Mathematical physics and stochastic analysis : essays in honour of Ludwig Streit

0. The lemma 2 is proved. 3. // o diagram with (n~,n+) external lines consists only from pairs of conjugated vertices then n~ = n+ and the diagram has on n+ connected parts. Each connected diagram has (1,1) external lines and is half-planar, i.e. it can be drown in the half-plane without self intersections. LEMMA

We will not present here the simple proof of this lemmata. Now the theorem follows from the above three lemma.

32

5

Entangled Commutation Relations

5.1

Stochastic limit for N point correlators in one-particle states

The following theorem has been proved in 1 3 . THEOREM

2. The stochastic limit

lim Ax(p,k,t)

A-»0

= B~{p,k,t),

lim A$(p,k,t)

X-yO

=

B+(p,k,t)

exists in the sense of the convergence of the matrix elements (ti = ±) lim < 0\c(q)Ax1{pi,kutl)...Ax-(Pn,kn,tn)c+(q')\0

>

= ((pukut1)...B<"(pn,kn,tn)c+(q,)*0)

(5.47) +

as distributions and the limiting operators B~ = B and B gled commutation relations B{p, k, t)B+ (p',k',t')

satisfy the entan­

= 2n6(t - t')8(p - p')5(k - k') ■ 6{E{p, k))n(p)

[n{p'), B*(p, k, t)) = (±)(S(p' -p)-

S(p' - p + k))B*(p, k, t)

[n{p),n(p')}=0 s

(5.48)

(5.49) (5.50)

+

Here *o * ^ e vacuum in the new Hilbert space, B(p,k,t)YliC (qi)^o = 0. We use the same notations for the creation and annihilation operators of cparticles in the original and in the new Hilbert spaces. n(p) is the operator density of the c-particles, n(p) — c+(p)c(p). If we set B(p, k, t) = b(t) ® B(p, k) where b{t)b+{t') =

2n6(t-t')

then we obtain the relations (1.2) B{p, k)B+(p', k1) = n(p)8(E(p, k))S(p - p')S(k - k') The more general theorem will be proved in the next section. Note that to get non-zero in the RHS of (5.48) we have to choose suitable dispersion relations, so that there are non-trivial solutions of the equation E(p, k) — 0.

33

5.2

Stochastic limit for N point correlators in n particles states

In this section the main result of this paper will be proved. Let us consider the matrix element N

<*;,n^A*m)

(5-51)

t=l

where A\ and A% are given by (4.31) and (4.32) and m

n

c+(f) = Jc+(k)f(k)dk

, c(f') = J'c(k)f'(k)dk

Lemma 2 claims that if diagram corresponding to (5.51) doesn't consist only of pairs of conjugated vertices then its contribution vanishes in the limit A —> 0 (in the sense of distributions). One can see that to select diagrams corresponding to (5.51) such that all their vertices form pairs of conjugated vertices one has to consider diagrams that satisfy the following requirements: a) n = m, b) If n ^ 1 then the diagram must be disconnected and it must include n connected parts. Each connected part corresponds to Ni point correlator of composite operators between one particle stats and

c) Each connected part contains only pairs of conjugated vertices. To describe all diagrams satisfying above requirements it is suitable to do the following. Let us first apply the Wick theorem to represent the product A 's in normal form with respect to only of operators c and c+ N

Y[A£i = Y, i=l

:

KpA£i}(a,a+,c,c+)

:c

(5.52)

P

Here : : c means the normal product with respect to operators c and c + , and summation on p involves all terms that appear in the process of application of the Wick theorem. Applying the Wick theorem to

34

nc(/f)^nc+(/,-) i

j

we have f[c(f'i)f[Ac):f[c+(fi)= t

«=i

(5-53)

j=\

symbol j means that we take into account the diagram with / contractions (c-lines). For example, A+Ax = Klt+- + K2,+(5.54) Ki,+- = I dpdp'dkdk'a+{k)c+{p - k)c+{p')c{p)c(p' #2,+- = f

k')a{k')v(p,k)v{p'k')

dpdkdk'c+{p-k)a+{k)a(k')c(p-k')v{p,k)v(p,k)

(see fig. 5)

_^

<~

Figure 5: Friedrichs diagrams representation of (5.54)

For the product of AA+ we have AxAt = Ki,-++ K3,-+

(5.55)

where Klt-+ = fdpdp'dkdk'v(p,k)v(p',k')c+(p)c+{p'

- k')c(p - k)c(p')a{k)a+ (*')

K2,-+ = J dpdkdk'v(p, k)v(p -k + k', k')c+(p)c(p -k + k')a(k)a(k')

35

Figure 6: Friedrichs diagrams representation of (5.55)

(see fig. 5) To select the non-zero vacuum average of (5.53) that consists from dia­ grams satisfying requirements a), b) and c) we take the factorized products

I J {
£ j^ji=Ni=i

1

1

1

(5-56)

1

Now (5.56) is a sum of products of Wick monomials containing a and a+ operators. Diagrams corresponding to (5.51) that survive in the limit A -> 0 consist from n continuous c-lines connected one annihilator operator, some number of vertices and one creator operators. These c-lines enter in different connected parts of the same disconnected diagram. Beside c-lines the diagrams contain a-lines, that connect only vertices that are crossed by the same c-line. Each of connected part is represented by a half-planar diagram. This consideration proves the following T H E O R E M 3. The stochastic limit of the matrix element (5.51) can be repre­ sented as follows

im(oiTTc(/n (oi n cifj) n A? n c+^)i°>=

A-K)

= *«m £p

j=i

£

N

m

<=i

j=i

<°lc(/i)S£- • • • ^

c+(/P(i))|0>-

pe n "partitions"

•<0|c(/J)B e 'i .. . e ^ c + ( / p ( 2 ) ) | 0 ) . . . (0\c(fn)B^

. .. ^

c + (/ p ( n ) )|0>

Here Vn is the group of transpositions, the sum over "partitions" includes the sum overall subsets ( i i , . . . , ^ , ) ( j i , . . . ,jk2)... (lly... ,lkn) of {1,...,N} preserving the order and fci + &2 + ... + kn = N.

36

Conclusion The theorem 3 from the previous section describes the stochastic limit of com­ posite operators in n-particle states. It gives the second quantized general­ ization of interacting commutation relations from the previous works 5 , 1 3 . It would be interesting to find an operator expression of the obtained fomula as well as the corresponding path integral representation. Acknowledgments. I.Ya.A. and I.V.V. are grateful to the Centro Vito Volterra Universita di Roma Tor Vergata for the kind hospitality. This work is supported in part by INTAS grant 96-0698, I.Ya.A. is supported also by RFFI-99-01-00166 and I.V.V. by RFFI-99-01-00105 1. L. Streit, A new look at functional integration. In: Capri 1993, Proceed­ ings, Advances in dynamical systems and quantum physics, 307-325. 2. L. Streit, White noise analysis and functional integrals, in: "Mathemat­ ical approach to fluctuations", T.Hida, ed., World Sci., 1993 3. I.Ya.Aref'eva and I.V.Volovich. Large N QCD and q-deformed quantum field theories, Nucl.Phys.B 462 (1996) 600 4. M. B. Halpern and C. Schwartz, The Algebras of Large N Matrix Me­ chanics, hep-th/9809197. 5. L. Accardi, Y.G. Lu and I.V.Volovich, Quantum theory and its stochastic limit, Springer Verlag (in press) 6. L. Accardi, S.V. Kozyrev and I.V. Volovich, Dynamical q-deformation in quantum theory and the stochastic limit, J. Phys. A: Math.Gen.32(1999)3485-3495 7. I.Ya. Aref'eva and I.V. Volovich, Quantum group particles and nonarchimedean geometry, Phys. Lett. B268(1991)179-187 8. I.Ya.Aref'eva and I.V.Volovich. Quantum group gauge theory, Mod.Phys.Lett.A6(1991) 893-907. 9. I.Ya. Aref'eva and I.V. Volovich, Noncommutative Gauge Fields on Poisson Manifolds, hep-th/9907114 10. R. Vilela Mendes, Geometry, stochastic calculus and quantum fields in a non-commutative space-time, math-ph/9907001 11. L. Accardi, Y.G. Lu and I.V. Volovich, Interacting Fock spaces and Hilbert module extensions of the Heisenberg commutation relations, Pub­ lications of HAS, Kyoto,1997 12. L. Accardi, Y.G. Lu, Comm. Math. Phys. 180 (1996) 605 13. L. Accardi, I.Ya. Aref'eva and I.V. Volovich, Non-Equilibrium Quantum Field Theory and Entangled Commutation Relations, hep-th/9905035 14. V.S.Vladimirov, Equations of Mathematical Physics, Academic Press, 1986

37

D E R H A M COMPLEX OVER P R O D U C T MANIFOLDS: DIRICHLET FORMS A N D STOCHASTIC D Y N A M I C S SERGIO ALBEVERIO Institut fur Angewandte Mathematik, Universitat Bonn, Wegelerstr. 6, D 53115 Bonn; SFB 256 (Bonn); BiBoS Research Centre (Bielefeld); CERFIM (Locarno); Ace. Arch. (USJ) ALEXEI DALETSKII Institut fir Angewandte Mathematik, Universitat Bonn, Wegelerstr. 6, D 53115 Bonn; SFB 256 (Bonn); Institute of Mathematics, Kiev (Ukraine) YURI KONDRATIEV Institut fur Angewandte Mathematik, Universitat Bonn, Wegelerstr. 6, D 53115 Bonn; SFB 256 (Bonn); BiBoS Research Centre (Bielefeld); Institute of Mathematics, Kiev (Ukraine) We define de Rbam complex over a product manifold (infinite product of compact manifolds), and Dirichlet operators on differential forms, associated with differentiable mesures (in particular, Gibbs measures), which generalize the notions of Bochner and de Rham Laplacians. We give probabilistic representations for corresponding semigroups, and study properties of the corresponding stochastic dynamics.

1

Introduction

Various questions of stochastic analysis on product manifolds (that is, on infinite products of compact manifolds), have received a strong interest in resent times. This interest is strongly motivated by applications to models of statistical mechanics. Various aspects of the study of Dirichlet operators associated with Gibbs measures on product manifolds, and corresponding semigroups, were considered by many authors, see ', ! for a detailed review of the literature. Let us remark that the Dirichlet operator of a differentiable measure, which is actually an infinite dimensional generalization of Laplace-Beltrami operator, has a natural supersymmetric extension. Namely, we can consider Dirichlet operators on differential forms over a product manifold M, extending the notions of Bochner and de Rham Laplacians. The study of the latter operators and corresponding semigroups on finite dimensional manifolds was the subject of many works, and leads to deep results on the border of stochastic analysis, differential geometry and topology, and mathematical physics, see

38

e.g. 1 7 , 1 8 . In an infinite dimensional situation, such questions were discussed in the flat case in 10 , n , 12 , 6 . A regularized heat semigroup on differential forms over the infinite dimensional torus was studied in 13 . The study of such questions on product manifolds reflects the non-trivial interplay of measure theory and differential geometry (which is essential in an infinite dimensional setting). The present paper represents a step in this direction. The structure of it is as follows. In Section 2 we introduce main notations and geometrical objects. Section 3 is devoted to the definition of the de Rham complex over M . In Section 4 we introduce Bochner-Dirichlet and de Rham-Dirichlet op­ erators in the de Rham complex, and prove an analog of the Weitzenbock formula. Then we formulate the main result which characterizes Markov and hypercontractivity properties of the corresponding semigroups. Section 5 is devoted to the development of a probabilistic technique needed for the proof of this theorem. In particular, we consider SDE on tensor bundles over M and give with their aid probabilistic representations of semigroups. Section 6 is devoted to a discussion of the obtained results in the framework of Gibbs measures. We give here only main ideas of proofs. More extended presentation see in6. Dedication. It is a great pleasure for the authors to dedicate this work to Ludwig Streit on the occasion of his 60th birthday. His work on the interface of mathematical physics and stochastic analysis was always a great inspiration for us, and we are very grateful to him. Acknowledgments. We are glad to thank K. D. Elworthy, M. Rockner and A. Thalmaier for useful and stimulating discussions. The financial sup­ port SFB-256, DFG-research projects and INTAS-project No.378 is gratefully acknowledged. 2

Setting

Let A, B be Banach spaces and % be a Hilbert space. We will use the following general notations: - (•, •) - pairing of .4 and A', A' being the dual space; ( ) ' ) « - t n e scalar product in H; - tne norm m

— II'IU

-4;

- C{A, B) - the space of bounded linear operators A -* B; -C(A) = C(A,A); - Cn(A,B) - the space of bounded n- linear operators A -> B\ - HS{H, B) - the space of Hilbert-Schmidt operators H-+B.

39 Let M be a compact connected smooth JV-dimensional manifold. Let us assume that M is equipped with the Riemannian structure given by the operator field G(x) : TXM -f T*M, (•, -)T.M =< G(x); • >. The distance on M which corresponds to this Riemannian structure will be denoted by p. Let us consider the integer lattice Z d , d > 1, and define the space M, which is an infinite product of manifolds Mk = M: M =Mzi := xk€ZjJkfk 9 x = (xk)k€z<< • M is endowed with the product topology. Given A C Z M9iHi

A

(1)

d

= (zk)kGA € M A

(2)

A

denotes the natural projection of M onto M . We denote by Vk the derivative of the function <j> w.r.t. the variable x k (which will be always identified with the vector field over M (gradient of ) via the Riemannian structure of M), resp. Levi-Civita covariant derivative of a vector or tensor field. Thus for a n— times differentiable w.r.t. Xk function <j> we have VJ#(x) e C^iiT^M^^M). The symbol Aktf> := TrV k 0 will denote the corresponding Laplace-Beltrami operator on functions. We will use the notations Vu := (Vku)kezd> A := I^ez* ^kLet 0 be the family of allfinitesubsets of Z d . We will denote by TCm{M) the space of m—times continuously differentiable real-valued cylinder func­ tions on M, FCm(M) := UA6nCT"(MA). (3) We also define the space TCm{M. -*• TM) of m—times differentiable cylin­ der vector fields on M with both domain and range consisting of cylinder elements: FCm{M ->TM) 3X = (XOkeA, A e n, X k € TCm{M. -+TMV). m

r

Let us remark that for u € TC {M) we have Vu € J C will use the notations < X(x),Y(x)

m_1

(4)

(M ->TM). We

>x= Y, (*k(*),*k(*))T„kM, divX = Yl div^X^, (5) kez* kez"4 dtVk meaning the divergence with respect to x k . The space M has a Banach manifold structure with the Banach space lb(Zd -*■ B.N) of bounded sequences Y = (yk)kez*,*k € KN, equipped with the norm imi„ := sup ||Yk||RW , (6) kez*

40

as the model. However, this norm being not smooth, one gets difficulties in using this manifold structure for the purposes of stochastic analysis. The way of overcoming of this difficulty was proposed in 3 , 4 . In these works, an analog of Riemannian structure on M was introduced. On a heuris­ tic level, the tangent space to M at the point x can be identified with the space x k 6 Z i T z k M . In order to define a differentiable structure on M , it is natural to consider some Hilbert subspace of x k 6 Z .iT x k M. Let h := h(Zd -* R^.) be the space of summable sequences p = (pk)kez,i of positive numbers. For a f i x e d p e l i let us define the space T P l , = I X 6 x kGZ< ,T xk Jlf : Y, P* H*kHr.kM < ° 4 , { kez<« J

(7)

equipped with the natural scalar product <*• y )p,» = £ **(*><> y " ) r . k M • (8) kez d The scalar product (X,Y) in the spaces T P)X wiU play the role of a Riemannian-like structure for M . The space M equipped with this struc­ ture will be denoted by M p . The bundle over M p with fibres TPiX will be called the tangent bundle of M p and denoted by TMp. The fibres TPiX will be denoted by T x Mp. The bundle T M P is not the tangent bundle to M p in the proper sense. Nevertheless T M P gives us the possibility to define analogues of various differentiable structures on M p . In 3 , 4 , the spaces C n (Mp -> B) of difierentiable mappings of M into a Banach space B, resp. C"*(Mp -> TMp) of differentiable vector fields over M , were introduced. That is, / € C x (Mp -> TMp), iff the matrix V/(ar) := (V,/ k (x)),

(9)

of partial (covariant) derivatives generates a bounded operator in T X M P , con­ tinuous in x. We will use the notation T M :=2 1 M 1 , where 1 is the weight sequence with elements Ik = 1. The space M p possesses the metric pp defined by

Pp{x,y)2 = 5Z P(*k,*k)9Jfc, kez d

(10)

which makes it a complete metric space. Let /i be a probability measure on M differentiable in the sense that the following integration by parts formula holds true: for any u G TCX (M -> R 1 ) ,

41

and any vector field X e TC1 (M ->TM) / J2 (Vku(x),X k (a;))T„ k M **(*) = - / / ? > ( * ) **(*), ^ kez* ^

(H)

with some /?£ € L 2 ( M , / J ) . /?£ is called the logarithmic derivative of n in the direction X. We assume that /3£ is given by 0x(«) = E (^W-■Xk(s))T. k M + dfoXkfr)), kez^

(12)

where /3M(x) = (/3k 0*0) e C 1 ( M P -»■ TM P ) for some weight sequence p£ We will call 0^ the (vector) logarithmic derivative of \i. 3

h.

Differential forms and de R h a m complex

The simplest differential forms over M are the forms with both image and domain consisting of cylinder elements. The space of m— times differentiable forms of the order m of such type will be denoted by .Ffi™. We will use the notation 7Tln := U m ^n™. Each v € .Ffi„ has the form v x

() =

v

53 ki

*i

k-(*)i

(13)

knCZi

where Ukx k„(«) € T ^ M A ... A T ^ M, and the sum is finite. Actually, each w G T£ln can be regarded as a differential form on the manifold MA for some finite A C Zd. For such forms, the symbols V* resp. A* will denote the covariant derivative resp. the Bochner Laplacian (Afcw(a:) := TrV^w(x)), w.r.t. X*. We will use the notation Vw = (Vfcw)fc€Zd, Aw = X) AjfcW. For w e TQ.n we have Vw e , H } n . We will use the notation <w{x),v(x)

>x=

^ ki

(win

k„(a;),Uk,

k.Wk^AfA.-.Ar^M.

k„&Zd

(14) Let / x b e a differentiable measure on M with the logarithmic derivative p. We introduce spaces of integrable forms L* iln as the completions of F£ln w.r.t. the norm \\-\\s given by

IMC = /

E («ki,...,k„(a0>t'ki i k,,...,k„6Z'

^ ( x ) ) ^ ^ MA...AT,k, Af

d/x(a;). (15)

42

In order to define de Rham complex over M, we need spaces of smooth forms. Given a Hilbert space )C, we define the space A"AC which is the n—fold antisymmetric tensor power of the Hilbert space AC. We introduce in the stan­ dard way the exterior multiplication A : A"AC x A m AC -»• A n + m A C ,

(16) 0 A tp:=^±I^.ASn+m((l> n'.ml

<8> VO,

ASn+m : ®n+mIC -* A n + m K. being the antisymmetrization operator, and creation resp. annihilation operators a*(Jfc):AnAC->An+1AC, k G K, (17) am(k):=kA

resp. a(k) : An+1AC -»• AnAC

(18)

(the adjoint operator). Let us define the bundle A T M with fibres AT Z M. Sections of this bundle can be considered as differential forms over M (we identify the space T^M with TmM). Each section w of A T M can be represented as a sum of sections with cylinder images, w(*) =

5Z k,

«ki.....k.0O. <^i,....k„(*)€T^MA...Ar x k t M,

(19)

k„€Z<<

converging at each x in the norm of AT^M. We define the (covariant) deriva­ tive Vu> as an infinite block-operator matrix, Vw(s) : = (Vjc^ kl> ...,k I> (a:))i )kl) ... )k „ eZ , ,

(20)

where VtWk, k n (i) e C(TX\M,Tx]Si MA...AT lk> M) (if exists). We can now define the space C 1 (M P -+ AnTM) of differentiable forms, requiring that Vw(s) G C(TXMP, A n T x M),

(21)

and is bounded uniformly in x G M. Similarly we define the spaces C m (M p -¥ ATM). We wiU use the notation ft™(M„) := Cm(Mp -¥ A T M ) .

(22)

43

We introduce the exterior differential dn^CMp^n^CMp),

(23)

dnwix) = (n + l)ASn+1 (Vw(z)),

(24)

setting

where ASn+i : (£) n+1 TXM -»• A"+1 TXM is the antisymmetrization opera­ tor, and Vw(x) € C(TXMP -> A"TXM) is identified with the element of (AnTsM)
k„(aO = ailu...tit,(x)hx\Sl A ... A hx^ , hx^ € T^ Af,

akl

k„€C

m

(Mp-^R1).

We have as usual dnWk1 k.(ar) = da k l

k„(z) A / i ^ A ... A hx\^ , (25)

dnw = '^2
kn,

where daki,...,k„ is the 1— form defined by the derivative of the function "k,

k„-

Let us observe that d„ can be considered as unbounded operator I/£ft„ -*• L^H„ + i. We denote by d* : l£fl„+i -)■ l£ft„ the adjoint operator. The following fact follows in a straightforward manner from the integra­ tion by parts formula. Proposition 3.1 Let us assume that ^ e C^Mp -*• TMP). Then the oper­ ator e£ is densely defined, its domain of definition contains the space . H V n , and for u> € .Ffi„+i we have d*w € flJ,(Mp). In what follows, we will also use the spaces L"(M -* A"TM,,/j) of integrable forms with values in A n TM„ with the norm given by (JIHIA-T.M.

*<*))*•

44

4

Dirichlet operators on differential forms

In this section we define Dirichlet forms and Dirichlet operators in spaces of differential forms over M, generalizing the notion of Bochner and de Rham Laplacians. Let n be a differentiable measure on M such that the corresponding log­ arithmic derivative &P belongs to C1 (M p -t TM P ) for some weight sequence p € h. We introduce two Dirichlet forms on l£ft„. For «,« 6 T£ln we define the pre-Dirichlet form £f? resp. £* associated with /x, setting £«(«,») := f ^ ( V k u ( x ) , Vkt;(x))T.M dn(x), J

(26)

k

resp. £*(u, v) := (du, d«) t jn„ +1 + (<**". d * u )/.jn„_ 1 •

(27)

Analogously to the classical pre-Dirichlet form £M on functions (see e.g. 7 ), the forms £j? resp. £j* have generators Hjf resp. H£ acting in L2fi„ on the domain Tiln as H*u(x) = - E Aku(x) - ^(/3 k (a;), Vku(x))A»T.M®T.kAf, k

(28)

k

resp. H£ = dd* + d*d,

(29)

and are therefore closable. We preserve the same notations for their closures resp. generators of the closures. Let us observe that in the case of a single manifold M and fi a Riemannian volume measure on it, the operator Hjf resp. Hj* coincide with the Bochner resp. de Rham Laplacian, see e.g. , 15. We will call them the BochnerDirichlet resp. the de Rham-Dirichlet operator associated with p. Our next goal is to find a relation between the operators H * and H^. / .\i=i,-.Af

/ ,\i=i

N

Let f e^J be an orthonormal basis in T^M. Then f e£ 1 is obviously the orthonormal basis for T Z M, (for any weight sequence q). We set (4)* := aV k )> 4 :=«(«£)• Let us introduce the operator R-W= E

E

[^i.«(^k)(o k )X(a k )*oL]

k € Z d i,j,M,l

in A n T x M,, where Rij,i is the curvature tensor on M.

(30)

45

Lemma 4.1 Rn(x) is a bounded operator in each space AnTzM,, q arbitrary. Because of the assumptions on /?", we have V/3**(x) G £(T x Mp). Because of the symmetry of Levi-Civita connection, the operator V/J^x) is symmetric w.r.t. the scalar product of TZM, and therefore belongs also to £(Tj.M p -i). Let us define the operator [V/?"(x)]An := V/3"(i)®lO...®l (31) +1 ® V0M(x) ®... ® 1 +... + 1 ®... ® 1 ® V0M(x) in A n T x M. We have [V0"(x)]An G £ ( A T I M p - i ) and [V0*(x)]An € £(A n T z M p ). The following result is an extension of the classical Weitzenbock formula. Theorem 4.1 For u G ?Sln H««(i) = H*u(x) + Rn{x)u(x) - [V/3"(x)]An u(x).

(32)

n

Let us introduce the operator R£(x) € £(A T I M p ), R>(x) = Rn(x)-[V0"(x)]An. (33) Formula (32) implies that R£(x) does not depend on the choice of the basis in (30). Both Hjf resp. H * are non-negative self-adjoint operators in L2ft„ and generate therefore strongly continuous contraction semigroups Tff(t) resp. T*(t) in it. The following result characterizes the properties of these semi­ groups. Theorem 4.2 Let us assume that 0" 6 C4(MP -► TMP). Then : 1) both Hjf and H* are essentially self-adjoint operators on TCPQA); 2) for any weight sequence q the semigroup T^(t) leaves invariant the space C{M -¥ A"TM,), and for v G C(M -¥ ATM,) we have II T J?(*)«(*)|IA-T.M 1 = T "W ll«HA-TM, (*)!

(34)

3) for the weight sequence q = p or q = p _ 1 the semigroup T*{t) leaves invariant the space C(M -t ATM,), and for v G C(M -> ATM,) tue /w«e ||T«(t) V (x)|| A „ T>Mq < e ^ T ^ t ) HHIA.TM, (*).

(35)

where r is such that r • (h,h)A„TmM < (R£(x)h, /I)A»T,M, for each h e AT*M, and x G M; 4) let us assume that there exists a constant ro is such that ro ■ (h, fe)A,TiM < (RJJ(X)/I,/I) A „ T M M for each h G AT^M, and x G M; then the semigroup

46 T£(*) leaves invariant the space C(M -+ ATM), and for v G C(M -4 A"TM) we have ||T?(*M*)|| A ,. T . M < e-tr" T„(t) ||t,|| A „™ (*).

(36)

The prooffollowsfrom a probabilistic representations of the semigroups Tjf(i) and T*(i), which will be given in Section 5. Let us recall that a semigroup T(t) (acting on functions on M) is called hypercontractive, if for all 1 < s < a < oo T{t) : L'(M,M) -*• L"(M,n),

(37)

and T(t) is a contraction when ^

> e" 2tA

(38)

for all t > 0 and some A > 0. Formula (34) emphasizes a certain Markov property of Tjf (t) and implies its hypercontractivity provided TM is hypercontractive. An analogous state­ ment for T*(£) requires positivity of R£. The following result is the corollary of Theorem 4.2. Theorem 4.3 Let us assume that in the framework of Theorem 4-2 r > 0 resp. r0 > 0. Then: 1) the semigroup T*(i) is Markov in the sense that for q = p orq = p~l resp. g= l

IW<Mz)| A „ T . M , < T„(«) IMU-™, (*);

(39)

2) the semigroup T*(<) is contractive in each L*(M -> A" TM.g,fi), where q = p - 1 resp. q = 1; 5^ «/TM(t) is Aj/peram£racttve, Men T%(t) is also hypercontractive (with the same X), in the sense that T*(t) : L'(M -► AnTMq,(i) -> L a (M -+ AnTM,,/x), where q=p~l 5

(40)

resp. q = 1, ond T^(t) is a contraction when (38) holds.

Probabilistic representations of semigroups

The aim of this section is to obtain probabilistic representations of the semi­ groups Tjf(t) and T*(i), and to prove with their aid Theorem 4.2. The construction of the diffusion process in M, which gives the stochastic dynamics associated with the classical Dirichlet form £M, is given in 2 - 4 . This process was constructed as the strong solution to an infinite system of SDE

47

d(k(t) = &«(*))# + P(&(t)) o dufcft), k € Z d ,

(41)

in the Stratonovich form on M, where w^, k € Zd, are independent Wiener processes in a given Euclidean space R n such that M c R " isometrically, and P(m) : R n -¥ TmM is the orthoprojector. We will also write this system in the form of one SDE #(*) = J ^ K W ) * + P(«*)) o dw{t)

(42) d

on M, where w is the cylinder Wiener process in /C := /2(Z -► R"), and P(a;) is the block-diagonal operator K. -> r x M with diagonal blocks Pkkfa) = P(xk). It is easy to see that A(x) £ HS(K.,TXMP) for each p eh. Theorem 5.1 (3, 4) Let &> € Cl(Mp -»> TM P ). TAen fAe SDE (42) has a unique solution £x(t) for any initial data x € M. £x{t) depends continuously (in the mean square sense) on the initial data x. The corresponding generator coincides on TC1 (M) with the Dirichlet operator of measure /*. Our next goal is to construct the parallel translation of differential forms along solutions of (42). For this, we need an analog of Levi-Civita connection onM. Let OM be the orthonormal frame bundle over M. The space OM has obviously the structure of a compact manifolds which fits into the framework of previous sections. We define the product manifold 0M:=x k 6 Z d0M.

(43)

This space will play the role of orthonormal frame bundle over M, with fibres O s M:=x k6Z
(44)

We denote by ir: OM -¥ M the corresponding projection. Given the Levi-Civita connection on OM, we equip OM with the product connection. This means the following. Let us consider the tangent bundle T(OM)p to OM. By the definition, for each z G OM T,(OM) p = © k 6 Z , VPLT^OM).

(45)

We define the horizontal tangent space ffT^(OM)p resp. the vertical tangent space VTz(OM)p as HTz(OM)p:= ©keZ<< JpZHTzk(OM), (46) VTz(OM)p:= © k 6 Z ,

^pZVTzk(OM),

48

where HT^OM) resp. VT^OM) is the horizontal resp. vertical tangent space of OM associated with the given connection. We have obviously the decomposition Tz(OM)p = HT,(OM)p © VTz{OM)p.

(47)

Let us observe that the corresponding covariant derivative of a vector field X G Cl(Mp -+ TM,,) coincides with the derivative dX defined in above. We denote by X the horizontal lift of the vector field X over M to OM. It follows directly from our definition of HTz(OM) that (*(*)) k = £ ( * k ) ,

(48)

where Y* is the horizontal lift of the vectorfieldY* := Xk(n(z\)) over M, A = Zd\ {k}. We have then obviously that X G C n ((OM) ->• T (OM) ) provided X G C n (M p -*• TMP). Let now £(t) be the Brownian motion with the drift a G C 1 (Mp -¥ TMP) described by SDE (42). We consider its horizontal lift ~f(t) G OM defined by the SDE dj(t) = a(j(t))dt + P( 7 (t)) o dw(t),

(49)

where P(z)h := P(z)h, with initial data 7(0) such that TT(7(0)) = £(0), which is by Theorem 5.1 uniquely solvable. As in the case of a single manifold M (see 17) we have n(g(t)) = £(t). Given n > 1 and a weight sequence q of positive numbers (not necessarily decreasing), we consider the bundle V := An TMq over M. Let v G V x and define by p(g)v the natural action of g G O {TXM) := xkeZ*0 (TX\M) on Vx, O (TmM) being the space of orthogonal linear transformations of TmM. We can now define the parallel translation p

W)--Vil0)-+Vm

(50)

along the solution £(t) of (42) as **«*)« = PWMO)- 1 )*, (51) where j(t) solves (49). The transformation P^ is obviously orthogonal. Let J be a continuous operator field on M, J(x) G C(VX). We define the mapping J : OM -> V€(0) by the formula J(*) := p ( « b ~ M - , ^ M * ) ) p ( * 0 . «b = 0(0), and consider the equation

(52)

<*?(*) =

(53)

J(l(t)Ht)dt

49 in Vf( 0 ). This equation is uniquely solvable for any initial data by general theory of SDE in Hilbert spaces (see e.g. 1 6 ), and defines a multiplicative functional of the process f(t) and consequently of £(t). Let us consider the semigroup T*'J(t) acting in the space C(M -* V'), which is defined by the expression < T^J(t)w{x),y

> = E < w(^{t)),PUt)Vv(t)

>,

(54)

where nv(t) is the solution to (53) with the initial condition y, and denote by Hi,J its generator. Proposition 5.1 Forut T(?$A -> V ) we have &<Ju{x) = l&u{x)+

< a ( i ) , V u ( i ) >x +Jm{x)u(x).

(55)

st

Proof. Let us first assume that a = 0 and J = 0. This case reduces obviously to the case of a single manifold, and the corresponding generator is equal to \ A by the finite dimensional theory 1T. In the general case, applying Ito formula to the function $(z,v) =< u(n(z)), P(ZZQX)V > on O M x V x , we obtain the additional first-order term V , * ( z , « ) [3(*)] + V.*(z,t>) [J(z)v] = < Vu(*(*)) [a(ir(x))],p(zz^1)v + < u(-w{z)),J(z)p(zZo1)v

> (56)

>,

which implies the result. The following statement follows in a straightforward way from the Ito formula and Gronwall inequality. Proposition 5.2 The semigroup T^'J(t) satisfies the following estimate: \\T*'J(t)v(x)\\{yl)m

< e tc T<(*) ||«|| v , (x),

(57)

v 6 C ( M -» V ) , where c is such that cl > J(x) for each x, as operators in V,. Remark 5.1 Let us consider the bundle W := A n T M , i , where the weight sequence q1 is such that q£ < ?k for each k. Then we have V x C W x for each x e M . Let us assume that there exists a constant c\ such that Cyl > J(x) for each x, as operators in Wx. Similar arguments show that the process n(t) satisfies also the estimate

lln(tM0||W{(1)=e*Mb?(0)llw.0,

(58)

50

which implies that T €,J (f) leaves invariant the space C(M -► W ) , and for v € C(M -► W ) we have ||T € ' J (*)«(*)|| (W , ) . < e te ' I*(t) || V || W , (x).

(59)

Now we can construct probabilistic representations for Bochner and de Rham semigroups and to obtain with their aid a proof of Theorem 4.2. Let us observe that on TQ,n we have H« = -H«,

(60)

where H € is the generator of the parallel translation along the paths of the "stochastic dynamics" process f associated with /i, defined by SDE (42). If ( - H ^ , ^ n „ ) is essentially self-adjoint, the semigroup T^(i) has a simple probabilistic interpretation. Indeed, in this case Tjf(i) is the unique semigroup with the generator (—H^,^fi„) , and therefore for w € C(M -»• AnTM) we have T«(t)u;(x) = E(P t M& (*))).

(61)

In order to obtain a probabilistic representation for the semigroup T^ associated with H*, let us observe that for w € FSln we have H«'- R -w = -H*w.

(62)

The corresponding semigroup T*' -R S (t) leaves invariant the space C(M -»• An TMp-i), and coincides on this space with T^, provided H* is essentially selfadjoint on ^ n „ . The latter fact holds true if /?" € C 4 (M P -> TM,) (in this case both semigroups T*(f), T* , - R - (t) leave invariant the space C 2 (M -> An TMp-i), which obviously contains in the domain of Hjf and H*). This implies Theorem 4.2. 6

Stochastic dynamics for lattice models associated with Gibbs measures on product manifolds

Let us consider a family of potentials U = {UA)A€{I, U\ 6 (?(AfA). Let ft(k) be the family of all sets A G fi containing the point k G Zd. We will assume the following: (Ul) £

sup |l/ A (x)| < oo

Aen(k)* 6 M

(63)

51

for any k € Zd. Let n be a Gibbs measure on the Borel er-algebra B(M), associated with the family of potentials U. We denote by G{U) the family of all such Gibbs measures. G{U) is non-empty under the condition (63), see e.g. 19, 18. Let us now assume that the family of potentials U satisfies (in addition to (Ul) the following conditions: (U2) Us € C 1 (M A ) for each A, and sup V |||V k tf A |||™ < oo, k

(64)

6zdA6n

where |||Vk[/A|||TM :=sup l6M ||V k £/ A (a;)|| TiikM ; (U3) U\ G C2 (A) for each A, and there exists C < oo such that SU

P E E IIIVjVkt/A|||TM®TM < C, (65) Jez< R 1 ) has the form TM(t)u(x) = E(u(£z(f))). This process is given by Brownian motion on M with the drift /3". We see that any Gibbs measure \i G G{U) under conditions (U1)-(U3) fits the framework of Section 4, and we have the Weitzenbock representation (32) for de Rham-Dirichlet operator H^ f i . However, Theorem 4.3 requires additional smoothness of potentials. Let us assume that U\ € C5(A) for each A, and keZ
SU

P E

k6Z 1

Elll V Ji V J' VkC/A IH® 4 ™ <00 '

' JiJj6Z''A6n

52

(66) 8U

P

£

V

V

£lH J'- J«

Vk:/A

8

HI® ™
where ll|Vj1...VjmVkyA|||®-.+iTi»#:= 8U

P liyi,...Vj m V k C/A(s)||T.S Af®-®T.; Af®T,UM-

This condition implies obviously that there exists a weight sequence p € h such that /?" 6 C 4 (M„->rM p ). We summarize our discussion in the following Theorem 6.2 Let the family of interactions U satisfies (Ul), (U2), (US) and (66), and \i e G{U). Then there exists a weight sequence p€ h such that the statements of Theorems 4-2 and 4-S hold true. Remark 6.2 An application of an approximation technique similar to the one developed in3,7 gives the possibility to prove the essential self -adjointness of operators Hjf and H * on FSln only under the conditions (Ul), (U2), (US), and therefore to avoid the additional condition (66) in Theorem 6.2. References 1. S. Albeverio: Some applications of infinite dimensional analysis in math­ ematical physics, Helv.Phys.Acta 70 (1997), 479-506. 2. S. Albeverio, A. Daletskii, Yu. Kondratiev: Infinite systems of stochastic differential equations and some lattice models on compact Riemannian manifolds, Ukr. Math. J. 49 (1997), 326-337. 3. S. Albeverio, A. Daletskii, Yu. Kondratiev: Stochastic analysis on prod­ uct manifolds, in the book: "Stochastic Dynamics", ed. H. Crauel and M. Gundlach, Springer 1999 (Proceedings of the conference "Random Dynamical Systems", Bremen, April 28 - May 2, 1997). 4. S. Albeverio, A. Daletskii, Yu. Kondratiev. Stochastic equations and Dirichlet operators on product manifolds, Univ. Bonn, Preprint No. 591 SFB 256 (1999). 5. S. Albeverio, A. Daletskii, Yu. Kondratiev: Stochastic analysis on prod­ uct manifolds: Dirichlet operators on differential forms, Univ. Bonn, Preprint No. 598 SFB 256 (1999). 6. S. Albeverio, Yu. Kondratiev: Supersymmetrie Dirichlet operators, Ukr.Math.J. 47 (1995), 583-592.

53

7. S. Albeverio, Yu. Kondratiev, M. Rockner: Uniqueness of the stochastic dynamics for continuous spin systems on a lattice, J.Func.Anal., 133, No.l (1995), 10-20. 8. S. Albeverio, Yu. Kondratiev, M. Rockner: Quantum fields, Markov fields and stochastic quantization, in the book: Stochastic Analysis: Mathematics and Physics, Nato ASI, Academic Press (1995) (A. Car­ doso et al.,eds). 9. S. Albeverio and M. Rockner: Dirichlet forms on topological vector spaceconstruction of an associated diffusion process, Probab. Th. Rel. Fields 83 (1989), 405-434. 10. A. Arai: Supereymmetric extension of quantum scalar field theories, in Quantum and non-commutative analysis, H.Araki et al. (eds.), 73-90, Kluwer Academic Publishers, Holland (1993). 11. A.Arai: Dirac operators in Boson-Fermion Fock spaces and supereym­ metric quantum field theory, Journal of Geometry and Physics 11 (1993), 465-490. 12. A.Arai, I.Mitoma: De Rham-Hodge-Kodaira decomposition in oo-dimensions, Math. Ann.,291 (1991), 51-73. 13. A. Bendikov, R. Leandre: Regularized Euler-Poincare number of the in­ finite dimensional torus, to appear. 14. Yu. M. Beresansky and Yu. G. Kondratiev: "Spectral Methods in Infinite Dimensional Analysis", NaukovaDumka, Kiev, 1988.[English translation: Kluwer Akademic, Dordrecht/Norwell, MA, 1995]. 15. L. Cycon, R. G. Froese, W. Kirsch, B. Simon: "Schrodinger Opera­ tors with Applications to Quantum Mechanics and Global Geometry", Springer, 1987. 16. Yu. L. Daletcky and S. V. Fomin: "Measures and Differential Equations in Infinite-Dimensional Space", Kluwer Academic, Dordrecht, Boston, London, 1991. 17. K. D. Elworthy: Geometric aspects of diffusions on manifolds, Lecture Notes in Math., 1362, Springer Verlag, Berlin and New York, 276-425 (1988). 18. V. Enter, R. Fernandez, D. Sokal: Regularity properties and Pathologies of Position-Space renormalization-group transformations, J. Stat. Phys. 2, Nos. 5/6 (1993), 879-1168. 19. H. O. Georgii: "Gibbs measures and phase transitions", Studies in Math­ ematics, Vol. 9, de Gruyter, Berlin, New York (1988). 20. L.Gross: Hypercontractivity and logarithmic Sobolev inequalities for Clifford-Dirichlet forms, Duke Math. J., 43 (1975), 383-386 .

54

REAL TIME R A N D O M WALKS ON P - A D I C N U M B E R S SERGIO ALBEVERIO Institut fur Angewandte Mathematik, Universitdt Bonn, Bonn, Germany WITOLD KARWOWSKI Institute of Theoretical Physics University of Wrocllaw, Wroclaw, Poland Dedicated to Ludwig Streit on the occasion of his 60 birthday The field of p-adic numbers is a complete metric space under a non-Archimedian metrics. For this reason it is a suitable mathematical framework for the description of hierarchical phenomena appearing in many disciplines. Here we summarize some investigation of the real time random walks on p-adic based on the properties of transition functions obtained by solving the Kolmogorov equations.

1

Introduction

The last two decades have seen a steady growing interest in p-adic analysis. A substantial part of mathematical investigations has received momentum from physical motivations. There have been two main physical ideas leading towards application of local fields and in particular the p-adic numbers 1 . One came from particle physics. It was based on the conjecture that the space time at Planck distances may have a structure better described by a field other than the one of the real numbers. This has given motivation to exten­ sive mathematical work which resulted in elaborating interesting structures going under the names of p-adic quantum mechanics and p-adic quantum field theory 2,3 . In this paper we will relate mainly to another idea coming from statisti­ cal physics, in particular in connectioin with models describing relaxation in glasses. The non exponential nature of those relaxations has been interpreted as a consequence of a hierarchical structure of the state space, which can in turn be put in connection with p-adic structures. In fact p-adic numbers can be interpreted as having a "tree structure". In the physical literature various models of "diffusion" or random walks on (finite or infinite) trees have been investigated 1,15 . In particular L. Brekke and M. Olson4 constructed a class of random walks on p-adic numbers and discussed the relation with relaxations in glasses. This and pure mathematical interest motivated systematic studies of ran-

55 dom walks indexed by continuous real time with the field of p-adic numbers as state space. This note is intended to present some methods and results ob­ tained in this direction. The Levy processes on p-adic numbers have been con­ structed by different methods see Evans 16 and Figa-Talamanca 5 , but we shall be mostly concerned with the approach developed by Albeverio, Karwowski7,8 and later extended by Karwowski, Vilela-Mendes 9 , Albeverio, Karwowski, Zhao 10 and Albeverio, Karwowski, Yasuda 13 . For the relation betrween Figa-Talamanca 5 and Albeverio, Karwowski 7|8 see Husssmann 6 . In Section 2 we give basic properties of p-adic numbers and show how the translation and rotation invariant Levy processes are obtained by solving the corresponding Kolmogorov equations. We also present spectral properties of the generator. In Section 3 we generalize this method to cover corresponding processes taking values in weighted state space. In Section 4 we discuss conditions for a process to be recurrent and also hitting and exit times for p-adic balls. In Section 5 we discuss a p-adic trace formula and its analogy with the Selberg trace formula. 2

Levy Processes on Qp

We begin with basic definitions and facts about p-adic numbers 14 . Let p > 1 be a prime number. A p-adic number a is associated with the formal power series oo

a=J2^iP^

(2-1)

i=N

where AT is an integer and a,- = 0 , 1 , . . . , p - 1. With addition and multiplica­ tion defined in the natural way for formal power series the set Qp of all p-adic numbers becomes a field. Given a G Qp, set i0 for the smallest value of i in the sum (2.1) for which at ^ 0. Then we put

NI P =p- i 0 .

(2-2)

The map a -+ ||o|| p defines a norm in Qp, which has the non Archimedian triangle property ||a + % ^ m a x { | | a | | p , | | 6 | | p } ,

(2.3)

56 and Qp with this norm is a complete separable locally compact totally dis­ connected space (with the cardinality of the continuum). The series (2.1) converges with respect to the || || p norm. Let a G Qp, Mel, then the set K(a>PM)

= {x € Q P ; \\a - x\\p ^ pM} M

is called a sphere (or a ball) of radius p consequences of (2.3) i) If x e K{a,pM)

then K{x,pM)

=

(2.4)

centered at a. We note the following K(a,pM)

ii) If at e Qp, 1=1,... ,p, \\a, - ak\\p = pM+\

k # I then ^

K{ahpM)

=

M+1

K(ak,p

)

M

in) Let /C be the family of all disjoint balls of radius pM. Then K.M is countable and writing K.M = {Ki}^ we have U Ki = Qp. iv) K(a,pM)

is open and compact.

Let £ denote the er-algebra generated by the family of all balls in Qp. Then the set function /z defined on the balls by M

(K{a,pM))

=pM,

a € Qp, M <E Z

(2.5)

can be uniquely extended to a measure on £ ) p also denoted by /*. It is a Haar measure for the additive group in Qp. Put G+ for Qp as an additive group and G. = {x € QP, \\x\\p = 1}

(2.6)

for Qp \ {0} as a multiplicative group. Then G. defines a group of automor­ phisms Qz, z E G* of G+ by Qza = za, a € G+. We put G for the semidirect product of C?» and G+ relative to 0 : G = G. x e G+

(2.7)

i.e. G = {ff€ [z,a]:zeG*,

a€G+}

(2.8)

and 5i52 = [z\z2,ziai

+02].

(2.9)

Defining action of G on Qp by 0Z = [z, a]x = zx + a,

x e Qp, g € G

(2.10)

57

we find that G is a doubly transitive group of isometries for Qp We also have (see 1 3 ): Proposition 2.1 /z is the Haar measure for the group G. There are different starting points possible for the construction of random processes. Before presenting the approach proposed in Albeverio et al7-8 we briefly mention two other a) Levy-Khinchine representation 12 ' 16 . Put {x} for the fractional part of x G Qp. The normalized additive character on Qp (G+) is defined by X{x)= exp(27ri{x}) and then the Fourier transform of a complex valued function <j> e L1 (Qp, fi) is defined by

m = f x&MzMdx) Any Levy process on Qp can be defined by the LeVy-Khinchine formula

*(t,fl = exp< tj\x(xO-lMdx)\ where v is a cr-finite measure on £

(2.11)

satisfying v({x € Qp; \\x\\p > pM}) <

oo, M € Z. v is called a Levy measure and $(t,£) is the Fourier transform of the transition function for a Levy process. b) The generator of a Markovian semigroup 17 ' 18 . Consider the real space L2(Qp\n). Then the notion of Dirichlet form and Markovian semigroup are well defined and their relations with the Hunt processes follow from general theory 19 . Vladimirov, Volovich, Zelenov and Kochubei 2,17 ' 18 have taken as the starting point the generators of the Marko­ vian semigroups Da, a > 0 defined on the space D(QP) of locally constant functions with compact support by the formula

Da

+w = g lp-C-i / w* - f) - *(*)] Wvw;0"1^) ■

(2-12)

Qy

R e m a r k 2.1 Recently H. Kaneko22 modified formula (2.12) and obtained new random walks which apparently do not belong to any class discussed in this note.

58

The construction proposed in Albeverio et al 7 ' 8 is based on the Kolmogorov equations. We begin by taking KM for the state space. Then the system of forward and backward Kolmogorov equations reads PKiKi (*) = -a{Kj)PKtKi

(*) + £

u(Kf, Ki)PKiK,

(t)

(2.13a)

resp.

PKtKi(t) = -a{Ki)PKiKi{t) + Y,^K^Kf)P^f^t>>

(2-13b)

/#* with t > 0, i,j € N and the initial condition P K ^ . (0) = dy. a(Kj) is interpreted as intensity of the state Kj and u{K^Kj) as the infinitesimal transition probability from the initial state K{ to the target state Kj. Till now K,M is just a countable set. Its non Archimedian structure must be imprinted in the coefficients u(Ki,Kj). We shall do it as follows We say that a sequence of real numbers A = {a(n)}„ 6 z belongs to the class A iff a) b)

a(n) ^ a(n + 1), lima(n)=0.

(2.14)

n—>oo

Let A G A. If for any M e Z and m € N put U(M) = a(M - 1) - a(M) and u{M,m) = (p - l ) - 1 ? - " " - 1 C/(M + m ) . MKif\Kj

= 0 then dist p (J^i, Kj) = pM+m for some m € N. Then we define «(lfi,iir i ) = «(M,m)

(2.15)

a(Kj) = Y,u(Ki,Kj)

(2.16)

Requiring further

we immediately obtain a(Kj) = a(M).

59

The equations (2.13) with the coefficients specified by (2.15) and (2.16) can be explicitly solved. Further by shrinking the initial ball to a point while keeping the target ball fixed one arrives at the formulas which can be extended to the transition function of a random walk with Qp as the state space 7 ' 8 . Namely oo

Pt(x, K(a,pM))

=p-1{p-

l)53p-»exp{-T M .H*}

(2.17a)

t=0

if x eK(a,pM)

and

M\\ Pt(x,K(a,pM))=

— p—m

P'1 (P - 1) 5 Z P~* exp{-TAf+m+< *} «=0

- exp{-T W + m _i t}

(2.17b)

if dist p (x, K(a,pM)) = pM+m. These formulas hold for all x,a € Qp, M G Z. We put here rn = ( p - l ) _ I [ p a ( n ) - a ( n + l)] ,

neZ.

(2.18)

The transition function Pt{x,B) (t > 0 , i £ Qp, B G £ ) determined by the formulas (2.17) can be seen as the integral kernel for a Markovian semigroup (Tt,t>0)mL2(Qp]ti). Set Tt — e~Ht. Then H is a positive self adjoint operator in this space. Its action on the indicator functions for the balls can be given explicitly, which provides sufficient information for the spectral analysis of H. It turns out that the spectrum of If is essential pure point

<*{B) = {M B E z • The corresponding eigenvectors are defined as follows. Given a e Qp and n € Z. Put a.1 = a and choose 0 2 , . . . , o p so that ||a* - a.j\\p = p" i ^ j i,j = l,...,p. Let further Xi(x) stand for the indicator function of K{a,i,pn). Then for p

any real numbers 6 1 , . . . bp such that £ bi = 0 we have _•

1

y

v

t=l

i=l

60

The Dirichlet form £ corresponding to H has only the jump part in the Beurling-Deny representation i.e.

W,9) =

I

(/(*) - /(»)) (g{x) - g(y)) J(dx,dy)

Q„xQp\d

for all f,g € D[£]. J is a symmetric positive Radon measure on Qp x Qp supported off the diagonal. J is uniquely determined by J(K{a,pN),K(b,pM)) if &stp(K{a,pN),K{b,pM))

= ^"-"^[ain =pn

- 1) - a(n)}

(n > maxM,N)

J(K(a,pN),K(a,pN))=pNa(N) Since G (2.7) is a group of isometries it follows that the transition functions (2.17) are G invariant. This results in the following Theorem 2.1 2 0 The class of random walks on Qp constructed above is iden­ tical with the class of all G« symmetric L6vy processes defined by (2.11). It naturally includes the processes defined by {Da,a > 0} (2.12) with the fol­ lowing relation between a and {o(n)} n 6 z

3

n(n\ P-1 P ^ a(n) = . p l-p-<*-i Random Walks with Weighted Target States

The procedure presented in Sec. 2 can be generalized to yield a wider class of random walks which are not necessarily LeVy processes. In Karwowski et al 9 a class of processes which are not necessarily LeVy processes has been constructed. Similarly as in Albeverio et al 7 ' 8 one begins by solving the Kolmogorov equations (2.13), but the coefficients u(Ki,Kj) are denned differently. Let p be a nonnegative locally Ll{y) function. For B 6 £ ) p we set PB = fp(x)n(dx). Taking {a(n)}„ 6 Z G A we put B

u(Ki,Kj)=pK.U(M

+ m)

where as before U(N) = a(N - 1) - a(N) and distp(Ki, K5) = pM+m, m G N. Since in our notation Ki is the initial state and Kj the target state we see that in this formulation the target states are weighted. The method of solving equations (2.13) developed in Albeverio et al 7,8 has been modified in

61

Karwowski et al9 to also cover this case. To formulate the results we shall need more notation. For any a G Qp and n e Z put p*=

j

p{x)p.{dx).

(3.1)

K(a,p") n

Note that if p ^ ||a||p then K(a,pn) = K(0,pn) and p£ = p°. For the target state Kj = K(b,pM) we define -WhM = a{Ki) = Y,*(Ki*Ki)

(3-2)

and - W ' = X > ( » + * " 1) " a(" + * + 1)K+* • Jfc=0 10

These quantities are finite

(3-3)

iff

«=1

The formulas for transition functions depend on whether / p{x)p,{dx) is finite or not. In the first case we normalize p so that / p(x)p,(dx) = 1. Then Q,

Pt (x,K(a,pM))

=^

{ l + f ; ( 1 - J - ) exp {Mfc.,}} P I i=M ™ <+ l / J

(3.4a)

if x€ K{a,pM) and Pt (x,K(a,pM))

= p%, \l + f ) ( 1 - - U exp {*W?+1}

-i-exp{
(3.4b)

if distp(x, K(a,pM)) =pn,n> M. If J p(x)n(dx) = oo then OP

P t (*, K(a,pM)) = p% f ; ( 1 - J - ) exp {tW?+1}

(3.5a)

62

if xtK(a,pM)

and

P^tfCa.p^-pdfM i - ^ r - ) « P {***+!} v ,,,„

Pi

«+!/

_JLexp{tVO|

(3.5b)

if dist p (i,i<:(a,p M )) = p", n > M. Remark 3.1 When / p(x)p,{dx) < oo thenPt{x,K{a,pM))

-+ paM as

t-too

i.e. paM, a € Q p , M e Z defines the invariant measure for the process. Remark 3.2 If a{N) = 0 for some N € Z then Pt(x,K(a,pM)) = 0 when­ ever distp(x,K(a,pM)) > pN i.e. the process is confined to the ball K{x,pN). This is a consequence of the non Archimedian distance property. Indeed if a(N) = 0 then U(n) — a(n - 1) - a(n) — 0 for all n > N which means that the process can not make a jump larger than pN. But the p-adic dis­ tance between K(x,pN) and any ball disjoint with it is longer than pN and this distance must be taken in one jump which is impossible. Remark 3.3 Ifa{N + 1) < a(N) = a(N - m) for allmGN then the process does not make jumps smaller than pN+1. In this case the process is equivalent to one with K.N as the state space. Remark 3.4 If p{x) = 1 then the resulting process is G invariant. However, given A G A the transition functions (2.17) and (3.5) are different. Never­ theless both formulations are equivalent. It can be seen by direct computations that (3.5) defined by p(x) = 1 and A £ A is identical with (2.17) defined by A' € A where oo

o'(M)=p"

1

a M

{p-l)- a{M)-J2 (

+ i)P~i ■

»=i

Similar arguments work in the converse direction. R e m a r k 3.5 The processes discussed in this section have been used in a model for turbulent cascades n . In order to consider Pt(x, B) as the integral kernel for a Markovian semigroup (Tt, t > 0) we have to take L2(QP, pp.) for the underlying Hilbert space. Then explicit formula for the action of the generator H on the indicator functions can be given9. It follows further that H has pure point spectrum cr(H) = {Wf}i6Z,o6Qp-

63

The eigenvectors are denned as follows. Given a G Qp and n G Z we define Xi(x)' symmetric case. Then

»=i

* = 1,. . . , p as in the G

»=i p

where c* G R, i = l,...,p

satisfy X) CtPJi' = 0-

Passing to the corresponding Dirichlet form we find that it is p\x symmetric and consists of the jump part only as should be expected. The jump measure J(dx, dy) is defined by J(K(a,pM),K(a,pM))

= -\waM

p%

and J{K(a,pM),K{b,pM)) if distpCK, (a,pM), K(b,pM)) 4

= \p%pbM[a{M + m-\)-a{M = pM+m,

+ m)}

meN.

The Global Properties

Now we shall list some global properties of the Markovian semigroups (Tt,t> 0) constructed in previous sections 10 . First {Tt,t > 0) is conservative i.e. Pt{x,Qp) = 1. This property holds generally provided in case />«> = / p{x)n{dx) < oo the function p is normalQP

ized so that />«> = 1Concerning the transience, resp. recurrence problem results have been obtaind in 10 , we summarize them here. The semigroup (Tut > 0) of the bounded operators in I?{Qv\pp) is irreducible iff a{n) > 0 for all n G Z (see Remark 3.2). Recall that a Markovian semigroup which is irreducible is either transient or recurrent. The simplest case is when p^ = 1 and a(n) > 0, n G Z. Then the constant function 1 G Lx{pp) and Ttl(x)

oo

= Pt(x,Qp)

= 1 so that fTtl{x)dt = oo o and (Tt, t> 0) is not transient. As it is irreducible it must be recurrent. oo

If Poo = oo then (Tt,t > 0) is recurrent iff / Pt{x, B)dt = oo for some B o

64

with / p(x)p,(dx) > 0. This by direct examination is seen to be equivalent to B

-T,e%Awr- = ^

(4-1)

where M is such that p°M > 0. If (r t , t> 0) is determined by (2.17) then it is recurrent12 iff °°

E

■ 1

P _ , - 7 T T = OO.

•„

°(»)

v This condition can also be obtained by and passing from A «=0 ' specifying This condition can also be obtained by specifying p = 1 and passing from A to A' according to Remark 3.3. We add a short comment on the probabilistic interpretation of the constants W^. Let & be the random walk denned by the transition function (3.4) or (3.5). Then TK(a,pM) = inf {t > 0;6 i K(a,pM)} is the exit time for the ball K(a,pM). We have Theorem 4.1 10 If a(M) > 0 and p% ^ poo then TK(a,PM) ** almost surely finite and its expectation is ExTK{p.,vM) — ~WM)~X> x e K(a,pM). If p = 1 or equivalently the transition functions are defined by (2.17) then

ExTK{a,p")

Denote by

=a(M)-1. TK^IPM^ the

hitting time for a ball K(a,pM) i.e.

rK{a>pM) = inf {* > 0;& 6 K(a,pM)}

.

Recall that recurrence is said to be positive or persisting if ExTK(a
Trace Formula

We shall close this review with a result which bears similarity with the trace formulas for the Laplace-Beltrami operator for hyperbolic spaces. We shall make our point using probabilistic interpretation13.

65

Consider the complex upper half-plane C+ with the Poincare metrics. Then SL(2, B)/ / {1, —1} is a group of isometries of the half-plane C+. Given a discrete subgroup T consisting of hyperbolic elements there is a fundamental domain F for T i.e. an open subset of C+. such that JiFn

72.F=0

if

71,72 GT

and

717^72

and

Then identification of the points x G F with 72, 7 G T defines a compact Riemann surface M. The Laplace-Beltrami operator A on C+ (resp. AM on M) generates a random walk Xt, t ^ 0 on C+. (resp. Rt,t ^ 0 on M). Let Pt(x,y) (resp. qt(x,y)) be the transition density for Xt (resp. for Rt). Define Qt(x,y) = ^2Pt{x,jy) -rer

x,y£M.

It follows that qt = qt i.e. qt is the transition density for Rt. Moreover Tre A M t = / pt(x,x)dx + ^ F

/ pt(x,'yx)dx F

-i*'

where e is unit of T, can be expressed in terms of lengths of closed geodesies in M i.e. the distances between x and 7 a;, 7 G T21. When looking for a p-adic analog of this formula we first observe that we do not have a diffusion but rather a family of jump processes. Thus instead of the Laplace-Beltrami operator on C+ we have a class of generators for the random walks on Qp. We already mentioned that G is a group of isometries for Qp. For certain discrete family Y of p-adic translations (r cannot be a group and we assume that 7 G T implies ||7|| p > 1) 2^(0,1) becomes a "fundamental domain" for Qp. Let Pt(x, K(a,pM)) x, a G Qp, M G Z be given by (2.17) i.e. we consider G symmetric processes. We define the transition density pt{x, y), t > 0, and assume it to be bounded on Qp x Qp.(this can be realized by a suitable choice of the sequence a(n),) Then qt(x,y) = pt{x,y) + ^2pt{x,n/ + y) 7er

x,y G K(0,1)

66

is a transition density for the process confined to K(0,1). generator of the process in K(0,1) then T r e - " " ' = ft (0,0) = p t (0,0) + $ > ( 0 , 7 ) •

If — H0 is the

(5.1)

If the process in Qp is denned by a(n) = p~" then the right hand side can be expressed in terms of ||7|| p i.e. the p-adic distances between 0 and 7. The process in K(0,1) can be viewed as obtained from the process on Qp by identification of the points x € A"(0,1) and x + 7, 7 G T. Conversely if — Ho generates a process on K(0,1) such that T r e - " 0 * = ft(0,0) < 00 then there is a family of processes on Qp such that (5.1) holds. References 1. R. Rammal and G. Toulouse, Ultrametricity for Physicists, Rev. Modern Phys. 58, 765-78 (1986) 2. V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, p-adic Numbers in Mathe­ matical Physics, World Scientific, Singapore (1993) 3. A. Khrennikov, p-adic Valued Distributions in Mathematical Physics. Kluver, Dordrecht (1994) 4. L. Brekke, M. Olson, p-adic Diffusion and Relaxation in Glasses, Preprint EFI Chicago (1989) 5. A. Figa - Talamanca, Diffusion on compact ultvametvic spaces. In: Noncompact Lie Gruops and Some of Their Applications, Ed. E.A. Tanner, R. Wilson, Kluwer, Dordrecht (1994) 6. S. Hussmann, Random walks on p-adic tree. SFB 237 Preprint No. 359 '97 (1997) 7. S. Albeverio, W. Karwowski, Diffusion on p-adic Numbers, pp. 86-99 in •.Gaussian Random Fields, Ed. K. Ito, H. Hida, World Scientific, Singa­ pore (1991) 8. S. Albeverio, W. Karwowski, A Random Walk onp-adics: The Generator and its Spectrum, Stochastic Processes and their application, 53, 1-22 (1994) 9. W. Karwowski, R. Vilela-Mendes, Hierarchical Structures and Asymmet­ ric Processes on p-adics and Adeles, J. Math. Phys. 35, 4637-4650 (1994) 10. S. Albeverio, W. Karwowski, X. Zhao, Asymptotic and Spectral Results for Random Walks on p-adics, Stochastic Processes and their Applica­ tions, 83,39-59 (1999)

67

11. R. Lima, R. Vilela-Mendes, Stochastic Processes for the Turbulent Cas­ cades, Phys. Rev. E 5 3 , 3536-3540 (1996) 12. K. Yasuda, Additive Processes on Local Fields, J. Math. Sci. Univ. Tokyo, 3, 629-654 (1996) 13. S. Albeverio, W. Karwowski, K. Yasuda, Trace Formula for p-adics, (to be published) 14. N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta-Function, Springer, New York, 2nd ed. (1984) 15. M. Schreckenberg, Long Range Diffusion in Ultrametric Spaces, Z. Phys. B60, 483^88 (1985) 16. S.N. Evans, Local Properties of LeVy Processes on a Totally Disconnected Group, J. Theoret. Probab. 2, 209-259 (1989) 17. V.S. Vladimirov, Generalized Functions over the Field of p-adic Numbers, Russian Math. Surveys 43, 19-64 (1988) 18. A.N. Kochubei, Parabolic Equations over the Field of p-adic Numbers, Math. USSR Izviestiya 39, 1263-1280 (1992) 19. M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter, Berlin 1994 20. S. Albeverio, Z. Zhao, On the Relation between different Construction of Random Walks on p-adic, (to be published) 21. D.A. Hejhal, The Selberg trace Formula for PSL(2,R), Vol. 1,2, Lecture Notes in Mathematics 548 resp. 1001, Springer-Verlag (1976 resp. 1983) 22. H. Kaneko, A class of spatially inhomogeneous Dirichlet spaces on the p-adic number field. Preprint (1999)

68 CHARACTERIZATION OF TEST F U N C T I O N S IN CKS-SPACE NOBUHIRO ASM* Graduate School of Mathematics Nagoya University Nagoya 464-8602 JAPAN IZUMI KUBO Department of Mathematics Faculty of Science Hiroshima University Higashi-Hiroshima 739-8526 JAPAN HUI-HSIUNG KUO t Department of Mathematics Louisiana State University Baton Rouge LA 70803 USA We prove a characterization theorem for the test functions in a CKS-space. Some crucial ideas concerning the growth condition are given.

1

Introduction

Let £ be a real countably-Hilbert space with topology given by a sequence of norms {| • | } £ 1 0 (see n . ) Let £p be the completion of £ with respect t o t h e norm | • | p . Assume the following conditions: (a) There exists a constant 0 < p < 1 such t h a t | • | 0 < p\ ■ |i < • • • < pp\ ■ \p <

(b) For any p > 0, there exists q > p such t h a t the inclusion m a p iq>p : £Q —►
69

By using the Riesz representation theorem to identify £0 with its dual space we get the continuous inclusion maps: £ C £p C So C S'v C £',

p > 0,

where £' and S' are the dual spaces of Sv and S, respectively. Condition (b) says that S is a nuclear space and so S C So C S' is a Gel'fand triple. Let fi be the standard Gaussian measure on S'. Let (L2) denote the Hilbert space of /^-square integrable functions on S'. By the Wiener-Ito theorem each tp in (L2) can be uniquely expressed as oo

oo

V = £/n(/„) = l > ® n : , / n ^ , n=0

fn£Sfn,

(1)

n=0

and the (L 2 )-norm \\
2

\ i/a

IMIo=[$>!|/n| J ■ Now, we describe the spaces of test and generalized functions on the space £' introduced by Cochran et al. in a recent paper 4 . Let {a(n)}£L 0 D e a sequence of numbers satisfying the follow ing conditions: (Al) a(0) = 1 and inf„> 0 a(n) > 0. (A2) l i m n _ H X ) ( 2 i j i ) 1 / B = 0. Let

\ V2

IMIp,a= (Y,nla(n)\fn\2p) \n=0

■

(2)

/

Let [Sp]a — {

0}. Its dual space [S]*a is the space of generalized functions. By identifying (L2) with its dual we get the following continuous inclusion maps:

[£]a c [Sp]a c (L 2 ) c [SPYa c [S]*a, p > o. If $ G [Sp]*a (the dual space of [£p]a) is represented by $ = E^=o( : ' <8n •'»F" ~>> then its [£p]*a-norm is given by / oo

II*H-»I/«=

,

\ V2

a

E^-i -P

•

^

70

This Gel'fand triple [£]a C (L 2 ) C [£\*a is called the CKS-space associated with a sequence {a(n)}£L 0 of numbers satisfying the above conditions (Al) and (A2). Several characterization theorems for generalized functions in [£]^ have been proved in the paper 4 . However, no characterization theorem for test functions in [£]a is given. The purpose of the present paper is to prove such a theorem. In addition we will mention some crucial ideas in order to get a complete description of the characterization theorems for test and general­ ized functions in our ongoing research collaboration project. We remark that similar results have been obtained by Gannoun et al. 5 . 2

Characterization theorems

For £ e £c (the complexification of £,) the renormalized exponential function :e(''t~*: is defined by oo

:e<,

^

:=

£^<:-®n:^n^-

(4)

n=0

For any p > 0, we have l|:c<-' I - > :|| P .a = G a ( | ^ ) 1 / 2 ,

(5)

where Ga is the exponential generating function of the sequence {a(n)}^_ 0 , i.e., Ga(z) = £

^ * » ,

Z € C.

n=0

By condition (A2) of the sequence {a(n)}£L 0 , the function Ga is an entire function. Equation (5) implies that :e^' ,a:_+ : is a test function in [£]a for any

Se£c. The 5-transform of a generalized function $ in [£]£ is the function S $ defined on £c by S*(fl = «*,:<:<■•*-:»,

£e£e,

where ((•, •)) is the bilinear pairing of [£]* and [£]aWe state three conditions on the sequence {a(n)}^L 0 as follows: (Bl) l i m s u p ^ ( ^ i n f r > 0 ^ - f

" < oo.

71 (B2) The sequence 7(71) = ^ffi,n

> 0, is log-concave, i.e.,

7(n)7(n + 2) < 7 ( n + l ) 2 ,

Vn > 0.

(B3) The sequence {a(n)}£L 0 is log-convex, i.e., a(n)a{n + 2) > a ( n + l ) 2 ,

Vn > 0.

Condition (Bl) is used in the characterization theorem for generalized functions. By Corollary 4.4 in 4 condition (B2) implies condition (Bl). Con­ dition (B3) implies condition (B2) to be defined below. The following characterization theorem for generalized functions in [£]£, has been proved in 4 . T h e o r e m 2.1 Suppose the sequence {a(n)}^L0 satisfies conditions (Al) and (A2). (I) Let $ G [€]*a. Then the S-transform F = S$ of $ satisfies the conditions: (1) For any £, n in £c, the function F(z£ + n) is an entire function of z € C. (2) There exist constants K,a,p>

0 such that

\F(0\
£££,

(6)

(II) Conversely, assume that condition (Bl) holds and let F be a function on Sc satisfying the above conditions (1) and (2). Then there exists a unique $ G [£]* such that F = S$. We mention that condition (2) is actually equivalent to the condition: there exist constants K, p > 0 such that \F{0\
S€£C.

This fact can be easily checked by using the fact that |£| p < pQ~p\^\q for any q > p. Having the constant a in condition (2) is for convenience to check whether a given function F satisfies the condition. By this theorem, if we assume condition (Bl), then conditions (1) and (2) are necessary and sufficient for a function F defined on £e to be the Stransform of a generalized function in [S\*a. As mentioned above, condition (B2) implies condition (Bl). Thus under condition (B2), conditions (1) and (2) are also necessary and sufficient for F to be the 5-transform of a general­ ized function in [£]*.

72

For the characterization theorem for test functions in [£]Q, we need to define the exponential generating function of the sequence { j^y}^=o : OO

j

Moreover, we need the corresponding conditions (A2) (Bl) (B2) for the sequence(^}^o: (A2) l i m „ - + o c ( ^ l ) 1 / r i = 0. (Bl) l i m s u p ^ ^ (n!a(n) inf r>0 % ^ ) '

/ n

< oo.

(B2) The sequence {^7^y}n^=o ' s log-concave. It follows from condition (A2) that the exponential generating function G\/a is an entire function. Note that by condition (Al), a(n) > c*o for all n, where Q 0 = inf„>o a(n). Hence by the Stirling formula l/n

n\a{n) J

/

<

t

-:— \n\aoJ

\ l/n

/

~

j

\a0

\ '/"

; \j2-nn)

e

n

> 0,

as n -> oo.

This shows that condition (Al) implies condition (A2). On the other we see t n a t hand, by applying Corollary 4.4 in 4 to the sequence {^y}^Lo 1)J condition (B2) implies condition (Bl). Moreover, it is easy to check that condition (B3) implies condition (B2). Now, we study the characterization theorem for test functions in [£]aFirst we prove a lemma. L e m m a 2.2 Assume that condition (Bl) holds and let F be a function on £c satisfying the conditions: (1) For any £, n in £c, the function F(z£ + 77) is an entire function of z G C. (2) There exist constants K,a,p>0

such that

\F(0\
(e£c.

Then there exists a unique ip € [£]l, such that F = S
^— )

< 1

(8)

73

and

ML < £ £ (»'*(») W % ^ ) (-2Hv„ll^)nn=0 ^

0)

'

Proof. We can modify the proof of Theorem 8.9 in n . Use the analyticity in condition (1) to get the expansion oo

n=0

Then use the Cauchy formula and condition (2) to show that for any £ i , . . . , £n in [Sp]a and any R > 0: l(/n,6®"-®€n-^|<^

^

—

Kl|-p"-|£n|-p.

Make a change of variables r = an2R2 and use the inequality n n < n!e n /v / 27r (see page 357 in n ) and then take the infimum over r > 0 to get

K/»,{,S-SC- - I2 < £«V« ( i n f % ^ ) l&lV-KnlV Now, suppose
2?ra

e

^nt o

r„

J||tp,,||HS.

Therefore oo

|M|2,Q = ^n!a(n)|/„| 2 n=0

n=0 ^

'

Note that the series co nverges because of the condition in Equation (8). ■ Theorem 2.3 Suppose the sequence {a(n)}£L 0 satisfies conditions (Al). (I) Let if E [£}a- Then the S-transform F = Sip of

0 there exists a constant K > 0 such that \F(Z)\
Z€£e.

(10)

74

(II) Conversely, assume that condition (Bl) holds and let F be a function on £c satisfying the above conditions (1) and (2). Then there exists a unique ¥ G [£]a such that F = Sip. We remark that condition (2) is actually equivalent to the condition: for any constant p > 0 there exists a constant K > 0 such that \F(0\
(e£c.

This fact can be easily checked by using the fact that |£|_, < p ? _ p | £ j _ p for a^y Q > P- Having the constant a in condition (2) is for convenience to check whether a given function F satisfies the condition. Proof. Let

= ((:e^:,v)),

£ € £C.

Since

0. Hence

|F(0|
Hence

|F(OI 0, choose q > p such that pq~v < y/a. Then \S\-q < P<-P\Z\-P

<

^\S\-P-

Therefore, we obtain

\F(t)\ < IMI,,aG1/a(|f|2_,)1/2 < IMI,.aG1/a(a|f|2_p)1/2. To prove the converse, assume that condition (Bl) holds and let F be a function on £c satisfying conditions (1) and (2). Let q > 0 be any given number. Choose a, p > 0 such that

(

C (r) \ '^n nla{n) inf — — — 1 < 1. r>0

T

)

This inequality can be achieved in two ways: (1) choose any a > 0 and then use the fact that limp-joo ||i P l ,||//s = 0, (2) choose any p > 0 such that i Pi9 is a Hilbert-Schmidt operator and then choose a sufficiently small number a > 0. (For the first way we can choose a — I and this is exactly the fact mentioned in the above remark.) With the chosen a and p, use condition (2) to get a constant K such that the inequality in Equation (10) holds. Then we apply

75

Lemma 2.2 to get the inequality in Equation (9) so that ||y|| 9 ,a < oo. Hence

0. Thus y? 6 [£]a and the converse of the theorem is proved. ■ 3 3.1

Examples and comments Four conditions

In the definition of CKS-space and the characterization theorems of general­ ized and test functions we have assumed several conditions on the sequence {<*(n)}£°=0: (Al), (A2), (Bl), (B2), (B3), (A2), (Bl), (B2). Recall that we have the f ollowing implications: (A1)=*(A2),

(B2)=»(B1),

(B2)=>(B1),

(B3) = » (B2).

Taking these implications into account we will consider below the four con­ ditions: (Al), (A2), (B2), (B3). 3.2

Examples

We give three examples corresponding to the Hida-Xubo-Takenaka, Kondratiev-Streit, and CKS-spaces. Example 3.1 For the Hida-Kubo-Takenaka space {£) C (L 2 ) C {£)* (see 6 9 10 l3 ,) the sequence is given by a(n) = 1. Obviously, this sequence satisfies the above four conditions. The corresponding exponential generating functions are Gtt(r)=er,

G1/Q(r)=eT.

Thus the growth conditions in Equations (6) and (10) can be stated as

|F(0I < Kea^l,

\F(0\ < Kea^'-'.

Theorems 2.1 and 2.3 are due to Potthoff-Streit 14 and (uc-Potthoff-Streit , respectively. Example 3.2 For the Kondratiev-Streit space (£)& C (L 2 ) C (£)*p (see 7 8 n ,) the sequence is given by a(n) = (n!)^. It is easy to check that this sequence satisfies the above four conditions. The corresponding exponential generating functions are 12

oo

n=0

1

v

';

oo

n=0

.. v

';

76

But from page 358 in u and Lemma 7.1 (page 61) in 4 we have the inequalities: exp (1 - / 3 ) r ^ |
< 2 /3 exp [(1 - / 3 ) 2 ^ 3 r ^ l .

(11)

On the other hand, from page 358 in n and the same argument as in the proof of Lemma 7.1 in4 we can derive the inequalities: 2-0exp

(l + 0)2~&r^]


< exp [(1 + / 3 ) r ^ l .

(12)

The inequalities in Equations (11) and (12) imply that the growth conditions in Theorems 2.1 and 2.3 are respectively equivalent to the conditions: • There exist constants K,a,p>0

such that

|F(0|
Z€£e.

• For any constants a, p > 0 there exists a constant K > 0 such that \F(0\
t e Sc.

These two inequalities are the growth conditions used by Kondratiev and Streit in7 8 (see also Theorems 8.2 and 8.10 in11.) Example 3.3 (Bell's numbers) For each integer k > 2, consider the k-th iterated exponential function exp t (z) = exp(exp(- • • (exp(z)))). This function is entire and so it has the power series expansion expk(z) =

^

Bk(n) nn ^—-^-z .

n=0

The k-th order Bell's numbers {bk(n)}%L0 are defined by

Obviously, this sequence {bk{n)}%L0 satisfies conditions (Al) and (A2). It has been shown recently in J that this sequence satisfies conditions (B2) and (B3). The corresponding exponential generating functions are given by G,,Ar) = Z^ob^r»

= 2%$,

Gift(--)=E:=os4yrn-

(13)

(14)

The function expk(r) in Equation (13) can be used to give the growth condition in Theorem 2.1 for generalized functions, i.e.,

77

• There exist constants K,a,p>

0 such that

|F(OI
Z6£c

On the other hand, we cannot express the sum of the series in Equation (14) as an elementary function. Thus the growth condition (2) in Theorem 2.3 is very hard, if not impossible, to verify. Hence it is desirable to find similar inequalities as those in Equation (12) for the sequence {bk(n)}^=0. 3.3

General growth order

Being motivated by Examples 3.2 and 3.3, we consider the question: What are the possible functions U and u such that the growth conditions in Theorems 2.1 and 2.3 can be respectively stated as follows? • There exist constants K, a,p > 0 such that \F(Q\
Z ££c.

• For any constants a, p > 0 there exists a constant K > 0 such that

\F(0\
£ e£c.

The answer to this question will be given in our forthcoming papers 2 . In particular, when a(n) = bk{n), condition (2) in Theorem 2.3 can be replaced by the following growth condition: 3

• For any constants a, p > 0 there exists a constant K > 0 such that \F(0\ < Kexp [2 J a | e

p

log*-i (a^| 2 _ p ) 1 ,

£ G £e,

where logj, j > 1, is the function denned by l o g ^ i ) = log(max{x,e}),

log^i) = logxOog^^x)),

j > 2.

References 1. Asai, N., Kubo, I., and Kuo, H.-H.: Bell numbers, log-concavity, and logconvexity; in: Classical and Quantum White Noise (A special volume in honor of Hida's 70th birthday,) L. Accardi et al. (eds.) Kluwer Academic Publishers (1999) 2. Asai, N., Kubo, I., and Kuo, H.-H.: Log-concavity and growth order in white noise analysis; Preprint (1999)

78

3. Asai, N., Kubo, I., and Kuo, H.-H.: General characterization theorems and intrinsic topologies in white noise analysis; Preprint (1999) 4. Cochran, W. G., Kuo, H.-H., and Sengupta, A.: A new class of white noise generalized functions; Infinite Dimensional Analysis, Quantum Probability and Related Topics 1 (1998) 43-67 5. Gannoun, R., Hachaichi, R., Ouerdiane, H., and Rezgui, A.: Un Thoreme de Dualite Entre Espaces de Fonctions Holomorphes a Croissance Exponentiele; BiBoS preprint, no. 829 (1998) 6. Hida, T., Kuo, H.-H., Potthoff, J., and Streit, L.: White Noise: An Infinite Dimensional Calculus. Kluwer Academic Publishers, 1993 7. Kondratiev, Yu. G. and Streit, L.: A remark about a norm estimate for white noise distributions; Ukrainian Math. J. 44 (1992) 832-835 8. Kondratiev, Yu. G. and Streit, L.: Spaces of white noise distributions: Constructions, Descriptions, Applications. I; Reports on Math. Phys. 33 (1993) 341-366 9. Kubo, I. and Takenaka, S.: Calculus on Gaussian white noise I; Proc. Japan Acad. 56A (1980) 376-380 10. Kubo, I. and Takenaka, S.: Calculus on Gaussian white noise II; Proc. Japan Acad. 56A (1980) 411-416 11. Kuo, H.-H.: White Noise Distribution Theory. CRC Press, 1996 12. Kuo, H.-H., Potthoff, J., and Streit, L.: A characterization of white noise test functional; Nagoya Math. J. 121 (1991) 185-194 13. Obata, N.: White Noise Calculus and Fock Space. Lecture Notes in Math. 1577, Springer-Verlag, 1994 14. Potthoff, J. and Streit, L.: A characterization of Hida distributions; J. Fund. Anal. 101 (1991) 212-229

79

N O N L I N E A R LIE ALGEBRAS IN Q U A N T U M PHYSICS A N D THEIR INTEREST IN Q U A N T U M FIELD THEORY J. BECKERS' Theoretical and Mathematical Physics, Institute of Physics (B5) University of Liege, B-4000 Liege 1 (Belgium) Solutions of the soliton type and their stability conditions obviously appear through polynomial interactions in toy models enlightening relativistic scalar field theories. We show that differential realizations of operators characterizing non­ linear Lie algebras connected with deformations of the angular momentum algebra are useful in the study of such polynomial interactions in quantum field theory.

A Ludwig, en reference a notre premiere rencontre lors du "Workshop" de I'Universite de Bujumbura (Burundi, octobre 1987) ! Onze ans, deja ! The use and interest of spectrum generating algebras [1] in quantum problems are very well known : in particular, they lead to solvable (or quasisolvable) Schrodinger equations and facilitates the determination of the eigen­ values of the spectrum and their associated eigenfunctions. The method has been often exploited, when, in general it is subtended by linear Lie alge­ bras such as, for example, the real forms sl(2,U) and su(2,C) of the simple Lie algebra A\ in the Cartan classification [2]. There are no reasons to ex­ clude nonlinear Lie algebras which have been studied very recently [3,4]. In fact, here we want to exploit the irreducible representations of such nonlin­ ear sl(2,3R)-Lie algebras in order to add further information in quantum field theory in the particularly well studied context of the scalar case. As a typical work in this scalar field theory, let us point out the remarkable Christ and Lee reference [5] leading to solitonic solutions but without any reference to a spectrum generating algebra and let us see how a (non-linear) cubic (Lie) algebra - the so-called Higgs algebra [6] - can help us to get more general solutions. In terms of the scalar field <j>(x,t) and its derivatives t and x in a ♦E-MAIL : [email protected]

80

(l+l)-dimensional space, the Lagrangian density takes the form C{4>,ux) = \{\-ct>l)-V{4>)

(l)

where V{<j>) refers to the interaction potential characterizing specific toy mod­ els very often called <j>4 or ^-models more specifically. Through the principle of least action in the stationary context and by varying (x), it is well known that it appears a stability condition which, after a simple change of variables of the type

z = f{x)

(2)

(when we restrict ourselves to scalar potentials seen as even functions of (j>), simply becomes d2

d

F(z)=u,2F(z)

.

(3)

Here Pk(z) obviously refers to polynomials of degree k in the variable z. Such an equation (3) is thus equivalent to a Schrodinger equation associated with a specific scalar field theory. Its general solutions are welcome through con­ siderations on nonlinear Lie algebras as we propose to show in the following. Let us first consider the linear sl{2, 8t)-algebra given by the commutation relations [J0,J±) = ±J±

,

(4)

[J+,J-]=2J0

,

(5)

where Jo is the diagonal operator and J± the scaling ones as is well known and, secondly, the nonlinear Higgs algebra [6] given by the commutation relations (4) but with eq. (5) replaced by [J+,J-] = 2Jo + 80Jg

(6)

where (3 refers to a possible family of such nonlinear algebras and can also be interpreted as a deformation parameter. This algebra (4), (6) is recognized as a specific W-algebra [7] and belongs to the so-called category of polynomial deformations already studied [3,8] with respect, in particular, to the determi­ nation of its irreducible representations through an angular momentum basis [3] or a monomial one [8], the latter one asking obviously for a realization of the operators (Jo,J±) as differential operators.

81

In terms of a single variable x and its powers N — 1,2,3,..., we have proposed [8] differential operators on the forms N

J + = x F(D)

dN , J_ = G{D) _

1 , J 0 = _ (D + c)

where D is the dilation operator x — with the properties ax dN N N [D,x ] =Nx , = » & ■ dxN

(7)

<8»

when we take account of the usual Heisenberg commutation relations. If we ask for Schrodingerlike Hamiltonian operators expressed in terms of generators belonging to (nonlinear) spectrum generating algebras and if we want to priviledge the category of Higgs algebras, we have noticed [8] that we have two families of realizations for N = 1 and 2 only respectively given in terms of real parameters (c\, c2, ..., C5) and (di, d2, ^3, dt) by J+ = x(ciD2 + c2D + cs) , J-=c4D

+ c5 , J0 = D + c

(9)

and 2

2

J+ = x (diD

+ diD + d3) , J-=d4

d2 1 -^s , J0 = -(D

+ c) .

(10)

Both families lead to a commutation relation of the type

[j+,j-] = f(i,j0,JZ,JZ)

,

(ii)

including obviously the Higgs character (6) and leading to Schrodingerlike Hamiltonians given by H = ai J+ + a2J- + o 3 J 0 + a 4 Jo + a 5

(12)

where a\, a2, ..., 05 are parameters chosen in such a way that this if-operator is selfadjoint as required. In particular, the above realization (9) then leads to a corresponding Hamiltonian

H^fW^

+^Ws

+ ^W

<13)

where the superscripts refer to the jV = 1-case and where P3(1)(x) = AlX3 + Bxx2 + Cix , P2(1)(x) = DlX2 + Eix + 1, Pll\x)

= Fix + G{U)

Here is the evident connection with the stability condition (3) and its Hamil­ tonian intimately connected with our information (13) and (14).

82

We thus conclude that nonlinear deformations of the angular momentum algebra sl(2,3?) have to deal with polynomial interactions of special interest in scalar field theories, these (nonlinear) algebras playing the role of (nonlinear) spectrum generating algebras for such developments. Let us end this discussion by considering the example of a 0 6 -model lead­ ing to a Hamiltonian in the N = 2-case : BW=Pi2\z)~i

+ Pi2)(z)±+Fi%)

(15)

with P4(z) = 4z(z - A){z - B)(z - C) ,

(16)

P3(z) = 2 [4z3 - 3z2(A + B + Q + 2z(AB + AC + BC) - ABC] (17) and P2(z) = -15z2 + 6z(A + B + Q + AB + AC + BC

(18)

6

so that the equation (3) is completely determined in this 0 -model. We thus get families of solutions of the corresponding Heun equation [9] and it is easy to show that we obtain in that way more general (solitonic) solutions in comparison with the old results coming from the Christ and Lee developments [5]. In particular, the solution F(z) = (A- zf2

(, - * ± £ ) , J2 = 2(S - Cf ,

(19)

exists not only for A = B = 1 and C = —\ [5] but also if A = 2(B + C) in general, etc.

References 1. "Dynamical Groups and Spectrum generating Algebras", A. Barut, A. Bohm and Y. Ne'eman, Eds., World Scientific (Singapore, 1989) 2. J.F. Cornwell, "Group Theory in Physics, an Introduction", Academic Press, London (1997) 3. B. Abdesselam, J. Beckers, A. Chakrabarti and N. Debergh, J. Phys. A 29 (1996) 3075 and references therein 4. N. Debergh, J. Phys. A30 (1997) 5239 5. N.H. Christ and T.D. Lee, Phys. Rev. D 1 2 (1975) 1606 6. P.W. Higgs, J. Phys. A12 (1979) 309

83

7. F. Barbarin, E. Ragoucy and P. Sorba, Nucl. Phys. B442 (1995) 425 8. J. Beckers, Y. Brihaye and N. Debergh, "On realizations of nonlinear Lie algebras by differential operators" (University of Liege preprint, August 1998) 9. A. Erdelyi (Editor), "Higher Transcendental Functions", McGraw Hill, New York, 1955).

84

C O N T I N U O U S PERCOLATION: T R E E A P P R O X I M A T I O N A N D MOST P R O B A B L E C L U S T E R S

PH. BLANCHARD Fakultat fur physik, Theoretische Physik and BiBos, Universitat Bielefeld, Universitatsstrasse, 25, D-33615, Bielefeld, GERMANY. E-mail: blanchard@physik. uni-bielefeld. de G. F. DELL'ANTONIO Dipartimento di Matematica, Universita di Roma, "La Sapienza", Piazzale Aldo Moro 5, 00185, Roma, Italy. E-mail: [email protected] D. GANDOLFO Centre de Physique Theorique, CNRS, Luminy, F-13288 Marseille Cedex 9, and PhyMat, DSpartement de Mathimatiques, UTV, F-83957 La Garde, France. E-mail: [email protected] M. SIRUGUE-COLLIN Centre de Physique Theorique, CNRS, Luminy, F-13288 Marseille Cedex 9, and UFR Sciences de la matiere, Universite de Provence, 3 Place Victor Hugo, 13001 Marseille, France. E-mail: [email protected] A new approach based on tree approximation is derived in order to estimate the critical parameters of continuous percolation of overlapping disks in R 2 when the centres of the disks are Poisson distributed. In the case of directed percolation, a lower bound is found for the critical density. The cluster structure near the percolation threshold is analyzed.

1

Introduction

Percolation theory on regular lattices has already a long standing and success­ ful history. On the contrary percolation in continuous systems and on random graphs has received less attention in spite of its great interest in applications. Indeed continuous percolation has been used to model such diverse phenomena as conductivity in disordered semiconductors, permeability of porous media, fracture network of rocks, spread of Aids epidemics ! and even quark-gluon plasma formation in QCD gauge theories 2 . We would like to stress that our approach here will refer to continuous systems without any introduction of lattice approximation. Let us recall the

85

main features of continuous percolation, i.e. modeling a random medium — in fact randomly distributed objects in homogeneous media — by using the features of stochastic geometry. The basic facts are quite elementary: a great number N of "small" objects are thrown at random in a large volume V; their distribution follows a Poisson law if the mean density p = y is sufficiently low. So let {xt}ie/cN be the points of a standard Poisson process with intensity p in R d , d > 2. We choose a geometrical object P with a distinguished point xo and a given orientation and for each i £ I we place a copy of P with xo s n. We could also consider objects of random shape and size but in the following we will only be concerned with the simplest case. Denoting Pi the object centered in Xi we are interested in the description of clusters of overlapping objects and particularly in the possible existence of an infinite cluster "spanning" the space domain. Let us declare two objects Pi and Pj adjacent if Pi f] Pj ^ 0. We say that two objects (or the two corresponding points Xi and Xj) are connected, Pi <—> Pj (or Xi <—> Xj), if there exists a chain of adjacent objects (or points) connecting these two objects (or these two points). A cluster is a maximal set {Pi}iGJciN (or {xi}i£j) of connected objects (or points). The aim of continuous percolation theory is to study the sizes and shapes of clusters for specified values of p and typical objects P. As in the case of lattice percolation in a infinite domain, the existence of an "infinite incipient" cluster is linked to the long range connectivity property of the random system. The study of the phase transition associated with the appearance of an infinite cluster is one of the main problem of percolation theory. Objects as diverse as disks, squares, needles in R 2 , spheres, cubes, ellipsoids, in R 3 have been used for typical models (see e.g. 5 ' 1 3 ).

Figure 1.

86

For general results on continuous percolation see 3 and also 4 ' 6 ' 7 . For recent results concerning the scaling properties of the infinite percolating cluster see n 12 ' . Percolation theory in continuous media is naturally related to classical problems of stochastic geometry, see e.g. 8 . Let us now state the framework in which we want to develop our approach. It follows from the scaling property of the Poisson process that the statistical properties of the system are invariant under the scaling transformations: p —» Xp,

v —► Xv,

VA €J0, oo[

where v is the volume of the elementary objects. Due to this property, the simplest invariant, dimensionless parameter we can use in order to describe the behaviour of a percolation model is the socalled filling factor 7 defined by 77 = pv. We can better understand the physical meaning of the parameter 77 by considering N objects of volume v thrown at random in a domain of volume V. In the limit of low density, the fraction of space not covered by these

\N

(V -

objects is lim ( ——— I . In the limit V —> 00, JV —> 00, TT = po < 00, we V—too \

V

J

have

(—y^-J - exP (~vV)

= exp -7?

( )-

Then the fraction of space (p occupied by the randomly distributed objects is clearly given by

00, there exists a critical value TJC (percolation thresh­ old) for which the overlapping disks form an infinite cluster "spanning the entire domain of volume V". The dimensionless factor T)c has been obtained by numerical simulations 13 ' 14 ?7C w 1.18


Notice that (pc is in some rough sense an indicator of how the "incipient spanning cluster" is branched, at least if one makes the assumption that there is only one such cluster. In order to find the main characteristics of the cluster distribution one defines the following quantities for 77 near T7C • the probability Px for a disk to belong to an infinite cluster,

87

• the correlation length £, maximum size above which the clusters are ex­ ponentially scarce, • the average cluster size Sci (mean size of the clusters containing one given disk ), Currently accepted wisdom tell us that there should exist a set of critical exponents describing the behaviour of these quantities near the percolation threshold, i.e. there should exist positive constants /3,7, v such that the fol­ lowing asymptotic behaviour holds: P 0 0 (T ? )

« (i - - )

>

v>Vc

Z(v) <* \v-Vc \~" Sci{rj) oc I r\-i]c I" 7 In R 2 for different types of objects (squares, sticks, . . . ) the percolation threshold r?c changes but the critical exponents are the same as those found for lattice percolation in dimension d = 2, i.e.

fl-A

-t

-H

^~36' V 3' 7 18" The same description can be used for spheres percolation in R 3 , in this case one has T7(d==3) = p^irR3. The corresponding numerical values are in this case: 7]c w 0.35 (
p w 0.41,

v ss 0.90,

7 w 1.8.

In R 3 , as in R 2 , lattice percolation and continuous percolation belong to the same universality class of critical phenomena: the class of random (uncorrelated) percolation. Let us just remark that in R the continuous percolation problem is trivial, clearly we have

88

• hopping conduction in doped semi-conductors, • localization of electrons in random potential generated by disordered sys­ tems, • mathematical biology, epidemiology, • deconfinement of strongly interacting matter at high density (transition to a plasma of deconfined, coloured quarks and gluons). In spite of its theoretical and practical importance, continuous percolation is essentially an open field for mathematical physics. Some exact general results have already been obtained (see e.g. 11>12) but almost all what is known in practice concerning continuous percolation has been obtained from numerical simulations (mainly from Monte-Carlo simulations or direct connectedness expansion) or by using heuristic arguments 4 . However no realistic model has been constructed which allows to have even a tentative answer to the problems which are of practical importance: • size and shape of typical clusters, • percolation threshold and the large scale phenomena such as the existence of an infinite cluster, • behaviour of the percolation process near the threshold: power laws, critical exponents, • generalization to others types of percolating objects (sticks, ellipsoids, squares, cubes, . . . ) , • problems tied to computer simulation e.g. the determination of the proper window in which to perform computations, • connection with scaling theory, renormalization group, universality, frac­ tal properties, • study of the properties of spanning probabilities on compact regions

\s,s}d. (see however 11,12 for the last two points). In fact, as we shall give examples in the following, we can think of dif­ ferent basic techniques in order to build more realistic models. Besides the general mathematical framework which is related to the general theory of "coverage processes" 10 or the relation between continuous percolation mod­ els and bond percolation on Vorono'i lattices, some techniques which appear relatively "natural" are:

89 • put in evidence geometrical and probabilistic constraints which give an insight on the behaviour of the system. • use the connection with the theory of branching processes by constructing a tree approximation (via, e.g. the method of generations). • use the results and experience of bond and site percolation on the lattice Zd, i.e. establish a relation between the continuous percolation problem on Rrf with an ordinary percolation process on Zd (long range percola­ tion). • use methods of statistical mechanics and relate phase transition in con­ tinuous Potts model with the existence of an infinite cluster in a corre­ sponding continuous percolation process . . . Concerning the last point, we would like to state some facts and results dealing with the random cluster representation of a Potts spin system when the spin positions are Poisson distributed in R d . This problem was initialy introduced by Klein 15 as an attempt to generalize Fortuin and Kasteleyn transformation 16 for Potts model in the case of systems in continuous spaces. Closely related are the studies of the Widom-Rowlinson model 17>18, (see also 19 ). Random cluster measures are natural generalizations of percolation, Ising and Potts models. Let Go, = {Vw,Eu) be the finite graph associated to some realization w of a Poisson distribution of density p in a finite domain A C R d . With E w = { 1 , 2 , . . . , q}v" and Q, = {0,1} E " we consider the probability measure TPu,(
K 1 -P( e ))<Me),o + P( e )<W),i *»(«)] , f f £ ^ , w ' e £2'

where p(e) = 1 — exp [—/37(e)] > 0 with inverse temperature j3 and it \ in i\ / 1 if | Xj - X j |< 2R J(e) = J ( | x i - x j | ) = | ( ) o t h e ; w i g e and <Ja(e) = Saiaj,We = (xi,Xj) = (i,j) € E. The first marginal of Pw(<7,w) leads to the distribution of the continuous Potts model nA(
-x-exp

-/3

£

J(e)(l-6aiai)

which reduces, when q = 2, to the Ising model with spins positioned on R rf , where (i, j) £ E means any edge for which \xt —Xj |< 2R.

90 The second marginal is the random cluster measure $ w

( ' ) = F " n [p(e)w'(e) <j -p( e )) i _ w ' ( e ) ] 9fc(a,,(e)) A

k u

e€E„

e

where q ( ^ " is the number of open components of u' (connected clusters in the sense that two points are connected iff [ x, — Xj \< 1R). For q = 1 we recover the continuous percolation model. This generalization of the continuous percolation model allows a geometric representation of the Potts model. The interested reader is encouraged to consult 20 . Our purpose here is to develop a model for continuous percolation based on tree approximation. 2 2.1

Model of generations Presentation

Consider the simplest model of percolating spheres (radius R, volume v
Figure 2. Notice that a path is a collection of overlapping spheres but the order in the sequence may not be uniquely defined. The percolation threshold pc can be denned as the infimum over L of the values p{L) of the density such that, for any box of characteristic length L, there exists, with probability one, a

91

spanning cluster, i.e. a cluster which connects two opposite sides of the box. In the limit of large volume, where translational invariance of the Poisson law holds, we can choose any point 0 in the percolation cluster and consider all the paths leaving O. The set of points belonging to such a graph can then be divided in generations Gk, (fc = 1,2,...), with Go = {O} in the following way: xk £ Gk iff {0,Xi,... ,xk,... ,xk+i,...} being on the same path, then necessarily

I *k -xk±i I Xk-Xk±i

|< 2R |> 2i?,Vi>2

Figure 3. This definition of generations is unique if the graph has no loop. Due to the Poisson distribution, for any sphere of volume vj the mean number of neighbours is given by: JE(#neighbours) = pv^l ,

p being the mean density of centres of sphere.

Then, for any k sufficiently large, we can define Fk by: E(#Gfc+1|#Gfc)
PcVd = % =

2.2

Vd — .

Directed percolation

This method of generations turns out to be particularly well suited to study the directed percolation problem of disks in R 2 (characterized by a mean direction vector e).

92 This model is defined by simply adding to the previous conditions (*), the additional constraint: (**) For a path (0,xi that

(xfc+i - x f c ) . e > 0 .

€ G i , x 2 € G 2 , . . . ) (see Fig. 4) the condition (*) implies / xi 6 Ci

\ x2£U(6)

=

C2\(C1nC2).

Figure 4. Moreover, the condition (**) implies x2 £ U($)f]PA = Ur(9) where P A is the half-plane limited by A (xi € A , e orthogonal to A). Then using some easy geometrical arguments and obvious symmetry simplifications, it is easy to calculate the mean value (U) of Ur{6). The same is true for any sequence of 3 successive points. In this way we can identify (U) and F. We obtain F = -;pfl2

and

r)c = 1.21.

93 To our knowledge, there was till now no known result concerning this type of percolating model which can be thought to model, for example, percolation of liquid in porous rocks under the influence of the gravitational field. Numeri­ cal simulations based on a new algorithm for cluster statistics in continuous percolation systems confirm the above result whose details will be developed in a forthcoming paper. We can point out that this result is not too far from the known numerical result for ordinary (undirected) percolation (TJC = 1.18). The case of directed percolation in R 3 can be handled in the same way. The corresponding value for the critical parameter is found to be r\c = 0.47 to be compared with r)c = 0.35 for ordinary continuous percolation. On a quite general basis, one of the interesting problems is to find the structure of the most probable types of clusters near the percolation threshold (p
3

The A:-cluster model

In 4 Alon, Balberg and Drory proposed a new heuristic percolation criterion for continuous systems, based on the comparison of two fundamental lengths of the system, namely the average bounding length Zi (defined as the mean distance between two connected centres) and the average distance lk between two centres each of which have at least k neighbours. They postulated that percolation, which can be regarded as the condition that the system be macroscopically connected, occurs when the condition lk = 2lx is fulfilled for some definite k depending on the space dimension. The mean distance lk of the centres of two fc-clusters is approximated using the effective density pk of such clusters which is given by the basic Poisson law :

■ exp(-B)

Pk

i=i

where

B — p2dVd

J

The results they obtained (k = 4 in dimension 2 and k = 2 in dimension 3), are in very good agreement with the numerical simulations. A better insight concerning the typical clusters near the percolation threshold can be obtained using a probabilistic model. First let us give some results on the cluster shapes.

94 3.1

Most probable cluster shapes d

In R we consider spheres with radius R, thrown at random with uniform density p. Their centres are distributed according to a homogeneous Poisson law so that the probability P n ( ^ ) to have n points in a volume V is IP„(V) = ^ p e x p ( - p V ) Two points xi and X2 are neighbours iff | x\ - x
Figure 5. The volume of the empty region is minimal if the centres of the spheres almost coincide, but this event has very small probabiUty. Nevertheless, for the same volume of overlapping spheres, the empty volume is minimal if the spheres are in a compact shape, so to say in a sphere of radius r 0 . Then there is a balance between the probabilities of the two following events (see Fig. 6): • to have n centres in a sphere of radius ro, • to have no centre in an annulus }ro, TQ + 2R).

Figure 6.

95 The probability to have a cluster of size n and radius r$ surrounded by a corona of depth 2R is (with TT^ the volume of the unit ball in dimension d): P(„, ro) =

KF

J>

exp(-p7rdr0d) exp [-fmd((r0

+ 2R)d - rft]

This probability is maximal for ro = fo such that: f0(f0 + 2R) d-\

P*d

For d = 2, we obtain r~o(n) / n K 2 R ' = - l + A /l + pwR For a given density (and a given filling factor rj) r"o(n) can be taken to be the radius of the most probable clusters with n spheres. 3.2

Structure of percolating clusters

To find out whether some types of clusters (with a fixed number of elements) have more prominence than others at the percolating threshold, we now intro­ duce a probabilistic model of percolating clusters. For this we use as building blocks the most probable clusters introduced above. Let O be the centre of a sphere in R d , O is an occupied point. We consider a spherical cluster i.e. an occupied point 0\ surrounded by k occupied points Ai,i = l,...,k, at the same distance IR from 0\ and uniformly distributed around 0\. Take two such clusters {0\,Ai\02,Bj} such as O1O2 = SR

OtAi = OiB, Oi02 = 5R

IR

6 and / are dimensionless parameters. Figure 7. and let Mk(l,8)R be the minimum of the mean square distance between one point of the set {Oi, A{} and one point of the set {O2, Bj}. We shall stipulate

96 that the two clusters percolate if Mk(l,5)R fixed I, a critical value Sc(k) for <S Mk(l,Sc(k)) The set of curves Mk(l,S/l) and 6c(k) satisfies

is less than 2R. This gives, for =2

can be obtained by numerical simulations

M fc (lA(fc)/i) = 2/J

(1)

The proportion of points with fc-neighbours decreases with k. They have a density Xk and if LkR is the mean distance between them we can write (LkR)dXk

« 1

(2)

Xk is given by the Poisson law, i.e. Xk — P^r exp(—a) and a = pTT(i(2R)d = 2dr\ being the mean number of neighbours of one occupied point. Percolation is supposed to appear first when Sc(k) « Lk.

(3)

We focus first on the case d = 2. Using dimensionless variables, the minimum of the mean squared distance between two elements of two different k-clusters is given by the set of curves fk(x) =

\Mk(l,x)}2

which depend only on one parameter x = f and are obtained numerically. If we choose for I the radius of the most probable cluster with fc + 1 elements, I is a function of k and rj:

h(v) = - i + ^ i +

^ -

Moreover the Poisson law can be put in the form:

Then the relations (1), (2), (3) lead to a system of two coupled equations:

{h{x)=^z

\ hk(v) = ,a[!„(„)]» For each value of fc we can determine the critical value T]c(k) given by the lowest intersecting point of the curves

Fk(v) = fk—I

4

lk(v)2\

97 and

4

F

11/2

"fc(77) = [wrm. The percolation critical value r\c is just given by T]C =mmr)c(k)

(4)

K

The value fco for which TJC minimum is attained in (4) can be regarded as a measure of the size of the building blocks of the percolation cluster. The results we obtain with these prescriptions are summarized below. 3.3

Results

• Dimension d = 2 — the curves do not intersect for fc < 3, are almost tangent for fc = 4, but intersect clearly for k w 5. — t)c(k) is almost stationary around k = 5 and very near the value obtained by numerical simulations TJC = 1.18. Moreover the heuristic hypothesis of Alon, Bradberg and Drory, is in reasonable agreement we the result we have obtained. However the percolation seems not to distinguish between fc-clusters for k between 4 and 6. — For k > 6 T]c(k) increases regularly and behaves roughly as ^(k + 1 ) for k —> oo. • Dimension d = 3 The situation is rather different. The clusters are smaller and there is no solution with one type of cluster, compatible with the known physical values of the percolation problem. This may be seen as evidence that in dimension 3 the percolating cluster is more structured and is composed of several types of "building blocks" placed according to an optimal strategy. In order to understand this structure we have to consider more collective phenomena, replacing Xk by the probability of clusters of size greater than or equal to k. Then for I w 0.75 and k = 2 we obtain r\c ~ 0.32 to compare with rjc « 0.35 obtained by numerical simulations.

98

Acknowledgments The BiBoS research center (Bielefeld-Bonn-Stochastic) of Bielefeld University, the Centre de Physique Theorique, CNRS, Marseille and the Mathematical Institute Guido Castelnuovo, Universita La Sapienza, Romal are gratefully acknowledged for kind hospitality and financial support. G.F.D.A. acknowl­ edges the generous support of the Alexander Von Humboldt Foundation. References 1. Ph. Blanchard, Modelling BJV/AIDS, in Fractals in Biology and Medicine, T. F. Nonnenmacher, G. A. Losa, E. R. Weibel Eds, Birkhaiiser, (1994). 2. H. Satz, Nucl. Phys. A 642, 130 (1998). 3. I. Balberg, Philos. Mag. B 55, 991 (1987). 4. U. Alon, I. Balberg and A. Drory, Phys. Rev. A 42, 4634 (1990). 5. E. J. Garboczi, K. A. Snyder, J. F. Douglas and M. F. Thorpe, Phys. Rev. E 52, 819 (1995). 6. U. Alon, I. Balberg, B. Berkowitz and A. Drory, Phys. Rev. A 43, 6604 (1991). 7. M. B. Isichenko, Rev. Mod. Phys. 64, 4, 961 (1992). 8. D. Stoyan, W. S. Kendall and J. Mecke, Stochastic Geometry and Appli­ cations, Akademie-Verlag Berlin (1989). 9. V. K. S. Shante and S. Kirkpatrick, Adv. Phys. 20, 325 (1971). 10. P. Hall, Introduction to the Theory of Coverage Processes, Wiley Series in Probability and Mathematical Physics (1988). 11. M. Aizenman, Nucl. Phys. B , 551 (1997). 12. M. Aizenman, A. Burchard, C. M. Newman and D. B. Wilson, to appear in: Random Structures and Algorithms, (1999). 13. G. E. Pike and C. H. Seager, Phys. Rev. Lett. B 10, 1421 (1984). 14. E. T. Gawlinski and H. E. Stanley, J. Phys. A 10, 205 (1977). 15. W. Klein, Phys. Rev. B 26,5, 2677 (1982). 16. P. W. Kasteleyn and C. M. Fortuin, J. Phys. Soc. J. (suppl) 26, 11 (1969). 17. D. Ruelle, Phys. Rev. Lett. 27, 1040 (1971). 18. L. Chayes and R. Kotecky, Comm. Math. Phys. 172, 551 (1995). 19. O. Haggstrom, Mark. Proc. Rel. Fields 4, 275 (1998). 20. G. R. Grimmet, Lectures on probability theory and statistics, Ecole d'ee de Probabilites de Saint Flour, (XXV), Lectures Notes in Mathematics, M. T. Barlow, D. Nualart Eds. Spinger, (1998).

99 R I G G E D HILBERT SPACE R E S O N A N C E S A N D TIME A S Y M M E T R I C Q U A N T U M MECHANICS* A. BOHM AND H. KALDASS University of Texas at Austin Physics Department Austin, TX 78712 E-mail: [email protected] The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is re­ viewed and contrasted with the standard Hilbert space (HS) theory of quantum mechanics. The main difference is in the choice of boundary conditions. Whereas the conventional theory allows for the in-states <j>+ and the out-states (observables) rl>~ of the S-matrix elements (ip~,(p+) = (V>ou',S0,n) any elements of the HS H, {ip~} = {+}(= W), the RHS theory chooses the boundary conditions : + e * - C U C * * , t/>~ € *+ C H C * * , where *_ (*+) are Hardy class spaces associated to the lower (upper) half-plane of the second sheet of the ana­ lytically continued S-matrix. This can be phenomenologically justified by causal­ ity. The two RHS's for states + and observables ip~ provide new vectors which are not in W, e.g. the Dirac-Lippmann-Schwinger kets IB*) 6 # * (solutions of the Lippmann-Schwinger equation with ±ie respectively) and the Gamow vectors \ER - iT/2±) e # £ . The Gamow vectors \ER - ir/2~) have all the properties that one heuristically needs for quasistable states. In addition, they give rise to asymmetric time evolution expressing irreversibility on the microphysical level.

1

Introduction

I am happy to be here to celebrate Ludwig Streit. My relation with Ludwig goes back to 1967 when we overlapped as post-docs at Syracuse University. I still remember that at Syracuse Ludwig gave me his unsolicited advise, that I did not heed, which had severe repercussions. He was already at that time much more clever than I. The other thing that I remember often is that sometime in the 1980s Ludwig worked on Gamow vectors and organized a meeting on resonances 1,2. I think the motivation for the meeting was that nobody really knew what resonance states were. At that time my paper on Gamow kets was out al­ ready 3 and I had written many chapters in a monograph 4 on that subject, but Ludwig did not invite me to that meeting. Then there is Ludwig's interest in the Rigged Hilbert Space 5 . I had been advocating its use in quantum physics for many years, but was not very "Based on a talk delivered at the International Conference on Mathematical Physics and Stochastic Analysis in honor of the 60th birthday of Professor Ludwig Streit.

100

successful. When I found out that Ludwig also was using it, I gained hope again that ultimately the Rigged Hilbert Space will make it into physics. Resonances and Rigged Hilbert Space are the subjects that I want to discuss here. Resonances and decaying states can really not be understood as au­ tonomous elementary particles in Hilbert space quantum mechanics because the Hilbert space mathematics does not allow state vectors characterized by both an energy ER, and a lifetime r (or a width T = ft/7*)- This is in con­ trast to the way experimentalists analyze their data and list their results as Breit-Wigner peak value and width (ER,T) for resonances (large values of Y/ER) and as (Ed,T) for decaying states if the (Breit-Wigner or Lorentzian) line shape cannot be resolved but the decay rate can be fitted to an exponen­ tial (small values of ^ - ) - Since experimental data are finite in number and there are always experimental tolerances and interference with background, one will never be able to say that the experiment has precisely established an ideal Breit-Wigner or an exact exponential. However, Breit-Wigner en­ ergy distribution and exponential time evolution have been observed so many times for such an enormous number of different systems in all areas of physics and chemistry that one can safely take them as the defining signature of a quasistable state and attribute observed experimental deviations to the ever present background and calculated mathematical deviations to an unjustified mathematical idealization. For the mathematical physicists this means that they have to provide the suitable mathematical idealization. How and why to do that is the content of our discussion here. 2

The fundamental calculational tools of quantum mechanics

In quantum theory one has states denoted by p or W , or by <j> for a pure state p = \(j>)(\,

(1)

and one has observables denoted by A(= A1), A, P (P2 =P),

or by V, if P = \^)(tf>\, for properties.

(2)

The vectors , 4>forma linear scalar product space $. The operators p, W and A, A, P are linear operators in <"">. The space $ is usually called Hilbert space, but it is mostly treated like a pre-Hilbert space. For each "kind" of quantum physical system one takes one particular space
101

(or 4>) is thus experimentally denned ("determined") by the preparation ap­ paratus. The quantum physical observables are observed or registered by a registration apparatus, e.g. a detector. The observable A (or ip) or A are thus experimentally defined by the registration apparatus. In experiments with quantum systems one measures ratios of large inte­ gers Ni/N or N(t)/N, e.g. as ratios of detector counts of the z-th detector JVj and counts of all detectors N or as ratios of detector counts N(t) in the time interval between t = 0 and t = t and in "all" time N = N(oo). This ratio of large numbers is interpreted as probability, e.g. as probability V{Pi) for a property Pi jj*l>(Pi)

(3a)

or as probability for the observable A(() at a time t

M«p(A(0)

(3b)

where the observables P* or A are experimentally given ("defined") by the detector. For a more general observable

X>P<

(4)

one obtains the average value (of the eigenvalues a*) finite

J.J

oo

'=1 finite

,=1

»,

(3c)

oo

The symbol « denotes the association of the experimentally measured quan­ tity on the left hand side with the theoretically calculated quantity on the right hand side. In quantum theory the probability of an observable A in the state W at time t is calculated as V(t) = V(\(t))

= Tr(A(*)W0) = Tr{AoW{t)).

(5a)

If the state is pure, given by the state vector <j>, and if the observable is a property given by the one-dimensional projector \tp)(ip\, or given by the "observable" vector V, then

P(0 = KVW0>l2 = KiK0WI2-

(5b)

102

The trace in (5) is calculated using any basis system of the space $; either any discrete basis

#^=53|t)<»|0;

(6a)

i

or any continuous basis (Dirac basis vector expansion) $ 9 0 = fd\\X)(X\4>)

(6b)

or any basis system consisting of discrete and continuous (generalized) eigen­ vectors of a complete system of commuting observables. Thus the trace is given by e.g. Tr(AW) = ^2(i\AW\i)

or Tr(AW) = f dX{X\AW\X).

(7a)

•*

i

For the special case (5b) the probability for the observably ip in the state

2

W)M*WI) = KV#)I

£<#)<#>

(7b)

i=0

or by (7c) if one uses a continuous basis of Dirac kets |A). The time evolution in (5a) and (5b) (dynamics of the quantum system) is given by the Hamiltonian operator H of the system; either as ^

= i[H,W(t))

(8a);

i h ^ - = H<j>{t)

(8b)

(j)[t = 0) = 0o ( Schroedinger picture), or by : ^

= -i[^,A(0]

(8c);

i h

<^

=

-

H m

.

(8d)

(Heisenberg picture) None of the above equations is mathematically precise until we define the space $ , the kets |A) or the integration dX in (6b) and (7c), and specify the initial-boundary conditions
103

3

Empirical reasons for time asymmetric quantum mechanics

The time t in (5) and in the operators A{t) and W(t) is usually allowed to take positive and negative values, i.e. one chooses time symmetric boundary conditions for the Schroedinger and Heisenberg equations of motion (8). This choice is not compelling from an empirical point of view for the following reason. An obvious expression of causality is that a state WQ (or <j>0) has to be prepared first before an observable A(t) or rp(t) can be measured in it. If one calls t = 0 the time at which the preparation of the state W (or <j>) is completed then the time translation in A(t) (or ij)(t)) in (5) makes physically sense only for t > 0. The registration counts N(t) in (3b), by which V(t) is measured, can be taken in the future t > 0, but not in the past. Thus, Ti(A(t)W) will have an experimental counterpart for t > 0 but not for t < 0. The experimental data for V(t) would give that V(t) = Tr(A(t)W)

« ^

for t> 0

(9a)

and V{t) = Tr(A{t)W) « 0 for t < 0

(9b)

because if the detector would click before the state is prepared we would discount this click as noise. Often, such as for stationary states and/or time independent observables, it does not matter at which time V(t) = Tr(A(t)W) = Ti(AW(t)) is calcu­ lated. But one cannot make it a general principle that time evolution of the observable A(t) (or of the state W(t)) must go into both directions. In fact the most natural description of the experimental situation (9) would be to require a time ordering in the probability formula V(t) = Tr(A(0W(0))

t> t0 = 0

(10)

and admit in the mathematical description only time translations of observ­ ables A(t) relative to the state W(to) by an amount t-to > 0. This would not only reflect the experimental situation, which forbids time translation of the registration apparatus relative to the preparation apparatus to a time t < t0. It would also incorporate the notion of causality into the mathematical theory on the operator level. The time translation is quite different from other transformations of the space-time symmetry group, e.g. rotations and space-translations. These sym­ metry transformations of space-time are experimentally realized as transfor­ mations of the registration apparatus (detector) relative to the preparation

104

apparatus (accelerator). Whereas space translations and rotations of the ap­ paratuses can be performed back and forth, the time at which the detector is activated can only be shifted into the future not backward past the time of preparation. Thus time translations of the apparatuses form only a semigroup whereas rotations and translations of the apparatuses form a group. Therefore it is natural to represent space translations and rotations by unitary groups of operators in the space of physical states. But it is unjustified to assume that time translations are also always represented by group operators in the space of states, i.e. to assume that in quantum mechanics only states with reversible time evolution exist. When we make the choices that give a precise mathematical meaning to the vectors <j> and ip and the operators W, A, A, A* in (1) and (2) we should therefore not restrict ourselves to those mathematical assumptions that dictate a unitary group evolution. 4

The Hilbert space idealization has reversible time evolution

Of the vectors <j>, tp we have so far assumed that they fulfill the mathemat­ ical axioms of a linear space with scalar product (\ip) and that W, A, ... are linear operators in this space. In addition, we needed some rules like (6) that permit the calculations in (7); we have not yet said anything about the convergence of infinite sequences and the meaning of integration in (7). In order to prove existence theorems mathematicians need their spaces to be complete (topologically)—they use spaces that also fulfill topological axioms. This gives infinite series, like the ones that occur in (6), (7) and (4) and on the right hand side of (3c), a well defined meaning of convergence. It also defines the meaning of such continuous operator functions of a parameter t as occur on the right hand side of (5) and (3b). These topological properties (the definition of open sets) cannot be directly inferred from experimental data as one can see by comparing the right hand side (calculated theoretical quantities) of (3b) and (3c) with the left hand side of (3b) and (3c) (measured experimental quantities). The left hand side of (3b) changes as a function of t only in integer steps of ^ , whereas the right hand side is presumed to be a continuous function of time. Similarly, the experimental expression on the left hand side of (3c) has a finite sum, in the theoretical expression on the right hand side the sum is infinite. Experiments necessarily involve a finite amount of data. Measured probabilities therefore are only approximate probabilities and cannot supply continuous functions. They cannot determine the meaning of continuous operator functions (like A(i)) or continuous vector functions (like ip(t)) or the meaning of convergence of infinite sequences as in (3c) or

105

in (7a) or other topological notions. The equality w between experimental and mathematical quantities has a meaning only within certain experimental errors and for large numbers N. Therefore, one has to make some—more or less—arbitrary mathematical idealizations in order to obtain complete math­ ematical structures, like linear topological spaces and topological algebras of operators. One such idealization is the Hilbert space (HS). This is obtained by ad­ joining to the linear scalar product space $ the limit element of infinite con­ verging (Cauchy) sequences, with the Hilbert space convergence (or Hilbert space topology TU) defined by vTA iff \\<j>v - (j)\\ -*• 0 f o r v - > oo

(11)

where the norm ||^>|| is defined by the scalar product ||| = \J(<j>,). Thus the infinite sequences of the discrete components {(i|V>)}£^i, { ( i ^ ) } ? ^ of the vectors tp,

(\) | A € Spectrum} are square integrable functions. However, the integrals ( l M ) = y"dA
(12)

are not Riemann but are Lebesgue integrals. This means the values of the functions V'(A) = (A|V>) at a particular point (or at all rational numbers) are not defined, which in turn means that the Dirac kets |A) cannot be defined. The Lebesgue integration also makes the interpretation of the probability density |(.E|V>)|2 as the energy resolution of the detector for the observable \ip){ip\ rather unintuitive, at least for those «/> 6 % that do not have a smooth function in the class of Lebesgue integrable functions {(E\ip)} which represent In the (complete) Hilbert space one can define self-adjointness and give the precise meaning of "spectral resolution" to equations like (4). One can make the hypothesis that observables are (not just hermitian or symmetric but) self-adjoint operators and one can prove existence theorems. One of these existence theorems is the Gleason theorem 7 which states that if V{Pi) is the function on the set of projectors {Pi} which fulfills the axioms of probabilities 1 then there exists a positive trace class operator (density x

For V(Pi) or 'P(A) to be a probability it has to fulfill the axioms of probability theory : V(P)>0,

-P(1) = 1, ([PU Pj] =

V{Pi + Pj) = 0,PiPi=0).

V{Pi)+V{Pj),

106

operator) p in U such that V{Pi) = Tr{Pip) = Vp{Pl). This operator p thus defines the state. Since the converse (for any positive trace class operator p, Vp(Pi) = Tr(Pip) fulfills the axioms of probability theory ') is simple to see, one was led to the conclusion 8 that there is one to one correspondence between quantum physical states £> density operators p

(13a)

and in particular between pure quantum physical states o

vectors 4> (up to a phase) in H

(13b)

Another existence theorem (Stone-von Neumann operator calculus 9 ) as­ serts that the Cauchy problem in quantum mechanics, e.g. in the form of the Schroedinger equation (8b) with the initial condition o € V. and a selfadjoint Hamiltonian H with domain D(H) dense in U, has a solution given by the one parameter group of (strongly continuous) unitary operators W{t) inH:2 4>{t) = U\t)4>0=e-iHt
-oo<<
(14a)

where 4>(t) depends continuously on the initial condition. This means U^t + r) = t/ f (<)£/* ( r ) , ^^-<j>={-iHYU^(t)4>

UH-t)

= (Ul)-1(t)=U(t)

= U\t){-iHY4>

for 4> eD(H).

(15) (16)

The same result holds for the time evolution of the observable vector ip in H xl>{t) = U(t)4>0 = eiHtip0

- co < t < oo.

(14b)

In terms of the operators W and A this is given by W{t) = U\t)W0U{t)

= e-iHtW0eiHt

\{t) = eiHtA0e~iHt

- oo < t < +00

-oo
(17a) (17b)

2 T h e operator £/'(£) is defined in all of H by the Stone-von Neumann calculus as U^(t) = / ^ ° e~'EtdP(E) not by the exponential series

n=0

which is defined (converges) only on a dense subspace A C D C 7i called analytic vectors.

107

The conclusion that one draws from these two existence theorems is the following : The choice to describe states in the HS theory is very restricted and can only be given by a density operator (positive definite trace-class) and (for a pure state) by a vector <j> 6 K. The time evolution of these states must be the time symmetric reversible group evolution (14). There are no states in the HS quantum mechanics of closed systems (which fulfill (8a) or (8b)) which have asymmetric (irreversible) time evolution 3 . The evolution of the quantum mechanical observables is also time symmetric, i.e. the observables A'(t - to) can evolve relative to the state W(t0) by an arbitrary amount t - t0 > 0 as well as t - t 0 < 0. For every E/(<2 — D (or more general state WD) which has been created at any finite time to / — °° (which we call to = 0) and then decays into decay products. This mathematical consequence is not surprising if one keeps in mind that in the HS theory the t-evolution-where t is the relative time between state and observable-is given by a reversible unitary group. One cannot just impose on the group evolution an arbitrary condition like causality, chopping off one half of the theory, and expect that what remains is still a consistent theory. The reversibility of quantum mechanics in HS and the violation of the causality principle is a consequence of the mathematical idealization given by the topology of the HS, i.e. by (11). It is not a consequence of the fundamental hypothesis of quantum physics as given by the interpretation (5) and by the dynamical equation (8). We shall therefore return to the quantum mechanical Cauchy problem (8) and modify the boundary conditions
Irreversibility in conventional quantum theory is considered to be "non-quantum mechan­ ical" and always thought of as being due to external influences upon "open" systems. It is described by an additional term on the right hand side of (8a) which is not given by the commutator with H of the quantum system.

108

since we have already a scalar product ( , ) in the linear space $ (and we need the scalar product to calculate such physical quantities like the probabilities \(<j>, ifr)\2), Banach space completion is no more an option. 5

The Rigged Hilbert Space idealization has irreversible semigroup evolution and Gamow vectors with exponential decay

We shall complete the linear scalar product space $ into a locally convex nuclear space with a topology stronger than the Hilbert space topology given by the scalar product. Specifically we shall choose a countable Hilbert space where the meaning of convergence, i.e. the topology T* is defined by a count­ able number of scalar products ( , ) p , p = 0 , 1 , 2 , . . . where ( , ) p= o = ( > ) is the scalar product of the HS. Convergence with respect to T*, <j>„ A 0, means : iff \\\\2V = {<$>„ - < p , < p u - o o

for every p = 0 , 1 , 2 , . . . . This topology is stronger than T « . The countable number of scalar products, i.e. the topology T*, is usually chosen such that the algebra of observables for the physical system under consideration becomes an algebra of r$ -continuous operators. If we denote the "^-completion of the linear space $ again by $ then we have the two complete topological spaces $ C % with $ dense in %. Taking in addition the space $ x of r*-continuous antilinear functionals F(4>) on # , and the space H* of r«-continuous functionals f(h) = (h, f) which are given by the scalar product, we obtain the Gelfand triplet or Rigged Hilbert Space (RHS) 10 $CH

= H* C $ x .

(19)

We shall use the Dirac notation for the r*-continuous functionals F(<j>) = (4>\F), because F(0) is an extension of the scalar product {h,f) to those F € $ x which are not in H. In these RHS's (one for each kind of quantum physical system) Dirac's formalism of kets (with a continuous set of eigenvalues) and the continuous basis vector expansion (6b) attain a mathematical meaning and the integrals in (7) are Riemann integrals. These RHS's also allow for time-asymmetric solutions of the quantum mechanical Cauchy problem (8). In the RHS formulation one can choose different subspaces of V. to distin­ guish between states and observables. We call $_ the space that describes the

109 states (called in-states in the scattering experiment) prepared by preparation apparatuses (e.g. accelerator). We call $ + the space that describes the ob­ servables (called out-states in scattering theory) registered by the registration apparatuses (e.g. detector). The two subspaces $_ and $+ are not disjoint. The HS formulation, in contrast, does not allow for this mathematical dis­ tinction into separate subspaces of states and observables. Thus there is one Hilbert space % and for each quantum mechanical (scattering) system two dense subspaces $ T and therewith two RHS's <J>_ C H C $ ! with the physical interpretation as in-states and

(20a)

$+ C H C $+ with the physical interpretation as out-observables.

(20b)

Mathematically $ T are defined by their realization as function spaces for their energy wavefunctions :

+ e $_ «• (+E\4>+) eSr\H2_|R+ il>-e*+&(-E\i>-)€Sr\Hl\R+

(2ia) (2ib)

where S denotes the Schwartz space of functions and "H2_, %\ denotes Hardy class functions in the lower and upper complex plane, respectively. (Here complex half-planes refer to the second Riemann sheet of the S-matrix). The ± in {±E\ refers to the ±ie in the Lippmann-Schwinger equation for the eigenkets oiH = H0 + V. The interpretation (20) of the mathematical spaces (21) can be inferred from the preparation -> registration arrow of time n . In addition to the vectors + e <J>_ C % and the ip~ G $ + C % denned by the experimental apparatus, the RHS formulation also provides elements outside of W, e.g. Dirac's scattering states \p) = \E,dp,4>J) G $ ± and the Gamow (decaying) states (or Gamow kets) V G = \ER - iT/2~) e $ £ . A Gamow ket is a generalized eigenvector of a self-adjoint Hamiltonian extended to $ x , with complex eigenvalue ZR = ER — iT/2, precisely,

(Hxp- \ER - tr/2-> = (i,- \H* \ER - tr/2-> =

(ER-iT/2)(ip-\ER-iT/2-)

for all xj)~ 6 $+ (the space of observed decay products). Here ER represents the resonance energy and T the width of the Breit-Wigner energy distribu­ tion. The superscript " in \ER - iT/2~) is inherited from the kets of the Lippmann-Schwinger equation, the subscripts on the corresponding space $.£ from the mathematicians' notation for Hardy class functions (21). Scattering theory in physics and Hardy class functions in mathematics were developed independently of each other. Except for this discrepancy in the notation for

110

the labels +", the Hardy class spaces $_ provide an excellent mathematical image for prepared states {
{ t )

= e-iHXt\ER-iT/2-)

= e~iE-le-rt/2\ER

-

iT/2~),

for t > 0 only. The Gamow kets are solutions of the Schroedinger equation (8b) but do not fulfill the Hilbert space boundary condition. Instead they fulfill the time asymmetric boundary condition ipG(t = 0) £ $ + . Whereas the first part of (23a) can be formally verified from (22), the derivation of the time asym­ metry t > 0 is highly non-trivial and requires specific properties of Hardy class functions. The semigroup eZ'H l is only defined for positive values of the time, t > 0, (H* is the operator H^ extended into 4>x defined by the first equality of (22)). There is another semigroup eZtH l,t<0 and another Gamow ket tj)G = \ER + iT/2+) G $ * with the asymmetric time evolution jG(t)

= e~_iHXt\ER+ir/2+) = e-iB^e+r^\ER for t < 0 only .

+

iV/2+),

The Gamow vectors are defined from the pole term of the analytically con­ tinued 5-matrix at the resonance position at zR = ER — iT/2 (and at zR = ER + iF/2 for (23b)) in the second Riemann sheet. From this one obtains, using the Hardy class property, the Breit-Wigner energy distribution for their wave function 3 ' 13 {-E\i,G) = iJfrbr——± £, - (iSR -

^ - , - 0 0 / / < E < +oo .

(24)

l-^)

The variable E extends over the physical values (upper rim of positive real axis first sheet = lower rim of positive real axis second sheet) and from — oo// to 0 in the second sheet. This is an idealized Breit-Wigner in contrast to the standard Breit-Wigner for which 0 < E < oo. The results (22) and (23) are derived from the pole term definition using the properties of the Hardy spaces (21) 3 ' 13 . If there are iV* resonances in the system, each occurring as a pole of the j - t h partial 5-matrix at the positions zRi = ERi — iIV2, then one obtains N Gamow vectors ipG.

111

The Gamow vectors %l)f are members of a "complex" basis vector expan­ sion 13 . In place of the well known Dirac basis system expansion (6b) given for the Hamiltonian H by /•+00

4>+ = / dE\E+)(+E\^) (25) Jo (where a discrete sum over bound states has been ignored), every state vector (f>+ £ $ _ can be expanded as

+ = - £ IVfW?l0+> + /

dE\E+){+E\<j>+)

(26)

(where —OOJI indicates that the integration along the negative real axis (or other contours) is in the second Riemann sheet of the 5-matrix). The "com­ plex" basis vector expansion (26) is rigorous. This allows us to mathemati­ cally isolate the exponentially decaying states tpf. It also allows us an easy approximation by omitting the background integral in (26), and just using N +

* =!>?>*'

ci = -<^?|0+).

(27)

t=i

Then one obtains the "effective" theories with finite complex Hamiltonian. For instance, for the K\ - K% meson system with TV = 2,
+ ^bL

(28) 14

and one obtains the Lee-Oehme-Yang theory . The finite dimensional ap­ proximations (27) have been successfully applied to many areas of physics, in particular to nuclear physics 15 ' 2 , which shows that to isolate the Gamow states can be a good approximation. Since (23a) implies the exponential law for the decay rate 6 Vv(t) = Tve~rt the width T of the Breit-Wigner distribution (24) and the lifetime fulfill the exact relation r = p. This has not been obtained before as an exact, precisely derived relation, though it has always been assumed on the basis of some "approximate derivations" 16 . Gamow vectors are ideally suited to describe resonances (the pair (23a) and (23b)) in a scattering process or quasistable particles (23a) that decay. Like the Dirac-Lippmann-Schwinger kets | S ± ) , from which they are con­ structed, they do not describe interaction-free in- or out- asymptotic states. In a theory that allows only asymptotic particles, they are therefore not ad­ mitted. Gamow states have all the properties that heuristically the unstable

112

states need to possess. In addition and unintended they give rise to an asym­ metric time evolution, which may not have been wanted but is in agreement with the empirical principle of causality. Acknowledgement We gratefully acknowledge valuable support of the Welch Foundation for the preparation of this paper and kind hospitality of Rui Vilela Mendes in Lisbon. References 1. Resonances-Models and Phenomena, S. Albeverio, L. S. Ferreira and L. Streit [Eds], Springer Lecture Notes in Physics 211, (Berlin, 1984). 2. M. Baldo, L. S. Ferreira and L. Streit, Nucl. Phys. A 467, 44 (1987); M. Baldo, L. S. Ferreira and L. Streit, Phys. Rev. C 36, 1743 (1987). 3. A. Bohm, J. Math. Phys. 22, 2813 (1981); Lett. Math. Phys. 3, 455 (1978). 4. A. Bohm, Quantum Mechanics, 3rd ed., sections XVIII.6-9 and XX.3, (Springer-Verlag, New York, 1994). 5. L. Streit in Stochastic Analysis and Applications in Physics, A. I. Car­ doso, et al. [Eds.], (Klumer, Dordrecht, 1994), page 415; M. de Faria and L. Streit in Stochastic Analysis on Infinite Dimensional Spaces, H. Kunita and H. H. Kuo [Eds.], (Longman, 1994), page 52. 6. A. Bohm and N. L. Harshmann in Irreversibility and Causality, p. 225, Sect. 7.4; A. Bohm, H. D. Doebner and P. Kielanowski [Eds.], (Springer, Berlin, 1998). 7. A. M. Gleason, J. Math. 6, 885 (1957). 8. J. von Neumann, Mathematische Grundlagen der Quantentheorie, (Springer, Berlin, 1931) (English translation by R. T. Beyer) (Prince­ ton University Press, Princeton, 1955). 9. M. H. Stone, Ann. of Math. 33, 643 (1932). 10. A. Bohm and M. Gadella, Dirac Kets, Gamow Vectors and Gel'fand Triplets, Lecture Notes in Physics 348, (Springer-Verlag, Berlin, 1989). 11. A. Bohm, I. Antoniou and P. Kielanowski, Phys. Lett. A 189, 442 (1994); A. Bohm, I Antoniou and P. Kielanowski, J. Math. Phys. 36, 2593 (1995). 12. E. P. Wigner, Symmetries and Reflections, (Indiana University Press, Bloomington, 1967), page 38. 13. A. Bohm, S. Maxson, M. Loewe and M. Gadella, Physica A 236, 485 (1997). 14. T. D. Lee, R. Oehme and C. N. Yang, Phys. Rev. 106, 340 (1957).

113

15. L. S. Ferreira.in Resonances, E. Brandas [Ed], Lecture Notes in Physics 325, (Springer, 1989), p. 201; C. Mahaux, \ p. 139; 0 . I. Tolstikhin, V. N. Ostrovsky and H. Nakamura, Phys. Rev. 58, 2077 (1998); V. I. Kukulin, V. M. Krasnopolsky and J. Horacek, Theory of Resonances, (Kluwer Acad. Publ., 1989). 16. e.g. M. L. Goldberger and K. M. Watson, Collision Theory, (Wiley, New York, 1964); Newton, Scattering Theory of Waves and Particles, 2nded., (Springer-Verlag, 1982), chapter 8.

114 Q-ISING N E U R A L N E T W O R K D Y N A M I C S : A COMPARATIVE R E V I E W OF VARIOUS A R C H I T E C T U R E S D. BOLLE AND G. JONGEN Instituut voor Theoretische Fysica, K.U. Leuven, B-3001 Leuven, Belgium E-mail: {desire.bolle, greetje.jongen}@fys.kuleuven.ac.be. G. M. SHIM Department of Physics, Chungnam National University Yuseong, Taejon 305-764, R-O. Korea E-mail: [email protected]. This contribution reviews the parallel dynamics of Q-Ising neural networks for var­ ious architectures: extremely diluted asymmetric, layered feedforward, extremely diluted symmetric, and fully connected. Using a probabilistic signal-to-noise ratio analysis, taking into account all feedback correlations, which are strongly depen­ dent upon these architectures the evolution of the distribution of the local field is found. This leads to a recursive scheme determining the complete time evolution of the order parameters of the network. Arbitrary Q and mainly zero tempera­ ture are considered. For the asymmetrically diluted and the layered feedforward network a closed-form solution is obtained while for the symmetrically diluted and fully connected architecture the feedback correlations prevent such a closed-form solution. For these symmetric networks equilibrium fixed-point equations can be derived under certain conditions on the noise in the system. They are the same as those obtained in a thermodynamic replica-symmetric mean-field theory approach.

1

Introduction

Artificial neural networks have been widely applied to memorize and retrieve information. During the last few years there has been considerable interest in neural networks with multistate neurons (see * and references cited therein) in order to function as associative memories for gray-toned or coloured patterns or to allow for a more complicated internal structure in the retrieval, e.g., a distinction between background and patterns. Here we review the dynamics of so-called Q-Ising neural networks (see the references in 2 ) for arbitrary Q. They are built from Q-Ising spin-glasses 3,4 with couplings defined in terms of patterns through a learning rule. For Q = 2 one finds back the Hopfield model 5 ' 6 , for Q -» oo one has an analog network (see 7 and references therein). One of the aims of these networks is to memorize a number of patterns and find them back as attractors of the

115

retrieval process. Consequently these networks are also interesting from the point of view of dynamical systems. Besides a learning rule one also needs to specify an architecture indicating how the spins (=neurons) are connected with each other. Several architectures have been studied in the literature for different purposes. From a practical point of view mostly perceptrons or, more general, feedforward layered net­ works are used since a very long time (see, e.g., 8 for a history). Hopfield 5 6 ' studied a fully connected network with symmetric couplings because it satisfies the detailed balance principle and hence a Hamiltonian can be de­ fined. Asymmetrically diluted models 9 were used because their dynamics can be solved exactly and because they can learn us something about the loss of information content when some of the synaptic couplings break down. In this contribution we review the study of the parallel dynamics of these types of network using a probabilistic approach (see, e.g., 1 0 ' H ) . In more detail, employing a signal-to-noise ratio analysis based on the law of large numbers (LLN) and the central limit theorem (CLT) we derive the evolution of the distribution of the local field at every time step. This allows us to obtain a recursive scheme for the evolution of the relevant order parameters in the system being, in general, the main overlap for the condensed pattern, the mean of the neuron activities and the variance of the residual overlap responsible for the intrinsic noise in the dynamics of the main overlap (sometimes called the width parameter). The details of this approach depend in an essential way on the architecture because different temporal correlations are possible. For extremely diluted asymmetric and layered feedforward architectures recursion relations have been obtained in closed form directly for the relevant order parameters 9 . 12 . 13 . 14 . This has been possible because in these types of networks there are no feedback correlations as time progresses. As a technical consequence the local field contains only Gaussian noise leading to an explicit solution. For the parallel dynamics of networks with symmetric connections, how­ ever, things are quite different Mo.n Even for extremely diluted versions of these systems 15>16'17 feedback correlations become essential from the second time step onwards, complicating the dynamics in a nontrivial way. There­ fore, explicit results concerning the time evolution of the order parameters for these models have to be obtained indirectly by starting from the distribution of the local field. Technically speaking, both for the symmetrically diluted and fully connected architectures the local field contains both a discrete and a normally distributed part. The difference between the diluted and fully connected models is that the discrete part at a certain time t does not involve the spins at all previous times t-l,t — 2,... up to 0 but only the spins at time

116

step t — 1. But in both cases the discrete part prevents a closed-form solution of the dynamics for the relevant order parameters. Nevertheless, the devel­ opment of a recursive scheme is possible in order to calculate their complete time evolution. In this way a comparative discussion of the parallel dynamics at zero temperature for the various architectures specified above is possible. Finally, by requiring the local field to become time-independent implying that some correlations between its Gaussian and discrete noise parts are ne­ glected we can obtain fixed-point equations for the order parameters. It turns out that they are equivalent to the fixed-point equations obtained through a thermodynamic replica-symmetric mean-field theory approach. At this point we remark that we do not aim for complete rigour in our derivations. From the point of view of rigorous mathematics, the Hopfield model and, in general, spin-glass theory is recognized to be an extremely difficult, if not imposible, field. For a recent overview of the modest results obtained, mostly concerning thermodynamics, we refer to 18 . The rest of this contribution is organized as follows. In Section 2 we introduce the model, its dynamics and the Hamming distance as a macro­ scopic measure for the retrieval quality. In Section 3 we use the probabilistic approach in order to derive a recursive scheme for the evolution of the distribu­ tion of the local field, leading to recursion relations for the order parameters. The differences between the various architectures are outlined. We do not aim for complete rigour and mostly concentrate on zero temperature. In Sec­ tion 4 we discuss the evolution of the system to fixed-point attractors. Some concluding remarks are given in Section 5.

2

Q-Ising neural networks

Consider a neural network A consisting of AT neurons which can take values Oi from a discrete set S — {-1 = s\ < s 2 < . . . < SQ = +1}. The p patterns to be stored in this network are supposed to be a collection of independent and identically distributed random variables (i.i.d.r.v.), {£? 6 S}, fi £ V = { 1 , . . . ,p} and i € A, with zero mean, E[^] = 0, and variance A = Var[£f]. The latter is a measure for the activity of the patterns. We remark that for simplicity we have taken the patterns and the neurons out of the same set of variables but this is no essential restriction. Given the configuration a\(t) = {
hi(crA(t)) = Y/Jij(t)oj(t) ;6A

(1)

117

with Jij the synaptic coupling from neuron j to neuron i. In the sequel we write the shorthand notation /iA,i(t) = hi(
J D

« = ^£tftf

J c

Z = 4iZGG

■^(0 = ^ £ with the {cij = 0,1},i,j Pr{aj - x} = (1 - C/N)SXt0 symmetric dilution, and c^ for asymmetric dilution. At zero temperature all rule

for

**>''

for

<*>•

^ + 1)^(0,

W (3) (4)

€ A chosen to be i.i.d.r.v. with distribution + (C/N)8Xti and satisfying Cy — Cji, cu = 0 for and Cji statistically independent (with ca — 0) neurons are updated in parallel according to the

<Ti(t) -> <Ji{t + l)=sk:

mine i [s|o- A (t)] = ti[sk\
(5)

We remark that this rule is the zero temperature limit T = / 3 _ 1 —> 0 of the stochastic parallel spin-flip dynamics defined by the transition probabilities « . , , , , > ,

c

o

U

^M _

exP[-/3£i(Sfc|
Pi{(Ji{t + l) = sk €S\
= = . E 8 € 5 e x Pl-/ 3 € <0 s l < T Mt))] Here the energy potential €i[s|crA] is defined by 19 uls\
f(.,

(6)

(7)

118

where 6 > 0 is the gain parameter of the system. The updating rule (5) is equivalent to using a gain function g 6 (),
g,,(hAM) Q

gb(x) = £

*k P [b{sk+i +sk)-x}-6

[b(sk + 8k-t) - x]]

(8)

t=i

with so = - c o and SQ+1 = +oo. For finite Q, this gain function g6(-) is a step function. The gain parameter b controls the average slope of g6(-)In order to measure the retrieval quality of the system one can use the Hamming distance between a stored pattern and the microscopic state of the network

i€A

This introduces the main overlap and the arithmetic mean of the neuron activities

igA

t€A

We remark that for Q = 2 the variance of the patterns A = 1, and the neuron activity a\(t) = 1. 3 3.1

Solving the dynamics Correlations

We first discuss some of the geometric properties of the various architectures which are particularly relevant for the understanding of their long-time dy­ namic behaviour. For a fully connected architecture there are two main sources of strong correlations between the neurons complicating the dynamical evolution : feed­ back loops and the common ancestor problem 20 . Feedback loops occur when in the course of the time evolution, e.g., the following string of connections is possible: i —y j —>■ k —>• i. We remark that architectures with symmetric connections always have these feedback loops. In the absence of these loops the network functions in fact as a layered system, i.e., only feedforward con­ nections are possible. But in this layered architecture common ancestors are still present when, e.g., for the sites i and j there are sites in the foregoing time steps that have a connection with both i and j .

119 In extremely diluted asymmetric architecture these sources of correlations are absent. This class of neural networks was introduced in connection with Q = 2-Ising models 9 . We recall that the couplings are then given by eq. (2) and that in the limit N —> co two important properties of this network are essential 9 ' 2 1 . The first property is the high asymmetry of the connections, viz. Pr{ C l J = c Jt } = ( § )

,

Pr{cy = l A C i i = 0 } = ^ l - ^ .

(11)

Therefore, the number of symmetric connections in the infinite configuration c = {cij}, i, j ^ i € N is finite with probability one, i.e. almost all connections of the graph GN(C) = {(i,j) ■ Cy; = l,i, j ^ i € N} are directed : Cy ^ Cj{. The second property in the limit of extreme dilution is the directed lo­ cal Cayley-tree structure of the graph Gn(c). By the arguments above the probability FJ. ' (c) that k connections are directed towards a given site i 6 A is

where jf/ = {cji = l , j £ A \ i } is the in-tree for i and |T> m, | its cardinality. This probability is equal to Pr{& = |l\ ( o u t ) | = \{Cij = 1, j € A \ i}\} for connections directed outward a given site i € A. In the limit of extreme dilution we get a Poisson distribution : lim FlA)(C) = ~e-c.

(13)

Hence, the mean value of the number of in (out) connections for any site i E A is E[\T^in)ioui)\] = C. The probability that two sites i and i' have site j as a common ancestor is obviously equal to C/N. From £[17^ '|] = C it follows that after t time steps the cardinality of the cluster of ancestors for site i will be of the order of Cl. The same is valid for site i'. Therefore, the probability that the sites i and i' have disjoint clusters of ancestors approaches (1 - Cl/N)c' ~ exp(-C 2 7A0 for N » 1. So we find that in the limit of extreme dilution : (i) Almost all (i.e. with probability 1) feedback loops in GN(C) are eliminated, (ii) With probability 1 any finite number of neurons have disjoint clusters of ancestors. So we first dilute the system by taking N -v oo and then we take the limit C -* oo in order to get infinite average connectivity allowing to store infinitely many patterns p.

120 This implies that for this asymmetrically diluted model at any given time step t all spins are uncorrelated and, hence, the first step dynamics describes the full time evolution of the network. For the symmetrically diluted model the architecture is still a local Cayley-tree but no longer directed and in the limit N —> oo the probabil­ ity that the number of connections Ti = {j £ A|cy = 1} giving information to the the site i 6 A is still a Poisson distribution with mean C = £[|7i|]. However, at time t the spins are no longer uncorrelated causing a feedback from t > 2 onwards I 7 . In order to solve the dynamics we start with a discussion of the first time step dynamics, the form of which is independent of the architecture. 3.2

First time step

Consider a fully connected network. Suppose that the initial configuration of the network {a-j(0)},z G A, is a collection of i.i.d.r.v. with mean E[
ml>0.

(14)

This pattern is said to be condensed. By the law of large numbers (LLN) one gets for the main overlap and the activity at t = 0 m^O) = Jim m\(0) £

^ [ ^ ( 0 ) ] = mj

o(0) = lim o A (0) = E[
(15) (16)

where the convergence is in probability 22 . In order to obtain the configuration at t = 1 we first have to calculate the local field (1) at t = 0. To do this we employ the signal-to-noise ratio analysis (see, e.g., 10 ' 12 ). Recalling the learning rule (3) we separate the part containing the condensed pattern, i.e., the signal, from the rest, i.e., the noise to arrive at

M-A(0)) = ^

£ ^(0) + V 5 ^ E &~m £ tf"i(o)

(17) where a = p/N. The properties of the initial configurations (14)-(16) assure us that the summation in the first term on the r.h.s of (17) converges in the limit N -¥ oo to

^^E^W"-1^)ieA\t

(18)

121

The first term £?m1 (0) is independent of the second term on the r.h.s of (17). This second term contains the influence of the non-condensed patterns causing the intrinsic noise in the dynamics of the main overlap. In view of this we define the residual overlap r " ( 0 = lim r£(<) = lim - ± = £ # o , - ( « )

/i£P\{l}.

(19)

jGA

Applying the CLT to this second term in (17) we find

„"5.,/f £ ««*«> = J S . ^ p E ^

£ ^ (0)(20)

= >/a-V(0,XZJ(0))

(21)

where the quantity M(0, V) represents a Gaussian random variable with mean 0 and variance V and where D(Q) = Var[r'i(0)] = a(0). Thus we see that in fact the variance of this residual overlap, i.e., D(t) is the relevant quantity characterising the intrinsic noise. In conclusion, in the limit N -> oo the local field is the sum of two independent random variables, i.e. hi(0)=

lim h A i i ( 0 ) i £ r o 1 ( 0 ) + >/aA/r(0,a(0)).

(22)

For a more rigourous discussion of the first time step for the underlying spinglass model we refer to 23 . At this point we note that the structure (22) of the distribution of the local field at time zero - signal plus Gaussian noise is typical for all architectures discussed here because the correlations caused by the dynamics only appear for t > 1. Some technical details are different for the various architectures. The first change in details that has to be made is an adaptation of the sum over the sites j to A for the layered feedforward architecture and to Tj, the part of the tree connected to neuron i, in the diluted architectures. The second change is that for the diluted architectures an additional limit C -> oo has to be taken besides the N -> oo limit. So in the thermodynamic limit C, N -> oo all averages will have to be taken over the treelike structure, viz. jj £ < G A -> £ £ i 6 7 v • Furthermore a = p/N has to be replaced by a = p/C. 3.3

Recursive dynamical scheme

The key question is then how these quantities evolve in time under the parallel dynamics specified before. For a general time step we find from the LLN in

122

the limit C, N —► oo for the main overlap and the activity (10) m\t

+ l) £

l^\{ai{t

+ \))0)),

a(t + l) £

(({*i(t + l))}))

(23)

with the thermal average denned as

( / ( . ( . + i))>,=V

/w

T^; ( ' ) ^T ) l

<M>

E a e s exp[±/?<7(M0 - for)] where hi(t) = lim^^oo h,\ti(t). In the above ((•)) denotes the average both over the distribution of the embedded patterns {ff} and the initial configurations {<7j(0)}. The average over the latter is hidden in an average over the local field through the updating rule (8). In the sequel we focus on zero temperature. Then, using eq. (8) these formula reduce to

m^t+l)

P

^ jUlgMt)))),

a(t + l) ^

((gUhiit)))) ■

(25)

As seen already in the first time step, we have to study carefully the influ­ ence of the non-condensed patterns causing the intrinsic noise in the dynamics of the main overlap. The method used to obtain these order parameters is then to calculate the distribution of the local field as a function of time. In order to determine the structure of the local field we have to concentrate on the evolution of the residual overlap. The details of this calculation are very technical and depend on the precise correlations in the system and hence on the architecture of the network as discussed before. For these technical de­ tails we refer to the relevant literature 1.2.12,14,15 } j e r e w e g j v e g^ extensive discussion of the results obtained. In general, the distribution of the local field at time t + 1 is given by M< +1) = tlml(t

+1) +Af(0,aa(t

+1)) + X(t)[F(hi{t) -gml(t))

+ Baai{t)} (26) where F and B are binary coefficients given below, which depend on the specific architecture. From this it is clear that the local field at time t consists out of a discrete part and a normally distributed part, viz. hi(t) = Mi(t)+M(0,V(t))

(27)

where Mi(t) satisfies the recursion relation M{(t + 1) = X(t)[F(Mi(t)

- Zlml(t))

+ Baai(t)} + ^ m 1 ^ + 1)

(28)

and where V(t) = aAD{t) with D(t) itself given by the recursion relation D(t + 1) = ^ j - ^ + LX2(t)D(t)

+ 2B2X{t)Cov[r»{t),

r»(t)}

(29)

123

where L and B2 are again coefficients specified below. The quantity x(t) reads Q-i

X(0 = £

h»(t)(b(sM

+ *k))(**+i - 8k)

(30)

it=i

where f-h»tt\ is the probability density of hf (t) = lim/v-»oo ^A t(0

w tn

'

(31) Furthermore, F 1 ^) is defined as (32) v

i6A

Finally, as can be read off from eq. (28) the quantity Mi(t) consists out of the signal term and a discrete noise term, viz. 1-2

t-\

Mi(t) = tlm'it) + B.axit - 1)^(1 - 1) + B2 £ a ! ! * ( < t'=0


ls=t'

Since different architectures contain different correlations not all terms in these final equations are present. In particular we have for the coefficients F,B,L,B\ and Bi introduced above FC SED LF AED

F BLBX 1 1 1 1 0 10 1 10 10 0 0 0 0

B2 1 0 0 0

(34)

with B indicating the feedback caused by the symmetry in the architectures and L the common ancestors contribution. At this point we remark that in the so-called theory of statistical neurodynamics 24,25 one starts from an approximate local field by leaving out any discrete noise (the term in <7j(£))- As a consequence the covariance in the recursion relation for D(t) can be written down more explicitly since only Gaussian noise is involved. For more details we refer to 26 . We still have to determine the probability density of f^m in eq. (30), which in the thermodynamic limit equals the probability density of fh{{t)This can be done by looking at the form of Mi{t) given by eq. (33). The evolution equation tells us that <7i(£') can be replaced by pt(/i,(<' — 1)) such that the second and third terms of M,(t) are the sums of stepfunctions of

124

correlated variables. These are also correlated through the dynamics with the normally distributed part of h{(t). Therefore the local field can be consid­ ered as a transformation of a set of correlated normally distributed variables xs, s = 0,.. .,t — 2,t, which we choose to normalize. Defining the correlation matrix W = (p(s,s') = E[a;sa:g<]) we arrive at the following expression for the probability density of the local field at time t

fhtw(y) = / II d x > d x t 6 {y - M<W - V<*AD(*)*t) x

1

.

with x = (xo,.. .xt-2,xt). simplifies to

exp (-lxW-1xT)

For the symmetrically diluted case this expression

dxt 2

6 y

1

exp (-\-xW-l-xr) V 2

fhi(t)(y) = / I I

(35)

-*

\~

£ml(()

_ a

x(t)ai(t)

- y/aa(t)

xt)

=0

^/det^jrWO

(36)

with x = ({i»}) = (a:t_2[t/2]i • • xt-2>xt)- The brackets [t/2] denote the integer part of t/2. So the local field at time t consists out of a signal term, a discrete noise part and a normally distributed noise part. Furthermore, the discrete noise and the normally distributed noise are correlated and this prohibits us to derive a closed expression for the overlap and activity. Together with the eqs. (25) for m 1 ((+1) and a(t+l) the results above form a recursive scheme in order to obtain the order parameters of the system. The practical difficulty which remains is the explicit calculation of the correlations in the network at different time steps as present in eq. (29). For AED and LF architectures this scheme leads to an explicit form for the recursion relations for the order parameters m"(t + 1) = SvAjlltHt

+ l)Jvzg(?(t

a(t + 1) = (ljvzg2{il{t D(t + 1) = ^ p ±

+^

+ l)ml(t) [(^Jvzzgtfit

+ l)ml{t) + \/aAD(t)

+ yfiAD(fj z)\\

+ %'(() +

*)))

(37) (38)

,/^AW)z)))} (39)

125

with Vz = dz(2ir)-1/2exp(-z2/2). For the AED architecture (L = 0) the second term on the r.h.s. of (39) coming from the correlations caused by the common ancestors is absent. For the LF architecture we remark that this explicit solution requires an independent choice of the representations of the patterns at different layers. At finite temperatures analogous recursion relations for the AED and LF networks can be derived 12,14 by introducing auxiliary thermal fields 27 in order to express the stochastic dynamics within the gain function formulation of the deterministic dynamics. Furthermore, damage spreading 9>28-29, i.e., the evolution of two network configurations which are initially close in Hamming distance can be studied 12 - 14 . Finally, a complete self-control mechanism can be built in the dynamics of these systems by introducing a time-dependent threshold in the gain function improving, e.g., the basins of attraction of the memorized patterns 3 0 _ 3 2 . For explicit examples of this dynamical scheme with numerical results we refer to 1,1S . By using the recursion relations the first few time steps are written out explicitly and studied numerically, e.g, for the Q = 2 and Q = 3 FC and SED models with equidistant states and a uniform distribution of patterns. These results are compared with the approximations studied in the literature JO.11^,17,24,25,33-38 by neglecting some feedback correlations for t > 2. In the whole retrieval region of these networks we find that the first four or five time steps give us already a clear picture of their time evolution. 4

Fixed-point equations

Equilibrium results for the AED and LF Q-Ising models are obtained imme­ diately by straightforwardly leaving out the time dependence in (37)-(39) (see 12 14 , ), since the evolution equations for the local field and the order parame­ ters do not change their form as time progresses (see eqs. (37)-(39)). This still allows small fluctuations in the configurations {
126

i.e., Mi(t) (recall (33)), only the <7<(t - 1) term is kept besides, of course, the signal term. This procedure implies that the main overlap and activity in the fixed-point are found from the definitions (10) and not from leaving out the time dependence in the recursion relation (25). We start by eliminating the time-dependence in the evolution equations for the local field (26). This leads to hi = gm1 + [ x a r ] - W ( 0 , a a ) + [ x a r ] _ 1 a x ^

(40)

ar

with \ = 1 - Fx being 1 for the SED and 1 - x for the FC model and hi = \imt-yoo hi(t). This expression consists out of two parts: a normally distributed part hi = N(£lm1,aa/[xar]2) and some discrete noise part. At this point some remarks are in order. First, the discrete noise coming from the correlations of the {<Ji(t)} at different time steps (here only the preceding time step is considered) is inherent in the SED and FC dynamics. Second, the so-called self-consistent signal-to-noise ratio analysis of the FC network con­ sidered in the literature 39 ' 40 starts from such a type of equation by assuming the presence of a term proportional to the output in the local field without any reference or argumentation based upon the underlying dynamics of the network. Employing this expression in the updating rule (8) one finds Oi=6b(hi + \xaT1«X°i)

(41)

where hi = ^ ( f - m ' . Q o ) is the normally distributed part of eq. (40). This is a self-consistent equation in Oi which in general admits more than one solution. These types of equation have been solved in the literature in the context of thermodynamics using a geometric Maxwell construction 39 , 40 . We remark that for analog networks the geometric Maxwell construction is not necessary: the fixed-point equation (41) has only one solution. For more technical details we refer to 2 6 . This approach leads to a unique solution b = b-[2xaT^X-

* = $&),

(42)

We remark that plugging this result into the local field (40) tells us that the latter is the sum of two Gaussians with shifted mean (see also 3 7 ) . Using the definition of the main overlap and activity (10) in the limit N -> oo for the FC model and limit C,N -> oo for the SED model, one finds in the fixed point m1 = (U1 j Vz

6i

(C1 m1 + V^AD z) \ \

(43)

127 fvz gf (('m1 + V^ADz)\\ . " < From (29) and (30) one furthermore sees that D = [xaT2a/A

(44)

(45)

with

l =

7S5((/P"6i(f'm' + ^ z ) ) ) '

(46

»

These resulting equations (43)-(45) are the same as the fixed-point equations derived from a replica-symmetric mean-field theory treatment in 4 1 ~ 4 4 . Their solution leads to capacity-gain parameter phase diagrams (see, e.g., 44 ). 5

Concluding r e m a r k s

An evolution equation is derived for the distribution of the local field governing the parallel dynamics at zero temperature of extremely diluted symmetric and asymmetric, layered feedforward and fully connected Q > 2-Ising networks. All feedback correlations are taken into account. In general, this distribution is not normally distributed but contains a discrete noise part. Employing this evolution equation a general recursive scheme is developed allowing one to calculate the relevant order parameters of the system, i.e., the main overlap, the activity and the variance of the residual noise for any time step. For the extremely diluted asymmetric and the layered feedforward architectures this scheme immediately leads to explicit recursion relations for the order parameters because the discrete noise part in the local field is absent. For the extremely diluted and the fully connected architectures equilibrium fixed-point equations for the order parameters are obtained under the condition that the local field becomes time-independent, meaning that some of the discrete noise is neglected. The resulting equations are the same as those derived from a replica-symmetric mean-field theory approach. Acknowledgments This work has been supported in part by the Research Fund of the K.U.Leuven (Grant OT/94/9) and the Korea Science and Engineering Foundation through the SRC program. The authors are indebted to S. Amari, R. Kiihn, G. Massolo, A. Patrick and V. Zagrebnov for constructive discussions. One of us (D.B.) thanks the Belgian National Fund for Scientific Research for financial support.

128 References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

16. 17. 18.

19. 20. 21. 22. 23. 24. 25. 26.

D. Bolle, G. Jongen and G.M. Shim, J. Stat. Phys. 9 1 , 125 (1998). D. Bolle, B. Vinck, and V.A. Zagrebnov, J. Stat. Phys. 70, 1099 (1993). D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35, 1792 (1972). S.K. Ghatak and D. Sherrington, J. Phys. C10, 3149 (1977). J.J. Hopfield, Proc. Nat. Acad. Sci. USA 79, 2554 (1982). J.J. Hopfield, Proc. Nat. Acad. Sci. USA 81, 3088 (1984). R. Kiihn and S. Bos, J. Phys. A: Math. Gen. 26, 831 (1993). J. Hertz, A. Krogh and R.G. Palmer, Introduction to the theory of neural computation Addison-Wesley, Redwood City (1991) B. Derrida, E. Gardner, and A. Zippelius, Europhys. Lett. 4, 167 (1987). A.E. Patrick and V.A. Zagrebnov, J. Stat. Phys. 63, 59 (1991). A.E. Patrick and V.A. Zagrebnov, J. Phys. A: Math. Gen. 24, 3413 (1991). D. Bolle, G.M. Shim, B. Vinck, and V.A. Zagrebnov, J. Stat. Phys. 74, 565 (1994). E. Domany, W. Kinzel, and R. Meir, J. Phys. A: Math. Gen. 22, 2081 (1989). D. Bolle D, G.M. Shim, and B. Vinck, J. Stat. Phys. 74, 583 (1994). D. Bolle, G. Jongen and G.M. Shim, Parallel dynamics of extremely diluted symmetric Q-Ising neural networks, to appear in J. Stat. Phys. 96 August 1999. T.L.H. Watkin and D. Sherrington, J. Phys. A: Math. Gen. 24, 5427 (1991). A.E. Patrick and V.A. Zagrebnov, J. Phys. A: Math. Gen. 23, L1323 (1990); J. Phys. A: Math. Gen. 25, 1009 (1992). A. Bovier and V. Gayrard, in Mathematical aspects of spin-glasses and neural networks, eds. A. Bovier and P. Picco, Progress in Probability, Vol 41 (1997). H. Rieger, J. Phys. A: Math. Gen. 23, L1273 (1990). E. Barkai, I. Kanter and H. Sompolinsky, Phys. Rev. A41, 590 (1990). R. Kree and A. Zippelius, in Models of neural networks, eds. E. Domany, J.L. van Hemmen and K. Schulten, Springer, Berlin (1991). A.N. Shiryayev, Probability (Springer, New York, 1984) A. Patrick J. Stat. Phys. 84, 973 (1996) S. Amari and K. Maginu, Neural Networks 1, 63 (1988). M. Okada, Neural Networks 9, 1429 (1996). G. Jongen, On the dynamics of spin-glass models of neural networks, Ph. D. thesis, K.U.Leuven, Belgium (1999).

129

27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44.

A.E. Patrick and V.A. Zagrebnov, J. Phys. (Paris) 51, 1129 (1990). B. Derrida, J. Phys. A: Math. Gen. 20, L271 (1987). B. Derrida and R. Meir, Phys. Rev. .438, 3116 (1988). D.R.C. Dominguez and D. Bolle, Phys. Rev. Lett. 80, 2961 (1998). K. Kitano and T. Aoyagi, J. Phys. A: Math. Gen. 31, L613 (1998). S. Grosskinsky, J. Phys. A: Math. Gen. 32, 2983 (1999). W. Kinzel, Z. Phys. B 60, 205 (1985). E. Gardner, B. Derrida and P. Mottishaw, J. Physique 48, 741 (1987). W. Krauth, J.P. Nadal and M. Mezard, J. Phys. A: Math. Gen. 21, 2995 (1988). H. Horner, D. Bormann, M. Frick, H. Kinzelbach and A. Schmidt, Z. Phys. 5 76, 381 (1989). R.D. Henkel and M. Opper, Europhys. Lett. 11, 403 (1990); J. Phys. A: Math. Gen. 24, 2201 (1991). D. Gandolfo, M. Sirugue-Collin and V.A. Zagrebnov, Network: Compu­ tation in Neural Systems 9, 563 (1998). M. Shiino and T. Fukai, J. Phys. A: Math. Gen. 25, L375 (1992). M. Shiino and T. Fukai, Phys. Rev. E 48, 867 (1993). T.L.H. Watkin, and D. Sherrington, Europhys. Lett. 14, 791 (1991). D. Bolle, D. Carlucci and G.M. Shim, Thermodynamic properties of ex­ tremely diluted symmetric Q-Ising neural networks, in preparation. D. Amit, H. Gutfreund and H. Sompolinsky, Ann. Phys. (N. Y.) 173, 30 (1987). D. Bolle, H. Rieger and G.M. Shim, J. Phys. A: Math. Gen. 27, 3411 (1994).

130 T H E RELATIVISTIC A H A R O N O V - B O H M - C O U L O M B PROBLEM: A PATH I N T E G R A L SOLUTION J. BORNALES Physics Department, MSU-Eigan Institute of Technology Iligan City 9200, Philippines C. C. BERNIDO Research Center for Theoretical Physics, Central Visayan Jagna, Bohol 6308, Philippines

Institute

M. V. CARPIO-BERNIDO Research Center for Theoretical Physics, Central Visayan Jagna, Bohol 6308, Philippines

Institute

The Dirac equation in (2+l)-dimensional spacetime for a particle interacting with a combined Aharonov-Bohm field and Coulomb potential is evaluated by the path integral method. To do this, a modified Biedenharn transformation is used to reduce the path integral to a form similar to the non-relativistic Coulomb problem.

1

Introduction

It is with gratitude and pleasure that we take this opportunity to contribute this work to the 60th birthday celebration in honor of Professor Dr. Ludwig Streit. Professor Streit has been scientific host to two of the authors (C. C. B. and M. V. C. B.) at the Research Center BiBoS in Bielefeld, Germany, during their terms as Alexander von Humboldt Research Fellows. Indeed, the research stays at BiBoS provided special periods of personal and scientific growth — a unique opportunity made memorable by a kind and supportive host. The subsequent visits of Professor Streit to Jagna, Bohol in the Philip­ pines have also enlarged his circle of friends to include Philippine students, such as the first author (J. B.), who now appreciates his generosity. Together we all wish Professor Streit all the best. The bonds that strongly couple Professor Streit's group at BiBoS and Bohol are primarily based on our investigation of various mathematical and physical aspects of Feynman path integrals. We thus briefly present here the highlights of a path integral solution to the relativistic Aharonov-Bohm-plusCoulomb problem. The details are contained in reference [1].

131

2

Green Function for a Relativistic Aharonov-Bohm-Coulomb System

The original Aharonov-Bohm system [2] is described by the magnetic vector potential, A — ($/27r/j) f, p2 = x2 + y2, for a magnetic flux $ trapped in a long impenetrable solenoid of radius R at the origin, and taken to lie along the 2-axis. To this system we add a Coulomb interaction potential, V = —k/p. The relativistic case for this problem continues to be investigated as shown in recent works [3-6]. Here we show a derivation, within the path integral framework, of the Green function for a spin 1/2 particle interacting with the combined Aharonov-Bohm-Coulomb potential. For the relativistic problem, we consider the Dirac equation (h = c — 1), (M-M)G(/f,p') = <*(/?'-A

(1)

r

for the Green function, G(p",/5 ) = (p"\G\j?), for a particle of mass p.. In Eq. (1), p = (p,
+ P[(k/p) + E\,

(2)

with, etp = a ■ p/p, and,

KABC

= 0

^

L

*-2

) +

2

(3)

in terms of, Lz = x pv - y px , and, a = e$/n . At this point, we make use of an approach used by Biedenharn [7] when he solved the 3-dimensional relativistic Kepler problem by casting it into a nonrelativistic Schroedinger form via a suitable similarity transformation. To this end, Eq. (1) is iterated giving, (n2-M2)g((rj)

= 6(fr-f?).

(4)

l

for a Green function, g{p",p ) - {p"\g\P), satisfying, G(f?\f?) = (n + MWJ). Following references [8-9], we can write the operator g as,

(5)

132 /•OO

g = (n2 - M 2 )" 1 = (t/2/*) /

exp(-iHA) dA,

(6)

Jo

where, H = {fj2 - M2)/2fj,. The Green function of the iterated Eq. (4) is just the matrix element of Eq. (6), i.e., 9(FJ)

= (i/2ft) f"{p\ezp(-iHA)\?) dA, (7) Jo in which the integrand ,(p"\exp(—iHA)\pf), can be identified as a quantum propagator for evolution in A- time with an effective Hamiltonian H. This integrand can then be evaluated as a path integral to get g((?',j?). From the result, the Green function G(/?',/?) for the linear Dirac equation can then be recovered. Introducing an operator in terms of KABC in Eq. (3), TABC

= -(0KABC

+ ikap),

(8)

the effective Hamiltonian can be written in the suggestive form,

2/j

2 , rABC (rABC + 1 ) Pp+ — f?

2kE

(9)

-VT

where, pj, = -[(8/dp) + (l/2/>)]2 and r)2 = E2 - fi2. We can relate TABC in Eq. (9) for this (2+l)-dimensional system to the Temple-Martin-Glauber operator [10-11] used by Biedenhaxn for the Kepler problem [7], and also by Berrondo and Mclntosh [12] for systems with magnetic monopoles. However, as in these earlier works, it is noted that \TABC,H] ^ 0> and a suitable similarity transformation S is required for diagonalization. For our case, this is found to be, S = exp[(i/3ap/2)

(10)

tanh(k/KABc)],

such that, *s = SHS-* = ^ where,

2 , ^ABCJ^ABC + 1) _ ^fcE _

2

(11)

133

TSABC = STABCS-1

=

-0\{KABC

~ *2)1/2|-

(12)

=

For some eigenstates, I7), we have, r ^ g ^ M 7|7)» where, 7 = ±(K2 - A;2)1/2, K? = [m ± (1/2) - (a/2)] 2 , m = 0,±1,±2,..., and a = e$/n. From Eq. (11), we also have TSABC(TSABC + l)|fl - f(£ + 1)|0 , where, £ — |7| + |( 5 ff n 7 _ !)• The diagonalization procedure, in fact, allows for a straightforward separation of the radial and angular components. Thus, we can then write, using, 1 = X) l£)(fl> the Green function as, g(FJ)

= (»/2/i) ^\its"\exp(-iHsA)\f?s) Jo

dA

= (i/2/*) / P ° ° I ] ( ^ | 0 ( p " | e x p ( - i i f c A ) | p ' ) ( e k ' ) dA J0

c

= I>"IO&(P",P')

(13)

(%>'),

where the radial part is given by, / (pn\exp(-iHcA)\p') Jo with an effective radial Hamiltonian of the form, 9dp">P') = (i/W

1

€ tf

+ l)

2kE

dA,

(14)

2

(15) " P P . With K taking positive or negative values, t h e angular function (
2

1\i

«-5 $ (
K+J

(16)

- K - j

where, **±*(V) = ( 1 / v ^ ) exp » U ± - + f J ¥> • At this stage, the radial part &(p",p') still remains to be determined.

(17)

134

3

The Radial Sum-Over-All Histories

We now proceed to evaluate the radial Green function by expressing the in­ tegrand as the path integral, (-iff{A)|p'> = K(p»,p'-A)

= Jexp[iS(p\p')}

Q(p-R)

D\p). (18)

The presence of the impenetrable solenoid is represented by Q(p—R), which is a step-function taking the value unity when the particle is outside the solenoid, p > R, and zero when p < R. The calculation can be carried out by slicing the time-like parameter A into iV-subintervals, i.e., A/N = EJ = Aj - Aj_i, (j = 1,2,..., JV) giving the propagator in the form, /v-i

N

/ TlexrfiSj)

0(/5j-i?)W27ri£j-)1/2 U

J=l

dPj

3=1

(19) with, pj = p{\j), po = P',PN = p", and A = £ ] . Ej. The corresponding action for each interval is, S - M (An)2 b >-2E^Pl)

^

+ 1

)] £ > | b c + 2/ipj p /

■*?'2M£j'

J +

(2Q\

(20)

where, Apj = pj — Pj_i, b = kE/p, and pj = (pjPj^ij1/2 is the geometric mean. We note that the radial path integral, with the Sj given above, is similar to that encountered for the non-relativistic Coulomb problem and, hence, path integrable [13,14]. This, in fact, was the motivation for the application of the Biedenharn transformation to the iterated Dirac equation for the system. Then, taking the limit, R -*• 0, for the radius of the solenoid, while maintaining a non-vanishing flux $ at the origin, the path integration yields the result, K{r",r')

= (l/2r'r") exp[i4ba] {—ipw) csc{ua) exp ?-liu){r"2 +rl2)cot{u)
x Ix+^[-ipMr"r'csc(ua)],

(21)

where, u = i2r]/p, X = 2f + | , r' 2 = p', r" 2 = p", a = A/4(p'p") 1/2 > and, I\+1 is a modified Bessel function. We can use this result to write the radial Green function (14) as,

135

&(p",p')

F

exp[—2pq] csch(q) exp[ir)(p" + p')coth(q)]

Jo

Ix+l[-2ir)(p'p")1/2csch(q)]dq,

*

(22)

where, p = -ikE/r) , ua - ir)A/(4p'p"n2)1/2, and q = T?A/(4p'p"M2)1/2Integrating over q, we further obtain the radial Green function in terms of the Whittaker functions, Matp and Wa>/3,

r(p+£ + i)

9(P",P') = 2ir](p'pny/2T(2^

+ 2)

M_p>s+h(-2ir,p»)

W_pt+h(-2ir,p').

(23)

The Green function G(p",(?) for the linear Dirac equation (1) can also be evaluated in the S representation using Eq. (5), where the operator Ms — SMS'1 = i&a.p [(8/dp) + ( | + i)lp + {kE/j)] - (KE/^), when acting on the K state. This can be used in Eq. (5) together with the recurrence relations for Whittaker functions. The energy spectrum for the Aharonov-Bohm-Coulomb problem can, however, already be obtained from the poles of the Gamma function in equa­ tion (23), i.e., when p + £ +1 = - n (n = 0,1,2,...), and this yields, 1-1/2

E = p, 1 +

2

,

(24)

(n + ^ m i f - f ) - ^ This energy spectrum agrees with that of reference [3]. For the special case where the flux, o = e$/7r, is zero, the result yields the energy spectrum of the relativistic (2+l)-dimensional hydrogen atom [15]. 4

Conclusion

In this paper, we have considered the relativistic problem for a fermion in­ teracting with a combined Aharonov-Bohm-Coulomb potential using a path integral approach to solve the (2+l)-dimensional Dirac equation. For our solution, a key role is played by a similarity transformation that casts the iterated Dirac equation into a form similar to the nonrelativistic problem for which path integration has been done earlier. This type of transformation

136

was used by Biedenharn [7] in his treatment of the relativistic Coulomb prob­ lem. In analogy to Biedenharn's work, we have also introduced an operator, FABC = —(0KABC + ikotp), of the Temple- Martin- Glauber type [10-11], used for (3+l)-dimensional problems. Appropriate diagonalization of this op­ erator allows the straightforward separation of the radial and angular parts of the path integral. Evaluation of the radial path integral reduces to the simple application of techniques used in the solution of the non-relativistic Coulomb problem. From the resulting Green function, we also obtained the energy spectrum of the Aharonov-Bohm-Coulomb system which reduces to the energy values for the (2+l)-dimensional relativistic hydrogen problem for the case when the magnetic flux $ = 0. We note here that a path integral treatment of the relativistic AharonovBohm-Coulomb problem has also been recently considered [4,6]. The approach adopted in reference [4], however, applied the Levi-Civitd transformation to handle the AB potential and, therefore, differs from the procedure presented here. The system tackled in reference [6], on the other hand, deals only with spinless particles which obey the Klein-Gordon equation. We also note that other systems can be investigated within the relativistic Aharonov-BohmCoulomb framework. This include a spin-i particle moving in the vicinity of a straight cosmic string with a flux $ inside its long lineal structure [16,17], a particle in the field of a massless spinning source [18], and a generalization of the gravitational anyon [19,20] to cases which involve fermions. Acknowledgments Two of the authors (C. C. B. and M. V. C. B.) wish to thank the conference organizers for the invitation to participate in this celebration in honor of Pro­ fessor Ludwig Streit. They are also grateful to the Abdus Salam International Centre for Theoretical Physics (Trieste) where part of this work was done dur­ ing an Associateship visit with support from the Japanese government. References [1] J. Bornales, C. C. Bernido, and M. V. Carpio-Bernido, Phys. Lett. A 260 (1999) 447. [2] Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485; An account of interesting basic theoretical and experimental features of the A-B effect is given in, M. Peshkin and A. Tonomura, The Aharonov-Bohm Effect (Springer Verlag, Berlin, 1989) LNP 340. [3] C. R. Hagen and D. K. Park, Ann. Phys. 251 (1996) 45;

137 [4] D. K. Park and S.-K. Yoo, Ann. Phys. 263 (1998) 295. [5] V. M. Villaba, J. Math. Phys. 36 (1995) 3332. [6] D.-H. Lin, J. Phys. A: Math. Gen. 31 (1998) 4785. [7] L. C. Biedenharn, Phys. Rev. 126 (1962) 845. [8] J. Schwinger, Phys. Rev. 82 (1951) 664. [9] B. S. DeWitt, Dynamical Theory of Groups and Fields (Gordon and Breach, New York, 1965). [10] G. Temple, The General Principles of Quantum Mechanics (Methuen, New York, 1948) p. 92. [11] P. C. Martin and R. J. Glauber, Phys. Rev. 109 (1958) 1307. [12] M. Berrondo and H. V. Mclntosh, Jour. Math. Phys. 11 (1970) 125. [13] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Poly­ mer Physics (World Scientific, Singapore, 1990). [14] R. Ho and A. Inomata, Phys. Rev. Lett. 48 (1982) 231; A. Inomata, Phys. Lett. A 101 (1984) 253; See, also, M. A. Kayed and A. Inomata, Phys. Rev. Lett. 53 (1984) 107. [15] S. H. Guo, X. L. Yang, F. T. Chan, K. W. Wong, and W. Y. Ching, Phys. Rev. A 43 (1991) 1197. [16] M. G. Alford and F. Wilczek, Phys. Rev. Lett. 62 (1989) 1071; M. G. Alford, J. March-Russel, and F. Wilczek, Nucl. Phys. B328 (1989)140. [17] Ph. Gerbert, Phys. Rev. D40 (1989) 1346. [18] Ph. Gerbert and R. Jackiw, Comm. Math. Phys. 124 (1989) 229. [19] Y. M. Cho, D. H. Park, and C. G. Han, Phys. Rev. D43 (1991) 1421. [20] C. C. Bernido, J. Phys. A: Math. Gen. 26 (1993) 5461.

138

STATISTICAL MANIFOLDS: T H E N A T U R A L AFFINE-METRIC S T R U C T U R E OF PROBABILITY THEORY G. BURDET AND PH. COMBE Centre de Physique Thiorique, CNRS-Luminy, Case 907, F 13288 Marseille cedex 9, France. E-mail: [email protected], [email protected] H. NENCKA University of Madeira, Center of Mathematical Sciences, PT 9000 Funchal, Madeira, Portugal. E-mail: [email protected] The methods of differential geometry applied to probability and statistics theories open a new domain of investigation the statistical manifold or Information Ge­ ometry. This framwork provides a geometrical description of statistical quantities and leads to a new approach of complex statistical problems. A pecular feature is that statistical manifolds are naturaly associated to a family of affine-metric geometries. The differential geometry contents of a particular statistical model, the finite probability manifold, is used to exhibit the relations of some dynamical proprties of the relative entropy and self-parallel curves. Applications are done to the learning process of stochastic neural networks.

1

Introduction

The efforts of Fisher 2 5 and Cramer 17 , to give a more rigorous presentation of statistics and to provide new mathematical tools to solve complex statistical problems, were successfull in the forties when C.R. Rao 33 linked statistics to differential geometry. Having in mind the basis given by Cramer and Fisher, Rao could show that Fisher matrix was in fact a Riemannian metric, giving the very first beginning of Riemannian geometry on spaces of probabilities. It was only thirty years after that was introduced by Efron 2 2 , 2 3 an affine connection into the geometry of parameter spaces to interpret the important notion of statistical curvature. Then Dawid suggested the interest of introducing other affine connections 2 0 , 2 1 . This idea to relate together different branches of mathematics, as differ­ ential geometry, probabilities, information theory and statistics, presents all the kind of difficulties due to translation of separated and complete mathe­ matical formalisms, but leads to a new branch of mathematics: the theory of statistical manifolds or Information geometry 3,n,i9,30,34_

139

The second section treats the construction of parametric statistical man­ ifolds. The a-Chentsov-Amari connections 3 ' 4 ' 15 are introduced and the aself-parallel curves 12,13 are discussed. Sec. 3 is devoted to the important case of (±l)-flat manifolds l 4 and some interesting properties are given. The role of (±l)-self-parallel curves in the minimization of the Kullback-Leibler relative entropy 28 ' 29 is studied . In Sec 4 the special case of statistical man­ ifolds of finite probabilities are considered and (±l)-self-parallel curves are constructed. Sec. 5 we apply these results to the learning of the Boltzmann machine 1>2'27. Finaly in Sec. 6, we give an explicit calculation for a toy model based on probabilities on four points. The (±l)-self-parallel curves orthogonal to submanifolds associated to the simplest Boltzmann machine are discussed. 2 2.1

Parametric statistical manifold, an overview Statistical manifold

Let (X, T, /i) be a measure space with fi a a-finite measure and consider a parametric family of probabilities, absolutely continuous with respect to the measure /z,

pfi(e) = {p9,oee,

P*<M},

d

where 8 e R is an open subset (or a family of open overlaping subsets) which defines uniquely the elements of the family. Because of the absolute continuity of P$, the Radon-Nikodym derivative dPg/dfj, exists and the family PM (0) can be realized in terms of random variables:

*M(e) = { » = ^ .

oee)

This realization is called the distribution or likelihood realization. The application ip : Pg >-> 0 defines a chart tp : 7^(0) —* ©• Under general smoothness conditions it is possible to equipe V^{&) with a fully symmetric 2-covariant tensor g: the Fisher metric 25,33 and a fully symmetric 3-covariant tensor t: the skewness tensor 3 - 4 ' 15 . The triplet S — {P^(6), g, t} or equivalentely S = { ^ ( 0 ) , g, t} is called a (parametric) statistical manifold on (X ,F,n). Any real one to one application allows to construct other realizations of the parametric family { P < j , # € 0 } a s a random variable spaces. One which being largely used is the log-likelihood: £ „ ( 0 ) = {ie = lnp e ,a.s.,

Pe

e V»(Q)} .

140

This realization is more adapted to information theory and it will be called information realization. 2.2

Tangent and cotangent plane

For sake of simplicity let us assume that the Pg and Pg- are equivalent for all 6 and 9', Pg = Pg' that is Pg < Pg> and Pg- -C Pg and that 6 is an open set of Rd with coordinates 9 = {91,92,..., 9d, } in the canonical basis of Rd. The tangent plane Tpg at a point Pg e P»(Q) is generated by the tangent vectors atPe,

A = Y.AkWw-

(1)

k

The derivative dkPg of Pg along the direction 9k is a signed-measure absolutely continuous w.r.t. Pg, with Radon-Nikodym derivative: ^=dkeg=dklnpe

(2)

wich verifies Ep.[dktg]=0.

(3)

The randon variables dktg, fc = 1, d are called in statistics components of the score vector and generate a (^-dimensional linear space. The tangent space Tpe at Pg € P^(Q) can be identified with the ddimensional Euclidean space Tg of random variables with vanishing expec­ tation:

Tg = | At e Rx :Ae = £

Ah

(P'We

\,

The space Tg can be equipped with the natural scalar product: =Ep,[AeBe\. The cotangent plane forms on Tpt

12

(4)

T£fl at Pg e P M (©), generated by the differential

d

i=l

can be also realized on the space Tg through the application d

A(a) = £ 0 , ( 0 ) 0 * ,

(6)

141

where

(a*)* = M

Y,

ekjl-Z-'j*(die)jl

... {d~£)i<... (ddl)jd,

the a' and dj verifying the duality relation Ep.[aidit\ = 8ij. 2.3

(7)

S as a Riemaniann manifold

Using the natural scalar product Eq. (4) on T$ we can define a metric tensor on P^(0) in the following way gip§(A,B)

= Ep,[AB],

or in ^-coordinates: 9\Pe (»i. dj) = 9ij = EPB[d{ee djle] = -Ep, [ 9 ^ 4 ] .

(8)

This tensor g is just the Fisher information metric 25 . The choice of this metric is unique under the condition of Markov invariance 32 . The Riemannian geometry is given by the Levi-Civita connection (covariant derivative) VA ■ TP$(P^(e))

-► Tp^P^Q));

B >-> yAB defined by

(9)

v^^Er*/^. k

or by the Christoffel symbols defined by

9\Pa(VaA,d™) = £ryfcflfcm = r«*.

(io)

k

Then r ijfc = 2 t^ffj* + 9j"5ifc - dk9ij],

(11)

with the the symmetry relation r « k = r,-ifc.

(12)

This connection is torsion free, indeed the torsion is defined by the following relation

T(A,B) = VAB-VBA-(AB-BA),

A,B€TPe,

142

or in ^-coordinates Tijk = 9\PAT(dudj),dk)

=Tijk

- f

jik

•

(13)

o It follows, from the symmetry of r , that f(A,B)

= 0,

(14)

Let us now introduce the curvature tensor R(A,B)C

= [V4, V B ] C -

\A,B[C■

Then the components, in the ^-coordinates, of the Riemann tensor are given by o Rijkt = 9\Pe (R(dit dj,di),dt) = 9tm(dif ,km - dj f

m ik

) + YimeTjkm-Tjmef

m ik

.

(15)

By contraction, we obtain the components of the Ricci tensor o

^^ o

.

and the scalar curvature o

A

o *

( 17 )

R

R = T, ife=i

The geodesies are curves s —► 7(5) on 5 , such that 7 := -£ is parallely transported along 7, therefore solutions of the equation 0

V,7 = 0. Which in the ^-coordinate system is rewritten as ...

0

*....

0k + Tij 6l03 = 0 . The geodesic curves minimize the Riemannian 5(i> 7) = Cst, corresponding to ^(7,7)-0.

(18) distance,

indeed

(19)

143

2.4

Affine metric geometries on S

Let us first define the fully symmetric "skewness" tensor t by t]Po(A,B,C)

= Ep,[AeBBC9],

A,B,Ce

TPo,

or in the ^-coordinates Ujk =

EPW

djt dk£].

(20)

a€R.

(21)

The family of affine connections Vg = at,

3 4 15

are the a-connections of Amari-Chensov ' ' . Such a-connections are non Riemannian if a ^ 0. Moreover, the couple (a, —a), a ^ 0 defines a pair of dual connections w.r.t. the metric g in the sense that AglPti(B,C)=g\,,B(VAB,C)+glPB(B,~VAC),

A,B,CeTPt.

(22)

In the same way as for the Levi-Civita connection we define the coefficients of an a-connection by

and the a-torsion f(A,B)

= VAB-VBA-{AB-BA),

A,BeTPe,

or Tijk = 9\Pe f{di,dj),dk)

= f ijk ~ f at.

(24)

Again the a-connections are torsion free f(A,B)

= 0.

(25)

The associated a-Riemannian tensor takes the form R ijke = 9em (% Tjkm-

dj f

m ik

) + f tmi f jkm - f

giving rise to the Ricci a-curvature tensor

fc=i

and to the scalar a-curvature defined by d

i

jmt

f

m tk

,

(26)

144

The a-self-parallel curves or "a-geodesics" are the curves s —► 7(s), with 7 := dj/ds being parallely transported along 7 with respect to the a-connection. Then: V,7 = 0.

(29)

In the ^-coordinate system the equation for a-geodesics takes the form ...

a k . .

ek + r 0 i V = o.

(30)

But the a-selfparallel curves, with a ^ 0 do not minimize the Riemannian distance, indeed fe9(i,i)

= <xt(i,i,y) -

(31)

then the a-self-parallel curves do not admit 5(7,7) = Cst, as a first integral. 3 3.1

Exponential families The manifolds

We consider in this section important (±l)-flat families of probability distri­ butions: the exponential families, dominated by the probability P 12 . Let P be a probability on (X,.F,/x), the probability P' is of exponential type with respect to P if the logarithm of the Radon-Nikodym derivative dP'/dP is given by an element of the linear space generated by the score vectors defined in the subsection 2.2. Let us denote by Tp the set of (centered) random variables over (X, T, P) which admit an expansion in terms of the scores under the following form Tp = {XERn

\ X-

Ep[X] = g-p\Ep[Xdi},d£)}

.

By direct inspection, one finds that the log-likelihood £ = In p of the usual parametric families of probability distributions belong to T p as well as the difference I — £* of log-likelihood of two probabilities of the same family. So we want to study the consequences of such empirical properties. Proposition 3.1 Being given an exponential family of probability distribu­ tions, the metric is generated by the (-l)-Hessian of the corresponding entropy VdlPH(P)

= -glP,

where H{P) = — Ep[£] is the entropy of the probability P

(32)

145

Indeed, for an exponential manifold one has V d£=-g-^-t(de),

(33)

and it is easy to verify that = -g[P+l-±^t(J)\P,

Vd]PH(P)

(34)

where J denotes the "entropy current" J = g-\dH).

(35)

Moreover, the a-curvature tensor of the torsionless a-connection is given by R(X,Y)d£=^—^-

{t{t(d£,X),Y)

-t(t(d£,Y),X)}

= ~R(X,Y)di.

(36)

An algebraic computation leads to the P r o p o s i t i o n 3.2 Let P and P* be two probability distributions of an expo­ nential family, then Vd|P

K(P;P')=g\P

(37)

Vd|p.Ar(P;P*)=S|p., where K(P;P*)

= EP[e-e*}

(38)

is the Kullback-Leibler relative entropy. More generaly, one verifies that: V d | P K(P;P*)=g\P Vd|P.K(P;P*)=5lp. 3.2

+

±±ZEP[(e-e*)de®d£]

(39)

+L^t(Ep[diP.t}).

The (±1)-self-parallel curves

To obtain properties relating (±l)-self-parallel curves to the relative en­ tropy ( 12 ), we consider the following identities ^ K ( P ( S ) ; P * ) = VdlpK(P;P*)(7s,7,)+ff(7«,grad|pK(P;P*)X40) | ^ ( P ; P*(t)) = V d| P . K(P; P*)(7t*,7,*) + 9\P> <7«,grad | P .K(P; P')X41)

146 where P belongs to 7. Let us suppose that 'y(s) is (-l)-self-parallel, using the proposition 3.2, one gets

£.j A - ( P ( * ) ; P ' ) = | , ( 7 . , 7 . ) .

(42) P i ds ' The lenght of the velocity field appeared at the right hand side and does not depend on P*, so the differential in P* is null X 1

d,p. (^K(P(s);n

= ^(d,P.*(P(s);P*)) =0.

(43)

Therefore, grad|p.i^(P(s);P*) is a vector field linear in s. Now, to express that the differential in P* of the relative entropy vanishes if P* = P , the origin of 7 is fixed at P(0) = P*. Hence, the set of points P(s), satisfying the relation grad | P .X)(P(s);P(0)) = - 7 0 s , belongs to the (-l)-self-parallel curve 7 of which s is the affine parameter and 70 the initial velocity. Finaly, one gets the interchanging the roles of P et P* and working with the identity (41) Proposition 3.3 The parametrized (±1)-self-parallel curves of a 1-flat sta­ tistical manifold are explicitly described by the gradients of the relative entropy, (-1) - self - parallel curves : (grad {P. K)(P(s)\ P(0)) = -70s (+1) - self - parallel curves : (grad | P A')(P , (0); P*(t)) = -%t

(44) (45)

Hence, a theoretical stand is given to the steepest gradient optimization method in the statistical manifold framework. 3.3

Minimizations of the Kullback-Leibler relative entropy

Let P be a current probability on a curve 7 in the statistical manifold <S. Being given P* e S, with P* ^ 7, the position of P which minimizes localy the Kullback-leibler relative entropy K(P;P") = Ep[£ - £*}, if it exists, is given by the solutions P(so) of the equation

±K(P(s);P*)=0 i.e. by the s — so solutions of 0(7.,(grad|Ptf)(P;P'))=O.

(46)

On the other hand, by proposition 3.2, the relation (40) can be written as

147

^K(P(s);P*)

= glP(is,js) +g(lss,gTadlpk(P;P*)).

(47)

Due to the properties of the Riemann metric, the first term in the left hand side is positive and from the minimization condition one deduces that the second term vanishes at so showing that the extrema with 7S ^ 0 are minima. Then, from relation (45), it follows the Proposition 3.4 Let 7* be a (1)-self-parallel curve with t as affine parame­ ter, crossing for t = 0 a curve 7 in P(so). Then, for any t, K(P(so); P*{t)) is minimizing K(P(s);P*(t)) locally, if the curves 7 and 7* are orthogonal, i.e. «/s(7 5o ,7o) = 0.

P*(t)

P(so) = P*(0) ^P(s) "courbe (l)-auto-parallele Figure 1. Minimization of K(P(s); P*(t)) w.r.t. the parameter s

As previously, the above analysis can be done by interchanging the roles of P and P*, so one gets by relation (44) the proposition Proposition 3.5 Let 7 be a (—1)-self-parallel curve with s as affine parame­ ter, crossing for s = 0 a curve 7* in P*(to). Then, for any s, K(P(s); P*(t0)) is minimizing K(P(s);P*(t)) locally, if the curves 7 and 7* are orthogonal, i.e. 1/5(70,7*0) = °These properties can be extended to congruences of curves, i.e. to submanifolds of 5, allowing for instance to define the orthogonal (±l)-projections minimizing the relative entropy localy. This is the geometrical version of the Pytagorian theorem of Csiszar 18,19 , see also Amari 3 ' 6 .

148

r Figure 2. Minimization of K(P(s); P'(t)) w.r.t. the parameter t

4

Statistical manifolds of finite probability distributions

4-1

The manifold

Let (X, V{X), P) be a probability space, where X = {xo, x i , . . . , x^} a finite space of cardinality | X |= d + 1 and 77(X) the algebra of all mesurable subsets of X, | V(X) |= 2 n + 1 . This algebra is generated by the family {{xo}, { x i } , . . . {xrf}} of the d + 1 one point subsets of X. The finite prob­ ability P, absolutely continuous with respect to the uniform measure is well defined by its distribution function: p(xk) = P[{xk}} = Pk,

Vxfc 6 X,

d

(48)

fc=0

The distribution p appears as a random variable p : X —* [0,1] which can be written: d

d

P = ^2pkSXk,

Inp = £ = ^2ekSXk,

k=o

4=lnp f c

fc=o

where

*-»(*) = {J

lfX = Xfc

'

otherwise.

Then the d-dimensional manifold S of finite probability distributions on d+ 1 points is defined by one of the natural coordinate systems {pi} or {9i},

149

i =

l,...,d, d

d

p = Y,PiSXi + ( i - EPtJ^xo. t=l

t=l

^ = E ^ x t + ln(l - I ) e fl 0, i=l

^.2

(49)

»i = l n ^ ,

i=l,...,d.

t=l

T/ie Fischer metric

The Fischer information metric is given by 9\P = ^,9ij(p)dpi

®dpj=y^(-il + —)dpi<8> dpj, .-.• \ P» Po/

a

where po = 1 - E Pi an< ^ 9ij(p) = ^p

[ ^ P < ^ P 3 ] (P)>

(50)

&n<

* i t s inverse

i=l

V = E »ij (p)^« ® m* = E (**« - MA de"> ® 9 ^ •

4-5

(51)

77ie skewness tensor

The skewness tensor is defined through the fully symmetric 3-covariant tensor: *\P

=

E*ijfc(P)dPi ® d Pj ® d P*

ij

E f^-^)^® *>»**■ ijk

(52)

Let us introduce the mixed tensor *\P = E AjdPi ® d Pj ® 9Pfc. f'

- V t

i.nkt

Hj ~ I, l*}k9 k

-

5ijSi

p,

Pi

Pi

'

l) Oxj

Pt

(53)

Pn

a

The components of the a-connection V takes the form,

T% = - ^ h ,

(54)

150

and the components, in the p's coordinates, of the Riemann tensor 1 - a2,

a

R hijk

4

, (9hj9ik - 9hk9u),

ah Rijk

1 - a2 = —j—(tj9ik

- Sk9ij),

(55)

those of the Ricci tensor 1 -a2 Rij = — : — {d-

(56)

l)9jk,

and the scalar curvature is given by a 1-a2 R = — T - d(d-\).

(57)

It follows that the a-scalar curvature is constant for any a and that the statistical manifold of probabilities on a finite set is (±l)-flat, the Riemannian geometry being obtained for a = 0. Finally, the a-self-parallel curves are solutions of the following system of differential equations 2\ ^

(

Pi =

1+a

Pt

d

-Pi

EPj

2

jr[Pi

-,

PO

pa = \-Y^Pj-

(58)

j=i

V

I)

It follows that the (-l)-self-parallel curves are linear in t, Pi(t)

= Pi(0) +

Pl(0)t

(59)

and the (l)-self-parallel curves correspond to Pi(0)exp(ff,t)

(60)

Pi(t) = (1 + £ Pi(0)(exp(Kit) - 1)) 3=1

where d

£ «-*$+£*$. *<w-'-i>«»-

(61)

151

5

Symmetric Boltzmann machines

5.1

The manifold

A symmetric Boltzmann machine L2.5.7-8,16,27,31 w j t n n n e u r o n s (B(n)) is a probabilistic formal neural network, the architecture being given by a non-directed simplicial labeled graph Q = (V, £), with set of vertices V = {u\ ,V2,...,vn} and set of edges £ = {e\, e2, • •., e m } . The graph is not neces­ sarily complete but we prescribe that each vertex has a high degree of connec­ tivity. Because the graph is simplicial and non-directed, between two vertices i/i, i/j there is at most one edge e = e^j = (vi, Vj). The weight Wij = W(ei}j) of the edge eitj is symmetric Wi, = Wa,

W« = 0.

On each vertex there is a probabilistic binary automata with state space {0,1}, then the configuration space of the network is X — {0,1 } n with cardinality d = 2 n . For sake of simplicity, we assume in the following that each neuron is an input and output neuron. Then the ambient probability space (S) is the family of finite probabilities on the X and depends on 2 n - 1 parameters. Moreover, the log-likelihood £ — lnp can be written as follows n

n

e(x) = ^e\>xtl+

£

ii = l

ii
e^xtlxl2 +

...+6Xi>-i«xllxi2...x,n-V(e),

(62)

with ii < i2 < . . . < in- The normalization constant $(0) being related to the entropy H by

T,ei^-^rii2-i'+H(P),

m = where ri™"1* =

Wr

= EP\xuxl2..

.x t ,.].

(63)

Let us notice that this parametrization is well adapted to describe higher order Boltzmann machines 9 . This family of probability is (±l)-flat and has the 0's as natural parametrization. The TJ'S define an other parametrization the "momentum parametrization" 5 .

152

Neurons are updated according to the state of the connected neurons. An individual neuron at vertex Vi is, at time t + 1, in the state 0 or 1 according to the probability law: (

1 with probability pl0 = _ flh ./ t » Xi(t + l)=l * l + e "Mt)) (64) 0 with probability 1 -pxp where ft is a characteristic parameter of the neurons and hi(t+l) is an effective field defined by 1

n

hi(t) = hi(x(t)) = Y,Wijxj{t),

(65)

i=l

to simplify, the threshold of neurons is chosen to be at zero. The updating procedure (64) does not define the dynamics of the net­ works, we need to precise how the states of the network are updated. In the following, we consider two extreme cases where the updating is sequential or parallel respectively. 5.2

Sequential dynamics

In a sequential Boltzmann machine l,2(Bseq(n)) one neuron is updated at each step of time along the probabilistic law (64), this neuron being chosen with the uniform law. This procedure determines a Markov chain x(t) on the set of configuration X = {0, l } n and is defined by the transition probability: ifu = u
(iPk(u) P[x(t

+ 1) = v\x(t)

= u] = <

l-Efi(u)

if« = «,

,

(66)

t=i

.0

otherwise,

The configuration u^ = { u i , . . . , 1-ujt,..., uj,}, Uj 6 {0,1} is obtained from u = {ui,...,Ufe,..., u<j} by the change of the state of the neuron x^ alone. The transition probability of this change is given by Pk(u) = P[x(t + 1) = u<*> I x(t) =u}= with AEk{u) = E(u^)

1+JAEk{u),

(67)

— E(u) where E(u) is the energy function 1

n

E(u) = - - J2

n w

u u

Hii =- E

WiiWi-

( 68 )

153 The transition probability (66) defines an irreducible Markov chain with Gibbs probability measure Gp(u) as unique stationary probability p-0E(u)

G

z

*(«) =-77m-.

w=

E

KP)

e 0E{u)

~

>

(6g)

«€{0,1}"

or taking the logarithm n W

Huiui ~ inZ(P) ■

\nG0(u) = -P Ys

(70)

i<j = l

The family of all Gibbs measures is a submanifold Q of <S, defined by the linear equations

f

q

=-pwiit

(n)

then Q is a (+l)-linear submanifold of S which inherits of the dually flat structure of S. By proposition of section 3, because Q is (l)-flat submanifold of <S there exists a unique (-l)-self parallel orthogonal projection Gp on Q of the probability P. The projection Gp is the best approximation of P by a probability in Q. By the Pytagorian theorem 5 if G is a Gibbs measure we have for the Kullback-Leibler relative entropy K(P; G) = K(P; GP) + K(GP; G). Then the minimal value for the Kullback-Leibler relative entropy is fixed by the projection property and is given by min K(P;G) =

K(P;GP).

Then the learning method proposed by Ackley, Hilton and Sejnowski 2 which consists to minimize the Kullback-Leibler relative entropy K(P;G), is reduced to a motion of the current point G toward Gp in Q. This learning procedure is implemented by the steepest descent gradient method which consists in the modification of the synaptic weight Wij in Wij + 6Wij with SWtj

= -lYjJ^dw^ {ki}

kl

=

^Y.9iJM{Ep\ukUt]

- Ea[ukut)},

(72)

{ki}

where t is a sufficiently small parameter. Let us remark that under this infinitesimal transformation the momentum coordinates 77^ = /3Ec[uiUj] of W^ in Q is changed in TJ% + 5t]%,

154 Sr^ = Y,9HMSWu.

(73)

{kt)

Then the gradient learning method for a B se ,(n)-machine consists to follow the (+l)-self parallel curve in Q from the current point G to the (-l)-self parallel orthogonal projection Gp of P e 5 on Q. This method provides a minimization algorithm of the Kullback-Leibler relative entropy K(P; G) in G. 5.3

Parallel dynamics

In a parallel Boltzmann machine l'vo {Bpar(n)) all the neurons are updated simultaneously, then, for thresholds taken at zero, the transition probability takes the form: n

U[x(t + l) = v\ x(t) = u] = n

.

n

M«) = 5ZW«U>-

1 + e0{l_2vi)hi{u),

( 74 )

1

t=l

i-

In this case the Markov chain defined by this transition probability has also a unique stationary probability measure:

nsW=

m S*lw")cosh i**0=W) § (eSM")+0 • (75)

and the log-likelihood

lnll0(u) = 0J2

E

»=i j=i

W

**i + E^cosh ( | £ i=i

j=i

^i)

+ ln-WTff,-

( 76 )

^

Let us remark that for a parallel dynamics the submanifold V of the output stationary measures has a bondary. Indeed, in this case the stationnary measure depends on the Wij by terms of the form z^ = e?Wij cosh § Wy which are bounded below by | (we discuss this aspect on a toy model in the next section). Moreover, for n > 2 the log-likelihood (76) cannot be written in the form of Eq. (62) where the B^ t , are linearly related. Then the submanifold V of the output stationary measures of a given parallel Boltzmann machine is not a (l)-linear submanifold of 5 and even not a (+l)-convex submanifold in the sense that (+l)-self parallel curve joining two elements of V is in V.

155

The uniqueness of the (-l)-projection is not automatically insured and needs a study of the (±l)-self parallel curves. The classical learning algorithm is always based on the minimization by the gradient method of the Kullback-Leibler relative entropy K{P-,IL,)=

£ P(«)ln^, ue{o,i}"

and leads to 5W 0x3

_

y ^

c

= e0 £

9

.iiju*K(P;n,) dwkl '

9'jM{EP\uk(t)ue(t

- 1)] - En[uk(t)ut(t

- 1)]

{kt)

+EP[uk(t - l) u / (t)] - En\uk(t - l)ue(t)}} .

(77)

which does not depend on the present configuration but rather on two consec­ utive points of the trajectory, then the geometrical interpretation is no longer simple and needs a further study to give a characterization of the learning trajectory. 6 6.1

A toy model The manifolds

Let us consider the simplest Boltzmann machine with two neurons. It will be used as a toy model. Clearly, the associated statistical manifold has specific properties due to the small dimension, but allow to an explicit calculus work out and gives some light on the learning of stochastic neural networks. In this model, the family of ambient probabilities are the probabilities on X = {0, l } 2 . We denote by x = {u2,ui} with u* € {0,1} an element of Q, and put p(0,0)=po,

p(0,l) = P l >

p(l,0)=p2,

p ( l , l ) = P3

(78)

with Po = 1 - P i - P 2 - P 3 Let £ be the logarithm of a probability of the ambient family t = ]np = e\ui+

e\ u2 + e\h2} u l U 2 - V(0),

W)

= - lnpo-

(79)

156 with * } = ! „ £ ! , 0? = l n ^ , tf11+fl? + ^ 1 ' a } = l n ^ (80) Po Po Po The B seg (2)-machine is characterised by the submanifold Bseq of probabilities, 1_ 3 + z'

Pi = P2 = PO P3 =

.

z = ew

(81)

3TI

or equivalently by the equations f Pi = P2 = P, 1 ps = 1 - 3p

o < P < < i , J2P*

=1

(82)

-

i=0

The Bp(ir.(2)-machine is characterised by the submanifold Bpar of probabilities, Pi =P2 = (1 + z ) 2 ' z — ew cosh W

(83)

P3 = (1 + zf it follows that po = 7T+IT7 a n c * t n e e Q u a t i o n s c a n

De

written 3

Pi = P2 = P, P = \^5i - p 3 ,

0 <J»4 < 1,

P3 > I

X ^ P . = 1-

(84)

i=0

Let us remark that the submanifold Bpar is bounded by p\ = P2 = §, P3 = | . 6.2

Tangent and orthogonal vector fields to the B submanifolds

Since the submanifolds Bseq and Bpar have the common characteristic p\ P2, then it is natural to introduce the coordinate system {b = p> ~^Pi, p EL £l | , P3}, in which the metric tensor takes the form

/ pT^i g = (db, dp, dp3) <8>

6 p'2-b3

V 0

-?rr^ 2p p2-62 _2_ Po

, _£ po

° .2. Po

\ / d6\ dp

*+*/ v<w

(85)

157

with p 0 = 1 - 2p - p 3 , and its inverse / f(l-2p) 3

J

-4(1-2p)

= (3&, dp, 9,P3^ \

-PP3

- | ( 1 -2p) ±p-6

-pp3

2

-t>P3

/M

\

(86)

-&P3

P3(l-P3)/

\<W

The restriction g\,b=0) of the metric g to the submanifold {b = 0} takes the form 0 9\u.=0] = (db> d P- dp3)®

0 a + JP

A

PO

\

fdb\ dp

(87)

PO

vo i i+i/

w

withpo = 1 - 2 p - p 3 . The tangent vector field V on {b = 0} is defined, up to a multiplicative factor, by V = Xdp + (j.dP3

The orthogonal vector field N to V in the ambient space N = adb + pdv + 7dP3 is defined, up to a multiplicative factor, by

which gives 7 P3 ... HP + A(l - p 3 ) (88) — = r — with r = P ' P Mp-5)-Aps" Then the vector field normal to the integral curve B of the vector field V — Xdp + jj.dPi is given, up to a multiplicative factor, by N ~ 0 (dp + r ^ dp3\ + adb,

(0,p,p 3 ) € B

(89)

Now, we are interested in the family of (±l)-self-parallel curves, (b{t),p(t),p3(t)) orthogonal to the integral curves Baeq and Bpar for t = 0. The (-l)-self-parallel curves normal to the integral curve B in the ambient space are given by Eq. 59 with 6(0) = 0,

158 b(t) = 6(0) t p ( t ) = p ( 0 ) t + p(0)

V3(t)=(r^t

,

(90)

+ l)p,(0)

and the (l)-self-parallel curves normal in the ambient space are given by Eq. (60) also with 6(0) = 0, b(t) = f $ exp {p(0) t [(2 + r ^ g ) ^

+ jjjy] } sinh * $

Pit) = f § exp {p(0) t[(2

+ jfo]} cosh %$ ,

+ r»ff)

^

(91)

rtt) =PMexp {m * [( 2 +i$?) ste + sfa]} where D

W

=

P

o(0)+ex

P

[(2

+

r^)^]

x {p(0)exp ( $ j j f ) 2cosh ^

+ p 3 (0)exp ( r $ j j f ) } '

Let us first remark that the vector field normal to the integral curve B in (0,p(0),p 3 (0)) € B, corresponds to 0 — p(0) and a = 6(0) arbitrary, V = Xdp 4- ndP3 takes the form, N ~ p(0) (dp + r £jg> 9 P 3 ) + 6(0) a6, 6.5

(0,p(0),p 3 (0)) € B

(92)

The submanifold Bseq

The restriction of the tangent vector V to the submanifold Bseq is given by A = I and fi — — 1, up to a multiplicative factor, V«, ~ \dp - dP3

(93)

then r,eg = 0 and

7, e g = 0.

(94)

The (-l)-self-parallel curves normal to the integral curve Bseq in the am­ bient space are given by Eq.(59) with 6(0) = 0, one gets 6 ( 0 = 6(0) t p(t)=p(0)t + 1\rz,

z = ew>0

(95)

159

(• I (-self-parallel curves

(I )-self-parallel curves

Figure 3. (±l)-self-parallel curves orthogonal to B,cq with 6(0) = 0

They correspond to line p - T ^ 6 = 5^7 in the surface P3(t) = 3 ^ . It follows that each distribution of probability p{u2,u\) in the ambient space has a unique (-l)-projection on Baeq. The (l)-self-parallel curves orthogonal in the ambient space are given by Eq. (60) also with 6(0) = 0,

w- e x p ga o ) 0 ^ h {(3+4(o)n P(t) =

e x p

\ l ^ ^

cosh{(3 + z)b(0)t}

z = ew,

(96)

*»(o - * exp\Wl\m where ^(<) = 3JT [ 1 + '"rWW'JWl

(2 exp{(3+ 2 )p(0)() cosh{{3+z)b(0)t} + z)] .

It follows that each distribution of probability in the open simplex 0 < p{u2,u\) < 1 has a unique (l)-projection on Baeq. However, iit is not true for the limit points p = p3 = 0 and p = 0.5, p$ = 0 corresponding to t —► TOO which have an infinity of orthogonal projection on Baeq. Let us note that for this probabilities the Kullback-Liebler relative entropy K(G; P(=poo)), where G € Bseq is diverging.

160

6.4

The submanifold Bpar

riction of V to 1the submanifold Bpar corresponds to A = 1 — The restriction M = -1> Vpar

^(1-^)dp~dj'3

YT**

an(

^

(97)

then Tpar = 1 + J^ ^

Z l = z= = eeWHCOSh ' c o sWh ^>>=; i = _ - I- ,, Z

(98)

and Tpar —

Ppar-

I"")

The (-l)-self-parallel curves orthogonal to the integral curve Bpar in the ambient space are given by Eq. (59) with 6(0) = 0 b{t)=b(0)t,

p{t)=p(o)t + wfar,

(ioo)

they lie on the surfaces 2

P + P3 = 7 — • 1+Z Therefore, due to the constraint z > ^, one deduces that the probability distributions which are such that P + P3 < | do not have a (-l)-projection on Bpar, but every others have a unique (-l)-projection on Bpar. The (+l)-self-parallel curves orthogonal to Bpar, in the ambient space, are given by Eq. (60) also with 6(0) = 0, exp{iii^p(0)t}

W) = *

Uiym)

j

sinh{i^-b(o)t}

exp{ii^P(0)(}

Pit) = *

U)»»to

;

,

(101)

cosh{^6(0)0

where D t

( ) = ( T T ^ ' [ 1 + 2 z e x p j ^ ^ ^ P W t j c o 8 h { i i i ^ 6 ( 0 ) t } + 2 2 exp{(l + z ) 3 i p i p ( 0 ) t } l .

161

N^.

* =

3

V

\

* " ' " ■

"'-..

'"-..,

s S

\

'""""■-■■

v

\ ® \ V

N N

*=■; z=0.5

r*:

N

i~.

r~.

71

N Tv

(-1 )-self-parallel curves

(1 )-self-parallel curves

Figure 4. (±l)-self-parallel curves orthogonal to Bpar with 6(0) = 0

they lie on the surfaces In

z(l-2p-p3)

vV362

z In-

P3 2vy^62

Here we are faced with a new situation, again there is an open subset in the ambient space for which probability distributions do not have a (+1)projection on Bpar, but probability distributions belonging to the complementatary subset admit a two-fold (+l)-projection on Bpar characterized by z > 1 and 2 < 1. This reflects in the fact that there are three kinds of limit points which project • (p = \, P3 = 0) —> on the whole B v -'part • (p = 0, P3 = 0) —> on the half part of Bpar corresponding to z > 1, • (p = 0, P3 = 1) —► which project on the other half where i < z < 1. (See Fig. 4 exhibiting graphs of some self-parallel curves, characterised by the initial condition 6(0) = 0, therefore lying in the plan 6(0) = 0 of this Euclidean drawing.) Remark. The learning of the B por (2)-machine by minimization of the Kullback-Leibler relative entropy allows only to reach a subfamily of the am­ bient probability distributions. This toy model has specific properties due to the low dimension of underlying space, nevertheless it shows that parallel computations can lead to non-converging domain for the algorithm, even if sequential one are converging.

162 Acknowledgments Two of us Ph. C. and H. N. are partially supported by the projects: PRAXIS XXI and pluriannual, Portugal. References 1. E. Aarts, J. Korst, Simulated annealing and Boltzmann machines. A stochastic approach to combinatorial optimization and neural computing (Wiley and Sons, 1989). 2. D. Ackley, G. Hilton, T. Sejnowski, A learning algorithm for Boltzmann machine, Cognitive Science 9, 147-169 ( 1985). 3. S.-I. Amari, Differential Geometrical methods in Statistics, Lecture Notes in Statistics 28, (Springer Verlag 1985). 4. S.-I. Amari, Differential Geometrical Theory of Statistics - Towards New Developments, in Differential Geometry Statistical Inference ed S.S. Gupta, IMS Lecture Notes Monographs series 10, 19-94 ( 1987). 5. S.-I. Amari, Information Geometry of Boltzmann Machine, IEEE Trans­ actions on Neural Networks 37, 260-271 ( 1992). 6. S.-I. Amari, Information Geometry , Contemporary Mathematics 203, 81-95 ( 1997). 7. R. Azencott, Synchroneous Boltzmann machines: Learning rules, in Pro­ ceeding of Neural Networks (les Arcs) NATO series , (Springer Verlag 1990). 8. R. Azencott,A. Doutriaux, L. Younes, Synchroneous Boltzmann machines and outline based image classification , in Proceeding of Neural Networks (les Arcs) INCC , (Paris 1990). 9. R.Azencott, Boltzmann machines: Higher-order interactions and syn­ chronous learning, in Stochastic models, statistical methods, and algo­ rithms in image analysis. Proc. Spec. Year Image Anal., Rome:Italy 1990, Lect. Notes Stat. 74, 14-45 (1992). 10. B. Appoloni, D. de Falco, Learning by parallel Boltzmann Machines, IEEE Transactions on Information Theory 37, 1162 ( 1981). 11. O. Barndorff-Nilsen, Parametric Statistical Models and Likelihood, Lec­ ture Notes in Statistics 50, ( Springer Verlag 1988). 12. G. Burdet, Ph. Combe, H. Nencka, Statistical Manifolds, Selfparallel Curves and Learning Processes, Progress in Probability 45, 87-99 ( Birkhauser 1998). 13. G. Burdet, H. Nencka, M. Perrin, An Example of Dynamical Behaviour of the Relative Entropy, Contemporary Mathematics 203, 97-104 ( 1997).

163 14. G. Burdet, H. Nencka, Equation of Self-Parallel Curve deviation on Sta­ tistical Manifolds, Methods of Functional Analysis and Topology 3, 4650 ( 1997). 15. N.N. Chensov, Statistical Decision and Optimal Inference, in Russian (Nauka, Moscow 1972), English translation AMS 53, (Providence, R.I. 1982). 16. Ph. Combe, H. Nencka, Information Geometry and Learning in Formal Neural Networks, Contemporary Mathematics 203, 105-1116 ( 1997). 17. H. Cramer, Mathematical Methods of Statistics, ( Princeton University Press 1946). 18. I. Csiszar, I-divergence geometry of probability distributions and mini­ mization problems, Ann. of Probab. 3, 146-158 (1975). 19. I. Csiszar, J. Korner Information Theory, Academic Press , (1981). 20. A. P. Dawid, Discussion on Pr. Efron's paper (1975), Ann. Stat. 3, 1231-1234 (1975). 21. A. P. Dawid, Further comments on some comments on a paper by Bradley Efron, Ann. Stat. 5, 1242 (1977). 22. B. Efron, Defining the curvature of a statistical problem, Ann. Stat. 3, 1189-1942 (1975). 23. B. Efron, The geometry of exponential families, Ann. Stat. 6, 362-376 (1975). 24. W. Feller, An introduction to probability theory and its Applications, 3rd ed.Wiley 1968. 25. R.A. Fisher, Theory of statistical estimation, Proc. Camb. Phil. Soc. 22, 700-725 (1925). 26. R.J. Glauber, Time-dependent statistics of the Ising model, J.Math.Phys. 4, 294-307 (1963). 27. G. Hilton, T Sejnowski, Learning and relearning in Boltzmann machine, in Parallel Distributed Processing: Exploration in Microstructures of Cognition eds. L.L. McClelland, D.E. Rumelhart, Vol 1, ch 7, 147-169 (MIT Press, Cambridge, 1986). 28. S. Kullback, Information Theory and Statistics, (Wiley, New York, 1959). 29. S. Kullback, R. Leibler, On Information and Sufficiency, Ann. Math. Stat. 22, 79-86 (1951). 30. M.K. Murray, J.W. Rice, Differential geometry and Statistics, (Chapman & Hall 1990). 31. P. Peretto, Collective properties of neural networks : a statistical physics approach, Biological Cybernetics 50, 45-62 (1984). 32. D.B. Picard, Statistical morphisms and related invariance properties, Ann. Inst. Statist. Math. 44, 45-61 (1992).

164

33. C.R. Rao, Information and the accurency attainable in the estimation of statistical parameters, Bull. Calcutta Math. Soc. 37, 81-91 (1945). 34. I. Vajda, Theory of statistical inference and information, (Kluwer 1989).

165

B R Y D G E S ' OPERATOR IN RENORMALIZATION THEORY PIERRE CARTIER Ecole Normale Superieure, 45 rue d'Ulm, F-75230 Paris Cedex 05, France CECILE DEWITT-MORETTE Center for Relativity and Department of Physics, The University of Texas at Austin, Austin, Texas 78712-1081 This note is a tribute to both Ludwig Streit and Sergio Albeverio. Given the schedules of the conferences in their honor and the publications of their festschrifts, as well as our current interest in their works, it is more appropriate for us to make our contributions to these celebrations a joint one. Our joint contribution to both Streit and Albeverio makes scientific sense because we show that the white noise calculus of Streit and the AlbeverioHoegh-Krohn functional integrals are two examples of the same scheme. The scheme is introduced in our paper "Scaling and Functional Integration" sub­ mitted recently to the Canadian Mathematical Society Proceedings which will publish the Albeverio Conference Proceedings. It is summarized here. The initial steps were presented at the Streit Conference in Lisbon. Following the work of D.C. Brydges, J. Dimock, and T.R. Hurd, we define a field f(x) for x in R4, or x in spacetime K 1,3 , as an integral of scale dependent fields over a scaling variable I £ [0, oo[. Let / be a functional integral over such fields of the type used in Quantum Field Theory, /^DxV-expQsfa))

^ X

(1)

which can be rewritten in terms of a gaussian integrator fio I =j

dpab)

• exp Q I/fa)) = (ua, exp Uu^j

\ .

(2)

For instance, if the covariance G is given by ~C< for x e R d , a i \x — y\ one decomposes it into scale dependent contributions. G(\x - y\) =

(3)

/»00

G(£) =

d*l-St

u(0

,

d* I = dill

(4)

Jo where St u(£) = £'"' u(f/£)

,

[u] = physical length dimension of u

(5)

166

and /•OO

dx k ■ jfe-M u(jfc) = - Cd .

/>0 Jo

(6)

The domain of integration of the scaling variable [0, oo[ can be broken into a OO

sum V , [2J^0) 2 J + 1 £Q[. The corresponding decomposition of the covariance, j=-oo + O0

G

(2>
(7)

221 V[Vto,V + lto{<

(8)

= 22

G

] — — oo

induces decompositions of the fields +00

^~

i=-oo

of the gaussian na, and of all quantities defined in terms of no, in particular the functional laplacian" A

G = ^~ f dx f dyG{x,y)

*€{1,*}>

(9)

the Bargmann-Segal transform Bc =MG* = e x p - A c ,

(10)

and its inverse the Wick transform ::G=exp(-iAc) .

(11)

Henceforth the suffix G is replaced by the interval defining the scale de­ pendent covariance; for example ^Ko,oo[ = ^ [ / o , * [ * A * [ * , o o [ -

(12)

A key integral in Quantum Field Theory can be written, with the notation of (2) I (to) = (/^„,oo[, Z) = (H[t,oo{ , H[io,t[ * Z).

(13)

The convolution H[ia,t[ * Z integrates out the contributions of the scale depen­ dent fields in the range [IQ, ^[; it substitutes to the integrand Z an ^-dependent "With our normalization the covariance is given by J x dudf)

f{x) f(y) = T~ < -*( I > y).

167

effective integrand; in the case of (1) (rewritten as (2)), it substitutes an ef­ fective action to the action 5. An important operator Pi introduced by Brydges provides the means for constructing a parabolic equation satisfied by the effective action. The operator Pi rescales the Bargmann-Segal transform so that all integrals are performed with an ^-independent gaussian; indeed let Pe = St/loB[toA

(14)

with S defined by (5) and B defined by (10), then
(15)

The scaling evolution equation defined by Pi is readily obtained from the definition of P/: t±{PtZ)=(s

+ \£}{PtZ),

P(0Z = Z

(16)

where t=fn

HhA-

(17)

Equation (16) determines the renormalization group flow equation. We are now in a position to bring together the functional integrals of Streit and Albeverio, as well as several other functional integrals. In (1) and (2), X is the domain of integration of the functional integral. Let ^"(X) be the space of functional, F : X -> K, integrable with respect to the chosen dimensionless volume element dnc\ then T(X) is the domain of the Bargmann-Segal transform BG : .F(X) -► Fock(X).

(18)

In some of Streit's works, JT(X) = L 2 (X,/i G );

(19)

for Albeverio-Hoegh-Krohn, if Fx(
with A a bounded measure on the dual X' of X, then Fx e J-(X).

(20)

168

Other cases include a space .F(X) of polynomials Fn(
exp(27ri{J,
(21) J=0

or a space ^"(X) of distributions whose Bargmann-Segal transforms are entire functional on the complexified Xc of X. In all cases the Brydges operator Pi rescales the convolution BQ = HG *■

169 F L U C T U A T I O N OF THE BOSE-EINSTEIN C O N D E N S A T E IN A T R A P

H. EZAWA Department of Physics, Gakushuin University, Mejiro, Toshtma-ku, Tokyo 171-8588, Japan E-mail: hiroshi.ezawaQgakushuin. ac.jp K. NAKAMURA Division of Natural Science, Meiji University, Izumi Campus Eifuku, Suginami-ku, Tokyo 168-8555, Japan E-mail: [email protected] K. WATANABE Department of Physics, Meisei University Hino, Tokyo 191-8506, Japan E-mail: [email protected] Y. YAMANAKA Waseda University Senior High School Kamishakujii, Nerima-ku, Tokyo 177-0044, Japan E-mail: [email protected] With modified Bogoliubov replacement ao -» -/No + a'0 we study the fluctuation of trapped condensate, No atoms in average, taking into account the interaction between particles inside and outside the condensate. It is indicated that the oper­ ator a'0 should in effect be of O(N0' ) for large No if the interaction is repulsive, based on some anticipation of numerical results still to be worked out.

1

Introduction

Since the first observation of the Bose-Einstein condensation (BEC) in 1995 : with 2 x 103 trapped atoms of 87 Rb at 170 nK, followed by the cases 2 ' 3 , 4 of 23 Na, 7 Li and down to 1 H, experimental and theoretical works have ex­ posed different aspects of this interesting quantum effect. Yet, basic questions remain to be answered. For the free Bose gas, the condition for the BEC is that its number density p be high and temperature T low such that pX3 > 2.162, where A is the thermal de Broglie wave lengthaA := \I2-KTI2/mk^T.

This condition derives from the

"This is not the de Broglie wavelength Ad = 2-nh/p for p 2 /(2m) = 3kBT/2.

170

requirement of maximum entropy. In order to see how the condition is affected by the particle interactions, we have to know the excitation spectrum of the system. The basic tool for the analysis has been the Bogoliubov prescription 5 to replace ao by \/No> where ao is the annihilation operator and 7V0 the number of the condensate particles. In 1967, we proposed6 to modify the Bogoliubov replacement such that ao—>\/N)+a' 0

(1)

by adding another annihilation operator a 0 to preserve the canonical commu­ tation relations, and studied the fluctuation of the condensate to find out the condition that a'0 be effectively small relative to ^/No. However, analysis was difficult because the Goldstone theorem dictates that the energy spectrum of the system with translation invariant interaction extends down to zero as the associated momentum goes to 0, leading to diverging expectation value of a'0 a'0, if we simply let the volume of periodic box be infinite as the momentum summation tempts us to. With the atoms trapped in a finite volume, we are free from the Goldstone theorem on the one hand, and on the other have no longer the representa­ tion question 7 of the canonical commutation relations compelling us to the Bogoliubov relacement. We wish to confirm that the replacement (1), and hence the original one, will serve as a valid method of approximation when the condensation takes place. 2 2.1

Hamiltonian Modified Bogoliubov replacement

In terms of the Bose field operator,

A(X)

= V a n " n ( ^ ) with a complete n

orthonormal system of one-particle states {un(x)} and annihilation operators {a„}, the Hamiltonian of the system is given formally by H := JtfA(x) [~A

+ v(x) -

Mj

<j>A(x)d3x

+ \ J A(x)4>A(x')V(x - x')ct>A(x)A(x')d3xd3x',

(2)

where v(x) is the trap potential and V(x) = V(—x) the interaction, the former varying much more slowly than the latter; /i is the chemical potential. Suppose the system has iV0 3> 1 condensate particles in the state u0(x) to suggest the replacement (1). Hereafter, we shall simply write ao for a 0 .

171

Then, the field operator takes the form,
(3)

where Jo n ti n (x). The system {u n (x)} is left to be determined later. n

2.2

Hartree Potential

If we put (3) in (2), then terms linear in
I {(x)l

+

± j 2

— ^

A

+ v(x)

+ VH(x)

- fl1

K{x)(Px

4>\(x)4>\{x')V[x-x')fl>K{x)4>Kix')#x(Px'

/ ■ A(x)d3x

(4)

by adding and subtracting the term, JA(x)vH{x)^>A{x)d3x, with vH(x) := N0 j V(x - x')ul{x')o9x'.

(5)

Now, we choose the system {u„(cc)} to be the eigenfunctions of h2 h=-—A

+ v(x) + vH(x),

(6)

that is to say that u„'s are the self-consistent solutions to hun(x) = enun(x), where n = 0,1,2, ••• in the linear order of increasing e n ; sometimes n is taken to mean a multi-index, arranged yet in the same order. We can take a system of real-valued functions {un(x)}, which will be better suited for later calculations. Then, we have H = E0 + ^ ( e n - /i)oJ,a„ + V^Vo(e0 - /i)(ao + a o) n

+

~rfr

5 2 ■^(mn(a}aman + a/am a n) + -rr /,m,n

5 2 LMmnaka\aman K,/,m,n

(7)

172

where E0 := N0e0 - l-N2 j u2{x)V{x

- x')ul(x')d3xd3x'

- i ] T Jnn,

(8)

and Lkimn ■■= N0 / u fe (x)u,(a:')V(x - x')um(x)un(x')d3xd3x', which has the symmetry, Lfc(mn = Llknm Jmn

3

=

M)Omn>

= Lmnkl. Klmn

=

(9)

Further,

Loimn.

(10)

Delta-Function Interaction

In view of the long wavelengths of the atoms under consideration compared with the range of their interaction potential V, people take the delta-function approximation, V(x-x')=g6(x-x'),

with g = — —

(11)

where M is the mass and a the scattering length of the atoms. Then, (5) turns out to be VH(X) = gN0uo2(x) and the eigenvalue equation for (6) becomes ( h2 1 | ~2MA + v(x) + 9N0u0*(x) | «„(*) = enun(x),

(12)

of which the one for the ground state n = 0 is the Gross-Pitaevski equation 8 . We remark that while this particular equation for n = 0 is a nonlinear Schrodinger equation, those with n ^ 0 are not, being linear Schrodinger equations with a given potential v{x) + gN0Uo2{x). Therefore, the eigenfunctions {un} constitute a complete orthonormal system. (9) becomes Lkimn - gN0 / uk(x)ul(x)um{x)un(x)d3x,

(13)

acquiring full symmetry under any permutation of the suffices. We have to anticipate that (11) may cause some troubles 9 . We shall be ready to truncate the matrix (Lkimn) when necessary.

173

3.1

Splitting the Hamiltonian

Let us now split the Hamiltonian (7) into four parts, 7i — Tic + %B + %BC + Eo, such that the first part involves only the operators a0, pertaining to the condensate particles, the second only those aj, (n ^ 0) of the particles outside the condensate, the third both. Note that ak means a^ and/or a\. We split each of the operator parts of the Hamiltonian further with respect to the powers in A := l/y/No, nr = \-1n{T-1)+Hi0)

+ \H{rl] + \2n™

(r = C,B,BC)

(14)

and correspondingly Ai:=At(0)+AM(1)+AV2) + -"Thus, the condensate part is given by

(15)

4 - 1 ) = (£o-M(°))(aJ + a0), ?4 0 ) = (eo - / i ( 0 ) ) 4 a 0 - M U) (a 0 + a o) + ^Joo(4 + a 0 ) 2 , HQ 1 ' = -/i ( 1 ) a 0 a 0 - M(2)(ao + a o) + ^oo(aoao + a o a o)'

( 16 )

n{c] = -H{2)alao - M(3)(a0 + <*o) + ^ooa^ao 2 , the out-of-the-condensate part by ?4 0 ) = E ' ( £ " - M ( 0 ) )4an + \Y,'jmn{al Tig)

= -fl^Y^'aldn n

+ am)(al + a„)

+ S ' / f l m n ( a J a m C + a/a^O,,) Imn

^B 2) = -fl{2}^2alian+

^2

(17)

L

ktmnala}aman,

klmn

and finally the interaction between the particles in and out-of the condensate by ^BC = S ' - M o J , + an)(al + a0) n

+ 2(aJ + a0)alnan + o o ^ a * J mn

+H n

Jn0 a

\o

a

n + ^ o M ^ n + an) + ad*ah]

(18)

277 174

+4c4a 0 aJ ri a n + o02oj„ajl) + 2 ^ J„ 0 (aJ aoan + a j a o 2 ^ ) n

Here and in the following,

E

l

means summation that excludes n = 0.

n

3.2

Some estimates

Since the mode functions un are normalized, they are of the order of (volume of the condensate) - 1 / 2 , so that the coefficients L^imn ~ 9P with p being the number density of the condensate. In the experiments reported l • 2 ' 3 , the trap potential v(x) can be assumed to be of harmonic oscillator type, and the corresponding numerical solutions to the Gross-Pitaevski equation are known 11 , so that vu(x) can be calculated. For the sake of orientation, however, we take + y2) + vz2z2}

v(x) + VH(X) = y {I/J. V

.

(19)

Then, the eigenvalue problem (12) has the solution, £nxnvnx

= l(nx

+ Uy + 1) U± +

in the triple index notation n = (nx,ny,nz),

( Uz + - J Vz \ h,

(20)

with

«».»,». (x) = NxNyNzHni (0Hny (v)Hn, (C)e-(«2+"2+«2)/2,

(21)

where £ = axx etc. and

a

' = \n?'

^

=

(^2^T)

Ci = *,y,*).

We have

J°° Hn(0Hm(Z)e-2t2dS = (-!)("—)/2 N / 2 ^^T r fm + n + 1 if n + m = even. Also /

Hk(0Hn(0Hm(0e-2i2dt oo

_ Kj2^r, +m-,T(n + m-k

+ l)r(k

+ n-m+l)T(m

+

k

-

n + 1

■n \ 2 ) \ 2 ) \ 2 if k-n — m =even. The integrals vanish otherwise. So, for instance, we obtain (22)

175 where

is a measure of the number density of the condensate since l/oy measure the extension of the condensate in the direction j , and ^nxnynz\mxmym.T

( r ( _ 1 ) (n J -m j ) /2

TT

(nj+m^-l)!!

\

even)

= <

0

(otherwise)

Table 1.

(^ -I- m _ iV| , " . 2("+ m )/ 2 \/n!m!

Case of (n,m) even-even.

T h e m a t r i x is s y m m e t r i c . Its lower-left p a r t is su ppressed a n d so is 1 for (0, 0).

0 2 4 6 8 10 12 14 16 18 20

2 0.353 0.375

4 0.153 0.271 0.273

6 0.070 0.173 0.225 0.226

8 0.033 0.104 0.165 0.196 0.196

10 0.016 0.060 0.113 0.155 0.176 0.176

12 0.007 0.034 0.074 0.115 0.146 0.161 0.161

14 0.004 0.019 0.047 0.081 0.113 0.137 0.149 0.149

16 0.002 0.010 0.029 0055 0.084 0.111 0.130 0.140 0.140

18 0.001 0.006 0.017 0.036 0.060 0.085 0.108 0.124 0.132 0.132

20 0.000 0.003 0.010 0.023 0.042 0.064 0.086 0.105 0.119 0.125 0.125

We note (n + m — 1)!! 1 ^„2/(s„\ ^, is Vn\m\ vitn as n —> oo. This number is maximum on the diagonal m = n, the maximum value decreasing with increasing n only very slowly (see Table 1). Take the case of 87 Rb atom, 6 having M = 86.90919 a.u. = 1.443162 0 x 10" 25 kg, which was used in the pioneering experiment of M.H. Anderson et al. 1 ; the parameters of (19) they gave were: vz = 2 7 T X 2 2 0 H Z = 7 . 5 X 1 0 2 S - \ b

It is radioactive, undergoing /?

u± = vz/y/8 = 2.7 x 1 0 2 s _ 1 ,

decay with the half life of 4.8 x 10 1 0 yrs.

(24)

176

so that az = 1.01 x 1 0 6 m _ \

a x = 6.1 x 10 5 m _ 1 .

(25)

9

We take a = 6 x 10~ m for the scattering length of Rb, finding g = &x 1 0 _ 5 1 J m 3 ,

gp = (1.4 x HT 34 J)iV 0

or, in unit of the quantum of the phonon, hv± = 2.8 x 10

-32

(26) J,

^ - = (5x 10-3)iV0

(27)

Bl/X

which is of the order of 10 for N0 ~ 2000, the case of ref. : . We shall find that the excitation energies of the out-of-the-condensate particles are enhanced by their interactions (see Table 2 below), somewhat reducing the measure (27) of the strength of the interaction. 4

Effective Condensate Hamiltonian (1)

We shall take the following steps to study the fluctuation of the condensate. 1. Diagonalize HB := W^' in (17). Let its eigenstate \n) € HB be the un­ perturbed state of the out-of-the-condensate particles; HB is their Hilbert space. 2. Take He ■— X~l~Hc~ +'HC as the unperturbed condensate Hamiltonian acting on the Hilbert space He of the condensate particles. 3. Use the perturbation theory of Ezawa and Luban (EL) 6 to construct the effective Hamiltonian operator A for the condensate such that (tfB + tfc + A#i+A 2 tf 2 )Vn = t/>„A,

ViV»n = l ,

(28)

where ipn = W„|n) with Un being an operator in the Hilbert space H B ® H C of the total system, by taking A//i + \2H
1

^ + H& + U^

+ nc1]

(29)

and //2:=^B

2 )

+^C

2 )

+^BC

(30)

4. Determine f/-"^ such that A'") at each order v of the perturbation calcu­ lation has no terms linear in a0. This is necessary to keep the average number 7V0 of the condensate particles fixed by guaranteeing that no change in the first scalar term in our T/NO + ao be needed when diagonalizing A.

177

5. A is the effective Hamiltonian for the condensate in the sense that the solutions of A„Xa = £naXa solve KlpnXa - IpnKXa

= ZnalPnXa

(31)

to give the eigenstates \pnXa and the eigenvalues £.naoi the total Hamil­ tonian H, the operator on the left-hand side of (28). This is basically a Born-Oppenheimer procedure 10 which treats the faster motion first, then proceeding to examine its effect on the slower motion. Now after the first step of solving HB\TI) = Wn\n), the EL method 6 proceeds to solve (28) perturbatively for ^ = (l + AW( 1 )+A 2 ^ 2 ) + .--)|n), An = \-lA
(32)

We now specialize to the ground state of the particles outside the con­ densate. In the lowest order, let |0) be the ground state of HB, and then (28) gives (HB + Hc)\0) = lOHA-'A*-1) + A<°>),

(33)

so that \(-V=H{c-l) A<°> = W0+n^

= (e0-ni0))(al

+ a0)

= W0 + (e 0 -M ( 0 ) )aS«o-M ( 1 ) (4 + ao) + ^ o o ( 4 + oo) 2 . (34)

By the requirement that A ^ should have no terms linear in a', we obtain (o) =_

M

£o,

^(1)=0

and hence A (-D =

o,

\(°) = W0 + ± J 0 o(a 0 + aj) 2 .

(35)

In terms of the canonical pair, Po = -i-j=(ao

-oj),

x0 = -y=(a0+a*0),

\p0,x0] = -i

(36)

we have A(0) = Jooxl,

(37)

which has continuous spectrum. If Joo > 0, or equivalently if g > 0, then the condensate has non-negative excitation energy and hence stable. The expectation values of % can be finite if the condenstate is in an appropriate wave packet state. To proceed to the higher order approximation, we have to finish the step 1.

178

5

Diagonalization of the Out-of-the-Condensate Hamiltonian

In order to diagonalize HB = %B in (17), define the canonical pairs, Pn ■= - « W y ( a n - 4 ) >

Qn ■= J-^-(an

+ a]n)

(n ^ 0),

(38)

with € n := £„ - £o, to obtain

n,m

n

n

with J n m := 2y/enemJnm, which is real and symmetric. We can diagonalize the potential energy part in (39) by an orthogonal transformation, r)k =

Y^TknPn, & = J 2 ' T k n ^

( 40 )

without affecting the form of the kinetic energy part nor the commutation relations. The eigenvalues u\ are all positive if g > 0. Thus,

"B = £ (U + y^ 2 ) - 1 £ ' ^ >

(41)

Therefore, by defining h = J y & + iyJ^-Vk,

such that

[bk,bj] = 6Ut

[bk,bi] = 0,

(42)

we achieve the diagonalization,

HB = £ u,k (b\bk + i ) - \£e™-

( 43 )

Table 2 gives the values of u\ for the case of (24) and (26) in unit of (hvx)2Table 2. The eigenvalues of the matrix (t\8nm + Jnm)/{hvj.)2 Those with superscript d are doubly denerate. 5.048d 13.86d 16.36" 20.54 21.07 31.69d 33.63d 35.22d 47.40 61.96 74.32d 140.2 204.6 Recalling (38), we can write (42) as bk = E

(cknan + s*„<4),

b[ = E

(cknal + sknan)

(44)

179 with

cn

: )=l(\f^±\F)T^

Skn )

2 \ \ en

^

\j oJk/

which satisfy 2_^ n

(CknCln - SknSln) = Skl,

^ n

(Cfc„S(„ - Sfc„Qn) = 0

(46)

and reciprocally 22(CknCkm - SknSkm) = &nm, k

2j(Cfc„Sfcm ~~ sknCkm) = 0. k

(47)

Therefore, (44) can be inverted: an = Y l ^ n

h

k

- Sknbl),

aj, = ^(CfcnfcJ. - Sknbk)-

k

6

(48)

k

Effective Condensate Hamiltonian (2)

We have already obtained the effective condensate Hamiltonian (35) in the lowest two orders. To proceed further, we use the formulas from EL 6 , A^=<0|fri|0>

(49)

and A<2> = H.P.{0\HiU^\0)

+ <0|tf2|0)

(50)

where H.P. stands for Hermitian part, and {Wo - Wm)(m\U<»\0) = (ml^lO) + [ff c ,HW ( 1 ) |0>].

(51)

For (49), we get from (29) A (1) = Joo{a02 a0+ a0a02)

(52)

after eliminating the terms linear in a0 by the choice ^

= £ ( 2 J £ - JS),

(53)

k

where *'

: =

/

,, Jmnskmslni

Jkl

: =

/

y

JmnskmClni

etc.

180

In (50), different states

\K)

contribute to the sum

}^{0\HI\K)(K\U^\0). K

Firstly, single particle states \k) = 6^10) give (fc|#i|0) = ] P Jno{-Sknal

+ 2(ckn - skn)ala0

+ cknal

n

+ X'Hckn - skn){al + <*>)}+&,

(flu ~

Wi^\0))

for which we determine the coefficients in the Ansatz, (fc|W(1)|0) := Aka\2 + Bkalao + Cka20 + Dka\ + Eka0 + Fk, by (51), or -uk (k\U^ |0> = <*|ZT1|0> + [HC, (k\uM |0>],

(54)

obtaining

vk ( 2

u>k

Dk=Ek = -A"1 -U-,

uk

)

Fk = - i - {/?, + £ 7 + - ^ 7 f c - | •

U>k

UJk (

U)k

U>k

)

where fik = — 2 ^ Ltmn(cki

— Ski)(cqm —

2sgm)sqn,

imn,

lk

(5g)

■= 2^, Jno(Ckn ± n

Skn)-

The contributions to («;|i?i|0),and(K|i/^1^|0) from normalized 2- and 3particle states, r)kib\b\\fy and r)hkib)fi\.b\\fy can be obtained similarly, where r)ki = 1 if k ^ I, and = 1/V2 if k = I, and similarly for 77/^. In (50), we also have

(0|tf2|0> = (4 ] T JS - // 2 >)aja 0 -

(3) M

(4 + a0)

k

+ Jooatfat - ( £ Jkl) (<4* + <*o)>

(57)

181

where, we note, / / 2 ' has been determined in (53). In this way, we obtain the second order effective condensate Hamiltonian,

+Lu{x0pl

+ PQZO) + M^oPo + Pozo)2,

(58)

up to a c-number after eliminating the terms linear in xo by choosing /x'3^ appropriately, where

Hi

2k ■= - E ^ ( * * + c * » + c ' ^ - E ^ - ^ ° ™ + E [ft + 2^) ~ -700' IS

mi:=" E ^-kclk + j^oo,

1

- E ZTTT^ + E fa - \JS) - ^oo, K3 ■= - Yl —A -1 c 2 t(c 3 fc + c'3k), k

L4 := - ^

(59)

K4 := - Y* — 4 * c 3 * + 7^00

Uk

W

k

k

4

— f c 4fc c; t + ^/| c 2*(c3fc + c'3k)\ + - Joo-

with c\k ■= Pk ~ %, c2k ■= V2~lk~, /3 4 J 0 0 \ _ 2J 00 ,

C3k t

= ( 2 - ^ r ) i k + wJt> T*1S

TOO

Ol /

Tf*S

a\ki ■= Jkl + Jkl ~ lKJkl

TSC* ^

1

C4k:=

+ Jkl) + -—;

UU

/

+

Joo _

. .

2lk ~~klk'

TOO

\Jkl

TSS \

~ Jkl),

T

d

(60) TCP

2ki '■- Jkl ~

The prime on Cjk (j = 3, 4) means to put J0o = 0 in them.

T*?S

J

ki-

182

7

C o n d e n s a t e Fluctuation

The result we have obtained so far for the effective condensate Hamiltonian is the sum of (37), (52) and (58): A<°>

+ A A « + A 2 A« = ^ j - / o + 9P4 + * ,

(61)

where we have put Joo = 9P and R is the sum of the products of the entries in the first and third lines, and the second and third lines in respective columns of Table 3. Since the coefficients in the second line as well as M4 here are yet to be determined by numerical computations, we simply assume that they have similar magnitudes ~ Joo — 9P- We assume further that g > 0 and Mi > 0, which is cruicial to the following considerations. Table 3. The terms in R 1/(2M2)

Pt>

magn.

K13

9P/23/2 K3 r3

N0-'/2

9P/2Z>2 L\2 zoPo + ■ 2 7V01/6

K4 x0 N-2/3 iV

0

Li

(xoPo + -Y 4

Then, in place of solving the Schrodinger equation for (61), we resort to a conventional uncertainty principle argument. Minimize (61) —R under the condition X0P0 = 1, then po = (gpNoMi)1'6,

xo = (gpN0M4)-1/6,

(62)

which leads to the estimates as given in Table 3, the entries in the third line having the magnitudes given in the fourth line, where A^ stands symbolically for gpN0M4. We see that

>

J

^ + _^ ( ^ + p S l 0 ) .(^)" j I + (»^)" } (63) dominates over any other term in (61) if TVo 3> 1; nevertheless it is much smaller than fuvk as to endorse our assumption of slow condensate fluctuation. (62) indicates also that aj ~ N0' , which is smaller than N0' , justifying our method and hence the Bogoliubov prescription. All these conclusions hinges on our cruicial assumption of Mi > 0, which, together with the other assumptions on the magnitudes of the coefficients in (61), has to be corroborated by numerical works. Besides, calculations are now in progress that incorporates "H^, in (18) into HQ in the step 1. in §4. So far, the effective condensate Hamiltonian in §6 appears not to be changed substantially.

183

Acknowledgments We are very happy to dedicate this article to Prof. Ludwig Streit on the occasion of his sixtieth birthday. He has been a constant source of inspiration on the interplay between physics and mathematics. One of the authors (H.E.) is grateful to Prof. J.R. Klauder and his group for discussions and warm hospitality during the memorable month of March 1999 at the University of Florida, where a part of this work was done. References 1. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Science 269, 198 (1995). 2. K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995); M.-O. Mewes et al., Phys. Rev. Lett. 77, 416 (()1996). 3. C.C. Bradley, C.A. Sackett, J.J. Toilet and R.G. Hulet, Phys. Rev. Lett. 75, 1687 (1995); C.C. Bradley, C.A. Sackett and R.G. Hulet, Phys. Rev. Lett. 78, 985 (1997) 78, 985 (1997). 4. D.G. Fried et al., Phys. Rev. Lett. 8 1 , 3811 (1998); T.C. Killian et al. Phys. Rev. Lett. 81, 3807 (1998). 5. N.N. Bogoliubov, J. Phys. (USSR) 11, 23 (1947). 6. H. Ezawa and M. Luban, J. Math. Phys. 8, 1285 (1967); H. Ezawa. J. Math. Phys. 6, 380 (1965). 7. H. Araki and J, Woods, J. Math. Phys. 4, 637 (1963). 8. L.P. Pitaevski, Zh. Exp. Theor. Fiz. 40, 646 (1961) [Sov. Phys. JETP 13, 451 (1961)]; E.P. Gross, Nuovo Cimento 20, 454 (1961); J. Math. Phys. 4, 195 (1963). 9. F.A. Berezin, The Method of Second Quantization, tr. by N. Mugibayashi and A. Jeffrey, Academic Press (1966). Chap. M. 10. M. Born and J.R. Oppenheimer, Ann. Physik 84, 457 (1927). 11. M. Edwards and K. Burnett, Phys. Rev. A 5 1 , 1362 (1995); F. Dalfovo and S. Stringari, Phys. Rev. A 53, 2477 (1996). T. Isoshima and K. Machida, J. Phys. Soc. Jpn. 66, 3502 (1997).

184

TIME D E P E N D E N T A N D N O N L I N E A R P O I N T INTERACTIONS RODOLFO FIGARI Dipartimento di Scienze Fisiche, University di Napoli "Federico II", Italy dedicated to Ludwig Streit The aim of the paper is to stress the effectiveness of point interactions for the construction of models of quantum systems whose dynamics is generated by time dependent or nonlinear hamiltonians. Examples of time dependent point interac­ tion hamiltonians which could have some importance in applications are given. In particular nonlinear interactions of range zero are analyzed in dimensions one and three. Results on local and global existence of the solution and conditions for the blow-up are given and an explicit blow-up solution is constructed in the critical case.

During the activities of BiBoS, at the end of '70s and the beginning of the '80s, research on various subjects of Mathematical Physics was flourish­ ing around the Zentrum fiir Interdisziplinare Forschung at the University of Bielefeld. Ludwig Streit, as director of several study groups held at ZIF during those years, was able to gather together well known experts from almost every field of mathematical physics and young researchers from almost everywhere in the world. An extremely warm social environment was a strong support for a very effective scientific work. 1

Introduction

Driven by new results obtained by S. Albeverio, R. Hoegh-Krohn, H. Holden, F. Gesztesy, M. Mebkhout, L. Streit and many others a new impulse was given to the analysis of singular perturbations of the laplacian, in particular point interactions, in non relativistic quantum mechanics. As stressed already in the title of the reference book on the subject, ([AGH-KH]), one of the main properties of point interactions is the fact that they are solvable, in the sense that one can write explicit formulas for the integral kernel of the resolvent, of the propagator and of other functions of such hamiltonians. The construction of the dynamics is particularly simple in the one dimensional case due to the fact that the Dirac delta is a small

185

form perturbation of the laplacian. For this reason one dimensional point interactions have been used in elementary courses and textbooks of quantum mechanics to exemplify a complete computation of spectral and scattering data for quantum model systems. In dimensions two and three the perturbative approach does not work and point interactions are most naturally defined using extension theory for sym­ metric operators. One starts with the laplacian denned on smooth functions vanishing on the interaction centers and then characterizes all the self-adjoint extensions. Such kind of hamiltonians are a set of non trivial perturbations of the laplacian indexed by a minimal set of parameters: the strengths and the po­ sitions of the interaction centers. As a consequence, the formulas for the propagator of the corresponding dynamics look particularly simple. This re­ mains true even if the dynamical and geometrical parameters depend on time, opening the possibility of analyzing the dynamics of quantum particles sub­ ject to linear or nonlinear time dependent forces. In particular Yafaev gave in ([Y2]) a complete analysis of the scattering of a quantum particle by an in­ teraction center with strength variable in time (see also ([DO]) and references therein). In the next section some new results on solutions of the Schroedinger equation with point interactions with strength and position depending on time in a preassigned way will be given. In the last section a class of nonlinear point interactions will be introduced in dimensions one and three together with some preliminary results on local and global existence of the solution of the corresponding evolution problem. Cases in which the solution blows up in finite time will be exhibited. 2

Time dependent interactions

In order to avoid unnecessary technical difficulties and involved notation only the case of one interaction center will be considered. Results valid in the general case and complete proofs will be given in ([DFT]). In order to clarify differences and common features, the one dimensional and the three dimensional cases will be presented in two different subsections. 2.1

One dimensional case

Point interaction hamiltonians in one dimension are a large class of self-adjoint operators corresponding, roughly speaking, to "potentials" 8, 5' and their combinations, placed in any point of the real line (for a complete characteri-

186

zation of any hamiltonian in the class see ([AGH-KH])). Only the 5 potential will be considered here. The hamiltonian Ha>v with a single point interaction of strength a, placed in y can be defined as the self-adjoint operator in L2(R) whose quadratic form is D(Fa,v) = H1{R) FatV(u)=

[ dx\Vu\2 + a\u(y)\*

(1)

JR.

Due to standard Sobolev inequalities, the value in one point of any func­ tion in Hl(R) is well defined and bounded by the H1 norm. This implies that the form (1) is a Kato-small perturbation of the quadratic form of the laplacian. As it is well known Ha
a >0 a<0

- for a < 0 the only eigenvalue is simple and the corresponding normalized eigenfunction is

tM*) = Yir e x p(fl x l) - for any a, to each momentum k e [0, oo), corresponds the generalized eigen­ function

WM) =

v b {ex^ikx) - ^ijfc[ e x p ( i | f c a : | ) )

The integral kernel of the propagator of the unitary group generated by Ha
187

»^2 =

Ha,yipt

= ( - A + a5) tl>t

one obtains the integral equation

V>t(z) = (UQ{t)iM (a?) -ia

f ds U0 {t-s,\x-

y\)tps(y)

(2)

Jo

where UQ is the free propagator in dimension one. From (2) one sees that the solution is completely determined by the free evolution oirpo and by i()t(y), which in turn, as a function of time, must satisfy the integral equation

My) = (U0(W0)

(y) -ia

f ds U0 (t - «,0)^.(tf)

(3)

Jo

It is easy to check that in fact (2) and (3) define a unitary flow in L2(R) whose generator is HaiV and therefore they may be used as a definition of This procedure is directly generalizable to the case of y and a varying with time. In fact, under suitable conditions on the regularity of y(t),a(t) and on the initial condition it is possible to prove that for each s e R the map ips ->• ipt induced by

M s ) = (U0(t - a)i>.) (x)-if

dr O(T) UO (t -T,\X-

y(r)\) ^ r (p(r))

J8

i>Mt)) = (U0(t - s)xps) (y(t)) -if

dr a(r) U0 (t - r, \y(t) - y(r)\) ^ r (y(r)) J8

(4)

defines a group of unitary transformations V(s, t),with V(s, s) = 1, which solves the Schroedinger equation idV{s^"=Ha(t)
(5)

Remark 1 The most general assumptions needed for the validity of the scheme introduced above will not be analyzed. It should be noticed that in dimension

188

one the form domain of the hamiltonians Ha^iy^ does not depend on time and coincide with ^(R). This permits to apply the general abstract theory about the solution of non autonomous evolution problems (see e.g.[Si]). Notice that, on the contrary, in the physical literature, assumptions were always chosen tailored on the specific model taken into consideration and on the techniques used. Remark 2 The problem of ionization of a one dimensional atom perturbed by a periodic force was recently considered in ([CLR]). The qualitative features shown by the computed ionization probability seem in good agreement with what is measured in transitions between states of large principal quantum number in hydrogen atoms. For preliminary results in three dimensions see

(PI). 2.2

Three dimensional case

The theory of self-adjoint extensions of symmetric operators in Hilbert spaces allows to classify all the non trivial extensions of the laplacian restricted to Co°(i? d \ {j/})- It is well known that for d > 3 only the trivial extension exists. In d = 2 and d = 3 there are infinitely many extensions Ha>v indexed by the value of a strength parameter a € f l (see [AGH-KH] for details) . The corresponding quadratic forms are recalled below. In dimensions two D(Fa,9) = {V 6 L2(R2) s.t. 3x e H'iR^qe

C;${x) = <j>x{x) + qGx{x - y)}

F«,vm = HV^II2 + AH^II2 - A|M|2 + [a + log A ) |,|>

(6 )

In dimension three ([T]) D(Fa,v) = {V- 6 L2(R3) s.t. 3

6 C;rp(x) = (x) + qG°3(x - y)}

i;,a,„(V') = l|V
(7)

A > 0 is the integral kernel of ( - A + A ) - 1 in dimension

Spectral properties and eigenfunctions of Hail) in three dimensions are listed below - the spectrum of Ha
189 a(#a,y) = [0,oo)

a>0

<x(H0ty) = {-167r 2 a 2 } U [0,oo)

a <0

- for a < 0 the only eigenvalue is simple and the corresponding normalized eigenfunction is

- for any a, to each momentum k € i? 3 , corresponds the generalized eigen­ function exp(r|fc||x|)> lk x) = e x +* ' ( 2 ^ 7 2 ( ? ^ • *> " a - i|fc| /(47 r) |x| Like in the one dimensional case, the solution of the Schroedinger equation .Wt .dipt „ , dt can be given in dimension three via an explicit formula for the propagator of HQ,V ([ST]) . The implicit characterization of the solution, given by formulas (2), (3) in d = 1, becomes in this case tft(z) = (U(t)rPo) (x)+i

f dsU(tJo

a, \x - y\) q(s)

(8)

where (U(a)iPo) (y) / da , = 4\/i7r / da s/t^a Jo Vt - s Jo where U(t) is the free unitary group defined by the kernel q(t) + tViwa

U{t.a-x<)

= e^(x-x>)

=

^

0)

m

To be precise formulas (8) and (9) hold for initial data^o € C£°(fl 3 \{2/}). The evolution of any other initial vector in L2(R3) is obtained by continuity. Formula (9) for the charges holds true for an initial function in the domain

190

of HQty if in the r.h.s. of (9) some oscillatory integrals are interpreted as boundary values of analytic functions. In dimension two the equation for the "charges" q(i) is considerably more delicate to handle due to the log singularity of the Green's function. Corre­ spondingly the generalization to dynamics generated by time dependent point interaction hamiltonians is more difficult. Here only the three dimensional case will be considered. Notice that the form domain in (7) changes if the position of the point y varies. As a consequence the Schroedinger problem corresponding to a moving point interaction is more complicate than in dimension one. Yafaev ([Yl]) solved the case where the strength parameter a was varying in time (in which case the operator domain is varying but not the form domain). In the following theorem the results in the general case when strength and position change in time are summarized. Theorem Ify(t) is regular curve in R3, a{t) a smooth function in R and f G C^iR3 \ {y(s)}) then there exists a unique i(),(t) € D(Fa^tytV^, t € R, such that *$' 2 3 is in the dual (with respect to L {R )) of D(Fa(t)tV(t)) and i < v(t), ^ ^

> = Ba{t)>y{t)(v(t),

V.W)

Mt)

6

D(FQ(tUit))

Ms) = f (10) where Ba(t),y(t) w the bilinear form corresponding to the quadratic form FQ(t)iV(t). Moreover tp,(t) has the following representation Mt)

= U(t -S)f

+ ij

dr U(t - r; • - y{r))q(r)

(11)

where the charges q(t) satisfy the Volterra integral equation 4

V^l J,

C(t,r)

=--

y/t-T

f

*" Jo

dz

v ^ f* ,JUo(r ~ s)f)(y(T)) \/t=T V-l Js (12)

JB

*

U(T

V ( l - z)z V

+{t-

T)Z,T)

+ B(T + {t-

T)Z,T)

191 +

B(T + ( f-r)z,r)-l\ 2(t - T)Z

)

W3{t,T) J0

2{t-T) 1

/■w(t.T)

B(t,T)=-7—r / v w(*,T)y 0

where B{t,r) 3

d*e , z

denotes the derivative with respect to the second argument.

Nonlinear Interactions

In this section it is described a class of nonlinear evolution problems with point interaction concentrated in a fixed point (which for sake of simplicity will be taken to be the origin) with a strength depending on the behaviour of the solution near the origin. Models of this kind, in space dimension one, appeared already in the physical literature ([J-LPS], [MA], [BKB]) to describe the resonant tunnelling of electrons through a double barrier heterostructure. The results presented in the previous section will be used to define models where the strengths of the interactions are coupled with the solution itself. Such models might be of interest in applied as well as in fundamental Quantum Mechanics. Let us start considering a one dimensional case introduced in ([ATI]). 3.1

One dimensional case

The nonlinear evolution problem is obtained by replacing a(t) in (4) with a function of the value at the origin of the solution itself. More precisely, for 7 G R, a > 0, let us consider a(t) = 7|V>t(0)|2<7

(13)

The resulting nonlinear integral equation is M<*) + *7 f dsU (t - s, \x\) hM0)| 2 'iM0) = (U(t)ip0) (x) Jo

(14)

192

As in (4) the evaluation of (14) in x = 0 gives a closed equation for i/>t(0)

MO) +il

f ds JO

]

> , ( Q ) | 2 g V , ( 0 ) = (U(t)iM (0)

(15)

V 47Tl(t - 8)

Equation (15) is a nonlinear Abel integral equation, involving only the time variable. The search for the solution of (14) is then reduced to the solution of (15). The following proposition refers to results obtained in ([ATI], [AT2]) about uniqueness and existence of the solutions. Proposition 1 For any ipo € H1 (R), there exists i > 0 s.t. problem (14) has a unique solution rpt € Hl{R), t € [0,1). Moreover if j > 0 or if j < 0, a < 1 the solution is global in time. In fact, using the contraction principle, equation (15) can be uniquely solved for small times in the space of continuous functions (see e.g. [M]). Ex­ ploiting the smoothing properties of the Abel operator ([GoVe]) and classical results about the free propagator one can prove local existence. The result on global existence is a direct consequence of Sobolev inequalities and of the conservation of the X2-norm and of the energy E{rl,t) = f dxmx)\2 JR

+ - J l - I ^ O ) ! 2 ^ 2 = EWo) V+

1

Notice that the evolution reduces to free propagation if the initial datum is odd. Only propagation of the even part is not trivial. According to the standard definition, one says that ipt is a blow-up solu­ tion if there exists to < oo s.t. \\ip't\\ -»• oo for t -»• toDue to the conservation of the energy, ^ is a blow-up solution if |Vt(0)| diverges for t -+ to- This means that, in this model, a blow-up solution tpt can be equivalently defined by the condition ||V>t||£°° -*■ oo for t -¥ toIf the nonlinear interaction is sufficiently attractive, for suitable initial datum, one can prove that blow-up occurs. Proposition 2 Assume 7 < 0 and a > 1. Then for any ipo 6 Hl(R) E(ipo) < 0 the solution ipt is a blow-up solution.

s.t. \\xtp0\\ < cc and

193 The proof exploits standard technical tools in the analysis of blow-up of so­ lutions of the nonlinear Schroedinger equation, in particular the computation of the second time-derivative of the moment of inertia I(t)

I(t) = f dxx2\Mx)\2 JR

and the uncertainty principle (see e.g. [RH]). The above results shows that the critical exponent of the nonlinearity is a = 1. The critical case is of special interest because of the existence of an additional symmetry (pseudo-conformal invariance). This symmetry can be expressed as follows: if $t (x) is a solution then

{ )s .-rf i/ i

a to>o

* ' ji^* " *A^)

'

(16)

is again a solution. Moreover, a stationary solution of the problem, i.e. a solution of the form Q(x)e,xt, is easily computed

*«(*)= « p ( - M + i | )

(17)

Combining (16) and (17), we explicitly construct a blow-up solution, namely , ,

x

_

1

(

\x\

.

\x\2

1

^

(18)

An advantage to have the explicit solution (18) is that one can directly check the properties of the solution near the blow-up (non existence of strong L2limit, L 2 -concentration phenomenon, local behaviour, etc.). 3.2

Three dimensional case

In the following some preliminary results on the three dimensional version of the nonlinear point interaction model will be discussed. The same problem in dimension two will be not considered here. There are in fact peculiar features (e.g. the absence of a critical exponent) indicating that the two dimensional model requires a separate analysis. We intend to approach this problem in further work. Notice that even in the linear three dimensional case the solution of the evo­ lution problem exhibits a singularity where the interaction is placed. This

194

means that the nonlinear interaction cannot be introduced as in (14). The value of the function at the origin should be replaced in this case by the co­ efficient q(t) of the singular part. This suggests the following formulation of the evolution problem (see (9))

Vt(z) = (U{t)iM(x)+iJ

q{t) +

lv£ f ^^M V - » Jo

y/t-8 2

<*(*) = f\Q(t)\ ",

dsU(t - a, \x\)q(a)

=

^r

d8 V-» JO

OWoKo) y/t — 3

7 € R, o > 0 (19)

The natural space where one looks for the solution of a standard nonlinear Schroedinger equation is if 1 , i.e. the form domain of the generator of the linear dynamics obtained setting to zero the exponent of the nonlinear term. This suggests to look for the solution of (19) in the form domain D(Fa>0) of Hato. It is easy to check that the following characterisation of D(Fa<0) is equivalent to the one given in (7) D(Fafi)

= {ue L2(R3) \u = <j>x + qGx,

<j>xeH\q£C}

One can prove the following result Proposition 3 Let Vo = <£o + 1GX G ^(^0,0), with x G H2 then there exists i > 0 s.t. problem (19) has a unique solution ipt G D(Fa 0 the solution is global in time. The local existence result is obtained following closely the proof for the one dimensional case. Notice however that in three dimensions we were not able to prove the local existence for every initial datum in D(Faio). This is only a consequence of technical difficulties, and we believe that a more carefull analysis of the r.h.s. of (19) should allow the extension of the proof to any initial datum in D(Fa>o). Global existence for repulsive interactions (i.e. 7 > 0) is a direct conse­ quences of the conservation of the L 2 -norm and of the energy which in this case takes the form vty — 11 v vti ""c + 1 where ipt =
195

One expects global existence also for weakly attractive interactions. The proof is not trivial because one is not working in H1 and standard Sobolev inequalities are not available. Consider finally the problem of the existence of blow-up solutions. Since the solution lives in £>(FOi0) (which is strictly larger than H1) it is necessary to modify the usual notion of blow-up. A solution ipt of problem (19) will be said to be a blow-up solution if there exists a io < oo such that

lim IIV&H = oo t—¥to

or equivalently Hm \q{t)\ = oo With this definition of blow-up one can prove Theorem If ^0 = 0* + q{0)Gx with £ € H2(R3), \\x^oh < o o 7 < 0 andCT> 1 then the solution of problem (19) is a blow-up solution. Acknowledgements With a delay of almost twenty years I want to thank S. Albeverio and L. Streit who put me in touch with point interactions (and much more). I want also to use this occasion to thank my co-workers G.F. Dell'Antonio and A. Teta, whose continuous help and friendship were the main motivation to continue to work on this subject, and my young colleagues R. Adami, G. Panati and F. Pisano on whom we can count for future results on point interactions. References [A] Adams, R., Sobolev Spaces, Academic Press, New York, 1975. [ATI] Adami, R., Teta, A., A Simple Model of Concentrated Nonlinearity, in "Mathematical Results in Quantum Mechanics", Eds. Dittrich, J., Exner, P., Tater, M., Birkhauser, Berlin, 1999. [AT2] Adami, R , Teta, A., A Class of Nonlinear Schroedinger Equations with Concentrated Nonlinearity, Preprint Universita di Roma "La Sapienza", 1999.

196

[AGH-KH] Albeverio, S., Gesztesy, F., Hogh-Krohn, R., Holden, H., Solvable Models in Quantum Mechanics, Springer-Verlag, New York, 1988. [BKB] Bulashenko, O.M., Kochelap, V.A., Bonilla L.L., Coherent Patterns and Self-Induced Diffraction of Electrons on a Thin Nonlinear Layer, Phys. Rev. B, 54, 3, 1996. [CLR] Costin, O., Lebowitz, J.L., Rohlenko, A., Ionization of a Model Atom: Exact Results and Connection with Experiments, Preprint Rutgers University ( mpjirc n.99-185), 1999. [DFT] DelFAntonio, G.F., Figari, R., Teta, A., Schroedinger Equation with Moving Point Interactions in Three Dimensions, in "Stochastic Processes, Physics and Geometry: New Interplays", Vol. 1, Eds. Gesztesy, F., Holden, H., Jost, J., Paycha, S., Rockner, M., Scarlatti, S., CMS Conference Proceed­ ings Series, 2000. [DO] Demkov, Yu.N., Ostrovsky, V.N., Zero-range Potentials and their Ap­ plications in Atomic Physics Plenum Press, New York and London, 1988. [GV] Ginibre, J., Velo, G., On a Class of Nonlinear Schrodinger Equations. I. The Cauchy Problem, General Case, J. Func. Anal., 32, 1-32, 1979. [GoVe] Gorenflo, R., Vessella, S., Abel Integral Equations, Springer-Verlag, Berlin Heidelberg, 1978. [J-LPS] Jona-Lasinio, G., Presilla, C , Sjostrand J., On Schrodinger Equations with Concentrated Nonlinearities, Ann. Phys., 240, 1-21, 1995. [M] Miller, R. K., Nonlinear Volterra Integral Equations, W. A. Benjamin Inc., 1971. [Me] Merle, F., Construction of Solutions with Exactly k Blow-up points for the Schrodinger Equation with Critical Nonlinearity, Comm. Math. Phys., 129, 223-240,1990. [MA] Malomed, B., Azbel, M., Modulational Instability of a Wave Scattered by a Nonlinear Center, Phys. Rev. B, 47, 16, 1993. [N] Nier, F., The Dynamics of some Quantum Open Systems with Short-Range Nonlinearities, preprint Ecole Polytechnique, 1998. [P] Pisano, F. Tempi di Ionizzazione di un Atomo Modello Perturbato Periodicamente, Tesi di Laurea in Fisica, Universita Federico II, Napoli, 1999 [RR] Rasmussen, J.J., Rypdal, K., Blow-up in NLSE - A General Review, Physica Scripta, 33, 481-497,1986.

197

[ST] Scarlatti, S., Teta, A., Derivation of the Time-dependent Propagator for the Three Dimensional Schrodinger Equation with One Point Interaction, J. Phys. A, 23, 1033-1035, 1990. [S] Schulman, L.S., Applications of the Propagator for the Delta Function Potential, in Path Integrals from meV to MeV (Bielefeld 1985), World Sci. Publ., Singapore, 1986 [Si] Simon, B., Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton University Press, 1971 [T] Teta, A., Quadratic Forms for Singular Perturbations of the Laplacian, Publications of the R.I.M.S., Kyoto University, 26, 803-817, 1990. [Y2] Yafaev, D.R., Scattering Theory for Time-dependent Zero-range Poten­ tials, Ann.Inst. H. PoincarS Phys.Theor 40, 343-359,1984. [W] Weinstein, M.I., NLSE and Sharp Interpolation Estimates, C.M.P. 87, 567-576, 1983.

198

T H E COLE-HOPF A N D M I U R A T R A N S F O R M A T I O N S REVISITED FRITZ GESZTESY Department of Mathematics, University of Missouri, Columbia, MO 65211, USA E-mail: fritzQmath.missouri.edu URL: http://www.math.missouri.edu/people/fgesztesy.html HELGE HOLDEN Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway E-mail: holdenQmath.ntnu.no URL: http://www. math. ntnu. no/ 'holden/ Dedicated with great pleasure to Ludwig Streit on the occasion of his 60th birthday. An elementary yet remarkable similarity between the Cole-Hopf transformation relating the Burgers and heat equation and Miura's transformation connecting the KdV and mKdV equations is studied in detail. In the special (1 + l)-dimensional case, our considerations apply to the entire hierarchy of Burgers evolution equa­ tions. 1

Introduction

Our aim in this note is to display the close similarity between the well-known Cole-Hopf transformation relating the Burgers and the heat equation, and the celebrated Miura transform connecting the Korteweg-de Vries (KdV) and the modified KdV (mKdV) equation. In doing so we will introduce an addi­ tional twist in the Cole-Hopf transformation (cf. (1.23)), which to the best of our knowledge, appears to be new. Moreover, we will reveal the history of this transformation and uncover several instances of its rediscovery (including those by Cole and Hopf). We start with a brief introductory account on the KdV and mKdV equa­ tions. The KdV equation [43] was derived as an equation modeling the be­ havior of shallow water waves moving in one direction by Korteweg and his student de Vries in 1895°. The landmark discovery of the inverse scatter­ ing method by Gardner, Green, Kniskal, and Miura in 1967 [20] (cf. also [21]) brought the KdV equation to the forefront of mathematical physics, and started the phenomenal development involving multiple disciplines of science "But the equation had been derived earlier by Boussinesq [7] in 1871, see Heyerhoff [37] and Pego [50].

199

as well as several branches of mathematics. The KdV equation (in a setting convenient for our purpose) reads KdV(V) = Vt - 6VVX + Vxxx = 0,

(1.1)

while its modified counterpart, the mKdV equation, equals mKdV(tf>) = 4>t- 6020x + 4>xxx = 0. (1.2) Miura's fundamental discovery [46] was the realization that if satisfies the mKdV equation (1.2), then V±{x,t) = (f>{x,t)2±(t>x{x,t),

(i,()6R2

(1.3)

both satisfy the KdV equation. The transformation (1.3) has since been called the Miura transformation. Furthermore, explicit calculations by Miura showed the validity of the identity KdV(V±) = (2(£ ± dx) mKdV(<£).

(1.4)

The Miura transformation (1.3) was quite prominently used in the construc­ tion of an infinite series of conservation laws for the KdV equation, see [47], [16, Sect. 5.1]. Miura's identity (1.4) then demonstrates how to transfer so­ lutions of the mKdV equation to solutions of the KdV equation, but due to the nontrivial kernel of (2 ± dx), it is not immediately clear how to reverse the procedure and to transfer solutions of the KdV equation to solutions of the mKdV equation. It was shown in [31] (see also [22], [23], [30], [33]) how to revert the process. Following a similar treatment of the (modified) Kadomtsev-Petviashvili equation in [34], we use here a method that considerably simplies the proofs in [22], [31], [30], [33]. Introducing the first-order differential expression P{V) = 2Vdx-Vx,

(1.5)

one derives mKdV(0) = dx (±(4>t - P(V±)1>)\ ,

(1-6)

where 0 = V*M

4>>0,

V±=4>2±<j>x.

(1.7)

Next, let V — V(x, t) be a solution of the KdV equation, KdV(V) = 0, and ip > 0 be a function satisfying rl>t = P(VW,

-i/>** + VTP = 0.

(1.8)

200

Then one immediately deduces that <> / solves the mKdV equation, mKdV(<^) = 0, and hence the Miura transformation has been "inverted". The KdV equation (1.1) and the mKdV equation (1.2) are just the first (nonlinear) evolution equations in a countably infinite hierarchy of such equa­ tions (the (m)KdV hierarchy). As we will indicate at the end of Section 2, the considerations (1.3)-(1.8) extend to the entire hierarchy of these equations. Next we turn to the the Cole-Hopf transformation and its history. The classical Cole-Hopf transformation [13], [42], covered in most textbooks on partial differential equations, states that uOM) = - 2 ^ ^ ,

(x,t)eRx(0,oo),

(1.9)

where tp > 0 is a solution of the heat equation 1>t ~ 4>xx = 0 ,

(1.10)

satisfies the (viscous) Burgers equation ut + uux - uxx = 0.

(1-11)

However, already in 1906, Forsyth, in his multi-volume treatise on differential equations ([19, p. 100]), discussed the equation (in his notation)

where a = a(x,y).

Hence there exists a function 6 such that 99 da 2 nad9 =d-*> " - ^ - a = 2 0 d - y -

a

,,1 0 , (113)

Assuming the function z satisfies z „ + 2cwa + 2/fey + 7 = 0,

(1.14)

an easy calculation shows that dL(ze»)+20^(ze°)=O.

(1.15)

Introducing new variables t = -y and u(x, t) = - 2 a ( x , y) as well as fixing 0 = 1/2, 7 = 0, and z = 1, one concludes that (1.12) indeed reduces to the viscous Burgers equation ut + uux - uxx = 0,

(1-16)

while (1.15) equals ( A - (ee)xx

= 0,

(1.17)

201

with solutions related by u=-20».

(1.18)

However, Forsyth did not study the ramifications of this transformation, and no applications are discussed. Shortly thereafter, in 1915, Bateman [1] introduced the model equation u ( + uux — vuxx = 0.

(119)

He was interested in the vanishing viscosity limit, that is, when v -¥ 0. By studying solutions of the form u — F(x+Ut), he concluded that "the question of the limiting form of the motion of a viscous fluid when the viscosity tends to zero requires very careful investigation". Only in 1940 did Burgers ([9, p. 8]) introduce6 what has later been called the (viscous) Burgers equation, as a simple model of turbulence, and did some preliminary investigation on properties of the solution. Taking advantage of the later rediscovered Forsyth transformation by Cole and Hopf, Burgers continued the investigations of what he called the nonlinear diffusion equation, focusing mainly on statistical aspects of the equation. The results of these investigations were collected in his book [11]. In 1948, Florin [18], in the context of applications to watersaturated flow, rediscovered Forsyth's transformation, which would become well-known under the name Cole-Hopf transformation only some 44 years later. Although the Cole-Hopf transformation had already been published in 1906, it was only with the seminal papers by Hopf [42]c in 1950d and by Cole [13] in 1951 that the full impact of the simple transformation was seen. In particular the careful study by Hopf concerning the vanishing viscosity limit represented a landmark in the emerging theory of conservation laws. Even though the Cole-Hopf transformation is restricted to the Burgers equation, the insight and the motivation from this analysis has been of fundamental im­ portance in the theory of conservation laws. Furthermore, Cole states the gen­ eralization of the Cole-Hopf transformation to a particular multi-dimensional system. More precisely, if ip = tp(x,t), (x,t) € Kn x (0,oo), satisfies the n-dimensional heat equation ipt ~ "A = 0,

v > 0,

(1.20)

fc Frequently Burgers' equation is quoted from his 1948 paper [10], but he had already introduced it in 1940. c With a misprint in the title, writing ut + uux = p.xx rather than ut + uux = fiuxx. d Hopf [42] states in a footnote (p. 202) that he had the "Cole-Hopf transformation" already in 1946, but "it was not until 1949 that I became sufficiently acquainted with the recent development of fluid dynamics to be convinced that a theory based on (1.19) could serve as an instructive introduction into some of the mathematical problems involved".

202

and one defines u = -2i/Vln(V»),

(1.21)

then u satisfies ut + (u ■ V)u - uAu = 0,

(1.22)

and the vector-valued function u = u(x, t) € R" has as many components (i.e., n) as the dimension of the underlying space. Observe, in particular, that u is irrotational (i.e., u = Vio for some w, or equivalently, curlu = 0). The multi­ dimensional extension was rediscovered by Kuznetsov and Rozhdestvenskii [44] in 1961. In this note we show the following relation, ut + uux - vuxx = 2v f --j-dx + -rf ) 0>t -

v^xx)

= -2vdx f ±(V>t - vi>xx)\ ,

(1.23)

whenever u — -2i/ipx/i> for a positive function tp. This clearly displays the nature of the Cole-Hopf transformation and closely resembles Miura's identity (1.4) and the relation (1.6). Even though identity (1.23) is an elementary observation, much to our surprise, it appears to have escaped notice in the extensive literature on the Cole-Hopf transformation thus far. While both the KdV and mKdV equations are nonlinear partial differential equations, the case of the Burgers and heat equations just considered is a bit different since it relates a nonlinear and a linear partial differential equation (see also [6, Sect. 6.4]). One can also extend the Cole-Hopf transformation to the case of a po­ tential term F in the heat equation, see, for instance, [38]. Here the relation (1.23) reads as follows, ut + uux - vuxx + 2vFx = -2vdx (-r(4>t - ^xx

- FipU ,

(1.24)

whenever u = -2i/5jln(i/)) for a positive function t/>. The case of Burgers' equation externally driven by a random potential term recently generated particular interest, see, for instance, [3], [4], [36], [38], [39], [40] and the refer­ ences therein. We also mention a very interesting application of the Cole-Hopf transformation to the pair of the telegraph and a nonlinear Boltzmann equa­ tion in [41], generalizing the pair of the heat and Burgers equation considered in this note. As in the KdV-mKdV context, the Burgers equation (1.19) is just the first in a hierarchy of nonlinear evolution equations, and we will show

203

in Section 3 that identity (1.23) extends to the entire hierarchy of Burgers equations. Equation (1.24) extends to the multi-dimensional case corresponding to (1.22) and one obtains

ut + a(u ■ V)u - i/Au + — V F = - — V (\ Ut - «/Atf - Fi>)) , a a \ip )

(1.25)

whenever a € K\{0} and u = -(2u/a)V\n(ip) for a positive function tp. Obviously there is a close similarity between the heat and the Burgers equation expressed by (1.23), and Miura's identity (1.4) relating the mKdV and the KdV equation. The principal idea underlying these considerations being that one (hierarchy of) evolutions equation(s) can be represented as a linear differential expression acting on another (hierarchy of) evolution equa­ tion^). As long as the null space of this linear differential expression can be analyzed in detail, it becomes possible to transfer solutions, in fact, en­ tire classes of solutions (e.g., rational, soliton, algebro-geometric solutions, etc.) between these evolution equations. In concrete applications, however, it turns out to be simpler to rewrite a relationship between two evolution equations, such as (1.4) and (1.23), in a form analogous to (1.6) and the sec­ ond relation in (1.23), rather than analyzing the nullspaces of (2 ± dx) and (-dx + (ipx/i*)) »n detail. These strategies relating (hierarchies of) evolution equations and their modified analogs is not at all restricted to the Burgers and heat equations and the KdV and mKdV hierarchies, respectively, but ap­ plies to a large number of evolution equations including the Boussinesq [28], and more generally, the Gelfand-Dickey hierarchy and its modified counter­ part, the Drinfeld-Sokolov hierarchy [27], the Toda and Kac-van Moerbeke hierarchies [8], [26], the Kadomtsev-Petviashvili and modified KadomtsevPetviashvili hierarchies [25], [34], etc. For simplicity we restrict ourselves to classical solutions throughout this note. The case of distributional solutions of Burgers' equation is considered, for instance, in [15]. Throughout this note we abbreviate by Cp'9(il x A), fi C K n ,A C K open, n € N, p, g 6 No (No =NU{0}) the linear space of continuous functions f(x, t) with continuous partial derivatives with respect to x — {x\,..., x„) up to order p and q partial derivatives with respect to t. C p ' ? (fi x A; R n ) is then defined analogously for f(x, t) € K n .

204 2

The M i u r a transformation

We turn to the precise formulation of the relations between the KdV and the mKdV equation and omit details of a purely calculational nature. Lemma 2.1 Let xjj = ip(x,t) > 0 be a positive function such that t\> 6 C4-°(R x R), ipt e C1-°(R x R). Define <j> = ipx/ip. Then 4> € C3'l(R x R) and mKdV(<£) = ±dx (^\(i>±l)t

- PiiVi)^1)),

(2.1)

where Pi(V) = 2Vdx-Vx

(2.2)

and V± = 2±(j>x.

(2.3)

Proof. A straightforward calculation. The application to the KdV equation then reads as follows. Theorem 2.2 Let V = V{x, t) be a solution of the KdV equation, KdV(V) = 0, with V 6 C 3,1 (R x R), and let xp > 0 be a positive function satisfying i/> 6 C2'°(R x R), ipt 6 C l i 0 (R x R) and *Pt = Pi {V)rl>, -i>xx + Vil> = 0,

(2.4)

with Pi 0 0 given by (2.2). Define <j> = ipx/ip and V = 4>2 - x. Then V = <j>2 + <j>x and 0 satisfies € C 4,1 (R x R) and the mKdV equation, mKdV(0) = 0.

(2.5)

Moreover, V satisfies V € C 3,1 (R x R) and the KdV equation, KdV(f) = 0.

(2.6)

Proof. A computation based on Lemma 2.1. Originally, Theorem 2.2 was proved in [31] (see also [22], [23], [30], [32], [33]) using supersymmetric methods. The above arguments, following [34] in the context of the (modified) Kadomtsev-Petviashvili equation, result in con­ siderably shorter calculations. The "if part" in Theorem 2.2 also follows from prolongation methods developed in [51]. A different approach to Theorem 2.2, assuming rapidly decreasing solutions of the KdV equation, can be found in Sect. 38 of [2]. Remark 2.3 The chain of transformations V -¥ - ► - - > V

(2.7)

205 reveals a Bdcklund transformation between the KdV and mKdV equations (V -> - 0 for all (x, t) G R2 in The­ orem 2.2. However, as proven by Lax [45], one can show that ip(x, to) > 0 for some t0 G K and all x G E actually implies i>(x, t) > 0 for all (x, t) € R2 (see also [31]). Moreover, in case V(x,to) is real-valued, we note that L{to)vj{x,to) — 0 has a positive solution tp(x,to) > 0 if and only if the Schrodinger differential expression L(to) = —dx + V(x,to) is nonoscillatory at ±oo (cf. [35]/ While the system of equations (2.4) always has a solution ip(x,t), (cf. Lemma 3 in [31]j, it is the additional requirement ip(x,t) > 0 for all (x, t) € K2 which renders (j> (and hence V) nonsingular. Without the condition xj) > 0, Theorem 2.2 describes (auto)Bdcklund transformations for the KdV and mKdV equations with characteristic singularities (cf. [32]). We conclude this section with a brief discussion of the KdV and mKdV hierarchies and indicate the extension of the basic identity (2.1) to these hierarchies. The higher-order (m)KdV equations can recursively be denned by KdVn(V) = Vt-2Xn+hx(V) = Q, n € N ) , mKdVn() = t- Yn+iA) = 0, n G No,

(2.8) (2.9)

where *o = l, Xn+l,x

= --Xn,xxx

+ 2VXn,x + VxXn,

neNo,

(2.10)

lo = l, Yn+hx = -^Yn,xxx-r22Yn,x

+ 2<j>x(d-1(4>Yn,x)+cn),

n€H,,

(2.11)

where c„, n G N, are the integration constants chosen to define Xn, CQ = 1, and we denote

{d;1f)(x) = jXdx'f(x').

(2.12)

It can be shown that Xn(V), Yn(<j>) are differential polynomials in V and (p, respectively (see, e.g., [17, Sect. 2.3]). In particular, one can show that 4>Yn is a total derivative and our notation in (2.11) implicitly assumes a homogeneous integration procedure with cn the only integration constant. Explicit computations then reveal XQ

= Co = 1,

206

J C i = V + Ci,

(2.13)

X2 = ~^VXX + -V2 + cxV + c 2 , etc., Vo = l, Y 1 =2<£ + d 1 ,

(2.14) 3

Y2 = -4>xx + 2<£ + ci2i^ + d2, etc., with c n , d„, n 6 N, integration constants, and thus, KdV 0 (V) = V i - 2 V s = 0 , KdVi(V) = Vt + Vxx - 6VVX mKdV o (0) = t- 2<j>x = 0,

(2.15) Cl2Vx

= 0, etc., (2.16)

2

mKdVi(0) = <j>t + xxx - 64> x = 0, etc. Given Miura's transformation (2.3), the analogs of (1.4) and (2.1), (2.2) then read KdV„(V±) = ( 2 ^ ± a , ) m K d V n ( ^ ) ,

neNo,

(2.17)

and mKdV„(0) = ±dx(2^1((^±1)t

- Pn(V±)1>±1)),

neNo,

(2.18)

where Pn(V) = 2Xn(V)dx-Xn,x(V), 3

n€^.

(2.19)

T h e Cole—Hopf transformation

Finally we return to relations (1.23) and (1.24). Since they are all proved by explicit calculations we may omit these details and focus on a precise formulation of the results instead. L e m m a 3.1 Let xp = ip(x,t) > 0 be a positive function with ip G C 3 '°(R x (0,oo)), ipt G Clfi(R x (0,oo)). Define u = -2vipx/i> with v > 0. Then ueC2'l(®-X (0,oo)) and ut + uux - vuxx = 2v I -~jdx + -r§ ) (ipt - wfyxx)

=-2vdx (Urpt - vTpxx)Y

(3.1)

(3.2)

The extension to the case with a potential term F in the heat equation reads as follows.

207

L e m m a 3.2 Let F G C1'°(IR x (0,oo)) and assume xp = xp(x,t) > 0 to be a positive function such that xp G C3'°{R x (0, oo)), xpt € C 1 , 0 (R x (0, oo)). Define u = -2uxpx/xp with v > 0. Then u G C2'l{R x (0,oo)) and ut + uux - vxixx + 2uFx = -2vdx (—{A- ^ u - Ftp) j .

(3.3)

We can exploit these relations as follows. T h e o r e m 3.3 Let F G C 1 '°(K x (0, oo)) and v > 0. (i) Suppose u satisfies u G C 2,1 (R x (0,oo)), and ut + uux - vuxx + 2uFx = 0

(3.4)

for some v > 0. Define

H x,t)

= exp(-±-J

dyu(y,t)j.

(3.5)

Then xp satisfies 0 < xp G C3'l{R x (0, oo)) and ^ M - v$„ - FiP) = C{t)

(3.6)

for some x-independent C G C(K). C»^ Lei V > 0 be a positive function satisfying xp G C 3 '°(R x (0, oo)), xpt G C 1,0 (K x (0,oo)) and suppose xpt ~ vxpxx -Fr/> = 0

(3.7)

u = -2i/%.

(3.8)

for some v > 0. Define

T/ien u G C 2 , 1 ( ! x (0,oo)) satisfies (3.4). R e m a r k 3.4 One can "sca/e away" C(t) in Theorem 3.3 (i) by introducing a new function xp. In fact, the function xp(x,t) = xp(x, t) exp(— / 0 dsC(s)) satisfies xp = vxpxx + Fxj).

(3.9)

R e m a r k 3.5 Using the standard representation of solutions of the heat equa­ tion initial value problem, xpt ~ vipxx = 0,

tp(z, 0) = ik{x),

(3.10)

assuming ipo G C(R),

ipo{x) < Cx exp(C 2 |x| 1 + 7 ) for \x\ > R

(3.11)

208

for some R > 0, Cj > 0, j = 1,2, and 0 < 7 < 1, f/iven 6y (cf. [12, Ch. 3], [14, Ch. V]; J dy exp(-(x - y)2/4ut)Mv)

rl>(x, t) = 2{Jt)1/2

> 0,

(3.12)

the corresponding initial value problem for the Burgers equation ut + uux — vuxx = 0,

u(x, 0) = uo(x),

(3.13)

reads iRdy(x-y)riexp(-(2^J0vdr1u0(r1)-(x-yn4ut)-1) HX

'

'

fR dy exp ( - (2^)-i /0V dnuoiv) - (x -

y)2(4ut)-') (3.14)

assuming UQ £ C(R) and u0(x) > 0 or \ I I Jo

=

dyu0(y)

0 ( | i | 1 + 7 ) for some 7 < 1.

(3.15)

|i|-+oo

Before turning to multi-dimensional extensions of the Cole-Hopf transfor­ mation we briefly discuss the Burgers hierarchy of nonlinear evolution equa­ tions and derive the analog of identity (3.2) for the Burgers hierarchy. The higher-order Burgers equations (choosing v = 1 for simplicity) can be defined recursively by [48], [49, Sect. 5.2], B„(u) =ut + 2Zn+hx(u)

=0,

n€No,

(3.16)

where Z0 = 1, Zn+i,x

=

(3.17)

Zn<xx ~ ^uZn,x ~ r ^ i Z , , ,

n € No•

Explicitly, one computes Zo = A) = 1, Zi = - - u + d , Z2 =--^ux

+-u2-C1-U

(3.18) + C2, etc.,

(3.19)

with integration constants c n , n € N, and thus B0(u)=ut-ux, Bi(u) — ut - uxx + uux — c\ux = 0, etc.

(3.20) (3.21)

209 Introducing i/> as in Lemma 3.1, that is, assuming u = -21>x/xl>,

(3.22)

the analog of relation (3.6) for the Burgers hierarchy now reads n+l

B n (u) = - 2 9 I ( i / ; - 1 ( ^ - ^ c n + 1 _ j ^ ) ) ,

neKfc.

(3.23)

j=\

Since clearly ut = -2dx{ip-lipt),

(3.24)

relation (3.23) is equivalent to n+l

Zn+l,x{u)

= dx{i>-lY^Cn+i-idixl>),

n€N,.

(3.25)

i=i

Since (3.25) is easily verified for n = 0,1 one can proceed inductively as follows. Assuming the homogeneous case where Cj = 0 for j = 1 , . . . ,n, and supposing (3.25) to hold in the homogeneous case, that is, Zj+1(u)

= rp-1dl+lxP,

j=0,...,n,

(3.26)

one computes z n +2.x(u) = d2x{i>-ld?+l4>)

+ip-l4>xdx(4>-1d?+1ip)

+ 1>-2(iH>xx ~ ^x){H>-ld^+lxl>).

(3.27)

Similarly, one calculates dx{ip~ld^+1ip)

= right-hand side of (3.27),

(3.28)

and hence Zn+2(u) =

T/T'^+V

(3.29)

in the homogeneous case, completing the proof. Given the analog of identity (3.2) for the entire Burgers hierarchy in the form of (3.23), one can derive the analog of Theorem 3.3. We omit further details along these lines but note that the Cole-Hopf transformation was used by Blaszak [5] to transform the nth order homogeneous Burgers equation B n (u) = 0 to the linear partial differential equation ipt — dx+1ip = 0. Identity (3.23) goes beyond this transformation as detailed in Theorem 3.3 in the case n= 1. The multi-dimensional extension of Lemma 3.2 reads as follows.

210 Lemma 3.6 Let F G C 1 ' ° ( l n x (0,oo)) and assume ip = ip(x,t) > 0 to be a positive function such that ip G C 3>0 (R n x (0,oo)), ipt € C 1 , 0 (R" x (0,oo)). Define u = ~(2u/a)\7ln(ip) with a G R\{0}, v > 0. Then u G C 2 - ^ ! " x ( 0 , o o ) ; l n ) and 2v + — VF =

ut +a(u- V)u-vAu

2v

(1 , A V I - ( ^ t - uAip - Ftp) ) .

(3.30)

Our final result shows how to transfer solutions between the multi­ dimensional Burgers equation and the heat equation. Theorem 3.7 Let F G C 1 -°(R n x (0,oo)), a G M\{0}, and v > 0. (i) Assume that u G C 2>1 (R n x (0,oo);R n ) satisfies u = V$ for some potential $ G C

3,1

(R

n

(3.31)

x (0,oo)) and

2v ut + a{u ■ V)u - i/A« + — V F = 0. a

(3.32)

De/ine tf = e x p ( - ^ * ) .

(3.33)

Then ip G C 3 l l (R n x (0,oo)) and ^{il>t-vAil>-Fil>)=C(t),

(3.34)

for some x-independent C G C((0, oo)). (ii) Let ip > 0 be a positive function satisfying ip G C 3 , 0 (R n x (0,oo)), ipt G C l i 0 (R n x (0,oo)) and suppose ipt ~ vAij) ~Fip = 0.

(3.35)

Define 2v u =

Vln(V>). a Then u G C 2 ' J (R" x (0,oo);R n ) satisfies (3.32).

(3.36)

Acknowledgments Supported in part by the Research Council of Norway under grant 107510/410 and the University of Missouri Research Board grant RB-97-086. We thank Mehmet Unal and Karl Unterkofler for discussions on the Burgers and (m)KdV equations, respectively.

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213

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214

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215

SOME MODELS OF NONIDEAL BOSE GAS WITH BOSE-EINSTEIN CONDENSATE Dedicated with admiration to Prof. Ludwig Streit on his 60

birthday

ROMAN GIELERAK Technical University of Zielona Gora 65-246 Zielona Gora Poland E-mail: gieleratMproton.ift.uni.tvroc.pl Using the functional integral techniques homogeneous limits of the perturbations of thermal states (describing nonrelativistic Bose Matter at the thermal equilibrium) by bounded cocycles are constructed rigorously. Additionally some elementary pro­ perties of these limiting states are discussed and in particular the preservation of the nonpurity in the critical case is proved.

1 1.1

Introduction The problem of Bose-Einstein condensate

One of the most spectacular achievements in the experimental low temperature physics of the past few years is the laboratory realization of the Bose-Einstein condensate (BEC) of cold atoms l l 2 , 3 . This bizarre quantum state of a Bose Matter is formed at nanokelvin temperatures and requires also high atomic densities. The BEC is formed when the quantum wave packets of atoms overlap at low temperatures and the atoms condense almost motionless, into the lowest quantum state. This means that the wave-lenght of the matter waves associated with the cold atoms 4 , the de Broglie waves, become comparable in size to the mean atomic distances in a cold and dense sample. The phenomenon of BEC was predicted by Bose 4 and Einstein 5 already in 1924-25. But even for the case of simplest ideal Bose Gas the complete mathematical proof took a long period of time ended successfully only in 70-ties by the elaborations of the Dublin group 7 ' 8 . Concerning the standard, gauge-invariant many body hamiltonians with realistic pair interatomic potentials the situation from the theoretical physics point of view is still very far from being clarified. It seems that the class of models which is closest to realistic systems and being tractable mathematically is that described by the so called model systems with diagonal hamiltonians 8 . In a series of papers MO,11,12 new mathematical technologies for study­ ing the nonrelativistic Bose Matter at thermal equilibrium and in nonzero

216

temperature were invented. They are based on the observation that the mo­ dular structure of the free Bose Matter has the so called stochastic positivity property 1 1 . It is this property which enables us to use certain functional inte­ gral/random field description of these genuine noncommutative structures. Ad­ ditionally those commutative analysis methods open the doors for applications of the methods of classical statistical mechanics for studying the perturbations of the free thermal structures by the so called thermo-field like perturbations. Such programme was initiated in 9 ' 1 0 and developed in certain directions 11 - 12 . The present contribution continues the analysis of9,10 for the so called gentle perturbations of the free Bose Matter. A new, purely commutative strategy for studying the standard many-body hamiltonians initiated in 13 has been extended recently in 1 2 , 1 6 , see also the activity 17 . 1.2

The thermal structure of the Ideal Bose Gas (IBG)

Let W be a *-algebra version of the Weyl algebra over the one-particle Hilbert space h = L2(Rd,dx) with generic elements denoted as Wj, f G h. The one-particle unitary dynamics U° in h is generated by h% = - A - / d , where - A is the (Friedrichs) Laplacean and ^ is the chemical potential. The natural lifting 5? of C/,° is defined as afws = Wvoj. The free thermal state wo on W is given by 18 uo(Wf) = exp-1-(f\coth^hM)

(1)

and the corresponding free thermal state wgr for IBG containing Bose-Einstein condensate w?(Wj)

= exp-ic|/(0)|2 " o ^ / ) ^

(2)

which is well defined, providing d > 3, and where / i s the Fourier transform of / G L1 n L2(Rd, dx), c > 0 is some constant depending on details of the thermodynamical passage and measuring to some extent the size of the condensed fraction of gas. Both states, w0 and wgr are invariant under the action of 3? (5° 1 fi = 0 in the case of WQ1"). Applications of the GNS construction lead to the well known Araki-Wood thermal modules ( ^ c r ) , ^ c r ) ) 5 r ° ' ( c r ) , £/°' (cr) ). We define the corresponding von Neumann algebras M0 (resp. M%) as weak closures of TT°(>V) ( resp. of 7r°'(er)(W)). Then the systems ( M , , a ? , u , 0 ) = .4o,

{Mc0r,a°t'cr,u>^)=Alr

217

form W - K M S systems (and where at are the corresponding extensions of (cr) f/,°' ). The presence of the Bose-Einstein condensate in A" is manifested throughout the nonpurity of AQ . The central decomposition of wgr is well known since Araki-Wood work 1S. In the present contribution we shall study the (homogeneous limits of) perturbations of A0cr throughout the unitary cocycles perturbations 18 and in particular the question on the preservation of the nonpurity of the limi­ ting thermal states (in the critical regime) will be settled up positively. The techniques employed are based on the functional integral approach invoked in 9,10,11

2

T h e main result

Let (x()oo be a net from Ca(Rd) (the space of continuous functions with compact support), such that Xi > 0 pointwise and \\mxc - 6 an d

in the weak sense. For x £ R , c > 0 and a g f l w e define We(a,x) = Wa.Xt(_x) where Wj is now the representative of Wj in the corresponding free thermal module. Let dp be a (complex in general) Borel measure on R, with compact support and such that dp(—a) — dp(a). For A C Rd being bounded region we define H^xjdp^JdxW^cx) (3) A

where X £ R is called coupling constant, and HlL = \ j dp(a) j dp(0) JdxJdy

W((a, x)T{x - y)Wc(p, y)

(4)

where T € L](Rd) and integrals are defined in the o--weak topology of the corresponding W*-algebras. From the simple estimates ||itf|| < |A|Var(p)|A| l|//ALH<|A||Var(p)|2||r||1|A| where Var(p) is the variation of p and |A| means the volume of A it follows that the unitary cocycles perturbations theory 18 for studying the perturbed

218

(by H*) free dynamics can be applied. For this goal we define: • the unitary cocycle T*' : t

(i

'»-!

A

Tf' ES 1 + £ . ' / *i J dt2 • • • J dtna^\H*) ">!

o o

■ ■ ■ *y«\H*)

o

where # stands for L or nL and the free dynamics a 0 acts as «?(#A) = A / dp(a) | and similarly for H%L, • the perturbed dynamics af'

dxW a U - x . ( ._ r )

:

*


er

= a°t< \A) + £ i" jdhjdh-. l^1

0

j dtn

0

(6)

0

[««;(crW).[---[«?;(cr)(^)^]---]] forAe^, • the perturbed thermal vacuum Qf:

n*i(er) = n ^ +

|

«/«,•■•«/«„

0<»i<-<«»<# e

-»,floi/#e-(.,-»»-)i/....i/#flW

(7)

where //Q r is the generator of a, in WQ anc ^ again the integrals and series are weakly convergent in H" . R e m a r k 1 From the Araki theorem 18,20 it follows that there is a holomorphic map: Rn + iTm

9

( Z l i . ..,*„) -

e"-».jy#e«(*--*-.)»o.

..H*Q(0eT) € W 0 cr)

where

rf/ 2 = { ( S i , . . . , s „ ) | o < S l <••■<«„
(5)

219

continuous on Rn + iTn

and additionally obeying the estimate (uniformly on

Rn + i7fn/2) || e i;nff 0// #

e i(».-i.-,)ir.

< \X\n(Var(p)r < \X\n(Var(p)f"

.Hfn(0cr)\\

..

■ |A|" for

# = L

■ ||r||M|A|r

for

# = n£

From the general theory 18 it follows that the systems

AiA*) = {rff\\M\\-l-a£*o,af*A',)} form a /?-KMS structures and such that the modular dynamics corresponding to the vector states w* given by fl* are exactly equal to a*' . In the following we shall sometimes drop out the superscript # in the notation and the following abbrevations will be used: / d(T, «, *)? = A

dxi A

■ dxn / dn • • •

drn / dp(ar)

/ dp(an),

etc.

A

Now we are ready to formulate main results of this contribution. Theorem 1 (The noncritical case) /. There exists small Ao (depending on /?, # , r,p) such that for all \X\ < X0 the unique thermodynamic limit limw*(A)=u/#(A) A

exists in the sense of weak convergence. The limiting state w*(A) is faithful I on Ao and is ergodic with respect to the (natural) action of the Euclidean motions on AQ. The limiting state is analytic in X and entire analytic on Ao2. If additionally dp(a) = dp(—a), A > 0 and T > 0 (pointwise) in the case # = nL, then the unique thermodynamic limit w#(A) = limw*(A) A

exists for all X > 0 and is faithfull and Euclidean invariant. Here is the main result: Theorem 2 (The critical case) There exists A0 (depending on p, # , T,...) such that the unique thermodynamic limit \\mu)er(X)=u)cr{X) A

220

exists for all\X\ < Ao. The limiting state wer (A) is faithfull, Euclidean invariant and satisfies: there exists a Borel measure dXTen(a, 0) on [0, oo) x [0, 2ir) and a family of faithfull and ergodic (with respect to the translations) states w r »(A), indexed by [0,oo) x [ 0 , 2 T ) such that oo 2 *

/-)(A) = y|dAre„MK,*(A) o o Moreover the measure dXren(r,0) 3

is not concentrated on a single point.

Proofs: the main ideas

For / , g e h such that f = f, g =~g and for any r € [0, fi/2) it follows by the straightforward computations that 9 , 1 0 : (Qicr)\Wfe-^r)Wg^r))

=exp-l-S0o«r)(T,f®g)

(8)

where ^ l ( c r ) ( r , / ® g) = Sg(r, f®g) + (cjf(x)dx and

■,/}

, ~

Sp0(rJ®g)

J g{x)dx)

(9)

irie-T(-*-ii)+e-V)-T)(-*-i*)

= (/|

1

_e^(_A_/l)

9)h

(10)

Extending S^'^CT\T, •, •) to R1 by reflection invariance and periodicity it fol­ lows that 5^'( cr ^(r, •, •) is stochastically positive and reflection positive on Kp = (—/3/2, (3/2) continuous kernel. Therefore, there exists a Gaussian pro­ cess £°'( cr ) with values in S'(Rd), which is faithfull, stochastically continuous, periodic and reflection positive, and satisfies

E(e ( c r ) ,)«o' ( c r ) ,-) = ^' (cr) (r,-,-)

(11)

The law of this process can be identified with the Gaussian random field fi^ on the space S'{Kp x Rd) such that E^orMr,

* M 0 , y) = S f ( e r ) ( r . x - y)

We define the following functionals of fi0 ^A (
(12)

.

f dp(a) f dx : eia^T^

:

(13)

221

Uf' = X f dr f dp(a) f dp(0) f dx f dyeia^T^T{x 0

A

- y)eip^T^

A

where ipc(r, x) = (ip * X<)(r> x) The finite volume Gibbsian perturbed measures dp.^cr' = (Z^r1

dfi^)

(14)

are defined as

exp V*{!p)dW\V)

and their corresponding canonical correlation functions PA (n,<xi,xi,...,Tn,an,xn) as />A

cr)

(r1,a1>x1,...,rn)a„,xn) = A n E ^ e « a - ^ r — ) . - . e i « ^ ( T " ' ^ )

Recall, that their thermodynamic limits in various ranges of couplings were controlled rigorously in previous publications 9 | ! 0 . Using definition of WA it follows by straitforward computations that Z^

= (Q^\^)n^

(15)

and for f E h "{Ar\Wl)

= (^ACr)\Wf^r))

= exp -l-S^r\0,f®

f)

• £ [ i /d(r>al«)yn«"m(/|BiX,1,r',(*i) n t e - i ^ - ' - ^ ^ - l ^ K a , * ) ? ]

(16)

>= i

(where SQ^ = (Xe ® Xt) * S o ) complicated) in the case of U%L:

7V,Af,X,>0

'

L

/fl

^ or ^ A

case

i

an

^ similarly (albeit more

222

I d(r\ a\ x 2 ) f d{a\ /?2, y 2 ) f j d(r 3 , a 3 , x 3 )f d(
" r •J] ; =1

exp(i/m;(• - xj»exp(i/m(/|/?jx € | i < ,,(. - 2/j)r(xj - y)) L

[ e x p ( - a j S * ( e r > * /(*})) - 1] [ e x p ( - / ? X f ° * /(»))) - l ] } M

f

•J]

exp(iJm(/|a? X ( ,,>.(- - *?»exp(i7m(/|/J? X c,,vj(- - 2/;)r(x? - y))

; = 1 >>

[exp(-a2S0^/(x2))-l]} • f[ [ ex P (t7m(/|a 3 x e | 1 > ; (- - x?» exp(i/m
- yj >r(x 3 - y3)

[eXp(-^35jf)*/(y3))-l]} •^((rSaSx1^,...,^3,/?3,^)]

(17)

where X«.ir(--*) = ^ T X « ( - - * )

(18)

and

Aer)((ri,«1,*1)fr,-,(^,^,»3)f) =

M

i

x

Formulae (16) and (17) provide a link between the analysis on the abelian sectors described in the earlier papers 9 ' 1 0 and the corresponding states wAcrJ on the whole algebra(s) of observables M." ■ This is why we propose to call them the reduction formula(e) for the state(s). The essential volume dependence of wA is that of the canonical corre­ lation functions entering in formulas (16) and (17). To control the thermody-

223 namic limits limu>A

we have to divide the analysis into two parts, the first

A

(easier) devoted to the noncritical case and the second dealing with the critical

3.1

The case of noncritical IBG

The following results have been proved in 9 : There exists a small Ao (depending on z,P,...) such that the unique thermodynamic limits limp A >{T, a, x)f - p\

'(T,a,x)f

A

exists for all N and |A| < Ao. The limiting correlation functions p^n \T, a, x)^ are Euclidean invariant, possess the cluster decomposition property and are analytic in the disc |A| < A0. Moreover /?A" —* p\ locally uniformly. See Prop. 3.2 and Prop. 2.4 in 9 . Applying the formulated results, the proof of the first part of Theorem 1 follows whence using the reduction formulae (16) and (17). If additionaly we assume that the measure dp is real and even, A > 0 and T > 0 pointwise (in the case of nL), then we can use Theorem 3.9 of9 to controll the limit limp A = p" . Modulo the cluster decomposition property, the limiting A

canonical correlation functions possess most of the properties as in the local perturbations case and this enables us to prove the second part of Theorem 1 similarly as above. 3.2

The case of critical IBG

The main difficulty here is that SQT has no long range decay and this is why the high temperature/low density methods (Kirkwood-Salsburg identities, cluster expansions) do not apply straightforwardly to study the limit(s) limp A r . A

However, such an analysis is possible in the pure phases. For this goal, let K^ be the circle of radius 2ir and let dAo be the spectral measure of the state u>Qr on K-I-K x (0,oo) i.e.

v% = JdX0(r,6)u«g where to"B are pure states on M"

(20)

characterised by

< i ( W » = e i c ' / V / 3 c o s f i V(o) c -is 0 '(o,/»/)

(21)

The explicite form of dAo is well known 18,19 . It was observed in 10 that the states ufg are stochastically positive and the underlying periodic processes

224

£T with values in S'(Rd) are Gaussian processes with covariances given by (informally): /,«

e-M(-A)

E # ■'>(«)# ■ % ) = ^

. e -(/3-|r|)(-A)

TT^PA)

(22)

and means E£r'\x)

= cl'1r1'2a»6

(23)

The corresponding random fields n®'gT on the space S'(Kp x Rd) are Gaussian and with second moments given by (informally): j

¥>((), x)
= S$(r, x - y)

(24)

d

S'(K„xR ) and means:
/

= c^2r1'2 cos 6

(25)

5'(A'«xft<')

The perturbed, finite-volume Gibbs measures d^' dft9\
J

are defined as:

= ( 4 r ' 8 ) ) _ 1 exp U*{
(26)

where Z™

=

j

eKPU*(
(27)

d

S'(KgxR )

The thermodynamic limits ,.

(r,«)

hm/4

J

(r,8)

= /4'

were controlled rigorously in 10 and in particular the existence of the thermodynamic limits of the correlation functions pKA' '((T, a, x)") defined as

fre\(T, a, *)») = A"

j

S'(KfixR")

H

eiaMTit'i)dft9)

j = 1

and for small values of couplig constant A were proved in 10 .

(28)

225

Similarly as in the noncritical case the following reduction formula (writtten only for the local perturbations only) holds: W

A

(W/) — similar as in (16) but with pj[' ' instead of p\ and with S^> instead of Sf ( c r )

(29)

and where wAr' ' is the state obtained from UJTQ by the unitary cocycle(s) perturbations as in the noncritical case. Using the reduction formula (29) and results of10 the existence of the unique limits ,nmw _,,(»•>») ilT1,,(--,«) u)x A A

as a weak limits on M" and for all (r, 0) G (0, oo) x Kp follows. In particular the limiting states w^r' are Euclidean invariant, pure and faithfull. Using the central decomposition (20) the following formula can be derived easily:

"Z(Wj) = j d\0(r,
(30)

where ofc'^, £1% are the corresponding vacuums given by the formula (7). From the reduction formulae (29) and (30) it follows easily that sup w%(Wf) < oo A

for any / € S(Rd). Using this observation it follows by a similar arguments to that presented in 1 0 that there exists limdA 0 (r, 6^ A

*

" = dA Pen (r, 0)

(31)

II^'AII

in the sense of measures. That the measure dA re „(r, 0) is not concentrated on a single point was proved in . Summarizing this discussion we have obtained the following result: "x{WS)=

j

dXren(r,0)^'s\(Wf)

(32)

if^x(O.oo)

for sufficiently small A and the measure dA ren {r, 0) is not concentrated on a single point which means that the limiting state w£r is not the pure state. Moreover the states u^ ' appearing in (32) are pure states on M'Q ■

226

4

Concluding remarks

From the Tomita-Takesaki theory it follows that there exists a canonical (mo­ dular) dynamics afT on MQ such that the constructed states ur"' are KMS states with respect to ajT. On the other hand from the corresponding abelian euclidean-time Green functions some (a priori) another W'-KMS structure(s) can be constructed 9 , 1 °. This yields the difficult (and therefore very intere­ sting) question whether these two a priori different W*-KMS structures do coincide. In the finite volume perturbation case it has been proved in 15 that these W'-KMS structures coincide. Concerning the infinite volume situation the following result has been proved in 1 5 : Theorem Let £* be the corresponding to the infinite volume limits of the per­ turbations considered in this contribution thermal processes. Then, these pro­ cesses are Markovian diffusions on the circle Kp providing A is sufficiently small. From the above theorem it follows that the corresponding thermal vacuum £lA constructed from the abelian euclidean-time Green functions (see 9,1 ° for details) has the following cyclicity property: the linear hull of the vectors

Wfe~T"? WgAQt f = J,g = g,re[0, 0/2) in the corresponding Hilbert space 7iA is dense. However, the still missing point of the identification of the canonical Tomita-Takesaki W'-KMS structure with those obtained in 9 , 1 0 is the question whether the va­ cuum QA is cyclic under the action of the time - 0 Weyl algebra W(h) in HA. Acknowledgements I would like to express my deep gratitude to Prof. Ludwig Streit for giving me opportunity throughout the study of his papers and discussions with him to understand the power of the functional integral methods in quantum physics. I appreciate very much the acceptance by Robert Olkiewicz to present some of the unpublished results from our joint work being still in progress. References 1. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cor­ nell, Science 269, 198 (1995) 2. K.B. Davies, M-O. Mewes, M.R. Anderson, N.J. van Druten, D.S. Durfee, D.M. Kurn, W. Ketterle, Phys. Rev. Lett. 75, 3966 (1995)

227

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

C.C. Bradley, C.A. Sackett, R.G. Hulet, Phys. Rev. Lett. 78, 985 (1997) S,Chu, Nobel Lecture, 8 December 1997, Stockholm and ref. therein A. Einstein, Sitzungsberichte der Preussischen Akad. D. Wiss. 1, 3 (1925) S.N. Bose, Z. Phys. 26, 178 (1924) M. van den Berg, J.T. Lewis, J.V. Pule', Helv. Phys. Acta59, 1271 (1986) T.C. Dorlas, J.T. Lewis, J.V. Pule', Commun. Math. Phys. 156, 37 (1993) and ref. therein R. Gielerak, R. Olkiewicz, J. Stat. Phys. 80, 875 (1995) R. Gielerak, R. Olkiewicz,/. Math. Phys. 37, 1268 (1996) R. Gielerak.L. Jakobczyk, R. Olkiewicz, J. Math. Phys. 39, 6291 (1998) R. Gielerak, J. Damek, Unbounded perturbations of the Bose-Einstein condensate, subm. to Rep. Math. Phys. R. Gielerak, A. Rebenko, Operator Theory: Adv. and Appl. 70, 219 (1994) R. Gielerak, J. Damek, Polynomial, thermofield perturbations of the ideal Bose gas, in preparation R. Gielerak, R. Olkiewicz, Gentle perturbations of the free Bose gas. III. The states and cyclicity of the thermal vacuum, in preparation R. Gielerak, Gibbsian approach to quantum many body problems, papers in preparation Yu. Kondratiev, E.W. Lytwynov, A. Rebenko, M. Rockner, G.V. Schepaniuk. BiBoS preprint 789/6/97 0 . Bratelli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, II, Springer, 1981 H. Araki, E.J. Wood, J. Math. Phys. 4, 637 (1963) H. Araki, Publ. RIMS 4, 361 (1968)

228

CONFIGURATION SPACES FOR SELF-SIMILAR R A N D O M PROCESSES, A N D M E A S U R E S QUASI-INVARIANT U N D E R DIFFEOMORPHISM GROUPS GERALD A. GOLDIN Rutgers University, SERC Bldg. Rm. 239, Busch Campus, 118 Frelinghuysen Road, Piscataway, NJ 08854, USA E-mail: gagoldinQdimacs. rutgers. edu In honor of Ludwig Streit

To describe a continuous unitary representation of the group of diffeomorphisms of physical space, we typically specify a quasi-invariant measure together with a unitary 1-cocycle on some configuration space. Lebesgue measure on the space of Appoint configurations describes JV-particle quantum mechanics, where noncohomologous cocycles (obtained from unitary representations of the symmetric group or the braid group) give the statistics of bosons, fermions, paraparticles, anyons, or plektons, and even suggest possibilities for quantum nonlinearity. Poissonian or more general measures on spaces of infinite but locally finite configurations describe the infinite-volume limit of a gas of particles. Then some cocycles of interest are as­ sociated with infinite permutations or braids. A class of quasi-invariant measures, obtained from self-similar random processes, requires spaces of configurations hav­ ing accumulation points. A program of research on this topic is discussed, including ongoing joint work with Moschella.

1

Diffeomorphism Groups, Local Currents, and Quantum Physics

Motivated by the desire to make use of gauge-invariant currents as coordinates in nonrelativistic and relativistic quantum field theory, Dashen and Sharp pro­ posed a singular local current algebra of mass and momentum densities.L When these are interpreted as operator-valued distributions, one obtains a natural, infinite-dimensional Lie algebra, modeled on spaces of C°° scalar functions and vector fields on the physical space. Exponentiation of this Lie algebra yields a group G—the natural semidirect product of the group of smooth scalar functions (under pointwise addition) with the group of diffeomorphisms (un­ der composition). 2 ' 3 Distinct quantum systems in the nonrelativistic limit are described by unitary representations of G. In such a representation, interpre­ tation of the self-adjoint generators as gauge-invariant mass and momentum density operators, obeying an equation of continuity, specifies the kinematics. 4 To establish notation, let X be the Euclidean manifold of physical space equipped with the (local) Lebesgue measure dx. Define as operator-valued

229 distributions the mass density Pop(x) and momentum density Jop(x), x € X, Pop(x) = m^(x)V'op(x), J ^ x ) = {h/2i) [< p (x) VVoP(x) - ( V ^ ( x ) ) ^ ( x ) ] ,

(1)

where tp0p{x) is a second-quantized, nonrelativistic field at fixed time t obeying the equal-time, canonical commutation (—) or anticommutation (+) relations [&, P (x,*),t// op (y,i)] ± = [ < , M ) , ^ p ( y , t ) ] ± = 0, [ ^ o p ( x , t ) , ^ ( y , t ) ] ± =
Pop{h)\

= 0,

[Popif),

Jop(g)] = »fiPop(g- V / ) ,

[JepiSi), Jopfa)] = -ihJop{[gu

ga]),

(3)

where [gi,g2] is the Lie bracket. Define U(sf) — exp[(ta/m)/J op (/)] and V(<j>f) = exp[(is/fi)J op (g)], a € R, where the flow * obeys d*{x)/ds = g(0£(x)) with initial condition f=0{x) — x . Existence of i^f(x) for all s is guaranteed if g has compact support in X, or if g(x) -► 0 (rapidly in all derivatives) when |x| -4 oo. It is also of interest, for X = R d , to permit g(x) to approach some generator of a general linear transformation as |x| -¥ oo; but here we content ourselves to take / , g with compact support. Then the diffeomorphisms of interest are likewise compactly supported—i.e., with : X -> X, we have ). The group law obtained by exponentiating (3) is U{h)V{fa)U{f2)V{4>2)

= U(fi + h ° *i)V(fcfc),

(4)

where 4>\4>2 — \- We thus take U and V in (4) to be a continuous unitary representation (CUR) of the semidirect product group G — V(X) x DiffX, where X is the manifold of physical space, V(X) is the additive group of com­ pactly supported, real-valued C°° functions on X, and DiffX is the group of compactly supported C°° diffeomorphisms of X. We endow G with a topology where all the one-parameter subgroups sf and <j>f (x) are continuous in «, and the group operations are jointly continuous in / and <j>. Our philosophy is now to study the inequivalent CURs of G given by (4), quite independently of whether or how these are derived from underly­ ing second-quantized fields. Each CUR requires interpretation as describing a

230

possible quantum system by means of the corresponding self-adjoint represen­ tation of (3). The latter informs us about the physical meaning of the theory, when we interpret pop and J o p as mass and momentum density operators: e.g., the spectrum of pop describes the particle content. An early and rigorous pre­ diction of anyon statistics in two space dimensions came from carrying out this program for DiffR2, and turned out to confirm a possibility conjectured by Leinaas and Myrheim and afterward suggested by Wilczek. 5 ' 6,7 This led to our understanding of how unitary representations of the braid group BN, includ­ ing its higher-dimensional represenations, induce CURs of the diffeomorphism group that describe exotic quantum statistics. 4 ' 8 ' 9 The continuity equation, embodying probability conservation, imposes an important constraint on the time-evolution (thus, on the choice of Hamiltonian) in a quantum theory obtained this way. For $ G H obeying Schrodinger's equa­ tion, the (scalar) mass density expectation value is p m (x,t) = ( ^ / ^ ( x ) * ) , while that of the (vector) momentum density is Jmom(x, t) = (*> Jop(x)*). Then we require that dpm/dt — —V • Jmom2

Fields Intertwining TV-Particle Representations

The iV-particle Bose or Fermi representations of (3) are given by ,£(/)*(•■•> ( x l t . . . x „ ) = m E 7 = i / ( x i ) * ( 8 , a ) ( x i , . . . x N ) , ^(g)*(s'a)(xi,...xw) =

(ft/2i)E7=i{g(xJ)-Vi*C-a)(x1,...xw) + Vi-[g(xi)*(8'a>(x1,...x,v)]},

(5)

where the superscripts "s" or "a" of the square-integrable wave functions * denote respectively symmetric or antisymmetric functions of the N variables. These representations, in which there occur no Schwinger terms, follow from (1) in the "s" (—) or "a" (+) nonrelativistic Fock representations of (2), where the current algebra acts irreducibly on iV-particle subspaces. So we know from the underlying field theory that each series, "s" or "a", N = 0 , 1 , 2 , 3 , . . . , describes a hierarchy of particles (more generally, of quantum excitations) of the same type. When exponentiated, these representations give the Nparticle representations of G (see below). But suppose we do not know about underlying fields, and have only a collection of representations of G satisfying (3) or (4), indexed by N, in Hilbert spaces HN- When are we justified in concluding that they belong to a series describing a hierarchy of excitations? Sharp and I proposed the following criteria. 10 Let pop(f) and Jop(g) be the direct sum over N of all the operators p£,(f) and J<^(g) respectively, in the given indexed collection. Then there should exist, intertwining the

231

representations, fields ip* : HN ->• ' H N + I and ip : "H/v+i -*■ (averaged with a test function h on X) ip* satisfies the brackets

\Pop(f),r(h)} = V*(P£=1 (/w,

UN

, so that

[Japfcww]=r(j£=l(g)h),

(6)

and ip satisfies the adjoint of (6). Note how, on the right-hand side, h is also interpreted as belonging to the Hilbert space for N = 1. Note also that only commutator brackets have been assumed. Since Eqs. (6) do hold in both the "s" and "a" Fock representations, they are reasonable requirements for the collection of representations indexed by N to be a hierarchy. In fact, they apply more generally than in these cases. At the level of the group, we have corresponding intertwining conditions, UN+1(fW(h)

=

r(UN=i(f)h)UN(f),

VN+1()r(h) = r{VN=l(4)h)VN(),

(7)

where again the adjoint of (7) gives the conditions for ip. One can visualize V>*(x) as creating a singular excitation labeled by x € X, and h as averag­ ing over such excitations to yield one that is more smooth. Then Eqs. (7) have a natural geometrical meaning. They assert first, that U and xp* act locally in X; and second, that if we create a new (single) excitation labeled by h and then transform the state vector under a diffeomorphism of X, we achieve the same outcome as transforming X by the diffeomorphism and then creating the transformed (single) excitation. The reason for the generality of language is that we are thinking here not only of bosons and fermions, but of anyons, vortices, or more complicated extended quantum objects. Sharp and I constructed nonrelativistic anyon creation and annihilation fields in a particle number representation, obeying the commutation relations above, and found that the fields ip and rp* themselves satisfy ^-commutation relations, where q is the anyonic phase shift. This is one of the places that the g-deformation enters quantum physics without having been introduced "by hand." 3

Configuration Spaces, Quasi-Invariant Measures, and Cocycles

Next we discuss CURs of G more broadly. In a quite general framework, a unitary representation of the diffeomorphism group DiffX can be written so as to act in a Hilbert space ~K = L£(A,W). Here we need a configuration space A, a continuous action of DiffX on A, and a measure / i o n A that is quasi-invariant under diffeomorphisms; i.e., diffeomorphisms must preserve the class of measure zero sets. Then "H is the space of /i-square-integrable functions #(7), for 7 G A, taking values in an inner product space W. The

232

representation V is given by [Vimh)

= X0(7)*(07) W ^ f (7),

(8)

where 07 indicates the action of the diffeomorphism <j> on the configuration 7; /i0 is the measure transformed by <j>; dn^/d/i is the Radon-Nikodym derivative of fifj, with respect to /z; and the X(t>(l) '• W ->• W are unitary operators in W. The latter are defined (V0 G DiffX) almost everywhere in A, and must satisfy (V0i, Al)x*{il) = X*i02(7)

(9)

almost everywhere in A. The quasi-invariance of /x is the condition that guar­ antees the existence of the Radon-Nikodym derivative in (8); the square root is the real 1-cocycle needed to make the representation unitary. While A together with n tell us the quantum configurations, the unitary 1-cocycle x gives in­ formation about (generalized) phases, including possible exchange statistics of particles. It is always allowed to set x(l) = 1; b u t additional, noncohomologous cocycles give us interesting, unitarily inequivalent representations. To rep­ resent the semidirect product group G = V(X) x DiffX in W, embed A in the (continuous) dual space V so that all the evaluation maps < 7, / > (/ 6 V) are measurable in 7 with respect to y,. Then, for * € ~H, take [C/(/)*](7) = e»<^> * ( 7 ) ,

(10)

which together with (8) defines the representation obeying (4). With X = Kd, the CURs of G that result from exponentiating (5) fit this picture very nicely. They are obtained by letting AN be the space of (unordered) iV-point subsets of X, and /x be the (local) Lebesgue measure on this space. Then Diff X acts on AN in the natural way: for 7 = {xi,..., x/v}, we have $ 7 = {^(x!),.. . ,$(x/v)}. The group action is transitive, and ft is quasi-invariant under it. Indeed the Radon-Nikodym derivative is given by the product (dfj,^/d/j,){j) = U^=i Jtt>(xj), where ^ ( x ) is the Jacobian of <j> at x. Furthermore, we embed A in V'{X) by defining < 7, / >= X)jLi /( x i)- This gives a class of CURs of the semidirect product group describing the quantum theories of iV indistinguishable particles. To specify a particular CUR in the class, we must now choose a unitary cocycle x^MWe can fix an arbitrary element 7 0 = {xj, ...,x°N} € AN, and associate noncohomologous unitary cocycles with inequivalent, unitary representations of the stability subgroup Diffyo X. The latter consists of all diffeomorphisms

233

<j> such that 07° - 7 0 . Note that for <j> 6 Diffyo X, Eq. (9) reduces directly to such a representation at 7 0 . Under the right technical conditions, the CUR of DiffX described by the cocycle can be obtained directly by inducing from a CUR of Diff^o X. The method is similar to Mackey's construction, n except that our groups (being infinite-dimensional) are not locally compact—hence we do not have Haar measure with which to work, and must obtain n and its good properties by other means. 12 Let us look a little more closely at Diffyo X, when 7 0 is an iV-point con­ figuration. A difFeomorphism <j> can leave 7 0 fixed by leaving each point x9 of 7 0 individually fixed; but for X = R d , d > 1, <j> can also permute the points. Thus we immediately have a natural, continuous homomorphism hyo from Diff^a X to the symmetric group SN- A unitary representation u of SN yields a CUR u o /i7o of Diff^o X, which in turn induces a CUR of DiffX. In this way, the iV-particle representations with different unitary cocycles describe not only Bose and Fermi particle statistics, but also parastatistics (induced from the higher-dimensional unitary representations u of SN)- The trivial repre­ sentation u = 1 gives the cocycle x<j>{l) = 1> which describing bosons. The alternating representation of SN leads to a 1-cocycle describing fermions. When d = 2 more possibilities exist. There is now a natural homomor­ phism from Diff^o R 2 to the braid group BN- Unitary representations of BN then provide CURs of Diff7o R 2 which induce CURs of Diff R 2 . In this way one-dimensional unitary representations of BN lead to the intermediate statis­ tics of anyons, and higher-dimensional representations of BN to plektons. 6 ' 9 ' 13 It is interesting to remark on what might seem a pathological feature. The measure zero set 2$ c A on which x(l) is undefined, or the measure zero set .20,,0a c A on which (9) fails, can depend on <j>, i, and fa in such a manner that there may actually be no configurations 7 on which \ is defined for all 4> G DiffX, and (9) holds for all fa,
234

DiffX. Lebesgue measure on A N is quasi-invariant under diffeomorphisms, but now the corresponding CURs of DiffX are reducible; diffeomorphisms alone do not connect the subspaces having different exchange symmetry. How­ ever, let us introduce a set of distinct particle masses rrij ^ m* for j ^ k, and define < 7 , / > = X) J = 1 mjf(xj)Then the mass density operator acts as multiplication of $(7) by < 7, / >; i.e., pNp,m„...,mN{fMxu

...,Xyv)

=

^ m j / ( x j ) $ ( X l ) ...,x„).

(11)

i=i

The corresponding CUR of the semidirect product group is now irreducible. We return to systems of distinguishable particles later. There are some different ways to go from CURs of G describing N parti­ cles to those describing infinitely many particles. One way is to put N bosons in a compact, rectangular region of volume V, impose periodic boundary con­ ditions on wave functions in the region, and then let N and V become infi­ nite while their ratio N/V approaches a finite constant (the mean density). More precisely, let V'N,V(/WN,V{4>) be the iV-particle Bose representation of G = T>(X) x DiffX, where X is now a torus of volume V. Let the (normalized) ground state wave function on this torus be CIN,V = V ~N/2, and write the gen­ erating functional Lsy{f) = (£lNy,UN,v{f)£lN,v)The limiting functional L„(f) =

lim LN,v(f) = exp{p/"[e''M-l]dx} (12) ;v,v-»oo, N/v-tp > 0 J is then the Fourier transform of a Poisson measure Up, where p > 0 is the Poisson parameter. To elaborate, we obtain \ip as a probability measure on the configuration space I'j^ of countably infinite but locally finite subsets of X. The measure is concentrated on the subset consisting of all configurations that have average density p. For 7 = {x.j\j — 1,2,3,...} e ^x , a n ^ f° r K (X), the configuration 7 defines a functional < 7, / > = £)£U / ( x j ) < °°. Then Lp{f)

= f

expi<7,/>dM/,(7).

(13)

Now for <j> € DiffX, its natural action on Tj^ is given by $ 7 = {(xj)}. Since <j> has compact support, this does not change the average density p.

235

The more important fact is that fip is quasi-invariant under this action—the Radon-Nikodym derivative (dn^/d^)(-y) is the infinite product of the Jacobians J0(xj), x.j € 7, but as <(> is compactly supported, only finitely many of the factors are unequal to one. Thus we have (for each value of p) a CUR of the semidirect product group G. This CUR describes the infinite free Bose gas. Another way to obtain the same representation is to start in the Fock representation of the canonical Bose fields ip^p, ipop. Make the substitutions ip*p -+ ^gP+^/pI and ipop -> ipop+VpI, which leave the canonical commutation relations manifestly invariant. Now Eq. (1) gives the self-adjoint representation of the current algebra associated with the Poisson measure. 14 ' 15 The configuration space I * , equipped with Poisson or Gibbs measure and carrying a CUR of the group DiffX, is of great current interest in both quantum and statistical physics, and the above development has been advanced considerably. 16,17 For the Bose case we have been discussing, x^(7) = 1- Evi­ dently, in more than one space dimension, one can obtain additional, inequivalent representations from the cocycles associated with the nontrivial represen­ tations of a group of infinite permutations, or (in two space dimensions) infinite braids. I am currently pursuing an idea in this direction, in collaboration with Kondratiev, that we believe might give us a nice description of an "infinite Aharanov-Bohm gas." Of long-term interest is also the question of how to construct measures that are quasi-invariant, under diffeomorphisms of X, on spaces of extended objects—configurations such as strings, loops, or ribbons embedded in X. A step in this direction is to consider the more elementary possibility of infinite configurations of points in X that are not locally finite, but have accumulation points. This is one of the key ideas behind my work with Moschella, which I shall describe next. 4

Measures from Self-Similar Random Processes

When X is the real line, the half-line, or the circle, there are some special techniques for obtaining CURs of DiffX and its central extensions. 18>19>20. Here, through a few elementary examples, I would like to explain a method that Moschella and I introduced 21 ' 22 . 23 > 24 for constructing measures quasiinvariant for DiffR.1. Let us work in the sequence space X°°, with X = R 1 . To start, draw a pair of points £1,2:2 6 R 1 from some nowhere vanishing probability density f(x), and for convenience label the points so that x\ < x-i. Now draw two more points from the uniform probability density on the interval [11,12]. and label these so that 13 < X4. Iterating this procedure, we have a Markov process in which X2m+i and a;2m+2 are drawn from the conditional

236 probability density hm+l

{x\X2m-l,X2m)

= hm+2

(x | X2m-1,

X2m)

=

8 am-i'»»-J ^^ F2m — X2m-l|

^ ^

where I[a,b] denotes the indicator function of the interval [a,b]. To adapt the method to the circle S 1 , begin by drawing X\ and x-i from a uniform density and continue with the shorter of the two resulting intervals of arc (which with probability one are of unequal length). It is not hard to show that with probability one there is precisely one cluster point for the sequence {XJ), to which it converges. The sequence of conditional probability measures for obtaining the Xj defines (using Kolmogorov's theorem) a probability measure H on the space X°° of infinite sequences (XJ). Now let 7 = (XJ), and consider the Radon-Nikodym derivative (d(i#/dfi)(y) of the transformed measure /i^ with respect to /i, at 7. If it exists, it can be expressed as an infinite product: (dfi^/dn)^) = rij^i uj,<)>(y) > where the jfth factor 1^,0(7) arises from the step in which the point Xj is drawn. Quasi-invariance of pi under diffeomorphisms means that this infinite product converges to a finite, non-zero limit. In general, supposing Xj to be selected from the conditional probability density fj (x | Xi,..., Xj-i), we have =

fM*s)\«*i),---M**-i))

J { )

(15)

fj(Xj\Xl,---,Zj-l)

The explicit expression for ^ ^ ( 7 ) in the present example, using (14), is «i=2m+l,0(7J =

777v T7x7 J
■

U&)

and similarly for j = 2m + 2. Since with probability 1 the sequence Xj con­ verges, we have Ujt<j> ->■ 1. Note how the nature of the conditional probability leads to this conclusion. We let the width of the probability distribution for each pair of points be established by the outcome in choosing the preceding pair. Then, as the Xj approach a limit, the first factor in (16) tends toward the reciprocal of the Jacobian. The self-similarity built into the configurations selected by the measure is what gives us the quasi-invariance. Now, tij-,0 -+ 1 is necessary but not sufficient to ensure that the infinite product converges to a finite, non-zero limit. It is also necessary that the rate at which Uj^ -¥ 1 be sufficiently great. In the case at hand, it is not hard to show that (with probability 1) £ j l i \XJ+I - xj\ < 00, which guarantees the desired convergence. So we do have a quasi-invariant measure, a CUR of DiffK1, and a quantum system that the representation describes.

237

Physically, this CUR may be interpreted as the quantum theory of a strictly confined cluster of infinitely many particles. Despite the fact that the Xj are ordered by the index j , the component particles in the cluster are not truly (i.e., not intrinsically) distinguishable. This is because the index la­ beling of the points in (XJ) coincides, with probability one, with the ordering obtained directly from the values of the Xj (using the order relation in R 1 ). We can simply label the odd points a;2m+i as we come in from the left, and the even ones X2m+2 as we come in from the right, and obtain the same in­ dices that occurred during the random process just described. However we can further introduce a mass distribution for the particle cluster. Choose a fixed sequence (rrij), with each entry m,j > 0 and X),-mi = M. For / G X>(RX) let < 7, / > = M - 1 E ~ i "»y/(xy) < oo. Then U(f)V (<(>)$ (i) = e l ' < w > l * ( 0 7 )

N

U^iAi)

(17)

gives a CUR of the semidirect product group G. The infinite free Bose gas in R 1 , described as in the previous section by a Poisson measure with parameter p, can also be constructed through a self-similar random process "reciprocal" to the one above—where the pairs of points are chosen outside rather than inside the intervals established by the preceding choices. We choose the first pair of points xi and xi to the left and the right of the origin (respectively) from exponential densities with parameter p. Then continue with densities Um+i and fim+z that vanish in the interior of the interval [af2m-i»^2m]» and are nonzero but fall off exponentially to the left and the right of the interval respectively (keeping p independent of j). There exist further interesting possibilities for quasi-invariant measures based on intervals. We can construct tensor product representations, for which the configurations are N tightly-bound (possibly overlapping) clusters. We can also consider configurations that are the union of infinitely many clusters, where the accumulation points distribute according to a Poisson measure with fixed parameter. In such examples, we begin to deal with the possibility that distinct ordered sequences may lead to the same unordered configuration. Measures on spaces of random Cantor sets, quasi-invariant for diffeomorphisms, were constructed by the present method, and independently by Neretin.24,25 As above, we draw x\ and x-i from nowhere vanishing densities fi(x) = h{x) on R 1 , or S 1 , labeling so Xi < x2, and draw x3 and au from a uniform density on [ari,^], labeling so that x3 < x 4 . Let U\ be the relative complement of [23,£4] in [si.sa], the disjoint union of a pair of intervals (of different lengths). Now draw £5 and XQ from the uniform density on the lower

238

of these intervals, and £7 and x% from the uniform density on the upper one. Again form the relative complement by removing the inner intervals, to obtain £/2 as the disjoint union of four intervals. At the mth stage we thus draw 2 m random points (x2m +1 ,x 2 '»+2>---,X2 m + 1 ) from a measurable set Z7m_i, obtain­ ing Um C Um-\ as the disjoint union of 2 m intervals. This procedure defines a measure fi on the space of infinite sequences (XJ). With probability one (by this construction) the set of accumulation points in R 1 of such a sequence is a random Cantor set, uncountably infinite but of Lebesgue measure zero. To demonstrate quasi-invariance of \i under a diffeomorphism , we again express the Radon-Nikodym derivative as an infinite product. The lemma that, with probability one, the total length of all the complementary intervals remaining at the mth stage converges to zero faster than the geometric sequence A m , for some real number A between 0 and 1, is sufficient to obtain the desired con­ vergence of the product. 26 Variations in this construction are possible, where the uniform densities are replaced by other suitable nonvanishing probability densities on the intervals, respecting the rate of convergence. Neretin's construction results directly in measures on random Cantor sets (regarded as subsets of the half-line), but they are obtained without passing through a sequence space. To relate the two methods, we need to be able to project measures on (ordered) sequence spaces to measures on spaces of (un­ ordered) configurations that are not necessarily locally finite, while preserving quasi-invariance under diffeomorphisms. Now the quasi-invariance of the measures in the examples so far discussed depends on the fact that orientation-preserving diffeomorphisms of R 1 respect the order of the values of points in R *. Then a region of positive measure—e.g., with X2m+i and *2m+2 selected to be between X2 m -i and xim—can never be mapped by a diffeomorphism into a region of zero measure—e.g., with a^m+i or X2m+2 outside that interval. For manifolds of dimension greater than 1, we cannot generalize the self-similarity in our construction by using uniform probability densities on compact spatial regions, since diffeomorphisms distort any preselected shapes of such regions. To overcome this, Moschella and I also considered nowhere vanishing probability densities. In one space dimension, suppose we draw a point XQ from a nonvanishing density /o, and then draw X\ from a unit normal density f\ with mean XQ. NOW draw X2 from a normal with mean x 0 (the mean remains fixed), and standard deviation o\ = n\x\ - Xo|, where /c > 0 is a fixed parameter. Continuing the process, draw x m + i from a normal density with mean Xo and standard deviation am = /c|x m - xo|. Again we have a self-similar random process, where smaller values of K describe more tightly bound systems. We have shown that in this situation there is a critical value KQ > 0; so that when

239 0 < K < Ko , (XJ) converges to XQ with probability one (condensed phase); and when K > Ko, \XJ\ -t oo geometrically with probability one (rarefied phase). The associated measures on the sequence space are again quasi-invariant under elements of the group Diff'R.1. Thus for all K ^ Ko, we have a unitary group representation and the possibility of its interpretation as a quantum system. In this construction, the use of normal densities is not really essential. We need only the scaling property, and applicability of the central limit theorem. The next major step is to obtain some examples in R d for d > 1. We conjecture that measures can be constructed through self-similar random pro­ cesses that generalize the preceding example. In R d we anticipate choosing the point i m + i fr°m a density that depends on the outcome for xo, establishing the mean, and on the preceding d points (xm-d+i, ■ ■ ■ , i m ) , establishing the covariance matrix. 5. Configuration Spaces with Accumulation Points Typically distinguishable particles in a manifold X are described by ordered or labeled points in X. Indistinguishable particles are described by (unordered) subsets or generalized subsets of X, where in a generalized subset we permit the same element to occur more than once. Correspondingly, measures quasiinvariant under diffeomorphisms can be constructed on spaces of iV-tuples or infinite sequences of points drawn from X, or alternatively on spaces of finite or countable subsets or generalized subsets. To relate the methods that Moschella and I are using to those of others, especially the cited papers by Albeverio et al. 18>17 and Neretin, 25 we need a framework for statistical physics with a configuration space larger than T ^ . This is a direction of our ongoing research, about which I shall mention just a few ideas. 27 We begin as usual with a smooth, noncompact manifold X, with all the needed nice topological properties. Let S x = { 7 C X : 7 is finite or countably infinite} .

(18)

That is, with N e x t = N U { « 0 } , we have \-y\ 6 N e x t for all 7 G E x . This is the largest "point" configuration space in which we now propose to work. At least in principle, E x allows the description of continuous, extended configurations such as submanifolds embedded in X (strings, membranes, and so on), since these can occur as sets of accumulation points of sequences. We take "gen­ eralized configurations," to be generalized finite or countably infinite subsets of X; i.e., elements of E x each of whose members x has a multiplicity n(x) which is 1,2,3,..., or N0. Denote the set of generalized configurations by £x.

240

Next we consider the natural projections from ordered iV-tuples and se­ quences to unordered generalized finite and countable configurations. Let XN be the space of JV-tuples drawn from X, where by convention X N~° contains one element, the empty sequence (); let X °° be the space of infinite sequences of elements of A"; and let S(X) = (Wfi=0XN) UX°°. Thus S(X) includes both finite and infinite sequences (XJ). We can now let p : S(X) -* Ex be the map which assigns to a given sequence (XJ) G S(X) the corresponding generalized set of points; and let p : S(X) -¥ Ex be the map which assigns to 7 = (XJ) the corresponding set 7 = {XJ}. Note that the image of X°° under p is actually all of Ex, as infinite sequences can consist of finitely many recurring points. Of course, this will typically occur with probability zero. Let X?° C X °° be the set of infinite sequences having exactly j accumulation points in X. Such sets will typically have positive measure. Note also that ac­ cumulation points in X depend only on 7, and not on the particular sequence 7Gp_1(7)-

Let Tx = {7 G Ex : |7 n K | is finite (V compact K C X). Then T™ = {7 € Tx : |7l = No}- Define E ^ = {7 € Ex : H = No}, and let E ^ c ^x consist of configurations that have exactly j accumulation points'in X. So we have T^ = E £ ° L 0 C E ^ , and p(Xf°) = E J ^ . Subsets of the space Ex of generalized configurations can be introduced along similar lines. Our goal is to move via p from quasi-invariant measures on S(X) or X°°, constructed through sequential, self-similar random processes, to corresponding quasi-invariant measures on configuration spaces of identical particles. This should also permit the subsequent consideration of interesting, nontrivial cocycles. I would remark at this juncture that a locally finite configuration 7 defines a functional on V{X) as a sum of evaluation functional, but this functional is not defined for a general element of Ex. Therefore measures on Ex quasiinvariant for DiffX, from which we obtain unitary representations of DiffX, may not always yield unitary representations of the semidirect product group. Now each of the quasi-invariant measures /1 on X°° described in the pre­ vious section is obtained, by a standard construction, through a compatible family of measures on cylinder sets with finite-dimensional Borel base. We wish to obtain from /J under these conditions a quasi-invariant measure on Ex or one of its subsets; i.e., to be able to essentially disregard the order of the entries in the sequence, and look only at the target points. To accomplish this, the idea is to introduce the largest a-field on Ex for which the projection map p is measurable, and obtain the projected measure p,. One nice result is that for any j , the set of all configurations with precisely j accumulation

241

points in a given Borel set of X belongs to the the a-field on Ex- Thus we can count the numbers of accumulation points. A second, essential result is that the construction respects the quasi-invariance of \x under DiffX. Acknowledgments I would like to thank the Alexander von Humboldt Foundation for its sup­ port of the continuation of this work and related research during my 199899 sabbatical year in Germany. I am also grateful to the Institut fur Angewandte Mathematik, Universitat Bonn, Germany, the Dipartimento di Scienze Chimiche, Fisiche, e Matematiche, Universita delPInsubria, Como, Italy, and the Arnold Sommerfeld Institut fur Mathematische Physik, Technische Uni­ versitat Clausthal, Germany, for hospitality during my recent visits. I have benefitted from interesting conversations connected to this work with S. Albeverio, Y. G. Kondratiev, T. Kuna, U. Moschella, and M. Rockner. References 1. R. Dashen and D. H. Sharp, Phys. Rev. 165, 1867 (1968). 2. G. A. Goldin and D. H. Sharp, Lie algebras of local currents and their representations. In 1969 Battelle Rencontres: Group Representations, Lecture Notes in Physics 6, ed. V. Bargmann, (Springer, Berlin, 1970), p. 300. 3. G. A. Goldin, J. Math. Phys. 12, 462 (1971). 4. G. A. Goldin, R. MenikofF, and D. H. Sharp, Phys. Rev. Lett. 51, 2246 (1983). 5. J. M. Leinaas and J. Myrheim, Nuovo Cimento 37B, 1 (1977). 6. G. A. Goldin, R. Menikoff, and D. H. Sharp, J. Math. Phys. 22, 1664 (1981). 7. F. Wilczek, Phys. Rev. Lett. 48, 1144 (1982). 8. G. A. Goldin and D. H. Sharp, Phys. Rev. D 28, 830 (1983). 9. G. A. Goldin, R. Menikoff, and D. H. Sharp, Phys. Rev. Lett. 54, 603 (1985). 10. G. A. Goldin and D. H. Sharp, Phys. Rev. Lett. 76, 1183 (1996). 11. G. W. Mackey, Induced Representations of Groups and Quantum Me­ chanics (Benjamin, New York, 1968). 12. G. A. Goldin, R. Menikoff, and D. H. Sharp, J. Math. Phys. 2 1 , 650 (1980). 13. G. A. Goldin, Parastatistics, ^-statistics, and topological quantum me­ chanics from unitary representations of diffeomorphism groups, in Procs.

242

14. 15. 16. 17.

18. 19.

20. 21.

22. 23.

24.

25. 26.

27.

of the XV. Int'l. Conf. on Differential Geometric Methods in Theoret­ ical Physics, eds. H.-D. Doebner and J. D. Hennig (World Scientific, Singapore, 1987), p. 197. G. A. Goldin, J. Grodnik, R. T. Powers, and D. H. Sharp, J. Math. Phys. 15, 88 (1974). A. M. Vershik, I. M. Gelfand, and M. I. Graev, Dokl. Akad. Nauk. SSSR 232, 745 (1977). S. Albeverio, Y. G. Kondratiev, M. Rockner, J. Funct. An. 154, 444 (1998); J. Funct. An. 157, 242 (1998). S. Albeverio, Y. G. Kondratiev, M. Rockner, Diffeomorphism groups and current algebras: Configuration space analysis in quantum theory, Rev. Math. Phys. (in press). G. Segal, Commun. Math. Phys. 80, 301 (1981). R. S. Ismagilov, Representations of Infinite Dimensional Groups, AMS Translations of Math. Monographs 152 (Providence, RI: American Math. Society, 1996). E. T. Shavgulidze, Proceedings of the Steklov Institute of Mathematics 217, 181-202 (1997). G. A. Goldin and U. Moschella, Diffeomorphism groups, quasi-invariant measures, and infinite quantum systems. In Symmetries in Science VIII, ed. B. Gruber, (New York, Plenum, 1995), p. 159. G. A. Goldin and U. Moschella, J. Phys. A: Math. Gen. 28, L475 (1995). G. A. Goldin and U. Moschella, Diffeomorphism group representations and quantum phase transitions in one dimension. In Xlth International Congress on Mathematical Physics, ed. D. Iagolnitzer, (Cambridge, MA: International Press, 1995), p. 445. G. A. Goldin, Unitary representation of diffeomorphism groups from ran­ dom fractal configurations. In Lie Theory and its Applications, eds. H. D. Doebner, V. Dobrev and J. Hilgert (Singapore, World Scientific, 1996), p. 82. Y. A. Neretin, Sbornik: Mathematics 187, 857 (1996). See also G. A. Goldin and U. Moschella, Random Cantor sets and measures quasi-invariant for diffeomorphism groups. Submitted for the Procs. of the 17th Workshop in Geometric Methods in Physics, Bialowieza, Poland, 1998 (1999, in press). G. A. Goldin and U. Moschella, Generalized configuration spaces for quantum systems. Submitted for the proceedings of the International Conference on Infinite-Dimensional (Stochastic) Analysis and Quantum Physics, Leipzig, January 18-22, 1999 (in press).

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STOCHASTIC PROCESSES A N D THE F E Y N M A N INTEGRAL Z. HABA Institute of Theoretical Physics, University of Wroclaw, Poland We discuss a representation of quantum dynamics in terms of Markov processes. It is shown that a holomorphic continuation of the Schrodinger wave function ^t(x) —* ^t(z) determines a complex Markov process qi(z). Feynman integration at a finite temperature is briefly discussed.

1

Introduction

An attempt to express the quantum dynamics by means of a stochastic process has a long history. We consider here some methods which relate the Feynman integral l to a probability measure (for some other approaches see 2 - 3 and references quoted there). As is well-known the Feynman integral cannot be expressed as a complex measure on an infinite dimensional space of real paths. It can however be represented as a distribution on a space of Wiener function­ a l ( Hida distribution; see 4 - 5 ) . We discuss in this paper another approach which allows to express the Feynman integral by means of a stochastic process running over a complexified configuration space (the process is a functional of the Brownian motion). Such an approach has been discussed earlier by Cameron 6 , Doss 7 and Azencott and Doss8. We shall denote by b(t) € R" the Brownian motion, which is the Gaussian process with the covariance E[bj{t)bk(s)] = 6jkmin(t, s) where j,k = 1,2, ..,n. The expectation value can be realized by the Wiener measure. A gener­ alization of the Feynman formula of Cameron and Doss 6 7 reads (we exclude some points from Rn, then the admitted class of potentials is much larger than the one considered by previous authors). Theorem 1 Assume that i/>(x) and V(t,x) (for each t 6 [0,r]) are analytic functions and e x p f - ^ / V(t-s,x

+ \ob(s))ds\tp(x

+\ob(t))

is integrable with respect to the Wiener measure for t € [0,r] and x € V,

244

where T> is a region in Rn, then ip(t,x)=E where

exp

-u>

--

/ V (t - s, x + Xab (s)) ds)ip(x

+ \
(1)

A = - L ( l + z) and a —

is the solution of the Schrodinger equation ihdtip(t,x)

= -—A4>(t,x) + V(t,x)1>(t,x) (2) 2m (with the initial condition il>) for < € [0, r] and x E D . In the proof of this theorem one shows through differentiation (apply­ ing elementary stochastic calculus) that the Schrodinger equation is satisfied pointwise. The assumption of integrability can be satisfied for a large class of potentials with V which is the whole of Rn with a possible exception of arbitrarily small neighbourhoods of a finite number of points (see a discus­ sion in 9 ). Analytic potential which is linearly bounded on the domain (z € C" : z = x + y + iy where x € V and y G Rn) will satisfy our assump­ tions. In particular, we discussed in 9 the rational functions of the form P(x)Q(x)-1 where P is a polynomial of degree k and Q is a polynomial of degree greater than k-2. Then, the set V will be Rn without small neighbourhoods of a finite number of points where the Brownian motion (multiplied by (1 + i) ) hits the complex roots of Q. Another class of admissible potentials includes rational functions of trigonometric and hyperbolic functions. For example we can treat V(x) = tan(z) if x ^ (2k + l)7r/2 in ipt(x). 2

Representation of the quantum dynamics by a stochastic process

We show in this section that quantum dynamics can be represented by a Markov process. First, we show how to construct the process as a solution of a stochastic differential equation. Then, we express the correlation functions of the process and its characteristic function by means of the solution of the

245

corresponding Schrodinger problem. We assume here that the ground state x is known ( for another classical construction of the process see 9 ) . We write the ground state x °f the Hamiltonian H in the form x = e x p ( - ^ ) (x is strictly positive, hence In x is well-defined) # e x p ( - - ) = £0exp(--) where h2 H = - ^ A

+

V

(3)

We consider the initial condition for the Schrodinger equation (2) in the form tp = x4> where both x and <j> are assumed to be analytic. Vt is the solution of the Schrodinger equation (2) with the initial condition tp = x if and only if 4>i is the solution of the equation dt4>t = -l-H4>t = £-&4>t--VS\/4>t

(4)

with the initial condition