Γ1 , Γ; X) and s ∈ I(Γ; X2 ), we have ˆ η) ΦX2 (is, b2 )U(e,η,X) (−is0 )ψ2 ∈ D(Γ; ˆ ⊂ eR+ + Σe which contains Γ ∪ U2 (s, X2 ). for each region Γ Now, as illustrated by Fig. 3, for a given point (z, is) ∈ Ge (X1 , X2 ; U1 , U2 ; I) there exists a conic region Γ(z, is) ∈ Con(e) with Γ(z, is) ⊃ Γ ∪ U2 (s, X2 ) such that D(Γ(z, is); η) is contained in the domain of ΦX1 (z, b2 ). Furthermore, for a given interval I, the cone Γ can be chosen to be small enough such that this holds for each (z, is) with Im z, s ∈ I. Since Γ1 is O(d − 1)-invariant, the result follows.
^ U2
^ Γ
e-component of the difference ^ U 1
ˆ such that region Uˆ1 ∩ Γ ˆ = ∅ and Uˆ2 ⊂ Γ. ˆ Fig. 3. There exists a cone Γ
Let V ⊃ L↑+ be a neighborhood of the identity in P+ (C). We may choose a cone C(Γ, V) ∈ Con(e) such that gC(Γ, V) ⊂ Γ for each g ∈ E(d) ∩ V. Note that the representation U can be extended to V by ˆ 1 , η) where Γ1 ⊂ C(Γ, V). unbounded operators with domain D(Γ In order to finish the proof of Lemma 3.8, we show the following statement: Lemma B.3. Let U1 , U2 be two bounded disjoint regions and let Γ1 ∈ Con(e) (ψ ,ψ ) such that Γ1 ⊂ C(Γ, V) is a proper inclusion. Then the function F(X11 ,X22 ,b1 ,b2 ) has (ψ ,ψ ) an extension Fˆ 1 2 which is holomorphic in (X1 ,X2 ,b1 ,b2 )
G(V; X1 , X2 ; U1 , U2 ; I) :=
[ g∈V
ˆ 1 ; η). for each ψ1 , ψ2 ∈ D(Γ
Gge (X1 , X2 ; U1 , U2 ; I)
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Proof. For a given neighborhood V ⊃ L↑+ of the unit element in P+ (C) an for a given cone Γ ∈ Con(e), there exists > 0 such that gU2 + e ⊂ Γ. We easily observe that the substitution ψj0 := T ()U (g)ψj Xj0 := exp(−iH)gXj g −1 exp(iH) yields
(ψ 0 ,ψ 0 )
(ψ ,ψ )
F(X10 ,X20 ,b1 ,b2 ) (z1 , z2 ) = F(X11 ,X22 ,b1 ,b2 ) (z1 , z2 ) 1
2
for each (z1 , z2 ) ∈ Ge (X1 , X2 ; U1 , U2 ; I) where H is the generator of translations in (ψ 0 ,ψ 0 ) e-direction. According to Lemma B.2, the function F(X10 ,X20 ,b1 ,b2 ) is holomorphic in 1 2 Gge (X1 , X2 ; U1 , U2 ; I) which implies the result. Acknowledgements I am grateful to Prof. Jakob Yngvason for supporting this investigation with many ideas. I am also grateful to Prof. Erhard Seiler and Prof. Jacques Bros for many hints and discussions during the workshop at the Erwin Schr¨odinger International Institute for Mathematical Physics in Vienna (ESI) this autumn. This investigation is financially supported by the Deutsche Forschungsgemeinschaft (DFG) which is also gratefully acknowledged. References [1] A. Ashtekar and J. Lewandowski, “Differential geometry on the space of connections via graphs and projective limits”, J. Geom. Phys. 17 (1995) 191–230. [2] D. Brydges, “A short course on cluster expansions”, in Les Houches 1984, Proc., Critical Phenomena, Random Systems, Gauge Theories, pp. 129–183. [3] W. Driessler and J. Fr¨ ohlich, “The reconstruction of local algebras from the Euclidean Green’s functions of relativistic quantum field theory”, Ann. Inst. Henri Poincar´e 27 (1997) 221–236. [4] W. Driessler and S. J. Summers, “Nonexistence of quantum fields associated with two-dimensional spacelike planes”, Commun. Math. Phys. 89 (1983) 221–226. [5] J. Feldman and K. Osterwalder, “The Wightman axioms and the mass gap for weakly coupled φ43 quantum field theories”, in Mathematical Problems in Theoretical Physics, ed. H. Araki, Berlin, Heidelberg, New York, Springer-Verlag. [6] J. Fr¨ ohlich, “Some results and comments on quantized gauge fields”, Cargese, Proc., Recent Developments in Gauge Theories (1979) 53–82. [7] J. Fr¨ ohlich, “On the triviality of λφ4 in d-dimensions theories and the approach to the critical point in d > 4-dimensions”, Nucl. Phys. B200 (1982) 281–296. [8] J. Fr¨ ohlich, K. Osterwalder and E. Seiler, “On virtual representations of symmetric spaces and their analytic continuation”, Ann. Math. 118 (1983) 461–489. [9] G. Gawedzki and A. Kupiainen, Asymptotic Freedom Beyond Perturbation Theory, Les Houches lectures, 1984. [10] J. Glimm and A. Jaffe, Collected Papers, Vol. 1 and Vol. 2: Quantum Field Theory and Statistical Mechanics, Expositions, Boston, USA, Birkh¨ auser, 1985. [11] J. Glimm and A. Jaffe, “A λφ4 quantum field theory without cutoffs I”, Phys. Rev. 176 (1968) 1945–1951; J. Glimm and A. Jaffe, “A λφ4 quantum field theory without
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[16] [17] [18] [19]
[20] [21] [22] [23]
[24] [25]
D. SCHLINGEMANN
cutoffs II”, Ann. Math. 91 (1970) 362–401; J. Glimm and A. Jaffe, “A λφ4 quantum field theory without cutoffs III”, Acta Math. 125 (1970) 204–267; J. Glimm and A. Jaffe, “A λφ4 quantum field theory without cutoffs IV”, J. Math. Phys. 13 (1972) 1568–1584. J. Glimm and A. Jaffe, “The Yukawa-2 quantum field theory without cutoffs”, J. Funct. Anal. 7 (1971) 323–357. J. Glimm and A. Jaffe, “Positivity of the φ43 Hamiltonian”, Fortschritte der Physik 21 (1973) 327–376. J. Glimm and A. Jaffe, Quantum Physics, a Functional Integral Point of View, Springer, New York, Berlin, Heidelberg, 1987. D. Hall and A. S. Wightman, “A theorem on invariant analytic functions with applications to relativistic quantum field theory”, Mat. Fys. Medd. Dan. Vid. Selsk. 31(5) (1957). R. Jost, The General Theory of Quantized Fields, American Math. Soc., Providence, Rhode Island, 1965. J. Magnen, V. Rivasseau and R. S´en´eor, “Construction of YM-4 with an infrared cutoff”, Commun. Math. Phys. 155 (1993) 325–384. J. Magnen and R. S´en´eor, “The infinite volume limit of the φ43 model”, Inst. H. Poincar´e 24 (1976) 95–159. K. Osterwalder and R. Schrader, “Axioms for Euclidean Green’s functions I”, Commun. Math. Phys. 31 (1973) 83–112; K. Osterwalder and R. Schrader, “Axioms for Euclidean Green’s functions II”, Commun. Math. Phys. 42 (1975) 281–305. A. M. Polyakov, “Gauge fields as rings of glue”, Nucl. Phys. B164 (1979) 171–188. R. Schrader, “A remark on Yukawa plus boson self-interaction in two space-time dimensions”, Commun. Math. Phys. 21 (1971) 164–170. R. Schrader, “A Yukawa quantum field theory in two space-time dimensions without cutoffs”, Ann. Phys. 70 (1972) 412–457. E. Seiler, “Gauge theories as a problem of constructive quantum field theory and statistical mechanics”, Lecture Notes in Physics 159, Berlin, Germany, Springer (1982) 192. E. Seiler and B. Simon, “Nelson’s symmetry and all that in the Yukawa2 and φ43 field theories”, Ann. Phys. 97 (1976) 470–518. R. F. Streater and A. S. Wightman, PCT, Spin and Statistics and All That, Redwood City, USA, Addison-Wesley (1989) 207 p. (Advanced book classics.)
CONTRACTIONS OF LIE ALGEBRA REPRESENTATIONS U. CATTANEO∗ CERFIM, Via F. Rusca 1 CH-6601 Locarno, Switzerland E-mail : [email protected]
W. F. WRESZINSKI† Instituto de F´ısica, Universidade de S˜ ao Paulo C. P. 66318, 05389-970 S˜ ao Paulo, SP, Brazil E-mail : [email protected] Received 16 May 1998 A theory of contractions of Lie algebra representations on complex Hilbert spaces is proposed, based on Trotter’s theory of approximating sequences of Banach spaces. Its main distinguishing feature is a careful definition of the carrier space of the limit Lie algebra representation. A set of quite general conditions on the contracting representations, satisfied in all known examples, is proven to be sufficient for the existence of such a representation. In order to show how natural the suggested framework is, the general theory is applied to the contraction of so(2) into the Lie algebra h(1) of the 3-dimensional Heisenberg group and to the related study of the limit N → ∞ of a quantum system of N identical two-level particles.
1. Introduction Contractions of Lie algebra representations have been investigated in mathematical physics since the first appearance of the concept of a Lie algebra contraction [1, 2] and continue to be a subject of active interest, particularly in connection with quantum groups (cf., for instance, [3–5]). In spite of this, the question of a suitable definition of a “limit carrier space” and of an appropriate contraction procedure for Lie algebra representations is still open. The problem was mostly tackled in the analysis of particular cases: the contracted representations were studied in terms of limits of matrix elements [6–8] and it is well known that it is insufficient to treat questions relating to symmetric or skew-symmetric operators by means of matrices, due to the pathological properties of infinite matrices obtained from unbounded operators [9]. On the other hand, in a previous attempt to define a general framework for contractions of Lie algebra representations [10] there was a flaw (cf. [6]) whose correction seems to require unwarranted restrictions. In this paper, we put forward a new theory of contractions of Lie algebra representations whose peculiarity is the construction of a limit carrier space, inspired by Trotter’s theory of approximating sequences of Banach spaces [11], and which ∗ Supported † Supported
in part by FNSRS Grant 20-32740.91. in part by CNPq and by CAPES. 1179
Reviews in Mathematical Physics, Vol. 11, No. 10 (1999) 1179–1207 c World Scientific Publishing Company
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is centered on the connected idea of limits of nets of operators in Hilbert spaces. It provides a natural framework for the applications considered so far (cf. [6, 7] and the references given therein) and is suitable to cover the recent applications to quantum groups. In Sec. 2, after a brief recollection of some basic facts concerning contractions of nets of Lie algebras and approximating nets of Hilbert spaces, we present the main results. In particular we show that, if customary practical conditions on the representations are satisfied, then, given a net (gι ) of real or complex Lie algebras contracting into ˆ g and, for each ι, a skew-symmetric representation πι of gι carried by a (complex) Hilbert space Hι , the net (πι ) generates a skew-symmetric representation π ˆ of ˆ g on a Hilbert space H approximated in Trotter’s sense by the net (Hι ). Moreover, if all gι are real and finite-dimensional, and if the matrix elements of the operators π ˆ (g) are bounded in a suitable way for all elements g of a basis of ˆ g, then π ˆ gives rise to a (strongly continuous) unitary representation on H of any ˆ whose Lie algebra is isomorphic to ˆg. simply connected Lie group G In Sec. 3, the applicability of our theory is tested on the well-known example of the contraction of so(2) (the Lie algebra of the group SU(2)) into the Lie algebra h(1) of the 3-dimensional Heisenberg group H(1). This contraction is relevant, for instance, in the study of the limit N → ∞ of a (quantum) system of N identical two-level particles, and this model shows how natural is, from the standpoint of physics, the proposed contraction procedure for Lie algebra representations. The theory also applies to the other examples which have been treated in [12]. In all examples of contractions of Lie algebra representations considered so far, it was sufficient to use sequences of representations. However, even in these cases, the “natural” contraction index was not often an element of N but of a different subset of R and it is better to utilize directly this parameter (cf., for instance, Sec. 3 and [12]). Therefore, we use nets of Hilbert spaces, of Lie algebras, of Lie algebra representations, and so on. If the contrary is not explicitly stated, we will use the symbol to denote the ordering of any directed set and, usually, we will omit specific mention of this ordering and speak of an index set I instead of (I, ). Throughout the paper, every Hilbert space will be tacitly understood to be a complex one, with inner multiplication linear in the second component, and we will denote: • by K either the field of real numbers R or the field of complex numbers C. Unless otherwise specified, every Lie algebra is over K; • by alg(V, µ) the (finite- or infinite-dimensional) Lie algebra with underlying vector space V and Lie multiplication µ. We assume that V is a Hausdorff topological vector space; • by h·|·iH (resp. k · kH ) the inner multiplication (resp. the norm) of a Hilbert space H; • by h·|·iι (resp. k · kι ) the inner multiplication (resp. the norm) of a Hilbert Q space Hι , where ι is an element of a directed set I, and by ι Hι the product vector space of the Hι with ι ∈ I; • by k · k any operator norm;
CONTRACTIONS OF LIE ALGEBRA REPRESENTATIONS
1181
• by sp(S) the linear span of an orthonormal basis S of a Hilbert space; • by D(A) the domain of a (bounded or unbounded linear) operator A in a Hilbert space H, and say that A is defined in D whenever D is a vector subspace of D(A). If A is an unbounded symmetric (resp. skew-symmetric) operator, then D(A) is dense in H. 2. Contractions of Nets of Lie Algebras 2.1. Limits of nets of Lie algebra structures The concept of a Lie algebra contraction was introduced in the seminal papers of Segal [1], and In¨ on¨ u and Wigner [2]. Since then, the notion has been precised in its general meaning and applied to particular cases by Saletan [13], L´evy–Nahas [14], and others [15, 16]. These authors have also determined necessary and sufficient conditions for the existence of contractions under various assumptions. We adopt the following general definition. Definition 1. A Lie algebra alg(V, µ ˆ) is called the contraction of a net (alg(V, µι )) of Lie algebras, indexed by a directed set I, if µ ˆ (g, g 0 ) = lim µι (g, g 0 ) ι
(2.1)
for all g, g 0 in V. If in addition, for each ι ∈ I, there exists an automorphism Γι of V such that 0 0 ˆ) is called the µι (g, g 0 ) = Γ−1 ι (µ(Γι (g), Γι (g ))) for all g, g in V, then alg(V, µ contraction of alg(V, µ) by means of the net (Γι ). We also say that we have a contraction of the net of Lie algebras (alg(V, µι )) (resp. of the Lie algebra alg(V, µ)) into the Lie algebra alg(V, µ ˆ). In the case of the contraction of the Lie algebra alg(V, µ) by means of the net (Γι ), every Γι is a Lie algebra isomorphism of alg(V, µι ) onto alg(V, µ). Notice that Definition 1 includes the trivial case of alg(V, µ ˆ ) isomorphic to alg(V, µι ) for all ι ∈ I. We have adopted here an obvious generalization of the usual definition of a Lie algebra contraction. The latter is the particular case of the contraction of a Lie algebra by means of a sequence of automorphisms of the underlying vector space. In order to construct a contraction of a net of Lie algebras with a common underlying vector space V, it is sufficient to give a net (µι ) of Lie algebra multiplications on V such that, for each pair gj , gk of elements of a basis (gj )j∈J of V, the net (µι (gj , gk ))ι∈I converges. On account of the linearity of the limits and of the ˆ of the contraction is then univocally bilinearity of the µι , the Lie multiplication µ defined by µ ˆ(gj , gk ) = limι µι (gj , gk ) for all j, k in J. ˆ) Remark 1. Let (ˆ γjkl )(j,k,l)∈J3 be the family of structure constants of alg(V, µ with respect to a basis (gj )j∈J of V. If, for each ι ∈ I, we denote by (γ(ι)jkl )(j,k,l)∈J3 the family of structure constants of alg(V, µι ) with respect to the basis (gj )j∈J , so that X γ(ι)jkl gl µι (gj , gk ) = l∈J
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U. CATTANEO and W. F. WRESZINSKI
for all j, k in J, then (2.1) is satisfied if and only if, for each triplet j, k, l of indices in J, we have limι γ(ι)jkl = γˆjkl . In fact, X γˆjkl gl µ ˆ (gj , gk ) = l∈J
and lim µι (gj , gk ) = lim ι
ι
X l∈J
γ(ι)jkl gl =
X lim γ(ι)jkl gl l∈J
ι
for all j, k in J. Notice that the sums in the previous formulas are actually finite, since almost all structure constants (i.e., all but a finite number) are 0. 2.2. Contractions of nets of Lie algebra representations We first define the limit carrier Hilbert spaces of our contractions of Lie algebra representations, using notions due to Trotter [11, Sec. 2]. The results of this subsection will show that this definition is natural and fruitful. Definition 2. Let H, Hι be Hilbert spaces and Pι a linear mapping of H into Hι , where ι is any element of a directed set I. The family (Hι ) is called a net of Hilbert spaces approximating H with respect to the net (Pι ), and we write (Pι )-lim Hι = H, if, for each ι ∈ I, (2.2) kPι k 6 1 and, for each φ ∈ H, lim kPι φkι = kφkH . ι
(2.3)
Condition (2.2) implies the continuity of all mappings Pι and condition (2.3), which refers to Banach spaces, means essentially that the Pι “become isomorphisms in the limit”. This last condition is suitable for Hilbert spaces too since, by reason of the polarization identity, we have limι hPι φ|Pι ψiι = hφ|ψiH for all φ, ψ in H. Remark 2. Let H, H0 be isomorphic Hilbert spaces and U a unitary mapping of H onto H0 . Then H = (Pι )-lim Hι if and only if H0 = (Pι0 )-lim Hι , where Pι0 = Pι U −1 . In fact, if H = (Pι )-lim Hι , we have kPι0 k = sup
φ0 ∈H0
kPι0 φ0 kι kPι U −1 φ0 kι kPι φkι = sup = sup = kPι k 6 1 0 −1 φk 0 0 kφ kH0 kU H φ ∈H φ∈H kφkH
and, for each φ0 ∈ H0 , lim kPι0 φ0 kι = lim kPι U −1 φ0 kι = kU −1 φ0 kH = kφ0 kH0 , ι
0
whence H =
(Pι0 )-lim Hι .
ι
The converse follows in the same way.
Q We say briefly that a family (Aι ) is a net of operators in ι Hι if Aι is an operator in a Hilbert space Hι for every element ι of a directed set I. Definition 3. Let (Hι ) be a net of Hilbert spaces indexed by a directed set I and Q approximating a Hilbert space H with respect to a net (Pι ). A net (φι ) ∈ ι Hι is
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(Pι )-convergent to φ ∈ H, and we write (Pι )-lim φι = φ, if limι kφι − Pι φkι = 0. A Q net (Aι ) of operators in ι Hι is (Pι )-convergent to an operator A in H, and we write (Pι )-lim Aι = A, if Pι D(A) ⊆ D(Aι ) for all ι ∈ I and (Pι )-lim Aι Pι φ = Aφ for all φ ∈ D(A). Q Notice that, for each net (Aι ) of operators in ι Hι , the relation Aφ = (Pι )-lim Aι Pι φ defines an operator A = (Pι )-lim Aι in H whose domain D(A) is the vector subspace of all φ ∈ H which satisfy this equality, i.e., the set of all φ ∈ H such that Pι φ ∈ D(Aι ) for all ι and the net (Aι Pι φ) is (Pι )-convergent. If (aι ) is a net of complex numbers converging to a, then aIdH = (Pι )-lim aι IdHι with D(aIdH ) = H, because lim kaι IdHι Pι φ − Pι aIdH φkι = lim k(aι − a)Pι φkι = lim |aι − a| kPι φkι = 0 . ι
ι
ι
In the following, we shall often be interested in vector subspaces of the domains Q Q of operators. We will say that a net (Aι ) of operators in ι Hι is defined in ι Dι when, for each ι, the operator Aι is defined in Dι . If R is a relation involving operators in a Hilbert space which are defined in some vector subspace D, we will say that we have R in D whenever R is satisfied when applied to all elements of D. In particular, we have A = (Pι )-lim Aι in every vector subspace of D(A). Q Remark 3. If (Pι )-lim Hι = H, a net (φι ) ∈ ι Hι cannot (Pι )-converge to more than one element of H for, if (Pι )-lim φι = φ and (Pι )-lim φι = ψ, then kψ − φkH = lim kPι (ψ − φ)kι = lim k(φι − Pι φ) − (φι − Pι ψ)kι ι
ι
6 lim kφι − Pι φkι + lim kφι − Pι ψkι = 0 ι
ι
Q and so ψ = φ. Analogously, a net (Aι ) of operators in ι Hι cannot (Pι )-converge to more than one operator in H since the equalities (Pι )-lim Aι = A and (Pι )-lim Aι = B require (Pι )-lim Aι Pι φ = Aφ = Bφ for every φ ∈ H for which the net (Aι Pι φ) is (Pι )-convergent. Q On the other hand, for each φ ∈ H, the net (Pι φ) ∈ ι Hι is (Pι )-convergent to φ and therefore, for each operator A defined in a vector subspace D of H, the net Q Q (Aι ) of operators in ι Hι defined in ι Pι D by Aι Pι φ = Pι Aφ is (Pι )-convergent to A: (Pι )-lim Aι Pι φ = (Pι )-lim Pι Aφ = Aφ . Remark 4. Let H = (Pι )-lim Hι and, for each ι, let Dι be a vector subspace of Hι and D a vector subspace of H such that Pι D ⊆ Dι . The set of all (Pι )-converging Q Q Q elements of ι Hι (resp. of all nets of operators in ι Hι defined in ι Dι which Q (Pι )-converge in D) is a vector subspace of ι Hι (resp. of the vector space of all Q Q nets of operators in ι Hι defined in ι Dι ) and the operation (Pι )-lim is a linear mapping of this subspace onto H (resp. onto the vector space of all operators in H defined in D). In fact, lim k(µφι + νψι ) − Pι (µφ + νψ)kι 6 |µ| lim kφι − Pι φkι + |ν| lim kψι − Pι ψkι = 0 ι
ι
ι
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U. CATTANEO and W. F. WRESZINSKI
for all µ, ν in C and all nets (φι ), (ψι ) in (Pι )-lim ψι = ψ, whence
Q ι
Hι such that (Pι )-lim φι = φ and
(Pι )-lim(µφι + νψι ) = µ(Pι )-lim φι + ν(Pι )-lim ψι ; the linear mapping (Pι )-lim is surjective by Remark 3. The result concerning nets of operators follows in the same way. Remark 5. If H = (Pι )-lim Hι and (Aι ) is a (Pι )-convergent net of symmetric Q (resp. skew-symmetric) operators in ι Hι , then A = (Pι )-lim Aι is a symmetric (resp. skew-symmetric) operator in H, provided D(A) is dense in H. In fact, if the operators Aι are symmetric, then hAφ|ψiH = limhPι Aφ|Pι ψiι = limhPι Aφ − Aι Pι φ|Pι ψiι + limhAι Pι φ|Pι ψiι ι
ι
ι
= limhPι φ|Aι Pι ψiι = limhPι φ|Aι Pι ψ − Pι Aψiι + limhPι φ|Pι Aψiι ι
ι
ι
= hφ|AψiH for all φ, ψ in D(A) (and analogously for skew-symmetric Aι ) since limhPι Aφ − Aι Pι φ|Pι ψiι = limhPι φ|Aι Pι ψ − Pι Aψiι = 0 ι
ι
as, for instance, limhPι Aφ − Aι Pι φ|Pι ψiι 6 lim kAι Pι φ − Pι Aφkι kPι ψkι = 0 . ι
ι
Remark 6. Let α be an order preserving bijective mapping of a directed set I onto a directed set Λ. If (Hι ), H, (Pι ), (φι ), φ, (Aι ) and A are as in Definition 3, then is a net of Hilbert spaces indexed by Λ and approximating (Kλ ) with Kλ = H−1 α (λ)
H with respect to the net
(Rλ ) = P−1
α (λ)
;
we have (Rλ )-lim φ0λ = (Pι )-lim φι = φ ,
where φ0λ = φ−1
(Rλ )-lim A0λ = (Pι )-lim Aι = A ,
where A0λ = A−1
α (λ)
,
α (λ)
.
In fact, if (xι ) is a net (in some topological space) indexed by I and converging to x, then lim x0λ = lim xι = x (ι ∈ I; λ ∈ Λ) λ
with x0λ = x−1 x0λ
α (λ) −1
ι
since, a neighborhood U of x being given, xι ∈ U for ι ι0 implies
∈ U for α (λ) ι0 , hence for λ α(ι0 ).
The following four lemmas establish general results concerning Hilbert spaces approximated by nets of Hilbert spaces and operators acting on them. Lemma 1. Let I be a directed set and, for each ι ∈ I, let Hι be a Hilbert space, Aι , Bι two operators in Hι , Dι an Aι -stable and Bι -stable vector subspace of D(Aι ) ∩ D(Bι ). Suppose that
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1185
(a) the net (Hι ) approximates a Hilbert space H with respect to a net (Pι ); (b) A = (Pι )-lim Aι , B = (Pι )-lim Bι are operators in H defined in an A-stable and B-stable vector subspace D of D(A) ∩ D(B) such that Pι D ⊆ Dι for all ι ∈ I; (c) limι kAι (Bι Pι φ − Pι Bφ)kι = lim kBι (Aι Pι φ − Pι Aφ)kι = 0 for all φ ∈ D. ι
In D, we have then AB = (Pι )-lim Aι Bι , BA = (Pι )-lim Bι Aι and therefore [A, B] = (Pι )-lim[Aι , Bι ], cA = (Pι )-lim cι Aι , where (cι ) is a net of complex numbers converging to c. Proof. For each φ ∈ D, and on account of the assumptions on D and Dι , the vectors Aι Bι Pι φ, Pι ABφ, Aι Pι Bφ are well-defined elements of Dι ; we have lim kAι Bι Pι φ − Pι ABφkι ι
= lim kAι (Bι Pι φ − Pι Bφ) + Aι Pι Bφ − Pι ABφkι ι
6 lim kAι (Bι Pι φ − Pι Bφ)kι + lim kAι Pι Bφ − Pι ABφkι = 0 ι
ι
and, analogously, limι kBι Aι Pι φ − Pι BAφkι = 0. It follows that AB = (Pι )- lim Aι Bι ,
BA = (Pι )- lim Bι Aι
and so [A, B] = (Pι )-lim[Aι , Bι ] in D [Remark 4]. Furthermore, as cIdH = (Pι )- lim cι IdHι in H and lim kcι (Aι Pι φ − Pι Aφ)kι = |c| lim kAι Pι φ − Pι Aφkι = 0 ι
ι
for all φ ∈ D, then cA = (Pι )-lim cι Aι always in D.
Lemma 2. Let I, Hι , Aι , H, A be as in Lemma 1, Dι an Aι -stable vector subspace of D(Aι ), and D an A-stable vector subspace of D(A) such that Pι D ⊆ Dι for all ι ∈ I. If, in addition, we have limι kAι (An−1 Pι φ − Pι An−1 φ)kι = 0 for all ι ∗ n n φ ∈ D and some n ∈ N , then A = (Pι )-lim Aι in D. Proof. For each φ ∈ D, the vectors Anι Pι φ, Pι An φ, Aι Pι An−1 φ are well-defined elements of Dι and we have Pι φ − Pι An−1 φ)kι lim kAnι Pι φ − Pι An φkι 6 lim kAι (An−1 ι ι
ι
+ lim kAι Pι An−1 φ − Pι AAn−1 φkι ι
= 0. Lemma 3. Let I be a directed set and suppose that, for each ι ∈ I, we have
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(a) a set Sι such that Sι ⊆ Sι0 whenever ι ι0 ; (b) a Hilbert space Hι of dimension Card(Sι ) with a given orthonormal basis (s) Sι = (ϕι )s∈Sι . S If H is any Hilbert space of dimension Card(S), where S = ι Sι , then the net (Hι ) approximates H with respect to the net (Pι ), where Pι is the continuous linear (s) mapping of H onto Hι defined by Pι φ(s) = φι (ι ∈ I; s ∈ S) in an orthonormal basis S = (φ(s) )s∈S of H, with ( (s) if s ∈ Sι ϕι (s) (2.4) φι = 0 if s ∈ S Sι . P (s) be an arbitrary element of H (so that zs = Proof. Let φ = s∈S zs φ (s) hφ |φiH ). The mapping Pι is well-defined for its linearity implies X X X zs Pι φ(s) = zs φ(s) zs ϕ(s) Pι φ = ι = ι , s∈S
s∈S
P
s∈Sι
P
2 |zs |2 6 hence Pι φ ∈ Hι because s∈S |zs | < +∞. Besides, Pι is surP s∈Sι (s) is any element of Hι , then φι = Pι φ˜ with jective since, if φι = s∈Sι zs ϕι P φ˜ = s∈Sι zs φ(s) ∈ H. By virtue of (2.4), we also have
X
2 X
2 (s) z ϕ
|zs | 2 s ι kP φk s∈S ι s∈Sι ι ι 2 ι sup X 6 1, kPι k = sup
2 = sup X 2 = 2
φ∈H kφkH kzk<+∞ kzk<+∞ |zs | zs φ(s)
s∈S
with kzk = 2
s∈S
P
|zs | , and
H
2
s∈S
lim kPι φk2ι = lim ι
ι
X
2
|zs | =
s∈Sι
X
2
|zs |2 = kφkH
s∈S
P for the net ( s∈Sι |zs |2 )ι∈I of positive real numbers is monotone increasing with P supremum s∈S |zs |2 by condition (a). It follows that H = (Pι )-lim Hι . In Lemma 3, we may choose H = `2C (S), i.e., pick the Hilbert space of all complex-valued functions f defined in S and satisfying X |f (s)|2 < +∞ , s∈S
with inner multiplication given by hf |hiH =
X
f (s)h(s) .
s∈S
In this case, a suitable orthonormal basis of H is the canonical basis S = (f (s) )s∈S of `2C (S), with f (s) (s0 ) = δss0 . Lemma 4. Let I, Sι , S, Hι , Sι , H, S, Pι be as in Lemma 3 and, for each ι ∈ I, let Aι , Bι be two operators in Hι such that
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(a) sp(Sι ) is an Aι -stable and Bι -stable vector subspace of D(Aι ) ∩ D(Bι ); (b) for each s ∈ Sι , we have X 0 aι,s,s0 φι(s ) (2.5) Aι ϕ(s) ι = s0 ∈F(A,φ(s) )
and Bι ϕ(s) ι =
X
0
bι,s,s0 φι(s ) ,
(2.6)
s0 ∈F(B,φ(s) )
where F(A, φ(s) ), F(B, φ(s) ) are finite subsets of S and aι,s,s0 , bι,s,s0 are complex numbers. / Sι , conIf the nets (aι,s,s0 )ι∈I , (bι,s,s0 )ι∈I , with aι,s,s0 = bι,s,s0 = 0 whenever s ∈ 0 (s) verge for every s ∈ S, respectively, to as,s0 when s ∈ F(A, φ ) and to bs,s0 when s0 ∈ F(B, φ(s) ), then the operators A = (Pι )-lim Aι , B = (Pι )-lim Bι in H exist and are defined by X X 0 0 as,s0 φ(s ) , Bφ(s) = bs,s0 φ(s ) , (2.7) Aφ(s) = s0 ∈F(A,φ(s) )
s0 ∈F(B,φ(s) )
and linearity in sp(S), so that sp(S) is an A-stable and B-stable vector subspace of D(A) ∩ D(B). Moreover, in this vector subspace we also have AB = (Pι )-lim Aι Bι , BA = (Pι )-lim Bι Aι , [A, B] = (Pι )-lim[Aι , Bι ], and cA = (Pι )-lim cι Aι , where (cι ) is a net of complex numbers converging to c. Proof. If φ ∈ sp(S), then there exists a finite subset F(φ) of S such that φ = P (s) (s) , hence Pι φ = s∈F(Pι φ) zs ϕι ∈ sp(Sι ) with F(Pι φ) = F(φ) ∩ Sι . s∈F(φ) zs φ It follows that, for each ι ∈ I, we have X X X 0 zs Aι ϕ(s) zs aι,s,s0 φι(s ) Aι Pι φ = ι =
P
s∈F(Pι φ)
=
X s∈F(A,φ)
s∈F(Pι φ)
!
X s0 ∈F
s0 ∈F(A,φ(s) )
X
zs0 aι,s0 ,s φ(s) ι =
s (φ)
a ˜ι,s φ(s) ι ,
s∈F(A,φ)
P ˜ι,s = s0 ∈Fs (φ) zs0 aι,s0 ,s , where F(A, φ) = s0 ∈F(φ) F(A, φ(s )) is a finite subset of S, a and, for each s ∈ F(A, φ), we denote by Fs (φ) the set of all s0 ∈ F(φ) such that P 0 aι,s )ι∈I converges to a ˜s = s0 ∈F(φ) zs0 as0 ,s for s ∈ F(A, φ(s ) ). Moreover, the net (˜ all s ∈ F(A, φ) and we get an operator A = (Pι )-lim Aι defined in sp(S) by X a ˜s φ(s) , Aφ = S
0
s∈F(A,φ)
i.e., by (2.7) and linearity, since
X
lim kAι Pι φ − Pι Aφkι = lim ι ι
s∈F(A,φ)
= lim ι
a ˜ι,s φ(s) ι
− Pι
X
s∈F(A,φ)
a ˜s φ
s∈F(A,φ)
X (s) (˜ aι,s − a ˜s )φι 6
ι
(s) ι
X
lim |˜ aι,s − a ˜s | = 0 ι
s∈F(A,φ)
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U. CATTANEO and W. F. WRESZINSKI
for all φ ∈ sp(S). In the same way, we define B = (Pι )-lim Bι in sp(S) by X
Bι Pι φ =
˜bι,s φ(s) ι
X
and Bφ =
s∈F(B,φ)
˜bs φ(s) .
s∈F(B,φ)
From Lemma 1, with Dι = sp(Sι ) and D = sp(S), we have AB = (Pι )-lim Aι Bι , BA = (Pι )-lim Bι Aι , [A, B] = (Pι )-lim[Aι , Bι ], cA = (Pι )-lim cι Aι in sp(S) since
X
(s) (˜bι,s − ˜bs )φι lim kAι (Bι Pι φ − Pι Bφ)kι = lim Aι ι ι
s∈F(B,φ)
= lim ι 6
X
ι
X
(˜bι,s − ˜bs )
s0 ∈F(A,φ(s) )
s∈F(B,φ)
X
X
s∈F(B,φ)
s0 ∈F(A,φ(s) )
aι,s,s0 φι(s )
0
ι
lim |˜bι,s − ˜bs | |aι,s,s0 | = 0 ι
and, analogously, limι kBι (Aι Pι φ − Pι Aφ)kι = 0 for all φ ∈ D. Notice that we can also prove the (Pι )-convergences, without to call on Lemma 1, by showing directly that limι kAι Bι Pι φ − Pι ABφkι = 0. In order to clarify the meaning of conditions (2.5) and (2.6), let us note that • for each ι ∈ I, the restriction Pι |sp(S) is a surjective mapping onto sp(Sι ): we have already remarked that Pι φ ∈ sp(Sι ) in the proof of Lemma 4; if P (s) φι ∈ sp(Sι ), then φι = s∈F(φι ) zs ϕι , where F(φι ) is a finite subset of Sι , P whence φι = Pι φ˜ with φ˜ = s∈F(φι ) zs φ(s) ∈ sp(S); • the right-hand sides of (2.5) and (2.6), without the condition on the convergence of the nets (aι,s,s0 )ι∈I and (bι,s,s0 )ι∈I , are required by the Aι -stability and Bι -stability of sp(Sι ): we have X
Aι ϕ(s) ι =
0
aι,s,s0 ϕι(s )
X
and Bι ϕ(s) ι =
s0 ∈F(A,φ(s) )∩Sι
0
bι,s,s0 ϕι(s ) .
s∈F(B,φ(s) )∩Sι
Before proving the main results of this paper, we recollect some basic notions concerning Lie algebra representations. Definition 4. A representation of a Lie algebra g on a Hilbert space H is an ordered pair (π, D(π)), where π is a mapping of g into the set of all operators in H T and the domain D(π) of π is a π(g)-stable vector subspace of g∈g D(π(g)) dense in H, such that, for all λ, λ0 in K, all g, g 0 in g, and all φ ∈ D(π), we have π(λg + λ0 g 0 )φ = λπ(g)φ + λ0 π(g 0 )φ (i.e., π is linear) and [π(g), π(g 0 )]φ = π(µ(g, g 0 ))φ , where µ is the Lie multiplication of g.
CONTRACTIONS OF LIE ALGEBRA REPRESENTATIONS
1189
If D0 is a π(g)-stable vector subspace of D(π), the representation (π 0 , D0 ) of g on the closure D0 of D0 such that π 0 (g)|D0 = π(g)|D0 for all g ∈ g is called the subrepresentation of (π, D(π)) with domain D0 . The representation (π, D(π)) is irreducible if it does not admit subrepresentations whose domains are vector subspaces D0 6= {0} of D(π) such that D0 6= H. The representation (π, D(π)) is symmetric (resp. skew-symmetric) if π(g) is symmetric (resp. skew-symmetric) in D(π) for all g ∈ g. It is quite usual to speak simply of the representation π, understanding the domain D(π). We will follow this convention only when H is finite-dimensional and so D(π) = H. Definition 5. Two representations (π, D(π)) and (π 0 , D(π 0 )) of a Lie algebra g on Hilbert spaces H and H0 , respectively, are (unitarily) equivalent if there exists a unitary mapping U of H onto H0 such that U D(π) = D(π 0 ) and π 0 (g)U φ = U π(g)φ for all g ∈ g and all φ ∈ D(π). ˆ = alg(V, µ Definition 6. Let g ˆ) be the contraction of a net (gι ) = (alg(V, µι )) of Lie algebras, indexed by a directed set I, and, for each ι ∈ I, let (πι , D(πι )) be π , D(ˆ π )) of gˆ on a a representation of gι on a Hilbert space Hι . A representation (ˆ Hilbert space H is called the contraction of the net ((πι , D(πι ))) of representations ˆ (g) = (Pι )-lim πι (g) in D(ˆ π) by means of the net (Pι ) if H = (Pι )-lim Hι and π for all g ∈ V, where the continuous linear mapping Pι of H into Hι is such that π ) ⊆ D(πι ). Pι D(ˆ π 0 )) of the net ((πι , D(πι ))) by means of the Two contractions (ˆ π , D(ˆ π )), (ˆ π 0 , D(ˆ 0 nets (Pι ), (Pι ), and with carrier Hilbert spaces H, H0 , respectively, are equivalent π 0 ) = U D(ˆ π ) and Pι0 = Pι U −1 , where U is a if H and H0 are isomorphic with D(ˆ 0 unitary mapping of H onto H . π , D(ˆ π )) and we We also say that the net ((πι , D(πι ))) is (Pι )-convergent to (ˆ write (ˆ π , D(ˆ π )) = (Pι )-lim(πι , D(πι )). π 0 )), given as in Definition 6, are equivIf two contractions (ˆ π , D(ˆ π )) and (ˆ π 0 , D(ˆ 0 alent via a unitary mapping U of H onto H [cf. Remark 2], then the representations π 0 )) are equivalent. In fact, for each g ∈ V and each φ ∈ D(ˆ π ), (ˆ π , D(ˆ π )) and (ˆ π 0 , D(ˆ ˆ (g)φ because we have π ˆ 0 (g)U φ = U π ˆ (g)φk = lim kPι0 (ˆ π 0 (g)U φ − U π ˆ (g)φ)kι kˆ π 0 (g)U φ − U π ι
6 lim kπι (g)Pι0 U φ − Pι0 π ˆ 0 (g)U φkι ι
+ lim kπι (g)Pι φ − Pι π ˆ (g)φkι = 0 . ι
Proposition 1. Let ˆ g = alg(V, µ ˆ ) be the contraction of a net (gι ) = (alg(V, µι )) of Lie algebras, indexed by a directed set I, and let (gj )j∈J be a basis of V. Suppose that we have a Hilbert space Hι and a representation (πι , D(πι )) of gι on Hι for every ι ∈ I, and that the net (Hι ) approximates a Hilbert space H with respect to a ˆ be a mapping of ˆg into the set of all operators in H which net (Pι ). Let π
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U. CATTANEO and W. F. WRESZINSKI
(a) is defined by π ˆ (gj ) = (Pι )-lim πι (gj ) (j ∈ J) and linearity in a π ˆ (ˆg)-stable ˆ ⊆ D(πι ) ˆ of T D D(ˆ π (g)) dense in H and satisfying P vector subspace D ι g∈ˆ g for all ι ∈ I; ˆ and (b) is such that π ˆ (gj )φl ∈ D ˆ (gk )φl )kι = 0 lim kπι (gj )(πι (gk )Pι φl − Pι π ι
(2.8)
ˆ for all j, k in J and all elements of an algebraic basis (φl )l∈L of D. ˆ is a representation of ˆg on H which is the contraction of the net Then (ˆ π , D) ((πι , D(πι ))) by means of the net (Pι ). Moreover, if the representation (πι , D(πι )) ˆ is is symmetric (resp. skew-symmetric) for all ι ∈ I, then the representation (ˆ π , D) symmetric (resp. skew-symmetric). ˆ by the linearity of Proof. For each g ∈ V, we have π ˆ (g) = (Pι )-lim πι (g) in D P ˆ and of the operation (Pι )-lim [Remark 4]. Moreover, with g = j∈J λj gj , πι , of π P P g 0 = j∈J λ0j gj , and φ = l∈L zl φl , where the elements λj , λ0j of K and the zl ∈ C are almost all 0, we obtain ! XX X X ˆ λj gj zl φl = zl λj π ˆ (gj )φl ∈ D π ˆ (g)φ = π ˆ j∈J
l∈L
l∈L j∈J
and, by (2.8), ˆ (g 0 )φ)kι lim kπι (g)(πι (g 0 )Pι φ − Pι π ι
=
X X
|zl | |λ0k | |λj | lim kπι (gj )(πι (gk )Pι φl − Pι π ˆ (gk )φl )kι = 0 ι
l∈L j,k∈J
ˆ It follows that [ˆ π (g), π ˆ (g 0 )] = (Pι )-lim[πι (g), πι (g 0 )] for all g, g 0 in V and all φ ∈ D. 0 ˆ is a π ˆ for all g, g in V [Lemma 1]. Since D ˆ (ˆg)-stable vector subspace of in D T ˆ is a representation of ˆg D(ˆ π (g)) which is dense in H, the ordered pair (ˆ π , D) g∈ˆ g on H and is the contraction of the net ((πι , D(πι ))) by means of the net (Pι ). In γjkl )(j,k,l)∈J3 are, respectively, the families of structure fact, if (γ(ι)jkl )(j,k,l)∈J3 and (ˆ g with respect to the basis (gj )j∈J , then, for each pair j, k of constants of gι and ˆ indices in J, we have X ˆ (gk )] = (Pι )-lim[πι (gj ), πι (gk )] = (Pι )-lim γ(ι)jkl πι (gl ) [ˆ π (gj ), π l∈J
=
X
(Pι )-lim γ(ι)jkl πι (gl ) =
l∈J
=π ˆ
X
γˆjkl π ˆ (gl )
l∈J
X
! γˆjkl gl
=π ˆ (ˆ µ(gj , gk ))
l∈J
ˆ for all g, g 0 in V by virtue of ˆ [Remark 1], hence [ˆ ˆ (ˆ µ(g, g 0 )) in D in D π (g), π ˆ (g 0 )] = π the linearity of π ˆ and the bilinearity of [·, ·] and µ.
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If, for each ι ∈ I, the representation (πι , D(πι )) of gι is symmetric (resp. skewsymmetric), then the operators π ˆ (g)(g ∈ V) are symmetric (resp. skew-symmetric) ˆ of ˆg is symmetric (resp. skew[Remark 5] and therefore the representation (ˆ π , D) symmetric). ˆ = alg(V, µ Remark 7. If g ˆ) is the contraction of a Lie algebra g = alg(V, µ) by means of a net (Γι ) and if (πι , D(πι )) is a representation of g on a Hilbert space Hι for a given ι, then (πι ◦ Γι , D(πι )) is a representation of gι = alg(V, µι ) on Hι , and therefore also of g, since Γι is a Lie algebra isomorphism of gι onto g. Here, µι is the Lie multiplication obtained from µ via Γι as in Definition 1. Proposition 2. Let ˆ g, gι , I, and (gj )j∈J be as in Proposition 1. Suppose that, for each ι ∈ I, we have (a) a set Sι such that Sι ⊆ Sι0 whenever ι ι0 ; (b) a Hilbert space Hι of dimension Card(Sι ) with a given orthonormal basis (s) Sι = (ϕι )s∈Sι ; (c) a representation (πι , sp(Sι )) of gι on Hι satisfying X 0 dι,s,s0 (gj )φι(s ) (2.9) πι (gj )ϕ(s) ι = s0 ∈F(j,s)
for all j ∈ J and all s ∈ Sι , where F(j, s) is a finite subset of S = dι,s,s0 (gj ) ∈ C, and ( (s) if s ∈ Sι ϕι (s) φι = 0 if s ∈ S Sι .
S ι
Sι ,
If, for each j ∈ J, each s ∈ S, and each s0 ∈ F(j, s), the net (dι,s,s0 (gj ))ι∈I , with / Sι , converges to ds,s0 (gj ), then there exists a repdι,s,s0 (gj ) = 0 whenever s ∈ resentation (ˆ π , sp(S)) of ˆ g on any Hilbert space H of dimension Card(S), where S = (φ(s) )s∈S is any orthonormal basis of H. This representation (1) is such that X 0 ds,s0 (gj )φ(s ) (2.10) π ˆ (gj )φ(s) = s0 ∈F(j,s)
for all j ∈ J and all s ∈ S, and is the contraction of the net ((πι , sp(Sι ))) by means of the net (Pι ), where Pι is the continuous linear mapping of H onto Hι defined in S by Pι φ(s) = φ(s) ι
(ι ∈ I; s ∈ S) ;
(2.11)
(2) is, up to equivalence, the unique contraction of the net ((πι , sp(Sι ))) by means of a net (Pι ), where Pι is defined by (2.11) on the elements of an orthonormal basis of a carrier Hilbert space of dimension Card(S). Moreover, if the representation (πι , sp(Sι )) is symmetric (resp. skew-symmetric) for all ι ∈ I, then the representation (ˆ π , sp(S)) is also symmetric (resp. skewsymmetric).
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U. CATTANEO and W. F. WRESZINSKI
Proof. From Lemma 3, we have H = (Pι )-lim Hι ; moreover, because of (2.9), then Lemma 4 (with Aι = πι (gj )) implies the existence, for each j ∈ J, of the ˆ (gj ) = (Pι )-lim πι (gj ) in sp(S) and which satisfies operator π ˆ (gj ) in H such that π (2.10). Let π ˆ be the mapping of ˆg into the set of all operators in H defined in sp(S) by the operators π ˆ (gj )(j ∈ J) and linearity. Apart from item (2), all the assertions ˆ = sp(S), to be proven follow now from Proposition 1, with D(πι ) = sp(Sι ) and D (s) because π ˆ (gj )φ ∈ sp(S) by (2.10) and ˆ (gk )φ(s) )kι lim kπι (gj )(πι (gk )Pι φ(s) − Pι π ι
= lim ι 6
X
X
s0 ∈F(k,s)
s00 ∈F(j,s0 )
X
X
s0 ∈F(k,s)
s00 ∈F(j,s0 )
00
(dι,s,s0 (gk ) − ds,s0 (gk ))dι,s0 ,s00 (gj )φι(s
)
ι
lim |dι,s,s0 (gk ) − ds,s0 (gk )kdι,s0 ,s00 (gj )| = 0 ι
for all j, k in J and all s ∈ S. Notice that, by using directly Lemma 4 and linearity, we can also prove that [ˆ π (g), π ˆ (g 0 )] = (Pι )-lim[πι (g), πι (g 0 )] in sp(S) for all g, g 0 in V without to call on Proposition 1. In order to show that the representation (ˆ π , sp(S)) is unique up to equivalence, suppose that we also have a representation (ˆ π 0 , sp(S0 )) of ˆg on a Hilbert space H0 of dimension Card(S) such that π ˆ 0 (g) = (Pι0 )-lim πι (Γι (g)) in sp(S0 ) for all g ∈ V. 0 0 (s) Here S = (φ )s∈S is an orthonormal basis of H0 and Pι0 is the linear mapping of H0 onto Hι defined, as Pι in (2.11), via S0 . Then, the unitary mapping U of H onto 0 H0 given by U φ(s) = φ (s) (s ∈ S) is such that sp(S0 ) = U sp(S), Pι0 = Pι U −1 , and so the contractions (ˆ π , sp(S)) and (ˆ π 0 , sp(S0 )) are equivalent. Let U be a (strongly continuous) unitary representation of a finite-dimensional real Lie group G on a Hilbert space H. The differential of U is the skew-symmetric representation (dU, H∞ (O)) on H of the Lie algebra Lie(G) of G defined by dU (g)φ = lim
t→0
U (exp(tg))φ − φ t
for all φ ∈ H∞ (O), where H∞ (O) is the vector subspace of indefinitely differentiable vectors for U , i.e., of all φ ∈ H such that the mapping x 7→ U (x)φ of G into H is of class C∞ . Definition 7. A skew-symmetric representation (π, D(π)) of a finite-dimensional real Lie algebra g on a Hilbert space H is integrable (to U ) if, for every simply connected Lie group G whose Lie algebra is isomorphic to g and for every isomorphism θ of Lie(G) onto g, there exists a (necessarily unique strongly continuous) unitary representation U of G on H such that π(g) = dU (θ−1 (g)) in D(π) for all g ∈ g. It is well-known that the integrability of (π, D(π)) to U requires the density in H of the vector subspace Hω (O) of analytic vectors for U , i.e., of all φ ∈ H such
CONTRACTIONS OF LIE ALGEBRA REPRESENTATIONS
1193
that the mapping x 7→ U (x)φ of G into H is real-analytic. It was shown by Nelson Pm [17] that if (gj )16j6m is a basis of g and if the operator j=1 π(gj )2 is essentially self-adjoint in D(π), then (π, D(π)) is integrable. We need another integrability criterion proven by Flato et al. [18]: the skew-symmetric representation (π, D(π)) is integrable if D(π) is a set of analytic vectors for each skew-symmetric operator π(gj ) (1 6 j 6 m), namely if, for 1 6 j 6 m, the series Sπ(gj )φ (t) =
∞ X kπ(gj )n φkH n t n! n=0
is defined and convergent for some t > 0 and all φ ∈ D(π). It follows that, if (φl )l∈L is an algebraic basis of D(π), then (π, D(π)) is integrable if, for 1 6 j 6 m, the series Sπ(gj )φl (t) is defined and convergent for some t > 0 and all l ∈ L since, with P φ = l∈L zl φl and the zl ∈ C almost all 0,
X
∞ ∞ π(gj )n z φ
n l l X X kπ(gj ) φkH n l∈L H n t = t n! n! n=0 n=0 6
X l∈L
|zl |
∞ X kπ(gj )n φl kH n t . n! n=0
This last condition is equivalent to the existence, for 1 6 j 6 m and each l ∈ L, of a positive real number v(j, l) not depending on n such that kπ(gj )n φl kH 6 v(j, l)n n!, so that the series Sπ(gj )φl (t) is convergent for 0 < t < v(j, l)−1 if v(j, l) > 0 (and for t > 0 if v(j, l) = 0). Proposition 3. Let ˆ g = alg(V, µ ˆ ) be the contraction of a net (gι ) = (alg(V, µι )) of finite-dimensional real Lie algebras of dimension m, indexed by a directed set I, π , D(ˆ π )) is a representation of ˆg and let (gj )16j6m be a basis of V. Suppose that (ˆ on a Hilbert space H and that (a) for each ι ∈ I, there exists a skew-symmetric representation (πι , D(πι )) of gι on a Hilbert space Hι which satisfies kπι (gj )n Pι φl kι 6 vι (j, l)n n! for 1 6 j 6 m, all n ∈ N, and all elements of an algebraic basis (φl )l∈L of D(ˆ π ), where Pι is a continuous linear mapping of H into Hι such that π ) ⊆ D(πι ) and v(j, l) is a positive real number not depending on n; Pι D(ˆ ˆ (gj )n−1 φl )kι = 0 for 1 6 j 6 m, all n ∈ N∗ , (b) limι kπι (gj )(πι (gj )n−1 Pι φl −Pι π and all l ∈ L. If (ˆ π , D(ˆ π )) = (Pι )-lim(πι , D(πι )) and if, for 1 6 j 6 m and each l ∈ L, the π , D(ˆ π )) is skew-symmetric and net (vι (j, l)) converges, then the representation (ˆ integrable. Proof. The representation (ˆ π , D(ˆ π )) is skew-symmetric by Remark 5. For 1 6 j 6 m, each n ∈ N, each l ∈ L and with v(j, l) = limι vι (j, l), we have
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U. CATTANEO and W. F. WRESZINSKI
kˆ π (gj )n φl kH = lim kPι π ˆ (gj )n φl kι ι
6 lim kπι (gj )n Pι φl − Pι π ˆ (gj )n φl kι + lim kπι (gj )n Pι φl kι ι
ι
= lim kπι (gj ) Pι φl kι 6 lim vι (j, l) n! = v(j, l)n n! n
n
ι
ι
π ) by condition (b) extended to all φ ∈ D(ˆ π) since π ˆ (gj ) = (Pι )-lim πι (gj ) in D(ˆ ˆ (gj ). It follows that by linearity [Lemma 2], therefore φl is an analytic vector for π D(ˆ π ) is a set of analytic vectors for each π ˆ (gj ) (1 6 j 6 m) and so, by the result of Flato et al. just quoted, the representation (ˆ π , D(ˆ π )) is integrable. Notice that, if Pι is surjective, then also the representation (πι , D(πι )) is integrable. n
n
Proposition 4. Let gˆ, gι , I, and (gj )16j6m be as in Proposition 3. Suppose that the conditions (a), (b), (c), and the requirement on the convergence of the nets (dι,s,s0 (gj ))ι∈I in Proposition 2 are fulfilled with J = [1, m]N . Moreover, assume that the least upper bound sups∈S Card(F(j, s)) (1 6 j 6 m) exists in R+ and that we have three mappings r, t, u of [1, m]N × S into R+ such that (a) the inequality (2.12) |ds,s0 (gj )| 6 r(j, s)(t(j, s) + u(j, s0 )) is satisfied for all s, s0 in S and 1 6 j 6 m; (b) the mapping r(j, ·) is bounded above in S and the least upper bounds 0 00 0 00 |u(j, s ) − u(j, s )| sup 0 00max |t(j, s ) − t(j, s )| , sup 0 max 00 s ,s ∈F(j,s)
s∈S
s ,s ∈F(j,s)
s∈S
exist in R+ for 1 6 j 6 m. Then the skew-symmetric representation (ˆ π , sp(S)) of ˆg on H defined in Proposition 2 is integrable. Proof. Again by the result of Flato et al., it is enough to prove that the elements of S, hence also those of sp(S), are analytic vectors for each operator π ˆ (gj ) (1 6 j 6 m). For 1 6 j 6 m, each n ∈ N∗ , and each s ∈ S, (2.10) and (2.12) imply kˆ π (gj )n φ(s) kH
X
=
ds,s1 (gj )
s1 ∈F(j,s)
X
6
X
6
|ds1 ,s2 (gj )| · · ·
r(j, s)(t(j, s) + u(j, s1 )) X
sn ∈F(j,sn−1 )
dsn−1 ,sn (gj )φ
X
(sn )
sn ∈F(j,sn−1 )
s2 ∈F(j,s1 )
s1 ∈F(j,s)
×
X
X
ds1 ,s2 (gj ) · · ·
s2 ∈F(j,s1 )
|ds,s1 (gj )|
s1 ∈F(j,s)
X
H
|dsn−1 ,sn (gj )|
sn ∈F(j,sn−1 )
X
r(j, s1 )(t(j, s1 ) + u(j, s2 )) · · ·
s2 ∈F(j,s1 )
r(j, sn−1 )(t(j, sn−1 ) + u(j, sn )) .
(2.13)
CONTRACTIONS OF LIE ALGEBRA REPRESENTATIONS
1195
On the other hand, by condition (b) and the assumption on Card(F(j, s)), there exist, for 1 6 j 6 m, positive real numbers h1 (j), h2 (j), l1 (j), l2 (j) defined by h1 (j) = sup Card(F(j, s)) , s∈S
s∈S
l1 (j) = sup s∈S
|t(j, s ) − t(j, s )| , 0
max
s0 ,s00 ∈F(j,s)
and l2 (j) = sup s∈S
h2 (j) = sup r(j, s) , 00
|u(j, s ) − u(j, s )| . 0
max
s0 ,s00 ∈F(j,s)
00
Therefore t(j, sk ) 6 t(j, s) + |t(j, s1 ) − t(j, s)| + · · · + |t(j, sk ) − t(j, sk−1 )| 6 t(j, s) + kl1 (j) , and analogously u(j, sk ) 6 u(j, s) + kl2 (j), hence X r(j, sk−1 )(t(j, sk−1 ) + u(j, sk )) sk ∈F(j,sk−1 )
6 h1 (j)h2 (j)(t(j, s) + (k − 1)l1 (j) + u(j, s) + kl2 (j)) = h1 (j)h2 (j)(t(j, s) + u(j, s) + l2 (j) + (k − 1)l(j))
(2.14)
for 1 6 k 6 n with s0 = s, where l(j) = l1 (j) + l2 (j). On account of (2.14), it follows from (2.13) that kˆ π (gj )n φ(s) kH 6 h1 (j)n h2 (j)n
n Y
(t(j, s) + u(j, s) + l2 (j) + (k − 1)l(j))
k=1
6 h1 (j)n h2 (j)n p(j, s)
n Y
(k − 1)q(j, s)
k=2
6 h1 (j)n h2 (j)n q(j, s)n (n − 1)! 6 v(j, s)n n! , where p(j, s) = t(j, s) + u(j, s) + l2 (j), q(j, s) = p(j, s) + l(j), and so the positive real number v(j, s) = h1 (j)h2 (j)q(j, s) depends on j and s, but not on n. We conclude ˆ (gj ). that φ(s) ∈ S is an analytic vector for π Remark 8. The integrability conditions of Proposition 4 refer to the representation (ˆ π , sp(S)) of ˆ g on H. The integrability of this representation may also be obtained by imposing, besides the existence of sups∈S Card(F(j, s)) (1 6 j 6 m), the following conditions on the representation (πι , sp(Sι )) of gι on Hι for every ι ∈ I: • there exist three mappings rι , tι , uι of [1, m]N × S into R+ such that, for 1 6 j 6 m, we have:
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U. CATTANEO and W. F. WRESZINSKI
(a) |dι,s,s0 (gj )| 6 rι (j, s)(tι (j, s) + uι (j, s0 )) for all s, s0 in S; (b) the least upper bounds h2,ι (j) = sup rι (j, s) , s∈S
l1,ι (j) = sup
max
s0 ,s00 ∈F(j,s)
s∈S
and l2,ι (j) = sup s∈S
|tι (j, s ) − tι (j, s )| , 0
max 00
s0 ,s ∈F(j,s)
00
|uι (j, s0 ) − uι (j, s00 )| ;
• again for 1 6 j 6 m and each s ∈ S, the nets (rι (j, s)), (tι (j, s)), (uι (j, s)), (h2,ι (j)), (l1,ι (j)), (l2,ι (j)), indexed by I, are supposed to converge to r(j, s), t(j, s), u(j, s), h2 (j), l1 (j), l2 (j), respectively. Proceeding as in the proof of Proposition 4, we see that conditions (a) and (b) imply kπι (gj )n Pι φ(s) kι 6 vι (j, s)n n! for 1 6 j 6 m, all n ∈ N, and all s ∈ S, whence the integrability of the representation (πι , sp(Sι )). By Proposition 3, the representation (ˆ π , sp(S)) is integrable, because limι vι (j, s) = v(j, s) and ˆ (gj )n−1 φ(s) )kι lim kπι (gj )(πι (gj )n−1 Pι φ(s) − Pι π ι
= lim ι
X
X
X
···
s1 ∈F(j,s) s2 ∈F(j,s1 )
(dι,s,s1 (gj ) · · · dι,sn−2 ,sn−1 (gj )
sn ∈F(j,sn−1 )
(sn )
− ds,s1 (gj ) . . . dsn−2 ,sn−1 (gj ))dι,sn−1 ,sn (gj )φ X
6
X
s1 ∈F(j,s) s2 ∈F(j,s1 )
...
X sn ∈F(j,sn−1 )
ι
lim |dι,s,s1 (gj ) . . . dι,sn−2 ,sn−1 (gj ) ι
− ds,s1 (gj ) . . . dsn−2 ,sn−1 (gj )kdι,sn−1 ,sn (gj )| = 0 for 1 6 j 6 m, all n ∈ N∗ , and all s ∈ S. 3. Examples Let so(2), e(2), h(1), r3 , and so(1, 1) be, respectively, the Lie algebras of the (real) Lie groups SU(2), E(2) (the Euclidean group of the plane), H(1) (the 3-dimensional Heisenberg group), the additive group of R3 , and SU(1, 1). Examples of contractions of nets of representations of so(2) into representations of Lie algebras isomorphic to e(2), h(1), r3 and of nets of representations of so(1, 1) into representations of a Lie algebra isomorphic to e(2) have already been treated elsewhere [12, Sec. 3] in a way that stems from the basic ideas of the theory of the previous section. In order to illustrate the applicability of this theory, we review and put in a new light what is maybe, from the standpoint of physics, the more interesting case of contraction of a net of representations of so(2), namely a contraction into
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CONTRACTIONS OF LIE ALGEBRA REPRESENTATIONS
the standard representation of h(1). To stick to customary notation, we use here the indices j, n, n0 instead of ι, s, s0 , and Nj instead of Sι . 3.1. Contraction of so(2) into h(1) Let (g1 , g2 , g3 ) be a basis of alg(V, µ) = so(2) such that [g1 , g2 ] = g3 ,
[g2 , g3 ] = g1 ,
[g3 , g1 ] = g2 ,
where [·, ·] = µ, let I = 12 N be the contraction index set ordered by 6 and, for each j ∈ 12 N, let Hj be the Hilbert space (isomorphic to C2j+1 ) of all polynomials of degree 6 2j in one complex variable. This space carries the standard irreducible so(2) of dimension 2j + 1 of so(2), unique up to skew-symmetric representation πj (m)
(unitary) equivalence [19, Chap. 3, Sec. 2]. If (ψj )m∈Mj is the orthonormal basis of Hj defined by z j−m (m) (3.1) ψj (z) = p (j − m)!(j + m)! with Mj = {−j, −j + 1, . . . , j}, then this representation is given by so(2)
πj
(m)
(gk )ψj
=
1 X
(m+m0 )
so(2)
cj,m,m0 (gk )ψj
(m ∈ Mj )
(3.2)
m0 =−1 (m+m0 )
= 0 if m + m0 = −j − 1 or m + m0 = j + 1, where, for for 1 6 k 6 3 with ψj 0 each m ∈ Mj and −1 6 m 6 1, 1 p − i (j + m)(j − m + 1) if m0 = −1 2 so(2) if m0 = 0 cj,m,m0 (g1 ) = 0 1 p − i (j − m)(j + m + 1) if m0 = 1 , 2 1p (j + m)(j − m + 1) if m0 = −1 2 so(2) if m0 = 0 cj,m,m0 (g2 ) = 0 1p − (j − m)(j + m + 1) if m0 = 1 , 2 0 so(2) cj,m,m0 (g3 ) = −im 0
if m0 = −1 if m0 = 0 if m0 = 1 . so(2)
Being finite-dimensional, the representation πj (n)
is integrable. (n)
(n−j)
, The sequence Nj = (ϕj )n∈Nj , where Nj = {0, 1, . . . , 2j} and ϕj = ψj is also an orthonormal basis of Hj and, for our purposes, is more suitable than the (m) family (ψj )m∈Mj defined by (3.1); on account of (3.2), we have
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U. CATTANEO and W. F. WRESZINSKI
so(2)
πj
(n)
(gk )ϕj
1 X
=
(n+n0 )
so(2)
cj,n−j,n0 (gk )ϕj
(n ∈ Nj )
n0 =−1 (n+n0 )
for 1 6 k 6 3 with ϕj
= 0 if n + n0 = −1 or n + n0 = 2j + 1, namely
p 1 p (n) (n−1) (n+1) (g1 )ϕj = − i n(2j − n + 1)ϕj + (n + 1)(2j − n)ϕj , 2 p 1 p so(2) (n) (n−1) (n+1) (g2 )ϕj = n(2j − n + 1)ϕj − (n + 1)(2j − n)ϕj , πj 2
so(2)
πj
so(2)
πj
(n)
(n)
(g3 )ϕj = −i(n − j)ϕj (j)
From the symmetric (and self-adjoint) operators Jk Hj , we get the operators
(j) J−
=
(j) J1
−
(j) iJ2 ,
(j) J+
so(2)
+
(j)
so(2)
(g1 ) + πj
(g2 ) ,
(j)
so(2)
(g1 ) − πj
(g2 ) ,
so(2)
(g3 ) .
J− = iπj J+ = iπj (j)
J0
= iπj
so(2)
so(2)
(gk ) (1 6 k 6 3) in
(j) iJ2 ,
(j)
J0
(j)
= J3 , i.e.,
(3.4)
It follows that so(2)
πj
so(2)
πj
so(2)
πj (j)
1 (j) (j) (g1 ) = − i(J+ + J− ) , 2 1 (j) (j) (g2 ) = − (J+ − J− ) , 2 (j)
(g3 ) = −iJ0 ,
(j)
(j)
and that the operators J− , J+ , J0 (j)
(j)
(j)
[J0 , J− ] = −J− ,
satisfy the commutation relations
(j)
(j)
(j)
[J0 , J+ ] = J+ ,
(j)
(j)
(j)
(n)
(j)
(n)
(j)
(n)
J+ ϕj J0 ϕj (j)
(0)
and J− ϕj
(j)
(2j)
= J+ ϕj
p (n−1) n(2j − n + 1)ϕj p (n+1) = (n + 1)(2j − n)ϕj =
(n)
= (n − j)ϕj
if n > 0 , if n < 2j ,
,
= 0, so that s (2j − n)! (j) n (0) (n) (J+ ) ϕj ϕj = n!(2j)! (j)
for all n ∈ Nj , where (J+ )0 = IdHj .
(j)
[J+ , J− ] = 2J0 .
By (3.3) and (3.4), we have J− ϕj
(3.3b) (3.3c)
= iπj
(j) J1
=
.
(3.3a)
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CONTRACTIONS OF LIE ALGEBRA REPRESENTATIONS
If we define an automorphism Γj of V by √ √ 2 2 g1 , Γj (g2 ) = √ g2 , Γj (g1 ) = √ 2j + 1 2j + 1
Γj (g3 ) =
2 g3 2j + 1
for every j ∈ 12 N, then there exists the contraction alg(V, µ ˆ) of so(2) by means of the net (Γι ). The Lie products of the basis elements (g1 , g2 , g3 ) of V in this contraction are [g1 , g2 ] = g3 , [g2 , g3 ] = 0 , [g3 , g1 ] = 0 , with [·, ·] = µ ˆ, hence alg(V, µ ˆ ) is a Lie algebra isomorphic to h(1) that we identify with h(1). Moreover, we have an irreducible skew-symmetric representation πj = πjso (2) ◦ Γj of so(2) on Hj [Remark 7] such that (n)
πj (gk )ϕj
=
1 X
(n+n0 )
cj,n,n0 (gk )ϕj
(n ∈ Nj )
(3.5)
n0 =−1 (n+n0 )
for 1 6 k 6 3 with ϕj n ∈ Nj and −1 6 n0 6 1,
= 0 if n + n0 = −1 or n + n0 = 2j + 1, where, for each
√ 2 so(2) c cj,n,n0 (g1 ) = √ 0 (g1 ) , 2j + 1 j,n−j,n
√ 2 so(2) c cj,n,n0 (g2 ) = √ 0 (g2 ) , 2j + 1 j,n−j,n cj,n,n0 (g3 ) =
2 so(2) c 0 (g3 ) . 2j + 1 j,n−j,n
(3.6a)
(3.6b)
(3.6c)
If we put cj,n,n0 (gk ) = 0 (−1 6 n0 6 1; 1 6 k 6 3) for all n ∈ N Nj , then (3.6) implies lim cj,n,n0 (gk ) = cn,n0 (gk ) (n ∈ N; −1 6 n0 6 1; 1 6 k 6 3) , j
where limj means limj→+∞ , with
√ n −i √ 2 cn,n0 (g1 ) = 0 √ n+1 −i √ 2 √ n √ 2 cn,n0 (g2 ) = 0 √ n+1 − √ 2
if n0 = −1 if n0 = 0 if n0 = 1 , if n0 = −1 if n0 = 0 if n0 = 1 ,
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U. CATTANEO and W. F. WRESZINSKI
0 cn,n0 (g3 ) = i 0
if n0 = −1 if n0 = 0 if n0 = 1 .
We may now apply Proposition 2, since Nj ⊆ Nj 0 whenever j 6 j 0 and (3.5) may be written in the form (2.9): X (n) (n0 ) dj,n,n0 (gk )φj (n ∈ Nj ) πj (gk )ϕj = n0 ∈F(k,n)
for 1 6 k 6 3, where F(k, 0) = {0, 1}, F(k, n) = {n − 1, n, n + 1} if n 6= 0, and dj,n,n0 (gk ) = cj,n,n0 −n (gk ), ( (n) ϕj if 0 6 n 6 2j (n) φj = 0 if n > 2j S 0 for all n ∈ N and all n ∈ F(k, n). Moreover, we also have j∈ 1 N Nj = N and 2 limj dj,n,n0 (gk ) = dn,n0 (gk ) for 1 6 k 6 3 and all n, n0 in N, with dn,n0 (gk ) = cn,n0 −n (gk ). Let H be any separable Hilbert space, N = (φ(n) )n∈N any orthonormal basis of H and, for each j ∈ 12 N, let Pj be the continuous linear mapping of H (n) onto Hj defined by Pj φ(n) = φj (n ∈ N). It follows from Proposition 2 that H = (Pj )-lim Hj and the net ((πj , sp(Nj ))) is (Pj )-convergent to a skew-symmetric representation (ˆ π , sp(N)) of h(1) on H such that 1 X
π ˆ (gk )φ(n) =
0
(n ∈ N)
cn,n0 (gk )φ(n+n )
n0 =−1 0
for 1 6 k 6 3, where φ(n+n ) = 0 if n + n0 = −1, namely √ 1 √ π ˆ (g1 )φ(n) = − √ i n φ(n−1) + n + 1 φ(n+1) , 2
(3.7a)
1 √ (n−1) √ nφ − n + 1 φ(n+1) , π ˆ (g2 )φ(n) = √ 2
(3.7b)
π ˆ (g3 )φ(n) = iφ(n) .
(3.7c)
By Proposition 4 the representation (ˆ π , sp(N)) is integrable, for |cn,n0 (gk )| 6 n + 1
(1 6 k 6 3; n ∈ N; −1 6 n0 6 1) ,
hence |dn,n0 (gk )| 6 n + 1 for 1 6 k 6 3, all n ∈ N, and all n0 ∈ F(k, n), so that |dn,n0 (gk )| 6 r(k, n)(t(k, n) + u(k, n0 )) is satisfied with r(k, n) = 1, t(k, n) = n+1, u(k, n0 ) = 0. Besides, the mapping r(k, ·) is constant, hence bounded above in N, and we have supn∈N Card(F(k, n)) = 3, 0 00 |t(k, n ) − t(k, n )| = 2, sup max 0 00 n∈N
n ,n ∈F(k,n)
sup n∈N
max
n0 ,n00 ∈F(k,n)
|u(k, n ) − u(k, n )| = 0 . 0
00
CONTRACTIONS OF LIE ALGEBRA REPRESENTATIONS
1201
The operators A− , A+ , A0 in H, defined in sp(N) by 1 π (g1 ) + π ˆ (g2 )) , A− = √ (iˆ 2 1 π (g1 ) − π ˆ (g2 )) , A+ = √ (iˆ 2
(3.8)
π (g3 ) , A0 = −iˆ i.e., such that 1 π ˆ (g1 ) = − √ i(A+ + A− ) , 2 1 π ˆ (g2 ) = − √ (A+ − A− ) , 2 π ˆ (g3 ) = iA0 , satisfy the commutation relations [A− , A0 ] = 0 ,
[A+ , A0 ] = 0 ,
[A− , A+ ] = A0
and A+ = (A− )∗ in sp(N). By (3.7) and (3.8), we have √ A− φ(n) = n φ(n−1) if n > 0 and A− φ(0) = 0 , √ A+ φ(n) = n + 1 φ(n+1) , A0 φ(n) = φ(n) and so
(3.9)
(A+ )n (0) φ φ(n) = √ n!
for all n ∈ N, where (A+ )0 = A0 = IdH ; furthermore, 1 (j) J , A− = (Pj )-lim √ 2j + 1 − 1 (j) J , A+ = (Pj )-lim √ 2j + 1 + A0 = (Pj )-lim
−2 (j) J . 2j + 1 0
By (3.9), every vector of sp(N) is cyclic for the representation (ˆ π , sp(N)) which is therefore irreducible and, by von Neumann’s uniqueness theorem [20], is the unique integrable irreducible skew-symmetric representation of h(1), up to equivalence. 3.2. Systems of N identical two-level particles and the limit N →∞ The contraction (ˆ π , sp(N)) of the net ((πj , sp(Nj ))) has the following interesting application which shows that the theory proposed in this paper is, in some sense, forced by physics. We were here inspired by [21] and [22]; in particular, [21] was the first application of Trotter’s theory to a problem in statistical mechanics.
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U. CATTANEO and W. F. WRESZINSKI
Let SN (C2 ) = (C2 )⊗s N be the N th symmetric power of the Hilbert space C2 (which is endowed with the canonical inner multiplication) and so, in particular, (C2 )⊗s 0 = C. For each N ∈ N, we interpret the N th power of the Hilbert space C2 , i.e., the 2N -dimensional Hilbert space TN (C2 ) = (C2 )⊗N , as the space of states of a (quantum) system of N identical two-level particles with the ground state ⊗N 0 if N 6= 0 |0iN = 1 1 if N = 0 , where
⊗N 0 0 0 0 = ⊗ ⊗ ···⊗ (N factors) . 1 1 1 1 Here, and in what follows, we stick to the customary abuse of language of calling “states” the vectors of a “space of states” which generate the rays representing the pure states, and we use Dirac’s bra-ket formalism. In analogy with spin- 21 systems, the (N + 1)-dimensional Hilbert space SN (C2 ) may be canonically identified with a vector subspace of TN (C2 ), namely the subspace of states of the system which have maximal “total spin” j = N/2, with |0iN = |N/2, −N/2i in standard angular momentum notation. We consider the limit N → ∞ only for these spaces SN (C2 ). (N ) In SN (C2 ), we define the total spin operators Sk (1 6 k 6 3) by N X 1 (l) I2 ⊗ · · · ⊗ I2 ⊗ σk ⊗ I2 ⊗ · · · ⊗ I2 if N 6= 0 (N ) 2 Sk = l=1 0 if N = 0 , (l)
where I2 is the 2 × 2-identity matrix and σk is a Pauli matrix (acting on the lth factor in SN (C2 )). From these operators, we get the lowering spin operator (N ) (N ) (N ) (N ) (N ) (N ) S− = S1 − iS2 and the raising spin operator S+ = S1 + iS2 in SN (C2 ) (N ) (N ) which, together with S0 = S3 , satisfy the commutation relations (N )
[S0
(N )
(N )
, S− ] = −S− ,
(N )
[S0
(N )
(N )
, S+ ] = S + ,
(N )
(N )
(N )
[S+ , S− ] = 2S0
.
If, for each n ∈ N with n 6 N , we consider the Dicke state |niN of SN (C2 ), which is defined by r (N − n)! (N ) n (S+ ) |0iN |niN = n!N ! (N )
with (S+ )0 = IdSN (C2 ) , then we have p (N ) S− |niN = n(N − n + 1) |n − 1iN p (N ) S+ |niN = (n + 1)(N − n) |n + 1iN N (N ) |niN , S0 |niN = n − 2 (N )
(N )
if n > 0 , if n < N ,
and S− |0iN = S+ |N iN = 0. The sequence (|niN )n6N is an orthonormal basis of SN (C2 ), the Dicke basis, since the S (N ) -operators satisfy the same commutation
CONTRACTIONS OF LIE ALGEBRA REPRESENTATIONS
1203
relations as the J (j) -operators of Subsec. 3.1 and they act in a Hilbert space of the same dimension (with j = N/2): in standard angular momentum notation, |niN = |N/2, −N/2 + ni. The state |niN is the (totally) symmetric N -particle state with n spins up. Let F(C) be the (symmetric) Fock space over C, i.e., let F(C) =
∞ M
C⊗N ≈
N =0
∞ M
C
N =0
with the annihilation and creation operators A, A† , the Fock basis (|ni)n∈N , and the vacuum |0i. This means that the operators A and A† in F(C) satisfy the commutation relations [A, I] = 0 ,
[A† , I] = 0 ,
[A, A† ] = I ,
where I = IdF(C) , and that √ A|ni = n |n − 1i if n > 0 and A|0i = 0 , √ A† |ni = n + 1 |n + 1i , I|ni = |ni , so that
(A† )n |0i |ni = √ n! for all n ∈ N, with (A† )0 = I. The Fock state |ni is an eigenstate with eigenvalue n of the number operator A† A. There is a strong relation between the Hilbert spaces and operators considered in this subsection and those of Subsec. 3.1. This correspondence can be specified as follows: (a) For each N ∈ N, we identify the (N + 1)-dimensional Hilbert space SN (C2 ) (n) with the Hilbert space Hj by putting N = 2j and |niN = ϕj . Therefore (N )
(j)
(N )
(j)
(N )
(j)
we have S− = J− , S+ = J+ , S0 = J0 and SN (C2 ) carries the integrable irreducible skew-symmetric representation πj of so(2) given by (3.5). (b) We identify the (symmetric) Fock space F(C) with the Hilbert space H by putting |ni = φ(n) . Then A = A− , A† = A+ , I = A0 and F(C) carries the integrable irreducible skew-symmetric representation (ˆ π , sp(N)) of h(1) given by (3.7). Taking account of Remark 6, it follows from (a), (b) and from the results of Subsec. 3.1 that F(C) = (RN )-lim SN (C2 ), |ni = (RN )-lim |niN , A = (RN )-lim √
1 (N ) S− , N +1
A† = (RN )-lim √
1 (N ) S+ , N +1
I = (RN )-lim
−2 (N ) S , N +1 0
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U. CATTANEO and W. F. WRESZINSKI
where the continuous linear mapping RN = Pj of F(C) onto SN (C2 ) is defined by ( |niN if n 6 N RN |ni = 0 if n > N . Moreover, for each z ∈ C, we have |zi = (RN )-lim |Ωz iN , where |zi is the Glauber (or canonical ) coherent state of F(C) [23] defined by 1 |zi = exp − |z|2 exp(zA† )|0i 2 and |Ωz iN is the Bloch (or spin, or atomic) coherent state of SN (C2 ) [23] defined by 2 !− N2 |z| z (N ) S+ exp √ (3.10) |0iN . |Ωz iN = 1 + √ N +1 N +1 In fact, lim N hf (z)|f (z)iN = lim
N →∞
N
N hf (z)|f (z)iN
=0
with |f (z)iN = |Ωz iN − RN |zi. In order to show this, we first notice that we have ∞ ∞ 1 2 X 1 n 1 2 X zn † n √ z |ni (A ) |0i = exp − |z| |zi = exp − |z| 2 n! 2 n! n=0 n=0 and hz|zi = exp(−|z|2 ) as well as
∞ X (|z|2 )n = 1, n! n=0
X N 1 1 √ z n |niN |ziN = RN |zi = exp − |z|2 2 n! n=0
and 2 N hz|ziN = exp(−|z| )
N X 1 (|z|2 )n n! n=0
for all z ∈ C and all N ∈ N. Analogously, we also have |Ωz iN =
1+
=
1+
|z| √ N +1 |z| √ N +1
2 !− N2 X n N z 1 (N ) √ (S+ )n |0iN n! N + 1 n=0 2 !− N2 X N 12 N n=0
n
z √ N +1
n |niN
and N hΩz |Ωz iN
=
1+
|z| √ N +1
2 !−N X N N n=0
n
|z| √ N +1
2 !n = 1.
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CONTRACTIONS OF LIE ALGEBRA REPRESENTATIONS
It follows then that N hf (z)|f (z)iN
=
N hΩz |Ωz iN
− 2<(N hΩz |ziN ) + N hz|ziN
= 1 +N hz|ziN − 2N hΩz |ziN , hence lim
N →∞
N hf (z)|f (z)iN
= 1 + lim
N →∞
N hz|ziN
− 2 lim
N →∞
N hΩz |ziN
6 0,
and therefore limN →∞ N hf (z)|f (z)iN = 0, since lim
N →∞
N hz|ziN
N X 1 (|z|2 )n = 1 N →∞ n! n=0
= exp(−|z|2 ) lim
and lim
N →∞
N hΩz |ziN
2 !− N2 |z| 1 2 = exp − |z| lim 1 + √ N →∞ 2 N +1 N 2 X N 1
×
n=0
n
r
(N + 1)n n!
|z| √ N +1
2 !n
2 !− N2 X 2 !n N |z| N |z| 1 2 √ > exp − |z| lim 1 + √ N →∞ 2 n N +1 N +1 n=0 N2 1 2 1 2 1 2 |z|2 |z| = 1 = exp − |z| = exp − |z| exp lim 1 + N →∞ 2 N +1 2 2 as
r 12 r N N (N + 1)n (N + 1)n (N − n)! N = > n n n! N! n
for all N ∈ N and all positive integers n 6 N . We have shown that the Glauber coherent states of F(C) are limits for N → ∞ of the Bloch coherent states of SN (C2 ) in the same sense that the Fock states of F(C) are limits for N → ∞ of the Dicke states of SN (C2 ). This result was first stated in [24] using a contraction of the Lie algebra u(2) of U(2) into a Lie algebra isomorphic to that of the harmonic oscillator group (for one degree of freedom). The Fock space F(C) appears in many physical models, where it is usually seen as the space of states of a harmonic oscillator. This interpretation is particularly important for applications in quantum optics, since a single photon mode of a free radiation field may be regarded as a dynamical system equivalent to a harmonic oscillator in which the Fock state |ni is an n-photon state. If the N identical twolevel particles of our system are free, the energy levels of the system are equally
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U. CATTANEO and W. F. WRESZINSKI
spaced and they give rise to a spectrum analogous to that of a harmonic oscillator but with a highest level. The properties of the two systems are similar for lowlying excitations and the harmonic oscillator may be considered as the limit case in which the number of energy levels of the system of identical two-level particles goes to infinity, i.e., the limit case for N → ∞. The contraction considered in this subsection has the following geometrical interpretation [25, Appendix A.2] based on a parametrization of the Bloch coherent states by the spherical coordinates θ and φ, with θ measured from the “south pole”: |Ωz iN = |θ, φiN . With θ z √ = tan exp(−iφ) (0 6 θ < π; 0 6 φ < 2π) , 2 N +1 we get |θ, φiN
θ θ (N ) exp tan exp (−iφ) S+ = cos |0iN 2 2 N
from (3.10) and we can regard the Bloch coherent state |θ, φiN as the point in the direction (θ, φ) of a sphere of radius N/2 without the “north pole”. This sphere is the Bloch sphere of radius N/2 [25] and |θ, φiN is the vector obtained by applying to the ground state |0iN (which coincides with the south pole) the operator representing the rotation taking the south pole into the point of the sphere in the direction (θ, φ) [24]. In this framework, the geometrical meaning of the limit process which gives rise to the contraction is that of letting the radius of the Bloch sphere tend to infinity while the small rotations of the sphere turn into translations of the tangent plane at the south pole. The Bloch sphere of radius N/2 corresponds to the phase space of our N -particle system and its tangent plane at the south pole corresponds to the phase plane of a harmonic oscillator. Acknowledgments One of us (U.C.) would like to thank the Instituto de F´ısica, Universidade de S˜ ao Paulo, for the hospitality extended to him. We are very grateful to L. Cattaneo for valuable discussions. References [1] I. E. Segal, “A class of operator algebras which are determined by groups”, Duke Math. J. 18 (1951) 221–265. [2] E. In¨ on¨ u and E. P. Wigner, “On the contraction of groups and their representations”, Proc. Nat. Acad. Sci. USA 39 (1953) 510–524. [3] L. L. Vaksman and L. I. Korogodskiˇı, “An algebra of bounded functions on the quantum group of the motions of the plane, and q-analogues of Bessel functions”, Sov. Math. Dokl. 39 (1989) 173–177. [4] E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini, Contractions of Quantum Groups, Lect. Notes Math. 1510, Springer-Verlag, Berlin, 1992. [5] N. A. Gromov and V. I. Man’ko, “Contractions of the irreducible representations of the quantum algebras suq (2) and soq (3)”, J. Math. Phys. 33 (1992) 1374–1378. [6] E. Weimar-Woods, “Contraction of Lie algebra representations”, J. Math. Phys. 32 (1991) 2660–2665.
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[7] E. Weimar-Woods, “The three-dimensional real Lie algebras and their contractions”, J. Math. Phys. 32 (1991) 2028–2033. [8] R. J. B. Fawcett and A. J. Bracken, “The classical limit of quantum mechanics as a Lie algebra contraction”, J. Phys. A: Math. Gen. 24 (1991) 2743–2761. [9] J. Von Neumann, “Zur Theorie der unbeschr¨ ankten Matrizen”, J. Reine Angew. Math. 161 (1929) 208–236. [10] U. Cattaneo and W. Wreszinski, “On contraction of Lie algebra representations”, Commun. Math. Phys. 68 (1979) 83–90. [11] H. F. Trotter, “Approximation of semi-groups of operators”, Pac. J. Math. 8 (1958) 887–919. [12] U. Cattaneo and W. Wreszinski, “Trotter limits of Lie algebra representations and coherent states”, Helv. Phys. Acta 52 (1979) 313–327. [13] E. J. Saletan, “Contraction of Lie groups”, J. Math. Phys. 2 (1961) 1–21. [14] M. L´evy-Nahas, “Deformation and contraction of Lie algebras”, J. Math. Phys. 8 (1967) 1211–1222. [15] H. D. Doebner and O. Melsheimer, “On a class of generalized group contractions”, Nuovo Cimento A49 (1967) 306–311. [16] G. C. Hegerfeldt, “Some properties of a class of generalized In¨ on¨ u–Wigner contractions”, Nuovo Cimento A51 (1967) 439–447. [17] E. Nelson, “Analytic vectors”, Ann. Math. 70 (1959) 572–615. [18] M. Flato, J. Simon, H. Snellman and D. Sternheimer, “Simple facts about analytic ´ Norm. Sup. (4e s´erie) 5 (1972) 423–434. vectors and integrability”, Ann. Sci. Ec. [19] N. Ja. Vilenkin, Special Functions and the Theory of Group Representations, Translations of Mathematical Monographs Vol. 22, Amer. Math. Soc., Providence (R.I.), 1968. [20] J. Von Neumann, “Die Eindeutigkeit der Schr¨ odingerschen Operatoren”, Math. Ann. 104 (1931) 570–578. [21] A. Wehrl, “Spin waves and the BCS model”, Commun. Math. Phys. 23 (1971) 319–342. [22] K. Hepp and E. H. Lieb, “On the superradiant phase transition for molecules in a quantized radiation field: The Dicke maser model”, Ann. Phys. 76 (1973) 360–404. [23] J. R. Klauder and B.-S. Skagerstam, Coherent States — Applications in Physics and Mathematical Physics, World Scientific, Singapore, 1985. [24] F. T. Arecchi, E. Courtens, R. Gilmore and H. Thomas, “Atomic coherent states in quantum optics”, Phys. Rev. A6 (1972) 2211–2237. [25] H. M. Nussenzveig, Introduction to Quantum Optics, Gordon and Breach, New York, 1973.
REDUCTION OF PRESYMPLECTIC MANIFOLDS WITH SYMMETRY ∗ ˜ A. ECHEVERR´IA-ENR´IQUEZ, M. C. MUNOZ-LECANDA ´ and N. ROMAN-ROY†
Departamento de Matem´ atica Aplicada y Telem´ atica Campus Norte U.P.C., M´ odulo C-3 C/Jordi Girona 1 E-08034 Barcelona Spain ∗ E-mail : [email protected] † E-mail : [email protected] Received 17 June 1998 Revised 5 October 1998 1991 AMS Subject Classification: 57S25, 58D19, 70H33 PACS: 0240, 0320 Actions of Lie groups on presymplectic manifolds are analyzed, introducing the suitable comomentum and momentum maps. The subsequent theory of reduction of presymplectic dynamical systems with symmetry is studied. In this way, we give a method of reduction which enables us to remove gauge symmetries as well as non-gauge “rigid” symmetries at once. This method is compared with other step-by-step reduction procedures. As particular examples in this framework, we discuss the reduction of time-dependent dynamical systems with symmetry, the reduction of a mechanical model of field theories with gauge and nongauge symmetries, and the gauge reduction of the system made of a conformal particle. Keywords: Presymplectic manifolds, Lie groups, momentum maps, symmetries, reduction.
1. Introduction The problem of reduction of dynamical systems with symmetry has deserved the interest of theoretical physicists and mathematicians, with the purpose of reducing the number of evolution equations, by finding first integrals of motion. In particular, geometric treatment of this subject has been revealed as a powerful tool in the study of this question. The pioneering and fundamental work on this topic has been carried out by Marsden and Weinstein [42] (see also [1, 34, 52]). They demonstrated that, for a free and proper symplectic action of a (connected) Lie group on a (connected) symplectic manifold (which is the phase space of an autonomous regular Hamiltonian system with symmetry), and a weakly regular value of the momentum map associated with this action, the reduced phase space has a structure of symplectic manifold and inherits a Hamiltonian dynamics from the initial system. Nevertheless, the problem of reduction can appear under many different aspects. Subsequently, other authors have investigated aspects of the theory of reduction for other particular cases. 1209 Reviews in Mathematical Physics, Vol. 11, No. 10 (1999) 1209–1247 c World Scientific Publishing Company
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Thus, for instance, if zero is a singular value of the momentum map (in a symplectic manifold) then the Marsden–Weinstein technique gives a reduced phase space which is a stratified symplectic space [49]. Starting from this result, reduction of time-dependent regular Hamiltonian systems with momentum mappings with singular value at zero is achieved in [33], where, using the extended phase space symplectic formalism, it is proved that the reduced phase space is also a stratified space but with a cosymplectic structure. Another approach to the problem of singular values can be found in [4] (see also other references quoted therein), where reduction of symplectic manifolds at singular values of the momentum mapping is considered, showing that, under certain conditions, the reduced space inherits a nondegenerate Poisson structure. However, research in this area is not yet complete. In the realm of momentum maps with regular values, the Marsden–Weinstein symplectic reduction scheme has been applied to many different situations. For example, reduction of time-dependent regular Hamiltonian systems is developed in the framework of cosymplectic manifolds in [3], obtaining a reduced phase space which inherits a structure of cosymplectic manifold. The study of autonomous singular Lagrangian systems can be found in [9] and, in particular, the conditions for the reduced phase space to inherit an almost-tangent structure are studied for certain kinds of degenerate Lagrangians. Some of the results here obtained are generalized to the case of non-autonomous singular Lagrangian systems and for a larger class of degenerate Lagrangians in [28]. Another approach to this question is made in [32], where the authors analyze the conditions for the existence of a regular Lagrangian function in the reduced phase space obtained after reduction, in such a way that the reduced cosymplectic or contact structure (and hence the reduced Hamiltonian function) can be constructed from it. Furthermore, there are other situations in reduction theory. So, for instance, the theory of reduction of Poisson manifolds is treated in works such as [31] and [39]. Reduction of cotangent bundles of Lie groups within semidirect products is considered in [40], with several applications to outstanding problems in mathematical physics. Concerning the subject of Lagrangian reduction, there are some works, such as [41], which consider the problem from the point of view of reducing variational principles (instead of reducing the almost tangent structure, as it is made in some of the above mentioned references). Finally, the study of reduction of nonholonomic systems can be found, for instance, in [6, 10, 36]. (Of course, this list of references is far to be complete.) The aim of this work is to apply the Marsden–Weinstein method to reduce presymplectic manifolds with Lie groups of symmetries acting on them. The interest of this topic lies in the fact that the geometrical description of many dynamical systems is given by means of presymplectic manifolds. One of the more frequent cases is the Lagrangian formalism of singular mechanical systems, where the phase space is the manifold TQ (Q being the configuration manifold of the system), endowed with the presymplectic form ΩL , which is constructed from the singular Lagrangian function L. Other typical examples are certain descriptions of non-autonomous mechanical systems (both in the Lagrangian and Hamiltonian formalism), where
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the phase space is a contact (cosymplectic) manifold. Certainly, these kind of systems could be reduced by first constructing an ambient symplectic manifold where the system is coisotropically imbedded, and then applying the symplectic reduction procedure to it [29]. But we give a reduction procedure that allows us to implement the Marsden–Weinstein technique directly for the initial presymplectic system. In particular, we construct comomentum and momentum mappings for presymplectic actions of Lie groups, analyzing the obstruction to their existence and studying some characteristics features of the level sets of the momentum map. Then, we prove that, for weakly regular values of this momentum map, and under the usual suitable assumptions, the reduced phase space inherits a presymplectic structure. Next we apply these results in order to reduce presymplectic dynamical systems with symmetry, showing that, if we consider together gauge and non-gauge (“rigid”) symmetries, and we reduce the system by all of them, then this procedure leads to the same results as if we first remove the gauge redundancy and then reduce the remaining “rigid” symmetries. Finally, we analyze three examples, namely: non-autonomous dynamical systems with symmetry (comparing then the results so obtained with those of some of the above mentioned references), a mechanical model for field theories, and the conformal particle. The paper is organized in the following way: The first part is devoted to the study of presymplectic group actions. Thus, in Secs. 2.1 and 2.2, we review some basic concepts on presymplectic manifolds and present the actions of Lie groups on them. In Secs. 2.3 and 2.4 we define the comomentum and momentum mappings for this kind of actions, studying the obstruction to their existence, the level sets of the momentum map and their reduction. The second part deals with symmetries of presymplectic dynamical systems. First, in Sec. 3.1, we review the basic features of this kind of dynamical systems. Section 3.2 is devoted to defining and analyzing the concept of symmetry for these systems and to establish the reduction procedure for compatible presymplectic systems. The reduction procedure for non-compatible presymplectic systems and its characteristic features is established in Sec. 3.5. This part ends with a comparative study between this reduction method and other different ways for reducing presymplectic systems, which is performed in Secs. 3.3 and 3.4. In the third part some examples are analyzed. In Secs. 4.1 and 4.2 these techniques are applied in order to make the reduction of non-autonomous systems with symmetry and, as a particular example, the dynamics of autonomous regular dynamical systems is obtained in this context. A further example is the complete reduction of a particular case of a mechanical model of field theories coupled to external fields (due to Capri and Kobayashi), which is investigated in Secs. 4.3 and 4.4. As the last example, the gauge reduction of the system of a conformal particle is discussed, in this framework, in Sec. 4.5. Finally, we discuss the results and compare them with those obtained in some of the works above mentioned. An appendix is devoted to a linear interpretation of the reduction theory. All the manifolds are real, connected, second countable and C∞ . The maps are assumed to be C∞ and the differential forms have constant rank. Sum over
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crossed repeated indices is understood. We will denote by X (M ), Ωp (M ) and C∞ (M ) the sets of vector fields, differentiable p-forms and functions in the manifold M respectively. Finally i(X)α will denote the inner product or contraction of X ∈ X (M ) with α ∈ Ωp (M ) and L(X)α the Lie derivative of the form α along the vector field X. Finally, along the work, quotient of manifolds by involutive distributions will be made; and then we assume that the corresponding quotient spaces are differentiable manifolds (conditions in order to assure this fact are stated in [2]). 2. Presymplectic Group Actions 2.1. Presymplectic manifolds: Previous statements Let us first recall that a presymplectic manifold is a couple (M, Ω) where M is a m-dimensional differentiable manifold and Ω ∈ Ω2 (M ) is a closed degenerate differentiable form in M . Let ker Ω := {Z ∈ X (M )|i(Z)Ω = 0} which is assumed to be a distribution on M (that is, it has constant rank). A vector field X ∈ X (M ) is said to be a Hamiltonian vector field (with respect to the presymplectic structure Ω) iff i(X)Ω is an exact 1-form; that is, there exists fX ∈ C∞ (M ) such that (1) i(X)Ω = dfX . We will denote by Xh (M ) the set of Hamiltonian vector fields in M . X ∈ X (M ) is said to be a locally Hamiltonian vector field (with respect to the presymplectic structure Ω) iff i(X)Ω is a closed 1-form. In this case, for every point x ∈ M , there is an open neighbourhood U ⊂ M and f ∈ C∞ (U ) such that i(X)Ω|U = df . We will denote by Xlh (M ) the set of locally Hamiltonian vector fields in M , and it is obvious that Xh (M ) ⊂ Xlh (M ). On the other hand, it is also immediate to observe that X ∈ Xlh (M ) if, and only if, L(X)Ω = 0. Finally, for every X ∈ Xlh (M ) and Z ∈ ker Ω, we have that [X, Z] ∈ ker Ω. f ∈ C∞ (M ) is said to be a presymplectic Hamiltonian function iff there exist a vector field X ∈ X (M ) such that (1) holds. We will denote by Xf the Hamiltonian vector field associated with f and by C∞ h (M ) the set of presymplectic Hamiltonian functions in M . If f is a presymplectic Hamiltonian function then L(Z)f = 0, for every Z ∈ ker Ω (and the same results holds for locally Hamiltonian functions in U ⊂ M ). Since ker Ω ⊂ Xh (M ), then, if X ∈ Xh (M ) and Z ∈ ker Ω, then fX = fX+Z and, conversely, if X, Y ∈ Xh (M ) and fX = fY , therefore a vector field Z ∈ ker Ω exists such that X = Y + Z. On the other hand, if f ∈ C∞ h (M ) and λ ∈ R then (M ) and X = X then there exists λ ∈ R Xf = Xf +λ and, conversely, if f, g ∈ C∞ f g h such that f = g + λ (remember that M is supposed to be connected).
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Let f1 , f2 ∈ C∞ h (M ) be presymplectic Hamiltonian functions and X1 , X2 ∈ Xh (M ) Hamiltonian vector fields for these functions. The Poisson bracket of these Hamiltonian functions (related to the presymplectic structure Ω) is the function {f1 , f2 } given by {f1 , f2 } := Ω(X1 , X2 ) = i(X2 )i(X1 )Ω = i(X2 )df1 = −i(X1 )df2 . It is trivial to prove that this definition does not depend on the Hamiltonian vector fields we have chosen. In addition, {f1 , f2 } ∈ C∞ h (M ) and i([X1 , X2 ])Ω = d{f2 , f1 }, in fact, i([X1 , X2 ])Ω = L(X1 )i(X2 )Ω − i(X1 )L(X2 )Ω = L(X1 )i(X2 )Ω = L(X1 )df2 = d{f2 , f1 } hence, Xh (M ) is a Lie subalgebra of X (M ). The same thing holds for Xlh (M ) and ker Ω is an ideal of both algebras. So we have a map (f1 , f2 ) 7→ {f1 , f2 } defined in C∞ h (M )/R which transforms it into a real Lie algebra. In order to prove the Jacobi identity, observe that, from the last equality, we obtain that Ω([X1 , X2 ], X3 ) = −{f3 , {f2 , f1 }}. Considering the map Ω] : X (M ) → Ω1 (M ) defined by Ω] (X) := i(X)Ω, for every X ∈ X (M ), its restriction Ω]h : Xh (M ) → dC∞ h (M ) goes down to the quotient Xh (M )/ ker Ω, which is a Lie algebra because ker Ω is an ideal of the Lie ˜ : Xh (M )/ ker Ω → C∞ (M )/R is bijective algebra Xh (M ), and hence the map Ω h and, according to the previous remark, a Lie algebra (anti) isomorphism. 2.2. Actions of Lie groups on presymplectic manifolds Let G be a Lie group (which we will assume to be connected), g its Lie algebra, (M, Ω) a presymplectic manifold and Φ : G × M → M a presymplectic action of G on M ; that is, Φ∗g Ω = Ω, for every g ∈ G. As a consequence, the fundamental vector field ξ˜ ∈ X (M ), associated with every ξ ∈ g by Φ, is a locally Hamiltonian vector field, ξ˜ ∈ Xlh (M ) (conversely, if for every ξ ∈ g, we have that ξ˜ ∈ Xlh (M ), then Φ is a presymplectic action of G on M ). In this case we have that, for every ˜ = 0 or, what is equivalent, i(ξ)Ω ˜ ∈ Z 1 (M ) (it is a closed 1-form). We ξ ∈ g, L(ξ)Ω ˜ the set of fundamental vector fields. denote by g Now, following the same terminology as for actions of Lie groups on symplectic manifolds [1, 34, 46, 51], we state:
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Definition 1. Φ is said to be a strongly presymplectic or Hamiltonian action ˜ is an exact ˜ ⊆ Xh (M ) or, what is equivalent, for every ξ ∈ g, i(ξ)Ω of G on M iff, g form. Otherwise, it is called a weakly presymplectic or locally Hamiltonian action of G on M . It is important to discuss when a presymplectic action is strongly presymplectic. The fundamental obstruction appears because the map Ω] is not an isomorphism and, as a consequence, we have the following sequence of Lie algebras: 0 −→ ker Ω −→ Xh (M ) −→ Xh (M )/ ker Ω −→ 0 but Xh (M )/ ker Ω ' Ω] (Xh (M )), then denoting Xh (M )/ ker Ω ≡ Bh1 (M ), we have that Bh1 (M ) ⊂ B 1 (M ) (where B 1 (M ) is the set of exact differential 1-forms in M ) and it is a strict inclusion. In an analogous way we have the sequence 0 −→ ker Ω −→ Xlh (M ) −→ Xlh (M )/ ker Ω −→ 0 but Xlh (M )/ ker Ω ' Ω] (Xlh (M )), then denoting Xlh (M )/ ker Ω ≡ Zh1 (M ), we have that Zh1 (M ) ⊂ Z 1 (M ) (where Z 1 (M ) are the closed differential 1-forms in M ), and this is also a strict inclusion. There is no problem with these exact sequences and the morphisms relating them, but it is not possible to identify Xlh (M )/Xh (M ) with H 1 (M ) (the first de Rham’s cohomology group of M ), like in the symplectic case. Nevertheless, we have Xlh (M )/Xh (M ) ' (Xlh (M )/ ker Ω)/(Xh (M )/ ker Ω) ' Zh1 (M )/Bh1 (M ) and we can write 0 −→
[g, g] −→ yΞ
0 −→ Xh (M ) −→
&
g yΞ Xlh(M ) yρ
−→
g/[g, g] ˜ yΞ
−→ 0
−→ Zh1 (M )/Bh1 (M ) −→ 0
(2)
%
Xlh (M )/ ker Ω ˜ is a Lie algebra homomorphism which makes the diagram commutative. where Ξ Then, the action is strongly presymplectic (that is, the image of g by Ξ is in Xh (M )) ˜ = 0. Obviously, if H 1 (M ) = 0, then Z 1 (M ) = B 1 (M ) and if and only if Ξ Xh (M ) = Xlh (M ), therefore Zh1 (M ) = Bh1 (M ). In particular, if (M, Ω) is an exact presymplectic manifold (that is, there exists Θ ∈ Ω1 (M ) such that dΘ = Ω) and Φ is an exact action (that is, Φ∗g Θ = Θ, for every g ∈ G) then Φ is strongly presymplectic. 2.3. Momentum mapping Let G be a Lie group, (M, Ω) a presymplectic manifold and Φ : G × M → M a presymplectic action of G on M .
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Definition 2. (1) A comomentum mapping associated with Φ [50] is a Lie algebra map (if it exists) J ∗ : g → C∞ h (M ) ξ 7→ fξ ˜ = dfξ ; or, what is equivalent, such that the following diagram such that i(ξ)Ω commutes g J ∗y
Ξ
−→
Xlh(M ) yρ
&
(3)
˜ −1 ◦d Ω
1 1 0 −→ R −→ C∞ h (M ) −−−−→ Xlh (M )/ ker Ω −→ Zh (M )/Bh (M ) −→ 0 .
(2) A momentum mapping associated with Φ is the dual map of a comomentum mapping; in other words, it is a map J : M → g∗ such that, for every ξ ∈ g and x ∈ M, (J (x))(ξ) := J ∗ (ξ)(x) = fξ (x) . As in the symplectic case we have: Proposition 1. A comomentum map and the dual momentum map associated with the presymplectic action Φ on (M, Ω) exist if, and only if, the action is strongly presymplectic. ˜ −1 ◦ Proof. In fact; by definition, if a comomentum mapping exists, then Ω ∗ −1 ˜ = Xh (M )/ ker Ω, and this d ◦ J = ρ ◦ Ξ (see (3)), but then Im (ρ ◦ Ξ) ⊂ Im Ω implies that Im Ξ ⊂ Xh (M ) and the action is strongly presymplectic. Conversely, if the action is strongly presymplectic: Im Ξ ⊂ Xh (M ), then we have that (ρ ◦ Ξ)(g) ⊂ Xh (M )/ ker Ω and then, for all ξ ∈ g, there exists a unique ˜ (except constants) fξ ∈ C∞ h (M ) such that i(ξ)Ω = dfξ , and this is a Lie algebra homomorphism. Therefore, if a comomentum mapping exists, the commutative part of the diagram (3) reduces to g J ∗y
Ξ
−→
Xh (M ) yρ
˜ −1 ◦d Ω
C∞ h (M ) −−−−→ Xh (M )/ ker Ω . As in the symplectic case, it is important to point out that, if a comomentum map J ∗ exists for a presymplectic action, and F : g → R is a linear map (that is, F ∈ g∗ ), then J ∗ + F is another comomentum map for the same action Φ. Moreover, a comomentum map is not necessarily a Lie algebra homomorphism. Then: Definition 3. The action Φ is said to be a Poissonian or strongly Hamiltonian action iff:
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(1) There exists a comomentum mapping for this action (and then also a momentum one). (2) It is a Lie algebra homomorphism. As a particular case, we have that, if (M, Ω) is an exact presymplectic manifold with Ω = dΘ, and the action Φ of G on M is exact, then: ˜ = −i(ξ)Θ, ˜ (1) A momentum mapping exists and it is given by J (ξ) = −Θ(ξ) for every ξ ∈ g. (2) The action is Poissonian. In fact, the first item is immediate. For the second one we have f[ξ1 ,ξ2 ] = −i([ξ˜1 , ξ˜2 ])Θ = −L(ξ˜1 )i(ξ˜2 )Θ = L(ξ˜1 )fξ2 = {fξ1 , fξ2 } . In other cases, local comomentum mappings can always be defined for every presymplectic action, although without necessarily being Lie algebra homomorphisms. In addition, we have that if G is a connected Lie group and Φ is a strongly presymplectic action of G on the presymplectic manifold (M, Ω). Then the following statements are equivalent: (1) The momentum mapping associated with this action is Ad∗ -equivariant, that is, for every g ∈ G, the following diagram commutes: M −→ g∗ J ∗ Φg y yAdg
(4)
J
M −→ g∗ . (2) The action is Poissonian. (The proof of this statement is the same as for the symplectic case and can be found in any of the above mentioned references.) 2.4. Level sets of the momentum mapping First remember that, if Φ is a strongly presymplectic action of a Lie group G on (M, Ω) and J is the momentum mapping associated to this action, then µ ∈ g∗ is a weakly regular value of J iff: (1) J −1 (µ) is a submanifold of M . (2) Tx (J −1 (µ)) = ker Tx J , for every x ∈ J −1 (µ). If Tx J is surjective, µ is said to be a regular value. Of course, every regular value is weakly regular.
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Taking into account that if a fundamental vector field belongs to ker Ω its Hamiltonian function can be taken to be zero, we have that: Proposition 2. If µ is a weakly regular value of J then µ(ξ) = 0, for every ˜ ∩ ker Ω. ξ ∈ g such that ξ˜ ∈ g From now on we will assume µ ∈ g∗ is, at least, a weakly regular value of J . So, we will denote by jµ : J −1 (µ) ,→ M the corresponding imbedding. Then, in order to make a description of J −1 (µ), we have that the constraints defining it are the component functions of J = µ. In fact, observe that, if {ξi } is a basis of g, {fξi } are the Hamiltonian functions associated to the fundamental vector fields {ξ˜i } by the comomentum map and {αi } is the dual basis in g∗ , then µ = µi αi , with µi real numbers, and we have that J −1 (µ) := {x ∈ M |J (x) = µ} = {x ∈ M |(J (x))(ξ) = µ(ξ) ,
∀ ξ ∈ g}
= {x ∈ M |(J (x))(ξi ) = µi } = {x ∈ M |fξi (x) = µi } that is, jµ∗ fξi − µi = 0, and then the constraints are ζi := fξi − µi . Notice that this is equivalent to saying that the expression of the momentum mapping is J (x) ≡ fξi (x)αi .
(5)
Bearing in mind a well-known result in the theory of exterior differential systems (see, for instance, [8]), we have that all the level sets of the momentum mapping can also be obtained as the integral submanifolds of a Pfaff system. In fact: Proposition 3. Let G be a Lie group and Φ a strongly presymplectic action of G on the presymplectic manifold (M, Ω). The connected components of the level sets of the momentum mapping J associated with this action are the connected maximal ˜ = 0, for ξ˜ ∈ g ˜. integral submanifolds of the Pfaff system i(ξ)Ω As a consequence of this proposition, we obtain that: ˜x⊥ . As a consequence, since Corollary 1. If x ∈ J −1 (µ) then Tx J −1 (µ) = g ⊥ −1 −1 ˜x , then ker Ω ⊂ X (J (µ)) (where X (J (µ)) denotes the set of vector ker Ωx ⊂ g fields of X (M ) which are tangent to J −1 (µ)). ˜ and Pfaff system i(ξ)Ω ˜ =0 If Ω = dΘ and the action is exact, then fξ = −i(ξ)Θ ˜ can be equivalently expressed as di(ξ)Θ = 0. From now on, we will assume the following: Assumption 1. The action Φ that we will consider will be Poissonian, free and proper and µ will be a weakly regular value of the momentum mapping associated to this action.
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Let Gµ be the isotropy group of µ for the coadjoint action of G on g∗ . Then we have: Theorem 1. Gµ is the maximal subgroup of G which lets J −1 (µ) invariant and so the quotient J −1 (µ)/Gµ is well defined and it is called the reduced phase space or the orbit space of J −1 (µ). Proof. For every x ∈ M such that J (x) = µ and g ∈ Gµ , we have J (Φg (x)) = (J ◦ Φg )(x) = (Ad∗g ◦ J )(x) = Ad∗g (µ) = µ then Φg (x) ∈ J −1 (µ), so J −1 (µ) is invariant under the action of Gµ and the quotient is well defined. The maximality of Gµ is a direct consequence of the equivariance of J . ˜µ are If gµ is the Lie algebra of Gµ then, as a consequence of this theorem, g −1 −1 ˜µ = g ˜ ∩ X (J (µ)). vector fields tangent to J (µ), and we have that g At this point, it is interesting to point out two different possibilities: ˜ ∩ ker Ω = {0} : In this case all the fundamental vector fields give constraints • g which are not constant functions. Then dim J −1 (µ) < m = dim M . ˜ ∩ ker Ω 6= {0} : Now, only those fundamental vector fields such that ξ˜ ∈ • g / ker Ω give constraints which are not constant functions. Then dim J −1 (µ) ≤ m. J −1 (µ) inherits a presymplectic structure Ωµ := jµ∗ Ω. We are going to char˜µ ⊂ X (J −1 (µ)) (and hence, for every acterize ker Ωµ . First of all we have that g ˜ J −1 (µ) ). Consider now ˜ µ , there exists ξ˜µ ∈ X (J −1 (µ)) such that jµ∗ ξ˜µ = ξ| ξ˜ ∈ g −1 the orthogonal presymplectic complement of X (J (µ)) in X (M ), that is, the set (X (J −1 (µ)))⊥ := {Z ∈ X (M )|(i(X)i(Z)Ω)(x) = 0 ,
∀ X ∈ X (J −1 (µ)) ,
∀ x ∈ J −1 (µ)}
= {Z ∈ X (M )|jµ∗ i(Z)Ω = 0} . Then, let ker Ωµ := {Yµ ∈ X (J −1 (µ))|i(Yµ )Ωµ = 0} and denoting by ker Ωµ the set of vector fields of X (M ) such that ker Ωµ |J −1 (µ) = jµ∗ ker Ωµ , it is immediate to prove that ker Ωµ = X (J −1 (µ)) ∩ (X (J −1 (µ)))⊥ . Therefore, we have the following result: ˜µx + ker Ωx , for every x ∈ J −1 (µ). Proposition 4. ker Ωµx = g Proof. For the proof see the appendix with the following identifications: E = ˜x , N = S ⊥ = g ˜x⊥ = Tx J −1 (µ), and S ∩ N = g ˜x ∩ Tx J −1 (µ) = g ˜µx . Tx M , S = g
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At this point, we can state the following result which generalizes the idea of the Marsden–Weinstein reduction theorem [42, 52] to presymplectic actions of Lie groups on presymplectic manifolds: Theorem 2. The orbit space J −1 (µ)/Gµ is a differentiable manifold. Then, if σ : J −1 (µ) → J −1 (µ)/Gµ denotes the canonical projection, there is a closed 2-form ˆ (that is, Ωµ is σ-projectable), and : ˆ ∈ Ω2 (J −1 (µ)/Gµ ) such that Ωµ = σ ∗ Ω Ω ˆ is symplectic if, and only if, for every x ∈ J −1 (µ), g ˜µx = ker Ωµx or, what • Ω ˜µx . is equivalent, ker Ωx ∩ Tx J −1 (µ) ⊆ g ˆ is presymplectic. In particular, for every x ∈ J −1 (µ), if • Otherwise Ω ˆ = rank Ω. ˜x ∩ ker Ωx = {0}, then rank Ω ker Ωx ⊂ Tx J −1 (µ) and g Proof. For the proof of the first part of this statement (J −1 (µ)/Gµ is a differˆ and the two items of the entiable manifold) see [1, 42] or [34]. The existence of Ω second part are a direct consequence of Proposition 4. 3. Symmetries of Presymplectic Dynamical Systems and Reduction 3.1. Review on presymplectic dynamical systems One of the most important features in the study of dynamical systems with symmetry is the so-called reduction theory. Next we want to apply the above results in order to state the main results on this topic concerning presymplectic dynamical systems, generalizing the ideas of the Marsden–Weinstein symplectic reduction procedure [42, 52]. We start by giving the background ideas on presymplectic dynamical systems. For further information on this topic one may see, for instance, [7, 13, 14, 23, 25, 43] (see also [19] as a pioneering work). A presymplectic locally Hamiltonian dynamical system is a triad (P, ω, α), where (P, ω) is a presymplectic manifold and α ∈ Z 1 (P ). If α is exact then α = dH for some H ∈ C∞ (P ), and then the triad (P, ω, H) is said to be a presymplectic Hamiltonian system (and this is the case we are going to consider, without loss of generality). Every presymplectic dynamical system has associated the following equation: i(XP )ω = dH; XP ∈ X (P ) which is compatible everywhere in P if, and only if, i(Z)dH = 0, for every Z ∈ ker ω. In this case XP ∈ Xh (P ); and the presymplectic dynamical system is said to be compatible. If the equation is not compatible, in the most interesting cases, there is a (maximal) closed regular submanifold jM : M ,→ P , for which a vector field XP tangent to M exists such that the following equation holds [i(XP )ω − dH]|M = 0
(6)
(and this is an equation for XP and M ). M is called the final constraint submanifold ∗ ω. This submanifold is obtained at and inherits a presymplectic structure Ω = jM
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˜ ´ A. ECHEVERR´ IA-ENR´ IQUEZ, M. C. MUNOZ-LECANDA and N. ROMAN-ROY
the end of a recursive algorithm which gives a sequence of submanifolds P ←- P1 ←- · · · ←- Pi−1 ←- Pi ←- · · · ←- Pf −1 ←- Pf ≡ M . Equation (6) can be pulled-back to M obtaining ∗ [i(XP )ω − dH] = i(X)Ω − dH = 0 jM
(7)
∗ H, XP ∈ X (P ) tangent to M , X ∈ X (M ) and jM∗ X = XP |M . Notice with H = jM that this is a compatible system because if XP ∈ X (P ) tangent to M is a vector field solution of (6), this implies that Eq. (7) holds for X. For the final submanifold M the vector field XP satisfying (6) is not unique, in general. Then the difference between two solutions is called a gauge vector field; and the points of M reached from another fixed one x ∈ M by means of an integral curve of a gauge vector field (passing through x) are the so-called gauge equivalent points or states. Under certain regularity conditions, it is proved that the set of gauge vector fields is just ker Ω (the necessary and sufficient condition for this is the following [7, 22]: the constraint functions locally defining M in P can be classified into first and second class; then there is a basis of the set of first class constraints whose differentials do not vanish along M ). In order to remove the redundancy of solutions, it is assumed that gauge equivalent states represent the same physical state. Geometrically this means that we must go from M to the quotient of M by the foliation generated by the involutive ¯ ≡ M/ ker Ω is a differdistribution ker Ω. It is assumed that the quotient space M ¯ is a submersion and M ¯ is endowed entiable manifold, the projection πM : M → M ∗ ¯ ¯ = Ω. (M , ω ¯ ) is called the manifold of with a symplectic structure ω ¯ such that πM ω ¯: real physical states, and Eqs. (6) and (7) project in a natural way to M
¯ =0 ¯ P )¯ ω − dH i(X
(8)
∗ ¯ H = H and for every XP ∈ X (M ) which is solution of (6), there is a πM where πM ¯P projectable vector field X ∈ X (M ), with jM∗ X = XP |M , such that πM∗ X = X ¯ is assured because is the unique solution of (8). Note that the existence of H i(ker Ω)H = 0, since the dynamical Eq. (7) on M is compatible. This is the socalled gauge reduction procedure. On the other hand, the following structure theorem for presymplectic dynamical systems plays a crucial role in some of the developments of this work:
Theorem 3. Let (M, Ω, H) be a compatible presymplectic dynamical system. Then: (1) There exists a symplectic manifold (M, Ω) such that j0 : M ,→ M is a coisotropic imbedding, and j0∗ Ω = Ω. (2) There exists a family Dlh (M, M ) of symplectic locally Hamiltonian vector fields in M tangent to M, which gives all the dynamical solutions of the equation j0∗ [i(Xβ )Ω − dH] = i(X)Ω − dH = 0 ;
Xβ ∈ Dlh (M, M ) .
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(3) The symplectic manifold (M, Ω) and the family Dlh (M, M ) are unique up to symplectomorphic neighbourhood equivalences between symplectic manifolds containing (M, Ω) as a coisotropic submanifold; and all these symplectomorphisms reduce to the identity on M . (That is, if ji : M ,→ (Mi , Ωi ), i = 1, 2, are two coisotropic imbeddings, then there exist two tubular neighbourhoods Ui of ji (M ) in Mi and a symplectomorphism ψ : (M1 , Ω1 ) → (M2 , Ω2 ) such that ψ ◦ j1 = j2 and ψ∗ (Dlh (M1 , M ) = Dlh (M2 , M )). The pair (M, Ω) is called an ambient symplectic manifold for (M, Ω) and (M, Ω, H) is called an ambient symplectic dynamical system for the presymplectic system (M, Ω, H). Outline of the proof. The first part of this statement (together with the symplectomorphic equivalence of coisotropic imbeddings) is the well-known coisotropic imbedding theorem [24, 35]. The symplectic manifold (M, Ω) is constructed as a tubular neighbourhood of the zero section of the dual characteristic bundle K ∗ ≡ (ker Ω)∗ , which is identified with M . The strategy consists in considering the vector bundle πK : K → M ; then, as K is a subbundle of TM , using a metric in M , we can split TM = G ⊕ K, and then TM K ∗ = TM ⊕ K ∗ = G ⊕ K ⊕ K ∗ . Denoting ∗ Ω + σ ∗ ΩK ; where ΩK is the symplectic form σ : TM K → K ⊕ K ∗ , we set Ω = πK canonically defined in K ⊕K ∗ . Then Ω can be extended to a tubular neighbourhood of M in K ∗ using Weinstein’s extension theorem [52]. In relation to the second part, the family Dlh (M, M ) is made of the vector fields Xβ = Ω−1 (dH + β), where H ∈ C∞ (M) is an extension of H to M (that is, j0∗ H = H) and β ∈ Z 1 (M) is a closed first class constraint one form, (that is, j0∗ β = 0 and j0∗ i(Z)β = 0, ∀ Z ∈ X (M )⊥ ). (See [14] for the details of this part of the proof.) Finally, the local uniqueness of the coisotropic imbedding is a straightforward consequence of the local uniqueness part of Weinstein’s extension theorem. From now on we will consider presymplectic dynamical systems (P, ω, H) with final constraint submanifold (M, Ω, H) which hold all these features. 3.2. Reduction of compatible presymplectic dynamical systems with symmetry Consider now a compatible presymplectic dynamical system (M, Ω, H); that is, such that the dynamical equation i(X)Ω − dH = 0
(9)
has solution X ∈ X (M ) everywhere in M . We are then able to introduce the concept of group of symmetries for a compatible presymplectic dynamical system (and its reduction) as follows (the case of non-compatible systems will be studied afterwards): Definition 4. Let G be a Lie group, (M, Ω, H) a compatible presymplectic dynamical system and Φ : G × M → M an action of G on M . G is said to be a symmetry group of this system iff
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(1) Φ is a presymplectic action on (M, Ω) ˜ = 0, for every ξ ∈ g. (2) Φ∗g H = H; for every g ∈ G or, what is equivalent, L(ξ)H The diffeomorphism Φg , for every g ∈ G, is called a symmetry of the system. ˜ are the so-called infinitesimal generators of The fundamental vector fields ξ˜ ∈ g symmetries. Obviously this definition is the same as the usual one for symmetries of symplectic dynamical systems. At this point, we first prove that gauge symmetries are symmetries of the presymplectic dynamical system. Proposition 5. Let (M, Ω, H) be a compatible presymplectic dynamical system. Then the vector fields of ker Ω are infinitesimal generators of symmetries of this system. Proof. First, for every Z ∈ ker Ω, by definition i(Z)Ω = 0 and hence L(Z)Ω = 0. On the other hand, since Eq. (9) is compatible, it implies that i(Z)dH = L(Z)H = 0 .
According to the terminology of the above section, we say that the vector fields of ker Ω are infinitesimal generators of symmetries of the system. Then, if we want to remove the symmetries, following a reduction procedure in order to get a symplectic dynamical system, we must suppose that the vector fields in ker Ω are contained in ˜ . So, from now on we will assume that: the distribution generated by g ˜ be the vector space made of the fundamental vector fields Assumption 2. Let g ˜. of the action Φ of the symmetry group G on M. Then ker Ω ⊂ C∞ (M ) ⊗ g Comments: • This assumption means that, if {ξ1 , . . . , ξh } ⊂ g is a basis of g, and Z ∈ ker Ω; 1 h then there exist {f i , . . . , f i } ⊂ C∞ (M ) such that Z = f i ξ˜i . ˜ is the submodule of X (M ) made of the vector • Observe that C∞ (M ) ⊗ g fields tangent to the orbits of the action of G (or, what is equivalent, if the ˜ has constant dimension, they are the sections of this distribution defined by g distribution). Therefore, the assumption means that the elements of ker Ω are tangent to these orbits. Hence, the leaves of the foliation induced by ker Ω are contained in those orbits. • Notice that the elements of ker Ω are infinitesimal generators of symmetries ˜ are not. but, in general, those of C∞ (M ) ⊗ g In the usual physical terminology, the vector fields of ker Ω are called infinitesimal generators of gauge symmetries. On the contrary, the vector fields of (˜ g) which do not belong to ker Ω are the so-called infinitesimal generators of non-gauge or rigid symmetries. Now, suppose that the action of the symmetry group G on the compatible presymplectic dynamical system (M, Ω, H) is Poissonian. Let J be the momentum mapping associated with this action, and µ ∈ g∗ a weakly regular value of J .
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Then the submanifold J −1 (µ), the form Ωµ = jµ∗ Ω and the function Hµ = jµ∗ H make a presymplectic Hamiltonian system (J −1 (µ), Ωµ , Hµ ). Then we have: Proposition 6. If X ∈ Xh (M ) is a vector field solution of Eq. (9), then: (1) X is tangent to J −1 (µ). (2) The dynamical equation i(Xµ )Ωµ − dHµ = 0
(10)
is compatible and its solutions are Xµ + ker Ωµ , where Xµ ∈ X (J −1 (µ)) is a vector field such that jµ∗ Xµ = X|J −1 (µ) . Proof. First we prove that if X ∈ X (M ) is a solution of the Eq. (7), then ˜ ˜ , defining ξ˜ ∈ g X ∈ X (J −1 (µ)). In fact, for every constraint ζ, with dζ = i(ξ)Ω, −1 J (µ), ˜ jµ∗ X(ζ) = jµ∗ (i(X)dζ) = jµ∗ (i(X)i(ξ)Ω) ˜ ˜ = −jµ∗ (i(ξ)dH) = 0. = −jµ∗ (i(ξ)i(X)Ω) Therefore i(Xµ )Ωµ − dHµ = jµ∗ (i(X)Ω − dH) = 0 and the second result follows.
In addition we have that: Lemma 1. ker Ω ⊂ X (J −1 (µ)). (That is, ker Ω lets J −1 (µ) invariant.) Proof. In fact, take the constraint functions {ζ} defining J −1 (µ) such that ˜ ˜ . Then, if Z ∈ ker Ω, we have that dζ = i(ξ)Ω, for some ξ˜ ∈ g ˜ = −i(ξ)i(Z)Ω ˜ L(Z)ζ = i(Z)dζ = i(Z)i(ξ)Ω =0 therefore Z ∈ X (J −1 (µ)).
˜ µx = ker Ωµx ; for every x ∈ J −1 (µ). Lemma 2. g ˜µx + ker Ωx , and AssumpProof. By Proposition 4 we have that ker Ωµx = g ˜x . On the other hand, g ˜µx is the maximal subspace tion 2 gives us that ker Ωx ⊂ g ˜ x being tangent to J −1 (µ); and ker Ωx is made of vectors which are tangent to of g −1 ˜µx , and the J (µ) (as a consequence of the above lemma). Therefore ker Ωx ⊂ g result follows. ˆ Now, the last step is to obtain the orbit space (J −1 (µ)/Gµ , Ω). Theorem 4. Consider the presymplectic Hamiltonian system (J −1 (µ), Ωµ , Hµ ), the quotient manifold J −1 (µ)/ ker Ωµ , and the canonical projection πµ : J −1 (µ) → J −1 (µ)/ ker Ωµ . Then the function Hµ and the vector field Xµ ∈ ˆ H), ˆ is X (J −1 (µ)) of Proposition 6 are πµ -projectable. Hence (J −1 (µ)/ ker Ωµ , Ω, a symplectic Hamiltonian system and
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ˆ Ω ˆ − dH ˆ =0 i(X) ˆ = Hµ and πµ∗ Xµ = X. ˆ where πµ∗ H
(11)
˜ µx = ker Ωµx , for every Proof. According to the last lemma, we have that g x ∈ J −1 (µ), and so J −1 (µ)/Gµ = J −1 (µ)/ ker Ωµ . Then, taking into account the ˆ is a symplectic manifold. first item of Theorem 2 we have that (J −1 (µ)/ ker Ωµ , Ω) Now, in order to see that Hµ is πµ -projectable it suffices to prove that L(ξ˜µ )Hµ = ˜µ ⊂ g ˜ . But this holds since H is G-invariant and then Hµ is Gµ 0, for every ξ˜µ ∈ g invariant. ˜µ , we have that On the other hand, for every ξ˜µ ∈ g i([ξ˜µ , Xµ ])Ωµ = L(ξ˜µ )i(Xµ )Ωµ − i(Xµ )L(ξ˜µ )Ωµ = L(ξ˜µ )dHµ = 0 since Ωµ and Hµ are Gµ -invariant, and then [ξ˜µ , Xµ ] ∈ ker Ωµ . But, as all the elements of ker Ωµ are of the form Zµ = f i ξµi , then we also have that [Zµ , Xµ ] ∈ ker Ωµ , for every Zµ ∈ ker Ωµ ; and therefore, Xµ is πµ -projectable. Finally, Eq. (11) follows immediately from (10). We can summarize the procedure in the following diagram jµ πµ ˆ H) ˆ . (M, Ω, H) ←− (J −1 (µ), Ωµ , Hµ ) −→ (J −1 (µ)/ ker Ωµ , Ω,
ˆ H). ˆ The final result is a reduced symplectic dynamical system (J −1 (µ)/ ker Ωµ , Ω, Observe that, making only one quotient, we have removed the symmetries of the action of G and the non-uniqueness arising from the existence of ker Ω; then obtaining a symplectic dynamical system where G acts by the identity. From now on we will refer to this reduction scheme as the complete presymplectic reduction procedure. 3.3. Other reduction procedures: Gauge reduction plus symplectic reduction We finish this study by comparing the reduction method of presymplectic dynamical systems with symmetry here presented with the other step-by-step reduction procedures. In particular, the reduction procedure that has been developed in the last section, removes both rigid and gauge symmetries. Now, in this section, we make these procedures successively, proving that we obtain the same result as above. Consider a compatible presymplectic dynamical system (M, Ω, H). Let G be a group of symmetries of the presymplectic dynamical system, g its Lie algebra ˜ the corresponding algebra of fundamental vector fields. We suppose that and g Assumption 2 holds. First, we apply the gauge reduction procedure obtaining the reduced phase ¯. ¯,ω ¯ (which is a symplectic dynamical system), with πM : M → M space (M ¯ , H) So the gauge symmetries have been removed. Next we must study under what ¯ and, therefore, the corresponding nonconditions the action of G goes down to M gauge symmetry can be removed by means of the standard symplectic reduction procedure of Marsden–Weinstein.
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Proposition 7. If Φ : G × M → M is a (strongly) presymplectic action, then ¯ : G× M ¯ →M ¯ such that Φ ¯ g (πM (p)) := πM (Φg (p)); there exists a “reduced action” Φ for every g ∈ G and p ∈ M. ¯ is well defined, we must prove Proof. In order to see that this reduced action Φ that, given p1 , p2 ∈ M belonging to the same leaf of the foliation defined by ker Ω, then Φg (p1 ) = Φg (p2 ), for every g ∈ G. If p1 , p2 are in the same leaf of this foliation, then they are connected by a (piecewise) regular curve, which is made of (pieces of) integral curves of vector fields belonging to ker Ω. But, if γ(t) is an integral curve of some Z ∈ ker Ω, then (Φg ◦ γ)(t) is an integral curve of Φg∗ Z, which is also a vector field in ker Ω since, as Φ is a presymplectic action, we have ∗ −1 ∗ i(Φg∗ Z)Ω = i(Φg∗ Z)(Φ−1 g ) Ω = (Φg ) (i(Z)Ω) = 0 .
¯ can be defined as is set in the statement. Therefore the action Φ
Now the problem is that, although the action Φ can be assumed to be free, the ¯ is not so in general; because Assumption 2 implies that the leaves reduced action Φ of the foliation induced by ker Ω are in the orbits of G and, then, the quotient by ker Ω leads to a non free action, in general. Hence we set the following hypothesis (which is implicitly assumed in the physical literature): ˜ be the vector space of the fundamental vector fields of Assumption 3. Let g the action Φ of the symmetry group G on M. Then there is a subalgebra G ⊂ g, ˜ ˜ ⊂g ˜ verifies that ker Ω ⊂ C∞ (M ) ⊗ G. such that the corresponding G Now, suppose that G ⊂ G (the subgroup having G as Lie algebra) is closed; and ¯ := G/G be the quotient group, which acts on M ¯ by the reduced action Φ. ¯ let G ¯ and by g ˜ M¯ the corresponding set of fundamental Denote by gM¯ the Lie algebra of G, vector fields. Then we have the projections g −→ gM¯ −→ 0 ¯
ξ 7→ ξ M and the duals ∗ ∗ 0 −→ gM ¯ −→ g
µ ¯ 7→ µ .
(12)
We have: ˜ is πM -projectable. Lemma 3. The set of fundamental vector fields g ˜ and for every Z ∈ ker Ω, we have that Proof. In fact, for every ξ˜ ∈ g ˜ Z])Ω = L(ξ)i(Z)Ω ˜ ˜ =0 i([ξ, − i(Z)L(ξ)Ω ˜ Z] ∈ ker Ω and the result follows. therefore [ξ, Therefore:
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Proposition 8. With the above assumptions, if the action Φ of G on M is ¯ of the quotient group G ¯ strongly presymplectic and free, then the reduced action Φ ¯ is strongly symplectic and free. on M ¯ is free. In fact; if g¯ ∈ G ¯ and p ∈ M , then Proof. First we prove that Φ ¯ ¯ Φg¯ (πM (p)) = πM (p), by definition. But, if Φg (πM (p)) = πM (p), for some g ∈ G, then πM (Φg (p) = πM (p), and hence g ∈ G. Therefore the isotropy group of a point ¯ on M ¯ is free. of the action of G on M is G, thus the action of G ¯ ˜ with Second we prove that Φ is strongly symplectic. In fact; for every ξ˜ ∈ g ˜ i(ξ)Ω = dfξ , we have that fξ is πM -projectable. In fact ˜ = −i(ξ)i(Z)Ω ˜ =0 L(Z)fξ = i(Z)dfξ = i(Z)i(ξ)Ω ¯ ) such that π ∗ f¯ξ = fξ is for every Z ∈ ker Ω. Then the function f¯ξ ∈ C∞ (M M ¯ ¯ ˜M¯ such that πM∗ ξ˜ = ξ˜M , since Hamiltonian for the fundamental vector field ξ˜M ∈ g ∗ ∗ ¯ ∗ ˜ = i(ξ)π ˜ ∗ ω ˜M¯ ω df¯ξ = dπM fξ = dfξ = i(ξ)Ω πM M ¯ = πM i(ξ )¯ ¯
ω , since πM is a submersion. and hence df¯ξ = i(ξ˜M )¯
¯ as a Now, we can define the reduced comomentum mapping associated with Φ Lie algebra linear map ¯) J¯∗ : gM¯ → C∞ (M ¯ ξ M 7→ f¯ξ ¯ ¯ is its such that i(ξ˜M )¯ ω = df¯ξ . The reduced momentum mapping associated with Φ ¯ → g∗¯ ; that is, for every ξ M¯ ∈ gM¯ and x ∈ M , dual map, J¯ : M M ¯ ¯ (J¯(πM (x)))(ξ M ) := J¯∗ (ξ M )(πM (x)) = f¯ξ (πM (x)) .
¯ Finally, it can be proved that if Φ is a Poissonian and proper action, then so is Φ. At this point, the standard symplectic reduction program is applied: µ), ωµ¯ , Hµ¯ ), for weakly regular values • First we construct the level sets (J¯−1 (¯ ∗ ¯,ω . Each one of them is locally defined in (M ¯ ) by the constraints µ ¯ ∈ gM ¯ ¯ {fξ }. ¯ µ¯ and its Lie algebra (gM¯ )µ¯ , for which • Second we take the isotropy group G ¯ µ¯ . µ)/G we have that (˜ gM¯ )µ¯ = ker ωµ¯ . Then we make the quotient J¯−1 (¯ ˆ is µ)/ ker ωµ¯ , ω ˆ , h) After that, the reduced symplectic Hamiltonian system (J¯−1 (¯ free of gauge and rigid symmetries and the following theorem proves that it coˆ H), ˆ which is the one obtained after the complete incides with (J −1 (µ)/ ker Ωµ , Ω, presymplectic reduction procedure (for µ and µ ¯ related as shown in (12)). Theorem 5. There exists a diffeomorphism ˆ ˆ H) ˆ −→ (J¯−1 (¯ µ)/ ker ωµ¯ , ω ˆ , h) ρ : (J −1 (µ)/ ker Ωµ , Ω, ˆ ˆ = ρ∗ h. ˆ = ρ∗ ω ˆ and H such that Ω
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Proof. First of all, we can construct a map τ : (J −1 (µ), Ωµ , Hµ ) → (J¯−1 (¯ µ), ωµ¯ , Hµ¯ ) verifying the relation jµ¯ ◦τ = πM ◦jµ (see the diagram below). Observe that, µ), by Proposition 8 and the relation between µ if p ∈ J −1 (µ) then πM (p) ∈ J¯−1 (¯ and µ ¯ (see (12)); hence τ is well defined and it is a surjective submersion (since so is πM ). Moreover, we have that ∗ ω ¯ = τ ∗ jµ∗¯ ω ¯ = τ ∗ ωµ¯ Ωµ = jµ∗ πM
and in the same way we obtain that Hµ = τ ∗ Hµ¯ . ˆ H) ˆ −→ (J¯−1 (¯ µ)/ ker ωµ¯ , Now there is an unique map ρ : (J −1 (µ)/ ker Ωµ , Ω, ˆ ω ˆ , h) such that ρ ◦ πµ = τ ◦ πµ¯ . So we have the diagram
complex presymplectic reduction
(M, Ω, H) x jµ (J 1 (µ), Ωµ , Hµ ) x πµ
π
M −→
¯,ω ¯ (M ¯ , H) x jµ¯
τ
(J¯−1 (¯ µ), ωµ¯ , Hµ¯ ) x πµ¯
−→ ρ
ˆ ˆ H) ˆ −→ (J¯−1 (¯ (J −1 (µ)/ ker Ωµ , Ω, µ)/ ker ωµ¯ , ω ˆ , h)
standard symplectic reduction.
The map ρ has the following properties: (1) ρ is well defined: Let p1 , p2 ∈ J −1 (µ) such that πµ (p1 ) = πµ (p2 ); we have to prove that πµ¯ τ (p1 ) = πµ¯ τ (p2 ). Since πµ (p1 ) = πµ (p2 ), it implies that p1 , p2 can be joined by a polygonal made of integral curves of vector fields of ker Ωµ (but we will take a single curve, since it suffices to repeat the reasoning a finite number of times). For every point p of the curve, if Zp ∈ Tp J −1 µ is tangent to this curve at p, as τ is a surjective submersion and τ ∗ ωµ¯ = Ωµ , we have that i(Zp )(Ωµ )p = i(Zp )(τ ∗ ωµ¯ )p = τp∗ i(Tp τ (Zp ))(ωµ¯ )πτ (p) = 0 . As a consequence Tp τ |ker (Ωµ )p : ker (Ωµ )p → ker (ωµ¯ )τ (p) is surjective and ker Tp τ ⊂ ker (Ωµ )p . Hence, the curve joining p1 and p2 can be covered by a finite number of open sets satisfying this property and, then, τ (p1 ) and τ (p2 ) are connected by a polygonal of integral curves of vector fields of ker ωµ¯ ; that is, πµ¯ τ (p1 ) = πµ¯ τ (p2 ). (2) ρ is bijective: τ maps the leaves of the foliation defined by ker Ωµ into the leaves of the foliation defined by ker ωµ¯ ; therefore ρ is injective. To see that ρ is surjective is trivial. (3) ρ is a diffeomorphism: πµ : J −1 (µ) → J −1 (µ)/ ker Ωµ is a surjective submersion, then there are differentiable local sections sµ : J −1 (µ)/ ker Ωµ → J −1 (µ) (that is, such that πµ ◦ sµ = Id); hence, locally, we have that ρ = πµ¯ ◦ τ ◦ sµ and then ρ is differentiable since so are πµ¯ , τ and sµ . (The choice of the local sections sµ is called gauge fixing in the physical literature.)
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Now we must prove that ρ−1 is differentiable. Taking into account the theorem of the inverse function, it is sufficient to prove that the tangent map Tπµ (p) ρ is an isomorphism for every p ∈ J −1 (µ). Then, let u ∈ Tπµ (p) (J −1 (µ)/ ker Ωµ ) such that Tπµ (p) ρ(u) = 0 and v ∈ (Tp πµ )−1 (u). By the commutativity of the diagram above we have that Tp (πµ¯ ◦ τ )(v) = 0; hence Tp τ (v) ∈ ker Tτ (p) πµ¯ = ker (ωµ¯ )τ (p) and therefore v ∈ ker (Ωµ )p because (Tp τ )−1 (ker (ωµ¯ )τ (p) ) = ker (Ωµ )p . But ker (Ωµ )p = ker Tp πµ , therefore u = Tp πµ (v) = 0, hence Tπµ (p) ρ is injective and, as a consequence, it is an isomorphism. (4) ρ is a symplectomorphism: ˆ and As Ωµ = πµ∗ Ω, ˆ = πµ∗ ρ∗ ω ˆ Ωµ = τ ∗ ωµ¯ = τ ∗ πµ∗¯ ω ˆ = πµ∗ ρ∗ ω ˆ = ρ∗ ω ˆ and, since πµ is a submersion, we have that Ω ˆ. hence πµ∗ Ω ∗ (5) The proof for Hµ = τ Hµ¯ is like in the last item. 3.4. Other reduction procedures: Coisotropic imbedding plus symplectic reduction Let (M, Ω, H) be a compatible presymplectic dynamical system and (M, Ω, H) an ambient symplectic system associated to it (see Sec. 3.1). If the presymplectic dynamical system exhibits non-gauge symmetries as well as gauge symmetries, under certain hypothesis, both can be removed applying the standard symplectic reduction procedure of Marsden–Weinstein to the symplectic dynamical system (M, Ω, H). On the other hand, we can apply the presymplectic reduction method here explained and then we will prove that both procedures also lead to the same final result. Let G be a group of symmetries of the presymplectic dynamical system, g its Lie ˜ ⊂ Xh (M ) the corresponding set of fundamental vector fields. First of algebra and g all, using the coisotropic imbedding theorem, it can be proved [14] that, for every presymplectomorphism Φg : (M, Ω) → (M, Ω), there exists a symplectomorphism Φg : (M, Ω) → (M, Ω) such that it reduces to Φg on M ; that is, Φg ◦ j0 = j0 ◦ Φg . Taking this into account, we will assume that: Assumption 4. The presymplectic action Φ : M × G → M can be extended to a symplectic action Φ : M × G → M which reduces to Φ on M × G, that is, Φg ◦ (j0 × IdG ) = j0 ◦ Φg (and it is also assumed to be Poissonian, free and proper). ˜ M ⊂ Xh (M) the set of fundamental vector fields for this Then, denoting by g ˜M , there ˜M ⊂ X (M ) and then, for every ξ˜0 ∈ g extended action, it is obvious that g 0 ˜ ˜ ˜ ˜ such that j0∗ ξ = ξ |M , and conversely. exists one ξ ∈ g Let J : M → g∗ be a momentum map associated with the extended action Φ. Once again, the standard symplectic reduction program can be applied: • First constructing the level sets (J−1 (µ), Ωµ , Hµ ), for weakly regular values µ ∈ g∗ of J. Each one is locally defined in (M, Ω) by the constraints {fξ } ˜M. which are the Hamiltonian functions of the vector fields of g
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• Second taking the isotropy group Gµ and its Lie algebra gµ , for which we ˜ µ = ker Ωµ , and constructing the quotient J−1 (µ)/Gµ . have g After that, the corresponding reduced symplectic Hamiltonian system (J−1 (µ)/ ˆ H) ˆ is free of symmetries. ker Ωµ , Ω, On the other hand, we can consider the momentum map J : M → g∗ which is compatible with J. This means that J is induced by J on M ; that is, J := J ◦ j0 . We can then apply the complete presymplectic reduction procedure constructing the level sets (J −1 (µ), Ωµ , Hµ ), for weakly regular values µ ∈ g∗ of J , and the quotient J −1 (µ)/Gµ . So we have the (commutative) diagram j0 (M, Ω, H) ←− (M, Ω, H) x x j j µ µ standard complete j0µ −1 −1 presymplectic symplectic (J (µ), Ω , H ) ←− (J (µ), Ω , H ) µ µ µ µ reduction 0 reduction . yπµ y πµ −1 ˆ0 −1 ˆ ˆ ˆ ˆ (J (µ)/ ker Ωµ , Ω, H) ←− (J (µ)/ ker Ωµ , Ω, H) Then, if µ ∈ g∗ is a weakly regular value for J and J , we wish to compare ˆ H) ˆ with (J −1 (µ), Ωµ , Hµ ) and the sets (J−1 (µ), Ωµ , Hµ ) and (J−1 (µ)/ ker Ωµ , Ω, −1 ˆ H) ˆ respectively. So we have: (J (µ)/ ker Ωµ , Ω, Theorem 6. With the conditions stated in Assumption 4 : (1) j0µ (J −1 (µ)) is a connected component of J−1 (µ), or a union of connected ∗ ∗ Ωµ = Ωµ and j0µ Hµ = Hµ . components of it. Moreover, j0µ −1 (2) ˆ0 (J (µ)/ ker Ωµ ) is a connected component of the quotient J−1 (µ)/ ˆ =Ω ˆ and ker Ωµ , or a union of connected components of it. Moreover, ˆ∗0 Ω ∗ˆ ˆ ˆ0 H = H. Proof. Let {ξi } be a basis of g and fξi ∈ C∞ (M) Hamiltonian functions for ˜ M . Let {αi } be the dual basis in g∗ and µ = µi αi (with the corresponding ξ˜i0 ∈ g µi ∈ R) a weakly regular value for J and J . First, we are going to prove that, if the submanifold J−1 (µ) ,→ M is locally defined by the constraints ζi := fξi − µi , then the constraints j0∗ ζi define locally the submanifold J −1 (µ) ,→ M. In fact; as J := J ◦ j0 we have that J(x) = J (x), for every x ∈ M , and, taking into account (5), this implies that j0∗ fξi (x)αi = fξi (x)αi and thus j0∗ fξi = fξi . Therefore: • If ξ˜i0 6∈ ker Ω (that is, it generates infinitesimal non-gauge symmetries), then the corresponding ξ˜i ∈ X (M ) is not in ker Ω and hence dj0∗ fξi = j0∗ dfξi = j0∗ (i(ξ˜i0 )Ω) = i(ξ˜i )Ω = dfξi . As dfξi 6= 0 then j0∗ fξi is not constant on M , but jµ∗ j0∗ fξi is; hence fξi are constraints for J −1 (µ) ,→ M necessarily, but not for M ,→ M. Conversely, every constraint function ζ ∈ C∞ (M ) for J −1 (µ) ,→ M can be extended to a function of ζ 0 ∈ C∞ (M) such that its Hamiltonian vector field does not belong to ker Ω.
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• If ξ˜i0 ∈ ker Ω (that is, it generates infinitesimal gauge symmetries), then the corresponding ξ˜i ∈ X (M ) is in ker Ω and hence 0 = i(ξ˜i )Ω = i(ξ˜i )(j0∗ Ω) = j0∗ (i(ξ˜i0 )Ω) = j0∗ dfξi = dj0∗ fξi . So fξi is constant on M and therefore it is a constraint for J −1 (µ) ,→ M, as well as for M ,→ M. Conversely, for every constraint function φ for M ,→ M, its Hamiltonian vector field necessarily belongs to ker Ω. On the other hand, since j0 is a coisotropic imbedding, we have that dim (ker Ω) = dim M − dim M ; and as dim (J−1 (µ)) (in M) is equal to dim M − dim g, we obtain that dim (J −1 (µ)) = dim M − (dim g − dim (ker Ω)) = dim M − dim g = dim (J−1 (µ)) . Hence we conclude that j0µ (J −1 (µ)) is a submanifold of J−1 (µ) and, as both have the same dimension, we can conclude that j0µ (J −1 (µ)) is open in J−1 (µ). So, it is a connected component of J−1 (µ), or a union of connected components of it (remember that both manifolds are closed, since they are defined by constraints). In addition, ∗ ∗ ∗ Ωµ = j0µ jµ Ω = jµ∗ j0∗ Ω = Ωµ j0µ ∗ and, in the same way j0µ Hµ = Hµ . Finally, the results for the reduced phase spaces follow immediately from here.
A particular case of this reduction procedure (coisotropic imbedding plus sym˜M . Then (J−1 (µ), Ω, H) = (M, Ω, H) plectic reduction) is when ker Ω = C∞ (M )⊗ g ˆ H) ˆ = (M ¯,ω ¯ In this case, this method is the general¯ , H). and (J−1 (µ)/ ker Ωµ , Ω, ized symplectic reduction studied in [29] and [37]. (See also [45] for a study on other features on this topic.) The successive steps of the three reduction procedures here analyzed can be summarized in the following diagram: coisotropic imbedding j0
gauge reduction πM
(M, Ω, H)
←−
(M, Ω, H)
− −−−→
¯,ω ¯ (M ¯ , H)
(J−1 (µ), Ωµ , Hµ )
=
(J −1 (µ), Ωµ , Hµ )
τ
π y µ
−→
(J¯−1 (¯ µ), ωµ¯ , Hµ¯ )
ˆ H) ˆ (J−1 (µ)/ ker Ωµ , Ω,
=
ˆ H) ˆ (J −1 (µ)/ ker Ωµ , Ω,
'
ρ
ˆ (J¯−1 (¯ µ)/ ker ωµ¯ , ω ˆ , h)
x jµ
standard symplectic reduction
x jµ
π y µ
complete presymplectic reduction
x jµ¯
π y µ¯
standard symplectic reduction
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(where the equalities mean that the imbeddings of the manifolds in the center column are (a union of) connected components of the corresponding manifolds in the left column). 3.5. Reduction of non-compatible presymplectic dynamical systems with symmetry The concept of group of symmetries can also be established for a non-compatible presymplectic dynamical system (P, ω, H) with final compatible system (M, Ω, H) (where M is the final constraint submanifold). Thus, from Definition 4 we state: Definition 5. Let (P, ω, H) be a non-compatible presymplectic dynamical system with final compatible system (M, Ω, H), G a Lie group and Ψ : G × P → P an action of G on P . G is said to be a symmetry group of this system iff (1) Ψ leaves M invariant; that is, it induces an action Φ : G × M → M such that Ψ ◦ (IdG × jM ) = jM ◦ Φ. (2) The induced action Φ is a presymplectic action on (M, Ω) (which is assumed to be Poissonian, free and proper); that is, for every g ∈ G, ∗ (Ψ∗g ω − ω) = Φ∗g Ω − Ω = 0 . jM
(Following a very usual terminology in physics, we will say that Ψ is a weakly presymplectic action on (P, ω, M ).) (3) For every g ∈ G, ∗ (Ψ∗g H − H) = Φ∗g H − H = 0 . jM Of course all the results discussed in the above sections hold for the compatible dynamical system (M, Ω, H) and the action Φ. In particular, let g be the Lie algebra ˜P the sets of fundamental vector fields on M and P (with respect ˜ M and g of G and g ˜M to the actions Φ and Ψ) respectively. Then, for every ξ ∈ g, there exist ξ˜M ∈ g ˜P , with jM∗ ξ˜M = ξ˜P |M , such that and ξ˜P ∈ g ∗ L(ξ˜P )H = L(ξ˜M )H = 0 , jM ∗ L(ξ˜P )ω = L(ξ˜M )Ω = 0 . jM
As the level sets of the momentum map associated to the action Φ are submanifolds of M , J −1 (µ) ,→ M , we may ask how are they defined as submanifolds of P . We have the following diagram: Ψg
(P, ω) x jM
−→
(M, Ω) x jµ
−→
(J −1 (µ), Ωµ )
Φg
(P, ω) x jM (M, Ω) x jµ (J −1 (µ), Ωµ ) .
Taking into account the above discussion, we have that the constraint functions ζiM ∈ C∞ (M ) defining J −1 (µ) in M can be extended to P as functions ζiP ∈ C∞ (P ) ∗ P ζi = ζiM since, if {ξi } is a basis of g, we have such that jM
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∗ ∗ ∗ P dζiM := i(ξ˜iM )Ω = jM (i(ξ˜iP )ω) = jM dζiP = d(jM ζi ) .
Hence the submanifolds J −1 (µ) ,→ P are defined in P by these constraints ζ P together with the constraints η defining M in P . As a particular case of special interest we have: Proposition 9. With the conditions of Definition 5, and assuming the following hypothesis: (1) Assumption 2 holds for (M, Ω). (2) There is a basis of constraint functions {ηi } defining M in P made of presymplectic Hamiltonian functions in (P, ω). (3) The presymplectic Hamiltonian vector fields Xηi ∈ Xh (P ) associated to these constraints are tangent to M. Then the momentum map J : M → g∗ can be extended to a map J : P → g∗ such that J = J ◦ jM and J −1 (µ) = J−1 (µ). Proof. First of all, from items 2 and 3 we have that Xηi expand locally the ˜ . Then, the map set ker Ω [7, 22] and, from item 1, we have also that Xηi ∈ g J : P → g∗ is defined in the following way: for every ξ ∈ g let fξM ∈ C∞ (M ) be the presymplectic Hamiltonian function of the fundamental vector field ξ˜M ∈ X (M ) and fξP ∈ C∞ (P ) its extension to P ; then, for every p ∈ P , (J(p))(ξ) := fξP (p) . Observe that if ξ˜M ∈ ker Ω then fξM is constant and therefore fξP ≡ η is a constraint for M in P ; whereas if ξ˜M 6∈ ker Ω then fξM is a constraint for J −1 (µ) in M . Thus J = J ◦ jM and J −1 (µ) = J−1 (µ) (see the proof of Theorem 6). It is important to point out that, in general, J is not strictly speaking a momentum map, because the action Ψ is not necessarily presymplectic for (P, ω). In any case, if Assumption 2 is assumed for (M, Ω), the reduction program follows in the same way as in the case of compatible presymplectic dynamical systems. 4. Examples 4.1. Reduction of non-autonomous systems with symmetry in the presymplectic formulation Non-autonomous dynamical systems can be geometrically treated in several ways (see, for instance, [1, 18, 20, 27, 30, 47] for a review on these formulations). Reduction of time-dependent systems with symmetry can be achieved by using the extended or symplectic formulation, and then by using, then, the usual reduction theory for symplectic systems with symmetry (see, for instance, [9] and [33]). Nevertheless, we will use reduction for presymplectic systems, (whose features we have just presented) because it has some advantages in relation to the symplectic case; for instance, singular time-dependent systems with symmetry can be treated in this formulation in a very natural way. Thus we need to use the presymplectic formulation of non-autonomous systems [15, 20]. The main characteristics of this formulation are the following: the dynamics
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takes place in a differentiable manifold P × R, where (P, ωP ) is either a symplectic manifold, if the non-autonomous system is regular, or a presymplectic one, if it is singular; and we have the natural projections τ :P ×R→P ;
t : P ×R → R.
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The dynamical information is entirely contained in a function h ∈ C∞ (P × R). Then, P × R is endowed with the following presymplectic structure: Ωh := τ ∗ ωP + dh ∧ dt which is exact if, and only if, so is ωP . So, we have the dynamics fully included in the geometry and, therefore, we can obtain the equations of motion stating that (P × R, Ωh , 0) is a presymplectic Hamiltonian system; that is, in the presymplectic equations of motion i(X)Ωh = dH, we take H = 0. Then the equations of motion are reduced to i(X)Ωh = 0 , X ∈ X (P × R) . Since dH = 0, these equations are compatible in P × R and, consequently, there exists solution X ∈ ker Ωh . On the other hand, if we want to yield the timereparametrization t = s we must add the equation i(X)dt = 1; (however, other possible reparametrizations having physical sense are also possible [5, 16]). We differentiate the following situations: • The Lagrangian formalism of non-autonomous systems: P is the tangent bundle TQ of the configuration space Q. Then, given a time-dependent Lagrangian function L ∈ C∞ (TQ × R), using the extensions to TQ × R of the natural geometric structures in the tangent bundle (the vertical endomorphism and the Liouville’s vector field ), we can construct the exact form ωL ∈ Ω2 (TQ × R), which plays the role of the form τ ∗ ωP , and the energy Lagrangian function EL ∈ C∞ (TQ) which plays the role of h in this formalism. Then Ωh ≡ ΩL = ωL + dEL ∧ dt (see [20] and [21] for a discussion on the construction of these elements). If the system is regular, then the form ωL is symplectic. If the system is singular then ωL is a presymplectic form. • The Hamiltonian formalism of non-autonomous systems: If the system is not singular, then (P, ωP ) is a symplectic manifold, P being the cotangent bundle T∗ Q of the configuration space Q (if it is hyper-regular ) or an open submanifold of it (if it is regular ) and ωP ≡ ω ∈ Ω2 (T∗ Q) being its natural canonical form, which is an exact form. Then, h ∈ C∞ (T∗ Q) is the time-dependent Hamiltonian function. If the system is singular, then (P, ωP ) is a presymplectic manifold, j : P ,→ T∗ Q being a submanifold of the cotangent bundle T∗ Q of the configuration space Q (really it is the image of TQ by the Legendre transformation) and ωP = j ∗ ω. Then, h ∈ C∞ (T∗ Q) is called the canonical time-dependent Hamiltonian function.
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Concerning the study of symmetries, time-dependent dynamical systems display some particular characteristics which are interesting to point out. Thus, from the geometrical and the dynamical point of view, a natural way of defining the concept of symmetry is the following: Definition 6. Let G be a Lie group, (P × R, Ωh ) a non-autonomous system and Φ : G × (P × R) → P × R an action of G on P × R. G is said to be a group of standard symmetries of this system iff, for every g ∈ G, (1) Φg preserves the forms τ ∗ ωP and dt; that is, Φ∗g τ ∗ ωP = τ ∗ ωP ;
Φ∗g dt = dt .
(2) Φg preserves the dynamical function h; that is, Φ∗g h = h. The diffeomorphisms Φg are called standard symmetries of the system. The first part of this definition is equivalent to that of cosymplectic action introduced in [33] and agrees also with the concept of standard canonical transformation for time-dependent Hamiltonian systems, which other authors have previously introduced see [5, 16]. As immediate consequences of this definition we have that: • If G is a group of standard symmetries of the non-autonomous system (P × R, Ωh ) then, for every g ∈ G, every standard symmetry Φg preserves the form Ωh ; that is, Φ∗g Ωh = Ωh . • G is a group of standard symmetries of the non-autonomous system (P × R, Ωh ) if, and only if, the following three conditions hold for every ξ ∈ g: ˜ ∗ ωP ) = 0 , (1) L(ξ)(τ
˜ = 0, (2) L(ξ)dt
˜ = 0. (3) L(ξ)h
• If G is a group of standard symmetries for the non-autonomous system (P × R, Ωh ) then it is also a symmetry group for the presymplectic Hamiltonian system (P × R, Ωh , 0) (in the sense of Definition 4). At this point, reduction of non-autonomous dynamical systems with symmetry (both in the Lagrangian or in the Hamiltonian formalism) is merely a direct application of the considerations we made above in order to reduce presymplectic systems with symmetry. 4.2. Autonomous dynamical systems As an example, we are going to analyze the time-independent dynamical systems as the particular case of non-autonomous regular systems which are invariant under time-translations. This study is identical for the Lagrangian and the Hamiltonian formalism and we will do it in general. Let (P × R, Ωh ) be a non-autonomous regular dynamical system (then P is either T∗ Q or TQ and dim P = 2r). G is the group of translations in time. The action Φ : G × (P × R) → P × R is effective, free and proper. The real Lie algebra ∂ and hence g∗ = {dt}. Thus, the set of g is spanned by the vector field ξ ≡ ∂t ∂ ˜ is generated by the vector field ξ˜ ≡ ∂t . fundamental vector fields g
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Suppose that the dynamical function h is time-independent, that is, ∂ ∂h = 0. h= L ∂t ∂t It is evident that this action verifies the conditions of Definition 6, and hence G is ∂ )Ωh = 0, a symmetry group for this system. Then the action is presymplectic: L( ∂t and, in addition, it is strongly presymplectic since the fundamental vector field is Hamiltonian (it is in fact an exact presymplectic action) and ∂ Ωh = dh . i ∂t In this way the momentum map is given by ∂ := h(x) (for every x ∈ P × R) (J (x)) ∂t and the level sets of this map, for every weakly regular value µ = µ0 dt ∈ g∗ , are J −1 (µ) := {x ∈ P × R|h(x) = µ0 } ∂ is tangent to all of (they are defined by the constraints ζ := h − µ0 and hence ∂t them). They are the hypersurfaces of constant energy in P × R. In each one, we have the presymplectic Hamiltonian system (J −1 (µ), Ωµ , 0), where Ωµ := jµ∗ Ωh = jµ∗ τ ∗ ΩP . Notice that, even though dim J −1 (µ) is even, Ωµ is ∂ , Xµ }, where Xµ ∈ X (J −1 (µ)) is the solution of presymplectic since ker Ωµ = { ∂t ˜µ is generated the dynamical equation i(Xµ )Ωµ = 0. So, in this case, since G = Gµ , g ∂ ˜µ ⊂ ker Ωµ . Therefore, applying the and we have that g by the vector field ξ˜ ≡ ∂t reduction theorems, we have the same situation as the first item in Theorem 2, and ˆ 0). This is hence this presymplectic system reduces to another one (J −1 (µ)/G, Ω, ˆ a (2r − 1)-dimensional differentiable manifold (and then Ω is a presymplectic form ˆ = 2r − 2) where the global coordinate t is avoided. The evolution with rank Ω equations are ˆ Ω ˆ = 0, X ˜ ∈ X (J −1 (µ)/G) . i(X)
Observe that the main advantage of this reduction procedure is that, in addition to eliminating the ignorable time-coordinate, it already gives the solution of dynamics directly on the corresponding hypersurface of constant energy. This is an advance in relation to the use of the symplectic reduction procedure of Marsden and Weinstein, applied for treating this same example but starting from the extended symplectic formalism of the non-autonomous systems. In this case, even though the reduced dynamical system is regular (and then symplectic), the symplectic reduction procedure removes time only (from the initial time-dependent system), but it does not give the dynamics on the constant-energy hypersurfaces which is obtained by projection, that is, after another step not included in the reduction procedure (see [33]). Nevertheless, a further reduction could be made by the residual part of ker Ωµ (that is those one generated by Xµ ) or, what is equivalent, make the reduction of the
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presymplectic Hamiltonian system (J −1 (µ), Ωµ , 0) by ker Ωµ . In this way we would have the situation of the second item in Theorem 2, and hence this presymplectic ˆ 0 , 0), where J −1 (µ)/ ker Ωµ is a (2r − 2)system reduces to (J −1 (µ)/ ker Ωµ , Ω ˆ 0 is a symplectic form. As a consequence, dimensional differentiable manifold and Ω there is no dynamics in this reduced system; that is, the orbit space is made of the dynamical trajectories of the initial time-independent dynamical system (for a fixed constant value of the energy). 4.3. A mechanical model of field theories: Description The following example we study is based on a mechanical model of field theories (coupled to external fields) due to Capri and Kobayashi [11, 12]. See also [17] for a deeper analysis. The general form of the Lagrangian of the system is L = ψ˙ ∗a mab ψ˙ b + ψ˙ ∗a cab ψ b − ψ ∗a c¯ab ψ˙ b − ψ ∗a rab ψ b where: — ψ a , ψ ∗b (a, b = 1, . . . , n) are scalar (complex) fields. In this mechanical model they will be interpreted as independent “coordinates” of certain 2n-dimensional configuration space Q. — mab , cab , c¯ab , rab are (time-independent) functional coefficients such that, in order L to be real, the matrices mab , rab are hermitian and c¯∗ab = −cba . In particular, if rank (mab ) < n, the Lagrangian is singular and this is the case of greatest interest to us. Lagrangians of this kind enable us to describe some relativistic bosonic field theories (after a (3+1)-decomposition), where the eventual coupling to external fields is tucked away in the coefficients cab and rab . In order to make the example more pedagogical, we will analyze the following simple case: a, b = 1, 2, 3 and 0 0 0 0 0 0 1 0 0 1 mab = 0 m2 0 cab = c¯ab = 0 i 0 rab = 0 1 0 . 2 0 0 m3 0 0 i 0 0 1 We can write the Lagrangian in its real form by the change ψ 1 = x1 + iy 1 ,
ψ 2 = x2 + iy 2 ,
ψ 3 = x3 + iy 3
ψ˙ 1 = u1 + iv 1 ,
ψ˙ 2 = u2 + iv 2 ,
ψ˙ 3 = u3 + iv 3
ψ ∗1 = x1 − iy 1 ,
ψ ∗2 = x2 − iy 2 ,
ψ ∗3 = x3 − iy 3
ψ˙ ∗1 = u1 − iv 1 ,
ψ˙ ∗2 = u2 − iv 2 ,
ψ˙ ∗3 = u3 − iv 3
and hence L = m2 ((u2 )2 + (v 2 )2 ) + m3 ((u3 )2 + (v 3 )2 ) + v 2 x2 − u2 y 2 + v 3 x3 − u3 y 3 − (x1 )2 − (y 1 )2 − (x2 )2 − (y 2 )2 − (x3 )2 − (y 3 )2 .
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Here, the configuration space is taken to be Q = R6 with local coordinates (xi , y i ) (i = 1, 2, 3) and TQ ' R12 with local coordinates (xi , y i ; ui , v i ), where ui , v i denote the generalized velocities corresponding to xi , y i respectively. Using the canonical structures of the tangent bundle TQ, the Lagrangian 2-form and the energy Lagrangian function are constructed: ωL = 2[m2 (dx2 ∧ du2 + dy 2 ∧ dv 2 ) + m3 (dx3 ∧ du3 + dy 3 ∧ dv 3 ) + dx2 ∧ dy 2 + dx3 ∧ dy 3 ] , EL = m2 ((u2 )2 + (v 2 )2 ) + m3 ((u3 )2 + (v 3 )2 ) + (x1 )2 + (y 1 )2 + (x2 )2 + (y 2 )2 + (x3 )2 + (y 3 )2 . The system is singular since ωL is presymplectic and ∂ ∂ ∂ ∂ , , , ker ωL ≡ ∂x1 ∂y 1 ∂u1 ∂v 1 (TQ, ωL , EL ) is a presymplectic dynamical system which is not compatible since ∂ ∂ 1 i dEL = 2x 6= 0 , i dEL = 2y 1 6= 0 . ∂x1 ∂y 1 So the constraints η1 := x1 = 0 ,
η2 := y 1 = 0
define locally a submanifold jM : M ,→ TQ where the vector fields which are solutions of the dynamical equation (i(X)ωL − dEL )|M = 0
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are the following: X|M = f 1 −
∂ ∂ ∂ ∂ ∂ ∂ ∂ + u2 2 + u3 3 + g 1 1 + v 2 2 + v 3 3 + F 1 1 ∂x1 ∂x ∂x ∂y ∂y ∂y ∂u 1 2 ∂ 1 3 ∂ (v + x2 ) 2 − (v + x3 ) 3 m2 ∂u m3 ∂u
+ G1
1 2 ∂ 1 3 ∂ ∂ + (u − y 2 ) 2 + (u − y 3 ) 3 1 ∂v m2 ∂v m3 ∂v
(15)
where f 1 , g 1 , F 1 , G1 are arbitrary functions. Now we consider two options: (1) If we look for solutions of the dynamics which are second order differential equations (SODE) then, in this case, we obtain such a solution taking the first two arbitrary functions to be f 1 = u1 and g 1 = v 1 . Therefore the stability of this vector field on the constraints η1 , η2 originates two new constraints (which are called non-dynamical constraints following the terminology of [44]) χ1 := u1 = 0 , χ2 := v 1 = 0 which, joined to the above ones η1 , η2 , define locally the submanifold jS : S ,→ TQ. Finally, the stability of the SODE vector field on the last constraints
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fixes the value of the remaining arbitrary functions to be F 1 = 0, G1 = 0. So the final constraint submanifold is S (local coordinates are (x2 , x3 , y 2 , y 3 , u2 , u3 , v 2 , v 3 ) and ΩS := jS∗ ωL = 2[m2 (dx2 ∧ du2 + dy 2 ∧ dv 2 ) + m3 (dx3 ∧ du3 + dy 3 ∧ dv 3 ) + dx2 ∧ dy 2 + dx3 ∧ dy 3 ] , ES := jS∗ EL = m2 ((u2 )2 + (v 2 )2 ) + m3 ((u3 )2 + (v 3 )2 ) + (x2 )2 + (y 2 )2 + (x3 )2 + (y 3 )2 . Observe that, in this example, (S, ΩS ) is a symplectic manifold. The SODE vector field tangent to S being the (unique) solution of the dynamical equation (i(X)ωL − dEL )|S = 0 is then X|S = u2 −
1 2 ∂ ∂ ∂ ∂ ∂ + u3 3 + v 2 2 + v 3 3 − (v + x2 ) 2 ∂x2 ∂x ∂y ∂y m2 ∂u 1 3 ∂ 1 2 ∂ 1 3 ∂ (v + x3 ) 3 + (u − y 2 ) 2 + (u − y 3 ) 3 . m3 ∂u m2 ∂v m3 ∂v
(See also [17] for a more detailed discussion on this analysis.) (2) If we look for solutions of the dynamics which are not SODE, then the stability of (15) on the constraints η1 , η2 fixes the value of the first two arbitrary functions to be f 1 = 0, g 1 = 0. So the final constraint submanifold is M (local coordinates are (x2 , x3 , y 2 , y 3 , u1 , u2 , u3 , v 1 , v 2 , v 3 ) and the coordinate ∗ ∗ ωL and EM := jM EL are the same as for ΩS and expressions of ΩM := jM ES respectively. Hence (M, ΩM ) is a presymplectic manifold with ∂ ∂ , ker ΩM ≡ ∂u1 ∂v 1 and the vector fields tangent to M being solutions of the dynamical Eq. (14) are X|M = u2 −
∂ ∂ ∂ ∂ ∂ + u3 3 + v 2 2 + v 3 3 + F 1 1 2 ∂x ∂x ∂y ∂y ∂u 1 2 ∂ 1 3 ∂ (v + x2 ) 2 − (v + x3 ) 3 m2 ∂u m3 ∂u
+ G1
1 2 ∂ 1 3 ∂ ∂ + (u − y 2 ) 2 + (u − y 3 ) 3 . ∂v 1 m2 ∂v m3 ∂v
4.4. A mechanical model of field theories: Symmetries and reduction Next we are going to study the symmetries of the systems, splitting the two cases considered in the above section; that is, we will apply the reduction procedure to the compatible dynamical systems (S, ΩS , ES ) and (M, ΩM , EM ).
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Both of them exhibit some non-gauge rigid symmetries which are rotations on Q whose infinitesimal generators are the following vector fields in Q x2
∂ ∂ − y2 2 , 2 ∂y ∂x
x3
∂ ∂ − y3 3 3 ∂y ∂x
and whose canonical liftings to TQ give the following fundamental vector fields: ∂ ∂ ∂ ∂ ξ˜1 = x2 2 − y 2 2 + u2 2 − v 2 2 , ∂y ∂x ∂v ∂u ∂ ∂ ∂ ∂ ξ˜2 = x3 3 − y 3 3 + u3 3 − v 3 3 . ∂y ∂x ∂v ∂u In fact, these vector fields generate infinitesimal symmetries for these presymplectic systems because both of them are tangent to S and M and they satisfy that ∗ L(ξ˜k )EL = 0 = jS∗ L(ξ˜k )EL jM ∗ jM L(ξ˜k )ωL = 0 = jS∗ L(ξ˜k )ωL
since L(ξ˜k )EL = 0 (k = 1, 2) and L(ξ˜k )ωL = 0, so both of them are presymplectic Hamiltonian vector fields in (S, ΩS , ES ) and (M, ΩM , EM ). 4.4.1. Reduction of the system (S, ΩS , ES ) Since (S, ΩS ) is a symplectic manifold, there are no gauge symmetries and the only symmetries to be taken into account are the rigid ones which have just been introduced. Denoting by G the corresponding group and by g its Lie algebra, then ˜ ≡ (ξ˜1 , ξ˜2 ). The action considered is in fact strongly presymplectic, since it is an g exact action in relation to the 1-form jS∗ θL = (2m2 u2 + y 2 )dx2 + (2m2 v 2 − x2 )dy 2 + (2m3 u3 − y 3 )dx3 + (2m3 v 3 + x3 )dy 3 . The presymplectic Hamiltonian functions of ξ˜1 and ξ˜2 in (S, ΩS ) are fξ1 = 2m2 (x2 v 2 − y 2 u2 ) − (x2 )2 − (y 2 )2 , fξ2 = 2m3 (x3 v 3 − y 3 u3 ) − (x3 )2 − (y 3 )2 . So a momentum map JS can be defined for this action and, taking into account the discussion in Sec. 3.5, for every weakly regular value µ ≡ (µ1 , µ2 ) ∈ g∗ , its level sets JS−1 (µ) are defined as submanifolds of TQ by the constraints η1 := x1 = 0 ,
η2 := y 1 = 0 ,
χ1 := u1 = 0 ,
χ2 := v 1 = 0
fξ1 := 2m2 (x2 v 2 − y 2 u2 ) − (x2 )2 − (y 2 )2 = µ1 fξ2 := 2m3 (x3 v 3 − y 3 u3 ) − (x3 )2 − (y 3 )2 = µ2 . The submanifolds (JS−1 (µ), ΩSµ ) are presymplectic and 6-dimensional. Next, the final step of the reduction procedure leads to the 4-dimensional quotient manifolds ˆ S ). (JS−1 (µ)/ ker ΩSµ , Ω
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4.4.2. Reduction of the system (M, ΩM , EM ) This compatible presymplectic system exhibits the above rigid symmetries as well as gauge symmetries, which are infinitesimally generated by the fundamental vector fields ∂ ∂ , ξ˜4 = ξ˜3 = ∂u1 ∂v1 (which generate ker ΩM ). Let G be the group of all these symmetries and g its ˜ ≡ (ξ˜1 , ξ˜2 , ξ˜3 , ξ˜4 ). The action considered is in fact also strongly Lie algebra, then g ∗ θL . The presymplectic, since it is an exact action in relation to the 1-form jM presymplectic Hamiltonian functions of the fundamental vector fields in (S, ΩS ) are fξ1 := 2m2 (x2 v 2 − y 2 u2 ) − (x2 )2 − (y 2 )2 , fξ2 := 2m3 (x3 v 3 − y 3 u3 ) − (x3 )2 − (y 3 )2 fξ3 := 0 ,
fξ4 := 0
where the constant value of fξ3 and fξ4 equal to 0 is just a possible choice for the constant Hamiltonian functions corresponding to the vector fields ξ˜3 and ξ˜4 respectively. So a momentum map JM can be defined for this action and, taking into account the discussion in Sec. 3.5, for weakly regular values µ ≡ (µ1 , µ2 , 0, 0) ∈ g∗ , −1 (µ) are defined as submanifolds of TQ by the constraints its level sets JM fξ1 := 2m2 (x2 v 2 − y 2 u2 ) − (x2 )2 − (y 2 )2 = µ1 fξ2 := 2m3 (x3 v 3 − y 3 u3 ) − (x3 )2 − (y 3 )2 = µ2 η1 := x1 = 0 ,
η2 := y 1 = 0 .
(Observe that η1 := x1 and η2 := y 1 are the presymplectic Hamiltonian functions −1 (µ), ΩMµ ) of ξ˜3 and ξ˜4 in (TQ, ΩL ), respectively.) Now, the submanifolds (JM −1 (µ)/ ker are presymplectic and 8-dimensional and the final quotient manifolds (JM ˆ M ) are 4-dimensional. ΩMµ , Ω Nevertheless, this quotient manifold is locally symplectomorphic to (JS−1 (µ)/ ker ˆ S ). In fact, instead of using the complete presymplectic reduction, we can ΩSµ , Ω apply to the system (M, ΩM , EM ), first, gauge reduction and, afterwards, standard symplectic reduction, then obtaining a quotient manifold which is symplectomorphic −1 ˆ M ) (see Sec. 3.3) and locally symplectomorphic to (J −1 (µ)/ (µ)/ ker ΩMµ , Ω to (JM S ˆ S ) (obviously). ker ΩSµ , Ω 4.4.3. Comment In this example we have just shown that whether or not non-dynamical constraints (that is, those arising in the stabilization algorithm from demanding that the vector field solution of the Lagrange equations to be a SODE) are taken into account in the reduction procedure is irrelevant, since, in any case, we obtain the same quotient manifold. In reality this must be a general property. In fact: let (TQ, ωL , EL ) be a singular (but almost-regular [22, 44]) Lagrangian system, (M, ΩM ) the final constraint submanifold when the SODE-condition is not considered and (S, ΩS ) the final
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constraint submanifold when the SODE-condition is considered, such that we have a group of rigid symmetries both for (M, ΩM , EM ) and (S, ΩS , ES ). Then, if Assumption 2 holds for (M, ΩM ) and (S, ΩS ), the complete presymplectic reduction procedure leads to the same reduced system for both systems. The reason for this feature lies in the following facts: As is proved in [44], the non-dynamical constraints defining S in M remove degrees of freedom in the leaves of the foliation generated by the vertical part of ker ΩL which, on its turn, is included in ker ΩM . As a consequence, it is also proved that ker ΩS ⊂ ker ΩM . But, as Assumption 2 holds, the final quotient in the complete presymplectic reduction is made by a foliation whose leaves contain those of ker ΩM or ker ΩS respectively. Therefore, when the reduction is made for (M, ΩM ), the degrees of freedom in the leaves of the foliation generated by the vertical part of ker ΩL are removed in the final quotient. However, when the reduction is made for (S, ΩS ), these degrees of freedom have been previously removed. 4.5. The conformal particle Finally, we consider the system of a massless relativistic particle with conformal symmetry. The original Lagrangian function was introduced by Marnelius [38] and, subsequently, Siegel used it for describing the behaviour of these kind of particles [48]. Recently, Gr` acia and Roca [26] have carefully studied the gauge transformations for this system. The configuration space of this system is Q = Rd+2 × R and is locally coordinated by the set (q a , λ) (a = 0, 1, . . . , d + 1), where λ is an unphysical parameter which is introduced in order to make the description of the system covariant, and it is responsible for the local scale invariance. At the Lagrangian level, the system is dynamically described by the Lagrangian function 1 L := gab (v a v b − λq a q b ) ∈ C∞ (TQ) 2 where g is an indefinite metric in Rd+2 with signature sign (gab ) = (1, −1, . . . , −1, 1). From here and using the canonical structures of the tangent bundle TQ, we construct the Lagrangian 2-form and the energy Lagrangian function: 1 ωL = gab dq a ∧ dv b ; EL = gab (v a v b + λq a q b ) 2 (v a denote the generalized velocities corresponding to the coordinates q a ). The system is singular since the generalized velocity u corresponding to the generalized coordinate λ does not appear explicitly in the Lagrangian function. Hence ∂ ∂ , ∂u }. So (TQ, ωL , EL ) is a presymplectic ωL is presymplectic and ker ωL ≡ { ∂λ ∂ )dEL 6= 0. Using some of dynamical system which is not compatible since i( ∂λ the known stabilization algorithms, we find that the final constraint submanifold jM : M ,→ TQ is defined by the constraints [26] 1 gab q a q b , η2 = gab v a q b , η3 = gab v a v b − λgab q a q b . 2 In this case we can take a basis of constraints made of presymplectic Hamiltonian functions, for instance η1 =
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1 1 gab q a q b , fξ2 = gab v a q b , fξ3 = gab v a v b . 2 2 The vector fields which are solutions of the dynamical equation fξ1 =
(i(X)ωL − dEL )|M = 0 are the following: X|M = v a
∂ ∂ ∂ ∂ +f + a +u ∂q a ∂v ∂λ ∂u
(where f is an arbitrary function). ∗ ∗ ωL , jM EL ) exhibits The compatible presymplectic system (M, Ω, H) = (M, jM point gauge symmetries which are infinitesimally generated by the following fundamental vector fields: ∂ ∂ a ∂ a ∂ a ∂ a ∂ ˜ ˜ ˜ ˜ ˜ , ,v −q ,v , . (ξ1 , ξ2 , ξ3 , ξ4 , ξ5 ) = q ∂v a ∂v a ∂q a ∂q a ∂λ ∂u Observe that ξ˜1 , ξ˜2 , ξ˜3 are the presymplectic Hamiltonian vector fields corresponding to fξ1 , fξ2 , fξ3 respectively and ξ˜4 , ξ˜5 ∈ ker Ω. They are all tangent to M and, ∗ Ω. hence, they make a local basis of ker jM It is interesting to note that the system is also invariant under rigid O(2, d) rotations. Nevertheless, it can be shown that there exist O(2, d) Lagrangian gauge transformations (see [26] and [38]) and hence, in this case, this group of symmetries is a closed subgroup of the gauge group G. ∗ ωL ) is strongly presymTaking all of this into account, the action of G on (M, jM ∗ θL = plectic (it is in fact an exact action in relation to the Lagrangian 1-form jM ∗ a b jM (gab v dq )). Thus, a momentum map J can be defined for this action such that M = J −1 (0). Therefore, the presymplectic reduction procedure is simply the well∗ ∗ ωL , jM EL ). known gauge reduction for the compatible presymplectic system (M, jM 5. Conclusions and Outlook We have made a study about actions of Lie groups on presymplectic manifolds and the subsequent reduction procedure. The main results and considerations discussed here are the following: • We have made the natural extension of the concepts of the theory of symplectic actions of Lie groups on symplectic manifolds to this case. • The existence of comomentum and momentum maps are analyzed, obtaining an obstruction similar to the symplectic case (but involving the set Bh1 (M )/ Zh1 (M ) instead of the first cohomology group H 1 (M )). • We have investigated the properties and characteristics of the level sets of the momentum map for weakly regular values, as a standpoint for reduction. As a particular result, the interpretation of these level sets as the maximal integral submanifolds of a Pfaff system allows us to simplify the proof of some results. We hope that this interpretation will be of interest with a view to extending the reduction procedure to field theories. • The reduction of presymplectic manifolds by presymplectic actions of Lie groups has been achieved for weakly regular values of the momentum map,
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•
•
•
•
•
•
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following the guidelines of the symplectic reduction theory of Marsden– Weinstein. With the usual hypothesis, the reduced phase space is endowed with a structure of presymplectic manifold, in general. The concept of symmetry for presymplectic dynamical systems is displayed. The reduction of compatible and non-compatible presymplectic dynamical systems with symmetry is made as an application of the theory just developed. These results hold both for presymplectic Lagrangian or Hamiltonian systems. When gauge symmetries are taken together with the non-gauge symmetries of the system, then the reduced phase space is endowed with a structure of symplectic manifold with dynamics of Hamiltonian type. The procedure of complete presymplectic reduction allows us to reach the orbit space in a straightforward way, in comparison with other step-by-step reduction procedures, namely, coisotropic imbedding plus symplectic reduction and gauge reduction plus symplectic reduction, which lead to the same final reduced phase space. The equivalence of all these methods is also proved. As an example, we have considered non-autonomous dynamical systems. Starting from the presymplectic formulation of these systems (which allow us to include also the singular case in a natural way), we have adapted the notion of symmetry, and then by applying the reduction procedure previously studied, results similar to those of other works that have analyzed this problem have been obtained. The main advantage of the formalism is that the treatment of the singular case is absolutely “on way”. As a particular case, the reduction of time-dependent regular dynamical systems is considered in the framework of time-invariant non-autonomous systems. In this case, the reduced phase space is a contact manifold since the level sets of the momentum map are the energy constant hypersurfaces and reduction removes the time coordinate from the initial system. In this way, in our opinion, this is a better result than those obtained applying the symplectic reduction techniques to the extended phase space of the system, since reduction then leads to a symplectic system in the reduced phase space, but does not directly give the dynamics on the constant-energy hypersurfaces. Another interesting example is the complete reduction of a particular case of the Capri–Kobayashi mechanical model for field theories coupled to external fields, exhibiting both gauge and non-gauge symmetries, in the Lagrangian formalism. It is shown that, under suitable circumstances, the existence of Lagrangian constraints arising from the search for dynamical solutions which are second order differential equations is irrelevant in the reduction procedure. Finally, we have also checked this method by applying it to a discussion of the gauge reduction of the conformal particle (in the Lagrangian formalism).
A. Linear Reduction In this appendix we wish carry out a quick review of the reduction theory, giving at the same time a linear algebraic interpretation of this theory for the general case of linear forms of arbitrary order.
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Let E be a linear vector space, with dim E = n, and a linear form α ∈ Λk E ∗ , with k ≥ 2. Let S be a subspace of E. Then take S ⊥1 := {u ∈ E|i(u)i(v)α = 0 ,
∀ v ∈ S} ≡ N
let j : N ,→ E be the natural inclusion and αN := j ∗ α. If v ∈ N ∩ S, then i(u)i(v)α = 0, for every u ∈ N , and therefore v ∈ ker αN ; that is, N ∩ S ⊂ ker αN ⊂ N . Then we have the projections π
π
1 2 N/N ∩ S −→ N/ ker αN = (N/N ∩ S)/(ker αN /N ∩ S) N −→
and there exist α1 ∈ Λk (N/N ∩ S) and α3 ∈ Λk (N/ ker αN ) such that αN = π1∗ α1 and αN = π3∗ α3 , where π3 = π2 ◦ π1 . Notice that ker α3 = {0}, because the space S, projected by π3 , “has been removed”. This is a “reduction” procedure in the sense that a subspace is removed from a vector space by a reduction of the dimension. Note that it is not useful to make the quotient E/S and then the projection E → E/S because the form α does not project onto the quotient unless S ⊂ ker α. Then, N = S ⊥1 is a subspace of E which can be reduced in such a way that the form α goes down to the quotient. As a particular situation, we can study the case k = 2. Then we can prove that ker αN = ker α + N ∩ S . In fact; let {e1 , . . . , ek } be a basis of S. If v ∈ ker αN , then i(u)i(v)α = 0, for every u ∈ N , and N ⊂ ker i(v)α. But N = ∩j=1,...,k ker i(ej )α, then we have that i(v)α is a linear combination of i(e1 )α, . . . , i(ek )α and, therefore, v ∈ (ker α + S) ∩ N . But, since ker α ⊂ N , the result follows. If in addition, α is a symplectic linear form; that is, ker α = {0}, then ker αN = N ∩ S and we have the unique projection π
N −→ N/N ∩ S ˆ = α, which is also a symplectic form. and a unique form α ˆ ∈ Λ2 (N/N ∩ S) with π ∗ α This is the result of the Marsden–Weinstein reduction procedure in the linear case. Acknowledgements We thank Dr. Xavier Gr` acia-Sabat´e (U.P.C.) for bringing the example of the conformal particle to our attention and for explaining to us some of its characteristics. We are grateful for the assistance of the referee, whose suggestions have enabled us to improve the final version of the work. We also thank Mr. Jeff Palmer for his assistance in preparing the English version of the manuscript. We are grateful for the financial support of the CICYT TAP97-0969-C03-01. References [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed., Addison-Wesley, Reading, 1978. [2] R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Addison-Wesley, Reading MA., 1983.
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[50] J. M. Souriau, Structure des Syst`emes Dynamiques, Dunod, Paris, 1969. [51] F. W. Warner, Foundations on Differentiable Manifolds and Lie groups, Scott, Foresman and Co., Glenview, 1971. [52] A. Weinstein, Lectures on Symplectic Manifolds, C.B.M.S. Reg. Conf. Ser. Math., 29 (1979).
UPPER BOUNDS FOR QUANTUM DYNAMICS GOVERNED BY JACOBI MATRICES WITH SELF-SIMILAR SPECTRA I. GUARNERI and H. SCHULZ-BALDES International Centre for the Study of Dynamical Systems Universit` a di Milano sede di Como, via Lucini 3 22100 Como, Italy and Instituto Nazionale di Fisica della Materia Unit˜ a di Milano, Via Celoria 16 20133 Milano, Italy Received 1 October 1998 We study a class of one-sided Hamiltonian operators with spectral measures given by invariant and ergodic measures of dynamical systems of the interval. We analyse dimensional properties of the spectral measures and prove upper bounds for the asymptotic spread in time of wavepackets. These bounds involve the Hausdorff dimension of the spectral measure, multiplied by a correction calculated from the dynamical entropy, the density of states, and the capacity of the support. For Julia matrices, the correction disappears and the growth is ruled by the fractal dimension.
1. Introduction One-particle Schr¨ odinger operators with almost-periodic potentials display a rich variety of spectral types, including singular continuous spectra. In the latter situation, there is compelling numerical evidence [8, 12, 14, 20, 23] that transport is typically sub-ballistic and anomalous. More precisely, this means that the second moment of the position operator asymptotically grows in time as t2β where β can take any value in [0, 1] depending on the specific model and on parameter values. Determining the transport exponent β from the Hamiltonian is an interesting and important task both from the mathematical and the physical viewpoint. Concerning the latter, let us point out that the metal-insulator transition in quasicrystals occurs by anomalous transport [13] and furthermore that the transport exponent enters in the anomalous Drude formula [25]. On a rigorous level, some connections have been established between dimensional properties of the local density of states (LDOS) and asymptotic transport properties. The asymptotic decay of the time-averaged staying probability at the initial site is ruled by the correlation dimensions of the LDOS [2, 8, 15, 19, 25]. As to the growth of the αth moment of the position operator, a general argument bounds it below by αd/D where d is the Hausdorff dimension of the LDOS and D the dimension of physical space [2, 9, 10, 19, 25]. Obtaining upper bounds and sharp estimates appears to require more detailed information than just the dimensional 1249 Reviews in Mathematical Physics, Vol. 11, No. 10 (1999) 1249–1268 c World Scientific Publishing Company
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properties of the LDOS. Improved lower bounds involving the structure of eigenfunctions have been heuristically derived and numerically verified in [17]. In general, there cannot be a tight connection between dimensional properties and the behavior of moments of the position operator; for example, there are models with spectra of arbitrarily small Hausdorff dimension and ballistic (β = 2) propagation [18], and also models with zero-dimensional spectra which display ballistic motion on arbitrarily large time scales [11, 18, 19]. On the other hand, information about eigenfunctions may be redundant at least in the case of Jacobi matrices, whose generalized eigenfunctions are given by the orthogonal polynomials of the LDOS. When investigating such issues on a given one-particle Hamilton operator, one is actually confronted with two different problems. First, one has to analyse the spectral measures; then, as a second step, comes the derivation of transport properties. These two problems are in principle different and have to be tackled separately. It is convenient to circumvent the first problem by constructing the Hamilton operator as the Jacobi matrix associated with a LDOS prescribed in advance as self-similar fractal measure. This strategy has already lead to numerical results [12] and to further numerical-theoretical analysis [20]. By the very same strategy, we obtain in this paper rigorous upper bounds for transport in purely spectral terms. As a prototype class of self-similar measures, we consider measures constructed by non-linear, disjoint iterated function systems (IFS). The spectral analysis of such measures can be performed relatively easily: the Hausdorff dimension of the measure is given by the quotient of the dynamical entropy and the Lyapunov exponent of the associated dynamical system. The corresponding Jacobi matrices are believed to be almostperiodic [20], but a rigorous proof only exists for the special case of Julia matrices, which are known to be limit-periodic [4]. The basic quantity considered here for the study of quantum transport is the minimal carrier of a wave packet originally localized at the origin. It is defined, at all times, as the radius of the smallest ball centered at the origin and carrying a fixed percentage of the time-averaged presence probability at the given time. Upper bounds on its algebraic growth in time translate into upper bounds for the growth of other quantities measuring the spatial extent of wave packets, such as inverse participation ratios [10], entropic widths [10] and (inverse) negative moments of the position operator; however, positive moments are out of reach. In our approach the dynamical spreading of wavepackets over increasing time scales is controlled by an appropriate renormalization dynamics which resolves the spectrum on accordingly decreasing energy scales. It is exactly the existence of a dynamical system generating the spectrum which makes the renormalization dynamics accessible. Our upper bound for the growth exponent of the minimal carrier is then given by the Hausdorff dimension of the spectral measure multiplied by a correction factor calculated from the dynamical entropy, the density of states (DOS) and the (logarithmic) capacity of its fractal support. The DOS enters our estimates because it controls the exponential growth of orthogonal polynomials via a formula of the Herbert–Jones–Thouless type. This is the only information about eigenfunctions we use.
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2. Statement of the Results In this section, we introduce the operators studied in this work, quickly review some general facts from the theory of orthogonal polynomials, and finally state our main results. 2.1. Spectral measures The self-similar fractal measures considered in this work are constructed by nonlinear, disjoint iterated function systems (IFS) [16]. The construction is as follows. Let I11 < I21 < · · · < IL1 be a finite sequence of pairwise disjoint closed intervals all contained in a closed interval I 0 . Let S be a smooth real function such that, for all l = 1, . . . , L, the restriction Sl of S to Il1 is bijective from Il1 to I 0 with smooth inverse Sl−1 (in particular, we assume the derivative of S to be bounded away from 0 and ∞ on the intervals Il1 ). We call codes the one-sided sequences of symbols taken from {1, . . . , L} and denote the set of codes of length N by ΣN L and the set of codes of infinite length by ΣL . For all N ∈ N, S −N (I 0 ) consists of LN closed, disjoint intervals ◦ · · · ◦ Sσ−1 (I 0 ) , IσN = Sσ−1 1 N
σ = (σ1 . . . σN ) ∈ ΣN L ,
which we call the intervals of the N th generation. We further assume that there exist positive constants a < 1 and b so that, for any N ∈ N, all intervals of N th generation satisfy |IσN | ≤ b aN where |I| denotes the length of the interval I. A non-negative (rescaled) Schwarzian derivative of S is a sufficient condition for this basic contraction property to hold (see Misiurewicz’ theorem in [7]). We note that the contraction property can hold even if the Se−1 ’s are not everywhere contractive, but are locally expansive. T Now J = N ≥0 S −N (I 0 ) is a fractal set which is invariant under S, i.e. S(J) = J. The dynamical system (J, S) is conjugated to the shift on ΣL by the coding map E ∈ J 7→ σ(E) ∈ ΣL . The associated pullback allows to construct invariant, ergodic measures for the dynamical system (J, S) as pre-images of shift-invariant, ergodic measures on ΣL . For measures µ on J obtained in this way, the pointwise dimensions log(µ([E − , E + ])) (1) dµ (E) = lim →0 log() exist µ-almost surely and are µ-almost surely equal to the information or Hausdorff dimension dimH (µ) of µ (see Appendix 1 for a proof and references to the literature). Moreover, E(µ) (2) dimH (µ) = Λ(µ) where E(µ) and Λ(µ) are the dynamical (Kolmogorov–Sinai) entropy and the Lyapunov exponent of the dynamical system (J, S, µ). We shall always suppose that the sequences (log(µ(IσN ))/N )N ∈N are bounded from below, uniformly in σ ∈ ΣL . This assumption holds true in all cases explicitely considered below.
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2.2. Jacobi matrices Once the measure µ on J is fixed, we construct the Hamiltonian as the Jacobi matrix of µ. Let Pn , n ≥ 0, denote the orthogonal and normalized polynomials associated to µ. They are a Hilbert basis B = (Pn )n∈N in L2 (R, µ) and satisfy a three term recurrence relation EPn (E) = tn+1 Pn+1 (E) + vn Pn (E) + tn Pn−1 (E), n ≥ 1, and EP0 (E) = t1 P1 (E) + v0 P0 (E) where vn ∈ R and tn ≥ 0 are bounded sequences. Therefore the isomorphism of L2 (R, µ) onto `2 (N) associated with the basis B carries the operator of multiplication by E in L2 (R, µ) into the self-adjoint finite difference operator H defined on `2 (N) by: H|ni = tn+1 |n + 1i + vn |ni + tn |n − 1i ,
n ≥ 1,
(3)
and H|0i = t1 |1i + v0 |0i. Then µ is the spectral measure of H associated to |0i, also called its local density of states (LDOS). That means, Z f (E) dµ(E) h0|f (H)|0i = J
for all continuous functions f . The vector |0i is a cyclic vector for H, therefore the measure µ dominates all spectral measures of H associated to other states in Hilbert space. The second important measure associated with the operator H is the density of states (DOS). It is defined by X 1 δE , (4) N = w– lim n→∞ n E, Pn (E)=0
whenever the weak limit of point measures appearing on the right-hand side exists. Conditions for this and interesting consequences thereof follow from the work of Stahl and Totik [26] which we shall review next. 2.3. Orthogonal polynomials We first need to review some basic notions from (logarithmic) potential theory (see, e.g., [1] or the appendix of [26]). The (logarithmic) capacity cap(K) of a compact set K ⊂ R is defined by Z Z 0 0 log(cap(K)) = sup dν(E) dν(E ) log(|E − E |) supp(ν) ⊂ K , where the supremum is taken over all Borel probability measures ν carried by K. The capacity of a Borel set Y is given by the supremum over the capacities of compact sets contained in Y . If cap(Y ) > 0, then there exists a unique Borel measure ωY for which the supremum is attained. It is called the Frostman equilibrium measure. Next, the Green’s function of Y with pole at infinity is the (unique) non-negative subharmonic function in C, harmonic in C\(Y ∪ {∞}), and satisfying gY (E) = 0 for all E ∈ Y \Y 0 where cap(Y 0 ) = 0. If cap(Y ) > 0, then Z (5) gY (z) = dωY (E) log(|E − z|) − log(cap(Y )) .
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We shall verify in Sec. 4 that the capacity of the sets J in Subsec. 2.1 is positive and that any of the self-similar measures µ on J is asymptotically regular in the sense of Stahl and Totik [26] (see Appendix 2 for a definition). By the results of [26] stated in Appendix 2, these facts imply the following: (i) The DOS is well defined and coincides with the Frostman equilibrium measure of J: (6) N = ωJ . (ii) The capacity can be calculated by the formula 1X log(tj ) . n→∞ n j=1 n
log(cap(J)) = lim
(7)
(iii) If we introduce the transfer matrices (by convention, t0 = 0) (z − vj )/tj+1 −tj /tj+1 , Φn (z) = An (z) · · · A0 (z) , Aj (z) = 1 0 then the following Herbert–Jones–Thouless type formula holds: lim
n→∞
1 log(kΦn (z)k) = gJ (z) , n
(8)
where the convergence is locally uniform in C\I 0 (a sequence of functions fn converges locally uniformly in an open set G ⊂ C to a function f if for all z ∈ G and sequences zn → z, one has fn (zn ) → f (z)). Moreover, (8) with the equality replaced by ≤ holds locally uniformly in C. Thus gJ can be seen as the Lyapunov exponent of the one-dimensional lattice Hamiltonian H. Moreover, the spectrum is characterized as the set of points where this Lyapunov exponent vanishes (except for a set of vanishing capacity). Note that the main modification with respect to the usual Herbert–Jones– Thouless formula with discrete Laplacian (tn = 1 for all n ≥ 1) is the capacity term in (5). 2.4. Growth exponents Under the dynamics e−ıtH , a wave packet initially localized on the state |0i spreads out over the basis B. To study this spreading, we introduce the minimal carrier as X Z T dt |hn|e−ıHt |0i|2 ≤ , n(, T ) = min n ∈ N n≥n 0 T and the corresponding growth exponents: β0+ () = lim sup T →∞
β0− ()
log(n(, T )) , log(T )
log(n(, T )) , = lim inf T →∞ log(T )
(9)
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as well as β0± = lim→0 β0± (). These transport exponents are linked to the growth exponents of the moments of the position operator in the basis B. For α 6= 0, the latter are defined by X log nα pn (T ) n≥0 , βα+ = lim sup α) log(T T →∞ (10) X α log n pn (T ) n≥0 , βα− = lim inf T →∞ log(T α ) RT where pn (T ) = 0 dt|hn|e−ıHt |0i|2 /T . We have βα+ ≤ β0− whenever α < 0 and βα− ≥ β0+ whenever α > 0. Furthermore [10], β0− is an upper bound for the entropic growth exponent in the basis B defined by X − pn (T ) log(pn (T )) n≥0 + . = lim sup βent log(T ) T →∞ Finally, we refer the reader to [10] for a definition of inverse participation ratios and a proof that also their growth exponents are bounded from above by β0− . The lower bound β0− ≥ dimH (µ) holds for any self-adjoint H and for any Hilbert basis {|ni} [2, 9, 10, 19]. In addition, for operators in the class (3), we have the a priori ballistic upper bound β0+ ≤ 1. One way of deriving this is observing that (3) is a bounded operator in the Banach space Xα (α > 0) of `2 (N)-vectors ψ such that kψkα = supn |hψ|ni| exp(αn) < ∞. Then at any time t, the minimal carrier can be estimated from |hn| exp(−iHt)|0i|2 ≤ exp(−2αn + 2tkHkα ), which directly yields the ballistic bound. 2.5. Main results Our main results give upper bounds on the exponent β0+ . Theorem 1. Let S be an analytic map and let Rc = |I 0 |/2 denote the spectral radius and Ec the center of the spectrum. Then 1 maxz∈ΓR gJ (S(z)) log inf , (11) β0+ ≤ R>Rc log(R) − log(Rc ) Λ(µ) where ΓR is the circle of radius R around Ec . Theorem 2. For any l = 1, . . . , L, let the branch Sl have an analytic continuation Sˆl given by a polynomial map of degree Dl . We set D = maxl=1,...,L Dl and γ = arcsinh(∆(4Rc )−1 ), where Rc = |I 0 |/2 is the spectral radius and ∆ is the size of the smallest gap at the first generation. Then 1 1 log max sup gJ (Sˆl (E)) + D . (12) β0+ ≤ Λ(µ) γ l=1,...,L E∈J Remark 1. When combined with the rigorous lower bound β0− ≥ dimH (µ) from [2, 9, 10, 19], these results prove that quantum transport in presence of singular
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continuous spectra can actually be anomalous (as it will become more explicit in the applications presented below). To our knowledge, the only other examples of quantum models in which the motion can be rigorously shown to be other than ballistic or localized are the following: for finite rank perturbation of a localized model, the moments of the position operator diverge at most logarithmically [24]; next, less pathological, the quantum motion is diffusive in the Anderson model with free random variables [25] (i.e. the growth exponent of the disorder averaged second moment of the position operator equals 1); the latter Hamiltonian model can be identified with the coherent potential approximation of the usual Anderson model, and a special case of it is the Wegner n-orbital model in the limit n → ∞ [21]. Remark 2. Neither of the bounds (11) and (12) is optimal in general. In the case of (12) this becomes particularly evident when J is a linear Cantor set, see Remark 4 below. Our present proof may possibly be improved at several places; nevertheless, it yields optimal results in the case of Julia sets, as shown in Subsec. 2.6 below. Remark 3. Using Eq. (2), the above bounds can be written as the product of the Hausdorff dimension of the measure times a correction factor involving the DOS, the dynamical entropy, and the capacity. 2.6. Applications and comments One class of shift-invariant, ergodic measures on ΣL are the measures for which the σn , n ∈ N, are independent random variables with same distribution Prob{σn = PL l} = pl , l = 1, . . . , L, l=1 pl = 1. The corresponding measures µ on J will be called P Bernoulli measures with weights pl . For such measures, E(µ) = − ` p` log p` . Our present applications only consider this class. First, we treat Julia matrices. A real Julia set is the fractal J associated to a polynomial mapping S of degree L satisfying the hypothesis imposed in the construction of J. Putting this into (11) and (5), we find that maxz∈ΓR gJ (S(z)) ∼ L log(R) at large R. Taking the limit R → ∞, we get the upper bound log(L)/Λ(µ) = dimH (µ) log(L)/E(µ) because of Eq. (2). The correction factor with respect to the Hausdorff dimension disappears if µ is the maximal entropy measure, for which E(µ) = log(L). This is the the Bernoulli measure with equal weights; the corresponding Jacobi matrices are called Julia matrices. Recalling the general lower bound β0− ≥ dimH (µ) [10], we obtain: Corollary 1. If µ is the Bernoulli measure with equal weights on a real Julia set J, then β0− = β0+ = dimH (µ) . Thus the dimensional properties of the LDOS completely determine the transport properties in Julia matrices. Formal arguments and numerical verifications go actually farther [20]: the scaling exponents βα± defined in (10) satisfy βα± = D1−α for α > 0 where (Dq )q∈R denotes the family of multifractal dimensions. The main
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reason is presumably the exact renormalization property of the orthogonal polynomials (hn|m ◦ Si = δn,Lm [3]). Our proof makes no use of this fact; however, using it considerably simplifies the proof, as explained in Remark 5 in Subsec. 3.2. Nor do we use the fact that the LDOS and the DOS of Julia matrices coincide (i.e., the Frostman equilibrium measure is the Bernoulli measure with equal weights [6]). As a second example, we consider the usual Cantor set C in [0, 1] where Sˆ1 (E) = Λ e E and Sˆ2 (E) = eΛ − eΛ E with eΛ > 2. Let us replace this in (12). We first note the maximum over branches becomes irrelevant because of symmetry, next that the supremum over E ∈ J = C is actually taken when E is the outer border of the Cantor set. Therefore we obtain from (12): Λ −1 Z ! 1 e −2 + Λ dN (E) log(e − E)−log(cap(C)) +1 . β0 ≤ log arcsinh Λ 4 (13) The DOS and the capacity can be calculated numerically by standard procedures. Nevertheless a crude estimate of cap(C) (see Remark 6 in Sec. 4) shows that the argument of the second logarithm in (13) grows linearly with Λ. This proves that β0+ tends to 0 in the limit Λ → ∞. Remark 4. The bound (13) for the case of linear Cantor sets does not depend on the weights pl , but there is numerical evidence that growth exponents do (for momenta at least) [12]. In addition, as the gaps shrink on decreasing Λ, the bound can become larger than 1, the a priori ballistic bound. Our estimates are therefore far from optimal in this situation. Apart from the appendices giving complementary results, the rest of the article contains the proof of Theorems 1 and 2. Section 3 concludes with Proposition 3 which, when combined with the results of Subsecs. 4.1 and 4.2, directly implies Theorems 1 and 2. All Ci (ξ, . . .) appearing below denote quantities only depending on constants ξ, . . . . 3. Proof of Upper Bounds In this section, we will bound β0+ under a hypothesis formulated in Subsec. 3.2. This hypothesis will be verified in the next section. 3.1. Resolving the spectrum In this subsection we state and prove a general result making no reference to the specific structure of the Hamiltonian. Then follows a corollary allowing to deduce upper bounds on the dynamics for the Jacobi matrices constructed in Subsecs. 2.1 and 2.2. Let |ψi be a vector in some separable Hilbert space H, µ its spectral measure, J = supp(µ), and (|ni)n∈N a Hilbert basis in the cyclic subspace of ψ. Suppose that, for any integer N , we have a finite covering of J by intervals ∆N j , j = 1, . . . , nN , with pairwise disjoint interiors and satisfying
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0 < lim
N →∞
1 log N
1 |∆N j(E) |
! = Λ < ∞ µ−a.e. ,
(14)
where, for given N and E ∈ J, ∆N j(E) is the interval containing E. Finally, let N N N χj (E) be the characteristic function of ∆N j . Using the notation |χj i = χj (H)|ψi, the following holds true: Proposition 1. Fix λ ∈ (0, Λ) and > 0. Let K ⊆ J be a compact set with µ(K) > 1 − . To a given time T, we associate N = N (T ) by T e(N −1)(Λ−λ) < √ ≤ eN (Λ−λ) ,
(15)
and set JN (K) = {j|K ∩ ∆N j 6= ∅}. Then there are T (λ, ) and C1 (Λ, λ) so that, for all T > T (λ, ), and for any family of indices F ⊂ N : X X Z T dt C1 (Λ, λ) 2N λ X 2 √ |hn|e−ıHt |ψi|2 ≤ 18 + e |hn|χN (16) j i| . 0 T n∈F
n∈F j∈JN (T ) (K)
Proof. Because log(1/|∆N j(E) |)/N is a measurable function of E, it follows from (14) and Lusin’s theorem that there is a N (λ, ) and a compact K ⊆ J with µ(K ) > 1 − so that −N (Λ−λ) (17) e−N (Λ+λ) < |∆N j |< e for all N > N (λ, ) and for all j ∈ JN (K ). Let us define K1 = K ∩ K ; then µ(K1 ) > 1 − 2. From now on, we shall assume N and T to be related to each other via (15). For 0 ≤ t ≤ T , we approximate e−ıHt |ψi by X N |ψT (t)i = e−ıtEj |χN j i j∈JN (K1 )
where EjN is any point in ∆N j . Then we have: X Z k |ψT (t)i − e−ıtH |ψik2 ≤ dµ(E) t2 |E − EjN |2 + 4
X
j∈JN (K1 )
j ∈J / N (K1 )
∆N j
µ(∆N j ).
N As |E − EjN | ≤ |∆N j | for j ∈ JN (K1 ) and E ∈ ∆j , using (17) we get that the latter expression is not larger than X 2 −2N (Λ−λ) µ(∆N +8, j )t e j∈JN (K1 )
which, for 0 ≤ t ≤ T , is not larger than 9 because of (15). Hence, X Z T dt X Z T dt |hn|e−ıtH |ψi|2 ≤ 18 + 2 |hn|ψT (t)i|2 . 0 T 0 T n∈F
(18)
n∈F
We now choose the points EjN so that the latter term can be easily estimated. Let ` be the integer part of (1 + exp(N (Λ + λ)). As all intervals ∆N j , j ∈ JN (K1 ), have
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length not less than exp(−N (Λ + λ)), everyone of them contains at least one integer multiple of 1/`. Let EjN be such a point. Then Z 0
T
dt |hn|ψT (t)i|2 ≤ T
Z
2π`
0
dt |hn|ψN (t)i|2 T
X
=
Z hn|χ∆N ihχ∆N |ni j l
j,l∈JN (K1 )
2π`
0
dt −ı(EjN −ElN )t e . T
The latter integral yields 2π`δjl . Recalling the definition of `, and K1 ⊆ K, we directly get Ineq. (16). We now restrict µ to be a fractal measure as constructed in Subsec. 2.1, (|ni)n∈N the basis of its orthogonal polynomials and H the associated Jacobi matrix. As N intervals ∆N j we choose the intervals Iσ of the N th generation. As shown by Eq. (33) in the proof of Theorem 3, in that case Eq. (14) holds with Λ given by the Lyapunov exponent Λ(µ). Further, we set K = J and F = (n)n≥n in Proposition 1. Hence we obtain: Corollary 2. Let µ and H be as constructed in Subsecs. 2.1 and 2.2. Fix λ ∈ (0, Λ(µ)) and > 0. To a given time T, we associate a generation index N by T e(Λ(µ)−λ)(N −1) < √ ≤ e(Λ(µ)−λ)N . If n = n(, N ) is chosen such that X X |h0|χIσN (H)|ni|2 ≤ 3/2 e−2λN ,
(19)
(20)
n≥n σ∈ΣN L
then there is a constant C2 (Λ, λ) so that, for all T larger than some T (λ, ), we have X Z T dt |hn|e−ıtH |0i|2 < C2 (Λ, λ) . (21) T 0 n>n
3.2. A bound of quasi-ballistic type The aim of this subsection is to determine n(, N ) such that bound (20) holds under the following hypothesis, which will be verified in Sec. 4. Let S be the isometric operator defined in L2 (R, µ) by Sφ = φ ◦ S. Hypothesis: There exist constants 0 < A ≤ B and C such that |hn|S|mi| ≤ exp(−An + Bm + C) .
(22)
Proposition 2. Assume (22) to be valid and let ξ > 1. Then there is a constant C3 (ξ, λ) so that the bound (20) holds if n(, N ) ≥ C3 (ξ, λ) 1−ξ eN ξ
log(B/A)
.
(23)
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We call this a quasi-ballistic bound because it is directly given by the growth rate of the matrix elements in (22). Apart from hypothesis (22), the essential ingredient of the proof is that the spectrum contains gaps at all scales (self-similarity). In the next subsubsection, we set the stage for the proof of Proposition 2. Subsubsections 3.2.2 and 3.2.3 give technical details which allow to conclude the proof in Subsubsec. 3.2.4. 3.2.1. A renormalization operator In order to estimate the sum on the left-hand side of (20) we introduce suitable operators Pl , l = 1, . . . , L, and S on the Schatten ideal L1 of trace class operators on L2 (R, µ). For ρ ∈ L1 , these are defined by Pl (ρ) = χIl1 (H)ρχIl1 (H) ,
Sρ = S ρ S† .
PL We call S the scaling operator, P = l=1 Pl the reduction operator, and R = P ◦ S the renormalization operator. One easily verifies that R is trace and positivity preserving. Let us further introduce the density matrices (positive and of unit trace) X χIσN (H)|0ih0|χIσN (H) . ρN = PL
σ∈ΣN L
Because χIσN (S(E)) = l=1 χS −1 (I N ) (E), we have R(ρN ) = ρN +1 and the expresσ l sion in (20) can be written as X X X |h0|χIσN (H)|ni|2 = hn|RN (ρ0 )|ni . (24) n≥n σ∈ΣN L
n≥n
We hence have to estimate R and for this purpose we introduce the norms kρka = sup ea(n+m) |hn|ρ|mi| ,
a > 0.
n,m≥0
For any a > 0, the set Xa = {ρ ∈ L1 | kρka < ∞} is a Banach space. Under S Xa for hypothesis (22), we shall show that P, S, R are continuous operators in a>a
some a, and we shall thereby estimate their norms, finally obtaining the proof of Proposition 5. The following lemma is based on the well-known Combes–Thomas argument. Improving the lower bound on θ defined below would lead to a better bound in Theorem 2, because it enters the estimates of Proposition 5. Lemma 1. Let I be any interval of the first generation and ∆ be the smallest gap in the first generation. Using the notation Γnm (I) = hn|χI (H)|mi, we define 1 1 ∆ log , θ0 (∆) = arcsinh . θ(I) = lim inf n,m→∞ |n − m| |Γnm | 4Rc Then θ0 (∆) ≤ θ(I) < ∞ .
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P 2 Proof. We have χI (E)Pm (E) = n≥0 Γnm (I)Pn (E) in L (R, µ). Suppose θ(I) = ∞. Then Γnm (I) decays faster than exponentially as n → ∞. Since the theorem in Appendix 2 sets an uniform bound on the exponential growth of Pn (E), E ∈ I 0 , the series is uniformly convergent in I 0 and its sum is analytic there. This cannot be true, because χI (E)Pm (E) vanishes in I 0 \ I. For θ ∈ R, we define the operator Uθ in `2 (N) by hn|Uθ |mi = e−θn δn,m and set Hθ = Uθ−1 HUθ and Qθ = Hθ − H. Let further Γ denote the circle of radius (|I|+∆)/2 around the center of I. By a standard argument, the operator (z −Hθ )−1 is bounded whenever kQθ (z − H)−1 k < 1. Since k(z − H)−1 k > 2/∆ for all z ∈ Γ and kQθ k ≤ max tj |eθ − e−θ | ≤ Rc |eθ − e−θ | , j≥1
this is guaranteed with a uniform bound in z ∈ Γ whenever sinh(|θ|) < Therefore we have e
θ(n−m)
Z
hn|χI (H)|mi = Γ
∆ . 4Rc
dz hn|Uθ−1 (z − H)−1 Uθ |mi = 2πı
Z Γ
dz hn|(z − Hθ )−1 |mi , 2πı
and the last expression is bounded by some constant. This determines a minimal rate of exponential decay. 3.2.2. Norms of the reduction operator It is easily seen that, if ρ ∈ Xa , 1/2
kPl (ρ)kb
≤ sup enb n≥0
X
|hn|χIl1 (H)|ki|e−ak kρk1/2 a .
k≥0
The proof of Lemma 1 shows that |hn|χIl1 (H)|ki| ≤ c(∆) exp(−θ0 (∆)|n − k|). A short computation shows that Pl is a bounded operator in Xa if a < θ0 (∆), and that 2c(∆) . (25) kPl k1/2 a→a ≤ θ0 (∆) − a 3.2.3. Norms of the scaling operator Hypothesis (22) is used at this very point. If ρ ∈ Xa , X 1/2 kS(ρ)kb ≤ sup ebn |hn|S|ki|e−ak kρk1/2 a . n≥0
k≥0
Using Hypothesis (22) for k ≤ B −1 (An−C), and |hn|S|ki| ≤ 1 for k > B −1 (An−C), one easily finds that, if a < B, then S is bounded from Xa to XaA/B , and that kSka→aA/B ≤
B eaC/B . a(B − a)
(26)
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Remark 5. If we restrict ourselves to Julia matrices (S is a polynomial of degree L and µ is defined as the Bernoulli measure with equal weights), a bound on the norm of S can be obtained more directly by using the renormalization property hn|S|mi = δn,Lm of the associated orthogonal polynomials [3]. It is then immediate that S has unit norm from Xa to Xa/L . This considerably simplifies the arguments in Subsubsec. 3.2.4. For the case of Julia matrices, it is even not necessary to go through the developments of Sec. 4. 3.2.4. Conclusion of the proof of the quasi-ballistic bound According to (24), we have X X n≥n
|h0|χIσN (H)|ni|2 ≤ kRN (ρ0 )ka
σ∈ΣL N
X
e−an
n≥n
≤
1 N kR (ρ0 )ka e−a(n+1) . a
(27)
We now choose a = (A/B)N a0 with a0 < min{B, θ0 (∆)}, so that the bounds (25) and (26) are applicable, and estimate kRN (ρ0 )ka by kρ0 ka0 kPka0 →a0 kSka0 →a0 A/B . . . kPka0 (A/B)N −1 →a0 (A/B)N −1 × kSka0 (A/B)N −1 →a0 (A/B)N . Substituting the bounds (25), (26), and kρ0 ka0 = 1 leads to kR (ρ0 )ka0 (A/B)N ≤ Q N
N
B A
2N 2
≡
4c2 (∆)B 2 Le2a0 C/B (θ0 (∆) − a0 )2 (B − a0 )2 a20
N
B A
2N 2 .
Now, in order for (27) with a replaced by a0 (A/B)N to be less than 3/2 e−2λN , it is sufficient that N 2N 2 +N ! 1 B QN e2λN B . n≥ log a0 A A 3/2 a0 For this, it is in turn sufficient that n satisfies (23) where the constant C3 (ξ, λ) may also depend on A, B, Q, and a0 , but not on and N . 3.3. Upper bound under hypothesis (22) Replacing (19) in (23), and using Definition 9 of the transport exponent β0+ , one gets: Proposition 3. Suppose that the orthogonal polynomials of µ satisfy (22). Then log(B/A) . β0+ ≤ Λ(µ)
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4. Asymptotic Properties of Orthogonal Polynomials In this subsection we prove two propositions which, when combined with Propostion 3, directly lead to Theorems 1 and 2. We shall use the main results of the work of Stahl and Totik [26] reviewed in Appendix 2. For this purpose, we first verify that the capacity of J is positive and that the LDOS is asymptotically regular in the sense of [26]. The definition of this property is recalled in Appendix 2. Let us point out that the measure µ on J need not be defined via an invariant and ergodic measure on the symbolic dynamics, but we do use the hypothesis imposed at the very end of Subsec. 2.1. Lemma 2. Let J be a set constructed as in Subsec. 2.1. Then cap(J) > 0. Lemma 3. The measures µ on J constructed in Subsec. 2.1 are asymptotically regular. Knowing that cap(J) > 0 and that µ is asymptotically regular, the theorem in Appendix 2 directly implies that formulas (6)–(8) in Subsec. 2.3 hold. Proof of Lemma 2 (Adapted from [1]). Let µ be a probability measure carried by J. Its potential is Z 1 0 . Φµ (E) = dµ(E ) log |E − E 0 | We show that, if µ is the Bernoulli measure with equal weights, then |Φµ (E)| ≤ C < ∞ for E ∈ J. This implies that cap(J) ≥ exp(−C). N the N th generation interval containing E ∈ J, we have the Denoting Iσ(E) S N −1 0 N \Iσ(E) ). Thus identity I = N ≥1 (Iσ(E) Φµ (E) =
XZ N ≥1
N −1 N Iσ(E) \Iσ(E)
dµ(E 0 ) log
1 |E − E 0 |
.
(28)
N −1 N \Iσ(E) , |E − E 0 | cannot be less than the gaps |GN | of As long as E 0 ranges in Iσ(E) N generation N adjacent to Iσ(E) . From Lemma 4 in Appendix 1, we get |GN | ≥ a/λN 0 where λ = maxE∈J |S (E)| and a is some positive constant. Inserting this in (28), we obtain N X 1 λ , (29) (L − 1) N log |Φµ (E)| ≤ L a N ≥1
which directly implies the result.
Remark 6. The proof of Lemma 2 yields a lower estimate for the capacity of the Cantor set C in [0, 1] generated with L = 2 and Sˆ1 (E) = ΛE, Sˆ2 (E) = Λ(1 − E), Λ > 2. In that case, |GN | = (Λ − 2)Λ−N , so Eq. (29) yields cap(C) ≥ (Λ − 2)Λ−2 . This crude estimate was used in the discussion in Subsec. 2.6. Proof of Lemma 3. Here we use the hypothesis that − log(µ(IσN ))/N has a uniform upper bound, say h. We shall use the criterion (37) and verify that the
UPPER BOUNDS FOR QUANTUM DYNAMICS
1263
sets on the left and right hand side of (37) coincide. For that purpose, we use the formula for the upper pointwise dimensions given in Theorem 3 in which the denominator is replaced by Eq. (33). As |IσN | ≤ baN with a < 1, we therefore obtain for any E ∈ J, dµ (E) ≤ h/ log(1/a) < ∞ which proves the lemma. 4.1. The analytic case Proposition 4. Let S be an analytic map, and let Ec and Rc = |I 0 |/2 be the center of the spectrum and its radius, respectively. For any R > Rc , the bound (22) holds for any η > 0 with R , B = max gJ (S(z)) + η , (30) A = log z∈ΓR Rc where ΓR is the circle of radius R around Ec and gJ denotes the Green’s function of J. Proof. Using analyticity of S and orthogonality of the polynomials, we first write Z X 1 Z Pm (S(z)) dµ(E)Pn (E) (E − Ec )k , dz hn|S|mi = 2πı Γ (z − Ec )k+1 k≥n
where Γ is some integration path around the spectrum with positive orientation. For given R > Rc , let us choose Γ to be ΓR . Furthermore, we apply the Cauchy– Schwarz inequality to the last term and bound (E −Ec ) by Rs for E in the spectrum. Thus summing up, we get n 1 Rc max |Pm (S(z))| . |hn|S|mi| ≤ R 1 − Rc /R z∈ΓR Finally we use (34) in order to get an upper bound on the growth of the orthogonal polynomials. For any η > 0, there exists an M (η) such that |Pm (S(z))| ≤ exp(m gJ (S(z)) + η) ,
(31)
for all m ≥ M (η). Hence (22) holds for any m ≥ M (η) and the choices (30). We can choose C = C(η) in such a way that it holds for all m ∈ N. 4.2. The polynomial IFS case Proposition 5. For any l = 1, . . . , L, let the branch Sl have analytic continuation Sˆl given by a polynomial of degree Dl . Set D = maxl Dl . Let furthermore Rc = |I 0 |/2 be the spectral radius and ∆ the size of the smallest gap at the first generation. Then, for any η > 0, the bound (22) holds with ∆ ∆ , B = max sup gJ (Sˆl (E)) + D arcsinh +η. A = arcsinh l=1,...,L E∈J 4Rc 4Rc
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Proof. We start from the following identity: hn|S|mi =
L Z X l=1
=
R
dµ(E)χIl1 (E)Pn (E)Pm (Sˆl (E))
L XZ X l=1 k≥0
dµ(E)Γk0 (Il1 )Pk (E)Pn (E)Pm (Sˆl (E)) ,
R
with Γk0 (Il1 ) defined as in Lemma 1. By orthogonality, the sum over k is restricted to k ≥ |n − Dm|. Using the bound (34) (with the same precautions as in the proof of Proposition 4), plus the Schwarz inequality, we obtain by Lemma 1 X e−θ0 (∆)k |hn|S|mi| ≤ L c(∆) exp m max sup gJ (Sˆl (E)) + η l=1...L E∈J
k≥|n−Dm|
and the announced result follows. Appendix 1. Pointwise Dimensions of IFS Measures
The lower and upper pointwise dimensions of a Borel measure µ at a point E ∈ R are defined by dµ (E) = lim inf →0
log(µ([E − , E + ])) , log()
log(µ([E − , E + ])) . dµ (E) = lim sup log() →0
(32)
Furthermore, the Hausdorff dimension dimH (µ) of µ is defined by the infimum of the Hausdorff dimensions of all Borel subsets ∆ ⊂ R satisfying µ(∆) = 1. It is equal to µ − esssupE∈R dµ (E) [2, 10]. In this appendix we give a proof that the pointwise dimensions of the IFS measures described in Subsec. 2.1 coincide and are constant µ-almost surely. Although this can be deduced from more general results in the literature ([22], but there probably exist other works), we give here a short, independent proof for the sake of completeness and also because it leads to a slightly more general result, used in Sec. 4. Theorem 3. If µ is any measure supported by J, then its pointwise dimensions are given by dµ (E) = lim inf N →∞
1 N
dµ (E) = lim sup N →∞
1 N
N − N1 log(µ(Iσ(E) )) , XN −1 log(|S 0 (S ◦i (E))|) i=0
N − N1 log(µ(Iσ(E) )) . XN −1 log(|S 0 (S ◦i (E))|) i=0
UPPER BOUNDS FOR QUANTUM DYNAMICS
1265
It immediately follows from the Breiman–Shannon–McMillan theorem and Birkhoff’s ergodic theorem [5], that the upper and lower pointwise dimension coincide, and are constant µ-almost surely whenever the measure µ is ergodic. This gives formula (2). For the proof of Theorem 3, we shall need several reformulations of the basic contraction properties of S stating that there exist finite positive constants b1 and N N a1 < 1 such that |IσN | ≤ b1 aN 1 . We denote the gap to the right of Iσ by Gσ . Lemma 4. There exist finite positive constants a2 , a3 , a4 and a5 < 1 such that for any N ∈ N and σ ∈ ΣL the following properties hold. 0 0 ∈ IσN , one has |(S ◦N )0 (EN )|/a2 ≤ |(S ◦N )0 (EN )| ≤ (i) For all EN , EN 0 )|. a2 |(S ◦N )0 (EN (ii) For any EN ∈ IσN , one has 1/a3 ≤ |(S ◦N )0 (EN )| |IσN | ≤ a3 . (iii) If I is either of the intervals of generation N adjacent to GN σ , then |I|/a4 ≤ | ≤ a |I|. |GN 4 σ (iv) |IσN | ≤ a5 |IσN −1 |. Proof. (i) It is sufficient to show that for any choice of Ej , Ej0 ∈ Iσj , j = Q −1 |S 0 (Ej )| ◦N 0 ) (EN ) = 0, . . . , N − 1, one has N j=0 |S 0 (Ej0 )| ≤ a2 . In fact, by the chain rule (S QN −1 0 ◦j ◦j N −j and this implies (i). Next we note that j=0 S (S (EN )) and S (EN ) ∈ Iσ |S 0 (Ej )| ≤ |S 0 (Ej0 )| + max |S 00 (E)| |Iσj | ≤ |S 0 (Ej0 )| + b2 b1 aj1 , E∈J
for some finite constant b2 . By one of the hypothesis in Sec. 2, |S 0 (E)| ≥ b3 and it follows that ! N −1 N −1 Y Y b2 b1 aj1 |S 0 (Ej )| b2 b1 ≤ 1 + ≤ exp . |S 0 (Ej0 )| b3 (1 − a1 )b3 j=0 j=0 (ii) follows directly from minE 0 ∈I 0 |(Sσ−1 ◦ · · · ◦ Sσ−1 )0 (E 0 )| |I 0 | ≤ |IσN | ≤ 1 N −1 −1 0 0 0 maxE 0 ∈I 0 |(Sσ1 ◦ · · · ◦ SσN ) (E )| |I | and (i). Concerning (iii), we denote the gap by G and the intervals to its left and right by I+ and I− . Let us suppose that the gap opened at generation N , notably that ◦ · · · ◦ Sσ−1 (I 0 ) is the disjoint union of there exists an σ such that IσN −1 = Sσ−1 1 N −1 I+ , G and I− . The situation where G opened at some earlier generation can be treated in an analogous way. Now I+ , G and I− are respectively the images under 1 ◦ · · · ◦ Sσ−1 of the intervals Il1 , G1l and Il+1 of the first generation the mapping Sσ−1 1 N −1 N −1 for some l = 1, . . . , L. Fix EN −1 ∈ Iσ . By the same argument as in (ii), we obtain |Il1 |/a1 ≤ |(S ◦N −1 )0 (EN −1 )| |I+ | ≤ a1 |Il1 |, a similar inequality for I− as well as |G1l |/a1 ≤ |(S ◦N −1 )0 (EN −1 )| |G| ≤ a1 |G1l |. From these inequalities we can easily deduce (iii). To prove (iv), we use the same notations and suppose that the interval of the N th generation is I− (the case I+ being again similar). Then |IσN −1 | = |I+ | + |G| + |I− |. The above estimates now lead to |G1l | |Il1 | N −1 |Iσ | ≥ 1 | + a2 |I 1 | + 1 |I− | . a21 |Il+1 1 l+1
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Proof of Theorem 3. For > 0, let N () be the smallest positive integer satN () N () N () isfying Iσ(E) ⊂ [E − , E + ]. Clearly µ(Iσ(E) ) ≤ µ([E − , E + ]) and |Iσ(E) | ≤ 2. N ()−1
Moreover, Iσ(E)
contains either [E − , E] or [E, E + ]. Let us show that there N ()−r
exists an r ∈ N (independent of ) such that µ([E − , E + ]) ≤ µ(Iσ(E)
). For this
N ()−s purpose, we note that Iσ(E) is an in s ≥ 1 increasing sequence of intervals N ()−s N ()−1 fying |Iσ(E) | ≥ a−s+1 |Iσ(E) | ≥ a−s+1 with a5 as given in Lemma 4. The 5 5 N ()−s
satislatter
of generation N () − s having possilemma moreover implies that the gap G N ()−s |≥ bly non-empty intersection with [E − , E + ] satisfies |GN ()−s | ≥ a−1 4 |Iσ(E) −1 −s+1 −1 −r+1 . Now we choose r such that a4 a5 ≥ 1. Then [E − , E + ] is covered a 4 a5 N ()−r N ()−r N ()−r by Iσ(E) and G and clearly we have µ([E − , E + ]) ≤ µ(Iσ(E) ) as well N ()−r
as |Iσ(E)
| > . Consequently N ()−r
log(µ(Iσ(E)
))
N ()
log(|Iσ(E) |/2)
N ()
≤
log(µ(Iσ(E) )) log(µ([E − , E + ]) . ≤ N ()−r log() log(|Iσ(E) |)
Now N () → ∞ as → 0 and therefore it suffices to show that lim inf N →∞
N − N1 log(µ(Iσ(E) )) N +r − N1 log(|Iσ(E) |/2)
= lim inf N →∞
N − N1 log(µ(Iσ(E) )) N −r − N1 log(|Iσ(E) |)
,
and that their common value is that in the statement, as well as a similar statement N for the superior limit. But because E ∈ Iσ(E) for all N ∈ N, Lemma 4(ii) implies that for M = N and M = N − r −
M |) log(|Iσ(E)
N
=
N 1 X 1 , log(|S 0 (S ◦i−1 (E))|) + O N i=1 N
(33)
and this concludes the proof. Appendix 2. The Theory of Stahl and Totik
The following theorem summarizes the main results of [26] used in this work. Let us recall that a sequence of functions fn converges locally uniformly in an open set G ⊂ C to a function f if for all z ∈ G and sequences zn → z, one has fn (zn ) → f (z). In a similar way, one defines a bound fn ≤ f to hold locally uniformly in G. Theorem [26, Theorems 3.1.1 and 4.2.1]. Let µ be a Borel probability measure on R with compact support J = supp(µ) and I ⊂ R be the smallest interval containing J. Furthermore (Pn )n≥0 and (tn , vn )n≥0 denote the associated orthogonal polynomials and coefficients of the Jacobi matrix. Then the following three assertions are equivalent: (1) the limit lim |Pn (z)|1/n = exp(gJ (z))
n→∞
holds true locally uniformly in C\I;
(34)
UPPER BOUNDS FOR QUANTUM DYNAMICS
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(2) the limit lim |Pn (E)|1/n = 1
n→∞
holds true for all E ∈ J\Y where cap(Y ) = 0; (3) the formula n 1X log(tj ) = log(cap(J)) lim n→∞ n j=1
(35)
(36)
holds. If the above assertions are satisfied, the measure µ is called asymptotically regular. This also implies that (34) holds locally uniformly in C when the equality is replaced by an inequality ≤ . Furthermore, if cap(J) > 0, then 1 n→∞ n
w- lim
X
δE = ω J .
E,Pn (E)=0
A sufficient condition for a measure µ to be asymptotically regular is the following equality: (37) cap E ∈ R|dµ (E) < ∞ = cap(J), where dµ (E) is the upper pointwise dimension of ν at E defined in (32). Acknowledgements The work of H. S.-B. was supported by the grant ERBFMRX-CT96-0010 of the European Community and the SFB 288 at the Technical University Berlin. We acknowledge pleasurable discussions with G. Mantica. Useful remarks by Prof. B. Simon are also acknowledged. Note Added in Proof Based on the results of the present work, Barbaroux and Schulz-Baldes have proven upper bounds also on the exponents βα , α > 0 (Annales Inst. H. Poincar´e, Nov. 1999). References [1] H. Aikawa and M. Ess´en, Potential Theory: Selected Topics, Lecture Notes in Mathematics 1633, Springer, Berlin, 1995. [2] J.-M. Barbaroux, J.-M. Combes and R. Montcho, “Remarks on the relation between quantum dynamics and fractal spectra”, J. Math. Anal. and Appl. 213 (1997) 698– 722. [3] M. F. Barnsley, J. S. Geronimo and A. N. Harrington, “Infinite dimensional Jacobi matrices associated with Julia sets for polynomials”, Proc. Am. Math. Soc. 88(4) (1983) 625–630. [4] J. Bellissard, “Stability and instability in quantum mechanics”, in Trends and Developments in the Eighties, eds. S. Albeverio and Ph. Blanchard, World Scientific, Singapore, 1985. [5] P. Billingsley, Ergodic Theory and Information, Wiley, New York, 1965.
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[6] H. Brolin, “Invariant sets under iteration of rational functions”, Arkiv f¨ or Matematik, Band (6) (1965) 103–144. [7] P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkh¨ auser, Basel, 1980. [8] R. Ketzmerick, G. Peschel and T. Geisel, “Slow decay of temporal correlations in quantum systems with Cantor spectra”, Phy. Rev. Lett. 69 (1992) 695–698. [9] I. Guarneri, “Spectral properties of quantum diffusion on discrete lattices”, Europhys. Lett. 10 (1989) 95–100; “On an estimate concerning quantum diffusion in the presence of a fractal spectrum”, Europhys. Lett. 21 (1993) 729–733; and I. Guarneri and G. Mantica, “On the asymptotic properties of quantum dynamics in the presence of fractal spectrum”, Ann. Inst. H. Poincar´ e 61 (1994) 369–379. [10] I. Guarneri, “Singular continuous spectra and discrete wave packet dynamics”, J. Math. Phys. 37 (1996) 5195–5206. [11] I. Guarneri, “On the dynamical meaning of spectral dimensions”, Ann. Inst. H. Poincar´e 68 (1998) 491–506. [12] I. Guarneri and G. Mantica, “Multifractal energy spectra and their dynamical implications”, Phys. Rev. Lett. 73 (1994) 3379–3383. [13] F. Hippert and D. Gratias, eds., Lectures on Quasicrystals, Les ´editions de physique, Les Ulis, 1994. [14] H. Hiramoto and S. Abe, “Dynamics of an electron in quasiperiodic systems. I. Fibonacci model”, J. Phys. Soc. Japan 57 (1988) 230–240; and “Dynamics of an electron in quasiperiodic systems. II. Harper model”, J. Phys. Soc. Japan 57 (1988) 1365–1372. [15] M. Holschneider, “Fractal wavelet dimension and localization”, Commun. Math. Phys. 160 (1994) 457–474. [16] J. Hutchinson, “Fractals and self-similarity,” Indiana Univ. Math. J. 30 (1981) 713– 747. [17] R. Ketzmerick, K. Kruse, S. Kraut and T. Geisel, “What determines the spreading of a wave packet?”, Phys. Rev. Lett. 79 (1997) 1959–1962. [18] Y. Last, private communication. [19] Y. Last, “Quantum dynamics and decomposition of singular continuous spectra”, J. Funct. Anal. 142 (1996) 402–445. [20] G. Mantica, “Quantum intermittency in almost periodic systems derived from their spectral properties”, Physica D103 (1997) 576–589; “Wave propagation in almostperiodic structures”, Physica D109 (1997) 113–127. [21] P. Neu and R. Speicher, “Rigorous mean field model for the CPA: Anderson model with free random variables”, J. Stat. Phys. 80 (1995) 1279–1308. [22] Y. Pesin and H. Weiss, “The multifractal analysis of Gibbs measures: Motivation, mathematical foundation, and examples”, Chaos 7 (1997) 89–106. [23] F. Pi´echon, “Anomalous diffusion properties of wave packets on quasiperiodic chains”, Phys. Rev. Lett. 76 (1996) 4372–4375. [24] R. del Rio, S. Jitomirskaya, Y. Last and B. Simon, “Operators with singular continuous spectrum: IV. Hausdorff dimension, rank-one perturbations and localization”, J. d’Analyse Math. 69 (1996) 153–200. [25] H. Schulz-Baldes and J. Bellissard, “Anomalous transport: A mathematical framework”, Rev. Math. Phys. 10 (1998) 1–46; H. Schulz-Baldes and J. Bellissard, “A kinetic theory for quantum transport in aperiodic media”, J. Stat. Phys. 91 (1998) 991–1027. [26] H. Stahl and V. Totik, General Orthogonal Polynomials, Cambridge Univ. Press, Cambridge, 1992.
THE CONTINUOUS SPIN RANDOM FIELD MODEL: FERROMAGNETIC ORDERING IN d ≥ 3* ¨ CHRISTOF KULSKE WIAS, Mohrenstrasse 39 D-10117 Berlin, Germany E-mail : [email protected] Received 10 August 1998 We investigate the Gibbs-measures of ferromagnetically coupled continuous spins in double-well potentials subjected to a random field (our specific example being the φ4 theory), showing ferromagnetic ordering in d ≥ 3 dimensions for weak disorder and large energy barriers. We map the random continuous spin distributions to distributions for an Isingspin system by means of a single-site coarse-graining method described by local transition kernels. We derive a contour-representation for them with notably positive contour activities and prove their Gibbsianness. This representation is shown to allow for application of the discrete-spin renormalization group developed by Bricmont/Kupiainen implying the result in d ≥ 3. Keywords: Disordered systems, contour models, cluster expansions, renormalization group, random field model.
1. Introduction The study of phase transitions in continuous spin lattice models has a long history. An important prototypical example of a random model in this class is the continuous spin random field model, where ferromagnetically coupled real valued spins fluctuate in randomly modulated local double-well potentials. In the present paper we study this model for weak disorder in dimensions d ≥ 3 proving ferromagnetic ordering. Our aim is more generally to describe an expansion method mapping multiple-well continuous spin models to discrete spin models with exponentially decaying interactions by means of a single-site coarse-graining. Then we make use of information about the latter ones. This transformation can be regarded as an example of a useful (and moreover non-pathological) single-site “renormalization group” transformation. While it is already interesting in a translation-invariant situation, it is particularly useful for non-translational invariant systems since it allows to “factorize” the degrees of freedom provided by the fluctuations of the spins around their local minima. It is ten years now since the existence of ferromagnetic ordering for small disorder at small temperatures was proved for the ferromagnetic random field Ising-model (with spins σx taking values in {−1, 1}) by Bricmont–Kupiainen [5], answering a question that had been open for long in the theoretical physics community. The ∗ Work
supported by the DFG Schwerpunkt “Stochastische Systeme hoher Komplexit¨ at”. 1269
Reviews in Mathematical Physics, Vol. 11, No. 10 (1999) 1269–1314 c World Scientific Publishing Company
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¨ C. KULSKE
“converse”, namely the a.s. uniqueness of the Gibbs-measure in d = 2 was proved later by Aizenman and Wehr [1]. For an overview on the random field model from the perspective of theoretical physics, see e.g. [19]. Given the popularity of continuous spin models it is however certainly desirable to have a transparent method that is able to treat the additional degrees of freedom present in such a model. Bricmont and Kupiainen introduced in [5] the conceptually beautiful method of the renormalization group (RG) to the rigorous analysis of the low temperature behavior of a disordered system, that turned out to be very powerful in this situation although there is no scale-invariance in the problem. The heuristic idea is: map the initial spin-system onto a coarse-grained one that appears to be at lower temperature and smaller disorder. Then iterate this transformation. This idea has to be implemented in a suitable representation of contours (that are the natural variables at low temperatures). (For a pedagogical presentation of such a RG in application to the proof of stability of solid-on-solid interfaces in disordered media, see also [8, 17].) An alternative treatment of disordered lattice systems with finite local spin-space was sketched by Zahradn´ık [22], however also using some iterated coarse graining. It is also clear that in the more difficult situation of continuous spins, spatial renormalization will be needed. However, continuous spins being more “flexible” than Ising spins make it difficult to cut the analysis in local pieces. It is then to be expected that the difficulties to control the locality of a suitably defined renormalization group transformation acting directly on continuous spins in a rigorous way would blow up tremendously compared with the discrete spin case of [5]. (The amount of technical work needed in their proof is already not small!) For an example of a rigorous construction of an RG-group for a continuous spin-lattice system, see [2, 3] for the ordered Heisenberg–Ferromagnet. (This might give some idea of the complexities of such a method.) Indeed, despite the conceptual beauty, technical difficulties have kept the number of rigorous applications of the RG to low-temperature disordered lattice spin systems limited. Moreover, usually a lot of technical work has to be repeated when extending such a method to a more complex situation, while it would be desirable to make use of older results in a more transparent way. We will therefore describe a different and more effective way to the continuous spin problem: (1) Construct a single-site “RG”-transformation that maps the continuous model to a discrete one. Obtain bounds on the first in terms of the latter one. In our specific φ4 double-well situation this transformation is just a suitable stochastic mapping to the sign-field. (2) Apply the RG group to the discrete model. As we will show, the discrete (Ising-) model in our case has a representation as a contour model whose form is invariant under the discrete-spin RG that was constructed in [5]. So we need not repeat the RG analysis for this part but can apply their results, avoiding work that has already been done. In the last few years there has been an ongoing discussion about the phenomenon of RG pathologies. It was first observed by Griffith, Pearce, Israel (and extended
THE CONTINUOUS SPIN RANDOM FIELD MODEL
1271
in various ways by van Enter, Fernandez, Sokal [13]) that even very “innocent” transformations like taking marginals on a sub-lattice of the original lattice can map a Gibbs-measure of a lattice spin system to an image measure that need not be a Gibbs-measure for any absolutely summable Hamiltonian. (See [13] for a clear presentation and more information about what pathologies can and cannot occur, see also the references given therein.) On the other hand, as a reaction to this, there has been the “Gibbsian restoration program” initiated by the late Dobrushin [11] whose aim it is to exhibit sets of “bad configurations” of measure zero (w.r.t. the renormalized measure) outside of which a “renormalized” Hamiltonian with nicely decaying interactions can be defined. This program has been carried out in [7] for a special case (again using RG based on [5]). Since we will be dealing with contour representations of finite volume measures that provide uniform bounds on the initial spin system we do not have to worry about non-Gibbsianness vs. Gibbsianness to get our results. Nevertheless, to put our work in perspective with the mentioned discussion, we will in fact construct a uniformly convergent “renormalized Hamiltonian” for the measure on the sign-field, for all configurations. In other words, there are no pathologies in our single-site coarse graining and the situation is as nice and simple as it can be. Let us introduce our model and state our main results. We are interested in the d analysis of the Gibbs measures on the state space Ω = RZ of the continuous spin model given by the Hamiltonians in finite volume Λ q X q X ˜ ∂Λ ,ηΛ EΛm (mΛ ) = (mx − my )2 + (mx − m ˜ y )2 2 {x,y}⊂Λ 2 x∈Λ;y∈∂Λ d(x,y)=1
d(x,y)=1
+
X x∈Λ
V (mx ) −
X
ηx mx
(1.1)
x∈Λ
˜ ∂Λ . Here we write for a configuration mΛ ∈ ΩΛ = RΛ with boundary condition m ∂Λ = {x ∈ Λc ; ∃ y ∈ Λ : d(x, y) = 1} for the outer boundary of a set Λ where d(x, y) = kx − yk1 is the 1-norm on Rd . q ≥ 0 will be small. Given its history and its popularity we will consider mainly the example of the well-known double-well φ4 -theory. As we will see during the course of the proof, there is however nothing special about this choice. We use the normalization where the minimizers are ±m∗ , the curvature in the minima is 1, and the value of the potential in the minima is zero and write (m2x − (m∗ )2 )2 , (1.2) V (mx ) = 8m∗ 2 where the parameter m∗ ≥ 0 will be large. We consider i.i.d. random fields (ηx )x∈Zd that satisfy (i) ηx and −ηx have the same distribution, t2
(ii) P[ηx ≥ t] ≤ e− 2σ2 , (iii) |ηx | ≤ δ, where σ 2 ≥ 0 is sufficiently small. The assumption (iii) of having uniform bounds is not essential for the problem of stability of the phases but made to avoid
¨ C. KULSKE
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uninteresting problems with our transformation and keep things as transparent as possible. ˜ ∂Λ ,ηΛ are then defined as usual through The finite volume Gibbs-measures µm Λ the expectations Z m ˜ ∂Λ ,ηΛ 1 m ˜ ∂Λ ,ηΛ (mΛ ) µΛ (f ) = m dmΛ f (mΛ , m ˜ Λc )e−EΛ (1.3) ˜ ∂Λ ,ηΛ ZΛ RΛ for any bounded continuous f on Ω with the partition function Z m ˜ ∂Λ ,ηΛ ˜ ∂Λ ,ηΛ (mΛ ) ZΛm = dmΛ e−EΛ .
(1.4)
RΛ
We look in particular at the measures with boundary condition m ˜ x = +m∗ (for all ∗ ,ηΛ d . x ∈ Z ) in the positive minimum of the potential, for which we write µ+m Λ To prove the existence of a phase transition we will show that, for a suitable range of parameters, with large probability w.r.t. the disorder, the Gibbs-expectation of finding the field left to the positive well is very small. Indeed, we have as the main result: Theorem 1. Let d ≥ 3 and assume the conditions (i), (ii), (iii) with σ 2 small enough. Then, for any (arbitrarily small) γ > 0, there exist q0 > 0 (small enough), δ0 , δ1 > 0 (small enough), τ0 (large enough) such that, whenever δ ≤ δ0 , q(m∗ )2 ≥ τ0 2 and q(m∗ ) 3 ≤ δ1 we have that # " const m∗ +m∗ ,ηΛN ≥ γ ≤ e− σ 2 mx 0 ≤ (1.5) P lim sup µΛN 2 N ↑∞ for an increasing sequence of cubes ΛN . Remark. Note that the quantity q(m∗ )2 gives the order of magnitude of the minimal energetic contribution of a nearest neighbor pair of spins with opposite signs to the Hamiltonian (1.1); it will play the role of a (low temperature) Peierls constant. Smallness of q (to be compared with the curvature unity in the minima of the potential) is needed to ensure a fast decay of correlations of the thermal fluctuations around the minimizer in a given domain. The stronger condition on 2 the smallness, q ≤ const(m∗ )− 3 , however is needed in our approach to ensure the positivity and smallness of certain anharmonic corrections. Let us now define the transition kernel Tx · · from R to {−1, 1} we use and explain why we do it. Put, for a continuous spin mx ∈ R, and an Ising spin σx ∈ {−1, 1} 1 (1.6) Tx (σx |mx ) := (1 + σx tanh(am∗ mx )) , 2 where a ≥ 1, close to 1, will have to be chosen later to our convenience. In other words, the probability that a continuous spin mx gets mapped to its sign is given by 12 (1 + tanh(am∗ |mx |)) which converges to one for large m∗ . The above kernel
THE CONTINUOUS SPIN RANDOM FIELD MODEL
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˜ ∂Λ ,ηΛ defines a joint probability distribution µm (dmΛ )T (dσΛ |mΛ ) on RΛ × {−1, 1}Λ Λ whose non-normalized density is given by m ˜ ∂Λ ,ηΛ
e−EΛ
(mΛ )
Y
Tx (σx mx ) .
(1.7)
x∈Λ
Its marginal on the Ising-spins σΛ Z m ˜ ∂Λ ,ηΛ T µΛ (dσΛ ) :=
RΛ
˜ ∂Λ ,ηΛ µm (dmΛ )T (dσΛ mΛ ) Λ
(1.8)
will be the main object of our study. To prove the existence of a phase transition stated in Theorem 1 we will have to deal only with finite volume contour representations of (1.8), as given in Proposition 5.1. Nevertheless, it is perhaps most instructive to present the following infinite volume result in the Hamiltonian formulation to explain the nature of the transformation. Theorem 2. Assume the hypothesis of Theorem 1 and let η be any fixed realization of the disorder. Suppose that µη is a continuous spin Gibbs-measure obtained ˜ ∂Λ ,ηΛ along a sequence of cubes Λ for some boundary condition as a weak limit of µm Λ d ∗ ∗ Z m ˜ ∈ {−m , m } . Then, for a suitable choice of the parameter a ≥ 1 (close to 1) in the kernel T the following is true. d The measure T (µη ) on {−1, 1}Z is a Gibbs measure for the absolutely summable Ising–Hamiltonian η HIsing (σ) = −
a2 (m∗ )2 X (a − q∆Zd )−1 x,y σx σy 2 x,y
− am∗
X X (a − q∆Zd )−1 ΦC (σC ; ηC ) , x,y ηx σy − x,y
(1.9)
C:|C|≥2
where ∆Zd is the lattice Laplacian in the infinite volume, i.e. ∆Zd ;x,y = 1 iff x, y ∈ V are nearest neighbors, ∆Zd ;x,y = −2d iff x = y and ∆Zd ;x,y = 0 else. The many-body potentials are symmetric under joint flips of spins and randomfields, ΦC (σC , ηC ) = ΦC (−σC , −ηC ), and translation-invariant under joint latticeshifts. They obey the uniform bound |ΦC (σC , ηC )| ≤ e−˜γ |C|
(1.10)
with a positive constant γ˜ . Remark 1. As in Theorem 1, γ˜ can be made arbitrarily small by choosing q0 , δ0 , δ1 small and τ0 large. More information about estimates on the value of γ and γ˜ can in principle be deduced from the proofs. Remark 2. By imposing the smallness of δ we exclude pathologies due to exceptional realizations of the disorder variable η (“Griffiths singularities”) in the
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1274
transformation T . (We stress that this does not simplify the physical problem of the study of the low-temperature phases which is related to the study of the formation of large contours.) Starting from the joint distribution (1.7) it is natural to consider the distribution of continuous spins conditional on the Ising spins; here the Ising spins σx will play the role of a second sort of external fields. Then, as it was explained in [7], possible pathologies in the transformation T would be analogous to Griffiths-singularities created by pathological Ising configurations. In this sense, Theorem 2 states that there are neither Griffiths singularities of the first type (w.r.t. η) nor the second type (w.r.t σ). The treatment of unbounded random fields would necessitate the analysis of so-called “bad regions” in space (where the realizations of the random fields are anamolously large). This should be possible but would however obscure the nature of the transformation T . Let us now motivate the form of Tx and comment on the structure of the Hamiltonian. Introducing quadratic potentials, centered at ±m∗ , Qσx (mx ) :=
a (mx − σx m∗ )2 + b 2
(1.11)
with b > 0 (close to zero) to be chosen later, we can rewrite the transition kernel in the form ωx e−Q (mx ) P Tx (σx |mx ) = (1.12) ¯ x (m ) . −Qω x ω ¯ x =±1 e The crucial point is that the joint density (1.7) contains a product over x over the quantities σx (1.13) e−V (mx ) Tx (σx |mx ) = e−Q (mx ) (1 + w(mx )) where, using (1.12), we can write the remainder in the form (1 + w(mx )) := P
e−V (mx ) ¯ x (m ) . −Qσ x σ ¯ x =±1 e
(1.14)
Now, if the initial potential V (mx ) is sufficiently Gaussian around its minima and the quadratic potential Qσx is suitably chosen, w(mx ) should be small in some sense. If w(mx ) were even zero, we would be left with σΛ -dependent Gaussian integrals that can be readily carried out. They lead to the first two terms in the Ising–Hamiltonian (1.9), containing only pair-interactions. This can be understood by a formal computation. The modification of the measure for “small” w(mx ) then gives rise indeed to exponentially decaying many-body interactions, as one could naively hope for. Q Expanding x∈Λ (1 + w(mx )) then leads in principle to an expansion around a Gaussian field.a However, one problem with this direct treatment is that resulting contour activities will in general be nonnegative only if w(mx ) ≥ 0 for all mx . But note that the latter can only be true for the narrow class of potentials such that a The author is grateful to M. Zahradn´ık for pointing out the idea to decompose e−V (mx ) into a sum of two Gaussians and a remainder term that should be expanded. However, contrary to [23] we write the remainder in a multiplicative form which allows for the transition kernel interpretation.
THE CONTINUOUS SPIN RANDOM FIELD MODEL
1275
V (mx ) ≤ Const m2x for large |mx |. Thus, w(mx ) will have to become negative for some mx e.g. for V compact support or in the φ4 -theory. While it is not necessary to have positive contour activities for some applications (see [4, 23]) it is crucial for the random model: A RG, as devised in [5], needs non-negative contour weights.b We are able to solve this problem and define positive effective anharmonic weights by a suitable resummation and careful choice of the parameters a, b of the quadratic potential Qσx ; these will be kept fixed. This choice is the only point of the proof that has to be adapted to the specific form of the initial potential V . Later the positivity of weights will also be used for the control of the original measure in terms of the Ising-measure (see Proposition 5.2). In Sec. 2 it is shown how non-negative effective anharmonic weights obeying suitable Peierls bounds can be defined. Section 3 finishes the control of the anharmonicity around the Ising model arising from the purely Gaussian theory (i.e. w(mx ) ≡ 0) in terms of a uniformly convergent expansion. Section 4 treats the simple but instructive case of the Ising field without the presence of anharmonicity, showing the emergence of (generalized) Peierls bounds on Ising contours. In Sec. 5 we obtain our final contour model for the full theory and prove Theorems 1 and 2. The Appendix collects some facts about Gaussian random fields and random walk expansions we employ. 2. Anharmonic Contours with Positive Weights We will explain in this section how (preliminary) “anharmonic contours” with “anharmonic weights” that are non-negative and obey a Peierls estimate can be constructed. We start with a combinatorial Lemma 2.1 and a suitable organization of the order of Gaussian integrations appearing to derive algebraically the representation of Lemma 2.3. We will make no specific assumptions about the potential at this point that should however be thought to be symmetric “deep” double-well. Our later treatment is valid once we have the properties of “positivity” and “uniform Peierls condition of anharmonic weights” that are introduced in (2.19) and (2.20). These are then verified for the φ4 -theory in an isolated part of the proof that can be adapted to specific cases of interest. We will have to deal with the interplay of three different fields: continuous spins mx (to be integrated out), Ising spins σx and (fixed) random fields ηx , subjected to various boundary conditions in various volumes. In some sense, the general theme of the expansions to come is: keep track of the locality of the interaction of these fields in the right way. For the sake of clarity we found it more appropriate in this context to keep a notation that indicates the dependence on these quantities in an explicit way in favor of a more space-saving one. Now, since we are interested here in a contour-representation of the image m ˜ ∂Λ ,ηΛ under the stochastic transformation (1.6), let us look at the measure T µΛ b Vaguely speaking, the method keeps lower bounds on the energies of all configurations, but also upper bounds on the energies of some configurations (that are candidates for the true groundstates). This can be seen nicely in the groundstate-analysis of the models treated in [8]. To do an analogue of this for finite temperatures, non-negative (probabilistic) contour weights are necessary in this framework.
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non-normalized weights on Ising-spins given by Z Y m ˜ ∂Λ ,ηΛ ˜ ∂Λ ,ηΛ (mΛ ) (σΛ ) := dmΛ e−EΛ Tx (σx |mx ) ZΛm RΛ
(2.1)
x∈Λ
˜ ∂Λ ,ηΛ so that we get the desired Ising-probabilities dividing by ZΛm =
P σΛ ∈{−1,1}Λ
˜ ∂Λ ,ηΛ ZΛm (σΛ ). To describe our expansions conveniently let us define the following quadratic continuous-spin Hamiltonians, that are made to collect the quadratic terms that arise from the use of (1.13) to the above integral. We write, for finite volume V ⊂Zd , q X q X ˜ ∂V ,ηV ,σV (mV ) = (mx − my )2 + (mx − m ˜ y )2 HVm 2 {x,y}⊂V 2 x∈V ;y∈∂V d(x,y)=1
d(x,y)=1
X aX + (mx − m∗ σx )2 − ηx mx . 2 x∈V
(2.2)
x∈V
Here and throughout the paper we shall write ∂G for the outer boundary inside Λ, i.e. ∂G = {x ∈ Λ ∩Gc ; d(x, G) = 1}. The notion “nearest neighbor” is always meant in the usual sense of the 1-norm. The fixed Ising-spin σV ∈ {−1, 1}V thus signifies the choice of the well at each site. From the point of view of the continuous fields it is just another parameter. With this definition we can write the non-normalized Ising-weights (2.1) in the form Z Y m ˜ ∂Λ ,ηΛ ,σΛ ˜ ∂Λ ,ηΛ (mΛ ) (σΛ ) = e−b|Λ| dmΛ e−HΛ (1 + w(mx )) . (2.3) ZΛm RΛ
x∈Λ
If the w(mx ) were identically zero, we would be left with purely Gaussian integrals over Ising-spin dependent quadratic expressions. This Gaussian integration can be carried out and yields Z m ˜ ,η ,σ m ˜ ∂Λ ,ηΛ ,σΛ − inf m ∈RΛ HΛ ∂Λ Λ Λ (mΛ ) (mΛ ) Λ dmΛ e−HΛ = CΛ × e (2.4) RΛ
with a constant CΛ that does not depend on σΛ (and ηΛ ). The latter fact is clear since σΛ (and ηΛ ) only couple as linear terms (“magnetic fields”) to mΛ while they do not influence the quadratic terms. Note the pleasant fact that no spacial decomposition of the Gaussian integral is needed here and no complicated boundary terms arise. Now the minimum of the continuous-spin Hamiltonian in the expression on the r.h.s. of (2.4) provides weights for an effective random field Ising model for the spins σΛ ; its (infinite volume) Hamiltonian is given by the first two terms in (1.9). The treatment of this model is much simpler than that of the full model; all this will be postponed to Sec. 4. There it is discussed in detail how this model can be transformed into a disordered contour model by a mixed low- and hightemperature expansion. However, since this model provides the main part of the
1277
THE CONTINUOUS SPIN RANDOM FIELD MODEL
final contour model that is responsible for the ferromagnetic phase transition some readers might want to take a look to Sec. 4 to understand the form of our final contour-representation in a simpler situation. Our present aim is to show how the anharmonic perturbation induced by the w-terms can be treated as a positive-weight perturbation of the purely Gaussian model. Let U = U + ∪ (−U + )⊂R, where U + is a suitable “small” neighborhood of the positive minimizer of the potential m∗ that will be determined later and that will depend on the specific form of the potential. The first key step to define non-negative activities is to use the following combinatorial identity on the set U = {x ∈ Λ; mx ∈ U }. Lemma 2.1. Let Λ⊂Zd be finite and connected. For any set U⊂Λ we can write Q the polynomial x∈Λ (1 + wx ) in the |Λ| variables (wx )x∈Λ in the form: Y
Y
Y
G:∅6=G⊂Λ
Gi conn.cp of G
x∈∂Gi
(1 + wx ) = 1 +
x∈Λ
"
X
1x∈U
Y
(1x6∈U + wx ) −
x∈Gi
Y
# 1x6∈U
.
x∈Gi
(2.5) The proof is given at the end of this section. Application of Lemma 2.1 gives us the expansion ˜ ∂Λ ,ηΛ (σΛ ) eb|Λ| ZΛm Z m ˜ ∂Λ ,ηΛ ,σΛ (mΛ ) dmΛ e−HΛ + = RΛ
×
X G:∅6=G⊂Λ
Y
Y
Gi conn.cp of G
x∈∂Gi
" 1mx ∈U
Y x∈Gi
Z RΛ
m ˜ ∂Λ ,ηΛ ,σΛ
dmΛ e−HΛ
(1mx 6∈U + w(mx )) −
Y
(mΛ )
# 1mx 6∈U
. (2.6)
x∈Gi
Note that the expression under the integral factorizes over connected components of G := G ∪ ∂G. To introduce the anharmonic (preliminary) weights we need a little preparation. To avoid unnecessary complications in the expansions it is important to organize the Gaussian integral in the following conceptually simple but useful way: We decompose the nonnormalized Gaussian expectation over the terms in the last line into an outer integral over m∂G and a “conditional integral” over mΛ\∂G given m∂G . The latter integral factorizes of course over connected components of Λ\∂G; in particular the integrals over Λ\G and G become conditionally independent. W.r.t. this decomposition they appear in a symmetric way. To write down the explicit formulae we need to introduce: Some notation. The V × V -matrix ∆V is the lattice Laplacian with Dirichlet boundary conditions on V ⊂Λ, i.e. ∆V ;x,y = 1 iff x, y ∈ V are nearest neighbors, ∆V ;x,y = −2d iff x = y ∈ V and ∆V ;x,y = 0 else. ΠV is the projection operator onto ΩV (in short: onto V ), i.e. ΠV ;x,y = 1x=y∈V . We also use the redundant but intuitive notations mΛ |V ≡ ΠV mΛ ≡ mV for the same thing. 1V is the vector in
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1278
RΛ given by 1V ;x = 1x∈V . For disjoint V1 , V2 ⊂Λ we write ∂V1 ,V2 for the matrix with entries ∂V1 ,V2 ;x,y = 1 iff x ∈ V1 , y ∈ V2 are nearest neighbors and ∂V1 ,V2 ;x,y = 0 else. −1 We write RV := (c − ∆V ) for the corresponding resolvent in the volume V . Here a and later we put c = q . For the sake of clarity we keep (at least for now) the dependence of all quantities on continuous spin-boundary conditions, random fields, Ising-spins, as superscripts. Then we have: Lemma 2.2. For any subset G⊂Λ the random quadratic Hamiltonians (2.2) have the decomposition m ˜
,m∂G ,ηΛ\∂G ,σΛ\∂G
∂Λ ˜ ∂Λ ,ηΛ ,σΛ m ˜ ∂Λ ,ηΛ ,σΛ (mΛ ) = ∆H∂G,Λ (m∂G ) + ∆HΛ\∂G HΛm
(mΛ\∂G )
˜ ∂Λ ,ηΛ ,σΛ (m0Λ ) . + inf0 HΛm
(2.7)
mΛ
Here the “fluctuation-Hamiltonians” are given by m ˜ ∂Λ ,ηΛ ,σΛ (m∂G ) ∆H∂G,Λ
=
1D ˜ ∂Λ ,ηΛ ,σΛ m∂G − mm , (Π∂G (a − q∆Λ )−1 Π∂G )−1 Λ ∂G 2 E ˜ ∂Λ ,ηΛ ,σΛ × m∂G − mm Λ ∂G
(2.8)
∂G
and the “conditional fluctuation-Hamiltonian” (i.e. conditional on m∂G ) m ˜
,m∂G ,ηΛ\∂G ,σΛ\∂G
∂Λ ∆HΛ\∂G
=
mΛ\∂G
1D m ˜ ∂Λ ,m∂G ,ηΛ\∂G ,σΛ\∂G mΛ\∂G − mΛ\∂G , a − q∆Λ\∂G 2 E m ˜ ∂Λ ,m∂G ,ηΛ\∂G ,σΛ\∂G × mΛ\∂G − mΛ\∂G .
(2.9)
Λ\G
As centerings are occuring: the “global minimizer” ηΛ m ˜ ∂Λ ,ηΛ ,σΛ ∗ + ∂Λ,∂Λ m mΛ = RΛ cm σΛ + ˜ ∂Λ q
(2.10)
and the “conditional minimizer” m ˜
,m∂G ,ηΛ\∂G ,σΛ\∂G
∂Λ mΛ\∂G
ηΛ\∂G ∗ + ∂Λ\∂G,∂G m∂G + ∂Λ\∂G,∂Λ m = RΛ\∂G cm σΛ\∂G + ˜ ∂Λ . (2.11) q The proof is a consequence of Appendix Lemma A.1(iii) which is just a statement about symmetric positive definite matrices. Lemma 2.2 can be seen as an explicit expression of the compatibility property for the Gaussian local specifications defined through the Hamiltonian (2.7) in the volumes Λ\∂G⊂Λ. Indeed, the
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THE CONTINUOUS SPIN RANDOM FIELD MODEL
Gaussian measure defined with the quadratic form (2.8) describes the distribution on Λ projected onto ∂G. (Since we will use this formula later for subsets of Λ it is convenient to make the Λ explicit at this point, too.) The Gaussian measure on Λ\∂G defined with (2.9) is the conditional measure given m∂G . We would like to stress the following decoupling properties of the conditional expressions. Equation (2.11) for the conditional minimizer decouples over connected components Vi of Λ\∂G since the resolvent RΛ\∂G is just the direct sum of the RVi ’s. So we have that ηVi m ˜ ∂Λ ,m∂G ,ηΛ\∂G ,σΛ\∂G ∗ + ∂Vi ,∂Vi m∂Vi + ∂Vi ,∂Λ m = RVi cm σVi + ˜ ∂Λ mΛ\∂G Vi q m ˜ ∂Λ ,m∂Vi ,ηVi ,σVi
=: mVi
(2.12)
is a function depending only on what is appearing as superscripts, namely random fields and Ising-spins inside Vi and continuous-spin boundary condition on ∂Vi . (The dependence on the global boundary condition m ˜ ∂Λ is of course only through m ˜ x for d(x, Gi ) = 1. We don’t make this explicit in the notation.) Also, the conditional fluctuation-Hamiltonian on Λ\∂G decomposes into a sum over connected components of its support Λ\∂G: X m ˜ ∂Λ ,m∂Vi ,ηVi ,σVi m ˜ ∂Λ ,m∂G ,ηΛ\∂G ,σΛ\∂G (mΛ\∂G ) = ∆HVi (mVi ) ∆HΛ\∂G i
where m ˜
∆HVi ∂Λ
,m∂Vi ,ηVi ,σVi
(mVi )
1D m ˜ ,m ,ηV ,σV mVi − mVi∂Λ ∂Vi i i , (a − q∆Vi ) = 2 E m ˜ ∂Λ ,m∂Vi ,ηVi ,σVi × mVi − mVi .
(2.13)
Vi
Putting together the connected components of Λ\G we can thus write m ˜
,m∂G ,ηΛ\∂G ,σΛ\∂G
∂Λ ∆HΛ\∂ ¯ G
(mΛ\∂G )
m ˜ ∂Λ ,m∂G ,ηΛ\G ,σΛ\G
= ∆HΛ\∂ G¯
X
+
(mΛ\G )
m ˜ ∂Λ ,m∂Gi ,ηGi ,σGi
∆HGi
(mGi ) .
(2.14)
Gi conn.cp of G
So, the sum over G’s in (2.6) can be written as X Z m ˜ ∂Λ ,ηΛ ,σΛ (mΛ ) dmΛ e−HΛ G:∅6=G⊂Λ
×
RΛ
Y
Y
Gi conn.cp of G
x∈∂Gi
" 1mx ∈U
Y
x∈Gi
(1mx 6∈U + w(mx )) −
Y x∈Gi
# 1mx 6∈U
¨ C. KULSKE
1280
X
=
m ˜
e
− inf m0 HΛ ∂Λ
,ηΛ ,σΛ
Λ
(m0Λ )
Z
m ˜ ∂Λ ,ηΛ ,σΛ
dm∂G e−∆H∂G,Λ
(m∂G )
G:∅6=G⊂Λ
Y
×
Z dmΛ\G e
1mx ∈U
−∆H
m ˜ ∂Λ ,m∂G ,η
Λ\G
,σ
Λ\G
Λ\G
(mΛ\G )
x∈∂G
Z
Y
×
dmGi e
m ˜ ∂Λ ,m∂G ,ηG ,σG i i i i
−∆HG
(mGi )
Gi conn.cp of G
" ×
Y
(1mx 6∈U + w(mx )) −
x∈Gi
#
Y
1mx 6∈U
.
(2.15)
x∈Gi
Now we note the pleasant fact that the Gaussian integral over Λ\G is independent of all of the superindexed quantities (since they appear only in the shift of the quadratic form), so that it can be pulled out of the m∂G -integral. It gives Z dmΛ\G e
−∆H
m ˜ ∂Λ ,m∂G ,η
Λ\G
,σ
Λ\G
(mΛ\G )
Λ\G
= (2π)
|Λ−G| 2
(det(a − q∆Λ\G ))− 2 . (2.16) 1
Let us look at the last two lines now. Conditional on m∂G we define anharmonic activities by the formula Z m ˜ ∂Λ ,m∂G ,ηG ,σG i i i m ˜ ∂Λ ,m∂Gi ,ηGi ,σGi −∆HG (mGi ) i := dmGi e IGi " ×
Y
(1mx 6∈U + w(mx )) −
x∈Gi
Y
# 1mx 6∈U
.
(2.17)
x∈Gi
m ˜ ∂Λ ,m∂G ,ηG ,σG = 1 for G = ∅. So we have obtained the following repreWe write IG sentation for the non-normalized Ising-weights:
Lemma 2.3. With the above notations we have ˜ ∂Λ ,ηΛ (σΛ ) = e eb|Λ| ZΛm
m ˜
− inf m0 HΛ ∂Λ
,ηΛ ,σΛ
Λ
X
×
(2π)
(m0Λ )
|Λ\G| 2
(det(a − q∆Λ\G ))− 2 1
G:∅⊂G⊂Λ
Z ×
m ˜ ∂Λ ,ηΛ ,σΛ
dm∂G e−∆H∂G,Λ
(m∂G )
Y
1mx ∈U
x∈∂G
×
Y
m ˜
IGi∂Λ
,m∂Gi ,ηGi ,σGi
.
(2.18)
Gi conn.cp of G
Let us pause for a minute and comment on what we have obtained. For the purely Gaussian model (i.e. the w-terms are identically zero) the contributions for G 6= ∅ vanish. So the above formula is a good starting point for the derivation
THE CONTINUOUS SPIN RANDOM FIELD MODEL
1281
of the signed-contour representation whose main contributions are provided by the minimum of the Gaussian Hamiltonians in the first line. The main other non-trivial m ˜ ∂Λ ,m∂G ,ηG ,σG . First of all, ingredient are the preliminary anharmonic activities IG the whole construction only makes sense, if we are able to prove a suitable Peierls estimate for them, to be discussed soon. They factorize over connected components Gi of the set G. The conditioning on m∂G has allowed us to have them local in the sense that they depend only on random fields and Ising-spins inside Gi . Note that such a factorization does not hold for the remaining integral over ∂G (that would mean: over connected components of ∂G), as it is clear from (2.8). Indeed, the fields m∂G fluctuate according to the covariance matrix in the total volume Λ. So to speak, their (stochastic) dependence is mediated by the Gaussian local specification defined with (2.8). Furthermore, the dependence of their mean-value in this local specification is (weakly) on all Ising-spins and random fields in Λ. Both kinds of dependence will have to be expanded later in Sec. 3 when the integral over ∂G is carried out. This will be done by enlarging the “polymers” G and performing a high-temperature expansion. Finally, the determinants provide only trivial modifications of the weights that we will obtain; they can easily be handled by a random walk expansion. Let us stress the following nice feature of the above representation: “Lowtemperature contours” (see Sec. 4) will be created only by the global energyminimum in the first line. Consequently there will be no complicated boundary terms for these “low-temperature” terms (that could be easily produced by a careless expansion). Our further treatment of the expansion will be done under the assumption of the following two properties: Positivity of anharmonic weights. m ˜ ∂Λ ,m∂G ,ηG ,σG ≥0 IG
(2.19)
for all connected G, and m ˜ ∂Λ ∈ U ∂Λ , m∂G ∈ U ∂G , ηG ∈ [−δ, δ]G , σG ∈ {−1, 1}G. Uniform Peierls Condition for anharmonic weights. m ˜ ∂Λ ,m∂G ,ηG ,σG ≤ |G| IG
(2.20)
for all connected G, and m ˜ ∂Λ ∈ U ∂Λ , m∂G ∈ U ∂G , ηG ∈ [−δ, δ]G , σG ∈ {−1, 1}G with > 0. Rather than trying to be exhaustive in the description of potentials that satisfy these conditions we will use the rest of this section to fix some properties that imply them and discuss in detail the explicit example of the φ4 -theory in Lemma 2.6. This should however indicate how the above two conditions can be achieved in concrete cases by suitable choices of the neighborhood U and the constants a and b occuring in the quadratic potential. The expansion will be continued in Sec. 3. Let us start by fixing the following almost trivial one-site criterion. It makes sense if we are assuming the nearest neighbor coupling q to be small.
¨ C. KULSKE
1282
Lemma 2.4. Suppose that w(mx ) ≥ 0 for mx ∈ U. (i) Assume that we have uniformly for all choices of superindices Z m ˜ ∂Λ ,m∂G ,ηG ,σG 2 a+4dq ) dmx e− 2 (mx −mx w(mx )1mx ∈U Z ≥
m ˜ ∂Λ ,m∂G ,ηG ,σG 2
dmx e− 2 (mx −mx a
)
1mx 6∈U .
(2.21)
Then we have the positivity (2.19). (ii) Assume that Z m ˜ ∂Λ ,m∂G ,ηG ,σG 2 a ) dmx e− 2 (mx −mx (w(mx )1mx ∈U + (1 + w(mx ))1mx 6∈U ) ≤ . (2.22) Then we have the uniform Peierls estimate (2.20) with the same . Proof. Since we always have −1 ≤ w(mx ) < ∞ the assumption 1mx ∈U w(mx ) ≥ 0 implies that Z Y m ˜ ∂Λ ,m∂G ,ηG ,σG (mG ) (1mx 6∈U + w(mx )) dmG e−∆HG Z ≥
x∈G m ˜ ∂Λ ,m∂G ,ηG ,σG
dmG e−∆HG
(mG )
Y
w(mx )1mx ∈U ≥ 0 .
(2.23)
x∈G
We reduce the estimation of the integrals to product integration by the pointwise estimate on the quadratic form 2 akvG k22 ≤ hvG , (a − q∆D G )vG iG ≤ (a + 4dq)kvG k2 .
(2.24)
This gives Z
m ˜ ∂Λ ,m∂G ,ηG ,σG
dmG e−∆HG ≥
YZ
(mG )
Y
w(mx )1mx ∈U
x∈G
dmx e−
m ˜ ,m ,η ,σ a+4dq (mx −mx ∂Λ ∂G G G )2 2
w(mx )1mx ∈U
(2.25)
x∈G
and, on the other hand, Z Y m ˜ ∂Λ ,m∂G ,ηG ,σG (mG ) dmG e−∆HG 1mx 6∈U ≤
YZ x∈G
This proves (i).
x∈G
dmx e
m ˜ ∂Λ ,m∂G ,ηG ,σG
−a 2 mx −mx
2 1mx 6∈U .
(2.26)
THE CONTINUOUS SPIN RANDOM FIELD MODEL
1283
The Peierls estimate (ii) follows from dropping the second product in the definition of I and using (2.24) to write m ˜ ∂Λ ,m∂G ,ηG ,σG IG Z Y m ˜ ∂Λ ,m∂G ,ηG ,σG (mG ) ≤ dmG e−∆HG ((1 + w(mx ))1mx 6∈U + w(mx )1mx ∈U )
≤
YZ
x∈G m ˜ ∂Λ ,m∂G ,ηG ,σG 2
dmx e− 2 (mx −mx a
)
((1 + w(mx ))1mx 6∈U + w(mx )1mx ∈U )
x∈G
(2.27) Next we compute how big the nearest neighbor coupling q and size of the random fields δ can be in order that any boundary condition in U yields a minimizer of the Gaussian Hamiltonian on G that is “well inside” U . We have: Lemma 2.5. Let 0 < A1 ≤ A2 and U + = [m∗ − A2 , m∗ + A2 ], U = U + ∪ ∗ −1 2m +A2 a 1 (−U + ) . Assume that q ≤ 2d − 1 and δ ≤ aA A1 2 . Then we have that m mx˜ ∂Λ ,m∂G ,ηG ,σG − m∗ σx ≤ A1
(2.28)
for all G, m ˜ ∂Λ ∈ U ∂Λ , m∂G ∈ U ∂G , ηG ∈ [−δ, δ]G , σG ∈ {−1, 1}G. ˜ ∂Λ ,m∂G ,ηG ,σG ˜ ∂Λ ,m∂G ,ηG =0,σG Note the linear dependence mm = mm + x x Proof. ηG RG q . Let us thus choose the condition for q s.t. x
m mx˜ ∂Λ ,m∂G ,ηG =0,σG − m∗ σx ≤ A1 . 2
(2.29)
This condition is in fact achieved for a one-point G = {x} and the boundary conditions having the “wrong sign” with modulus m∗ + A2 as we will formally see as follows. Let us assume that σx = −1 and write this time for simplicity ∂G for the boundary in Zd (including possible sites in the outer boundary of Λ in Zd ). Then we have, due to the positivity of the matrix elements of RG that ˜ ∂Λ ,m∂G ,ηG =0,σG mm x
≤ −RG;x,x cm∗ +
X
RG;x,y cm∗ + (RG ∂G,∂G 1∂G (m∗ + A2 ))x . (2.30)
y∈G\{x}
We employ the equation RG (c1G + ∂G,∂G 1∂G ) = 1G to write the last line of (2.30) as (2.31) m∗ − 2RG;x,xcm∗ + A2 − A2 (RG c1G )x . We note that RG;x,x is an increasing function in the sets G 3 x (which can be seen by the random walk representation, see Appendix (A.9)). Further (RG 1G )x is an increasing function in G. So the maximum over G of (2.31) is achieved for G = {x}.
¨ C. KULSKE
1284
1 2d With R{x};x,x = c+2d the value of (2.31) becomes −m∗ + (2m∗ + A2 ) c+2d which gives the upper bound m ˜
mx ∂Λ
,m∂G ,ηG =0,σG\{x} ,σx =−1
+ m∗ ≤ (2m∗ + A2 )
2d . c + 2d
(2.32)
In the same way we obtain m ˜
mx ∂Λ
,m∂G ,ηG =0,σG\{x} ,σx =−1
+ m∗ ≥ −A2
2d . c + 2d
(2.33)
Equating the r.h.s. with A1 /2 gives the r.h.s. of the condition on q stated in the hypothesis. For the estimate of the random field term note that 0 ≤ RG;x,y ≤ RZd ;x,y and P 1 y∈Zd RZd ;x,y = c which give us X A1 δ ηy δ X . (2.34) R RZd ;x,y = ≤ G;x,y ≤ q q a 2 d y∈G
y∈Z
At this stage the treatment has to be made specific to the concrete potential and we specialize to our example, the φ4 -theory with potentials given by (1.2). The following Lemma summarizes how we can produce positivity and an arbritrarily small anharmonic Peierls constant. More specific information can be found in the proof. Lemma 2.6. For fixed 0 > 0 we put U + = [m∗ − (0 m∗ ) 3 , m∗ + (0 m∗ ) 3 ] . 1
1
(2.35)
Then we have (i) For any value of 0 , m∗ , q, δ there exists a choice of parameters a and b such that the anharmonic weights obey the positivity (2.19). Furthermore there exist strictly positive constants a(m∗ , 0 ), b(m∗ , 0 ), q0 (m∗ , 0 ), and δ0 (m∗ , 0 ) such that the following is true. (ii) For all q ≤ q0 (m∗ , 0 ) and δ ≤ δ0 (m∗ , 0 ) we have the Peierls estimate (2.20) with a constant (0 , m∗ ) that is independent of q, δ. 0 when(iii) If 0 is small enough this constant obeys the estimate (0 , m∗ ) ≤ 10 ∗ ∗ ever m ≥ m0 (0 ) is large enough. The above constants can be chosen like (2 + 0 3 m∗ − 3 )2 ∼1 4 2
1
a(m∗ , 0 ) =
−1 2 a(m∗ , 0 ) 1 200 − 3 m∗ 3 + 9 , q(m , 0 ) = 2d ∗
and b(m∗ , 0 ) ∼ e−const m
∗2 3
with m∗ ↑ ∞.
1
a(m∗ , 0 ) (0 m∗ ) 3 δ0 (m , 0 ) = 20 (2.36) ∗
1285
THE CONTINUOUS SPIN RANDOM FIELD MODEL
Proof. We will take time to motivate our choices of the parameters that are made to ensure the validity of the assumptions of Lemma 2.4. Let us write the neighborhood U + in the form U + = [(1 − 1 )m∗ , (1 + 1 )m∗ ] and show why the choice of 1 given in (2.35) comes up. The zeroth requirement on a and b we have to meet is w(mx )1mx ∈U ≥ 0. So, let us choose the Gaussian curvature a > 1 to be the smallest number s.t. we have, for all mx ∈ U + , that the Gaussian centered around m∗ is dominated by the true potential i.e. e−
a(mx −m∗ )2 2
≤ e−V (mx )
(2.37)
with equality for mx = (1 + 1)m∗ . This amounts to a = we have on U + for the Gaussian centered around −m∗ e−
a(mx +m∗ )2 2
≤ e−
(2+1 )2 (2−1 )2 +1−(1+1 )2 8
(2+1 )2 , 4
as in (2.36). Then
m∗2 −V (mx )
e
(2.38)
which gives us the estimate −1 (2+1 )2 (2−1 )2 +1−(1+1 )2 m∗ 2 8 1 + w(mx ) ≥ eb 1 + e−
(2.39)
on U + . Any choice of eb bigger than the denominator thus ensures w(mx ) 1mx ∈U ≥ 0. To have property (i) in Lemma 2.4. we have to choose eb even bigger. Obviously it is implied by R inf
mx ∈U
w(mx ) ≥ R +
m ˜ ∂Λ ,m∂G ,ηG ,σG 2
dmx e− 2 (mx −mx
dmx e
a
)
1mx ∈U /
m ˜ ,m ,η ,σ − a+4dq (mx −mx ∂Λ ∂G G G )2 2
.
(2.40)
1mx ∈U
But note that we always have ˜ ∂Λ ,m∂G ,ηG ,σG | − m∗ | ≤ m ˆ max (m∗ , δ, q, a) | |mm x
(2.41)
with a constant m ˆ max (m∗ , δ, q, a) that is finite for any fixed m∗ , δ, q, a and that is estimated by Lemma 2.5. So the trivial choice (2+1 )2 (2−1 )2 +1−(1+1 )2 ∗ m∗2 8 eb(m ,δ,q,a) : = 1 + e− R
×
b) 1 dmx e− 2 (mx −m mx ∈U / 1+ sup R a+4dq 2 − (m − m b ) x 2 1m ∈U m b |≤m b max (m∗ ,δ,q,a) dmx e b :|m a
2
! (2.42)
x
gives some finite number and ensures the positivity of the anharmonic activities. This proves (i). Let us now turn to quantitative estimates on the Peierls constant. To start with, the above definition of b is of course only useful if b will be small. Now, the r.h.s. of (2.42) is small whenever the centering of the Gaussian integrals is “safe” inside U and the neighborhood U is big enough to carry most of the Gaussian integral. We
¨ C. KULSKE
1286
apply Lemma 2.5 with A2 = 1 m∗ and A1 = A102 . The hypotheses of the Lemma then give us the conditions q ≤ q0 and δ ≤ δ0 with a q0 = 2d
20 +9 1
−1 ,
δ0 =
a1 m∗ . 20
(2.43)
Then we have R R
m ˜ ∂Λ ,m∂G ,ηG ,σG 2
dmx e− 2 (mx −mx a
dmx e
)
1mx ∈U /
m ˜ ,m ,η ,σ − a+4dq (mx −mx ∂Λ ∂G G G )2 2
1mx ∈U h √ 91 m∗ i r a 10 P |G| ≥ a + 4dq i. h ≤ √ ∗ a 1m 1 − P |G| ≥ a + 2dq 910
(2.44)
∗ 2
This shows that b ∼ e−const · (1 m ) tends to zero rapidly if 1 m∗ is getting large. Let us now see what Peierls constant we get according to Lemma 2.4 (ii). This will explain why the neighborhood U + should in fact be of the form (2.35). Our choice of U and a yields that we have, for all mx ∈ U + , that e−V (mx )+
a(mx −m∗ )2 2
This gives 1 + w(mx ) ≤ eb+1 (mx −m Z dmx e
∗ 2
)
≤ e1 (mx −m
m ˜ ∂Λ ,m∂G ,ηG ,σG
Z ≤ e
)
.
(2.45)
. From this we have
−a 2 mx −mx
b
∗ 2
2 (1 + w(mx ))1mx ∈U σx ) +1 (mx −m∗ )2
m ˜ ∂Λ ,m∂G ,ηG ,σG 2
dmx e− 2 (mx −mx a
r
m ˜ ,m ,η ,σ a1 2π (m ∂Λ ∂G G G −m∗ )2 e a−21 x a − 21 r a3 m∗ 2 1 2π ≤ eb e 100(a−21 ) a − 21
=e
b
(2.46)
and hence Z
m ˜ ∂Λ ,m∂G ,ηG ,σG 2
dmx e− 2 (mx −mx a
"r ≤ 2e
b
)
3
∗2
(1 + w(mx ))1mx ∈U
a m 1 2π e 100(a−21 ) − a − 21
r
# √ 91 m∗ 2π P |G| ≤ a . a 10
(2.47)
Indeed, the l.h.s. is O(31 m∗ 2 ) + O(1 ) and thus imposes the condition that 31 m∗ 2 be small! This estimate essentially cannot be improved upon. It determines the dependence of the Peierls constant on 1 and m∗ .
1287
THE CONTINUOUS SPIN RANDOM FIELD MODEL
Finally, the integrals over U c are much smaller: Indeed, for the bounded part of U we estimate Z (1−1 )m∗ m ˜ ∂Λ ,m∂G ,ηG ,σG 2 a ) dmx e− 2 (mx −mx (1 + w(mx )) c
0
Z
(1−1 )m∗
≤
m ˜ ∂Λ ,m∂G ,ηG ,σG 2
dmx e− 2 (mx −mx a
)
e−V (mx )+
a(mx −m∗ )2 2
0 ∗
m ˜ ∂Λ ,m∂G ,ηG ,σG 2
) = e− 2 (m −mx ∗ Z (1−1 )m m ˜ ∂Λ ,m∂G ,ηG ,σG ∗ )(mx −m∗ ) −V (mx ) × dmx e−a(m −mx e . a
(2.48)
0
We have for the last integral Z (1−1 )m∗ m ˜ ∂Λ ,m∂G ,ηG ,σG ∗ )(mx −m∗ ) −V (mx ) dmx e−a(m −mx e 0
Z
(1−1 )m∗
≤
dmx e
a1 m∗ 10
(mx −m∗ ) −V (mx )
a1 m∗ 10
(mx −m∗ ) −
e
0
Z
(1−1 )m∗
≤
dmx e
e
(mx −m∗ )2 8
.
(2.49)
0 ∗
1m which is outside the range The maximizer of the last exponent is mx = m∗ + 2a10 of integration (due to our choice of the 10 before (2.43)). Estimating for simplicity the integral by the value of the integrand at (1 − 1 )m∗ just gives
Z
(1−1 )m∗
m ˜ ∂Λ ,m∂G ,ηG ,σG 2
dmx e− 2 (mx −mx a
)
(1 + w(mx )) ≤ m∗ e−( 8 − 10 )(1 m 1
a
∗ 2
)
.
0
(2.50) For the unbounded part of U c where m ≥ m∗ (1 + 1 ) we have with our choice of a that 1 + w(mx ) ≤ 1. This gives us Z ∞ m ˜ ∂Λ ,m∂G ,ηG ,σG 2 a ) dmx e− 2 (mx −mx (1 + w(mx )) (1+1 )m∗
r ≤
√ 91 m∗ ∗ 2 2π P G≥ a ≤ e−const(1 m ) a 10
(2.51)
Collecting the terms gives our final estimate on the Peierls constant "r r a3 m∗ 2 1 2π 2π b e 100(a−21 ) − ≤ 2e a − 21 a r a ∗ −( 18 − 10 )(1 m∗ )2
+m e
From here the lemma follows.
+3
# √ 91 m∗ 2π P G≥ a . a 10
(2.52)
¨ C. KULSKE
1288
Q P Q Proof of Lemma 2.1. We expand x∈Λ (1 + wx ) = 1+ Λ0 :∅6=Λ0 ⊂Λ x∈Λ0 wx . Let A(Λ0 )⊂(Λ\U)\Λ0 denote the maximal set amongst the sets A⊂(Λ\U)\Λ0 that are connected to Λ0 . (We say that a set A is connected to a set Λ0 iff, for each point u in A, there exists a nearest neighbor path inside A ∪ Λ0 that joins u and some point in Λ0 .) Equivalently, this A(Λ0 ) is the unique set A⊂Λ\Λ0 s.t. x 6∈ U for all x ∈ A and x ∈ U for all x ∈ ∂(Λ0 ∪ A). We collect terms according to the sets G = Λ0 ∪ A(Λ0 ). Denoting by Gi the connected components of G and by Li = Λ0 ∩ Gi we have then X Y Y Y Y (1 + wx ) = 1 + 1x6∈U 1x∈U wx Λ0 :∅6=Λ0 ⊂Λ x∈A(Λ0 )
x∈Λ
= 1+
X
Y
G:∅6=G⊂Λ
Gi conn.cp of G
x∈∂(Λ0 ∪A(Λ0 ))
X
Y
x∈Λ0
1x6∈U
Li :∅6=Li ⊂Gi x∈Gi \Li
Y
1x∈U
x∈∂Gi
Y
wx .
x∈Li
(2.53) Adding and subtracting the term for Li = ∅ we have Y Y Y Y X 1x6∈U wx = (1x6∈U + wx ) − 1x6∈U Li :∅6=Li ⊂Gi x∈Gi \Li
x∈Li
x∈Gi
(2.54)
x∈Gi
which proves the lemma. 3. Control of Anharmonicity
We start from the representation of Lemma 2.3 for the non-normalized Ising weights. We assume positivity and Peierls condition for the anharmonic (I-) weights as discussed in Sec. 2 and verified for the φ4 -potential. Carrying out the last remaining continuous spin-integral we express the last two lines in (2.18) in terms of activities that are positive, obey a Peierls estimate and depend in a local way on the Ising-spin configuration σΛ and the realization of the random fields ηΛ . We stress that all estimates that follow will be uniform in the Ising-spin configuration and the configuration of the random field. The result of this is: Proposition 3.1. Assume that the anharmonic I-weights (2.17) satisfy the Positivity (2.19) and the uniform Peierls Condition (2.20) with a constant . Suppose that is sufficiently small, q is sufficiently small, a is of the order one, q(m∗ )2 sufficiently large. Suppose that δ ≤ Const m∗ and |U | ≤ Const m∗ with constants of the order unity. Then, for any continuous-spin boundary condition m ˜ ∂Λ ∈ U ∂Λ and any realization of the random fields ηΛ ∈ [−δ, δ]Λ , the non-normalized Ising weights (2.1) have the representation |Λ|
˜ ∂Λ ,ηΛ (σΛ ) = e−b|Λ| (2π) 2 (det(a − q∆Λ ))− 2 e ZΛm X ˜ ∂∂Λ G × ρ¯m (G; σG , ηG ) G:∅⊂G⊂Λ
1
m ˜
− inf m0 HΛ ∂Λ Λ
,ηΛ ,σΛ
(m0Λ )
(3.1)
1289
THE CONTINUOUS SPIN RANDOM FIELD MODEL
where the activity ρ¯ appearing under the G-sum is non-negative and depends only on the indicated arguments. ρ¯ factorizes over the connected components Gi of its support G, i.e. Y ˜ ∂∂Λ G ˜ ∂∂Λ Gi (G; σG , ηG ) = ρ¯m (Gi ; σGi , ηGi ) (3.2) ρ¯m i ˜ ∂∂Λ G and we have ρ¯m (G = ∅; σG , ηG ) = 1.
ρ¯ has the “infinite volume symmetries” of : (a) Invariance under joint flips of spins and random fields ρ¯(G; σG , ηG ) = ρ¯(G; −σG , −ηG ) if G does not touch the boundary (i.e. ∂∂Λ G = ∅) (b) Invariance under lattice shifts ρ¯(G; σG , ηG ) = ρ¯(G + t; σG+t , ηG+t ) if G, G + t⊂Λ do not touch the boundary. We have the uniform Peierls estimate ¯ ˜∂ G ¯ ¯ σG¯ , ηG¯ ) ≤ e−α|G| ∂Λ (G; ρ¯m
with α = const × min
log 1q
1 1 log , log q
log m∗
(3.3) !d
.
Remark 1. Note that the first line of (3.1) gives the value for vanishing anharmonicity (i.e. w(mx ) ≡ 0). Remark 2. For any fixed Ising-spin σΛ and realization of random fields ηΛ the sum in the last line is the partition function of a non-translation invariant polymer model for polymers G. Note that there is no suppression of the activities ρ¯ in the above bounds in terms of the Ising-spins. From the point of view of the polymers G the Ising spins and random fields play the similar role of describing an “external disorder”. Proof of Proposition 3.1. To yield this representation we must treat the last two lines of (2.18). We cannot carry out the m∂G -integral directly but need some further preparation that allows us to treat the “long range” parts of the exponent by a high-temperature expansion. Depending on the parameters of the model (to be discussed below) we will then have to enlarge and glue together connected components of the support G. For any set G⊂Λ we write Gr = {x ∈ Λ; d(x, G) ≤ r}
(3.4)
for the r-hull of G in Λ. Then we have, under the assumptions on the parameters as in Proposition 3.1. ∗
m such that the following is Lemma 3.2. There is a choice of r ∼ Const log log( 1 ) q
true. For each fixed subset G⊂Λ, continuous-spin boundary condition m ˜ ∂Λ ∈ U ∂Λ ,
¨ C. KULSKE
1290
fixed Ising-configuration σΛ ∈ {−1, 1}Λ and random fields ηΛ ∈ [−δ, δ]Λ we can write Z m ˜ ∂Λ ,ηΛ ,σΛ m ˜ ∂Λ ,m∂G ,ηG ,σG (m∂G ) dm∂G e−∆H∂G,Λ 1m∂G ∈U ∂G IG = (2π)
|∂G| 2
X m p ˜ ˜ σ ˜ , η ˜ ) , (3.5) det(Π∂G (a − q∆Gr )−1 Π∂G ) ρ ∂∂Λ G˜ (G, G; G G ˜ G⊂Λ ˜ G: ˜ Gr ⊂G
˜ where the activity appearing under the G-sum depends only on the indicated arguments and obeys the uniform bounds 0≤ρ with
m ˜∂
˜ ∂Λ G
˜
¯ G| ˜ σ ˜ , η ˜ ) ≤ e−α| (G, G; G G
1 1 α ¯ = Const × min log , log q
log 1q
(3.6)
!d
log m∗
.
˜ i.e. ˜ i of the set G, It factorizes over the connected components G Y m m ˜ ˜ ˜ σ ˜, η ˜) = ˜i, G ˜i; σ ˜ , η ˜ ) . ρ ∂∂Λ G˜ i (G ∩ G ρ ∂∂Λ G˜ (G, G; G G Gi Gi
(3.7)
i
˜ = ∅) ρ is invariant under joint flips of ˜ not touching the boundary (i.e. ∂∂Λ G For G spins and random fields and lattice shifts. Remark. Later it will be convenient to have the determinant appearing on the ˜ r.h.s.; in fact it could also be absorbed in the activities under the G-sum. Proof of Lemma 3.2. Let us recall definition (2.8) of the “fluctuation-Hamiltonian” (involving the global minimizer (2.10)) which gives the Hamiltonian of the projection onto ∂G of an Ising-spin and random-field dependent Gaussian field in Λ. Our first step is to decompose this projection from Λ onto ∂G into a “low temperature-part” and a “high temperature-part”. For fixed G we will consider definition (2.8) where Λ will be replaced by Gr ; for r large enough the resulting term “low-temperature”- term is close enough to the full expression, so that the rest can be treated by a high-temperature expansion. We write ∂B A := {x ∈ B; d(x, A) = 1} for the outer boundary in a set B⊂Zd . Recall that, with this notation ∂A = ∂Λ A, so that ∂Zd (Gr ) = ∂∂Λ (Gr ) ∪ ∂(Gr ). Then the precise form of the decomposition we will use reads: ∗
m Lemma 3.3. With a suitable choice of r ∼ Const log we have log( 1 ) q
(m ˜∂
Gr ,0∂Λ Gr ),ηGr ,σGr
m ˜ ∂Λ ,ηΛ ,σΛ ∆H∂G,Λ (m∂G ) = ∆H∂G,G∂Λr X +
C⊂Λ C∩∂G6=∅;C∩(Gr )c 6=∅
(m∂G )
HT ¯ ∂G,G H r (m∂G , σGr , ηGr ; C, σC , ηC ) , (3.8)
THE CONTINUOUS SPIN RANDOM FIELD MODEL
1291
where the functions appearing under the C-sum depend only on the indicated arguments and obey the uniform bound HT −α|C| ˜ H ¯ (3.9) ∂G,Gr (m∂G , σGr , ηGr ; C, σC , ηC ) ≤ e uniformly in m∂G ∈ U ∂G and all other quantities for the C’s occuring in the sum in (3.8). Here α ˜ = const log 1q . Remark. Note that the first part (“low temperature-part”) decomposes of course over the connected components (Gr )i of Gr , i.e. (m ˜∂
∆H∂G,G∂Λr =
Gr ,0∂Λ Gr ),ηGr ,σGr
X
(m ˜∂
(Gr )
(m∂G )
,0∂
(Gr )i ),η(Gr )i ,σ(Gr )i
i Λ ∂Λ ∆H∂G∩(G r ) ,(Gr ) i i
(m∂G∩(Gr )i ) .
(3.10)
i
Proof of Lemma 3.3. The l.h.s. and the first term on the r.h.s. of (3.8) differ in two places: The matrix and the centerings. We expand both differences using the random walk representation. The decomposition of the matrix into the matrix where Λ is replaced by Gr and a remainder term can be written as (Π∂G RΛ Π∂G )−1 = (Π∂G RGr Π∂G )−1 X − ∂∂G,Λ\∂G R(· → · ; C)∂Λ\∂G,∂G ,
(3.11)
C⊂Λ\∂G C∩(Gr )c 6=∅,C∩G2 6=∅
where the Λ × Λ-matrix R(· → · ; C) has non-zero entries only for x, y ∈ C that are given by |γ|+1 X 1 . (3.12) R(x → y ; C) = c + 2d paths γ from x to y Range(γ)=C
For the proof of this formula see the Appendix (A.8) ff. and (A.13). where more details about the random walk expansion can be found. Simply from the decomposition of the resolvent RΛ = RGr + (RΛ − RGr ) and the random walk representation for the second term follows the formula for the centerings X ˜ ∂Λ ,ηΛ ,σΛ m ˜ ∂Λ ,ηGr ,σGr = m + m(C; ¯ σC , ηC ) (3.13) mm r Λ G C⊂Λ C∩(Gr )c 6=∅
with ˜ ∂Λ ,ηG mm Gr
r ,σGr
:= RGr
ηGr + ∂Gr ,∂Λ m cm∗ σGr + ˜ ∂Λ q
and “high-temperature” terms given by the matrix product ηΛ ∗ + ∂C,∂Λ m m(C; ¯ σC , ηC ) = R(· → · ; C) cm σΛ + ˜ ∂Λ . q
(3.14)
(3.15)
¨ C. KULSKE
1292
From the bound on the resolvent (A.12) we have uniformly −|C| a . |m ¯ x (C; σC , ηC )| ≤ Const(m∗ + δ) 1 + 2dq a −|C|/2 2dq ) m∗ Const log for log( q1 )
This quantity is in turn bounded by, say, (1 + with r :=
∗
log m . Const log(1+ a 2dq )
So we have r ∼
(3.16)
if we have that |C| ≥ r small q.
To write both type of summations over connected sets C in the same form we note that X ∂∂G,Λ\∂G R(· → · ; C1 )∂Λ\∂G,∂G C1 ⊂Λ\∂G C1 ∩(Gr )c 6=∅,C1 ∩G2 6=∅
=
X
∂∂G,Λ\∂G R(· → · ; C2 \∂G)∂Λ\∂G,∂G 1C2 \∂G conn. (3.17)
C2 ⊂Λ C2 ∩∂G6=∅;C2 ∩(Gr )c 6=∅
which gives us the same range of summation for both sort of terms. The expansion then produces triple sums over connected sets C. Collecting terms according to the union of the occuring C’s we obtain the desired decomposition with ¯ HT r (m∂G , σGr , ηGr ; C, σC , ηC ) H ∂G,G q Gr ¯ σ∂G ), ∂∂G,Λ\∂G R(· → · ; C\∂G)∂Λ\∂G,∂G 1C\∂G conn. = − h(m∂G − m 2 Gr Gr × (m∂G − m ¯ σ∂G )i + qh(m∂G − m ¯ σ∂G ), (Π∂G RGr Π∂G )−1 m(C; ¯ σC , ηC )i X Gr −q h(m∂G − m ¯ σ∂G ), ∂∂G,Λ\∂G R(· → · ; C1 \∂G) C1 ,C2 ⊂Λ;C1 ∪C2 =C Ci ∩∂G6=∅;Ci ∩(Gr )c 6=∅
¯ 2 ; σC2 , ηC2 )i + × ∂Λ\∂G,∂G 1C1 \∂G conn. m(C
q 2
X C2 ,C3 ⊂Λ;C2 ∪C3 =C Ci ∩∂G6=∅;Ci ∩(Gr )c 6=∅
× hm(C ¯ 2 ; σC2 , ηC2 ), (Π∂G RGr Π∂G )−1 m(C ¯ 3 ; σC3 , ηC3 )i X q − hm(C ¯ 2 ; σC2 , ηC2 ), ∂∂G,Λ\∂G 2 C ,C ,C ⊂Λ;C ∪C ∪C =C 1 2 3 1 2 3 Ci ∩∂G6=∅;Ci ∩(Gr )c 6=∅
× R(· → · ; C1 \∂G)∂Λ\∂G,∂G 1C1 \∂G conn. m(C ¯ 3 ; σC3 , ηC3 )i (3.18) ˜ ∂Λ ,ηGr ,σGr Gr with the short notation m ¯ σ∂G = mm . The bounds are clear now from Gr ∂G
the bounds on the resolvent, the choice of r and the (trivial) control of the Ci -sums, i.e. provided by X e−α(|S1 |+|S2 |+|S3 |) = (3e−α + 3e−2α + e−3α )|C| ≤ e−const α|C| . all subsets S1 ,S2 ,S3 ⊂C ∪i Si =C
(3.19)
1293
THE CONTINUOUS SPIN RANDOM FIELD MODEL
To proceed with the proof of Proposition 3.1 and high temperature-expand the ¯ HT -terms we use the subtraction of bounds-trick to ensure the positivity of the H resulting activities. We thus write for fixed G e
−
P C⊂Λ C∩∂G6=∅;C∩(Gr )c 6=∅
¯ HT r (m H ∂G ,σGr ,ηGr ;C,σC ,ηC ) ∂G,G
Y
=
−
P
e
C⊂Λ;C conn. to (Gr )i C∩∂G6=∅;C∩(Gr )c 6=∅
˜ e−α|C|
(Gr )i conn. cp. of Gr
P C⊂Λ C∩∂G6=∅;C∩(Gr )c 6=∅
×e
˜ ¯ HT r (m n(Gr ,C)e−α|C| −H ∂G ,σGr ,ηGr ;C,σC ,ηC ) ∂G,G
, (3.20)
where n(Gr , C) is the number of connected components of Gr that are connected to C (i.e. have (Gr )i ∩ C 6= ∅). The exponential in the last line can then be clusterexpanded and gives
P C⊂Λ C∩∂G6=∅;C∩(Gr )c 6=∅
e
X
=
˜ ¯ HT r (m n(Gr ,C)e−α|C| −H ∂G ,σGr ,ηGr ;C,σC ,ηC ) ∂G,G
ρHT ∂G,Gr (m∂G , σGr , ηGr ; K, σK , ηK )
(3.21)
K⊂Λ;K=∅ or K∩∂G6=∅,K∩(Gr )c 6=∅
−α|K| ˜ with 0 ≤ ρHT . Here we use the convention ∂G,Gr (m∂G , σGr , ηGr ; K, σK , ηK ) ≤ e HT that ρ∂G,Gr (m∂G , σGr , ηGr ; K = ∅, σK , ηK ) = 1. Note that the resulting activities factorize over connected components of K ∪Gr ; this is due to the (trivial) fact that the number n(Gr , C) that enters the definition of the contour activities depends only on those components of Gr that C is connected to. We put
ρ
geo
r
(∂G, G ) :=
Y
−
e
P
C⊂Λ;C conn. to (Gr )i C∩∂G6=∅;C∩(Gr )c 6=∅
˜ e−α|C|
(3.22)
(Gr )i conn. cp. of Gr
and note that 1 ≥ ρgeo (∂G, Gr ) ≥ e−|G
r
˜ |e−const α
.
(3.23)
We can finally carry out the integral on ∂G to get the form as promised in the proposition. In doing so it is convenient to pull out a normalization constant and introduce the normalized Gaussian measures on ∂G corresponding to the Hamiltonian on the r.h.s. of (3.8), given by Z
(m ˜∂
∂Λ µ∂G,G r
R :=
Gr ,0∂Λ Gr ),ηGr ,σGr
(dm∂G )f (m∂G )
(m ˜∂ r ,0∂ Gr ),ηGr ,σGr Λ ∂Λ G
(m∂G ) f (m∂G ) dm∂G e−∆H∂G,Gr . (m ˜∂ r ,0∂ Gr ),ηGr ,σGr G R Λ ∂Λ (m0∂G ) dm0∂G e−∆H∂G,Gr
(3.24)
¨ C. KULSKE
1294
So we can write Z m ˜ ∂Λ ,ηΛ ,σΛ m ˜ ∂Λ ,m∂G ,ηG ,σG (m∂G ) 1m∂G ∈U ∂G IG dm∂G e−∆H∂G,Λ = (2π)
|∂G| 2
X
ρgeo (∂G, Gr )
K⊂Λ;K=∅ or K∩∂G6=∅,K∩(Gr )c 6=∅
Z ×
p det(Π∂G (a − q∆Gr )−1 Π∂G )
(m ˜∂
∂Λ µ∂G,G r
Gr ,0∂Λ Gr ),ηGr ,σGr
(dm∂G ) 1m∂G ∈U ∂G
m ˜ ∂Λ ,m∂G ,ηG ,σG × ρHT . ∂G,Gr (m∂G , σGr , ηGr ; K, σK , ηK )IG
(3.25)
This has in fact the desired form (3.5) with the obvious definition ρ
m ˜∂
˜ ∂Λ G
˜ σ ˜ , η ˜ ) := ρgeo (∂G, Gr ) (G, G; G G Z (m ˜ ∂∂Λ Gr ,0∂Λ Gr ),ηGr ,σGr (dm∂G ) 1m∂G ∈U ∂G × µ∂G,G r m ˜ ∂Λ ,m∂G ,ηG ,σG × ρHT ∂G,Gr (m∂G , σGr , ηGr ; K, σK , ηK )IG
(3.26) ˜ r on the r.h.s. Note that these activities factorize over connected with K = G\G ˜ components of G. In view of the trivial bound (3.23) on the geometric activity (3.22) and the normalization of the measure, the bounds follows from the HT-bounds and the bounds on the anharmonic activities I. The value of the “Peierls constant” α ¯ is ˜ }, assuming that both terms in now clear from α ¯ = Const min{(2r + 1)−d log 1 , α the minimum are sufficiently large. To finish with the proof of Proposition 3.1 is now an easy matter. Using the formula for the determinant from Appendix (A.3) we can write 1 det Π∂G (a − q∆Gr )−1 Π∂G det(a − q∆Λ\G ) =
det(Π∂G (a − q∆Gr )−1 Π∂G ) 1 × × det(a − q∆G ) . det(a − q∆Λ ) det(Π∂G (a − q∆Λ )−1 Π∂G )
(3.27)
Remember that the correction given by the middle term on the r.h.s. stems from the lack of terms with range longer than r in the quadratic form of (3.24) that we had cut off. The random walk representation then gives the following expansion whose proof is given in the Appendix. Lemma 3.4. −2 det(Π∂G (a − q∆Gr )−1 Π∂G ) = e det(Π∂G (a − q∆Λ )−1 Π∂G )
P C⊂Λ C∩∂G6=∅;C∩(Gr )c 6=∅
where 0 ≤ det (C) ≤ e−α|C| with α ∼ const log 1q .
det (C)
,
(3.28)
1295
THE CONTINUOUS SPIN RANDOM FIELD MODEL
Next we use subtraction of bounds as in (3.20) to write e
−
P C⊂Λ C∩∂G6=∅;C∩(Gr )c 6=∅
det (C)
X
= ρgeo,det (∂G, Gr )
ρdet ∂G,Gr (K) ,
(3.29)
K⊂Λ;K=∅ or K∩∂G6=∅,K∩(Gr )c 6=∅
where 1 ≥ ρgeo,det ≥ e−|G ˜ ∂Λ eb|Λ| ZΛm (σΛ ) = (2π)
×
r
˜ |e−constα
−const α|K| ˜ and 0 ≤ ρdet . So we get ∂G,Gr (K) ≤ e
|Λ| 2
− inf
0
H
m ˜ ∂Λ ,ηΛ ,σΛ
(m0 )
Λ m Λ Λ (det(a − q∆Λ ))− 2 e X X m |G| p ˜ ˜ σ ˜, η ˜) (2π)− 2 det(a − q∆G ) ρ ∂∂Λ G˜ (G, G; G G 1
˜ G⊂Λ ˜ G: ˜ Gr ⊂G
G:∅⊂G⊂Λ
× ρgeo,det(∂G, Gr )
X
ρdet ∂G,Gr (K) .
(3.30)
K⊂Λ;K=∅ or K∩∂G6=∅,K∩(Gr )c 6=∅
˜ K (collecting terms that give the same G ˜ ∪ K) to This can be summed over G, G, yield the claims of Proposition 3.1. 4. The Effective Contour Model: Gaussian Case It is instructive to make explicit the result of our transformation to an effective Ising-contour model at first without the presence of anharmonic potentials where the proof is easy. In fact, as we will explain in Sec. 5, the work done in Secs. 2 and 3 will then imply that a weak anharmonicity can be absorbed in essentially the same type of contour activities we encountered in the purely Gaussian model. We remind the reader that in the purely Gaussian case the Ising-weights ˜ ∂Λ ,ηΛ ˜ ∂Λ ,ηΛ ,σΛ ))(σΛ ) are obtained by normalizing exp(− inf mΛ ∈RΛ HΛm (mΛ )) by (T (µm Λ ˜ x = m∗ for its σΛ -sum. For simplicity we restrict now to the boundary condition m all x (that is everywhere in the minimum of the positive wells). We will now express the latter exponential as a sum over contour-weights. To do so we use the following (by now standard) definition of a signed contour model, including +-boundary conditions. Definition. A contour in Λ is a pair Γ = (Γ, σΛ ) where Γ⊂Λ (the support of Γ) and the spin-configuration σΛ ∈ {−1, 1}Λ are such that the extended configuration (σΛ , +1Zd \Λ ) is constant on connected components of Zd \Γ. The connected components of a contour Γ are the contours Γi whose supports are the connected components Γi of Γ and whose sign is determined by the requirement that it be the same as that of Γ on Γi . A contour model representation for a probability measure ν on the space {−1, 1}Λ of Ising-spins in Λ is a probability measure N on the space of contours in Λ s.t. the marginal on the spin reproduces ν, i.e. we have X N ({Γ}) . (4.1) ν({σΛ }) = Γ σΛ (Γ)=σΛ
¨ C. KULSKE
1296
Recall that, in the simplest low-temperature contour model, arising from the standard nearest neighbor ferromagnetic Ising model, N ({Γ}) = Const × ρ(Γ) is proportional to a (non-negative) activity ρ(Γ) that factorizes over connected components of the contour and obeys a Peierls estimate of the form ρ(Γ) ≤ e−τ |Γ| . There is a satisfying theory for the treatment of deterministic models with additional volume terms for activities that are not necessarily symmetric under spin-flip, known as Pirogov–Sinai theory. For random models then, while the activities will be random, there also have to be additional random volume-contributions to N ({Γ}), even when the distribution of the disorder is symmetric, caused by local fluctuations in the free energies of the different states. The fluctuations of these volume terms are responsible for the fact that, even in situations where the disorder is “irrelevant”, not all contours carry exponentially small mass but the formation of some contours (depending on the specific realization) is favorable. It is the control of this phenomenon that poses the difficulties in the analysis of the stability of disordered contour models and necessitates RG (or possibly some related multiscale method). To write down the Peierls-type estimates to come for the present model we introduce the “naive contour-energy” (i.e. the d−1-dimensional volume of the plaquettes separating plus- and minus-regions in Zd ) putting X
Es (Γ) =
X
1σx 6=σy +
{x,y}⊂Γ,d(x,y)=1
1σx =−1
(4.2)
x∈Γ,y∈∂Λ d(x,y)=1
again taking into account the interaction with the positive boundary condition. Then the result of the transformation of the purely Gaussian continuous spin model to an effective Ising-contour model is given by the following: Proposition 4.1. Suppose that q is sufficiently small, q(m∗ )2 sufficiently large, a is of the order 1 and δ ≤ Const m∗ with a constant of the order 1. Then there is a σΛ -independent constant KΛ (ηΛ ) s.t. we have the representation e
+m∗ 1∂Λ ,ηΛ ,σΛ
− inf m ∈RΛ HΛ Λ
(mΛ )
P
= KΛ (ηΛ )e X ×
C⊂V + (σΛ )
Gauß SC (ηC )−
ρ0 (Γ; ηΓ )
P
C⊂V − (σΛ )
Gauß SC (ηC )
(4.3)
Γ σΛ (Γ)=σΛ
for any σΛ , with V ± (σΛ ) = {x ∈ Λ; σx = ±1}. Here Gauß (ηC ) are functions of the random fields indexed by the con(i) ηC 7→ SC Gauß Gauß (−ηC ) = −SC (ηC ) nected sets C⊂Λ. They are symmetric, i.e. SC and invariant under lattice-shifts. For C = {x} we have in particular am∗ ηx . SxGauß (ηx ) = a+2dq (ii) The activity ρ0 (Γ; ηΓ ) is non-negative. It factorizes over the connected components of Γ, i.e. Y ρ0 (Γi ; ηΓi ) . (4.4) ρ0 (Γ; ηΓ ) = Γi conn cp. of Γ
1297
THE CONTINUOUS SPIN RANDOM FIELD MODEL
For Γ not touching the boundary (i.e. ∂∂Λ Γ = ∅) the value of ρ0 (Γ; ηΓ ) is independent of Λ. We then have the “infinite volume properties” of (a) Spin-flip symmetry, i.e. ρ0 ((Γ, σΛ ); ηΓ ) = ρ0 ((Γ, −σΛ ); −ηΓ ). (b) Invariance under joint lattice shifts of spins and random fields. Peierls-type bounds. There exist positive constants β˜Gauß , β s.t. we have the bounds s ˜ (4.5) 0 ≤ ρ0 (Γ; ηΓ ) ≤ e−βE (Γ)−βGauß |Γ| uniformly in ηΓ ∈ [−δ, δ]Γ where the “Peierls-constants” can be chosen like q(m∗ )2 a2 , 2 (a + 2dq)2 − q 2 1 = Const × min log , qm∗ 2 q
β=
β˜Gauß
log 1q log m∗
!d
− m∗ δ .
(4.6)
The non-local random fields obey the estimate ˜
Gauß (ηC )| ≤ δm∗ e−β0 |C| |SC
(4.7)
for all |C| ≥ 1 with β˜0 = Const × log 1q . Remark 1. This structure will be familiar to the reader familiar with [5] or [8] (see p. 457). Indeed, the above model falls in the class of contour models given in (5.1) of [5] (as written therein for the partition function). This form was then shown to be of sufficient generality to describe the contour models arising from the random field Ising model under any iteration of the contour-RG that was constructed in [5]. (The additional non-local interaction W (Γ) encountered in [5] is not necessary and can be expanded by subtraction-of-bounds as in (3.20), giving rise to enlarged supports Γ, as it was done in [8].) Remark 2. There is some freedom in the precise formulation of contours and contour activities, resp. the question of keeping information additional to the support and the spins on the contours. [5] speak of inner and outer supports, while in [8] it was preferred to define contours with activities containing interactions. The latter is motivated by the limit of the temperature going to zero (making the interactions vanish). Since we do not perform such a limit here, we present the simplest possible choice and do not make such distinctions here, simply collecting all interactions from different sources into “the support”. Remark 3. The magnitude of β ∼ Const qm∗ 2 is easily understood since it gives the true order of magnitude of the minimal energetic contribution to the original Hamiltonian of a nearest neighbor pair of continuous spins sitting in potential wells with opposite signs. This term appears again in the estimate on β˜Gauß (up to logarithmic corrections) together with a contribution of the same form as β˜0 . The latter comes from a straight-forward expansion of long-range contributions. The last
¨ C. KULSKE
1298
term in (4.6), m∗ δ, is a trivial control on the worst realization of the random fields; it could easily be avoided by the introduction of so-called “bad regions”. These are regions of space where the realizations of the random fields are exceptionally (and dangerously) large in some sense and, while comparing with [5] or [8], the reader might have already missed them. Indeed, a renormalization of the present model will immediately produce such bad regions in the next steps. Of course, we could have started, here and also in the presence of anharmonicity, with an unbounded distribution of the ηx . In the latter case we would have to single out regions of space where the behavior of our transformation to the Ising-model gets exceptional (i.e. because we lose Lemma 2.5.) We chose however not to treat this case here in order to keep the technicalities down. Proof. An elementary computation yields the important fact that the minimum of the quadratic Hamiltonian (2.6) with any boundary condition m ˜ is given by − inf
mΛ ∈RΛ
˜ ∂Λ ,ηΛ ,σΛ HΛm (mΛ ) = −
−
a2 (m∗ )2 a(m∗ )2 hσΛ , RΛ σΛ iΛ + |Λ| 2q 2 am∗ hη + η˜∂(Λc ) (q m), ˜ RΛ σΛ iΛ q
1 hη + η˜∂(Λc ) (q m), ˜ RΛ (η + η˜∂(Λc ) (q m))i ˜ Λ 2q q X + m ˜2 (4.8) 2 x∈Λ;y∈∂Λ y −
d(x,y)=1
with η˜∂(Λc ) (m) ˜ := ∂Λ,∂Λ m ˜ ∂Λ denoting the field created by the boundary condition. We subtract a term that is constant for σΛ (and thus of no interest) and write am∗ ˜ ∂Λ ,ηΛ ,σΛ ˜ ∂Λ ,ηΛ ,1Λ hηΛ , RΛ 1Λ iΛ (mΛ ) − inf H m (mΛ ) − inf HΛm mΛ mΛ q =−
a2 (m∗ )2 am∗ (hσΛ , RΛ σΛ iΛ − h1Λ , RΛ 1Λ iΛ ) − hηΛ , RΛ σΛ iΛ 2q q
− am∗ h˜ η∂(Λc ) (m), ˜ RΛ (σΛ − 1Λ )iΛ .
(4.9)
The first term on the r.h.s. gives rise to the low-temp. Peierls constant; the next term is a weakly nonlocal random field term (suppressed by the decay of the resolvent) and the last term the symmetry-breaking coupling to the boundary. As P in Sec. 3 we use the random walk representation RΛ = C⊂Λ R(· → · ; C) (see Appendix (A.11)) and decompose according to the size of C’s. As the first step for the contour representation we associate to any spin-configuration σΛ ∈ {−1, 1}Λ a preliminary (or “inner”) support in the following way. Choose some finite integer r ≥ 1, to be determined below, and put Γ+ Λ (σΛ ) := {x ∈ Λ; ∃y ∈ Λ s.t. d(x, y) ≤ r where σx 6= σy } ∪ {x ∈ Λ; d(x, ∂Λ) ≤ r + 1 where σx = −1} .
(4.10)
1299
THE CONTINUOUS SPIN RANDOM FIELD MODEL
The second term makes this definition Λ-dependent by taking into account the interaction with the boundary leading to the (desired) symmetry breaking for contours touching the boundary. For given σΛ the activities ρ0 (Γ; ηΓ ) to be defined will be non-zero only for supports Γ ⊃ Γ+ (σΛ ). The range r will be chosen below in such a way that the terms corresponding to interactions with range larger than r have decayed sufficiently so that they can be high-temperature expanded in a straightforward way. This choice then also determines the value of the Peierls-constant for the low-temperature contributions. Keeping the small C’s of diameter up to r define the (preliminary) “low-temperature activities” ˜ ∂Λ (σΛ ) ρLT,m
P
:= e
C⊂Λ;diam(C)≤r
a2 (m∗)2 2q
(hσC ,R(·→·;C)σCi−h1C ,R(·→·;C)1Ci)+am∗h˜ η∂(Λc) (˜ m),R(·→·;C)(σC −1C)i
.
(4.11) Note that the “inner support” (4.10) can be trivially rewritten as [ [ C∪ C Γ+ (σΛ ) =
(4.12)
C conn. to ∂Λ diam(C)≤r;σC 6=1C
C⊂Λ;diam(C)≤r σC 6=1C and σC 6=−1C
which shows that it is just the union of all connected C’s with diameter less or equal r that give any contribution to the sum occuring in the exponent of (4.11). So we can rewrite m ˜ ∂Λ ,ηΛ ,σΛ
e − inf mΛ HΛ =ρ e
LT,m ˜ ∂Λ am∗ q
(σΛ )e
∗ m ˜ ,η ,1 (mΛ )+inf mΛ HΛ ∂Λ Λ Λ (mΛ )+ am hηΛ ,RΛ 1Λ iΛ q am∗ q
P C⊂V + (Γ)
C⊂Λ;diam(C)>r σC 6=const
P
e
C⊂Λ;diam(C)≤r σC 6=const
hηC ,R(· →· ;C)σC i
hηC ,R(· →· ;C)1C i− am q
P
e
P
C⊂Λ;diam(C)>r C∩∂(Λ)c 6=∅
h
∗
P
C⊂V − (Γ)
hηC ,R(· →· ;C)1C i
i
∗ a2 (m∗ )2 (hσC ,R(· →· ;C)σCi−h1C ,R(· →· ;C)1Ci)+ am hηC ,R(· →· ;C)σC i 2q q
am∗ h˜ η∂(Λc ) (m),R(· ˜ →· ;C)(σC −1C )i
.
(4.13)
The terms in the first line depend only on quantities on Γ+ (σΛ ) and factorize over its connected components. They will give contributions to the activities ρ0 . The terms in the second line are the small-field contributions to the vacua given by Gauß (ηC ) := SC
am∗ hηC , R(· → · ; C)1C i . q
(4.14)
The terms in the last two lines are small (since only C’s with sufficiently large diameter contribute) and only non-zero for C’s intersecting with Γ+ (σΛ ) or touching the boundary. They can be expanded. Let us see now what explicit bounds we get on the low-temperature activity (4.11). Keeping only C’s made of two nearest neighbors x, y = x + e we have the upper bound
¨ C. KULSKE
1300
X C⊂Λ;diam(C)≤r
≤−
a2 (m∗ )2 (hσC , R(· → · ; C)σC i − h1C , R(· → · ; C)1C i) 2q
a2 (m∗ )2 q
X
R(x → y; C = {x, y})1σx 6=σy .
(4.15)
{x,y}⊂Γ+ (σΛ ),d(x,y)=1
Computing R(x → x + e; C = {x, x + e}) =
1 c + 2d
∞ X
k=1,3,5,...
1 c + 2d
k =
1 (c + 2d)2 − 1 (4.16)
with c = a/q we get an upper bound on the l.h.s. of (4.15) of X
−2β
1σx 6=σy
{x,y}⊂Γ+ (σΛ ),d(x,y)=1
where β is given by (4.6). Applying a similar reasoning on the boundary term, 1 , gives the bound thereby using that R(x → ; C = {x}) = c+2d X
am∗ h˜ η∂(Λc ) (m), ˜ R(· → · ; C)(σC − 1C )i
C⊂Λ;diam(C)≤r
≤ − q(m∗ )2
2a a + 2dq
X
1σx =−1 .
(4.17)
x∈Γ+ (σΛ ),y∈∂Λ d(x,y)=1
Since the modulus of the prefactor in the last line is larger than 2β we get an energetic suppression of +
+
˜ ∂Λ (σΛ ) ≤ e−2βEs (ΓΛ (σΛ ),σΛ ) ≤ e−βEs (ΓΛ (σΛ ),σΛ )−β(2r+1) ρLT,m
Using
P y
−d
|Γ+ (σΛ )| Λ
.
(4.18)
RΛ;x,y ≤ 1/c for the next term in (4.13) we have immediately am∗ q
X
hηC , R(· → · ; C)σC i ≤ m∗ δ|Γ+ Λ (σΛ )| .
(4.19)
C⊂Λ;diam(C)≤r σC 6=const
This finishes the Peierls estimate for the low-temperature contributions. Let us come to the treatment of the “high-temperature parts” in (4.13) now, proceeding algebraically at first. Using subtraction-of-bounds as in Sec. 3 (3.20) we get the high-temperature expansion "
P
e
C⊂Λ;diam(C)>r σC 6=const
a2 (m∗ )2 2q
= ρ˜geo (Γ+ (σΛ ))
# ∗
(hσC ,R(· →· ;C)σC i−h1C ,R(· →· ;C)1C i)+ am hηC ,R(· →· ;C)σC i q
X K⊂Λ;diam(K)>r or K=∅ σK 6=const
ρHT1 (K, σK , ηK )
(4.20)
1301
THE CONTINUOUS SPIN RANDOM FIELD MODEL
if the terms in the exponential on the l.h.s. are sufficiently small. To control them we just use the bound (A.12) X
R(x → y ; C) ≤
y∈Zd
1 c
2d c + 2d
|C|−1 .
(4.21)
This gives the deterministic bound upper bound on the first two terms in (4.13) of a2 (m∗ )2 |hσC , R(· → · ; C)σC i − h1C , R(· → · ; C)1C i| ≤ e−α|C| 2q
(4.22)
if we have a log(am∗ 2 ) 1 − α ≤ log 1 + 2dq (|C| − 1) log 1 +
a 2dq
−
1 e
(4.23)
a which is in turn bounded by α0 := 12 log(1 + 2dq ) − 1e for the C’s in the above sum if we put ∗2 ) log m∗ log(am + 1 ∼ 4 . (4.24) r = 2 a log 1q log 1 + 2dq
Remember here that we are interested in the regime of 1q small and m∗ 2 even larger. Assuming (4.24), with |ηx | ≤ δ the random field contribution is estimated by δ −α0 |C| am∗ |hηC , R(· → · ; C)σC i| ≤ e , q am∗
(4.25)
δ Gauß (ηC ) are obtained in the where can use that am ∗ ≤ Const. The estimates on SC very same way. In passing we verify that all activities constructed so far are invariant under joint flips of spins and random fields (inside Λ). The boundary terms can be expanded similarly giving P
e
C⊂Λ;diam(C)>r C∩∂(Λ)c 6=∅
−2
am∗ h˜ η∂(Λc ) (m),R(· ˜ →· ;C)(σC −1C )i
P C⊂Λ;diam(C)>r C∩∂(Λ)c 6=∅
=e P
×e −2
=e ×
C⊂Λ;diam(C)>r C∩∂(Λ)c 6=∅
P C⊂Λ;diam(C)>r C∩∂(Λ)c 6=∅
am∗ h˜ η∂(Λc ) (m),R(· ˜ →· ;C)1C i am∗ h˜ η∂(Λc ) (m),R(· ˜ →· ;C)(σC +1C )i am∗ h˜ η∂(Λc ) (m),R(· ˜ →· ;C)1C i
X K⊂Λ;diam(K)>r or K=∅ K∩∂(Λ)c 6=∅
ρHT2 (K, σK , ηK )
(4.26)
¨ C. KULSKE
1302
This gives m ˜ ∂Λ ,ηΛ ,σΛ
e − inf mΛ HΛ ×e
+2
∗ m ˜ ,η ,1 (mΛ )+inf mΛ HΛ ∂Λ Λ Λ (mΛ )+ am hηΛ ,RΛ 1Λ iΛ q
P
=ρ
C⊂Λ;diam(C)>r C∩∂(Λc )6=∅
LT,m ˜ ∂Λ
×e
am∗ q
(σΛ )e
am∗ h˜ η∂(Λc ) (m),R(· ˜ →· ;C)1C i
am∗ q
P C⊂V + (Γ)
P C⊂Λ;diam(C)≤r σC 6=const
hηC ,R(· →· ;C)σC i
hηC ,R(·→·;C)1C i− am q
X
× ρ˜geo (Γ+ (σΛ ))
∗
P
C⊂V − (Γ)
hηC ,R(·→·;C)1C i
ρHT1 (K, σK , ηK )
K⊂Λ;diam(K)>r or K=∅ σK 6=const
X
×
ρHT2 (K1 , σK1 , ηK1 )
(4.27)
K1 ⊂Λ;diam(K1 )>r or K1 =∅ K1 ∩∂(Λ)c 6=∅
which proves the desired representation (4.3) with the obvious definition ρ0 (Γ; ηΓ ) := ρ
LT,m ˜ ∂Λ
(σΛ )e
am∗ q
P C⊂Λ;diam(C)≤r σC 6=const
hηC ,R(· →· ;C)σC i
ρ˜geo (Γ+ (σΛ ))
X
×
K0 ,K1 ⊂Λ;K0 ∪K1 ∪Γ+ (σΛ )=Γ;diam(Ki )>ror Ki =∅ σK 6=const, K1 ∩∂(Λ)c 6=∅ 0
× ρHT1 (K0 , σK0 , ηK0 )ρHT2 (K1 , σK1 , ηK1 ) .
(4.28)
The form (4.6) of the Peierls constant β˜Gauß is now clear from β˜Gauß = Const min{β(2r + 1)−d , α0 } − m∗ δ, assuming that both terms in the minimum are sufficiently large to control the entropy in (4.28) and the slight modification in the exponential bounds on ρHT1 arising from the subtraction of bounds. 5. The Final Contour Model
Proof of Phase Transition
We put together the results of Secs. 3 and 4 to obtain the contour representation of the full model. It is of the same form as the Gaussian model of Sec. 4, while a modifaction of the Peierls constant β˜ accounts for the anharmonic contributions. More precisely we have: Proposition 5.1. Assume that the anharmonic I-weights (2.17) satisfy the Positivity (2.19) and the uniform Peierls Condition (2.20) with a constant . Suppose that is sufficiently small, q is sufficiently small, a is of the order one, q(m∗ )2 sufficiently large. Suppose that δ ≤ Const m∗ and |U | ≤ Const m∗ with constants that are sufficiently small. ∗ 1∂Λ ,ηΛ ) on {−1, 1}Λ have the contour representation Then the measures T (µ+m Λ T (µ+m Λ =
∗
1∂Λ ,ηΛ
1
)(σΛ )
e +,ηΛ Zcontour,Λ
P C⊂V + (σΛ )
SC (ηC )−
P
C⊂V − (σΛ )
SC (ηC )
X Γ σΛ (Γ)=σΛ
ρ(Γ; ηΓ ) (5.1)
1303
THE CONTINUOUS SPIN RANDOM FIELD MODEL
with the contour-model partition function P X P S (η )− C⊂V − (Γ ) SC (ηC ) +,ηΛ Λ Zcontour,Λ = e C⊂V + (ΓΛ ) C C ρ(Γ; ηΓ ) .
(5.2)
Γ ∗
+,ηΛ with a trivial For the partition function (1.4) we have ZΛ+m 1∂Λ ,ηΛ = CΛ+,ηΛ Zcontour,Λ constant containing the contributions of Gaussian fluctuations that satisfies, a.s.
Eη 2 1 1 1 log CΛ+,ηΛ = − 0 [(a − q∆Zd )−1 ]0,0 − [log(a − q∆Zd )]0,0 − b + log(2π) . 2 2 2 Λ↑Z d |Λ| (5.3) The quantities appearing in (5.1) are as follows. lim
(i) ηC 7→ SC (ηC ) are functions of the random fields indexed by the connected sets C⊂Λ that are symmetric, i.e. SC (−ηC ) = −SC (ηC ). In particular we am∗ ηx . They obey the uniform bound have Sx (ηx ) = a+2dq |SC (ηC )| ≤ m∗ δe−αfinal |C| log
(5.4)
1
for all C with αfinal = const min{log 1q , log 1 ( log mq∗ )d }. (ii) The activity ρIsing (Γ; ηΓ ) is non-negative and depends only on the indicated arguments. It factorizes over the connected components (as in (4.4)). For Γ not touching the boundary it does not depend on Λ and has the infinite volume symmetries of (a) invariance under joint flips of spins and random fields and (b) invariance under lattice shifts. ˜ β s.t. we have the Peierls-type bounds: There exist (large) positive constants β, ρIsing (Γ; ηΓ ) ≤ e−βE uniformly in ηG . Here β =
q(m∗ )2 a2 2 (a+2dq)2 −q2
1 β˜ = Const × min log , qm∗ 2 q
log 1q
s
˜ (Γ)−β|Γ|
(5.5)
is the same as in (4.8) and
!d
log m∗
1 , log
log 1q
!d
log m∗
− m∗ δ .
(5.6)
Proof. Assuming the control of the anharmonicity, summarized in Proposition 3.1, the proof is easy. For any fixed σΛ we can cluster-expand the last sum in (3.1). Dropping now the dependence on the boundary condition m ˜ ∂Λ = +m∗ 1∂Λ in the notation we have X X log ρ¯(G; σG , ηG ) = ¯ (C; σC , ηC ) G:∅⊂G⊂Λ
C:∅⊂C⊂Λ
X
=
¯(C; 1C , ηC ) +
C⊂V + (σΛ )
+
X C⊂Λ;σC 6=const
X
¯(C; −1C , ηC )
C⊂V − (σΛ )
¯(C; σC , ηC ) ,
(5.7)
¨ C. KULSKE
1304
where the sum is over connected sets C and we have the bounds |(C; σC , ηC )| ≤ e−const α|C| with α given in Proposition 3.1. Together with the representation (4.3) for the purely Gaussian model this gives X m ˜ ,η ,σ − inf m ∈RΛ HΛ ∂Λ Λ Λ (mΛ ) Λ ρ¯(G; σG , ηG ) e G:∅⊂G⊂Λ P
= KΛ (ηΛ )e
C⊂V + (σΛ )
P
e
C⊂Λ;σC 6=const
Gauß (SC (ηC )+¯ (C;1C ,ηC ))−
X
¯(C;σC ,ηC )
P
C⊂V − (σΛ )
Gauß (SC (ηC )+¯ (C;1C ,ηC ))
ρ0 (Γ; ηΓ ) .
(5.8)
Γ σΛ (Γ)=σΛ
Note that the C’s in the exponential in the last line are in particular connected to Γ. Using subtraction-of-bounds as before we can expand those terms and, as we did before in Secs. 3 and 4, rewrite the last line in terms of a new (and final) contour summation as P X X ¯(C;σC ,ηC ) ρ0 (Γ; ηΓ ) = ρ(Γ; ηΓ ) . (5.9) e C⊂Λ;σC 6=const Γ σΛ (Γ)=σΛ
Γ σΛ (Γ)=σΛ
The values of the Peierls constants for the final activities on the r.h.s. follow from the statements of the Propositions 3.1 and 4.1 with a slight loss due to the control of entropy. Finally, to see the statement for the free energy, we start from (3.1) and recall the construction of the activities in the purely Gaussian case, using the explicit expression (4.11) for the energy minimum in the Gaussian model in terms of the resolvent we obtain, with some trivial control on boundary terms, using the SLLN applied on the random fields the desired formula lim
Λ↑Z d
1 1 ˜ ∂Λ ,ηΛ ,1Λ log CΛ+,ηΛ = − lim E inf HΛm (mΛ ) d |Λ| Λ↑Z |Λ| mΛ ∈RΛ − lim
Λ↑Z d
=−
1 1 log det(a − q∆Λ ) − b + log(2π) 2|Λ| 2
1 Eη02 (a − q∆Zd )−1 0,0 − [log(a − q∆Zd )]0,0 2 2
−b+
1 log(2π) . 2
(5.10)
The following result provides control of the original measure in terms of the coarse-grained one up to two corrections: Proposition 5.2. Assume the conditions of Proposition 5.1 and suppose that m ˜ ∂Λ ∈ (U + )∂Λ . Then we have ∗ 2 m∗ m ˜ ∂Λ ,ηΛ ˜ ∂Λ ,ηΛ ≤ (T (µm mx 0 ≤ ))[σx0 = −1] + e−const α + e−const(m ) , µΛ Λ 2 (5.11)
1305
THE CONTINUOUS SPIN RANDOM FIELD MODEL
where
1 1 α = const × min log , log q
!d
log 1q log m∗
is given in Proposition 3.1. Remark. The first error term on the r.h.s. accounts for the anharmonicity, the next one for the Gaussian fluctuations. Proof. We carry out the transformation that led to Lemma 2.3 while carrying through the indicator function 1mx ≤ m∗ to get 0
Z RΛ
dmΛ 1mx
∗
m 0≤ 2
= e−b|Λ|
X
m ˜ ∂Λ ,ηΛ
e−EΛ
m ˜
e
2
(mΛ )
− inf m0 HΛ ∂Λ
,ηΛ ,σΛ
Λ
(m0Λ )
σΛ
"
X
×
(2π)
|Λ\G| 2
(det(a − q∆Λ\G ))− 2 1
G:G⊂Λ G3x0
Z ×
m ˜ ∂Λ ,ηΛ ,σΛ
dm∂G e−∆H∂G,Λ
X
+
(2π)
|Λ\G| 2
(m∂G )
m ˜ ∂Λ ,m∂G ,ηG ,σG 1m∂G ∈U ∂G IG;x 0
(det(a − q∆Λ\G ))− 2 1
G:G⊂Λ ∂G3x0
Z ×
m ˜ ∂Λ ,ηΛ ,σΛ
dm∂G e−∆H∂G,Λ
X Z
+
(m∂G )
1m∂G ∈U ∂G 1mx
m ˜ ∂Λ ,ηΛ ,σΛ
dm∂G e−∆H∂G,Λ
(m∂G )
m∗ 0≤ 2
m ˜ ∂Λ ,m∂G ,ηG ,σG IG
1m∂G ∈U ∂G
G:G⊂Λ G63x0
Z ×
dmΛ\G e
−∆H
m ˜ ∂Λ ,m∂G ,η
Λ\G
,σ
Λ\G
Λ\G
(mΛ\G )
1 mx
m∗ 0≤ 2
# m ˜ ∂Λ ,m∂G ,ηG ,σG × IG
(1)
(5.12)
(2)
with IG;x0 = IG;x0 − IG;x0 (superscripts are dropped now) where we have defined Z (1)
IG;x0 := Z (2) IG;x0
:=
m ˜ ∂Λ ,m∂G ,ηG ,σG
dmG e−∆HG
(mG )
Y
(1mx 6∈U + w(mx ))1mx
m∗ 0≤ 2
x∈G m ˜ ∂Λ ,m∂G ,ηG ,σG
dmG e−∆HG
(mG )
Y x∈G
1mx 6∈U 1mx
m∗ 0≤ 2
.
(5.13)
¨ C. KULSKE
1306
We use the same notations without the subscript x0 on the l.h.s. to denote the 1 2 − IG . Note integrals without the 1mx ≤ m∗ on the r.h.s. so that we have IG = IG 0 2 that it is not clear anymore that IG;x0 is positive for any sign σx0 and dominated by IG . To bypass this little inconvenience we argue as follows. Let us slightly enlarge b in Sec. 2 by putting a factor 2 in front of the fraction of integrals in Definition 2.42. This leaves b very small and all subsequent arguments based on a fixed choice of b remain valid. Going back through Lemma 2.4, we see that this definition implies (2) (1) that even IG ≤ 2−|G|IG (which can be seen as a strengthening of the positivity of IG ). But from this we have in particular that (1)
(2)
(1)
(1)
IG;x0 = IG;x0 − IG;x0 ≤ IG;x0 ≤ IG ≤ 2IG .
(5.14)
We use this estimate on the last G-sum in (5.12) and bound the second G-sum in (5.12) by the corresponding expression without the indicator. Carrying out the m∂G -integral as described in Sec. 3 we get from this the bound X
(2π)
G:G⊂Λ G3x0
|Λ\G| 2
Z ×
(det(a − q∆Λ\G ))− 2 1
m ˜ ∂Λ ,ηΛ ,σΛ
dm∂G e−∆H∂G,Λ
X
+
(2π)
|Λ\G| 2
(m∂G )
m ˜ ∂Λ ,m∂G ,ηG ,σG 1m∂G ∈U ∂G IG;x 0
(det(a − q∆Λ\G ))− 2 1
G:G⊂Λ ∂G3x0
Z ×
m ˜ ∂Λ ,ηΛ ,σΛ
dm∂G e−∆H∂G,Λ
≤ 2 · (2π)
|Λ| 2
(m∂G )
(det(a − q∆Λ ))− 2 1
1m∂G ∈U ∂G 1mx X
m∗ 0≤ 2
m ˜ ∂Λ ,m∂G ,ηG ,σG IG
˜ ∂∂Λ G ρ¯m (G; σG , ηG ) .
(5.15)
G:x0 ∈G⊂Λ
Using the positivity of the activities in the last line we can use the usual Peierls argument on the fixed-σ contour model appearing in (3.1) that controls the anharmonicity. So we estimate X ˜ ∂∂Λ G ρ¯m (G; σG , ηG ) G:x0 ∈G⊂Λ
≤
X
˜ ∂∂Λ G ρ¯m (G0 ; σG0 , ηG0 )
G0 :x0 ∈G0 ⊂Λ
≤ e−const α
X
X
˜ ∂∂Λ G ρ¯m (G; σG , ηG )
G:G⊂Λ ˜ ∂∂Λ G ρ¯m (G; σG , ηG ) ,
(5.16)
G:G⊂Λ
where the first sum is over connected sets G0 and we have used Proposition 3.1 for its estimation. To treat the first G-sum in (5.12) we note that the expectation outside the anharmonic contours is given by the one-dimensional Gaussian probability:
1307
THE CONTINUOUS SPIN RANDOM FIELD MODEL
Z
m ˜ ∂Λ ,ηΛ ,σΛ
dm∂G e−∆H∂G,Λ = (2π)
|Λ\G| 2
(m∂G )
1 mx
m∗ 0≤ 2
(det(a − q∆Λ\G ))− 2 1
−1 m ˜ ∂Λ ,m∂G ,ηΛ\G ,σΛ\G 1 mx × N mΛ\∂G ; a − q∆D Λ−G x0 ,x0
R∞
−
∗
m 0≤ 2
(5.17)
(x−a)2 2σ2
φ(x) √ . We use the uniform control on with the notation N [a; σ ](φ) = −∞ 2πσ2 the expectation value given by Lemma 2.5 and the fact that the variance occuring in (5.17) is of the order one, in any volume. If σx0 = +1 we have from this, uniformly in all involved quantities that −1 m ˜ ∂Λ ,m∂G ,ηΛ\G ,σΛ\G ∗ 2 D 1mx ≤ m∗ ≤ e−const(m ) ; a − q∆Λ−G (5.18) N mΛ\∂G 2
e
0
x0 ,x0
2
so it can be pulled out of the m∂G -integral. For σx0 = −1 we use the trivial bound 1 to write X Z m ˜ ∂Λ ,ηΛ ,σΛ (m∂G ) dm∂G e−∆H∂G,Λ 1m∂G ∈U ∂G G:G⊂Λ G63x0
Z × ≤
dmΛ\G e
m ˜ ∂Λ ,m∂G ,η
Λ\G
,σ
Λ\G
Λ\G
(mΛ\G )
1 mx
m∗ 0≤ 2
X |Λ\G| ∗ 2 1 e−const(m ) 1σx0 =1 + 1σx0 =−1 (2π) 2 (det(a − q∆Λ\G ))− 2 Z ×
≤
−∆H
G:G⊂Λ G63x0 m ˜ ∂Λ ,ηΛ ,σΛ
dm∂G e−∆H∂G,Λ
(m∂G )
m ˜ ∂Λ ,m∂G ,ηG ,σG 1m∂G ∈U ∂G IG
X |Λ\G| ∗ 2 1 e−const(m ) + 1σx0 =−1 (2π) 2 (det(a − q∆Λ\G ))− 2 Z ×
G:G⊂Λ m ˜ ∂Λ ,ηΛ ,σΛ
dm∂G e−∆H∂G,Λ
(m∂G )
m ˜ ∂Λ ,m∂G ,ηG ,σG 1m∂G ∈U ∂G IG .
(5.19)
Now it is simple to put together (5.12), (5.16)–(5.19) and rerunning the next steps of the transformation yields the claim. Applying the information of [5] we obtain the main result of the paper. Proof of Theorem 1. We apply statement Theorem 2.1 [5] on the measure ∗ 1∂Λ ,ηΛ ). Indeed, this is justified from Proposition 5.1 which implies that T (µ+m Λ this measure is contained in the class of contour measures described in [5] Chap. 5 ˜ β, αfinal “Flow of the RGT”, Paragraph 5.1. We note that of the three constants β, (controlling the exponential decay of the activities in terms of the volume resp. in terms of the naive contour energy, and the decay of the non-local fields) the constant β˜ is the smallest.
¨ C. KULSKE
1308
So Statement 2.3 from [5] gives in our case that for d ≥ 3, β˜ large enough and σ small enough we have that 2
P[T (µ+m Λ
∗
1∂Λ ,ηΛ
)[σx0 = −1] ≥ e−const β ] ≤ e− ˜
const σ2
.
(5.20)
We apply Proposition 5.2 and note that the two correction terms given therein ˜ are also controlled by e−const β (with possible modification of const.) From this in particular the estimates of Theorem 1 follow. Remark. We have not given an estimate on the value of γ as a function of q and m∗ . This would of course follow from a more careful estimate of the best value ˜ as a function of q and m∗ of the “anharmonicity-constant” (which is entering β) (see Sec. 2) and is left to the reader. Finally, Theorem 2 for the φ4 -theory follows immediately from: Proposition 5.3. Assume that the anharmonic I-weights (2.17) satisfy the Positivity (2.19) and the uniform Peierls Condition (2.20) with a constant (that is sufficiently small). Let µη∞ be any continuous spin Gibbs-measure obtained as a ˜ ∂Λ ,ηΛ along a sequence of cubes Λ for some (not necessarily positive) weak limit of µm Λ d continuous-spin boundary condition m ˜ ∈ UZ . d Then the measure T (µη∞ ) on {−1, 1}Z is a Gibbs measure for the absolutely summable Ising–Hamiltonian η (σ) = − HIsing
a2 (m∗ )2 X (a − q∆Zd )−1 x,y σx σy 2 x,y
− am∗
X X (a − q∆Zd )−1 ΦC (σC ; ηC ) , x,y ηx σy − x,y
(5.21)
C:|C|≥2
where the interaction potentials ΦC (σC , ηC ) = ΦC (−σC , −ηC ) obey the uniform bound |ΦC (σC , ηC )| ≤ e−const α|C| for all C with
1 1 α = const × min log , log q
log 1q log m∗
!d
as in (3.3). Remark. Note that it follows in particular that the interaction will be the same e.g. also in continuous spin Dobrushin-states [11] (that are believed to exist) one could construct using the boundary condition +m∗ in the upper half-space and −m∗ in the lower half-space. σ ¯
Zd (σV ) the usual restriction of (5.21) to the finite Proof. Denote by HIsing,V volume V , obtained by keeping the sums over sets {x, y} and C that intersect V
1309
THE CONTINUOUS SPIN RANDOM FIELD MODEL
and putting the spin equal to σ ¯Zd for x 6∈ V . Following [7] it suffices to show that, for each σ ¯Zd we have that m ˜
lim P
lim
Λ2 ↑Zd Λ1 ↑Zd
ZΛ1∂Λ1
,ηΛ1
m ˜
σ ˜V
ZΛ1∂Λ1
σ ¯ d ,η Z
(σV , σ ¯Λ2 \V )
,ηΛ1
(˜ σV , σ ¯Λ2 \V )
= P
e−HIsing,V (σV ) σ ¯ d ,η Z
σ ˜V
e−HIsing,V (˜σV )
,
along (say) sequences of cubes where Z m ˜ ∂Λ ,ηΛ Y 1 1 m ˜ ∂Λ1 ,ηΛ1 −E (mΛ1 ) ZΛ1 (σΛ2 ) := e Λ1 Tx (σx mx ) . RΛ 1
(5.22)
(5.23)
x∈Λ2
This is clear, since (according to our assumption of weak convergence) there is a σΛ2 \V ). subsequence of cubes Λ1 s.t. the inner limit exists and equals (T (µη∞ ))|(σV |¯ Summing Proposition 3.1 over the spins in Λ1 \Λ2 we have then m ˜
ZΛ1∂Λ1
,ηΛ1
(σΛ2 ) = e−b|Λ1 | (2π) X
×
e
|Λ1 | 2
(det(a − q∆Λ1 ))− 2 1
− inf m0
Λ1
m ˜ ∂Λ ,ηΛ ,ˆ σ ;σ 1 Λ1 \Λ2 Λ2 1 (m0Λ1 ) 1
HΛ
σ ˆΛ1 \Λ2
X
×
m ˜ ∂∂Λ
ρ¯
1
G
(G; σG∩Λ2 , σ ˆG∩Λ1 \Λ2 , ηG ) .
(5.24)
G:∅⊂G⊂Λ1
From here the proof is easy, given the explicit formula (4.8) for the minimum and the absolute summability of the polymer weights, uniformly in the spins and random fields. For the convenience of the reader we give a complete proof for the simplest case of vanishing anharmonicity w(mx ) ≡ 0, and vanishing magnetic fields ηx = 0; it illustrates the way boundary terms are entering. Using (4.8) we have indeed m ˜
ZΛ1∂Λ1
,ηΛ1
(σΛ2 ) X
:= Const
e
a2 (m∗ )2 2q
h(σΛ2 ,σΛ \Λ ),RΛ1 (σΛ2 ,σΛ \Λ )iΛ1 1 2 1 2
σΛ1 \Λ2
×e
e+ am∗ h˜ η∂(Λ c ) (m),R ˜ Λ1 (σΛ2 ,σΛ1 \Λ2 )iΛ1 1
.
(5.25)
Now, using the exponential decay of the resolvent, m ˜
ZΛ1∂Λ1
,ηΛ1
(σV , σ ¯Λ2 \V )
= Const × e
a2 (m∗ )2 h(σV , σ ¯Λ2 \V ), RΛ1 (σV , σ ¯ Λ2 \V )iΛ1 2q + am∗ h˜ η
Λ1 V Λ1 ∂(Λ1 c ) Λ2 \V ×e X a2 (m∗ )2 h(σ ,¯σ a2 (m∗ )2 hσΛ \Λ ,RΛ1 σΛ \Λ iΛ1 V Λ2 \V ),RΛ1 σΛ1 \Λ2 iΛ1 + 2q 1 2 1 2 × e q
(m),R ˜
(σ ,¯ σ
)i
σΛ1 \Λ2
×e
+ am∗ h˜ η∂(Λ c ) (m),R ˜ Λ1 σΛ1 \Λ2 iΛ1 1
¨ C. KULSKE
1310
= Const × e
a2 (m∗ )2 2q
h(σV ,¯ σΛ2 \V ),RΛ1 (σV ,¯ σΛ2 \V )iΛ1 ±Const|Λ2 |e−α
0 dist(Λ ,Λc ) 2 1
−α0 dist(V,Λc ) 2
× e±Const|V |e X
×
e
a2 (m∗ )2 q
σΛ1 \Λ2
×e+ ×e
h¯ σΛ \V ,RΛ1 σΛ \Λ iΛ1 2 1 2
a2 (m∗ )2 2q
hσΛ \Λ ,RΛ1 σΛ \Λ iΛ1 1 2 1 2
+ am∗ h˜ η∂(Λ c ) (m),R ˜ Λ1 σΛ1 \Λ2 iΛ1 1
.
(5.26)
The terms in the last sum do not depend on σV so that we get m ˜
P
ZΛ1∂Λ1 σ ˜V
,ηΛ1
(σV , σ ¯Λ2 \V ) m ˜ ∂Λ1 ,ηΛ1 ZΛ1 (˜ σV , σ ¯Λ2 \V
)
−α0 dist(Λ2 ,Λc ) 1 ±Const|V
= e±Const|Λ2 |e ×P
e
a2 (m∗ )2 2q
σ ˜V
e
−α0 dist(V,Λc ) 2
|e
h(σV ,¯ σΛ2 \V ),RΛ1 (σV ,¯ σΛ2 \V )iΛ1
a2 (m∗ )2 2q
h(˜ σV ,¯ σΛ2 \V ),RΛ1 (˜ σV ,¯ σΛ2 \V )iΛ1
(5.27)
with uniform constants. Taking first Λ1 ↑ Zd (using that RΛ1 |Λ2 → RZd |Λ2 ) and then Λ2 ↑ Zd we get in fact the desired result in our special case. The (random) non-Gaussian case follows easily from the cluster expansion of the G-sum in (3.1). Indeed, since we have uniform exponential decay of the activim ˜∂ C ˜ ∂∂Λ G , the cluster expansion gives us quantities ΦC ∂Λ (σC ; ηC ) that obey a ties ρ¯m uniform bound of the form as desired s.t. we have m ˜∂ X P ∂Λ C (σ ;η ) ˜ ∂∂Λ G C C ρ¯m (G; σG , ηG ) = e C conn.:∅⊂C⊂Λ ΦC . (5.28) G:∅⊂G⊂Λ
The Gibbs potential in (5.21) is then given by the value of ΦC for polymers C that are not touching the boundary. With estimates on boundary terms as in (5.26) the claim (5.22) follows. Acknowledgements The author thanks A. Bovier and M. Zahradn´ık for interesting discussions and suggestions. This work was supported by the DFG, Schwerpunkt “Stochastische Systeme hoher Komplexit¨ at”. Appendix For easy reference we collect some formulae about quadratic forms and the random walk expansion of determinants and correlation functions we use. We start with: Lemma A.1. Let QΛ be symmetric and positive definite. Let V ⊂Λ and write, with obvious notations, QV QV,Λ\V . (A.1) QΛ = QΛ\V,V QΛ\V,Λ\V
1311
THE CONTINUOUS SPIN RANDOM FIELD MODEL
Then we have the following formulae: (i) Q−1 Λ =
(QV −QV,Λ\V Q−1
Λ\V,Λ\V
−Q−1
Λ\V,Λ\V
1 1 −Q− (QΛ\V,Λ\V −QΛ\V,V Q− QV,Λ\V )−1 V V
QΛ\V,V )−1
(QV −QV,Λ\V Q−1
Λ\V,Λ\V
QΛ\V,V )−1
!
1 (QΛ\V,Λ\V −QΛ\V,V Q− QV,Λ\V )−1 V
(A.2) (ii) det QΛ = det ΠV Q−1 Λ ΠV (iii) For any zΛ we can write
−1
× det QΛ\V
(A.3)
1 hmΛ , QΛ mΛ iΛ − hmΛ , zΛ iΛ 2 E −1 1 D zΛ mV − mzΛΛ , ΠV Q−1 m = Π − m V V Λ Λ 2 V V Λ D E 1 zΛ\V ,mV zΛ−V ,mV mΛ−V − mΛ\V , QΛ\V mV − mΛ\V + 2 Λ 1
zΛ , Q−1 − (A.4) Λ zΛ Λ , 2 where the “global minimizer” mzΛΛ = Q−1 Λ zΛ is the minimizer of the total energy, i.e. 1 1 (A.5) mΛ 7→ hmΛ , QΛ mΛ iΛ − hmΛ , zΛ iΛ . 2 2 We write QV = ΠV QΛ ΠV , QΛ\V,V = ΠΛ\V QΛ ΠV . The “conditional minimizer” z
Λ\V mΛ\V
,mV
= Q−1 Λ\V (zΛ\V + QΛ\V,V mV )
(A.6)
is the minimizer of the function mΛ\V 7→
1 1 h(mΛ\V , mV ), QΛ (mΛ\V , mV )iΛ − h(mΛ\V , mV ), zΛ iΛ 2 2
(A.7)
for fixed mV . Remark. The quadratic forms on the diagonal of the r.h.s. of (A.2) are automatically positive definite. The proofs are easy and well-known computations and will not be given here. Next we collect some formulae and introduce notation concerning the random walk representation. Lemma A.2. Denote by R the (non-normalized) measure on the set of all finite paths on Zd (with all possible lengths), defined by |γ|+1 1 (A.8) R({γ}) = c + 2d for a nearest neighbor path γ of finite length |γ|. Then we have RΛ;x,y = R(γ from x to y; Range(γ)⊂Λ)
(A.9)
1312
¨ C. KULSKE
where Range(γ) = {γt ; t = 0, . . . , k} is set of sites visited by a path γ = (γt )t=0,...,k of length |γ| = k. Proof. Write ∆D Λ = 2d − TΛ where TΛ;x,y = 1 iff x, y ∈ Λ are nearest neighbors and TΛ;x,y = 0 otherwise. Then t+1 ∞ X 1 RΛ;x,y = (c + 2d − TΛ )−1 = (TΛt )x,y (A.10) x,y c + 2d t=0
which proves (A.10). We will also use the obvious matrix notation
(A.11) (R(· → · ; C))x,y = R(γ from x to y; Range(γ) = C) P so that one has the matrix equality RV = C⊂V R(· → · ; C) for any volume V . We need to use a bound on its matrix elements at several places. Let us note the simple estimate |C|−1 X 1 2d R(x → y ; C) ≤ R(γ starting at x, length(γ) ≥ |C| − 1) = . c c + 2d d y∈Z
(A.12) We will use these notations at many different places. As an example, let us prove formula (3.11). Indeed, we have (Π∂G (c − ∆Λ )−1 Π∂G )−1 = c − (∆∂G + ∂∂G,Λ\∂G RΛ\∂G ∂Λ\∂G,∂G ) = c − (∆∂G + ∂∂G,Λ\∂G RGr \∂G ∂Λ\∂G,∂G ) − ∂∂G,Λ\∂G (RΛ\∂G − RGr \∂G )∂Λ\∂G,∂G = (Π∂G (c − ∆Gr )−1 Π∂G )−1 − ∂∂G,Λ\∂G (RΛ\∂G − RGr \∂G )∂Λ\∂G,∂G = (Π∂G (c − ∆Gr )−1 Π∂G )−1 X − ∂∂G,Λ\∂G R(· → · ; C)∂Λ\∂G,∂G . C⊂Λ\∂G C∩(Gr )c 6=∅,C∩G2 6=∅
(A.13) Here we have used Lemma A.1(i) in the first and third equality and Lemma A.2 in the last one. Finally we give the: Proof of Lemma 3.4. The random walk representation of the determinant is obtained writing 1 TV log det(c − ∆V ) = log det (c + 2d) 1 + c + 2d 1 TV (A.14) = |V | log(c + 2d) + Tr log 1 + c + 2d
THE CONTINUOUS SPIN RANDOM FIELD MODEL
1313
and expanding the logarithm. Using (A.3) we can then write log
det(Π∂G (a − q∆Gr )−1 Π∂G ) det(Π∂G (a − q∆Λ )−1 Π∂G ) = log
=
detGr −∂G (c − ∆Gr −∂G ) detΛ (c − ∆Λ ) detGr (c − ∆Gr ) detΛ−∂G (c − ∆Λ−∂G )
∞ X 1 t=1
1 (TrGr (TGr )t − TrGr −∂G (TGr −∂G )t t (c + 2d)t
− TrΛ (TΛ )t + TrΛ−∂G (TΛ−∂G )t ) .
(A.15)
It is not difficult to convince oneself that we have that TrGr (TGr )t − TrGr −∂G (TGr −∂G )t − TrΛ (TΛ )t + TrΛ−∂G (TΛ−∂G )t X =− #{γ : x 7→ x; Range(γ)⊂Λ; Range(γ) ∩ ∂G 6= ∅; x∈Λ
Range(γ) ∩ Λ\Gr 6= ∅; |γ| = t} .
(A.16)
So we get the form (3.28) putting 2det(C) :=
∞ X 1 t=2
X 1 #{γ : x 7→ x; Range(γ) = C; |γ| = t} . t t (c + 2d)
(A.17)
x∈C
From this the bounds of the form det (C) ≤ e−const(log c)|C| are clear, assuming that c is large. References [1] M. Aizenman and J. Wehr, “Rounding effects of quenched randomness on first-order phase transitions”, Commun. Math. Phys. 130 (1990) 489–528. [2] T. Balaban, “A low temperature expansion for classical N-vector models. I. A renormalization group flow”, Commun. Math. Phys. 167 (1) (1995) 103–154. [3] T. Balaban, “A low temperature expansion for classical N-vector models. II. Renormalization group equations”, Commun. Math. Phys. 182 (3) (1996) 675–721. [4] C. Borgs, J. T. Chayes and J. Fr¨ olich, “Dobrushin states for classical spin systems with complex interactions”, J. Stat. Phys. 89 (5/6) (1997) 895–928. [5] J. Bricmont and A. Kupiainen, “Phase transition in the 3d random field Ising model”, Commun. Math. Phys. 142 (1988) 539–572. [6] J. Bricmont and A. Kupiainen, “High-temperature expansions and dynamical systems”, Commun. Math. Phys. 178 (1996) 703–732. [7] J. Bricmont, A. Kupiainen and R. Lefevere, “Renormalization group pathologies and the definition of gibbs states”, Commun. Math. Phys. 194 (2) (1998) 359–388. [8] A. Bovier and C. K¨ ulske, “A rigorous renormalization group method for interfaces in random media”, Rev. Math. Phys. 6 (3) (1994) 413–496. [9] D. Brydges, “A short course on cluster expansions”, in Critical Phenomena, Random Systems, Gauge Theories (Les Houches 1984) (eds. K. Osterwalder and R. Stora) North Holland, Amsterdam (1986).
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¨ C. KULSKE
[10] R. L. Dobrushin, “Gibbs states describing a coexistence of phases for the threedimensional Ising model”, Th. Prob. and its Appl. 17 (1972) 582–600. [11] R. L. Dobrushin, Lecture given at the workshop Probability and Physics, Renkum, August 1995. [12] R. L. Dobrushin and M. Zahradnik, “Phase diagrams of continuum spin systems”, Math. Problems of Stat. Phys. and Dynamics, ed. R. L. Dobrushin, Reidel (1986) pp. 1–123. [13] A. C. D. van Enter, R. Fern´ andez and A. Sokal, “Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of gibbsian theory”, J. Stat. Phys. 72 (1993) 879–1167. [14] R. Fernandez, J. Fr¨ ohlich and A. Sokal, Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory, Springer, Berlin, Heidelberg, New York, 1992. [15] H. O. Georgii, “Gibbs measures and phase transitions”, Studies in Math. Vol. 9, de Gruyter, Berlin, New York, 1988. [16] J. Glimm and A. Jaffe, Quantum Physics: A functional Integral Point of View, Springer, Berlin-Heidelberg-New York, 1981. [17] C. K¨ ulske, Ph.D. Thesis, Ruhr-Universit¨ at Bochum, 1993. [18] R. Kotecky and D. Preiss, “Cluster expansion for abstract polymer models”, Commun. Math. Phys. 103 (1986) 491–498. [19] T. Nattermann, “Theory of the random field Ising model”, in Spin Glasses and Random Fields, ed. P. Young, World Scientific, available as cond-mat preprint 9705295 at http://www.sissa.it, 1997. [20] R. H. Schonmann, “Projection of gibbs measures may be non-gibbsian”, Commun. Math. Phys. 124 (1989) 1–7. [21] M. Zahradn´ık, “An alternate version of Pirogov–Sinai theory”, Commun. Math. Phys. 93 (1984) 559–581. [22] M. Zahradn´ık, “On the structure of low temperature phases in three dimensional spin models with random impurities: A general Pirogov–Sinai approach”, Mathematical Physics of Disordered Systems, Proc. of a workshop held at the CIRM, eds. A. Bovier and F. Koukiou, IAAS-Report No.4, Berlin (1992). [23] M. Zahradn´ık, “Contour methods and Pirogov Sinai Theory for continous spin models”, preprint Prague (1998), to appear in the AMS volume dedicated to R. L. Dobrushin.
VARIATIONAL PRINCIPLE AND THE DYNAMICAL ENTROPY OF SPACE TRANSLATION HAJIME MORIYA∗ Research Institute for Mathematical Sciences Kyoto University, Kyoto Japan Received 4 June 1998 The dynamical entropy of space translations is used in the variational calculus of the pressure. It is shown that for quantum spin systems on a lattice of an arbitrary dimension, the pressure obtained in this way coincides with the usual one.
1. Introduction In this note we consider ν-dimensional quantum lattice systems. Let Zν be the ν-dimensional lattice, and Ak (k ∈ Zν ) be copies of Md (C), the d × d full matrix algebra. For any bounded region Λ ⊂ Zν , the local algebra AΛ is given by ⊗x∈Λ Ax . It follows from the definition that, for Λ ⊂ Λ0 , we have AΛ0 = AΛ ⊗ AΛ0 \Λ , where Λ0 \ Λ is the complement of Λ in Λ0 . We shall identify AΛ with AΛ ⊗ 1Λ0 \Λ in AΛ0 \Λ . The definition also implies that if Λ1 and Λ2 are disjoint, the elements of AΛ1 commute with those of AΛ2 . The C∗ -algebra A under our investigation is S the norm completion of the normed algebra A∞ = Λ AΛ , the union of all local algebras AΛ associated with finite regions Λ ⊂ Zν . We note that the group Zν of space translations is a subgroup of the automorphism group of A and we denote the action of this group by A ∈ AΛ 7→ σ~x A ∈ AΛ+x , x ∈ Zν . For each finite subset Λ of Zν , we are given an element Φ(Λ) ∈ AΛ , called an interaction potential satisfying the following conditions: (1)
Φ(φ) = 0,
(2)
σ~x Φ(Λ) = Φ(Λ + x) : translation invariance,
(3)
Φ(Λ)∗ = Φ(Λ) : self-adjointness.
Given a potential Φ and a finite subset Λ ⊂ Z, the internal Hamiltonian confined to Λ takes the form X Φ(X) . UΛ ≡ X⊆Λ
∗ Present
address: Department of Mathematics Faculty of Science and Technology Science University of Tokyo, Noda Japan.
1315 Reviews in Mathematical Physics, Vol. 11, No. 10 (1999) 1315–1328 c World Scientific Publishing Company
1316
H. MORIYA
In the following, we consider two different kinds of Banach space of interactions. For the first one, we impose the growth condition on Φ as follows: (4)
kΦk◦ ≡
X kΦ(Λ)k < ∞. |Λ|
Λ30
With this norm, the potentials under consideration form a real Banach space B◦ . For the next one, we recall the notion of the standard potential [2]. Let ω be a product state of A, namely ω(aa0 ) = ω(a)ω(a0 ) for a ∈ AΛ , a0 ∈ AΛc for any finite subset Λ. Let Γω Λ be the conditional expectation from A onto AΛ . Namely, for each ω a ∈ A, ΓΛ (a) is defined as the element of AΛ satisfying ω(Γω Λ (a)b) = ω(ab) for all b ∈ AΛ . We assume (4)0
Γω I (Φ(Λ)) = 0 unless I ⊇ Λ : standardness .
Finally, for each finite subset Λ ⊂ Zν we assume the existence of the surface energy connecting inside and outside of Λ, i.e. nX o (5)0 WΛ ≡ Φ(X) : X ∩ Λ 6= ∅, X ∩ Λc 6= ∅ ∈ A , where on the right-hand side the finite sum over X ⊂ Λ0 is first taken and the van Hove limit Λ0 % Zν is then taken, this limit is assumed to converge in norm. Total interaction Hamiltonian of Λ is defined by HΛ = UΛ + WΛ . We then define kΦk ≡ sup{kHn k : n ∈ Zν }(= kHn k) , where Hn = HΛ for the one-point set Λ = {n} and kHn k is independent of n ∈ Zν due to the assumed translation invariance. With this norm, the potentials under consideration form a real Banach space B. For each Φ ∈ B◦ , let us define AΦ ∈ A by AΦ ≡
X Φ(Λ) . |Λ|
Λ30
By definition, AΦ is linear in Φ and kAΦ k ≤ kΦk◦ . For Φ ∈ B, the following estimates have been obtained [2] X kUΛ k ≤ kHΛ k ≤ kHn k ≤ |Λ| · kΦk ,
(1)
(2)
n∈Λ
lim
Λ%Zν
kWΛ k = 0, |Λ|
where the limit of Λ % Zν is the van Hove limit.
(3)
VARIATIONAL PRINCIPLE AND THE DYNAMICAL ENTROPY OF SPACE TRANSLATION
1317
Let TrΛ denote the canonical trace of AΛ , where the term “canonical” means that TrΛ (e) = 1 for each minimal projection e ∈ AΛ . For each finite subset Λ ⊂ Zν , the partition function is defined by ZΛ (βΦ) ≡ TrΛ (e−βUΛ ) . Let PΛ (βΦ) ≡ known.
log ZΛ (βΦ) , |Λ|
then for Φ ∈ B◦ or Φ ∈ B the following basic results are
P (βΦ) ≡ limν PΛ (βΦ) exists in the van Hove limit , Λ%Z
|P (Φ − Ψ)| 5 kΦ − Ψk◦
or |P (Φ − Ψ)| 5 kΦ − Ψk ,
P (Φ) is a convex function of Φ . We call P (βΦ) the (thermodynamic) pressure of Φ at β. If ω is a state of A, ωΛ denotes its restriction to AΛ for each finite subset of Λ ⊂ Zν . The energy and entropy of this finite system Λ are given by EΛ (ω) = ω(UΛ ) , SΛ (ω) ≡ −TrΛ DΛ log DΛ . Here DΛ denotes the density matrix of ωΛ with respect to the canonical trace TrΛ on AΛ . Let A∗+,1,inv be the set of translation invariant states. We may define the energy density and the entropy density of the translation invariant state ω ∈ A∗+,1,inv to be eΦ (ω) ≡ limν Λ%Z
EΛ (ω) , |Λ|
s(ω) ≡ lim ν Λ%Z
SΛ (ω) . |Λ|
It has been shown that for Φ ∈ B◦ or Φ ∈ B, eΦ (ω) exists and |eΦ (ω)| ≤ kΦk◦
or |eΦ (ω)| ≤ kΦk ,
ecΦ+dΨ (ω) = c · eΦ (ω) + d · eΨ (ω) : linearity , eΦ (λω + (1 − λ)µ) = λeΦ (ω) + (1 − λ)eΦ (µ) : affinity . Furthermore, especially in the case of Φ ∈ B◦ we have eΦ (ω) = ω(AΦ ) . From the strong subadditivity property, it has been shown that the limit entropy density for the translation invariant state exists and is given by the infimum of over all finite subsets Λ ⊂ Zν . It has also been shown that the function ω ∈ A∗+,1,inv 7→ s(ω) is affine and upper semi-continuous with respect to weak∗ -topology. The “Gibbs Variational Principle” is used to define the equilibrium states (see for example [5]), that is for any Φ ∈ B◦ or Φ ∈ B, P (βΦ) is obtained by the following variational equality: P (βΦ) =
sup
ω∈A∗ +,,inv
[s(ω) − β · eΦ (ω)] .
(4)
1318
H. MORIYA
Furthermore, if ρ ∈ A∗+,1,inv satisfies P (βΦ) = s(ρ) − β · eΦ (ρ) ,
(5)
then ρ is called an invariant equilibrium state for Φ at inverse temperature β. Denote the set of all invariant equilibrium states as IβΦ . In our paper, we use the dynamical entropy of space translations hω G(~σ ) to be defined in the next section instead of entropy density s(ω). Our main theorem is the following: Theorem 1.1 (Main Theorem). Let Φ in B◦ or Φ in B, then P (βΦ) = sup [hω G(~σ ) − β · eΦ (ω)] . ω∈A∗ +,,inv
Narnhofer obtained this theorem for exponentially decreasing potentials, i.e. for some r > 0 X er|Λ| kΦ(Λ)k < ∞ . kΦkr ≡ Λ30
We obtain the above result for wider classes of potentials. Moreover, we simplify the proof. (We will mention the Narnhofer’s proof later.) Our strategy to prove the theorem is similar to those of the usual Variational Principle obtained in [20] and [19]. To make use of the methods in these original versions, we just need the well-known basic properties of the dynamical entropy such as (i) covariance property, (ii) scaling property in automorphisms, (iii) affinity in states, which we will recall in the next section. 2. Dynamical Entropy We will recall the definition of the dynamical entropy introduced by Connes, Narnhofer, Thirring in [6]. For this purpose, we first explain the quantum relative entropy. Let A be a finite-dimensional C∗ -algebra and let ψ1 and ψ2 be states on A. The density matrices corresponding ψ1 and ψ2 are denoted by D1 and D2 , respectively. The relative entropy of ψ1 and ψ2 is defined by S(ψ1 |ψ2 ) ≡ Tr(D2 (log D2 − log D1 )) . We then introduce the notion of an abelian model defined in [6]. Let A be a unital C∗ -algebra with a state ψ. Let C be a finite-dimensional abelian algebra with a state µ and a positive identity preserving map P : A 7→ C such that µ ◦ P = ψ. We call any triple (C, P, µ) which satisfies the above condition an abelian model for Pr (A, ψ). Let {ei }ri=1 be the set of minimal projections of C such that i=1 ei = 1. An abelian model then induces a decomposition of the state ψ into a convex combination of states ψˆi with weights µi (≡ µ(ei )) as follows: ψ(A) =
r X i=1
µi ψˆi (A)
A ∈ A,
VARIATIONAL PRINCIPLE AND THE DYNAMICAL ENTROPY OF SPACE TRANSLATION
1319
where ψˆi are uniquely determined by P(A) =
r X
ψˆi (A)ei .
i=1
“The information gain by the measurement” is defined by Eµ (P) =
r X
µi S(ψ|ψˆi ) .
i=1
Then “the entropy defect” is defined by sµ (P) = S(µ) − Eµ (P) , Pr where S(µ) denotes the abelian entropy − i=1 µi log µi . From now on we shall assume A to be a nuclear C∗ -algebra. We denote by CP1 (A) the set of all completely positive unital maps from finite-dimensional C∗ algebras into A. Let {γ1 , . . . , γk } be a finite subset of CP1 (A) with γj : Nj 7→ A(j = 1, . . . , k), where each Nj is a finite-dimensional C∗ -algebra. Let C1 , . . . , Ck be C∗ subalgebras of C and let Ej : C 7→ Cj (j = 1, . . . , k) be µ-preserving conditional expectations. (Cj , ρj , µj ) gives an abelian model for (Nj , ψ ◦ γj ), where ρj = Ej ◦ P ◦ γj and µj = µ|Cj . Then the quadruple (C, Ej |kj=1 , P, µ) is called an abelian model for (A, ψ, {γ1 , . . . , γk }). Now the entropy of this abelian model is defined by S(µ) −
k X
sµj (ρj ) .
j=1
The supremum of the entropies of all abelian models for (A, ψ, {γ1 , . . . , γk }) is denoted by Hψ (γ1 , . . . , γk ). We shall call Hψ (γ1 , . . . , γk ) the entropy of completely positive unital maps {γ1 , . . . , γk }. Let θ be an automorphism of A such that ψ is invariant under θ. For γ ∈ CP1 (A), the existence of the following limit has been shown by using the subadditibity of Hψ (γ1 , . . . , γk ) and the θ invariance of ψ: hψ,θ (γ) = lim
k→∞
1 Hψ (γ, θ ◦ γ, . . . , θk−1 ◦ γ) , k
(6)
where θi denotes the i-fold repetition of θ. The dynamical entropy of θ with respect to ψ is defined by (7) hψ (θ) = sup hψ,θ (γ) . γ∈CP1 (A)
The dynamical entropy for a single automorphism has been generalized to a group generated by several commuting automorphisms [11] as follows. Let G(~θ) be a group of automorphisms generated by commuting automorphisms {θ1 , . . . , θν }. We write θ~x for (θ1 )x1 ◦ · · · ◦ (θν )xν where ~x = (x1 , . . . , xν ) ∈ Zν . We consider a set of finite completely positive unital maps X = {γ1 , . . . , γm }, γi ∈ CP1 (A) for 1 ≤ i ≤ m. Let F be the family of all finite subsets of CP1 (A), and by F1 we denote the set of all ‘elementary’ sets X = {γ} containing only one element. Let Λ be a finite subset of Zν . For X ∈ F and G(~θ), we set
1320
H. MORIYA
XΛ (~θ) ≡
m [ [
θ~x ◦ γj ∈ F .
x∈Λ j=1
Given ~k ∈ Zν , let Λ0 (~k) be the rectangular parallelepiped of sides (ki )νi=1 with its lower corner at (0, . . . , 0), i.e. Λ0 (~k) ≡ {x ∈ Zν |0 5 xi 5 ki − 1, i = 1, . . . , ν}. Let {Λ0 (~kn )}n be a monotone increasing sequence of rectangular parallelepipeds with their lower corner ~0 such that Λ0 (~kn ) % Nν as n → ∞. In the following we use the simple notation Λ0,n for Λ0 (~kn ). It is easy to see that the following limit exists as infimum for the same reason as the 1-dimensional case. 1 Hψ XΛ0,n (~θ) . hψ,G(θ) ~ (X) = lim n→∞ |Λ0,n | The dynamical entropy for the commuting automorphisms group G(~θ) with respect to an invariant state ψ is defined by (8) hψ G(~θ) = sup hψ,G(θ) ~ (X) . X∈F
It is easily shown that the above supremum can be attained even if we take X in F1 . We then recall the following fundamental properties of dynamical entropy which has been also proved in [11] for the dynamical entropy of Zν -actions by using almost the same way for the 1-dimensional case in [6]. (i) Covariance: hψ G(~θ) = hψ◦ϑ (ϑ−1 ◦ G(~θ) ◦ ϑ) ,
(9)
where ϑ is an arbitrary automorphism of A, and ϑ−1 ◦ G(~θ) ◦ ϑ ≡ {ϑ−1 θi ◦ ϑ|i = 1, . . . , ν}. (ii) Scaling property of automorphisms: (10) hψ G(~θ) = p · hψ (Gp (~θ)) , θ) is a subgroup of G(~θ) with finite index p ∈ N. where Gp (~ (iii) Affinity in states: hλψ1 +(1−λ)ψ2 (G(~θ)) = λhψ1 (G(~θ)) + (1 − λ)hψ2 (G(~θ)) ,
(11)
for 0 ≤ λ ≤ 1, and G(~ θ)-invariant states ψ1 , ψ2 . Note that for the 1-dimensional case, (ii) implies hψ (θp ) = p · hψ (θ) . Now we shall go back to our setting of ν-dimensional quantum lattice systems. S The C∗ -algebra A(= Λ AΛ ) under our investigation is a UHF algebra. We consider ν-dimensional translation group of automorphisms G(~σ ) of A generated by σ1 , . . . , σν , where σi is the 1-shift in the direction of the i-coordinate axis. We assume that ψ is a translation invariant state. The non-commutative Kolmogorov– Sinai theorem proved in [6] holds also for the multi-dimensional cases. Therefore
VARIATIONAL PRINCIPLE AND THE DYNAMICAL ENTROPY OF SPACE TRANSLATION
hψ (G(~σ )) = lim hψ,G(~σ) (An ) , n→∞
1321
(12)
where {An } is an arbitrary increasing sequence of finite dimensional unital subalgebras of A such that An % A in norm, and we always take the convention that a subalgebra An is standing for the inclusion map An 7→ A. In the proof of our theorem we shall compare Hψ with the von Neumann entropy S(ψ). It has been shown that Hψ is always dominated by the entropy S of ψ restricted to the algebra generated by the arguments of Hψ [6]. More precisely if {N1 , . . . , Nk } is a set of finite dimensional subalgebras of A and they generate a W finite dimensional algebra N = ki=1 Ni , then Hψ (N1 , . . . , Nk ) ≤ SN (ψ) .
(13)
(As above, N1 , . . . , Nk in Hψ denote the inclusion maps Ni 7→ A (i = 1, . . . , k), and especially we call Hψ (N1 , . . . , Nk ) the entropy of subalgebras {N1 , . . . , Nk }.) We recall one of the sufficient condition under which the equality holds in (13). Let Λ1 , . . . , Λk be pairwisely disjoint finite subsets of Zν . We set Λ ≡ Λ1 ∪ . . . ∪ Λk and Λc ≡ Zν \ Λ. Let φ be a product state φΛ1 ⊗ · · · φΛk ⊗ φΛc on A(= AΛ1 ⊗ · · · ⊗ AΛk ⊗ AΛc ). We have Hφ (AΛ1 , . . . , AΛk ) =
k X
S(φΛi )(= SΛ (φ)) .
(14)
i=1
The detailed estimates of difference between H and S under several kinds of assumptions have been studied in [6]. 3. Proof of Main Theorem Let ω be a translation invariant state on A. Since eβΦ (ω) = βeΦ (ω), it suffices to prove the theorem for β = 1. From the variational inequality, P (Φ) ≥ s(ω) − eΦ (ω) . It is easily obtained that the entropy density always dominates the dynamical entropy of space translation as follows. Let {Λm } be an increasing sequence of finite subsets of Zν such that Λm % Zν in the van Hove limit. From the non-commutative version of Kolmogorov–Sinai theorem we have hω (G(~σ )) = limm→∞ hω,G(~σ) (AΛm ), and the algebra generated by {σ~x AΛm } (x ∈ Λ0,n ) is equal to AΛm +Λ0,n so that by (13) we have 1 1 Hω (AΛm , . . . , σ~x AΛm , . . . , σ~kn AΛm ) ≤ SΛ +Λ (ω) . |Λ0,n | |Λ0,n | m 0,n
(15)
When n → ∞, the right-handside of the inequality converges to the entropy density s(ω) thus we obtain hω G(~σ ) ≤ s(ω). Now we know P (Φ) ≥ hω G(~σ ) − eΦ (ω) . (16) We will then construct states ρε with hρε (G(~σ )) − eΦ (ρε ) ≥ P (Φ) − ε for an arbitrary positive number ε. Together with the above inequality (16), this proves our theorem.
1322
H. MORIYA
Given positive integer a, let Λ0 (a) be the cube of side a with its lower corner at (0, . . . , 0), i.e. Λ0 (a) = {x ∈ Zν |0 ≤ xi ≤ a − 1, i = 1, . . . , ν}. Let ϕcΛ denote the local Gibbs state in Λ, that is, ϕcΛ (A) ≡
TrΛ Ae−UΛ TrΛ e−UΛ
(A ∈ AΛ ) .
∗ ∗ ∗ σm ~ is defined by the relation σm ~ ϕ(A) = ϕ(σm ~ (A)), (A ∈ A, ϕ ∈ A+,1 ). c ∗ c ∗ We define ϕa ≡ ⊗~n∈Zν σa~n ϕΛ0 (a) ∈ A+,1,a−inv which is a periodic state invariant cca which is translation under (aZ)ν . From ϕca we construct an averaged state ϕ invariant: X σ ∗ ϕc ~ n a cca ≡ ∈ A∗+,1,inv . ϕ |Λ0 (a)| n∈Λ0 (a) ~
Our ρε is this state for sufficiently large a. First, we shall show the following lemma to estimate the energy term. cca and ϕcΛ (a) be as above, for Φ in B0 or in B. Then Lemma 3.1. Let ϕ 0 cca ) − |Λ0 (a)|−1 ϕcΛ (a) (UΛ0 (a) ) = 0 . lim eΦ (ϕ 0
a→∞
Proof. We prove this lemma for the case B◦ and B separately. (i) Φ ∈ B◦ : In the following we take the methods in [20] (and also in [18]), and adapt cca ) = ϕ cca (AΦ ). By definition them to the present situation. We know eΦ (ϕ c ca and AΦ , we have of ϕ ! X X ϕc (Φ(Λ + ~n)) X σ ∗ ϕc X Φ(Λ) a a ~ n c ca (AΦ ) = = ϕ |Λ0 (a)| |Λ| |Λ0 (a)| · |Λ| ~ n∈Λ0 (a)
=
~ n∈Λ0 (a) Λ30
Λ30
X |Y ∩ Λ0 (a)| ϕc (Φ(Y )) , |Λ0 (a)| · |Y | a
(17)
Y
because a given Y ⊂ Zν can be written as Λ + ~n with Λ 3 0 and ~n ∈ Λ0 (a) exactly |Y ∩ Λ0 (a)| times. 0 (a)| = 1 for Y ⊂ Λ0 (a) and We note that |Y ∩Λ |Y | X |Λ0 (a)|−1 ϕca (Φ(Y )) |Λ0 (a)|−1 ϕca (UΛ0 (a) ) = Y ⊂Λ0 (a)
=
X Y ⊂Λ0 (a)
Thus it suffices to prove " lim
a→∞
X
Y 6⊂Λ0 (a)
|Y ∩ Λ0 (a)| c ϕ (Φ(Y )) . |Λ0 (a)| · |Y | a
# |Y ∩ Λ0 (a)| kΦ(Y )k = 0 |Λ0 (a)| · |Y |
(18)
VARIATIONAL PRINCIPLE AND THE DYNAMICAL ENTROPY OF SPACE TRANSLATION
or equivalently
1323
X kΦ(Y )k lim = 0. a→∞ |Λ0 (a)| · |Y | n∈Λ (a) ~
(19)
0 Y 3~ n Y 6⊂Λ0 (a)
b 0 (a − 2r) be the cube with side a − 2r and Fix a positive integer r. Let Λ with the same center as Λ0 (a). We have X n∈Λ0 (a) ~ Y 3~ n Y 6⊂Λ0 (a)
=
kΦ(Y )k |Λ0 (a)| · |Y | X ˆ (a−2r) n∈Λ ~ 0 Y 3~ n Y 6⊂Λ0 (a)
kΦ(Y )k + |Λ0 (a)| · |Y |
"
ˆ 0 (a−2r)| · ≤ |Λ0 (a)|−1 |Λ
X ˆ (a−2r) n∈Λ0 (a)\Λ ~ 0 Y 3~ n Y 6⊂Λ0 (a)
X Y 30 diam(Y )≥r
kΦ(Y )k |Λ0 (a)| · |Y |
# kΦ(Y )k ˆ 0 (a−2r)| · kΦk◦ . +|Λ0 (a)\Λ |Y | (20)
ˆ
0 (a−2r)| = 1, the lim supa→∞ of (20) is bounded by Due to lima→∞ |Λ|Λ 0 (a)| P kΦ(Y )k Y 30 for all r and so we obtain the desired result by taking |Y | diam(Y )≥r the limit r → ∞. (ii) Φ ∈ B: From (3), for ε > 0 there exists a positive number aε such that
kWΛ0 (a) k <ε |Λ0 (a)|
for a ≥ aε .
We fix a(≥ aε ) and we take a sufficiently large finite subset Λ in Zν . We denote σ~l(Λ0 (a)) by Λ~l(a) for ~l ∈ Zν . For each ~n ∈ Λ0 (a), we divide Λ into disjoint regions as follows. Let Λ1−~n (2) be the union of all translates ~ ∈ Zν ) which are contained in Λ, and let ∂Λ−~n be its {Λ−~n+am ~ (a)}(m complement in Λ. Thus Λ = Λ1−~n (2) ∪ ∂Λ−~n . For the sake of a shorter notation, we assign a numbering i(m) ~ = 1, 2, . . . , q 1 ( 2 ), where q is the total number for all m ~ ∈ Zν satisfying Λ−~n+am ~ (a) ⊂ Λ−~ n (a) = Λ with i = i( m), ~ so that Λ1−~n (2) = of such m, ~ and we denote Λ−~n+am i ~ S Sq q k n (2) ≡ i=1 Λi . Now we set Λ−~ i=k Λi . By the assumed standardness of the potential, we have UΛ1−~n (2) = UΛ1 + Γω Λ1
−~ n
=
q X i=1
UΛi +
2
( ) (WΛ1 )
q X i=1
Γω Λi
−~ n
2
+ UΛ2−~n (2) = · · ·
( ) (WΛi ) .
1324
H. MORIYA
From this equality and the assumption, we obtain
q q
X X
1 −1 1 −1 ω UΛi = |Λ−~n (2)| ΓΛi (2) (WΛi ) |Λ−~n (2)| UΛ1−~n (2) − −~ n
i=1
i=1
≤ |Λ1−~n (2)|−1
q X
k(WΛi )k
i=1
≤
2)|
|Λ1−~n (
−1
q X
! |Λi |
· ε = ε . (21)
i=1
From inequality (2) and the standardness of the potential, )k ≤ |∂Λ−~n | · kΦk . kUΛ − UΛ1−~n (2) k = kΓω Λ (H∂Λ−~ n
(22)
From the definitions of ϕca and of UΛi , σ~n∗ ϕca (UΛi ) = ϕcΛ0 (a) (UΛ0 (a) ) for every i(1 ≤ i ≤ q). Hence we have −1 ∗ c |Λ| σ~n ϕa (UΛ ) − |Λ0 (a)|−1 ϕcΛ0 (a) (UΛ0 (a) ) −1
≤ |Λ|
|σ~n∗ ϕca (UΛ
+ |Λ1−~n (2)|−1
− UΛ1−~n (2) )| + 1−
|Λ1−~n (2)| |Λ|
2)|
|Λ1−~n (
−1
∗ c σ~n ϕa
UΛ1−~n (2) −
q X i=1
! |σ~n∗ ϕca (UΛ1−~n (2) )| ≤ ε + 2
! UΛi
|∂Λ−~n | kΦk , |Λ| (23)
∂Λ
where we have used (22), (21) and (2). For every ~n ∈ Λ0 (a), Λ−~n → 0 as Λ % Zν in the van Hove limit and since Λ0 (a) is a finite subset this convergence is uniform cca we obtain in ~n ∈ Λ0 (a). From the definition of ϕ cca ) − |Λ0 (a)|−1 ϕcΛ (a) (UΛ0 (a) )k ≤ ε . keΦ (ϕ 0 Since we take ε → 0 as a → ∞, we obtain Lemma 3.1.
Finally, we consider the dynamical entropy term. Let Gaν (~σ ) be the subgroup of G(~σ ) which is generated by σ1a , . . . , σνa . By definition σ~n∗ ϕca is in A∗+,1,a−inv for each ~n, thus we can define hσ~n∗ ϕca (Gaν (~σ )). By (10) and (11), we have hϕbc (G(~σ )) = a
=
1 h c (Gaν (~σ )) aν ϕba 1 X 1 aν
~ n∈Λ0 (a)
|Λ0 (a)|
hσ~n∗ ϕca (Gaν (~σ )) .
(24)
By (9), hσ~n∗ ϕca (Gaν (~σ )) = hϕca (Gaν (~σ )) .
(25)
VARIATIONAL PRINCIPLE AND THE DYNAMICAL ENTROPY OF SPACE TRANSLATION
1325
Combining (24) and (25), we have hϕbc (G(~σ )) = a
≥
1 hϕc (Gaν (~σ )) aν a 1 hϕc ,G ν (~σ) (AΛ0 (a) ) , aν a a
(26)
where the inequality follows from the definition of the dynamical entropy (8). Since ϕca is a product state, it follows from (14) that SΛ0 (a) (ϕcΛ0 (a) ) 1 . hϕc ,G ν (~σ ) (AΛ0 (a) ) = aν a a |Λ0 (a)|
(27)
Thus from (26) and (27), hϕbc (G(~σ )) ≥
SΛ0 (a) (ϕcΛ0 (a) )
a
|Λ0 (a)|
.
(28)
There exists a size a such that |P (Φ) − PΛ0 (a) (Φ)| ≤
ε
(29)
by the existence of the infinite-volume pressure and cca ) − |Λ0 (a)|−1 ϕcΛ (a) (UΛ0 (a) )| ≤ |eΦ (ϕ 0
ε 2
(30)
by Lemma 3.1. Now we use the following identity: PΛ0 (a) (Φ) =
SΛ0 (a) (ϕcΛ0 (a) ) |Λ0 (a)|
− |Λ0 (a)|− ϕcΛ0 (a) (UΛ0 (a) ) .
(31)
Combing (28), (29), (30) and (31), we obtain cca ) ≥ P (Φ) − ε , hϕbc (G(~σ )) − eΦ (ϕ a
which proves our theorem.
4. Discussions In this section we shall review the related topics. From the definition [6], the procedure of the calculation of the entropy of completely positive unital maps Hψ (γ1 , . . . , γk ) is to choose finite decompositions of the state ψ as closely as possible to the optimal decomposition. For the special cases such as classical states and product states, the optimal decompositions are obvious. But in general, it is hard to find the optimal decompositions for given states. In comparison with the Narnhofer’s proof [13] we avoid the difficulties in taking the optimal decompositions of the states by using the approximation as we have shown. As regards our main result, it should be noted that we do not claim that hρ G(~σ ) = s(ρ) for translation invariant equilibrium states ρ ∈ IβΦ . The entropy density always dominates the
1326
H. MORIYA
dynamical entropy of space translation thus we know hρ G(~σ ) ≤ s(ρ). There are known examples for which these two entropies take the same value. For the classical case, s(ω) = hω G(~σ ) for every translation invariant state ω. For the quantum case it has been shown that this equality holds for the Generalized Markov Chains ∗ in [16](see also [3]). So far, no examples ω ∈ A+,1,inv have been found to satisfy the strict inequality hω G(~σ ) < s(ω). This problem is connected to the continuity properties of hω G(~σ ) in ω ∈ ∗ A∗+,1,inv . It has been shown that for any fixed X ∈ F , ψ 7→ hψ,G(θ) ~ (X) is weak upper semi-continuous function on A∗+,1,inv [14]. Unfortunately the weak∗ upper semi-continuity property of hψ G(~θ) = supX∈F hψ,G(θ) ~ (X) is not known as yet. If ∗ we assume that hω G(~σ ) has the weak upper semi-continuity, we can derive the affirmative answer to this question as follows. From our main theorem, P (βΦ) = sup [hω G(~σ ) − β · eΦ (ω)] . ω∈A∗ +,,inv
By the assumption of the weak∗ upper semi-continuity of hω G(~σ ) in ω ∈ A∗+,1,inv , we can apply the general result of the Legendre transform to the present case and we obtain for every ρ ∈ IβΦ hρ G(~σ ) = P (βΦ) + β · eΦ (ρ) . (32) Thus we have
hρ G(~σ ) = s(ρ)(= P (βΦ) + β · eΦ (ρ)) .
(33)
Moreover, we have the following strong result under the same assumption that hω G(~σ ) has the weak∗ upper semi-continuity. Let ω ∈ A∗+,1,inv be an arbitrary translation invariant state. Let {Λm } be an increasing sequence of finite subsets of Zν such that Λm % Zν in the van Hove limit. Λ0 (a) is the finite subset as defined before i.e. {x ∈ Zν |0 ≤ xi ≤ a − 1, i = 1, . . . , ν}. We denote the restriction of ω to ∗ ∗ AΛ0 (a) by ωΛ0 (a) . We define ωa ≡ ⊗~n∈Zν σa~ n ωΛ0 (a) ∈ A+,1,a−inv which is a periodic ν ca which is state invariant under (aZ) . From ωa we construct an averaged state ω translation invariant: X σ ∗ ωa ~ n ∈ A∗+,1,inv . ω ca ≡ |Λ0 (a)| ~ n∈Λ0 (a)
We then repeat the discussions in the proof of our main theorem. By the same arguments as in (24), we have hωba (G(~σ )) = =
1 h (Gaν (~σ ))) aν ωba 1 X 1 aν
~ n∈Λ0 (a)
|Λ0 (a)|
hσ~n∗ ωa (Gaν (~σ )) .
(34)
By (9), hσ~n∗ ωa (Gaν (~σ )) = hωa (Gaν (~σ )) .
(35)
VARIATIONAL PRINCIPLE AND THE DYNAMICAL ENTROPY OF SPACE TRANSLATION
1327
Combining (34) and (35), we have hωba (G(~σ )) = ≥
1 hω (Gaν (~σ )) aν a
(36)
1 hω ,G ν (~σ) (AΛ0 (a) ) , aν a a
(37)
and we note the following equality SΛ0 (a) (ωΛ0 (a) ) 1 . hω ,G ν (~σ) (AΛ0 (a) ) = aν a a |Λ0 (a)| Thus hωba (G(~σ )) ≥
(38)
SΛ0 (a) (ωΛ0 (a) ) . |Λ0 (a)|
As a → ∞ the right-hand side converges to the entropy density s(ω), and by definition ω ca → ω in the weak∗ topology. Thus our assumption yields (39) hω G(~σ ) = lim sup hωba (G(~σ )) ≥ s(ω) . a Since the inverse inequality always holds, we conclude hω G(~σ ) = s(ω) .
(40)
The difficulty of this problem is connected to the fundamental difference between the quantum state and the classical probability in the description of composite systems. To explain this feature, we will raise one simple example and then consider the meaning of the difference between the algebraic entropy H and the von Neumann entropy S. Let |s >≡ √12 { 10 ⊗ 01 − 01 ⊗ 10 } and ψ1,2 be its vector state on M2 (C)⊗M2 (C). The restriction of ψ1,2 to M2 (C)⊗1 denoted by ψ1 is a mixed state 1 0 whose density matrix is 2 1 . We easily see that S(ψ1 ) = log 2 > S(ψ1,2 ) = 0. 0
2
The former is a mixed state on M2 (C) with greater uncertainty than the latter which is a pure state on M4 (C). In contradiction to the classical case, SΛ (ψ) is not generally monotone in regions Λ for the quantum case. This property comes from quantum mechanical correlations between regions. However the algebraic entropy has the monotone property in regions, that is for every X, Y ∈ F the inequality Hψ (X) ≤ Hψ (X ∪ Y ) holds, and this is one of the crucial points of [6]. For the above example, we have Hψ1,2 (M2 (C) ⊗ 1) = Hψ1,2 (M2 (C) ⊗ M2 (C)) = 0. Let ρ ∈ A∗+,1,inv be a pure state, i.e. ρ does not have non-trivial decompositions in ∗ A+,1 . From the definition we have hρ G(~σ ) = 0. Now the following question arises. Whether or not s(ρ) is equals to 0? There are subclasses of states on quantum spin chains (1-dimensional quantum lattice systems) called “quantum Markov chains” [1]. It has been shown in [10] that the entropy density of a purely generated C∗ finitely correlated state vanishes. (See also [8] and [9] for the reference of the C∗ finitely correlated states.) They are defined by giving an explicit formula for every finite volume expectation in terms of certain auxiliary objects. Roughly speaking,
1328
H. MORIYA
they are constructive states and their quantum correlations are not so strong. If there exists the translation invariant pure state which has non-vanishing entropy density, it can be said that this state has strongly long range quantum mechanical correlations. Acknowledgments The author is mostly grateful to Prof. Araki for his continuous help and encouragement. He thanks Prof. Ojima for his reading of the manuscript and his suggestions. He also thanks Prof. Choda, Prof. Hiai, Prof. Matsui, Prof. Nachtergaele, Prof. Ohya and Prof. Petz for useful discussions. References [1] L. Accardi and A. Frigerio, “Markovian cocycles”, Proc. Roy. Irish Acad. Sect. A 83 (1983) 251–263. [2] H. Araki, “On KMS states of a C∗ dynamical system”, Lecture Notes in Math. 650 (1978) 66–84. [3] O. Besson, “On the entropy of quantum Markov states”, Lecture Notes in Math. 1136 (1985) 81–89. [4] F. Benatti, Deterministic Chaos in Infinite Quantum Systems, Trieste Notes in Physics, Springer, 1993. [5] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics II, 2nd ed., Springer, 1997. [6] A. Connes, H. Narnhofer and W. Thirring, “Dynamical entropy of C∗ algebras and von Neumann algebras”, Commun. Math. Phys. 112 (1987) 691–719. [7] A. Connes, H. Narnhofer and W. Thirring, “The dynamical entropy of quantum systems” in Recent Developments in Mathematical Physics, eds. H. Mitter and L. Pittner, Springer, 1987, pp. 102–136. [8] M. Fannes, B. Nachtergaele and R. F. Werner, “Finitely correlated states on quantum spin chains”, Commun. Math. Phys. 144 (1992) 443–490. [9] M. Fannes, B. Nachtergaele and R. F. Werner, “Entropy estimates for finitely correlated states”, Ann. Inst. H. Poincar´ e 57 (1992) 259–277. [10] M. Fannes, B. Nachtergaele and R. F. Werner, “Finitely correlated pure states”, J. Functional. Anal. 120 (1994) 511–534. [11] T. Hudetz, “Spacetime dynamical entropy of quantum systems”, Lett. Math. Phys. 16 (1988) 151–161. [12] R. B. Israel, Convexity in the Theory of Lattice Gases, Princeton Univ. Press, 1979. [13] H. Narnhofer, “Free energy and the dynamical entropy of space translation”, Rep. Math. Phys. 25 (1988) 345–356. [14] H. Narnhofer and W. Thirring, “Dynamical entropy of quasifree automorphisms”, Lett. Math. Phys. 14 (1987) 89–96. [15] H. Narnhofer and W. Thirring, “Dynamical entropy and the third law of thermodynamics”, Lett. Math. Phys. 15 (1987) 261–273. [16] Y. M. Park, “Dynamical entropy of generalized quantum Markov chains”, Lett. Math. Phys. 32 (1994) 63–74. [17] M. Ohya and D. Petz, Quantum Entropy and Ite Use, Springer, 1993. [18] B. Simon, The Statistical Mechanics of Lattice Gases, Princeton Univ. Press, 1993. [19] D. W. Robinson, “Statistical mechanics of quantum spin systems. II”, Commun. Math. Phys. 7 (1968) 337–348. [20] D. Ruelle, “A variational formulation of equilibrium statistical mechanics and the Gibbs phase rule”, Commun. Math. Phys. 5 (1967) 324–329.
REVIEWS IN MATHEMATICAL PHYSICS Author Index (1999)
Albeverio, S., Kondratiev, Yu.G. & Röckner, M., Diffeomorphism groups and current algebras: configuration space analysis in quantum theory André, N. & Shafrir, I., On nematics stabilized by a large external field Antoine, J.-P., Bagarello, F. & Trapani, C., Topological partial *-algebras: basic properties and examples Avramidi, I.G., Covariant techniques for computation of the heat kernel Bagarello, F., see Antoine Carpi, S., Quantum Noether’s theorem and conformal field theory: a study of some models Cattaneo, U. & Wreszinski, W.F., Contractions of Lie algebra representations de Monvel, A.B. & Sahbani, J., On the spectral properties of discrete Schrödinger operators: the multi-dimensional case Derezi½ski, J. & Gérard, C., Asymptotic completeness in quantum in field theory. Massive Pauli–Fierz Hamiltonians Dickson, R., Gesztesy, F. & Unterkofler, K., Algebro-geometric solutions of the Boussinesq hierarchy Dito, G., On generalized abelian deformations Dörrzapf, M., The definition of Neveu– Schwarz superconformal fields and uncharged superconformal transformations Echeverría-Enríquez, A., MuñozLecanda, M.C. & Román-Roy, N., Reduction of presymplectic manifolds with symmetry
1(1999)
1
Fioresi, R., A deformation of the big cell inside the Grassmannian manifold G(r, n) Gallavotti, G., Gentile, G. & Mastropietro, V., Mel’nikov’s approximation dominance. Some examples Gentile, G., see Gallavotti Gérard, C., see Derezi½ski Gesztesy, F., see Dickson Gover, A.R. & Zhang, R.B., Geometry of quantum homogeneous vector bundles and representation theory of quantum groups I Guarneri, I. & Schulz-Baldes, H., Upper bounds for quantum dynamics governed by Jacobi matrices with self-similar spectra Guerrini, L., Formal and analytic rigidity of the Witt algebra Guha, P., The τ functions of the AKS hierarchy and twistor correspondence Hagedorn, G.A. & Joye, A., Molecular propagation through small avoided crossings of electron energy levels Hausser, F. & Nill, F., Diagonal crossed products by duals of quasi-quantum groups Helffer, B., Remarks on decay of correlations and Witten Laplacians II – analysis of the dependence on the interaction Ichinose, W., On convergence of the Feynman path integral formulated through broken line paths Irac-Astaud, M. & Rideau, G., Deformed harmonic oscillator algebras defined by their Bargmann representations
6(1999)653
3(1999)267
8(1999)947
3(1999)267 5(1999)519
10(1999)1179
9(1999)1061
4(1999)383
7(1999)823
6(1999)711
2(1999)137
10(1999)1209
1329
1(1999) 25
4(1999)451
4(1999)451 4(1999)383 7(1999)823 5(1999)533
10(1999)1249
3(1999)303
8(1999)981
1(1999) 41
5(1999)553
3(1999)321
8(1999)1001
5(1999)631
1330
JakÓi, V. & Molchanov, S., Localization for one dimensional long range random Hamiltonians Johnson, G.E., Interacting quantum fields Joye, A., see Hagedorn Juyumaya, J., A new algebra from the representation theory of finite groups Knauf, A., Number theory, dynamical systems and statistical mechanics Kondratiev, Yu.G., see Albeverio Kostrykin, V. & Schrader, R., Scattering theory approach to random Schrödinger operators in one dimension Külske, C., The continuous spin random field model: ferromagnetic ordering in d ≥ 3 Mastropietro, V., see Gallavotti Matsutani, S., Immersion anomaly of Dirac operator on surface in R3 Michéa, S., 3D singletons and their boundary 2D conformal field theory Molchanov, S., see JakÓi Moriya, H., Variational principle and the dynamical entropy of space translation Müger, M., On soliton automorphisms in massive and conformal theories Muñoz-Lecanda, M.C., see Echeverría-Enríquez Nill, F., see Hausser
AUTHOR INDEX
1(1999)103
7(1999)881 1(1999) 41 7(1999)929
8(1999)1027
1(1999)
1
2(1999)187
10(1999)1269
4(1999)451 2(1999)171
9(1999)1079
1(1999)103 10(1999)1315
3(1999)337
10(1999)1209 5(1999)553
Popov, A.D., Self-dual Yang–Mills: symmetries and moduli space Rideau, G., see Irac-Astaud Röckner, M., see Albeverio Román-Roy, N., see Echeverría-Enríquez Sahbani, J., see de Monvel Schlingemann, D., From Euclidean field theory to quantum field theory Schrader, R., see Kostrykin Schulz-Baldes, H., see Guarneri Shafrir, I., see André Suris, Y.B., Integrable discretizations for lattice system: local equations of motion and their Hamiltonian properties Suzuki, M., General formulation of quantum analysis Trapani, C., see Antoine Unterkofler, K., see Dickson Woon, S.C., Analytic continuation of operators applications: from number theory and group theory to quantum field and string theories Wreszinski, W.F., see Cattaneo Xia, J., Two-dimensional coulomb interactions in a magnetic field Zhang, R.B., see Gover
9(1999)1091
5(1999)631 1(1999)
1
10(1999)1209 9(1999)1061 9(1999)1151
2(1999)187 10(1999)1249 6(1999)653 6(1999)727
2(1999)243
3(1999)267 7(1999)823 4(1999)463
10(1999)1179 3(1999)361
5(1999)533
