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6
J. Martin Lindsay
In view of Exercise H, the functor n : T (S) —» . F ( r ) descends to a map e : L° (X) —> L° (r) between the respective algebras of measure equivalence classes, which also maps IP (X) into Lp (F), for 0 < p < oo. In particular, the following Hilbert space relation holds {EV,
e$) = exp {(/?, ip),
2
for tp,ij) £ L (X). The Hilbert space Q := L2 (T) is called Guichardet space. Here are some useful subspaces of L° (T):
Ga := UzL°(T):\\f\\{ay.= K. :=
f|
1
Ga la>0
K' := M &„. £D £
:=
[j^a*\f(o)fj'\oo\
ft, 2
{ / £ i ( r ) : there is NeNs.t.f
(a) = 0 for #<x > TV}
:= Lin {e^, : tp G D} := £z,2(r),
where a > 0 and Z) C L2(X). Thus 5 := Q\ is Guichardet space and, for a > 1, <7a is the domain of the operator ^/o on 5, where N is the number operator. Exercise I. Show that both Qan, and £ are contained in K. Exercise J . (a) The injective map tp i—* ev is continuous L2 (X) —+ I/2 (F), with continuous inverse. (b) {e
^symm. (§^) descends to a {natural) isometric isomorphism a n d £,/, is the multiple of the identity (/ ip) I. Also let % = (av + a v ) >P*> = i (at> ~ av) > a n d cf = ( / £ + /<*>) • Exercise O. Check that: (a) Each of the above operators leaves the subspace K. invariant. (b) The following hold on K: Kk\\2 ,A1>), ( t f € W , ||tf|| = l), then we can express all probabilities at later times t in terms of the wave function T/>(zi,...,x nd ;i) := (Uf1ip)(xu...,xnd) ip o / from the algebra of functions on R into the algebra of functions on fi whereas an observable A can be looked upon as the homomorphism ip —> p for all a. li a y ao we have P < (w, Pau) < (u, Paou) 0, there exists n € T such that T(p, n) > 1 — e. We therefore obtain: Proposition 2.6. fib = hpp, hj = hcont, and hs = {0}. Moreover, h = h^ © haWithin this context, notice that our notion of ^-scattering states coincides with that of outgoing states. To analyze the work of Pearson in our framework, one should first introduce the core filter T of projections 7rr, (r > 0), defined as multiplication by the characteristic function of a ball of radius r centered at the origin of the Euclidean space K n . The notion of bound state introduced by Pearson through equation (0.4), is indeed stronger than ours, since that property clearly implies'that the pure state p = \ {x) < +00} is a-weakly dense in M+; it is faithful: for x in M+ , o is an increasing strongly rightcontinuous filtration of subspaces of [/(Aoo). A stop time in this context is a map S : Borel([0,oo]) -> A%?' (projections in A^) such that S([0,i\) € At for all i > 0 and 5([0, h]) < S{[0, t2}) for tx < t2. Let X = (Xa) be a complete L 2 -martingale, so there exists an element Xoo € A& such that Xa = E ct (X 0O ). Then the stopped X-process is defined as f XaS(da), C
L2(r
^®n^-^(n))-
(L8)
If < is a measurable total order on X, then the linear isomorphism A : T (r) —» ■^antisymm. (§) descends to an isometric isomorphism L2
(r)-0^o^ntisymm.(^n)-
(1-9)
Integral-sum kernel operators 7 In the nonatomic situation we are discussing here Xn\X^ is m n -null, so that (1.8) reveals Guichardet space over X as symmetric Fock space over L2 (X). On the other hand (1.9) shows that a measurable order on X induces an isomorphism between Guichardet space and antisymmetric Fock space. (Note that this isomorphism sur vives the transition to measure spaces X with atoms). The link between these is provided by the linear isomorphism T, which descends to an isometric isomorphism L2
(r)-©n^2PQ)-
(i-io)
With the measure space understood, integration with respect to the symmetric measure will be denoted by / r .. -da, Jr .. .dto, etc. The fundamental property of the symmetric measure is contained in the next result. Lemma 1.6 (J-J2)- Let f : Td —* C be measurable. Then the function F-.T^C,
ff^V,
,
f(alt...,ad)
is measurable; moreover if f is either non-negative or integrable, then J ... I f(a1,...,ad)da1...dad
= / F (a) da.
Exercise L: Verify the / - ^ - r e l a t i o n when f = n^ <8> ■ ■ ■ <8) ^v>d2. INTEGRAL-SUM CONVOLUTIONS
From now on the underlying measure space will be one of the Lebesgue spaces [0, T] or R + , and will be denoted L 2.1. Duality Transforms. Let L2(Wi), which we shall call Wiener space, denote the L 2 -space of the standard Wiener measure on { £ : / —> K| £ is continuous and C (0) = 0}. Any element F of L? (W/) may be decomposed into its components in each of the homogenous chaos subspaces and each of these components is expressible as a multiple Wiener integral:
F = f° + J2„
[■■■ [
fn(t)dWh...dWtn
(2.1)
^"^V Jo
FU 2 = £ > n l l / n | | 2
(2-2)
2
where f0 = E [F] 6 C and, for n ) l , / „ 6 l (/"). I.E. Segal referred to the resulting natural isomorphism between Wiener space and symmetric Fock space (2.1) as a duality transform. Invoking the isomorphism (1.10), and heeding (2.2), gives us an isometric isomorphism between Wiener space and Guichardet space: F i—> / , where / is defined by /„ (t) = / ( { i j , . . . , tn}). This leads to the following elegant and useful statement of Wiener-Ito decomposition (2.1): F=jf{a)dWa.
(2.3)
8
J. Martin Lindsay
Having invented the noncommutative integration theory in which a trace functional on a von Neumann algebra replaces integration with respect to a measure, Segal also obtained a duality transform between the £ 2 -space of the Clifford von Neumann algebra over L2 (I) with its natural trace, denoted L2 (C/, r) and antisymmetric Fock space over L2{I). Invoking the isomorphism (1.9), this duality transform is nicely expressed as follows: F = J
f(a)dCa,
Here F 6 L2 (Ci, r) and {Ct ■ t £ / } is the Clifford process. An Ito-Clifford theory of stochastic integrals was extensively developed by C. Barnett, R.F. Streater and I.F. Wilde in the early 1980's. Since random variables may be multiplied as well as added, it is natural to wonder what the product on Guichardet space, corresponding to pointwise multiplication in Wiener space under the duality transform, should look like. This question has a beautiful closed form answer, discovered by H. Maassen. It is given by the integralsum convolution: {f*wg){l)
= T]
I' diof{a\Juj)g{LO\Ja),
(2.4)
*■—'aC7 J
called the Wiener product. 2.2. Formal Derivation. Let us formally multiply two L2-Wiener functionals given in terms of their Wiener-Guichardet representation (2.3) J f(a)dWa
j
using the Ito-rules: {dWtf = dt;dWsdWt and /? = T\CT, the Ito rules give /// J J i{an/3=0}
g(r)dWr
= dWtdWs.
Putting UJ = a D r, a =
f{aUu)g(uU/3)dWadwdW0,
and the f-^2 Lemma gives / / Yl
f(aUu)g(LoUa)dujdW1,
in other words (since the summation is over a finite number of terms, and so can be done either before or after the w-integration)
J(f*w9)(l)dW1. 2.3. Basic Estimate. Let / and g be versions of elements of Qz. Since the map (a,uj) i—> / ( o i U u ) is measurable, the / - £ ) Lemma gives JdaJ
du\f{a\Jw)\2
= JdaJ2aCa\f^)\2
= J da2#° \f (a)\2 < oo.
Integral-sum kernel operators 9 It follows that, for almost all a G T, the map w i—> / ( a U w ) is square-integrable. Now g shares these properties, and the map (a,u>, j3) i—> / ( a U w ) j ( w U | 3 ) is mea surable. It follows that, for almost all a 6 F, the map w i—> f (aUui) g(uDa) is integrable for all a C a. Applying the Cauchy-Schwarz inequality (twice), the J - J ^ Lemma (three times), Fubini's Theorem, and the binomial relation (x + y)*a = J2aca x*ay^a therefore gives
J da lj2aca
(aL>") g (UJ Ua)\
h^f
<
f da2*c,Y,aCa{
<
j da2#"J2aCa
=
J da JdP2*a2*13
=
jda
=
Jdo3#"\f(a)\2
f
dwf{a\Jio)g{u\Ja)\
Jd0Ji\f
(aUuji)f
Jdu2\g(iV2Ua)\2
J^lfiaUujJl2
I'chj12#a\f(aUoJl)\2
f duj2\g [u2\J p)\2
Jdp
I' diu22*e \g(u2U P)\2
Jdr3*T\g(T)\2
< oo. Thus (2.4) defines a square-integrable function on T. Moreover, linearity of the integral shows that the (measure equivalence) class of / *w g depends only on the classes of / and g. Exercise M. Recall the space IC' defined in Section 1.3, and modify the above argument to obtain the following. Proposition 2.1. Let f € Qa and g € Gb where ab > 1. Then f*wg element of IC', and satisfies
ll/^fl|l ( e )
2
is « well-defined
(2-5) ^(a-c)(b-c).
Corollary 2.2. Let f,g 6 L°(r). Then
(a) f,g€Gs=> f*wge G\ (b) / e Ua>i Sa, g e ic => f *w g e g. Proposition 2.3. (IC, *w) is an abelian associative unital algebra, with identity <5g ; moreover £ is a unital subalgebra. The last part follows from the relations
An equivalent version of (2.5) generalises nicely to products of more than two ele ments, and permits easy verification that the estimates (2.5) are optimal, with expo nentials being maximisers.
10 J. Martin Lindsay Theorem 2.4. Let fi 6 G(Vi-i) for i— 1, •. •, n, where vlt..., vn > 1 and vx 1 + ... + v~l < 1. Then g := fi *w ■ ■ ■ *w fn € G(v-i), where v = (wf1 + . . . + v~^) , and n
(2-6)
Ni(.-i)
Moreover, equality is realised when ft — eVi and ipi = v^tp some
(for i = 1 , . . . ,n) for
This theorem is proved by an induction on n, together with the easily verified identity
« ( , , _ ! ) = exp {a2 ( « - l ) | M | 2 } . If the formal argument, from which (2.4) is derived, is applied to L2-Clifford elements, using the Ito-Clifford rules: (dCt) = dt;dCsdCt = -dCtdCs, the resulting Clifford product is given by f*c9(-y)
= V]
/ dwp (a,a,ui) f (a U w) g (u Ua)
where p (a,/3,u) = (—1)" w '" ' and n : F 2 —> N was introduced in the example in Section 1.1. Further examples of integral-sum convolutions *p are obtained from functions p : T3 —► C satisfying a functional equation which assures associativity; interest ing noncommutative algebras (JC,*P) result. Returning to duality, and the formal derivations of the Wiener and Clifford prod ucts, here is the rigorous statement. Theorem 2.5. Let fi € G(Vi-i), for i = l , - - - ,n, where V\,--- ,vn > 1. Then, if v : = ( „ r i + . . . + „ - i ) - i > 2, (a) g := ft *w ■ ■ ■ *w fn e G and
j g (a) dWa = f ft (*) dWa--- j fn (a) dWa. (b) h := fi*c
■■■ *c fn &G, h satisfies (2.6) and
f h (o) dCa = fft (o) dCa--- f fn (a) dCa. The simplest proof of (a) uses the fact that for ip € L? (I), fr ev (a) dWa is the stochastic exponential exp {/ ip (s) dWs — f / V 2 } , to verify (a) for / i , . . . , / n £ £', the continuity of the multilinear map ( / l , • • ■ > fn)
£ G(vi-1)
X . . . X G(vn-\)
'
> / l *W ■ ■ ■ *W fn £
G(v-1)
which is implicit in (2.6); and the density of £ in each Gc- To prove (b) one must delve a little (but not much!) deeper into the representation of the Clifford algebra. We shall touch on this later. As a consequence of the theorem we have the following result.
Integral-sum kernel operators 11 Corollary 2.6. (a) The duality transform f i—> Jrf(o)dWa restricts to an al gebra homomorphism from (/C, *ty) into np
/ dtuf («i U a 2 U w ) j ( w U t t 2 U a 3 ) ,
*—'|a|=ff J
the sum being over all disjoint partitions a = a.\ U Q 2 U a3. On exponentials the Poisson product gives
where r\v := exp{f(
^
W (/)} .
Proposition 2.7. (/Co, *p) is a united abelian associative algebra, with identity 8$; the isometric isomorphism f i—> J r / (a) dll^ mapping Q to L2(Pi) restricts to an algebra homomorphism from (/Co, *p) into f)p<00 Lp (Pi)-
3. Q U A N T U M W I E N E R
INTEGRALS
Notational convention. We shall often not distinguish between / and Vx, thus aUt abbreviates a U {t}, and L2 (I) is sometimes viewed as a subspace of L2 (F). In this section we shall take a simple-minded view of what quantum Wiener inte grals should be. This will serve as good preparation for tackling full-scale integralsum kernel operators in the next section; it also serves as a link with the algebraic structures on Guichardet space that we have already considered.
12 J. Martin Lindsay Define operators on Q by the formulae
Kk) (a) = Y] (nxk) {a)
c
= E ^ X W*(<0
{avk) (a) =
/ dtip(t)k {a U t)
(t^k) (cr) =
j dtip (t) k (cr)
(/;*) w)
= E , (-i)B('^M*)fc(
(UQ (a) =
dt{-l)n(t',AJt)V{t)k{a\Jt)
j
and maximal generosity of domain. Here
< \Wf f du{l + #u)
IMf
< h\\2JdLo#uj
{av)* *P
n a
x *v
= a'v, "X.
—
\k{u)\\
\k(w)\2,
(nx)* = n^,
(i^)* = £?,
Tt-yOiu) ~T~ ^x^p
= at>nx + a*x>P
a* A; = <pofc, (a* + n x ) A; = x • A, /*fc = >po_k, q^k =
(a* + n x + oy) k = cvk =
ip*w k,
k, ip*ck.
X*P
The operators a',0,, and n\ are the usual Boson Fock space creation, annihila tion and number operators, viewed on Guichardet space, and the operators /* and fv are the usual Fermion Fock space creation and annihilation operators, viewed on Guichardet space; both courtesy of Proposition 1.5. From the point of view of
Integral-sum kernel operators 13 quantum probability the last three relations (with what has already been said), fully justify the terminology quantum Wiener integrals, and the notations
/ \ p d A \ I' ipdN, I' cpdA, j ipdT, I\>dF*, j
The idea is to represent operators on Q as mixed multiple quantum Wiener inte grals. In other words we seek a sensible meaning for X = jfj
x (a, 0,7)
dA^dNfidAj.
The action of such an operator may be formally obtained in the same way as we derived the Wiener product, and you derived the Clifford and Poisson products, but now using the quantum Ito relations: dA\ dNt dAt dt dAt dNt dA*t dNt (with all other products vanishing), and dQt = (dA*t + dAt). Exercise P . The result is (Xk)
(a) = > •i—'|a|=<7 J
dux
(<*! , C*2J U) k(oj
Ua2
Uaa).
(4.1)
In terms of the following transform of the (integral-sum) kernel x:
x' (a, 0,7) = J2aQ0
x
(a' a>7)'
the action has the alternative description (Xk) (a) = Y,a
a
j
duj x
' ("- s > u)k(io\Ja).
(4.2)
Notice how this includes all the products considered so far: x(a,0,u>) = f(aU0)60(oj)
=>
x(a,0,u>) = f(aU0Uuj)
=>
Xk =
=>
Xk =
x>(a,0,w)=69(a)f(0)69(<j) x'(a,0,u) = q(a,0)69(uj)f(a) x'(a,0,uj)=P(a)0,Lj)f(aUoj)
Xk=f»k
= > Xk = =4> Xk =
f*Pk f-k foqk f*pk.
Here are some examples of new operators expressible as mixed quantum Wiener integrals.
14 J. Martin Lindsay Example. Integral kernel operators. x1 (a, P, w) = y (a, u>) 5$ (/?) =4> Xk (a) = / dio y (a, ui) k (w). Thus, in particular, Hilbert-Schmidt operators on Q are included. Notice how the contribution of the number process mediates in this example; one cannot obtain integral kernel operators in the form z(a,r)dA*adAT.
/ /
This squares with the presence of the number process in the representation of HilbertSchmidt-valued martingales in terms of quantum stochastic integrals. Example. Fock Weyl operators. In terms of the normalised exponential vectors w^ := exp(—| J \tp\2)£
ljM2^jwlfi(a)59(p)w^(w).
Then Xw^, = exp{-i Im / ipip}w^+lp, which may be recognised as the action of a Weyl operator in the Fock representation of the canonical commutation relations: W{ip)W{il>) = exp{-i lm{tp, x/j)}W(tp + ip) 4.1. Basic Estimate. Let x : T3 —* C and k : T —> C be measurable functions. The conditions
I
L
dio\x(a,0,ui)k(ujDPU'y)\
< co for a.a. (a,@, 7) £ T3,
du}\x'{a,P,u)k{ojl)PU'y)\
< 00 for a.a. ( a , £ , 7 ) € T3,
/r are equivalent, and imply that (4.1) and (4.2) are defined and equal for a.a. a G T; the resulting a.e. defined function will be denoted Xk. Moreover, if h : T —> C is measurable, and
///'
dadf3dj
\h (aU P) x' (a, f3,y) k (P U -y)\ < 00,
then the function a 1—> h (a) (Xk) (a) is integrable, and j dah{a)(Xk){a)
= j77dadpd-y
h{aU P)x' (a,/3,7) fc(/? U 7 ) .
Exercise Q. (a) x = 0 a.e. if and only if x' = 0 a.e. (b) If x = 0 a.e. or k = 0 a.e. then Xk is defined and zero a.e.
Integral-sum kernel operators 15 We therefore have a partially defined bilinear map
L°
(r 3 ) x L° (r) —» L° (r),
(x', k) >—+ xk,
and trilinear functional L°(T) x L°(r 3 ) x L°(T) —> C,
(ft, x', fc) >—» f
h(Xk).
Our key tool in moving towards operators is the family of norms indexed by three positive numbers a, b and c: \\x'\\a,b,c : = { /
da
, , . s a h r (a,/3,7) =
/ ^ess.sup^K^Ja,/?^)!
H
a:'(a,/?,7) =,
T h e o r e m 4 . 1 . Le* p > a > 0, g > c > 0, Ze* ft, A; € L ° ( r ) and /ei x € Then
L°{T3).
With the techniques developed so far, it is not hard to verify this inequality. We may write the basic estimate as an operator norm inequality:
\\xk\\(a+bir^\\xX^2jk\\{c+b2) for a, b\, 62, c > 0. In particular, for any e > 0,
ll^ll(e)
(4.3)
leave K, invariant; in particular they are densely defined on Q. Insisting that there is b > 0 such that ||z'|| a ), c < 00 for all a,c > 0 ensures that the adjoint operator is also densely defined on Q and leaves K. invariant too, since its integral-sum kernel is (p, a, T) 1—> x (r, a,p)*. Two further special cases of interest are: ll^'lladc x
\\ '\\
<
°°>
wriere
a < 1 =>
n\—TT,—r < °°> where a, c < 1 = >
X is densely defined on Q, X is bounded on Q.
4.2. Uniqueness of the kernel. There are various senses in which an integralsum kernel operator has a unique kernel. Here is one. Let « be the linear span of {1E : E C I, compact}. Proposition 4.2. Let x G L° (T3) be locally integrable in its third argument, in the sense that for E C I compact,
L
doj \x(a,/3, u))\ < 00 for a.a.
(a,/3).
IT(B)
Then Xk is defined for k G £K, and Xk = 0 for all k e £K = > x = 0 a.e.
16 J. Martin Lindsay Alternative forms of determining sets for integral-sum kernel operators arise from restricting re, or from replacing SK by finite particle spaces. 4.3. Reconstruction of kernel from operator. There is a systematic way of as sociating a decreasing sequence {Fu} of finite measure sets of T (I) to each element = {jj w of T such that p| n >i ^ - The association depends on a Vitali system for the Lebesgue measure space I. Theorem 4.3. Let X be an integral-sum kernel operator, and suppose that its kernel is locally integrable. Then x> (p, a, r) = Jim {r L e b . (F
for a.a. (p,a,r)
1^,
(xi^)
,
£ T3.
Example. Let I = R+ and let X be the shift operator on L° (F), given by Xk(a)
= k{a + 1),
where { s i , . . . , sn} -f t := {si + t,..., sn + t}, and 0 + t := 0. Then X leaves each Qa invariant and is a nice unitary operator on each of these Hilbert spaces, however the sequence f lP(n) (Xl„(„) ) is eventually zero, so the shift cannot be an integral-sum kernel operator. 4.4. Algebras of integral-sum kernel operators. If X and Y are operators with integral-sum kernels x and y respectively, then how about the operator product XY1 Exercise R. By systematic application of the quantum Ito relations, and the J"-^ Lemma, give a heuristic derivation of the following formula for the kernel x*y of XY at
{P,(T,T):
Y^
/ dw x
(Q 2 ,
A U /?2 U a 3 ,
7I
U W U 73) y (QI U W U a 3 , 71 U f32 U #,, 72)
(4.4)
where the sum is over all partitions of p, a and r into a, /3 and 7 respectively. The special case in which x and y vanish except when their middle argument is the empty set is important. In this case x* y is of the same form, and the formula simplifies to x*y{p,T)=Y\
Y\a
[dux(atpUu>)y(uUa,P).
*■—'act* ^—'PCT
J
(4.5)
'
Note also how the action of operator X on vector k is contained in (4.4): V (P, cr,T) = k (p)<5B (er) <50 (r) = > x * y (p, 0, 0) = (Xk) (p). If we put K.m = \x € L°(r 2 ) : f[ dady\xa,c(a,-y)\2 where x o c ( a , 7 ) = x(a,y)/^a#ac#~>
< 00 for all a,c < 00 j
(c.f. (4.3) ), then we may summarise as follows.
Integral-sum kernel operators 17 Theorem 4.4. (/C'3',*) is a unital associative algebra with involution; the corre sponding algebra of integral-sum kernel operators leaves K, invariant. Moreover, in a natural way (/C'2', *) may be viewed as a unital subalgebra of (/C'3', *) Formula (4.5) and a version of Theorem 4.4 were found by H. Maassen, its extension (4.4) was given by P.-A. Meyer. 4.5. Four argument integral-sum kernels. There are several reasons for extend ing our mixed quantum Wiener integrals to include time, so that we are considering operators of the form:
X = jf (I x {a, 0,7, S) dA'a dN0 dA^ dT5. We shall demonstrate just one of these. It should be noted at once that uniqueness is lost in the extension (cf. the next subsection). However this is offset by the fact that the convolution of such integral-sum kernels becomes purely combinatoral. Theorem 4.5. Let X and Y be four-argument integral-sum kernel operators, with sufficiently regular kernels x and y then XY has integral-sum kernel x*y where the convolution is given by x*y{p,
a, T, u) = Y^x
(a2,ftUftUa3,
7i U (52 U 73, 5{)y («i U <52 U a3, 71 U (32 U f33, -y2, S3)
in which the sum is over all partitions ofp, a, r and ui into a, 0, 7 and S respectively. The verification is a minor perturbation of Exercise R. The resulting algebra of operators on Q has the following subalgebra, defined in terms of its integral-sum kernels: Lin {vv : v = (Vl,
(I) x £°° (/) x L2 (I) x L 1 (/)} ,
where The product on L2 (I) x L°° (I) x L2 (I) x L1 (I) corresponding to the four-argument convolution is y « ^ = ( ^ +
18 J. Martin Lindsay 4.6. Matrix-valued kernels. The next stage is to amalgamate the four arguments into one. This is done by moving to matrix-valued kernels and exploiting the fact that p, a, T and w are mutually disjoint for almost all (p, a, r, UJ). Thus our operators are now written j X{a)-dKa,
or A(X),
where X{a) 6 B{C © C)® #ff , dKa = d\Sl ® • • • <8> dASn ior a = {s1 < ■ ■ ■ < sn}, dAs combines the quantum stochastic differentials into a matrix, and ■ denotes tensor contraction. The product formula now recovers its former elegance: X * Y{a) = Y,X(ai
u
"2;a)A(a 2 ) ff)Y(a2 U a 3 ; a),
(4.6).
the sum being over partitions a = axUo^Uas. Here, for a C a, X(a; o) is the result of embedding X(a) into 5(CeC)® #CT according to how a lies in a, and A(a) = [™]®#<*. The beauty of this formula is that it remains valid in multidimensional noise, with X{a) now taking its value in B(C © k)®#<7, for the carrier Hilbert space k. Exercise S. Deconstruct (4.6). CONCLUSION
Further topics of interest barely touched upon here include: extension to multidimensions ([Der], [Att]), which also allows finite temperature representations of the canonical commutation relations; extension to operator-valued integral-sum kernels admitting an initial algebra] stochastic calculus of integral-sum kernel operators, and the solution of linear quantum stochastic differential equations using integral-sum kernels ([LM3], [Sch], [LWi]; the Guichardet-space approach to Azema martingales ([Pal]); gaussian fields and their symmetries ([Maj], [Pa2]); deformations of canon ical commutation and anticommutation relations ([LP1]); the Guichardet-space ap proach to non-causal (quantum) stochastic calculus ([L4]); the adapted stochastic derivative, and the SDCP approach to quantum stochastic calculus ([AL 1], [AL2]).
BIBLIOGRAPHICAL N O T E S
The cohomological analysis of Section 1.1 was carried out in [LP 1] and [LP2], where it was applied to the deformation of the Wiener and Poisson products described in Sections 2.2 and 2.3, and also to deformations of the canonical and anticanonical commutation relations. Symmetric measure spaces were introduced in [Gui], The totality of exponential vectors of indicator functions is proved in [PaS]; [Bha] contains an indirect proof, and [Ske] gives a short direct proof. The nontotality under restriction to indicator functions of intervals is folklore ([FeL]). The key role played by the integral-sum lemma was emphasised in [LM 1]; a proof may be found in [LP1]. Due to its ubiquity it could be called the Fundamental Lemma of Guichardet space analysis. It turned out to be (a special case of) a wellknown result in point process theory under the general heading of Campbell's The orem ([MKM]). Antisymmetric duality was introduced and analysed in [Se2] as an
Integral-sum kernel operators 19 example of the Segal-Dixmier noncommutative integration theory ([Se 1], [Dix]). ItoClifford stochastic integration was worked out in a series of papers beginning with [BSW]. The integral-sum convolution representing the product of Wiener space ran dom variables in Guichardet space was found, along the way to representing solutions of Hudson-Parthasarathy quantum stochastic differential equations as integral-sum kernel operators on Guichardet space, in the key original paper [Maa], which was inspired by [Ber] and [EvL]. Its antisymmetric extension to Clifford-duality was given in [Mel], [LM2], see also [LMe]. The basic estimate appears in [LM 1], and its refinements in [LI], [LMe]. Application to both gaussian ([Nel]) and Clifford ([Gr 1], [Gr 2]) hypercontractivity, is given in [L2] and [LMe]. Gross' conjecture on the best possible constants in the Clifford case (the same as for the gaussian case) was finally proved, using different methods, in [CaL]. Integral-sum kernel operators of the form ff z(a, T)dA"a dAT were introduced in [Maa]; extension to 3-argument kernels was done in [Mel] and [Me2]. The basic estimates for their action were established in [L 1] and [BeL], Uniqueness of the ker nel of such operators was proved in [L3], and in the multidimensional case described in [Der] uniqueness was established in [Att], Reconstruction of the kernel from the operator was carried out in [BeL], The product formula was extended to 4-argument kernels in [LI]. The product formula for matrix-valued single argument kernels is stated in [CEH] and applied to noncommutative functional Ito formulae and chaotic expansions in the universal enveloping algebra of a finite dimensional Lie algebra in [HPu], where it is also proved. It has been extended to infinite dimensional noise in [LWi] where it is also shown that injectivity of (mixed, multiple) quantum sto chastic integration is recovered at the level of processes. In turn injectivity gives a characterisation of the generators of (regular, Fock-adapted) *-homomorphic Markovian cocycles on a C*-algebra, extending the finite dimensional ([Eva]) and von Neumann algebra ([MoS]) cases. The book [Me 3] devotes considerable space to integral-sum kernel operators.
REFERENCES
[Att]
S. Attal, Problemes d'unicite dans les representations d'operateurs sur l'espace de Fock, in [Sem26], pp. 619-632. [AL1] S. Attal and J.M. Lindsay, Quantum Ito formula: the combinatorial aspect, in, "Contribu tions in Probability. In Memory of Alberto Frigerio," Proceedings, Udine, September 1994, ed. C. Cecchini, Forum, Universita degli Studi di Udine, 1996, pp. 31-42. [AL2] S.Attal and J.M.Lindsay, Quantum stochastic calculus with maximal operator domains, Ann. Probab. (to appear). [BSW] C.Barnett, R.F.Streater and I.F.Wilde, The Ito-Clifford integral, 3. Fund. Anal. 48 (1982) no. 2, 172-212. [BeL] V.P. Belavkin and J.M. Lindsay, The kernel of a Fock space operator II, in [QP 8], 87-94. [Ber] F.A. Berezin, "The Method of Second Quantization," [transl. N.Mugibayashi and A. Jeffrey] Pure and Applied Physics 24, Academic Press, New York ■ London 1966. [Bha] B.V. RajaramaBhat, Cocycles of CCRflows, Mem. Amer. Math. Soc. 149 (2001), no. 709. [CaL] E.A. Carlen and E.H. Lieb, Optimal hypercontractivity for Fermi fields and related non commutative integration inequalities, Comm. Math. Phys. 155 (1993) no. 1, 27-46.
20
J. Martin
[CEH]
Lindsay
P.B. Cohen, T.M.W. Eyre and R.L. Hudson, Higher order Ito product formula and gener ators of evolutions and flows, Internat. J. Theoret. Phys. 34 (1995), 1481-1486. [Der] A. Dermoune, Formule de composition pour une classe d'operateurs, in [Sem 24], 371-401. [Dix] J. Dixmier, Formes lineares sur un anneau d'operateurs, Bull. Soc. Math. France 81 (1953), 9-39. [EvL] D.E. Evans and J.T. Lewis, Dilations of irreversible evolutions in algebraic quantum theory, Comm. Dublin Inst. Adv. Studies Ser. A 24 (1977). [Eva] M.P. Evans, Existence of quantum diffusions, Probab. Theory Related Fields 81 (1989) no. 4, 473-483. [FeL] J.F. Feinstein and J.M. Lindsay, Exponentials of indicators of intervals are not total, Un published note (2000). [Gr 1] L. Gross, Existence and uniqueness of physical ground states, J. Fund. Anal. 10 (1972), 52-109. [Gr 2] L. Gross, Hypercontractivity and logarithmic Sobolev inequalities for the Ciifford-Dirichlet form, Duke Math. J. 42 (1975) no. 3, 383-396. [Gui] A. Guichaxdet, "Symmetric Hilbert Spaces and Related Topics," Lecture Notes in Mathe matics 261, Springer-Verlag, 1972. [HPa] R.L. Hudson and K.R. Parthasarathy, Quantum Ito's formula and stochastic evolutions, Comm. Math. Phys. 93 (1984) no. 3, 301-323. [HPu] R.L. Hudson and S. Pulmannova, Chaotic expansion of elements of the universal enveloping algebra of a Lie algebra associated with a quantum stochastic calculus, Proc. London Math. Soc. 77 (1998), 462-480. [L 1] J.M. Lindsay, On set convolutions and integral-sum kernel operators, in, "Probability The ory and Mathematical Statistics, Volume II," Proceedings, Fifth Vilnius Conference, 1989, eds. B. Gregelionis, Yu. V. Prohorov, V. V. Sazanov and V. Statulevicius, VSP, Utrecht 1990, pp. 105-123. [L2] J.M. Lindsay, Gaussian hypercontractivity revisited, J. Fund. Anal. 92 (1990), 313-324. [L 3] J.M. Lindsay, The kernel of a Fock space operator I, in [QP 8], pp. 271-280. [L4] J.M.Lindsay, Quantum and non-causal stochastic calculus, Probab. Theory Related Fields 97 (1993) no. 1-2, 65-80. [LM 1] J.M. Lindsay and H. Maassen, An integral kernel approach to noise, in [QP 3], pp. 192-208. [LM2] J.M.Lindsay and H. Maassen, Duality transform as *-algebraic isomorphism, in [QP5], pp. 247-250. [LM 3] J.M. Lindsay and H. Maassen, Stochastic calculus for quantum Brownian motion of nonminimal variance—an approach using integral-sum kernel operators, in, "Mark Kac Sem inar on Probability and Physics," (Amsterdam, 1987-1992), CWI Syllabi 32 Math. Cen trum, Centrum Wisk. Inform., Amsterdam, 1992, pp. 97-167. [LMe] J.M.Lindsay and P.-A. Meyer, Fermionic hypercontractivity, in [QP7], pp. 211-220. [LP1] J.M. Lindsay and K.R. Parthasarathy, Cohomology of power sets with applications in quan tum probability, Comm. Math. Phys. 124 (1989) no. 3, 337-364. [LP2] J.M.Lindsay and K.R.Parthasarathy, Rigidity of the Poisson convolution, in [QP5], pp. 251-262. [LWi] J.M.Lindsay and S.J.Wills, Homomorphic Feller cocycles on a C*-algebra, J.London Math. Soc. (to appear). [Maa] H. Maassen, Quantum Markov processes on Fock space described by integral kernels, in [QP 2], pp. 361-374. [Maj] P. Major, "Multiple Wiener-Ito Integrals. With applications to limit theorems," Lecture Notes in Mathematics 849, Springer, 1981. [MKM] K.Matthes, J.Kerstan and J. Mecke, "Infinitely Divisible Point Processes," [transl. B. Simon] Wiley Series in Probability and Mathematical Statistics, Wiley, 1978. [Me 1] P.-A. Meyer, Elements de probabilites quantiques I-V, in [Sem 20], pp. 186-312. Me 2] P.-A. Meyer, Elements de probabilites quantiques VI-VIII, in [Sem21], pp. 34-80.
Integral-sum kernel operators
[Me 3] [MoS] [Nel] [Pa 1]
[Pa 2] [PaS] [QP2] [QP3] [QP5] [QP7] [QP8] [QP 10] [Sen] [Sel] [Se2] [Sem20] [Sem21] [Sem24] [Sem26] [Ske]
21
P.-A. Meyer, "Quantum Probability for Probabilists," Second Edition, Lecture Notes in Mathematics 1538, Springer-Verlag, 1995. A. Mohari and K.B. Sinha, Quantum stochastic flows with infinite degrees of freedom and countable state Markov processes, Sankhya Ser. A 52 (1990), 43-57. E. Nelson, A quartic interaction in two dimensions, in, "Mathematical Theory of Elemen tary Particles," Proceedings, Dedham, Mass., 1965, M.I.T. Press, 1966, pp. 69-73. K.R. Parthasarathy, Azema martingales and quantum stochastic calculus, in, "Probability, Statistics and Design of Experiments," Proceedings, R.C. Bose Symposium, Wiley Eastern, New Delhi 1990, pp. 551-569. K.R. Parthasarathy, Maassen kernels and self-similar quantum fields, in, "Hommage a RA. Meyer et J.Neveu," Asterisque 236 (1996), pp. 227-247. K.R. Parthasarathy and V.S. Sunder, Exponentials of indicator functions are total in the boson Fock space T(L2[Q, 1]), in [QP10], pp. 281-284. "Quantum Probability and Applications II," eds. L. Accardi and W.von Waldenfels, Lecture Notes in Mathematics 1136, Springer-Verlag, 1985. "Quantum Probability and Applications III," eds. L. Accardi and W.von Waldenfels, Lecture Notes in Mathematics 1303, Springer-Verlag, 1988. "Quantum Probability and Applications V," eds. L. Accardi and W.von Waldenfels, Lecture Notes in Mathematics 1442, Springer-Verlag, 1990. "Quantum Probability and Related Topics VII," ed. L. Accardi, World Scientific, 1992. "Quantum Probability and Related Topics VIII," ed. L. Accardi, World Scientific, 1993. "Quantum Probability Communications X," eds. R.L. Hudson and J.M. Lindsay, World Scientific, 1998. M. Schiirmann, "White Noise on Bialgebras," Lecture Notes in Mathematics 1544, Springer-Verlag, 1993. I.E.Segal, A noncommutative extension of abstract integration, Ann. of Math. (2) 57 (1953), 401-456; Correction, Ann. of Math. (2) 58 (1953), 595-596. I.E. Segal, Tensor algebras over Hilbert spaces II, Ann. of Math. (2) 63 (1956), 160-175. "Seminaire de Probabilites XX 1984/85," eds. J. Azema and M. Yor, Lecture Notes in Mathematics 1204, Springer-Verlag, 1986. "Seminaire de Probabilites XXI," eds. J.Azema, P.-A. Meyer and M. Yor, Lecture Notes in Mathematics 1247, Springer-Verlag, 1987. "Seminaire de Probabilites XXIV 1988/89," eds. J.Azema, P.-A. Meyer and M. Yor, Lecture Notes in Mathematics 1426, Springer-Verlag, 1990. "Seminaire de Probabilites XXVI," eds. J.Azema, P.-A. Meyer and M. Yor, Lecture Notes in Mathematics 1526, Springer-Verlag, 1992. M. Skeide, Indicator functions of intervals are totalising in the symmetric Pock space r(L 2 (R + )), in, "Trends in Contemporary Infinite Dimensional Analysis and Quantum Probability," Volume in honour of Takeyuki Hida, Istituto Italiano di Cultura (ISEAS), Kyoto 2000.
SCHOOL OP MATHEMATICAL SCIENCES, UNIVERSITY OF NOTTINGHAM, UNIVERSITY PARK, NOTTINGHAM NG7 2RD, UK
E-mail address: jmlSmaths.nott.ac.uk
Quantum Probability Communications, Vol. XII (pp. 23-58) © 2003 World Scientific Publishing Company
QUANTUM PROBABILITY APPLIED TO T H E D A M P E D H A R M O N I C OSCILLATOR HANS MAASSEN ABSTRACT. In this introductory course we sketch the framework of quantum proba bility in order to discuss open quantum systems, in particular the damped harmonic oscillator.
CONTENTS
1. The framework of quantum probability 2. Some quantum mechanics 3. Conditional expectations and operations 4. Second quantisation 5. Unitary dilations of spiraling motion 6. The damped harmonic oscillator References
23 31 41 48 52 53 57
1. T H E FRAMEWORK O F QUANTUM PROBABILITY
Noncommutative probability theory (or 'quantum probability') generalises Kolmogorov's classical probability theory in a way that allows the inclusion of quantum mechanical models. For a discussion of its motivation we refer to [KiiM] in this se ries. Basic sources on quantum probability outside the present series are [Bia], [Dav], [Hoi], [Gud], [Mac], [Mey], [Neu], [Par], [Var], An independent introduction is given here. M a k i n g p r o b a b i l i t y n o n c o m m u t a t i v e . In the last two decades a succesful strat egy has become popular in mathematics: the generalisation of classical mathematical structures by noncommutative algebraic constructions. The most widely known ex ample where this strategy was applied is doubtlessly the noncommutative version of geometry, as explained in the imaginative book of A. Connes ([Con]). There the clas sical structures of a topological space and of a differential manifold are the pillars on which the K-theory of C*-algebras and a variety of cohomological algebras are built. Another application is the field of 'quantum groups', where the classical structure of a Lie group leads into new areas in the theory of Hopf algebras. However, the oldest case by far is von Neumann's and Segal's 'noncommutative integration theory', which has developed into noncommutative measure theory and probability theory. The general strategy consists of the following three steps. 23
24
Hans Maassen (1) Encode the information contained in the classical structure into an appropri ate algebra of functions on it. (2) Characterise the resulting algebra axiomatieally. One of the axioms will be commutativity. (3) Drop the commutativity axiom.
Classical probability. Let us apply this strategy to the structure of a probability space. We remind the reader that a probability space is a triple (fi, E, P), where fi is a set, E is a cr-algebra of subsets of $7, containing Q. itself, and P is a
P ( ( 5 \ T) U (T \ Sj) = 0 .
This simplification is a gain rather than a loss. Finally the probability measure P is regained by putting P : E -> [0,1] : p . -
Quantum probability applied to the damped harmonic oscillator 25 Let 7i be a Hilbert space and let Ai, A2, A3, ■ ■ ■ be a sequence of bounded operators on H. This sequence is said to converge to a bounded operator A in the strong operator topology if for all ip G 7i: lim || Anij - A/> || = 0 . n—>oo
It increases to A if, moreover, Aj < Aj+i in the sense that Aj+i — Aj is a positive operator. A von Neumann algebra A is an algebra of bounded operators on some Hilbert space H which is closed in the strong operator topology. We shall always assume that A contains the identity operator 1, and we only consider separable Hilbert spaces. A state on a von Neumann algebra A is a linear functional ip : A —> C mapping 1 to the number 1, and which is positive in the sense that tp(A*A) > 0 for all A G A. If
= Xtl>.
Without loss of generality we may assume that || X || = 1. We must show that X = MS for some / € L°°. Put / := XI. Then for all g G L°° we have Xg = L2-lim fxg = I2-limMgfx
= M9 {i2-lim
fx ■ l ) = MgX\ = Mgf = fg .
Now let the event Ee for e > 0 be defined by Ee:=
{ w G f i | |/(u/)| 2 > 1 + e } .
Then, since || X || < 1,
P(£ E ) = II 1E. II2 > II XlBc ||2 = || flE. ||2 = f \f\2dP > (1 + e)V(Ee) , and it follows that f"(Ec) = 0. Since this holds for all e > 0, we have | / | < 1 almost everywhere with respect to P. So / G L°°(fl, E,P"). Finally, since the operators X and Mf are both bounded and coincide on the dense subspace L°° of TL, they are equal. □
26 Hans Maassen Step 3 . We now drop the commutativity requirement to arrive at the following definition. Definition. By a noncommutative probability space we mean a pair (A,
containing the same information as X itself. The probability distribution of X is given by
Px:=WoJx:S^W'(X-1(S)). Let us see what our program does with this structure. In Step 1 the embedding Jx is replaced by the mapping j x : L°°(Q', E', P x ) -
L°°(fi, E, P) : / -
/ oX ,
the natural extension of the map 1$ — i > E(S) := l*-i(s) to an (injective) *-homomorphism L ° ° ( 0 ' , E ' , P x ) -» L°°(ft,E,P). Still the projection E(S) stands for the classical event X~1(S) that the random variable X takes a value in 5 e E'. In Step 3 this is now generalised to the following notion. Definition. By a generalised random variable on a noncommutative probability space (.4, tp) we mean a *-homomorphism from some other von Neumann algebra B into A mapping l g to 1^. The probability distribution of j is the state ip := tp o j on B. We denote this state of affairs briefly by
Quantum probability applied to the damped harmonic oscillator 27 If B is commutative, say (B, ip) = L°°(Q.',E',¥'), to take values in fl', and j can be written
then the random variable j is said
j(f) = f f(X)E(dX) , where E denotes the projection-valued measure given by E(S) := j(ls),
(SeZ').
In the particular case that fl' = K, j determines a unique self-adjoint operator on the representation space 7i of A: Theorem 1.2 (Spectral Theorem, von Neumann). There is a one-to-one correspon dence between self-adjoint operators A on a Hilbert space H and projection-valued measures E : E(K) —> B(7i) such that A = f XE(dX) . JR
When E(S) € A for all S in the Borel cr-algebra E(R), then A is said to be affiliated to A- Moreover: Theorem 1.3 (Stone's Theorem). There is a one-to-one correspondence between strongly continuous unitary representations 11—> Ut of the abelian group R into A and self-adjoint operators A affiliated to A such that Ut = eitA . Here the right hand side is to be read as the strongly convergent integral eit.A
._ j eME(dX)
_
JR
where E is given by the spectral theorem. If we put e t : 1 - t C : i H eztx, then the connection with j can be written 3{eL) = eUA . So altogether we can characterise a real-valued random variable or observable in any one of four ways: (1) by a self-adjoint operator A affiliated to A; (2) by a projection-valued measure E in A; (3) by a normal injective *-homomorphism j : L°°(R, E(M),P) —> A; and (4) by a one-parameter unitary group (£/i)*eR in A. Interpretation of quantum probability. It is a surprising fact that nature — at least on small scales — appears to be governed by noncommutative probability. Quantum probability describes manipulations performed on physical systems by certain mappings between generalised probability spaces called operations. (The same has been said about classical probability see, e.g. [Kam].) These mappings will be treated in some detail in Section 3. The generalised random variable which we just saw is such a mapping. It represents the operation of restricting attention to a subsystem. Another such mapping is the conditional expectation, describing the
28 Hans Maassen immersion of a physical system into a larger one. Yet other operations are the time evolution and the transition operator: they represent the act of waiting for some time while the system evolves on its own, or in interaction with something else respectively. At the end of a chain of operations we land in some probability space (A,
Quantum probability applied to the damped harmonic oscillator 29 As E* = E and tr E = 1 we may write E
%y 3 = \2 \x l+ V . iy ^~ \ — z\ - with(x,y,2)eIR v i»i y
Then det i? = 0 implies that i((l-z2)-(x2 + /)) = 0
==>
a'2 + ? / 2 + z 2 = l .
So the one-dimensional projections in M2 are parametrised by the unit sphere S2. Notation. For a = (ai, a2, 0,3) G R 3 let us write ff fl
(
:=
n 4- in
= aiCJl
„
+ a2ff2 + a3(73 '
where o-!, 02 and CT3 are the Pauli matrices "0 ll ai
'-
[1 o\ '
f0 -i] a2
'-[i
\1
oj ■
ff3:=
0 "
[0 -i_ •
3
We note that for all a, b € K we have a(a)a(b) = (a,b)-l+ia(axb)
.
(1.1)
Let us write £ ( « ) : = A(l + <7(ffl)),
(|| a || = 1 ) .
(1.2)
In the same way the possible states on M2 can be calculated. We find that ip{A) = tr{pA) where p = p(a) := ±(l + <x(a)),
|| a || < 1 .
(1.3)
The situation is summarised by the following proposition. Proposition 1.4. The states on M 2 are parameterised by the unit ball in M3, as in (1.3), and the one-dimensional projections in M 2 are parametrised by the unit sphere as in (1.2). The probability of the event E(a) in the state p(b) is given by tr(p(b)E(a))
= \(l + (a,b)).
The events E(a) and E{b) are compatible if and only if a = ±6. Moreover we have for all a E S2: E(a) + E{-a) = 1 , E{a)E(-a) = 0 . Interpretation. The probability distribution of the quantum coin toss or 'qubit' is given by a unit vector b in three-dimensions. For every a on the unit sphere we can say with probability one that of the two events E(a) and E(—a) exactly one will occur, E(a) having probability | ( 1 + (a, b)). We therefore have, for each direction a, a classical coin toss with probability for heads equal to | ( 1 + (a, b)). The coin tosses in different directions are incompatible. Particular case: the quantum fair coin is modelled by (M 2 , | t r ) . The quantum coin toss is realised in nature: the spin direction of a particle with total spin \h behaves in this way.
30 Hans Maassen Positive definite kernels. In this section we introduce a useful tool for the con struction of Hilbert spaces, used heavily in quantum probability. Let cS be a set and let A" be a kernel on S, that is a function S x S —> C. Then K is called positive definite if for all n € N and all n-tuples ( A i , . . . , An) € C" we have n
n
Theorem 1.5 (Kolmogorov's dilation theorem). Let K be a positive definite kernel on a set S. Then up to unitary equivalence there exists a unique Hilbert space H and a unique embedding V : S —> Ti such that VXiyeS : (V(x),V(y))
= K{x,y)
(1.4)
and \/V(S)=H .
(1.5)
A map V : S —> Tt is called a (Kolmogorov) dilation if (1.4) holds. It is called minimal if (1.5) holds. Proof. Consider the space C of all functions S ^ C with finite support. Then C becomes a pre-Hilbert space if we define the (pre-)inner product
AK^A(I)1/(I).
xes This map vanishes on N: for A £ M we have 2
|t/oAf =
^A(x)K(z) 165
= J2Y,Mx)K(x,y)X(y)
= (A,A) £ = 0 .
165 !/€5
So f/o may be considered as a map C/M —> W. By the same calculation we find that UQ is isometric. Since \j V(S) is dense in H and \j VK{S) is dense in Tin, U0 extends to a unitary map U : HK —> H mapping VK{X) to V(x). O Examples 1.6. (a) Let S be any set and let K(x,y) := SXiV. Then Tt = l2(S) and V maps the elements of <S to the standard orthonormal basis of H.
Quantum probability applied to the damped harmonic oscillator 31 (b) Let S := Hi x Tt2, the Cartesian product of two Hilbert spaces H\ and 7i2Let K((ipi,f2),(Xi,X2)) ■= {ipi,Xi) ■ ii>2,X2) ■ Then H = Hi®H2, the tensor product of Hi and H2, and V(tpi,ip2) = i>i®ip2(c) Let <S be a Hilbert space; call it AC for the occasion. Let K(4>,x) '■= (i>,x)2Then H is the symmetric tensor product AT
: = C © AC © i (AC ® s AC) e g (AC ® s AC ® s £ ) ©
and V(^>) is the so-called exponential vector or coherent vector
Exp (VO : = 1 © V1 © (V> ® VO ® W ® ^ ® V') © ' ' • (e) Let S = E and let if : R x R -> C be given by K{s, t) := e-v\s-t\+Ms-t)
(V>0,UJER).
t
The Kolmogorov dilation of this kernel can be cast in the form H = L2{R, 2r]dx) ;
V : t >-► vt G L 2 (R) :
vt{x) :=
efo-iw)(x-0
0
if a; <
t;
ifa;>t
2. S O M E QUANTUM MECHANICS
Quantum mechanics is a physical theory that fits in the framework of noncommutative probability, but which has much more structure. It deals with particles and fields, using observables like position, momentum, angular momentum, energy, charge, spin, isospin, etc. All these observables develop in time according to a certain dynamical rule, namely the Schrodinger equation. In this section we shall pick out a few elements of this theory that are of partic ular interest to our main example: the damped harmonic oscillator considered as a quantum Markov chain. P o s i t i o n a n d m o m e n t u m . Let us start with a simple example: a particle on a line. This particle must have a position observable, a projection valued measure on the Borel tr-algebra £(R) of the real line R: E : £(R) -► B(H) . The easiest choice (valid when the particle is alone in the world and has no further degrees of freedom) is H := L 2 (R) ; E(S) : $ -» Is • i> ■ In this example the Hilbert space H naturally carries a second real-valued random variable in the form of the group (Tt)teR of spatial translations: (Ttil>){x) := 1>(x - ht) ,
(2.1)
32 Hans Maassen according to the remark following Stone's theorem (Theorem 1.3). This second ob servable is called the momentum of the particle. The constant h~ is determined by the units of length and of momentum which we choose to apply. The associated self-adjoint operators are Q and P given by (Qip)(x) = xip(x) ; (Pil>)(x) = -ih-^Mx)
.
(2.2)
Just as we have Tt = e~ltP, it is natural to introduce Ss := els® whose action on His Ssi>{x) := eisxij(x) . (2.3) The operators P and Q satisfy Heisenberg's canonical commutation relation (CCR) [P, Q] = -ih-l.
(2.4)
A pair of self-adjoint operators (P,Q) satisfying (2.4) is called a canonical pair. Representations of the canonical commutation relations. What kinds of canonical pairs are there? Before this question can be answered, it has to be reformulated. Relation (2.4) is not satisfactory as a definition of a canonical pair since the domains on the left and on the right are not the same. Worse than that, quite pathological examples can be constructed, even if (2.4) is postulated to hold on a dense stable domain, with the property that P and Q only admit unique self-adjoint extensions ([ReS]). In order to circumvent such domain complications, Weyl proposed to replace (2.4) by a relation between the associated unitary groups (Tt) and (Ss), namely: TtSs = e~ihstSsTt ,
(s,teR).
(2.5)
It was von Neumann's idea to combine the two into a two-parameter family W(t)S):=e%siTtSs,
(2.6)
2
forming a 'twisted' representation of K , as expressed by the Weyl relation: for all s,t,u,v £ M, W{t, s)W(u, v) = e " f ('"-»")w(t + u,s + v) . (2.7) This relation captures the group property of Tt and Ss together with the relation (2.5). Formally,
W(t,s)=eiW-tP)
.
We shall call the representation on L 2 (R) of the CCR given by (2.1), (2.2), (2.3) and (2.6) the standard representation of the CCR. Here and in the rest of the text we shall follow the quantum probabilist's convention ([Mey]), namely that k=2. Theorem 2.1 (von Neumann's Uniqueness Theorem). Let (Wit, s)) t R be a strongly continuous family of unitary operators on some Hilbert space H satisfying the Weyl relation (2.7). Then H is unitarily equivalent with L 2 (R) ®K, such that W(t,s) corresponds to Ws(t,s) ® 1, where W$ is the standard representation of the CCR.
Quantum probability applied to the damped harmonic oscillator 33 Proof. Let W : R2 -> U{U) satisfy the Weyl relation (2.7). For each integrable function / : R2 —» C with J f \f(t, s)\dt ds < oo, define a bounded operator A(f) on H by the strong sense integral /*O0
OO
/
/
f(t,s)W(t,s)dt ds .
oo J —oo
We find the following calculating rules for such operators A(f) and their kernels f: A(f) + A(g)
=
A(/+S);
^(/)*
=
Mf),
i4(/)A(5)
=
^(/*ff).
_
where/(t,s):=/(-i,-s);
Here the 'twisted convolution product' * is defined by /-DO
CO
e- i ( t "- s u ) /(t-w>s-t;)ff(u,t;)dud?; .
/
/
oo J — oo
Moreover we claim that an operator (on a nontrivial Hilbert space) can have at most one kernel: A(f) = 0
=»
W = {0}or/ = 0.
(2,8)
Indeed, if A{f) = 0 then we have for all a, b G R, OO
0 = W(a,b)*A{f)W(a,b)=
/
/"OO
/
e 2 i ( a s - M ) /(i,s)W(4,s)dids
Applying the linear functional A — i > (ip,Aip) with tp,ip & H to both sides of this equation, we find that for all ip,ip £H the (integrable) function (t,S)^/(i,S)(^W(t,S^> has Fourier transform 0. By the separability of H, either W(t, s) = 0 for some (t, s), (that is H = {0}), or f(t, s) = 0 for almost all (i, s). The key to the proof of uniqueness is the operator /-OO
OO
E:=l
/
/
e-^t2+s')W{t,s)dtds
■oo J ~oo
It has the remarkable property that for all a,b G R, £W(t,s)-E is a scalar multiple of E: EW(a,b)E
= e~i(ai+b2)E.
Indeed, £ has kernel g(t,s) := Je"5*-* + s ), and the product W(a,b)E h{u,v) := i e - * < — H ■ e-h{(«-u?Hb-v?)
(2.9) has kernel
34 Hans Maassen So EW(a, b)E has kernel (g*h){t,s)
I'e-i(tv-su)g{t-u,s-v)h{u,v)dudv
=
I
=
— /" /' e -i("'-™) e -K( t - u ) 2 + ( s -'') 2 )e- i ( a ''- H e-K( a ~ , ') 2 +( i , - l ') 2 )du(i'y
= J_ e -K a2+(,2 ) e -5( t2 + s2 ) / f =
-Ie-5(o2+<>2)e-§(t2+s2) /" f
=
I e - I ( a 2 + " 2 ) . e-H<2+*2)
iu2+v2) (u iv){t+is+a+ib
e-
e -
Uudv
e-^-^+i^+^Y-iv-Ht+is+a+ib))2dudv
= r-K^ 2 ), (M) , which proves (2.9). We conclude that E* = E (since g = g), E2 = E (putting a = b = 0 in (2.9)), and that EAE = CE, where A is the von Neumann algebra generated by the Weyl operators. So £ is a minimal projection in A- Denote its range by KL. Then we have for all
= = =
(W{t,s)E
Therefore the map V : R 2 x K -> H : ((*, s), tp) >-» e^t2+s2)W(t,
s)y
is a Kolmogorov dilation (cf. Section 1) of the positive definite kernel K : (R2 x K) x (R 2 x K) -» C,
({t, s),
.
(2.10)
By explicit calculation you will find that E$ is the orthogonal projection onto the one-dimensional subspace spanned by the unit vector Q(x) := ^ 7 ( 1 ) , where 7(2;) := —7=.e 2
.
So in the standard case the dilation is Vs : R 2 -► L 2 (R) : (t,s) >-► e i ( ' 2 + s V 5 V C T fi(s; - 2i) . By Kolmogorov's Dilation Theorem, there exists a unitary equivalence U : L2(R) ® K. —> H such that for all a, b e R and ip £ !C: U(Ws(a, b)Q ®i>)= W(a, b)ip , and therefore for all a, b 6 R: W(a,fo) = f / ( H / S ( a , 6 ) ® l ) t / - 1 ,
Quantum probability applied to the damped harmonic oscillator 35 provided that the range of V is dense in H. Let C denote the orthogonal complement of this range. Then C is invariant for the Weyl operators; let W0(t, s) be the restriction of W(t, s) to £. Construct E0 := A0(g) in terms of W0 in the same way as E was constructed from W. Then clearly EQ < E, but also E0 J_ E. So E0 = A0(g) = 0 and by (2.8) we have C = {0}. □ Exercise. Calculate the minimal projection Es in the standard representation. Energy and time evolution. The evolution in time of a closed quantum system is given by a pointwise strongly continuous one-parameter group (at)tew. of ^auto morphisms of the observable algebra A. Like in the case of a particle on a line, for a finite number n of (distinguishable) particles in d-dimensional space we take A = B(H) with 7i = L2(Rnd). Since all automorphisms of this algebra are implemented by unitary transformations of Ti, the group (at) is of the form at{A) = UtAUf1 . It is possible to choose the unitaries so that t H-+ Ut is a strongly continuous unitary representation R —> U(H). We denote its Stone generator by H/h: Ut = eltHlh . The self-adjoint operator H corresponds to an observable of the system of parti cles, called its energy. The operator H itself is known as the Hamilton operator or Hamiltonian of the system. As the Hamiltonian commutes with the time evolution operators, energy is a conserved quantity: at(H) = UtHUf1 = H . The nature of a physical system is characterised by its dynamical law (a term of Hermann Weyl, see [Wey]). This is an equation which expresses the Hamiltonian in terms of other observables. For n interacting particles in Kd in the absence of magnetic fields the dynamical law takes the form nd
# = V-
..
Pf +
V(Q1,Q2,...,Qnd)
for some function V : (M.d)n —> R, called the potential. The positive constants m*, k = 1, • ■ • , n are the masses of the particles. (Incidentally we put k(j) := 1 + \(j — l)/d], where [ ] denotes integer part, in order to attach the same mass to the coordinates of the same particle.) Free particles. If V = 0, then Ut factorises into a tensor product of nd one-dimensional evolution operators, all of the form Ut = itH'h = e i 5 ^ p 2 . Since the Hamiltonian H = P2/2m now commutes with P, momentum is conserved: at(P) = P .
36 Hans Maassen On a formal level the time development of the operator Q is found by solving the differential equation
i«(Q) = &M1
= ^FMQ)),
(2.ii)
a solution of which is at{Q)=Q
+ —P. m According to the Uniqueness Theorem the canonical pairs (P, Q) and (P, Q + —P) are indeed unitarily equivalent. So we expect the evolution of the Weyl operators to be the following: at(W(x,y))
= at (e-ixP+i^)
= e-^+W+^J
= e-i(x-£v)P+iyQ
= W(x_±y
\
y
m
Proposition 2.2. Let P := —ih-j^. denote the momentum operator on H := L2 and let W : R 2 -► U{H) be given by (2.7). Let Ut := e'5^Rp2 . Then UtW(x,y)Ut-1
=
wfx--^y,y
Proof. From the definitions of Tt and EQ it follows that for all measurable sets B c K and all t e R: TtEQ{B)Trl = EQ{B + ht). By the uniqueness theorem irreducible representations of the CCR have the symmetry Q —» P, P —> —Q. So we also have the exchanged imprimitivity relation SVEP{B)S-1
VSe2(R)Vy6R :
= EP(B + hy) .
Hence for all y, t G E, r
l x2 e- ^ Ep{d\)\
S~l
— CO
/
CO
/
e'^^-W
EP{d\)
-co
= EL, ■ T_XvV ■ .
Its
e-^v
Multiplying by Ut on the left and by Sy on the right we find UtW{Q)V)U-1
= USyUr1
= e-i^T.^Sy
= W ( - l y , y) .
As Tx commutes with Ut we may freely add (2;, 0) to the argument of W, and the proposition is proved. D
Quantum probability applied to the damped harmonic oscillator 37 By imposing some state ip on A = B(L2(R)), all stochastic information on the model (.4, f,at) can be obtained from the evolution equation at(Q) = Q + ^P- For example, at large times t the random variable \at{Q) approaches ^P in distribution, provided that tp does not favour large Q values too much. So a position measurement at a late time can serve as a measurement of momentum at time 0. This puts into perspective the well-known uncertainty principle for position and momentum at equal times. The Schrodinger picture and the Schrodinger equation. The type of description of a system given so far, namely with random variables moving in time, and the state ip given once and for all, is called the Heisenberg picture of quantum mechanics. In probability theory this is common usage, and we shall adopt it also in quantum probability. However, quantum mechanics is often thought of in a different way, where one lets the state move, and keeps the operators fixed. This is close to Schrodinger's 'wave mechanics', and is therefore called the Schrodinger picture: If we take for
.
This wave function satisfies the Schrodinger equation, a partial differential equation reflecting the dynamical law: ._ d ,, -ih-Q-ip{x1,...,xnd;t) nd
= ^
. „2
-7)
« ^~2 VKzi> • • ■ > x„d; t) + V(xu ..., xnd)ip(xu
..., rnd; t) .
If E is an orthogonal projection in H, then the probability of the associated event can be calculated in the Schrodinger picture by
=
(U-^EU^iP)
xnd) (Eipt) (xu ..., xnd)dxu
..., dxnd .
Rnd
The harmonic oscillator. A harmonic oscillator is a canonical pair (Q,P) of observables that under time evolution (a«)teR performs a rotation such as &t(Q) = Qcost + Ps'mt ; at(P) = - Q s i n i + Pcost . Since rotation in the plane is symplectic (preserves the area two-form), this evolution respects the canonical commutation relation QP — PQ = ih-1. So by the Uniqueness Theorem it determines (up to a time-dependent phase) a group of unitary transfor mations (Ut)tsR of the Hilbert space on which it is represented. (For example, C/» is a
38 Hans Maassen unitary transformation of L2(M) that sends Q into P and P into —Q in the standard representation (2.2). This is the Fourier transform.) Making a formal calculation as in (2.11) by differentiating the equality ^HlhAe-itH'h
at(A) = we find that a Hamiltonian of the form H=\(P2
+ Q2)
(2.12)
can be expected to generate such a rotating evolution. The textbook treatment of the harmonic oscillator (e.g. [Han]), follows the elegant algebraic reasoning of Dirac, who rewrote the Hamiltonian (2.12) as H = \{Q-
iP)(Q + iP) + ii[P, Q] =: ha*a + \h • 1 .
The operators a and a* are then seen to lower and raise the eigenvalue of H, and are called the annihilation and creation operators. Here we choose to proceed more analytically, seizing the opportunity to introduce techniques which will be useful again later on for the treatment of free quantum fields and the damped oscillator. Our goal is to describe H and Ut explicitly. Heisenberg's matrix representation. First we note that, since at has period 27r, the differences between spectral points of H must be multiples of h. On the grounds of (2.12) we suspect that H is bounded from below, so let us try sp(tf) = hN + c. We take as our Hilbert space H.H := I2 (N, ^ ) with the Hamiltonian given by (Htf){n) = (hn + c)ti(n) . The subscript 'H' indicates that on this space we wish to stage matrix mechanics of the Heisenberg type. If we define on Tin the 'product' or 'coherent' vectors n(z):=(l,z,z2,z3,---),
{z G C) ,
then our intended time evolution takes the form UtHTT(Z) = eitc^Tr (eV) .
(2.13)
Now we want to represent a canonical pair (P, Q) in this space, or equivalently, Weyl operators W(z), that rotate in the same way: UtW(z)Uf1 = W(eltz). We note that oo — n
(7r(W),7r(V)) = ^
^
n
= e5\
71=0
so that we have here another dilation of the positive definite kernel (2.10) used in the proof of the Uniqueness Theorem. An irredicible representation of the CCR is close at hand. Put: W„(Z)TT(U)
= e~*u-^\(u
+ z) ,
(z,ueC).
These operators satisfy the Weyl relation WH(w)WH(z)
= e-ilm{Wz)WH{w
+ z) ,
(2.14)
Quantum probability applied to the damped harmonic oscillator 39 the same as (2.7) if we identify W(t, s) with Wjj(t+is). UtW{z)U-1
Clearly we have also obtained
= W (e{tz) .
(2.15)
Let us summarise, again replacing h by 2. Proposition 2.3. The Heisenberg representation of the Harmonic oscillator is given by
l2(N,-);
nH =
n\ {HHd){n) = (2n + l)tf(n);
^ ( z ) = e^7r(euz) ;
W^WTT(M) = e-™-s|z|27r(M + z) .
In concrete terms, on the standard orthonormal basis,
<5=
"0 1 0 0 0 .. .1 1 0 V2 0 0 ... 0 \ / 2 0 V3 0 . . . 0 0 \/3 0 v/4 . . . ' 0 0 0 \/4 0 . . .
P =
[0 1 0 0 -1 0 v/2 0 l 0 —y/2 0 \/3 7 0 0 —\/3 0 0 0 0 —A
0 0 0 VI 0
..." ... ... ... ■ ...
These matrices satisfy = 2i-l and | ( Q 2 + P2) = H ,
QP-PQ where
"1 3
0 5
H=
7
0
9
Proof. It only remains to check the matrices for Q and P. We note that eiyQ7r(u) = VMi2/)7r(u) = e i " u _ 5»V( u + iy) , and we find by differentiation Qir(u) = wr{u) + TT'(U) .
Taking the coefficient of un the matrix of Q is found. The matrix for P is found in the same way. The choice of the ground state energy c = \h — 1 in the definition of H fixes the relation with Q and P correctly. D
40 Hans Maassen The Gaussian representation. Here is another useful representation of the harmonic oscillator algebra on a Hilbert space. Let Tic := L 2 (R,7), where 7 is the standard Gauss measure on M: 1
_ 2i 2
y(dx) := 'y(x)dx := —j=e
* dx .
V27T
Define for z £ C the vector e{z) by 1,2
e{z) : x 1—> ZX-^Z' e
Then e(z) with z € C is a total set in WG. (Actually, z € iR is already sufficient by the uniqueness of the Fourier transform.) Again we find (e(z),e(u))
= e™ ,
(z.tieC).
Proposition 2.4. There exists a unitary map UHG '■ T~(-H —> ^ G such that for all
zeC UHGn(z)
=e{z) .
This map sends the vector en := (0, ■ ■ • , 0,1, 0, • • ■) into the n-th Hermite polynomial, where these polynomials are given by the generating function
Yjznhn{x) = &zx-\z* Consequently, this version of the Hermite polynomials satisfies
f
.
hn(x)hm(x)j(dx)
= -r<5nm n.
J —c
Proof. The map n(z) \—» e{z) extends to a unitary map since the linear spans of the ranges of 7r and e are dense and both n and e are minimal dilations of the positive definite kernel (z, u) 1—» ezu. □ Let us carry over the relevant operators with this unitary transformation. We find:
(eis^i>) (x) = eisx^(x\ (e-itp^)(x)=^(x-2t)(^-~^j
(Qa^)(x) = x^{x) ; ,
HGi> = (2NG + 1)V =
~2Q^*P
(PG^)(x) = ixr/,(x)-2rP'(x); +2
x
^ + i> ■
The Schrodinger representation. Finally we get to the standard Schrodinger repre sentation (2.1), (2.3) and (2.6) of the harmonic oscillator by dividing away a factor ^y{x). Let Hs := L2(R) and define
UGS :nG^ns:
(UGSi>)(x) :=
^Mx)^(x)
Quantum probability applied to the damped harmonic oscillator 41 The problem of damping. A damped harmonic oscillator is an evolution (Tt)t>o on the real-linear span of a canonical pair (P, Q) that has the form Tt(Q) = e~vt (Qcosojt + Psmut) Tt{P) = e "
,J
,
(-Q sin ojt + P cos tot) ,
fa
> 0) .
(2.16)
(We apologise for a clash of notation: Tt is not related to translations.) This spiralling motion in the plane compresses areas by a factor e~2r,t, so that for t > 0 the operators Tt(Q) and Tt(P) disobey the canonical commutation relation, and Tt cannot be extended to an automorphism of B(H). Yet this damped oscillatory behaviour occurs in nature, for instance when an atom is loosing its energy to its surroundings by emission of light. So it would be worth while to make sense of it. There are two basic questions related to this model. Question 1. How should Tt be extended to B(H)7 Question 2. Can (Tt)t>o be explained as part of a larger whole that evolves by *-automorphisms of the form at(A) = UtAU^1, where Uf1 satisfies a Schrodinger equation? Spirals and jumps. In Heisenberg's matrix mechanics atoms were supposed to move in a mixture of two ways. Most of the time they were thought to rotate according to the evolution Uf* as described above, but occasionally they made random jumps down the ladder of eigenvalues of the energy operator H. Each time an atom made such a jump, it emitted a quantum of light whose (angular) frequency u> was related to the size E of the jump by E = tku . The probability per unit of time for the atom to jump was given by Fermi's 'Golden Rule', formulated in terms of the coupling between the atom and its surroundings, and it is proportional to the damping rate rj. In the following sections we shall describe this behaviour as a quantum Markov process. Both jumps and spirals will be visible in the extension of our Tt to the atom's full observable algebra. This will be our answer to Question 1, for which we shall need the notion of completely positive operators. Our answer to Question 2 will be a reconstruction of the atom's surroundings: a dilation. There we shall see how the atom can absorb and emit quanta. 3. CONDITIONAL EXPECTATIONS AND OPERATIONS
We shall now give a sketch of the operational approach to quantum probability which was pioneered by Davies, Lewis and Evans ([Dav], [EvL]). Conditional expectations in finite dimension. In this section we choose for definiteness: A ■= Mn, the algebra, of all complex n x n matrices, and f : A -> C :
A H-> tr (pA) ,
where p is a symmetric n x n matrix with strictly positive eigenvalues and trace 1, so that cp is faithful.
42 Hans Maassen Let A be a symmetric n x n matrix with the spectral decomposition
A=
]T
aEa.
aesp(A)
The orthogonal projections Ea, a € sp(yl), form a partition of unity. Measuring the observable A means asking all the compatible questions Ea at the same time. Precisely one of the answers will be 'yes'> as stipulated in the interpretation rules. If the answer to Ea is 'yes', then A is said to take the value a. This happens with probability
Yl av{Ea) = ¥>\ Y2 assp(^l)
aEa
yaesp(^l)
) = fW ■ J
So the state
/3gsp(B)
If we first measure B and then A in each trial, in the limit of increasingly many trials we obtain a probability measure P on sp(A) x sp(B). By the discussion of interpretation of quantum probability in Section 1 the probabilities are given by P({(a,P)})
=
It is then natural to define the conditional probability P[A = a\B = 0] as that proportion of the trials that have yielded B — (3 which turn out to give A = a later: 1
P]
'
-
EQesp(^(l(a./3)})
f{FP)
'
The associated conditional expectation is naturally defined as E(A\[B = /?]):=
£
a p[>i =
a|B
= /31 =
a&p(A)
2(^)i ^
/3j
Note that this is a function, / say, of 0. Seen as a quantum random variable this conditional expectation is described by the matrix f(B): E(A\B):=f(B)=
£
f(m=
£
^ S )
Note that
F /
, .
(3.1)
Quantum probability applied to the damped harmonic oscillator 43 Remark. In general we do not have
(3.2)
The left hand side is the expectation of A after measuring B. The right hand side is the expectation of A without any previous operation. The fact that these two expectation values can differ is typical for quantum probability. Let us give a simple counterexample to (3.2) here: Let A := M2, choose A 6 (0,1), and put 1 1 B = \ P ~ [0 1-X\' |o 0 1 1 It is readily checked that tp(A) = A,
so that the equality (3.2) holds if and only if
£(B) =
{Y/F0\Vcsp(B)},
Pev and that B is the linear span of the projections Fp. The following is a finite dimensional version of Takesaki's theorem ([Tak]) on the existence of conditional expectations onto von Neumann subalgebras. T h e o r e m 3 . 1 . Let B = B* g Mn and let B be the *-algebra generated by 1 and B. Let ifi : Mn —> C : A — i > tr (pA) with p strictly positive and tr (p) = 1. Then the following are equivalent. (a) There exists a linear map P : Mn —> B such that V^ e W „V F 6 £ ( e ) :
cp(FAF) = y{FP(A)F)
.
(3.3)
(b) There exists a linear map P from Mn onto B such that (i) P maps positive definite matrices to positive definite matrices.
(ii) P(l) = 1; (iii) ip o P = ip; (iv) P2 = P. (c) Bp = pB. If these equivalent conditions hold, then the linear maps P mentioned in (a) and (b) are the same. It is called the conditional expectation onto B compatible with ip.
44 Hans Maassen Proof, (a) = > (b): suppose P : Mn —> B is such that (3.3) holds. Let A > 0 and decompose P(^4) as J2pesp(B) apF& w i t n ^a G £(^)- Then apip(Fp) = ip(Ff)P{A)) = ip{F0P{A)Fp) = tp(FpAF0) > 0. So ap > 0 for all /? and P{A) > 0. Putting ,4 = 1 in (3.3) we find that for all 0 6 sp(P):
11^)11* < Mil* •
(3-4)
Given A € Mn, define numbers » ^ e C and b@ > 0 by /3Ssp(B)
/3Ssp(B)
Then from the positivity property (b)(i) it follows that VA6C:
P((\-1-A)*(\-1-A))>0.
This implies that for all (5 6 sp(B) and all A € C, \\\2-(\a0
+
Xad+bp>Q,
from which it follows that M
2
< b/3 ,
that is
P(.4)*P(yl) < P(A*A) .
Applying tp to the last inequality and using (iii) yields the statement (3.4). So P is an orthogonal projection Mn —> B, that is for all A € Mn, A - P{A) ± „ B ■ This means that for all A e M„:
Quantum probability applied to the damped harmonic oscillator 45 Therefore, defining P{A) by the r.h.s. of (3.1), and putting F = Y2aev ^0 V C sp(B):
^
^^
= Y,
=
w
^tn
^(FeAFp)
=
pev
□ Operations in finite dimension. Let A and B be finite dimensional von Neumann algebras, and let A* and B* denote their duals. A linear map T : A —> B defines by duality a linear map T* : B* —+ A* ■ The map T* maps states into states if and only if T is positive, that is maps positive elements of A to positive elements of B, and is identity preserving. The map T is said to be n-positive if T ® id maps positive elements of A ® Mn to positive elements of B ® Mn: (4;)",=i > 0
(T(^))I\=1 > o
1
7 is called completely positive if it is n-positive for all n € N. In that case T*®id maps states on B <8> M n to states on .4 ® Mn. T is called identity preserving if T(l^) = lg. Definition. An operation T : A —> B is a completely positive identity preserving map. Adjoints of operations will also be called operations. The idea is that any physical procedure which takes as an input a state on some quantum system described by B, and which turns out a state on a quantum system described by A must necessarily be of the above kind. Not all operations in the sense of the definition can actually be performed, but certainly nothing else is physically possible. Indeed any physical operation on a quantum system A should also define a physical operation on A ® 72., where 72. stands for some quantum system not affected by the operation. The existence of such an 'innocent bystander' outside our quan tum system A should never lead to the prediction by quantum theory of negative probabilities. The following example shows that complete positivity is strictly stronger than positivity. Let a b a c T:M2^M2 c d b d Then T(A*A) = T{A)T(A)* > 0 for all A, but "l 0 T
0 0 0 0
0 0 0 0
l" 0 to 0 1
1 0 0 0
0 0 0 0 10 10 0 0 0 1
i.e. it maps a one-dimensional projection to a matrix with eigenvalues 1 and — 1.
46 Hans Maassen Operations on quantum probability spaces. A quantum probability space {A,
(A,B) ^ tp(A*B) .
States which are given by density matrices on this space are called normal states on .4, and the set of all normal states is denoted by A*. When we write T : (A,
T'IP
ensures that T" maps normal states to normal states. This property is only relevant for infinite dimensional von Neumann algebras. When speaking of operations between quantum probability spaces we shall always imply that the state is preserved. Quantum stochastic processes. Let us now consider the category QP whose ob jects are quantum probability spaces and whose morphisms are operations. Lemma 3.2 (Schwartz's inequality for completely positive operators). LetT : (A,
.
Proof. Let A be represented on H. By the positivity of T ® id M 2 we have for all AeA, (iP(BT(A)ij,(T®id)(^
-
1
Q
~Q^®T(A)i,)>0.
Writing this out we obtain (ip, (T(A*A) - T{AyT(A))ip)
> 0.
□ Corollary 3.3. IfT : (A,y) -> (B,ip) then for all A e A ip(T{A)*T(A))
< tp(A*A) .
This inequality states that T is a contraction between the GNS Hilbert spaces of (A,(p) and {B,ip). Lemma 3.4. T : (A, ip) —> (B, ip) is an isomorphism in the category QP if and only ifT:A—>Bisa *-isomorphism. Proof. (Exercise:) Apply Schwartz's inequality to T and to T _ 1 .
D
A random variable (cf. Events and random variables, in Section 1) is an injective *-homomorphism j : {A, ^) -> {A,
Quantum probability applied to the damped harmonic oscillator 47 A quantum stochastic process ([AFL]) is a family (j't)tgT of random variables indexed by time T. Here, T is a linearly ordered set such as Z, R, N or R+. If T = R or R + we require that for all A € A the curve t <—> it{A) is strongly continuous. If T is a group, say Z or R, then the process is called stationary provided that jt = Tt o jo for some representation t H-> Tt of T into the automorphisms of (A,
■
■■jt2(E2)jtl(E1))
is the probability that Ei occurs at time i 1 ; E2 at time t2, ..., and En at time tn. Note the double role played here by the time ordering: Unless some of the questions jtk{Ek) recur, that is they lie in jt(A) for different values of t, they must be asked in the order dictated by the times t^. Stochastic process interpretation. In a classical stochastic process (Xt)tej the ran dom variable Xt is a different one for different times £, so the events concerning Xt change in time accordingly. If the process is stationary, Xt and Xs differ by an auto morphism of the underlying probability space. These observations generalise to the noncommutative situation. Conditional expectations and transition operators. If we are to describe an open quantum system such as the damped harmonic oscillator by an internal dynam ics, say Tt : A —> A, without reference to its surroundings, we need to be able to keep track of an observable A which starts in A at time zero, during its motion away from the algebra A at positive times. That is, we need its conditional expectation. In view of the discussion of conditional expectations in finite dimensions, we give the following general definition. Definition. Let j : (A, ) be a random variable. The conditional expecta tion (if it exists) is the unique morphism P : {A,
= P(j(B,)Aj(S 2 )) •
In particular VFes(A)VAeA ■■ tp{j{F)Aj{F))
=
The second line indicates the connection with Theorem 3.1. Markov processes. Let us now apply the above notion to an open quantum system.
48 Hans Maassen Two-time-probabilities. Suppose that for all s 6 T there exists a conditional expec tation Ps with respect to j s . Then the probability for F to occur at time s and E at time t > s can be written as 0(js(F)Jt(E)js(F))
=
= v(FTs,t(E)F)
,
where Tst = Ps o j t is an operation on (A,
\s
.
and moreover the Markov property holds: t<s
=*.
P^qtisiA))
C jt{A) .
(3.5)
Proposition 3.6. Let (jt : (.4,
=j.
r
I\tTt,u = %,U .
In particular, if the process is stationary, then Tt := T 0|t = TSiS+( satisfies TsTt = T,+l
(M>0).
In the latter case, {Tt)t>o is known as the dynamical semigroup induced by the stationary Markov process. Conversely, the process (jt)tet is called a Markov dilation ([Kum]) of the dynamical semigroup (Ti) teT . The situation is symbolised by the commutative diagram (3.6)
(A,
P
L
(A,?)— *{A,
A quantum model of n harmonic oscillators is obtained by taking the n-fold tensor product of the representation z i-» W(z) := exp(i(Imz)Q—i(Rez)P) of the canonical commutation relation (CCR) over C. This turns out to be equivalent to a single representation of the CCR over C n . An infinity of harmonic oscillators is obtained by replacing C with an infinite dimensional separable Hilbert space AC. It depends on the spectrum of the time evolution on K (discrete or continuous), whether a countable infinity of oscillators is obtained or a continuum, that is a quantum field. In our dilation of the damped harmonic oscillator we shall need a quantum field. As in the case of a single oscillator we have the choice between different concrete representations: we may emphasise the field aspect of the construction, like in the Gaussian representation of the harmonic oscillator, or the particle aspect of it, like in its matrix representation. (The Schrodinger representation on L 2 (R) as in (2.2),
Quantum probability applied to the damped harmonic oscillator 49 (2.6) has no analogue in infinite dimension, since there exists no Lebesgue measure on K°°.) The following definition generalises the Weyl relation (2.14) over C to that over a general complex Hilbert space K.. If K, is the L2-space of some measure space (X, fj,), then K may be considered as the 'quantisation' of X, and the construction below as its 'second quantisation'. We refer to Mark Fannes' lectures in these volumes. T h e functor F. Definition. Let K, be a complex Hilbert space. A representation of the Canonical Commutation Relations {CCR) over K, is a map W from /C to the unitary operators on some Hilbert space H such that for all / , p € /C: W(f)W(g) = e-am^W(f
+ g) ,
(4.1)
and t — I ► W(tf)ip is continuous for all / E )C,ip e H. The map is called a vacuum representation if there is a unit vector tt € H such that
A vacuum representation is called cyclic if the linear span of the vectors W(f)Q is dense in Tt. A cyclic vacuum representation of the CCR over K. can be constructed by a gener alisation of the method used several times in the harmonic oscillator in Section 2: Let 7r be a minimal Kolmogorov decomposition of the positive definite kernel K, x K, —> C mapping (/, g) to e^'9\ and on the total set of 'coherent vectors' n(g), define
W(fMg)
:=e-<'rf>-$»/UV(/ + ff) ,
(f,ge/C).
Then put fi := 7r(0), and all the requirements in the above definition are met. On the other hand, given any cyclic vacuum representation of the CCR over K. with vacuum vector ft, a Kolmogorov decomposition ir' of the above mentioned kernel is obtained by putting ir':f^el^fU2W(f)n. Indeed, W ) , TT'G?)} = e3(» / H2+lloII2' • eam <'•»>(Q,W(-f + g)n) = e5(ll/l|
2
+llsl|2) . e *n e 4U'-9ll 2
= e<'*> . Thus all cyclic vacuum representations of the CCR over a Hilbert space K, are unitarily equivalent. However, this can not be concluded from von Neumann's uniqueness theorem, since the latter breaks down for infinite dimesional K,. In this case there are indeed many inequivalent (non-vacuum) representations, for instance those associated to positive temperatures ([BrR]). Since el'h9h^@92) = e{!u9l) ■ e{h'92\ a representation of the CCR over a direct sum K,\ © K-2 of Hilbert spaces is isomorphic to the tensor product of the representations of the CCR over Kx and /C2-
50 Hans Maassen Definition. Let T0(K.) denote the linear span of the operators W(f), f € /C in some representation of the CCR over /C. Let T(/C) be its strong closure. On the von Neumann algebra T(/C) we assume by default the vacuum state
Thus F(X) = (r(/C), IC2, let r 0 ( C ) : To(/Ci) —> ^ ( A ^ ) be given by := e K l l c / l l 2 - i m i V ( C / ) .
T0(C)(W(f))
(4.2)
Proposition 4 . 1 . The operator Y0(C) ftos o unique strongly continuous extension to an operation T(C) ofT{K). Proof. Cf. for instance [Tee].
□
Remark. Second quantisation is a functor Y from the category of Hilbert spaces with contractions to the category of quantum probability spaces with operations. Fields. From the Weyl relation (4.1) it follows that A 1—> W(Xf) is a strongly contin uous unitary representation of R. By the spectral theorem there exists a self-adjoint operator $ ( / ) on T(fC) such that W(Xf) = eiA*(/> . The Weyl relations then imply that [$(/), $( s )] = 2ilm(/,g) - 1 , and in the vacuum state ip& the random variable $ ( / ) has normal distribution with mean 0 and variance || / ||2. The random variables $ ( / ) and $(
M
Bt
)
if * > 0,
-\-mm)
if*
(43)
a stochastic process (Bt)teR i s defined with compatible normally distributed indepen dent increments having variance
n=0
1
Quantum probability applied to the damped harmonic oscillator 51 with 7r(/) given by the exponential vectors. Given a contraction C : K, —* K. let J-~(C) be the contraction T{K,) —» ^{K) mapping ir(f) to ir(Cf) for every / 6 AT. This map can be written oo n=0
Given an orthogonal projection P on K. let a self-adjoint operator dJ:(P) on ^(/C) be defined by ei\dF(P)
■- jr
lei\P\
r
This operator dJ (P) is interpreted as the random variable that counts for how many particles the 'question' P is anwered 'y es '- I n particular the total number of particles N equals d F ( l ) . If K, = L 2 (R, 2r]dx), the Fock space .F(/C) can be written as L 2 (A(R), //,,) where A(K) is Guichardet's space over R ([Gui], [Maa] — see also the lectures by Martin Lindsay in these volumes). A(R) := { < x C K | # ( a ) < oo }; and fin is the measure on A given by *,({0» = i . Unido) = (2rj)n dsids2 ■ ■ ■ dsn if a = {sx, s2, ■ ■ ■ , sn}. The coherent vectors are represented as the functions 7r(/) : A(R) —> C given by
S£CT
Indeed,
JA(R) oo
= E(
= £
27
p
?)
n
/
(79)(si)---(f9)(sn)ds1---dsn
^T" /
=Ei? n
2
n=0 - V
^
5 l
) ■ •' (79)(sn)dSl
■ ■ ■ dsn
W / ( a M * H =e-/-ex,
/
In this concrete representation the number operator dJr(P), where P is the mul tiplication in L 2 (R) by Is, counting the number of particles in the region S C K, is itself a multiplication operator, multiplying by the number #(
52 Hans Maassen This is seen by the following calculation: for all A € R, / € K, and a € ( e ^ ^ V ( / ) ) ( a ) = 7r(e^/)(er)
= J]e"M')/(t) t€tr
= exp f i A ^ l 5 (t) j ■ 7r(/)(a) = e i A # ^ s > • TT(/)(<7) . 5. UNITARY DILATIONS O F SPIRALING MOTION
In preparation for the solution of the physical problem of damping posed at the end of Section 2, we now consider embeddings of the spiraling evolution (2.16) into a unitary one. Let us describe the spiral by Ct : C -* C : z >-► e ( -" +iu;)i 2 ,
(t > 0).
(5.1)
T h e o r e m 5.1 ((Sz. Nagy, Foias 1953; special case.). Up to unitary equivalence there exists a unique Hilbert space K, with a unit vector v and a one-parameter group of unitary transformations Ut on K, such that the span of the vectors Utv, t 6 R is dense in K, and (v, Utv) = e ( - " + i ^ , (t>0). Proof. Existence: Take K. := L 2 (R, 2ndx), let Ut be the shift to the right, and [O ye(v^w)t
v ;
if x>0, if a; < o .
Then we arrive at Example 1.6 (e) at the end of Section 1. Uniqueness follows from Theorem 1.5. □ The structure (IC,J,Ut), unitary dilation of (Ct)t>o-
illustrated in the diagram below, is called a minimal
C
e (
"^C
(5.2)
In practice several — unitarily equivalent — minimal unitary dilations of (Ct)t>o can be useful. If K. = L 2 (R) and Ut is the shift, then we speak of translation dilations of (Ct)t>o- They differ only in the shape of v G L 2 (R), which must satisfy ( A 6 R ) ' ^ " ( ^ ' ' Particular solutions are v±{\) := l/(\—u±irj). Here v+, which occurred in the proof of Theorem 5.1, leads to the incoming translation dilation and w_ to the outgoing translation dilation:
w
0
(x
Quantum probability applied to the damped harmonic oscillator 53 The former is more useful for the study of incoming fields and particles, the latter for outgoing ones. We shall have occasion to employ both below. The unitary equivalence of these two unitary dilations, asserted by Theorem 5.1, is implemented by the scattering operator S which in terms of the Fourier transform F can be written as + lV s(X):=X-W . \ — u — irj Apart from these two translation dilations, the interaction dilation (IC,J,Ut), where K. = L2{—oo,0]©C©L2[0,oo), J : z H-> Offiz©0, and Ut describes a more complicated coupling of C to an incoming and an outgoing channel, is physically more enlightening, but too cumbersome to treat here. We refer to [KiiS] for a thorough treatment.
S:=FMsF-\
6. T H E DAMPED HARMONIC OSCILLATOR
Equipped with the notions introduced in Sections 1, 3, 4, and 5 we are now in a position to answer the questions posed at the end of Section 2. We act with the second quantisation functor F of Section 4 on all four corners and all four arrows of the dilation diagram (5.2) of Section 5. The corners become quantum probability spaces (Section 1), and the arrows become operations (Section 3):
r(c)^ir(c) r(j)
(6.1)
r(j-)
r(/C) ^ l T(K.) The answers to Questions 1 and 2 can now be read off. (1) F(C) = B(H), where U = UH = Z2(N, i ) or equivalent^ U = UG = L2(R, 7) is the Hilbert space of the harmonic oscillator in the Heisenberg or in the Gaussian representation. The damped time evolution Tt : B(H) —> B(TC) is now given by Tt := r(C ( ) = T (e{~,>+iu)t) . Then Tt(W(z))
= e^e~2*l-1)W2W(e<--ri+^tz) ltp
.
(6.2)
%s
By substituting W(t + is) = e- + Q a n d differentiating with respect to t and s respectively, we indeed obtain the equations (2.16). (2) The diagram shows how Tt is embedded into a larger whole, where the time evolution is a one-parameter group of *-automorphisms, that is reversible. Here j := T(J) is an injective *-homomorphism, P = T(J*) is a conditional expectation. By Theorem 5.1 this is the only quasifree dilation, that is in the range of the functor F. It is automatically Markov. In this Section we shall discuss four aspects of the construction: the stochastic behaviour of the oscillator (spirals), its driving field (a quantum Brownian motion), the jumps between the levels of the oscillator (a death process), and the outgoing quanta (a point process). A complete picture would include the outgoing field and
54 Hans Maassen the scattering of incoming particles as well. This can easily be achieved using the tools developed here. Stochastic behaviour of the oscillator. By the functorial character of T we can split Tt as Tt = r(e ( -' ) + i u ; ) t ) = r(e i u ")r(e-" ( ) . The operator r(e _ I " ( ) is the automorphism at studied in Section 3. So let us now look at the 'dissipative' part r(e~vt). Proposition 6.1. For 0 < c < 1 the operator T(c) leaves invariant the abelian subalgebras generated by 1 and any of the operators xP — yQ with (x,y) £ l 2 . In particular its action on the algebra Q ■= { f(Q) | / e L°°(R) } is given by r(c)(/(Q)) = - = = = / e-^=^f(cQ / 2 V 27r(l - C ) J-oo
+ x)dx.
(6.3)
Proof. Obviously, T(c) leaves the linear span of { W{\z) IA G R } invariant, and thus also its strong closure by Proposition 4.1. Putting f(Q) = elyQ the r.h.s. of (6.3) equals /
-.
/-no
VMi - c2)
^2
e'w^e'^dx
\
■ eicyQ = e^(l-c2)y2W(icy)
,
which is equal to the l.h.s. by the definition (4.2) of F. The theorem follows from the strong continuity of T(c). □ We recognise the semigroup Tt of transition operators restricted to Q as the tran sition operators of a diffusion on R with a drift towards the origin proportional to the distance to the origin. The driving field. Let us consider the dilation of the semigroup Tt for w = 0. We take the second quantised incoming translation dilation of Section 5, and substitute it into the diagram (6.1). Let Bt be the Brownian motion given by (4.3), and let Qt denote the embedded oscillator &(vt). Proposition 6.2. The embedded oscillator Qt satisfied the integral equation Qt-Qs
= -V f Qudu + Bt~Bs.
This is the integral version of the stochastic differential equation dQt = -vQtdt + dBt . So we find an embedded Ornstein-Uhlenbeck process in our Markov dilation.
(6.4)
Quantum probability applied to the damped harmonic oscillator 55 Proof. ([LeT]) The following equality between functions in K holds: Utv- - Usv_ = -rj
vudu + l[S|t] ,
(s
Acting with $ on both sides of the equation yields (6.4).
D
Quanta. We now concentrate on another abelian subalgebra of F(C), namely the algebra of all diagonal matrices in Heisenberg's matrix mechanics. In terms of the operator N denoting the number of excitations of the oscillator, this algebra can be written as M ■= { f(N) | / 6 l°°(N) } ~ Z°°(N) • This time we need not put ui = 0. Let (fn denote the state Z°°(N) —> C : / — i > /(n). Proposition 6.3. The diagonal algebra N is invariant for Tt and N
Vn(Tt(s
))
{l-e-^(l-8))n.
=
This is the probability generating function of a pure death process [GrS] with the generator " 0 0 0 0 1 - 1 0 0 2 - 2 0 L = 2n 0 0 0 3 - 3
Proof. (Sketch. Cf. [Tee] for the detailed proof.) We can write sN as a weak integral over the operators W(z): = i f
e-5i^WV(z)A(dz)
(6.5)
where A denotes the two-dimensional Lebesgue measure on C. This relation can be checked by taking matrix elements with respect to coherent vectors. Application of Tt to both sides of (6.5) yields for all u, v € C, {n{u),Tt{sN)K{v))
_
„5«(l-e-2»'(l-s))
Since this expression is not sensitive to the relative phase of u and v, the operator Tt (sN) lies in Af. The statement is proved by comparing the coefficients of (uv)n on both sides. D Emitted quanta. Finally, let us see what happens outside the oscillator while it is cascading down its energy spectrum. Since we are interested in outgoing quanta at positive times, let us now consider the outgoing translation dilation of (Tt)t>o and represent T(/C) on the Fock space L 2 (A(R)). We denote the number operator dJr(Pvt) counting the excitations of the oscillator at time t by Nt.
56
Hans Maassen
From Proposition 6.3 it follows that the diagram (6.1) can be restricted to the subalgebra M ~ Z°°(N)
p
r(£) -^i r(/c) Here j : = I V ) : / - /(iV 0 ) ; P : = T(J*) : X >-► ( ( « 8 " , X / " ) ) ^ 0 e Z°°(N). However, since for different times t and s the functions vt and vs are neither parallel nor orthogonal, the one-dimensional projections Pn and PVt do not commute. And since for A , / i £ R
t h e number operators iVt and A^s do not commute either. So the embedded algebras jt(Af) with t € R do not generate an abelian subalgebra of T(K,), as was the case for t h e algebras jt(Q) above. For every t G R let us consider the following three number operators. Nt := d^F(PVt), Mt := dT ( M1([ Kt := dF [ Mi{_^
the number of quanta in the oscillator, I
the number of quanta that have not yet left the oscillator, J
the number of outgoing quanta that have left the oscillator.
Note that the number Mt — Nt of incoming quanta is not given by a multiplication operator, but the number Kt of outgoing quanta is. This is due to the fact that we are considering the outgoing translation dilation of Tt. Note furthermore that the operators Mt and Ks (s, t G R) all commute. For positive times the number Mt — Nt of incoming quanta has expectation 0 in the states of the form i) o P (i) £ ^ W ) which we consider. So we may expect that replacing Nt by Mt would lead to an embedded classical Markov chain. Proposition 6.4. For t > 0 we have the following commuting diagram involving abelian von Neumann algebras.
M rJ^f(Mo)
»N p=r(j")
L°°(A;/i„)-^-L~(A,,g
58
Hans
Maassen
[KiiM] B. Kiimmerer and H. Maassen, Elements of quantum probability, in [QPIO], pp. 73-100. [KiiSJ B. Kiimmerer and W. Schroder, A new construction of unitary dilations: singular coupling to white noise, in [QP2], pp. 332-347. [LeT] J.T. Lewis and L.C. Thomas, How to make a heat bath, in, "Functional Integration and its Applications," Proceedings, International Conference, London 1974, ed. A.M. Arthurs Clarendon Press, 1975, pp. 97-123. [Maa] H. Maassen, Quantum Markov processes on Fock space described by integral kernels, in [QP2], pp. 361-374. [Mac] G.W. Mackey, "The Mathematical Foundations of Quantum Mechanics," A Lecture-Note Volume, W.A. Benjamin, 1963. [Mey] P.-A. Meyer, "Quantum Probability for Probabilists," Second Edition. Lecture Notes in Mathematics 1538, Springer-Verlag, 1995. [Neu] J. von Neumann, "Mathematical Foundations of Quantum Mechanics, [transln. of the origi nal German edition.] Princeton Landmarks in Mathematics, Princeton University Press, 1996. [Par] K.R. Parthasarathy, "An Introduction to Quantum Stochastic Calculus," Birkhauser-Verlag, Basel 1992. [QP2] "Quantum Probability and Applications II," eds. L. Accardi and W. von Waldenfels Lec ture Notes in Mathematics 1136, Springer, 1985. [QPIO] "Quantum Probability and Related Topics X," eds. R.L. Hudson and J.M. Lindsay, World Scientific, 1998. [ReS] M. Reed and B. Simon: "Methods of Modern Mathematical Physics I: Functional Analysis," Second Edition. Academic Press, 1980. [Tak] M. Takesaki, Conditional expectations in von Neumann algebras, J. Fund. Anal. 9 (1972), 306-321. [Tee] S. Teerenstra, "Wave-particle duality in the damped harmonic oscillator," Masters Thesis, University of Nijmegen. [Var] V.S. Varadarajan, "Geometry of Quantum Theory," Second Edition. Springer, 1985. [Wey] H. Weyl, "The Theory of Groups and Quantum Mechanics," [transl. from the second German edition, by H.P. Robertson.] Dover Publications, 1950. DEPARTMENT O F MATHEMATICS, UNIVERSITY OP NIJMEGEN, TOERNOOIVELD 1, 6525 NIJMEGEN, THE NETHERLANDS.
E-mail address: [email protected]
ED
Quantum Probability Communications, Vol. XII (pp. 59-138) © 2003 World Scientific Publishing Company
Q U A N T U M PROBABILITY A N D STRONG Q U A N T U M MARKOV PROCESSES K R PARTHASARATHY
CONTENTS
0. Introduction 1. A comparative description of classical and quantum probability 2. The role of tensor products of Hilbert spaces 3. Some basic operators on Fock spaces 4. Prom urn model to canonical commutation relations 5. Stochastic operators on C*-algebras 6. Stinespring's theorem 7. Extreme points of the convex set of stochastic operators 8. Stinespring's theorem in two steps 9. Construction of a quantum Markov process 10. The central part of minimal dilation 11. One parameter semigroups of stochastic maps on a C-algebra 12. Noncommutative stop times 13. Markov process at simple stop times 14. Minimal Markov flow at simple stop times 15. Strong Markov property of the minimal flow for a general stop time 16. Strong Markov property under a smoothness condition 17. A quantum version of Dynkin's localization formula Acknowledgements References
0.
59 61 69 74 78 83 85 92 94 97 102 103 111 114 116 118 127 134 137 137
INTRODUCTION
The first chapter of these notes (Sections 1-4) presents a comparative description of elementary classical and quantum probability, a brief account of tensor products of states and observables, Fock spaces, creation, annihilation and second quantization operators and concludes with a central limit theorem for normalised sums of exchange operators in the L 2 -space of random variables arising from urn models. The canonical commutation relations for free quantum fields emerge naturally as a consequence of this central limit theorem. The topics covered here were more-or-less chosen by the organisers of the Summer School. 59
60 K. R. Parthasarathy The main emphasis of these notes is on quantum Markov processes and the strong Markov property. It is now widely accepted that irreversible dynamics in quan tum theory is mediated by contractive semigroups of completely positive linear maps (or, equivalently, stochastic operators) on a C*-algebra ([Kra], [AFL]). These are known as quantum dynamical semigroups and they play the role of Markov transition probability semigroups in classical probability. There exists an extensive literature on the dilations of such semigroups to quantum Markov processes. (For example, see [AFL], [Kiil], [Kii2], [Me2], [P], [Sau], [Sch], [Vin].) Here we follow the approach of Bhat and Parthasarathy ([BP1], [BP2]), and associate a canonical time indexed family of *-homomorphisms of a C""-algebra A to any given family {T(s, t), 0 < s < t < oo} of stochastic operators on A obeying the Chapman-Kolmogorov equation T(r, s)T(s, t) = T{r, i) for all r < s < t. These *-homomorphisms are not unital and they arise as a natural extension of the well-known Stinespring's theorem ([Sti]). It is a simple application of the GNS principle. The same *-homomorphisms evaluated at the identity yield an increasing family of projections. Truncations by these projections can be interpreted as a filtration of conditional expectations. When T(s,t) depends only on the difference t — s, the conditional expectations give rise to a Markov flow. One of the most interesting open problems of this subject is the con crete realization of this canonical Markov flow associated with the semigroup {Tt}. Quantum stochastic calculus as outlined by Hudson and Parthasarathy ([HuP]) and the free calculus of Speicher ([Spe]) lead to quantum stochastic differential equations and a wide variety of Markov flows described by such equations ([Bia], [Me 2], [P]). Here we do not touch on the methods of quantum stochastic calculus. Recently Bhat has shown that in the case when the C*-algebra A coincides with the algebra of all operators on a Hilbert space the minimal Markov flow associated with a quantum dynamical semigroup {Tt} can be realized through unitary cocycles on a Fock space ([Bh2]). The last chapter (Sections 12-17) deals with the strong Markov property of minimal flows. The strong Markov property is stability under 'random shifts' and therefore involves the introduction of the notion of stop time. Hudson introduced the idea of a quantum stop time as a spectral measure which is Fock-adapted and showed that quantum Brownian motion in a boson Fock space enjoys a strong Markov property ([Hud]). Barnett and Lyons ([BaL]) and Barnett and Thakrar ([BaT]) developed a theory of stop times with respect to a filtration of von Neumann algebras and intro duced the fruitful idea of stopping a quantum process at a stop time. In particular they established a quantum version of Doob's optional sampling theorem for operator martingales. There also exists a detailed analysis of stop times in Fock space and some applications due to Parthasarathy and Sinha ([PSi]) and Applebaum ([App]). Here our aim is to present a notion of stop time with respect to the filtration of the minimal Markov flow associated with a quantum dynamical semigroup and associate *-homomorphisms at a stop time. Following Attal and Parthasarathy ([AtP]) we define a stop time r as a spectral measure on [0, oo] for which the projection r([0, £]) signifying the event of stopping at or before time t commutes with the observables of the flow from time t onwards. Using a result of Enchev ([Enc]) we construct a
Quantum probability and strong quantum Markov processes 61 *-homomorphism at time V for the Markov flow and exhibit a strong Markov prop erty. However, we are unable to define, in general, a Markov shift by time V on the path algebra of the flow. Obviously, there are many loose ends. We conclude these notes with a quantum version of Dynkin's localization formula for defining the characteristic operator ([Dyl], [Dy2]).
I. Quantum Probability 1. A COMPARATIVE DESCRIPTION OF CLASSICAL AND QUANTUM PROBABILITY
Let (fi,T,P) be a finite probability space with Q, = {1,2,... ,n}, T the boolean algebra of all subsets of U and P({i}) = Pi- The expectation E / of any complexvalued random variable on fl can be expressed as
E/ = £/Wft =
Tr
/(I) 0
0 /(2)
0
0
/(")
Pi 0
V2
0
0 0
0
0
Pn (1.1)
J=l
w h e r e
whose
u' = (VPT e * \ V S e * » , . . . , Jp-n e*») n
in the Hilbert space C with scalar product n
where w ' = ( w i , . . . , w„), v' = ( « i , . . . , v„). Thus E / in (1.1) can be expressed as E/ = Tr/p=(u,/u)
(1.2)
where the random variable / is viewed as the diagonal matrix operator with diagonal entries f(i) and the distribution P as the diagonal matrix operator p with entries piThe same expectation can also be viewed as the scalar product on the right hand side of (1.2) where the unit vector u involves the square roots of the probabilities multiplied by arbitrary phase factors. Note that p in (1.2) is a positive semidefinite matrix with unit trace. Quantum probability, or equivalently, probability as encountered in problems of quantum mechanics deals with the situation when / and p in (1.2) are replaced by not necessarily diagonal matrices or operators and the Hilbert space is not necessarily finite dimensional.
62 K. R. Parthasarathy We first examine the situation when the finite set £2 = { 1 , 2 , . . . ,n} is replaced by the n-dimensional complex Hilbert space H ( = C"). Denote by B(TC),V(H), S(7i),0{H) respectively the *-algebra of all operators, the lattice of all orthogo nal projection operators, the convex set of all positive operators of unit trace and the real linear space of all hermitian operators on Ti. Exercise l . A . (i) The set of extreme points in the convex set of all probability distributions on Vl has cardinality n. (ii) The set of extreme points in the convex set S(H) (where H = C") consists of all one dimensional projections. It is a manifold of dimension 2(n - 1). Exercise l . B . The linear space of all real-valued random variables on fi has di mension n whereas the real linear space 0(TL) of all hermitian operators on Ji has dimension n2. Exercise l . C . (i) Let jFfc be the collection of all /c-point subsets of Q. Then
#Tk = (n\ and T = [JTk. k=o
^ '
The permutation group (or symmetric group) Sn on fi acts transitively on Tk and Tu can be identified with the homogeneous space Sn/Sk x <Sn_t. (ii) Let Tk(7~L) be the set of all /c-dimensional orthogonal projections in TL. Then n
p(n) = \Jvk(H). k=0
If Un(7i) = Un is the Lie group of all unitary operators on 7i then lAn acts transitively on V\&H) by
(u, E) -^ UEU-\ v e un, Ee rk{n) and Vk (fi) can be identified with the homogeneous space Un/Uk xUn^k. is a manifold of dimension 2k(n — k).
Vk(Tt)
Elements of V(H) are called events. If P 1 ; P 2 , . . . , Pk are events then their lattice maximum Pi V P 2 V ■ • ■ V Pk is interpreted as the event of occurence of one of the P^s. Their minimum Pi A P2 A ■ • • A Pk is the event that all the Pi's occur. Denote the identity projection by 1 and the null projection by 0. They are interpreted as the certain and null events. For any event P its orthocomplement 1 —P is the complement of the event P . If Pi < P 2 in the sense that P 2 — Pi is also a projection we say that the event Px implies the event P 2 . Note that 0 < P < 1 for every P G V(H). If Pit i = 1, 2, 3 are events it is not necessary that Pi A (P 2 V P 3 ) = (Pi A P 2 ) V (P x A P 3 ). (This can be verified by taking the projections on three distinct lines through the origin in IR2.) Thus V, A, <, 1 — •, 0 and 1 play the role of U, n, C, set complement, 0 and Q respectively in quantum probability. Elements of 0(7i) are called real-valued observables. They play the role of realvalued random variables in £1 Whereas real-valued random variables on Q constitute
Quantum probability and strong quantum Markov processes 63 a commutative algebra, 0(H) is a real linear subspace of the *-algebra B(TL) which is not commutative. Any real observable A admits a spectral decomposition k
A = ] T A ^
(1.3)
where Ai, A2,..., A^ are all the distinct eigenvalues of A and E* is the eigenprojection of A corresponding to the eigenvalue Aj. Compare this decomposition with that of any real-valued random variable / on 0. as k
1=1
where {Ai, A 2 ,..., A*,} is the range of / , E{ is the inverse image of {Ai} under / and \E denotes the indicator function of the subset E C fi. Compare the relations EfEf = SijE* with lBflEi = SijlBf. Also compare the relations
£>(A0#,
(1.4)
i=l k
(L5)
i=l
We say that the observable A assumes the values Ai, A 2 ,..., A^ and the event that A assumes the value Aj is the projection Ef. Note that an observable on H = C n cannot assume more than n values. It is instructive to note that a random variable / can be interpreted as the homomorphism
64 K. R. Parthasarathy by *
{A)p = Y, ^ = Tr PAIn view of (1.4) the fc-th moment of A in the state p is given by (Ah)p = Tr /?/lfc and the characteristic function of the distribution of A in the state p is given by (eitA)p = Tr peitA, t e R as a function of t. Note that {e*1/4} is a one parameter unitary group and the char acteristic function is continuous and positive definite with value 1 at t = 0. The variance (or dispersion) of A in the state p, denoted VP(A), is given by Vp(A) = {A2)p-(A)2p
Trp(A-m)2
=
where m — (A)p and the convention that a scalar m times the identity operator is denoted by the same symbol m. Exercise l . D . (i) The distribution of an observable A in a state p is degenerate at a point if and only if VP(A) = 0. (ii) VP(A) = 0 if and only if the range of p is contained in the eigensubspace of A corresponding to the eigenvalue (A)p = TrpA. (iii) If p is a pure state determined by the one dimensional projection on the subspace CM, where u is a unit vector in H, then VP(A) = 0 if and only if u is an eigenvector of A. Thus, given any non zero observable A there always exists a pure state p in which VP(A) / 0, or equivalently, A has a nondegenerate distribution. Compare this with the situation in classical probability. Exercise l . E (Uncertainty principle). Let A,B £ (D(H), p 6 S(H), {A)p = m, {B)p = rri, z = reiB, r € R, 9e[0, 2TT). Then 0 < {{A + zBY(A =
r^
) p
+
+ zB)}p
2rU^±^\ ) cosd -(^EA) \ sin^+(^)if cos o - i
Minimizing over r first and then over 6, 2 Putting A = A-m,
I„
\
2%
B = B - m!', VP(A)VP(B) > (
AB-BA\2
IAB + BA'"2 ) + ( £ } .
(1.6)
P
If A and B do not commute with each other then AB — BA ^ 0 and there will be states in which the first term on the right hand side of (1.6) does not vanish. This
Quantum probability and strong quantum Markov processes 65 shows that the variances of both A and B cannot be too small. In quantum theory, the position and momentum observables of a particle q and p obey the commutation rule qp — pq = ih where h is a positive universal constant. In that case ( 2£ § E2 ) = 7and
VP(qK(P) > J This implies that in no state can p and q be measured with total precision. This is called the uncertainty principle. However, in this case q and p are unbounded selfadjoint operators and care is needed in writing such inequalities. Exercise l . F . (i) Let n = 2, H = C 2 , 00 =
[10] 0 1
r o i l
,01 =
1 0
r n —i =
,0-2
i
0
r 1 n 1 ,03 =
0 -1
In the standard orthonormal basis T eo
ei
each <7j is a hermitian operator. The elements a^i = 1,2,3 are called Pauli spin matrices. They are observables assuming the values ± 1 . They obey the following multiplication and commutator tables: Oi<Jj
0;,ff,' — OiOi
01
02
03
01
00
ias
-102
01
02
-J03
00
lO\
02
03
io2
—10 \
00
03
(ii) If t = (ii,t 2 ,t 3 ),
01
0 -2iaz 2ia2
02
03
2J03
-2za 2
0 —2icr1
2i<j\
0
UeR
ei(tl<Ti+t2a2+t3a3)
=
cos\\t\\+i
sin lit II
(tiffi + f202 + t3a3)
where ||t|| = {t\ + t\ + if)1'2. (iii) In the state p0 = \o~a, the observable t\ai+t2a2 with equal probability \. (iv) Note that Tr / 9 0 e i(flffl+ ' 2t72)
assumes the values ± \ A ? + *:
'V^
cannot be the characteristic function of a pair (£,77) of real valued random variables. An open problem: Characterise functions tp(t\, t2,..., tjt) offcreal variables t\, t2,..., i* which admit a representation of the form v(ti,-..,in)
where X\,X2,'
= Tr / oe ft - x
■ ■ ,Xk are observables in the Hilbert space Ti. = Cn.
66 K. R. Parthasarathy Exercise l . G . If Air-■
,Ak e 0(7i)t
p € S(H) the matrix ((cry)) defined by
(Tij = TrpAiAj -
TrpAiTrpAj
is positive semidefinite. If TrpAi = rrii, Ai — Wj = Ai} then oy = TrpAiAj ■ ((cry)) is called the covariance matrix of the observables Ai, ■ • ■ ,A^. Even though each Ai assumes only real values the covariance cry between Az and Aj can be complex. In particular their correlation
can be a complex number. Exercise l . H (Bell's inequality [Bel], [BGP], [Gup]). A random variable £ on a prob ability space is called a spin variable if it assumes values ± 1 with equal probability | . If {£,, i — 1, 2, ■ • • , m) are m spin variables with m > 3 and ay = E££j then for any i < j < k 1 + EiSjCTij + £j£k&jk + EkZiVki > 0
where €i,£j,£k assume any values from the two point set {—1,1}. However, this is not sufficient for an m x m positive semidefinite matrix ((ay)) with ait = 1 for every i to be the correlation matrix of m spin variables when m > 5. Exercise 1.1 (J.C. Gupta [Gup]). The set of all m x m spin correlation matrices is a convex set C. A matrix ((ay)) € C is an extremal element if and only if crji = 1 for every i,
atj = (-l)K ! .j} nT l for i ^ j
where {1} C T C {1,2, • • • ,m} is arbitrary. Thus C has 2"" 1 extreme points. Exercise l . J . Let ((<%)) be any 3 x 3 positive semidefinite matrix with a„ = 1, i = 1,2,3. Then there exist unit vectors u 1 , it2, v? in R 3 such that Cy = (u1 ,u3) for all i,j. Define Ai = u\(Jx + u\a2 + u\a3 where u1 = (ul^l/ul) and ai, 02.03 are Pauli spin matrices. In the state p = A
= cry for all i, jf € {1, 2, 3}
Comparing with Exercise l.G we conclude that spin observables in quantum proba bility violate Bell's inequality. Exercise l . K . Let X = (2p—1)03 —2-^/p(l — p)a : where 0 < p < 1 and cr^i = 1,2,3 are the Pauli spin matrices. In the pure state determined by the unit vector e0 = Q the observable X assumes values 1,-1 with probabilities p,l —p respectively. Exercise l . L . Let U = C 2 n
X =
03
0
0
a3
0
0
y = •■•
0
cr3
ax 0
0 a2
0 0
••• •■•
0 0
0
0
■••
0
an
Quantum probability and strong quantum Markov processes 67 where 0-3
=
" 1 0
0 -1
,Ctj =
Xj
yj
,Xj,Vj e R , 4 + ^ = 1-
Then X and Y are observables taking the values ± 1 only whereas X + Y is an observable assuming the values ±-^/2(l + Xj),j = 1,2, ■•■ , n. (If £,77 are classical random variables asuming the values ± 1 only then £+77 is a random variable assuming the values {-2,0,2}). Exercise l . M (Dirac notation). If Ti is a Hilbert space denote the elements of Ti by \u > when u € Ti and by < u\ when u € Ti is considered as an element of Ti* (i.e., as a linear functional on Ti). For any u, v € Ti, define the operator |u) (v\ by putting |W){I>|«J = (v,
w)u.
This is a rank one operator on Ti with range Cu. An operator P on H is a one dimensional orthogonal projection if and only if P = |u)(u| for some unit vector u eTi. The operators {\u)(v\,u,v G 7i} satisfy the following: (i) \u){v\ is linear in u and conjugate linear in v; (ii)(\u)(v\)* = \v)(u\ (hi) |Ul)(wi| | u 2 ) ( W 2 | " - | M n ) ( « n | = { n " = l 1 ( u < . u « + l ) } l u l ) ( u n l ;
(iv) If u 7^ 0,v ^ 0 the range of \u)(v\ is the one dimensional subspace Cu;
(V) || \u)(v\ II = ||W|| ||V||; (vi) For any T e 3{Ti), T\u)(v\ = \Tu)(v\, \u){v\T = \u){T*v\ ; (vii) If P is a projection and {ei, e2-..} is any orthonormal basis for the range of P then P = J2i\ei){ei\; (viii) An operator T in Ti is of finite rank k if and only if there exists an or thonormal set {u\, v.2, ■ ■ ■ , Wfc} in Ti such that {Tu\, TU2, • • • , Tuk} is linearly independent and k
r = £12^X14 (ix) TrT|w)(v| = (TJ.TU) for any T e £(ft). Exercise l . N . Let dim7i = n and let e 0 ,ei,--- ,e„_i be an orthonormal basis for Ti. In Dirac notation consider the (ladder) operators: n-2
Ln = ^ | e j ) ( e ; - + i | , L ; =
n-1
J2\ej)(ei-i\-
Then Lne0 = I^e,- = LnL; =
0, L n ej = e^-i for 1 < j < n - 1, ej+i for 0 < j < n - 2, L;e n _ x = 0, l - | e „ _ 1 ) ( e „ _ i | , L;L„ = 1 - | e 0 ) ( e 0 | .
Furthermore, the following properties hold:
68 K. R. Parthasarathy (i) (L„ + L*n)ke0 = J2kr=o Ck(r)er, 0
where co(0) = 1,
c w ( r + l) + c n ( r - i ) c/t-i(l)
ck{r) =
if r > 1, ifr = 0.
(ii) The recursion relations in (i) imply
{
0
if fc is odd,
( m + 1 ) - ^ ) iffc = 2 m < n - l . ' (iii) The first n — 1 moments of the observable ^(Ln + L^) in the pure state |eo)(e 0 | coincide with the first n — 1 moments of the density function
fyT^x*
/(*)
if|i|
(iv) The distribution of \{Ln + L£) in the pure state |eo)(eo| converges weakly as n —> co to the Wigner's semicircular distribution in (iii). Exercise l . O . In the above Dirac notation consider the observable n-l
W» = X>|e,-)(e,-| and define the observable Mn = s/Nn + lLn + L*nsjNn + l. Then Mn has the matrix in the basis {eo, ei, ■ • • , e n _!} 0
v/I
0
0 0 0
V2
0
0 0 0
0 0 0
v/n-1
0
Suppose MkneQ = ^ 4 ( r ) e r , 0 < k < n - 1. Then d„(0) = 1 and dk+i{r)
vV + 1 4 ( r + 1) + y/rdk(r - 1) 4(1)
if r > 1, if r = 0.
These recurrence relations imply that 4 ( 0 ) = 0 if A; is odd and 4 ( 0 ) = 1-3.5. • • • (2m— 1) if k = Ira < n — 1. Thus the distribution of the observable Mn in the pure state leo)(eo| converges weakly as n —> oo to the standard normal distribution.
Quantum probability and strong quantum Markov processes 69 2. T H E ROLE O F TENSOR PRODUCTS O F H I L B E R T SPACES
If (fij, Fi, Pi), i = 1, 2 , . . . , k are finite probability spaces concerning independent statistical experiments they can be considered together as a single probability space ( J ] i X ( l 2 x - - - x % T\ x Ti x • ■ • x Tk, Pi x P2 x • • • x Pk), namely their direct or cartesian product. The discussion in Section 1 shows that, in elementary quantum probability, ( f l j , ^ , ^ ) may be replaced by {Hi, V{Hi), pi), i = 1,2, ...,k and they can be considered together as {Hi ® H2 ® ■ • ■ ® Hk, V{Hi ® H2 ® ■ ■ ■ ® Hk), pi ® p2 ® ■ ■ • ® pk) where ® denotes tensor product. The justification for such a procedure lies in the fact that for Et 6 P{Ht), l
PiE{.
i=l
Note that pi ® ■ ■ ■ ® pk is also a state, called the product of the pis. We may ignore the states pi and consider V{Hi ® ■ ■ ■ Hk) as the quantum probabilistic analogue of the product boolean algebra T\ x T2 x • • • x Tk. Exercise 2.A. (i) If dim H = rii then dim Hi ® ■ • ■ ® Hk = nin2 • • ■ nk and k
k
i=l
i=l
dim 0{Hi ® ■ ■ ■ ® Hk) = [ | dim 0{H{) = ]J nt2 (ii) If, however, 7Y; is a real Hilbert space and 0{Hi) denotes the real linear space of symmetric operators on Hi for every i then dim 0{Hi ® ■ ■ ■ ® Hk) = -nin2 ■ ■ ■ nk{nin2 ■■■nk + l) whereas
n dim
o^)=nn-^-
Thus k
dim 0{Hi ® ■ ■ ■ ® Hk) > Y[ dim
0{H{)
i=l
if, for some pair i, j , rii > l,rij > 1. (If we choose a real Hilbert space model for quantum probability putting systems together seems to lead to more observables.) Suppose V{H) describes the events concerning the behaviour of a particle. What should be the Hilbert space for examining events concerning n such identical or indistinguishable particles? To begin with consider the n-fold tensor product H®n =
70 K. R. Parthasarathy Ti <8> • • • ® T~L. If «i <8> u2 ® • • • ® un is a product vector and u € iSn is an element of the permutation group acting on {1, 2 , . . . , n} consider the correspondence Ml ® U2 ® • • ■ ® Un —» UCT(i)
On the set of such product vectors the correspondence is scalar product preserving. Such product vectors constitute a total set in 7i®n, i.e., their closed linear span is H®n. Hence there exists a unique unitary operator U~l in H®n such that U~lUi
®U2®---®Un
= MCT(i) (g> Ua{2)
® ■ • • <8> MCT(n).
It is clear that f/CTlUa2 = Uaia2 for all a\,a2 £ <Sn. In other words, we obtain a unitary representation of the group Sn- This group has a dual space, namely, the set of all equivalence classes of irreducible unitary representations. Denote it by Sn. If n is an irreducible unitary representation denote by x-n its character, i.e. x-n{a) = Tr 7rCT. Let d(n) = X-n^d) denote the dimension of the representation n. Put
crgSn
Prom the famous Schur orthogonality relations it follows that P^ is an orthogonal projection in 7i®" which commutes with Ua for every a and
wesn Denote the range of Pn by ft£°. Then
where it is identified with its equivalence class or type in Sn. Each Hi? is invariant under Ua, a € Sn. Any irreducible subspace in 7ii (invariant under all Ua) is of the type n, i.e., vectors in H„ transform according to the type w under cr-action. Any one of the Hilbert spaces Hn , ir 6 >Sn is a 'good candidate' for describing n identical particles. There are two very elementary irreducible representations for Sn, denoted n+ and w_. They act in the one dimensional space C: n+(a)z it-(a)z
= zV a € Sn, z £ C ; = sgn(cr)z V a £ Sn, z € C
where sgn (a) denotes the signature of a, which is equal to ± 1 according as a is an even or odd permutation. Then p(n)
.=
p
=
!
Y" U a€S„
P™ := P*.=*^
^(sgna)^. aeSn
Quantum probability and strong quantum Markov processes 71 The range of P+ in 7{®n consists of all vectors fixed by every Ua. It is denoted by and called the n-fold symmetric tensor product of copies of 7i. The range of P_ is denoted by and called the n-fold antisymmetric tensor product. 71
n
It is a hypothesis (called Pauli's exclusion principle) that only ft® and H® are relevant for describing n identical particles and the remaining types TO? do not occur (in nature). Prom a purely mathematical point of view there is no reason for rejecting any of the model spaces 7"A- ■ When H®71 is chosen we say that the particles under consideration obey boson statistics. If is chosen we say that the particles obey fermion statistics. What is the Hilbert space when one has to deal with situations in which there is an indefinite number of identical particles? This is the case when particles are created and annihilated. With a view to consider events concerning such systems we introduce three kinds of Hilbert spaces: oo
rfr(«) = 0w*" n~0 oo
TS(H) = 0 H ® 1 1 n-0 oo
ra(w) = 0 # , respectively called the free (or Maxwell-Boltzmann), boson and fermion Fock space over H. The subspaces H®n,7i(i>n and H®n are called n-particle subspaces in each case. When n = 0, these are to be interpreted as the one dimensional Hilbert space C and the vector 1©0©0©..., denoted $, is called the vacuum vector. The corresponding pure state determined by $ is called vacuum state. Denote by E± the projections in Tb(H) defined by E± = 1ffiP£] © Pi 2 ) © • ■ ■ Let !%{H), T°(H) and Tl{H) be the dense linear manifolds generated by all the finite particle vectors in the corresponding Fock spaces. Exercise 2.B. Define multiplications (u,v) —> u®v,uv, Fock spaces r ^ ( H ) , r ° ( H ) , r ° ( 7 i ) respectively by
u A v in the finite particle
OO
M®U = 0
J2 P(r)U®P(s)V,
n—0 r+s=n
=
E+U
u Av =
£Lu ® v
UV
where P^ is the projection on the r-particle subspace of Yf^fi). that l ® u = u®l = u. Then the following hold: (i) r° r (7i) is an associative algebra; (ii) r°(7i) is a commutative and associative algebra;
Here we assume
72 K. R. Parthasarathy (iii) Tl(H) is an associative algebra in which u/\v
= {-l)mnv
A u if u € H ® m , v £
H®n.
Exercise 2.C. Suppose {ej,j = 1,2,....,} is an orthonormal basis in H. Then the three sets (0 {$,ei, ®e i 2 ® ••■ ®e i m : ir = 1, 2, ...;r = 1,2,..., m; m = l , 2 , . . . } /
\ i/2
("){*. ( ^ i ) « • ■ ■ < * : l < i 1 < i 2 < - - . < i i < l + dimW,r1 + . - . + rfc = m , m = 1,2,...}, (iii) { $ , ( m ! ) 1 ^ ! A ei2 A • ■ • A eim : 1 < ix < i2 < ■ ■ ■ < im < 1 + dim Ti, 1 < m < 1 + dim H } are respectively orthonormal bases in TfT(Ti),Ts(H) being as in the previous exercise.
and r a (7i), the multiplications
Exercise 2.D. If dim U = N < oo then dimW(D,
=
dimH®"
=
(7V+;-1), ( (n) I 0
i f n
^N' otherwise.
Exercise 2.E. For any u € Ti write e(u) = 1 © u © -7=
l|e(u + ^) - e(n) - Z>^J||
\H-+°
\)v\\
=
Q
where Du is an operator from TL into VS(H); (ii) For any finite set {ui,u2,... ,un} C H the set {e(uj), e(u 2 ), • • •, e(wn)} is linearly independent. This is a consequence of the identity (e(u), e(v)) = exp(u, v) for all v., v 6 Ti; (iii) There is a Hilbert space isomorphism U : Ts{Tii © Ti2) -> T s (Hi) ® Ts(Ti2) satisfying (7 e(u © v) = e(it) <£> e(t>) for all u 6 Tij, v £ 7^2Exercise 2.F. Let /j, be the standard normal distribution in K. Consider the complex Hilbert spaces L2(n) and Tfr(C) = TS(C) = C © C © C © • ■ ■ For any z € C, V2!
vn!
Quantum probability and strong quantum Markov processes 73 In L2(fj) consider the generating function of the (orthogonal) Hermite polynomials {Hn} given by oo
„
z nV
= £ >»(*)■ n—u
There exists a unitary isomorphism U : TS(C) —> L2(fi) satisfying (Ue{z))(x) = ezx-^2 V z e C . In particular [7(0,0,..., 1, 0,0 ...) = Vn!#n, n = 0,1, 2,... where, on the left hand side, 1 occurs in the nth position. Let n be the Poisson distribution on {0,1,2, • • • } with mean value A so that M(W) = e - A ^ , j = 0,1,2, .... In L2(/u) consider the generating function of the Charlier-Poisson polynomials {nn(\, x)} defined by 1+ ^ j
= £
^ 7 r n ( A , s ) , s = 0,1,2,...
Then there exists a unitary isomorphism U : F S (C) —* L2{p) satisfying [Ue{z)]{x) = e~VAz I l + - p 1 f o r a l l z G C , x = 0,1,2,... Exercise 2.G. For vectors ui, u 2 , . . . , um, Vi,v2,..., {uiAu2A---Aum,
Vi Av2A---
vm e 7i,
/\vm) = —
det{((ui,Vj})).
Exercise 2.G shows that it is more convenient to define the operation A by Ui A u2 A • ■ • A um = \fm\ P_U\
®u2®---®um
for all w; g Ti. We shall adopt this convention hereafter. Then (ui A « 2 A ■•• Aum,Vi
Av2 A-- • Avm) =
det{((Ui,Vj))).
Exercise 2.H. Let dim 7i — n < oo and let ei, e2,. ■ ■, e n be an orthonormal basis for Ti. For any set 5 = {i\ < i2 < ■ ■ ■ < ik} C {l, 2,..., n) define eiS\
^ '
=
f eii A ei2 A • • • A eifc if k =£ 0, [1 otherwise.
Then the family {e(5), S C {1,2,..., n}} is an orthonormal basis for F a (H). If P is the probability measure of n i.i.d. symmetric Bernoulli random variables fj, 1 < z < n assuming the values ± 1 with equal probability | then the correspondence &1&2''' Cijt l—* eii A • • ■ A eik, i\ < ■ ■ ■ <, ik for k ^ 0 and 1 — i > 1 for A; = 0 extends to a unitary isomorphism between L2(P) and F a 7i).
74 K. R. Parihasarathy 3. S O M E BASIC OPERATORS ON F O C K SPACES
Let Ti be a complex Hilbert space. For any u 6 Ti define the operator a(u) in r fr (7^) as follows: put a(u)$ = 0,
a(u)viL<&V2®...®vn
Extend this linearly to T^H). cJ(u)$ = u,
= (u,Vi)v2®...®vn.
(3.1)
Similarly define the operator al(u) by
a^(u)vi ® v2 ® . ■ • ® vn = u(g> «! ® w2 <8> ■• • ®vn
(3.2)
and extend it linearly to T°r(TC). Note that a(u) and a^u) are adjoint to each other on the domain of finite particle vectors and a(u)(J(v) =
(u,v).
In particular, for any ip € ^°T(Tt)
\\a\vW=\\v\\2\m2. This shows that aJ(v) closes to a bounded operator of norm ||t>|| on TiT(H). So does the operator a(v). Denote the closures of a(v) and a\v) by the same symbols. This yields the following proposition. Proposition 3.1. For any v 6 Ti there exist bounded operators a(v),a^(v) on Tit(Ti) satisfying (3.1) and (3.2). The correspondences v —> a(v) and v —> a*(v) are antilinear and linear respectively. They satisfy the relations a(u)a*(v) = (u, v) for all u,v
eH-
IfU is a unitary operator on Ti then there exists a unitary operator T(U) on T!T(Ti) satisfying T(U)$ = $,
T(U)ui ® ... ® u„ = f/ui ® Uu2 ® • • • ® Uun, n > 1,
/or aM Wj e ?i. For an?/ C/,Vg W(7i), u 6 W r ( I / ) r ( V ) = r(C/V),r([/)a(u)r((7)- 1 = a(Uu), and r ( [ / ) a t ( « ) r ( ( / ) - 1 = Exercise 3. A. For any unit vector / i \ i u \\k
a\Uu).
u^Ti (
0
if k is odd,
( • • ( ^ k ^ h - j o . + U-'P ift-^
M
Thus the observable "W+a (") has, in the vacuum state $, the Wigner's semicircular law as its distribution with its fe-th moment given by the right hand side of (3.3). (Hint: Use the recursion relations of Exercise l.M.) We call a(u), a}(u) and T(U) respectively the free annihilation, creation and second quantization operator associated with u G Ti, U £ U{Ti).
Quantum probability and strong quantum Markov processes 75 We now move on to the boson (or symmetric) Fock space Ts(fi) and define the operators a(u),a?(u) for any u E Ti as follows: put o(u)$ = 0, a{u)v®n =
V^{u,v)v®(n-1),
o f (u)$ = u, n
a?(u)v®n = {n+ 1)" 1 / 2 ] T v®r ® u ® w® (n - r) , » £ ? i . r=0
Observe that vectors of the form v®" span the n-particle space in Ts(7i). Extend a(u),oJ(u) linearly to T°(?i) and note that a(u) and a)(u) are adjoint to each other on T°(TC). Denote their closures by the same symbols. If £(H) denotes the linear manifold generated by {e(u),u 6 ft}, namely, the set of all exponential vectors in Ts(7i) then £{7i) is in the domain of all the operators {a(u),o)(u),u G 7i} and a(u)e(v) =
(u,v)e(v),
a\u)e{v)
= -^-e(v + eu)\e=0 ae for any v £ Ti. (Compare the second relation with (i) in Exercise 2.E.) On the domain of finite particle vectors one has the canonical commutation relations (CCR): [a(u),a)(v)\ = (u,v), [a(u),a(v)\ = [at(u),<J{v)] = 0. For any unitary operator U on H there exists a unique unitary operator T(U) satis fying
r(t/)$ = $,. r(L/)u®n = ([/«)«", uen, n = 1,2,.... One has the relations Y{U)T{V) = Y{UV),
T(U)a{u)T{U)-1
= a{Uu), V G U{H),
ueJi.
Exercise 3.B. For any unit vector u £7i /i / / \ t/ wh^\ I 0 ($ 1 (a(u) + at(U))fc$> = | 1 3
5
_
2 m
_
1
if k is odd .ffc = 2 m
In the vacuum state $, the observable determined by the closure of a{u) + o)(u) has the standard normal distribution. The operators a(u),oJ(u),T(U) are respectively called the boson annihilation, cre ation and second quantization operators associated with u € H, U € U(H). Now consider the fermion or antisymmetric Fock space r a (7i) and define a(u),a^(u) as follows: n
a(u)$ = 0, a(u)vi Av2A...Avn
= Y^(—l) j ~ l (u,Vj)vi
A v2 A . . . Adj A ...
where~indicates omission of the term, a*(u)vi Av2A...Avn
=
uAviAv2A...Avn.
Avn
76 K. R. Parthasarathy (Note that we have adopted the definition: = v / n!Pi n) (wi ® .. - ® vn.)
v1A...Avn
Extend a(u) and at(u) as linear operators on FaJH). Then a(u) and aJ(u) are adjoint to each other on Tl(Tt) and the following holds: {a{u)o)(v) + a\v)a{u)}ijj
= (u, v)ip for V £ r°(ft).
This shows that ||a(u^||2 + ||at(W)^||2<||«||2||^||2. Thus the closures of a(u) and at(u) are bounded operators on TJH). Denoting their closures by the same symbols we obtain the relations: a(u)a\v)
+ a}(v)a(u) = (u, v).
It is straightforward to verify that a(u)a(v) + a(v)a(u) = 0. This leads to the following proposition: Proposition 3.2. In the fermion Fock space F^H.) there exist bounded operators {a(u), o){u), u € H] satisfying the following: (i) a(u)<J> = 0; a{u)v\ A . . . A vn = 2" = 1 (— ^Y~l{utvi)vi A ■ ■ ■ A vj A . . . vn where " over Vj means that Vj is omitted) (ii) at(«) is the adjoint ofa(u) anda^(u)$ = u, cft{u)v\f\.. -Avn = uAviA.. .Avn; (iii) a(u)a(v) + a(v)a(u) = 0, a{u)a){v) -\-o){v)a(u) = (u,v) for allu,v € H\ (iv) IfU is any unitary operator in H there exists a unique unitary operator T(U) on r a ( 7 i ) satisfying T(U)4> = $, r(C/)wi A . . . A vn = Uvi A Uv2 A ... A Uvn for all Vi € H, n = 1,2,
Furthermore F(U) satisfies the relation
r(?7)a(w)r(t/)- 1 = a(Uu) for all u € H. (v) F{U)F(V) = F(UV) for all U, V € U{H). The operators a(u), a*(u), F(U) in F^H) are respectively called the fermion anni hilation, creation and second quantization operator associated with u € H,U 6 U(Ti). Relations (iii) of Proposition 3.2 are called canonical anticommutation re lations (CAR). Exercise 3 . C . (i) For any unit vector u G H the fermion annihilation and creation operators a(u) and a!(u) satisfy I*. i r \
s
,K
v ;
u \\k*\ *• ; / '
f 0 if k is odd, ^ 1 if & is even.
In the vacuum state $ , the observable a(u)+aJ(u) has the symmetric Bernoulli distribution with probability | at each of the values ± 1 .
Quantum probability and strong quantum Markov processes 77
{$, (a(u) + a t (u))(a(w) + af(w))<E>) = (u, v) for u,v £ H. In particular, the spin observables a(ui) + af(ui), i = 1, 2 , . . . , n with ||MJ|| = 1 for every i have, in the state $, the correlation matrix (((UJ,«,-))). Thus any n-th order positive semidefinite matrix with complex entries and diagonal entries equal to unity can be realized as the correlation matrix of n spin observables. (See Exercise l.G, Exercise l.J.) Exercise 3.D. In the free and fermion Fock spaces over 7i there is no proper subspace invariant under all the creation and annihilation operators. (In the case of boson Fock space the same holds but the statement must be more carefully formulated since the creation and annihilation operators are not bounded.) Exercise 3.E. Let H be a bounded selfadjoint operator in H. and let Ut = eltH, t £ R. Then T(Ut), the boson second quantization of Ut, as a function of t is a one parameter unitary group. By Stone's theorem there exists a selfadjoint operator \{H) in Ts(7i) such that T(Ut) = e _ i t A ( H '. Consider the unit vector ip(v) = e - 1 / 2 W 2 e ( u ) , v 6 H. Then (iP(v), r(Ut)i>(v)) = exp
- l)(v,
PH(dx)v)
R H
where P is the spectral measure of H on the real line. As a function of t, it is the characteristic function of a mixed Poisson distribution. This can be interpreted as the distribution of the observable X(H) in the pure state ip(v). (See Section 2.1, [P]). Exercise 3.F. For any u g H, U 6 U(ri) there exists a unique unitary operator W(u,U) in TS(H) satisfying W(u, U)e(v) = e~1/2M2-(u'Uv)e(Uv
+ u) V v e H.
These operators satisfy the relations W(u, U)W(v, V) = W(u + Uv, uV)e-iIm{u-Uv).
(3.4)
The correspondence (u,U) —> W(u, U) is a strongly continuous projective unitary representation of the semidirect product of the additive group H and the multiplica tive unitary group U(H). This is called the Weyl representation of H(sjU(TC). When U = V = 1, (3.4) is called the Weyl commutation relations. Exercise 3.G. For any fixed u,{W(tu, l ) , t 6 R} is a strongly continuous one pa rameter unitary group. Write W(t,u) = e~ltp^u\t G R where p(u) is the selfadjoint Stone generator of this one parameter group. Write a(u) = |(— p(iu) +ip(u)). Then v®n is in the domain of a(u) and a(u)v0n
= y/n(u, vjv®^'1),
a(u)$ = 0.
Compare this with the definition of the boson annihilation operator associated with u.
78 K. R. Parthasarathy 4. F R O M URN MODEL T O CANONICAL COMMUTATION RELATIONS
Consider n distinguishable balls marked 1,2, ...,n and (d + 1) cells numbered 0,1, 2 , . . . , d. Place each ball at random in one of the (d+1) cells successively in the order 1, 2 , . . . , n. Then the resulting configuration may be identified with a partition E = (E0,Eu...,Ed)
(4.1)
of the set {1, 2 , . . . ,n} where Ea denotes the set of ball numbers in the a- th cell. Some of the Ea's can be empty. Thus there are (d+ l ) n possible configurations, each occuring with probability (d+ l ) ~ n . Denote by Hn the (d+ l) n -dimensional Hilbert space of all complex-valued random variables defined on the underlying probability space of such an experiment. Introduce the family {up(i),0 < a, (3 < d, 1 < t < n} of operators on Hn by putting (UaJt)f)(EQ,Elt...,Ed) f(Eo, 0 where E0, Ex,...,
Et,.-.,
Ea U {*},..., E0\{t},
...Ed)
if t e Ep, (4.2) otherwise
Ed are as in (4.1). When a = (5 we have K(t)f)(Eo,
£ ! , . . . , £ „ ) = lEa (t)f(E0,
Eu...,Ed).
These operators satisfy the following: v${ty = ui{t),
(4.3)
[ttf(s),«?(*)]=0if s ^ t ,
(4.4)
ua0(t)u](t) = %u}(t)
(4.5)
where * denotes adjoint and [-, ■] the commutator. Relation (4.5) implies the com mutation relation [uap(t),u>(t)]=6?u}(t)-SlvZ(t) (4.6) which yields a representation of the Lie algebra gl(d + 1) for every t 6 {1, 2 , . . . , n}. Operators arising from such representations with different t's commute. A partition E of the form (4.1) can be thought of as a pair of partitions, one of the set { 1 , 2 , . . . , t} and another of the set {t + 1 , . . . , n} into (d + 1) parts. Thus Tin can be viewed as a tensor product of the (d+ 1)'-dimensional Hilbert space Ht and a (d + 1)'"""^-dimensional Hilbert space Hf. Thus Tin = T-Lt® Til for each t and can be expressed as a tensor product of n copies of the (d+1) - dimensional Hilbert space H\. The operator Up(t) of (4.2) is of the form v°(t) = 1<8>1<8>...®1®1$®1<8>...®1 where u% on the right hand side is in the t-th position of the n-fold tensor product and w^,0 < a, (5 < d is a basis in the standard (d + l)-dimensional irreducible representation of gl(d + 1). We now introduce an "initial" Hilbert space i) and couple it with Tin by putting
Hn = f)®'Hn = f ) ® n t ® n \
\
Quantum probability and strong quantum Markov processes 79 In statistical terms this yields a 'filtration' of Hilbert spaces 7it = \]®7it with t being viewed as time and Ho = f). The space rj can be imagined as the space of random variables concerning a statistical system S. Balls 1,2,... are placed successively at random in one of the d + 1 cells. Random variables concerning the system S and the placement of the first t balls make the Hilbert space Ht. Now consider a family X = {X(t),0 < t < n} of operators in Hn satisfying the property X(t) = Xt ® ll where Xt is an operator in Ht and 1' is the identity opertor in Tit. We say that X is an adapted process. We now define a family of basic adapted processes in terms of the operators u%{t) in (4.2) as follows: U${Q) = 0; U$(t) = ^
lo ® «|(a), 1 < t < n,
(4.7)
l<s<(
lo denoting the identity operator in h. It is important to note that for any fixed 0 < s < t < n the operators {Ujj(t) - Up{s),Q < a,(3 < d + 1} constitute a representation of gl(d + 1). For any adapted process X we define its difference scheme by
(AX)(t) =
X(t)-X(t-l).
The relation (4.5) can now be expressed in terms of Up(t) and the operation A as (AU$)(t)(AUl)(t)
= *?(AE/?)(t)
(4.8)
(This looks like a toy ltd formula). From (4.7) and the subsequent remark it follows that for any adapted process X there exists a unique family of adapted processes £■1,0 < a, 0 < d satisfying (AX)(t)
= E%(t - l)(AE#)(f), t > 1
(4.9)
where the Einstein convention is adopted. Thus X(t) = X(0) + "£EI(s - l)(AU£)(a).
(4.10)
s=l
If random variables in classical probability theory are viewed as operators of multi plication by them in the L? of the underlying probability space of a statistical system and observables in quantum probability are looked upon as operators then Equations (4.8) - (4.10) appear as discrete time analogues of identities in the Wiener-Ito-Doob stochastic calculus. This suggests the possibility of developing a quantum stochastic calculus. To make such a passage to the case of continuous time we shall investigate the asymptotic behaviour of the basic processes {Up,Q < a,(5 < d} after an appro priate normalization as n —> oo. In order to strengthen further this analogy we solve a "simple stochastic difference equation": {AX)(t)
= Eg{t-l){AU2){t)X(t-l),
X(0) = X0®1°
(4.11)
where XQ is an operator in the initial space fj. This can be expressed as x(t)
=
(i + £ ; ( t - i K ( t ) ) i ( t - i )
=
(1 + E%{t - l K ( t ) ) ( l + E$(t - 2)ul(t - ! ) ) . . . ( ! + ££(0) U £(1))X(0).
80 K. R. Parthasarathy Thus we have solved (4.11). Since ^2av%(t) = 1 it follows that X{i) is unitary Vi if and only if XQ is unitary and ((Ep(t) + 5fj) is unitary in h ® Tit ® Cd+1. This can be imagined as the solution of a discrete time-dependent Schrodinger equation in the presence of noise. Suppose X, Y are adapted processes in 7in satisfying (AX)(t)
= E%{t - l ) t ^ ( t ) ,
(AK)(i) = Ff(t -
l)U2(t)
where Efi and Fp are adapted processes. Then by elementary algebra we have (AXY)(t)
= X(t - l)(AY)(t) + (AX)(t)Y(t
- 1) + E«{t - l ) i £ ( i - l ) ( A [ £ ) ( i ) .
This is a summation by parts formula where the third term is the analogue of Ito's correction in the toy set up. The basic processes {Up} are essentially operators in Tin and we wish to study their asymptotic behaviour. In order to study a sequence An of operators in changing Hilbert spaces 7in we take the following approach. Choose and fix a "parameter space" 0 . Imbed 6 in Hn by a map un : 0 —» Hn so that the range of un is total in Tin. Suppose there exists a Hilbert space Tioo and an imbedding u^ : 0 —> H.^ such that the following hold: (i) Range of Uoo is total in Ti^ (ii) (un(01)}un(82)) -*• (uoo(#i),«oo(02)) as n -> oo for all 6i,02 € 0 ; (iii) 3 an operator Aoo in W„o such that lim {un{6i),Anun(6?))
= {u^i),
AaDuoa{e2))
n—>oo
for all ^ , 0 2 £ 8 . Then we say that j4 n converges to A^ with respect to the chosen sequence of parameterizations {«„, n = 1,2,..., oo}. For the problem on hand we choose the parameter space 0 = (C[0, l])d. For any f = (fi, V?2, • • •, ¥><*),
( e „ M ) ( £ 0 , S i , ■ • •, Ed) = (d + l)"/ 2 J ] I ] ^ ( r / n j n - 1 ^ where E = (E0, E\,...,
(4.12)
£<j) is a partition of the form (4.1). By elementary algebra n
d
(en(v), «»M> = U.0- + n_I ^2(^i)(rM)r=l
( 4 - 13 )
i=l
This shows that the natural choice for TL^ is the boson Fock space r s (L 2 (0,1] <8>
9
6(C[0,f.
In order to analyse the asymptotic behaviour of Up we have to introduce an ap propriate normalization. This turns out to be given by A%(t,n):=l
( n^Ugdnt}) n-^Ugdnt}) [ Up([nt])
if a = j3 = 0 if a = 0,0 / 0 or a ^ 0,/? = 0 otherwise
Quantum probability and strong quantum Markov processes 81 for t G [0,1]. The appropriateness of this normalization is arrived at by the following heuristics. We imagine that cell number 0 is not available for investigation. It is a kind of black box. The operator v!a{t) defined by (4.2) symbolises the transfer of ball number t from cell a to cell 0. If a — 0 and 0 =£ 0 then we say that it is the creation of ball or particle number t in cell 0. If a ^ 0 and 0 = 0 it is a transfer to the black box and so we call it the annihilation of ball t from cell a. If a ^ 0,0 y£ 0 it is an exchange. The asymptotics is done in such a manner that all but a finite number of balls (particles) lie in cell 0. Observe that n^Ugdnt]) is the operator of multiplication by n _ 1 times the number of balls from { 1 , 2 , . . . , [nt]} lying in cell number 0. If the cells 1, 2 , . . . , d contain only a finite number of balls in the limit then A°(n, t) is expected to converge to tl for every t > 0. Indeed, with our parametrization it turns out, rather remarkably, that lim {en(
(4.14)
where (A^(t)} are the basic operators describing the noise system in the boson Fock space stochastic calculus. Indeed, A°(t) =
tl
Aj(i)
=
a(l[0,t] ® et)
A°(0
=
at(lM®ej)
where e i , e 2 , . . . , e^ is the standard orthonormal basis. The operators Aj(t), i,j
^ + 1 [M1 ® 1) =
fe e
1 <
£i,j°^W
for any hermitian matrix ((%))• The operators {Ap*(t)} obey the canonical commu tation relations (OCR) Aa0(s)A](t) - A](t)Aag(s) = 5?A}(s At) ~ 6}A°(s A t)
(4.15)
where 6% = 0 if a = 0 or 0 = 0 and 6% = Sp* otherwise. Exercise 4.A. Define S+ = { 0 , 1 , 2 , . . . , r } , 5_ = {r + l,...,d} parity matrix: ff„_[l 13
~ \ 0
and the associated
if a eS+,0eS_ otherwise.
Put Gn(t)
= ( ~ l ) s where E =
£
<(s),
aeS-,s
va0(t) = { G , , ( t - l ) } ^ ( t ) , t 6 { l , 2 V?(t) = X > £ ( s ) , £ £ { l , 2 , . . . , n } .
n},
82 K. R. Parthasarathy Now for t € [0,1] define
C n-^Unt})
if a = j3 = 0
E£(i;n)=| n ^ ^ V ^ n * ] ) 1. ^/f(Ml)
if a ^ 0,0 = 0 or a = 0,/J ^ 0 otherwise.
Then as in (4.14) lim <e„(v»),S|(t,n)e„M> = <e(V)> £?(i)eW0>
(4.16)
where {E^(t)} are operators in r s (L 2 [0,1] ® Cd) satisfying the (Eyre-Hudson) Lie super commutation relations: £?(a)EI(*) - {-iT^mmts)
= ^ E ] ( s A t ) - ( - i y * * 6 £ S ( s A t).
(4.17)
Compare this with the CCR (4.15), (see [EyH].) When S_ = { 1 , 2 , . . . , d} we get the canonical anticommutation relations (CAR). If {X(t)} is a family of selfadjoint operators commuting with each other in a Hilbert space K, then define the functions for any unit vector u G £ , f i, £ 2 , . . . , tn in the index set of the family {X(t)}. Then V"i,t2.-,tn ^s a continuous normalised positive definite function in W1. By Bochner's theorem it is the Fourier transform of an n-dimensional probability measure /x" t2... („ • For any fixed u these finite dimensional distributions are consistent and hence, by Kolmogorov's theorem, determine a stochastic process. We say that the family {X(t)} of observables executes this stochastic process in the pure state u. Now we define Q,{t) = (A40(i) + A?(t))~ (4.18) Nlit) = (Al(t) + vft(AJ(i) + A?(*)) + \t)~ for 0 < t < 1, 1 < i < d, A > 0, where ~ denotes closure on the domain which is the linear manifold generated by the exponential vectors. Call this domain £. Then [Qi(ti),Qj(t2)] = o, [JVJ(tl).^(*2)]=0, and each of the operators Qi(t),N\(t) is selfadjoint. In the vacuum state e(0), {Qi(i),Q2(i),... ,Qd{t)} executes a d-dimensional standard Brownian motion whereas {Nl(t),..., N£(t)} is a family of d independent Poisson processes of intensity A. Thus we have obtained a quantum probabilistic (functional) central limit theorem for the urn model. A quantum stochastic calculus for the basic processes {A^(i)} is developed in the lectures of R L Hudson.
Quantum probability and strong quantum Markov processes 83
II. Quantum Markov Processes 5. STOCHASTIC OPERATORS ON
C*-ALGEBRAS
7
Let (X, £), (Y, J ) be Borel spaces. Suppose P(x, F), x e X, F € T is a function o n l x j f satisfying the following: (1) P(x, •) is a probability measure on the cr-algebra T for each x in X; (2) P(-, F) is an if-measurable function for each F 6 T. Then P(-, •) is called a transition probability function. It describes the transition from a state x e X to a state y e F CY with probability P(£, F) in a Markov chain or a random walk. If A and B denote respectively the algebra of complex-valued bounded measurable functions on (X, £) and (Y, J-) respectively then P determines uniquely a backward transition probability operator or a stochastic operator T from B into A by (Tg)(x) = j
g(y)P(x, dy), g G B, x e A.
(5.1)
Then T has the following properties: (i) T is linear, T\ = 1; (ii) Tg > 0 whenever # > 0; (iii) If gn > 0, g > 0 and gn\ g pointwise then Tp„ j Tg pointwise; (iv) If ((gij)) is any d x d matrix with entries from B and ((ffy(j/))) > 0 in the sense of positive semidefiniteness for every y e V then (((Tgij)(x))) > 0 for every a; € X. Conversely, any map T : B —* A satisfying (i) - (iii) automatically satisfies (iv) and uniquely determines a transition probability function P satisfying (5.1). We now replace the commutative algebras A and B in the discussion above by not necessarily commutative C* - algebras A C B(H),B C B(fC) where H and K. are complex Hilbert spaces and for any Hilbert space H, B(H) denotes the C* - algebra of all bounded operators on H. We always assume that the identity operator belongs to A as well as B. Motivated by the properties (i) - (iv) above we introduce the following definition. Definition. A linear map T : B —» A is called a (backward) transition probability operator or simply a stochastic operator if the following hold: (i) T l = 1; (ii) If ((Yij)) is any finite square matrix with entries from B such that {(Yij)) considered as an operator on the direct sum )C © • • • © K is nonnegative then (.(T(Fy)) is a nonnegative operator on Tt ® ■ ■ ■ @ TL (the number of copies of TL and YZ in the direct sums being the order of the matrix). (iii) If Yn,Y 6 B are nonnegative and Yn | Y as n —> oo (i.e., (u,Ynu) T (u,Yu) for every u € K.) then T(Yn) | T{Y) as n -> oo. If T l < 1, T l 7^ 1 and T satisfies (ii) and (iii) then T is called a substochastic operator. Exercise 5.A. If TJ : B —> A, i = 1,2, are stochastic operators then for any 0 < p < 1, pT\ + (1 — p)T2 is also a stochastic operator. If A,B,C are three unital
84 K. R. Parthasarathy C* - algebras, 7\ : C —* B, T2 : B —> .A are stochastic operators then T2 o 7\ : C —> .4 is also a stochastic operator. Example 5.1. Let A = B = B(H), and let { L a } a € / be a family of elements in ,8(7^) satisfying ^ | | L Q u | | 2 = ||w||2forallueft Define r(A-) = ^ L * X L Q , X G 5(W). a
Then T defines a stochastic operator. T can be expressed as T(X) = C*(X ® 1)£ where £. : 7i -^ H
n
L XL
T{X) =J2 * i
= J^iuuXui)
i=l
\ei){ei\, X G B(W)
i=l
defines a stochastic operator. If X = £ ^ Xi|ej){ei[ then T ( X ) = $3™=12/i|ei)(ej| where j/i = S j = i P y x 3 a n d Py = \{ui>ej)\2- Here ((py)) is a stochastic matrix and (Ui,ej) can be interpreted as the transition amplitude for transition from the state i to the state j . Conversely, if ((Py)) is any nxn stochastic matrix one can choose Uj = J2k \f¥jkelVik&k where the angles tpjk are arbitrary. Then T as defined above and restricted to the al gebra of diagonal operators in the basis {e^} yields the classical transition probability operator induced by the stochastic operator ((Py))Example 5.2. In Example 5.1 let La be a projection for every a € I. Then T 2 = T. Denote by A the smallest von Neumann algebra generated by all the La,a G / . If A' denotes the commutant of A then T(X) G A' and T{AXB) = AT{X)B for all A, B G A'. If p G S{H) is a state such that p £ A' then TipT(X) = Tip(X). Thus T can be interpreted as a conditional expectation map from B(H) into .A' (given A') in the state p. Exercise 5.C. Let B = B{Hi) ® B{H2) = B(Hi ® H2) where Hi,W 2 are Hilbert spaces. Let A = B(Hi) ® 1. For any state p on T^j i-e-i a nonnegative operator on H2 of unit trace define T°(X) G B{Hi) through the relation {u,T°{X)v) for all u,v G 7itX
=
^Vi{u®uuXv®Vi) u
G S, where p = £)fp«|Uj)(' il >s t n e spectral resolution of p. Put Tp(X) =
T°p(X)®ln2.
Then Tp : B —> A is a stochastic operator, T 2 = T p and T ^ A X B ) = ATP(X)B
for all , A, B G A X G B.
Quantum probability and strong quantum Markov processes 85 If pa is a density matrix in TL\ (i.e., an element of S(Hi))
then
0
Tr(p 0 ® p)X = Trp0Tp (X) = Tr(p 0 ® p)Tp(X). Thus Tp is a conditional expectation map from B onto A in the state p0
T(X) = / u9gxu:d9 G
where dg indicates the normalised Haar measure on G. Then T is a stochastic operator from B(H) to Aa satisfying T2 = T, T{AXB) = A T ( X ) B , X € 23(70- If p £ A n « S ( H ) then TrpX = TrpT(X). In other words T is a conditional expectation given Aa in the state p. Exercise 5.E. Let dim ft < oo, A C 23(7-0, a * subalgebra, p £ A'nS(H). Suppose p is A - faithful, i.e., for any X £ A, T r p X ' X = 0 implies X = 0. Then for any X 6 23(70 m i ze.4 T r p ( ^ — Z)*(X — Z) is attained at a unique element denoted T(X) e A. The map X -> T(X) satisfies the following: (i) T is stochastic and T 2 = T; (ii) T(.AXB) = AT{X)B for all i . B e A X e 23(70; (hi) TrpX = T r p r ( X ) for all X e B{H). Remark. Let Ji = L 2 (K), 2? = 23(70, A = the abelian von Neumann algebra gener ated by the unitary operators {Ut} given by
(Utf)(x) = f(x + t),teR. Suppose T : B —> .A is a stochastic map satisfying T{AXB)
= AT(X)B
for all A, Be
A, X 6 23.
For any bounded measurable function ip denote by the same symbol
Let A C 23(7i) be a unital G*-algebra and let T : A —> 23(/C) be a stochastic operator. Denote S = {{X,u)\X e A, uEK}
86 K. R. Parthasaruthy and define Ky on S x S by KT((X,u),(Y,v))
=
(u,T(X*Y)v).
Proposition 6.1. K? is a positive definite kernel i.e., Y^CiCjKTdX^Ui),
(Xitv,j))
>0
(6.1)
for any finite sequence {c*} of scalars and elements (Xi,Ui) in S. Proof. By definition the left hand side of (6.1) can be expressed as
Y.WiKTWXjfrj)
=
{®ciuu{{T{X*Xj)))®cjuj).
Since {(X*Xj)) is a positive operator and T is stochastic it follows that is a positive operator and hence (6.1) holds.
((T(X*Xj))) □
It follows from the GNS principle (see Proposition 15.4 in [P]) that there exists a Hilbert space K and a mapping A : S —> K such that (i) KT((X,u),(Y,v)) = (X(X,u),X(Y,v)); (ii) {X(X,u), (X,u) e S} spans K\ (iii) The pair (/C, A) satisfying (i) and (ii) is unique modulo a unitary isomorphism. Proposition 6.2. For any A e A there exists a unique operator jr(A) on K. such that (i)jT{A)X{X,u) = KAX,u); (ii) \\jT(A)\\ < ||A||; (iii) The correspondence A —> jr{A) is a unital *-homomorphism from A into B(IC); (iv) / / An, A 6 A are nonnegative and An ] A as n —► oo then J T ( A 0 | jr(A) in K. Proof. Observe that A(X, u) is linear in each of the variables X and u. Furthermore || 5 2 A(AX i l U i )|| 2 = ^ ( U i . T ^ A ' / U C j h - ) .
(6.2)
Since A M < ||A|| 2 i, ||A|| 2 - A*A = B*B for some B in A and it follows that {{X^B'BXj)) > 0 and therefore ({XTA'AXj))
< ||A|| 2 ((T(X*X,))).
By the stochasticity of T we conclude that ((TWA'AXj))
< ||A|| 2 ((T(X*X,))).
Now (6.2) implies ||^A(AXi)Wi)H2
=
((BuiAinXlA'AXj)))®^)
i
< WAlfiQuiMnxtXi)))®*)
= wAw^faTwx^) = pini ][>(*, tor.
Quantum probability and strong quantum Markov processes 87 This proves the existence of the operator jr(A) in K, satisfying (i) and (ii) because vectors of the form \(X, u) span IC. Property (iii) is immediate. To prove (iv) consider nonnegative elements An, A in A satisfying 'An f A as n —> oo'. Let ip = ^2i \{X{, Uj), a finite sum. Then by definition
WJAAnW) = ^ ( t i i . r w ^ w = (em, ((rfAjAj,-))) e «,) Since ((J^MnXj)) is an increasing sequence of operators and T is stochastic it fol lows that ((T(X?AnXj))) is also increasing. Thus j r ( A i ) is increasing in n. Since X*AnX increases to X* AX it follows that T{X*AnX) increases to T(X'AX) as n —> oo. By polarisation T(X*A n Y) converges weakly to T(X*AY) as n —> oo. Hence lim n _ t0O (^,; r (A n )V') = (4>,jT(A)ip). Hence j:r(Ai) T JT(A) as n -» oo. D Proposition 6.3. T7*e map V : u —* A(l,u) /rom /C into /C is an isometry and T(X) = V*jT(X)V
for all X € A.
Proof. From Proposition 6.2 we have for any u, v € IC, X € A (u}V*jT(X)Vv)
= {Vu,jT(X)Vv)
=
{u,T(X)v}.
□ Definition. Let yl be a unital C*-algebra of operators on H. A map j : A —* B(IC) is called a representation (of .4 in K.) if the following hold: (i) j is a unital *-homomorphism; (ii) for any An,A £ A, An > 0, ^4 > 0 such that An | A as n —> oo, J ( J 4 „ ) t j'(-A) as n —> co.
Now we summarise the results of this section so far as a theorem. Theorem 6.4 (Stinespring [Sti]). Let T be a stochastic operator from the unital C*algebra A into B(JOj. Then there exists a triple (IC,jr,V) consisting of a Hilbert space IC, a representation jx of A in K, and an isometry V : K, —> fC satisfying the following: (i) T{X) = VjT(X)V for all X e A, (ii) {JT(X)VU,
X €: A,U € IC} is total in IC.
V (fc'>JT' V) *s another such triple satisfying the properties (i) and (ii) above then there exists a unitary isomorphism T : IC —> K! such that V = TV and TjT(X) = j'T(X)T for all X. Proof. Only the last part requires a proof. This is immediate from the uniqueness of the GNS construction modulo a unitary isomorphism. □ The triple (IC, jr, V) in Theorem 6.4 is called a Stinespring dilationoi the stochastic operator T. K, may be expressed as K, @ k and jr{X) as a matrix 3T{X)
T{X) S(X)
6i(X) S(X)
X eA
(6.3)
where T(X) : IC -> IC, 8{X) : K, -> k, 8\X) : k -> K, S{X) : k -> k are bounded operators. Note that S is a stochastic operator from A into B(k). The sets {T(X)u®
88
K. R- Parthasarathy
5(X)u,X G A,u G K,}, {6(X)ur X € A,u £ IC} are respectively total in K © k and k. The fact that jx is a representation also implies <5+(X) = 5(X*)*, ) T(XY)-T(X)T(Y) = tf(X)6(Y), I S{XY)-S(X)S(Y)=6{X)tf(Y), ( 5{XY) = 6{X)T{Y) + S(X)8{Y). J
, l
, °^j
Clearly, a representation of A in K, is a stochastic operator but the converse is not true. The map 5 in (6.3) measures the deviation of T from being a representation. T h e last equation suggests that 6 may be called an (5, T) - derivation. It is fruitful t o compare (6.4) with the structure maps that appear in Evans-Hudson flows. (See Section 28 in [P].) If T(X) S'^X) XeA MX) S'(X) S'(X) is another representation in K, @ W with the properties described above then there ex ists a unitary isomorphism U : k ~> k' such that 6'(X) = US(X), S'(X) = US(X)U~l for all X eA. E x a m p l e 6.5. Let X,y be compact metric spaces with their associated Borel aalgebras £, T respectively and let P(x, F), x G X, F G T be a transition probability measure such that the correspondence x —>■ P(x,-) is weakly continuous. Put A = C{X),B = C{y) and
(TiP)(x) = JiP(y)P(x,dy). Let n be a probability measure on X with positive mass for every open set. Define t h e probability measure u> on X x y by OJ(E x F)=
fj.(dx)P(x,F) JE
and write u{F)=w(X xF). Assume that v has positive mass for every open set in y. Define the Hilbert spaces Ti = L » , £ = L2{n),)C = L 2 H and note that Tt and K, can be considered as subspaces of /C. View A and E as C*algebras of multiplication operators in K, and 7i respectively. Then T is a stochastic operator from B to A- For ip G B define the operator jr{ip) in K, by (JT(ip)f){x,y)
=
f(x,y)tp(y).
Denote by V the inclusion isometry from /C into K,. Then V*jr{ip)V is multiplication by Tip in /C and {K,]T, V) is a Stinespring dilation of the stochastic operator T. If /C = /C © k where k = K.1- in /C then in the matrix decomposition picture (6.3)
(T{f)f)(x) = T^)(i)/(i),^eB,/e/c, (SWf)(x,v) = / W W » ) - W ) ( i ) ) , ^ 6 , / € K .
Quantum probability and strong quantum Markov processes 89 Note that S(ip)f is m k. Finally, {S(1>)g){x, y) = g(x, y)V(y) - Jg{x,
y')iP(y')P(x,
dy').
E x a m p l e 6.6. Let A = B(TC) and let T : A —> B(K) be a stochastic operator. Consider a Stinespring dilation (K.,JT,V) for T. Since jr is a representation of B(fi) it follows from a well-known theorem that K, can be chosen to be H ® h = 0 a e / 7i Q , 7i a = H,I is an index set of cardinality equal to dim h and jT(X)
= U'{X ® 1)C7 = [T 0
XaU a
with X Q = X for every a, U being a unitary operator in Ti ® h. The isometry V : AC —> AC has the property
rp , (i®i)w = r(i),ieB(w).
(6.5)
Put W = UV. Then we may as well take
ael
and consider (H C§> h, j r , W) as an equivalent Stinespring dilation. Express
W = 0LQ a
where La : K. —> H is an operator for each a G i" and X^-^a-ka becomes
=
^- Then (6.4)
T(X) = Y,L*aXLa. ael
This Stinespring dilation has the property that {(X ig> 1)W^M, U G /C,X G S(7i)} is total in the direct sum © a 7 ^ Q , that is
{^XLau,ueK.,X
eB(7i)}
a
is total in 0 a W a . We claim that this is possible if and only if the family {La} satisfies the following property of 'tempered linear independence': for any family {ca} of scalars satisfying £^Q |c a | 2 < oo and Ylac<*La = 0 it is true that ca = 0 for all a. To prove necessity consider the relation for any fixed v G -Ti: (@acav, ®aXLav)
= (v, X 2^ caLau) = 0 V X
which implies (Bacav = 0 or ca = 0 for every a. Conversely, let (®ava, ®aXLau) = 0 V X 6 B(H), u 6 K. and ®ava £ 0 a fixed element. Choose X = \vao)(v\ where v G H. Then we get (v, 2_.(va, vao)Lau) a
= 0 for v G H,u e K.
Htt is
90
K. R. Parthasarathy
If we write ca = (va,vao) then E Q l c c«| 2 < I K o l P E J K I I 2 < oo. Thus Eac<*L<* = 0 a n d hence cQ0 = ||uO0||2 = 0 or vao = 0. Since CXQ is arbitrary it follows that ®ava = 0, proving the claim. If K, is separable it follows from the identity £ a L*aLa = / that at most a countable number of the La's are nonzero operators, or equivalently, dim h is finite or countably infinite. If dim Tt = m, dim K, = n are finite then the (tempered) linear independence property of the La's implies that dim h < mn, or equivalently, the number of La's is a t most mn in the Stinespring dilation. Now, a final remark. Suppose
T(X) = J2 LlXL« = E a
M XM
0
?
fi
for two sets {La},{Mp} of operators from K, —» Tt obeying the tempered linear independence property. Then there exists a unitary matrix ((7^)) with scalar entries such that Ma = 2_\ lapLp for every a. & This follows from the uniqueness part of Stinespring's theorem and the fact that a unitary operator Y :'H®\\ —»7Y(g>h commuting with all operators of the form X <8> 1 is of the form 1 ® ((7a/3)) where {(jap)) is a unitary matrix in an orthonormal basis { e a } for h. We express the discussion in the above example in the form of a theorem. T h e o r e m 6.7. Let T be a stochastic operator from B(H) into B(fC). Then there exists an index set I and a family {La, a £ / } of operators from K, into Ji satisfying the following: (ii) T(X) = £ t t L*aXLa for all X e B{H); (iii) If {ca, a € A} is a family of scalars such that ]T Q \ca | 2 < 00 and J2a caLa = 0 then ca = 0 for all a; (iv) If {Ma, a € A} is another family of operators from K, into Tt satisfying (i) - (iii) with La replaced by Ma for every a then there exists a unitary matrix ((lap)), oc,P S I such that Ma = Y^ IcpLp for every a. All the sums are to be understood in the weak operator topology. Remark. Theorem 6.7 can also be expressed in a coordinate free form. If T is as in Theorem 6.8 there exists an isometry W : /C —> Tt ® h for some Hilbert space h such t h a t the following hold:
(i) T(X) =
W(X®l)W;
(ii) If c £ h is an element for which the operator Wc : K. —> Tt defined by the relation («, Wcv) = (u ® c, Wv) for all u e Tt, v G £ ,
Quantum probability and strong quantum Markov processes 91 vanishes then c = 0; If W : K —> H. ® h' is another isometry satisfying (i) and (ii) with h, W replaced by h', W there exists a unitary operator U : h —* W such that W = {1H ® U)W. Note that condition (ii) is equivalent to the statement: W'1-H ® \c)(c\ = 0 only if c = 0. Example 6.8. Let h, k be Hilbert spaces and let C : h —+ k be a contraction operator. Denote by ko the range of the operator (1 — CC*)5 and define the isometry V : k —> hffik0 by Then there exists a unique isometry T(V) : F s (k) —> r s (hffik 0 ) satisfying T(V)e(ip) = e(Vtp) for all
Tc(x) = r(V)*(x®i)r(v) where 1 is the identity operator in F s (k 0 ) and F s (hffiko) is identified with F s (h) ® r s (k 0 ). Then Tc is a stochastic operator from B(Vs(h)) into £?(Fs(k)). Elementary computation shows that {e{
- CCT)1>).
When X = W(f) is the Weyl operator associated with / e h in F s (h) this yields the identity Tc(W(f)) = W(Cf)e^c^'~^^ for all / e h . (6.6) If C/ is a unitary operator in h and F((7) is its second quantization in F s (h) then by simple computation we have (e(
CC'W))
or TC{T(U)) = F(C(7C* + 1 - CC*)) (6.7) where the right hand side is the second quantization of a contraction. Furthermore, for X e S(r s (h)) (X ® l)r(V)e(y>) = { X e ( C V ) } ® e((l - C C * ) M and hence {{X
92 K. R. Parthasarathy (iii) There exists a unique stochastic operator Tc : S(F a (h)) —> S(F a (k)) satisfying Tc{a\h)
■ ■ ■ a\fk)a(9i)
■ ■ •
■ ■ ■ a\Cfk)a{C9l)
■ ■ ■ a(Cge)
for all / i , - - - > A , S i . - " ,ft € h (iv) There exists a representation j of S(r a (h)) in F a (h © k0) satisfying jV(/)) = at(/©0)forall/€/i; (v) The triple (T a (h © ko), j,T(V)) erator Tc-
is a Stinespring dilation of the stochastic op
Exercise 6 . B . There is a natural analogue of the previous example and exercise in the case of free Fock space with the free creation operators. 7. E X T R E M E POINTS O F T H E CONVEX SET O F STOCHASTIC OPERATORS
Let A C B(H) be a unital C-algebra and let S(A, K.) denote the convex set of all stochastic operators from A into B(1C), Ti,K, being fixed complex Hilbert spaces. We shall examine some properties of the extremal elements of this convex set. Proposition 7.1. Let T,TUT2 6 S{A,K.),0 < p < 1, T = p7\ + (1 - p)T2 and T =^ Ti. Suppose (/C © k, j T , i) is a Stinespring dilation of T so that i : /C —* K, © k is the inclusion embedding and 3T(X)
=
T{X) 6(X)
Then there exists an element p € JT{A)' the following: (i) p>0, p=
" 1 rt* a
p
tf{X) S(X)
, X e A.
(the commutant of the range of jr) satisfying
, a^O;
(ii) TX(X) = T(X) + a*5{X),
X GA
Proof. Let K.\ = K, @ k%, j T l be a Stinespring dilation of T\ in K,\ with 3TAX)
=
Ti{X) 5{(X) 5i{X) SX(X)
,XeA.
Write \i{X, u) = Tx{X)u © 5 i ( A > , A(X, u) = T(X)u © <5(X)w for X € A, u € XT. For any finite number of pairs (Xi, Ui) € A x K.,i = 1,2, have from the stochasticity of T2 = p Tlf1, || ^
Ax^u,) ||2 =
^(ui.T^X^K-)
< p" 1 Ys^nX'X^)
= p-11| ^ X ( X u U i )
Quantum probability and strong quantum Markov processes 93 Since the sets {Xi(X>u)}, {\{X,u)} are total in K. ® ki and K, © k respectively it follows that there exists an operator J:/C©k—> IC(Ski satisfying JX(X,u)
=
\i(X,u),
(X,u)eAxJC,
\\J\\ < P-*. Put p = J*J. Then p commutes with every jr(X).
Let
7 a a /? be the matrix decomposition of 5 in /C © k. Identifying the entry in the position 11 of pjT(X) we get the relation I \ ( X ) = 7 T ( X ) + a*<S(X). Since T ^ l ) = T 2 (l) = 1 and 5(1) = 0 we have 7 = 1 . Since T\ / T it follows that a cannot be 0. □ 5 =
Proposition 7.2. An element T G iS(.4, K) is extremal if and only if every hermitian element £ in JT(A)' of the form a*
£=
(7.1)
P
is 0.
i-Voo/. Sufficiency: suppose T is not extremal. Then T = pTi + (1 — p)T 2 , 0 < p < l,Ti ^ T2,Tl,Ti € 5(^4,/C). By Proposition 7.1 there exists p G JT{A)',P> 0, such that " 1 a* a^O. P=
a 0
Then l-p = is an element of
0 —a —a 1 - / ?
7^0
JT{A)'.
Necessity: suppose T is extremal and £ G JT(A)' is a hermitian element of the form (7.1). There exists a constant c > 0 such that 1 ± c£ are nonnegative elements in JT{A)'■ Then the matrix operators
(l±c£)
T{X)
5\X)
5(X)
S(X)
t 1 ± c ^ 2 [ <5(X) 5(X)
(lief)5
have the entry T(X) ± ca*8(X) in the (1,1) position. If we define T^X) ca*5(X),T2{X) = T{X) - ca*5(X) then TUT2 are stochastic and T = Since {
= T(X) + i ( 7 \ + T 2 ). ^ 0. Now 0. The set □
Corollary 7.3. Any representation of A in B(IC) is an extremal stochastic operator. Proof. Immediate.
□
Example 7.4. Let A = B(H) and let T : A -+ B(tC) be a stochastic operator. Then the Stinespring triple (K.,JT,V) has the following form :
94 K. R. Parthasarathy (1) K = H®£2(I) for some index set / and JT(X) = U*(X®1)U for some unitary operator U in /C; (2) V : K -> K is an isometry satisfying VU*(X ®l)UV = T(X)\ (3) If W = J/V then the only vector c € £2(I) satisfying (u ® c, Wv) = 0 for all u € ft, v 6 K. is the null vector. From property (1) it follows that any element of jr(A)' has the form U*(\-n ® A)U. Now Proposition 7.2 implies that T is an extremal stochastic operator if and only if the only element A 6 B(£2(I)) satisfying the relation W*(lH ® A)W = 0 is the null operator. Thus we conclude the following : T 6 <S(.A,/C) is extremal if and only if there exists an isometry W : K —> ft ® I2 (I) for some index set / satisfying the following: (a) T(X) = W{X ® \)W for all X e A; (b) W*(1H
(") XXA^ = 1;
(iii) The operators L*Lj, 1
+ D*2XD2, X e 13(H)
where " V P l O O O l
1
o v^ ° ° o o VP ° O O O ^ p J
[ v ^
n
'
=
0
0
0"
e
° yfi " °e i8 ° ° ° \/9 " ° ' [ 0
0
0
y/q
0 < Pi < 1, 0 < p < 1, qi = 1-pi, g = l - p , P i T^P- For any 9 G (0, 27r)\{7r}, T is an extremal stochastic operator on B(C 4 ). Note that T is not of the form X —> U'XU for some unitary operator U. In other words extremal stochastic operators need not be representations. 8. STINESPRING'S THEOREM IN T W O STEPS
Let Ai C B(ICi) be unital C*-algebras, i = 0,1,2, and let S, T be stochastic maps: A2^> A\-^> AQ. Define B((*2lXi,u), (y2)y1,i;))= (u,S(X{T(X^Yi)Y{)v) where X 2 , Y"2 6 A2, Xr, Yx e Au u, v € /Co. P r o p o s i t i o n 8 . 1 . B is a positive definite kernel.
Quantum probability and strong quantum Markov processes
Proof. Consider (X^,X^,Ui),i stochastieity of T and S
95
varying in a finite set {1,2,...}. We have from the
{((x®yxP)) > o, ((T((x®yxP))) > o,
{({xPrmxpyx^xP)) > o, ((saxPymxPyx^xP))
> o.
Thus K © uj © • • • , {{SixP'TlxP'X^X®)))*!
© «a ©•■•)> 0. D
Corollary 8.2. There exists a Hilbert space H and a map X : A2 x A\ x /C0 -» H satisfying the following: (1) {\{X2,Xl,u)\X2 e A2,Xi eAuue K,0} span H; (2) u —> A(l, l,u) is an isometry from /C0 into H;
(3) {\(x2,xuu),
A(y2>y1,«)> = {u,s(x1*T(X|y2)y1)«>;
(4) 7/"Hi is £/ie subspace generated by {X(l,X,u), projection F\ : K —* Hi is given by
X € Ai, u € /Co} i/ien the
F1A(X2,X1]W) = A(l,T(X2)X1,w); (5) 7/Fo : 7i ~^7io is the projection on the subspace HQ spanned by {A(l, 1, u), u e /C0} i/ien F0X(X2,Xuu)
= A(l,l,5(T(X 2 )X 1 )u).
Proo/. (1) and (3) follow from GNS. (2) follows from (3). To prove (4) observe that (X{X2,Xuu)
- X{l,T(X2)Xltu), X{l,Yuv)) = (u,S(XiT(XZ)Yi)v) - (u,S((T(X2)X1yY1)v)
= 0.
To prove (5) note that (X(X2,Xuu), A(l,l,«)> = (u,S(X*1T(XZ))v); (X(l,l,S(T(X2)X})u), \{l,l,v)) = (S(T{X2)X1)u,v). D Proposition 8.3. There exists a unique operator j2(X) in H. for any X & A2 such that (i) j2(X)X(X2,Xuu) = X(XX2,Xl,u), XeA2; (ii) j2 is a representation of A2 in Ji; (iii) ||j 2 (x)|| < ||x||.
96
K. R. Parthasarathy
Proof. We have as in the proof of Stinespring's theorem (for one step) | | ^ A ( X X « , X^\u)\\2
=
"£{Ui,S(X?"T(X^X*XX^)X^)u3)
i
i
< IIXIHI^A^',^,^!! 2 . i
□ Proposition 8.4. There exists a unique operator j°(X)
in Tii,X
S A\ satisfying
fi{X)X(l,Xuu)=\{l,XX1,u). j° is a representation of Ai in Hi, where TLi is as in (4) of Corollary 8.2. Proof This is nothing but the Stinespring dilation for the stochastic map 5 : A\ —> B(Ko). □ Proposition 8.5. (1) j2(X2)j°(Xi)u = X(X2,Xi,u) where u and A(l, l,u) are identified; (2) If ji(Xi) = Ji(Xi)Fi then ji is a *-homomorphism from A\ into B(H) with J'i(l) = Fv, (3a) Flj2{X2)F1 = ji(T{X2)) for X2 € A2; (3b) FohiXz^XJFo = S(T(X2)X1) for X{ 6 A, i = 1,2; (3c) FoJ 2 (X 2 )F 0 = S o T(X 2 ) /or X 2 e A2. Proof. Straightforward.
D
Proposition 8.6. Let Z C A\ denote the centre A\ C\AX of Z\. Then there exists a representation kx of A\ in B(H) satisfying kl{Z)X{X2,X1,u)
=
\(X2,ZXuu).
Proof. ||^A(X«IZX«,ui)||2
=
^(Ui,5(X«*Z*T(X«*^))Z^Vi)
= ^(m, 5(z*z(xf)*r(x«,^))xp')))«i) < ||Z|p Y.^Sixfnxf
X^X^u,)
i
(Here we have used the operator inequality {(Z'ZXij))
< WZW'dXij)) for ((Xi3)) > 0, Z £ Zlt Xi} € A,). D
Quantum probability and strong quantum Markov processes 97 R e m a r k . The Hilbert space H in Corollary 8.2 admits the decomposition Ji = /C0 © ki © k2 so that Hi = /C0 © ki, and the maps jf.j'i and j2 of Propositions 8.3 8.5 admit the following matrix decompositions: S(X) 5\(X)
Ax) =
S(X) 5\(X) 0
h{X) =
St(X) S^X) Si(X) SX(X) 0
J!(T(X))
MX)
=
5\(X)
inWj = / C 0 © k i , 0 0 0
52{X)
T2(X)
l e i
in ft = /Co © kx © k2, X E Ai.
in Hj © k2, l £
A2
so that X —> j2(X) is the Stinespring dilation of j® o T : A2 —> B(Hi) and jf is the Stinespring dilation for S : A\ —> B{K,0). The two-step Stinespring dilation now paves the way for constructing the Stine spring dilation for a chain of stochastic maps obeying the Chapman-Kolmogorov equations. We take this up in the next section. 9. C O N S T R U C T I O N O F A QUANTUM M A R K O V PROCESS
Let R = [0, oo) or ! Z+ = {0,1,2,...}, let At, t € T be a unital C*-algebra of operators in the Hilbert space K,t, and let T(s, t) : At —» As, s < t be a stochastic operator such that the Chapman-Kolmogorov equations hold: T(r, s)T(s, t) = T(r, t)Vr<s
• • • Yn)v)
eV.
Now consider arbitrary (s,X,u), ( t , Y , u ) € V. Define r = s U t ordered as a decreasing sequence and consider (r, X, u), (r, Y, v) 6 V by putting Xi
=
Xk 1
if Tj = Sk for some k, otherwise
Yi
=
Yk if tj = tk for some k, 1 otherwise
98 K. R. Parthasarathy Now define
___ L((s, X , u ) , (t, Y, v)) = L ( ( r , X , u ) , (r, Y, v)) We call L the c/iain kernel associated with T = {T(s, t)}. Important remark : If (s, X, u), (t, Y, v) eT>, s» = tj for some i, j and Xi = Yj = 1 then L ( ( s i , . . . , s „ , X 1 , . . . , X m , u ) , ( i i , . . . ,tn,Yi,... ,Yn,v)) may be written L{(si,...
,Si,,...
,sm, X\,...
,Xi,...
,Xm,u),
( t i , . . . ,tj,...
,tn,Yi,...,Yj,...
,Yn,vj)
where^means omission. Proposition 9.1. The chain kernel L associated with T is positive definite. Proof. Consider ( t « , X^', ut) e V, i = 1,2,..., N. Without loss of generality we may assume t ^ to be independent of i, thanks to the remark above. Let t ^ = t = {*i > t2 > ■ ■ ■ > in}, X « = {X\, X'2,..., Xln}, X) e Atj ■ Then for scalars cu c2,..., cn, ^ c t c , L ( ( t , X « , U i ) , (t,XW,u;)) = ^ g ^ u ^ O , *)(*** ■■■{T(t3,t2)(x£{T(t2,t1)(Xi'X{)}Xi}---Xi)uj}.
(9.1)
Write 7(1)
^ij
_
—
vi'
A
l
A
vi
l.
4 + 1 ) = r(o,tn)(4n)). ( ( 4 )) is a positive operator and the stochasticity of T{t2,t\) implies that ([Z\j)) is positive. Inductively, we conclude that ( ( • £ , • " ) ) is positive. Express the right hand side of (9.1) as {(Bam, ( ( 4 " + 1 ) ) ) ® Wi) > 0. D Now consider the Gelfand pair (H,\), A : T> —> H satisfying (1) (A(s,X > u ), X(t,Y,v)) = L((S,X,u), (t,Y,v)); (2) {A(s,X,u)} spans H. Proposition 9.2. A(t, X, u) is linear in each of the variables Xt € Ati and u G ICQ. IfXi = l then A(ii, t2,... ,tn, X\,...,
Xn,u) = \{t\,...
Proof. Straightforward from GNS .
,U,... ,tn, Xi,...,
Xit...
,Xn,u) □
Quantum probability and strong quantum Markov processes
99
Definition. Denote by Ht the closed linear span of {A(r, X, u), (r, X, u) £ V, r^ < t} and Ft the projection in Ji onto the subspace Ht- Then Ft t 1 as 1f oo. The family {Ft, t € T} of projections is called the filtration associated with the Gelfand pair (H,\). Proposition 9.3. For s = {s x > s 2 > • ■ • > sn} FtX(s,X,u)
A(s,X, u) = I X(t,s^si+i,
ift>Si ...,s„,X?,Xi,Xi+1,...,Xn,u) if si > s2 > • ■ • > sf_i > t > Si > ■ ■ ■ > sn
where X° = T(t, ai _!)(- • • {T(s3, s2)({T(s2,
s^X^X,)}
■ ■ ■ XM).
Proof. If t > si, A(s, X, u) € H t . If si > ■ ■ ■ > Si-i > t > Si > ■ ■ ■ > sn consider any element of the form A(r, Y, v) with r\ = t. Assume, without loss of generality, that r = (t,Si,si+1,...,sn). Then
t)({T(t,
ai_0(.
• • {T(s3, s2)(X;{T(s2,
Sl)(^)})}
• ■ • )}Yi)}Y2) • • • )«>
=(A(r, X ' . u ) , A(r,Y,«)> where X'1=T(t,Si-1)(---T(s3,s2)({T(s2,s1)(X1)}X2)---Xi„1),
X'2 = Xiy ..., X'n_i+2 = Xn
a P r o p o s i t i o n 9.4. There exists a unique representation j° : At —+ B(Ht) satisfying j?(X)\((t,
si,...,
sn), (Y0, Yu..., Yn),u) = A((i, slt...,
s^, (XY0, Ylt..., Yn), u).
Proof. Consider a vector of the form V = X ) A(*>fii>■ ■ •' a»> Yo\y^\
•••■^ U )
(9.2)
i
where t > Si > ■ ■ ■ > sn, Y0 G A, ^r e A r , r = 1. 2 , ••., n,ut € IC0. Such vectors are dense in 7it. If jf is denned as in the proposition we have \\j°(XW = £ > , T ( 0 , *„)(• • ■ Y®{T(Sll t)(Y®'X'XY®)}Y® ■ ■ • )u3). Note that ((Y^"X*XY^)) < \\X\\\(Y^'Y^)). Thus
wft(x)n2 < \\xr \w-
This shows that j°{X) extends to an operator of norm not exceeding \\X\\ in TitClearly, X —> jf(X) is a representation of At□
100 K. R. Parthasarathy Corollary 9.5. If ip is of the form (9.2) then
ty,j?(X)1,) = (<Bui, ((4?)))©"*> where Z$ = T(0, sn)(Y®'
■ • • T(s2, sl)(Yii>'{T(s1,t)(Yii)'X
Y®)}Y®)
■ ■ ■ Y&)
Proposition 9.6. Define jt ■ At —> B(H) by Jt{X)=ft(X)Ft
where Ft is the filtration projection at time t. Then j t is a *-homomorphism from At into B(7i). In particular, jt(l) — Ft. Proof. We have to only check that j°(X)Ftj°(Y)Ft on vectors of the form X(t, Si,..., s„, Ao, AY,..., sn, Ai E AH,u eK.0. Proposition 9.7. For s < t, Fsjt(X)Fs
= j°(XY)Ft. This is easily tested An, u) where t = SQ > S\ > ■ ■ ■ > □
= js(T(s,t)(X)),
X G At.
Proof. Consider s0 = s > Si > ■ ■ ■ > sn, Yt G ASi,Y[ s
,sn,l,Y0,
■■■ ,Yn,u),
= (X(s,s1,...,sn,Y0,Y1,...,Yn,u),
G /C0. Then for
si,..., sn, Y0',..., Y„, u'))
X(t,s,sir--
={«, T(0, sn)(Y: ■ ■ - (Y*{T(Sl,s)(Y*{T(s,
G ASi,u,u'
, sn, X,Y^
t)(X)}Y')}Y{)
■ ■ • ,Y^,u'))
■ ■ ■ Y>')
j.(T(s,t)(X))Ms,Su...,8nX>---,r>'))-
Thus we have proved the result for s < t. For s = t it is obvious.
□
We now summarise. T h e o r e m 9.8 ([BP 1],[BP2]). Let {T(s,t)} be a family of stochastic operators obey ing the Chapman-Kolmogorov equations for the family {At} of unital C*-algebras of operators in JCt, t G T. Then there exists a triple (H, {Ft}, {jt}) satisfying the following: (i) Ft is a projection in Ji and Ft { 1 as t f oo; (ii) j t ■ At —> B(Ti) is a *-homomorphism and j«(l) = Ft; (iii) Fsjt(X)Fs = j,{T(s,t)(X)) fors
=
X'(t,X,u)
=
JtAXi)---3tn(Xn)ui j'tl(X1)---Ji(Xn)u,(t,X,u)eV
the correspondence T : A(t, X, u) —> A'(t, X, u) is scalar product preserving and the rest follows. □
Quantum probability and strong quantum Markov processes 101 Definition. The canonical triple {7i, {Ft}, {jt}) in Theorem 9.8 is called the minimal Markov dilation for the family {T(s, t)} of stochastic operators. Basic problem : Realize the minimal Markov dilation homomorphisms {jt} through quantum stochastic differential equations or other concrete means. Proposition 9.9. Let {jt} be as in Theorem 9.8. For t\ < t2 > t3 one has ■i (X \o (Y\i
(X\-i
^1{XiT(t1,t2){X2)jt3{X3)
if
t3
Proof. First we consider the case t3 < t\ < t2. jtl(X1)jt2(X2)jt3(X3)
= = =
Jtl{X1)Ftljt2(X2)Fhjt3(X3)
jtl(X1)jtl(T(t1,t2)(X2))Jti(X3) jtl(XiT(tut2)(X2))Jt3(X3).
Then the remaining case t\
□
Corollary 9.10. For any sequence Sj, s2,..., sn inT and operators Xi, X2,..., ASi there exist tj > t2 > ... > t^ < u\ < • ■ ■ < ui such that k + I
= JtMi)
■ ■■Jtk(Ak)(Jue(Be)jut_1(Be-i)
Xn, Xt £
■ ■■jul(B1)Y
for some At e Ati, Bj £ Auy Proof. This is obtained by repeating the argument of Proposition 9.9 and using in duction on the length of products. D Remark. In Corollary 9.10 we have reduced a time unordered product as a product of two operators, the first being a time decreasing product and the second a time increasing product. Proposition 9.11 (Covariance of minimal dilation). Let At C B(JCt),Bt C B(Ct) be unital C*-subalgebras, Kt,Ct,t > 0 being Hilbert spaces. Suppose S(s,t) : At —> Aa, T(s, t) : Bt —> Bs,s < t are stochastic operators satisfying the Chapman-Kolmogorov equations andat : At —>■ Bt are unital *-homomorphisms so that one has the following commutative diagram for all 0 < s < t < oo. S{s,t)
A i
-^ _
Bt 1
T(s,t)
Let there exist an isometry U : /Co —* £Q such that U*ao(X)U = X for all X 6 AQ. If{Hs,Fs,js) and (HT,FT,jT) are the minimal dilations of{S(-,-)} and {T(-, •)} respectively then there exists an isometry V : 7is —> HT satisfying Tjf(X)
= iJWt{X))T s
for all XeAt,
t>0. T
Proof. Define the vectors X (ti,t2, •■■i n ,X b --- ,Xn,u), X (ti,t2,--,tn,Yx,--,Yn,v) for ti, > t2 > ■ ■ ■ > tn, Xt G A^, Yi e &U, u e ^-o, v € Co, as in the proofs of Proposi tion 9.3 - Theorem 9.8, corresponding to {S(-, •)} and {T(-, •)} respectively. Consider the correspondence r : A s ( t ! , - - - ,tn,Xu---
,Xn,u)^\T(tu---
,tn,atl(X1),---atn(Xn),Uu)
102 K. R. Parthasarathy for all £1 > t2 > ■ ■ ■ > tn > 0,u G IC0, Xi G Atr Then the commutativity of the diagram in the proposition implies that F is scalar product preserving and hence T extends to an isometry from Hs into TLT. We have for t > tx > ■ ■ ■ > tn > 0, X, X$ G At,Xt G Ati, i = 1, 2, • • • , n and u G K.0,
rj?(x)\(t,t1,---,tn,x0,x1,--=
,xn,u)
T(A (t, ti, ■ ■ ■ ,t„,XXo,Xi,
■ ■ ■ , Xn,u)
T
=
A (Mi,--- . ^ . ^ ( H o ) , ^ ^ ) , -
=
j?(at{X))\T(t,tu---
,tn,at{X0),atl(Xi)r--
=
jl{ot(X))T\{t,tu---
,i„,Xo,Xi,---
,atn(Xn),Uu) ,crtn{Xn),Uu) ,Xn,u). D
Example 9.12. Let A = Cb(R+), the C*-algebra of bounded continuous functions on R +> B = Ch(Rk) where k > 1), and (S(s,t)g)(r)
= Eg(\x +
B(t)-B(s)\)
where x G IR* is any point such that |x| = r and B(i) is the £;-dimensional stan dard Brownian motion. Note that the right hand side is invariant under orthogonal transformations and hence one can choose x = (r, 0,0, • • ■ ,0). Put (T(s, i)/)(x) = E / ( x + B(t) - B(s)) on B
a{
(p{\x\),
K,t = L {p(r)dr),
Ct = L2(Kfc) for all t
where dx\dx2 • ■ ■ dxk = p(r)drdw} k 1
p{r) = ci(r ~ ,dw is the uniform distribution on Sk~l and finally
(tfo/)(x) = /(|x|), / e tct. Here |x + B(i)| is the Bessel process starting at |x| with dimension parameter k. 10. T H E CENTRAL PART O F MINIMAL DILATION
Proposition 10.1 ([BPl], [BP2]). Let Zt C At denote the centre of At and let {T(s,t)} be as in Section 9. Then there exists a unital *-homomorphism kt : Zt —> B(H), (Ti, {Ft}, {jt}) being the minimal dilation, satisfying the following: (i) kt(Z)Ft=jt{Z), ZeZt; (ii) [k.(Z),jt(X)] = 0 V Z G Zs, X € At, s < t; (iii) [kt,(Zi), kh{Z2)} = 0 for Zt e Zh, Z2 6 Zt2, tlthe
T.
Proof. Define kt(Z)\(si,s2,---
,sn,X1,---
,Xn,u)
= A(si,--- ,st,---
,sn,Xi,---
,ZXit- ■■ ,Xn,u)
Quantum probability and strong quantum Markov processes 103 for Si > s2 > ■ ■ ■ > sn,Si = t,Xi G ASi,Z € ZSi, where we may, without loss of generality, assume that t = s4 for some i. Then k
| | £ A ( S 1 I . . . ,Sn,x[
\--. >zx\k\--
,xik\Uk)\\2
112
= ^(A(s1,..-,s„,X1(fc),-..,
ZXlk\-..x£\uk),
k,£
\(su---,sn,xii\---,ZX?\---,Xie\ut)) = "£(uk,T(0,
sn)(Xto
• ••
T(si+1,
Y
Si){ZX?
■ ■ ■ {xf°{T{s2,
8l)
k,e
(xikrxlO)}x?)-zxSO)...xP)ut). Since Z G ZSi, {ZX\k)T
■ ■ -T(S3! s2)(xik)'T(s2,
s a ) ( x f ]'x[e))Xie))
■ ■ ■ ZXf
=
Z'ZYke(i)
where for t h e fixed i, ((Yke(«))) is a positive operator with entries from ASi — At and Z*Z eZt. We have ((Z*ZYM(i))) < \\Z\\\(Yke(i))). Thus
\\Y,Ksi,---,sn)x(k\---,zx?\---x<£\uk)
112
<||Z||2||^A(Sll.-.,Sn,xW,...,Xf,Ufc)||2. k
Hence kt(Z) extends to an operator on H with norm not exceeding \\Z\\. Clearly, kt is a unital *-homomorphism or a representation of Zt. If £' > t we may assume that st = t and s^ — t' with i! < i. Then Jt'(X)kt(Z)X(si,s2,--= A(s i S --=
kt{Z)jv{X)\{su---
,Si',--- ,Si,--- ,sn,Xu--,Xn,u) ,SniXXi,T(si.,8i.-1)(---{T{S2,Sl)(X1)X2}---}X1,-1), Xi>+i, • • • , ZXi, • • • , Xn, u) so that ,sn,Xu---
,Xn,u)
if*' >t.
The same arguments apply when £' = t. Property (iii) follows by straightforward verification starting from definitions. □ R e m a r k . The family {ks(Z), Z G Zs, s G T} yields a commutative process with the feature that "observing ks(Z), Z G Zs for all s < t does not interfere with the Markov process {jt>(X),t > t, X G A ' } " 11. O N E PARAMETER SEMIGROUPS O F STOCHASTIC MAPS ON A C*-ALGEBRA
Prom now on we consider a fixed unital C*-algebra A of operators on a Hilbert space /C and a one parameter semigroup {Tt} of stochastic maps on A so that TsTt = Ts+t. Putting T(s,t) = Tts we apply Theorem 9.8 and Proposition 10.1 to construct W> Ft,jt, h satisfying the following: (i) H is a Hilbert space with /C C 7i\ (ii) {Ft} is a filtration of projections;
104 K. R. Parthasarathy (iii) jt : A —> /B(7Y) is a *-homomorphism with jt(l) = F t ; (iv) kt : Z —> B(7i.) is a unital *-homomorphism from the centre Z of A satisfying h(Z)Ft (v) (vi) (vii) (viii)
=
jt(Z);
{kt(Z),Z € -Z.t € T} is abelian; [ks{Z),jt(X)} = 0 for all s < t, Z 6 Z, X e A {jh(Xi) ■ ■ ■ jtn{Xn)u, ti > ■ ■ ■ > tn, Xi e A, u 6 H0 = K.) is total in H The range Ht of Ft is spanned by Oil (*i) • • • Jt„(Xn)u, t =
t1>t2>--->tn,XieA,u€H0=lC}
Proposition 11.1. Let t = {t\,t2,--- ,tn) be any n-length sequence (n being arbi trary), X = (Xu X2, ■ ■ ■ , X n ) , Xi € A, j(t,X)=jtl(X,)Jt2(X2)---jtn(Xn). Then there exists an element e(t, X) € A such that (i) F(0)j(t,X)F(0) (ii) F{s)j(t+s,X)F{s)
=e(t,X)F{0) = j s ( e ( t , X ) ) V s € T, wheret+s
= (^+s,i2+s, • • •
,tn+s).
Proof. By following Proposition 9.9 and Corollary 9.10 we express j(t,X)=j(r,A)j(s,B)* where r = {n > r 2 > • • • > rk}, s = {si > s2 > • • • > «f}, ^ < st and j ( t + a,X) = j ( r + a,A)j(s + a,B)* V a G R. We have Fi0)jri (Ax) ■ • • j r k {A^j^BDJ,^
{BU) ■ ■ ■ j S l (B*)F{0)
=
F(0)jrMi)
■ ■•3rk{Ak)jn{TSl_l.ai{r
=
F(0)jri(Aj)
• • ■jrk_1(Ak-1)jrk(AkT,t-rk(Tat_l-,l(-
■■TS7.S3(B;TSI.S2(B*)
= Trk{Trk-l-rk(' ' ' Tr2-r3(Tri-ra(A1)A2)A3) = e(t,X)F ( 0 ), say.
■ ■ ■ )B*t)Fm
■ ■ )B*e)F{0) ■ ■ ■ )Ak-1)[AkTSt-rk(TSe_1-St(-
■ ■ )B£]}F(0)
Note that j{t + a, X) = j ( r + a, A)j(s + a, B)* and ^ ( a ) J n + a ( ^ l ) ' ■■3rk+0.{Ak)jSe+a{B*t) =
F{a)jrk+a(Trk_1.Tk(-
=
ja(r r »(---)) = J«(£(t,X)).
■ ■■
jsl+a{B{)F{a)
■ ■ (Tr2.r3(Tri_r2(A1)A2))A3)
■ ■ ■ )Ak^)AkTH_Tk{-
■ ■ ))F{a)
□ We write
fc(t, z) = kh(zx)kt2(z2) ■ ■ ■ ktn(zn),Zi e A For s e t , write h(t, s, X, Z) = j h (Xi) ■ ■ • j t i i _ t (Xi^k^
(Z1)jtii+1 (Xil+1)
■ ■ • kti2 (Z2) ■ ■ ■
Quantum probability and strong quantum Markov processes 105 where X = (X1, X2, ■ ■ ■) with Xir = Zr € Z, sx =th,s2 ik-
= U2!- ■ ■ , ilt < i2 < ■ ■ ■ <
Proposition 11.2. Every /i(t,s,X,Z) can be expressed as fc(s',Z')j(t',X') where s' U t' is a permutation oft.
Here s' can be assumed to be nonincreasing.
Proof. Consider n tX)k (7\-l ^(X)it2(Z) if *i < t 2 , -\kt2{Z)jtl{X) if t! > t 2 Repeating this argument we get the required result. The second part follows from the commutativity of the operators {kt(Z), t £ T, Z € Z). D JtAX)K{Z]
Proposition 11.3. For any (t,s, X, Z),t D s , X D Z, ZSi € ZH, ¥(t, s, X, Z) 6 A such that (i) F{0)h(t,s,X,Z)F(O) = e[t,s,X,Z)F(0); (ii) F(a)h(t + a, s + a, X, Z)F{a) = ja{e(t, s, X, Z))
there exists
Proo/. This is done by using Proposition 11.2 and using the same arguments as in Proposition 11.1. □ P r o p o s i t i o n 11.4 ([AtP],[Bh3]). Let VQ denote the unital *-algebra generated by {ks(Z),jt(X),Z € Z, X G A,s,t € T}. Then for each a € T there exists a unique unital *-homomorphism 6a : VQ —> VQ such that Oa(jt(X)) = jt+a{X), Oa(ks(Z)) = ks+a(Z)
V X 6 A, Z € Z
and ea0b =
0a+bVa,b£T.
Proof. Let A = ^ / i ( s n , r n , ~Kn,Zn),rp = ^ n
j(vk,
Yk)wk.
k
We have the obvious inequality (A^,^)<||/1||2||V||2. Here L.H.S
=
]P(/i(sp,rp,Xp,
Zp)j{vk,Yk)wk,h(sq,rq,Xg!Zq)j(ve,Ye)we)
k,£,p,q
= £ > ( t P ) z;)j(up, x;)j(Vfc, Yk)wk, k(tg, z;)j(u,, x;)j(v«, Y , H > = ^(tUt,i(v;;1Yi;)j(u;Ix;)fc(t;)z;*)A(tSIz;)j(u„x;)i(v/,Y,K) =
2 ^ ( f t , Jo(e(a.k,e,p,qt D M,P,9! Jk,t,p,q, Kk,e,p,q))wt) k,£,p,q
whereas R.H.S = | | ^ | | 2 ^ ( W f c , i o ( e ( K I v , ) 1 Y ^ Y < ) ) W , )
106 K. R. Parthasarathy So b« P g, Jkipq, Kkipq)))kt matrix p.q
matrix(11.1) Now consider
6d{A) = Bd = Yl h(s" + d^n + d, Xn, Z„) k
Then (BdV,BdV)
=
E <&• F <^ v *+ d > Y^«+d>
X
P )fc(t;+d>
K)x
fc(t, + d, Z;)j(u ? + d, X'q)j(ve + d, =
/
;((k,3d(£(&kepq,
bklpq, Jklpq,
Yt)F^t)
Kkl-pq))£,t)
Since jd is a *-homomorphism (11.1) implies that
(BdV,Bdr,) < |H| 2 £>,j d ( £ K,v,,Y£,Y<))&> = \\A\\2\\Vf. k,e
Vectors of the form rj are dense in ft. Thus \\BM
< \\A\\ \\v\\ for all V € H and ||B d || < ||A||.
This at once implies that 9a is well-defined on Vo V a > 0, #a is a *-horaomorphism on Vo,9a{jt{x)) = jt+a{X), 9a{kt{Z)) = kt+a{Z). In particular, 0O(1) = 6a{k0(l)) = ka(l) = 1. Clearly 0O0„ = 0 a+6 . D Remark. Self-adjoint elements of A can be interpreted as bounded observables con cerning a system which evolves in time owing to interaction with a 'bath' or 'noise'. For such an observable X € A,{jt(X),t > 0} may be considered as the 'evolution of the observable X\ This means that {jt} may be viewed as the trajectory of a process. Self adjoint elements of Vo can be viewed as observables determined by the trajectory at a finite number of time points. So the closure of V under an 'appropri ate topology' can be considered as the * algebra containing observables concerning the whole trajectory. The shift homomorphisms {9a, a € T} of Vo (or their extensions to the closure of Vo) constitute the 'Markov shift semigroup'. Note that we have also included the central homomorphisms kt for technical reasons. Propositions 11.1 and 11.2 can be expressed as F(a)9a(0F(a)
= ja(F{0)£F(0))
V(£V0,aeT
(11.2)
This may be called as the Markov property of the process {H,Ft,jt,kt,t £ T}. Sometimes we write Ft as F(t). We view F(0)£F(0) as an element of A acting in /Co = Ho C H.
Quantum probability and strong quantum Markov processes 107 Vo has a natural family of seminorms:
liciu := \MOI>\\ + \mem\,t e T, $ e n, e € v0.
(n.3)
.4 sequence {£„} in Po is said to converge to £ G S('W) in the intrinsic sense if llfn — £|| —» 0 as n —> oo and lim
| | ^ m - ^ n | l w = 0.
m,n—*oo
This convergence is determined by a topology with neighbourhoods of the form
{£ : HCIU <e},s>
0
at 0, where {||£||f,v, = lim„_oo ||f„||t,,j,£„ € 7>o provided each <9t with domain "P0 is closable. The closure V of Vo under the family (11.3) of seminorms would then be a 'candidate' for the 'path algebra' of the Markov process. V is clearly a *-algebra of operators contained in B{7i). This intrinsic topology is stronger than the strong topology but weaker than the operator-norm topology. The unital *-homomorphisms 9a on Vo extend naturally to P . If we denote them by 8a again we still have 0a6b = 9a+i, and F(a)0a(S)F(a) = j o ( F ( 0 ) ^ ( 0 ) ) ^ G V. For the present the closability of 9a with domain Vo is an open question. E x a m p l e 11.5. Let h be a complex Hilbert space and let Ct : h —> h, t > 0 be a strongly continuous one parameter semigroup of contraction operators. Put HQ = F s (h), the boson Fock space over h. Following Example 6.8 define Tt = Tct the associated stochastic operator on B{7io) satisfying e^'ll2-ll'll2>W(Ct/),/Gh
Tt(W(f))
=
Tt(T(K))
= T(CtKC;
+ 1 - CtC*t)
for any contraction operator K in h, where W{f) is the Weyl operator associated with / and T(K) is the second quantization of K. Note that Y{K) is also a contraction. Let {Ut, t G R} be the one parameter group of unitary operators in a Hilbert space of the form h © k, constituting the minimal Nagy dilation of the contraction semigroup {Ct}- Then Ut has a matrix decomposition: r
U,
Ct Mt
L^ A
'**°
(1L4)
where Lt : k —+ h, Mt : h —> k, Dt : k —> k are bounded operators. Furthermore the set {Ut(f © 0), / G h, t G R} is total in h © k. Denote by bt the subspace spanned by {U3(f © 0 ) , / G h,0 < s < t} and put H = Ts(h © k) which is identified with "Ho ® F s (k). Define the representation kt of B(Ho) by putting kt{X) = T{Ut)(X ® l)T(Uty,
X G B(W0)
Denote by Ht,H the subspaces of TL spanned by {e(f),f G hj}, {e(f),f G Ut>oht} respectively. Let F(t) be the projection into Ti.t in the Hilbert space "H. Thus we claim that the following properties hold: (i) kt(W(f)) = W(Ut(f®0));
108 K. R. Parthasarathy (ii) For any / , uit u 2 , • ■ ■ , un G h, t, Si, • • • , s n e R
*t(W(/)M]C ^(«i © °)) = c e (^(/ © 0) + E f«(«i © 0)) i
i
for some scalar c depending on / , t, Si, Ui, 1 < i < n; (iii) A;((X) leaves % invariant for all t > 0, X e
© 0)), Va = e(J2uSi(vi
i
© 0))
i
where the Si's vary in [0,s]. Then ipi,ip2 G Hs and elementary algebra using (11.4) shows that both the expressions (ipi,kt(W(f)ip2)) and (ipi,ks(Tt-s(W(f)))ip2) coin cide with
(^1:^2)exp{-|||/||2 - Y^(Ct^ftVi) + X > , < W ) } . i
i
Since vectors of the form ^ span 7is the proof of (iv) is complete. From (i) and (ii) it follows that vectors of the form ksl(W(fi))kS2(W'(/b)) • • ■ kSn (W/(/n))e(ff © 0), t > Si > • • • > sn > 0, g, / i , • ■ ■ , /„ G h span the subspace Ht for every t. If we define j,(A-) =
fc.(X)FWl
X€B(no),s>0
it follows from the discussion above that (Ti.,jt,F(t),t > 0) is the minimal dilation Markov process associated with the semigroup {Tt}. Note that B(Ho) is not commutative but nevertheless the minimal dilation is real ized from unital *-homomorphisms {kt}. Our arguments can be adapted to the cases when the boson Fock space F s (h) = Ho is replaced by T a (h) or Ffr(h). E x a m p l e 11.6. Let G be a locally compact group, V : x —> Vx a strongly continuous unitary representation of G in a Hilbert space h and 5 : G —> h a continuous 1-cocycle for the representation V, so that 5(xy) = 5(x) + Vx5(y) for all x,y G G. Suppose there exists a continuous function ip on G satisfying {Six'1),
5{y)) = ip(xy) - ip(x) - ip(y),
x,yeG.
Then {e**, t > 0} is a 1-parameter multiplicative semigroup of positive definite func tions on G assuming the value 1 at the identity element of G. Denote by L2{G) the Hilbert space of square integrable functions with respect to a fixed left invariant Haar measure on G and put ft = L 2 ( G ) ® r s ( L 2 ( R + ) ® h ) .
Quantum probability and strong quantum Markov processes 109 For 0 < a < b < oo, let E W « = e i ( ^ ) I m * W ^ ( l [ a , 6 1 ® *(*), l[a,6I ® Kr + (1 - 1 M ) ® 1) where W(u, .A) denotes the Weyl operator associated with a vector u and a unitary operator A Then for each interval (a, 6], x —> U(a,b],x is a unitary representation of G and (e(0), U(axbixe(0)) = exp(6 - a)ip(x). The operator t/(0|i,]i:c 'lives' in the Fock space Ts(L2[a, b] ® h) and f o r 0 < a < 6 < c < oo, U(a,c\,x — U(a,b\,x ® ^(6,e],i- ^n t n e Hilbert space "H define for any / € -^(G) the operator W ) = / f{x)Lx
® C/[0,t],ida;, £ > 0
where L denotes the left regular representation of G. Then / —> A*(/) is linear, &*(/)* = &«(/*) where /*(a;) = /(a;- 1 ) and kt(f)kt(g) = kt(f * g),* denoting the convolution in Ll(G). We have the inequality
IM/)II < ll/lb where || ■ ||o> denotes the G*-norm in the G*-algebra C*{G) which includes the convolu tion algebra Ll(G). The correspondence / —> kt{f) extends uniquely to a representa tion of the G*-algebra C* (G) whose identity element can be identified with the Dirac measure Se. If F(t) denotes the projection on the subspace L 2 (G)®r s (L 2 [0, £]®h)®e[t, where ejt is the vacuum vector in Ts(L2([t, oo)) <8> h) then F(s)kt(f)F(s) = ks(Tt-.sf) where
Ttf =
e^f,t>0,feL\G)
The semigroup {Tt} is, indeed, a semigroup of stochastic operators on the G*-algebra
C{G). If u € L2(G) then for 0 < tx < i 2 < ■ ■ ■ < tn we have Af B (/„)At„_ I (/„-i)---fc, 1 (/i)u®e(0) =
/ /n(a:n)/n-i(a; n -i)---/i(a;i)ii n ...x I M®C/'[o,( 1 ],x n ... a:i et 1 ]
(11.5)
®^lti,«2],in-Mefc,t2] ® ■ • • ® %,-!,*„],*» eIt»-i.*nI ® e[t„da;i • • • cfe„ where e^jj denotes the vacuum vector in T(L2([a, b}}
+ l[tr,-utn\ ® 6(xn)),
0 < ii < ti < ■ ■ ■ < tn, xi, x-2, ■ ■ ■ xn 6 G. Thus Ti' includes every vector of the type e(lJB1 ® 5{xi) + lEi
110 K. R. Parthasarathy of positive definite functions on any locally compact group which is not necessarily commutative. Exercise 11.A. Let (Bi(t),B2(t),t > 0 be a two dimensional standard Brownian motion whose probability measure is denoted by P. In L2(R), let q,p be the standard Schrodinger pair of momentum and position observables satisfying the commutation relation [q,p] = i. Let H = L2(R)
e~iSo
=
BidBi^pB^t)eiqB2(t)_
Then {U(t)} is a unitary operator valued random process satisfying the stochastic differential equation dU = {ipdBl + iqdB2 - -{p2 + q2)dt)U Put kt(X) = U(tyXU(t),
Tt(X) = EPkt(X),
X e
B(L2(R)).
If F(t) is the projection in H determined by the conditional expectation given the Brownian motion up to time t define jt(X)
= kt(X)F(t),
X e
Then {Tt} is a stochastic semigroup in B(L2(R)) TMp))
B(L2(R)) satisfying
= (Pt
for any bounded Borel function tp on R where {Pt} is the transition probability semigroup of standard Brownian motion on R. The triple (7i,jt, F(t)),t > 0 is the minimal Markov dilation of the semigroup {Tt}. Exercise 1 1 . B . Let a^i = 1,2,3 be the Pauli spin matrices in C 2 and let Nj,i = 1,2,3 be independent Poisson processes with intensities Xi,i = 1,2,3 respectively. Denote their probability measure by P and consider the 2 x 2 unitary matrix-valued random process U(t) = {-l)So^dN2+!0t(N1+N2)dN3aNl{t)amt)aN,{t)^
> Q
Then 3
dU = (^fa
- l)dNi)U, U(0) = 1.
Consider the Hilbert space H = C 2 <8> L2(P) and put kt{X)
=
jt(X)
=
U{tf{X
® l)U(t),
X 6 B(C2)
kt(X)F(t)
where F(t) is conditional expectation given Ni, N2, N3 up to time t. Then {H, j t , F(t), t 2 0} is a Markovian dilation of the stochastic semigroup {Tt} on B(C2) with infinites imal generator 3
£(X) = ^XjicTjXffj
- X), X e B(C2).
Quantum probability and strong quantum Markov processes 111 A careful analysis of the Poisson processes Ni,N2, N3 shows that this is a minimal Markov dilation of {TJ.
III. Strong Markov Processes 12. NONCOMMUTATIVE STOP TIMES
Let A C B(fC) be a unital C*-algebra and let {Tt},t > 0 be a strongly continuous 1-parameter semigroup of stochastic operators on A. Recall that a conservative Markov flow with transition semigroup {Tt} consists of {Ti,jt,F(t),t > 0} where Ti is a Hilbert space, j t is a *-homomorphism from A into B(H),jt(l) = F(t) is a projection increasing in t, F(t) f 1 as 11 oo and the following Markov property holds: F(s)jt{X)F(s)
= j,(Tt-.(X))
for all X G A, t > s > 0.
Definition. A stoptime r for the flow {Tt, j t , F(t)} is a spectral measure on the closed interval [0, oo] satisfying the condition [r([0, s]), jt(X)}
= 0 for all s < t, X 6 A.
(12.1)
r is said to be finite if r({oo}) = 0. If p is a density matrix in Ti with spectral resolution J2jPi\'4>j)(i'j\ where pj > 0 for every j and J2jPj = 1 then T ' s s a id t o be finite in the state p if (^,T({OO})V>J) = 0 for every j . T is said to be bounded if the support of r is contained in a bounded interval, r is said to be discrete if there exist 0 < S\ < s2 < ■ ■ ■ such that sn | oo as n | oo and the support of r is contained in the countable set {si, s2, ■ ■ ■ , oo}. If r is discrete and the support is a finite set then it is said to be simple. The projection r([0,i]) is to be interpreted as the event that the stopping of the Markov process occurs at or before time t. Thus r({t, oo]) is the event that the stopping has not occurred up to time t. r({oo}) is the event that the process continues for ever. Condition (12.1) is to be interpreted as the fact that the event of stopping at or before time s does not interfere with the process {jt{X), t > s,X € ^4}. Sometimes it is suggestive to denote T(E) by lTeE for any Borel set E C [0,oo]. Thus l r < t , lT=t, lT>t, ■ ■ ■ denote r([0, t]), T({<}), r([t, oo]), ■ • • respectively. A net { r a } of stop times is said to converge to a stop time r if for any u 6 Ti and for any continuity point t0 of the map t —* r([0,£])u the net {rQ([0, to])u} converges to T([0, to])u as a —> oo. Two stop times r, T' are said to commute with each other if the projections r([0, a]) and r'([0, b}) commute with each other for all a > 0, b > 0. For two commuting stop times r, r ' we say that r < r ' if lT
r r({0}) ^UU,sjj-<
ifa
T([0]tn])
i f t n
s < T O ]
1
if s = oo.
112 K. R. Parthasarathy Clearly, TE and r commute with each other and TE>T. subsets then TE > TF-
If E c F C (0, oo) are finite
Proposition 12.1. For any two commuting stop times T\,T2 there exists a unique stop time T\ A r 2 (called the minimum of T\ and r^j satisfying 1TIAT2<« = 1TI<( + 1TS<( ~ lri
(12.2)
Furthermore, one has T\ A T% < rt,i = 1,2 and if T is any stop time such that T < Ti, i = 1, 2 then r < T\ A T2. Proof. The projection-valued function ln>ilT2>£ is right continuous and decreasing in t. Clearly, 1 *~ ITIXI-T^M
=
1TI<( + 1T2<* — lri
is a right continuous increasing function of t and hence there exists a unique spectral measure T\ A T% on [0, 00] satisfying (12.2). The right hand side of (12.2) commutes with jt'(X) for all t' > t, X € A. Hence Ti A T2 is a stop time. We have lT1/\T2>t — l T 1 > t l T 2 > t , so lTlAr2>t < I n x , « = 1,2. Thus T\ A T2 < 7j,« = 1, 2. If T is another stop time satisfying r < Titi = 1,2, then l r > t < lTi>«,« = 1,2, so lT>t < lTl>hlr2>t lTlAT2>i- Hence r < n A r 2 . □ Exercise 12.A. For any two commuting stop times Ti,T2 there exists a unique stop time T\ V r2 (called the maximum of Ti and r 2 ) satisfying 1TIVT2<( = ^n
Furthermore, T\ V T2 > r*, ? = 1, 2 and if r is any stop time such that r > r,, i = 1, 2 then r > Ti V r 2 . For any t e [0, 00] the spectral measure r degenerate at the point t is a stop time satisfying r({t}) = 1. Denote this "deterministic" stoptime by t itself. If r is an arbitrary stop time and 0 < t < 00 is fixed then the spectral measure r+t defined by (T+t)(E) =
T(E-t),ECR[0iOO
is also a stop time. Proposition 12.2. If {Pa} is a monotonic decreasing net of projections in a Hilbert space TL then there exists a projection P such that s.limP a = P. a
Proof. Let u 6 Ti be fixed and p — inf a (u, Pau). Then for any e > 0, there exists a0 e D, the directed set in the definition of the net with the partial ordering >-, such that (u, Paau)
□
Quantum probability and strong quantum Markov processes 113 Proposition 12.3. Let {ra},a € D be a monotonic decreasing net of mutually commuting stop times for the Markov flow (H,jt,Ft,t > 0). Suppose that the map t —> 3t{X) is strongly continuous for every X e A Then there exists a stop time r such that lim ra([0, t})u = T([0,t])u a
for every continuity point t of the map s —► T([0, S])U, U € 7i being any fixed element. Proof. Since {r 0 } is monotonic decreasing in a it follows that for any fixed t, {T Q ([0, t])}aeo is a monotonic increasing net of projections. Considering the decreasing net {1 — TQ([0,£])},a £ D and applying Proposition 12.4 we conclude the existence of a pro jection P(t) such that s.limQTQ([0,Z]) = P(t) for every t. Clearly Pfa) < P(t2) whenever t\ < t2. Once again by Proposition 12.2 it follows that Q(t) = P(t + 0) is a well-defined projection in the strong sense. Hence there exists a spectral measure T on [0, oo] such that T([0,£]) = Q(t) for all t. Let u € H and t0 be a continuity point of the map s —> T([0, s])u. Then for any t' < t0 we have {u,Q(t')u)
< (u,P{t0)u)
<
(u,Q(t0)u).
Letting t' increase to £0 and using the continuity at t0 we get (u, P(t0)u) = (u, Q(to)u). Thus Um||r a ([0,t 0 ])u-Q(*oHI = 0. a
In other words ra converges to T as a f oo. Since ra is a stoptime it follows that [P{s), jt{X)} = 0 for s < t, X e A. Hence [P(s + 0),jt(X)) = 0 if s < t. Allowing t | s and using the strong continuity of Jt{X) in t we conclude that [P(s + 0),jt(X)} = 0 for s < t. Thus r defined above is a stop time. □ Corollary 12.4. Let T be a stop time for the Markov flow (7i, {jt}, {F(£)}) which satisfies the condition that the map t —» jt{X) is strongly continuous for every X £ A. Then the net {TE} of stop times determined by all finite subsets E C (0,oo) converges to T. Proof. Since {TE} is a monotonically decreasing net of stop times it follows that there exists a stop time T' such that {TE} converges to T'. Let t0 be a point of continuity for the map t —* T'([0, t])u where u is a fixed element of H. Then we have limr £ ([0,io])M = r'([0,£o])M. On the other hand if t0 € E then TE{[0,to})u = T([0,£ O ])M- This shows that r([0,i 0 ])u = r'([0, to\)u. Since the set of discontinuity points of the map t —> r'([0, t\)u is at most countable the required result follows. □ Proposition 12.5. Let T be a stop time and let En C (0,oo),n = 1,2, ••• be an increasing sequence of finite sets such that En = {tni
114 K. R. Parthasarathy lim tnkn = oo 71—>00
Then TE„ converges to r as n —► oo. Proo/. Let u £TL and let t 0 be a continuity point of the map t —> r([0, t])u. Then for all sufficiently large n we have r
£„([0,£ 0 ]) = T([0,tnj]),tnj
< t0 < tnj+i
for some j between 1 and fc„_i. Since t„j+i—tnj —> 0 it follows that lim n T([0,t n j\)u — T([Q,to])u. In other words TE„ converges to r as n —> oo. D 13. M A R K O V PROCESS AT SIMPLE S T O P TIMES
Let A, {Tt}, {Ti, {jt}, {F(i)}), t > 0 be as in the preceding section. For any simple stop time r define the operators
jT(X) = 5>({s})j,(X). XeA,
(13.1)
s
F(T)=JT(1)
= "£T({S})F(S).
(13.2)
s
Note that the summations take place only over the support of T which is a finite subset of R + . Clearly, jT(X) is linear in X and jT{Xy = Y
Js(X*)r({s}) = Y,<{s})UX*) s
= jr(X*)-
s
Furthermore, the fact that r is a spectral measure implies jT(X)jT(Y) = Y,^(X)r({s})r({s'})jAY) s,s'
= ^r({ S })i s (Xy) = jT(XY). s
In particular, F(T) is a projection in H. P r o p o s i t i o n 1 3 . 1 . Let Ti,T2 be two commuting simple stop times. Then F{T{) and Ffa) commute with each other and F(TI)F(T2) = F(T% A T J ) . Proof. Since j$(l) = F(s) is also nondecreasing in s we have from (12.1)
F(T,)F(T2) = fc + Yl + Y) \s=s' l
= Y
s<s'
-n=*K=* {s) + Y ^(s)lr1=slr2=,' + Y f
s
—
/
F{>)TI{{8})T2(W}W)
$>s'/ F
s<s J\*-Tl=S*-T2=S
•* ^ri=S^-T2>S
l
n^K=s'F{s')
s>s'
~T J-Ti >S ^T2=S )&
\$)
S
=
£ ] 1T1AT2=S-F(S)
=
f(nAT2).
S
D
Quantum probability and strong quantum Markov processes 115 Exercise 13.A. If TI,T 2 are two simple commuting stop times then F ( T I V T 2 ) = F(TI) V F(T2) where V on the right hand side denotes maximum in the lattice of projections. (Hint: Use the relation (TiVT 2 )({s})-Ti({s})-r 2 ({s}) + (TiAr 2 )({s}) = 0.) Proposition 13.2. For any simple stop time r and t > 0 F(r)jT+t(X)F(r)
= jT(Tt(X)),
X e A.
(13.3)
Proof. By the definition of r+t we have JTU(X)
=
^(r+i)({S})is(X) S
=
"£T({S
-t})js(X) = J2r({s})Js+t(X).
s>t
F(r)jT+t(X)F(r)
=
s>0
Y^r({s})F(s)js+t(X)F(S) s>0
= E T (M)j.WW)=i T (T ( (A-)). O Corollary 13.3. Let r be a simple stop time. Define ft(X) = jT+t(X), Then (Ti,{j't(X)}, group {Tt}.
{F'(t)}),
X€A,
F'(t) = F(r+t) = j T + ( ( l ) .
t > 0 is a Markov flow with the same transition semi
Proof. This is immediate from the fact that (r+t)+s = r+(t + s) for any s > 0, t> 0.
□ We remark that Corollary 13.3 is the so-called strong Markov property with respect to simple stop time. Our aim is to extend this property to more general stop times. Example 13.4. Let (H, {jt}, {F(t)}),t > 0 be the minimal Markov dilation of the semigrouup {Tt} and let {kt} be the central part of the minimal dilation as defined in section 10. Choose and fix a projection P in the centre Z of A. Let E = {0 < S\ < S2 < ■ ■ • < sn < co} be a finite partition of R+. Since {kt(P)} is a commuting family of projections in H we can define a spectral measure TE with support {0, S\, s 2 , . . . , sn} as follows: TE({0}) TB({sm}) rE({sn})
= k0(l-P)}TE({Sl}) = k0(P)kSl(l-P), = k0(P)kSl(P)---kSrn_1(P)kSm(l-P)iim
Then Y17=o TE({SI}) = 1 where s0 = 0. It follows from Proposition 10.1 that TE({SJ}) commutes with js(X) for all s > s, and X G A- Thus TE is a simple stop time for the minimal Markov flow. Note that 1TB>S = k0(P)kSl(P) ■ ■ ■ ks (P) for j > 1.
116 K. R. Parthasarathy If t is fixed, E = {0 < s1 < s2 < ■ ■ ■ < sn < t < oo}, E = {0 < s-i < s2 < ■ ■ ■ < &j_i <s<Si<---<sn
In this section we restrict ourselves to the minimal flow (Ti, {jt},{F(t)}), t >0 associated with a strongly continuous semigroup {Tt} of stochastic operators on A. We shall also consider the unital homomorphisms {kt} of the centre Z and the Markov shift homomorphisms {0t} of the C*-algebra V generated by VQ. (See Remark after Proposition 11.4.) For any simple stop time r define kr(Z) = J2 r({s})ks(Z),
ZeZ,
(14.1)
£ E V.
(14.2)
S
*r(0 = £
-r({s})8s(0,
Recall that jT and F(T) are defined as in (13.1) and (13.2). Proposition 14.1. For s
kt(Z)} = 0.
Proof. Consider a vector tp in TL of the form ip = js^X^js^Xz)
■ •■i 8m (X m )ti, u€7io,XieA,
sj > s 2 > • • • > sm.
Suppose sp >t > Sp+i. Then by property (iv) at the beginning of Section 11 T({s})kt(Z)1>
= T({a})j. 1 (X 1 ) ■ ■■hp(XPMZ)jSp+1(Xp+1) = T ( { S } ) J S I ( X ( ) ■ - ■jSp{Xp)jt{Z)jSp+1(Xp+1) = =
JSl(Xi ■ ■■3sp(Xp)jt{Z)T{{s})jSp+1{Xp+1) jtl{Xi)
=
■ ■ -jsm(Xm)u ■ ■■jSm(Xm)u ■ ■■jSm{Xm)u
x
■ ■■JsP( p)kt(Z)T({s})jSp+1(Xp+1)
■ ■■jSm(Xm)u
kt(Z)r({s})i,.
Since vectors of the form ip are total in H the proof is complete. CorollEiry 14.2. The map kT : A —» B(7i) defined by (14.1) is a unital
□ *-homomorphisn
Proof. This is similar to the discussion concerning jT preceding Proposition 13.1.
□
Proposition 14.3. The map 9T : V —> B{H) defined by (14.2) is a unital *homomorphism. Proof. Consider £ € V0 C V of the form Z=
kri(Z1)---krp(Zp)j,l(Xi)---j*(Xq).
Then 8S(QT({S})
=
kr^siZi)
=
r ( { s } ) / c n + s ( ^ ) ■ ■ ■ krp+s{Zp)jSl+s(Xi)
■ ■ ■ krp+s{Zp)jSl+s(Xi)
■■■
JS,+S(XQ)T({S})
■ ■ ■ jSq+s(Xq)
=
r({s})B.(t).
This implies that 9T is a unital *-homomorphism from Vo into B(H). Closing in the intrinsic topology we see that 6T has the same property on the algebra V. D
Quantum probability and strong quantum Markov processes 117 Definition. For any simple stop time T define the map E T : B(H) —» B(H) by E T (X) = F(T)XF(T). E T may be looked upon as the "conditional expectation" of X given the knowledge of the minimal flow up to the stop time. P r o p o s i t i o n 14.4. E T # T (0 = # T E 0 (£) for
£eV.
Proof. Let f £ P 0 . Then s
=
s'
s"
^F(S)0s(£)F(s)r({S}) = £ s
j.(EoOr({s}) = 0T(EoO-
s
Closing under the norm topology we get the required result.
D
Definition. A stop time r is called a V-stoptime if r([0, s]) £ P for every s. Proposition 14.5. Let TI,T2 be simple stop times and let r 2 be also a V-stop time. Then $n $T2 = @TIOT2 where T\ O T2 is the simple stop time satisfying r1oTJ({t})= Y, ^WUrtW)))(s,s'):s+s'=t
Proof. We have for any ( 6 P
OrA,(0 = E>(Mtf}))n(W) s
= E E W(0T2({S'}))n({s}) s
s'
= E E ^+s'(0^(r2({S'}))r1({S}) S
S1
= I>(o| t
E
».(^(s'))n({s}) 1 •
^(s,s'):s+s'=t
J
Put »?({*}) =
E
^(^({^'}))n({s}).
Clearly ??({*}) is a projection for every t and non zero only if t £ {s+s'|s £ supp TI, S' £ supp T2}. Also «j({<})*u(0 = ^ ( O f C t o ) for t < u. Thus ?] is a simple stop time. Putting n = T\ O r 2 the proof is complete. □ P r o p o s i t i o n 14.6. Lei T i ^ 6e simple stop times and let r 2 fee also a V-stop time. Then E 0 #TlEo#T2 = Eo0TlOT2. Proof. By Proposition 14.4 and Proposition 14.5 we have Eo&rjEo&rj = E o E T l # T 1 # T 2 = E 0 # T l oT2-
n
118 K. R. Parthasarathy R e m a r k . Under the conditions of Proposition 14.6 we note that This is a generalized strong Markov property for Markov shifts induced by simple stop times. If T\ = T and r 2 = t (the deteministic stop time) then r of = T + t and the generalized strong Markov property becomes ETeT+t = 0TEo0* =
jTE0et.
When applied to an element X 6 A this becomes ErjT+t(X)
=
jT(Tt(X)).
which is the strong Markov property described in Proposition 13.2. P r o p o s i t i o n 14.7. Suppose 7i,T2, ■ • • is a sequence of simple V-stop times. Define Tn = Ti o r 2 o • • • o rn. Then {rn} is an increasing sequence of simple V-stop times. Proof. For any a > 0 we have (norOdO.a])
=
5]r 1 ({ S })0 s (T 2 ([O 1 o]-s)) S
=
£n({s})#s(r2([0,a-Sl))
= ndo.o]). Thus Ti o T2 > Tj and the result follows by induction.
D
P r o p o s i t i o n 14.8. Let^ Fn = F{rn)Jn(X) = j?n{X) for X e A and S„(X) = Eo9Tn(X)\n0The (H, {jn}> {Fn}) is a Markov flow with transition operators S[m,n) = Sm+i o Sm+2 o ■ ■ ■ o Sn for m < n. Proof. This is immediate from Proposition 14.6, and the above remark.
□
R e m a r k . Note that when all the T,-'S coincide with a fixed simple "P-stop time T we obtain a discrete time Markov flow with transition semigroup {Sn} where S(X) = EQeT(X)\Ho,X€A. 15. S T R O N G M A R K O V PROPERTY O F T H E MINIMAL FLOW F O R A GENERAL S T O P TIME
We restrict outselves to stop times for the minimal Markov flow {H, {jt}, {F(t)}), t > 0 associated with a strongly continuous semigroup {Tt} of stochastic operators on a C*-algebra A. For a fixed stop time T and any partition of R + by a finite set E = {s\ < s2 < ■ ■ ■ < Sfc} consider the associated discrete stop time TE as defined in Section 12. By Corollary 12.4, {TE} is a monotone decreasing net of stop times which converges to r. Then {F(TE)} is a monotone decreasing net of projections in TL which, by Proposition 13.1, converges strongly to a projection. We define F ( r ) = s-limF(T E ).
(15.1)
Quantum probability and strong quantum Markov processes 119 If TI, r2 are two commuting stop times then F ( T I ) F ( T 2 ) = F(r x A r 2 ). If n < r 2 then
Ffa) < F(r2). Proposition 15.1. Let En = {s n l < s n 2 < • • • < snkn} sequence of finite subsets such that
C R+ be any increasing
limmax(s„i, sn2 - snU sn3 - sn2, ■■■ , snkn - s„fc„_i) = 0, n
limsnfcn = oo. n
Tnen /or any stop time r, sIimF(r£J = F(r). n
Proof. Let u € 7i and e > 0 be fixed. By (15.1) there exists a finite set E = {ii < i 2 < • • • < tk} C K + such that (u, F(T)U)
< (u, F(TB)U)
< (u, F(T)U)
+ e.
(15.2)
Using the strong (right) continuity of F(t) in t and the strong right continuity of r([0, t]) in t we choose a sufficiently large n 0 , E = {s\ < s2 < ■ ■ ■ < sk} C Eno, such that ti < Si < t2 < s2 < ■ ■ ■ < Sk-i
(15.3)
(u,T((ti,aJ])u><^
(15.4)
for i = 1, 2, ■ ■ ■ , A;. We have ft-i
F(TE)
= r([0, ti])F(t!) + £
r((ti, £ i+ i])F(£ i+1 ) + r((tk, oo])
i=i
fc-i
=
F f a ) + £ ( F ( i J + 1 ) - F(U))T({ti, oo]) + (1 - F(tfc)M(**> oo]). *=i
Similarly, fc-i
F(TE)
= F(Sl) + £ ( F ( s i + 1 ) - F ( S i ) ) r ( ( a i ) oo]) + (1 - F(sk))T(sk,
oo]).
i=l
We are now ready to estimate the difference |(u, F(TE)U) we have \(u,F(t1)u)-(u,F(s1)u)\<^.
— (u, F ( T E ) U ) | . From (15.3) (15.5)
We express (F(si+1) =
- F(s i ))r((s, ! oo]) - (F(ti+1) - F f e J M f e . o o ] )
(F(si+l)
- F(ti+1) + F(ti+i) - F(Si))r((Si,
oo])
- ( F ( t i + i ) - F ( 5 j ) + F( S i ) - F f e J M f e , oo]) =
(F( S i + 1 ) - F(U+i))T((8it
oo]) - (F( S i ) - F(U))T((U,
-(F(ti+1-F(ai)M(ti)ai])
oo])
Quantum probability and strong quantum Markov processes 121 Proof. Note that X{TB)
=
{r([0,t 1 ]) J F(i 1 ) + ^ r ( ( < i , t i + 1 ] ) F ( i i + 1 ) } m ( T ) + ^ ] T ( ( i i , i m ] ) / i=l
=
j
h(s)ds
, t
'i
F(rE)m(T)
+ f TE([s,T])h(s)ds. Jo Taking the limit over E and using Proposition 15.1 we get the required result.
□
Proposition 15.2 enables us to define X(T) as X{T) = F{T)m(T)
+ f Jo
T([s,T})h(s)ds
which can be interpreted as x(r) = m(T) + / h(s)ds. Jo We now state a special case of a result of O. Enchev on the decomposition of a Hilbert quasimartingale into its martingale and finite variation parts. Theorem 15.3 ([Enc]). Let t —> x(t) be a right continuous map with left limits from R + into a Hilbert space Ji and let {F(t)},t > 0 be a right continuous increasing family of projections satisfying the following: (i) x(t) € Jit (the range of F(t)) for every t; (ii) \\F(s)x(t) - x(s)\\ < fsg{u)du for all s
□
Suppose T is a bounded stop time with support in the interval [0, T]. For any finite set E = {*! < t2 < ■ ■ ■ < tk] C [0, T], X e A, u € H0 define the vector k
\{rE,X,u)
+J2T((ti'ti+i1)X(U+i,X,u)
= T([0,tl])X(t1,X,u) i=l
where tk+i = T and X(t, X, u) is defined as in Section 9. Proposition 15.4. Let £ denote the infinitesimal generator of the strongly contin uous semigroup {Tt} with domain V{£) C A. Then the net {\{TE,X, U)} of vectors in Ji over the directed set of all finite subsets of [0, T] converges to a limit for every X £ V(£),u € Jio- The limit denoted by X(r,X,u) satisfies the following: (i) X(T,X,
U) G JiT, the range of F(T);
122 K. R. Parthasarathy W\\X(T,X,U)\\<\\X\\\\U\\.
Proof. We have from Proposition 9.3 F{s)X(t,X,u)
=
X(s,T^,(X),u).
Thus \\F(s)X(t,X,u)-X(s,X,u)\\
=
\\X(s,Tt.s(X)
=
\\js(TUX)
<
||Tt_.(X)-A-||||ti||
= || f
S
-
X,u)\\
~ X)u\\
rr(£(X))dr|||H|<||£(X)||W(t-S).
Jo Hence x(t) := X(t,X, u) satisfies the conditions of Theorem 15.3 and admits the decomposition \{t,X,u)
/ h(s)ds Jo where m is an .F-martingale and ||/i(s)|| < ||£(X)|| ||u||. A(r, X,u) is well-defined as the limit X(T,
= m{t)+
By Proposition 15.2,
X, u) = lim X(TE, X, u). E
Simple algebra shows that F{TE)X(TE,X,U)
=
X(TE,X,U)
and hence, in the limit, F(T)A(T,X,U) =
X{T,X,U).
Furthermore \\X(TB,X,U)\\2
=
J2 ( A fe+i. * , u), T({tu ti+1])\(ti+1,
X, u) + (A(«i, X, u), T([0, t^XfaX,
u))
USE
=
Y^{U,JU+1{X*X)T((U,U+I])V.)
+ (u,j tl (A-*A-)r((0, *!])«>
<
| | X | | 2 ( ^ > , r((t i ) i i + i])«) + <«, r([0, *!])«)) = ||X|| 2 || W || 2 . i
Taking limits we get
||A(r,X jU )|| 2 <||X|| 2 |H| 2 .
□ Proposition 15.5. For every X e A, u 6 Tio and stop time r with support C [0, T] there exists a vector A(T, X, u) in 7iT, linear in X for fixed u and linear in u for fixed X and satisfying (i) A(T, X, u) = lim s X(TE, X, u); (II)\\X{T,X,U)\\<\\X\\\\U\\.
Quantum probability and strong quantum Markov processes 123 Proof. For any e > 0, choose Xe £ V(£) such that | | X - X £ | | < e. Clearly X(TE, X, u) is linear in each of the variables X and u and \\\(TE,X,u)-X(TE,Xe,u)\\
<e\\u\\
for any finite subset E C [0,T]. Then limEl,E2\\HTEi,X,u)
-
\(TEI,X,U)\\
<
2e\\u\\+limEuE2\\X(TEl,Xe,u)
=
2£||u||
-
X(TE2!XC,U)\\
where E\, E2 vary over the directed set of all finite subsets of [0, T]. Thus A(r, X, u) is defined as the required limit (i) satisfying the inequality (ii). □ Proposition 15.6. Let T\ > T2 > ■ • • > rn be bounded stop times with support in [0, T],Xi, X 2 , • • • , Xn 6 A, u € Ho- Then the vector A(ri, T2, • • • , r„, Xj, • • • , X„, u) is inductively well-defined by A(TI,T2,--- J » , X I , - - -
=
,Xn,«)
lim{{T 1 [0,ti])j tl (X,) + y)Ti((* i l * i + i])j t , + 1 (A- 1 )}A(T 2 ) "- , r n , X 2 , - " , X B ) u )
where E = {ti < t2 < ■ ■ ■ < t^}. Furthermore (i) A(ri,r 2 ,--- , r n , X i , - - - ,Xn,u) € HTl; (ii) ||A(ri,r 2 l --- , ••■ ,TntXu--- ,Xn,u)\\ < ||A"i|| ||A(-r2, • - - , r n , X 2 , ■•• , X n , u ) | | . Proof. Suppose A has been defined for r 2 > • • ■ > r n , X 2 , ■ • • , X n , u. Define x(t) = Ti([0tt})jt(X1)X(r2,---
,Tn,X2,---
, X B , u ) , X i 6 2>(£).
Write F1(S)
=
T2([0,S])F(S).
Then, for s < t, we have F1{s)x(t)
= F(s)T2([0,s])T2([0,t})jt(X1)X(T2,---,Tn,X2r-=
,Xn,u)
F(5)j t (X 1 )T 2 ([0, S ])F(T 2 )A(r 2l --- ,Tn,X2,---
=
F(s)jt(X1)F(s)T2([0,s})F(T2)X(r2,---
=
T 2 ([0,s])j s (T t _ s (X 1 ))A(r 2 ,--- ,rn,X2,---
,Xn,u)
,rn,X2,---
,Xn,u)
,Xn,u)
Thus WF^xit)
- x(s)\\
=
^ ( [ O . s J ^ ^ . ^ X O - X O A ^ , - - - ,T n ,X 2 ,--- ,X n , W )||
<
(t - s^CiXJW
||A(T 2 , • ■ • , r n , X 2 l - - - ) X n , U ) | | .
124 K. R. Parthasarathy In other words, conditions of Theorem 15.3 are met and hence by Proposition 15.2, HTii T 2>''" i Tn, XX) • • • , Xn, u) is defined as Umn([0,ti])ar(t 1 )+ V r i ( ( ( i , ( 1 + 1 j ) i ( ( i + 1 ) =
lim{T 1 ([0,i 1 ])T 2 ([0,t 1 ])j il (X 1 )A(r 2 ,--- ,rn,X2,---
,Xn,u)
E
+ ^Tl{{ti,ti+l})T2{[0,ti+l])ju+1{Xl)\{Ti,=
]im{T1([0,t1])jtl(Xi)*{T2,---
,rn,X2r--
■ ■ ,Tn,X2, ■ ■ ■ ,Xn,U)} ,Xn!u)
hi
+ '^2Tl{{ti,ti+l\)jti+1(X1)X(T2,=
■ ■ ,Tn,X2,---
,Xn,u)}
lim> 1 E (X 1 )A(r 2 , • • ■ , r„, X2, ■ ■ ■ , Xn, u)
for Xi e ~D(£). Such a definition fulfills properties (i) and (ii) of the proposition for X\ 6 V(C). When X\ e A the proof is completed as in Proposition 15.5. □ Proposition 15.7. For stop times r > T\ > ■ ■ ■ > rn with supp r C [0,T], X,Y,Xi,--, Xn £ A and u e Ho HT, T, TI, • ■ ■ , r„, X, Y,Xir--
, Xn, u) = A(r, TU ■ ■ ■ , r n , XK, Xi, ■ • ■ , ^Tn, w).
Proo/. Let e > 0 be fixed. Choose a finite set i?0 C [0, T] such that
P ( r , r, Ti, • • • , rn, X, Y, Xi, ■ ■ ■ , Xn, u)-jTB{X)X{r>
n , • • • , r„, Y, Xu ■ ■ - , Xn, u)\\ < e
for all finite subsets T of [0, T] containing E0. This is possible in view of Proposi tion 15.6. Since jTE is a homomorphism we have <
P ( T , T , T I , - - - ,Tn,X,Y,Xu---
,Xn,u)
- JTB(XY)\(TU---
,Tn,Xu---
I|A(T,T,T 1 ,--- ,Tn,X,Y,Xu---
, Xn,u)
- jTE(X)X(T,n,
• ■ • ,Tn,Y,Xlr--
+ \\JTE(X){X(T,TU--<
,Tn,Y,Xu---
,Xn,u)
~jrE(Y)X(rlr--
,X„,u)\\
,rn,Xu---
,Xn,u)\\ ,Xn,u)\\
(l + \\X\\)e
for all finite sets E D E0. This completes the proof.
D
Definition. Let TiT denote the closed linear span of the family {A(r,Ti,--- , T„, X,Xi,--,Xn,u), T > T\ > ■■■ > rn, X,Xt £ A,u £ 7io\ where r is a fixed stop time, TI,T2,--, T„ are varying stop times. Denote by F(T) the projection on the subspace HT- If ri < T2 then F ( T J ) < F{T2). F(T) < F(T). Define the operator jf(Y) on ?lT by putting j°(Y)X(r,rir--
,rn,X,Xir--
,Xn,u)
=A(r,Ti,--- ,r n , F X , A"l7 • • •
and extending by linearity and continuity. Note that by Proposition 15.7. ]°T(Y) =
limjTE(Y)\fiT,YeA
,Xn,u)
Quantum probability and strong quantum Markov processes 125 and j°{Y) is a well-defined operator. Furthermore the correspondence Y —» Jr(Y) is a unital *-homomorphism from ^4 into B{HT). Put
jT(Y)
=
Then the correspondence Y —> jT(Y) >(1) = F(T) < F(r).
J*(Y)F(T).
is a *-homomorphism from .4. into B(7i) with
Proposition 15.8. For any Borel set E C K + ) T(E) and F(T) commute with each other.
Proof. Let G = {t\ < £2 < • • • < tk) C K+ be a varying finite set. Then
F(T)T(J5) t-i
=
lim{r([0, t 1 ])F(t 1 )r(£) + ^ r ( ( i „ £ i + 1 ])F(i < + 1 )r(F) + r((tk,
OO])T(E)}
i=l
Jfc-1
=
Um{F(ti)r( J B n [0, £1]) + ] T F(U+i)r{E
n (£;, t i + i]) + r((f t , 00]) n F )
t=i fc-i
=
r ( F ) lim{r([0, t i D F f a ) + V r((t i , t i + 1 ] ) F ( t i + 1 ) r ( F ) + r((£,, 00])
=
r(£)F(r).
2= 1
D
Proposition 15.9. L e i r fee a bounded stop time with support C [0,T]. F o r r > rx > • • ■ > TVi, X, X\, ■ ■ ■ , Xn e A and u G Ho one has
F(r)A(T + t,Ti,--- . T n ^ . X j , - - -
,Xn>u) = A(T,Ti,-.- . T n . T t W . X ! , - - -
,Xn,U).
126 K. R. Parthasarathy Proof. Let E = {ti < t2 < ■ ■ ■ < h} C [0,T]. Varying E we see that the left hand side of the equation above is equal to H m ^ r X r + t X ^ i D ^ W A f a , " - ,Tn,Xu---
,Xn,u)
E
+ I ] ^ W ( T + t)((*i,ti+i]b"«i+1(X)A(r1)--- ,Tn,Xu--.
,Xn,u)}
USE
= ]im{F(T)T[0,h - t])jtl(X) + J2 F(T)T(ti - Mi+i -
A)k+AX)}
ti£E
=
]im{F(T)T{[0,t1-t])F(ti-t)jtl{X)F(t1-t)\(T1)--+ ] T F(T)T({U
X(n, •• ■ ,Tn,Xi, ■ ■ ■ ,Xn,u) ,T)n,Xu---,Xn,u)
- t, U+i - t])F{U+i - t)ju+1 {X)F{ti+i - 0 )
UeE A ( T I , - - - ,T„,XU---
=
\imF(T){T([0,h
- t])jtl.t(Tt(X))
,Xn,u)}
+ £ > ( & - t,ti+i ~ *]) USE
=
jti+1-t(Tt{X))}X{T1,---,Tn,X1,---,Xn,u) F(T)A(T,n,." . r ^ T K X ) , ^ , - - - , ! „ , « ) .
D Corollary 15.10. Lei r be a bounded stop time and let F(r),jT above. Then __ _ F(r)iT+t(Z)F(r)=>(Tt(X)).
be as in the definition
Proof. Since j T ( l ) = ■F'M we have for r > Ti > ■ • ■ > r n F(r)J : + ( (X)F(r)A(T, n , ■ • • , r n i 7, y l t • • • , y„, u) =
F(r)7 T + t (X)A(r, n , ■ • • , r n i y, Fj, ■ • ■ , Yn, u)
=
F(T)X(T
=
F(r)F(T)A(T + t , r , T 1 , . - , r „ , ^ . y , ^ , ■ ■ ■ , y n , u )
=
F{T)X(T,T,TI,---
+ t, r , n , • • • , r n , X, y, y ^ • • ■ , Yn, u)
,Tn
,y„,u)
= J T r t (x))A(r,ri,--- .T-n.y.y!,--- ,y„,«). D Proposition 15.11. Let r be any bounded stop time. Define Jt(X)=3T+t(X),G(t)
= F(T + t).
Then the following hold: (i) Jt(l) = G(t) is an increasing family of projections; (ii) G(8)Jt+s(X)G{8) = J,{Tt(X)) fors,t>0,XeA Proof (i) is immediate from definitions. To prove (ii) observe that by Corollary 15.10 G(s)Jt+s(X)G(s)
= F(T + s)J{T+s)+t(X)F(T
+ s) = ~jT+s{Tt(X)) =
Js(Tt(X)).
Quantum probability and strong quantum Markov processes 127
□ 16. STRONG M A R K O V PROPERTY UNDER A SMOOTHNESS CONDITION
In the preceding section we defined the *-homomorphisms jT on the initial algebra A exhibiting the strong Markov property. However, jT(l) = F(T) turned out to be a projection smaller than F(T), the filtration projection at stop time T. We would also like to define the shift homomorphism 9T on the path algebra for an arbitrary stop time. We are unable to execute such a programme solely under the assumption of strong continuity for the semigroup {Tt}. We shall define jT as a * homomorphism under a smoothness condition. But, in order to define kT and 6T we need the additional condition of separability of H. We begin with a definition. Definition. Let C be the infinitesimal generator of the strongly continuous semigroup {Ti},£ > 0 of stochastic operators on the unital C*-algebra .A of operators on 7i0. We say that £ satisfies the condition S ([AtP]) if there exists a dense *-subalgebra AQ C T>(C), the domain of £ , satisfying (i) AQ is invariant under the action of {Tt}; (ii) The map t -► C{X{rt{Y)X2) is locally bounded for every XUX2, Y e A Throughout this section we assume that the condition S is fulfilled by {Tt} and (H, {jt}, {F(t)}) is the minimal Markov dilation of {Tt}. Proposition 16.1. Let u e H0,X,Yi e A0,i = 1,2, • • • ,n,tx > t2 > ■ ■ ■ > tn > 0 and tpt = jt{X)\(t1,t2,--,tn,Yi,--,Yn,u) where A is defined as in Section 9. Then \\F{s)ipt - V J < c(t
-s)forallO<s
where c is a positive constant depending
onT,ti,Yi,X.
Proof. First let 8 = mini(£; — ti-i). Suppose t — s > 5. Then \\F{sWt - A\\ < 2\\X\\ ||A|| < 2
=
F{s)jt(X)F{t)X{t1,---
=
FisMXTt^-tRy^Tt^-t^RY^
,tn,Yu---
,Yntu) ■■■
Ry2Ttl^{Y{))
A(t/b, ••■ ,tn,Yk,--Since tk < s and A(tfc, • • • , tn, Yk, • ■ ■ ,Yn,u) F(s)A A
G 7is we have from the Markov property
= UTt-sLxTtk_^t(Y))X(tk,-
where Y ~ RYk^Ttk_2-tk
■ ■ ■ RY2Ttl-t2(Y1).
= js(X)F{s)X{tlr--
,Yn,u)
■ ■ ,tn,Yk,---
,Yn,u)
On the other hand
,tn,Yu---
,Yn,u)
= i.(i^r tt _ 1 _,(y))A(t fc ,--- ,tn,Yk,--- ,Yn,u).
128 K. R. Parthasarathy Thus
< \\X(tk,---,tn,Ykr--,Yn,u)\\
\\Tt^LxTtk_^t(Y)
-
LxT^.^Y)]].
Furthermore ||r(_iLxTtl_I_t(y) -
LxTtk_ls,(Y)\\
< ||(rt_,-id)Lxrtt_1_t(y)|| + ||A:|| ||rtfc_1_t(y)-rtt.1_,(y)|| < || f V ^ A T ^ ^ y ) ) * ! ! + ||X|| || f"'1 STX(Y)dr\\ Jo
Jtk-i-t
< (sup||£(xrr(y))|| + ||x||||£(y)||)(t- a ). r
C a s e 2: tk+\ < s < tk < t < tk-\- As before, using Markov property we have F{a)1>t =
F(a)jt(A-)F(t)A(ti,---)tfc_i1tfc,--.)tn)y1)...iyniu)
= F( S )j t (ir ll ,. i (r'))A(i fc ,( w ,.. ,tn,Yk,--- ,Yn,u) = F(s)jtk(JRnrt_ttLxTti_1_i(y))A(i,+lj--- ,t„,yfc+1>--- ,y„,u) = F(s)j.(rt,_,iiyJbrt_tlbLxrtlb_1_t(y))A(tfc+1,--- A.n+i,--- ,y»,u) whereas Thus \\F(a)ilH-fl).\\
< ||A(i i + i,--- ,tn,Yk+1,---
,Yn,u)\\
WT^-sRY^t-^LxTt^^tiY)
-
LxTtk-sRYlTt^tkTtk.l-t{y)\\-
Since tk — s
IKr^-uQ^Mi < (t-s)\\c{A)\\, \\(Tt_tk-id)(A)\\
<
(t-s)\\£(A)\\iorAeA0.
Hence, using the fact that LxRyk = RykLx we get \\F{s)ik-il>.\\
< ||A(t t + i,---
,tn,Yk+u---^Yn,u)\\x
{WT^Ry.LxT^iY) < <
+ ( | | n | | I I ^ X T ^ ^ ^ H + linil \\X\\ -s) + c2(t - s){\\C{XTtk_^t(Y)Yk) c(t - s)
Cl(t
for some constants c\, c^, c of the required type.
-
LxT^RyJ^-WW +
\\£(Ttk_^t(Y))\\)(t-s)} X£(Ttk_^t(Y)Yk)\\ D
Corollary 16.2 ([AtP]). Let r be any bounded stop time with support in an interval [0, T]. For any finite set E C [0, T] let TE and jTE be defined as in Sections 12 and 13. Then there exists a unique *-homomorphism j r : A —> B(H) satisfying the following:
Quantum probability and strong quantum Markov processes 129 (i) jT(X) = s.llm.EJTB{X) for all X G A, where the limit is taken over the directed set of all finite subsets of [0,T]; (ii) jT(l) = F ( r ) ; (iii) F{r + s)jT+s+t{X)F{r + s) = jT+s(Tt(X)) for all s,t>0 and X E A. Proof. First, let X e A0. For Yu ■ ■ ■ , Yn E A0,u E K 0 and h > t2 > ■ ■ ■ > tn the vector ipt = j' t (X)A(ii, ■ • ■ , tn, Y\,- ■■ ,Yn,u) satisfies the conditions of Theorem 15.3. By Proposition 15.2 tyP3TB(X)\(tut2,---
,tn)Yi,---
,Yn,u) = lixnipTB
E
E
exists. Furthermore jTE is a *-hornomorphism satisfying ||jTE(-X')|| < ||X|| and J T E ( 1 ) = F{TE)Since vectors of the form A(t 1 ,t 2 ,--- ,tn,Y\,--- ,Yn,u) with u in HQ,YI,--,Yn e AQ and U's in R+ constitute a total set in TL it follows that s.liniE jTE(X) exists for all X £ A and jT thus defined as in (i) is a *-homomorphism satisfying (ii). The last part follows from its validity for each TE and by taking limit over E. □ P r o p o s i t i o n 16.3. Let r be an arbitrary stop time. Then the following hold: (i) JMX)T([0, t]) = r([0, t])jTM[X) for all (ii) The correspondence X —► jTM(X)T([0,t\) (iii) JTM(X)T([0,
S]) = JMX)T([Q,
s])for0<8
(iv) There exists a *-homomorphismjT Proof. Let S = {tut2,
■ ■ ■ ,tk},0
{r(0, h})jtl (X) + T((tu t2])jtl(X)
teR+)XeA; is a *-homomorphism\ oo;
such that jT(X) = s.limt^co
jTAt(X)T([0,t]).
< h < t2 < ■ ■ ■ < tk < t. Then J{TM)E(X)T([0,
+ ■■■ + T((tk.utk])jtk(X)+T((tk,
oo])jt (X)}T([0,
Note that r((i fc ,oo]) = r((tk,t\) + 1 - r([0, t]) commutes with jt(X). hand side of the equation above, in each of the summands we have
t}) t])
On the right
r(A)jtr(X)T([o,t}) = jjxMAnflM]) =
r(An[0,t})jtr(X)
=
T([0,t})r(A)Jtr(X)
where A = [0,ii] when r = 1,= (t r _i,i r ] when 2 < r < fc,= (ifc,oo] when tr = t. Hence j{TAt)B{X)T({0,t})
= T([0,t])j (tAt)ls (X)
Now taking limit over the directed set of all finite subsets of [0, t] we get (i). Property (ii) follows from (i) and the fact that j T h t is a *-homomorphism. To prove (iii) consider E as above with tp = s. Then J(TM)E(X)T([0,S})
=
j ( r A t ) E (X)r([0,f p ])
=
T{[0,t1\)3t1(X) + --- + T((tp_us})Js(X)
= J(TAS)F(A>(M)
130 K. R. Parthasarathy where F = {ii,*2, • • • ,tP-i}- Taking strong limit over E we obtain (iii). To prove (iv) observe that, for any ip eti, property (iii) implies H E J|jTA£(*)T([0, m
- jrAs(X)T([0,
sM\
S,t~*OQS
=
lim" JjrM{X)M[Q,
t))f - r([0, s])}tf||
s,t—►oos
<
||X||
Urn"
jT([0,t))1>-T([0,s])rl>\\
s,t—*oos
=
0.
Here we have used the fact that hro.t^00T([0,t])rp = r([0, oo))ip. This shows that jT(X) is well-defined and j ' T is a *-homomorphism. D Proposition 16.4. Let r be any stop time. Then s. lim F(T A t)r([0, oo)) = F(T)T([0,
OO)).
Proo/. Let ip e H be such that T ( [ 0 , O O ) ) ^ = ^ . Let e > 0 be arbitrary. Choose to > 0 such that ||T([*O, OO))V|| < £, \\F(T A t)if> - F{T A t0)1>\\ < e.
(16.1)
There exist 0 < T\ < r2 < ■ ■ ■ < r*. < oo such that for any 0 < t\ < t2 < ■ ■ ■ < tn < oo satisfying {tu ■ ■ - , £„} D {ru ■ ■ ■ , rfc} ||F(r)V - {T([0, *i])F(t!) + • • • + r ( ( V i , in])^*™) + r((t n > oo))}V|| < e. (16.2) Here we have used the fact that T({OO})IJJ = 0. Similarly there exist 0 < Si < s2 < ■ ■ ■ < se < t0 such that for any 0 < pi < p2 < ■ ■ ■ < pm < t0 and {pi,P2, • • • ,Pm} 3 {si,s2,-- ■ ,se} \\F(T A toty - {r([0,p 1 ])F(p 1 ) + • • • + r((p m _!,p m ])F(p m ) + r((p m , oo])F(t0)}^|| < e (16.3) Let {*i, *2, • - • ,tn} D in,--- ,rh} U {si,--- ,se} U {i 0 } and tq = t0 for some q > 1. Note that by (16.1) !!{r((i g ,t, + 1 ])F(V 1 ) + --- + r ( ( 4 - i , i n ] ) F ( i n ) + r(i n ,oo])}^|| < ||r(t 0 ,oo))^|| < e. Hence (16.2) implies | | F ( r ) ^ - { r ( [ 0 , i 1 ] ) F ( i 1 ) + --- + r((Vi,to])i 7 (to)}^|| < 2 e .
(16.4)
On the other hand (16.3) implies \\F(T At0)i> - { T ( [ 0 , i ^ t i ) + ■ ■ • + r((i,_2, t , _ i ] ) F ( V i ) + r((Vi.«o])^(*o) + r((t 0 , oo])F(i 0 )}^|| < e and ||T((to,Oo])F(io)^||<||T((t0lOo))^||<£.
Thus ||F(r A to)V> - {r([0, txDFfa) + ■ • • + T((*,_I. f o D W M I < 2e. Now (16.1), (16.4) and (16.5) imply \\F{T)i>-F{T/\t)iP\\
□
(16.5)
Quantum probability and strong quantum Markov processes 131 T h e o r e m 16.5 ([AtP]). Let {Tt},t > 0 be a strongly continuous semigroup of sto chastic operators on a C'-algebra A of operators on a Hilbert space TC0, whose in finitesimal generator satisfies the condition S (see the beginning of this section). Let (TL, {jt}, {F(t)}), t > 0 be its minimal Markov dilation and let r be any stop time. Then the *-homomorphism jT defined by Proposition 16.3 satisfies the following: (i) jT(X) = s.lim t _ 0O
jTAt(X)T([0,oo));
(ii) J T ( 1 ) = .F(T)T([0 I OO));
(iii) F(r)jT+t(X)F(r) for allt>0,
=
jT(Tt(X))
X e A.
Proof. Property (i) is immediate from Property (iv) of Proposition 16.3 and the facts that ||j T A t (X)|| < ||X|| and s.lim t _, 00 r([0, £]) — r([0, oo)). Property (ii) is immediate from the fact that jrAc(l) = F(rAt) and Proposition 16.4. To prove (iii) first observe that (T + £)([0,OO)) = T([0,OO)) and (r + t)Aa = {T A (a- £)} + £ for all a > t. Hence
by Corollary 16.2 and property (i) of the theorem F(T)jT+t(X)F(T) = F(r)r([0, oo))j ( T + t ) (X)F(r)r([0, oo)) =
s. lim F(T A (a-
t))j{rA(a.t))+t(X)F(T
A (a - £))T([0, oo))
a—*oo
=
a. lim
JTA(a-t)(Tt(X))T([0,oo))
a—»oo
=
Jr(Tt(X)).
a Corollary 16.6. In Theorem 16.5 put Jt(X) = jT+t(X),G(t) Then (i) G(s)Jt+s(X)G(s) = Js(Tt(X)) (ii) G(t) Tr([0,oo)) as t T oo.
for all
= F(r + i)r([0,oo)).
s,t>0;
We shall now assume the additional condition that the Hilbert space 7i in the minimal dilation (7i, {jt}, {F(t)}) of {Tt} is separable. Our next proposition describes the *-homomorphism kT : Z —> B(7i) for a bounded stop time, where Z C A is the centre of A. P r o p o s i t i o n 16.7. Let TL be separable and let r be a bounded stop time with support in the interval [0,T]. Then there exists a unique unital *-homomorphism kT : Z —■> B(TC) such that kT(Z) = s. lim kTE(Z) for all Z € Z, E
where the limit is taken over the directed set of all finite subsets of [0, T]. Proof. Owing to the separability of TL, the set of discontinuity points of the map t —> T([0, £]) in the strong operator topology is at most countable. Denote this countable set by C. Now consider a vector ip € TL of the form i> = A(£i, t2,. ■., £„, Xi, X2,...,
Xn, u) = fti ( X i ) . . . jjt
(Xn)u
132 K. R. Parthasarathy where t\ > t2 > ■ ■ ■ > tn belong to R + \ C , Xt e A and u e HQ. Such vectors constitute a total set in Ji. For any Z € Z and finite set E C [0,T] we express n
\+l
kTB(Z)tp = lTB>h kTB(Z) + Y^ i=l
where i n + 1 is defined to be 0. We have the relations l
rE>U kTB{Z)tl> = lTE>tlkTE(Z)F{ti)ij
=
lrB>tJrE(Z)lp,
1
«i+i
— jti(Xi) ■ ■ ■Jti(Xi)Ui+1
lim kTE(Z)ip = l T > tl jT(Z)tp + Y^JhiXi)
■ ■ ■ jti{xi)1ti+1
...,tn,
i=l ■^t+i) • • ■ i Xn,
u).
Since ||/c TE (Z)|| < \\Z\\ and vectors of the form tjj are total in TC we conclude that s. liiriE kTB(Z) exists. Since each kTE is a unital *-homomorphism it follows that the limit kT is also a unital *-homomorphism. D Proposition 16.8. Let 7i be separable and let r be a stop time such that T({CO}) commutes with jt{X) for all X 6 A,t > 0 as well as kt(Z) for all Z € Z,t > 0. Then there exists a unique *-hom,orphism kT : Z —> B(TL) such that kT(Z) = s. lim r([0,oo))kTM(Z),
Z € Z.
t—*oo
In particular, kT(l) — T([0, OO)). Proof. Choose and fix any vector tjj of the form in the proof of Proposition 16.7 where ti > t2 > ■ ■ ■ > tn are continuity points of r. Then by the proof of Proposition 16.7. we have r([0,oo))fc T A i (Z)^
=
lrAt>t,T([0, 00))jTM(Z)l/> + X > (
X
l ) ■ ■■Jti(Xi)lti+l
i
xJTAt{Z)X(ti+i,...
,tn, Xi+i,...,
Xn,u).
Taking the limit as t —> oo we obtain lim r([0, oo))krM(Z)rl> t—VOO
= lT>tjT{z)ip + '^itl{x1)...
jti(Xi)iti+l
..,tn,xi+i,...,xn,u).
i
Once again the totality of vectors of the form ip and the fact that r([0,oo))fcTAt is a *-homomorphism for each t imply the required result. □
Quantum probability and strong quantum Markov processes 133 Proposition 16.9. Suppose Ti. is separable and condition S holds. Let r be a stop time with support in a bounded interval [0,T], Let V\ denote the C*-algebra gen erated by {jt(X),kt(Z),t > 0,X € A, z e Z}. Then there exists a unique unital *-homomorphism 9T : V\ —> B(Ti) satisfying the following: (i) 9T(A) = s.limE@TB(A) for all A e Pi, where the limit is taken over the directed set of all finite subsets of [0, T] and 9TE is defined by Proposition 14.3; (ii) 8T(jt(X)) = jT+t(X), 9T(kt(Z)) = kT+t(Z) forXeA,Z<= Z; (ii) F(T)9T+t(A)F(T) = jT(F(0)et(A)F(0)) for all A e VX. Proof. By Proposition 14.3, 9TE is a unital *-homomorphism from V\ into B{Tt). By Proposition 14.3 and Proposition 14.5, properties (ii) and (iii) hold when r is replaced by T~E- Now consider an element A of the form A = kSl {Zx) ...k,m (Zm)jtl (Xi) ■ • • j t n (Xn), Zi eZ,X,e
A.
We have 9TB(A)
= kTE+sl(Zi)-
■ • krE+Sm(Zm)jTE+tl(Xi)
■ ■ -JTE+tniXn),
F(TE)9TE+t(A)F(rE) = jTE{F(0)9t(A)F(0)). Taking limits over E and using Corollary 16.2 and Proposition 16.7 we conclude that 9T{A) = s. lim.E9TE(A) exists and 9T(A) = kr+s^Zx) ■ ■ ■ kT+Sm(Zm)jT+tl(Xx) F(T)9T+t(A)F(r)
=
■ ■■jT+tm{Xm),
jT(F(0)9t(A)F(Q)).
Since linear combinations of elements of the form A are norm dense in Vi,9T(A) is well-defined for every A 6 Vi,6T is a unital *-homomorphism and properties (ii) and (iii) hold. □ Theorem 16.10 ([AtP]). Let {Tt} be a strongly continuous semigroup of stochastic operators on A, whose infinitesimal generator satisfies the condition S. Let the Hilbert space Ti. of its minimal dilation {T~t,{jt},{F{t)}) be separable. Suppose r is a stop time such that T({OO}) commutes with jt(X) for all X £ A and kt(Z) for all Z £ Z and allt > 0. Let V\ be the C*-algebra generated by {jt(X),kt(Z),t >0,XeA,Ze Z}. Then there exists a *-homomorphism 9T : V\ —* B(7i) satisfying the following: (i) 9r(A) = s. l m w , 6TAt{A)T([0, oo)) for AeVi; (ii) eT(jt(X)) = jT+t(X);9T(kt(Z)) = kr+t(Z); (iii) F(T)0T+1(A)F(T)
+ j T (F(O)0 t CA)F(O)).
Proof. Consider an element A € V\ of the form A = ksl{Zx) ■ ■ ■ kSm(Zm)Jtl{X1)
■ ■ ■ jtn(Xn),
Z{ eZ,Xte
A.
Then 9TM(A)T([0,OO)) = =
kTAt+sl(Zi)
■ ■ •/CrAt+s m (2' m )jrAt+t 1 (^l) • ' ■j f TAf+t n (^7iM[0, OO))
fc(r+Sl)A(£+sl)(Zi)(r
+ Sl)([0, OO)) • • • k(T+sm)A(t+sm)(Zm){T
+ S m )([0, CO))
i( T + f l ) A ( ( + t l ) (X 1 )(r + ti)([0, oo)) • ■•j(r+t„)A(t+tn){Xn)(T + t„)([0, oo))
134 K. R. Parthasarathy Letting i ^ o o and using Proposition 16.9 and Theorem 16.6 we get s. lim 0TAt(y4)r([O,oo)) = /c T+sl (Zi) • ■ ■ kT+Sm{Zm)jT+tl{Xi) £—♦00
■ ■ ■ jT+t„(Xn).
Since linear combinations of elements of the form A are norm dense in V\ it follows that s.lim(;_,oo#7-A«(A)T([0,oo)) exists for all A G V\ and 8T defined by (i) is a *homomorphism fulfilling (ii). To prove (iii) observe that by Proposition 16.10 F(T A a)0TAa+t{A)F{r
A a)r([0, 00)) =
jTAa{F(Q)et{A)F(0))T([0,00)).
Letting a —> 00 the proof becomes complete.
□
17. A QUANTUM VERSION OF DYNKIN'S LOCALIZATION FORMULA Consider a classical strong Markov process with homogeneous transition probabil ities Pt(x,E),x G £,E C £ where £ is a metric space and E is a Borel set in £. Write ^ = {/|/ a bounded real valued Borel function on £},
(Ttf)(x) = I f(y)Pt(x,dy) | | / | | = sup |/(:c)|, X
B0 = {/|/ G B, lim(T t /)(z) = f(x) for all x G £}, VA = {/|/ G B0, t~l\\Ttf - / | | is bounded in t > 0, 3# G Bo such that lim ^
^
~
f
^
= g(x)(0T all
xe£},
Af = g for every } eVA. Assume that the Markov process has right continuous trajectories. Let T be any stop time for the Markov process satisfying the condition EXT < 00 where E^ denotes expectation with respect to the process starting from x at t = 0. Then a well-known theorem of Dynkin asserts the following: If {X(t)} denotes the trajectory of the Markov process then for any / G VA rr
/>oo
E,/(X(r))
f(x) + Ex / (Af)(x(s))ds = f(x)+Ex / lT>s(Af)(X(s))ds. (17.1) Jo Jo The importance of this formula lies in the fact that the right hand side is an expression which involves the trajectories only up to time r as well as the generator A. For example, if U is a neighbourhood of x, and Ty is the exit time for the trajectory starting at x and Af is continuous at x then the value of Af at x can be expressed
(»)1.«P.
,1,2)
Thus the value of Af at x depends only on the values of / in an arbitrarily small neighbourhood of x. It would be desirable to establish quantum versions of (17.1) and (17.2) at least for the minimal Markov process (H, {jt}, {F(t)}) associated with the semigroup {Tt} of stochastic operators on a C*-algebra A C B(Ko). We shall prove here an analogue
Quantum probability and strong quantum Markov processes 135 of (17.1) but we have no idea of how one can even formulate (17.2) when TIJ is a selfadjoint operator. It the random variables f(X(r)) and (Af)(X(s)) are viewed as multiplication operators and denoted respectively by jT(f) and js{Af) then (17.1) can be interpreted as F(0)j T (/)F(0) = f + F(0){ /
lT>sj,(Af)ds}F(0)
Jo
where F ( 0 ) denotes the projection E. This suggests the possibility of the following version of (17.1) for the minimal Markov dilation (H, {jt}, {F(t)}) of {Tt} mentioned in the preceding paragraph: /■oo
F(0)jT{X)F{0)
= X + F(0){ / lT>sJs(jr(X))ds}F(0) (17.3) Jo for X £ £*(£), the domain of the infinitesimal generator £ of {Tt}. Applying (17.3) to a vector u € Ho, the range of F(0) (which coincides with the Hilbert space Ho on which A acts) we can express (17.3) in the notation of Section 16 (see Proposi tion 15.4). /•CO
F ( 0 ) A ( T , X, u)=Xu
+ F(0) /
l T>5 A(s, C(X), u)ds.
Jo
We shall first establish such a result for bounded stop times. T h e o r e m 17.1 ([AtP]). Let {Tt} be a strongly continuous semigroup of stochastic operators on a C*-algebra A of operators on Ho and let (H, {jt}, {F(t}}) be a minimal Markov dilation of {Tt}. Suppose T is a bounded stop time with support in the interval [0,r], Then F(0)A(r, X, u) = Xu + F(0) / lT>sX(s,C{X),u)ds Jo for all X € V(C),u € Ho, where \{r,X,u)
(17.4)
is defined as in Section 16.
Proof. Let E = {tx, < t2 < ■ ■ ■ < tn} C [0, T] with tx > 0, tn < T. We have F(0) r + Jti
!
T{{ti,T])X(s,C{X),u)ds
= F(0) (' " " Ju
F(U)r({U,T\)X{s,C(X),u)ds
=
F(0) f *+I T((U, T\)F{tiMs,L(X),
=
F(0) [ '+1 Ju
=
F ( 0 ) T ( ( I , , T ] ) A ( ^ f'+1
u)ds
T((tuT})X(ti,T^ti(C(X)),u)ds Ts.t,(C(X))ds,u)
Ju = F(P)T((U, T\)KU, Tk+1-u(X) - X, u) = F(0)r((t i ,T])F(t i )(A(t i + i I X,u) - X{t{,X,u)) =
F(0)T((U,T})(X(ti+1,X,u)
-
\{thX,u)).
136 K. R. Parthasarathy Adding over i we conclude that F{0) f J o
lTE>sX(s,C(X),u)ds
=
F(0)J2T((ti,U+1})\(ti+1,X,u)-\(0,X,u) i=o F(0)X(rE,X,u)-Xu
=
where, at i — 0 under the summation sign r([0, ii]) operates and at i = n, r((£ n ,T]) operates. This proves (17.4) when r is replaced by the discrete stop time TE. Taking limit over the directed set of all such finite subsets E we obtain (17.4). □ Proposition 17.2. Let r be an arbitrary stop time. Then for any 0 < s < t < oo and X £ A ||A(T A t, X, u) - A(r A s, X, u)|| < 2||X|| \\T((S, OO))U||
Proof. Without loss of generality we may assume that r has support in a finite interval [0, T] and 0 < s < t < T. Choose a finite set t\ < t2 < ■ ■ ■ < tn such that 0
A t, X, u) - \(TE A S, X, u)
= r((U, T})jti(X)u
+ {T((U, U+I}) jti+1 (X)u + ■■■ + T((tt_ 2l * * - I ] ) J V . ( * > } +
T((tk_uT])jtk(X)u.
We have M((U,T\)iu{X)u\\
= \\jti(X)T((thT})u\\
< H^llllrC^.TDull
and ||r((iilii+i])jtj+1(X> + - - - + T ( ( i f c - 2 , 4 - i ] ^ 1 ( ^ ) « + r((tfc_i,T])^(X)U||2 =
( E \\ju+AX)r((tr,tr+1])u\\2}
+
\\jtk(X)T((tk-UT\)u\\'
r=i \\X\\2\\T((U,T\M2-
<
Since U = s we get ||A(T B A i, X, u) - \{TE A S, X, u)|| < 2||JC|| ||r((s, T})u\\. Taking limit over the directed set of all finite subsets of [0, T] we get ||A(r A t, X, u)-X(r
A s,X,u)||
< 2||X|| ||r((s,T])u||.
□ Corollary 17.3. For every u 6 Ho satisfying the condition T({OO})U there exists a vector X(T,X,U) € H such that X(r,X,u)
= 0 and X € A
= lim X(T At,X, u). t—>oo
Proof. Immediate.
□
Quantum probability and strong quantum Markov processes
137
R e m a r k . Suppose r is a n a r b i t r a r y s t o p t i m e a n d u € Tio satisfies t h e condition T ( { O O } ) U = 0. Prom (17.4) in T h e o r e m 17.1 we have F ( 0 ) A ( r A £ , X , u ) = X u + F(0) f
lT>s\(s,C{X),u)ds
Jo for every t > 0. By Corollary 17.3, t h e left h a n d side converges as t —> oo to t h e vector F ( 0 ) A ( T , X, u). We do n o t have good necessary a n d sufficient conditions for t h e existence of t h e limit lim t _, 0 0 fQ l T > s A ( s , C(X),u)ds. If /0°° | | r ( ( s , oo])u||ds < oo, it is easy t o see t h a t /■DO
/ Jo a n d therefore
/>CO
| | l T > , A ( s , £ ( X ) , u ) d s | | < \\£(X)\\
/ Jo
| | T ( ( S | o o ] ) u | | d s < oo
roo
F ( 0 ) A ( T , X,u)
= Xu + F ( 0 ) /
lT>sA(s,
C{X),u)ds.
Jo However, w e feel t h a t this should hold u n d e r much weaker conditions. ACKNOWLEDGEMENTS
I wish t o t h a n k Stephane A t t a l for providing me an o p p o r t u n i t y to present these lectures a t t h e Institut Fourier in t h e beautiful city of Grenoble w i t h its charming m o u n t a i n s a n d rivers. Many of t h e ideas in these notes arose from collaborations with B.V. R a j a r a m a B h a t and S t e p h a n e A t t a l as well as my lectures a t t h e Indian Sta tistical I n s t i t u t e , Delhi Centre, addressed t o Debashish Goswami a n d P a r t h a s a r a t h y C h a k r a b a r t h y . Financial s u p p o r t from t h e Indian National Science Academy in t h e form of a C.V. R a m a n Professorship is gratefully acknowledged. T h a n k s t o Stephen Wills for proof-reading a major p a r t of these notes. REFERENCES
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Parthasarathy
[BP 2]
B.V. Rajarama Bhat and K.R. Parthasarathy, Markov dilations of nonconservative dynami cal semigroups and a quantum boundary theory, Ann. Inst. H. Poincare, Probab. Statist. 31 (1995) no. 4, 601-651. [Bia] Ph. Biane, Calcul stochastique non-commutatif, in, "Lectures on Probability Theory," Ecole d'Ete de Probabilites de Saint-Flour XXIII, 1993, ed. P. Bernard, Lecture Notes in Math ematics 1608, Springer, Berlin 1995. [Dy 1] E.B. Dynkin, Infinitesimal operators of Markov processes, [in Russian] Teor. Veroyatnost. i Primenen. 1 (1956), 38-60. [Dy 2] E.B. Dynkin, "Markov Processes I & II," Springer-Verlag, Berlin 1965. [Enc] O. Enchev, Hilbert space-valued quasimartingales, Boll. Un. Mat. Ital. B (7) 2 (1988) no.l, 19-39. [EyH] T.M.W. Eyre and R.L. Hudson, Representations of Lie superalgebras and generalized bosonfermion equivalence in quantum stochastic calculus, Comm. Math. Phys. 186 (1997) no. 1, 87-94. [Gup] J.C. Gupta, Characterization of correlation matrices of spin variables, Sankhya, Ser. A. 61 (1999), no. 2, 282-285. [Hud] R.L. Hudson, The strong Markov property for canonical Wiener processes, J. Fund. Anal. 34 (1979) no. 2, 266-281. [HuP] R.L.Hudson and K.R.Parthasarathy, Quantum Ito's formula and stochastic evolutions, Comm. Math. Phys. 93 (1984) no. 3, 301-323. [Kra] K.Kraus, General state changes in quantum theory, Ann. Physics 64 (1971), 311-335. [Kii 1] B. Kiimmerer, Markov dilations on W*-algebras, J. Fund. Anal. 63 (1985) no. 2, 139-177. [Kii 2] B. Kiimmerer, Survey on a theory of noncommutative stationary Markov processes, in [QP3], pp. 154-182. [Me 1] P.-A. Meyer, Quasimartingales hilbertiennes, d'apres Enchev, in, "Seminaire de Probabilites XXII," eds. J.Azema, P.-A. Meyer and M. Yor, Lecture Notes in Mathematics 1321, Springer-Verlag, Berlin 1988, pp. 86-88. [Me 2] P.-A. Meyer, "Quantum Probability for Probabilists," Second Edition. Lecture Notes in Mathematics 1538, Springer-Verlag, Heidelberg 1995. [P] K.R. Parthasarathy, "An Introduction to Quantum Stochastic Calculus," Monographs in Mathematics 85, Birkhaiiser Verlag, Basel 1992. [PSi] K.R. Parthasarathy and K.B. Sinha, Stop times in Fock space stochastic calculus, Probab. Th. Rel. Fields, 75 (1987) no. 3, 317-349. [PSu] K.R. Parthasarathy and V.S. Sunder, Exponentials of indicator functions are total in the boson Fock space r(L 2 [0,l]), in [QP10], pp.281-284. [QP3] "Quantum Probability and Applications III," eds. L.Accardi and W.von Waldenfels, Lecture Notes in Mathematics 1303, Springer, 1988. [QP10] "Quantum Probability Communications X," eds. R.L. Hudson and J.M. Lindsay, World Scientific, 1998. [Sau] J.-L. Sauvageot, Markov quantum semigroups admit covariant Markov C* dilations, Comm. Math. Phys. 106 (1986) no.l, 91-103. [Sch] M. Schurmann, "White Noise on Bialgebras," Lecture Notes in Mathematics 1544, Springer-Verlag, Berlin 1993. [Spe] R. Speicher, Stochastic integration on the full Fock space with the help of a kernel calculus, Publ. Res. Inst. Math. Sci. 27 (1991) no.l, 149-184. [Sti] W.F. Stinespring, Positive functions on C*-algebras, Proc. Amer. Math. Soc. 6 (1955). [Vin] G.F. Vincent-Smith, Dilations of a dissipative quantum dynamical system to a quantum Markov process, Proc. London Math. Soc. (3) 49 (1984) no. 1, 58-72. INDIAN STATISTICAL INSTITUTE (DELHI C E N T R E ) , 7 SJS 110 016, INDIA
E-mail address: [email protected]
SANSANWAL MARG, N E W DELHI -
Quantum Probability Communications, Vol. XII (pp. 139-172) © 2003 World Scientific Publishing Company
LIMIT P R O B L E M S FOR Q U A N T U M D Y N A M I C A L S E M I G R O U P S — INSPIRED BY SCATTERING THEORY ROLANDO REBOLLEDO PREFACE. There is an extensive literature on Scattering Theory, the case of the so called "hamiltonian dynamics" being a well known subject for anyone interested in Mathematical Physics. Thus, I want to say it loudly, these lectures do not pretend to teach that subject even though it provided an important motivation for the current research. These pages are aimed mostly at providing an insight on some problems of the theory of "quantum open systems" which are inspired by scattering theory. The text is divided into five sections. The introductory section contains some well-known results from Scattering Theory just quoted here as motivation for the subsequent sections. Section 1 compares the behaviour of two quantum dynamical semigroups at a large time. Most of the results of this section are new and are published here for the first time. The classification of dynamics is the core of Section 3. These results have been obtained and published as a joint research with Luigi Accardi, Claudio Fernandez, Humberto Prado and myself. The ergodic properties of quantum Markov flows and semigroups are studied in Section 3, where I recover some old results of Frigerio, Dang Ngoc and others, introducing a notion of quantum recurrence due to myself. The final section is dedicated to explaining the convergence towards the equilibrium as it emerges from a joint work done with Franco Fagnola and published recently in the journal Infinite Dimensional Analysis and Quantum Probability. It is easy to understand that these notes have much of a collective work which I decided to organise as an expository review for the use of young researchers interested in this field. I am greatly indebted to all the coauthors mentioned before as well as to many other colleagues with whom I have had very fruitful discussions. In this respect it is worth mentioning that I learned about the work of Dang Ngoc thanks to Alain Guichardet. On the other hand, Kalyan Sinha became interested in the subject of contiguity (introduced in Section 1), when he visited our group in October 1998, and we started a research for extending the results presented in the current notes. Finally, I want to express my gratitude to Jan van Casteren, for his careful reading of a previous version and for his suggestions to improve the English text.
CONTENTS
0. Introduction 1. Comparison of the time behaviour of two semigroups 2. The classification of states 3. Ergodic properties of quantum dynamical semigroups 4. Convergence towards the equilibrium Acknowledgements References 139
140 142 151 157 163 171 171
140
Rolando
Rebolledo
0. INTRODUCTION
Mathematical Scattering Theory may be considered as a branch of Perturbation Theory. Indeed, given two different dynamics, one is interested in determining whether the limiting behaviour of one of them (the perturbed dynamics) may be equivalent to that of the other (the free dynamics). In Physics, the free dynamics is commonly associated with a semigroup of operators which represents, for instance, quantum particles not interacting with one another. On the contrary, the perturbed dynamics is given as a representation of a more complex system which includes in teractions. Collisions between particles is a prototype of interaction, however, this can occur between particles that repel one another, such as two positive (or negative) ions, and need not involve direct physical contact. As a result, from the mathematical point of view, two different kinds of problems arise: (1) The comparison between two given dynamics, to determine whether they have the same limiting properties, when time goes forward to ±oo; (2) The classification of all limiting dynamics. Similar questions have inspired some developments in the theory of (commutative) stochastic processes too, which serve as a guide for current research in quantum stochastic analysis. This allows to expect that this emergent theory will need to consider, as a part of its own development, the qualitative study of quantum dynamics motivated by Scattering Theory. We illustrate briefly the kind of properties which have been established in the customary setting of Scattering Theory in Quantum Mechanics, without pretending to exhaust the subject. We deliberately choose the simplest versions of well known results which may be found in any of the major reference books like [LaP],[ReS],[Yaf], [Pea],[Per],[AJS] and [New]. Wave operators. We are given a complex separable Hilbert space h and two selfadjoint operators HQ and H. They define two unitary groups Uo(t) = e~UH° and U{t) = e'ltH. Given a vector
= \\U*(t)U0(t)cf> - W+cf>\\.
Limit problems for quantum dynamical semigroups — inspired by scattering theory 141 So that, the problem of the existence of the wave operator is equivalent to that of finding the conditions under which the limit of U*(i)Uo(t) exists as a bounded operator. Two main different approaches have been followed to solve this problem. One, due to Birman, makes use of spectral analysis and the properties of trace class operators. The other, due to Cook, uses elementary properties of integration on Hilbert spaces. Indeed, if W(t) = U*(t)U0{t), it suffices to write \\{W(t)-W(s))4>\\<
[ \\(iHe%Hue-iHou
-
eiHuiHQe-iH
and since H and U(t) commute and the latter is unitary, one obtains \\{W{t) - W{s))4>\\ < f \\{H - H0)U0{u)cf>\\du. This is the famous Cook's inequality from where it is now easy to give sufficient conditions for the limit of W(t) to exist. Indeed that will be the case if the integral on the right hand side of Cook's inequality is well defined on the whole real axis. Later we will look for generalisations, of both the dynamics and the concept of wave operator, in the framework of Quantum Stochastic Calculus. Classifying the dynamics. Suppose a Hamiltonian H is given at the outset acting on h, which determines a unitary group Ut = exp(-iHt), t e t . Within the above framework, Stone's Theorem relates the abstract Schrodinger equation on Hilbert spaces, i ^
= Hip(t),
(0-1)
associated to the Hamiltonian H, with the time evolution of the state of a physical system represented by tp(t) = Uttp, where tp G h. In what follows we review the classification of states given in this particular framework by the spectral analysis for the operator H and by the approach given in [Per] and [Pea], which is based on the corresponding Scattering Theory for the above equation. Finite rank projections. The Hamiltonian H induces a spectral measure \iv for the pure state |y)(v|. According to Lebesgue's Theorem any Borel measure on the real line decomposes as a sum of three components: the atomic or pure point measure, the Lebesgue absolutely continuous, and the singular continuous part. This property yields a decomposition into mutually orthogonal (Ut;t > 0)-invariant subspaces of the given Hilbert space h = bpp(H) © h ac (iJ) © hsc(.ff), where h p p (/f), (respectively, hac(.H') and h s c (#)), denotes the space of all those vectors ip for which fi^ is pure point (resp. Lebesgue absolutely continuous, resp. singular continuous). Moreover, introducing bcmt(H) = bac(H) © hsc(.ff), the Wiener Tauberian Theo rem implies that (p G hcont(H) if and only if for any finite dimensional orthogonal projection w, lim - / \\ire-iHs
(0.2)
142 Rolando Rebolledo This vector ip is called an outgoing state in Scattering Theory. piHs
,
Here 7^(-) =
p—iHs
On the other hand, ip 6 hpp(H) if and only if for any e > 0 there exists a finite rank projection 7r£ and to > 0 such that
inf Ke-^VII > 1 - e,
(0.3)
£>to
These are the bound s t a t e s of Scattering Theory according to Perry (see [Per]) who proved that (0.2) and (0.3) provide an orthogonal decomposition of the set of all pure states. The evolution of supports. In the particular case of the Hilbert space L 2 (R n ), another classification of states is also available. Namely, the solutions of (0.1) are classified according to the evolution of their support in R". We let 7iy be the projection given by the multiplication with the characteristic function of the ball of radius r in R n . In the spirit of the work [Pea] written by Pearson; a bound state is defined to be a unit vector ip e L 2 (R n ) for which given any e > 0, there exists r = r(e) > 0 such that ||(/-7rr)e-iffV||<e
(0-4)
for all t e R. In contrast with the notion of a bound state, tp is called a scattering state whenever lim II 7rr e-iHtip\\ = 0,
(0.5)
and it is an absorbed state if for all r > 0, lim || (1 - 7rr) e-iHt
Setting the basic framework. The main concept to be used throughout the re mainder of these notes is that of a Quantum Dynamical Semigroup. Introduced by physicists during the seventies, Quantum Dynamical Semigroups (QDS) are aimed at providing a suitable mathematical framework for studying the evolution of open systems. Typically, a quantum open system involves a dissipative effect modeled through the mutual interaction of different subsystems. One commonly distinguishes between at least the "free system" and the "reservoir". The description of the evolu tion of an important class of open system, was achieved in the seventies by giving the explicit form of a Liouvillian map (or semigroup generator) ruling the dynamics. This was done, in the norm continuous case, by Gorini, Kossakowski, Sudarshan ([GKS]) and Lindblad ([Lin]), and further extended by Davies in [Da2]. Moreover, some important limit properties of open systems, like the approach to equilibrium, reveal their nature within the formalism of QDS.
Limit problems for quantum dynamical semigroups — inspired by scattering theory 143 In general a QDS can be defined over an arbitrary von Neumann algebra, as follows: Definition. A Quantum Dynamical Semigroup (QDS) of a von Neumann algebra M is a weakly*-continuous one-parameter semigroup {Tt)t>o of completely positive linear normal maps of M into itself which preserve the identity. In addition, it is assumed that % coincides with the identity map / . The class of semigroups defined over the von Neumann algebra M = B(b) of all bounded operators over a given complex separable Hilbert space h, is better known and we will remain in this terrain. In particular, as was mentioned before, several results on the form of the infinitesimal generator of these QDS are available (see eg. [Lin], [Da2], [Hoi]). We denote C the infinitesimal generator of the semigroup T, whose domain is given by the set D(C) of all X £ B(W) for which the w*-limit of t~l(%{X) — X) exists when t —> 0, and we define L{X) to be this limit. Moreover, in the above framework, a Markov Process is associated to each given quantum dynamical semigroup. The space h contains the "initial data" of the Quan tum Markov process determined by T . To define the corresponding flow, we need a bigger space which is the tensor product H = h ® r ( L 2 ( R + ; C ) ) where the first factor is the initial space and the second, the Fock space associated to L2(R+; C). In a more general setting, one may replace L 2 (R+; C) by L 2 (R+; k), where k denotes a complex separable Hilbert space, but at this stage I prefer to keep notations and structures as simple as possible. The exponential vector on r ( L 2 ( K + ; C ) ) , associated to a function / € L 2 (R + ;C), is denoted e(/), and £ denotes the dense set of those elements in the Fock space. Furthermore, we introduce a canonical projection E defined by Eu®e{f)
=u<8>e(0).
(1.1)
For simplicity we write the vector u
= Es\3t{X)),
j0(X) = X®I,(Xe
M).
(1.2)
144 Rolando Rebolledo The flow may be represented by means of a Quantum Markovian Cocycle V = (Vt;t > 0) in B{H), such that each Vt is a unitary contraction and jt{X) = Vt*(X ® I)Vt, {X 6 M, t > 0). As a result, the semigroup T is represented as Tt(X) = EV;{X
® J ) l / t £ , (X G M , t > 0).
If we denote, for all X £ B(H), 9t(X) = r{at)XT{at)\ where T(crt) is the second quantisation of the right shift operator (see the lectures of F. Fagnola), the cocycle property reads as follows
vt+s = vses(vt),
(s,t>o).
It has been fundamental to the theory the discovery and resolution of Stochastic Differential Equations (SDE) satisfied by Markovian Cocycles, (see for instance Fagnola's lectures in these volumes). We will state some specific SDE for markovian cocycles later, in terms of the canonical quantum noises introduced by Hudson and Parthasarathy: A(t)ue(f) A\t)ue(f)
= ( / f(s)ds)ue(f), Jo
(Annihilation process),
(1.3)
= - « e ( / + el[o,t])|£=o, (Creation process),
(1.4)
A(t)ue(/) = - i j-ue(exp(iel[ 0 ,t])/)| £ =o, (Gauge process),
(1.5)
2
forall/€L (R+;C),t>0. The wave map for two open systems. To prove the existence of a wave operator connecting two quantum dynamics is a problem in noncommutative integration the ory. In this section we will provide an illustration of this statement by building up the wave operator for two Markovian Cocycles. However, due to the dissipative nature of quantum open systems, in numerous cases such a construction could lead to a trivial wave operator. Indeed, think of two semigroups decreasing weakly to 0, they will have eventually the same behaviour in the weak topology. However, inspired from Classical Probability, other notions of asymptotic equivalence of two QDS may be introduced. This is a subject which I am currently investigating. Some of these new results are reviewed here. In particular, a notion of wave map for two QDS reveals more adapted to their property of complete positiveness. We begin by a most elementary and general result which interprets the product of two dynamics, leading to the construction of a wave operator, as an integration by parts identity. This makes use of the theory of Quantum Semimartingales as developed by S. Attal in these volumes. The wave operator for quantum semimartingales. To simplify this part of the expo sition, we restrict ourselves to the case k = C and adopt Attal's notations for the noises: a\ = A\t), a- = A(t), a°t = A(i), a? = tl.
Limit problems for quantum dynamical semigroups — inspired by scattering theory 145 Let 5 = {f, — ,0, x } and consider an adapted process (Wt) of bounded operators on the Fock space which admits the representation
wt = Y,fH^MAt>% SeSJo
(i.6)
on the whole of the space. Intuitively, for this family to have a limit W^ (in the sense that ||(W ( —W / 00 )e(/)|| —► 0 for all e ( / ) € £), it suffices that the representation holds as an integral on the whole interval [0, oo[. Following Attal's lectures, this means the following Proposition 1.1. The family (Wt) has a limit W^o if there exists a positive real function g which is integrable over [0, oo[ and the following inequalities are satisfied ■) G S, Ere(f) = fr --= e(/l[o,r]),
\\Wtfr-Wsfrf
< \\fr\\2 / g(u)du
(1.7a)
WUr-WJrtf
< ||/ r f fg(u)du
(1.7b)
J S
EsWtfr-WJr\\
< \\fr\\j
g{u)du.
(1.7c)
The proof is ommitted since it is a simple modification of results in Attal's lectures. Now, let two quantum semimartingales, regular in the sense of Attal, be given: V° and V1. Then the product Wt = V°Vtu is also a regular quantum semimartingale. Therefore, the limit of V°Vtu exists whenever this process satisfies conditions (1.7). The limit W^ corresponds to the wave operator associated to V° and V 1 . Writing integrals in differential form, the integrability of d(V°Vtu) over the whole interval [0, oo] may be analyzed via a simple integration by parts formula: d(Vt°Vtu) = dVt° V?* + V° dVtu + dVt° dV^, which may be rigourously interpreted giving the decomposition of dVt\ (i = 0,1) in terms of the fundamental noises leading to Proposition 1.1. Notice that the cocycles are continuous in the weak topology, hence no discontinuous terms appear in the integration by parts formula. As an illustration of this method, below we quote an example of two cocycles satisfying a particular type of differential equation, due to Fagnola and Sinha. The wave operator for quantum cocycles and the wave map. We come now to con sider two Markovian Cocycles. The existence of Quantum Markovian Cocycles has been analyzed by several authors. We refer to [Par], [Mey] and [Fa2] for a more de tailed study on the connection with quantum stochastic differential equations (EvansHudson flows). Consider cocycles V° and V 1 , associated to quantum dynamical semigroups T ° and T 1 respectively, and, as in the previous subsection, define Wt = V^V^*. We introduce a completely positive map D,t : H —> H by fi((y) = W?YWt, for all t > 0.
146 Rolando Rebolledo If we write $j(Y) = EVfYVjE, for all Y € H, where E is the projection (1.1), then Tj{X) = ty{X
= $}(Slt{X®I))
{t>0,XeM).
Definition. If the strong limit of (Wt) exists as t —» oo, we say that the wave operator W = W(V°, V 1 ) for the cocycles V° and V1 exists and is defined by W = limt Wt. We say that the wave map Q, : /3(7i) —> B('H) for the semigroups T°, T 1 exists if there is a bounded operator VK 6 B{Ti) such that
ft(X®I)
= W*{X®I)W,
and it satisfies
w- lim ($°(X ® /) - ^ ( a p f ® *))) = 0, t—»oo
where w-lim denotes the limit in the weak topology. We will see later under which conditions the existence of the wave operator for the two cocycles implies the existence of the wave map. We first analyze the existence of wave operators for the two cocycles in a particular case. In what follows, we do not consider the most general case of SDE for Quantum Cocycles, since the results (due to Fagnola and Sinha [FaS]) illustrate the general method contained in the preceding subsection, and could be improved in some specific cases. Consider two cocycles (V^), k = 0,1, which satisfy the stochastic differential equa tions: dVk = Vk ({Sk - I)dA{t) + Lk*dA(t) - LkdA\t)
+ {iHk - \Lk*Lk)dt),
(fc = 0,1), (1.8) are supposed to
with the initial conditions Vk = I, where the operators (Sk, Lk,Hk), be bounded in h; Sk is unitary and Hk self-adjoint,(fc = 0,1). An application of the construction of the stochastic integral (in quantum proba bility) yields the following result.
Theorem 1.2 (Fagnola-Sinha). The wave operator associated to the above cocycles exists and it is an isometry, provided that the stochastic differential equations (1.8) have unique unitary solutions and the following integrability conditions are satisfied: oo
1
IKS1*!.1 - S°*L°)Vauue(0)\\2ds
< oo,
/•OO
/ || (iiH1 - H°) - \L°*L° - \Ll*L} + L^SfiS^L1) Jo for all u in a dense domain in h.
Vsuue{0)\\ds < oo,
Contiguous semigroups and wave maps. The notion of wave map will be studied in connection with that of contiguity which is inspired by probability theory. We first introduce the definition of contiguous (classical) markovian kernels. Assume (E, £) to be a measurable space. Given a classical markovian kernel P : (E, £) —» [0,1], we denote Pf{x) — JE P{x, dy)f{y), for any x e E and any bounded
Limit problems for quantum dynamical semigroups — inspired by scattering theory 147 £ -measurable function / . On the other hand, we write P*(i or fiP for the action of P on a positive measure JJL, fj,P(dy) = JEfi(dx)P(x,dy). Moreover, the n-fold iteration of the kernel P is denoted Pn, (n > 0). So that (P n ) nS N is a discrete time semigroup which is an example of a discrete-time quantum dynamical semigroup. Definition. Given two markovian kernels P, Q over (E,£), we say that (Qn)n is contiguous to (Pn)n whenever Pn(x, An) —> 0 for all x E E implies that Qn(x, An) —> 0 for all x € E. We write (Qn)n < (P„)„. The property of contiguity is closely related to absolute continuity. Indeed, we can always decompose the kernel Q as follows (see e.g. [Nev]): Qn(x,A)=
[ Pn(x,dy)ZZ(y)
+Qn(x,{ZZ
= 00} n A),
(1.9)
JA
for all A £ £, x e E. Hence the following criterion, which is an easy extension of a well known result of Le Cam, holds in the above framework. Lemma 1.3. With the above notations, (Qn)n < {Pn)n if and only if both conditions below are satisfied: (i) For all x g E, lim sup / Pn(x,dy)ZZ(y)l{z*>N]{y) N^oo
(ii) For all x€E,
=0
JE
n
lim n JE Pn(x, dy)Z£(y) = 1.
Proof. Let x G E be fixed. Since for all A £ £ and all n G N, (1.9) yields
IPn{x,dy)Zxn{y)<\
0< JA
it follows that the positive random variables are Pn(x, -)-almost surely finite. More over, Qn({Z* > N}) = / Pn(x,dy)ZZ(y)llz.>N}(y)
+ Qn(x, {Z*n = 00}),
JE
and Qn{x, {N
00}) = J
Pn(x,dy)ZZ(y)l{zz>N}(y)-
Now, suppose that (Qn)n is contiguous to (P„)«. Therefore, Qn(x, {Z* = 00}) —> 0 and limivlimsup n (2„(a:, {TV < Z% < 00}) = 0. This implies (i). Condition (ii) follows easily from (1.9)too since 1 = Qn(E) = / Pn(x, dy)Z*{y) + Qn(x, {Z*n = cx^}). JE
Conversely, suppose that (i) and (ii) hold, and let (An)n be any sequence for which Pn(x,An) —> 0. Condition (i) implies that fAnPn(x,dy)Z%(y) -> 0. By (ii), this is equivalent to Qn(x,An) —> 0. □
148 Rolando Rebolledo Notice that condition (i) of the above lemma (which is indeed uniform integrability), implies that for any x G E, the sequence (Zx)n of random variables is tight. This means that there exists at least a subsequence which converges in distribution. Now assume (Pn)n (respectively (Qn)n) to be aperiodic and recurrent in the sense of Harris with invariant measure n (resp. v). Then, it is well known (see [Rev]) that for any probability measure r\ over (E,£), the sequence (r)Pn)n (resp. {r)Qn)n) converges in the norm of the total variation to fj. (resp. v). As a consequence, we obtain T h e o r e m 1.4. (Qn)n < {Pn)n if and only if v is absolutely continuous with respect to Proof. Indeed, if (Q„)„ < (Pn)n then, for all x € E, there exists a subsequence {Z*)k which converges in distribution to a random variable Zx and E^{ZX) = 1. Hence, given any x € E and A 6 £, we write the decomposition of the subsequence Qnk following (1.9): Qnk(x,A)
+ Qnk(x, {Zxk = oo} nA).
= f Pnk(x,dy)Z*k(y) JA
The second term on the righthand side converges to 0 because of condition (ii) of the previous lemma. Moreover, from condition (i) of the same lemma, together with the convergence in distribution of (Zxk)k to Zx and convergence in total variation norm of Pnk{x, •) to fi, we derive / Pnk{x,dy)Z*k{y)-,
f
fi(dy)Z'(y),
JA
JA
as k —> oo. Since Qnfc(a:, A) —> v(A), we finally obtain
u(A) = J Zx(yMdy). This relationship proves that, given any two points x\, x-i e E,
f Z*>(y),JL(dy)= [ JA
Z*>(y)p(dy).
JA
Therefore, there exists an element Z € L 1 ( £ , f ,/i) such that all the variables Zx are ^-almost surely equal to Z. Thus,
u(A) = J Z(yUdy), for all A 6 £ and v is absolutely continuous with respect to \i. Conversely, assume that Pn(x, An) —* 0 for all x 6 E and v is absolutely continuous with respect to /z. Then, v{An) —> 0 since / JA„
Z{y)fi{dy)-
f
Z(y)Pn(x,dy)^0,
JAn
where Z is the Radon-Nikodym derivative of v with respect to fi. The proof is then completed by noticing that Q(x, An) — v{An) —> 0. □
Limit problems for quantum dynamical semigroups — inspired by scattering theory 149 C o r o l l a r y 1.5. Assume that (Pn)n (respectively (Qn)n) is aperiodic and recurrent in the sense of Harris with invariant measure p (resp. v). Then the wave map fl of (Pn)n and (Qn)n) exists if and only if (Qn)n < (Pn)n. In this case the wave map is given by f2(/) = Zf = Z1?2 fZ1^2, for all bounded and measurable functions f, where Z is the Radon-Nikodym derivative of v with respect to \x and the expectation ofti(l) is one. Proof. Indeed, if (Qn)n
- PrSl(f)(x)\
< [Qnf(x) - 1//I + \vf -
Pnn(f)(x)\,
for all x 6 E. Notice that (Qn(x, •))„ converges to v (resp. (Pn(x, -)) n converges to p) in total variation and that v is absolutely continuous with respect to p,. It follows that \Qnf(x) - Pnn(f)(x)\ -» 0, (1.10) and fl is the wave map of (Pn)n and (Qn)nConversely, if the wave map exists and JE Q(l)(y)[i(dy) = 1, then PnQ(lAn)(x) —> 0 for any sequence (An)n of measurable sets for which Pn(x, An) —> 0. Hence, property (1.10) implies that Qn(x, An) -> 0 and (Qn)n < (Pn)n. D We now turn to the noncommutative framework. Definition. Given two quantum dynamical semigroups T° and T 1 , they are repre sented by V(X) = EVr(X ® [)V}E = $\(X ® / ) , and since they preserve the identity, the cocycles V, i = 0,1, are isometries. We say that T ° is contiguous to Tl (and we write T° « T 1 ) if EVt°*(wt)Vt°E con verges weakly to 0 for any family of projections (irt)t of B(Tt) for which EVtu (w^Vj1 E has 0 as a weak limit. The definition above states roughly that T°
= * ° ( P / (X)) + $°(Ql(X)) u
l
u
since Vt Vt = / . Put Wt = V°Vt We finally obtain ,
= EVrVtlVt0tP?(X)VW*VtE
+ $?(<#(X)),
and Qt(Y) = W*YWU for all operator Y e B(H).
7?(X) = QKSltiPXX))) + $°(QXX)).
(l.H)
150 Rolando Rebolledo Definition. In the above framework, we say that {Qt,$}) is uniformly tight if for any family of projections (wt)t of B(7i) for which EV^*{-Kt)V^E tends weakly to 0 as t —> oo, <&at(p.(jp\-Klp1t)) —> 0 in the weak topology. L e m m a 1.6. T°
+ EV°*q\*tq\V?E,
(1.12)
is true. Assume first that T°
we also have w-limt_oo EVt *qlntqlVt°E and (i) holds. Moreover,
= 0,
(1.13) 1
1
1
= 0. So that w-limt_100$t (nt(p( 7rtpt )) = 0
i = Tt°(i) = ^t(nt(Pl(i))) + ^(Qi(i)), and w-limt_,0o $°(Qj (/)) = 0 because of the contiguity hypothesis. This proves that w-lim t _ O 0 $J(n i (P( I (/))) = / , but it is worth noticing that under condition (i), the previous equality is equivalent to (ii) because of (1.13). Conversely, if we assume (i) and (ii), given any family of projections irt G B(Tt) for which EV^*-ntV^E —> 0 weakly, then the first term in equation (1.12) goes to 0 in the weak topology. Moreover, from (ii) one obtains w-lim^oo $°(Qj(J)) = 0 which implies w- lim EV?*q\mlXE = 0. t—*oo
Using this limit in (1.12) again, we finally conclude w- lim EV°*TTtV°E = 0. i—*oo
D Corollciry 1.7. Assume that (fij,$() is uniformly tight, that the wave map f2 for T°, T 1 exists. If in addition 0 preserves the identity, then the quantum dynamical semigroup T° is contiguous to T 1 . Proof. The proof is straightforward using the above lemma and the definition of the wave map. Indeed, since H(J) = I and w- lim (*}(«,(/)) - / ) = w- lim(*?(/) - &t{Sl(I)) = 0, t—»oo
t—»oo
it follows that conditions (i) and (ii) of Lemma 1.6 hold.
□
Assume now that both semigroups converge to the equilibrium in the following sense. For any X G M, Ttl(X) converges weakly to a completely positive limit T^(X) given by a representation T^X) = EVl*(X ® I^E, {i = 0,1).
Limit problems for quantum dynamical semigroups — inspired by scattering theory 151 Definition. With the above assumptions, we say that 7^, is absolutely continuous with respect to 7^, if the range of V°E is included in that of V1E. Theorem 1.8. We assume the hypothesis on the convergence towards the equilibrium as stated before and furthermore that (fi t , $J) is uniformly tight. Then T ° <3 T1 if and only if T^ is absolutely continuous with respect to T^. Moreover, in this case the wave map exists and £l(X ®I) = W*(X® I)W, where W = V°VU. Proof. Suppose first that T°
lim T?(X) = w- lim $ J ( ^ P O ) = * L ( « ( * ® -0), t—*oo
t—*oo
one obtains condition (ii) in Lemma 1.6 by taking X = I.
□
It is worth noticing that to prove the necessity of the absolute continuity in the previous theorem, we do not need the assumption on uniform tightness. On the other hand, the theorem obtained for Harris chains in the commutative case did not explicitly include a uniform tightness condition because a stronger topology was used to approach the equilibrium. 2. T H E CLASSIFICATION O F STATES
This section is aimed at analyzing the asymptotic behavior of a quantum dynamical semigroup by estimating the average time (the upper Cesaro limit) spent by its flow on a given state. This idea goes back to Ruelle (see eg. [Ru 1], [Ru2]), among others, and has been developed in the framework of Scattering Theory for closed systems (the Enss method). As an example, we provide a comparison and an extension of earlier results of Perry [Per] and Pearson [Pea] regarding the classification of pure states for a given hamiltonian dynamics as we mentioned in the Introduction. We choose the general setting of a von Neumann algebra M. to extend the former classification of states under the action of a given quantum dynamical semigroup T. The earlier results depended on a given filter of projection operators in M. This filter-dependent classification is also given in our framework, however we develop a new weak classification for the states of M. which makes no use of a specific filter. We show that in the case of the algebra of all the bounded operators on a given Hilbert space, the weak classification does not depend upon the choice of specific filters of projections. This provides a more intrinsic classification of quantum dynamics. Let M denote an arbitrary von Neumann algebra on a Hilbert space h. Then M+ denotes the cone of all positive elements of M. The predual M.* of M. is the subspace of M* consisting of all the a-weakly continuous functionals of M. Then a positive functional / € M* is normal if and only if / belongs to M, (see eg. [BrR], Lemma 2.4.19.). For such a normal element / of the predual there exists a trace-class operator pf such that
Limit problems for quantum dynamical semigroups — inspired by scattering theory 153 Notice that T can also be defined as a map T : Xi(h) x 2j(h) —» C since the quantum dynamical semigroup is defined over all M. The space Xi(h) x M. is endowed with the weakest topology that makes the map (p, X) — i > tr pX continuous, which turns out to be the product of the weak and the weak*-topologies. Proposition 2.1. The functional T is invariant under the action of the quantum dynamical semigroup . Moreover, T is weakly lower semicontinuous on Ii(h) x M+. Proof. The first property follows directly from the invariance under translations of the Lebesgue measure because sT(p,%(X))
=
sT(p,X).
Secondly, each sT is jointly continuous in S x M, so that for each T > 0, the map fT = sups T /, T'>T
is lower semicontinuous. Since the family (/ T ;r > 0) is decreasing and positive on Xi(h) x M+ it follows that its monotone limit is also lower semicontinuous. □ Furthermore, if we denote Q(X) = T(X, X), the following lemma follows immedi ately from the properties derived for the mean sojourn time. Lemma 2.2. The map p >-> \/Q(p)
is a convex function on the set of all states p.
Definition. We say that a family T of projections is a core filter if it satisfies the following properties: (1) T is a weak totally ordered family of projections, which converges weakly to the identity I on h; (2) The corresponding sets S{T) and S\(T) are weakly dense in S. We review below two important examples of core filters The core filter of Perry. Let T be the collection of all those elements of M. which are projections irn of finite rank n. We notice that T can be identified as a subset of Mt as well, since its elements have a finite trace. In this framework we say that an operator r\ E ^ ( h ) has ^"-compact support if there exists an element i r „ 6 f such that 7r n 7? = 77.
In such a case, we say that r\ is supported by 7rn. The set of all trace class operators with compact support is denoted T\fi{F). We call S\{T) the set of all positive elements in M* with trace < 1 which are linear convex combinations from elements of T. It is clear from their definition that both «S(JF) and Si(T) are included in T\C{T). The following lemma, from which Property 2 follows, is obtained using the sepa rability of h. The proof is straightforward and it is ommitted (the details are given in [AFPR]). Lemma 2.3. S\{T} is strongly dense in S(M*). Moreover, given any p 6 S, there exists a sequence rjn E 5(^ r ) which converges to p and r\n <w p, for all n > 1.
154 Rolando Rebolledo The core filter of Pearson. This is the filter T of the projections 7iy, (r > 0), defined as multiplication by the characteristic function of a ball of radius r centered at the origin of the Euclidean space Rn. This family clearly satisfies properties 1, 2 of core filters. Definition. Assume T to be a given core filter and p a fixed state. p is T—bound if for any e > 0 there exists nn E T such that T{P, 7TB) > 1 - 6.
p is ^-scattering
if T(p,xn)
for all irn € T. p is ^-singular
= 0,
if
0 < inf{T(p, 7rn) : nn € .F} < sup{T(p, nn) : TT„ e F } < 1. We denote by S* (resp. <S,f, resp. <Sf) the set of all F-bound (resp. scattering, resp. singular) states. T h e o r e m 2.4. The set of all states S can be decomposed as a disjoint union Proof. Since by definition <Sf and S^ are clearly disjoint, it suffices to remark that the complement of their union coincides with S^. □ Proposition 2.5. If F and Q are core filters such that F C Q, then Sf S°CSZ andSfCS*.
C Sfj,
Proof. Straightforward consequence of the definitions.
D
In our framework the classification provided by Perry in [Per], is based on the filter T of finite rank projections. Indeed, the manifold of pure states, which corresponds to the set of extremal points of S, is isomorphic to the Hilbert space h given at the outset. As we have seen, the filter T induces a partition of S into three convex subsets: <Sjf, S£, S^; whose sets of extremal points are respectively associated to the subspaces hb, hj and hs of h. Equation (0.2) in Section 0, implies that a pure state p = \ip)(ip\ satisfies
Limit problems for quantum dynamical semigroups — inspired by scattering theory 155 W e a k classification. The scope of this section is to obtain a filter-independent classification of states. Before proceeding we state a further property satisfied by the functional T. Proposition 2.7. Given any state p, Q(p) = T(p, p) = lim„ T(p, rjn) for all sequence (rjn\n > 1) in *5i(^r) such that r\n -<„ p and r\n converges weakly to p. Proof. The existence of such a sequence r/n is justified by Lemma 2.3. Since r\n <w p it follows T(p)r]n) < T(p,p), so that limsupT(p,?? n )
n
Moreover, T is weakly lower semicontinuous, hence T{p, p)
T{p,Vn),
n
and the proof is complete.
□
We should notice that the above proposition still remains true for any core filter T. Lemma 2.8. For any projection TT £ T and p & S, T(p,rr) = sup T(p,jy), where C(n) is the set of all r\ € S\(T) supported by n and such that \\TJ\\ < 1. Proof. Given p € S and 7r € T, we obtain sT(p,Tr) > sT(p,i])) for all r] € C(n) and any T > 0, thus T(p,Tc)>T{p,V). In addition, C{ir) is a convex weakly compact set, hence the lower semicontinuous map T) — i > T(p, 7?) attains its maximum on it. □ Definition. A state p is weakly bound (resp. scattering) if 0 < Q(p) < 1 (resp.
Q(p) = o). Theorem 2.9. A state p is weakly bound (resp. scattering) whenever there exists a core filter T for which p is T-bound (resp. T-scattering). Furthermore, the space is decomposed in a disjoint union S = owb U o w s , where <Swb (resp. iSws) is the set of all weak bound states (resp. weak scattering states). Proof. The decomposition of the set of states in a disjoint union <S = <Swb U <SWS is a direct consequence of the definition. Furthermore, assume T to be a core filter, then Proposition 2.7 holds and for any p € S, Q(p) =
limT(p,Vn), n
where r]n € S\{T) converges weakly to p and r\n -<w p.
156
Rolando
Rebolledo
Assume first that p £ Sf. Then for any n > 1, there exists nn 6 T such that T(p,nn) > 1 — 2 _ n . Moreover, we can choose a subsequence of (rjn)n, denoted as the whole sequence for brevity, such that r/n 6 C(-Kn) and T(p, i]n) > 1 — 2~l-n~1h So that, in the limit T(p,p) = 1. By a similar argument, if p € S% then T(p, rjn) = 0 for all n, and T(p, p) — 0. D Extremal points of iSwb and iSws are pure states which can be identified by their supports on h, defining corresponding subspaces h wb and hws of the given Hilbert space. Moreover, from Theorem 2.9 and Proposition 2.6 it follows that hb C hwb and hs C hws, which yield to the following proposition. Proposition 2.10. The space h is decomposed in a direct sum h = h wb © h ws . Spectral t y p e measures. If Vt denotes a unitary group and u,t{p) = V£pVt, then it is straightforward to verify that for all pure states p — \4>){<j>\ , we have %r{p%t{p)) = \{ck,Vt(j))\\ Since the function t H-> (0, Vt<j)) is of positive type there exists a positive finite Borel measure p such that the Fourier transform p(t) of p satisfies pit)={^Vt<j>).
(2.2)
Moreover p, is the spectral measure of the infinitesimal generator associated to the unitary group Vt. Thus according to Wiener's theorem (see [ReS] Theorem XI.114.) we obtain
l i m i / tr (pZ (p%t(p))dt \p(t)\2dt = J^\pp({x})\2 t(p))dt = T lim i^ ff r ,^0 T J Jo 0 xt^i m ^ ° ° 1 Jo ~ ° * Jo
T
We therefore obtain the following property. Proposition 2.11. Let p be a pure state in a von Neumann algebra M.
Then,
(1) p is weakly scattering if and only if the measure pp is nonatomic. (2) A weakly bound pure state p has a measure pp with a nontrivial atomic part. (3) Under Hypothesis (H) of the previous section, the flow Ttt(p) converges to p if and only if pp is a pure point measure. In the general case where p is an arbitrary state in M. we decompose p a s a linear convex combination of pure states pk that is p — Y^k=iPkPk- Then a direct computation shows that tr(pT» ( (p))=
J2 l
K&>WI 2 PiP,-.
Limit problems for quantum dynamical semigroups — inspired by scattering theory 157 Thus according to equation (2.2) above we get that there exists a finite collection of finite Borel measures {pk} where each ji^ depends on p^ such that
tr(p%t(p))
=
J2 \(h,Vt
> £rfl/&(*)l2k
We therefore have a lower bound for the mean quantum sojourn time: rIim
i T t r (p%t(p))dt > £ £ > * ( { * } ) |2pi
(2-3)
Proposition 2.12. On the other hand if p is a weakly scattering state, then the measure p,p has no atomic part. 3. E R G O D I C PROPERTIES O F QUANTUM DYNAMICAL
SEMIGROUPS
Ergodic theorems on von N e u m a n n algebras. During the seventies several au thors proved ergodic theorems for semigroups acting on a von Neumann algebra. Namely, in 1977 Frigerio ([Fr 1]) started a systematic study of the large time asymp totic behavior of semigroups of completely positive linear maps of a von Neumann algebra. He pursued this investigation in [Fr 2] while Watanabe ([Wat]), working inde pendently, published various ergodic theorems for semigroups of linear positive maps of a von Neumann algebra into itself in 1979. In all the earlier references, a crucial hypothesis was the assumption of the existence of a faithful normal invariant state. Later, in 1982, Frigerio and Verri ([FrV]), succeeded in obtaining an ergodic-type theorem with no assumption on the existence of a faithful normal invariant state. Indeed, their result establishes the convergence of the Cesaro mean for a suitable reduced semigroup of completely positive maps. We follow the articles of Frigerio and denote by .F(T) the set of fixed points of T in M.. If the existence of a faithful, normal, stationary state u is assumed, then F(T) becomes a von Neumann subalgebra of M. Moreover, this subalgebra is globally invariant under the modular automorphism group a", in the theory of Tomita and Takesaki (see [BrR], [Gui]), since a" and Tt commute. Therefore, there exists a faithful normal conditional expectation ET{-r^ which satisfies CE1: CE2:
ET{-r^ : M —> T{T) is linear, w*-continuous and completely positive, ET(?\I) = I, where I is the unit of M,
CE3: u o £^ T > = u, and E^r\aE^T\b)) M.
= EF(r\a)E^r\b),
for all a, b e
The above characterisation contains the Ergodic Theorem for QDS. Indeed ET{^ is unique since, given any other map E which satisfies CE1, CE2, C E 3 , it follows that E^m =Eo Ey^ = £ ^ r > o £ = £ . More precisely,
158
Rolando Rebolledo
Theorem 3.1. If ui is a faithful, normal state which is invariant under T, then there exists a unique normal conditional expectation Er<-7"> onto F(T). In addition, ET^ o Tt = £/^T> for allt > 0; for any element a € M, Er(-T)(a) belongs to the w*-closure co(T(a)) of the convex hull of the orbit T(a) = (7^(a)) t > 0 . Moreover, invariant states under the action of the predual semigroup (Ttt)t>o, are elements of the form
a = lim / n
fd\xn. Jo Given any probability measure fi on the positive real line (endowed with the Borel cr-field) one obtains (from C E 3 and the first part of the theorem): J°°Tt(a)dis(t)\
= u o ^
fr%{a)d^{t)\
w(E^T\a)),
=
where the integrals of operators are understood in the weak* sense. Due to the faithfulness of w and the above lemma, the latter equation shows that EJ7^(a) r> cannot belong to the complement of co(T(a)). So that E^' (a) 6 co(T(a)) and it is the unique element which belongs to -F(T). Finally, if ip is an invariant state defined on T(T), it follows easily that ip o ET^ is an invariant state on M. Let us prove the converse property. Indeed, given any invariant state ip on M. and an element a £ M, write E^7^(a) as in Lemma 3.2: ET{T)(a)
= w*-lim f
Tt(a)dnn(t).
Therefore, il){E^r\a))
= w'-lim /\p(T t {a))dn n {t)
= ^(o).
So that it suffices to take ip to be the state-restriction of I/J to T(T) proof. It is worth noticing that co(T(a)) n T{T) = {ET{?\a)} computed, for instance through the equivalent formulae: E^T\a)
= w*- lim i
/
Tt(a)dt
/•oo
=
w*-limA / A
~^°
Jo
e-xtTt(a)dt.
to conclude the □
and EF<:T\a)
can be
Limit problems for quantum dynamical semigroups — inspired by scattering theory 159 Definition. According to the classification of noncommutative dynamical systems developed by Dang Ngoc ([Dan], [Gui]), the QDS T is finite if there are enough normal invariant states, that is, if for every non-zero positive element X of M, there exists a normal state u in the space of fixed points J~{Tt) of the predual semigroup, such that u>(X) > 0. The QDS is infinite if it is not finite. Moreover, a given projection % 6 T{T) is said to be finite if the induced semigroup Ttn(X) = TrTt(X)n, X € TTMTC, is finite, where irMir is the algebra of elements of the form X = nX0n, X0 € MWe recall here for easy reference the following result of Dang Ngoc ([Dan]). For a given state tp, we denote by S(
160 Rolando Rebolledo Proposition 3.5. The map Z >—> E°\Z) sets up a bijective correspondence between 1 and F(T). Moreover, given X G F(T), Z G I may be chosen as Z = wMim
jt{X).
Proof. Indeed, i? 0 ' is a normal, faithful, completely positive conditional expectation and if Z is invariant, denoting X = E°^(Z), then for all t > 0, Tt(X) = E°\jt(X))
= E°\X
®I) = X.
Therefore, X £T(T). On the other hand, if X G .F(T), the Markov property (see (1.2)) implies that
MX) = E*(jt(X)),
(t>s),
so that (jt{X))t is a bounded martingale and the limit w*-limt j t ( X ) exists (see for instance Exercise 28.10, p. 246 in the book of Parthasarathy [Par]). Call Z this limit and notice that ES\Z) = js(X), for all s > 0. So that X = E°\Z). D Given an event or projection 7r of M, we denote JOO(TT) = w*- lim - /
js{n)ds,
whenever this limit exists. It is worth noticing that the latter limit exists as a bounded operator on TL as soon as the conditional expectation E^7"1 (n) exists. Indeed, the family of all Cesaro means of the flow is uniformly bounded in norm, so that it is relatively weak* compact. Moreover, since E 0 ' is faithful, the set of limit points of the above Cesaro mean family is reduced to a single element and E:F(-'1">(ir) — E^fJ^-K)). As a result, the operator j^ir) exists at least when the QDS is finite. We denote by R(w) the r e c u r r e n c e subspace of w, defined as the elements x G H for which joo(n)x = x, that is, R(ir) = Ker(j00(7r) — I). Intuitively, if R(ir) coincides with H, then % is "visited" by the flow infinitely often. Notice that joo(n) is an invariant element since JE^'CJ'OOM) = £^ T '(7r) 6 ^F(T). Definition. We say that the event 7r G M is recurrent if joo{if) — I\ it is transient if
lZ(ir) = 0. In general, joo(ir) does not need to be a projection. However, each jt(n) does, and if the w*-limit of jt(n) exists then it coincides with joo("") being a projection. After a result of Frigerio and Verri ([FrV]), the above limit exists in particular when T{T) is trivial. Theorem 3.6. If w is a faithful, normal state which is invariant under T, the fol lowing three statements are equivalent: (l)T{T)=CI; (2) J = CI; (3) Every event -K G M is either recurrent or transient.
Limit problems for quantum dynamical semigroups — inspired by scattering
theory
161
Proof. The equivalence between (1) and (2) follows easily from Proposition 3.5. Now, assume that (1) holds. If TT is any event, then £^ T '(TT) = E°)(joo(ir)) 6 T(T). Hence 1 yields E^^^TT) = U(ET^(TT))I = UJ(TT) I. As a result, one has two possibilities: either U>(TT) = 1 and n is recurrent, or UJ(TT) = 0 and the event is
transient. Finally, we prove that (3) implies (1). For this, we recall that -F(T) is a von Neumann algebra so that, any element of it being a linear combination of self adjoint operators, the projections in T(T) span a norm-dense subspace of TiT). Therefore, it suffices to prove that projections in T(T) are trivial if (3) holds. Take an arbitrary projection TT € T(T). Assume first that j 0 o( 7r ) = I- Since E°]Ut(Tr))j= T = ET{T\TT), one obtains that / = E°\(jZ(ir)) = TT. On the other hand, if JX(TT) = 0, the above argument implies n = 0. Therefore, in both cases TT is trivial. D Recurrence and sojourn time. We keep denoting a faithful normal, stationary state by ui. Even though, its existence will not be assumed unless explicitly stated. We recall here the definition of a mean sojourn time as stated in the preceding section. Given a normal state ip € Mt and a projection w G A4, the M e a n Sojourn Time of the QDS on IT, measured on the state
T(tp,
TT)
=
= V o E°\JZ(n))-
Theorem 3.7. The cones Sbd and Ssc are disjoint. Proof. Given ui e 5 b d , the conditional expectation E:FI-'T'\LO) exists. Indeed, since the orbit %(u>) is weakly compact, there is a unique element in co(%(w)) r\F(Tt), which is the conditional expectation E^^'^ui) (rephrasing the proof of Theorem 3.4). Therefore T(w,7r) = u o E^1-1"1 (TT), for all events TT. AS a result, for any TT 6 ^ ( T ) , we have T(U>,TT) = OJ(TT) and there exists at least one TT < I such that T(U),TT) > 0.
Therefore, u £ Ssc.
D
162 Rolando Rebolledo It is worth noticing that for a bound state ui, its conditional expectation E^r*\u>) provides an invariant state for the QDS. In particular, invariant states belong to iS bd . Proposition 3.8. A state tp is scattering if and only if its support projection S(tp) belongs to F(T) . If the QDS has a faithful scattering state, then there is no invariant state and all finite rank events are transient. On the other hand, if the QDS is finite there is no scattering stateProof If tp is scattering, then for all TT < / , tp o %(n) = 0 for almost all t > 0 and hence, for all t > 0 since t H-► tp o Tt(ir) is a continuous map. So that, given any 7r 6 J-{T), tp(rr) = 0, which implies that S(tp) is orthogonal to T[T). Conversely, if S(tp) is orthogonal to T{T), then given any finite rank projection IT, S(ip) is orthogonal to all elements of the convex closure co(T(tp)). It follows that tp o Tt{rr) = 0 for almost all t and all finite rank projection TT. Therefore, T(tp, TT) = 0 for all n < I. If there is a faithful scattering state tp, then for all TT 6 JF(T),7T < / , we obtain 7r = 0, since tp(n) = 0, (TT < I). Therefore e = 0 and there is no invariant state. In addition, for all n < I, JOO{K) = 0 and all events are transient. If the QDS is finite, then e — I and one cannot have a state with projection support orthogonal to e. So that there is no scattering state. □ Proposition 3.9. If u is a normal faithful bound state, then T(ui, TT) = u o ET^ Therefore, ir < I is transient if and only T(w, n) = 0.
(n).
Proof. The equality T(CO,TT) = u o E^'1"1 {-K) has been derived in the proof of Theorem 3.7. Furthermore, notice that j'oo(7r) is 0 if and only if UI(E:F^(TT)) = 0, which is equivalent to T{u>, TT) = 0.
□
Definition. The quantum flow is Harris recurrent if there exists a normal, faithful, invariant state UJ on M such that all events 7r for which CJ(TT) > 0 are recurrent. Proposition 3.10. If there exists a normal faithful invariant state and every event tr > 0 is recurrent, then the flow is Harris recurrent. Proof. It suffices to remark that faithfulness implies that all TT > 0 are exactly the projections for which W(TT) > 0. □ Theorem 3.11. / / the flow is Harris recurrent, the algebra of fixed points F{T) trivial, the QDS is finite and there is no scattering state.
is
Proof. It suffices to notice that under the hypothesis any event n is either recurrent (UI(TT) > 0) or transient (UI(TT) = 0). Theorem 3.6 then yields the conclusion. □ The closed quantum system. If the semigroup is given by Tt(X) = eitHXe-itH,
(3.2)
where H is a self-adjoint operator of B(b) and X € #(h), then our classification of states is an extension of the one used in Scattering Theory (see eg. [Per] p. 15). It
Limit problems for quantum dynamical semigroups — inspired by scattering theory 163 turns out that pure bound states ip in the sense of Perry lead to pure bound states IV'K^'I in our (weak) sense. Moreover, pure states that leave any compact subset in the time mean, in Perry's sense, correspond to our pure scattering states. Some other authors use the wording outgoing state for the same concept. Within this framework the crucial role played by the Wiener Tauberian Theorem is well-known (see [ReS], Theorem XI. 114). Indeed, to simplify notations, given a unit vector ip hi h, we write \ip)(ip\ for the projection in the direction of ip and the (tracial) pure state uty(-) = tr (\ip)(ip\ • ) induced by ip € h. Denote by £,(dx) the spectral measure of H and /i^(dx) = tr (\ip) (ip\t;(dx)) = (ip, £(dx)ip}. As in Scattering Theory, we define h pp (i7) (respectively, hcont(H), h ac (i7), hsc(H)) as the space of ip 6 h for which fiy is a pure point (respectively, continuous, absolutely continuous, singular continuous) measure and refer to these as the pure point (respectively continuous, absolutely continuous, singular continuous) spectral subspaces. Therefore, as we recalled in the Introduction, the following characterisation holds (see [Per], Theorem 1.2): T h e o r e m 3.12. Under the above hypothesis on ip and H, (1) ip E h pp (i7) if and only if for every e —* 0 there is a finite rank projection -wt such that sup||(/-7r£)e-i(/Vll<e-
(3.3)
(2) ip € hcontiH) if and only if for any finite rank projection w,
lim - I* \\ire-ltHipfdt = 0.
T
-"x> T Jo Equation (3.3) in the first part of the above theorem yields tr
(3.4)
(Ttt(\1>)(tl>\)irt)>l-e,
for all £ > 0, e > 0. This implies that the orbit {%,t(\ip)(ip\))t>0 is weakly relatively compact in /A*. So that, if ip 6 h pp , then \ip)(ip\ is a bound state in our sense. On the other hand, equation (3.4), implies that T(|^»)(^>|,7r) = 0 for all IT < I. Therefore, if ip £ h cont , it follows that \ip)(ip\ is a scattering state in our sense. If hcont (H) is trivial, there exists an orthonormal basis (e„) of h consisting of eigen vectors of H. A state is then faithful and invariant if and only if it is associated to a positive trace-class operator p of the form p ~ 5^ n Pn|e ra )(e n | with pn > 0 for all n and YlnPn = 1- In that case a simple computation shows that w*-limsup t eltH-Ke~ltH is a finite rank projection if % < I, so that -K cannot be recurrent unless TT = I. 4. C O N V E R G E N C E TOWARDS THE EQUILIBRIUM
The equilibrium is represented by stationary states of a QDS. And the analysis of the evolution of an open system towards the equilibrium is related to ergodic theorems for QDS. In such a framework, the best general reference for the approach to equilibrium is that of Frigerio and Verri, [FrV] (see also the references therein). In what follows, I explain an improvement of the main result of [FrV] obtained in a joint work with F. Fagnola (see [FR2]), by establishing conditions on the convergence
164 Rolando Rebolledo towards the equilibrium based upon the explicit form of the generator of a QDS (without assuming norm continuity). Some basic hypotheses: (HI) The QDS has a faithful normal stationary state p. (H2) The infinitesimal generator of the semigroup induces a bilinear form T£.{X), (X € 13(h)), given by OO
(v, FC(X)u)
= {Gv, Xu) + J^{Lkv,
XLku)
+ (v, XGu),
(4.1)
*=i
where u, v S D(G); G, Lk, (k > 1) are operators which satisfy the following hypotheses: (H2.1) The operator G is the infinitesimal generator of a strongly continuous con traction semigroup on h and D(G) C D(Lk), for all k. (H2.2) (u,^ r £(/)u) = 0, where / denotes the identity operator, u,v E D(G). On the other hand, given G and the Lk satisfying the hypotheses (H2.1) and (H2.2), and a domain D C D(G), it is possible to build up a QDS solving the equation: (v, %(X)u) = (v, Xu) + f (v, TC(Ts(X))u)ds, (u, v 6 D). (4.2) Jo However, this equation does not determine a unique semigroup. Uniqueness in this sense is equivalent to the property of preservation of the identity as has been proved in [Fal] and [Dal]. Moreover, the results obtained in [ChF] allow us to easily conclude preservation of the identity in our case. The predual space of B(b) is 2i(h), the space of trace-class operators on h. We remark that Tt (respectively £) is the dual of an operator %t (resp. £«) defined on Ii(h), which is called the predual of Tt (resp. £ ) . When there is only one semigroup which satisfies the equation (4.2), it coincides with the dual of the so called minimal quantum dynamical semigroup introduced by Davies ([Dal], Sections 2 and 3). More precisely, under the hypothesis (H2), T is the unique solution to (4.2) if and only if the linear manifold V, generated by the one-dimensional projections \u)(v\ where u, v E D, is a core for £». As a result, a useful criterion to characterise the domain of the generator is ob tained. Lemma 4.1. Suppose that the Hypothesis (H2) holds as well as one of the equivalent conditions recalled before. Then the domain of C is given by the elements X E B(h) for which the map (v, u) i-> (v, FC(X)u) is norm-continuous on the orthogonal sum Hilbert space. The interested reader is referred to [FR 2] where the proof is given in full detail; it is ommitted here for the sake of brevity . Suppose that the semigroup T has a faithful stationary state p. Then there exists a conditional expectation X — i > %x,{X) in the sense of Umegaki, defined on the von Neumann algebra of invariant elements under the action of T. We recall an early result of Frigerio and Verri ([FrV], Theorem 3.3, p.281) proved in this framework.
Limit problems for quantum dynamical semigroups — inspired by scattering theory 165 Denote by M(T) the set of elements X e B(h) for which %(X*X) and Tt(XX*) = %{X)Tt(X*), for alH > 0
=
Tt{X*)Tt(X)
T h e o r e m 4.2 (Frigerio-Verri). / / the semigroup T has a faithful stationary state p and the set of fixed points ofT coincides with N(T), then w*- lim %{X) =
TM(X),
forallXeB(b). We will apply this theorem to a particular class of semigroups, which we introduce in the next section. Natural quantum dynamical semigroups. A Quantum Markovian Cocycle (QMC) can be associated to our quantum dynamical semigroup. There is a need for some additional hypotheses which are listed below: (H3) There exists a domain D which is a core for both G and G*; (H4) For all u € D, its image R(n; G)u under the resolvent of G, belongs to D(G*) and (nG*R(n; G)u\n > 1) strongly converges. Under the hypotheses (H3) and (H4) it follows that the minimal quantum dynam ical semigroup is connected with a unique contractive cocycle (see e.g. [Fa 2] and Fagnola's lectures) V = (Vt;t > 0) through the relationship %{X) =
EVtXV;E
(with E as usual given by (1.1)) and the equation:
dVt = Vt IJT[LldAk(t)
- LkdA\{t)\ + G*dt) ,
(4.3)
inD. The dual cocycle V = (Vt; t > 0), is given by Vt = 7ltVt*7lt , where TZt denotes the unitary time reversal operator on the Fock space, (t > 0), (see eg. [Mey], ch.VI, 4.9). In addition, the equation satisfied by V is obtained by replacing Lk by -Lk and G* by G in equation (4.3). This dual cocycle is associated to another semigroup T given by: %{X) = EVtXV*E, (t > 0). (4.4) The property of preservation of the identity by the semigroup and that of being an isometry for the cocycle, are related. This follows from a known result of Fagnola ([Fa2], Theorem 5.3) quoted below for further easy reference. P r o p o s i t i o n 4.3. Under the hypotheses (H2), (H3), (H4), the cocycle V (respectively the dual cocycle V) is an isometry if and only if the semigroup T (resp. T) preserves the identity. From now on we further assume, (H5) Both semigroups, T and T, preserve the identity. Definition. We say that a quantum dynamical semigroup is natural if it satisfies hypotheses (H2) to (H5).
166 Rolando Rebolledo Attaining t h e equilibrium. We begin by establishing a property of the space M{T). Proposition 4.4. Given a natural quantum dynamical semigroup, the space N(T) is a von Neumann algebra contained in the generalised commutator algebra of L = (Lk, LI; k > 1), denoted by L'. Proof. Let X be any element in N{T).
Then for all fixed t > 0,
%{X*X) = Tt(X')Tt(X).
(4.5)
The quantum flow j t associated to the cocycle V is introduced through the equation jt(X) = VtXVt*. This is a homomorphism and we obtain jt(X*X) = jt(X*)jt(X), which yields the identity Ejt{X*X)E
= Ejt{X*)jt(X)E.
(4.6)
The left-hand sides of both (4.6) and (4.5) are equal. Moreover, the right-hand side of (4.5) can be written Ejt(X*)Ejt(X)E; it follows that EJt(X*)EJt(X)E Subtracting gives Ejt(X*)E±jt(X)E onal to E. Thus, by postivity,
= Ejt(X*)jt(X)E.
E1Jt(X)E ±
and since Af(T) is self-adjoint, E jt(X*)E 2
(4.7)
= 0, where Ex denotes the projection orthog
2
Hence, for any / in L (R+;£ (N)),
= 0, = 0 too .
u,v€h,
0 = {Kte(f),Jt(X)ue(0))
and all t > 0: = (Vtve(f),
XVtue(0)),
and the equation for V yields 0 =
/ [(V,Gve(f),XV,ue(0)) Jo
+ (V.ve(f),
XVsGue(0))
d
+ Y,(VsLkveU\
XVsLkue(0))]ds
it=i
+ J2 I [(VsVe(0),XVsLkue(0)}
-
(V,Llve{0),XV.ue(0))}fk{s)ds,
k=iJo
where we have assumed that the component functions of / vanish for k > d. In particular, if a continuous / is chosen and the derivative of both members of the above equation is performed: 0 =
(VtGve(f),XVtue(0)) d
+
+
(Vtve(f),XVtGue(0))
~ J2{VtEkve(f),XVtLkue(0))
d
+ ^{Vtve(0),XVtLkue(0))
-
{VtLlve(0),XVtue(0))}fk(t).
Limit problems for quantum dynamical semigroups — inspired by scattering theory 167 Now, for all k > 1 fixed, we choose a function / such that /fc(0) ^ 0, and make t —■* 0. Then the previous equation becomes (v, XLku)
- {L*kv, Xu) = 0
From (4.8) w e first deduce that Xu € D(Lk) u G D(G). In addition,
(v,XLku) =
(4.8)
for all k > 1, u G D, thus, for all
(v,LkXu),
from which we obtain for all k > 1, XLk C LkX, since Lk is the closure of its restriction to the domain D(G). Similarly, X*Lk C LkX* is proven and it follows (see e.g. [Yos], Theorem 3, p.195) that
L;X = (x*Lky D {Lkx*y D XL\ for all k > 1 a n d the proof is over.
D
It is an interesting problem to determine under which conditions the equality of 7V(T) and L' holds. To answer this question, we first notice that from the equality C{I) = 0 it follows that G can be decomposed as a sum oo
fc=i
where the series weakly converges in D and if is a symmetric operator defined on that domain. Theorem 4.5 (Fagnola-Rebolledo [FR2]). A natural quantum dynamical semigroup which satisfies ( H i ) converges in the sense that w*- lim Tt(X) =
T^X),
t—*oo
for all X G B(h), algebra CI.
whenever the generalised commutator L' is reduced to the trivial
Proof. This result follows easily from the Theorem of Frigerio and Verri, since N{T) is contained in L'. D Proposition 4.6. Under the above hypotheses, if in addition the closure of H is self-adjoint, then the set J-{T) of fixed elements for the semigroup is given by F(T) = {Lk,Lt,H;
k>l}'.
168 Rolando Rebolledo Proof. Since JF(T) C Af(T), and N{T) is contained in L', it follows that T(T) So that, for all X G T{T) and any u,v G D, 0 =
(v,FC(X)u) CO
OO
Jb=l
k^l
OO
OO
+ | ^ { L f c W , XLku)
- I ^ ( w , XL£L fcM )
t=i
fc=i
-(iHv,Xu)~
{v,iXHu).
We now study the right-hand side of the above equation. Since XLk first two terms cancel and the computation (Lkv,XLku)
C L'.
= (X*Lkv,Lku)
= {LkX*v,Lku)
=
C L^X, the
{X*v,L*kLku),
shows that the third and fourth terms cancel as well. From the above we deduce that XH C HX. Therefore, X belongs to {Lk, L*k, H; k > 1}'. Conversely, if X £ {Lk,L*k,H; k > 1}', the equation for (w,J"/!(X)u) gives 0 and from Lemma 4.1 we obtain that X is a fixed point of T. □ Proposition 4.7. For any natural quantum dynamical semigroup which satisfies (HI) and either (a) H is bounded; or (b) H is selfadjoint and eitH(D) C D(G), then, N(T) = {Lk,L\; k>iy. Proof. Consider first the case (a) of a bounded operator H. Then X G D(£), and {v,TC{X)u)
= {v,i(HtX-XH)u),
(4.9)
so that, %(X) = eimXe~im.
(4.10)
Since L' is a '-algebra, it contains both X* and X*X. those elements, and Tt{X*)Tt{X)
= eiHtX*Xe~iHt
=
Then (4.10) holds as well for Tt(X*X),
for all t > 0. Therefore, X G AA(T). Now, in case we have hypothesis (b), H being self-adjoint and exp(itH)(D) C D(G), take u,v € D. The algebra M{T) is invariant under the action of the semigroup by its own definition. Then, -^-(v, e~isHTt.s(X)eisHu) as
=
(iff e* 4 ^,
%-s{X)eisilu)
+{eisHv,Tt_s(X)(iH)eisH) =
0.
-
(e™Hv,FZ{Tt^s{X))eisHu)
Limit problems for quantum dynamical semigroups — inspired by scattering theory 169 It is worth noticing that the hypothesis (b) on the core D implies that eltHu G D(H). Prom the a b o v e equations it follows that (v,e-isHTt_s(X)eisHu) is constant i n s. Therefore X £ J V ( T ) .
D
The corollary which follows is easily derived from the propositions and theorem above. Corollary 4 . 8 . For any natural quantum dynamical semigroup which satisfies (HI), convergence towards the equilibrium holds if {Lk,L*k,H;k > 1}' = {Lk,L*k;k > 1}' and either (a) H is bounded; or (b) H is selfadjoint and eitH{D) C D(G). Remark. T h e sufficient condition obtained for proving the convergence towards the equilibrium is necessary, at least for a wide class of operators H, as we state in the following t h e o r e m . Theorem 4 . 9 . Under above notations, given a natural quantum dynamical semi group which satisfies (HI) and for which H is a self-adjoint operator with pure point spectrum and either (a) H is bounded; or (b) H is selfadjoint andeitH(D) C D(G). Then % converges towards the equilibrium if and only if {Lk, L\, H; k > 1}' = {Lk, L\; k > 1}'.
(4.11)
Proof. From t h e corollary before, (4.11) is a sufficient condition for the convergence towards the equilibrium. We will prove below that it is a necessary condition as well. Indeed, the assumed hypotheses imply that eitHXe-itH,
Tt(X) =
for all X E A/"(T). For any two different eigenvalues A and /J, of H, choose corre sponding eigenvectors v and u in h. Then, (e-itHv, Xe~UHu)
=
eit{li-x\v,Xu),
converges as t —+ co. Therefore, [v, Xu) = 0, and X commutes with H. Conse quently,
{Lk,Ll,H;k>l}'
=
{Lk,Ll;k>iy.
n Remark. Theorem 4.5 provides a complement to Theorem 1.8 of Section 1. That is, given two natural quantum dynamical semigroups T°, T 1 which satisfy the hy potheses of Theorem 4.5, if in addition (Qt,$j) is uniformly tight, then T° o T 1 if and only if T£ is absolutely continuous with respect to 7^,.
170 Rolando Rebolledo Examples. The asymptotic behavior of the Jaynes-Cummings model in quantum optics. In [FRS] and [FR1] the quantum dynamical semigroup associated to master equations in Quantum Optics (see [JaC]) has been obtained. The initial space is h = ^2(N) endowed with the creation (resp. annihilation) operator at (resp. a), and the num ber operator denoted N. In addition, the coefficients G and Lk (k = 1 , . . . ,4) are given by the expressions L\ = jia, 1/2 = Aaf, L3 = Rcos(
R a ^ ^ ^ l , G = ~ ^ L l L vaa< r~:
,
k
where the parameters
Poo = y j 7 r n | e n ) ( e n | , n=0
where (7Tn)n>o is the sequence defined by TO = C,
n
A^ + n, tfW^v/A:) A ft, -r MU i ^ v K 31 p
.
(n^1).
and c is a suitable normalisation constant. The remaining hypotheses showing that the semigroup is natural have been checked in [FRS] as well. Now, to verify the hypotheses of Theorem 4.5, it suffices to study the action of operators on the standard basis (e m ;m > 0) of h. In particular, it brings about a recurrence relationship among the elements of the basis from which it follows that (er,Xem) — 0 for all elements X of the generalised commutator algebra of Lic,L*k, (* = 1,...,4). Corollary 4.10. The semigroup of quantum optics introduced before approaches the equilibrium in the sense of the w* -topology, as t —» 00. As a trivial consequence of the above corollary, the ergodic means of the semigroup converge in the w*-topology. This result had been stated in [FR 1] with a different direct proof. A class of examples with a non-trivial fixed point algebra. Keeping the notations on spaces and operators of the above example, consider a quantum dynamical semigroup with generator given by In = a(N), Lk = 0, (k ± 1), H = (3{N), with a and j3 given functions, a assumed to be injective and (3 real-valued. So that, any faithful state which is a function of the number operator is an invariant state. In
Limit problems for quantum dynamical semigroups — inspired by scattering theory
171
addition t h e algebras {Lk,L*k,H; k > 1 } ' a n d {Lk,L*k; k > 1 } ' coincide with {N}' if and only if t h e s u p p o r t of /3 is included in t h a t of a. Therefore, the hypotheses of t h e Corollary are satisfied, whenever the s u p p o r t of (3 is included in t h a t of a and t h e semigroup converges towards t h e equilibrium.
ACKNOWLEDGEMENT
T h e Grenoble S u m m e r School on Q u a n t u m Probability was a great success, not only for t h e high scientific level of all t h e lectures, b u t for t h e friendly atmosphere developed a m o n g t h e participants. T h e kind hospitality of our colleagues of t h e University of Grenoble, especially t h e chief organiser, Stephane A t t a l , stimulated b o t h scientific collaboration and friendship. I want to express my deep g r a t i t u d e to all of t h e m . This research has been partially s u p p o r t e d by t h e program " C a t e d r a Presidencial en Ciencias 1999" a n d F O N D E C Y T grant 1990439.
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[AcC]
L. Accardi and C.Cecchini, Conditional expectations in von Neumann algebras and a the orem of Takesaki, J. Fund. Anal. 45 (1982), 245-273. [AFPRJ L. Accardi, C. Fernandez, H. Prado and R. Rebolledo, Mean quantum sojourn time, Open Syst. Inf. Dyn. 6 (1999) no. 3, 227-240. [AJS] W.O. Amrein, J.M. Jauch and K.B. Sinha, "Scattering Theory in Quantum Mechanics," Benjamin, New York 1977. [BrR] O. Bratteli and D. Robinson, "Operator Algebras and Quantum Statistical Mechanics I," 2nd Edition, Texts and Monographs in Physics, Springer-Verlag, 1987. [ChF] A. Chebotarev and F. Fagnola, Sufficient conditions for conservativity of quantum dynam ical semigroups, J. Funct. Anal. 118 (1993) no. 1, 131-153. [Dan] N. Dang Ngoc, Sur la classification des systemes dynamiques non commutatifs, J. Funct., Anal. 15 (1974), 188-201. [Dal] E.B.Davies, Quantum dynamical semigroups and the neutron diffusion equation, Rep. Math. Phys. 11 (1977) no. 2, 169-188. [Da2] E.B.Davies, Generators of dynamical semigroups, J. Funct. Anal. 34 (1979), no.3, 421432. [Eva] D.E.Evans, Irreducible quantum dynamical semigroups, Comm. Math. Phys. 55 (1977) no. 3, 293-297. [Fa 1] F. Fagnola, Unitarity of solutions to quantum stochastic differential equations and conser vativity of the associated quantum dynamical semigroup, in [QP 7], pp. 139-148. (Fa 2] F. Fagnola, Characterization of isometric and unitary weakly differentiable cocycles in Fock space, in [QP8], pp. 143-164. [FR1] F. Fagnola and R. Rebolledo, An ergodic theorem in quantum optics, in, "Contributions in Probability. In memory of Alberto Frigerio," Forum, Udine 1996, pp. 73-85. [FR 2] F. Fagnola and R. Rebolledo, The approach to equilibrium of a class of quantum dynamical semigroups, Infin. Dimens. Anal Quantum Probab. Relat. Top. 1 (1998) no. 4, 561-572. [FRS] F. Fagnola, R. Rebolledo, and C. Saavedra, Quantum flows associated to master equations in quantum optics, J. Math. Phys. 35 (1994) no. 1, 1-12. [FaS] F. Fagnola and K. Sinha, Scattering theory for unitary cocycles, in, "Stochastic Processes," A Festschrift in Honour of G. Kallianpur, Springer-Verlag, 1993, pp. 81-88. [Fr 1] A. Frigerio, Quantum dynamical semigroups and approach to equilibrium, Lett. Math. Phys. 2 (1977) no. 2, 79-87.
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Rolando
Rebolledo
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FACULTAD DE MATEMATICAS, PONTIFICIA UNIVERSIDAD CATOLICA DE CHILE, CASILLA SANTIAGO 22, CHILE
E-mail address: rrebollei5mat.puc.cl
306,
Quantum Probability Communications, Vol. XII (pp. 173-194) © 2003 World Scientific Publishing Company
A S U R V E Y OF O P E R A T O R A L G E B R A S JEAN-LUC SAUVAGEOT
CONTENTS
0. Complex Banach algebras 1. C*-algebras 2. von Neumann algebras 3. Modular theory 4. Conditional expectations References
173 174 180 186 191 194
0. C O M P L E X BANACH ALGEBRAS
0.1. Spectrum in Banach algebras. Let B be a complex Banach algebra with unit Is, thus B is a complex Banach space equipped with an algebra structure such that ||66'|| < ll&llll&'H V6,6' e B. We suppose moreover that | | 1 B | | = 1. Definition. The spectrum of an element b in B is defined by sp(6) = {z € C : Z\B — b is not invertible}. Note that, for any z with \z\ > |j6||, {zlg — &)_1 = z~l Y^=oz~nbn exists, which proves that the spectrum of 6 is contained in the closed disc with center 0 and radius ||6[|. Note also that the spectrum depends only on the algebra structure, however the following two lemmas are true only in Banach algebras. Lemma. For all b in B, sp(6) is a compact nonempty subset
ofC
Definition. The spectral radius of an element b of B is given by p(b) = sup \z\. 26sp(6) n
Lemma. One has p[b) = limn_^00 ||6 |[". In particular, p(b) < \\b\\. 0.2. Holomorphic functional calculus. Let P be a polynomial with complex coefficients. For b in B, one defines P(b) = anbn + ■ ■ ■ + aib + a0lB if P(z) = anzn -\ h a.\z + aoLet Q(z) = j3(z — z-i)... (z — zm) be another polynomial, with roots z\,...,zm outside the spectrum of b. Then Q(b) = (5{b — Z\1B) ■ ■ ■ {b — Zmls) is an invertible element of B, and one can define
~(b) = PibYQib)-1173
174 Jean-Luc Sauvageot In other words, there is a natural definition of f(b) for / a rational function with poles not belonging to the spectrum of b. Now, let / be a holomorhic function defined in a neighbourhood Q of the spectrum of b. Let 7 be any curve in Q\sp(b) making one turn around sp(6) in the direct sense: the Cauchy integral formula says /( z o) = 5 I - ff(z)-+-dz
Vz0 e sp(6)
and one defines analoguously
f(b) =
^-Jf(z)(zlB-b)-ldz.
Check that f(b) does not depend on the choice of 7 as specified above. One gets a holomorphic functional calculus which is additive [/(&) + g(b) = (f + g)(b)}, multi plicative \f{b)g(b) = (fg)(b)], continuous [if/„ —> f uniformly on a neighbourhood of sp(b), then fn(b) —► f(b)], and coincides with the natural definitions above in the case of a polynomial or a rational function. These four properties together characterise the holomorphic functional calculus. 1. C-ALGEBRAS
1.1. Definition and first spectral properties. Definition. A C*-algebra A is a complex involutive Banach algebra which satisfies what we shall call the spectral condition \\a'a\\ = H
2
VaeA
Explanations and comments: We do not assume that A has a unit; A is equipped with an involution, i.e. a map A B a 1—> a* 6 A which is additive, antilinear [(Aa)* = Aa*,A € C,a G A], involutive [(«*)* = a,a 6 /I], inverting the products [(aft)* = b*a*,a,b 6 .4] and isometric. In order to understand the spectral condition, assume A has a unit (if not, see next subsection) and first take a in A selfadjoint (a = a*); the property can then be written !l a2 ]| = ll a l| 2 ; so> by an obvious induction, ||a 2 "|| = ||a|| 2 " Vn € N and ||a|| = p(a) by the second lemma in Subsection 0.1 above. For general a, apply the previous result to a*a and get ||a|| 2 = ||a*o|| = p(a*a). In both cases, the norm can be computed from a spectral radius, and depends only on the involutive algebra structure. Let us point out two consequences: (a) there is only one norm on A which makes it a C*-algebra (b) any unit preserving *-morphism (i.e. morphism of involutive algebras) be tween two C*-algebras with unit A and B is continuous and a contraction, because it decreases the spectral radius. What if A has no unit?
A survey of operator algebras 175 1.2. Adding a unit. Lemma. If A has no unit, there exists one and, up to isomorphism, only one C*algebra with unit A which contains A as a codimension 1 ideal. Sketch of a proof. As a vector space, A = A © C. One identifies A with its image A © 0, and sets 0^ ffi l c = 1^, so that any element of A can be written in a unique way a + Al^ with a in A and A in C. The laws of involutive algebra are given by (o + Al^)* = o* + Alyj, and (a + Al^)(6 + /il%) = ab + Xb + /j.a + X/J.1^,
a, 6 e .A, A, ^ e C.
The norm is \\a + MA\\ =
sup
||a& + A&||
6(=AI|6||=1
(the main difficulty is to show that it coincides with the original norm for elements of A in A). For a in A, one can now define sp'(a) and p(a) as the spectrum and the spectral radius of a in A, and all what has been said before remains true: ||a|| = p{a*a)z for any a (\\a\\ = p(a) if a = a*) and so depends only on the *-algebra structure; any ^-algebraic morphism between two C*-algebras is a contraction; moreover, it is an isometry as soon as it is injective. 1.3. First examples: abelian C*-aIgebras. Example I. Let Y be a compact space, and let C(Y) be the algebra of complex-valued continuous functions on Y with the uniform norm, and obvious involution / >—> / . Then C(Y) is a C*-algebra. Example II Let Y be a locally compact space, and let CQ{Y) be the algebra of complex-valued continuous functions on Y which vanish at infinity, with the uniform norm and involution />—»/. Then CQ(Y) is a C*-algebra. ['Vanishing at infinity' means: for any e > 0, {y e Y : \f{y)\ > s} is compact.] These two examples are universal: Theorem (Gelfand's representation theorem). Let A be a commutative
C'-algebra.
I. If A has a unit, then there exists one and, up to a homeomorphism, only one compact space Y such that A is isomorphic to C(Y). II. If A has no unit, then there exists one and, up to a homeomorphism, only one locally compact space Y such that A is isomorphic to CQ(Y). This representation theorem explains why nonabelian C*-algebras are heuristically considered as noncommutative compact or locally compact spaces, and their theory can be called noncommutative topology.
176 Jean-Luc Sauvageot 1.4. Continuous functional calculus in C*-algebras. /. Let Abe & C*-algebra with unit (no longer abelian). For a in A and / holomorphic in a neighbourhood of sp(a), one can define f(a) (see 0.2). Now take a to be a normal element of A, which means that a commutes with its adjoint: aa* = a*a (note that any selfadjoint element is normal). Then B = C*(a, a*, 1A), the closed involutive subalgebra of A generated by a and the unit, is an abelian C*-algebra with unit, so that, by the previous representation theorem, B is isomorphic to a C*-algebra C(Y), with Y a compact space. Moreover, in this particular case, one can take Y to be the compact space sp(a), and chose the isomorphism n in such a way that n(a) is the inclusion sp(a) 3 Z H z € C (which implies that ir(P(a)) is the function sp(a) 3 z — i ► P(z) 6 C if P is a polynomial). For / a continuous complex-valued function on the spectrum of a, define / ( a ) := 7r - 1 (/) £ A: one gets a continuous functional calculus for the normal elements of A, which is characterised by the following two properties: (i) it coincides with the obvious definition for polynomials, and with the Banach algebra holomorphic calculus if / extends as a holomorphic function in a neighbourhood of sp(a); (ii) one has ||/(a)|| = sup 26sp(a ) \f(z)\, for all normal a in A and / in C(sp(a)). Note that those two properties and Stone-Weierstrass density theorem allow us to view f(a) as the limit in A of a sequence {Pn(a)} of polynomials in a, as soon as {Pn} is a sequence of complex polynomials which tends uniformly to / on the spectrum of a. II. Let A be a C*-algebra without unit. All that has been said above applies in A: for a normal element a in A and / a complex valued continuous function on sp'(a), one can define f(a) in A. The only question is: when does / ( a ) belong to A ? If P is a polynomial, it is clear that P(a) belongs to a iff the constant term of P is zero, i.e. P(0) = 0. If / is such that /(0) = 0, / can be uniformly approximated on sp'(a) by polynomials which vanish at 0, hence f(a) € A. Conversely, if /(0) ^ 0, one has / = g + /(0).l, where 1 is the constant function equal to one and g vanishes at 0; as g(a) is in A, f(a) = g(a) + / ( 0 ) 1 ^ , does not belong to A. In other words, one can define f(a) in A for (and only for) a a normal element in A and / a function on sp'(o) which vanishes at 0. One can also, as above, consider the sub-C*-algebra B = C*(a,a*) generated by a in A, which is abelian (a is normal), hence isomorphic to CQ(Y), where Y is a locally compact space which, in this case, can be taken to be sp'(a)\{0}; the image of a being again the identity function sp'(a)\{0} B z — i » z € C. Again / ( a ) is the reciprocal image of / by this isomorphism. (Note that 0 is 'the point at infinity' of sp'(a)\{0}, and that continuous functions 'vanishing at infinity' on sp'(a)\{0} are nothing but continuous functions on sp'(o) which vanish at 0). As particular cases of this functional calculus, consider
A survey of operator algebras 177 (i) for a selfadjoint element a of A, sp(a) or sp'(a) are subsets of K and one can define \a\, a+ and a_ as the images of a under the functions t \—> |£|, and 11—> sup(i,0), t — i > sup(—£, 0). One has a = a+ — a_ and |a| = a + + a_ with a + a_ = a_a+ = 0. (ii) for any element a of A, a*a has a spectrum which is contained in the real positive half-line R + , and one can define again the 'modulus' of a: \a\ = (a*a)5. 1.5. More examples: 13(H) and its sub-C*-algebras. Example III. Let H be a complex Hilbert space (with scalar product (77, £) linear in £ and antilinear in rf). For any T in 13(H), the Banach algebra of all bounded operators on H, equipped with the uniform norm ||T|| = s u p ? g / / ^ | | = 1 \\T£\\, one defines the adjoint operator T* by the formula and it is well known that ||T*|| = ||T|| and \\T*T\\ = ||T|| 2 , which makes B(H) a C*-algebra. Example III'. Note that for H = Cn, we have 13(H) = Mn(C), the algebra of n x n square matrices with complex entries: the operator norm and the usual involution (which to each matrix associates the conjugate of the transposed matrix) makes M„(C) a C*-algebra. Moreover, any finite dimensional C*-algebra is a direct sum of matrix algebras. Example IV. Let A be a closed involutive subalgebra of B(H). Then A is a C algebra. This example is universal: we shall see that any C*-algebra is isomorphic to a C*-subalgebra of a 13(H), but not always in a canonical way. Let us give two classical examples. Example IV.A. Let T(H) be the space of all finite rank operators on H: it is a ^-invariant ideal in B(H), and its closure £C(H) := algebra of compact operators in H is a C*-algebra, as a ^-invariant closed ideal in B(H). It plays the role of an infinite dimensional matrix algebra. Example IV. B. Let G be discrete group and consider the Hilbert space
e(G) =
{Z:G^C:YJU9)\2<™}geG
For h in G, there is a bounded operator X(h) on l2(G) defined by the formula (A(/i)0(ff) = f(ff-1ft) 9,heG. The map G B /i 1—» X(h) G B(£2(G)) is called the 'left regular representation' of G. It is a group morphism from G into the group of unitary operators on H (an operator U in B(H) is unitary if U"U = UU* = lH)The C*-algebra C*T(G) generated by \(G) in B(f(G)) is called 'the reduced C*algebra' of the group G and plays a crucial role in harmonic analysis. For instance,
178 Jean-Luc Sauvageot if G is abelian, then its dual group G is compact and the reduced C*-algebra of G is isomorphic to the C*-algebra C(G) of continuous functions on G. This duality theorem is a generalisation of the Fourier transform. C*(G) can play the role of a 'dual' of G when G is no longer abelian. 1.6. Order Structure, s t a t e s , and t h e G N S c o n s t r u c t i o n . 1.6.1. Positive elements and order in A. Theorem. For A a C*-algebra, define A+ = {a 6 A : 36 € .4s.t.a = b*b}. Then A+ is a closed convex cone. [It is a nonobvious fact, not true in general complex involutive Banach algebras, that a sum b\bi + b\b^ can be written 6J63.j An element in A+ is called a 'positive element' of A. For an abelian C*-algebra C(Y) or Co(Y), positive elements are continuous functions with values in R + . Any *-morphism from a C*-algebra A into a C*-algebra B maps A+ in B+. As a positive element is selfadjoint, it can be viewed as a positive element in an abelian C*-algebra (see Subsectionl.4) and one can check that its spectrum is a subset of R + . The converse is true: a normal element of A with spectrum in R+ is positive. (Similarly, a normal element is selfadjoint iff its spectrum is a subset of R). One can also check that the elements a+, a_ (a self adjoint) and \a\ constructed at the end of 1.4 are positive: so that any selfadjoint element is a difference of two positive elements which commute (their product is 0). Moreover, any element a in A can be written as a linear combination a _ O±2L. _J_ £2ril 0 f two se if adjoint elements, hence as a linear combination of four positive elements: A+ linearly generates A. Those four elements no longer commute, unless a is normal. Order structure. For a and b in A, one writes a < b iff b — a belongs to A+. An important inequality is a*b*ba < ||&||2a*a for all a and b in A. Note that, for 0 < a < b in A, one has 0 < a 2 < b?, but in any one word C*algebra, one can find a and b satisfying 0 < a < b, such that the inequality a2 < b2 is no longer true. For example in M2(C), consider the positive definite matrices "2 l' "1 0" and b = 0 0 1 1 1.6.2. Dual order structure and states. Definition. Let A* be the topological dual of A. Then one defines the convex cone of positive linear forms: A*+ = {ui 6 A" : ui(a) € M+ Va £ A+}, and the convex set of states on A: S(A) = {OJ € A*+ : \\u>\\ = 1}. Important facts: 1. As an easy consequence of the Hahn-Banach Theorem, an element a of A belongs to A+ if and only if it takes positive values on A*+: a £ A+ <=> u/(a) € R + Vw € A*+. Moreover, for a in A+, \\a\\ = supuieS^u(a). Similarly, an element a of A is self adjoint if and only if it takes real values on positive linear forms, and again ||o|| = supweS(yi) |w(a)|.
A survey of operator algebras 179 2. If A has a unit, a linear form u> positive if and only if ||ai|| = OJ(1A). In this case, S(A) = {w e A* : \\UJ\\ < 1 and UI(1A) = 1} is a a-weakly closed convex subset of the unit ball of A*, hence is a-weakly compact. 3. If A has no unit, S(A) is no longer cr-weakly compact (however the convex set of positive linear forms with norm less than 1 is a-weakly compact). A linear form UJ is positive if and only if it extends to a positive linear form u on A with w(l;j) = ||w||. In particular, a state on A can be extended as a state on A. Remark. Let £ be a vector in the Hilbert space H: then B(H) 3 T — i > (£, Tf) 6 C is a positive linear form on 13(H) with norm ||£|| 2 , and a state if ||£|| = 1. Let n be a nondegenerate *-morphism of the C*-algebra A into B(H) . Nondegenerate means that the linear span of n(A)H is dense in H: if A has a unit, this morphism must send I A onto 1H; HA has no unit, then it extends to a *-morphism of A into B(H) which sends the unit of A onto 1#. In both cases, if £ is a vector in H, then A 3 a t-> (£,ir(a)£) is a positive linear form on A with norm ||£|| 2 . The Gelfand-Naimark-Segal construction asserts that any positive linear form on A can be obtained that way. 1.6.3. GNS construction. Theorem. Letui be apositive linear form on A. Then there exists a triple (fl B ,ir u ,4,) (essentially unique) where Hu is a Hilbert space, nu is a ^-algebraic morphism from A into B(HW), and £w is a vector in Hw with norm \\u>\\i such that (a) £w is cyclic for ixu, that is 7ru(y4)£w is dense in Hu; (b) u(a) = (£w, 7ru(a)£w) for all a in A. 'Essentially unique' means that if (H'u, w^, ^'u) is another triple with the same prop erties, then there will be a unitary operator U from Hu onto H'u such that U(,u — £'w and Uitu(a)U* = n'^a) for all a in A. Representations. One calls representation of A in a Hilbert space H a nondegenerate *-morphism n from A into B(H) [for 'nondegenerate', see the Remark in Subsec tion 1.6.2]. If A has a unit, nondegeneracy implies ^(1^) = 1#, and if A has no unit, that 7r extends to a representation 7? of A with n(1^) = 1# . A representation is said to be faithful if it is injective. As remarked in Subsection 1.2, it is an isomorphism between A and its image, a sub-C*-algebra of B(H). Corollary. Every C*-algebra has a faithful representation, hence is isomorphic to a sub-C*-algebra of a B(H). Proof of the corollary. Let X be any subset of S(A) such that o £ i + and w(a) = 0 Va; 6 X implies a = 0. As ||a|| = supugS(A)u>(a) (see Important Fact 2 in Sub section 1.6.2), such a subset exists; one could chose X = S(A), but in the case when A is countably generated, X can be chosen countable and we shall get a separable Hilbert space.
180 Jean-Luc Sauvageot Set, with notations of the theorem, H = 0 w € X Hu, and n = (B^ex77"'- f° r a m A, if w(a) = 0, then n(a*a) = 7r(a)*Tr(a) = 0 and so 7rw(a*a) = 0, which implies uj(a*a) = 0, for every UJ in X, hence a*a = 0 and so ||a|| = ||a*o||2 = 0, thus a = 0. Proof of the theorem. Assume first that A has a unit. One can make A a pre-Hilbert space with the positive sesquilinear form (6, a) >-> ui(b*a). Let Hu be the Hilbert space obtained after separation and completion, and let A be the canonical linear map from the pre-Hilbert space A into its completion Hu: one has (A(6), A(a)) = ui(h*a) for all a and b in A. From the inequality b*a*ab < ||a||26*b (see Subsection 1.6.1), we deduce uj(b*a*ab) < ||a||2w(6*6), which can be written ||A(a6)|| < ||a||||A(6)||, for all a and b in A. Now, for any a in A there exists a bounded linear operator iru(a) on Hu, with norm less than ||a||, which is characterised by 7rw(a)A(6) = A (aft), V6 G A. One can check easily that 7rw is a representation of A. Define £u = A(l /4 ) and you get the theorem. If A has no unit, apply the previous construction to the canonical extension of to as a positive linear form with the same norm on A (see Subsection 1.6.2). 2. VON N E U M A N N ALGEBRAS
2.1. S o m e topologies on
B(H).
2.1.1. Three natural topologies. One can define on B(H) the uniform topology, as sociated with the operator norm, which makes 3{H) a C*-algebra (see Example III in Subsection 1.5); the strong topology, associated with the family of seminorms {T H^ |[T^||} fe H: a net {Tj} converges strongly to T iff X;£ converges to T£ for any £ in H; and the weak topology, associated with the family {T — i > \(r),T£)\}$iVef[ of seminorms: a net {%} converges weakly to T iff (77, T£) converges to (T?,T£) for any £, rj in H. It will be necessary to consider a fourth topology, explained below in Subsection 2.1.4. 2.1.2. The ideal ^(H) of trace class operators. For T in B(H)+, (£,T|) £ R+, V£ G H (see Subsection 1.6.1), and the sum in R + U {+00}
one has
TY(r) = ^ ( e i , r e i > , ie/
does not depend on the choice of orthonormal basis {ej} i s / of H. It is called the trace of the operator T. Moreover, just as in the finite dimensional case, one has Tr(T*T) = Tr(TT*), VT € B{H). Lemma.
1. ForT in B{H), the following are equivalent: (i) Tr[(T*T)5] < +00 {for the definition of (T*T)i, see Subsection 14); (ii) T is a linear combination of positive operators with finite trace. For such T one then defines its trace by TY(T) = ^ < e i , T e i ) , iei
A survey of operator algebras 181 the series being absolutely convergent and the sum not depending on the choice of orthonormal basis. 2. Cl{H) = {T € B(H) : Tr[(TT)5] < 00} is an ideal in B(H), i.e. a vector space such that ST e CX{H) and TS E £>{H), VS1 e B(H), T € C\H). Moreover, one then has Tr(ST)=Tr(TS). 3 . T n Tr[(T*T)i] is a norm on C^H), which will be denoted ||.|| 1; and Cl{H) equipped with this norm is a Banach space. Moreover, one has |Tr(5T)| < \\S\\B{H)\\T\\^S
e B(H),T
e
C\H).
( Which implies that, for fixed S in B(H), the map T 1—> Tr(TS) is a contin uous linear form on the Banach space CX(H), with norm at most \\S\\.) The unbounded functional Tr, defined either on B(H)+ or on ^{H), is called the canonical trace on B(H). 2.1.3. Fundamental duality theorem. Cl(H)* = B(H). This means that, as a Banach space, B(H) is a dual space, and more specifically, that the map from B{H) into the topological dual {Cl(H), ||.||i)* of Cl{H), which to S in B(H) associates the continuous linear form T H-> Tr{TS) 1
on £ (if), is bijective and isometric. 2.1.4. Definition of the a-weak topology on B{H). It is the a(B(H), £ 1 (H))-topology. Note that the unit ball of B{H) is a-weakly compact. As we have already noted it is weakly compact, so the weak and the cr-weak topologies must agree on bounded subsets of B(H). The cr-weak topology is sometimes called ultraweak, but it is never theless stronger than the weak topology. Note also that the cr-weak dual of B(H) is C}(H): for any cr-weakly continuous linear form w on B(H), there exists a (unique) trace class operator T such that Tr[(T*T)2] = || w || and Tr(TS') = to(S)
VS* <E B(H).
2.2. von N e u m a n algebras. 2.2.1. von Neumann bicommutant
theorem.
Theorem. Let A an involutive subalgebra of B{H) containing the identity operator 1H. Then the following involutive algebras are equal: (a) (b) (c) (d)
the the the the the
strong closure of A in B(H) weak closure of A in B(H) a-weak closure of A in B(H) bicommutant A" of A. Moreover, the unit ball of A is strongly dense in unit ball of its a-weak (— weak = strong) closure.
182 Jean-Luc Sauvageot Comments. The last assertion is known as the Kaplansky's density theorem. Let us explain the word 'bicommutant'. If C is any subset olB(H), its commutant C is the set of all operators in H which commute with every element of C: it is a a-weakly closed subalgebra of B{H) which contains 1#; moreover, if C is ^-invariant, it is a a-weakly closed involutive subalgebra of B(H). The bicommutant C" is the commutant of C: it is a cr-weakly closed subalgebra (*-subalgebra if C is ^-invariant) of B(H) containing C and 1H, and the bicommutant theorem says that, in the *invariant case, it is the smallest one. 2.2.2. Definition of von Neumann algebras. A von Neumann algebra M is an invo lutive subalgebra of B(H), for some Hilbert space H, which satisfies one of the four equivalent properties: (a) M is strongly closed and contains 1H (b) M is weakly closed and contains 1// (c) M is cr-weakly closed and contains 1# (d) M is equal to its bicommutant M". Note that M is given with (at least) four topologies: a-weak, weak, strong and uniform. The unit ball is compact for the first two. With respect to the last one, it is a C*-algebra (see Example IV in Subsection 1.5) and inherits all the properties (functional calculus, order and dual order structure, GNS representation and so on) of a C*-algebra. But M has only two dual spaces: M*= dual of M with respect to the uniform norm, and M , = predual of M, provided by the next subsection: 2.2.3. Predual of a von Neumann algebra. L e m m a . Let to be a linear form on M. The following are equivalent: (i) w is strongly continuous, (ii) ui is weakly continuous, (iii) ui is a-weakly continuous. Moreover, if u is positive, the three are equivalent to (iv) if {xi} is a decreasing net in M+, then \irmo{xi) = u)(]im{xi)) Note that, by the compactness of closed balls, any decreasing net in M+ has a cr-weak limit which is also a lower bound, and a strong limit. Definition. The predual M, of M is the space of strongly (weakly, a-weakly) con tinuous linear forms on M. As a norm closed subspace of M*, it is a Banach space. Theorem. (M,)* = M. This means that the (natural) map from M into (M„)* which, to x in M, associates the linear form u i-> ui(x) on M*, is bijective and isometric. [It is a general fact that, if E is a Banach space and F a a-weakly closed subspace of the topological dual E*, then F is the dual of the Banach space of a-weakly continuous forms on F, which can be identified with the quotient Banach space EjFx, where F x = {e e E /(e) = 0 V/ e F} is the annihilator of F in E. Apply with E = Cl{H), E* = B{H) and F = M].
A survey of operator algebras 183 From a concrete point of view, this means that for any w in M» sa £ 1 (if)/M J - and every e > 0, one can find T in Cl(H)} with ||T|| < ||w|| + e , such that w(x) = Tr(Tx) for all x in M.) In other words, a von Neumann algebra is a C*-algebra which, as a Banach space, is a dual, and then comes equipped with a cr-weak topology. (By a theorem by Sakai and Dixmier, this provides an abstract characterisation of von Neumann algebras among C*-algebras). Terminology. In view of property (iv) of the previous lemma, c-weakly continuous positive linear forms on a von Neumann algebras are called normal positive linear forms. Normal states are norm one er-weakly continuous positive linear forms. 2.3. Examples. 1. Any B(H) is a von Neumann algebra, with predual Zl(H). With H = Cn, the matrix algebra M n (C) is a von Neumann algebra (its predual is M„(C) with the trace norm). 2. Let (X, fi) be a measure space. As the pointwise product of an L°° func tion with an L2 function is an l? function, one can imbed L°°{X,[t) into B(L2(X,/J.)) just by pointwise multiplication of functions. One checks that L°°(X, y) is equal to its commutant in B(L2(X, /z)), which makes it a von Neumann algebra (see also Section 3, Picture 1, Application 1). Its predual is LX(X,/i), with the usual duality (f,g) = f fgd/j,, f €
L°°{X,y),geL\X,y)Conversely, any abelian von Neumann algebra is isomorphic (as a von Neumann algebra, which means as C*-algebra with a cr-weak topology) to an L°°(X,fi) for some measure space (X,fi). This explains why nonabelian von Neumann algebras are considered as noncommutative L°° spaces, their preduals as noncommutative L1 spaces, and their theory as a noncommutative measure theory. 3. Let G be a discrete group. We have met (see Example IV.B, in Subsec tion 1.5) the left regular representation g t—> X(g) from G into the unitary group of £2{G). The von Neumann algebra X(G)" generated by those unitaries is the von Neumann algebra of the left regular representation of G. Let 5e be the norm one element in £2(G) defined by <5e(e) = 1 and 5e(g) = 0 if g 7^ e (e is the unit in G). Then the normal state r on A(G)" defined by T{X) = (Se, xSe) has two main properties: for x in X(G)", one has T(X*X) = 0 iff x = 0 for any x and y in A(G)", r(xy) = r(yx). This is an example of a faithful normal tracial state which we meet in next section (see Section 3 below, Picture 1). 2.4. Theory of representations. 2.4.1. The GNS section. Definition. Let K be & Hilbert space.
184
Jean-Luc Sauvageot
A normal representation of a von Neumann algebra M in B(K) is a *-algebraic morphism 7r from M into B{K) which is continuous when M and B(K) are equipped with their respective tr-weak topologies. This means that the transposed map will send the predual of B{K) into the predual of M. A sufficient condition for a *-algebraic morphism to be normal is that it respects monotone limits (property (iv) of the lemma in Section 2.2.3). Let f be a vector in K: the linear map M 3 x >-* (f, 7r(x)f) will be a normal positive linear form on M with norm ||£|| 2 . Conversely, any normal positive linear form on M can be obtained that way: L e m m a (GNS construction). Let OJ be a normal positive linear form on M. Then there exists a triple (Ku,iru,£J) {essentially unique) where Kw is a Hilbert space and 7ru is a normal representation of M into B(KJ) £w is a vector in Kw with norm ||u;||2 such that 1. £UJ is cyclic for iru: 7rw(M)£w is dense in Kw 2. cu(x) - (£,u,nw(x)£,J) for all x in M. [Apply the GNS construction for positive linear forms on C*-algebras, (Subsection 1.6.3). T h e normality of w implies that iru respects monotone limits, hence is normal.] 2.4.2. Examples of representations. Example 1: Ampliation. Let H and K be two Hilbert space. The tensor product Hilbert space H®K is the Hilbert space obtained by completing the algebraic tensor product of H and K for the positive definite sesquilinear form m
n
j~l
t=l
i,j
One checks that, for S in B(H) and T in B(K), one can define a bounded operator S ® T on H ® K, with norm ||S , ||||T||, by the formula {S®T)(£®ri)
=
Sli®Tri.
Let now M be a von Neumann algebra in B(H). Then the map M B x^
x®lK
e B(H
is a faithful normal representation of M into B{H ® K). Such a representation is called an ampliation of M. ['Faithful' means injective: cf. 1.6.3]. Example 2: reduction on a projection in the commutant. Let M be a von Neumann algebra in 13(H). Let P be an orthogonal projection in H which commutes with M: Px = xP, Vz e M. Then x(P(H)) c P(H) for all x in M, and one can define the map M 3 J H
X\Pln)
6
B(P(H))
which to x associates its restriction to P(H): it is a normal representation of M.
A survey of operator algebras 185 Example 3: unitary representation. Let M be a von Neumann algebra in B(H). Let K be a Hilbert space and U : H —» K be a unitary operator (isometric and onto, or equivalently UU* = lfc, C ' C = Iff). Then the map M 3 x >-+ L W € B(K) is a faithful normal representation of M in B(K). Remarks. 1. This catalogue exhausts all possibilities: any normal representation of a von Neumann algebra can be obtained by making first an ampliation, then a re duction on a projection in the commutant, and finally a unitary equivalence. 2. Note that Examples 1 and 3 are faithful representations. A faithful normal representation of M is a homeomorphism of M onto its image for the aweak topologies. (Which can be no longer be true for the strong or the weak topologies). 2.5. Functional calculus revisited. 2.5.1. Complex measures. Let Y be a locally compact space and B be its Borel a-field, i.e. the a-field of subsets of Y generated by the open subsets. A complex measure on Y is a map fj,: B —> C such that n
n
with absolute convergence on the right hand side, for any countable disjoint family {Bn} of Borel subsets of Y. A complex measure is a finite linear combinations of bounded positive measures, and you can integrate at least bounded Borel functions with respect to it. 2.5.2. Spectral measures. A spectral measure on Y is a map E from the Borel a-field B into the set of orthogonal projections of a Hilbert space H, such that E(Y) = Iff and E([JBn) n
= ^£(Sn) n
with strong, weak or a-weak convergence on the right hand side (the three are equiv alent), for any countable disjoint family {Bn} of Borel subsets of Y. Note that {E{B)}BZB must be a commuting family of orthogonal projections in H, because of the relation E(B U B') = E(B n B') + E(B\B') + E(B'\B), and the fact that if a sum of orthogonal projections is again a projection, they must commute. Note also that, for any £ and n in H, the map B 3 B >-> (77, E(B)£) is a complex measure on Y, so that, for any bounded Borel function / from Y to C, there exists a bounded operator E(f) on H, with norm less than the uniform norm of / , such that (V, E{f)t)
= / f(y)(n, E(dy)0
for all £ and 7? in H.
JY
Moreover, one has £ ( / ) * = £ ( / ) , E(fg) Borel on Y.
= E(f)E{g)
for / and g bounded and
186
Jean-Luc Sauvageot
Spectral decomposition theorem. Let T be a normal operator in B(H). exists a (unique) spectral mesure E on sp(T) such that
T= I
Then there
zE(dz).
If T is selfadjoint, E can be considered as a measure on R with support included in t h e interval [-||T||, ||T|| ]. If T is positive, E can be considered as a measure on R + with support included in [0, ||T|| ]. For any bounded Borel function / on the spectrum of T, one can define the operator f(T) = E(f) in B(H). When / is continuous, f(T) coincides with the operator obtained from the continuous functional calculus in B(H) or in any sub-C*-algebra containing T, as defined in Subsection 1.4 2.5.3. Borel functional calculus in von Neumann algebras. Let M be a von Neumann algebra in B(H). For any normal element x on M, one gets a spectral measure E on the spectrum of a; and, for any complex-valued bounded Borel function / on sp(a;), a bounded operator f(x) in B(H). We already know that f(x) belongs to M whenever / is continuous (functional calculus in M as a C*-algebra). We show that f{x) remains in M for any / . Let S be an operator in the commutant M' of M. Let £ and rj be two vectors in H. The two maps, from the space of bounded Borel functions on sp(:r) into C, / -> (v,E{f)S0
and / -> (r),SE(f)S)
=
(S*V)E(f)0
are integrations with respect to complex measures, which provide the same result whenever / is continuous. By a classical unicity argument, the two complex measures (r/, E(.)S£) and (S*i],E(.)£) on sp(a;) must coincide, which implies, for any bounded Borel / on sp(x), any £ and n in H, {r,,E(f)S0
= {S%E(f)$
= {r),SE{f)0
,
i.e. E(f) commutes with every element S of M'. In other words, E(f) belongs to M" = M. In particular, taking / = 1B, every spectral projection E(B) of x belongs to M: a von Neumann algebra is generated by its orthogonal projections (in contrast with the case of C*-algebras - there are C*-algebras with unit in which the only projections are 0 and 1^). In conclusion, we see that the holomorphic functional calculus in complex Banach algebras, and the continuous functional calculus in C*-algebras, extend to a Borel functional calculus in von Neumann algebras. 3. M O D U L A R THEORY
M o d u l a r t h e o r y : P i c t u r e 1. Finite case - M is a von Neumann algebra and r is a normal faithful tracial state on M: 'normal state' means a positive u-weakly continuous linear form on M with,norm 1 (see Subsection 2.2.3); 'faithful' means T(X*X) = 0 ■!=> X = 0, X £ M; 'tracial' means r(xy) = r(yx) for all x, y € M.
A survey of operator algebras 187 Let (HT, nT,^T) be the triple provided by the GNS construction for r (see Lemma in Subsection 2.4.1), and let us change the notations: set L2(M,T) = HT note that the normal representation 7rr is faithful: if itT{x) = 0, then irT(x*x) = 0, T(X*X) = {^.T^T{X*X)^T) = 0 and x = 0. As remarked at the end of Section 2.4, 7rT is an isomorphism from M onto 7rT(M), SO that we shall identify M with its image 7rT(M) in B(L2(M, T)). The GNS construction properties can be written as follows: M£T
is dense in L?{M,T),
and r(x) = (£T,X£T}
VK 6 M.
Note now that the tracial property can be interpreted as ll^r|| 2 = T(X*X) = T{XX")
= ||x*^T||2
Vx G M,
so that there exists an antilinear operator J which is isometric and involutive, char acterised by J(xiT) = x*iT Va; 6 M. ['Antilinear' means additive, and such that J(A£) = AJ£, for A in C and £ in L 2 (M, r ) ; 'involutive' means J 2 = 1L2(M,T)-] One can show easily the inclusion JMJ C M': let xi, a;2 and a; be three elements of M, and compute JxiJx2x£T
— Jx\x*x?£T — X2Xx\^T
= I2^£I£*£T =
x2Jx\Jx^T.
As M£ T is dense, we get Jx\Jx2 = x2Jx\J for all xj, 0:2 in M, which is the result. As a matter of fact fact, one can prove the much stronger result: Theorem (Commutation theorem). M' =
JMJ.
In the Hilbert space L2(M,T), the von Neumann algebra M and its commutant are canonically antiisomorphic (the map y — i > Jy*J provides an isomorphism of M', not with M, but with the opposite algebra). Applications (i) Let M = ^(YJ/J,), where \i is a probability measure on the space Y, and consider the linear form M 3 / 1—> p ( / ) : it is a normal faithful tracial state, and the Hilbert space L2(M, fi) is naturally identified with L2(Y,fi), with M acting by pointwise multiplication as seen in Example 2 of Subsection 2.3. The involution J is nothing but the natural involution / K-> / of complex valued I? functions, so that JMJ = M. The commutation theorem provides then M = M'. (ii) Let M = A(G)" and r the natural tracial state on M explained in 2.3.3: the involution J is given by (J£)(ff) = £(ff_1)i £ m ^ 2 (G), ff in G, and for any h in G, one has JA(/i)J = p(/i), where p (G 3 h i-> p(ft) e unitary group of 12{G)) is the right regular repre sentation: (p(h)S)(g) = t{gh),
Vfc,seG,
£ e ^ 2 (G).
188 Jean-Luc Sauvageot At the von Neumann algebra level, we get JX(G)"J = p{G)", i.e. \{G)" = p(G)" and p{G)' = X(G)". Modular theory: Picture 2. Semifinite case - M is a von Neumann algebra and T is a normal semifinite faithful trace on M, that is T is a map from the positive cone M+ of M into the extended real half-line [0, oo] which is additive and (positive) homogeneous, that is T(X + y) = T{X) + r{y), T(XX) = Xr{x) for x,y € M+,
XeR*+,
which is semifinite: {x € M+ : T(X) < +00} is a-weakly dense in M + ; faithful: for x in M+ , T(X) = 0 <S=> x = 0; a trace: r{x*x) = T{XX")^X € M; and normal: r(limxj) = limr(xj) in [0, +00] for every bounded increasing net {xt} in M+. The associated L2 space. Note first that the set J T = {x e M : T(X*X) <
+00}
is a ^-invariant ideal in M. It is *-invariant, because of the trace property T(X*X) = T(XX*); a vector space, because of the inequality (x + y)*(x + y) < 2(x*x + y*y); a left ideal, because of the inequality y*x*xy < ||i|| 2 y*y; and an ideal, because a *-invariant left ideal is also a right ideal. The quadratic form x 1—> T{X*X) on Ir can be polarised into a sesquilinear form on I T , which makes it a pre-Hilbert space: let L2(M,T) be the completed Hilbert space and let AT be the canonical map from the pre-Hilbert space into its completion. Then AT : IT —> L2(M,T) is characterised by three properties 1. it is linear; 2. its image is dense; 3. ||A T (a;)|| 2 is equal to T(X*X) for every x in Tr. Just as in the GNS construction (Subsection 1.6.3), the inequality y*x*xy < ||:r||22/*2/ for all x in M and y in 2T, provides for every l i n l a bounded operator wT(x) on L2(M,T) with norm less than ||a;|| , characterised by irT(x)AT(y) = AT(zy) \/y €lT. One can show that wT is a normal and faithful representation of M, and as in the previous case, we shall identify M with its image nT(M) in B(L2(M,T)). We write simply xhr{y) = AT(xy). For every y in XT, one has l|AT(2/*)||2 = T(yy*) = T ( ^ ) = ||A T ( 2/ )|| 2 and we get, as in the finite case, an antilinear isometric involution J on characterised by JAT(y) = AT(y*) And again we get the commutation theorem JMJ = M'.
Vy€lT.
L2(M,T),
A survey of operator algebras 189 [Let us prove the easy inclusion JMJ C M', just as in Picture 1: let Xj, x2 and x be three elements of M, with x in XT, and compute 3x\ Jx2AT(x)
— Jx\JAT{x2x)
=
JxiAT(x*xl)
= J AT{x\X* x*2) =
AT{x2xx\)
=
x2AT{xx\)
= x2J AT(x\X*) = x2Jx\AT(x*)
=
x2JxiJAT(x)\
concluding as in the previous case.] Application. Let M = B(H) and let r = Tr, the canonical trace (see Subsec tion 2.1.2). Then one can identify L2{M,T) with the Hilbert space tensor product H®H, as defined in Example 1 in Subsection 2.4.2, where H is the conjugate Hilbert space of H, with B(H) acting by ampliation T —> T ® 1-JJ , and involution J char acterised by J(£ ® rj) = r\ ® £ for £ and r/ in i7. The commutation theorem can be written
{ r « i ? , T e B{H)}' = {i fl ®s,Se £(#)}. One can also adapt the two applications given in Subsection 3: For the abelian case, by replacing the probability measure ji by a a-finite measure. For the group example, by replacing the discrete group G by a locally compact unimodular group. Modular theory: Picture 3. a-finite case - Let M be a von Neumann algebra and w b e a faithful normal state on M. Normal state is defined in Subsection 2.2.3; faithful means: for x in M + , UJ(X) = 0 <=> x = 0 (see Picture 1). As in Picture 1, the GNS representation of LU is faithful. One again writes L?{M,uS) = Hw and identifies M with its image under 7^, so that M is considered as a von Neumann algebra acting in L2(M,UJ),
with
OJ(X) = (^,X^)
VxeM
and M£ u dense in L2(M,UJ). Note that the map M B x i-» x£„ € L2{M,u) is injective. Consider now the densely defined antilinear operator S on L2(M,ui), with domain M£ w , such that 5(s£ u ) = z*£w ,xeM. One can prove the non-trivial property: L e m m a . The operator S is closable. Then A = S*S (where S is the closure of 5) is a linear densely defined positive selfadjoint operator in L 2 (M, a;), and just as for bounded positive operators (see Section 2.5) it has a spectral decomposition A = J R XE(d\), the only difference being that the spectral measure E no longer has bounded support. A has a positive square root A5 = JK \2E(d\), and the polar decomposition of S is S = JA* where J, as in the previous pictures, is an antilinear isometric involution of L2(M,ui). The commutation theorem again reads JMJ = M'.
190 Jean-Luc Sauvageot But the presence of the modular operator A introduces a new feature here. For every t in R, the operator A l i = / R \ltE(d\) is a unitary operator in L2(M, UJ), and the fundamental theorem of Tomita's modular theory is: T h e o r e m . For every t in K, AitMA-it
=
M
Hence there is a one-parameter group ou = {afjtm M, called the modular group of w, defined by o?(x) = A ^ x A - "
tel,
of normal automorhisms of
x e M.
What one gets here is a dynamics, which measures the degree of noncommutativity in M, as read through the state UJ. Note that UJ is <7"-invariant: UJ o a" = UJ for all t€R. The state and the dynamics are related by the so called Kubo-Martin-Schwinger (or KMS) condition: For any x and y in M, there exists a (necessarily unique) continuous function F on the closed strip {z 6 C 0 < Im2 < 1}, holomorphic on its interior, open strip, such that uj{o?{x)y) = F(t) and uj{yo^{x)) = F(t + i) for all t in R. The modular group ou is characterised by the invariance of u> and the KMS con dition. Modular theory: Picture 4. General case - M is a von Neumann algebra and (j> is a normal semifinite faithful weight on M, that is
A survey of operator algebras 191 Just as in the GNS construction (Subsection 1.6.3), the inequality y*x*xy < \\x\\2y*y for all x in M and y in 2^, provides for every x in M a, bounded operator 71-4,(2;) on L2{M,4>) with norm les than ||a;|| , characterised by TT(j>(x)A^(y) — A^xy), for all y € X^. One can show that -K^ is a normal and faithful representation, and as in the previous cases we shall identify M with its image n^,(M) in B(L2(M, $)). We write simply xA^y) = A^xy). Note that the map x 1—> A^,[x) from X$ into L2(M,4>) is injective (because <> / is faithful). One can also prove that A^Z^nTJ;) is dense in L2(M, 0), and get a densely defined operator S with domain A^X^ D X^) by the formula
s(Ai(x)) = A^x") xex^nx;. From now on, things go just as in the previous Picture: S is closable and S = JAi, where A = 5*5 is the modular operator and J the antilinear isometric involution associated with <j>. The commutation theorem remains valid JMJ = M' in B{L2(M,<j>)). One again has: Theorem. For every t in R, A " M A - " = M. Hence there is a one-parameter group a^ = {o~t}teR of normal automorhisms of M, called the modular group of 4>, defined by at{x) = AitxA-it
feR
x€M.
4> is ff*-invariant: 4>{af{x)) = 4>{x) for all i e R , and x G M+. The state and the dynamics are still related by the KMS condition: For any x and y in T^ f i l l , there exists a (necessarily unique) continuous function F on the closed strip {z £ C : 0 < Imz < 1}, holomorphic on its interior, such that 4>{ot(x)y) = F(t) and ^yof(x))
= F{t + 1)
for all t in R. Again, the modular group a* is characterised by the invariance of 4> and the KMS condition. Final comment on this section. Von Neumann algebras on which there exist a normal faithful tracial state (respectively a semifinite trace) are called finite (resp. semifinite) von Neumann algebras. There exists a normal faithful state on every von Neumann algebra acting in a separable Hilbert space. There exists a normal faithful semifinite weight on any von Neumann algebra. 4. CONDITIONAL EXPECTATIONS
Throughout this section, M is a von Neumann algebra and A7 is a sub-von Neumann algebra (a-weakly closed *-subalgebra containing the unit).
192 Jean-Luc Sauvageot Definition. A normal conditional expectation from M onto 7V is a linear map E from M onto TV which is positive (E(M+) C N+), normal (u-weakly continuous), unit preserving (E(1M) = ljv), and which satisfies the 'conditional expectation property': E(y'xy) = y'E{x)y
Vx € M, y, y' 6 TV.
Note that those conditions imply E(y) = y for any y in TV, hence E2 = E. Construction: Cases 1 and 2. We refer here to the finite and semi-finite cases of Pictures 1 and 2 in the previous chapter. Let r be a normal faithful semifinite trace on M. Moreover, if r is not bounded (i.e. if r is not a state, or a multiple of a state), one must assume also that the restriction of r to TV+ is again semifinite: {y € TV : r(y) < +00} is a-weakly dense in TV+. For making the situation clear, we shall denote by T' the restriction of T to TV, and shall apply the construction in 'Picture 2' of the previous chapter both to (M,T) and to (TV, T'). We first get the ideal TT in M, the canonical inclusion A r from TT into L 2 (M, r ) , the action of M on L2(M, r ) which allows us to consider M as a von Neumann algebra in B(L2(M,T)), and the involution operator JT such that JTMJT = M' in B(L2(M,r)). We then get the ideal 1T> in TV, the canonical inclusion AT> from IT< into £2(TV, r'), the action of TV on L2(N, r') which allows us to consider TV as a von Neumann algebra in B(L2(N,T')), and the involution operator JT< such that JT>NJT, = TV' in B(L2(N,T')). [Note that TV is allowed to act in two different Hilbert spaces: in L2(N,T') by the GNS construction for T', and in L2(M,T) as a subalgebra of M: this means that there is somewhere a natural isomorphism between two isomorphic images of TV, for which no specific symbol is used.] XTi is a subspace of J T , as a pre-Hilbert space: there is an obvious isometry W from L2(TV, r') into L2(M,T) characterised by WkT,{y)=K{y)
VyeXT>-
One computes, for y in 1T> F\I*,, WJT,AT,(y)
= WAT,(y*) = AT(y*) = JTK(y)
=
JrWhT,{y)
= AT(yy') = yAT{y") =
yWAT:(y')
and, for y in TV and y' in TT> , WyAT,(y')
= WAT,(yy')
which can be summarised as WJT, = JTW, and yW = Wy
Vj/ 6 TV,
where this last equality has a left hand side which refers to the action of TV in L2(M, T) while its right hand side refers to the action of TV in L2(TV, r ' ) . Apply those two properties, and the commutation theorem for M, for computing, with x in M and y in JV , W*xWJT,yJTl = W*xJTyJTW = W*JryJTxW
=
Jr.yJr.WxW
which means that, for every x in M, W*xW , as an operator in L2(N, r'), com mutes with JT'NJTi. By the commutation theorem applied to TV, this means that
A survey of operator algebras 193 W*xW commutes with N' and thus belongs' to N" = N. What we get is a a-weakly continuous linear map E : x ^ W*xW from M into N which is unit preserving: E(1M) = W*W = 1L2(N,T') = lw! posi tive: E{x*x) = W*x*xW e N n B(L2(N,T'))+ = N+, Vz 6 M; and a conditional expectation: E(y'xy) = W*y'xyW = y'W*xWy = y'E(x)y, x e M, y,y' € N. Conclusion. Under the assumptions made in this section, there exists a normal con ditional expectation from M onto N. One can check more or less easily that the one constructed above satisfies T(E(X)) = T(X) for all x € M+, and that it is the only one which satisfies this invariance property. C o n s t r u c t i o n : Cases 3 a n d 4. We refer here to Pictures 3 and 4 of the previous chapter. Let
S4, =
J^l
such that J^MJ^ = M' in B(L2(M,4>)) , and A^ leads to the construction of the modular group a^. We then get the left ideal 2^ in N, the canonical inclusion A,/, from %f, into L2{N,ip), the action of N on L2(N,ip) which allows us to consider N as a von Neumann algebra in B(L2(N, ip))t and then the closed operator S$ with its polar decomposition such that J^NJ^ = N' in B(L2(N,ip)) , and A^, leads to the construction of the modular group a*. Just as in the previous case, there is an obvious isometry W from L2(N,ip) into L2{M,
Vj/ e 2^.
and one easily gets the equality WS+ = S4W which passes to closures: WS$ = S$W. One could not go further without the third assumption made above (af(N) = N Vf € 1 ) , which has a strong consequence: the restriction to N of the modular group a^ is a one parameter group of normal automorphisms of N
194
Jean-Luc
Sauvageot
which leaves tp invariant, a n d has the K M S p r o p e r t y with respect to ip: as noticed at t h e end of P i c t u r e 3 or 4, it must coincide with t h e m o d u l a r g r o u p a*. Prom this conclusion, one can prove t h a t the isometry W satisfies A.1JW = WA^ for all t in R, t h u s A\\V
= W A J and
finally
J+W =
WJ^.
O n e can continue j u s t as in t h e previous section, and get E : M B x i—> E(x) = W*xW 6 TV a conditional expectation from M onto AT, and t h e only one which satisfies
[Di 1] J. Dixmier, "Les C*-algebres et Leurs Representations," Gauthier-Villars, 1969, (english translation, by F. Jellet, "C*-aIgebras," North Holland, 1977). [Di 2] J. Dixmier, " Les Algebres d'operateurs Dans L'espace Hilbertien (Algebres de von Neu mann)," (2nd Edition), Gauthier-Villars, 1969, (english translation, by F. Jellet, "Von Neu mann Algebras," North Holland, 1981). [DuS] N. Dunford and J.T. Schwartz, "Linear Operators," Parts I, II, III, Reprinted in Wiley Classics Library (1988), Interscience Publishers. [Pel] G.K. Pedersen, "C*-algebras and Their Automorphism Groups," Academic Press, 1979. [Pe2] G.K. Pedersen, "Analysis Now," Springer-Verlag, 1989. [Rud] W. Rudin, "Functional Analysis," McGraw-Hill, 1973. [StZJ S.Stratila and L. Zsido, "Lectures on von Neumann Algebras," Abacus Press, 1979. [Tak] M. Takesaki, "Theory of Operator Algebras, I," Springer-Verlag, 1979. ALGEBRES D'OPERATEURS ET REPRESENTATIONS, INSTITUT DE MATHEMATIQUES, BP UNIVERSITE PIERRE ET MARIE CURIE, F-75252 PARIS CEDEX 05
191,
Quantum Probability Communications, Vol. XII (pp. 195-207) © 2003 World Scientific Publishing Company
QUANTUM STOP TIMES KALYAN B SINHA
CONTENTS
0. Introduction 1. Notations 2. Theory of integration with stop times 3. Strong Markov Property 4. Stopped Processes 5. Pre-5 and Post-5 algebras 6. Remarks References
195 197 198 201 203 204 205 207
0. INTRODUCTION
For the standard Brownian motion in classical probability theory, a stop time T is a random variable defined on the space of trajectories, taking values in [0, co] and satisfying the condition that the event (r < t) belongs to the cr-algebra of the Brownian motion up to time t for every t. In the language of bosonic Fock space H = r s (L 2 (K + ; k)), this will translate into the following: a quantum stop time S is a spectral measure on R + U {oo} such that S[0, t] £ B(Tit) for every t. It u is a typical Brownian trajectory, then one has the Dynkin-Hunt property that the randomly shifted motion: (6rio)(t) = w ( t + r ) — w(r) is again a standard Brownian motion restricted to {r < oo}. In the Fock space, there exists a pair of commuting Brownian motions {Q{t)} and {P(t)} satisfying [Q(s), P(t)} = imin(s, t). Hudson in [Hud] constructed the random-shifted pair {QT(t) = Q(t + T) - Q(T)} and {PT(t) = P(t + T) -
P(r)}
such that [QT{s),PT(t)\ = imin(s, t), where we have now written r = J0°° \S(d\), a self-adjoint operator in the case when S({oo}) = 0. The stop time S is said to be finite if ^({co}) = 0 and for two stop times St and 52, we say Si < S2 if ^([O, t}) < Si([0,t]) for all t. The deterministic stop time for which S({t}) = / and S{E) = 0 if t ($ S, will be denoted by t. Then S At, defined by f 5[0,a] if a < i (5At)[0,a] = 1 li a>t is again a stop time and it is clear that S At increases to S as t —► oo. It should also be obvious that by its definition either 5({0}) = 0 or 1 so that every nonzero 195
196 Kalyan B. Sinha stop time has no spectral mass at 0. Here we shall therefore assume, without loss of generality, that 5({0}) = 0, Examples. (a) Let {ui(t)} be the standard Brownian motion with Wiener measure P, and let r be a classical stop time with respect to the Brownian filtration Tt, that is {u> : T{UI) < t] 6 Tf Then, considering the standard isomorphism /, from L2{¥) onto H = r(L 2 (R+)), we can say that ( J O I O T " 1 ) ^ , t] € B(Ht) for every t, where 1 is (multiplication by) the indicator function. Thus 5 = / o l o r _ 1 is a stop time. (b) Let S be the unit sphere and let oj(t) be the standard Brownian Motion, in 3-dimensional Euclidean space. Then T{LJ) = mi{t > 0 : u>(t) € S}, the first hitting time, is one such stop time as in (a). (c) Similarly a classical stop time associated with a Poisson process can be realized as a quantum stop time in an obvious way. Probably more instructive is the example with the number process A(£), defined below. Let Pj(t) denote the spectral projection of A(t) corresponding to its eigenvalue j (j — 0,1, 2, ■ • •) so that the range of Pj(t) is the subspace L2[0, t]3s m ® H\ where H^ym denotes the j-fold symmetric tensor product of a Hilbert space H. Define quantum stop times Sj (j = 1, 2, • • •) by oo
^•([M) = YlPr^
iit<00
and
'
$({«>}) = ^ F r ( o o ) = projection onto 0 L 2 [ O , t ] ^ m r=0
r=0
The projection Sj([0, i\) is to be interpreted as the event of at least j counts in the time interval [0, t] or equivalently, the associated stop time is the one at which the jth count takes place. (d) Consider the following in the setup of Example (a). As we have said earlier, there are two commuting (but mutually non-commuting) Brownian families {Q(t)} and {P(t)} in the Fock space U = r(L 2 (R+)). In fact, they are related by a unitary transformation F(i), the so called Wiener transform: p(t) =
r(i)Q(t)T(-i)
Consider the (classical) stop time r with respect to classical Brownian filtra tion as in Example (a) and pull it back in Fock space, once as S considering the Brownian motion being realized as {Q(t)} and again as S' when the Brownian motion is realized as {P(t)}. Thus S'([0,t}) = T(i)S([0,t])T(-i)
=
r(tlt)S{[0,t])T(-ilt).
Then S and S' are two non-commuting quantum stop times.
Quantum stop times
197
1. NOTATIONS
Let h = L?(M.+ ; k), k a separable Hilbert space, and set U = r s ( L 2 ( R + ; k)) = r s (h); thus H = Ht® H\ 2
where
2
Ht = r s (L ([0, t]\ k)) and « ' = r s (L ([t, 00); k)); e (/)
= i © / e 4=/® 2 ® • • • © -4=/®n ® • • V2! vn!
ft = l[o,tj/ and / ' = l[ t | 0 0 ]/ so that e
(/)
= e
(ft) ® e ( / ' ) and {e(/) : / e h } constitutes a total set in H.
We shall often make the identification e(0)£(s) = e(O s )f(s), where £(s) 6 W W(f, U)e(g) = e - 5 ' l / H 2 - ^ ^ e ( / + Ug), W(f, U)W{g, V) = e- j
Im
< ^ > W ( / + £/
f,geh,Ue
U(h).
The creation, number (gauge) and annihilation operators: aHf)e(g)
=
±e(g
+
ef)\e=0,
HT)e(g)
=
-ile(e^)|g=0l
a(f)e(g)
=
(f,g)e{g),
with the corresponding processes given by: A*(0 = a^lfo^j ® e^, A}(i) = A(l [0 , (]
[0 , ... f(t
. — S)
if t < s .... 11
t>S
so that #*#s = / and #,;#* = 1[S]00) (as multiplication operator) and its second quan tisation t(ds) is defined by: F(6s)e(f) = e(6sf) so that r(0 s ) is an isometry and T(9S)T{9*S) = r(l[ Si00 )). Also we shall have occasion to discuss Fermionic processes in Fock space: Hf)
= / J{s)f{s)dA{s), F\f) = / J{s)f{s)dtf{s), Jo Jo where J(s) = F(R(s)), R(s) being the reflection operator: -f{t)
(*w»w= l/(t)
iit<s
iH>s .
Then | | F ( / ) | | = ll/H and F(f)F^g)
+ F\g)F(f)
=
(see R..L. Hudson's lectures in these volumes). Also, r(6,)W(f,
eiv) = W[B,f, ew>f)T{6s), and T(6s)F(f)
=
F(6tf)T(9.)t
198 Kalyan B. Sinha for all / G h and ip € L§?(R + ), so that el,p is considered as the unitary operator of multiplication. We also write Wt = W{f,ei,f>,t) = W{ft,eiVt) and note that it satisfies the quantum stochastic differential equation dW = (fdA] + (e^ - l)dA - eivfdA
-
~\ffdt)W.
We note the factorisability of W: W(f, e * b) = W(f, e * o)W(/l [ o , 6 1 l e ^ M i ) for 0 < a < b < oo and also that ||e(0) - W{f)e{g)\\2 = ||e(0) - e -V2||/ll 2 -(/, s > e(/ + ^ )( |2 =
i + e-ll/lP-2Re{/,9>+!l/+sll2 _ 2Re(e-5ll/ll2-)
=
1 + e^l 2 - 2 e - 5 l l / l l 2 e - R e < ^ > c o s I m ( / j 5 )
=
( e M 2 _ i) + 2(1 - e-Hl^ll2) + 2e"» W a ( l - e~Re<^>) 2. T H E O R Y O F INTEGRATION WITH S T O P TIMES
Here we shall consider mostly integrals of improper Riemann type. These will be useful in the later sections both for proving the strong Markov property of Fock space—which contains as special cases that same property due to Dynkin and Hunt, Hudson and Applebaum ([App]), and also for studying stopped processes. Since the estimates involved are simple, and can be found in [PaS], we shall often content ourselves with just giving a few key steps and motivating the results. We shall also only consider H = F(L 2 (K. + )) i.e. Fock space with k = C However, it will be clear from the proofs that the case of an arbitrary separable Hilbert space k can be handled with little change. Let S be a stop time in Ti, let / , g G L 2 (R + ), let ^ : [0, oo] —> 7i be such that £(s) G W for all s, and let W(s) = W(fs), x(s) = e(gs)£(s), 0 < s < oo. We wish to define Jroooi W(s)S(ds)x(s). If S is a discrete stop time with support {ti,t2, ■ ■ ■ } U {oo}, then the above integral can be defined as the sum £>fo)S(fo})zfe) + W(f)S({co})e(g), i whenever it exists. Consider a finite interval [a, b] C K + with a, b points of continuity of S and a partition V of [a, b], i.e., V = {a = to < ti < t2 ■ ■ ■ < tn < tn+\ = b, tj points of continuity of S} . We now want to study the Riemann sum n
R{V) = Yw&M[tj,ti+i])x{tj+l)
(2.1)
3=0
under successive refinement of partitions. This we do by a series of lemmas. Lemma 2.1. Let s, t, s', t', u, v! be points of continuity of S such that s
Quantum stop times 199 Proof. The factorising property of W implies that W(u) 1W{v!) € B(H[U:Ui\) and ^([Sji]) commutes through W(u) _1 VV(u'). The result follows. D Lemma 2.2. Let V and V1 be two partitions of [a,b] such that P' is finer than P'. Then \\R(P) - R(P')\\2 < c{Ui(S(P))+w'(6(P))}, where c is a constant depending only on f, g and £, and 5{P)
—
UJ((5)
=
u'{6)
= sup{ f (|/| 2 + \g\2), a<s
i
max(tj+i—tj),
s u p { | | £ ( s ) - £ ( i ) | | 2 , a <s < t
t-s<5},
t-s<6}.
JS
Proof. Let V be given by tj = tj(p) < tj(i) • • • < tj(nj+l) = tj+x. Then, by Lemma 2.1, \\R(V) =
II z2
R(P)\\2 W t
( Hr+l)){S[tj(T),
tj(r+l)]x(tj(r+l))
-W(tj{r+1))
^(tj+^Sltjfr),
W t
=
\\J2
=
Y2\\S{[tnr),
tj(T+i)]x(tj(r+i)
~ W {tj{r+l))-lW
( Hr+l))S{[tjir),tj{r+l)]){x(tJlT+1))
2
{tj+l)x{tj+l)}\\
2
tjlr+1)))e(gtj(r+1))\\2\\t{tj{r+1)) -W{tj{r+l))-'W{tJ+M9\t}i,+l),tj+Ami+i)\\2
(2-2)
On the one hand, \\S{[tj{r),tj{r+1)))e{gtj{r+l))f
<
(e(g),S([tj{T),ti(r+1)])e{g))t
and on the other hand the second term in the summand on the R.H.S. of (2.2) is bounded by
m t i w ) - ^ +1 )ii 2 +2|ie(i j+1 )n 2 iie(o) - w(t i(r+1) )- i ^(t J - +1 ) C (^ (r+1)iti+ll )n a < 2w^6(P))
+ 4M<J'(6(P))
, 2
where we have set M = sup{||£(s)|| , a< s
O
Lemma 2.3. Suppose that £ is strongly continuous. Then the Riemann sum R{P) converges strongly to a unique limit, denoted Ja W(s)S(ds)x(s), as 5{P) —> 0. For x{s) = e(
f
W(s)S(ds)y(s))
Ja
r<°-»(Z(s),r,(°)M9),S(ds)e(h)).
(2.3)
200 Kalyan B. Sinha Furthermore, if a, b, c, d are points of continuity of S such that [a, b] n [c, d] = 0, then {I
W{s)S(ds)x(s),
f W(s)S(ds)y(s))
Ja
= 0.
(2.4)
Jc
Proof. The convergence of R{V) follows easily from Lemma 2.2. Now, if we denote by R(V,x) and R(V,y) respectively the Riemann sum in (2.1) and the one in (2.1) with x replaced by y, then by Lemma 2.1 we have (R(V,x),
R(P,y))
= £(eK.J,SaMm])eK+1)>(£(Vi)>
lfe+i)>
3=0
^-^"'^m^Uit^M^gls^^Mh))-
=
Since s —> e"^"'^(£(s),r](s)) is continuous by hypothesis, the R.H.S. of the above equality converges to the Riemann integral in the R.H.S. of (2.3) while by the first part of this lemma, the L.H.S. of the equality converges to the L.H.S. of (2.3). The relation (2.4) follows from a similar argument using Lemma 2.1. □ Next we define the improper Riemann integral over K + . Theorem 2.4. Let S,W,g,h,^,x,rj,y
be as in Lemma 2.3. Suppose further that
e - W ||£(s)|| 2 \\S(ds)e(g)\\2 < oo.
/ Jo
(2.5)
Then lim
/ W(s)S(ds)x(s)
= /
; a|0,t l°>6Too Ja
W(s)S(ds)x(s)
J0
exists where a, b move through the points of continuity of S. If we assume furthermore the integrability condition (2.5) for r\ then /•oo
W(s)S(ds)x{s), o
/ Jo
W(s)S{ds)y{s)) e
'^(mMs))(e(g),S(ds)e(h)).
(2.6)
o
Proof. Immediate, using Lemma 2.3, and left as an exercise.
□
The result above can now be extended to maps £ which are only assumed to be measurable, and satisfy (2.5). However, since we shall not make much use of this extension we refer the reader to [PaS] for the details. If S is a finite stop time, then the above definition suffices. If however, S is not finite, then we define / W{s)S(ds)x{s)
= j
W{s)S{ds)x(s)
+
W(f)S({oo})x(oo).
It is also easy to see that the relation (2.6) remains valid for these extended integrals.
Quantum stop times 201 3. STRONG M A R K O V P R O P E R T Y
As was mentioned in the beginning, Fock space Tt admits the factorisation Tt ~ Tts ® Tis for every s £ R + , where Hs can be thought of as the space up to the deterministic stop time s, and Tts as the space which is the range of the shift isometry r(# s ). Can we do this for an arbitrary (finite) stop time 5? For this, we need to define an isometry Us to replace T{9S). Lemma 3.1. Let S be a finite stop time. isometry Us on Ti.
Then v K-> JQ S(ds)T(9s)v
defines an
Proof. Setting / = g = h = 0, x(s) = T(9s)u and y(s) — T(6s)v with u,v € Tt, in Theorem 2.4, (2.6) simplifies to /"OO
(/ Jo
/'OO
S(ds)T(9s)u,
/ Jo
S(ds)F{es)v)
= {u,v).
The result follows.
□
We now define, for a given finite stop time S, the Fock space up to S, denoted Us, as the closed linear span of vectors of the form /■OO
{ / S(ds)e(gs)v(s) :geH,ip€ L£(R+)}, Jo and the Fock space beyond S, denoted by Hs, as the range of the isometry Us. T h e o r e m 3.2 (Strong Markov property of Fock space). There exists a unique unitary isomorphism Js from TLs <8> Hs onto H such that JS(J for all g eh,
S(ds)e(gsMs)
® Usu) = J S(ds)e(gs)lp(s)T(es)u
(3.1)
ip 6 L°° and u £ Tt.
Proof. By (2.6) we have for all u, v 6 Tt, f, g £ h and all bounded functions ip±,
(f S(ds)e(fs)^1(s)®Usu,
JS(ds)e(gs)y2(s)r(es)v) S(ds)e(g))(u,
v)
S(ds)e(gs)v2(s)®Usv).
J
By totality therefore, (3.1) defines an isometry Js from TLs ®Tis into Ti. The proof will be complete if we can prove that Js is surjective. For this we show that any exponential vector e ( / ) can be approximated strongly by a linear combination of vectors of the form on the R.H.S. of (3.1). Indeed, e(/)
S(ds)e(fs)e(fs)
=
J
=
J S(ds)e(fs)T(8s)e(V:f)
=£
f ^
S(ds)e(fs)r(6s)e(9;f),
202 Kalyan B. Sinha where 0 = a0 < ai < ■ ■ ■ < aj < ■ ■ ■ are points of continuity of S and a;- —> oo as j —> oo. Now each integral f°3+l S(ds)e(fs)T(9s)e(9lf) is a strong limit of sums of the form £*, //fc*+1 s(ds)e(fs)F{9s)e(9*sJ) as max (sk+1 - sk) —> 0 where a3- = s0 < Si < ■ ■ ■ < sn < sn+i = dj+i is a partition of [a,j,fflj+i]-Elements of this form are in the range of the isometry Js. □ Next we consider the shifted or post^S Weyl and Fermionic operators. Theorem 3.3. Let S be a finite stop time, and let Us be the operator constructed in Lemma 3.1, viewed as a unitary operator 7i —> Hs■ For f Eh, set Ws(f) Then Ws{f)
= UsW(f)(Us)-\
UsF(f)(Usr\
=
is a unitary operator in Hs and satisfy Ws(h)Ws(f2)
= e -*i-» W s ( / i
(e(0), Ws(f)e(0)) while Fs(f)
Fs(f)
=
+
/a))
= exp(-i||/||2),
is bounded operator in Hs with Fs(f!)Fs(f2) s
s
F (h)F *(f2)
+ F^f^ih)
= 0,
+ F^if^F^h)
= (flt / a ) .
Proof. The proof is clear from the definition once we understand Ws(f) as operators acting on 7is ■ Remark. If we set / = elp.tji
(3.2)
and
Fs(f) □
tnen
UsW(f) = J S(ds)T(9s)W(elm) = J S(ds)W(ellt^t])T{et) =
J W(el[s,t+s])S(ds)T(es)
= ( ^ H/(el [ s , t + s l )5(ds))C/ 5 .
Thus formally differentiating with respect to e at e = 0, we get
[/sQ(i)([/s)-x = J[Q(t + s) - Q{s)]S(ds). Similarly, by setting / = ieljo.t], one gets J 7 s P ( i ) ( [ / s ) - 1 = f[P(t + s) - P(s)]5(ds). These are precisely the post-S canonical Wiener processes as discussed in [Hud] while (3.2) has been introduced in [App]. Since the processes {Q(t)} and {P(t)} are both isometrically isomorphic to the operator of multiplication by the standard Brownian motion ui(t) in L 2 (Wiener), the above also implies the Dynkin-Hunt property that Brownian motion renews itself at every finite stop time S.
Quantum stop times 203 4. S T O P P E D P R O C E S S E S
First we shall consider left-stopped Weyl processes. Theorem 4 . 1 . Let S be a stop time, / 6 h. There is a unique unitary operator S o W(f), called stopped Weyl process, determined by S°W(f):e(g)~Js(ds)W(fs)e(g), with adjoint determined by (S o W(f))'
: e(h) » J
W(-fs)S(ds)e(h).
Proof. Since W(fs)e(g) = e((f + g),)Z(s) where £(a) = exp - I | | / s | | 2 (fs,g)e(g°), and £ is bounded, S o W(f) is well-defined on the exponential domain and (2.6) applies: (SoW(f)e(g), =
SoW(f)e(h))
/" e -((/+3) a .(/+")>( e -|IIMI 2 - e (^))x (e(f + g),S(ds)e(f h
=
=
e-(f+s,f+
)(J(e(f
+ g), S(ds)e(f
+
+ h))
h)))e^
(e(g),e(h))
Thus S o W(f) extends uniquely to an isometry on Ti. Similarly, the map e{h) \-* f W(—fs)S(ds)e(h) determines an isometry on H which is adjoint to S o W(f). □ Corollary 4.2. Let S and W(f)
be as before. Then
/>oo
SoW(f)
= I+ Jo
/>oo
S([s,oo])dW(fs)
and(SoW(f)y
= I+ Jo
S([s,oo})dW(-fs),
where the integrals on the R.H.S. are quantum stochastic integrals w.r.t. the Weyl processes W(fs) and W(—fs) respectivelyProof. Recall the definition of the stop time S f\ t and define X(t) = (S A t) o W(f), 0 < t < oo. Then by (2.6) {e(g),X(t)e(h)) =
( J S A t(ds)e(gs)e(gs),J
=
J{e(g),SAt(ds)e(f
SA
t(ds)W(fs)e(hs)e(hs))
+ h))L(s),
where L(s) =
ex P {-i||/ s || 2 -(f„h)-
(9s J)}-
204 Kalyan B. Sinha Thus, on integrating by parts, (e(g)tX(t)e(h)) r
(e(g),S(ds)e(f
+ h))L(s) + (e(g),S([t, oo])e(/ +
h))Lds(t)
o t
=
(e(g),e(f
+ h))L(0)+ Jo
(e( 5 ), S([s, oo])e(/ +
h))L{s)'{t)dg
=
(e(g), e(h)) + J (e(g), S([t, oo])e(/ + h))l{t){-\\f\2
=
(e(g), e{h)) + | ' ( e ( 5 ) , [ / d ^ _ JdA
- fh -
gf}dt
_ I|/|*d s ]S([ 5 ) oo])W(/ s )e(/i)),
in other words X(t) ^I
+J
[fdA* - JdA - ^\f\2ds]S([s,
oo))W(f.)
= / + J
S([s,
oo])dW(ft).
It can be shown that X(t) —> S o W ( / ) strongly as £ —> oo and hence S o W(f) = / + J0°° S([s, oo])dW(fs). The other relation follows similarly. D The above corollary encourages us to define a stopped process for a more general adapted process X(t) which has a differential: dX = E\dA* + E2dA + E3dA + E4dt, as /•oo
SoX
= X(0) + / S([s, Jo
oo])dX(s).
5. P-RE-S AND POST-5 1 ALGEBRAS
We have seen that given a finite stop time S, the Fock space Ti is isometrically isomorphic to Hs ®T~ls, and therefore B(H) is isomorphic to B(Hs) ®B(HS). Also since the set of Weyl operators {W(f) : / £ h} generate B(H), it follows easily from Theorem 3.3 that {Ws(}) : f £ h}, or equivalently by the remark following Theorem 3.3, the family {(P(t + S) - P{S)), {Q(t + S) - Q{S))\t > 0}, generates B{7is). This is exactly the post-S algebra as defined in [Hud]. Now what is a good candidate for the pre-S algebra? Here one has a few surprises. Let
As = {JS(ds)W(f.)ip(s) ■ / e h,p e £°°(R+)}". The integrals here are defined (using (2.6) again) as bounded operators with norm at most j|v?|JooLemma 5.1. Let S be a stop time. Then
as well as to As
Proof. Observe that
= J
S(ds)e(ft)V(sMs)
to conclude the first part while the second part follows from the definition of As by setting / = 0. □
Quantum stop times 205 T h e o r e m 5.2 ([AcS]). Let
(5.1)
Now, let Pa be a spectral projection of the self-adjoint operator Xa in 7ia so that (5.1) is also valid with Pa replacing Xa. For all a> ao > 0, S[(a, oo)) ^ 0 so that there exists a non-zero ( € Tta such that S([a, co))C ^ 0 and hence {BS([a, oo))C : B e B(Ha)} generates the whole of Ha. This implies that, for A, B € B(Ha), (PaA~
APa)BS([a, oo))C
= PaAB S([a, oo))C - APaB S([a, oo))C = (P„A5 - AB Pa)S{[a, oo))C = 0. Now by letting B run over B(H) we get P aJ 4 = APa for all ^4 6 B(Ha), which means that Pa = 0 or 7", or equivalently, E 0 (X) s AT0 = c(o)J for some constant c(a), for all a > ao > 0. By taking the limit a f oo, we see that X = cl for some real constant
c.
□
The above theorem shows that if the stop time has support near oo, then As — BCH). This, of course, rules out the 'natural' candidate As for pre-5 algebra. On the other hand, we do not have a good characterisation of B{Tis)6.
REMARKS
Stop times have also been considered for nitrations of abstract (finite) von Neu mann algebras (Aa)a>o (see e.g. [BaL] and [BaT]). Let „4oo = (|J A*)" a n ^ assume that Aoa is finite, and that
206 Kalyan B. Sinha where the integral is defined as a Riemann integral, just as in the Fock-space case. Here instead we consider an integral over maps: fEa(-)S(da). For a given partition V of [0, oo], we define ^s(T) by
i
Then the following properties can be deduced: (i) Es(7->) is a self-adjoint projection on L2(Aoo), (ii) if V\ is finer than V2, then l&s(Vi) > E S (p 2 ), (iii) if S, T are stop times such that S < T (that is S{[0, t]) > T([0, i\) for all t), then E$(p) < ET(J>) ■
From (i) and (ii) it follows that the family {E S (pj}, indexed by partitions, forms a decreasing net of projections on L2(Aoo), and hence the infimum exists, let it be denoted E$. Theorem 6.1. Let X = (Xa) be a complete L2 -martingale and let S be a stop time. Then X$ s J XaS(da) exists and is equal to Es(Xeo). Proof. Straightforward from the above discussion.
□
Theorem 6.2 (Optional Stopping). Let S andT be stop times such that S
□
For more details on these aspects, see [BaT]. There have been attempts at stopping vector-valued processes (more precisely semimartingales) in Fock space as opposed to operator processes (see [AtS]). An adapted process of vectors Z = (Zt) in H is a regular semimartingale if Z admits a (unique) decompositon : Zt = mt + at, where mt is a martingale and at = / . hsds with hs eHs and j;\\h.\\ds < 00 for all t. Using the L2- representation of martingales (see S. Attal's lectures) we can say that every regular semimartingale will have the representation : Zt = JQ £sduj(s) + J0 hsds with fQ \\(,s\\2ds < 00. Now we can stop such a semimartingale with respect to a given stop time S exactly as before. Theorem 6.3. Z and S be as above. Then the stopped process Z$ = j S(ds)Z3 exists as a Riemann integral and equals f S([s, 00])£(s)du>(s) + f S([s, oo])hsds. If (yt)t>o is a process of vectors in H, adapted to the future then, as in Section 2, one can define an integral J S(ds)Zsys exists and is equal to J S(ds)Es(Zs)ysOnce we have stopped vector processes as above, one can of course stop operator processes as well. If (Xt)t>0 is an adapted operator processes in Tt, then consider the adapted vector process {Xte(ft))t>o which, under certain conditions on X, will be a regular semimartingale of vectors and we can apply the above constructions. However, the net outcome turns out to be exactly the same as in [PaS], briefly explained in Corollary 4.2 and the subsequent discussion.
Quantum stop times
207
REFERENCES [AcS] [App]
L. Accardi and Kalyan Sinha, Quantum stop times, in [QP4], pp. 68-72. D. Applebaum, The strong Markov property for Fermion Brownian motion, J. Fund. Anal. 65 (1986) no. 2, 273-291. [AtS] S. Attal and K.B. Sinha, Stopping semi-martingales in Fock space, in [QP 10], pp. 171-186. [BaL] C. Barnett and T. Lyons, Stopping non-commutative processes, Math. Proc. Camb. Phil. Soc. 99 (1986) no. 1, 151-161. [BaTJ C. Barnett and B. Thakrar, Time projections is a von Neumann algebra, / . Operator The ory 18 (1987) no. 1, 19-31. [Hud] R.L. Hudson, The strong Markov property for canonical Wiener processes, J. Funct. Anal. 34 (1979) no. 2, 266-281. [Nel] E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1974), 113-116. [PaS] K.R. Parthasarathy and K.B. Sinha, Stop times in Fock space stochastic calculus, Probab. Theory Related Fields 75 (1987) no. 3, 317-349. [QP4] "Quantum Probability and Applications IV," eds. L. Accardi and W.von Waldenfels Lecture Notes in Mathematics 1396, Springer-Verlag, 1989. [QP 10] "Quantum Probability Communications X," eds. R.L. Hudson and J.M. Lindsay, World Scientific, 1998. INDIAN STATISTICAL INSTITUTE (DELHI CENTRE), 7 SJS SANSANWAL MARG, 110016 N E W DELHI, AND JAWAHARLAL NEHRU CENTRE FOR ADVANCE SCIENTIFIC RESEARCH, BANGALORE
E-mail address: kbsflisid.ac.in
Quantum Probability Communications, Vol. XII (pp. 209-235) © 2003 World Scientific Publishing Company
F R E E CALCULUS ROLAND SPEICHER
CONTENTS
0. Basic definitions and facts 1. Combinatorial aspects of freeness: the concept of cumulants 2. Free stochastic calculus References
209 211 226 234
0. B A S I C DEFINITIONS AND FACTS
Let me first recall the basic definitions and some fundamental realisations of freeness. For a more extensive review, I refer the reader to the lectures of Biane in these volumes or to the books [VDN], [Vo4] (see also the survey [Vo3]). 0.1. Definitions. 1) A (non-commutative) probability space consists of a pair (A,
A i C A are called free, if we have ip(ai . . . % ) = 0
whenever OieAjd),
j(l)^j{2)^...^j(k),aadip(oi)=Q
{i =
3) Random variables xlt..., xn G A are called free, if A\,..., is the unital algebra generated by xit
l,...,k).
An are free, where Ai
A canonical realisation of free random variables is given on the full Fock space. 0.2. Definitions. Let Ti be a Hilbert space. 1) The full Fock space over H is the Hilbert space
r{n):=cu®@n®n, n>l
where fi is a distinguished unit vector, called the vacuum. 2) The vacuum expectation is the state A^
(Q.,AQ). 209
210 Roland Speicher 3) For each / G H we define the (left) annihilation operator 1(f) and the (left) creation operator I* (f) by
i(f)n = o, i(f)fx ® • • • ® /„ =
r(/)n = /, r(/)/x ® • • • ® /„ = / ® /i ® ■ • ■ ® /„. One should note that, with respect to whether we denote by /(/) the annihilation or the creation operator, we follow the opposite convention to Voiculescu. 0.3. Proposition. Let Hi and 7i 2 be Hilbert spaces and put H := H\&Hi- Consider the full Fock space over H and the corresponding creation and annihilation operators 1(f) and l'(f) for f € H. Put A := C*(l(f) | / 6 K:),
A2 := C*(l(f) | / e H2).
Then A\ and .A2 are free with respect to the vacuum expectation. If we replace the C*-algebras by von Neumann algebras or by *-algebras, then the analogous statements are also true. The proof consists of directly checking for the case of *-algebras and then extending the assertion to uniform or weak closure by approximation arguments. Whereas freeness is just modelled according to the situation on the full Fock space hence its appearance in this context is not very surprising - there is another realisation of freeness in a totally different context: freeness can also be thought of as the mathematical structure of N x N random matrices in the limit N —» 00. We will not need this connection in our considerations, but it is always good to keep in mind that all our constructions also have some meaning in terms of random matrices. Such random matrices XN are N x N matrices whose entries are classical random variables and usually one is interested in the averaged eigenvalue distribution of these matrices corresponding to a state given by the averaged normalised trace. The following theorem is due to Voiculescu [Vo2] (see also [S2]). 0.4. Theorem. 1) Gaussian random matrices Let Aw = N~1tr(XN) and ^ = N~ltr(YN), where
^ =
ffl=„
^ = ($°)."=i
"=1.2,.-
are sequences of random symmetric N x N matrices with •
Free calculus 211 Then the sequence (\N,HN) limit as N —> oo.
converges in distribution to a semicircular family in the
2) R a n d o m l y r o t a t e d m a t r i c e s Let XN = AN and YN = UNBNUN where AN and BN are (nonrandom) symmetric N x N matrices and UN is a random unitary matrix from the ensemble QN = (U(N), normalised Haar measure). Suppose that the eigenvalue distribution of AN converges to a compactly supported measure, that is
N 1
~ E
/w - / M f o r / e °i>(R)>
A e.v. of AH
for some such meassure /j,, and likewise for BN- Then, with respect to the state v(') : = (w* r (')) n /vi XN a n d YN become free in the limit as N —* oo. 1. COMBINATORIAL ASPECTS O F FREENESS: THE C O N C E P T O F CUMULANTS
'Freeness' of random variables is defined in terms of mixed moments; namely the defining property is that very special moments (alternating and centered ones) have to vanish. This requirement is not easy to handle in concrete calculations. Thus we will present here another approach to freeness, more combinatorial in nature, which puts the main emphasis on so called 'free cumulants'. These are some polynomials in the moments which behave much better with respect to freeness than the moments. The nomenclature comes from classical probability theory where corresponding objects are also well known and are usually called 'cumulants' or 'semi-invariants'. There is a combinatorial description of these classical cumulants, which depends on partitions of sets. In the same way, free cumulants can also be described combinatorially, the only difference to the classical case is that one has to replace all partitions by so called 'non-crossing partitions'. In the case of one random variable, we will also indicate the relation of this combi natorial description with the analytical one presented in the course of Biane; namely our cumulants are in this case just the coefficients of the jR-transform of Voiculescu (in the classical case the cumulants are the coefficients of the logarithm of the Fourier transform). Thus we will obtain purely combinatorial proofs of the main results on the i?-transform. This combinatorial description of freeness is due to myself ([SI], [S3], [S 4]; see also [Nic]); in a series of joint papers with A. Nica ([NSl], [NS2], [NS3]) it was pursued very far and yielded a lot of new results in free probability theory. I will restrict here mainly to the basic facts; for applications one should consult the original papers or my survey [S5]. A recent fundamental link between freeness and the representation theory of the permutation groups Sn in the limit n —> oo, which rests also on the combinatorial description of freeness, is due to Biane ([Bia]). 1.1. Definitions. A partition of the set S := { 1 , . . . , n} is a decomposition w=
{Vi,...,Vr}
212 Roland Speicher of S into disjoint and non-empty sets V$, i.e. Vi^i
(i = l , . . . , r )
and
S = {j.=
We denote the set of all partitions of S by V(S). For 1 < p, q < n we write p ~-n q
Vi.
The V, are called thefeZocfcsof %.
if p and g belong to the same block of w.
A partition 7r is called non-crossing if the following does not occur: There exist 1 < Pi < <7i < V2 < 92 < w with Pi ~7T P2 A
?1 ~7T 92-
The set of all non-crossing partitions of { 1 , . . . , n} is denoted by NC(n). We denote the 'biggest' and the 'smallest' element in NC(n) by l n and 0 n , respectively: l n : = {(l,...,n)},
0n:={(l),...»}.
Non-crossing partitions were introduced by Kreweras [Kre] in a purely combinato rial context without any reference to probability theory. 1.2. Examples. We will also use a graphical notation for our partitions; the term 'non-crossing' will become evident in such a notation. Let 5 = {1,2,3,4,5}. Then the partition
. „ „ ,_ 12345
K
TT = {(1,3,5), (2), (4)}
=
is non-crossing, whereas
LLLLI ,
7r = {(l,3, 5), (2, 4)}
=
1 H-* 1
is crossing. 1.3. Remarks. 1) In an analogous way, non-crossing partitions NC(S) fined for any linearly ordered set 5; of course, we have WC(Si) = NC(S2)
if
#S1 =
can be de
#S2.
2) In most cases the following recursive description of non-crossing partitions is of great use: a partition 7r is non-crossing if and only if at least one block V £ n is an interval and 7r\V is non-crossing; i.e. V € n has the form V = (k, k + 1 , . . . , k + p)
for some 1 < k < n and p > 0, k + p
and we have TT\V € NC{1,...,
k - 1, k + p + 1 , . . . , n) = NC{n - (p + 1)).
Example: The partition
, oo^ccyoqin
, U II U {(1,10), (2,5,9), (3,4), (6), (7,8)}
Free calculus 213 can, by successive removal of intervals, be reduced to {(1,10), (2,5,9)}={(1,5), (2,3,4)} and finally to {(1,5)}={(1,2)}. 3) By writing a partition it in the form IT = {Vi, ■ ■ ■ ,Vr} we will always assume that the elements within each block V* are ordered in increasing order. 1.4. Definition. Let (.A, ip) be a probability space, i.e. A is a unital algebra and
(n e N)
(indirectly) by the following system of equations: tp(a1...an)
=
^
kn[au...,an]
(alt...
,an € A),
7reNC(n)
where kv denotes a product of cumulants according to the block structure of IT: Jc T [ai,...,a„] := fcyjai, • • • ,an\...
kVr[ai,...,
an]
for TT = {Vi,...
,VT} € NC(n)
and kv[ai,...,an]:=k#v(aVl,...,an)
for
V = (vh ... ,v{).
1.5. Remarks and Examples. 1) Note: the above equations have the form
= kn(ai,,..,an)+
and thus they can be resolved for the kn(ai,..., 2) Examples:
k a
A i> ■ ■ • >an],
^
an) in a unique way.
• n =1 ip(ai) = k\[ai] = fci(oi), thus ki(ai) = (p(ai). • 71= 2
ip(aia2) = ku[ai,a2} + k\ i[a 1 ,a 2 ] = k2{ai,a2) + &i(a1)fc1(a2), thus k2(a1,a2)
=
• n = 3 ip(a1a2a3) = kuj[alta2,a3]
+ k[ u [oi,a 2 ,a 3 ] + ku
\[alta2,a3]
+ k\_Li[a1,a2,a3] + k\ \ i[ai,a2,a3] = k3(alta2,a3)
+ k1(ai)k2(a2,
a3) + fc2(ai, a2)fci(a3)
+ k2(au a3)ki(a2) + A;1(ai)&i(a2)fc1(a3),
214 Roland Speicher and thus k3{au a2, o3) = ip(aia2a3) - ^(oi)
- v?(aia 2 )^(a 3 a4) - c/p(aia4)
4) The kn are multi-linear functionals in their n arguments. The meaning of the concept 'cumulants' for freeness is shown by the following the orem. 1.6. Theorem. Let (A,y>) be a probability space and consider unital subalgebras A\,..., Am C A. Then the following two statements are equivalent: i) Ai, ■ ■ ■ ,Am are free. ii) We have for all n > 2 and for all at e A^i) with 1 < j ( l ) , . . . ,j(n) < m: fc„(ai,...,
a„) = 0
if there exist 1 < I,fc< n with j(l) ^
j(k).
1.7. Remarks. 1) This characterisation of freeness in terms of cumulants is the translation of the definition of freeness in terms of moments - by using the rela tion between moments and cumulants from Definition 2.4. One should note that in contrast to the characterisation in terms of moments we do not require that j ( l ) v^ i(2) 7^ ■ ■ • ¥" j{n) o r (at) = 0. Hence the characterisation of freeness in terms of cumulants is much easier to use in concrete calculations. 2) Since the unit 1 is free from everything, the above theorem contains as a special case the statement: fc„(oi,...,
an) = 0
if n > 2 and a* = 1 for at least one i.
This special case will also present an important step in the proof of Theorem 2.6 and will be proved separately as a lemma. 3) Note also: for n = 1 we have fci(l) = V(l) = 1Prooj. (i) ==> (ii): If all oj are centered, i.e.
= kn(a,i,...,an)+
^
k^at,...
,an],
TriENC(n)
because at least one factor of kn for ir ^ ln is of the form fcp+i(oi, a i + i , . . . , at+p)
with p + 1 < n,
and thus the assertion follows by induction. The essential part of the proof consists in showing that on the level of cumulants the assumption 'centered' is not needed and 'alternating' can be weakened to 'mixed'. Let us start with getting rid of the assumption 'centered'. For this we will need the following lemma - which is of course a special case of our theorem.
Free calculus 215 Lemma. Let n > 2 and a j , . . . , an € A. Then we have: there exists a l < t < n with en = 1
=>
fcn(oi,...,
o„) = 0.
Proof. To simplify notation we consider the case an = 1, i.e. we want to show fcn(ai,...,an_i,l)
= 0.
We will prove this by induction on n. n = 2 : the assertion is true, since fc2(a, 1) =
]P
Av[ai,...,a„_i, 1]
TrgiVC(n)
= An(ai,...,an_i,l)+
2j
M a i>---> a n-i>l]-
According to our induction hypothesis only such ? r / l n contribute to the above sum which have the property that (n) is a one-element block of w, i.e. which have the form 7r = crU(n) with a € NC(n - 1). Then we have kv[ai,...,an-i,l]
=
fca[ai,...>a„_i]fc1(l)
= ka[au..
.tan^],
hence
=
fc„(ai,...,a„_i,l)
+
^
M a i > . . . ,a n _i]
a€iVC{n-l)
= ^ ( a ! , . . . , ^ - ! , ! ) + v?(oi...a„_i). Since Vj(a!. . . a n _ i l ) = v>(ai. ..o„_i), we obtain kn(ai,...
, o n _ 1 , l ) = 0.
D Let n > 2. Then this lemma implies that we have for arbitrary a i , . . . , an € A the relation fcn(ai,... ,a n ) = kn{ax - y ( a i ) l , . . . ,a„ - <^(a n )l), i.e. we can center the arguments of our cumulants kn (n > 2) without changing the value of the cumulants. Thus we have proved the following statement: Consider n > 2 and at G Aj(i) (i = 1 , . . . , n) with j'(l) 7^ j(2) ^ • ■ • =£ j(n). Then we have fen(ai,...,a„)
= 0.
It remains to weaken the assumption 'alternating' to 'mixed'. For this we will need the following lemma.
216 Roland Speicher L e m m a . Consider n > 2, a i , . . . , an € A and 1 < p < n — 1. Then we have fcn-l(al!
• • ■ i a p - l , a p a p + l i a p + 2 > • • • , 0>n)
= kn{a\,...
, ap, a p + i , . . . , an) +
2_^
kw[a\,...,
a p , a p + i , . . . , an\.
xeNC(n)
Examples: /c 2 (aia 2 ,a 3 ) = k3{aua2,a3)
+ h{a-i)k2{a2)a3)
+
k2(ai,aa)k1(a2,)
k3(a,i, a2a3, a 4 ) = fc4(ai, a2, a 3 , a 4 ) + fci(o2)fc3(ai, ^3, ^4) + k2(alta2)k2(a3,a4)
+ fc3(ai, a 2 , a4)A:i(a3).
Proof. For 7r G NC(n) we denote by 7r| p=p+ i € NC(n — 1) that partition which is obtained by identifying p and p + 1, i.e. for n = { V i , . . . , K } we have 7r| p=p+1
=
{ K , - - . , ( 1 4 U Vt)\{p + 1 } , . . . , K } ,
ifpeyfcandp + l 6 ^ .
(If p and p + 1 belong to different blocks, then 7r| p=p+1 has one block less than 7r; if p and p + 1 belong to the same block, then the number of blocks does not change; of course, we identify partitions of the set { 1 , . . . ,p,p + 2 , . . . , n} with partitions from NC(n — 1); the property 'non-crossing' is preserved under the transition from n to 7I"|p=p+l-)
Example: Consider
.„„...
rr={(l,5,6),(2) l (3 ) 4)}
=
LLbLLI
Then we have
12 3 4 5
7r|5=6 = {(l,5),(2),(3,4)}
=
LUzLi
and
12 3 4 6
7r|4=5 = {(1,3,4,6), (2)}={(1,3,4,5), (2)}
=
HJJJ.
With the help of this definition we can state our assertion more generally for ka for arbitrary a € NC(n — 1): Assume that our assertion is true for all I < n, i.e. fc/_i(ai,.. , , a p a p + i , . . . ,a ( ) = =
fci^Jai,... 2_^
,apap+i,... fc^fai,...,
,a ( ] ap, a p + i , . . . , a;].
*lp=p+l = 1 !-l
Then it is quite easy to see that we have for arbitrary a G NC{n — 1) with a / l„_i: a
p+l> ■ • • ) an\-
,eNC(n)
We will now prove the assertion of our lemma by induction on n. n = 2: The assertion is true because fc1(a1a2) = ^(0102) = (
kx(ai)ki(a2).
Free calculus
217
n — 1 —> n: Let the assertion be proven for all I < n, which implies, as indicated above, that we have also for all a £ NC(n — 1) with a ^ 1„_! ka[ai,...,
apap+x,...,
an\ =
y ^
« w [ai,..., a p , a p + i , . . . , a n ] .
iSNC(n) »lp=p+l="
Then we have ^ n - l l f l l i ■ • • > Opflp+l) • • • ! An)
=
2J
K[ai,...,apav+i,...,an)
tr£NC(n-l)
=
2J
z2
«E»C(n-l)
irEWC(n)
= v?(ai • • ■flpflp+i■ ■ • an) -
^[ai,.. .,ap,ap+1,...,an]
y2
kw[ai,...,ap,ap+i,...,an]
7r€«C(n) *ip=p+l?iln-l
=
M°i>-- -iapiOp+1)--- ,an] -
2J
22 Jr
=
^2
k^ai,.
..,ap,ap+i,...,an]
£1
lp=p+l? n-l
h[alt...,ap, ^ p + l ) • • • : &n\
7r
lp=p+l=1n-l
= Kn(a,i,
. . . , d p , G.p+1, • • ■ , CLn) +
/ y K w [fli, . . . , flip, flp+i, . . . , fflnJ. #»r=2,p/„p+l
D By using this lemma we can now prove our theorem in full generality: Consider n > 2 and a, £ A,-(;) (i — 1 , . . . , n). Assume that there exist k, I with j(k) ^ j'(0We have to show that kn(ai,...,On) = 0. If j ( l ) ^ j'(2) ^ • • • 7^ j( n )> then the assertion is already proved. Thus we can assume that there exists a p with j(p) = j(p+l), implying that apap+i £ Aj(p). In that case we can use the above lemma to obtain kn\a\,...,
ap, a p + i , . . . , an) = fcn_i(ai,..., a p a p + i , . . . , o„) —
y ^
k^lai,...
,ap,ap+i,...
,an\.
ireNC(n)
To show that this vanishes, we will again use induction on n: The first term fc„_i(ai,..., a p a p + i , . . . , an) vanishes by induction hypothesis since two of its argu ments are lying in the different algebras A,(/c) and Aj(i). Consider now the summand kw[a.i,..., Op, Op+i, • •., an] for n £ NC{n) with n = {Vi, V2},
where p £ V\ and p + 1 £ V2.
218 Roland Speicher Then we have k% = Avj&v2, and by induction hypothesis this can be different from zero only in the case where all arguments in each of the two factors come from the same algebra; but this would imply that in the first factor all arguments are in Aj(p) and in the second factor all arguments are in Aj(p+i). Because of j(p) — j(p + 1) this would imply j(l) = j(2) = ■ • • = j(n), yielding a contradiction with j(l) ^ j(k). Thus all terms of the right hand side have to vanish and we obtain kn(ai,...,
ap, a p + i , . . . , an) = 0.
(ii) ==> (i): (ii) gives an inductive way to calculate uniquely all mixed moments; according to what we have proved above these mixed moments must be calculated in the same way as for free subalgebras; but this means of course that these subalgebras are free. □ 1.8. N o t a t i o n . For a random variable a € A we put kl :=
kn(a,...,a)
and call (&£)n>i the (free) cumulants of a. Our main theorem on the vanishing of mixed cumulants in free variables specialises in this one-dimensional case to the linearity of the cumulants. 1.9. Proposition. Let a and b be free. Then we have kl+b = kan + kbn
foralln>l.
Proof. We have C
= kn(a +-b,...,a
+ b) = kn(a,...,
6) = A£ + kbn,
a) + kn(b,...,
because cumulants which have both a and b as arguments vanish by Theorem 2.6.
□
Thus, free convolution is easy to describe on the level of cumulants; the cumulants are additive under free convolution. It remains to make the connection between mo ments and cumulants as explicit as possible. On a combinatorial level, our definition specialises in the one-dimensional case to the following relation. 1.10. Proposition. Let (mn)n>i and (kn)n>i be the moments and free cumulants, respectively, of some random variable. The connection between these two sequences of numbers is given by 77ln =
f
t
KWi
w<ENC(n)
where K ■= k#Vl ■ ■ ■ k#Vr
for
it =
{Vu...,Vr}.
Example: For n = 3 we have "^3 = fciu + k\ u + ku i + &LLI + h 11
= kz + Zkxk2 + k\. For concrete calculations, however, one would prefer to have a more analytical de scription of the relation between moments and cumulants. This can be achieved by translating the above relation to corresponding formal power series.
Free calculus
219
1.11. Theorem. Let (m„) n >i and (fc„)n>i be two sequences of complex numbers and consider the corresponding formal power series oo
oo
+ Yjmnzn,
M(z):=l
C{z):=l
J^Kzn.
+
n—l
n=\
Then the following three statements are equivalent: (i) For all n 6 N mn =
]P
K =
ir€NC(n)
k v
# * ■ ■ ■ k*v*-
^ ir={Vi,...,Vr}eJVC(n)
(ii) For all n G N (where we put m0 := 1) n
mn = ^2
Yl
S=l h
ksmix...mi3.
«»6{0,l,...,n-») ilH Hs=n-s
(iii) C[xM(z)] = M(z). Proof. We rewrite the sum mn =
y ^ kx n£NC(n)
in the way that we fix the first block V\ of 7r (i.e. that block which contains the element 1) and sum over all possibilities for the other blocks; in the end we sum over
mn = Yl s=l
]C **■
Y, Vs with #Vj = s
•£»
If Vi = ( « ! = 1,V 2 , ■•■.«s),
then 7r = { V i , . . . } € NC(n) can only connect elements lying between some Vk and Vk+i, i.e. 7r = {Vi, V2,..., VT} such that we have for all j' = 2 , . . . , r: there exists a k with Vk < Vj < Vk+i ■ There we put vs+1 := n+ 1. Hence such a 7r decomposes as 7T = V i U T T i U - ' - U T T s ,
where ?f^ is a non-crossing partition of {VJ + 1, u,- + 2 , . . . , t;j + 1 — 1}. For such 7r we have
220 Roland Speicher and thus we obtain n
mn = y ^ S=\
/
/
J
\—Vl
ksk^l ... /cjs
y
jr=V1U?1U--Uir5
ij£NC(Vj+l
=X> s=l n =
E
"j+i-D
(
E
**)■■■(
l=vi
y
j
s—l n
Ks
/
777.U2— ui— 1 ■ • ■
y
fc
O
m
n—vs
l=vi
= E
E
S=l
E
Tf,eNC(v, + l,...,n)
ksmil...mi,
(ik~vk+i-vk-l).
ii,...,i»e{0.1,...,n-»} • !+••■+•»+»=!■
This yields the implication (i) ==> (ii). We can now rewrite (ii) in terms of the corresponding formal power series in the following way (where we put m0 := k0 := 1): oo
M{z) = 1 + J^ n=l oo =
1
+E n=I
zn
mn
n
E s=l >!
OO
kszsmilz11
^ i,£|0,I
...miazu
n-»}
OO
= i+^Ms(Em^)s s=l
t=0
= C[zM(z)]. This yields (iii). Since (iii) describes uniquely a fixed relation between the numbers (kn)n>i and the numbers (m n ) n >i, this has to be the relation (i). □ If we rewrite the above relation between the formal power series in terms of the Cauchy transform
and the /^-transform oo
R{z):=J2k^zn> n=0
then we obtain Voiculescu's formula. 1.12. Corollary. The relation between the Cauchy transform G{z) and the .R-transform R(z) of a random variable is given by G[R{z) + ~]=z.
Free calculus 221 Proof. We just have to note that the formal power series M(z) and C{z) from The orem 2.11 and G(z), R(z), and K(z) = R(z) + - are related by: G(z) =
\M{\)
and C(z) = 1 + zR(z) = zK{z),
thus
K(z) = ^ - .
This gives
K[G{z)] = J^CWz))
= ^ C [ i M ( i ) ] = ^M^)
= ,,
thus iC"[G(z)] = z and hence also G[R(z) + -] = G[K{z)} = 2. D 1.13. Remark. It is quite easy to check that the cumulants k" of a random variable a are indeed the coefficients of the /^-transform of a as introduced by Voiculescu: Remember that the distribution of a was modelled by the canonical variable (special formal power series in an isometry I*, see [VDN]) oo
b = r + YJh+iii e(6(0,r). Then we have oo
t=0
J2
T{r^...^)ki{1)+1...kiln)+1,
i(l),...,i(n)e{-l,0,l,...,n-l}
where /
_1
is identified with I*,
rl
=
r
and k0 := 1. The sum is running over tuples ( i ( l ) , . . . , i(n)), which can be identified with paths in the lattice 1?:
i —k
i = —1
=
diagonal step upwards: I .
i= 0
==
horizontal step to the right: [ „
=
diagonal step downwards:
(I < k < n — 1)
222 Roland Speicher We have now 1,
if i(l)+ ■■■ +i(m) < 0 V m = l , , n and i(l) + • ■ •-M(n) = 0 otherwise
r(rw...r^) = < 0, and thus
E
mn =
fo(l)+l • • • k{(n)+l-
i(l),...,i(„)6{-l,0,l n-1} i(l)+-+i(m)<0 Vm=l n i(l)+-+i(n)=0
Hence only such paths from (0, 0) to (n, 0) contribute which stay always above the x-axis. Each such path is weighted in a multiplicative way (using the cumulants) with the length of its steps. Example:
)h
y
v
V\
\ \ \
The above summation can now be rewritten in terms of a summation over noncrossing partitions leading to the relation from Proposition 2.10. We will leave the proof of this lemma to the reader. Lemma. There exists a canonical bijection NC(n) <—» { ( i ( l ) , . . . ,t(n)) \i(m) e { - 1 , 0 , 1 , . . . ,n - 1}, i(l) +
\-i(m)<0
Vro = l , . . . , n ;
i(l) + • ■ • + i(n) = 0
}.
It is given by 7ri-^n = (i(l),...,z(n)) where #14 — 1, if m is the last element in a block V? -1, otherwise Note that a block consisting of one element gives a horizontal step; a block consist ing of k (k > 2) elements gives k — 1 upward steps each of length 1 and one downward step of length k—1. An example for this bijection is 123456 I I = ir = {(1,6), (2,4,5), (3)} is mapped to
=^2\
/*
\ : ±s
/ 11= ( - 1 , - 1 , 0 , - 1 , 2 , 1 ) = Now note that with this identification of paths and non-crossing partitions the factor kn = &t(i)+i • ■ • &i( n )+i
Free calculus
223
for n =
n = (i(l),...,i(n))
{Vu...,Vr}
goes over to K-TT
'.
K:: # V i
»«•
Consider the above example:
JlZ\k^
TT =
{(1,6),(2,4,5),(3)}
n =
-7 7
-Ss»
thus K = kn = k1k3k2 = k#(3)fc#(2,4,5)fc#(i,6)This correspondence leads of course to the relation as stated in Proposition 2.10 ; thus the coefficients of the .R-transform of Voiculescu coincide indeed with the free cumulants as defined in 2.8. Note that in this way we obtained easy combinato rial proofs of the main facts on the iJ-transform - namely, its additivity under free convolution and the formula relating it to the Cauchy transform. Finally, to show that our description of freeness in terms of cumulants also has a significance apart from dealing with additive free convolution, we will apply it to the problem of the product of free random variables: Consider a i , . . . , an, b\,..., bn such that { a i , . . . , an} and {&i,..., bn} are free. We want to express the distributions of the random variables a\b\,..., anbn in terms of the distributions of the a's and of the b's. 1.14. Notation. 1) Analogously to k^ we define for Tr={V1,...,Vr}eNC(n) the expression ■ ,an]
: = VvAaii
■ ■ ■ » a n ].ip■Vr[alt...
,a
where ipv[au...,
for
■ ip(av
V=(v1>.
,Vl).
Examples: V U j [ a i i « 2 , f f l 3 ] = <^(«l«203)
Then we write a < IT
if each block of a is contained as a whole in some block of n, i.e. a can be obtained out of 7r by refinement of the block structure. ]5xQ>7Tl7)lc'
{(1), (2,4), (3), (5,6)} < {(1,5,6), (2,3,4)}
224 Roland Speicher With these notations we can generalise the relation ip{ax... an) =
^2
kn[ai,...,On]
rr6/VC(n)
in the following way. 1.15. Proposition. Consider n € N, a € NC(n) and a\,..., ipa[a1,...,an}=
^
fcfffai,..
an £ A. Then we have
.,an].
7r£«C(n) TT
Proof. Each 7r<(T={y1)...lK} can be decomposed as TT = 7TX U--- U7rr
where
^ e JVC(K)
(i=l,...,r).
In such a case we have of course n*n
n/71-j . . . fCfly.
Thus we obtain (omitting the arguments) TriSWC(Vj)
Kr£NC(Vr)
Tr = 7T]U---U7Tr
7T<(J
□ Consider now {a1,...,an},{b1,...,bn} free. We want to express alternating moments in a and b in terms of moments of a and moments of b. We have ip(aibia2b2. ■ ■ anbn) =
^
K[ai, h, «2, b2, ■ ■ ■, an, bn\.
ireNC(2n)
Since the a's are free from the 6's, Theorem 2.6 tells us that only such 7r contribute to the sum whose blocks do not connect a's with 6's. But this means that such a w has to decompose as 7r = 7riU7T2
where
^ <E NC{1,3, 5 , . . . , In - 1) 7T2 e 7 V C ( 2 , 4 , 6 , . . . , 2 n ) .
Thus we have (p(aib1a2b2 . . . anbn) =
^
kni [at
TriUn2eNC(,2n)
=
^ meNCiodd)
(fcn-Jai, 02,--.,»»] *
^2 "2£WC( OTe n)
K2[hMi--
■ A]
Free calculus 225 Note now that for a fixed ix\ there exists a maximal element a with the property 7Ti U a G NC(2n) and that the second sum is running over all 7r2 < o. 1.16. Definition. Let 7r £ NC{n) be a non-crossing partition of the numbers 1 , . . . ,n. Introduce additional numbers 1 , . . . , n, with alternating order between the old and the new ones, i.e. we order them in the way 1122... nn. We define the complement K{n) of n as the maximal a € NC(l,... property 7rU(7 6
tn)
with the
NC(l,l,...,n,n).
If we present the partition ■K graphically by connecting the blocks in 1 , . . . ,n, then a is given by connecting as much as possible the numbers 1 , . . . ,n without getting crossings among themselves and with n. (This natural notation of the complement of a non-crossing partition is also due to Kreweras [Kre]. Note that there is no analogue of this for the case of all partitions.) With this definition we can continue our above calculation as follows:
=
^ [kwi[ai,a2, nieNCIn)
=
fc
■ ■ ■ ,an] ■ ^ »2ewc(n)
kn2[b1,b2,...,bn]J
*a
X^
a a
Tl[ l) 2i
a
■■■ > n] '
Thus we have proved the following result. 1.17. Theorem. Consider {ai,...,
an}, {&!,..., bn}
free.
Then we have ip(aibia2b2 ... anbn) =
^
/c 7 r [a 1 ,a 2 ,..., o„] • fK(n)[h, b2, ■ ■ ■, bn].
7r€NC(n)
Examples: For n = 1 we get ip(ab) = ki(a)tp(b) = tp(a)tp(b); n = 2 yields V5(a161a262) = ki{al)kl{a2)Lp{bib2)
+ k2(au a2)
=
- v?(a1)
226 Roland Speicher 2. F R E E STOCHASTIC CALCULUS
In this lecture, we will develop the analogue of a stochastic calculus for free Brow nian motion. Free Brownian motion is characterised by the same requirements as classical Brownian motion, one only has to replace 'independent increments' by 'free increments' and the normal distribution by the semi-circle law. In the same way as classical Brownian motion can be written as at + a*t with at and a* being annihilation and creation operators, respectively, on the Bosonic Fock space, the free Brownian motion has a canonical realisation as lt + I* for lt and If being (left) annihilation and creation operators on the full Fock space. Thus, instead of developping a sto chastic calculus for free Brownian motion St = h + 1%, one could also split St into its two summands and develop a free stochastic calculus for lt and I*, in analogy to the Hudson-Parthasarathy calculus for at and a*. This was done by Kiimmerer and Speicher [KiiS]. The free stochastic calculus with respect to St, which is due to Biane and Speicher [BiS], however, has some advantages and we will here restrict to that theory. In our presentation we will put the emphasis on two main points: • appropriate norms: at a linear level all stochastic calculi have formally the same structure, the main point lies in establishing the integrals with respect to appropriate norms; in contrast to all other known examples, the free calculus has the very strong feature that one has estimates with respect to the uniform operator norm; i.e. the free stochastic integrals can be defined in I? with p = oo. • ltd formula: on a formal level the difference between stochastic calculi lies in their multiplicative structure; at least formally, a stochastic calculus is charac terised by its Ito formula; for free stochastic calculus this is very similar to the Ito-formula for classical Brownian motion, however, due to non-commutativity there is a small, but decisive difference. 2.1. Definition. A free Brownian motion consists of • a von Neumann algebra A • a faithful normal tracial state r o n y t • a filtration (At)t>o ~ i-e. At are von Neumann subalgebras of A with As C At
for s
• a family of operators (St)t>o with - St = S* e At - for each t > 0, St has semicircular distribution with mean 0 and variance t - for all 0 < s < t, the increment St — Ss is free from As - for all 0 < s < t, the increment St — Ss has semicircular distribution with mean 0 and variance t — s Usually, we will call (St)t>o the free Brownian motion. In the same way as the classical Brownian motion can be realised on Bosonic Fock space, the free Brownian motion has a concrete realisation on the full Fock space -
Free calculus 227 as follows by Proposition 1.3. Note, however, that for the development of our free stochastic calculus we will not need this concrete realisation but just the abstract properties of (£t)t>o2.2. Theorem. Let k ■= KMo,i)),
I* = '*(l(o,t))
be the left annihilation and creation operators, respectively, for the indicator function of the interval (0,i) on the full Fock space T{Ti) for H = L 2 (0, oo). Put T[A]:=(n,AQ), and St := lt+ l*t. Then (St)t>0 is a free Brownian motion with respect to the filtration At ■■= vN(Ss
\s
2.3. Remark. According to the connection between freeness and random matrices there is also a random matrix realisation of free Brownian motion: Consider random matrices
*,<»>:= ^ ( % « J = „ where • Bij(t) are classical real-valued Brownian motions for all i,j • the matrices B\ ' are symmetric, i.e. 5y(t) = Bjt(t) for all i,j • apart from the symmetry condition, all entries are independent, i.e. {Bij(-) 1 < i < j < o°} are independent Brownian motions. Consider now the state p:=Eo(ltr), where E denotes the expectation with respect to the probability space specified above, and ^ t r is the normalised trace on N x N matrices, Then we have St
=
lim
B[N\
i.e. for all n G N and i j , . . . , tn > 0 we have: r[Stl...5j=
UmE[ltr(BW...i#0)]N—>oo
iV
We will not use this realisation, but it shows that our free stochastic calculus can also be viewed as the large N limit of stochastic calculi with respect to N x N hermitian matrix-valued Brownian motion.
228 Roland Speicher 2.4. Remarks. 1) Note that we have the non-commutative L p -spaces associated with our free Brownian motion. Namely, LP(A), for 1 < p < oo, is the completion of A with respect to the norm \\A\\LP :=
T[\AP]V*.
For p = oo, we put IIXIU- := p l l ,
i.e.
L°°(A)=A.
In the concrete realisation of St on the full Fock space, we can identify L2(A) with the full Fock space ^{Ti) and we can embed A into the full Fock space by the injective mapping A C T{H) A^
Ail.
2) For our later norm estimates it will be important that we can obtain the operator norm as the limit p —> oo of the L p -norms: For A £ A one has \\A\\ = lim H^Hip = lim p—>00
m ll2m
T\{A*A)
]
.
m—too
2.5. Remarks. 1) Let At, Bt be adapted processes. Then we want to define the stochastic integral
j AtdStBt. In contrast to the stochastic theories considered in the other courses, we have to face now the new phenomenon of two-sided integrals. In the usual cases, adaptedness of the process implies that the differentials commute with the process (or anti-commute in the fermionic case), thus a two-sided integral can always be reduced to a one-sided one and there is no need to consider two-sided integrals. But in our case there is no such reduction. Adaptedness implies that the differential and the process are free, but this does not result in any commutation relation. Thus we should consider as the most general integral the two-sided one. Note that one could of course restrict oneself to one-sided integrals in the beginning, but then a meaningful form of Ito formula would result automatically in two-sided integrals. 2) Since f AtdStBt is bilinear in At,Bt, it is natural to consider more general linear combinations
YsJAdStBl for adapted processes A\ and B\. We will also write this as
/'
UtUSt,
with
Ut :=Y^Ai®B'teA®
Aop
and call U = (Ut)t>o a biprocess. (Aop is the opposite algebra of A, i.e. with the same linear structure and the order of multiplication reversed; it is quite natural to consider U as an element in this space, since At multiplies from the left, whereas Bt multiplies from the right.) 3) The definition of the integral proceeds now as usual: First define the integral
Free calculus 229 for simple biprocesses, prove some adequate norm estimates for such cases and then extend the definition with respect to the involved norms. 2.6. Definition. Let Ut = At ® Bt be a simple adapted biprocess, i.e. there exist 0 = t0 < ti < ■ ■ ■ < tn < oo such that 1
(Ai®Bi
U
\o
tn< t.
(Adaptedness means here of course: A,,Bi define the integral
6 Ati-)
For such a simple biprocess we
UtUSt = / AtdStBt := ^2MStl+1 ~ Sti)Bi. ■*
i=o
For simple adapted biprocesses of the general form Ut = Yii A\ ® B\ we extend the definition by linearity. As usual, it is quite simple to obtain the isometry of the integral in L 2 -norm. 2.7. Proposition (Ito isometry). For all adapted simple biprocesses U and V, one has T[JutUSt
■ {JvtUStY] = (U,V) := J(Ut,Vt)L,9L2dt.
Proof. By bilinearity, it is enough to prove the assertion for processes Ut = A
and
Vt =
C®D-lltlM(t).
Then the left-hand side is equal to T[A(Sh-St0)BD*(St3-St2)C*}. Note that by linearity it suffices to consider the cases where the two time intervals are either the same or disjoint. In the first case we have r[A(Stl
- St0)BD*(Stl
- Sh)C']
= T[AC'}T{BD*]r[(Stl
-
Sto)2}
= T[AC7*]r[B£>*](t1-to) (because {^4, B, D", C*} is free from the increment Stl — Sta), whereas in the second case one of the increments is free from the rest and thus, because of the vanishing mean of the increment, the expression vanishes. But this gives exactly the assertion.
□
2.8. Notation. We endow the vector space of simple biprocesses with the norms (1
Wt\\LP(T®T°v)dt] (For p = oo, LCO{T®T°'9) is the von Neumann algebra tensor product of A and >40p.) The completion of the space of simple biprocesses for these norms will be denoted by Bp. The closed subspaces of Bp generated by adapted simple processes will be denoted by B*
230 Roland Speicher Thus we have shown that the map U — i > f Ut$dSt can be extended isometrically to a mapping But it is now an essential feature of the free calculus, which distinguishes it from all other ones, that one can even show a norm estimate for p = oo, i.e. the stochastic integral a bounded operator for a large class of of (bi)processes. In particular we do not have to think about possible domains of our operators and the multiplication of such integrals (as considered for the Ito formula) presents no problems. 2.9. Theorem (Burkholder-Gundy inequality). For any simple adapted biprocess U one has ||
[ut$dSt\\<2V2\\U\\Boo.
2.10. Corollary. The stochastic integral map U >-* f Ut$dSt can be extended con tinuously to a mapping In particular, the stochastic integral of an adapted bounded biprocess from B^ is a bounded operator. Proof. Let us just give a sketch of the proof. We restrict ourselves here to biprocesses of the form Ut = At®Bt- The extension to sums of such biprocesses follows the same ideas. Put
f UtUSt= (' AtdStBt
Ms
Jo Jo We want to obtain an operator norm estimate for Ms by using
||M S ||= lim
r[(M;Ms)m]1/2m.
m—*oo
This means that we must estimate the p-th moment of our integral for p —> oo. This is much harder than the case p = 2, but nevertheless it can be done by using again the crucial property r[A(Sh
- St0)B(Stl
- St0)C] = T[AC\T[B](tr - t0),
if {A, B, C} is free from the increment Stl — Sto. By using also the Holder inequality for non-commutative ZAspaces one can finally derive an inequality of the form m-l
r[\Ms\2m} <2mYJ
.s
{r[\Mt\2k\ ■ r[|M t | 2 m - 2 - 2 f c ] • \\Atf
■ \\Bt\\2)dt.
k=oJo
Note that the structure of this inequality resembles the recursion formula for the Catalan numbers c„, n-l
Cn =
y]ckCn-k-l.
Free calculus 231 From the above implicit inequality one may derive by induction the following explicit one r[\Ms\2m}
T Jo We now take the 2m-th root and note that
\\At\\2\\Bt\\2dt)m.
lim cli2m = 2. m—too
Thus we obtain ||Ms||<2V2(^S||A(||2||Bf||2dt)1/2 = 2 ^ ( ^ | | C / , | | | „ ( r ^ ) d t ) 1 / 2 . Setting s = oo gives the assertion.
D
Let us now present the Ito formula for the free calculus. First, we will do this on a formal differential level. As stated above, if we work with biprocesses from B^, then our integrals are bounded operators and multiplication presents no problem. We will show that the Ito formula holds even with respect to the operator norm. On a formal level, the Ito formula makes the difference between different stochastic calculi. At first sight, free Brownian motion (St)t>o has the same Ito formula as classical Brownian motion {Bt)t>0, namely dBtdBt = dt
and
dStdSt = dt.
In our non-commutative context, however, this does not contain all necessary infor mation, since we must now also specify dStAdSt
for A € At.
In the classical case, A commutes with the increment dBt and we have there dBtAdBt
= AdBtdBt
= Adt.
But for free Brownian motion we have a different result. 2.11. Theorem (Ito formula - product form). For a free Brownian motion {St)t>o we have the following Ito formula dStAdSt
= r[A]dt
for A e At.
Proof. Let / C [0, oo) be an interval and consider decompositions into disjoint subintervals Ii, I = U/j. For an interval / we denote by S(I) the corresponding increment of the free Brownian motion, i.e. S{I):=St-Ss
for
/=[s,t[.
The main point now is to show that (with A denoting the Lebesgue measure)
J2S(Ii)AS(Ii)^T[A]X(I), i
where we take the usual limit with width max; A(/j) of our decomposition going to zero. As said above, we want to see that this convergence holds even in the operator norm. We will sketch two proofs of this fact, one using the abstract properties of freeness, whereas the other works in a concrete representation on full Fock space.
232 Roland Speicher 1) The assertion follows from the following two facts about freeness: • ([NS1]) Let { s i , . . . , s„} be a semicircular family, i.e. each st is semicircu lar and S i , . . . , sn are free; then, for a random variable a which is free from {$1,... ,sn}, SidSi,.. .,sna$n are free • ([Vo 1]) Let Xi,... ,xn be free random variables with T[XJ] = 0 for all i = 1 , . . . , n; then Um + ■ • • + xn\\ < max Hud + \
*•—'
2C£r[\xi\2})1/2
»=1
2) We realise S(I) on the full Fock space as S(I) = 1(1) +1*(I); then we have to esti mate the four terms '£l(Ii)Al(Ii), EKh)Al*(I{), ^^(lAAl^i), and £ r(Ii)Al*(Ii). Three of these terms tend to zero by simple norm estimates, the only problematic case is
Y,Kh)M\Ii)-*T[A]\(I). i
(This corresponds of course to the Ito formulas for lt and I*, namely the only non zero term is dltAdl^ = T[A]dt, see [KiiS].) To prove this later statement, one can model A by the sum of creation and annihilation operators on the full Fock space as A = ^an(l + l*)n for I = 1(f) with / being orthogonal to L2(I), hence I and l" are free from all 1(1i), l*(h). Then one has to expand this representation of A and bring it, by using the Cuntz relations l(f)l*(g) = (f,g)l, into a normal ordered form A = J2 Pn,mFnlm'■ Finally, note that, again by the Cuntz relations, only the term for n = m = 0 contributes to the sum in our statement. □ 2.12. Example. The Ito formula contains also the germ for the combinatorial dif ference between independence and freeness - all partitions versus non-crossing parti tions. For a classical Brownian motion (Bt)t>o we can calculate the fourth moment r[Bf\ with the help of the Ito formula as follows: d(BAt) = 3B?dBt + QB2tdt, thus
which yields T[B}]
= 3i 2 .
In the case of the free Brownian motion we obtain in the same way d(Sf) = dStSf + StdStS? + SfdStSt + SfdSt + dStdStStSt
+ dStStdStSt
+
dStStStdSt
+ StdStdStSt
+ StdStStdSt
+
StStdStdSt
= dStSzt + StdStS? + SfdStSt + SfdSt + 3Sfdt + T[S?]dt, and so
Free calculus 233 yielding = 2t 2 .
r^}
This difference between the fourth moments in the classical and free case reflects the fact that there are 3 pairings of 4 elements, but only 2 of them are non-crossing. The Ito formula can also be put into a functional form to calculate the differential of a function f(St) for nice functions / - not just for polynomials. The main message of the classical Ito formula is that we have to make a Taylor expansion, but we should not stop after the first order in the differentials, but take also the second order into account using dBtdBt = dt, hence df(Bt) = f{Bt)dBt
+
= f'(Bt)dBt
+
l
-f"(Bt)dBtdBt l
-f"(Bt)dt
There exists also a free analogue of this; whereas the first order term is essentially the same as in the classical case, the second order term is different; one of the two derivatives is replaced by a difference expression. 2.13. Theorem (Ito formula - functional form). Let / be a sufficiently nice function (for example, a function of the form f(x) = f elxyn(dy) for a complex measure fj, with f \y\2\/j.\(dy) < oo). Then we have df(St) = df(St)ldSt where df(X)
+
\AJ(St)dt,
is the extension of the derivation n-l
k=a and A f / denotes the function dx J
x-y
where vt is the distribution of St, i.e. a semicircular distribution with variance t. Proof. One has to check the statement for polynomials by using the product form of the Ito formula. All expressions in the statement make sense also for nice functions and the statement extends by continuity. For a polynomial f(x) = xn we have
dS^ = J2StdStst~h~1+ k=0
Y, M>0 k + l
^dStSltdStS^
234
Roland
Speicher
T h e first t e r m gives directly the first term in our assertion (this is j u s t a nonc o m m u t a t i v e first derivative), whereas t h e second yields
J2 k,l>0 k+l
S^dStSltdStSrk~l~2 = Yl S^T[S[]dtS^'k~1'2 k,l>0 k+l
= Y, srl'2rislt}dt = *,!>o
fl-1
j2mSr1r[srm-1\dt,
m=l
A:+Z
which can be identified with t h e second t e r m in our s t a t e m e n t . One should note: t h e fact t h a t t h e t r a c e and not t h e identity acts on t h e expression between two differentials results finally in t h e unusual form of t h e second order t e r m in t h e functional form of t h e Ito formula; it is not a non-commutative version of t h e second derivative, b u t a m i x t u r e of derivative a n d difference expression. □ REFERENCES
[Bia] [BiS] [Kre] [KiiSJ [Nic] [NS 1] [NS 2] [NS3] [SI] [S 2] [S 3] [S 4]
[S 5]
[Vo 1] [Vo 2] [Vo 3]
P. Biane, Representations of symmetric groups and free probability, Advances Math. 138 (1998), 126-181. P. Biane and R. Speicher, Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, Probab. Theory Related Fields 112 (1998), 373-409. G.Kreweras, Sur les partitions non-croisees d'un cycle, Discrete Math. 1 (1972), 333-350. B.Kummerer and R. Speicher, Stochastic integration on the Cuntz algebra 0&,, J. Fund. Anal. 103 (1992), 372-408. A. Nica, /J-transforms of free joint distributions, and non-crossing partitions, J. Fund. Anal. 135 (1996), 271-296. A. Nica and R. Speicher, On the multiplication of free ra-tuples of non-commutative random variables (with an appendix by D. Voiculescu), Amer. J. Math. 118 (1996), 799-837. A. Nica and R. Speicher, i?-diagonal pairs—a common approach to Haar unitaries and cir cular elements, in [Vo4], pp. 149-188. A. Nica and R. Speicher, Commutators of free random variables, Duke Math. J. 92 (1998), 553-592. R.Speicher, A new example of 'independence' and 'white noise', Probab. Theory Related Fields 84 (1990), 141-159. R. Speicher, Free convolution and the random sum of matrices, Publ. Res Inst Math Sci, Kyoto 29 (1993), 731-744. R. Speicher, Multiplicative functions on the lattice of non-crossing partitions and free con volution, Math. Ann. 298 (1994), 611-628. R. Speicher, Combinatorial theory of the free product with amalgamation and operatorvalued free probability theory [Habilitationsschrift, 1994], Mem. Amer. Math. Soc. 132 (1998), no. 627. R. Speicher, Free probability theory and non-crossing partitions, Sem. Lothar. Combin. 39 (1997), Art. B39c, 38pp. (electronic); also available under http://www.rzuser.uni-heidelberg.de/~L95. D. Voiculescu, Addition of certain non-commuting random variables, J. Fund. Anal. 66 (1986), 323-346. D. Voiculescu, Limit laws for random matrices and free products, Invent. Math. 104 (1991), 201-220. D. Voiculescu, Free probability theory: random matrices and von Neumann algebras, in "Proceedings of the International Congress of Mathematicians, 1994" Birkhauser, Basel 1995, pp. 227-241.
Free calculus
235
[Vo4] D.-V.Voiculescu (ed.), "Free Probability Theory", Fields Institute Communications 12, American Mathematical Society, Providence, 1997. [VDN] D.V. Voiculescu, K.J. Dykema, and A. Nica, "Free Random Variables," CRM Monograph Series 1, American Mathematical Society, Providence, 1992. DEPARTMENT OF MATHEMATICS AND STATISTICS, QUEEN'S UNIVERSITY, JEFFERY HALL, KINGSTON, ON K7L 3N6, CANADA
E-mail address: speicherSmast.queensu.ca
Quantum Probability Communications, Vol. XII (pp. 237-260) © 2003 World Scientific Publishing Company
CONTINUOUS KERNEL PROCESSES IN Q U A N T U M PROBABILITY WILHELM VON WALDENFELS ABSTRACT. We present in this paper a theory of quantum stochastic processes based on the notion of kernels, a notion well established in quantum field theory and introduced to quantum probability by H. Maassen. As we restrict ourselves to continuous kernel processes we shall need only the tools of classical calculus and the proofs become rather easy. For the solution of a linear quantum stochatic differential equation an explicit formula is obtained. Under the usual conditions the solution can be extended to a strongly continuous unitary cocycle. It is indicated how the white noise equation can be obtained as a limit of the coloured noise equation.
CONTENTS
0. Introduction 1. Discrete kernels 2. A basic l e m m a arid heuristic considerations 3. Measurable kernels 4. Calculus of kernels 5. A d a p t e d processes 6. A linear differential equation 7. T h e singular coupling limit Acknowledgements References
0.
237 239 241 243 247 251 253 256 259 259
INTRODUCTION
Let A be a finite dimensional algebra over t h e complex n u m b e r s . T h e r e are different ways to represent it. If A h a p p e n s t o be a m a t r i x algebra, one could describe t h e matrices corresponding to t h e different elements of A- If A has a basis e\,..., en t h e n we can introduce t h e s t r u c t u r e coefficients C £ with
k
In a d u a l way one could define a convolution in C n by (x *y)k =
237
YJxiyJCij
238
Wilhelm von Waldenfels
for x, y € C n A similar situation arises in the algebra of canonical commutation relations. We are thinking here of the algebra of creation and annihilation operators at and a\ with t E K and [as,a\\ = 5(s -t). Usually one considers the operators a(f) = J f(t)atdt and a\f) = j f(t)a\dt as operators on the Fock space. This corresponds to the representation by matrices. We choose a generalised basis given by awaT with a — {si < ■ ■ ■ < sm} and r = {t^ < ■ ■ ■ < tn} and a]aaT = a\x...almatl
...atn
The structure coefficients are given by a ttri a ri a tff2 a T2 =
at(<JI+/3l)aft+7"2<5(a2, ft)
^
This corresponds to the following convolution {x*y)(a,r)=
]T
d-yx(ai,f3i+j)y(a2
+ y,p2)
«1 +Q2 =17, /3l +02=T
where a, r, a, (5, 7 are finite subsets of R. The study of this convolution and its applications is the main object of this paper. Representing an operator in the form X = /
dadrx{a,
is a very old idea of quantum field theory. The order to put creation operators before annihilation operators is physically very intuitive and mathematically very appealing. It is called normal ordering. For then the structure coefficients are positive. The function x is called the kernel of X. It is, however, very difficult to characterise the functions x , which define an operator X and the operators X, which are defined by a function x ([At lj). The use of kernels in quantum probability goes back to Maassen ([Maa]), who introduced the convolution in the way described above. Hence the kernels were called Maassen kernels. There are many generalisations due to Meyer, Belavkin, Lindsay, Maassen, Parthasarathy, Attal, Accardi, Lu and Volovich (e.g. [Mey], [Bel], [LiM], [LiP], [Atl], [At 2], [ALV]). See the lectures by J. M. Lindsay in these volumes. In the second section we study discrete kernels. We discuss the algebra generated by finitely many creation and annihilation operators, and determine the structure coefficients and the convolution. We will not need discrete kernels later on. But the study of discrete kernels shows some essential apects of the theory and is analytically much simpler. In the second section we prove the so-called integral-sum lemma, we study creation and annihilation operators indexed by the elements of an interval, give the stucture coefficients and determine the form of the convolution. In the third section we define
Continuous kernel processes in quantum probability 239 the convolution in a direct way, avoiding the analytical difficulties created by the unbounded generalised operators used in Section 2. Prom there on we proceed in a mathematically rigorous way. We do not attempt, however to show very general results. In the fourth section we go back to Maassen's first paper, where continuity and differentiability are used as in standard calculus. We introduce the notion of a continuous kernel, which means that (a, T) — i > x(<7, r)
is continuous in the space of all subsets of the real line with a H r = %. Similarly a continuous kernel process is a function (t,
which is continuous in all (t, a, r) with a D r = 0 and t ^ a n r. So the function x may have jumps. Stochastic integration may be introduced by the means of Riemann sums. Defin ing differentiability in a similar way, we can show the analogue of the fundamental theorem of calculus and establish a generalisation of Ito's formula on the differentiatiation of a product. Applying this to adapted processes we establish the Ito table for forward and backward adapted processes. It is easy to solve a linear differential equation as a kernel process. The process turns out to be forward and backward adapted. Under the usual assumptions on the coefficients the kernel process can be extended in a unique way to a unitary cocycle, which is strongly continuous. In the last section we indicate the theorem, that the solution of a linear stochastic differential equation, using coloured noise instead of white noise, converges to the corresponding white noise equation as a kernel process, if the correlation function becomes a J-function. The Ito term arises in a natural way. The main ideas of this paper were already established in a paper studying the quantum stochastic process of the inverse oscillator ([Wl]). 1. D I S C R E T E KERNELS
A Weyl algebra ID(V, H) is an algebra say over the complex numbers, where V is a vector space and H is a skew symmetric bilinear form. The algebra is generated by any basis of V and has the defining relations uv — vu = H(u, v) for u,v e V. We restrict ourselves to the algebra ron generated by a,i,a\,i = l,...,n commutation relations hold a^a] - a t ^ = 5ij
where the
aidj — a,j(ii — ajat — a]aj = 0 The algebra ro„ forms a *-algebra in a canonical way. Denote by SOT the set of all multisets or multi-indices of n elements, i.e. all functions {!,..., n} —> N = { 0 , 1 , 2 , . . . }. The Fock space Tn is the complex Hilbert space
240
Wilhelm von Waldenfels
spanned by \a), where a runs through all elements of
{i})
a\\a) = Vi7~+T\a + {i}) where \a — {i}) = 0 if i does not belong to a, i.e. a* = 0. Introduce the notations
a\ = ot\\ • ■ ■ an\
# a = at -i
h an
Denoting by 0 the empty multiset, i.e. the function identically 0, we have |<*> = -^=|0>.
(1.1)
One obtains
™" =
E SA^Mfj.li/
(l-2)
Hence the structure coefficients are given by a^a^a^a^^
V
J^La^+^a^+M.
(1.3)
We have the matrix representation
0^, = E > ^ ± M ± 3 1 | t t + 7 ) ( / 9 + 7 i .
(1.4)
7
Definition. A kernel is a complex valued function on DJl x 2JI. The operator, or better the formal element of the Weyl algebra ron associated to a kernel x is given by
Associate to X the matrix (AIXlMH^-l^AlatVV)
(1.6)
then due to equation (1.4) (\\X\fi) = £
/Mj: ^-pxA_7,M_r
(1.7)
As this triangular system of equations can be solved in unique way, we obtain the following theorem: Theorem 1.1. To any kernel x is associated a matrix {\\X\fi} in a unique way, and vice versa.
Continuous kernel processes in quantum probability 241 Example 1: The unity element has the representation 1 if a = (3 = 0 0 otherwise Example 2: The vacuum projector has the representation
10x01 = £ L - ~ a t a a a Definition. Let x and y be kernels. Their convolution z = x * y is given by O~\T\ „ l„ Ifl Ifl r„i a: "i,/3i+7Z/Q2+7,/92
E
(I-**)
Ql +Q2=CT, P l + P 2 = T , 7
provided the sum converges absolutely. For the proof of the following proposition it is essential, that the coefficients of the convolution are positive. Proposition 1.2. Let x and y be two kernels and let X and Y be the associated matrices. If the convolution z of x and y exists, then the product Z of X and Y exists, i.e. (X\ZM =
Tr,(X\X\p)(p\Y\n) p
converges absolutely for fixed A and /j,, and Z is the matrix associated to z. 2. A BASIC LEMMA AND HEURISTIC CONSIDERATIONS We discuss in this section the Weyl algebra generated by {at,a\ :t€ defining non trivial relation
[as,aj] =
1} with the
S(s-t).
This Weyl algebra is familiar from Quantum Field Theory. As the algebra consists of integrals not of sums of generators, it is generated in a generalised way. Let / C R be an open subset. Denote by f2(J) the set of all finite subsets of / . 00
fi(/) = |J
sin(i), fio(/) = {0}, nn(i) = {oue n(i) ■. #w = «},
tl=0
where # denotes cardinality. We identify Q,n{I) with the set Let duj denote the measure on f2(J) which has 0 as an atom of measure 1, and which equals the Lebesgue measure on f2„(7) for n = 1, 2, • • •. Had we introduced the set of all finite submultisets of / instead of the set of all finite subsets, the submultisets with multiple points would form a set of measure 0, so we can restrict ourselves to proper subsets. We have
/ f(w)
242
Wilhelm von Waldenfels
Definition. Assume that u>i, ■ ■ -ujn € Q(I). Then , 1
w
n
( u\U ■■■ U u>„ if UiHLJj = $ for i ^ j , \ not defined otherwise.
=
So is defined almost everywhere. An important analytical tool is the so-called integral-sum lemma which has been known to specialists in stochastic point processes for a long time as a special case of Campbell's theorem ([MKM]) and is stated explicitly in [FF1], [FF2], [LiM], [Lin]. Lemma 2.1 (W,). Let f : Q,(I)n — i > C be integrable, then da
^2
f(ai,---
,an) =
■■■
da1---danf(cr1,---an)
ai+~-+an=o
Proof. Let D = {{
,an) Y2 m
I "I J
f{o-x,---
,on)doi---don.
JDn{#(Ti+-+#cr„=m}
Assume that 0\ + ■ ■ • + an = r = {tj < • • • < tm} and I3■, — {i : 1 < i < n : i, € a3}. Denoting a3 = t^, we obtain
Y2 m
Yl Ixj
J'■" / dh---dtmf{th,---
,tin)
|-/„={l,-,m}
=
Y2 \#r=m
dr
Yl /("i.--- >Q")ajH
\-an=r
□ The following discussion is heuristic, and we use the formalism of Quantum Field Theory without establishing the rigorous definitions of the unbounded generalised operators. This formalism is easier for calculations, algebraically more transparent but not so useful however for rigorous mathematical considerations. Therefore we will not use it any more later on and shall switch to the notion of convolution. Define (0 for#a^#r 5„tT= I 1 for a = r = % (2.1) { 5{si - h) ■ ■ ■ 8(sn - tn) fore-= { s l l . . . , s n } , T = {*!,...,*„} We have aaa(0=
Y, ai+a2=a,/3i+ft=/3
a101aai6a2!p2.
(2.2)
Continuous kernel processes in quantum probability 243 Prom there we obtain the structure coefficients aWlaT>aW2aT2=
a)^+^ah+n5{a2,f32).
J^
(2.3)
cti +02=1-1.01 +02 =02
We denote I") = atQ|0>,
(2-4)
and obtain ( 0 | a Q a t » = <5Q/?. Definition. A kernelis a [continuous, measurable, generalised,...)
function
x : 0(7) x fi(Z) -► C. We associate to it the quantity X = fj
dad(3x{a,l3)a)aa0
The matrix element associated to X is (a\X\p)=
£
i(ai,A)^A
(2-5)
ai+c«2=Q, l 3i+/?2=/3
So if a and [3 are fixed and a O (5 = 0 and x(a, (3) has a well defined value, then (*\X\P)=x{a,p).
(2.6)
This allows to calculate x from X. (This is carried out rigorously in [BeL].) The convolution of two kernels x and y is given by ]P
dtux(aUT1+ui)y(a2
+ oj,T2)
"l+"2 T1+T2=T
So if to x and y are associated the quantities X and Y, then to their convolution is associated the product XY. This, again is only a formal statement. In the next section we shall establish it in a more rigorous way. It is hidden in the associativity theorem. 3. M E A S U R A B L E KERNELS
Definition. A measurable kernel is a measurable function fi(/)2 <-+ C. Definition. Let x and y be measurable kernels. Their convolution x * y exists if ^2
dw\x(auTi+oj)\\{y(o2+^,T2)\
<,i+«2=<, J T]+T2=T
for almost all o, r and if this is the case we define (x*y)(a,r)=
^ TI+T2=T
/ d w i ^ n + w l ^ + w.rj)
244
Wilhelm von Waldenfels
Theorem 3.1 (Associativity). (zi
*---*xn)(
= ]T) '" I djn '''dllnrf723'''dl2n''' rf7n-1>n «i-t
t-a n =
xi(ai,j3i
+ 7 i 2 + --- + 7in)
X2 {a2 + 7l2i fa + 723 + ' • • + 72n)
Xn-l{an-l
+7l,n-l H
£n (Q„ + 7i,„ H n
/■
/
■■■ a If- —H»„=»
H7n-2,n-i;/?n-l + 7n-l,n)
h 7„-i,„; /?«) (
n
i~\
\
<*>« I I ^ ( aJ + X ) T*J> & + I ] Tw ■
n l
J= l
\
t=l
i=J + l
/
More explicitly assume that the righthand side exists almost everywhere and denote it by M(xi,--- ,xn). Then, for 1 < k < n, the quantities M(x\,--- ,Xk) and M(xk+i, ■ ■ • ,xn) exist a.e., the convolution exists, M{xu ■■■ ,xn) = M(xx, ■■■ ,xk)* M(xk+1 ■■•xn) and M(x\, 12) = £1 * £2-
Proof. Assuming that all Xi > 0, and introducing
7=
Yl
T«
l
we obtain M(xu ■ ■ ■ , xn){a, T) = / d^( I ■ ■ ■ I J
J
J k
x
Y[
l
a
J2 X) I I A i+Yl "i+"2=" A j = l T l+ T 2~ r n
d
7y
d
II
^
k+l
n
T«;fa+ J2 TW + J2 7*
V
t=l
/
k
«=7+l
7
j—\
i=A+l
;
n
\
n *>• (^+iz «' + ]c ^ & + x IK
where
Continuous kernel processes in quantum probability 245 A = { ( a i , • • ■ , an, Pi, ■ ■ ■ ,Pn, ai-\
('Yij)i
'■
h ah = ai, Q„+I H
h an
=
a2,
Pi + ■ ■ ■ + Pk = n , Pn+l + ---+Pn = T2,
J2
7
«J
=
l} ■
l
Define the set A = {(»7i,--- ,Vn,di,---
,dn) ■■Vi + --- + Vk = o-i; %+i H
H ?7» = o-2 + -y;
tf 1 + • • ' + $k = Tl + 7; ^fc+l + • ' ■ + tin = T2} and a mapping p : A - i A given by ' ctj
for 1 < j < k k
■ a,-:
+ Y^
7JJ
for fc + 1 < j < n
i=i
, _ M + £ Tw for 1 < 3 < k Vj ~~ \
i=k+l
(Pj
for k + 1 < j < n
The mapping
= 3 _
'
for j = l,--- ,fc for j = / c + l , - - - ,n
7 y = i?i n 7?j-
Hence M ( z i , - - - ,a;n)(cr,r)_
= /d7 / • ■ • / n ^
J
J
**
Ki<j'
/
n
d7y
(fe+l
ri+T2=r
X
-
J-1
II J K + £ j=k+i
T
V
i=fc+i
7i
J' di + E 7^ t=j+i
y
1+T2=T
D
246
Wilhelm von Waldenfels
We define e(a,T) = < „ ., . (3.1) v v (^ 0 otherwise. ' Then x*e = e*x = x, for all measurable kernels — the convolution being always defined. Define xT(a, r) = x(r,a), then (a: * y)T = yT * xT.
Let £ be a measurable function on fl(/) and let £ be a kernel. We define products
(z*£)(w)=
53 I dTX^Ttfto+T)
(£*s)(w) =
£)
(3.2)
and f dT^i+T)x(r,w2)
(3.3)
Finally if £ and 77 are two functions ctions on £1(7) £!(/) we define
t*V = Jdu£(t»)ri(u)
(3.4)
We obtain £*x*i]=
da dfidy
f(a + 7) z(a,/3) viP + l)
(3.5)
Assume now that £ and 77 have disjoint support, that is there exists 5 c l such that £(a) = 0 for a $£ 5, and 77(a) = 0 for a $£ Sc, then £*x*r)=
/ / dad/?
^{a)x(a,j3)r]{(3),
so it is directly giving the bilinear form defined by x. This is the integral version of equation (2.6). Define for a € £1(1)) ,
e
N
f 1 for a = 0
^ = {ofor^0
.
.
^
If £ is a function on u)(I) define \ (a, r) = (,(0) e (r), then for a kernel x (x * f)(w) = (a; * £)(w, 0), and (£ * x)(u>) = (C * x)(0, LO). Let
/ : 7 —» C be measurable. Define:
so that
^(/)(a,r)={ 0 'M 1 : = { 5 } ' T = 0'.
(3-8)
For real / we write A*(f) as usual. If a; is a kernel, and x(a, r) = 0 unless #(<x U T ) = 1, then x = AT{xW) + A (x
( s ) = x {{s}, 0) and x™ (s) = a; (0, {s}).
Continuous kernel processes in quantum probability 247 If x and y are two kernels which vanish unless #(cr U r) = 1, then (x * y)(0, 0) =
| d 7 1(0,7) 2/ (7. 0) = / <*s a;'2' ( s j i / ^ s ) .
The associativity theorem implies the following version of Wick's theorem. T h e o r e m 3.2 (Variant of Wick's theorem). Let X\, ■ ■ ■ , xn be n kernels with the property that Xi{o~,r) = 0 unless # c U r = 1. 7/cr U r = {£i < • • • < tm} and (n — m) = 21, then 1
(xj * • • • * xn)(a, T) = J2 ^ ( i j ( ^ ( i ) ) ' ' ' xim)
(*•»(•»)) I I
/
3=1
J
>7,C
f
ds X
S ) M 4 o ) ( s )-
Here r\ runs through all injective maps r\ : a U T —> {1, • • • ,n}, and where ( runs through all pair partitions
C = {{«(i)
= 2 ana!
<«(*) = ^1) (*) = ^w(0.{<})If n(i) G r, £/ien i9(i) = 1 and
<$w = 4&w=*,««*}. 0). Ifn — m is odd, then ( i ! * ■ • • * x n ) (cr, T ) = 0
Pnoo/. We proceed by induction. Assume at first, that a U r = 0. Go back to the main formula of the associativity theorem. Then a.\ = • • • an = Pi = • • • = / ? „ = 0. Unless everything vanishes, one has # (712 + ■ • • + 7i n ) = 1. So for one 2, 2 < i < n, there is # 7 K = 1 and hence %j = 0 for 2 < j < n, j / j . The kernel X; appears with the argument 2,(71,4 + • • ■ + 7i-i,i;7t,i+i + • • • + 7i,n)- So everything vanishes, unless 72,j = • • • = 7,-1,; = ju+i = ■ • • = 7i,„ = 0- We are left with / d-ju xf] (7H) xf] (71;) j (x2 * •••Xi-i *Si+i * ••• *xn) (0,0) This shows, that we can prove the formula in the case o = r = 0, by induction with respect to n. Assume now that a U r 7^ 0 and for simplicity that 77(1) €
First some notations. Let S be a set, A C 5 and B C S d , then A®B:=
{(a,b) : a £ A, b = (6X • • • , bd) G B : a^ bu-■ ■ a.jL bd},
and : = > 1 ( 8 ) J 4 < 8 > - - - ® , 4 = {(ai, • ■ • , a d ) € ^ ;
a, ^ a, for i 7^
j}.
248
Wilhelm von Waldenfels
If Ac V(S)d, A®B
where V(S) is the set of all subsets of S, then = {{a,f3):
a € A, 0 = (J3U--- ,0d) € B : a n (A U ■ • • U Pd) = 0},
and A®d = {(au ■ ■ ■ , a„) € Ad : en n a,- = 0 for j ^ j } . If ^ C 5 and B C P{S)d, then A® B = {(a,0):aeA,l3=(J3u---,Pd)€B:
a tfaU-'-U
Pd}.
So the reason for using ® instead of x is to take out the diagonals. Definition. Let I C R be an open subset. A continuous kernel in I is a continuous mapping Q.(I)®2 —► C, where f2(/)®2 inherits the topology o/K resp. R n in the usual way. A C° kernel process is a continuous mapping x : I ® fl(/)®2 —> C such that for fixed (er, T) € fl(/)®2 the function 11—> xt(a, r) is locally integrable. Recall the definitions of A#(f), Example: At(",r)
see equation (3.8)
= A[lhooA){tr,T)=io
^
AU N A\,, M \ fl ^ ( ^ r ) = At(l1_oo,tl)(aIr)=|0
if a = ^
{S},T
= 0, s < t
are C° kernel processes. The cases t' = t, s = t are not defined as they do not belong to/
< u, < t', andti,Ui <£ trUr. Lei<5 = max(i l —i l _ 1 ).
* ( J 4 t i - A i - 1 ) j (<7,T)-» ^ z ( (cr,r\{£}) / te]to,ti[nT
U i
^T^-^^W)V i
/
Proo/. Let A4 H ^ ' V ' ]
and
£ xs(a\{S},r) 86]to,tl[ncr
A
^ ( t ) = ^ ( U j ) , then
{xUi*A(Ai))(cr,T)=
Yl ^ ( ^ A W ) £€inA» Choose 8 so small that there is at most one element of r in any of the A;. Denote for t € T by i(t) the index i such that t G A,. Then ( ] £ xUi * ^ ( A i ) ) (o-, T) =
J2 xui(t){o-, r\{t}) tei-n]s,t[
-> ] T xt(o-, r\{*}), t
Continuous kernel processes in quantum probability 249 as \Ui(t) — t\ < 5 and s i—+ xs(o, r\{t})
is continuous in s = t.
Definition (Skorokhod - integral). Let Sal. I xt*dAt) ^
s
<JS
Then
(cr,r) = ^ terns
'
dA\*xt)(o-,T)= '
D
Y] seans
xt(a,r\{t}), xs(a\{s},t).
Remark. The proposition shows that for continuous processes the Skorokhod integral can be defined as a Riemann integral. The condition of local integrability of t i—> Xt{o-,r) was not needed. Definition. Let x : I ® n(/)® 2 —-> C be a C°-process. We say that x is a Cl process if for t £ cr U r each of foXt(a>T)> (R±x)t(a>T)
= xt±o{o- + {t},T),
and (R±x)t(o-,T)
= xt±0{a,T + {*}),
exists and defines a C° process. Remark. If x is C1 then xt±o(°~, T) exists for all t & I. Theorem 4.2 (Maassen). Let x be aCl process and let [to, *i] C / , then /■*!
xtl-0 - xto+0 = /
dA\ *ft+
J]ioM
gt*dAt +
htdt 11 Jt00
J]to,ti[
with ft = Dlx = Rl+itx — Re_tx,
gt = D[x = Rr+tx — RLitx,
and ht = -r;xt.
Definition. For the statement of the theorem we use the abbreviated version dxt = dA\ *ft + gt* dAt + htdt Proof. Assume that to = so < Sj < • • • < sn < sn+i = t\, and (cr + T)f]}t0,ti{= {si,--- , s„}. Then xtl-0(cr,T)
-xto+0(a,T)
n+1
n
= ^2(xSi_0(o-, r) - xSi_1+0) (o-, r) + Y^(xsi+ ~ ^_o)(ff. T) n+i
rsi
= J2 / i=l
J
n
( {Dlx)H{a
dtht(a,r) + J2
«»-i
i=l
— {si],r)
(Drx)Si(a,T\{Si}
if Sj 6 cr j if
Si
er
J
D Lemma 4.3 (Maassen). Let I C R be open and relatively compact, and let z / ® f i ( 7 ) —> C be a continuous (t,ui)-integrable function. Assume that zt{uj) = -j- zt(w),
and (Dz)t(w) = zt+0(u> + {t}) - zt-0(ui + {t})
250 Wilhelm von Waldenfels exist and are continuous, and (t,ui) iniegrable, and that 11—► / zt (ui)dui, and 11—> / (Dz)t(ui)du> are continuous. Then t i—► / zt(ui)duj is continuously differentiable and -— / zt(oj)duj = / zu(t)dui + / (Dz)t(ui)dw. Proof. Assume that w C I, [to, ti] C I and i0l*i $ w- Then ZtM-**(")
=
zt{w)duj+ Jt
°
Y,
{Dz)t{oj\{t}).
Kwn[t0,ti]
Using the ^T-lemma we obtain f
] T (Dz)t(u\{t})dw = f ^ tewn[to,ti)
dtduj{Dzt)(uj). D 1
Theorem 4.4 (Generalisation of the Ito formula). Let xt and yt be C processes. Then xt * yt is a C1 process under the assumptions stated below and d{xt * yt) = dxt *yt + xt* dyt + I{x * y)t where I(x * y)t is the ltd term given by I(x * y)t = (RT+x)t * (Rl+y)t - (Rr_z)t * (Rl_y)t. More explicitly d{xt * yt) = dA\ * ft + gt* dAt + ht dt, with ft = {Dlx)t *yt + xt* {Dly)u gt = (DTx)t *yt + xt* {Dry)t, and ht = xt*yt + xt*yt + I{x * y)t. Furthermore {Rl+{x * y))t = (Rl+x) *yt + xt* {Rl+y)t, and similarly for R[_ and R±. The assumptions of the theorem are, that all convolutions exist and form Cl pro cesses. Proof. Let a, T € Q®2 and let 7o C /, be a bounded open interval such that Io n (cr U T) = 0. Then (t,w)i—>zt{u) = ] T
xt{o-i,Ti+u)yt{o-2 + uj,T2)
Continuous kernel processes in quantum probability 251 fulfills the assumptions of the preceding lemma. Hence / zt(u>)duj =
(xt*yt)(cr,T)
is continuously differentiable and -j, ixy * Vt)(o-, T) = / zt(uj)doj + (Dz)t(ui)duj = (± * y)t{ff, T) + (X* y)(a, r ) + I(x * y)t(a, r ) Assume t0 £ h {xt*yt)(a
a
n d t > to. Then
+ {t0},T)=
\ duj
{{xt{vi + {to},n
^
+
uj)yt(a2+UJ,T2)
+aJt(oi, n + u))yt(a2 + {to} + OJ, T2)) Denote the integrand by zt(u>). Using the definitions and the proof of the previous lemma we obtain rt
Zt{u) =
/ ds ZS(OJ) +
^2
Dsz(uj\{t})+zt0+0(uj),
/ duizt{u) = ds dwzs(uj) + ds du>Dsz(u>) + J Jto J Jto J J
du; zto+0(uj).
Now / zs(ui)dw = (x * y)s(a + {t},r) + (x * y)s(a + {t}, t), and fdoj{Dz)s(w)
= I{x * y)s(a + {*},T).
By assumption both functions are locally integrable in s so, letting t J. to we obtain R+(x * V)u>{v, r)=
j dw zto+0(uj) = ((R+x)t * yto)(ff, r ) + (xto * (Rl+y)to(a, r ) .
□ 5. A D A P T E D PROCESSES 2
Definition. Let x : fi(/) —> C be a kernel. We say that x is supported by I0 C I if x{a, T) = 0 when a U r (jL IQ. R e m a r k . Let x and y be two measurable kernels supported by two disjoint sets A and Ac, then x * y always exists and (x*y)(a,T)
= (y*x)(a,r)
= x(cr D A, r n A) y (a D AC,T n Ac).
Definition. Let zt(<7, T) be a kernel process. We call xt forward adapted, if it is supported by ] — oo, t] and we call it backward adapted, if it is supported by [t, oof.
252
Wilhelm von Waldenfels
The forward Ito-integral is approximated by the Riemann sum £
xti * (A# +1 - A*) = ^(A*+1
- A*) * z t i ,
a n d the backward Ito-integral is approximated by
£ ^ * (A* - A*,,) = ^ ( 4 - AU * *uHence we use for adapted processes the notation / xt * dAf = / dAf * xt, where A* denotes either A or A^. Proposition 5.1. If x is forward adapted, then xmajCT(a,T\maxT) ( / xt * dAt j (a, r) = \ V]Wi[ / [0
L/ Al *") (a ' r)= {o
£maxo-(
if t0 < max
If x is backward adapted, then f
,,\/
J)toM[ f
/
s
J ,.f
if t0 < min r < m i n a < tx else
[°
\ /
s _ J xmina(a\
I
{°
"-At * xt\
J\toM
I Xmmrio-, r \ min r)
(O-,T) = <
miner, r)
if i 0 < miner < m i n r < t i else
-
Proof. Immediate.
□ l
r
Remark. If x is forward adapted, then (R _x)t = (R _x)t = 0, and if x is backward adapted, then (Rl+x)t = {Rr+x)t = 0 for all t. We use the assumptions of the generalised Ito formula of the last section (see Theorem 4.4). T h e o r e m 5.2 (Ito-formula). If x and y are adapted processes, then d(x * y)t = dxt *yt + xt* dyt + dxt * dyt where the product dxt * dyt is calculated with the help of the Ito-table
dA dt
dA dA* dt 0 edt 0 0 0 0 0 0 0
Continuous kernel processes in quantum probability 253 So only the coeficient of dAt * dA] survives and is to be replaced by edt. Here e is defined in the following way: + 1 if x and y are both forward adapted —1 if x and y are both backward adapted 0 if one of x and y is forward adapted, and the other is backward adapted. 6. A LINEAR DIFFERENTIAL EQUATION
We use kernels with values in B(H) instead of C where H is a Hilbert space. Theorem 6.1. Let I C K be a bounded open interval. Let Lt : / —> B(H), i = 1, 2, 3, be three norm continuous functions and consider the differential equation dtUls = Lx{t)Ul * dAt + L2(t) dA\ * Ul + L3(t)Uldt, with
Ul — e, that is rrsi
^
I * <7 = T = 0
U T)
^ =\Q
else
•
Then this differential equation has a unique solution,
namely
Ul(a, T) = VI L^QVt-r ^ ( V i ) • ■ ■ Lei (h) V?, ifaUr = {t1<-< *„} c ] s , t[, and Ul(a, r) = 0 if o U r <£ [s, i\. (The cases s,t G a U T being not defined as usual). The points s,t might be arbitrarily chosen in I with s
jtvi
= L3(t)v;,
v/ = i
and 1 2
for ti € T for ti e a
So Uls is forward adapted w.r.t. t and backward adapted w.r.t. s. We have ds Ul = -
Ul LY (s)*dAs
- &A\* Ul L2 (s) - Ul L3{s) dt.
l
Proof. At first we see that dt U s (0,0) = L3(t) Uls (0,0), so U\ (0,0) = VSK Use instead of a, T the set of pairs £ = {(ii, £i), • ■ ■ , {tn, £*,)}, then
Ul ( 0 = Y^ Lei(t) U? (Z\{tit
£i})
l{U€]sA} + f
L3 (f) Ul' (0 dt'
Js
i=i
and U^ (£) = 0 for # £ ^ 0. Let # £ = 1, then
Ul (£) = L£l (t) U? (0, 0) l { t l 6 l . i ( [ } + f
I
Jo
L3(t')Ul'
L3(t>) Ul'(0 dt'
(Odt'iit^}s,t[.
254
Wilhelm von Waldenfels
In this case Ul (?) = Kf £/* (?) = 0. Proceed by induction and show that Uls (a, r ) = 0 if o U T £ ] s , i[. Assume that (a U T) n ] s, £ [= {tj < ■ ■ ■ < tk}- Then Ul (?) = £
L
^
=I
« (*) ^
^ \ {**- e< » + f
L
3 ( 0 Ut (?) ^
L3(t')Ul'(Odt' = 0,
b y induction hypothesis and because U* (?) = 0. So f j is forward adapted w.r.t. t a n d backward adapted w.r.t. s. So ^ ( 0 = Len (Q Ul" (? \ {*„, £„}) + /
L3 (f) Ul' (?) dt'
Jtn
f/sf
for i > tn and equals 0 for t < tn. So (?) = V^ LEn (£„) Uln (? - {£„, £„}). Proceed by induction to arrive at the formula of the theorem. □ T h e o r e m 6.2. For r < s < t we have Uls * f/rs = U$. Proof. Uls and U* are supported by disjoint sets }s, t[ and ]r, s [. So the convolu tion is no problem. Differentiate: ds (Ul * U?) = 0, so Ul * U' is independent of s and equals U* . □ T h e o r e m 6.3. Define Ul by the equation dt Ul = -Ul L^t) * dAt - dA\ * Ul L2(t) - Ul (L3(t) - L^t) L2(t)) dt, Then Ul * Ul = Ul * Ul = e. Proof. Choose a closed bounded interval IQ (Z I and define C = sup{||L 1 (i)||, ||L 2 (£)||, ||L 3 (t)|| + \\L,(t)\\
\\L2(t)\\}.
i
Then, for s,t e IQ, we have
\Pl(o,r)\\<
C#^ec^l{aUTC]0A},
and Y,
f duWUKaun+^W
\\Ul(o2 + co,r2)\\
T1 +T 2 =T
TI
=
+T2=X
C « e « M
W c | M B
as f dujC2*"l{uj
c j ^ h ^ C
2
" ^
=
e2C^s\
Uss = e.
Continuous kernel processes in quantum probability 255 From this estimate we can deduce that we can apply Theorem 5.2. One checks inmediately that ds{U\ * £/*) = 0, and dt (Ul * U\) = 0 and one deduces from there the theorem. □ We shall use the Hilbert space L2 (f2 (I), H) with the scalar product (77IC) = J
du,(v(uMw))
Call L0 (Q(I),H) C L2 (Cl(I),H) the subspace formed by the functions £ : fi(7) -> U such that £ is continuous and ||£(w)ll is bounded and £ is supported by a bounded subinterval of I. T h e o r e m 6.4. Assume that Lj(t) = -L2(t)1 fine the operator
and L3{t) + L\(t) = Lx(t) L2{t).
De
The operator is well-defined on L0(Q.(I),Ti.). It can be extended to a unitary operator on L2 (Q(I),H) in a unique way. We define for s < t, W* = (WS')T, and obtain a unitary family of operators (W*)s,tei with W/ = 1, for r,s,ts
Wl Wrs = Wlr)
I. Furthermore (t, s) i—> W* is strongly continuous.
Proof. We want to show that {W^IW^) = (T?|0 for all £,r? e L0(Q(I),H). Now the candidate of this expression is (rj * (E/,)* * U\ * £). We have to check, that
IMI * 11(^11 * M * Hill < oo. Now this expression is
JIJll^a
+ ^Hm^^llUlll^rnmr
+
^lldadTd^
and by an inequality used in the previous theorem for U and U, we see that this expression is bounded by / / /
I M U | £ | | o o l { C T + W C / „ } 1{T+Uc/O}l{aurc]0,t[} C
<
llllloolhlloo [ duliuClo]
f dr
=
I K I U M U e|/o1 e~c(«-°) < oo.
J2
2
™ ' ^
e4C<'->
dodrdiU
Mrcm]C2#TeiC^
As W(T, a) = U(a, r)* by the last theorem, W * U = U * W = e we obtain by using associativity (Theorem 3.1)
Define W^ = (W)l * £, for £ e LQ (fi(7), H), and obtain (W^r) \ W& = (T7|£). So W\ can as well be extended in a unique way to a unitary operator on L2 (0(7), H).
256
Wilhelm von Waldenfels
As Ujt * jjt * £ = jjt ^ Vit ^
=
VKsf
^
for
^
l
e LQ
(Q^^ ?^
we see t h a t
wt
Ws< = 1, and that (W^ = W s = (W\)- . We have still to prove the continuity. Assume again that f £ L0 (£2(1),H).
K - C U
2
wt
=
1
Then
^ *([/<-e)+ * ( ^ " e) * ?,
and
w-.)(^)-{, 0! ""»!{*ir T + :> l ai . Hence IKtf] - e)(<7,r)|| < C#<"+T> ec<«-> l { # ( f f + T ) > 1 } , and
(IK^-c^lhll^-elD^r) T2+T2=r
l{
1{T2+T2+U;^0}
+T
< l { f f + T ^ } (2C)# ( - )l { < T + T C ] s , t ! } + (2C)#<" > ( e ^ * - ) - 1) l { f f + T C w[}. Hence
lien * l l ^ - en * wut - en * yen < U\\l JJj
dadrdcj
l^+ac/.jl^+rcMdl^-eMI^-elD^r)
< llellL e|7»l ((e4C(ts> - 1) e 4 ^ " 8 ' + e6C(£-°» (e2C
Recall the definition of the Stratonovich integral in the theory of integration with respect to Brownian motion. Choose a continuous function g of compact support and define coloured noise by
F(t) = f g{t - s)dB. where Bt is Brownian motion. In order to calculate an integral of the form
1=
J...ff(t1,...,tn)dBtl...dBtn
one replaces dBt by F(t)dt, white noise by coloured noise, and writes 1(9) =[■•■[
/&>•• ■ • - i " ) F ( t l ) ' ' ■ F(tn)dt! . dt-n
As F(t) has continuous trajectories, the integral can be defined pathwise, whereas dBt is very irregular. Go with g —> 6 and obtain / = lim/(
Continuous kernel processes in quantum probability 257 We use the notations of Section 2 and consider the stochastic differential equation ATI1 ^■={LQF{t) + Ll^{t))Ul, where
F(t) = j g{t-
s)asds
is coloured quantum noise. Byy letting g —> 7<5 we oobtain the quantum stochastic differential equation dt
(-yL0at + 7L10J + nLoL^Ug,
where 7 = f g(t)dt and « = fo° dt f*™ dsg(t — s)g~(—s). White noise is a limit of coloured noise. Whereas white noise is analytically very difficult but gives explicit and nice results, coloured noise is analytically much easier, but it is difficult to work with. Recall the definition of A(f) and A^(f)t given in equations (1.5) and (3.8). These quantities are the kernels of the operators a(f) and a!(f) with « ( / ) = [ f(t)atdt,
andat(/)= j
f{t)a\dt
Call T the Fock space L 2 (fi(R)) and let Tm be the subspace of vectors £ with £(w) = 0 for #w > m. Then a(f) maps fm into Tm-i,
IK/)£I|2 < yMWfhUh, whereas a)(f) maps Tm into Tm+\
and
l l ^ / ) ^ < ^^r^FTiiy||2||e])2We can formulate and prove this result easily in the language of kernels. The map f 1—> A(f) * £ takes Tm into ^" m _i, and satisfies
\\A(f) Hh < V^II/IWKIk f
£ i-» i4 (/) * £ maps Tm into J^m+i, and satisfies
\\A\f) *zh < v^TT||/||2||ei|2. The coloured noise F(t) is given by the kernel A{gt) with gt{s) = #(£ — s), which we denote again by F(t). Consider the stochastic differential equation ^
= (LoF(t) + L ^ - i ) ) * ^ = K{t) * [/.',
E£ = e,
where L 0 and L\ are bounded operators on a Hilbet space 7i. The formal solution of the differential equation is U\ = Yl^Lo U^ with U(0) = e and U(n) = /
# ( s n ) * • • • * K(Sl)dsn
Js<si<-<sn
Denote T = L2(fi(]R, £ ) ) . If f 6 J"m then
II^W*ll|2<2cy|2V^Ti||cil2,.
...dSl.
258
Wilhelm von Waldenfels
C = max{||Lo||, ||£i||}. Hence
\\U(n) * £||a < -, V(m+l)...(m
+ n)(2C\\gUt - S)TU\\2.
Since Yl™=o \\U^ * £||2 < oo, the solution exists. Now replace g by ge, where ,^
l
(t
&(*) = - * ( and denote the corresponding U and K by Ye and Ke respectively. Then
ll&l|2 = ^ | M | 2 , and for s | 0 the solution fails. We have however the following theorem. Theorem 7.1. Assume that g € L\ nZ^OR). As £ I 0 the kernel Ue converges to the solution of the quantum stochastic differential equation dUl —^
= (>yL0at+iLial
+
KLQL^UI,
or in kernel notation dtul = (~fL0dAt + 7lqcL4j + KLQLX e di) * uls, ■with 1=1
g(t)dt, and « = /
k(t)dt.
We have 2Re/c
k(t)dt. ■oo
Outline of the proof. For fixed <7, r we have U<-n\alT)=
f
(K£(sn)...K£(sn))(a,T)dsl...dsn.
Following Theorem 3.2 we can write
where rj : U U T —> { 1 , . . . ,n} is an injection and £ is a pair partition of the set { 1 , . . . , n} \ (a U r ) . We assume that o U r = {t\ < ■ ■ ■ < tm] and that n-m is even, say 21. (If n — m is odd, the expression vanishes anyhow.) We call (rj, C) regular, if r\ is order preserving and ( contains only pairs of nearest neighbours. If this is not the case 7r is called irregular. If(?7, () is irregular, then by some complicated estimates Aj,<,£ —> 0 as £ J. 0. Assume now that (77, Q is regular. We see that { 1 , . . . , n} is split into consecutive intervals: { 1 , . . . ,n} = lo U O i } UXi U {j2} U • • • U lm-i
U {jm} U l m
where each #Zj is even, say # 2 , = 2Zj. We have t) ■ a U T -> {J'I , . . . , jn} : r){i) = j , ,
Continuous
and £ consists of partitions of 2i,...,
probability
259
ln into neighbouring pairs. We have m
^4rj,C,£ = (LQLI)
kernel processes in quantum
L,j( m )... I/,>(i){LgLi) °B I i E
with #(z) = 0 if tj 6 T and #(i) = 1 if U £ a, and B,,e = f
dvx... dvmgt^
gt(1) («i - h) x
{vm -tm)...
Js
M(lie(w2, VI)M( 0IE (T;I, S),
where Mi(v, u)= I
k£(u2i - «2i-i) ■ • • M " 2 - ui)dui...
du2i,
J U
with = / 9e{t ~ s)ge(-s)ds
= -k{ e
and
k(t)=fg(t-s)g(-s)d. It is easy to see that R
^JH.m)
-DTT.E -> 7
.Ml) (ft - W
'..-7
r-;
"
lm\
•■•
((*! ~ S ) K )'° n
Jo'
Hence
X X ( " W ) -> e ^ ^ ' - y ^ L ^ ) . . .7fl(1)£»(i)eKZ",L,(',-*). But this is exactly the solution of the differential equation as stated in Theorem 6.1. For the details of this complicated proof see [W2]. ACKNOWLEDGEMENTS
These notes arose out of lectures given at the Grenoble Summer School, and also a course taught at the Pontificia Universidad Catolica in Santiago de Chile where they were completed. The last section was the subject of a talk given at the Summer School on Quantum Probability in Trento in 1999. The author wants to thank S. Attal (Grenoble) , R. Rebolledo (Santiago) and L. Accardi (Rome) for kind hospitality and fruitful discussions. Support received from the Catholic University of Chile, DIPUCvisiting fellowships program, and the Presidential Chair "Qualitative Analysis of Quantum Dynamical Systems" is gratefully acknowledged. REFERENCES [AFL] [ALV]
[At 1]
L. Accardi, A. Frigerio and Y.G. Lu, On the weak coupling limit problem, in [QP 4], pp. 2058. L. Accardi, Y.-G.Lu and I.V. Volovich, A white-noise approach to stochastic calculus, Re cent developments in infinite-dimensional analysis and quantum probability, Acta Appl. Math. 63 (2000) no. 1-3, 3-25. S. Attal, Problemes d'unicite dans les representations d'operateurs sur l'espace de Fock, in "Seminaire de Probability XXVI," Lecture Notes in Mathematics 1526, Springer-Verlag, Heidelberg 1992, pp. 619-632 .
260
Wilhelm von
Waldenfels
[At 2]
S. Attal, Non-commutative chaotic expansion of Hilbert-Schmidt operators on Fock space. Comm. Math. Phys. 175 (1996), no. 1, 43-62. [Bel] V.P. Belavkin, A quantum non-adapted Ito formula and non stationary evolution in Fock space, in [QP4], pp. 137-180. [BeL] V.P. Belavkin and J.M.Lindsay, The kernel of a Fock space operator II, in [QP8], pp. 87-94. [FF 1] K.H. Fichtner and W. Freudenberg, Point processes and normal states of Boson systems, Reihe Naturwissenschaftlich-Technisches Zentrum NTZ, Leipzig 1986. [FF 2] K.H. Fichtner and W. Freudenberg, Characterisation of states of infinite Boson systems, Comm. Math. Phys. 137 (1991), 315-357. [IoW] P.D.F.Ion and W.von Waldenfels, "Zeitgeordnete Momente des weissen klassischen und des weissen Quantenrauschen," in, "Probability Measures on Groups," Proceedings, 1981, Lecture Notes in Mathematics 928, Springer-Verlag, Heidelberg 1982, pp. 212-235. [Lin] J.M.Lindsay, Quantum and non-causal stochastic calculus, Prob. Theory Relat. Fields 97 (1993), 65-80. [LiM] J.M. Lindsay and H. Maassen, An integral kernel approach to noise, in [QP 3], pp. 192-208. [LiP] J.M. Lindsay and K.R. Parthasarathy, Cohomology of power sets with applications in quan tum probability, Comm. Math. Phys. 124 (1989) no. 3, 331-364. [Maa] H. Maassen, Quantum Markov processes on Fock space described by integral kernels, in [QP2], pp. 361-374. [MKM] K.Matthes, J.Kerstan and J.Mecke, "Infinitely Divisible Point Processes," Wiley 1978. [Mey] P.-A. Meyer, "Quantum Probability for Probabilists," Lecture Notes in Mathematics 1538, Springer-Verlag, Heidelberg 1993. [QP2] "Quantum Probability and Applications II," eds. L.Accardi and W.von Waldenfels, Lecture Notes in Mathematics 1136, Springer, 1985. [QP3] "Quantum Probability and Applications III," eds. L.Accardi and W.von Waldenfels, Lecture Notes in Mathematics 1303, Springer, 1988. [QP4] "Quantum Probability and Applications IV," eds. L.Accardi and W.von Waldenfels, Lecture Notes in Mathematics 1396, Springer, 1989. [QP8] "Quantum Probability and Related Topics VIII," ed. L. Accardi, World Scientific, 1992. [W1] W. von Waldenfels, Continuous Maasssen kernels and the inverse oscillator, in "Seminaire de Probability XXX," Lecture Notes in Mathematics 1626, Springer-Verlag, Heidelberg 1996, pp. 117-161. [W 2] W. von Waldenfels, An example of the singular coupling limit, in "Stochastic Analysis and Mathematical Physics," Proceedings, Santiago, 1998, Trends Math., Birkhauser Boston 2000, pp. 155-166. INSTITUT FUR ANGEWANDTE MATHEMATIK, UNIVERSITAT HEIDELBERG, I M NEUENHEIMER FELD 294, 69120 HEIDELBERG, GERMANY
E-mail address: wilhelm.waldenfelsat-online.de
BIBLIOGRAPHY FOR QPC XI & XII In this bibliography references have been checked and updated—in particular the latest editions of books have been listed. Full bibliographical details have been given for the benefit of those without easy access to either of the databases MathSciNet or Zentralblatt. The complete series QP-PQ, and volumes from 1984/85 onwards of the series Seminaire de Probabilities, are listed in two bibliographies to this bibliography and are referred to as [QP m] and [Sem n] respectively. The square-bracketed names are of the authors citing the given reference. L. ACCARDI
Noncommutative Markov chains, in, Proceedings, International School of Mathematical Physics, Camerino 1974, pp. 268-295. [Cecchini] On the quantum Feynman-Kac formula, Rend. Sem. Mat. Fis. Milano 48 (1978), 135-180 (1980). [Fagnola] Topics in quantum probability, Phys. Rep. 77 (1981) no. 3, 169-192. [Cecchini] A note on Meyer's note, in [QP3], (1988) pp. 1-5. [Fagnola] L. ACCARDI, R. ALICKI, A. FRIGERIO AND Y.G. LU
An invitation to the weak coupling and low density limits, in [QP6], (1991) pp. 3-61. [Fagnola, Holevo] L. ACCARDI AND C. CECCHINI
Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Anal. 45 (1982) no. 2, 245-273. [Cecchini, Rebolledo] L. ACCARDI, F. FAGNOLA AND J.QUAEGEBEUR
A representation free quantum stochastic calculus, J. Funct. Anal. 104 (1992) no. 1, 149197. [Fagnola] L. ACCARDI, C.FERNANDEZ, H. PRADO AND R. REBOLLEDO
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QUANTUM PROBABILITY SERIES
[QP 1]
[QP2]
[QP3]
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[QP4]
[QP5]
[QP6] [QP7] [QP8] [QP 9]
[QP10] [QPll]
[QP12]
[QP 13]
[QP14]
[QP 15]
283
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[Sem20] "Seminaire de Lecture Notes [Sem21] "Seminaire de Lecture Notes [Sem22] "Seminaire de Lecture Notes [Sem23] "Seminaire de Lecture Notes [Sem24] "Seminaire de Lecture Notes [Sem25] "Seminaire de Lecture Notes [Sem26] "Seminaire de Lecture Notes [Sem27] "Seminaire de Lecture Notes
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