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n^<'n^ ~~* F 1, the product is taken according to the multiplicity of the mass-points of 2(G, v) I \9f( (XtT>y(t, •))(v) and ({£}) > 1> By m a x ^ v? o r briefly maxxy> we denote the mass-point of tpix with maximal distance from XQ, i.e. for all tp € M and i f G w e have (3/ = max(p)if anrf on/j/ i/ (y € y>r and p(y,xo) = max />(<,Xo))Now we want to examine integrals for families of operators as considered first time by Maassen in [11]. Definition 4.1 Let X = (Xt)teG be a measurable family of operators on M. X is called non-anticipating with respect to i o if there exists a measurable family of operators (Bt)t£G with Bt € JC(ML,) such that Xt — Sc(l ® Bt)Vc. Remark. If G = [0,oo), XQ = 0, Lt = [0,t) and for each t G G the operator Bt has a kernel bt this definition coincides with Maassen's definition of non-anticipating kernel processes. Observe that even for bounded operators Bt the operator Xt may be unbounded or not exist at all as a linear densely defined operator. So one should require at least that all Bt are bounded. Proposition 4.2 Let X = (Xt)t^G be non-anticipating and let x be from G. (i) For all 9 € M such that j of M. we have , Q Given taking (arbitrary) value m ' two observables X, 0 (which, together with (i), is actually independent of j > no!). L e m m a ( 3 . 3 0 ) : For a periodic sequence {Ai} n € ]N as above with period p and PerronFrobenius eigenvalue /? of Tj = [Aj —» Aj+P] for j > ra0, we have lim„_,O0 i # ( . 4 „ ) = J log/? (cf. [34] for the Connes-St0rmer entropy [6]). P r o o f : Denote the resp. dimension vector of An = © J £ J = 1 Miu.i by a„ = ( HI x(a,co and dy = Y[ 3C is a product of Hida differential operators. This white noise cal culus for Fock space operators is currently being developed by Huang, Obata and oth ers (see [Hua], [HOS]). The importance of preservation integrals, which are in a sense lost in the distribution-theoretic view, has been re-affirmed by recent results of Attal (see [At 1]). §1. Notations and Preliminaries For any set S let rs be its finite power set : {cr c S: #a < «■}. rs has the countable partition \Jn7f0r„{S) w h e r e rn(s)'-= lCT c S : #cr = n). [a c S : #a « n) will be denoted r^„(S), and rx(S) will be identified with S itself, so that X c f j . Let S(n) = {seS": st * Sj if i * /}, Sm = {0} and let 0 :U„SO1c(«) -» * \a\ :=CTjU... ucr^ ( 2) are measurable with respect to the product a-algebras. Finally, a non-atomic measure space m on (5,3) determines a measure Fm on (r 5 r9) by 17 .-» i0(t/) + X n S l A® A, counit e : A —» 1 and antipode S : A —> A, all now morphisms in C. A is a homomorphism provided A ® A has the tensor product algebra structure obtained in Proposition 2.1. This axiom is shown Figure 2 (a). The other axioms are as usual for Hopf algebras, now written as morphisms in C. In concrete cases (such as our examples in later sections) one can write the product in A ® A explicitly (with a slight abuse of notation) as (a ® b)(c ® ®d) d) = a*(6 ® c)d. t2 / id the naive definitions • = • o ^A,A or ■ = ■ o ^ A \ d° n °t work. This is because these two candidates for an opposite product do not define Hopf algebra structures on A (with the same A) in the sense of Definition 2.2. This is also easily seen by means of the diagrammatic notation. We need a notion of an opposite Hopf algebra A°p (provided by an opposite product) before we can declare a braided-commutative one as a Hopf algebra for which A = J4° P . The strategy of braided groups is to introduce a weaker notion of commutativity that is a property of objects on which the Hopf algebra acts rather than one of the Hopf algebra itself. Even for ordinary Hopf algebras one can view any Hopf algebra as commutative with respect to some class O of comodules. For details of comodules (which correspond for A = L°°(G) to representations of G), see [15, Sec. 1], Briefly, a. (right) comodule (V,/?y) is a vector space V and a map /3v "■ V —> V ® A such that (f3v ® id) o f)v = (id ® A) o f}v and (id ® e) o fiy — id. For a comodule in a category C, V should be an object of C and 0v a morphism. It should be thought of as a representation of the underlying group-like object of which A is the "ring of functions". Definition 2.4 [18][20] A Hopf algebra A in C is braided-commutative with respect to a class of (right) comodules O if (id®-)o/3y (id®-)o/?v = (id®-)o(2v,A°*,l„4°/3v (id®-)o(2vu°*/M°/3v onV ®A for all right comodules f3v in O. Here QV,A = $A,V O ^V,A- The axiom is shown in Figure 2 (b).
ta
king
a
point in the Cartesian product to the set consisting of its
coordinates. However we shall abbreviate the cumbersome notation T m , for the sym metric measure, to /i. For a finite collection {£,: i e A ) of mutually disjoint subsets of S, let E% denote the following subset of F: *{XieX
Et)
(1.1)
with the convention E0 := {0}. Thus the elements of Ex are finite sets a whose ele ments may be labeled {J,-: ieX] {£, : i s A}, [Ej -.jefi]
in such a way that SjeEj
are mutually disjoint, then %E^
metric product o is given by
for each i. Note that if
= XE^XE^
where the sym
88
/•*(o) = V — a denoting the complement of a in a:
f(a)g(a) a\a.
Let !$o denote the collection of subsets of r of the form (1.1) where each £, belongs to 31, and define 2>0 := linear span \XEX- EX 6 #(>}■ The notion of local integrability we shall use here is the following:
tlcxn ■■= geL°(r):f
\g(a)\da
< <*> VneN,Ee9l\.
(1.2)
The operators we are considering are given by the prescription: x(al,a2,a))k(6)va2[uo!i)da>
(X^kMa) = f Y Jp *-i\a\ = a
(see [L] Definition 2.1) or, equivalently, by
u«
x' (a, 5, co) k{m u 5) da a a
where x':(a,/5,y)^Jjac/}x{a,a<7). For £ ; t e # o , if x(a,f), -)XE^')
e
^!
(1.3)
for a.a. (a,/J) and each subset f of A , then
S
X3XE, ' well-defined. This follows from the identity XEx{
c
^ ZEt(fi»ZExuffl>
Thus if JC is locally integrable in its third argument, then X3k is at
least well-defined for ke2)0.
Moreover, by [L], Lemma 2.2, null set variations of x
and k only result in null set variations of X$k , so that X3 may be viewed as a (par tially defined) linear operator on L°(r). The number operator N is the multiplication operator given by Nk(a) = #Gk(a), and for functions <j>: N -> C,
89
function spaces defined in [L], (2.5), may be described as Ka = {keL\r):\\k\\(a):=\\aNlh\\
<<*>}.
(An unqualified norm will always mean the L2(r)- norm.) The scale of Fock spaces {Kc: c > 0} has been exploited in [B] for analysing a generalisation of (0.1) to operator-valued x. §2. From kernel to operator For a measurable function x : .T3 —> C, a,b,c > 0 let *a,b,c ■ («.Ar) •-» x(a,p,r)lyla*ab'»Pc*r i
IKIk&.c = {// s "PK6,c(a.Ay)| 2 dad/J where x is the transform of x given by (1.3). Recall the identity [17 3 h(avP)x'{a,p,Y)k(Pvy)dcxdpdy
= j h(a){X3k)(&)da,
(2.1)
which is valid when either side is defined ([L], (2.4)). In particular a locally integrable kernel always defines a quadratic form on 3)Q. In this section we give conditions for a function x to be the kernel of a reasonable operator. The basic estimate is Lemma 2.1: Let p > a > 0, q > c > 0, h,ke L°(r) and xeL°(r3). j" \h(a)(X3k)(a)\d0^M(p)Ukq)\W\\a,^FW^Fu
Then (2-2)
Proof: Let b=^l(p-a)(q-c). (2.1), several applications of Minkowski's inequality, Cauchy-Schwarz and the ^-Lemma, give J|*(0)CX3*)(o)|dff H jjf = HI
dadpdy\h(ccvP)x'(a,p,y)k(Pvy)\ dadpdy^*aJp~^a~*l3\KavP)x^btC(a,p,y)k{Pvy)\
${/dady3a#a(p-a)^|/!(au^)|2}1/7d7^#''{//dad^(?-C)^Kfe,c(a,Ay)^u7)|2}1/2
90
*= {jdap*a\h(c)\2}V2jdy^jdp
(q-cfP\k(puy)\2jda
sup|<6,c(a,fi),/)|2}
W\(p){jdyc#rjdp(q-cfP\k(Puy)\2}m\\x'\\a,b,c
^
= \\h\\w\\khdx'\Ub,cm (2.2) may be re-written as an L2- operator norm estimate: \\{a+bxTNI2X,(b1+c)-NI2\\ for a,bi,bi,c
< |*'Uv5S.c
(2.3)
> 0, from which the next result is immediate.
Proposition 2.2: Let x e L°(.T3 ) satisfy ||x'|| a ;, c < °°, for some a, b, c > 0. Then X3 defines bounded operators from M(C+be~l) t 0 ^(.a+be)'1' f° r most ||x'|| a bx. (i)
eacn
* > 0, with bound at
In particular,
if a < 1 then X3 is densely defined on L2: Dom(X3)D#(c+fe2/(1_a));
(ii) if a < \,c < 1 and ||x'|| fl ,,/(i-a)(i-c),c < °° then X is bounded on L2. In the next section we shall merely impose a local integrability condition on the ker nel, in the sense of (1.2), so the next result is relevant. Proposition 2.3: Let x e L°(r3)
satisfy x'a x> l e L 2 ® L20C ® L20C for some a < 1, then
2
X3 is densely defined on L (F): Dom(X3 ) D J ) 0 . Proof: If keD0
then, for some n e N and £ e & , |*| « ||^|U^„, wherein is the indi
cator function of rin(E). j
Thus, for each g e
L2(r),
\g(cr)(X3k)(a)\da « /// « 1*11 P I - / dy{//
« i*mi*i-c.
dadpdy^#a^*P\g(aup)x'aAj{a,p,y)k(pKjy)\J[^-*P Apda\x'aAA{a,p,y)\2Xn{P^y){\-ay#P\XI2
91 where C is a constant depending only on n, a, E and an integral of x'a j j which is local in the second and third argument. Thus X^k e L2(r). ■ §3. From operator to kernel First we relate matrix elements of an integral-sum operator with integrated values of its kernel. Proposition 3.1: Let EX,EM,EV e 3 0 and let x e Lloc (r 3 ), then is
(i)
X
(»)
XEX{X*XEV)
3XEV
well-defined is integrable
and, if all the sets are mutually disjoint, then (iii)[
x' = \
(XtfE
)-Y
[
*'■
(3-1)
Proof: If x is locally integrable, then so is x', so (i) and (ii) follow from remarks in §1 and the identity (2.1). Mutual disjointness of the sets implies the (a.e.) identity ZE^Jfiiup)
ZE„Jftvr)
=£
a
^
XE^ioc)
XEJP)
so the result follows from (2.1).
XE^JY)
■
In order to actually recover the kernel of the operator it is necessary to make some assumption on the measure space — namely that it supports a Vitali system. We do not wish to make too technical a diversion and therefore settle for a particular type which is easier to visualise. Definition 3.2: A system of summable sets 9t = {F,, , o : ne N,ie N"} is a net of partitions for M if (9ti) {F; : J'I eN} is a partition of S and, for each n & 2 and I'I,...,«'„-i» {F^
, n : i„ e N) is a partition of F„
,nl;
(91 ii) the linear span of [XF ■ F e 91} is dense in LJ(M);
92 (91 iii)
for every null set E of M, and every e > 0, there is a set F which is a countable union from 91 such that F z> E and m(F) < e;
(91 iv)
lim sup m(F:
,• ) = 0.
Notation: Given a net of partitions 91, the sets {Ft: i e N"} are called o/nfn rank and for A: e S, F^n) denotes the unique set in 91, of nth rank, which contains x. For < r e r let F £ ° be defined as in §1 if {F^n): s e a] are mutually disjoint, and put F™ = {0} otherwise. Notice that, by cr-finiteness and (91 iv), for almost all aeT\{0},
F^1'
eventually gets off the ground, and (also by (91 iv)) / Z ( F £ ° ) -> 0 as n -> ~ for a.a. a * 0 .
(3.2)
Proposition 3.3: Suppose that x e Li0c.(F3) and that the measure space M supports a net of partitions 91 = {F,-: i e ( J n ^ " } -
Then
*'(/>,CT,T)= lim//(FW C T U T )- 1 f
(Xtfj^) * puc
for a.a. (p,cr, r ) e r 3 . Proof: By non-atomicity and the previous remark we may suppose that p, a and x have no points in common and the sets Fg,
with x running through p u c u i , are = ^(F p ( ^ CTUT ).
distinct for sufficiently large n — in particular ^(F^xF^xF^)
By Proposition 3.1 and the Lebesgue-Vitali Theorem or, more particularly, de Possel's theorem (see e.g. [ShG]), the absolute difference between x'(p,a,
t) and the
above limit is, for almost all (p,0, T), lim Y
{n(Fg)jn\F{n,p,co,T))}
n->-- t "<»SF ff
where F(n,p,co,x) = F^xF^xF^. n -> °° for a c cr, a / 0 ,
$
i\
(3.3)
lF(n,p,co.T)
By (3.2), for almost all cr, /i(F^ n) ) -> 0 as
and so another application of the Lebesgue-Vitali
Theorem shows that the limit (3.3) is zero for almost all (p,c, r). This completes the proof.
■
93 This gives a further uniqueness statement for integral-sum operators (c.f. [L]). Corollary 3.4: Suppose that the measure space M supports a net of partitions. If x s i-Joc.Cr3) and Xrf = 0 V / e S 0 , then x = 0 a.e. One might hope for a converse for Proposition 3.2 of the form: If X: 3)0 —> L° is a linear operator and if
(p,a,r) t-> lim /, , / r W
\1
' ppua uff
«&£,)
exists a.e. and defines a function x ' e L ^ (r3) then X = X 3 . The following example shows that this is false: Example 3.5: Let M be the Lebesgue space R, and let X be a second quantised shift thus Xk(o) = k(a+t) where t * 0 and a+t := {«! + {,...,*„ + {} when a = {$i 2
Then X is defined on all of L (r),
sn}.
and leaves each Ka invariant, but for almost all
(p, a, T), the terms of the sequence \
(XZFM^)
W
' U eventually be zero.
We end by summarising what is required of a converse. For an operator X to be an integral-sum operator with locally integrable kernel, there must exist a sequence of absolutely continuous measures (
«tw(EiX£^£v) = for Ex,E^,Eve
= I
Ezutia
(XXEV)
such that
and, for n = #n * 1.
™
(^AuflJ^^^^wuvj
# 0 and {£,-: i e A u / i u v} mutually disjoint.
94 References [B]
V P Belavkin: A quantum non-adapted Ito formula and generalised stochastic integration in Fock space, J. Fund, Anal. 102 no.2 (1991) 414-417.
[L]
J M Lindsay: The kernel of a Fock space operator I, Quantum Probability & Related Topics VIII (this volume).
[ShG]
G E Shilov and B L Gurevich: Integral, measure and derivative: a unified approach (transl. R A Silverman) Prentice-Hall, N.J. (1966). Mathematics Department University Park Nottingham NG7 2RD England
Q u a n t u m Probability and Related Topics Vol. VIII (pp. 95-97) © 1 9 9 3 World Scientific Publishing Company
AN APPLICATION OF FIXED POINT THEORY TO QUANTUM SDE by Andreas Boukas Department of Mathematics, American College of Greece 6 Gravias Str., Aghia Paraskevi, Athens, Greece GR 15342 Abstract. The existence and uniqueness of solutions of a certain class of quantum stochastic differential equations in Boson Fock space is established with the use of Tichonov's fixed point theorem. 1. INTRODUCTION Let HQ be a separable (initial) Hilbert space, let r(L2([0,co),c)) be the Fock space over the space of square integrable complex-valued functions on [0,oo), let A+={A+(t)/taO} and A=(A(t)/U0) be respectively the creation and annihilation processes of [1], and let A ,A ,A eL(H ®r(L ([0,a),c))), the space of bounded linear operators from H ®r(L2([0,a>),c)) whose identity is denoted by I. The existence and uniqueness of solutions of the quantum stochastic differential equation (QSDE) dX(t)=AiX(t)dA+(t)+A2X(t)dA(t)+A3X(t)dt with initial condition X(0)=I, was proved in [1] with the use of Picard's method of successive approximations. In an effort to establish the same result via the usually "cleaner" fixed point theory methods we consider adapted processes X={X(t)) (in the sense of [1]) with t ranging over some finite interval and with domain of interest for each X(t), H ®span{y(f)/feB}, where B is the unit ball of L2([0,a>),c). The initial condition is generalized to X(to)=XQ where t^O and XoeL(H0®r(L2([0,co),c))) but we assume, for technical reasons, that {A.=A.®I/i=l,2,3} is a commuting family with each A.eL(H ) compact and self-adjoint, where I is the identity of 2 L(r(L ([0,o>),c)). The solution obtained lives on a finite interval [t ,t +5] but can be continuously extended by reapplying our method to starting time t +5. The fixed point theorem suitable for our class of QSDE is that of Tichonov ([2]) which states that "a compact convex nonempty subset of a locally convex topological vector space has the fixed point property". 2. THE TOPOLOGICAL SET UP Let {A.=A.®IeL(Ho®r(L2([0,oo),c)))/i=l,2,3}
be a commuting
family
with each A compact self-adjoint, let U={u }°° be an orthonormal basis of i
n n=l
H consisting of common eigenvectors of A ^ and A 3 ([3]), and let D=(f }m be a dense subset of B. If T > 0 is arbitrary then for an adapted n n=l
process X={X(t)/tQststo+T) we define for each n,k,N,KeiN =
{1,2,3,...}
96
n.^ ( X ) -
t
'^K^v^^1
^ 0
(2 1}
-
0
provided that the right hand side of (2.1) is finite. For i=l,2,3 and n,k,N,Ke(N
,A T ,K(V> * I'M n X . W
(2 2)
-
since the norm of a compact self-adjoint operator in a Hilbert space is equal to the supremum of the set of i t s eigenvalues. Let M M v7 (||A 1 || 2 +||A 2 | 2 ) 1 / 2 +T||A 3 H|X O || and choose 5e(0, T ] such that 5Msl. I f we define S & £ {adapted X={X(t)/t^t Q +5)/t -» <X(t)u®y(fN),uk®y(fK)> (2.3) is bounded for all n,k,N,Ke[N} G M
(XeS/for all n,k,N,Ke!N and t,t'e[t„,t +5],
o JX)s5M and (2.4)
|<(X(t)-X(t'))un8y(fN),uk«y(fK)>|S(M-||Xo|)|t-t'|1/2} where
p
=
pf as in (2.1), then we can prove the following:
Theorem 2.1. (S,{ ,p }) is a metrizable locally
convex
topological
n,N k,K
vector space. Proof. It is easy to see that S is a vector space p :S—>[0,co) is a semi-norm. Moreover, if p 11, N
K , K,
ii iH
n,k,N,KelN then for u,veH
and that each (X)=0 for all
KiIs.
and f,geB we may choose {f.}™, {g }°° . c D A A —1
0
|j p— 1
such that f — > f and g — > g in the L -sense, as A,u — > », to conclude that for every te[t ,t +5] 00
<X(t)u®y(f),v®y(g)>=
T
n
k
A,p
n
A
k
IJ
Thus X(t)=0 on Ho®span(y(f)/fsB) for all te[t0,t0+5] and so (S>{ n N P k K )) is a semi-normed vector space which equipped with the topology induced by the countably many seminorms p becomes a metrizable locally convex topological vector space whose topology can therefore be described by sequential convergence. Theorem 2.2. G is a compact convex nonempty subset of S. Proof. The zero process is in G so G*0. By the triangle inequality G is convex. If { X ^ c G then {t«[t0,tB+fi]~-> <Xx(t)up®y(fN) . u ^ y ^ ) } ^ is, for each n,k,N,K€(N, a uniformly bounded and equicontinuous family of complex valued functions. By the Ascoli-Arzela theorem and the diagonal process we can select a subsequence, say {X }™ , again, such that for all n,k,N,KeiN
(t — >
<Xx(t)un®y(fN),uk®y(fK)>}"=i
converges
uniformly
on
97 [tn'tn+S] to a continuous function. For each te[t ,t +51 let U
0
0
<X(t)u n »y(f N ),u k ®y(f K )> M
0
lj.m<Xx(t)un®y(fN),uk®y(fK)>
(2.5)
and extend to H0®span{y(f)/feB) by 00
<X(t)u®y(f),v®y(g)> = £
n , k=l
with the infinite sum converging by (2.4) and Bessel's inequality. Thus defined, X={X(t)/tQststo+5} is a weakly continuous adapted process to which (X A >™ =1 converges in the topology of S. Moreover, for each
IA.K(XK.A.K<XAK,A.K
n,k,N,K6N,
upon letting X — > *> implies
n NPk K
which
£
W 5 M . Finally, for t,t'e[t ,t+5],
K(X(t)-X(t'))un«y(f|1),uk«y(f|c)>|-lJ«|<(XJk(t)-Xx(t'))un.y(fM),uk.y(fK)>|* £
(M-|X0ll)-|t-t'|1/2 since {X^}"^ c G. Thus XeG and so G is compact.
3. THE EXISTENCE AND UNIQUENESS THEOREM Theorem 3.1. For each (t0,X0)e[0,eo)xL(H0er(l/([0,o.),c))) the initial value AiX(t)dA+(t)+A2X(t)dA(t)+A3X(t)dt
dX(t)
(3.1) X(t )=X„ , t stst +5 ' 0'
0
0
0
where 5,A ,A ,A are as in section 2, has a unique solution X which belongs to the set G defined in (2.4). Proof. For XeG and for te[t ,t +51 define L o o J (PX)(t) = X0+JAiX(s)dA+(s)+A2X(s)dA(s)+A3X(s)ds . (3.2) PX is an adapted process which, by the way 5 was chosen and by the first fundamental theorem of the Hudson-Parthasarathy calculus ([1], Thm. 4.1) along with the triangle and Holder's inequalities, belongs to G. Moreover, if {X }™ c G and X^ — > X in the topology of Thm. 2.1, as X — > », then in view of (2.2) we have for each n,k,N,KelN n.NPk.K(PXx-PX)-C^tl|A1||2+||A2||z)1/z51/2+||A3||5]nNPkK(XA-X).
(3.3)
which implies that P:G — > G is continuous. By Tichonov's fixed point theorem P has a fixed point XsG which solves (3.1). By (2.2) and Gronwall's inequality such a solution is unique. REFERENCES [1] HUDSON, R.L., PARTHASARATHY, K.R.: Quantum Ito's formula and stochastic evolutions, Comm. in Math. Phys. 93, 301-323 (1984). [2] SMART, D.R.: "66 Fixed Point Theorems", Cambridge University Press (1974). [3] VON NEUMANN, J.: Zur Algebra der Funktionaloperationen und Theorie der Normalen Operatoren, Math. Ann. 102, 370-427, (1929-1930).
Quantum Probability and Related Topics Vol. VIII (pp. 99-108) ©1993 World Scientific Publishing Company
Matrix elements stability of quantua stochastic differential equations by ANDREAS BOUKAS Department of Mathematics, American College of Greece, 6 Gravias Str., Aghia Paraskevi, GR-15342, Athens, Greece
Abstract The stability of the zero solution of the quantum stochastic differential equation dX(t)=Ai(t)X(t)dB(t)+A2(t)X(t)dt, in the presence of Fock space white noise B, is defined and studied with the use of the Hudson-Parthasarathy quantum stochastic calculus. An explicit formula is obtained for the solution of the associated initial value problem in the case when A , A are compact self-adjoint adapted processes. A variation of parameters formula is obtained in the case of constant A„. 1. Introduction The Hudson-Parthasarathy Fock space T over L2([0,oo),c) is defined (along with the associated quantum stochastic calculus, familiarity with which is assumed by the reader) in [1], [2] as the Hilbert space completion of (span E, <,>) where E={y(f)/feLz([0,oo),c)} is a family of exponential vectors with
HA (t)il
en
<»
and sup HA (t)n
8r)<"
f°r eacn
T>0
>
and
the maps
t->A (t) and t-»A (t) are strongly measurable, where L(HQ®r) is the algebra of bounded linear operators: H ®r -» H »r, then the initial value problem dX(t) = A^tJXftJdBftJ+A^tJXftJdt (1.1) X
( V = Xo
associated with the quantum stochastic differential equation (SDE) dX(t) = A^tJXftJdBftJ+A^tJXftJdt
(1.2)
can be shown, by modifying the proof in [1] for X(0) = I, to have a unique strongly continuous solution. That is, for each tQaO and for each
100 X eL(H ®l") with X = X ® I , where I is the identity of L(l"), there exists a 0
0
0
0
unique adapted process X=(X(t)/t£to) defined on spanE, such that for each ueHo, f6L2([0,oo),c) and tatfl X(t)u«y(f)-[X0+J
A^sJXtsJdBtsJ+J
A.,(s)X(s)ds]u®y(f)
(1.3)
and the map t->X(t)y(f) is strongly continuous on [tQ,oo). Here, and in what follows, u and v denote arbitrary elements of Ho and y(f), y(g) denote arbitrary exponential vectors. Since in Fock space stochastic analysis an adapted process X is usually considered through its matrix elements <X(t)u®y(f),v®y(g)>, we define the stability of X in the following way adapted from the classical theory of ordinary and stochastic differential equations ([5], [7]). Definition 1.1. Let (X(t,tg,Xo)/Uto) be the solution of (1.1) and let DCH0XHQXL2([0,<»),C)XLU[0,^),C}. The zero solution of (1.2) is: (i) stable on D if and only if for every e>0, t±0 and (u,v,f,g)&D there exists 5>0 depending on them such that for all i>t ,HJf n<5 implies | <X(t,tg,X0)wy(f),v*y(g)>\<e. (ii) uniformly stable on D if and only if it is stable on D and 5 is independent of t . o (iii) globally asymptotically stable on D if and only if it is stable on D and for each (t ,X ) we have Urn <X(t,t ,X )u»y(f),v»y(g)> = 0 for all t-*o
(u,v,f,g)<sD. The stability of (1.2) in the case of constant compact self-adjoint A , A is considered with the help of the spectral resolution theorem for such operators. The case of constant A is considered via a variation of parameters formula by treating A (t)X(t)dB(t) as a perturbation. The case of nonconstant A , A is handled with the use of Gronwall's inequality. The results obtained constitute quantum stochastic analogues of stability results known for linear systems of first order ODE ([5], Ch. 1.1) as well as for classical SDE ([7]). In what follows, subscripts to norms are omitted since the underlying normed space is in each case obvious. All "time" integrals of complex-valued functions are in the sense of Lebesgue. All integrals of vector valued functions are in the sense of [1], [2]. In particular, if feL2([0,oo),c) and geL1([0,»o),GJ then "f> 2 -[J | f ( t ) | 2 d t ] 1 / 2 and ilgll=f l g ( t ) l d t . For an operator A, the o o notions of p o s i t i v i t y (A^O) and positive definiteness (A>0) as well as the associated notions of negativity and negative definiteness, defined
101 through -A, are as in [3]. Finally if A = A»I e L(Ho®r) then iiAH-ilAii ([8], Prop. 43.1). 2. The basic Leans Lema 2.1. (Variation of parameters formula). If AeL(Hg<s>r) is a constant adaoted orocess and p=(p(t)/t*t } is an adaoted orocess for which \>(s)dB(s) makes sense then the solution
of
l
o dX(t) = p(t)dB(t)+A
X(t)dt
(2.1) *<*„> " Xo is given by X(t) = exo((t-t0)A)X+
J exo((t-s)A)p(s)dB(s).
(2.2)
Proof. By (2.2) we have X(tQ) = X and
dX(t)=d(exp((t-to)A))Xo+d(exp(tA))|texp(-sA)p(s)dB(s}+
+exp(tA)d(f exp(-sA)p(s)dB(s))+d(exp(tA))d(| exp(-sA)p(s)dB(s))
=A exp((t-t0)A)X0dt+A exp(tA)dt J exp(-sA)p(s)dB(s)+p(t)dB(t)+
'o +A p(t)dt dB(t) = A X(t)dt+p{t)dB(t)
since dt-dB(t)
Thus the unique solution of (2.1) is given by (2.2). Leaaa 2.2. (i) for which MT(t) =
te[tg,T],ueHg
If
t ,Te=[0,<») and (AT(t)/t±t0)
J AT(s)dB(s) ,2 ,
and feL2([0,<*),£) \\MT(t)uay(f)\\2
0.
m
is an adaoted process 2
and J \\AT(s)u»y(f)\\ ds make sense for
then
s 2 exo(49fr)\
MT(s)u*y(f)rds.
s
(2.3)
102 (ii)
Let
t»0 and let
{A(t)eL(Her)/U0}
processes for which N(t) =
and (X(t)/UtQ}
be adapted
J A(s)X(s)ds, J \\A(s)\\ds, and
*o [ «A(s)u\\X(s)u9y(f)u*ds for each UtQ, A(t)*0
make sense for tttg,
ueHg and f
If
or A(t)*0 then
2
uA(s)nds) f WA(s)\u\X(s)wy(f)\\2o\s.
\\N(t)uey(f)\\ sexo(\
(2.4)
0 % Proof, (i) By Theorem 4.2 of [2], as in the derivation of (5.2) of [2], for to^tsT,U€Ho, and feL ([0,oo),C) we have
Jtl|MT(t)u8y(f)||2=2Re
using
the
integrating
*
factor
.t 2
exp(-4j lf(w)l dw). s
(ii) By Theorem 4.2 of [2], for each U t Q , usHQ and feL2([0,co) ,C) we have ^
||N(t)u®y(f)||2
2Re
(2.5)
Without loss of generality we may assume A(t)aO. If A(t)sO then the right hand side of (2.5) is written as -2Re
2Re
s iiA(t)lillN(t)u®y(f)ll2+llA(t)llHX(t)u®y(f)ii2 and the proof is completed by using the integrating factor exp(-f iiA(w)lldw).B s
Lena 2.3.: If A ,A are constant adapted processes and (Y f family of adapted processes such that for tet dYJt)
-
L
w -*
is a
AiYk(t)dB(t)+A2YJt)dt (2.6)
103 where LeL(H*r)
satisfies
Iim Hi wy(f)\l k
then
lim
= 0 for all
ueH. °
feL2([0,<»),c)
my(g)> = 0
(2.7)
k
uniformly (U,v,f,g)
on
comoact
subsets
of
[t ,«),
for
all
HoxHgxL2([0,a),C)xL2([0,
e
Proof: By Lemma 2.2, i f t « [ t D , p ] , pat fl , then ilY k (t)u®y(f)ii 2 =E3iiY k (t 0 )u®y(f)ii 2 +6exp(4iifii^)-| iiAiH2-iiYk(s)u®y(f)n2ds+ +2p J" HA2ll2llYk(s)u®y(f)ll2ds s 3llY k (t 0 )u®y(f)n 2 +c-J HYk(s)u«y(f)ii2ds
where c = max(6 exp(4iifn2)iiA II 2 , 2piiA II 2 ). Thus by Gronwall's inequality ( [ 5 ] ) , HYk(t) u®y(f)n 2 s 3HLku«y(f)n2 exp(cp) -> 0 as k -> «, . Leaaa 2.4. If
A=A
A=A®I
e L(H»r)
B
are constant
adapted
processes such that A,A commute and are compact and self-adjoint then the matrix elements of the solution of (1.1) are given for each tat £0 by <X(t,to,X0)u*y(f),v»y(g)> f<X„u>e„>
=
t-i
0
=
<e*y(f)>™y(9)> n
exD[i\\
(f-g)(s)ds+(t-t)u]
where (e }w , is a/? orthonormal basis for Hn consisting n n=l
n
of eigenvectors of
"
A, and A corresponding to eigenvalues (\ f 1
(2.8) 0
nj
n
2
, , (u f
pn=l
, respectively.
p n=l
Reaark. The existence of an individual basis of eigenvectors for each of A and A is a well-known fact for compact self-adjoint operators while the existence of a common basis for commuting Aj, A ? follows from a general spectral resolution theorem of Von Neumann ([3],[4]). Proof
of leaaa.
For n=l,2,...
define
Fourier
projection
like
104 operators E (X„):H«r -» H®r by v
nv
0'
0
0
En(X0)uaY <en,X0u>en9y I f X is the solution of dXn(t) AjX^DdBftJ+A^JtJdt
(2.9)
(2.10) and Y
[(A -X I)+(A -n I)]X ,
n
I n
Z
n
c
n
[ ( A - A I ) - ( A - u I)]X ,
n
i n
W
n
c
n
[ - ( A - X I ) + ( A - u I)]X I n
n
il
^
n
where I is the identity of L(H®l") then for K e{Y ,Z.W }, 0
n
n
n
n
dKp(t) = AjK^tJdBftJ+A^JtJdt K (tj nv
0.
0'
Thus K (t)=0 for a l l tet and so (Y +Z ) ( t ) = (Y +W )(t)=0 for a l l Ut . Hence, for each tat and n = l , 2 , . . . AX ( t ) = XX ( t ) ,
AX ( t ) = MX ( t ) .
(2.11)
For each ueH and yel", X U8Y
o
IEA)u8Y
<2-12>
n =1
By applying Lemma 2.3 to {Yk = X
1*}"
we see that
n =1 00
<X(t)u®y(f),v®y(g)>
£ <Xp(t) u®y(f),v®y(g)>
(2.13)
n =1
uniformly on compact subsets of [t0,«>), where X(t)=X(t,t ,X ). By (1.3), (2.9), (2.10), (2.11), for tst0 and nal <Xp(t)u®y(f),v®y(g)> = <XQu,en> <en®y(f),v®y(g)> exp[iX n J (f-g)(s)ds+(t-t Q )u 1 . 4
o
(2.14)
105 from which (2.8) follows in view of (2.13). B 3. The stability results Proposition 3.1. Let A=A®I, A=A»I e L(H«r) 1
processes such that AjfA2 let D = HnxHnxD„ where 0
0
1
2
2
commute and are compact
and self-adjoint
and
0
0={(f,9)eL!([Ora),C)xL2([0,a),C)/ (i)
be constant adapted
0
If A*0 then the zero solution
Im(g-f)eL!([0,«),!R)}.
VI =
of (1.2) is uniformly stable on D.
(ii) If ~A<0 then the zero solution stable on D.
of (1.2) is globally
asymptotically
Proof, (i) With the notation of Lemma 2.4, A s0 implies u aO for all n. Thus for tat 0 , exp(un(t-t0))*l and so for (f,g)eD0 using IIA II = suplXJ n
we obtain lexp(iX
f (f-g)(s)ds+(t-t„)u I * exp(iiA,iiiii|Mi). n J
(3.1)
1 1
O P
By Lemma 2.4, (3.1), and Bessel's inequality we have for (u,v,f,g)eO l<X(t,t ,X )u®y(f),v®y(g)>|siiXoiiiiu®y(f)iiiiv®y(g)iiexp(iiAiiniiJiiii) from which the result follows by taking in Definition 1.1 ( i i ) 5=E[«u®y(^) niiv®y(g) Mexp( HA1H n^n) ] - 1 . (ii)
As in the proof of
( i ) , A <0 implies pn<0 for a l l
n. Thus for
(f,g)eD , t aO and n = l , 2 , . . . we have l e x p ( i X n | (f-g)(s)ds+(t-t 0 )M ri )lsexp(iiA 1 Hiii|)ii 1 )exp(u n (t-t 0 ))^0,t->a..
(3.2)
Letting f (t)=<X u,e ><e ®y(f),v®y(g)>exp(iX f (f-g)(s)ds+(t-t )u ) and
g =KX u.e >IKe « y ( f ) , v®y(g)>lexp(iiA iiiimii) we see that by (3.2) f n
0
n
n
1
as t-»», If (t)|sg for all tat and n=l,2
(t)-»0 n
1
and
106 CO
y g sexppiA Hiiim )HX iiiiuiiiiy(f)iiiiv®y(g)n
t- n n =1
1
1
0
00
convergence theorem, lim £ f (t) = £ t->co n = 1
Thus
by
Lebesgue's
bounded
00
lim fjt)
0
which, in view of
n = 1 t-Ho
(2.8), implies <X(t,to>Xo)u®y(f),v®y(g)> ■* 0 as U for all (u,v,f,g)eO and (to,Xo)€[0,oo)xL(Ho®r). Combining this with (i) we see that the zero solution of (1.2) is globally asymptotically stable on D. B let
Proposition 3.2. Let A = A ®IeL(H®r) be a constant adapted process, (A^tJ/teO) be a locally bounded measurable adapted process such that
| UjftJ^dt^, and let D = HgxHgxLl([Orm),c)xL?([0,*),c). o (i) If A*0 then the zero solution of (1.2) is uniformly stable on D. (ii)
If there exists fe(0,w) such that A +pisO, where I is the identity of L(H ) , then the zero solution of (1.2) is globally asymptotically stable on D. Proof, (i) By the Lumer-Phillips theorem for contraction semigroups
([6]), A sO implies
sup Hexp(tA Jllsl. tao By Lemmas 2.1, 2.2 (i), for tat , ueHo and feL2([0,oj),c) we have HX(t,to,Xo)u®y(f)ll2=;2llx/-lly(f)ll2+ +4exp(4Hfll2)f llA i (s)ii 2 HX(s,t (j ,X o )u®y(f)li 2 d]s 4 o and by Gronwall's inequality,
iiX(t,t o ,X o )u«y(f)iisv7iiX o iiiiy(f)iiexp(2exp(4iifii 2 )J iiA^s)n2ds) o from which the result follows by applying the Cauchy-Schwartz inequality to <X(t,t 0 ,X o )u®y(f),v®y(g)> and taking in Definition 1.1 (ii) 5=e[/?iiu®y(f)iiilv®y(g)llexp(2exp(4Hfil2)
iiA (s)H2ds)] .
o ( i i ) For every taO, Hexp(t(A,,+BI))llsl and so llexp(tA )llsexp(-Bt). By Lemmas 2 . 1 , 2.2 ( i ) , for tat , usHQ and f € L2([0,oo),c) we have HX(t,to,Xo)u®y(f)H2s2 exp(-2B(t-t o ))ilX 0 il 2 llu®y(f)n 2 +4exp(4iifll 2 )-
107 pt
■J exp{-2p{t-s))iiA i (s)n z iiX(s,t o ,X o )u®y(f)ii 2 ds.
(3.3)
t
B
Multiplying both sides of (3.3) by exp(2B(t-t 0 )), applying Gronwall's inequality to the result and then multiplying by exp(-2B(t-t )) \ve obtain llX(t,t 0 ,X o )uey(f)ll 2 s2exp(-2p(t-t o ))llX 0 ii z lly(f)ii 2 exp(4exp(4iifii^.00
•J iiA1(s)uzds)->0 as t->oo, which by the Cauchy-Schwartz inequality 0
implies <X(t,t o ,X o )u®y(f),v®y{g)>-»0 as t-*» for a l l (u,v,f ,g)eO and (t 0 ,X o )e[0,oo)xL(H o «r). Since by part ( i ) the zero solution is uniformly stable on D the proof is complete. B Proposition 3.3 Suooose that (A^tj/teO) bounded measurable
adapted
Drocesses such
and (A ( t ) / t * 0 J are
locally
r°° \\A (t)w' 7
that
~dt and
0
HA (t)»dt<m. of
If for each UO, A^t^O
(1.2) is uniformly
stable on
or A2(t)s0 then the zero solution
D=HgxHg^r([0,'o),ic)xL'!([0,w),c)
Proof. By (1.3) and Lemma 2.2, for usH ,feL 2 ([0,oo),c) and t * t aO we 0
have HX(t,t 0 ,X 0 )u®y(f)ll 2 s3iiX 0 il z lly(f)ll z +| [6exp(4iifn|)iiA^s)n 2 +
+3exp(piA 2 (s)iids)HA 2 (s)ii]iiX(s ) t o ,X o )u®y(f)ii 2 ds.
(3.4)
0
Applying Gronwall's inequality to (3.4) we obtain r r00 HX(t,t o ,X 0 )u«y(f)n 2 s 3llXoll2liy(f)ii2exp|6exp(4llfn2)J nAl(s)H2d s+ 0
+3exp(J°°iiA2(s)iids)|0,|iA2(s)iids] 0
(3.5)
0
In view of (3.5) the proof is completed by applying the Cauchy-Schwartz inequality to <X(t,t o ,X o )u®y(f),v®y(g)> and by taking in Definition 1.1 ( i i )
108 5 = e|V3'exp(3exp(4ilfll*)f llA1(s)ll2ds + -f-expfj" llA 2 (s)llds]| llA2(s)ilds 0
■!iu®y(f)iiiiv®y(g)lll
0
.
0
B
AcknoMledgeaent: The author wishes to thank Professors R.L. Hudson and V.P. Belavkin for helpful discussions. References [1]
HUDSON, R.L., PARTHASARATHY, K.R.: Quantum Ito's formula and stochastic evolutions, Comm. Math. Physics 93, 301-323, (1984). [2] HUDSON, R.L.: Quantum stochastic calculus in Fock space: a review, Fundamental aspects of quantum theory,
[3] [4] [5] [6] [7] [8]
ed V. Gorini and A. Frigerio,
111-124, Plenum (1986). DUNFORD, N., SCHWARTZ, J.T.: "Linear Operators", Interscience Publishers 1966. VON NEUMANN, J.: Zur Algebra der Funktionaloperationen und Theorie der Normalen Operatoren, Hath. Ann. 102, 370-427, (1929-1930). BURTON, T.A.: "Stability and Periodic Solutions of Ordinary and Functional Differential Equations", Academic Press, 1985. YOSIDA, K.: "Functional Analysis", Springer, 1980. ARNOLD, L.: "Stochastische Differentialgleichungen. Theorei und Anwendung.", R. Oldenbourgh Verlag, 1973. TREVES, F.: "Topological Vector Spaces, Distributions and Kernels", Academic Press, 1967.
Quantum Probability and Related Topics Vol. VIII (pp. 109-118) ©1993 World Scientific Publishing Company
A non-commutative Markov equivalence theorem by Carlo Cecchini
Dipartimento di Matematica e Informatica deH'Universita di Udine Via Zanon 6 - 33100 Udine - Italy
110 1.
Introduction
In [6] several notions of markovianity were introduced and studied for states on a Neumann algebra A with respect to a given localization (cfr. also [1] and [3]). By this we denote a triple (Ai, K% A3) of von Neumann subalgebras of A which mutually commute, have trivial intersection and whose union generates A. The aim of this paper is to continue the study of this subject by introducing still one more notion of markovianity, giving the relations with the previous ones and proving for it the generalization of the classical Markov equivalence theorem.
2.
Notations and preliminaries
Let A be a von Neumann algebra, acting standardly on a Hilbert space H, Q be a cyclic and separating vector in H for A associated with a normal faithful state co on A and Jn (or simply J when no confusion arises) the canonical isometrical involution for Q. For a G A, we set u>a (•) = co (a+ ■ a). When using this notation we shall implicitely assume co (|a| ) = 1, so toa is a normal state on A. Whenever 9 is a normal state on A such cp s ato for some a £ R * w e denote by [cp/co] the extension to the point i/2 of the Connes' cocycle in A for
111
We recall that M(e) and F(e) are von Neumann subalgebras of A, and we have M(e) 2 Ao 2 F(e). Let now A be a von Neumann algebra generated by two von Neumann subalgebras Ai and A2. In [4] we defined the (A2, Ai) stochastic coupling associated to a normal state w on A as the completely antipositive weak operator continuous contraction X.2,1: A2 -» Ai defined by the equality Ji &1 (a2)+ Q = Ei a2 Q. In the above formula Q is a vector associated to co, Ei the projection on Hi, the closure of {ai Q: ai e Ai} and Ji an Q preserving isometrical involution for Ai on Hi. We recall that &1 ( k | 2 ) = |[(coa2)l/coi]|
a26A2.
In this paper we shall consider (as in [10], 38) a linearly ordered index set T (typically, a set of reals). We shall say, for U, V C T, that U < V if for each r G U; s £ V we have r < s. We shall consider a von Neumann algebra A (which will be assumed to act standardly on a Hilbert space H) containing for each U C T a von Neumann subalgebra Atj. We shall assume that the mapping U -» Au satisfies the following properties: 1.
I f U n V = OthenAijnAv = CI
2. 3.
IfUCVthenAuCAv Let M be an index set and UaeM U a = U. Then Au is generated by UaGM MlJa.
Let U, V, W be disjoint subsets of T such that U < V < W. Then we say that the triple (Au, Av, Aw) defines a localization for MuUvUw- It is useful to interpret U as "past", V as "present" and W as "future". In the following we shall label all objects introduced in this section by the indices rather than by the corresponding von Neumann algebras. For example, if co is a state on A we shall write cov for O)|AV (V C T) instead of 0)Av; EU'V will denote the co (cou) preserving conditional expectation from Au to A v f U C V C T ) , >A B shall denote the (AA, AB) transition coupling for coAUR etc. Whenever we use the states Wa (a € A) we shall drop the co in the related objects (e.q. ea instead of Eua, X.
instead of JtA oa, etc.).
112 3.
Some notions of markovianity
In this section we shall take T = {1, 2, 3} with the usual order and omit in our notations the brackets for the singleton subsets of T. We shall assume all the states we consider to be normal and, in order to simplify technically our treatment, COJ to be faithful on Aj. The last one is, however, only a technical hypothesis. We start by recalling two definitions from [6]. 3.1 Definition A state co on A is called strongly markovian with respect to the localization (Ai, A2, A3) if X 3 ' {1,2} = X.3,2. 3.2 Definition A state co on A is called symmetric markovian with respect to the localization (Ai, A& A3) if [(lOala3)2/W2] = [(a>al)2/i02] [(
selfadjoint subset of R (E* 1 ' 2 *' 2 ).
= R (E* 1 - 2 *- 2 ) O R (E { 1 , 2 >' 2 ) + is the biggest
113 b) => c) We have (cfr. [2], th. 4.3): E 2 a 3 Q = E 2 E { i,2 } a3 Q = E 2 J{i,2} X 3,{1,2} (a+3) Q = E { u } a3 Q, and then our claim follows by the density of {a3 Q: a3 S A3} in H3. c) => a) We have: E 2 J ^ } X.3l{1,2} (a3) Q = E 2 E{in = j{U}x3>{U}(a2)Q)
a+3 Q = E 2 a+3 Q = E n > 2 } a+3 Q = -
and, again by [2], 4.3, we get our claim. c) o d) We have: a.*1-3*-2 (ai a3) Q = J 2 E 2 (ax a 3 ) + Q = J 2 E 2 E { i, 2} a+x a + 3 Q = = J 2 E 2 a + iE{i > 2 }a + 3Q and X1'2 (ai) X3'2 (a3) Q = J 2 E 2 a + i E 2 a+2 £2; so c) is immediately equivalent to d). d) <* d') As J^ 1 " 3 }- 2 is a positive mapping, we have: X3-2 (a3) X1'2 (31) = [X1-2 (a+!) X3'2 (a + 3 )] + = [ \ ^ ^
(a+x a + 3 )] + =
2
= X^ (aia3). The converse follows by interchanging the indices. 3.4 Definition
We shall call markovian with respect to the localization (Ai,
A2, A3) any state on A satisfying the conditions in th. 3.3. 3.5 Proposition a) b) c)
markovianity with respect to the localization (Ai, A% A3) implies markovianity with respect to the localization (Ai, A^ A 3 ). strong markovianity implies markovianity. if F (E* 1 ' 2 *' 2 ) = R (e^1,2^2) then markovianity implies strong markovianity.
d)
symmetric markovianity implies markovianity.
Proof a)
is an immediate consequence of the equivalence of d) and d') in th. 3.3.
b) and c)
follow from th. 3.2 (cond. e).
114
d)
we have by th. 3.4 in [6]: X*1-3*-2 (ai a3) = |[(o)ai a3)2 / oi2]|2 = |[(<0al)afo>2] [(coa3)2/o02]|2 = = |[(C0al)2/a)2]|2 |[(Wa3)2/0)2]|2 = ^
(ai) X 3,2 (a 3 ).
3.6 Theorem The strong markovianity of a state co on A with respect to the localization (Ai, A2, A3) implies its symmetric markovianity with respect to the same localization. Proof: By [6], 3.2 E ^ ^ a l does not depend on ai E Ai; this implies - X3,2, and one of the requirements in [6], 3.4 is therefore satisfied. Using this and 3.3 d) and d1), since co is (Ai, A2, A3) markovian by 3.3, we get: < h [(0)al)2/«02]+ X 3,2 (a 3 ) [((0al)2/C02] Q, a 2 Q > = = < h X.3,2 (a 3 ) [(C0al)2/fO2] O, 32 [(C0al)2/C02] Q > =
= co (|ai|2 a2 a3) = < h X {1,3} ' 2 (|ai|2 a3) Q, a2 Q > = = < h X3,2 (a3) X1-2 (|ai|2) Q, a2 Q >, which implies X3'2(a3)|[(coai)2/co2]|2Q = = X 3 ' 2 (a 3 )X 1 ' 2 (|ai| 2 )Q = = [(C0al)2/C02]+ X 3 ' 2 (a 3 ) [(C0al)2/Q12] Q-
If supp ai = I, then [(coai)2/'02] Q is a separating vector for A2, and [(C0ai)2/c02]+ (and therefore [(0)31)2/(02]) commutes with X3' 2 (a3) for all a3 S A3. For the general ai E Ai we notice that the sequence an = (ai + 1/n I)/]|ai + 1/n I|| tends in norm to ai; this implies (ct)an)2 ~* ((0ai)2 in norm, and [(coain)2/co2] -» [(C0ai)2/(02] in the strong operator topology. 32 Now the commutation of X ' (a3) with [(0031)2/0)2] for all ai E Ai, a3 E A3 follows by continuity. This means that for all a3 E A3 X3' 2 (a3) commutes with the von Neumann algebra A 2 generated by the set {[(0)31)2/012] : ai E Ai}. From [5], it follows that there is an 002-preserving norm one projection from A2 to Ax2. Let us consider now the smallest von Neumann algebra A 2 which contains X (33) for all a3 £ A3 and onto which there is an o>2-preserving norm one projection from A2.
115 A 3 2 is generated by the operators {0*2 (X3,2 (a3», as G A3, t £ R}, if 0*2 is the automorphism group for (02 on A2. We have, for a3 S A3, a*2 G A*2, 1 6 R: 0*2 (V5,2 (a3)) a J 2 = o l 2 (X3-2 (a2) o ^ (ax2)) = = o l 2 (a-l2 (a^) X3-2 (a2)) = a x 2 (0*2 (X3-2 (a2)), since A 2 is invariant under aX2- This proves that A*2 and A32 commute. On the other hand, since A32 is C02 expected in A2, it contains [(1033)2/0)2] for each a3 £ A33; so for each ai G Ai, a3 G A3 [((1)83)2/102] and [((0ai)2/a)2], and our proof is complete. 3.8 Corollary Let A2 be a finite von Neumann algebra. Then strong markovianity, symmetric markovianity and markovianity for (0 with respect to the localization (Ai, A2, A3) are equivalent. Proof We need only to prove that markovianity implies strong markovianity. As A2 is finite, M (e^1-2*'2) = F (e {1,2} ' 2 ) (cfr [9]), and then our result follows by [6]. 3.9 Examples a)
if A is abelian, then, as already remarked in [6], the three equivalent notions of markovianity are equivalent to classical markovianity
b)
let A be matrix algebras (i = 1,2,3), Ai = Ai ® I ® I, A2 = I ® A2 ® I, A3 = A2 ® I ® I, and u) a state on M. Let [
We plan to deal in another paper with the structure of markovian chains.
4.
A Markov equivalence theorem
4.1 Lemma The u) is markovian on A with respect to the localization (Ai, A2, A3) if and only if (*)E{i,2}E{23} = E2E{23}-
116 Proof Let (*) be satisfied. Then E{i,2} E3 = E{i,2} E{23} E3 = E2 E{23} E3 = E2 E3, which implies by 3.3, c) markovianity. Let now a; £ Aj, i = 2,3, (o markovian. Then E{i,2} a2 33 Q = 32 E{i,2} 33 Q = 32 E2 33 Q = E2 a2 33 Q, snd our clsim follows from the density of the linear combinations of vectors of the form 32 a3 Q in H{23}4.2 Lemma Let I = {1,2,3,4} and (0 a state on A such that
a)
u>{l,2,3} is markovian on A f i ^ } with respect to the localization (Ai, A2, A3).
b)
u) is markovian on A with respect to the localization (A{ir2}, A3, a4). Then co is markovian on A with respect to both the localizations (Ai, A{2,3}, A4) and (Ai, A2, A{3,4}).
Proof We have, using E{1,2,3} E4 = E3 E4 = E{2,3} E3 E4 = E{2,3} E{1,2,3} E4 = E{2,3} E4. We prove our second claim by noticing that: E{1,2} E{3,4} = E{1,2} E{1,23} E{3,4} = E{1,2} E{2,3} E{3,4} = E2 E{2,3} E{3,4} = = E2 E{3,4}The second equality above follows from lemma 4.1 and markovianity with respect to (Ai, A{2,3}, A4), and the third from lemma 4.1 again and markovianity of u){i,2,3} with respect to (Ai, A2, A3). 4.3 Theorem (Markov equivalence theorem, cfr [10], 3.8.1.A)
The following Markov properties are equivalent for a state (u on a von Neumann algebra A:
a)
markovianity on Az with respect to the localization (Au, Av, Aw) for all
b)
markovianity on Az with respect to the localization (Au, Av, Aw), where
U, V, W C T with U < V < W , U U V U W = Z V = {s),U={r:r<s,r€T}, W= {t:t>s, t£T} forallsET c)
markovianity on A z with respect to the localization (Ay, Av, Aw), where V = {s}, W = {t}, U is any finite set in T such that U < s and Z = U U V U W, for all s, t £ T with s < t.
117 Proof Clearly a) =* b) ^ c). To prove c) => a) let us consider first the situation in which U is finite and V contains only one element. Then by lemma 4.2, we obtain markovianity with respect to the localization (Au, Av, Awn), where W„ is any finite subset of W. As Aw is generated by the union of all the subalgebras Mwn, we get our claim. In our next step we assume only V = {t}, t S T. We note that Eu u {t} = sup {Eun U {t}: U n C U, U„ finite}. So, for all aw S Aw: l|Eu U {t} aw Q|| = sup {||Eun U {t} aw Q|| :U„CU, U n finite}. As by our first step Eu n u {1} aw Q = Et aw Q for all U„ C U, U„ finite, the above supremum is simply ||Et aw Q||. Since Et s Eu u {t}, this implies Eu u {t} aw Q = Et aw Q, and our claim is proved. To complete our proof, let us consider first the case in which V has a last element t. We consider a localization (Au', Av\ Aw), where U' = U U V/{t}, V = {t}. Our state is markovian with respect to this localization by our previous step. On the other hand, for aw £ Aw:
Euuv aw S2 = Eu'UV a W Q = Ev' aw Q = Ev Ev' aw Q = Ev Euuv aw Q = = Ev aw Q and our claim follows. Finally, if V has no last element, let V = Vi U V2, with Vi < V2 and Vi endowed with a last element. Take a localization (Au, Av', Aw'), with V = Vi, W' = V2 U W. As seen in our previous step, Euuv' aw' Q = Ev' aw' Q for all aw' S Aw'- By th. 3.3 we can interchange U and V and get EvuW au Q = = Ev' au Q for all au S Au- By application of Ev on both sides, we get Ev au Q = Ev Evuw au Q = Ev Ev' au Q = Ev au Q = Ev u w au QThe above equality gives the markovianity of w with respect to the localizanon (Aw, Av, Au) and hence for the localization (Au, Av, Aw).
118 Aknowledgement This paper was completed during a visit at the University of Heidelberg, with the financial support of SFB123. The author would like to thank Prof. W. von Waldenfels for his most kind hospitality. Bibliography 1.
L. Accardi, Noncommutative Markov claims. International School of Mathematical Physics, Camerino (1974), 268-295.
2.
L. Accardi C. Cecchini, Conditional expectations in von Neumann algebras and a theorem of Takasaki. J. Funct Anal. 45 (1982), 245-273.
3.
L. Accardi - A. Frigerio, Markovian cocycles. Proc. Royal Irish Acad. 83 A (1983), 251-263.
4.
C. Cecchini, Stochastic couplings for von Neumann algebras. Quantum probability and applications IV. Springer Verlag Lecture Notes 1396 (1989), 128-142.
5.
C. Cecchini, Fixed points subalgebras and invariant states for u conditional expectations. To be published in the Lith. J. of Math.
6.
C. Cecchini, Markovianity for states on von Neumann algebras. Preprint.
7.
C. Cecchini - D. Petz, State extensions and a Radon Nikodyn theorem for conditional expectations on von Neumann algebras. Pac. J. Math. 138 (1989), 9-23.
8.
C. Cecchini D. Petz, Classes of conditional expectations over von Neumann algebras. J. Funct. Anal. 92 (1990), 8-29.
9.
R.V. Kadison, Remarks on the type of von Neumann algebras of local observables in quantum field theory. J. Math. Phys. 4 (1963), 1511-1516.
10. M. Loeve, Probability theory. Van Nostrand (1963).
Quantum Probability and Related Topics Vol. VIII (pp. 119-122) ©1993 World Scientific Publishing Company
A Note on the Cohomology of Quantum Flows P.K.Das Indian Statistical Institute Calcutta Marcb.,1992 Let A be a *-algebra and let C be a second *-algebra which is also a 2-sided ^-*-module so that there are actions A x C -* C, C x A -+ C satisfying (ac)* = c V
in addition to the usual linearity,associatrvrty and compatibility relations of a two-sided action. By defitk>n,a 0-chain is an element w of C satisfying w* + u> + w*w = u> +- w* + ww' = 0.
88
The 0-chains form a gronp under the composition Wl * W3 = W] +■ Uj + W1W2.
A 1-cocycle is a linear *-map X : A -» C satisfying the identity \(xy) = X{x)y + xX{y)+\{x)\(y).
(2)
It may be verified that given a 0-chain w, the map bw : A -+ C given by {6w)[x) = w*x + xw + w'xw is a 1-cocycle.Denote the set of 0-chains by C{A,C) and of 1-cocycles by Zl{A, C), and let Bl{A, C) be the range of the map 6. Zl{A, C) » not a group.but the group C°{A, C) acts on Zl(A, C) through (A,w)-» A", where Au(z) = A(x) +■ u*[x + \{x)) + (s + A(*))w + w*(x + A(z))«.
(3)
(1)
120
Now, let 0 be the *-algebra generated by differentials of integrator pro cesses satisfying a closed Ito multiplication table,for example, that given by (2.3)[Das and Sinha] with the involution
wr=<%. From now on we let A be a unital *-subalgebra of B(#0) which may be called a system algebra and let C be the tensor product *-algebra C = A ® 0 on which A acts through the multiplication in A that is, va € A, b 6 A, dA e r> a[b ®dA) = ab® dk, {b ® dk)a = ba ® dA. A family of linear bounded maps A% : A -* A a called structure map if (i) (») (tii)
AC(7) = 0 AJJ(*)- = Atf**) Ajj(iy) = xAg(y) + A{j(z)y + Aj;(x)Aj(y),
(4)
where x, y 6 ^,and (iv) if there exists for each v > 0 constants o„ > 0,a countable index Bet J„ and a family {•£>£}>€/* °f bounded operators in H0 such that for x € >f, tt€tf»
E*rJ2>Mt2 < af«llJ.
(5)
We define a quantum stochasticflowon A as a family {jt}i>o of contrac tive *-homomorphism from A into B(#) satisfying for each xE A: (i) j>(x) = x (ii) ;'((/) = / and j,(x) is an adapted process, (iii) there exist structure maps AJ obeying (4) such that jt(x) satisfy the q.s.d.e. dj((x) = jt(\t(x))dk;(t). (6) A theorem proved by Mohari and Sinha [Th.3.5,Mohari and Sinha ] ensures the existence of a quantum Btochastic flow j on A satisfying (6) if the structure maps A{J satisfy (4) and (5). Now by writting each element A in the form
A(x)= g X>{x)®dA; where each A{J maps A to A, we see that the set Zl(A,C) of 1-cocyles is co-extensive with the Bet of solutions to the structure relations (4)and(6). Thus quantum stochasticflowsare generated by elements of ZL(A, C).
121
To interpret the set of 0-chains, we write an element w e C In the form CO
£
w =
u£«®dA*><«£ * €G * ;
JM«=0
then it may be verified that the defining relation (1) is equivalent to the following relations among the u# :
«)* + <4i + E K ) X = < + K)* + V <shH T = o.
(7)
E El The structure maps corresponding to 6w are given by K(x) = « ) ' x + xw> + J>*)**u£.
(8)
provided each w{| is bounded,the conditions (7) are necessary and sufficient for the existence of a unique unitary-operator valued process u satisfying the stochastic differential equation du{t) = (wj ® l)u(t)^A;. The inner quantum stochasticflowcorresponding to the bounded struc ture maps (8) is then given by ;,(*) = u{t)*z ®lu(f). Thus quantum stochastic flows are generated by elements of Zl(A, C) and inner flows by elements of Bl(A, C). The group action (3) can be interpreted as follows : theflowgenerated by Xu is an inner perturbation of that generated by A in the following sense.Denoting the corresponding flows by f and j respectively we have j»{x) = u^*j{x)u^ where u^ is the solution of duW(x) = jt{v;)u(x)dA». In the case of bounded w* and AJ the existence.uniqneness and unitarity of a solution of this quantum stochastic differential equation was proved by Das and Sinha recently.
122
REFERENCES 1. R.L.Hudson Quantum Stochastic Flows .Prob. Theo. and Math. Stat. proceedings; Vilnius 1989,vol l,1990.pp512-525. 2. P.K.Das and K.B.Sinha Quantumflowswith infinite degrees of free dom and their perturbations . To appear in Q.P(7):Proceeding of Q.P(Delhi)( 1990); World Scientific, 1992. 3. A.Mohari and K.B.Sinha Quantum Stochastic Flows With Infinite Degrees of Freedom and Countable State Markov Processes. Sankhya : The Indian Journal of Statistics 1990,vol 52 A,Pt.l,pp. 43-57.
Quantum Probability and Related Topics Vol. VIII (pp. 123-142) ©1993 World Scientific Publishing Company
CHEBOTAREV'S SUFFICIENT CONDITIONS FOR CONSERVATIVITY OF QUANTUM DYNAMICAL SEMIGROUPS
Franco Fagnola Universita di Trento, Dipartimento di Matematica
I
38050 Povo (TN)
Italia
A b s t r a c t . We expose Chebotarev's sufficient conditions for the conservativity of a quantum dynamical semigroup with some extensions and simplifications needed in the applications to quantum stochastic calculus. 1. I n t r o d u c t i o n If B(h) is the Banach space of bounded operators on a Hilbert space h a quantum dynamical semigroup (q.d.s.) T = (Ti)<>o on B(h) is a semigroup of completely positive linear maps on B(h) with some continuity properties (see Definition 2.1) satisfying the condition Tt(I) < I for all t > 0. Here / denotes the identity operator. The q.d.s. T is conservative (or Markovian, or identity preserving) if Tt(I) = I. When T is conservative and uniformily continuous (see Definition 2.2) the canonical form of its infinitesimal generator £ 1
°°
C{X) = i [X,H] - -XC + J2 L)XL,
1
~ 2CX>
( L1 )
J=I
with C = Yy'Li L^jLj and H selfadjoint operator, is well known (see Theorem 2.3). As shown by Frigerio [9] and Davies [5], the infinitesimal generator has the same form also in many cases in which T is conservative and ultraweakly continuous. Given such an operator on B(h) one can construct a q.d.s. with this infinitesimal generator (see [2], [10], [5]). However the q.d.s. may not be conservative (see the example 3.3 in [4]). In [2] Chebotarev gave a necessary and sufficient condition for conservativity of a q.d.s. which is a quantum analogue of Feller's condition of non-explosion of a classical process (see also [10], [11]). This condition is practically impossible to check in many interesting examples. Extending some sufficient conditions for non-explosion of Markov jump processes [1], Chebotarev gave also sufficient conditions for the conservativity of a q.d.s.
124 with infinitesimal generator of the form (1.1) and Li ^ 0 only for a finite number of indices /. These are essentially the lower relative boundedness with respect to C 1 / 2 of the commutator of H and C and the lower boundedness of the operator C2 — ^ 3 ^ ! L'C'Li. These conditions are very easy to check when the operators H. Li are differential operators. However some operator domain problems and other technical problems related to the use of quadratic forms is left open in Chebotarev's paper [2]. In this note we extend Chebotarev's sufficient conditions to the case of infinite nonzero L\ under the assumption (see the condition C) that the quadratic form oc
(v,u) — i► >
{Liv,
Liu)
1=1
is closed. We weaken also the assumptions on the set of vectors on which one should check some quadratic forms inequalities (see Theorem 3.2). Moreover we simplify some steps of the proof. These extensions and simplifications are necessary to apply this conservativity result to quantum stochastic calculus [3], [6], [7], [8], [10]. Some applications of the main theorem of the present note have been given in [3]. The application to the realisation of classical diffusion processes and Azema-Emery martingales as quantum flows in the Boson Fock space will be studied in the forthcoming paper [7]2. Quantum dynamical semigroups Let h be a complex separable Hilbert space endowed with the scalar product (•,•) and norm ||-||. Consider a dense linear submanifold T> of h. Let B(h) denote the Banach space of bounded operators on h. The uniform norm in B(h) will be denoted also by D'H—. Let / be the identity operator on h. Definition 2.1. A quantum dynamical semigroup on B[h) is a family T = (Tt)t>Q of operators in B(h) with the following properties: i) %{X) = X, for all X eB(h), ii) Tt+,{X) = Tt {Ta(X)), for all s, t > 0 and all X 6 B(h), Hi) Tt(I) < I, for all t > 0, iv) (complete positivity) for all t > 0, for all integer n and all finite sequences (X})]=1, (Yi)f=1 of elements of B(h) we have n
£
Yt'Tt{XrXi)Yi > 0,
(2.1)
125 v) for every sequence ( I „ ) „ > o of elements ofB(h) converging weakly to an element X of B(h) the sequence (T ( (X n )) n > 0 converges weakly to Tt(X) for all t > 0, vi) (ultraweak continuity) for all trace class operator p on h and all X G B(h) we have \im+Tr(pTt(X))
=
Tr(pX).
Let us remark that, as a consequence of properties iii), iv) for all X € B{h) we have the inequalities between bounded operator on h © h 0<
Tt(I) Tt(X*)
Tt(X) \ ( I Tt(X*X)J - \Tt(X*)
Tt(X) \ Tt(X*X)J
Hence we obtain the identity Tt(X*) = Tt(X)* and inequality
Tt(XYTt(X)
< Tt(X*X).
Since T( is a positive operator because of property iii) we have the inequalities
Tt(X*X)<\\X\\lTt(I)<\\X\\l. Therefore, for all X £ B(h), we have
l|T«WIL
(2-2)
Let us recall also that the ultraweak topology and the weak topology induce the same topology on the spheres of B(h). Definition 2.2. A quantum dynamical semigroup T is said to be uniformly con tinuous if lim sup | | T < ( X ) - X | | o o = 0. We refer to [12] for the proof of the following characterisation of the infinitesimal generator of a uniformly continuous q.d.s.. T h e o r e m 2.3. (Gorini, Kossakowski, Sudarshan, Lindblad) A bounded operator C on B(h) is the infinitesimal generator of a uniformly continuous quantum dy namical semigroup such that Tt(I) = I for all t > 0 if and only if there exists a bounded selfadjoint operator H and a sequence ( L ^ g j of bounded operators in h such that the series 2~ISi L'Li is strongly convergent and C(X) = i[H, X] + - Y 1
1=1
{-L\UX
+ 2L\XU
-
XL\U)
126 for all X e 8(h). Letting 1 °°
G = ~Y,L*Ll-iH ~ 1=1
the infinitesimal generator is given by C(X) = XG + Y, L*XLi + G*X. 1=1
The condition Tt{I) = I for all t > 0 implies then oo
G + G' + J2L'L' = °i=i
In the applications of quantum stochastic calculus one a has q.d.s. T with unbounded generator of a similar form. The operators G, Li often satisfy the following assumptions A - The operator G is the infinitesimal generator of a strongly continuous con traction semigroup (P(t))t>o in h and T> is a core for G. The domain of the operators (£i)(Si contains T> and for all u,v £ T> we have oo
(o, Gu) + (Gv, u) + Y^ (Ltv, L,u) = 0.
(2.3)
B - The operator G2 is densely defined and its domain contains V. We consider a q.d.s. T with infinitesimal generator £ given by oo
(v, C(X)u) = (v, XGu) + (Gv, Xu) + Y^ (Liv, XL,u)
= 0
(2.4)
satisfying the equation (v,Tt(X)u) + f
= {v,Xu) +
Uv, T3(X)Gu) + (Gv, Ts(X)u)
+ JT (L,v, T.(X)L,u)\
ds
(2.5)
for all X eB(h) and all u, v E V. For all t > 0 denote Tt(I) by T(t). The following proposition can be proved as in [6].
127 Proposition 2.4. Let us suppose that conditions (A), (B) hold and let (T(t))t>0 be a strongly continuous family of positive contractions on h. The following con ditions are equivalent: i) for all v, u 6 V we .have (v,T{t)u)
= {v,u) + I
UGv,T{s)u)
+ {v,T{s)Gu)
+
^{liv,T{s)Liu)\ds,
ii) for all v,u 6 T> we have (v,T(t)u)
= (P(t)v,P(t)u)
+V
/ {L,P(t - s)v,T(s)L,P(t
- s)u) ds.
Definition 2.5. Suppose that the conditions (A), (B) hold. The semigroup T is called conservative (or MarkovianJ if the only solution of the equation (2.6) is given by T(t) = I for all t > 0. Let us consider the operator Q on B(h) defined by /•OO
(v,Q(X)u)=
°°
exp(-s)y2{L,P(s)v,XLiP{s)u) ds Jo
Using (2.3) and integrating by parts we can estimate this by \\X\\ /•OO
/ JO
(2.6)
1=1
times
J
yoo
e x p ( - ^ ) — ||P(s) U || 2 ds
e x p ( - s ) Y, \\LiP(s)u\\* ds = - I /^l
(2.7) /•OO
2
= ||U|| - / Jo
exP(-s)\\P(s)u\\
2
ds
Hence Q(X) belongs to B(h). Moreover the map Q : B(h) i-> B(h) is normal and monotone. The following theorem has been proved in [2]. (See also [6], [10]; the parameter A used there can be chosen to be 1.) T h e o r e m 2.6. (Chebotarev) Suppose that the conditions (A) and (B) hold. Then the following conditions are equivalent: i) the semigroup T is conservative, ii) the decreasing sequence (Q"(I))n>i of positive operators in B(h) converges strongly to zero, Hi) the equation Q(X) = X has no positive solutions in B(h), iv) the equation £(X) = X has no positive solutions in B(h). The above conditions are difficult to check in many cases. Therefore it is inter esting to give simpler easily verifiable conditions.
128 3. Sufficient conservativity conditions In this section we will give a simplified version of Chebotarev's sufficient conditions for conservativity. Let us first fix some technical preliminaries: Proposition 3.1. For all I > 1. the operator Li has a unique extension as an operator with domain coinciding with D{G) satisfying, for all v,u 6 D(G) oo
(v, Gu) + (Gv, u) + J2 (Liv> L'u) = 0.
I3-1)
(=i
moreover, for all u 6 D(G), the following map is continuous t^L,P{t)u.
(3.2)
Proof. Since V is a core for G, for all u G D(G) there exists a sequence of elements of V such that lim un = u,
(un)n>\
lim Gun = Gu.
n—+oo
n—►oo
Using (2.3) it is easy to see that the sequence (Liun)n>i is Cauchy in h for all / > 1. Therefore L\ has an extension to the domain of G that we will denote always by L{. The formula (3.1) can be easily checked. The uniqueness of the extension can be proved using again (3.1). Moreover, for all s,t > 0 and all u £ D(G), the identity oo
J2 \\L,P(t)u - LiP(s)uf
= -23fe {(P(t) - P(s))u, G(P(t) -
P(s))u)
i=i
shows the continuity of the map (3.2), for all I > 1. □ We introduce now a transformation of the operators G, ( £ i ) o i which leaves the infinitesimal generator invariant. This allows to skip some technical problems without loss of generality. Let us consider the operators with domain D(G) G =
G-\l 2
- _ / / *"-\ Lt-i
if ; = i if / > 1
Differentiating with respect to t it is easy to see that a strongly continuous family of positive contractions on h satisfyies the condition i) of Proposition 2.4 if and only if it satisfies the equation (v, T(t)u) = {v,u) + J
I (6v, T{S)U)
+ (v, T(s)Gu}
+ ] T (L,v, T(S)L,U}
j da.
129 Therefore from now on we will denote G and L\ by G and L\ respectively and we will suppose that L\ = I. Under this assumption it follows then from (3.1) that for all t > 0, u e D(G) we have ~ \\P(t)u\\2 = 25fe (P(t)u, GP(t)u) < -
\\P(t)uf
hence we have the inequality ||P(t)||<exP(-t/2).
(3.3)
We shall consider operators G, Li satisfying also the following assumption: C There exists a selfadjoint operator C with domain containing V such that D is a core for C and, for all u, u 6 T>, t > 0 we have oo
Y2 (Liv> L'u)
= («. Cu) = - {v, Gu) - (Gv, u),
(3.4)
P(t)u G D(C)
(3.5)
Clearly, since we supposed L\ = 7, for all u £ D(C) we have the inequality
(u,Cu) >\\uf.
(3.6)
Therefore C has a positive bounded inverse C - 1 ; the inequality (3.6) implies then OKC'SI
(3.7)
Moreover, for all e > 0, we can consider bounded operators CE = C(7 + e C ) - 1 = ( < ? - * + e l ) ~ \
(3.8)
For all u S D(C) we have lim Ccu = Cu.
(3.9)
£—0 +
The result we shall prove is contained in the following which is essentially due to A. Chebotarev.
130 Theorem 3.2. Suppose that the assumptions (A), (B), (C) are satisfied and there exists a positive constant b such that: i) for all u G V we have 23fe (Gu, C~lu)
> -b\\uf
,
(3.11)
ii) for all u € V we have 23fe (Cu, Gu) < - \\Cu\\2 + b (u, Cu),
(3.12)
iii) for all u £ V and all t > 0 we have oo
(Cu, Cu) - ^
{L,u, CcL,u) > -b (u, Cu).
(3.13)
Then a quantum dynamical semigroup satisfying the equation (2.5) is conservative. The above inequalities are more easily verifiable under some additional domain assumptions on the operators G, (Li)fl1, C. Corollary 3.3. Suppose that the assumptions (A), (B), (C) are satisfied and there exists a positive constant b such that the following conditions hold: a) the linear manifold C(V) coincides with V, b) for all I > 1, L,(T>) C D{C), c) for all u € V we have 23fe (Cu, Gu) < - \\Cu\\2 + b (u, Cu),
(3.14)
d) for all u £ V we have CO
(Cu,Cu) -^2(Liu,CL,u)
> -b(u,Cu).
(3.15)
(=i
Then a quantum dynamical semigroup satisfying the equation (2.5) is conservative. Proof.
The conditions b), d) and the inequality (Liu,CLiu)
>
(Liu,CeLiu)
for all £ > 0, 7 > 1, u € V imply that the condition iii) of Theorem 3.2. Since C(T>) = V, for all u € T> the vector C~xu belongs to T> and the condition c) gives 23te {u,GC-lu)
< -||«||2+6(u,C_1u).
(3.16)
131 Using the identity (3.4) we have {u,GC-lu)
= -\\u\\2
-{GuX'-'u).
Thus, from (3.16), we obtain the inequality 29te(u, GC~xu)
> -\\uf
-b(u,C-lu)
> - ( 6 + 1 ) ||u|| 2
Therefore the conditions of Theorem 3.3 are satisfied with the positive constant 6+1. □ Remark.
Let H be the operator with domain V defined by
iH = G + \c. Because of (3.4) H is symmetric. Then (3.14) is equivalent to the following in equality i({Hu,Cu)
- (Cu,Hu))
>
-b(u,Cu).
Thus, roughly speeking, (3.14) is an assumption on the commutator of H and C. In a similar way, if Li — 0 for all / > 2, then, under some more technical conditions, Cu = u + L'LiU for all u G T>. It is easy to see that (3.15) is an assumption on the commutator of L\ and L\.
132 4. Chebotarev's inequalities for completely positive maps In this section Q will denote an arbitrary completely positive map Q : B(h) — t > B(h) (not necessarily the map Q defined by (2.6)). Lemma 4.1. Let A,B,C be elements of B(h) such that B and C have bounded inverses. Suppose that the bounded operator on h © h
U S) is positive. Then the following inequality holds BC^B* Proof.
< A.
(4.1)
The following identities B*(B-1)"
= ( B _ 1 B ) * = I=(B(B-1))'
=
(B~l)*B*
imply that B* has the bounded inverse (B~1)*. Moreover the positivity assump tion implies that A and C are selfadjoint. The inequality (4.1) follows then from the positivity of the operator on h © h (B~l A(B-1)* ly 0
- C-1
0 \ _ (-B~l C) ~ \ 0
C~x\( A I ) \B*
B\f-(B")-1 Cj { C~l
0\ l)-D
Lemma 4.2. Let Q be a completely positive map on B(h) and let X be a positive invertible element of B(h) with bounded inverse X~l Suppose that Q(X^1) and Q(I) have also bounded inverses. We have then the inequality Q(I)Q(X-1)-lQ(I)
Since Q is completely positive, the operator on h © h
(Q{X) \Q(I)
Q{I) \_ ( Q(X 1 / 2 X 1 / 2 ) Q(X-'))-\Q{X-'l2Xll2)
QiX'^X-^2) \ Q{X-ll2X-^2))
is positive. Therefore it suffices to apply Lemma 4.1 with A = Q(X), B = Q(I),
C = Q(X~1).
□
133 Lemma 4 . 3 . Let Q be a completely positive map on B(h) and let X be a posi tive element of B(h). Suppose that the operators X, Q{I), Q(X) have bounded inverses. Then the following inequality holds Q ((I + X-1)-1) Proof.
< (Q(7)- ] + QiX)-')-1
Denote by B the positive square root of Q(I)1/2
.
(4.2)
We have then
Q ( ( / + - Y - 1 ) - I ) = Q ( / - ( / + A')- 1 ) = B2 - Q((I + X)~l) = B(I-B-1Q((I
X)-1)B-y)B
+
Using Lemma 4.2 we obtain the inequality B2 (Q(I + X))" 1 B2 < Q {(I + X)-1) . Therefore we have Q ((I + A " 1 r 1 ) < B (i -B(Q(I)
+ Q(X))-1
+ B-lQ{X)B-l)~1^
= B(I-(I
1
1
= B(I+BQ{X)- B)~ 1
= (Q(I)This proves the inequality (4.2).
B) B
+
B
B 1
1
Q(X)- )-
□
T h e o r e m 4.4. (Chebotarev) Let Q : B(h) t-* B(h) be a completely positive map and, for all n > 1, let (A'A..)JJ=1 be a family of positive bounded operators. Suppose that the operators Xi,... ,Xn, Q(I), Q(Xi),..., Q(Xn) have bounded inverses. Then the following inequality holds Q ((I + AT 1 + . . . + A" 1 )" 1 ) < (Q(I)-1
+ Q ( A 1 ) - 1 + ■ ■ • + S ( A n ) " 1 ) - 1 (4.3)
Proof. The inequality (4.3) follows from Lemma 4.3 when n = 1. Suppose it has been established for an integer n > 1. Consider the bounded operator Y defined by F"1 = / + A - 1 + . . . + A - 1 . For all A £ B(h) let A = Y-l'2XY-1'2,
Q{X) =
QiY'^XY1'2).
134 Clearly Q is a completely positive map on B(h), hence, applying Lemma 4.3 to the map Q and (4.3) for the integer n n and the map Q we get the inequalities 1 Q((/ + Xr11-+ ... + X-J,)A7J 1 r11)) = Q ) G((((/ /++-Y * *n+1 « ))~ *) eftz+xr
< (Qiir' +
Qix^r1)'1
= (Q(Yr1 +
Q(xn+1riyl
< (/+Q(.Y1)-1 + ... + Q(.Yn+1)-1)"1 This completes the induction argument.
□
5. Proof of Theorem 3.2 Let Q be the completely positive linear map on B(h) defined by (2.6) and let Q denote Q(I). Let us first prove some properties of Q. Lemma 5.1. The operator Q has a positive bounded inverse satisfying the fol lowing inequalities \l
/<<9_1<2/.
(5.1)
Moreover Qu — u if and only if u = 0. Proof.
From the inequalities (2.7) and (3.3) we obtain
\IMI2 = IMI2 ( i - Jo°° «p(-2*)d») < <«, Qu) < \\u\\2 Therefore Q has a bounded inverse Q^1 and (5.1) holds. Moreover it follows from (2.7) that Qu = u if and only if P(t)u = 0 for all t > 0, i.e. u = 0. □ Lemma 5.2. Let R be the positive selfadjoint operator on h defined by Ru =
Q(I-Q)~1u
for all u G D{R). For all e > 0 tiie bounded operators Rc, Ce defined by R£ = R(I + eR)'1,
C€=C{I
+ eC)~1
satisfy the inequalities {2 + e)-'I
<£"';,
(l+e)_1I
(5.2)
135 For all e > 0 and all n > 1 the operators Qn(Re), and the following inequalities hold (2 + £ ) - " / < Qn(R,)
Qn(Cc) have bounded inverses
(1 + £ ) - " / < Qn(Ce) <e~nI
<£-"/,
(5.3)
Proof. The operator R can be defined by functional calculus using the spectral family of Q. Then (5.1) yields the inequalities Rc = (el + R-1)'1
= (eI + Q~1 -I)'1
> (el + Q " 1 ) " 1 > ( 2 + £ ) - '
Moreover, because of (5.1), R has a bounded inverse. Hence, for all e > 0, the operators Rc, Cc are bounded from above by e~lI. The operator C£ is bounded from below by (1 + £ ) _ 1 J because of (3.6). This proves (5.2). Now (5.3) follows easily by induction. D Lemma 5.3. For all positive bounded operator X and all e e (0,1/2) we have (Q-1 + X)_1 Proof.
< c / + ( / + R;1 +
X)_1
Let us first compute
( Q - ' + X ) " 1 = (I + R^1 + X - el)~l
= (I + R-i+X)'1
(i - e (I + R;1 +
Xf'Y'
Using the inequality (5.2) we find the lower bound I + R;' + X > ( 3 + £)(2 + £ ) _ 1 For all £ € (0,1/2) and all t > (3 + e) (2 + e)~l we have ( l - £ t - 1 ) - 1 < 1+et
(5.4)
In fact the right-hand side is an increasing function, the left-hand side is a decreas ing function of t and (5.4) holds for t = (3 + e) (2 + e) because of the elementary inequalities (3 + £)(2 + £ ) < ( 2 £ + l ) ( 6 - 3 £ ) = (4 + 2 £ ) ( 3 - 3 £ / 2 ) < ( 2 + 4£ + £ 2 ) ( 3 - £ - £ 2 ) which hold for all £ S (0,1/2). Therefore, using (5.4), we obtain the inequality
(Q-i+Xy^^
+ RT'+xy^I 1
= ei+(i+R:
+ e^ +
R^+X))
1
+ xy
which completes the proof. D We shall prove now the inequality needed to prove Chebotarev's sufficient con ditions.
136 Theorem 5.4. For each integer n > 0 and each finite sequence (e,t)JL 0 of elements of (0,1/2) such that £ £ = o e * 5
1
/2
we have
an+\i) < / £ > +1 1 + JlQk(R**f) *:=0
V
fc=0
(5-5' I
Proof. The inequality (5.5) holds for n = 0 because of (5.1). Suppose that it has been established for an integer n — 1. Using (5.1) and Chebotarev's inequality (4.3) we have n (I)) S n + 1I)(/)= = Q(Q S(Q"(/))
1
i=0
1 :/$>t++ [Q
k=l
\
Applying Lemma 5.3 with e = £o we get the inequality
e»+'(j) < / £ e t + (/ + B-1 + £ Q * (A.*)"1) k=0
V
This completes the induction argument.
/fc=l
/
D
Proposition 5.5. Suppose that t i e conditions (A), (BJ and (C) hold and there exists a positive constant 6 such that 2%e(Gu,C-1u)
> -b\\uf
(5.6)
for all u &T>. For all e > 0 we have then ( / - Q ) > ( 6 + 1)-1C-1, fl£<(6
(5.7)
+ l)C £ .
(5.8)
Proof. Since V is a core for G, (5.6) holds for all u S D(G). the inequality C - 1 / 2 < 1, we have
> f«
Moreover, using 2
t°° °° f°°f°° II ii 1/22 I I I2 I e:> {u,(I-Q)u) 2 ds > exp(-s)\\P(s)is)u\\ ixp(-s)>)||c< , 1 / 2 P( P(.s)J| I< c {u,(I-Q)u)= =■ Jo exp(-s)\\P{s)u\\ ds> JoI exp(P(5)u||
exp(-s)||P(s)u|| ds>
/
exp(-s)|c-
P(s)u||
ds
137 Integrating by parts and using the inequality (5.6) we have °°
II2
n
/
exp(-s) C-1/2P(s)u
ds = (u,C~lu) + / exp(-s)23fe (GP(s)u.C'1 Jo >{u,C~lu)-b
P(s)u) ds
exp(-s)\\P(s)u\\2ds Jo
= ( u , C _ 1 u ) -b(u,(I
-Q)u)
Hence we obtain the inequality (6+l)(u,(7-Q)«)>(u,C'-1u) and (5.7) follows. Moreover, using (5.1), for all £ > 0, we get the inequality R'1
=£
+
Q-\l-Q)>e
+ (I-Q)>e
+ (b+ \)~xC~l
> (b + l)-*(e + CT 1 )
This implies (5.8). D Proposition 5.6. Suppose that the conditions (A), (B), (C) hold. Then: a) for all I > 1, the operator L\ has an extension with domain containing D(C), b) for all sequence (u n )„>i of elements ofD that converges to a vector u such that the sequence ( C u n ) n > i converges to Cu we have lim L[Un = Liu, n—*oo
c) the domain D(G) of G is contained in the domain D{CXI2) of C 1 / 2 , d) for all v 6 D{G), u 6 D(G) n D(C) we have CO
V (L,v, L,u) = (v, Cu) = - (v, Gu) - (Gv, u).
(5.9)
Proof. Since V is a core for C, for all u € D(C) there exists a sequence (u„)„>i of elements of V such that lim un = u, n—*oo
lim Cun = Cu. n—*oo
Therefore, by the identity (3.4), for all integers n,m, we have oo
£ 1=1
\\Lt(un - um)\\2 = | | C 1 / 2 K - «™)ir = " 2 ^ («„ - ti m , G(«„ - u w ) ) .
138 As in Proposition 3.1 this shows that the operators Li have as an extension to D{C). Using the above identity one can prove also b). Consider now a sequence (u„) n >i of elements of V such that lim un = u, n—-oo
lim Gun = Gu. n—'OG
For all integers n, m, the identity (3.4) yields |c1/2(un-um)||
= -2dte(un-um,G(un-urn)).
(5.10)
Therefore the sequence (C 1 / , 2 u n ) n >i is Cauchy. Since the operator C 1 ' 2 is selfadjoint, u belongs to the domain D(CX'2) of C1'2 and the sequence ( C 1 ' 2 u n ) n > i converges to C1?2u. This proves the statement c). Moreover, for all v, u 6 D(G), we have (c1/2v, C^2u\ = - (v, Gu) - {Gv, u). This implies that (5.9) holds for all v G D(G), u € D(G) 0 D{C).
D
P r o p o s i t i o n 5.7. Suppose that the conditions (A), (B), (C) are satisfied and that the inequality (3.12) holds for all u e T>. Then (3.12) holds for all u 6 D(G2)C\D(C). Proof.
Let H be the operator with domain coinciding with V defined by iH = G + ^C.
Since C is selfadjoint, (3.4) implies that H is symmetric. inequality (3.12) yields 29te(Cu,iJ7u} <
For all u 6 V the
b(u,Cu).
Therefore, for all u £ f , we have \\Guf
= \ \\Cuf
- SRe {Cu, zHu) + \\Huf
> \ \\Cu\\2
which implies \\Cu\\2 < 4 | | G u | | 2 + 2 6 | c 1 / 2 u | 2 .
(5.11)
Clearly, form (3.12), for all £ > 0 and all u 6 V we have 23fe (Cu, Gu) < - \\Ceu\\2 + bl&Pu,
Cll2u\
(5.12)
139 Let u be an element of D(G2) l~l D(C) and let (u„)„>i be a sequence of elements of T> such that lim un = u,
lim Gun = Gu.
n—*oo
n—.00
Then, by (5.10) and (5.11), we have lim C 1 / 2 u „ = Cll2u,
sup | | C u J | < oo.
(5.13)
Because of (5.9), for all v £ D(G), we have lim (Cun,v)
=
(Cu,v).
n—foo
Therefore, since u g D(G2), using (5.13) we obtain lim (Cun,Gun)
= lim (Cun,Gun
n—*oo
- Gu) + lim (Cu„,Gu)
n—-oo
= (Cu.Gu) .
n—'■oo
Writing the inequality (5.12) for the vector un and letting n tend to infinity, using (5.13), we obtain the inequality 23te (Cu, Gu) < - \\Ceu\f + b (C1/2u,
C1/2u\
.
2
Letting e tend to 0 we obtain (3.12) for all vectors u € D(G ) D D{C).
□
Proposition 5.8. Suppose that the conditions (A), (B), (C) are satisfied and there exists a positive constant b such that the assumption ii) of Theorem 3.2 holds. Then, for all u E £>, we have / Proof.
exp(-t)\\CP(t)fdt<(u,(C
+ bI)u).
(5.14)
Let us note first that, for all « 6 D, we have ~
\\P{t)u\\2 = -23fe {CP(t)u,
GP(t)u).
Moreover the function CP{-)u is measurable because it can be approximated by continuous functions (C e P(-)«)e>o- Since u g 2? C D(G2), due to Proposition 5.7, for all r > 0, the inequality (3.12) holds for the vector P(t)u g D(G2) D D(C), hence r°° f00 d2 2 J^ e x p ( - t ) \\CP(t)u\\ dt < j o e x p ( - t ) ^ \\P(t)u\\2 dt + b I
Jo
exp(-t)(P(t)u,CP(t)u)dt.
Integrating by parts we can easily compute the first integral /•oo
/
J2
e x p ( - r ) ^ \\P(t)uf
dt = (u, C«) - (ti,Qii)
The second one can be computed using (2.6) and (3.4). Thus we obtain (5.14).D
140
Proposition 5.9. Suppose that the conditions (A), (B), (C) and the assumptions i), ii) of Theorem 3.2 hold. Then, for every e > 0, for every integer n > 1 and every u € T>, we have (u, Qn(Cc)u) n
(u,Q (RE)u)
< («, (C + nbl) u)
(5.15)
< ( 6 + l ) ( u , ( C + n6J)u)
(5.16)
Proof. Let us first notice the following property of the regularisation of positive operators used in Lemma 5.2. For all e, rj > 0 and all positive bounded operator X with bounded inverse we have (A',), = (,?! + X f T
1
= ((£ + ^ 7 + X - 1 ) - 1 = X £ + „ .
Therefore, by Chebotarev's inequality (4.3) and the inequalities (5.1), we have Q(Xe+n)
=
Q((eI+X;iy1)
< (eQ-1 + QiX.rT1 1
(5 17)
-
1
<(£/+S(.Y7,)- )- =(Q(X„))£ We prove now (5.15) by induction. By the same argument of Proposition 5.6 one can see that the condition iii) of Theorem 3.2 holds for all vector u 6 D(C). Using (5.9) and (3.13), for all u £ V we easily obtain the inequalities (u,Q(Ce)u}<
/ < /
exp(-i)
S^(L,P{t)u,C,L,P{t)u)dt
exp(-t) (||CP(i)u|| 2 + 6 (P(t)u, CP{t)u)^j dt
(5.18)
= {u,(C + bI)i Suppose now that (5.15) has been established for a certain integer n > 1 and let us prove it for the integer n + 1. Using (5.17) and (5.18) we obtain (u,S"+1(C2£)u) = (u,Q"(Q(C2l))«) <
+ bI)u)
+ (n +
l)bI)u)
This completes the induction argument. Now (5.16) follows from (5.15) and the inequality (5.8). □
141 Proof, (of Theorem 3.2) For all e > 0 and all integer n consider a finite sequence (ejt)!t=o such that e0 + .. .+en = e. The inequality (5.5) and Chebotarev's inequality (4.3) applied to the completely positive map X i—v (u,Xu) with u € T> yield the inequality for all nonzero u
(u, Q"+\l)u)
< e H | 2 + (\\u\\-3 + £
k=0
Using (5.16) for all u S V we have then
(u, Qn+1(I)u)
< e \\uf + | [| U |r 2 + (b+ I ) " 1 Y, (C +
kbl)u)~l
The divergence of the series oo
^ ( « , ( C + fc6J)«)_1 k=0
implies then (u,Qn+1(I)u)<e\\u\\2
lim
Therefore the condition ii) of Theorem 2.6 is satisfied and the q.d.s. satisfying the equation (2.5) is conservative.
□
142
References [1] Chebotarev, A.M.: Sufficient conditions for the regularity of jump Markov pro cesses. Theory Probab. Appl. 33, No.l (1990). [2] Chebotarev, A.M.: The theory of the conservative dynamical semigroups and its applications. Preprint MIEM n.l. March 1990. [3] Chebotarev, A.M., Fagnola, F., Frigerio, A.: Towards a stochastic Stone's the orem. To appear in: Stochastic Partial Differential Equations and Applications III, Pitman Research Notes in Mathematics. [4] Davies, E.B.: Quantum dynamical semigroups and the neutron diffusion equa tion. Rep. Math. Phys. 11 (1977), 169-188 [5] Davies, E.B.: Generators of Dynamical Semigroups. J. Funct. Anal. 34(1979), 421-432. [6] Fagnola, F.: Unitarity of solutions of quantum stochastic differential equations and conservativity of the associated semigroups. To appear in: Quantum Prob ability and Related Topics VII. [7] Fagnola, F.: Realisation of diffusion processes and Azema-Emery martingales as quantum flows in Fock space. In preparation. [8] Fagnola, F.: Characterisation of isometric and unitary weakly differentiable cocycles in Fock space. Preprint UTM n.358 October 1991. [9] Frigerio, A.: Quantum dynamical semigroups and approach to equilibrium. Preprint, 1977. [10] Mohari, A.: Quantum stochastic calculus with infinite degrees of freedom and its applications. Ph.D. Thesis I.S.I., New Delhi 1991. [11] Mohari, A., Parthasarathy, K.R.: A quantum probabilistic analogue of Feller's condition for the existence of unitary Markovian cocycles in Fock space. Pre print, I.S.I., Delhi Centre. (1991). [12] Parthasarathy, K.R.: An Introduction to Quantum Stochastic Calculus. I.S.I., New Delhi 1990.
Quantum Probability and Related Topics Vol. VIII (pp. 143-164) ©1993 World Scientific Publishing Company CHARACTERIZATION OF ISOMETRIC A N D UNITARY W E A K L Y D I F F E R E N T I A B L E COCYCLES IN FOCK SPACE Franco Fagnola Dipartimento di Matematica, Universita di Trento I - 38050 POVO (TN) ITALY
Abstract. We prove a characterization theorem for weakly differentiable (con tractive, isometric and unitary) Markovian cocycles in the Boson Fock space analogue to Stone's theorem on strongly continuous groups of unitary operators. 1. Introduction A Markovian cocycle is a family (V(t))t>o of contractive operators with a time shift covariance and localisation properties. This notion was introduced by Ac cardi in [1] to construct Feynman-Kac perturbations of quantum Markovian semi groups and exploited by Accardi, Frigerio and Lewis in [2] to construct quantum stochastic processes. Unitary Markovian cocycles in the Boson Fock space with norm continuous re duced semigroup can be obtained solving a quantum stochastic differential equa tion of the form dV(t) = '£v(t)Liid\i V(0) = I (1.1) ■',;' 3
where A - are the basic integrators of Hudson-Parthasarathy Boson Fock space quantum stochastic calculus and the operators L'j are bounded operators playing the role of the infinitesimal generator. Conversely it has been shown by Hudson and Lindsay [12] that Markovian cocycles with norm continuous reduced semi group are all solutions of equations of this form. Clearly norm continuity is not satisfied by the most important semigroups. Thus it is interesting to weaken this assumption to establish a quantum stochastic analogue of Stone's theorem on one-parameter group of unitary operators. Journe [13] investigated the strongly continuous case and showed, under a regularity condition, that a cocycle can be reconstructed from the infinitesimal generator by a recursive procedure. However the cocycle obtained by this procedure may not satisfy any q.s.d.e. when the operator L'j have { 0 } as common domain. This cannot happen when the cocycle V satisfies a weak differentiability condition introduced by Accardi, Journe and Lindsay in [3] (see Definition 2.3 here). In this case the cocycle V satisfies a q.s.d.e. of the form (1.1) and, moreover, the operators £ ' have the property (A) (see Definition 2.6, Theorem 2.4). If V is
144 isometric it has t h e properties (B) (see Section 5), and (C) (the conservativity of the associated q u a n t u m dynamical semigroup, see Section 5). If V is u n i t a r y a n d its dual cocycle V (see [13]) is also weakly differentiable t h e n V has t h e properties (A),(B),(C). In this paper we establish a q u a n t u m stochastic Stone's theorem for weakly dif ferentiable cocycles proving the converse result: given t h e operators
Lljsatisfying
t h e property (A), there exists a unique solution of (1.1) which is a weakly differen tiable cocycle. Since every contractive solution of (1.1), when unique, is a weakly differentiable cocycle this generalizes also all previous existence results on q.s.d.e. of t h e form (1.1) with unbounded coefficients (see [4],[7],[8],[11],[15],[16],[17]) which have been obtained recently and applied t o t h e construction of stochastic processes in the Boson Fock space. Moreover we show t h a t conditions ( B ) , (C) on t h e operators L'j (resp. (B) and (C) on t h e operators £!•) are necessary and sufficient in order V to be isometric (resp. unitary). T h u s isometry (and by dualisation unitarity) of a contractive cocycle satisfying (1.1) can be obtained from t h e characterization theorem of Chebotarev [4] showing t h a t conservativity of a q u a n t u m dynamical semigroup is equivalent to a q u a n t u m analogue of Feller's classical condition or from quite general and easily verifiable sufficient conditions given also in [4] (see also Mohari [15] for a more detailed study of q u a n t u m dynamical semigroups arising from classical countable s t a t e Markov chains). T h e proof is done approximating t h e operators L'j by b o u n d e d operators using the resolvent family of the reduced semigroup (and multiplying the coefficients of creation and annihilation operators by a real number rj £ (0,1) as in D a P r a t o [6], in t h e case of classical s.d.e., and Frigerio [11]). We construct a solution of (1.1) as a weak limit of solutions of the approximating equations using t h e equicontinuity m e t h o d suggested by Frigerio as described by t h e author in [8]. This solution is unique by a result of Mohari-Parthasarathy [16]. T h e n , using some results on t h e minimal solution of the classical equation for a q u a n t u m dynamical semigroup (Langevin equation), we show t h a t the unique solution of (1.1) gives a dilation of the classical minimal solution and give necessary and sufficient conditions for isometry. Conservativity of the associated q u a n t u m dynamical semigroups for unitarity of the solution of (1.1), whose importance was pointed out in [4],[5] and [9], is necessary and sufficient for unitarity of t h e solution also for q.s.d.e. of t h e form (1.1) driven by "free" integrators as shown by Mancino [14] using a t i m e reversal property for cocycles in the full Fock space proved in [10]. "Free" integrators, however, have a different Ito table. T h e new notation of Belavkin-Parthasarathy (see [19],[16]) for Boson Fock space q u a n t u m stochastic calculus we use here allows to find easily a necessary a n d
145 sufficient conditions for contractivity of solutions t o (1.1) and t o deal with infinite integrator processes. T h e characterisation theorem of weakly differentiable unitary cocycles obtained in t h e present paper can be easily applied to the realisation of a large class of classical Markov processes as inner q u a n t u m flows in Fock space. This will be done in a forthcoming paper.
2. N o t a t i o n s a n d preliminaries We recall here some notations a n d results in Boson Fock space q u a n t u m stochas tic calculus as exposed in [16], [18], [19]. Let h, k be two complex separable Hilbert spaces a n d let { e; | / G S } be an orthonormal basis in k.
Let Ms
denote the
dense linear submanifold of Z 2 ( R + )
= 0 for
all t > 0 for all b u t a finite n u m b e r of indices k and let M°s be the submanifold of continuous functions in Ms-
For all Hilbert space K, let T(K.) denote the Boson
Fock space over K,. Let S = S U { - c o , + 0 0 } and consider t h e Hilbert spaces k = Oe_oo 0 k 0 Oe+oo, 2
■H=h®r(L (R+)®k),
h®k®T(L2(R+)®k).
H =
In t h e following all t h e operators denned on a tensor product factor of "H will be identified with their canonical extension to the whole space. Let F be t h e unique u n i t a r y o p e r a t o r in k defined by Fet = t\ a n d let E-oo,
if
/ £ 5,
E, E+tx
Fe+ao
= e_oo,
Fe-oo = e + 0 0 .
be t h e orthogonal projections of k onto C e
respectively. We denote by IB t h e Belavkin
algebra associated with (h, k)
IB = [ L G B(h ® k) I L u e _ 0 0 = L*ue+00 T h e m a p L 1-+ Lb where Lb = FL*F
= 0 for all u G h j
is an involution of IB.
Let us introduce t h e integrator processes in H, {A'm \ l,m € S} defined by A+°°(*) = AmW = AL^*) Al^W Al„(*) where A+,A, k).
A+( X ( o,t) ® K)),
if
rn G S, I = + 0 0
A(x(o,<)®|em)(e,|),
if
m, / G 5
if
Z G 5, m = - 0 0
if
^ = +oo, m = - o o
= A( X (o,t) ® (e/|), =
0
otherwise
A denote creation, annihilation and gauge operators in
T(L2(R+)®
Let D be a dense linear submanifold of h and let Ds, T>s and T>s be the
146
dense linear submanifolds of h ® k, H and H generated by { uei | u 6 D, I E S }, { ue(/) \u e DJ e Ms}, { ueie(f) \ueD,le S,f € Ms } respectively. We recall the definition of (Ds, Ms)- adapted process as a family { L(t) | t > 0 } of operators in Ti. satisfying the following conditions: the domain of L(t) contains T>s for all it > 0, the map t H» L(t)fe(/) is measurable for each f € Ds, f £ Ms, for each £ 6 D s , / £ Ms, t > 0 we have the tensor product factorization -- {L(t)fe( {L(t)(e(x J W ( / ) =-e(X(.,+oo)/)(0 ,<)/)} L(m
= 0,
I(t)ue_00e(/) = 0
for all u, v £ D, f, g G .Ms, ? G 5. With every L € I(Ds,Ms) it is possible to associate the family { L'm | l,m 6 § } of (Z), .A/f s)-adapted regular processes in W defined by { l > e ( » X ( i ) t t e ( / ) ) = ( W e / e ( S ),i(0ue m e(/)> . Moreover one can define the stochastic integral Ai(t) AIW ==
£
L'm{t)dKT(t) / ££ m(*)dAr(o Jo l,m£S l,me§
on the domain T>s as a (D,.Ms)-regular adapted process. The following fundamental formulas of Boson Fock space quantum stochastic calculus and results on q.s.d.e. can be proved as in [16], [18]. Proposition 2.1. Let L,M € 1(DS, Ms) and let X0, Y0 be operators on h with domain containing D. Consider the processes X(t) = X{t) = X0 + + Ax(t), Ai(t),
Y(t) = = Y0 ++
A\M(t).
For each v,u E D, f,g 6 Ms the following formulas hold: (ve(g),X(t)ue(f)) (ve(g),X(t)ue{f))
=
(ve(g),X (ve(g),X00ue(f)) ue(f))
+ ff (vie-n (v(e-oo + + g(s))e(g),L(s)u(f{s) g(s))e(g),L(s)u(f(s) + Jo
+ e+ao )e(f)) + +00)e(f)}
ds
(2.2)
147 (Y(t)ve(g),X(t)ue(f)) +
I
{
{{Y(S)
=
+ FM F
(Y0ve(g),X0ue(f))
^ >(e~°°
~ (YisMe-cc
+ 9(SM9),
+ g(s))e(g), X(s)u(f(s)
(X(s) + L(s))u(f(s) + e+00)e(f)) + e +00 )e(/)) 1 ds
(2.3)
Proposition 2.2. For eaci L £ JB there exists a unique (h, Ms)-regular adapted process X satisfying the q.s.d.e. dX(t) = X(t)d\i(t),
X(0) = I.
The process X is unitary if and only if L + Lb + LbL = Q,
L + Lb + LLb = 0.
For each t > 0 let 6t be the right shift and let pt be the time reversal on the interval [0,i] defined on L2(R+) ® k by
(*
0
tfs
(*/)(*)-|
/(l)
ifz>t.
Let St, 1Zt be the operators in T (L 2 (R+) ® k) defined respectively by the second quantizations of 0t, Pt S,e(f) = e(8tf),
Kte(f) = e(ptf)
for all / £ i 2 ( R + ) ® &■ The operators St, 8t are isometric and the operators 1Zt, Pt are unitary. Remark that, for all s,t > 0 we have SaSt = <Ss+i!
Kj'v, = J
For all s > 0 and all bounded operator X g S(W) the operator S„XS* maps ft ® T(i 2 (s, oo) ® A;) into itself. The canonical extension to ri will be denoted by S3XS*. Definition 2.3. An (h, jVf s)-reg-uiar adapted contractive process V in H is called a cocycle if, for all s,t > 0, we have V(t + s) =
V(s)SsV(t)Sl
A cocycle V is (Ds,-M°s)-weakly differentiable if for all u 6 D the function t i-» {ve(g),V(t)ue(f))
148 belongs to C 1 ( R + ; C ) for all v £ h and all functions Let us remark t h a t (Ds,Ms)-weak
g, f €
M°s.
differentiability for a contraction cocycle
V implies weak and hence strong continuity for b o t h V and V*. T h u s we can associate with V t h e strongly continuous contraction semigroup P (called also the reduced semigroup) on h denned by (v,P(t)u)
= (ve(0),V(t)ue(0)).
(2.4)
By a well-known fact in semigroup theory the infinitesimal generator G of P can be defined as the set of u G h such t h a t t h e limit Urn t " 1 (P(t)u <-*o+
- u)
exists in the strong or equivalently weak topology of h. Hence
(Ds,-M%)-weak
differentiability of V implies t h a t D is contained in the domain of G. T h e o r e m 2 . 4 . Let V be a (D s, M°s)-weakly G be respectively
the strongly
continuous
with V by (2.4) and its infinitesimal DS,G
dV(t)
= V{t)dAL(t),
e
associated
domain
containing
Ds such that
the
V(0) = / .
(2.5)
properties:
= 0, (w,Gv)
w
Let P and
| w G h, u G D, v 6 D(G) }
to DS,G and, for all u g D, v G D(G), Lwe-X
on h
in T>s
The operator L has the following i) L can be extended
cocycle.
semigroup
Let D$:G be the
L on h® k with domain
cocycle V satisfies the q.s.d.e.
ii) for all x £ Ds,G
generator.
= { we-oo + uf + ve+00
There exists an operator
differentiable
contraction
E+00L{ve+00
w G h, we have
+ uf) = 0
(2.6)
= (we-00,Lve+00)
(2.7)
have (Fx,Lx)
+ (Lx,Fx)
+ (FLFx,Lx)
< 0.
(2.8)
P r o o f . T h e first statement can be proved by the same technique of T h e o r e m p. 62 [3]. T h e operator L with domain Ds is defined by (2.6) and, for all w G h, u G D, g,f
G k,
(u-e-oo, Lue+00) (wg, Lue+00)
= lim t " 1 (ioe(0), (V(t)
- 7)ue(0)) = lim t'1
= lim i " 1 (w(e(gX(o,t))
~ e(0)), (V(t)
1
(iDe-oo.Lu/) = l i m * " (we(0),(V(t) (wg, Luf)
= lim f1
(w, (P(t)
-
I)u)
- J)ue(O))
- 7 ) u ( e ( / X ( 0 , ( ) ) - e(0)))
(to(c( W (o,t)) - e(0)), (V(t) - I)u(e(fX(0,t))
- e(0)))
149 Thus to prove i) it is sufficient to show that the right-hand side limit in the second formula exists for all u G D(G). This can be done by the same argument of the proof of Lemma 2.1 in [13]. To prove ii) consider v G D(G), u G D, f G M% and compute the norm t~\\V(t)(ve(0) + ue(f))\\2 using (2.3) which makes sense also for v G D(G) because of part i). Since V is a cocycle, the above function is decreasing. Therefore its derivative at t = 0 is negative and (2.8) follows. D Moreover, from [16] Propositions 4.3, 4.4, we have the following Proposition 2.5. The following facts hold: i) if D is a core for G, then V is the unique solution of (2.5), ii) ifV is isometric then, for all x,y G DS,G we have (Fy, Lx) + {Ly, Fx) + (FLFy, Lx) = 0.
(2.9)
Definition 2.6. Let P be a strongly continuous contraction semigroup on h and let G be the infinitesimal generator. Suppose that D is contained in the domain of G and is a core for G. An operator L on h® k with domain containing Ds satisfies the condition (A) if it has the properties i), ii) of Theorem 2.4. Clearly operators L satisfying the condition (A) are the natural candidates as infinitesimal generators of (Ds, -M^-weakly differentiable cocycles. Proposition 2.7. (Journe, Prop.l p.294) Let V = (V(t))t>0 the process V = (V(i))t>o defined by
v(t) = ntv*(t)nt
be a cocycle. Then
(2.io)
is a cocycle. The process V is called the duai cocycle associated with V. The following result follows from Theorem 2.4 and Proposition 2.7. Proposition 2.8. If a (Ds, Ms)-weakly differentiable cocycle V with associated operator L as in Theorem 2.4 satisfies the q.s.d.e. (2.5) in T>s and the dual cocycle V is (Ds,M%)-weakly differentiable then V satisfies the q.s.d.e. dV(t) = V(t)dkt(t),
V(C) = I
in T>s and its associated operator L satisfies the condition (y,Lx)
=
(FLFy,x).
(2.11)
150 for all x,y € Ds- In particular, Remark.
if L is bounded,
then L = L
It should be noted t h a t the (Ds, M°s)-weak\y
difFerentiabilty of V
and V implies t h a t t h e domains of G and G* have D as a common dense inter section. 3 . C h a r a c t e r i z a t i o n o f w e a k l y differentiable c o n t r a c t i v e c o c y c l e s In this section we give a general existence theorem for contractive solutions of q.s.d.e.
with unbounded coefficient.
This result will be used to construct
contractive cocycles with a given infinitesimal generator. P r o p o s i t i o n 3 . 1 . Let L £ IB and let V be the unique reguiar (h, M process satisfying
the
s)-&dapted
q.s.d.e. ( dV(t)
=
V(t)dAL(t) (3.1)
I V(Q) = I Tiie following
conditions
i) the process V is
are equivaient:
contractive,
ii) the following inequality
holds LF + FL* + LFL*
Hi) the following inequality
<0
(3.2)
holds FL + L*F + L*FL<0
Proof.
(3.3)
T h e existence of a unique regular (h, Ms)-adapted
(3.1) follows from Proposition 2.2. For all finite sets {UJ)J-=1 denote by ( t h e vector in ri given by X ^ _ i ujeUi)the q.s.d.e. dU(t) = dh.Li{t)U{t)
\W{tW
= Uf
process satisfying C h, ( / j ) ^ = 1 C
Ms
T h e adjoint process U satisfies
therefore by (2.3) we have
+ f (U(r) £ ^ ( e ^ + / i ( r ) ) e ( / i ) , Jo . (LF + FL* + LFL*)U(r)
£
„ ; ( « _ „ + / i ( r ) ) e ( / J - ) ) dr 3
Therefore condition ii) implies t h a t , for all t > 0, the operator U(t) and V(t)
are
contractions. Conversely, if condition i) holds by the cocycle property of V, we can show t h a t t h e function t -»
\\U(t)(\\2
151 is decreasing. Therefore, for all (fj)j=1 function fj, we have
- + mWfMFL f&MfiUFL ((XS'( E u i(ee-—
such that t is a continuity point for every
+ L*F L*F ++L*FL) L*FL)££ «,•(«,_. «,-(«,_.++ fj(i)Xfi)\ fjVMfi))
= £ai(Dff(t + W|f-- \mmf)
This shows that i) implies ii). The equivalence of i) and iii) can be shown in the same way considering the dual cocycle V. In fact the contractivity of V is equivalent to that of V. D In order to deal with q.s.d.e. with unbounded operators L on h ® k let us first prove the following technical lemma inspired by the regularity conditions of [6] Th. 4.4 for coefficients of classical s.d.e. and by [11] Th. 4. The parameter r\ will be useful to study isometry of cocycles in Section 5. Lemma 3.2. Let L e B satisfying the inequalities (3.2),(3.3). For all TJ £ [0,1] consider the operators on h®k /<"> = ft)
•qE-^ + +E + + r/E+ TJJS-O,, r,E+00,
DO!
L « = IWLIM £<»> ft) LI^ + (1 (i --
2 )E.0000LE LE+00 rir2) )E+x,
The operator L"' is an element of IB and satisfies the inequalities (3.2), (3.3). Proof. Clearly L ^ ' is an element of IB. The operators ft> and F commute. Moreover we have the following identities ELE+00 = 0, IMLI ft)LIMFE FE+ = 0, ooL00 E--oo — E-00xLE Ffti)L*ft) >L*/<"> = +00Fft +00L*E^ M M M M M = LEU LEV = LFL* = 0. Lft*>FI Lft^FI L* L* = = LEILEI FI EL* Fft >£X* = Hence we obtain the inequality M LMF F L
M 4. LL^FL^* + FL
= = M M = = I (LF (LF + + FIZ FL* + LFL*) IM + (1(1 --,»)£_ ^E-^LFooCLf + + F L 'FL^E^ LF//)/*"' Ji!-, < --(( l1 - rt^E-ULFL^E-o, ri^E-ooiLFL^E-*, --= - ( 1 -Vrj^E^LEL'E^ ) £ - .^LEL'E-n < 00 < =
This completes the proof.
-
(
!
■
□
Let L be an operator on h ® k satisfying the condition (A) and let G be the associated operator defined by (2.6). For all strictly positive integer n, let R(n; G) denote the bounded operator (nl - G ) _ 1 ([20] Ch.IX, Th.l p.240). It is to be noted that the adjoint operator G* is the infinitesimal generator of the dual contraction semigroup and we have R(n; G*) = R(n; G)*. Moreover we have the well-known relations for all w € h, v g D(G) ([20] Ch.IX, Th. p.246) nR(n; G)u> G)w = lim n.R(n; = w,
n—»oo n—^00
lim nGR(n; G)v = Gv. Gv.
n—*oo n—»oo
152 Proposition 3.3. Let L be an operator on h®k satisfying the condition (A). For each strictly positive integer n let In, Ln be the operators on h ® k with domain T>s defined by In = nR(n;G)E-00
+ E + nR(n-G)E+00,
Ln = I*LIn.
(3.4)
Then the operator Ln has a bounded extension which is an element of IB satisfying the inequalities (3.2) (3.3) and its uniform norm is estimated by ||L„||<2(n + 4 - A + l ) .
(3.5)
Moreover, for all £ G Ds, we have lim L„i = Li
(3.6)
n—*oo
Proof. The operator Ln satisfies the inequality (2.8). In fact it is sufficient to write (2.8) for vectors of the form Inx with x € Ds (since Inx £ DS,G by well-known propeties of the operators R(n; G)) and use the commutation of the operator F with the operators In and /*. To prove that Ln is bounded, because of (2.6), we need only to estimate the norm of Ln£ when f is a vector in h ® k of the form Ylj=i ui (fi + e+°o) with uj £ D, fj 6 Ms for all j . In this case we have ||£„e|| < \\ELnEZ\\ + WE^LnEZW + \\ELnE+aoZ\\
+ ||£_ 0 0 L 1 ,B f o o e||
(3.7)
The norm of E-ooLnE+oe can be estimated using (2.7). We have in fact, for each v,u £ D, (ve-oo, E-ooLnE+ooUe+oo) = n2 (R(n; G)v, GR(n; G)u) Using the identity GR(n; G) = nR(n; G) - I and the contractivity of nR(n; G) ([20] Ch.IX Th.l, Cor.2 p.240-241) we can estimate the norm of GR(n; G) by 2. Therefore we obtain the inequality | ( u e _ 0 0 , £ _ 0 0 L n £ + 0 0 u e + 0 0 ) | < 2n ||v|| ■ ||u|| which implies that \\E^LnE+00\\
< In.
(3.8)
Taking a vector i £ Ds in the range of E from the inequality (2.8) for Ln we obtain immediately the estimates \\E + ELnE\\
\\ELnE\\<2.
(3.9)
153 Consider now a vector x = ue+oc, yields
with u e D.
T h e inequality (2.8) for
2n 2 3fe (R{n; G)u, GR(n; G)u) + \\ELnE+00ue+00\\2
Ln
< 0
Hence, using again the equality GR(n; G) = nR(n; G) - I, a n d t h e contractivity of nR(n; G) we get t h e inequality \\ELnE+aoue+ao\\
<4n||ue+00||
which implies \\ELnE+ao\\ To estimate t h e n o r m of E-^L^E
< 2v^.
(3.10)
consider t h e inequality (2.8) for L„ with
vectors x of t h e form XQ + Ave+oo with XQ finite linear combination of vectors in Ds in t h e range of E, v £ D and A G R . Denote a = -23fe (ve^00,E^00LnE+oave+00) c = -23fe (x0,ELnEx0) b = 3fe {(ve_x,
- \\ELnEx0f
E-^LnExo)
\\ELnE+00ve+00\\2
-
= - \\{E + ELnE)x0f
+ ((E +
+ ||x 0 || 2
ELnE)x0,ELnE+cove+oo}}
From t h e inequality (2.8) for Ln it follows t h a t a and c are nonnegative. Moreover t h e estimates (3.8), (3.9) yield a<4n|M|2,
c<||x0||2
(3.11)
Formula (2.8) for Ln with x = Xo + Aue+oo gives us t h e inequality aA 2 - 26A + c > 0 for all A 6 R . This implies t h a t |6| < 2-^ac hence we get | 3 t e ( u e _ 0 0 , . E _ 0 0 L n . E x o ) | < 2^/ac + \((E +
ELnE)x0,ELnE+oove+oo)\
Therefore multiplying v by a complex number of modulus 1 such t h a t the scalar p r o d u c t in t h e right-hand side becomes real, from (3.9), (3.10), (3.11) we obtain t h e inequality | ( t , e _ o o , £ _ o o I n £ x 0 ) | < 6 V E | M | • ||co|| for all v 6 D which implies WE-^LnEW
< 6v^
Using (3.9), (3.10), (3.11), and the above inequality from (3.7) we obtain (3.5). This shows t h a t L„ is bounded.
T h e n Ln
e B because of (2.6).
Now (3.6)
154 can be proved by the same decomposition of Ln£ used to obtain (3.7) using the properties of the operators (R(n; G*))n>o, the inequality \\{ELnE+oa
- ELnE+00)ue+00\\2
= \\EL(nR{n; G)u -
u)e+00\\
< -29fe {nR(n; G)u - u, G(nR(n; G)u - u)) and the closedness of G.
□
The operators (Ln)n>0 will be used in the following proposition to find a se quence of cocycles approximating the cocycle generated by L by the equicontinuity method of [8]. Proposition 3.4. Let L be an operator on h®k with domain containing Ds such that there exists a sequence (£ n ) n >o of elements of IB satisfying the inequalities (3.2), (3.3) and the condition (3.6). For all n G N, r\ G [0,1] let LM
=lMLnI(-"\
LM = /("'i/("»
(3.12)
and let V„W be the unique reguiar (h, Ms)-adapted contractive process satisfying the q.s.d.e. dV^\t) = vM(t)dALM(t), V«(0) = / in h. The following facts hold: i) for all u G D, f G Ms, there exists a positive constant c(u, / ) such that, for all n G N and a/i 77 G [0,1] we have II (V<"\t) - V('»(i)) «e(/)|| 2 < c(u, / ) / ' (1 + ||/(r)|| 2 ) dr
(3.13)
Js
ii) for each 7? G [0,1] there exists a subsequence (Vn*')k>o and an (h, contractive process V^ such that
w-
lim V£\t) = VM(t)
for all t > 0. Moreover, for each u G D, f € Ms J (yW(t)
- V<">(s)) ue(/)|| 2 < c(u, f) f
Hi) for each J? G [0,1] t i e (h, Ms)-adapted q.s.d.e. in T>s dVM{t)
Ms)-adapted
we have the inequality (l + ||/( r )|| 2 ) dr
(3.14)
process V^l is regular and satisfies the
= VM(t)dALM(t),
V « ( 0 ) = I.
(3.15)
155 Proof. For all r\ £ [0,1], n £ N and each u e D, f e Ms by the Ito formula (2.3) , the Schwartz inequality and the contractivity of Vn we have \\(v<*Ht)-VW(s))ue(f)f< < exp(||/|| 2 ) J* (\Hf(r)
+ e + 0 0 )|| 2 + 2 | i « u ( / ( r ) +
e+oo)|Q
dr
A straightforward computation yields | 4 " ) U ( / ( r ) + e + 0 0 ) | 2 = \\ELnu(f(r)
+ rje+00)\\2 + \\E^Lnu{r,f{r)
+ e+ao)\\2
< 2 h | L n U e + 0 0 | | 2 + ^|/ ; (r)| 2 ||L„ U e i || 2 ) Therefore, denoting by c'(u, / ) the constant
we have | | i # > u ( / ( r ) + e + c o ) | 2 < 2c'(uJ)
( l + ||/(r)|| 2 )
and (3.13) follows taking c(u,f) = 5exp(||/|| 2 )c'(u,/). The inequality (3.13) implies that for each y 6 H , u 6 D, f G Ms, the sequence of functions on R+
is equicontinuous and equibounded. Hence by the Ascoli-Arzela theorem ([20] Th. p. 85) there exists a subsequence uniformly convergent on every bounded interval of R+. The diagonalisation argument and the separability of 7i allow to find a subsequence {Vnl )fc>o weakly convergent to an [h, A / fs)-adapted con tractive process VM. The inequality (3.14) follows from (3.13) and the lower semicontinuity of the norm with respect to the weak convergence. This implies, in particular, that V'*" is strongly continuous. Moreover, for all n 6 N, by (2.2) we have (ve(g),V^(t)ue(f)) + f
=
(ve(g),ue(f))
(«(c_oo + 9{r))t{g), V « ( r ) £ « U ( / ( r ) + e + 0 0 ) e ( / ) ) dr
Letting k tend to infinity and using the weak convergence of contractions Vnk (s) and the strong convergence of L„ku(f(r) + e + 0 0 ) to L^u(J(r) + e + 0 0 ) we obtain iii). D
156
Proposition 3.5. For each rj £ [0,1], the regular (h,Ms)-adapted contractive process V^ in Proposition 3.4 is the unique solution of the q.s.d.e. (3.15). The process V^ is a (Ds,Ms)-weakly differentiable cocycle. Moreover, for all t € R + , we have w- lim VM(t) = Vw(t) Tl —
l -
Proof. Uniqueness can be proved as in [16] Proposition 4.3. The cocycle prop erty for V ^ ' follows from uniqueness. Weak differentiability follows from (2.2). We can show as in the proof of Proposition (3.4) ii) that, for all sequence (T7„)„>O converging to 1, there exists a subsequence (?7n,);>o such that the limit w-
Urn Vr(«"',(*)
exists for all £ > 0 and is a regular (h, Als)-adapted contractive process satisfying (3.15) with 7] = 1. Therefore it coincides with V^(t) by the first statement.
□
Thus we have proved the following Theorem 3.6. Let L be an operator on h® k satisfying the condition (A). Then L is the infinitesimal generator of the unique (D5, M°s)-weakly differentiable cocycle satisfying the q.s.d.e. dV(t) = V(t)dAL(t),
V(0) = I.
(3.16)
4. Some results on the classical differential equation for a completely positive semigroup In this section we shall suppose that the operator L satisfies the condition (A). The following propositions give some results on the classical Feller-Kolmogorov equation for a completely positive semigroup we will need in the following. We are interested in the existence and properties of families T^ = (T^(t))t>0 of strongly continuous contractions on h satisfying one of the following equations: (v,T^\t)u)
= (v,u) + j
I
(v,TM{s)Gu} (4.2)
+ v2 (ELve+00,TM(s)ELue+00)
v,TM(t)u}
=
+ (Gv,T™(S)U}
1 ds
(P(t)v,P{t)u)
+ r)2 j
(4.3)
(ELP{t - s)ve+00, T^\s)ELP{t
- s)ut+00)
ds
for all u,v 6 D, T) € [0,1]. Let us recall that for contractions on h strong and weak continuity are equivalent. The following proposition shows that equation (4.3) is a weak form of equation (4.2).
157 Proposition 4.1. Let L satisfy the condition (A). Then, if TM satisfies (4.2), it also satisfies (4.3). Proof. Since D is a core for G and (2.8) holds, if TM satisfies (4.2) for each v,u € D then, it also satisfies (4.2) for each v,u in the domain of G. Moreover, for all such u, v, the function (v,T ( '»(t)u)
t i
is differentiable. Therefore, for every s e [0,<), and all u , » £ i ) w e have Ys(P{t
- s)v,TM(s)P(t
- s)u\ =
= r,2 (ELP(t
- s)ve+00,TM(s)ELP(t
-
s)ue+00}
Hence we obtain (4.3) integrating on [0,t]. D Proposition 4.2. Let L satisfy the condition (A). Then, for all rj G [0,1], there exists a strongly continuous family T*7'' of positive contractions on h satisfying the equation (4.3). Moreover, if r/i < 772 < 1, then, for all t > 0, we have T (m)( f )
< T^it).
(4.4)
Proof. We follow the iterative method for the construction of the minimal solution of a Feller-Kolmogorov equation exposed in [4]. Let us define a sequence T^ by
Ti"\t) = P(typ(t) (u,Tn'!?1(t)u)
= (P(t)u,P(t)u)
+ if f (ELP{t
(45)
- s)ue+oa,Tn"\s)ELP(t
- s)ue+00}
ds
where u,v € D. We will prove by induction that, for all n £ N, T„ is a strongly continuous family of positive contractions on h. Moreover, for each 77 £ [0,1], t > 0, the sequence of contractions {Tn (t))n>o is increasing. This fact clearly holds for n = 0. Suppose it holds also for an integer n. Because of condition (A) and the strong continuity of T„ ' the argument of the right-hand side integral in (4.5) is a continuous function. Therefore the integral makes sense. Moreover, using the inequality 0 < T„ ' < I and (2.8) we can majorize it by r/2 / | | i P ( t - s ) i t e + 0 0 | | 2 ds < -2r729te / (P(s)v,P(s)Gu) Jo Jo
= -Tj*J*±\\P(s)ufds = r,2 (IMI2 - \\P(t)uf)
ds
158 Hence the left-hand side of the above inequality can be majorized by
r,2M\2-(l-r,*)\\P(t)uf<\\uf This implies by polarization that the right-hand side of (4.5) defines a contraction r n + i M o n h- Clearly the family T ^ \ is strongly continuous. Moreover, since Tp\t) - T^%(t) > 0 for every t > 0, we have (u,(T«1(0-T(")(t))«) = - T(n*\(s)) ELP(t - s)ue+00)
= rf j * (ELP{t - s)ue+00, (T™(S)
>0
Finally, for each rj £ [0,1], t > 0 and each u £ D, let TM(t)
= sup T|')((). n
Since the sequence T ^ r ) converges strongly to T^(t) it follows that TM isfies (4.3). If 0 < r)i < T)i < 1 then we can show by induction that
for all n £ N and all t £ R + . Hence we obtain (4.4).
sat
□
Proposition 4.3. Let L satisfy the condition (A) and (B0). The following facts hold: i) Let 77 = 1. If T is a strongly continuous family of positive contractions solving (4.3), for all t > 0 we have Tw(t)
< T{t) < I
(4.6)
ii) For each t > 0 we have s - lim TM(t)
= T (1) (<).
Hi) For all r\ £ [0,1) T ^ is the unique solution of (4.3). Proof. For every rj £ [0,1] let S^ be another strongly continuous family of positive contractions solving (4.3) and let T„ be as in the proof of Proposition 4.2. It easily seen by induction that
r^oo < sM(t) < i for all n £ N and all t > 0. In particular, for rj = 1, i) follows.
159
Because of (4.4) the strong limit lim,_n- T^^t) exists, is less than Tw(t), defines a solution of (4.3) with 77 = 1. Therefore ii) follows from i).
and
If r) 6 [0,1) putting RW(t) = SM(t) - T<">(t) we obtain a strongly continuous family of positive contractions on h satisfying (u,RM(t)u}
=r]2 I (ELP(t
- s)ue+oa,R(i\s)ELP(t
- s)ue+00\
ds
(4.7)
Then from (4.4), (4.6) we obtain (u,RM(1)u)
< r,2 (||u|| 2 - \\P(t)u\\2) < rf \\uf
which implies that the positive operator R^v\t) is bounded from above by n2I. Using this fact to majorize the right-hand side of (4.7) by the same argument we can prove by induction on n the inequality 0 < R("\t) for all n G N. This proves that R^^t)
<:
2n V
I
= 0. D
The following condition is necessary in order T(t) = I for all t > 0 to be a solution of (4.2) and (4.3) Bo - for all v,u & D, we have 23fe (v, Gu) + (ELve+oo,ELue+0o)
=0
Corollary 4.4. Let L satisfy the conditions (A) and (B0) and let r) = 1. Then T(t) = I forallt>0 is the unique solution of (4.3) if and only ifTw(t) = I for allt>0. In this case the equation (4.2) has the unique positive solution T(t) = I for all t > 0. Moreover the equation (v,T(t)u\=
I
{{v,T(s)Gu)
+ {ELve+00,T(s)ELuc+oa)
+
(Gv,T(s)u)\ds (4.8)
has the unique positive solution T(t) = 0 for all t > 0. Proof.
In fact, if T satisfies (4.8), then (7 +T(f))(> 0 satisfies (4.2).
□
The same arguments of [4] Th. 3.4 (see also [9]) allow to prove the following theorem giving a necessary and sufficient condition for uniqueness of the solution of (4.3). This condition is the quantum analogue of Feller condition used in [16].
160 Theorem 4.5. (Chebotarev) Let L satisfy the conditions (A) and (Bo)- For each A > 0 let B* denote the set of positive operators X G B(h) such that (v, XGu) + (ELve+00, XELue+00)
+ (Gv, Xu) = A (v, Xu)
for all v, u G D. The following conditions are equivalent: i) the equation (4.3) has the only solution T(t) = I for all t > 0, ii) for each A > 0 we have B~^ = {0}.
5. Characterisation of isometry of cocycles satisfying a q.s.d.e. As a first step we will show that, under our assumptions, the vacuum expec tation of a contractive cocycle satisfying a q.s.d.e. coincides with the miminal solution of the associated Feller-Kolmogorov (or Lindblad) equation. In the fol lowing V^ will be denoted simply by V Proposition 5.1. Let L satisfy the conditions (A), (B0) and let T^1' be the minimal solution of (4.3), with r\ = 1 constructed in Proposition 4.2. For each u G D we have then (V(t)ue(0),V(t)ue(0))
= /u,Tw(t)u\
.
Proof. Since F™ satisfies the q.s.d.e. (3.15) in T>s, using (2.3) it easy to compute, for every 17 € [0,1], u G D, t > 0, s G [0,i), ^ (vM(s)P(t
- s)ue(0),VM(s)P{t
-
s)ue(0)\
and show that the bilinear form on h given by (v,u) i-> (V^(t)v,V^(t)u) defines a family of positive contractions on h satisfying the equation (4.3). Hence, from Proposition 4.3 iii) , for each T) G [0,1) and ( > 0 w e have: {V">(r) U e(0), V<">(<M0)) = /u,T^(t)u\
.
Because of (4.4), (4.6) and the lower semicontinuity of the norm with respect to weak convergence we have the inequalities |W"'(t) U e(0)| 2 < ||V(r)ue(0)|| 2 < liminf ||v<">(rW0)|| 2 for all IJ 6 [0,1], u G h, t > 0. Therefore, using Proposition 4.3 ii), we get / u , T W ( r ) u \ = lim
(u,TM(t)u\
= lim_ ||v<">(t)ue(0)|| =
(V(t)ue(0),V(t)ue(0))
161
This completes the proof. D To study the isometry of the cocycle V we need the following assumption on L strenghtening (B 0 ) which has been proved to be necessary in [16] Theorem 4.4. B - for each x, y 6 Ds we have (V, Lx) + (FLFy, x) + {FLFy, Lx) = 0.
(5.1)
We will consider also the following condition C - for all * > 0 we have Tw(t) = I. Proposition 5.2. Let L satisfy the conditions (A), (B). Let V be the unique regular (Ds, M°s)-weakly differentiable cocycle satisfying the q.s.d.e. (3.16). The following conditions are equivalent: i) for each v,u 6 D and each t >0 we have (V(t)ve(O), V(t)ue(0)) = (v, u) , ii) the process V is isometric. Proof. Clearly we need only to prove that the first condition implies the second one. By Proposition 5.1, condition i) implies that T^\t) = I for all t > 0. Hence, by Corollary 4.4, the equation (4.3) has only the solution T(t) = I for all t > 0 and the equation (4.8) has only the trivial positive solution zero. Using these facts, following the proof of Th. 2.4 in [9], we will prove by induction that for each TV £ N , n, m £ N with n + m = TV, v,u £ D, g,f 6 Ms we have (V(t)vg®m,V(t)uf®")
=
6m,n(v,u)(g,f)n
This clearly holds for TV = 0 because of condition i). Suppose it holds also for all integers not greater than a fixed TV > 0 and take n, m such that n + m = TV + 1 . Then by (5.1), writing the Ito formula (2.3) of Boson Fock space quantum stochastic calculus using /-fold tensor product vectors, we obtain the equation (V(t)vg®m,
= 6m,n (v, u) (g, / ) " + / ' { (V(*)vg*m,
V(t)uf®")
+ (V(s)ELve+aog®m,
V(s)ELue+00f®n)
+ (V(S)Gvg®m,
V(s)Guf®») V(s)uf®n)
} ds
Replacing / by f/(g, f) if (g, f) / 0, we obtain the strongly continuous family of positive contractions on h defined by (v,T(t)u)
=
(V(t)vg®m,V(t)uf®n)
which satisfies the equation (v,T(t)u}
=Sm>n{v,u)
+ I j (v, T(s)Gu) + (ELve+ocn T(s)ELue+00)
+ (Gv, T(s)u) 1 ds
Therefore, by Corollary 4.4, we have T(t) = 8mjnI. This completes the proof. The above results are summarized in the following
□
162
Theorem 5.3. Let L satisfy the conditions (A) and (B). There exists a unique regular (Ds, Ms)-weakly differentiable cocycle V satisfying the q.s.d.e. dV(t) = V(t)dAL(t),
V(0) = I.
The following conditions are equivalent: i) the process V is isometric, ii) for each v,u G D and each t > 0 we have {V(t)ve(0),V(t)ue(0)}
= (v,u) ,
Hi) condition (C) holds, iv) for each A > 0 we have B* = {0}. The characterisation of unitary cocycles which are {Ds,Ms) -weakly differ entiable toghether with their dual cocycle can be obtained from a dualisation argument with the help of the following proposition. We shall refer to condition (A) as to condition (A) with L, P, G replaced by L, P*, G* respectively. Proposition 5.4. Let L satisfy the conditions (A) and let L satisfy condi tion (A). Let V and Z be the unique (Ds, M°s)-weakly differentiable cocycles satisfying the q.s.d.e. dV(t) = V{t)dAL{t),
V(0) = I,
dZ(t) = Z(i)dA~(t),
Z(0) = I
in Vs in
Vs
Suppose that: i) the domain of the operator L* contains Ds, ii) we have L C (FL*F), Hi) for all u 6 D and all n > 1, R(n; G)u belongs to the domain of G* and the sequence (G*R(n; G)u) n >i converges. Then Z coincides with the dual cocycle VofV Proof. Remark that, because of conditions (A), (A), and assumptions i), ii) the operator G coincides with the adjoint G* of G. Consider the operators {Ln)n>0 defined in Proposition 3.3 and the contractive cocycles Vn which are the dual cocycles of Vn. These satisfy the q.s.d.e. dVn(t) = Vn(t)dAL>n(t),
Vn(0) = I.
We want to show that, under the conditions i), ii) the sequence (Lb) strongly to L on Ds- In fact, for all £ e Ds we have Lbni = ELbnEZ + E-«,LiE{
+ ELbnE+00£ +
= {ELETZ + nR(n; G*)FE+OQL*Et
converges
E_00LbnE+00^
+ {E^LnEf
FZ +
E^LbnE+00
163 Clearly the first two terms converge strongly when n tends to infinity. The vector ■E+oof can be written in the form ue+co with u € D, for all n, m 6 N, hence we have \\(E-00LbnE+00 - E^LlE+vc) i\\ = \\{Gn)*u (Gm)'u\\. Since (G„)*u = n2R(n; G*)G*R(n; G)u, the limit of the fourth term exists by condition iiij. The third term can be written in the form ELbnE+x,
= {E-^LnE)*
F = -(E
+ ELE)*
ELnE+00F
where the second identity follows from the equality (5.1) for the operator Ln. This shows that the sequence ( £ ' ) > 0 is strongly convergent. To show that the limit coincides with L on Ds consider y 6 Ds, x 6 D(L), we have then, because of condition ii) lim (y,Lbnx) = lim (LnFy,Fx)
= (LFy,Fx)
= (y,FL*Fx)
=
(y,Lx)
By Propositions 3.4, 3.5 the sequence Vn converges weakly to Z this implies that Z satisfies the condition Z(t) = 1ZtV(t)Tlt for all t > 0 and completes the proof. D
References [1] Accardi, L.: On the quantum Feynman-Kac formula. Rend. Sem. Ma.tema.tico e Fisico di Milano. Vol. XLVIII (1978). [2] Accardi, L., Frigerio A., Lewis J.T.: Quantum stochastic processes. Publ. R.I.M.S. Kyoto Univ., 18, 97-133 (1982). [3] Accardi, L., Journe, J.L., Lindsay, J.M.: On multidimensional Markovian cocycles. In: Accardi, L., von Waldenfels, W., (eds.) Quantum Probability and Applications IV. Proceedings Rome 1987 (Lect. Notes Math. Vol. 1396, pp. 59-67) Berlin Heidelberg New York, Springer, 1989. [4] Chebotarev, A.M.: The theory of the conservative dynamical semigroups and its applications. Preprint MIEM n.l. March 1990. [5] Chebotarev, A.M., Fagnola, F., Frigerio A.: Towards a stochastic Stone's the orem. Trento, Italia, Preprint (1990). [6] Da Prato, G.: Some results on linear stochastic evolution equations in Hilbert space by the semigroup method. Stoch. Anal. Appl., 1 57-88 (1983). [7] Fagnola, F.: On quantum stochastic differential equations with unbounded coefficients. Probab. Th. Rel. Fields, 86, 501-516 (1990).
164 [8] Fagnola, F.: P u r e birth and pure d e a t h processes as q u a n t u m flows in Fock space. Sa.nkb.ya., 5 3 , Series A (1991). [9] Fagnola, F.: Unitarity of solutions to q u a n t u m stochastic differential equations and conservativity of the associated q u a n t u m dynamical semigroup. Q u a n t u m Probability and Related Topics VII. (1991). [10] Fagnola, F., Mancino, M.: Free noise dilation of semigroups of countable s t a t e Markov processes. Q u a n t u m Probability and Related Topics VII. (1991). [11] Frigerio, A.: Positive contraction semigroups on B (7i) a n d q u a n t u m stochastic differential equations. In: Clement, P., Invernizzi, S., Mitidieri, E., Vrabie, I. (eds.)
Semigroup Theory and Applications.
Proceedings, Trieste 1987. pp.
175-188. New York and Basel: Marcel Dekker, Inc. 1989. [12] Hudson, R.L., Lindsay, J.M.: On characterizing q u a n t u m stochastic evolutions. Math.
Proc. Camb. Phil. Soc,
1 0 2 , 363-369 (1987).
[13] Journe, J.-L.: Structure des cocycles markoviens sur l'espace de Fock.
Probab.
Th. Rel. Fields 7 5 , 291-316 (1987). [14] Mancino M.: Q u a n t u m stochastic differential equations driven by free noises and dilation of Markovian semigroups with u n b o u n d e d generators. To appear in: Rend. Sem. Mat.
Univ.
Padova.
[15] Mohari, A.: Q u a n t u m stochastic differential equations with u n b o u n d e d coeffi cients and dilation of Feller's minimal solution. Preprint, I.S.I., Delhi Centre. To appear in
Sankya.
[16] Mohari, A., Parthasarathy, K.R.: A q u a n t u m probabilistic analogue of Feller's condition for t h e existence of unitary Markovian cocyclyes in Fock space. Pre print, I.S.I., Delhi Centre (1991). [17] Mohari, A., Sinha, K.B.: Q u a n t u m stochastic flows with infinite degrees of freedom and countable state Markov processes. Sankhya,
5 2 , Series A , 43-57
(1990). [18] Parthasarathy, K.R.: An Introduction
to Quantum
Stochastic
Calculus.
New
Delhi 1990, Birkhauser-Verlag, Basel (1992). [19] Parthasarathy, K.R.: Realisation of a class of Markov processes t h r o u g h unitary evolutions in Fock space. Preprint, I.S.I., Delhi Centre (1990). [20] Yosida, K.: Functional
Analysis.
delberg, New York 1978.
Fifth Edition. Springer-Verlag. Berlin, Hei
Quantum Probability and Related Topics Vol. VIII (pp. 165-188) ©1993 World Scientific Publishing Company
ON Q U A N T U M RESONANCE CLAUDIO FERNANDEZ AND ROLANDO REBOLLEDO
ABSTRACT. In a preceding research [4], [5], we characterized resonance for pure states by means of Shannon entropy of their spectral measures. In the present article we extent this investigation to general states p as considered in Quantum Probability [9]. Together with resonance phenomena we investigate the complementary behaviour, that is, the characterization of non resonant states. Namely, we establish that res onant states are not smooth in the sense of Kato [6] and derive main properties of smooth states.
1.
INTRODUCTION
Resonance phenomena appear in many fields in Physics. They are usually con sidered as unstable limit behaviour of a family of perturbations of a given system. Many attemps to give a precise formulation of Quantum Resonance have been done. One of the most common approach is via poles of a certain analytic continuation of the Hamiltonian resolvent (see [7]). Since there is no unique way to construct such an analytic continuation the latter approach has an intrinsic weakness. In this paper we use a phenomenological viewpoint (see the literature in Scattering Theory, for example [8]) and formulate the phenomenon in term of physical quantities that can be defined in a precise mathematical way. Two main physical facts have to be considered in this direction: when the system attains a resonant state it seems to stay there for a very long time (say, the sojourn time tends to infinity); furthermore, Key words and phrases. Entropy functionals, quantum probability, resonance, smooth operators. This research has been supported by DIUC Research Program "Analytic contributions to Math ematical Physics" and FONDECYT grants nr. 807-91 and 815-91.
166
a high concentration of the energy is observed. In our model this two facts are a consequence of the definition we provide for a resonant perturbation. In Quantum Mechanics a system is described by states and their evolution repre sented by a group of operators [1]. Our problem is then connected to the Theory of Perturbations of operators and also with Probability Theory. So that a natural framework to analyze this kind of phenomenon is Quantum Probability. That is the formalism used throughout this paper. For the sake of completeness we state below main notions and properties from Quantum Probability that will be used. For this we follow [9]. 1.1. Hilbert space notations. H will denote a complex Hilbert space. The cus tomary symbol (.,.) will be used for the scalar product on H. In what follows we use T* to mean the adjoint of an operator T. We will use some customary notations for some remarkable families of operators: B(H) is the algebra of all linear bounded operators on H; V{H) is the space of orthogonal projectors; K(H), the group of all unitary operators on H\ Ti{H) and Xoo (H) denote the trace class (or nuclear) and compact operators on H respectively. 1.2. Observables. Given a measurable space (0, ^F), an fi-observable is a mapping £ : T —> V(H) that satisfies f(f2) = / , where / denotes the identity in V(H) and for any countable collection of pairwise disjoint elements (Fi) in Tt it holds
£(U^) = £eW) i
(i.i)
i
In other words, we make no difference between observables and spectral (vector) measures associated to selfadjoint operators. Any selfadjoint operator X on H is identified with the observable f : B(R) —> V(H) defined by its spectral measure £ which satisfies the representation formula
167
X = jx£(dx)
(1.2)
This is simply Von Neumann's spectral theorem. The space of all selfadjoint operators (or observables) is denoted 0(H). 1.3. States. The trace of any nuclear operator T is given by oo
trT=^(en,ren),
(1.3)
n=0
where, (e„) is any orthonormal basis of the Hilbert space H. A positive operator p e Ti(H) of unit trace is called a state.Positivity means here that (u, pu) > 0 for all u e H. We let denote S(H) the convex set of all states on H. Any extreme point of S(H) is called a pure state. A state p is pure if and only if p = |<^)(v| for some unit vector ip in "H. The pure state |y)(v?| will be written as ip to simplify notations. For any observable £ its distribution in the state p is the probability measure /A over (H, T) defined by the expression 4(E) = trpt(E)
(1.4)
for each E 6 T, where "£r" means the trace. For a pure state p = |^)(v|, expression (1.4) reduces to M$(£) = < ¥ > , « £ »
(1-5)
The expectation of a selfadjoint operator X E Xi (H) or, similarly, t/ie expectation of the trace class observable f, is given by trpX Given any a-finite measure A on (Q, !F), a state p and an observable £, the measure / 4 can be decomposed in three parts by Lebesgue's Theorem, namely, a measure absolutely continuous with respect to A, another singular continuous and a third one atomic with respect to A. Fixing an observable f, this provides a decomposition
168 of S(H) the convex set of all states on H: we have three convex subsets <SQC(£) (respectively S3C($), respectively <Spp(£)) that correspond to all states p such that / J | is absolutely continuous with respect to A (resp. singular continuous, resp. atomic). Extremal points of each convex subset above can be identified with an element in H (pure states) thus leading to a decomposition of the latter in three closed subspaces 'tao'Xso'tpp •
1.4. T h e D y n a m i c s . In what follows the dynamics is described by a group (U{t); t e R) of unitary transformations or, equivalently, by the Hamiltonian H defined over a domain D(H) in H that generates the group. So that U(t) = exp(-itH).
The
spectral measure of H is our observable £. The space £1 is taken to be the real line endowed with Lebesgue measure. We refer to (H,p,t)
i-> 6t(H,p) = exp(-UH)pexp(itH)
from 0(H) x S(H) x R
into S(H) as the Quantum Flow. We write 6 = (6t\ t e R). The decomposition of the set of states introduced in 1.3 is then referred to H. So that we say the Hamiltonian is absolutely continuous if S(H) = <Soc(^), where f is the spectral measure of H. 2.
W E A K TOPOLOGY ON THE S E T O F STATES
We now turn to weak topologies. This material has been presented in [11]. It is well known that B(H) (respectively T\ (H)) is a Banach space with the customary norm || • || for bounded operators (resp. ||T||i = tr\T\, where \T\ = (T*T)?). The set X00(W) is a closed subspace of B(H) for the strong topology. 2.1. Definition. The weak topology over the set Zi('H) is the finest among all topologies for which all mappings T K-» trTX are continuous, for all X e I^CH).
Therefore,
a sequence (Tn) converges weakly to T in Xi (H) if and only if, \imtrTnX
= trTX,
(2.1)
169
for all X G 2„(W). We write Tn^T. We have the following main result: T h e o r e m 1. The set C of all positive nuclear operators p such that trp < 1 is a weakly compact convex set. Proof. The convexity property of C is inmediate; let us prove it is weakly compact. Since the Hilbert space is separable, the same property holds for X^H) in the strong topology of bounded operators. Thus, assume (Xk) to be a dense sequence in 2oo(W).
Consider (p„) to be any sequence of states. We will prove that it admits a weakly convergent subsequence whose limit belongs to S(H). Remark that for all n, k e N, \trpnXk\ < \\Xk\\. Therefore, there exists a subsequence (n^,) of indexes such that trp^Xo
(2.2) converges
to a limit X(X0) as n°m | oo. Let No denote the set of all indexes n°m. Since, \trP<X,\
< \\X%\\,
we obtain the existence of a sequence n „ of indexes in A^ such that trp^Xi
(2.3) converges
to X(Xi) as nlm \ oo over N0. We denote TVi the subset of all nlm e N0. By induction we construct a sequence (p,,*,)
sucn
that:
fr^-A(X,),
(2.4)
for all /
(2.5)
Prom (2.4) and (2.5) it follows both the linearity of X and its continuity on the set (Xk). Since this sequence is dense in l^H),
we can extend the functional A to
170
the whole space l^H)
in a continuous linear functional. By Schatten's Theorem [9]
(4.14), it follows that there exists a unique nuclear operator p such that trpX = X(X),
(2.6)
for all X e loo(H). From (2.5) it also follows that \trpX\ < \\X\\. To finish the proof, let us show first that p before is the weak limit of a convergent subsequence of (p n ). From this we will in particular derive that p is positive and trp < 1. Given any X e XooCW), we have: \trpX - trpn^X\
<
\trp(X-X,)\
+
\trpXi-trpnitX,\
+
\trpn*jX
-
X,)\.
Now if e > 0, we choose first I such that \\X — X j | < § and then take k > I and nj; such that \trpX — trpnk\ < | . This proves that pn*^+p. Finally, given any u £ H, we have (u,pu) = trp\u){u\ and since \u){u\ e Xoo(H), we obtain that {u, pu) = lim(u, pn(k)u),
(2.7)
therefore, (u,pu) > 0. Furthermore, notice that trp = T,'S=0trp\ei){ei\, where (d) is any orthonormal basis. Then, (2.7) and Fatou's Lemma imply trp
=
5Zliminfirp n ( A ) |e i )(e i | I
K
<
lim t inf^*rp n ( t ) |ej)(e i |
<
l.
i
Therefore p e C and the proof is complete.
□
171
In connection with the analysis of weak topology for states, we will introduce a concept that is important in its own because it deals with Information Theory. 2.2. Definition. Given a state p, and a finite dimensional subspace H' of H we let trn,p denote the trace trpP where P is the projection over H'. Now we call the Geometrical Information
Coefficient of order n (n > 1), the
expression F(p, n) = s\xp{trwp : dirnH1 = n}.
(2.8)
T(p,n) is bounded by 1, and gives an idea about how "spread" the state p is. In [13], Thirring uses this concept to introduce an ordering in the set of states, the ordering of "chaoticity". We prefer to call it "complexity": we say that a state p is more complex than another state p' if r(p,n)
(2.9)
for all n > 1. We use the symbol p' -< p to express this partial order. 2.3. Remark. For any n > 1, the mapping p t-► T(p, n) is lower semicontinuous in the weak topology, since it is the supremum of weak continuous functions. Notice T(-,n) can be extended to C for all integer n, in particular this procedure leads to a maximal element for the order ~<: 0 £ C will be more complex than any other element inC. Moreover, T(-,n) is convex for all n > 1. Given any convex combination p = api + (1 — a)p2, (0 < a < 1) of elements in C, it follows that r f o n ) < aT{pun)
+ (1 - a)T(p2,n).
(2.10)
Given any state p call C{p) = {f/ e C : p -< p'}.
(2.11)
172 This is a convex set and it is weakly closed because of the lower semicontinuity of I \ Since C(p) is included in C, we conclude it is a compact convex set from Theorem 1 even if limit points could have a trace less than 1. Notice that p is an extremal point of C(p) and any other extremal point will be of the form VpV, where V is an isometry . Indeed, given any other extremal point p\ we have that T(p\ n) — Y(p, n) for all n. By ordering eigenvalues Af of p by their magnitudes, we obtain F(j>, n) = A, + . . . + A £ , and similarly for p'. Then from the equivalence of geometrical information it follows Af = Af for all i. This means there exists an isometry V such that p' = VpV*. These are main properties of C(p) we will use in the sequel. However we finish this section by a criterion for weak compactness of a set of states that is interesting in its own. 2.4. Definition. A family K of states is tight if for any e there exists a finitedimensional subspace H( of H such that sup trpPt>
1-e,
(2.12)
peK.
where Pt is the projection over 7i€. Notice that K = {p} satisfies (2.12). Take any e > 0. Given an orthonormal basis (e n ), choose N(e) > 1 such that £
{en,pen)<e,
(2.13)
n>JV( £ )
then the space H€ spanned by the finite set ( e i , . . . ,eN(e)) makes the job for (2.12). T h e o r e m 2. A set K. of states is relatively weakly compact if and only if it is tight. Proof. Assume first that K is relatively weakly compact. Then for any projection P over a finite dimensional space the continuous mapping p H-> tr pP is uniformly bounded on fC, and (2.12) follows.
173 To prove the converse we use Theorem 1. K. is included in C and it is weakly relative compact. We only need to prove that limit points are states. Take a weakly convergent subsequence pn € K. and denote p its limit. Given any e > 0, the definition of weak convergence and (2.12) imply that tr pPt>\Since t is arbitrary it follows that tr p = 1.
e.
(2.14)
□
2.5. Definition. We say that a family K. of states is uniformly complex or chaotic if for all e > 0 there exists n(e) e N such that r(p,n(e))>l-e,
(2.15)
for all p e K.. We derive the following corollaries Corollary 1. A family K. of states is weakly relative compact if and only if it is uniformly
complex.
This is straightforward from the definition and Theorem 2. Corollary 2. Assume K to be a family of states that is less complex (or chaotic) than a given state p. Then K is relatively weakly compact. Proof. It suffices to remark that r(p,n)
(2.16) □
To use weak topologies in the analysis of asymptotic behaviour of both states and quantum dynamics, we need to work with couples (H,p) where H € 0(H) Hamiltonian and p € S(H).
is a
174 2.6. Definition. On the set 0{H) x S(H) we introduce the weakest topology for which the Quantum Flow 9t is weakly continuous for all t € R. For !simplicity, we also call it the weak topology on 0(H) x S(H) (abbreviated w) . Thus {Hn, p n )-^(//°°, p°°) if and only if for alU e R, we have tr exp{-itHn)pnexp(itHn)X
-* tr exp(-it//°°)p 0 0 exp(iti/ o t>)X,
(2.17)
for all X e loo(H).
3.
SOJOURN T I M E S
Our aim now is to analyze the time spent by a system in a given state. To do this we introduce the following definition: 3.1. Definition. The sojourn time of a positive observable X on a given state p is defined by r(p,X)= where pt = U(t)pU(-t),
TOO
/
J—oo
TOO
tr pXtdt = /
Xt = U(t)XU(-t),
J—oo
tr
ptXdt,
(3.1)
t e R.
Recall that tr pXt represents the mean or expectation of Xt in the state p. So that r(p, X) measures the mean value of the total time spent by X on the state p. The functional r(p,X)
takes values in [0,oo]. Moreover, (p, p') t-► r{p,p') is a
quadratic form on S(7i) and p — i ► Jr(p,p)
is convex. Denote r(p)= r(p,p) the
sojourn time of p in the state p. T h e o r e m 3. The mapping p i-+ r(p) is weakly upper semicontinuous. If the state p' is more complex than p, then r(p') < r(p). r(p) = sup
r(p').
Moreover, (3.2)
175 Proof. To prove the upper semicontinuity,take pn that converge weakly to p. Notice that for another fixed state p1, T(p,p') < liminf r ( p V ) ,
(3.3)
by Fatou's Lemma. So that r(-,p') is upper semicontinuous for all state p' and the property follows for T(-). To finish the proof, it suffices to remark that JT(-) is weakly upper semicontinuous and convex, then it attains maximum values on extremal points of the compact convex set C(p). As we discussed in Remark 2.3, p is extremal and all other extremal points are isometrically equivalent to p so they have the same sojourn time.
□
4. S P E C T R A L E N T R O P Y
We fix a Hamiltonian H with spectral measure £ and assume p € SaJ^).
Therefore
/i£ introduced in (1.4) is absolutely continuous with respect to Lebesgue measure. Denote pp(x) its Radon-Nikodym derivative. (To avoid complications we identify the class of functions pp with any representative in it). We will introduce a notion of spectral entropy for this state. 4 . 1 . Definitions. The spectral entropy of p is defined if pp(x) \ogpp(x) is integrable by the expression oo
/
pP(x)\ogpp(x)dx
(4.1)
-oo
where we adopt the convention OlogO = 0. For a pure state p = |^)(vl £ Sac we simply write pv the density P\v)\v\. Similarly we write r(
/
-OO
\(
(4.2)
176 Notice we have used the classical definition of Shannon entropy for a probability density function on the real line. It includes a dependence upon the group (U(t)) (or equivalently, on the Hamiltonian) describing the dynamics of the system. More over, the existence of the density pp and the integrability condition with respect to Lebesgue measure depend on the evolution of the state p under U(t). Furthermore, it is interesting to compare the entropy functional for two particular cases. Take first p(x) = 1/n if x e [0, n] and p(x) = 0 otherwise. In this case S{p) = logn and tends to oo as n grows. This is a "flat" density. If on the contrary we consider q(x) = n if a: € [0,1/n] and q(x) = 0 otherwise, then §(q) = — logn which tends to - c o as n grows. This density q approaches a "delta" function, that is it concentrates the probability in one point. It is natural to wonder if r{p) < oo and if §0fy) exists for a wide class of states p. The next proposition answers this question. P r o p o s i t i o n 1. The set DC(H) = {p e Sac '■ Pp has compact support} is dense in Sac-
Furthermore DC(H) is included in D(H), §(pp) is well defined and r(p) < oo, for allpeDc(H). Proof. First of all we notice that p i-> pD is linear from S^ into L ^ R ) . Therefore DC(H) is a convex set and its extremal points are pure states ip for which pv has compact support. Remark that given any p e S^, then pn = f ( [ - n , n ] ) p f ( [ - n , n ] ) / i r [£([-n,n])p] is a sequence in DC(H) that approaches p in norm as n —► oo. This proves the density of DC(H). Moreover, §(p p ) is obviously well defined if pp has compact support. Notice that given a pure state ip in DC(H) we can apply Plancherel's Theorem:
177 equation (4.2) is equivalent to T(
f°° J—oo
« \%{t)\2=
f°° / p^xfdx
J—oo
(4.3)
where p9 is the Fourier Transform of pv. Now, given any state p e DC(H) it can be written as a convex combination of pure states in DC(H): />=X^CilVi)(Vi|, i
(4.4)
where Cj > 0 for all i and 52i Ci = 1. Since \JT(-) is convex we then obtain from (4.3) T(J>)
and the proof is complete
< oo,
(4.5)
□
Call D3(H) the set of all p e S^ for which p^logp^, is integrable. This set is dense since DC(H) C
DS(H).
Proposition 2. Under the above hypotheses the following inequalities hold for any initial pure state
DS(H):
1 < r(
< r(
(4.6)
for all real number A. Proof. First of all consider two probability measures with densities p, q with respect to Lebesgue measure, and with finite entropy. By the elementary inequality log a; < x — 1, we have p
p
or, equivalently, 1 1 , 1 1 + log- < - + log-. p p q
178 Therefore, r°° S{p) < / J-oo
1 p(x)\0&-7-rdx q{x)
(4.7)
By choosing q in (4.7) as the density of a Gaussian distribution of mean m and variance a2, that is a(x) ? (
1
,
1 zxu( ~ ^ ^
(x m)2 - ) 2a» ]
we obtain S(p) < g log(27rc72) + - = log a + - C M & r ) + 1) = l o g ( a v / 2 ^ )
(4.8)
Thus we derive the conclusion that any density p with a first moment m and finite variance
J—oo
x2pv{x)dx
= \\Hipf,
(4.9)
Therefore,
S0v) < iog(||/M|V^)
(4.10)
As the reader can easily verify, inequality (4.8) is invariant by a change on the mean m. The same holds in (4.10) if H is replaced by the operator H — ml, where m is the mean of H in state \
(4.11)
But it is well known that the mean m gives the minimum value to the deviation \\(H - M)
179 Now consider the sojourn time associated to the state
/•OO
>
/
log
pip(x)fj,tp(dx)
J — oo
Then, logr(v>) > - S ( J V )
(4.12)
From (4.11) and (4.12) we finally derive the announced inequality: 1 < T(
< T(
D We remark that if pv is a Gaussian density with mean m = {ip, Hip) = A and variance a2 = \\(H — \)tp\\2, then the equality is attained in the second inequality in (4.6). Proposition before leads to our main "uncertainty principle" T h e o r e m 4 . For any state p G DS{H) and any real number A it holds 1 < r(p) exp(S(p)) < T(p)^2Tretrp(H-\I)2.
(4.13)
Proof. Assume p = J2i Ci\tpi)(ipi\ a convex combination of pure states in DS(H). Since the function x i-» —a: log a; is concave and p t-* pp is linear, it follows that p i-» S(pp) is concave. Therefore E o S & v . ) < Sfo,),
(4.14)
t
From (4.6) it follows that -Ecilogr(^)<E^(P^)-
(415)
180 Now since — \ logr(-) is a convex function, we get ( 4 - 16 )
- l o g r ( p ) < - J ^ c j log T(VO. i
after a multiplication by 2. Hence, from (4.14), (4.15), (4.16) it follows
l
(4.17)
To get the second part of inequality (4.13) we remark that if p e DC(H) then pp is a density that has moments of any order. The mean m is given by m = tr pH and the variance is a2 = tr p(H — ml)2.
By the same argument used to derive (4.11) we
get S(p„) < log(v/27re tr p(H - ml)2),
(4.18)
for all p e DC(H), but indeed (4.18) is trivially satisfied if tr p(H — ml)2 is infinite, thus it also holds in the more general case p € DS{H). 2
(4.18) by any A € R since tr p(H - ml)
Also m can be replaced in
< tr p(H - XI)2.
Prom (4.16) and (4.18) we obtain (4.13) and the proof is complete.
□
4.2. Energy width. Consider the family (qt; e > 0) of Cauchy densities on the real line, defined by
(419)
*M-h*h i e R and e > 0.
Assume p € S^. Denote m = tr pH We define the energy width AE(p) of p as the infimum over the set of all e > 0 such that oo
/ -oo
I
gt(x - m)pp{x)dx > —-
(4.20)
Z7TC
The next proposition gives the relationship between the energy width and the entropy of the state p.
181 P r o p o s i t i o n 3 . Under the above hypotheses, the following inequality holds: exp(S(p p )) < 7r[A£(p) + tr (p{H - mI)2)(AE(p))-1]
(4.21)
Proof. We use (4.7) applied to densities pp and qt(- — m) to derive °°
/
1 pP(x)log^ -dx, oo qt(x — m)
(4.22)
where m = tr pH. Now, by Jensen's inequality (used with the concave function log) we obtain oo
l
pp{x) log —
/
■oo Remark that
,oo
rdx < log /
qt(x — m) oo
/
-oo
l
pp{x) log —
J-ao
pp{x){x — rrifdx = tr p(H -
^dx.
(4.23)
qt[x — m) ml)2,
so that (4.21) follows if we put e = A(p) and we take exponentials in both sides of (4.22) and (4.23).
□
5.
RESONANCE
We are now in position to characterize resonance as a limit behaviour of the Quan tum Flow. 5.1. Definitions. A perturbation of the flow in (H°°,pco) n
n
sequence ((H ,p );
n e N) such that n
n
The perturbation (H ,p )
n
e 0{H) x S{H) is a
n
(H ,p )^(H°°,p°°).
is resonant if pn e Ds(Hn) limsupS(p£.) = - o o ,
and (5.1)
n
where p"„ is the spectral density of p" obtained with the spectral measure of the Hamiltonian Hn.
182 5.2. Remark. Given a perturbation (Hn,pn) of (H°°,p°°) it follows TOOCP00)
(5.2)
by a simple application of Fatou's Lemma. Theorem 5. / / ((//",/>"); n £ N) is a resonant perturbation of the flow, then liminf n T„(p") = oo. Proof. This theorem follows straightforward from (4.13) and resonance's definition.
Q
The theorem before says that a resonant perturbation "sojourns a long time" in a state with higly concentrated energy. Thus, our definition of resonance agrees with the phenomenological analysis in scattering theory (see [8]). As we refered in the introduction, this intrinsic characterization avoids ambiguities connected to definitions of resonance via poles commonly used in physics literature. The relationship between this approach and that of Lavine is illustrated by an application of (4.21) as follows. In general AE(p)
< \\(H — ml)
2.3), for a pure state p = \tp)(
is close to yjtr pn{Hn - mnI)2,
then (4.21) says that the exponential of n
the spectral entropy is dominated by AEn(p ).
In this case if the energy width goes
to zero, the state is resonant. And (4.13) gives a finer estimate of the convergence to +oo of sojourn times. In other cases it is not possible to compare explicitly Lavine's criterion on resonance and ours. Pure resonant states very often appear as "asymptotic eigenvectors" of a given sequence of Hamiltonians (//"). We say that
183 sequence (//") if there exists A € C, called asymptotic eigenvalue such that
lim8up||(/r-A/M| = 0.
(5.3)
n
Prom (4.6) it follows
Corollary 3. For any p which is a convex combinaton of pure states \ipi)(
We follow [6] to introduce the following concept connected with a Hamiltonian H. 6.1. Definition. A closed operator A on the Hilbert space H is H-smooth if for any
/
\\Aexp(-itH)
(6.1)
■oo
where c > 0 is a constant. Then the smallest constant c for which (6.1) holds is called the //-norm of A and is denoted \\A\\HWe say a perturbation ((//", pn) : n £ N) of (^ 00 ,p 00 ) is smooth if p°° is H°°smooth. 6.2. Remark. To check //-smoothness of a closed operator A it suffices to prove that condition (6.1) holds for all ip in a dense subset. Furthermore in [6] (or [12]) it is proven that A is //-smooth if and only if satisfies:
■ u p l ' ^ f
(6.2)
\b - a\
Proposition 4. The set £(//) of all H-smooth states is convex; extremal points of E(//) are H-smooth pure states and DC(H) C £(//)• Moreover, r(
184 7/ the Hamiltonian is absolutely continuous, then £ ( / / ) is dense in the set of all states in the sense of the strong topology of operators. Proof. The set of 77-smooth operators is a vector space and E ( # ) is the intersection of this space with the convex set S(H). This remark proves the first two statements. To prove DC(H) C E ( # ) it suffices to prove that pure states in DC{H) are smooth and for a general state the result will follow by convexity. Notice the numerator in (6.2) is equal to fi(p([a,b]) for a pure state A = p = \f){
/
\{ip,exp(-itH)
< c < oo.
(6.3)
■oo
The last statement follows from the density of DC(H) in .S^.
□
We then have the following main result on smooth perturbations: T h e o r e m 6. Suppose (Hn,pn)
is a perturbation of(H°°,p°°)
in 0(H) x S(H) The
perturbation is smooth if sup sup ^ . n
u
'
(6.4)
\0-a\
a
Proof. To simplify notations, we write nn (respectively p,°°) measure p,C (resp. p,iZ. By weak convergence we have fin(B) = tr fTC{B)
- tr p ° ° r (B) = /i°°(B),
(6.5)
for all borelian set B in the real line. Therefore, p,°° satisfies sup
a
fi°°([a, 61)
V \0-a\
I
< 0
°-
(6-6) v
'
185 Use then convexity of the set of all states: p°° is written as a convex combination c
Si i|Vi)(Vil of pure states. The same holds for measure p,°°: it is a convex combi nation Y,i CiH? of measures fif (■) = (
Therefore, p°° is also
D
Corollary 4. Given a perturbation (Hn,pn)
of (H°°,p°°) assume the following two
hypotheses hold: (1) pn e SadC)
for allneN
and
(2) The sequence of spectral densities (p!}n) is Lebesgue-uniformly integrable. Then the perturbation is smooth. Proof. Both hypotheses imply in particular that (p"„) is uniformly bounded in L 2 (R) and (6.4) holds.
□
6.3. R e m a r k . Smoothness avoids resonances in the case of pure states. Indeed, Proposition 4 and Remark 5.2 imply that a smooth perturbation {{Hn,pn)\n n
n
e N)
n
of {H°°,p°°) is not resonant if p = \
A P P L I C A T I O N TO TUNNELLING RESONANCE
In what follows, we shall deal with the tunnel effect, that is, the penetration of a quantum particle through a classicaly forbidden region. We consider this model as an example of the characterization of the resonance phenomena given in the previous sections. It is well known that a particle incident on a potential barrier from one side will tunnel through the other side and, as remarked in [14], [8], resonance will occur if the
186 particle encounters a second barrier and if it has a suitable energy. Moreover, the set of these resonant energies is discrete and it is contained in the absolutely continuous part of the spectrum. 7.1. Formulation of the model. Let a, b and h be real numbers such that h > 0 and 0 < a < b. For each positive integer n, let Vn(x) be a smooth function of compact support satisfying Vn(x) = 0, for \x\ < a and Vn(x) = h, for b < \x\ < b + n. Thus, Vn(x)
is a potential with two symmetric barriers of heigth h and width at least n.
As n converges to infinity, Vn(x) approaches to the potential V°°(x) which has two barriers of the same height but with an infinite length. For each n (including n = co), we denote Hn the selfadjoint realization of —j^ + Vn(x)
on the Hilbert space H = L 2 (R). For n finite, the operator Hn is unitarily
equivalent to the Laplacian — ^
in H and therefore it is absolutely continuous and
its spectrum consists of the interval [0, oof. On the other hand, the spectrum of H°° in [0, h] is discrete, while its absolutely continuous part is [h, oo[. Let ip be an eigenvector of H°°, with ||<^|| = 1. Take p = |v>)(¥>|, so that (Hn,p) is a perturbation of (H°°,p) by the construction of potentials made before. We claim that this perturbation is resonant in the sense of definition 5.1. From the eigenvalue equation, it follows that for \x\ > b,
— \\x\),
where A is the eigenvalues associated to
= - < / + V"V-A
=
{\T-V)tp.
187
It follows that
\\(H»-\)
2c*r{Vn-VY{
< 2c?h [
b+n Jb—
™
e-2VK=^(b+n)
By (4.6), this gives a n explicit u p p e r b o u n d on t h e entropy of t h e density p£ associated t o t h e p u r e s t a t e (p, with respect t o t h e Hamiltonian T h e o r e m 7.
The entropy
ofp"
m ) In particular,
the perturbation
Hn
verifies
$ kg [ — ^ j e - V ^ M ^ ] (Hn, |y>)(^>|) is
(7.1}
resonant.
Proof. It follows immediately from above calculations and (4.6).
Q
REFERENCES
1. Cohen-Tanoudji, C. - Diu, B. - Laloe, F. (1977) Quantum Mechanics, John Wiley and Sons. 2. DelIacherie,C.-Meyer,P.A. (1983) Probabilites et Potentiel. Theorie Discrete du Potentiel Chap. IX-XI. Hermann, Paris. 229p. 3. Fernandez, C.-Lavine, R. (1990) Lower Bounds for Resonance Widths in Potential and Obstacle Scattering. Comm. in Math. Phys. 128, 263-284. 4. Fernandez, C.-Rebolledo, R. (1991) Entropy functional and resonant pure states. Informe TScnico P.U.C.-F.M. 91/15. (To appear in Notas de la Sociedad de Matemdtica de Chile). 5. Fernandez, C.-Rebolledo, R. (1992) Sur l'entropie spectrale et la resonance quantique. C.R.Acad.Sci.Paris (to appear). 6. Kato, T. (1966) Perturbation Theory for Linear Operators. Springer-Verlag, N.Y. 7. Kukulin, V.I.-Krasnopol'sky, V.M.-Horacek, J. (1989) Theory of Resonances. Kluwer Ac.Publishers, 355p. 8. Lavine, R. (1978) Spectral density and Sojourn times, in Atomic Scattering Theory ed. J.Nutall, Univ. Western Ontario. 9. Parthasarathy, K.R. (1988) 7.5.7. Lectures on Quantum Stochastic Calculus. Indian Statistical Institute, Delhi Centre (mimeographed notes), 331 p. 10. Peierls, R. (1976) Surprises in Theoretical Physics. Princeton University Press. 11. ReboUedo, R. (1992) Complexity and Quantum Dynamics. Inf.T^cnico P.U.C./F.M.-92/1. (Lec tures delivered at CIMPA School on Dynamical Systems and Frustrated Systems). 12. Reed, M.^Simon, B. (1978) Analysis of Operators. Vol. IV, Acad.Press, N.Y., 396 p.
188 13. Thirring, W. (1983) A Course in Mathematical Physics 4- Quantum Mechanics of Large Systems.Springer-Verlag, New-York, Wien, 290 p. 14. Zakhariev, B.N.-Suzko, A.A. (1990) Direct and Inverse Problems. Potentials in Quantum Scat tering, Springer-Verlag, Berlin, 223 p. FACULTAD DE MATEMATICAS, PONTIPICIA UNIVEBSIDAD CAT6LICA DE CHILE, CASILLA 306,CORREO 22,SANTIAGO
Quantum Probability and Related Topics Vol. VIII (pp. 189-210) ©1993 World Scientific Publishing Company
QUANTUM STOCHASTIC INTEGRALS ON A GENERAL FOCK SPACE Wolfgang Freudenberg (Jena)
Abstract We introduce quantum stochastic integrals with respect to creation and annihilation operators on a general symmetric Fock space. Conditions for existence and boundedness of these integrals are given. It is shown that for local operators from a quasilocal algebra the quantum stochastic integrals exist. In the special case of non-anticipating kernel processes on an interval [0, T\ these operators coincide with the integrals introduced by Maassen.
1
Introduction
The aim of this paper is to discuss quantum stochastic integrals with respect to creation and annihilation operators on a general (symmetric) Fock space M. — L2(M, F) where M is the set of all finite counting measures on a Polish space G which is assumed to be equipped with a locally finite diffuse measure v. The <7-finite measure F restricted to configurations in a bounded area A 6 G is just the (non-normalized) Poisson measure with intensity measure v. We consider the densely defined operators Vc from M into M ® M resp. Sc from M ® M into M given on their domains by Dc*{
£,¥>-£) = £^2> (ui&p-'p)
S U(
(Vi,V2 ( v i , V 2Ge AM) f)
WeM). WeM).
In previous papers (cf. [1,2,3,5]) we considered operators of the type <SC(A® B)VC and applied them to the description and construction of locally normal states of boson systems. In the case that B is an integral operator and A = 1 is the identity the kernel of B is called by Maassen an integral kernel of Sc(l ® B)VC. For non-anticipating families (&t)tg[o.T] of such integral
190
kernels on the positive half-line in [11] there are defined quantum stochastic integrals with respect to creation and annihilation operators. Motivated by these definitions we introduce for a general family (Xt)t^G of operators on M and sets A € Q integrals / A X dA and / A dA* X by I XdA ¥(¥>) = / (XtVc*(6t,
-))(
(1)
and / dA* X*(
(2)
The aim of this paper is to discuss existence and possible domains of defini tion for operators of the type (1) and (2). For instance if the operators Xt are bounded and sup f e A \\Xt\\ < oo then / A X dA and / A dA* X are densely defined linear operators. Observe that we don't need any smoothness con ditions with respect to the family {Xt)teG except certain assumption of measurability. Further, we look for conditions ensuring that the integrals of operators of the form Sc(At® Bt)Vc exist and are again of the type SC(A®B)VC. We will see that non-anticipating processes Xt = Sc(l®Bt)Vc have quantum stochastic integrals of the same form as in [11] without the need of requiring the exis tence of kernels. We introduce families of so-called weakly non-anticipating operators for which still the quantum stochastic integrals exist (Proposition 4.4). These operators are combinations of local measurements with position measurements outside and seem to represent a reasonable class of integrands for quantum stochastic integrals. For any bounded area A G Q the operators from the local algebra A^ — C(M.\) ® 1AC where Ai\ is the Fock space over A can be written in the form X = Sc(l\c ® B)VC where B € C(M\) and l\c is the identity operator on configurations outside A. We will prove that for all A € B and families (Xt)t^G with the property that for all t € A the operator Xt belongs to the local algebra A\ and sup t e A \\Xt\\ < oo the quan tum stochastic integrals JXX dA andJ A dA* X exist (Theorem 3.7). So the results of the present paper may give a contribution to the study of quan tum stochastic integrals for operators on quasilocal algebras. An extensive study of such integrals finally could lead to a quantum stochastic calculus for infinite boson systems.That's why we also tried never to leave the general Fock space over G also when we touched problems of adapted processes. This caused to introduce a somewhat artificial notion of non-anticipating processes. In the present paper we consider only the operators J. X dA and J. dA* X.
191
But one should also define quantum stochastic integrals J.dAX, LX dA* and the integrals / A X dM and / A dj\f X where M is the number operator. However, especially the integrals / A dA X and J. X dA* seem to behave quite complicated. The assumption that the measure v on the phase space G has to be nonatomic is not necessary for most things in this paper (except the results and notions in section 4). So one could also treat lattice systems. However, in this case combinatorical problems lead to an unpleasant eyesore in the proofs. That's why we restricted ourselves here to the non-atomic case. As mentioned above in the case of non-anticipating kernel processes on the line the integrals (1) and (2) are the same as in [11]. Also in [7] one can find these integrals for special integrands. After having finished this paper we noticed a (still unpublished) paper by Lindsay [9] where quantum stochastic integrals are introduced in the same way as above. The intersection of the results in both papers is non-void but their union is much bigger than the contents of the single papers. So the papers may complement each other. 1 would like to thank K.-H. Fichtner for fruitful discussions and suggestions on the subject of this paper.
2
Basic Notions and Notations
2.1
The Symmetric Fock space
Let (G,p) be a complete separable metric space. By Q we denote the aalgebra of Borel subsets of G and by B the Borel sets bounded with respect to the metric p. Further, let v be an arbitrary locally finite diffuse measure on [G,G], i-e. v{B) < oo for B G B and u({x}) = 0 for x G G. G will rep resent the phase space of the considered quantum systems. In applications G usually will be an Euclidean space R.d,d > 1 and v the corresponding Lebesgue measure. Since we will deal only with the symmetric Fock space and to avoid using operators of symmetrizations we identify point configurations (xi, •■•,£„) from G with the counting measures tp = £ " = 1 SXj where Sx is the Dirac measure in x. Denote by MA the set of all finite configurations in A, i.e. the set of counting measures of the form
192
We equip MA with the cr-algebra FA generated by all sets of the type {
<* L { %h.LMp-h -
i 7 , A=XKW-
1
-^ n
-
r
nn
xr
v
n>\
A"
1> ' " * . , ) . (d[xi,---
■ ) XJ) nJ)
fV P (K
FA))
(3)
._.
where \Y denotes the indicator function of the set Y and 9 the empty configuration in MG, i.e. 9 £ MG,9(G) = 0. Let us remark that for A £ B the measure FA is a finite one, FA ( M A ) = exp{i/(A)} and F\(-)-exp{-i/(A)} is the Poisson point process with intensity measure v. Definition 2.1 Let A 6 Q. The Hilbert space M\ the (symmetric) Fock space over A.
=
£-2(MA, FA)
is called
In the case A = G we omit the index A in all of the notations above. Sometimes we also consider F \ to be a measure on [M, T\ concentrated on MA- A similar definition of the symmetric Fock space (in the case that G is an interval on the real axis and v the Lebesgue measure) one can find for instance in [6] and [11]. One easily checks (cf. [1, Remark 2.5] that Ms. is isomorphic to the usual definition of the Fock space in physics books.
2.2
Compound Malliavin Derivatives and Skorohod Inte grals
Definition 2.2 Let Vc : M —> M2 be given by VcV(
(y>i,2 € M,tf £
Dom(Vc))
where
Dom(Vc) := {* € M : j |tf(^)|VG>F(d^) < oo}. The operator Vc will be called compound Malliavin
derivative.
Definition 2.3 Let Sc : M2 —► M be given by
ScU(
U
(&V-
(f£M,U
£ Dom(Sc))
where Dom(Sc)
:= {U £ M2 : If
\U(yu
<SC wil be called compound Skorohod integral.
< oo}.
193
In the above definition
Dom(Vc)
For / £ L2(G,is) the function*/ : M - > C defined by
*/(v) ■= II A*) is called coherent function of f. Hereby x £
= exp{2||/|| 2 } < oo.
For coherent functions one gets the simple well-known relation D ^ y = tf, ® $ ,
and
<SC*/ ® *«, = * / + 9
(f,g €
L2{G,u)).
Definition 2.5 Let V : M —► L2(G,v) ® M be given by Dtf (x, ip) = * ( ^ +
(
where
Dom(V) := {* £ M : / [*(
(
£
Dom(S))
where Dom(S) := {U £ L2(G,u)®M: S is called Skorohod integral.
I I \U{x,ip)\\(.G)v(dx)F{dtp)
< oo}.
194
Using the isomorphism between the Fock space and Wiener space one can prove that V and S are isomorphic to the "usual" Malliavin derivative and Skorohod integral. Since we are interested in this paper only in operators on the Fock space we will not touch these problems here and refer e.g. to [8, 12, 13, 14, 15, 16, 17, 4]. Observe that V and S are "restrictions" of the operators Vc and Sc, i.e. we have that V restricted to Dom(Vc) is equal 1L2(G,L>)®M'DC and S can be identified with <SC1Z,2(G,I/)®A
O p e r a t o r s of t h e T y p e Sc{Ai
® A2)VC
For a set Y of configurations denote by Oy the operator of multiplication by the indicator function of Y, i.e. 0YV(ip)
■■=XY(
(*
€M,(f€M).
In [1, 2, 3, 5] we considered in detail operators of the type SC{A\ ® A-i)Dc and we discussed applications of these operators to the characterization of normal and locally normal states of boson systems (they are denoted in the above mentioned papers by S(Y,A)). Let us consider the special case that B £ C{M) and Y = M, i.e. Oy = 1. Provided SC{1® B)VC is a densely defined (usually unbounded) operator on M we get for each 9 from the domain of this operator and all ip G M
{S%1 ® B)V^)Cfi) - E ( 5 ( 2)C *)(^ - & •))(£)• If in addition B is an integral operator with kernel b we may continue
(Sc(l ® fi)Dc*)(v) = J^ J F{d(p)b{
= |W)^^.0%-H^-
(4)
Operators of this type represent the basic tool of Maassen's kernel calcu lus on the Fock space over R. In [11] the function b is called the integral kernel of the operator <SC(1® B)VC and in [11, 10] there are discussed quan tum stochastic integrals with respect to families of such integral kernels and applications to quantum stochastic differential equations and the character ization of quantum noises.
195
Starting with Sc(Oy ® B)VC erators B and operator from operators play ensuring SC(OY
a bounded operator B G C{M) and Y G T the operator usually will be unbounded. There even exist bounded op sets Y such that SC(OY ® B)VC does not lead to a linear M to M for any dense domain in M. Since this class of a crucial role in this paper we will give now some conditions ® B)VC to be bounded resp. densely defined.
Proposition 2.7 (i) Let A, A' eG, Be £(MA),Y X := SC{0Y
G-FA<, A n A' = 0. Then
® B)VC £ £ ( A W < ) C £ ( M ) ,
and one has \\X\\ < \\B\\. (ii) For all Yi,Y2 G T the operator Sc(Oyl ® Oy2)T>c is a densely defined operator. The proof of Proposition 2.7 (except the assertion ||A"[| < ||5||) one can find in [2, section 3]. We will show this inequality at the end of this section. Let us remark that each local operator in a quasilocal algebra is an operator of this type. To make this more precise, consider the quasilocal algebra A = U A 6 B ^ A w ^ e r e A\ = C(M\) ® OMKC- Locally normal states w over the quasilocal algebra are determined by the expectations u>(B) for all local operators B from some A\. But these operators can be written in the form SC(OMAC ® B)VC with B € C(M\). Denote by Mm the set of configurations having at most m points, i.e. Mm :- {
OM™BOM™}-
Proposition 2.8 Let B G Af .Then for each Y G T we have that Sc(0 ® B)VC is a densely defined operator on M. The set M* := U m e N - ^ m is contained in the maximal domain of definition of this operator. Observe that M* is already dense in M. Since in the proof we need some lemmas given below we postpone the proof of Proposition 2.8 to the end of this section. We still express the number operator, creation and annihilation operators in the form Sc(0 ® B)VC. Setting Mn := {
196
Proposition 2.9 [2, 3] (i) LetM :M —► M be defined by Sc(l ® 0Ml )VC with Dom(Af) := {* e M : f ip2(G)\V(p)\2F(d
< oo}.
A/" is
and A*j £ C(M) be the operator A*f9{
XMtitfffrMv)-
Then a*, : A4 —► .M gn'wen by
a}
:=Sc{l®A))Vc
with Dom(a*,) = Dom(V) — Dom{J\f^l2) is the creation operator corresponding to f, i.e.
«/*()=
lf(x)9(v-6x)
(in) Let f £ L2(G,v) and Aj £
f7iT}9(Sx)u(dx),
Then aj : M —► M given by af
:=Sc(l®Af)Vc
with Dom(af) — Dom(D) = Dom(Af1^2) corresponding to f, i.e.
is the annihilation
operator
ajV(
197
2.4
Some Useful Lemmas
At the end of this introductory section we collect some lemmas on integration in the Fock space which will be used very often in the sequel. For A e G and
—> C be a measurable function.
J / K
I h(x,
JM JA
JM
—> C be a measurable function.
f h{x,
(provided the integrals exist). L e m m a 2.13 Let A € G and let h : M —► C be a measurable Then JJh(
function.
(provided the integrals exist). The proof of these lemmas can be found in [1] and [2, sections 11 and 12]. Lemma 2.11 is also contained in [10].
198 Proof of Proposition 2.7. As mentioned above because of the results in [2, section 3] we only have to prove the relation \\X\\ < \\B\\. Observe that the assumptions B — OMA BOMA and Y C T^ imply that for all * <E M and
= =
/
JM
XY{
I XY(
< \\B\\2 I
I
\(BVc9(
XY(^)mv>i + ^2)\2F(d
JMAC JMA
<
\\B\\2 f
/
\9(
JMAC JMA
<
\\B\\2 f \Hf)?F{dv) JM lM
= \\B\\2\m2.
Consequently, we get ||X|| < \\B\.
O
Proof of Proposition 2.8. Let * be from Mf, B € A1. There exists an m £ N such that <J> € Mm and OMMBOM™ — B. Using several times Schwarz' inequality, Lemma 2.11 and Lemma 2.13 we get \\SC(OY
® 5)PCVI|2
= / 53 xy(v-^)(BP e *(v»-vi,-))(vi) XY
x
53
=
I 53
^
- V2){BVc^{V -
53
53
XY(
X (BVcV(
199
%h.LMp-h %h.LMp-h JJJJ^ ^ + V3)|(5P *( + , -))(Vi +
-
n
fV P FA)) xr 1> ' " ■ ) XJ) n J ) (K (3) i 7 , A=XKW* . , ) . (d[xi,--v ^ <* n L { n A" . _ . c 4 x (BV V(ipn>\+
<
n>\
VJ
-
i 7 , A=XKW-
-^ n
n>\
7,
i A=XKW-
<
2m
-
-
V3
Vl
2
3
n
xr
v
A" -
- ^ n C A"
n>\
1> ' " * . , ) . (d[xi,---
■ ) XJ) nJ)
fV P (K
FA))
(3)
._.
n
xr
v
1> ' " * . , ) . (d[xi,---
._.
■ ) XJ) nJ)
fV P (K
FA))
(3)
2
uH 2
< 2 2m ii5n 2 y y i * ( ^ + ^ i 2 ^ ^ , y > 2 ] ) =
2 2 m p | | 2 I \9{ip)\*7*WF(difi) < 2 3m ||5|| 2 ||*|| 2 < oo.
Hereby we used the fact that for * <E Mm we have VcV(
a
3
Stochastic Integrals on t h e Fock Space
3.1
Definitions
For a family (Xt)teG °f operators on M and A € Q we want to define stochastic integrals J A XdA and / A dA*X that can be interpreted as inte grals with respect to annihilation and creation operators. First we will give quite a formal definition assuming the existence of such integrals and by specialization to concrete cases we will motivate the definition. Definition 3.1 Let (Xt)t^o be a family of linear densely defined operators on A4. (Xt)teG l 5 called measurable if (i) D := flteG Dom(Xt)
is dense in M and
(ii) for all $ € Dand $ € M the mapping 11—► ($,Xtty) mapping from G into C.
is a measurable
Remark. It is easy to check that condition (ii) ensures that there exist measurable versions of the integrands in the integrals defined below. Espe cially, for all * € D and
200
t i—► Xt^if C.
— St) may be assumed to be measurable mappings from G to
Definition 3.2 Let X = (Xt)tea be a measurable family of operators de fined on a dense subset D of .M such that for a set A £ Q and for all W £ D [ (XtV9(t,-))(
(
The operator f. XdA from M into M defined on
( | XdA*)(p) := J (*|2>*(V))(VM*)
(5)
is called quantum stochastic integral of X with respect to annihilation. Definition 3.3 Let X = (Xt)tsG be a measurable family of operators de fined on a dense subset D of M such that for a set A £ Q and for all $ £ D
L
Xt9(tp - 6t)
defines an element of M. on D by
(
The operator JA dA*X from Ai into A\ defined
( / dA*X*){
j
Xt9(
(6)
is called quantum stochastic integral of X with respect to creation. The integrals in Definitions 3.2 and 3.3 are correctly denned. This can be seen by considering first step functions. Proposition 3.4 Let A £ Q, X a measurable family of operators on A\ and suppose JA XdA and J. dA*X are operators defined on dense domains D rep. D. Then JAXdA and JAdA*X are mutually disjoint, i.e. for all * £ D and $ £ D we have ( / XdA9,9)
= (9, I
dA*X$).
201
Proof. Using Lemma 2.12 we get
( / XdA *, $) = / ( I XdA * )(y>)$(^) F(d
JM
=
JA
I [ (XtV9(t, ■))&)*&) v(dt)F{dip) JM J A
= 11 {XtVnt,-)){v)*W)F{dv)v(dt) JA J M
= j (xtvv(t,-),*)v(dt) = j (v*(t,-),x;*)v(dt) JA
=
11
JAJM
JA
*(v> + Tt)Xt*{
= I [W)X:*(
=
j *{y) / X*t*{y - 6t)
JM
JA
JA
a 3.2
Domains of Definition
Suppose that for an $ € M and A € Q we have that J^XdAV € M. Then using several times Schwarz' inequality an easy calculation shows that || / XdA n2
< "(A) /
JA
/ \Xt9(
Analogously, if for $ € M and A e Q gets using Lemma 2.12 || f dA*Xn2< 7A
(7)
JM JA
I
JA dA*X * is square integrable one
[{
(8)
JM JA
Consequently, / A XdA is a well-defined operator if A € B and Dom( ( XdA) C {* € M : I JA
I |Jf«»(p + St)\2v(dt) F(dip) < oo}
JM JA
is dense in M. Analogously, one defines a maximal domain of definition for the integral L dA*X by the right-hand side of (8). If the operators Xt are bounded one can give more explicit domains of definition.
202
Proposition 3.5 Let X = {Xt)t£G be a measurable family of bounded oper ators on M. Then for all A € Q such that i/(A) < oo and sup <eA ||Xtj| < oo the operator J. XdA is a densely defined linear operator whose maximal domain of definition contains the set £ : = { * € M : I
(9)
JM
Observe that E is just the maximal domain of definition of the operator V\ being the restriction of V to A, i.e. V\ty(x,tp) — X A ( ^ ) * ( V + &x)- So one has the relation Dom(V) C Dom(V\) — E. If A/A denotes the number i ii operator on M\ we also have E = Dom{MK' ). Proof. Let 9 6 E. From the estimation (7) one gets using Lemma 2.12 || / XdA *|| 2 < i/(A) / / \Xt9(ip + St)\2u(dt) F(d
=
V(A)
j \\Xm{t,-)fv{dt)
J\
2
< KA) / ||JT«||2|]P*(t,.)||M*) J\
2
<
KA)sup||X t || / / \9(
<
KA) s u p | L Y t | | 2 / v (A)|»( ¥ .)| 2 F(d V )) teA JM
a Proposition 3.6 Let X = (Xt)t£G be a measurable family of bounded oper ators on M. Then for all A 6 £ such that i'(A) < oo and sup ( g A ||Xt|| < oo the operator J"A dA*X is a densely defined linear operator the maximal do main of which contains the set E. Thus we have that in the case u(A) < oo and sup t £ A ||X<|| < oo the operators JA dA*X and f. XdA could be defined on the same dense domain E given by (9). Proof. For
\\n2
[
JM
/(¥>(A) + l)|*«¥(¥>)lM«fcW¥>) J\
[ \\Xt\\2v{*)+ J\
I I v{k)\XtV{
(10)
203 The second term on the right-hand side of (10) can be estimated by
< = < <
[ /
[ \Xt*(
I
I
I \Xt*{v
I
I \\XtV9(s,
JA J M J \
J \ JM J \
^(A)sup||X t || 2 /
+
6s)\2v{ds)F{d
-)|| 2 v{ds)u{dt) < i/(A) sup \\Xt\\2 ( ||2?*(«, -)\\2u{ds) v'(A)|*(y>)|2JP(dv>)
(11)
Combining (10) and (11) we finally get the estimation || / d y l * X * | | 2 < K A ) s u p | | X t | | 2 ( | | * | | 2 + / ./A
t€A
^
v>(A)|¥(vO|2F(dv>)).
JM
'
0 Now, consider the local algebras ,4A = £ ( A 4 A ) ® !A C introduced in sec tion 1. Since each operator from A\ is of the form <SC(1A<: ® B)VC we have that ,4A Q A\ where AK := {X € C{M) : X = SC(0Y
® £ ) P C with F € .FA<=, 5 G
C(MA)}-
We have the following proposition: Theorem 3.7 Let A € B and let (Xt)teG be a measurable family such that Xt = SC(OY, ® Bt)Vc G A~K for all t € A and sup t G A ||J3 t || < oo. Then the operators J. dA*X and J . XdA are densely defined linear operators the maximal domain of which contain the set E. The proof of Theorem 3.7 is an immediate consequence of Propositions 2.7, 3.5 and 3.6. Observe that all of the above integrals could also be defined on the same (dense) domain Dom{V) of the MaUiavin derivative because Dom(T>) C E.
4
Integrals for Non-Anticipating Operators
In the sequel let XQ be an arbitrarily chosen fixed point from G. By Lx we denote the ball with center xo and radius p(x,xo), i.e. Lx :— {t € G : p(t,xo) < p(x,xo)}- Since the measure v was assumed to be diffuse we have
204
that F — a.a. configurations tp G M consist only of points with different distances from xo, i.e. F({
— p(s,xo)})
— 0.
Hereby t 6
u
with respect to xo,
dA* X * given by (3.3) defines an element
fI dA* dA* X X ** == SScc(l (l $$ B BXX)V )VCC W f
JLX
(12) (12)
with BxV(
II dA*BV(
JLXX JL
m a x , ifi
" m a x j >fi )
(ii) If s u p t € L j \\Bt\\ < oo then Bx is bounded and belongs to
(
205
(Hi) If Bx € A* then JL dA* X is a densely defined linear operator. (iv) If for all t £ A the operator Bt has a kernel bt then Bx has a kernel bx fulfilling ^ X (Vl,¥'2) = 6m aX ^i(¥'l-^m a x l¥ , 1 ,V2)XM Ll \{«}(Vl)
(
Proof. For all * G e M such that / ^ dA* XV is an element of M we have /
dA'XV(>p) dA*X9(
f
JL JLXX
Xt9(
JL JLXX
•L
E
JL
*
(BW(
£
0MLt(ip)(BtV^(
JLX
c LX \{e}(^) 5 max I tV *(ip
XM
E
=
■))(
tCv-St ifiCifi—St
-¥',-)(V,-*maxIv)
c 5 C (1S® (l®B Bxx)VccV(
= =
This proves (i). The corresponding representation (14) of the kernel bx in the case (iv) is immediately checked. Now assume s u p t € i \\Bt\\ < oo. We will prove that in this case the operator Bx is bounded. Using Lemma 2.12 we get for all $ € M
n2
X \\B \\Bxn2
= =
// JM
1 \{9)
\Bm!iXl^(ip-SmaXxV)\2F(dV)
JMLx Lx\{9)
= =
I1 I1
\B \BtHf t9(
St)\2ip(dt)F(dtp) 6t)\Mdt)F(df)
JMLl JLX
= = <
JMLX
1
JLX
1
JMLX JM JLXX LX JL
22 v{dt) 1i \*{
JM JMLx LX
JL JLXX
<
2
\Bt*{
22
v(L *,(!*) | f l tt|||| |||*|| | * | |22.. x) sup S u P | ||fl t€Lz f t€L,
The assertion (iii) is an immediate consequence of Proposition 2.8.
□
206
P r o p o s i t i o n 4.3 Let X = (Xt)t£G be non-anticipating and let x be from G.
with respect to XQ,
(i) For aUty € M such that JV X dA * given by (3.2) defines an element of M we have ( XdAV >LX it*
= SC(1®BX)VCV
(15)
with
Bxy = I BdAV
(16)
JLX (ii) / / s u p t g L | | 5 t | | < oo then Bx is densely defined and its maximal do main contains Dom(D\). (Hi) If Bx G A* then j
L
X dA is a densely defined linear operator.
(iv) If for all t G A the operator Bt has a kernel bt then Bx has a kernel bx fulfilling "max*
* m a x i ip?
)XMLX\M^)
(v> e M) (17)
Proof. First observe that for * G M and (f\,
(18)
(As an operator on A4 VcoV obviously is not defined for all 9 but e.g. on the dense domain M*). For all
(/ JLX
XdA9)(
= i £(*tP c (P*(V))(v-0,•))(£>(*) = I £(fl*z>(Pc*(p-<M)(*,-))(£M
(5tP(P c *(^-^-))(t,-))(^)Kd<)
= Sc(l® [ BdA\vcV(
207
The assertion (ii) is a consequence of Proposition 3.5, and (iii) follows from Proposition 2.8. We still have to prove (iv). Assume all operators Bt are integral operators with kernel bt. Then we get / JLxX Jh
BdA9(
= =
j/
J•f lax
(£ttP¥(V))(ViM
=
bt(
= JM JLXX
= =
/
/
JM JM JL JLxx
/
JM
bt(V\,V2-&t)y{
This proves the statement concerning the kernel of BBx.x.
D □
Remark. Quite formally (without taking care about domains of defini tions) one could illustrate the relation between both integrals in the following way: If Xt = Sc(l ® Bt)Vc then Xf = Sc(l ® 5 ? ) P C . Consequently, we have fL dA*X* = <SC(1 ® {B*)xX)V )VcC what implies because of Proposition 3.4 j L XdA = Sc(l
® ((B*)x)*)vc.
Combined with the relation
(¥,(fl*) x ¥) *) = < /
JLXX JL
BdA*,9)
this gives Bx = ((B*)x)*. From Proposition 4.2 we know that
(B*)x¥(v>) = \m(
208
integrals may have very strange domains of definition. On the other hand we saw in section 3 that for local operators from the quasilocal algebra A natural and weak conditions imply the existence of the stochastic integrals as densely defined operators. The operators Bt above are also local — they all belong to C{MLX). SO replacing 1 by 1(LX)C o r ®Yt w ' t n Yt £ -^(i,)' the quantum stochastic integrals will be well-behaved densely defined op erators (Theorem 3.7). But one could consider also operators of the form Xt = Sc{0Yt ® Bt)Vc with Yt e F(Lty and Bt € C(ML,)- This type of op erators representing a combination of non-anticipating local measurements Bt with position measurements Oyt outside Lt seems to be the appropriate class of operators for quantum stochastic calculus. Let us call families of such operators weakly non-anticipating. There is some hope that for weakly non-anticipating integrands one can derive handable ltd formulae. In any case finally we will see that the integrals for such families of operators exist. Proposition 4.4 Let (Xt)teG be a measurable weakly non-anticipating family of operators on M and assume that for an x € G we have s u p t e i i | | 5 ( | | < oo. Then JL X dA and L dA* X are densely defined op erators on M whose maximal domain of definition contains Dom{V\).
Proof. From Proposition 2.7 we know that Xt € C(M) and \\Xt\\ < \\Bt\\. Con sequently, we have s u p t € L i ||X t || < oo. The assertion follows now directly from Propositions 3.5 and 3.6. □
References [1] K.-H. Fichtner and W. Freudenberg. Point processes and the position distribution of infinite boson systems. J. Stat. Phys., 47, 959—978, 1987. [2] K.-H. Fichtner and W. Freudenberg. Characterization of states of infi nite boson systems I. On the construction of states of boson systems. Commun. Math. Phys., 137, 315—357, 1991. [3] K.-H. Fichtner and W. Freudenberg. Remarks on stochastic calculus on the Fock space. In L. Accardi, editor, Quantum Probability and Related Topics, pages 305 - 323, Singapore, New Jersey, London, Hong Kong, 1991. World Scientific Publishing Co.
209
[4] K.-H. Fichtner and G. Winkler. Generalized Brownian motion, point processes and stochastic calculus for random fields. Preprint, 1990. [5] W. Freudenberg. Characterization of states of infinite boson systems II. On the existence of the conditional reduced density matrix. Commun. Math. Phys., 137, 461—472, 1991. [6] A. Guichardet. Symmetric Hilbert Spaces and Related Topics, volume 231 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, Heidel berg, New York, 1972. [7] R.L. Hudson and K.R. Parthasarathy. Quantum Ito's formula and stochastic evolution. Commun. Math. Phys., 93, 301 - 323, 1984. [8] K. Ito. Multiple Wiener integrals. J. Math. Soc. Japan, 3, 157 - 169, 1951. [9] J.M. Lindsay. On generalised quantum stochastic integrals. 1991.
Preprint,
[10] M. Lindsay and H. Maassen. An integral kernal approach to noise. In L. Accardi and W. von Waldenfels, editors, Quantum Probability and Applications III, volume 1303 of Lecture Notes in Mathematics, pages 192 -208, Heidelberg, 1988. Springer-Verlag. [11] H. Maassen. Quantum Markov processes on Fock space described by integral kernels. In L. Accardi and W. von Waldenfels, editors, Quan tum Probability and Applications II, volume 1136 of Lecture Notes in Mathematics, pages 361 - 374, Berlin, Heidelberg, New York, 1985. Springer-Verlag. [12] D. Nualart and E. Pardoux. Stochastic calculus with anticipating inte grands. Prob. Theory Rel. Fields, 78, 535 -581, 1988. [13] D. Nualart and M. Zakai. Generalized stochastic integrals and the Malliavin calculus. Prob. Theory Rel. Fields, 73, 255 - 280, 1986. [14] A.V. Skorohod. On a generalization of a stochastic integral. Probab. Appi, 20, 219 - 233, 1975.
Theory
[15] D.W. Stroock. The Malliavin calculus. A functional analytic approach. J. Fund. Anal., 44, 212 - 257, 1980.
210
[16] M. Zakai. The Malliavin calculus. Ada Applicandae Mat., 3, 175 - 207, 1985. [17] M. Zakai. Malliavin derivatives and derivatives of functionals of the Wiener process with respect to a scale parameter. Ann. Probab., 13, 609 - 615,1985.
Wolfgang Freudenberg Friedrich-S chiller-Uni versi t at Mathematische Fakultat UHH, 17. OG D-O-6900 Jena Germany
Quantum Probability and Related Topics Vol. VIII (pp. 211-222) ©1993 World Scientific Publishing Company THE STATISTICAL INVARIANTS IN QUANTUM PROBABILITY AND DE FINETTI'S COHERENCE PRINCIPLE
Angelo Gilio, Universita' "La Sapienza", Roma.
A B S T R A C T . The concept of statistical
invariant,
years in the field of Quantum Probability, de Finetti's coherence case,
the
principle.
statistical
is examined from the
It is
invariants
conditions for conditional
introduced in the
argued
are
that,
nothing
probability
in
but
viewpoint
the
the
of
Kolmogorovian
particular
assessments:
recent
coherence
subjective
theory
provides the most suitable framework to study them with full generality. Then, using some results on conditional probabilities, a few examples of statistical invariants are examined.
1. INTRODUCTION In the last ten years, with the aim of explaining some classical paradoxes, &
new
approach
about
Probability, has See
also
Accardi
Probability are a) The
checking
the
foundations
of
quantum
theory,
the
been developed (see Accardi, 19S1, 1984, 1985, and
Fedullo,
1982).
Some
general
Quantum
1986,
points
in
1990.
Quantum
the following. of
the
predictions
of
the
theories
is
based
on
some
conditional probabilities derived from empirical data; b) If conditional probabilities
are
estimated
by
frequencies
in
mutually
incompatible experiments, then the application of classical probability to the
statistical
data
is
not
warranted.
Then
the
existence) of a Kolmogorovian probabilistic model for
existence
these
data
be
Invariants;
unplausible
properties need to be introduced to explain the difficulties
non
cannot
postulated, but must proved evaluating the so-called Statistical c) By adopting the Quantum Probability approach, no new
rules
(or
arising
physical in
the
interpretation of the experimental data. The fact that
one
cannot
postulate
the
existence
probabilistic model, when the data come from different claimed in a paper of van den Berg,
Hoekzema
and
of
a
Kolmogorovian
experiments,
Radder
(1990).
is
also
See
also
Scozzafava (1992) for a critical analysis of the (misleading) tendency to give frequencies the same "status" of probabilities. The aim of this paper is to show that the notion of
statistical
is naturally related to the concept of coherence introduced by de his subjective theory of probability (see de Finetti, 1974). known, using the approach allows
coherence to
principle
introduce
a
as
the
probability
unique P
on
As
axiom, arbitrary
invariant Finetti
it de
is
in well
Finetti's
families
of
212 (conditional
or
conditioning
unconditional)
events
have
events,
zero
comparison of de Finetti's and
even
probability
if
some
(see
Kolmogorovian
(or
approaches
conditional probability). Each time we estimate from
satisfied. This
condition
is
some
precisely
language is called
we
what
invariant
most general one to fully exploit aspect, in Sect. 2
numerical
give
a
this
(1990)
to
the
the
condition
in
the
all) for
notion
experimental
conditional probabilities, we need to check consistency of assessment. This process produces
possibly
Scozzafava
a of
data
some
probabilistic
which
Quantum
must
be
Probability
and the subjective framework is the concept.
concise
principle and of the betting and penalty
review
In of
order de
criteria.
Then,
relationship between the two concepts of coherence and
to
deepen
Finetti's in
this
coherence
Sect.
statistical
3,
the
invariant
is examined and, finally, in Sect. 4 some examples are considered.
2. COHERENCE PRINCIPLE, BETTING AND PENALTY CRITERIA The coherence principle of de Finetti can following
methods:
the
betting
application of these methods to
scheme the
be
or
case
of
based
the
on
one
penalty
conditional
studied by many authors: for the betting scheme see
of
the
two
criterion. events
Holzer,
The
has
(19B4),
been
(1985),
Regazzini (1985), (1987); for the penalty criterion see Silio, (1990), (1992). A concise review of both methods follows. C 2 . a 3 . P e n a l t y criterion. Denote by
Vi an arbitrary class of conditional
events and by P a real function defined on 9C. Then, given a family y =
CE |H 11 i
E | H , ...,E 2'
2
|H 5 £ 9C , consider
w i t h p. = P(E. | H . ) , i = l , . . . , n .
As i t
is well
F i n e t t i ' s p e n a l t y c r i t e r i o n , t o the p o i n t 3> i t
a
known,
the
i s associated a
2
basis
of
We can interpret geometrically the
values
of
same 5J
symbol.
using
the
atoms generated by the family {E , ..., E , H ,..., H } such that 1 n l n
H2 v — v H
, h = 1,2, ...,s , the generalized "
de
loss
concept
atom introduced in Gilio (1990). Denoting by C ,C ,...,
are obtained in K
n
2Hi(ErpJ-
We denote the events and their indicators by the
generalized
on
p ), 1
atom Q
=
C
hi
h2'
of
those ■= H v h i
(oi , a
h
n
C
oi ) ' hr,
from the C 's putting 1 , if C
■= E ^ H
0 , if C
£E..Hi ,
1=1,2,....n hi
.
(1)
h=l,2,...,s p., if C Then to each constituent C given by
£ H
c
there corresponds through <1) a value L
of
£
213
L
=
■! I".
h
1 = 1
P I = «4.
v
■
<2)
■*
In Gilio (1990) the following definition
and
results are
given . (2.1).
Definition.
The real function
P: 9C —» R
is a coherent conditional 3? s 9C , putting
probability on
SK if, for each n and for each finite family
J1 = (p , p ,
, p ) , with p. = P(E. | H ) , there does not exist a point P
such that L, < L, for each h and with
*
L
< L h
*
for at least
Dne
* P
index h
:
h
i. e., such that SJ < St , SI * St . Denoting, respectively, by 3 the convex hull of paints Q
<= R n , relative to h
the family 3? and paint P, and, for each J c; vl,2,...,n), by 3 the convex hull of generalized atoms relative to the family 5s" = {E |H : j <= J} and point J
P
J
= (p. : j e J ) , the following results can be proved. J
j
Theorem.
If
P
is coherent then, for each J a {1, 2, ..., n ) , P is
coherent. (2.3). Theorem.
(2.2).
If
P
is coherent, then
P <= 3 .
Note that P e 3 amounts to say that there exist s non-negative real X ,X ,...,X s , with
T Xh
1, such that P =
V.-4
(2.4). S X
h4
Theorem.
= 1 , such that P
s- positive
2 ' 2
numbers
A , A , — , A
, with
) X ft, then P is coherent.
Theorem. The p r o b a b i l i t y assessment
<= 3 f o r every J £ U , 2 , . . . , n ) . C 2 . b 3 . B e t t i n g scheme. For each integer
ElH,
h"
h4 h h
(2.5). P
Q
V-,-4.
If there exist
h
^ \
numbers
...,E
^
i s coherent if
if
n and f o r each family 3? = {E |H ,
|H 1 £ 9£ , given n a r b i t r a r y real numbers
n'
and only
r»
s , s , — , 5 1
2
n
, we
consider the random quantity S defined as n
B =
Es.H.IE-p.1, \.
L
L
\.
L= 1
which can be interpreted as the random gain corresponding to a combination
of
n bets on E |H ,...,E |H , with stakes s , ..., s . Puttina3 H
= H
o
1
s/H 2
v» . . . v Hn and denoting by GlH the restriction of G to °
H , we have the following Definition (see Holzer, 1984, 1985, Regazzini, 1985, o
1987) . i2.&)
Definition.
probability on family
The real function P: SK —* D? is a
9C if, for each n, for every s ^ ...,5^ and
for each
finite
? £ St , it is min GlH • max GlH
<
0 .
(3)
214 Denoting
1
H , .... H I
, C the atoms generated by the events E , — ,
by C , C , 2
a
E ,
I
and contained in H , the corresponding values o
n
n
g ,9 ,...,g 1
2
of
G
a
represent the possible values of G|H Then, condition (3) is equivalent to the following one. Min g • Max g < 0 . t-i n h h Notice that, using the generalized atoms, from (1) it follows g
= J] s. (01 -p.) ,
h
1,2,...,!=.
v = 1
It is equivalent
to use the penalty criterion or the betting
scheme
distinct proofs of this fact are given in Gilio, 1989a, 1990). As it
(two
is well
known, if the real function P defined on 5f is coherent, then P satisfies axiomatic properties of a conditional probability, while the converse
all
is in
general not true (see Gilio, Spezzaferri, 1992, for an example). Moreover
the
following Proposition can be proved (see Halzer, 1985, or Regazzini, 1985). (2.7).
Proposition.
¥
Given a family
r
=
-CE IH ,E |H ,...,E |H ) and
*
probability assessment JP
1I 1
n'
2 ' 2
a
n
(p ,p ,...,p I on 5 , consider an arbitrary class of
conditional events IK =2 5^. Then, there exists a (possibly not unique) coherent extension P of JP to 9C if and only if P is coherent. In particular it can be 9C %X $£ 1 where £ is the algebra generated by the events E1, ..., E n , H 1, ..., H n and X £ SK , with 0 e X, in which case P is a (coherent) conditional probability on 8 X X. We recall that, given a conditional probability
P
on
3
X
X
,
some
sufficient conditions for coherence of P are the following (see, respectively, Lehman,
1955, Regazzini,
1983, Holzer,
1984, Gilio,
1989b),
from
the
P
the
"strongest" to the "weakest": (a) X
'&
coaplete;
SVC0), in which case P is called
(b) X u C0) is a subalgebra of S; (c) X 15 an additive class; (d)
X is a P-quasi additive class. Finally, a necessary and sufficient condition for coherence
of
is
fallowing one (see Rigo, 1988). nPtEjHJ
=
nPlEjH.^J ,
l=>*
where E. •= 3, H. ■= X, v
t
E. £ H. H 1
(4)
i= l
L 1*1'
, i = l, 2,. .. ,n, (H '
'
'
'
n-H
H ). 1
Condition (4) has been introduced in a paper of Cs4szAr (1955), 3. STATISTICAL INVARIANTS AS COHERENCE CONDITIONS In Accardi (1986), p. 11, the notion intuitively defined in the following way:
of
statistical
invariant
215 "By analogy invariant"
uith
~ for
probabilistic
model-
realization admit
the
a given
geometrical set
a mathematical
is a necessary
a description
invariants,
of statistical constraint
and sufficient
within
the given
call
we
data,
uith
on the given
condition
for
probabilistic
of
describing
within
this
set
of
p
assessment &
P
whose
data
to examine
probabilistic
estimated
on
model
the basis
experimental data. But this problem can be looked on as that of existence of a probabilistic model
given
<s
of data,
to
model."
the given
conditional probabilities, which has been
to
set
Notice that in the above definition the actual problem is possibility
"statistical
a
respect
compatible
with
a
of
some
studying
the
given
probability
(p , p , ..., p ) , relative to a family of conditional
{E | H ,.. ., E |H 3. And, as shown by Proposition
assessment 3> can be described within a (possibly
(2.7),
the
not unique)
the some
events
probability
probabilistic
model P if and only if S> is coherent. Therefore the mathematical constraints corresponding to a given statistical invariant are nothing but some particular coherence conditions on the set of statistical data at hand. From this point of view, the most general statistical invariant for a set of statistical data is represented by the necessary and sufficient given in Theorem (2.5), while Theorems (2.3), necessary
condition and a sufficient
(2.4) provide
condition
respectively
one. Also notice that Theorem
be looked on as an algorithm providing a statistical invariant,
a
(2.4) can
corresponding
to a sufficient mathematical condition on the given set of data. Theorem (2.2) points
out
the obvious
statistical data can be described
within
circumstance a
that
probabilistic
if
a
model,
set of the same
happens for each subset of the given set. We observe that the subjective theory provides the mast
general
to study the notion of statistical invariant: in fact, within we can
directly
assign
(and check
coherence
of) a
set of
probabilities, even if some (or possibly all) conditioning
framework
this
approach, conditional
events
have
zero
probability. In other words, to investigate
the existence
of
a
probabilistic
describing a given set of statistical data, the subjective the compound
not necessarily require the use of
model
methodology
probability
theorem.
does In
Accardi (1985), p. 13, it is claimed that "The theory independent
of statistical
solution
model can be now decided statistical
invariants
of the problem: (at
least
provides
the existence in principle)
data."
It seems natural to conclude that:
for
the first
or not of uniquely
time a
in
a
model
Kolmogorovian terms
of
the
216 (i)
the study of statistical
invariants
can
be
naturally
embedded
in
de
Finetti's subjective theory; (ii)
the (more general) problem to investigate is: the existence or not
of
a
"de Finettian" model describing the given set of statistical data. In the next Section we apply the previous considerations to the analysis of some statistical invariants. 4. EXAMPLES OF STATISTICAL INVARIANTS (4.1).
experiment".
Feynman's "two-slit
existence of a Kolmogorovian model for
In
the
Accardi
(1985),
probabilities
p.
9,
P(A|C),
the
P(A|BC),
P(A|B=C) is studied, obtaining the following statistical invariant: P
0 <
<
,
(5)
P(A|BCC)
P(A|BC)
Apparently, (5) cannot be used if P(A|BC)
P(A|B°C). Moreover, what
about
the case in which the ratio is equal to 0 or 1 ? Using Theorem (2.5), with 9>
(A | C, A | BC, A|B C C), P=
C
q = P(A|BC) , r = P(A|B C> , 0 < p, q, r < 1,
in
(p,q,r), p = P(A|C),
the subjective
framework
condition (5) can be generalized to one of the following equivalent forms. Min Cq,r] < p < Max
In fact, for |J| a.
y
q| « |q
i"| ,
(6')
|P
r\
r\
(6">
< |q
1 , the condition
0 < p,q,r < 1 . For |J| CA|C, A|BCJ ;
(6)
|P
P
<= 3
. is trivially
satisfied
since
2, we have the three cases: b.
IT
{A | C, A|B e C) ;
{A|BC, A|B C C) ,
,_. F
and the generalized atoms are respectively: a.
(1,1) ,
(l,q) ,
(0,0),
b.
(1,1) ,
(l,r) ,
(0,0),
(0,q) ; (0,r) ;
c.
(l,r) ,
(q,l) ,
(0,r),
(q,0) .
Then, we can easily verify that in all cases it is P
<= 3
p, q, P € t O , n . Finally, for | J| = 3 we have f
, for all
values
j
-i
= F = {A | C, A | BC, A | B°C) and
the generalized atoms are the points (1,1,r) ,
(l,q,D ,
(0,0,r) ,
(0,q,0) .
Then, the condition ? s 3 amounts to the existence of a quantity
x e [0, ID ,
such that p = qx + r(l-x), i.e. to the three conditions (6), (6'), (6"). (4.2).
As another example to
show
that
de
Finetti's
approach
general than that of Kolmogorov, consider the family
is
more
ABIH, H l O ) .
3> = (p , p , p , p ), with P(A|H) ,
p 2 = P(B|H> ,
p 3 =P(AB|H) ,
p^ = P(H) ,
(7)
217 let us search a statistical invariant for data (7). If P(H)
0 the compound probability theorem cannot be used.
Yet,
in
the
subjective framework the following result can be proved. Proposition.
Provided 0 < p. < 1, i = 1,2,3,4, there exists a probabilistic
model compatible with the set of statistical data
(7)
if
and
only
if
the
following condition is satisfied: Max CO, p +p -1) < 1
p, <
Min {p , p 3 . r
3
2
L
Proof. Assume that 3> is coherent. Then J>,
(8)
*2
(p ,p ,p,) is coherent
3
too.
The
3
1 2
generalized atoms relative to 9> :A|H, B|H, AB|H3 and J> are the points (1, 1, 1) , Q (1, 0, 0) , Qj = (0, 1, 0) , (0, 0, 0) Applying Theorem (2.3) to P
it follows (8). a
Conversely, assume p ,p ,p ,p
> 0
and
Max {£>, p +p -1) < p
< Min ip , p }.
The generalized to 3? (1, atoms 1, 1, relative 1) , Q (1,and 0, J> 0,are 1) the points (0, 1, 0, 1) , Q Q, (0, 0, 0, 1) ,
= 1 and X.
P
Yx Q 1-
4
5
(p -p,)p , X
( P i -P 3 ) P i , x3
3P,
it can be verified that 3>
3
1 2
2
3
1
*4
4
4'
. Hence, from Theorem
, with X +X +\,+ 1 2 3 p (1-p -p +p ),
(2.4)
it
2
3 '
follows
that 3> is coherent. The particular assessments p _, = Max (0, p +p -13 or p , = Min ip , p 3 1 2 3 * 1 * *2 p 0 for some subscript h can 3 be obtained as limiting cases of the previous h one. Moreover, these limiting assessments are coherent since, in the finitely additive framework, coherence satisfy the closure property with respect to the limit. For example, letting p
tend to 0 , which implies X__ tend to 1 , we obtain
the limiting (admissible) assessment 3> = (p , p , p,, 0) . Therefore, conclude that, for all p
we
can
<= [0,13, the statistical invariant for data (7)
is
given by (8). (4.3).
Given two observables
X, Y
taking
only
two
(arbitrary)
Cx , x 3 , Cy , y 3, put (X=x ) = A, (Y=y ) = B . Moreover, l'
2
1
2
set of statistical
1
data
for
assume
values
that
the
1
the
family
¥ =
is
by 3> , where P(B A) P(B A ) (9) B~> = P(A(p±,B) P z , pJt p jP(A Notice that to (9) there correspond the transition probability matrices
given
1- Pl P(X/Y) =
^3
P(Y/X)
I P. We can prove the following result. Proposition.
Provided 0 £ p. 5 1, i = 1,2,3,4, there exists a probabilistic
218 model compatible with the set of statistical data
(9) if and only
if
the
following condition is satisfied:
p p^ U(1-pJd-pJ -pJd-pJ
= = Pi^P"
<10)
Proof. The generalized atoms relative to & and $> are the points Q i = (l,P2,l,Pi) , Q 2 = (Pi,t,0,pj , Qj = (0,p2,p3,l) , Q 4 = ( P l ,0,p r 0) . Using Theorem (2.4), each choice of four positive numbers X ^ X ^ X ^ X ^ , with X X
+ +X X + +X X., , i+X X -== 11, , generates generates a sset et 3 4. 2 3 4
i 1
p
X //<X ( X +X,1 , 1 1 3 3 '' 1 1
1 1
p = X / ( X +X ) , p 22 22 4 4 3 22
X X / ( X +X )> ,, 1 1 2 ' 1 1 2
p, 3
F r
Pp
4 4
= / ( X +X )> , = X , /<X 3 3 4 J J 4
wh:ich satisfies (10) admits a probab ilistic model. model (10) and, and. , , being being coherent, coherent , admits which a probabilistic by the the given by values for for p p , ,p p, 1.PP 3 i s given It can can be verified verif ied that the of values the range of is J are obtained obtained terval ars in closed interval C0,1] C0,1] (notice that the extremes of the interval ■
as limiting poin ts). points). kX ,, putt ing X. 3 = For each given set set of of statistical dat data p±,P ,p2,P iPjiP > °°, i putting X^ = kX^ a P 1 3'P. > such positive X. , 6 li-p ) /p , for = S\X , with k (1=6 k (1-p )/p , 6 (1-p )/p for all positive X , X such P )/P 2 2 . •
X 4.
4 that
2 (l+k)\ + (1+61X ( l + k ) X 1 + <1+<5)X
that
1
(i)
X . X i1 2
(ii)
ii tt ii ss
1
1
i «, 2 2 it can ke verified that: 1 1 , , i t can be v e r i f i e d t h a t :
22
'
g e n e r a t e tthe h e vvalues alues
p
generate = =
1 i
, p
2
1
11
22
llim im p p = 0 , , =0 X —>o X 1 —>o F i n a l l y , 1from X , / ( X , ' ' from X,/(X, 3 3 Finally, ' ' k
k
3 (1-p ) / p r
1
11
11 2
Z
3 J
3 ,
4 6
6
l '
(( ll -- pp3, ))//pp3.
2 1 1 2
llim im p p = = 1 1 .. X X2 —*o —*o 2 ) = kX / ( k X + 6 X ) , +X since 1 + SX2 ) , since + X4) = kX1/(kX
(1-p1 )/p1 ,
2
:
= p o rr X X //XX X /(X X )) = , ff o X X+ +X rp.,
X /(X +X ) <= X X +X = ((0,1) 0 , 1 ) , with
and with and with
1
1 1 (1-p )/p , (1-p2 )/p 2 , 2 2
2
J
< I - P 3 ) / P 3 ,, (1-p,)/p..
X2 /X1 2
J
1
3
3
<10) fallows \,/<\, + +X \ ) >= = p p if and and only only if if the the statistical statistical invariant invariant (10) follows X,/<X, if 3 4 4. ' satisfied. 3 satisfied. Notice that condition (10) can also be obtained as a particular case front Notice (10) can be obtained as (H a particular case from, = H (4) , p u tthat t i n g condition H = Bc .also H ) , = A C B= =A .
it it is is
(4) ,AB= putting E , E3 2
1
= A
,'
H
2
H = Ac , A HC B E
AB , 1 '
4
'
3 =
'
H
4
Bc , H, = A , H
2
3
B ,
B , (H5
4
AB = , E , AB , E ACB . Reaark. The condition (10) is trivial ly • =atis fied when 3 ' 4
E
2
E
'
i
H1 ) , E = A = B =
5
1
,
1
P(X/Y) = P(Y/X>. In
P = I - P Z = p3 i - p 4 P the standard case of symmett'ic particular forcondition Rexark. ,The (10) is trivially satisfied when P(X/Y) = P(Y/X>. In ! transition probabi1 particular, for p ities = 1-pis=obtained. p 1-p p the standard case of symmetric (4.4). two observables lue X, tal ing (arb itrar•y) va x ,..., x }, transition Given probabilities is obtained.
c y t(4.4). . y , . — ,y 5 consider the stochastic mat -ices P =
2'
* t '
where},
Cy ,y ,...,y }, consider stochastic Pi ==l ,(p P(Y=y |X=x.) p , theP(X=x |Y=y ) matrices = q , . . .) , t ,; Q d = (q l , . .,). ,, m .where 1
2
jt
is
I n G i l i o and S p e z z a f e r r i
is
coherent to
vj
m
P(Y=y|X=x.) p , P(X=x |Y=y ) = q , i=l,...,t; d= l,...,m. I n G i l i o J and a f e r r i (1992)L ' i t Ji s shown t h a t the assessment ( P , Q) ' tS p e z z Lj JL (1992)
it
i s shown t h a t
the assessment
(P, Q)
i f and o n l y f o r each n < min ( t , m) t h e f o l l o w i n g c o n d i t i o n ,
(4), is satisfied.
related
219 p.
where i
. q. . ■ 11 1 2
■ p.
. q. . n n
p. . q . ■ i j J i In n n
<= {l,...,t> , j
h-H J h
J
n 1
2J1
h h
J
(11)
l i
<= {1,. . ,m3 , k=l,...,n, and i *i , j ?=i k
Notice that (11) is satisfied if
t
m
and
h
p. M
k
h
for ks*h.
(symmetric
case).
Hence we obtain the following result. Proposition. x ,
Given two observables
, x ), Cy , y , — ,
2
Jtl
1
y )
X , Y
taking (arbitrary)
ix ,
values
if the set of statistical data
for X , Y
is
m
2
represented by two stochastic matrices
P = (p ) , Q = (q ) , with p <-J
q . <■]
ji-
JL
for each pair (i,j), then P and Q admit a probabilistic model. (4.5).
This example
(see
Accardi and Fedullo
refers to three observables X , Y , Z , <x,,i x 2 3 . ^ r y2'i <^zi, zzJ - We P"t A=(X
x ) , i
(1982) , Theorem 2, p. 165)
assuming
B=(Y = y ) ,
'
i
two
(arbitrary)
C=(Z
z > .
values
I
Assuming simmetry and denoting the transition probability matrices by
P(X/Y) with
( £ ^ ] , P(Y/Z, = (^ ^ ] , P(Z/X,
(£ 7 ) ,
0 < p,q,r < 1, in the quoted paper the following statistical
invariant
is obtained: |p + q - 1| < r < 1
|p - q| .
(12)
We will derive the same result examining the example in coherence. Using for convenience tabular
notations,
we
the
framework
have
the
following
events and respective probabilities: A|B
A|B C
B|A
B|A° 1
■ B|C
C|B
B|C C
C|B° I ,
A|C
C|A
A|C C
C|A C J
fp P
p
1-p
1-p "
q
q
1-q
1-q
[ r
r
1-r
1-r
The generalized atoms relative to 3? and P are the following points: 1
Q
=
1
i-p
1
1
1-q
1-q
1
1
1-r
1-r
p
0
1
q
q
0
r
0
1
■ 1
i-p , Q
=
1-p ' 0
. ««
1-r , •p
p
0
q
0
r
0
1
1-p
q
0
1
1-q
r
0
1
1-r
0
p
1-p
1
1
1
1-q
1-q
0
r
1-r
1
0
l
i-p
0
q
1-q
1
1
l
1-r
1-r
'
1-p , Q3
•
Q
a
3
•o
p
1-p
q
o
1
1-q
r
r
0
0
P
0
0 '
q
q
0
0
r
r
0
0
0
P
1-q
1
1-r
1
-
of
1 <
.
220 From Theorem (2.3) we have that, in order J> to be coherent, it is necessary e a that 3> e 3 , that is: J> \ X Q , with Sx 1 , X ,-•-» \ non-negative. h=l
h=l
Therefore, the following conditions must be satisfied X +X l
X +X ?
3 H
5
X +X <5
2
3
*
K
4
7
=
a
1
3
1
5
2
5
<S
=
p*(X +X +X +X >
=
q-(X +X +X +X >
8
5
q> (X +X,+X +X >
l 4
i
p-(X,+X +X +X )
a
X +X X +X
p- (X +X +X +X )
p- (X +X +X,+X ) F
2
7
1
d
7
2
B
5
<S
q- iX +X +X +X )
q- (X,+X +X +X )
!>• (X +X,+X +X ) i 3 = 7
r«(X +X +X 7 +X )
^
2
4
a
<5
1
(X +X +X +X ) a
2
4
=
B
<5
2
3
4
r- (X +X +X +X ) 5
7
<S
B
Then, recalling that 0 < p,q,r < 1, it follows (i)
X +X
= X +X 7
1 2
Hi)
X +X
(Hi)
X +X
a
,
X ,+X
X +X
3
X +X
4
5
X,+X
X +X
3
4
1 5
X +X
, X +X
a
p
7
,
q = 2(X +X ) ;
1
2
i
s
'
r = 2(X +X,)
X +X
2 4
(X +X ) ;
,
<S
l
3
Using the above relations we obtain the system: ' 2X +2X X,+X 3 4
= p ,
2X +2X
X +X
q ,
X +X
4
2
J
4
5
C
S
7
8
X
1 ,
> 0
we obtain
X
X +(r-q>/2 ,
= -X +q/2 , 5
X
X +(p-q)/2 ,
X,
5
2
3
-X +(l-p)/2 ,
(13)
8
Solving (13) with respect to p, q, r-, X
1
X,+X 3 7
X +X
X +X
X +X + X , + X +X +X +X +X 1
2X +2X
X
5
= -X +(l-p-r+q>/2 , 4
5
X +(p+r-l)/2 ,
-X +(l-r)/2 ,
and from the non-negativity of the X '= it follows the system (l-p-r)/2 < X (q-r)/2 < X
< q/2 ,
0 < X
< (l-p)/2 ,
< (l-p-r+q)/2 ,
(q-p)/2 < X
5
< (l-r)/2 , 5
which is equivalent to the statistical invariant (12). Notice that if we include in the family 5" the events ABC ,
ABC C ,
AB C C ,
A C BC ,
AB = C C ,
A C BC = ,
ACBCC ,
it can be verified that P(ABC)
X
, 1
P(ABC C ) = X
, 2
from which i t
f o l l o w s , f o r example: F'fAB ) = X,+X , P(A B) = X +X P(AB ) , 3 4 5 a ' and, in particular: P(A) P(B) PIC) X +X +X,+X 1/2 1 2
3
4
ACBCCC,
221 Then, using the additive property, we can extend the probability assessment y as a conditional probability P defined on 8 x 9f , where S is
the
algebra
generated by the events A, B, C and Sf = {A, B, C, A c , B c , C c , M
is
P-quasi
additive (see Gilio, 1989b). Therefore P is coherent and the assessment P,
as
a restriction of P, is coherent too. Notice that, putting C±=
(1,0,0) ,
the set of
points
Cz
(0,1,0) ,
(p,q,r)
of
the
C
= (0,0,1) ,
unitary
cube
C^ = (1,1,1) ,
satisfying
(12)
is
the
tetrahedron with vertices the points C 's and with faces the four triangles ¥
: C C C 1
;
¥
: C C C 2
12.3'
;
I , : CC,C 3
1 2 * '
|
I
1 3 *
: C C,C 2
*
Then, to the points (p,q,r) satisfying (12), such that
,
3*
0 < p,q,r < 1 , we
can add the limiting (coherent) points belonging to the segments: C C , CC, , CC, , C C , C C , C,C , i 3 '
12'
p 3
14'
3-i'
24'
characterized, respectively, by the conditions: r
0 , p+q = 1 ; P
l.q
q = 0 , p+r = 1 ;
= r;
q
p = 0 , q+r
1 , p = r ;
r =
1 ;
l , p = q .
5. CONCLUSIONS In this paper the notion of statistical invariant,
introduced
in
Quantum
Probability, has been examined and its relation to the coherence principle
of
subjective probability has been outlined. It has been shown that a statistical invariant is a condition of coherence that, when satisfied, allows to describe statistical data
within
a
probabilistic
model.
As
such,
statistical invariant can be developed with full generality in
the
notion
the
of
framework
of subjective theory of probability.
REFERENCES Accardi, L. (1981). Probabilita' Fasc. 4, 485-524.
e teoria
quantistica,
Physis,
Anno
Accardi, L. and Fedullo, A. (1982). On the statistical meaning numbers in Quantum Theory, Lettere al Nuovo Cimento 34, 161-172. Accardi, paradoxes.
L. (1984). The probabilistic roots of the quantum Diner et al. (eds) Dordrecht, K'luwer Acad. Publ.
XXIII
of
-
complex mechanical
Accardi, L. (1985). Hon-Kolmogorovian probabilistic models and Quantum Quad. 11. 8, Dip. di Matematica, II Univ. degli Studi di Roma.
Theory,
Accardi, L. (1986). Foundations of Quantum Mechanics: A Quantum Probabilistic Approach, Quad. n. 17, Dip. di Matematica, II Univ. degli Studi di Roma. Accardi, L. (1990). Quantum Probability and the Foundations of Quantum Theory, Centro Matematica V. Volterra, Quad. 11. 58, Univ. degli Studi di Roma II. van den Berg, H., Hoekzema, D., Radder H. (1990). Accardi on Quantum Theory and the "fifth axiom" of probability, Philosophy of Science 57, 149-157.
222 CsiszAr,A. (1955). Sur la structure des espaces de probability ficta Mathematica Academiae Scientiarum Hungaricae 6, 337-361. De Finetti, B. (1974),, Theory Gilio,
A.
(19B9a).
of Probability,
Probabilita'
conditionnelle,
John Wiley, New York.
condizionate
C -coerenti,
Rendiconti
di
Matematica, Serie VII , Vol. 9, Roma, 277-295. Gilio, A. (1989b). Classi quasi additive di eventi e coerenza di condizionate, Rendiconti dell'Istituto di Matematica dell'Univ. Vol. XXI, Fasc. I, 22-38. Gilio, A. (1990). Criterio di penalizzazione valutazione soggettiva della probabilita'. 3, Serie 7*, 645-660. Gilio, A. (1992). C -coherence
probabilita' Trieste,
di
e condizioni di coerenza nella Boll. Un. Mat. Ital., Vol. 4-B, n.
and extensions
of
conditional
probabilities,
Bayesian Statistics 4, eds. J. M. Bernardo, J. 0. Berger, A. P. Dawid F. M. Smith, Oxford Univ. Press, 1-8. Gilio, A. and Spezzaferri, F. (1992). Knowledge integration for probability assessments, Proc. of The 8th Conf. on Uncertainty in Intelligence, Stanford, California, July 17-19, 1992. In press.
and
A.
conditional Artificial
Holzer, S. (1984). Su.11a nozione di coerenza per le probabilita' subordinate. Rendiconti dell'Istiti_ito di Matematica dell 'Universita' di Trieste X V I , 46-62. Holzer, S. (1985). On coherence and Ital., Vol. IV, Serie VI, 441-460.
conditional
Lehman, R. S. (1955). On confirmation Symbolic Logic 20, 25 1-262. Regazzini, E. (1983). Sulle Editrice CLUEB, Bologila.
and rational
probabilita'
Regazzini, E. (1985). Finitely Mat. Fis. Milano, 55, 69-89. Regazzini, E. (1987). De Finetti's of Statistics 15, 845'-864.
additive
coerenti conditional
coherence
prevision,
Boll.
betting, nel
The
senso
di
Scozzafava, R. (1990) . Probabilita' condizionate: de Finetti Scritti in omaggio a 1.. Daboni, Ed. LINT, Trieste, 223-237.
inference,
o
of
Finetti.
Rend.
Rigo, P. (1988). Un teoreaa di estensione per probabilita' f in itamente additive, Atti della XXXIV Riunione Scientifica della Siena, 2-1, 27-34.
Mat.
Journal de
probabilities.
and statistical
Un.
Sem. Annals
condizionate S. I. S., Kolmogorov?
Scozzafava, R. (1992),. Observed frequencies are not probabilities, Atti XXXVI Riunione Scient ifica della S. I. S., Pescara, Vol. 2, 45-50.
della
Q u a n t u m Probability and Related Topics Vol. VIII (pp. 223-236) ©1993 World Scientific Publishing Company
Some Remarks on Trace of Operator Logarithm and Relative Entropy Pumio Hiai Department
of Mathematics,
Ibaraki
University
Mito, Ibaraki 310, Japan A b s t r a c t . In the setup of semifinite von Neumann algebras, we first present an additivity formula for the trace of operator logarithm, and second prove an inequality for the relative entropy under the hyperfiniteness assumption. Introduction The multiplicativity of matrix determinant implies the following trace formula for ma trix logarithm: Tr log XY
= Tr log X 1 / 2 Y X 1 ' 2 = Tr(log X + log Y)
(0.1)
for positive (definite) matrices X and Y When M is a finite von Neumann algebra with a faithful normal finite trace r, the (positive-valued) determinant A(x) of an invertible operator x in M was introduced in [7] by A ( z ) = expT(log|x|). Then it was proved among others in [7] that A(xy) = A(x)A(y) for any invertible x,y € M. This formula implies t h a t if x, y are positive invertible in M, then
A ^ 1 ' V 1 / 2 ) = Mv1"*1")'
= A(x)A(y),
that is, r(log x1/2yx1/2)
= r(log x + log y).
(0.2)
Let us note by the way that some expositions on determinant theories for infinite dimen sional operators are found in [10, 20]. In the first half (§1) of this paper we establish (0.1) and (0.2) in the setting of semifinite von Neumann algebras. More precisely, let M be a semifinite von Neumann algebra with a faithful normal semifinite trace r, and L1{M;T) be the noncommutative Z'-space associated with (M;T). We prove that if x and y are positive selfadjomt operators such that l o g i . l o g y e L1{M\T), then l o g i 1 / 2 ^ ! 1 / 2 G L1{M;T) and equality (0.2) holds. On t h e other hand, when X and Y are positive matrices, we proved in [15] the following logarithmic trace inequalities: -TTXlogYr/2XpYp/'2
< T r X ( l o g X + logY) < -TrXlogXp/2YrXp/2
(0.3)
224 for all p > 0. Umegaki's relative entropy [25] for nonnegative matrices X and Y is defined by S(X,Y)=TrX(logX-\ogY). Inequalities (0.3) give lower and upper bounds of S(X, Y) with Y being replaced by Y~l In particular when p = 1, the second inequality in (0.3) gives S(X,Y)
(0.4)
for every nonnegative matrices X and Y (the right-hand side is oo if the support of X is not dominated by that of Y). This was first proved in [14]. The right-hand side of (0.4) was introduced in [3] as a variant of the relative entropy (in a more general setting); so we write it as Sj}$(X,Y). In the latter half (§2) of this paper we prove S(x, y) < SBS(Z> 2/) ^ o r e v e r v nonnegative x,y € Ll{M; r ) when a semifinite von Neumann algebra M is hyperfinite (i.e. AFD). But we have to be careful in denning S&s{x,y). When x is dominated by y (i.e. x < \y for some A > 0), the operator xll2y~1x1l2 € M is given by x^y-'x1!2
^s-limx1'2^
+
1 1 2
£)-
x/,
and so 5 B S ( I , y) is denned by
SBS(x,y) = For general nonnegative x, y £ Ll(M\r) SBS(x,
rix^ilogx^y^x^x1'2). we can define
y) = l i m r ^ ^ l o g x1/2(y
+
ex)-lxll2)x112).
Furthermore we make clear the relation between the quantity Sft${x,y) a n d the relative operator entropy introduced in [9]. Although we confine ourselves to the semifinite case in this paper, the inequality for the relative entropy in §2 can be extended to more general cases. The relative entropy S(u, f) for positive normal functionals u> and ip of a general von Neumann algebra was introduced in [1, 2], and in [23] for positive functionals of an abstract *-algebra. Also the quantity 5 B S ( W I V ) w a s defined in [3] for positive functionals ui and
1. Formula for trace of o p e r a t o r l o g a r i t h m Let M be a semifinite von Neumann algebra on a Hilbert space H and r be a faithful normal semifinite trace on M. A densely defined closed opreator x affiliated with M is said to be r-measurable if there exists, for any e > 0, a. projection e in M. such that
225 eH C V(x) and r(e J -) < e where e-1 = 1 - e. We denote by M the set of all T-measurable operators affiliated with M, which becomes a complete Hausdorff topological *-algebra in the measure topology [17, 21]. For 1 < p < oo, the noncommutative L p -spaces L"{M;T) is defined by LP(M;T) = {x G M T(\X\P) < oo}, which becomes a Banach space with the i ^ - n o r m ||*|| p = T{\x\r)1l'' (see [4, 17, 19]). The trace r extends to a linear functional on
V-{M;T).
We denoteJ>y Mas. (resp. M+) the set of all selfadjoint (resp. positive selfadjoint) operators in M. For x G Msa. and a Borel subset E of R, let eE(x) denote the spectral projection of x corresponding to E. T h e generalized s-numbers /it(x) of x G M are defined by lit{x) = inf{s > 0 : T(e ( , | 0 0 ) (|a|)) < <}, < > 0. A complete exposition on generalized s-numbers is found in [6]. We also define the spectral scale Xt(x) of x G Msa. by \t(x)
= inf{s G R : r ( e ( , i 0 o ) ( « ) < <},
0 < < < r(l).
Properties of A ( (z) are analogous to those of fit{x) (see [18], [11, §6], [13, p. 5]). T h e main result of this section is t h e following: T h e o r e m 1.1. (1) If x,y are positive selfadjoint Ll(M;r), then log x^yx1'2 e L1{M\T) and
operators such that logx,logy €
r(log Z 1 ' V 1 / 2 ) = r(log z + logy).
(1.1) l
(2) If x, y are positive invertible operators in M such that log x, log y G L (M\ T), then logzy,\ogxxl2yxll2 £ L 1 ( X ; T )
and
r(log xy) = r(log xl/2yx1/2)
= r(log a; + log y).
(1.2)
Here let us note in (1) that the assumption logx,logy G Ll(M\ r ) contains e^0y(x) = {0}(2/) = 0- When z is a positive selfadjoint operator with e{0}(a:) = 0, we get l o g s € L1(M;T) if I — l , i _ 1 — 1 G i 1 ( A i ; r ) . Also note in (2) that logzy can be defined by power series or by analytic functional calculus becuase the spectrum of xy is included in [a, /?] for some 0 < a < 0 < oo. In particular when M = B(W) with the usual trace r = Tr, V-{M;T) is the space of all trace class operators on Ji. Hence we have: e
Corollary 1.2. If X,Y G B(W) are positive and log X, logY are in the trace class, then \ogXY and l o g X ^ F X 1 / 2 are in the trace class and TT log XY = Tr log X^^X1'2
= Tr(\og X + log Y).
The proof of Theorem 1.1 will be presented in the rest of this section. The next lemma follows from [7] as mentioned in Introduction (see also [5]).
226 L e m m a 1.3. Assume r ( l ) < oo. If x,y € M are positive invertible, (1.1) hold.
then
equality
Let us define for x, y e M+ and n e N xn = xeEn(x)
+ eEn(x)±,
Vn = yeE„(y) + eEn(y)±
where _ En=
ri " i -,——
\-n n + \i
U
\n +1
i n
(L3)
>
l ,n .
1
Then: L e m m a 1.4. If x,y are positive selfadjoint operators such that log x, log y € M, then x,y € A4 + , logz 1 '' 2 !/! 1 / 2 € .M, and logz„ ynxn converges in measure to logx^-^yx1/2 Proof. Since x = exp(logi) and z _ 1 = exp(—logs), we get x,x~l analogously y, !/ - 1 6 .A/i + . Since r e
( [ £ , n ] ( a 0 X ) = r(e(r.,cx>)(l log*D) ^ 0
<E M+ by [22] and
(« -» oo)
and
||(^ 2 - ^ ' • ^ t o H = ||(1 - I 1/2 )e ( ^,^ i) ( a: )|| < (
j
-1
-+0
(n-i-oo),
we have xj —> z 1 / 2 in measure and similarly y n —>)/ in measure, so that xn xll2yxll2 — 1 in measure. Therefore it follows from [22] that
o«g*y W ) + = ioZ((xwj>
-1)+ +1)
converges in measure to ( l o g x ^ V 1 / ^ = l o g U ^ V 1 / 2 - 1 ) + + 1). Moreover
(\0&xlJ*ynX^)_
= (logx-^y-xx-^)
+
converges in measure to (\o%xll2yx^)_
= (log*-1/V1*-1/,)+,
yn%n
— 1 —►
227 because a;"1 and y'1 are denned in (1.3) for z _ 1 and y'1 Hence we obtain the desired conclusions. I
in place of x and y, respectively.
Proof of Theorem 1.1. (1) Let xn and yn be as in (1.3) (note x, y e M+ by Lemma 1.4). Since e£„(x) = e [ l o g n ± i | l o g n ] ( | l o g a ; | ) and (logx)eEn(x)±,
logi-loga„ =
it follows that T(eEn(x)) < oo and T(logz n ) —* r ( l o g z ) . Analogously r(eEn(y)) •""(logj/n) -*■ r(logy). Now let en = xn = xeE„(x)
< oo and
eE„(x)VeEn(y),
+ (e„ - eEn(x)),
yn = yeEn(y)
+ (e„ -
eB„(y)).
Then r(e„) < oo and x„,y„ are positive invertible in enM.en. Hence Lemma 1.3 implies that T n{\°?,x\J2ynx\l2) = T„(log£„ + l o g yn) where T„ = r\enMen.
Since ,1/2
1/2 _ = 1 / 2 S
il/2+e-L
we get r(loga;n/2j,„x„/2) = r n ( l o 6 i y ^ n 5 i / 2 ) as well as r(loga; n ) = T„(logi„) and r(logj/„) = r n ( l o g y n ) . Therefore r ( l o g x + logy) = lim r ( l o g z „ + logj/ n ) = lim T(logz„ / 2 2/„x„ / 2 ). n-+oo
In the following let us prove that \o%x1l2yxll2
n-+oo
e LX{M\ r) and
r ( l o g I 1 / 2 y x 1 / 2 ) = lim T ( l o g z n / 2 y n z „ / 2 ) , n—*oo
which completes the proof of (1). The case r ( l ) < oo. For every 0 < t < T ( 1 ) we have by [11, §6]
A^**/W)=io g \ t (x n "y nXn i 2 )=io S M*y v*„/2) and by [6, Lemma 2.5]
M * i ' W ) =M*IW = Pt/ziZnht/iiyn)
< te/2(«n/2)^/2(^2))2 = A(/2(l„)A(/ 2 («/„),
228 so that \t(logxlJ2ynxlJ2)
< logA t / 2 (z„) + logA ( / 2 (j/„) = A ( / 2 (logx„) + A i / 2 ( l o g y n ) .
Also - A ( ( l o g ^ 2 s / n ^ 2 ) = AT(1)_t_o(-logxy22,n:l;y2) = AT(1)_(_0(logI-1/22/-1^1/2) < ^(r(l)-l)/2-o(log^1)
= -\r(l)+t)/2(logX„)
+-^(r(l)-l)/2-o(log2/n1)
- A( r ( 1 ) + i )/ 2 (logS/„).
Furthermore it is easy to see that |A((logx„)|<|A((logi)|,
0
The above estimates show that
|At(log*yV*y2)|
0<<
where /(<) = l A !/2( l o S x )l + l A (/2(log2/)| 4- | A ( r ( 1 ) + t ) / 2 ( l o g i ) | + | A ( T ( 1 ) + 0 / 2 ( l o g J/)|. Since by [6, Lemma 3.4] \t(\ogxlll>ynx\l2)
=
\ogMz1J2ynx1J2)
converges a.e. to
Miog^/V^HiogM^/V1/2) and / is integrable on ( 0 , r ( l ) ) (see [12, Proposition 1.1]), the Lebesuge convergence the orem shows that At(log e 1 / 3 ^ 1 ' 2 ) is integrable on ( 0 , r ( l ) ) , i.e. l o g ! 1 / 2 ! / ! 1 ' 2 € X.^A'fj-r), and that
Tflogz 1 'V) =
[r\oogz1l*yx1,t)dt Jo
= lim [Tl n ^°°jo
Xt(logx^ynxlJ^)dt
= lim r f l o g s y V " ^ ) . n—t-oo
TAe case r ( l ) = oo. Since r ( e ( 0 o o ) ( | l o g a i ' 2 j / n x V 2 | ) ) < oo, for every < > 0 we have by [13, p. 5] ^ ( ( l o g a i ' V * ; / * ^ ) = \t(Hx^ynx^) = l°gHt(xlJ2ynxlJ2)
= logA,^ <
2
^/
2
)
log[il/2(xn)n,/2(yn)
= l o g A ( / 2 ( i „ ) + logA 1/2 (j/„) = /*
229 and
2 1 = Moos**fi ((logx- ltlKW*M MOogzi'v*;/ *i /2 )-)-).) = 'v*n-1/2)+) x 2 Mi ((log x l v
t
< ^M
Mf/2((logj/)-). = Mt/2((logi)_) Mty2((log:t ) - ) + Mt/2((log3/)_). Furthermore ftt/2({\ogx)±) 1.4 and [6, Lemma 3.4]
and /x ( / 2 (log y)±) are integrable on (0, oo). Since by Lemma
l 22
2
Mt((log. t l v2/n*V M(l°g*V *V2)±) c n Vn
1 >, ((log las -> -* It, MQ°K* WU) W)±)
a.e. a.e.,
the Lebesgue convergence theorem shows that fit((\ogxll2yx1/2)±) (0,oo), hence l o g x 1 ' 2 ^ 1 ' 2 e L ^ X j r ) , and that
are integrable on
2 ^11 /22 ) ^r O l o og gx 1^' V / )
= /
7o
l
12 /*^(((log ( ( l ox'Vyx g ^ 'V). 1 / 2.)dt )^
) + )d<-^ 2/na; y »)+)* /i( ((iog :c y /:3, n:t y )_)d<} {J™ ^((iog^/ MVoz*HVJ M(i°g*n -)*} W)Jo
= j™ = lim o {^ n—►«
/•OO
2
2 ^W(((log* ( l o g * 1 '/V , , *)1+")
00
2
2
/•oo
00
2
2
2 2 = = lim lim TTU( ol og zgJ«, /W^4* /; /2 ))-. TI-+OO n—►oo
(2) When I , J £ M are positive invertible, since (xy)n = x1l2{x1l2yxll2)nx~ll2
for all
n € N , computation by power series implies that -1/2
l o gi yz ^=x x^ ^O^ ol oggz*^ll*yx V 1 l/l32)x) * - 1 / 2 log Hence (2) immediately follows from (1). I
Theorem 1.1 leads us to extend the determinant theory in [7] to the semifmite case. Let us set the domain ~D(A) by n d log logfx) L ^ M i rr ) } ,, V(A) = = {{xEM: \x\ €6l}(M; 2>(A) z e . M : e {e{0} (M) = = e { o}(KI) 0 } ( K I ) = 0 aand 0 }(\x\) and define the (positive-valued) determinant A(x) of x € X>(A) by A(x) == exp T(log|l|). r(log | i | ) . A(x)Then we have:
230 P r o p o s i t i o n 1.5. (1) A ( u x z u 2 ) = A(z) for every z e V(A) and unitaries ultu2 (2) A(z*) = A(z) for every x £ V{A). (3) A ^ i - 1 ) = A ( z ) " 1 for every z G D(A). (4) Ifx,y€ V(A), then zy <= X»(A) and A(zy) = A(x)A(y). (5) A(z) > A(?/) for every i , j £ £>(A) such that x > y > 0.
e
M.
Proof. (1) is easily seen. For z , j / e £>(A) let x = v\x\ and y = w\y\ be the polar decompositions of x and y, respectively. Then v and w are unitaries in A4. Since x* = \x\v' and z _ 1 = | z | _ 1 u * , (2) and (3) follow from (1). It is immediate that e{ 0 }(|ij/|) = e{0}(|
A(xy)2
and
= expr(log|z2/| 2 ) = e x p r ( l o g | z | 2 + l o g | y | 2 ) = A(z) 2 A(j/) 2 .
Hence (4) is proved. For x,y € V(A) with * > y > 0, since x~ll2yx~xl2 < 1, it follows from (3) and (4) that A ( z ) - 1 A(j/) = A ^ - 1 / 2 ^ " 1 / 2 ) < 1, implying (5). I
2. Inequality for relative entropy As in §1 let M be a semifinite von Neumann algebra with a faithful normal semifinite trace r. L e m m a 2 . 1 . Let x,y € M+. If x < \y for some X > 0, then there exists a unique a 6 M such that z 1 ' 2 = ay1'2 and ae^(y) = 0. Moreover a satisfies the following:
(l)\\a\\<W2, (2) rM3(y + e ) " 1 / 2 -> a strongiy* as £ J. 0, (3) *ll2(y + e ) - V / 2 / aa* as £ I 0. Proo/. The assumption x < Xy means that V(yxl2) C © ( z 1 / 2 ) and ||ar 1/2 CI| 2 < -Ml{yl/2). So a bounded operator a is uniquely defined by a(y 1 / 2 ^) = z 1 ' 2 ^ | e V(y1'2), and ae{0}(2/) = 0. It is immediate to get a e M and ||a| < A 1 / 2 . We have as £|0 *1/2(y
+ e ) ~ 1 / 2 = ayU2(y
+ e)"1/2 -
(z 1 / 2 (y + £ ) - 1 / 2 ) * = (y + e)-^2y^2a' -
1/2
(y + e ) - ^
1/2
-
ae(o,cc)(y) = a strongly, e ( 0 i 0 o ) (j,)«' = a ' strongly,
= aj/(j/ + £ ) _ 1 « * /
aa',
so that (2) and (3) hold. I For x,y € M+ such that z is dominated by y (i.e. x < Xy for some A > 0), we write aa* = x1'2y~1xll2 in view of (3) above.
231 Lemma 2.2. Let x,y 6 L1(M;T)+ l 2
2
l 2
X 2
X l {\o%X^I y^x l )x l
& L\M,T)
(= M+nL1(M;r)). an
Ifx is dominated byy, then
d so T ^ / Z Q o g j ; l / 2 r 1^/2)^1/2)
e
(_oo|00).
Proof. By definition, x^logx1/W^x1/
2 =
„ i / V ( l o g aa'Jay 1 /*
Using the polar decomposition of a we easily see that a*(logaa*)a = a'aloga'a. When x < Ay with A > 1, since a*a — 1 < a*a log a*a < A log A, we get « - » = 2/1/2(a*a - l),, 1 ' 2 < ^ " ( l o g x ^ V V ) . 1 " < (Alog\)y, implying the conclusion. I Lemma 2.3. Let x,y,y € L1(M; T ) + . 1/x is dominated by y and 3/ < y, then r(x 1 / 2 (logx 1 / 2 y- 1 x 1 / 2 )x 1 /2 ) >
rix^Qogx^y^x1/2^1'2).
Proof. Let 5 be defined for x, y as a for x, y in Lemma 2.1. Since xll2(y + e ) - 1 ! 1 ' 2 > x ' (j/ + e ) - 1 ! 1 ' 2 for e > 0, Lemma 2.1(3) implies aa* > aa*. By the proof of Lemma 2.2 we have 1 2
rCx^Oogx^V1*1'2)*1'2) = T W O log a*a)y1l2) = limr(y1l2(a*a\o$(a*a
+e^y1'2)
= limiV/VOogCaa* + e))a2/1/2) = limr(x 1/2 (log(aa* + e))x1'2) e{0
e[0
> limr(x 1/2 (log(55* +e))x 1 / 2 ) = T(X 1 / 2 (1O S X 1 / 2 J/- 1 X-'- |X The third equality in the above follows because a*a\og(a*a + e) = a*(log(aa* +e))a. Also the inequality follows from the operator monotonity of logi. I Umegaki's relative entropy [25] for x, y € L1(M; r)+ is defined by S(x,y) = r(x(logx-logj/)). On the other hand, in view of Lemmas 2.2 and 2.3 let us define for x, y e
= sup{r(x 1 / 2 (logx 1 / 2 j|- 1 * 1 / 2 )^ 1 / 2 : V < V G L1{M\T),X
L1(M;r)+
is dominated by y},
which is in (—00,00]. This quantity is a version of that introduced in [3]. In fact, SBS(*> y) is given as follows:
232 P r o p o s i t i o n 2.4. For every x, y € Ll(M; SQS(x,y)
r)+,
= \imT(xll2(\ogxll2{y
+ ex)-xxll2)xll2).
(2.1)
Proo/. Let a (€ (—oo, oo]) be the right-hand side of (2.1), which exists by Lemma 2.3. What we have to show is that a > T^ilogx1'2^*112)*112)
(2-2)
holds for every y € LX(M; T ) + such that y < y and x < Xy for some A > 0. For any e > 0 we get by Lemma 2.3 T(x1'2(\o!.x1l2(y
+ exr1x1l2)x1l2)>T(x1/2(lol>x1l2(y
+
exr1x1l2)x1l2),
so that a>limr(z1/2(logx1/2(ji + £ i ) - 1 i 1 / V / 2 ) . do
(2.3)
Let a and ac be defined for x, y and x,y + ex, respectively, as in Lemma 2.1. Since y + ex < (1 + eX)y, we get 5 e 5* > (1 + eA) _ 1 55* by Lemma 2.1(3). Hence as in the proof of Lemma 2.3 T ^ O o g z ^ ^ + Ez)- 1 * 1 / 2 ):* 1 / 2 ) = l i m r ( x 1 / 2 ( l o g ( 5 £ 5 ; + «))z 1 / 2 ) > l i m r ( x l / 2 ( l o g ( l + eXy^aa*
+
6))x1/2)
= -T(X) log(l + eA) + lim r(i 1 / 2 (log(55* + S ) ) z 1 / 2 ) = -r(o;)log(l+£A) + r(i1/2(logi1/2j/-1a:1/2)3:l/2), so that limr( I 1 / 2 (log : C 1 '' 2 (jf +
1 1 2 1 2 1 2 1 2 1 1/2 )a:1/2). £a;)- a: / )x / )>T(x / (logx / y- a:
This together with (2.3) implies (2.2) required. I By Proposition 2.4 and the proof of Lemma 2.2 we see that SBs(x,y) > r(x-y) holds for every x,y € LX(M; T ) + . The main result of this section is the following:
233 T h e o r e m 2 . 5 . Assume that M is hyperfinite.
Then (2.4) (2.4)
S(x,y)<SBS S{x,y) <(x,y)SBS{x,y) holds for every x, y e LX(M; T) + .
For the proof of Theorem 2.5 we first state the next lemma which was given in [14]. L e m m a 2 . 6 . Assume every x,y e M+.
that M is finite dimensional.
L e m m a 2 . 7 . Assume that M is hyperfinite invertible, then inequality (2.4) holds.
Then inequality
(2.4) holds for
and r ( l ) < oo. If x,y E M+ and y is
Proof. Let {Afj} be an increasing net of finite dimensional subalgebras of M and £j be the conditional expectation of M onto Mj with respect to r . By the martingale convergence of relative entropy (see [2, 16]) we have S(x,y)=limS( S(x,y) = £j(x),e\imS(£j(x), j(y)). £j(y)).
(2.5) (2.5)
Let us prove that SSBS{X,V) BS(x,y)
= =
\imSBS(e \imS j(x),e j(y)). BS(ej(x),£j{y)).
(2.6) (2.6)
ii
The martingale convergence theorem [24] says that £j{x) —► x and £j(y) —► y both strongly and in L'-norm. Hence a,- = ej(x)1^2£j(y)~1^2 —► a = x1/2y~1?2 strongly*, which implies that a*-a,j log a*,a.j —* a*a log a*a strongly. Therefore (2.6) is seen because M. 5SBkSw( £f al(«).
1/2 2 ^r(( 2£ j/()21//)2 ( a (*aa; ;a j l ol og ga ,aT a-ja) f, i) (y) ^ )111 ' 2 ) = r(ej T(ej(y)(a*ajloga*c (y)(a*j ai log a* a,•))
= =*(»)) = =*(»))
: — - t► - r (T(2/(a*aloga*a)) y ( a * a l o g a * a ) ) == SBSa(x,y). ■,y)5 B s(
Then inequality (2.4) follows from Lemma 2.6, (2.5) and (2.6). I Proof of Theorem 2.5. By Proposition entropy (see [2]) it suffices to prove the case assume e(0,oo)(j/) = 1- Let e„ = e[x/ ni „](j/), r ( e „ ) < oo, en / 1, and■*„< \y„ < Xnen. in enMen. Applying Lemma 2.7 in enMe„
2.4 and the lower semicontinuity of relative when x < \y for some A > 0. Furhter we may xn = e„xe„, and y„ = ye„ for n 6 N . Then Hence x„, y„ e ( e „ X e „ ) + and yn is invertible with r\e„Men we have
S{xn,y„) < SBBs(x„ s(x„,y 2/n) < ,»n)n). £(*»>
234 Since \\xn — x\\i —> 0 and \\yn — y\\i —* 0 in Ll{M\ T), we have also S(x,y) < liminf
S(xn,yn)
n—»oo
by the lower semicontinuity of relative entropy. Therefore, to show (2.4) it suffices to prove that SBS(Z,V)
= lim SBs(xn,yn)-
(2.7)
n—t-oo
Let a be as m Lemma. 2.1 and. ctn — a;n yn
&n. Then for every f 6 e«W we get
l l W ' O l l 1 = H4 /2 £ll 2 = («.«»*, 0 = ll* 1/2 £ll 2 = N » 1 / 2 0 l l 2 , so that a^an — ena*aen because yll2e„'H = enH- This shows that a^an —* a'a strongly and hence a*a n loga*a„ —» a*aloga*a strongly. Now we obtain (2.7) in the same way as (2.6). I The relative operator entropy introduced in [9] is defined as follows: When x,y e M are positive invertible, S(x, y) = ^ ( l o g z - ^ V " 1 7 2 ) * 1 7 2 = -* 1 / 2 (l°g ^ V V ' V '
2
For general x, y € M+, S(x,y) = - s - do limz^Oogx^ + s r V / V ' 2 , whenever the strong limit in the right-hand side exists. See [8, 9] for properties of relative operator entropy. For convenience we here state the following: 1° If x, y € M+ and x is dominated by y, then S(x, y) exists and -S(x,y)
^^(log^VV'V'
2
,
where x1l2y~1x1!2 is in the sense defined after Lemma 2.1. 2° If x, y, y e M+ with y < y and if S(x, y) exists, then S(x, y) exists and S(x,y)<S(x,y), Proposition 2.8. Ifx,ye
M+ D Ll(M\ r) and S(x, y) exists, then SBS(X,V)
=
T(-S(x,y)).
235 Proof. For every e > 0, since y < y + ex < y + e\\x\\, it follows from 2° that S(x, y + ex) exists and xll\\og
x'l*{y + £ | | * | | ) - V / 2 ) x i / 2 < -S(X,
Therefore -S(x, y + ex) /" -S(x, 2.2 and Proposition 2.4. I
y + £x)<
-S(x,
y).
y) as £ [ 0. So the desired equality holds by 1°, Lemma
In particular when M = B ( f t ) with the usual trace r = Tr, we have: Corollary 2.9. Let X and Y be positive trace class operators. Then SBS(X, if and only if S(X, Y) exists and is in the trace class. In this case,
Y
) < °°
SBS(X,Y)=Tr(-S(X,Y)). Proof. T h e ' i f part is seen by Proposition 2.8. Let us show the 'only if' part. By 1° and Proposition 2.4 we get oo > S B s ( * , y ) = \hnTr(-S(X,Y
+ eX)).
Moreover by 2°, — S(X, Y + eX) is increasing as e I 0. Hence there exists a trace class operator Z such that -S{X, Y + eX) / Z as e | 0. This shows that S(X, Y) exists and is in the trace class. ■
References 1. H. Araki, Relative entropy of states of von Neumann algebras, Publ. RIMS, Kyoto Univ., 11 (1976), 809-833. 2. H. Araki, Relative entropy for states of von Neumann algebras II, Publ. RIMS, Kyoio Univ., 13 (1977), 173-192. 3. V. P. Belavkin and P. Staszewski, C*-algebraic generalization of relative entropy and entropy, Ann. Insi. H. Poincare Phys. Theor., 3 7 (1982), 51-58. 4. J. Dixmier, Formes lineaires sur un anneau d'operateurs, Bull. Soc. Math. France, 8 1 (1953), 9-39. 5. T. Fack, Sur la notion de valeur caracteristique, J. Operator Theory, 7 (1982), 307-333. 6. T. Fack and H. Kosaki, Generalized s-numbers of r-measurable operators, Pacific J. Math., 1 2 3 (1986), 269-300. 7. B. Fuglede and R. V. Kadison, Determinant theory in finite factors, Ann. of Math., 5 5 (1952), 520-530. 8. J. I. Fujii, M. Fujii and Y. Seo, An extension of the Kubo-Ando theory: solidarities, Math. Japon., 3 5 (1990), 387-396.
236 9. J. I. Fujii and E. Kamei, Relative operator entropy in noncommutative information theory, Math. Japon., 34 (1989), 341-348. 10. I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monographs, Vol. 18, Amer. Math. S o c , 1969. 11. F. Hiai, Majorization and stochastic maps in von Neumann algebras, J. Math. Anal Appl., 109 (1985), 74-82. 12. F. Hiai and Y. Nakamura, Majorizations for generalized s-numbers in semifinite von Neumann algebras, Math. Z., 195 (1987), 17-27. 13. F. Hiai and Y. Nakamura, Closed convex hulls of unitary orbits in von Neumann alge bras, Trans. Amer. Math. Soc, 323 (1991), 1-38. 14. F. Hiai and D. Petz, The proper formula for relative entropy and its asymptotics in quantum probability, Comm. Math. Phys., 143 (1991), 99-114. 15. F. Hiai and D. Petz, The Golden-Thompson trace inequality is complemented, preprint. 16. H. Kosaki, Relative entropy of states: a variational expression, / . Operator Theory, 16 (1986), 335-348. 17. E. Nelson, Notes on non-commutative integration, J. Funct. Anal., 15 (1974), 103-116. 18. D. Petz, Spectral scale of self-adjoint operators and trace inequalities, J. Math. Anal. Appl., 109 (1985), 74-82. 19. I. Segal, A non-commutative extension of abstract integration, Ann. of Math., 57 (1953), 401-457. 20. B. Simon, Trace Ideals and Their Applications, Cambridge Univ. Press, Cambridge, 1979. 21. M. Terp, Lp spaces associated with von Neumann algebras, Notes, Copenhagen Univ., 1981. 22. O. E. Tikhonov, Continuity of operator functions in topologies connected with a trace on a von Neumann algebra (Russian), Izv. Vyssh. Uchebn. Zaved. Math., 1987, no. 1, 77-79; translated in Soviet Math. (h. VUZ), 31 (1987), 110-114. 23. A. Uhlmann, Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory, Comm. Math. Phys., 54 (1977), 21-32. 24. H. Umegaki, Conditional expectation in an operator algebra, II, Tohoku Math. J., 8 (1956), 86-100. 25. H. Umegaki, Conditional expectation in an operator algebra, IV (entropy and informa tion), Kodai Math. Sem. Rep., 14 (1962), 59-85.
Quantum Probability and Related Topics Vol. VIII (pp. 237-261) ©1993 World Scientific Publishing Company
Quantum Topological Entropy: First Steps of a "Pedestrian" Approach Thomas Hudetz Institut fur Theoretische Physik Universitat Wien Boltzmanngasse 5, A-1090 Wien
1
Introduction and Discussion
The purpose of this paper is to introduce a notion of topological entropy for automor phisms of arbitrary (non-commutative, but unital) nuclear C*-algebras A, generalizing the "classical" (see (2.1) below) topological entropy for a homeomorphism T : X —> X of an arbitrary (possibly connected) compact ElausdorfF space X, where the generalization is of course understood in the sense that the latter topological dynamical system (with 7Laction) is equivalently viewed as the C*-dynamical system given by the T-induced auto morphism of the Abelian C*-algebra A = C(X) of (complex-valued) continuous functions on X. In the same sense, also the Connes-Narnhofer-Thirring (CNT for short, [1]) entropy for a. (/'-dynamical system (with respect to a single invariant state) is a generalization of the ''classical" measure-theoretic (Kolmogorov-Sinai) dynamical entropy; and it was originally designed for this very same class of nuclear C*-algebras, but only recently it has been extended to the class of quasi-local algebras in quantum statistical mechanics (by Park and Shin [2]). In both cases, the key concept is that of completely positive unital maps 7 : B —» A of finite rank (i.e. from any finite-dimensional unital C*-algebra B) into the given C*-algebra A, which is the '"natural" extension of the special case of an inclusion *-homomorphism 7 = ig : B —» A of a finite-dimensional C*-stiialgebra B C A; and this extension is really necessary for a possible application of the CNT entropy theory to (nuclear) C'-algebras A without sufficiently many finite-dimensional subalgebras, in particular the projectionless (or '"connected", cf. [3] and e.g. [4]) C*-algebras A. The latter class of C - a l g e b r a s contains as Abelian algebras A exactly the C*-algebras A = C(X) of all possible connected compact spaces X; and even then every finite partition of unity a = {A{ € A+ | X); Ai = 11 G .4} determines a (completely) positive unital map 7a : Ba —* A from a certain finite-dimensional Abelian C*-algebra Ba into A (simply defining 7a(ej) = Ai with the minimal projections e; € Ba), but on the other hand this partition of unity a uniquely determines an open cover U{o) = {Ui C X\ |J; Ui = X} of X, simply defining the U{ to be the cozero-sets of At £ A = C(X), i.e. X \ Ui = Aj*(0). In turn, by Urysohn's lemma, any finite open cover of X has always a subordinate partition a of the unit 11 6 C(X) = A, and thus the set of open covers of the form U{a) is a cofinal subset of the set O(X) of all finite open covers of X, w.r.t. the natural partial order on 0(X); cf. Sect. (2.2) below. - This correspondence -ya >-> U{a) was used in [5] to "pull back" the classical definition of topological entropy (see still (2.1) below) to the category
238 of Abelian C'-algebras A 3 B (together with the "auxiliary" Ba) and positive unital maps (in particular the 7„ : Ba —> A and the *-automorphisms 8 : A —> A), and we shall once more reformulate this simple redefinition in Section (2.3) below. The challenging extension of this classically equivalent definition to the category of non-Abelian (nuclear) C"*-algebras A 9 11 and (completely) positive unital maps is of course motivated by the above-mentioned CNT generalization of the measure-theoretic dynamical entropy: Our simple basic idea is to consider the (completely) positive unital maps 7 : B —> A of finite rank also as the "proper" non-commutative generalization of finite open covers for a compact space X\ in much the same way as already the finite—dimensional C*-su6algebras B C A (if any exist) are viewed as the non-Abelian generalization of finite (measurable or clopen Borel-) partitions of X, originally in the C N T - and related work (see in particular [7, 6], and cf. e.g. [8, 5] for a review), but even so in Lindblad's later counter-proposal to it [9]. From the latter work, however, we take the essential idea of how to define the "joint topological entropy" H(fi, ■ ■ ■ , 7„) of several such positive unital maps -/k : Bk —* A (k = 1 , . . . , n) for non-commutative .A, generalizing the "topological" entropy H(Ui V . . . V W„) of the common refinement of several open covers Uk of a space X (see Def. (2.1) below for its classical definition) in the sense explained above: I f ( 7 1 , . . . ,7„) will be defined (in Def. (3.8)) by a variational expression, formally similar to the CNT definition of ^ ( 7 1 , • • • ,7„) for a. particularly chosen state w on A [1]; but now the supremum is taken over (finite) "square root" decompositions a/, C Bk of the identity (in the sense of Lindblad's "op erational partitions of unity'' = OPUs [9]) in each of the preimage—algebras Bk, respec tively, and the functional to be maximized depends only on the n - t u p l e of "image-OPUs" (7i[ a i]2, • • • )7n[c«n]2) C A"- Here 7*[*]2 '■ Bk —> A+ denotes the "quantum mechanically" non-linear application of the originally linear map 74 : Bk —> A (to be defined in Def.(3.6) below), and in the special case of the inclusion *-homomorphism 7* = iBt ■ Bk —> A of a finite-dimensional suialgebra Bk C A this coincides again with the usual linear ap plication on the positive cone B£. In this case we thus have tBt[<**]2 = iBt(otk) for our OPUs a.k C B^ with positive elements, which we can in turn identify with a* = iBk(ctk) as OPUs in the auialgebras Bk C A; and then the "entropy" functional, maximized to give H(iBl,..., IB„) = H(B\,..., Bn), depends more precisely only on the possible n-fold prod ucts (in A) of the elements in ctk, always one for each k and ordered in k. This is exactly the same construction as used by Lindblad [9] to define the "ordered" refinement (or "compo sition", in his terminology) of the OPUs a* "associated" with the resp. finite-dimensional subalgebras Bk C A; and evidently by this construction the functional H{B\,... ,Bn), or more generally ^ ( 7 1 . . . ,7„), is not necessarily symmetric in its arguments Bk resp. 74, in contradistinction to the CNT functional H„ with state w. In addition, we have to require in Def. (3.8) of ^ ( 7 ! , . . . ,7„) that the OPUs ak C B£ be determined by the respective map ik (not only by the counting index, i.e. for fk = Ji also ak{ik) = ai{ji) is fixed in the optimization) in order to retain at least the invariance (3.11,iii) under repetitions of only one single map: . # ( 7 1 , . . . ,71) = #(71); which makes a further difference to the vari ational procedure in the CNT definition of #0,(71, ■ ■ • ,7„) where this (and even "better") invariance is obtained without such an o priori condition referring to repeated arguments.
239
On the other hand, however, the main advantage of our definition of H(fi,..., -yn) seems to be t h a t there is a. very "natural" step-by-step physical interpretation for it, and thus also for t h e usual "dynamical" entropy h(9,f) of Def. (3.17) below. We cannot discuss this interpretation here in an accessible and readable manner within the space allotted for the introduction, but we refer to a forthcoming "physical" study [11] (and also the "comprehensive" thesis [10]). We only want to announce here that we regard this physical interpretation for the quantum "topological" entropy h(6,-y) of Def. (3.17) as a "partially" satisfactory answer to the second part of "Problem 4" posed by Emch [12] (there for the quantum dynamical entropy w.r.t. a single 0-invariant state u>, e.g. the CNT entropy fc„(0,7) as comparatively used in Sect. (2.2),(2.3) below), namely to obtain at least a "sufficient characterization" of the entropy h(9,7), " . . . in such a manner that the resulting entropy . . . " h(9), here defined in Def. (3.20), " . . . b e physically accessible to empirical testing". On the other hand, the "first" part of this "Problem 4" in [12], namely "to obtain a necessary and sufficient characterization of the . . . " here approximating net r in Def. (3.20) of h(9), " . . . in such a manner that the resulting entropy h(8) be mathematically adapted to the notion of K-flow" (in our case, of course that of C""-algebraic K-flow as e.g. in "Axioms A" of [12]) seems still to be widely open, as of this writing (cf. Problem (3.22) below). Unfortunately, our definition of the entropy h(9,i) as proposed here seems to be not even "mathematically adapted" to define a notion of entropic K-systems (and "memory loss", see [13] and cf. e.g. [12] but also [14]), in view of the "no-go" part of Prop. (3.18,iv) below (cf. also the additional "no-go" part of (3.21,ii), related to the main problem (3.22) and its "classical roots" in Prop. (2.10)). Taken together, this is what makes our present approach (only mathematically) "pedestrian"; but on the other hand, if one could succeed in finding a definition of quantum topological entropy which is closely "mimicking" the C N T construction [1] and presumably also related to the recent conceptual reformulation of the latter by Sauvageot and Thouvenot [15] (and thus mathematically much more "adapted" than the present approach), in turn the second part of the analogous "Problem 4" as posed by Emch [12] (and cited above) would pop up again, analogous also to Lindblad's explicit criticism of the CNT entropy as being "difficult to interpret in a physically meaningful way" [9].
240
2
Classical T h e o r y a n d C * - a l g e b r a i c R e f o r m u l a t i o n
2.1
The classical theory of topological entropy
For the convenience of the reader and to fix our notation, we first recall the definitions and general properties of the "classical" topological entropy as in the original paper by Adler, Konheim and McAndrew [16], which contains already most of the essential parts of the classical theory. Throughout this section we use the standing notation (X, T) for a topological dynamical system, given by a compact Hausdorff space X (not necessarily metric) and a homeomorphism T : X —> X. By definition, any open cover of X possesses a finite subcover, and we can restrict ourselves to the latter from the very beginning, denoting by O(X) or simply O the set of finite open covers of X. D e f i n i t i o n ( 2 . 1 ) : For U,V e O(X) we define their join U V V = {U HV\U V} £ 0{X), and the action of T on U £ O by T~\U) = {T~\U)\U £ U}.
eU,V
£
(i) N(U) = min{cardW|W C U : U* € O(X)} denotes the cardinality of a minimal subcover W o f W e O . The "topological" entropy oiU £ O(X) is defined by H{U) =
logJV(W). (ii) The entropy of T w.r.t. U £ O is defined as h(T,U) = limn^o., \H{U V T~lU V . . . V T~n+1U). (iii) The topological entropy of T is h(T) = sup W€C ,
h(T,U).
A cover V £ O is said to be a refinement of a cover U, U -< V, if VV £ V 3U £ W : V C U; and obviously (O, y) is a partially pre-ordered set, but even a (pre-)
(a) U < V => N{U) < N{V) <=> H(U) <
H(V).
(b) N(U) ■ N{V) > N{U V V ) < ^ . H(U V V) < H(U) + l
(c) N(T~ U) ad (ii)
= N{U) <^
(d) it -< V =!» h(T,U) (e) h(T,U)
H{T- U)
< h(T,V),
=
H{V).
H(U).
foUows from (a).
< H{U), follows from (b) & (c).
(f) h{T,U) = h(T~\U), ad (iii)
l
follows from (c).
(g) h(T) = ]imue0h(T,U) as a net limit (corollary of d); and in particular for a refining sequence {U„ £ C} n £ lN of open covers, i.e. Un -< Un+1 Vn and such that W £ O 3n £ IN : V < Un, as a suinet: h{T) = l i m , ^ h(T,Un). (h) Let $ : X —> X' be a homeomorphism onto a compact (Hausdorff) space X', then h(T) = h($oTo $ " 1 ) , follows from (c).
241 (i) h(T) = h[T~l),
follows from (f).
As used already in [16], (g) simplifies the computation of h{T) for metric spaces (X, d), with where each sequence {V* £ °(x)}keTN diameters d(Vk) = m a x n e V , d(Vk) shrinking to zero, d(Vk) -► 0, gives a refining sequence {Un\Un = VJJ=1 Vfc}nerN by Lebesgue's covering lemma. T h e o r e m ( 2 . 3 ) : In addition, the topological entropy (2.1,iii) has the following special properties refering to the varying (non-conjugate) topological dynamical system (X, T):
(i) h(Tk) = k ■ h(T) vfc g m, (ii) Let T = Tx x T2 be the product homeomorphism of T,- : X{ -> X; (i = 1, 2) acting on X = .Xi x X2 (with product topology). Then h(T) = h(Ti) + h(T2). (iii) Denote (again generally) by M{X) = M the convex compact space of (Borel) prob ability measures on X (with the weak-* topology), and by TM denote the induced affine homeomorphism TM : M -> M defined by TM(fi) = / i o T~L V/J £ M(X). If h(T) > 0, then h(TM) = oo (cf. [17, 18]). (iv) Denote by M(X,T) C M(X) the (non-empty) convex compact set of Tif-fixed points fi = Tjif(/i), i.e. T-invariant probability measures y. on X. Then h(T) = sup^gjf/jf:T)hf,(T), where h^T) denotes the measure-theoretic (Kolmogorov-Sinai, KS) entropy of T w.r.t. [i (cf. originally [19, 20]).
2.2
Motivations for t h e C*-algebraic formulation
Now, this latter KS entropy was generalized finally by Connes, Narnhofer and Thirring (CNT for short, [1]) to the entropy h^O) of a *-automorphism 8 on a (unital) C'-algebra A with invariant state u> = woO £ SA, the state space of A (cf. e.g. [8, 5] for a "historical" review and see in particular [7, 6]). Remember that, in the most general case, this CNT entropy is defined at first by h„(6) = sup 7 € C T > 1 (^ fc„,(0,7), where CVi(A) denotes the set of completely positive unital maps 7 : B —» A of finite rank (i.e. from any finite-dimensional unital C*-algebra B) into the given C*-algebra A, and where ^ ( 0 , 7 ) is the "entropy" [1] of 8 e * - Aut(.4) w.r.t. 7 £ CPi(A), formally analogous to (2.1,ii &; iii) above. Actually, this supremum taken over CVi(A) can already generally be restricted to the set Ki(A) C CVi(A) of all completely positive unital maps 7 : Af„(C) -» A (Vra £ IN!) from any matrix algebra into A: h^d) = sup 7€X;i (^) hw(8,i); and for an Abelian C*-algebra A = C(X), X as above, and 6T G * - Aut(.A) induced by T : X —> X as above (simply by 8T(A) = AoT V.A £ A = C(X)), we can restrict even further as stated in (i) of the following P r o p o s i t i o n (2.4)[5]: Let A = C(X) T : X —» X (as above) be fixed.
for some X and 9T £ * - Aut(^4) for some
242 (i) Denote by Vi(A) the set of all positive (as well known, here "completely-" is redun dant!) unital maps from finite-dimensional Abelian C*-algebras into A. Then the CNT entropy ft„(Sr) = *uV-,nVi(A) hw(fiT,~t), VUJ £ SA with w = w o 8T(ii) For /x £ M(X) denote by u>M £ 5 ^ the integral state wM = / x d/i. Then the CNT entropy h^8T) = h^T) V/i £ M(X,T), with then obviously invariant wll=w)io0T (and the KS entropy on the r.h.s.). Both (i) and (ii) are not at all surprising, but also not quite "tautologically n trivial; and in particular (ii) merely points out that, for the whole category of Abelian C*-algebras and their *-auto(or -isomorphisms, the CNT entropy is really nothing but the "pullback" of the KS entropy (for the category of compact spaces X and homeomorphisms) via the Gelfand isomorphism A : A —» C(X) and Riesz' representation of states on A. On the other hand, however, (i) & (ii) lead to a new "interpretation" (or equivalent redefinition) of this entropy h^T) for T : X —» X and u £ M(X, T) above, in terms of "fuzzy partitions" (cf. [21, 22]) of the unit 11 £ C{X) and their "entropy" (cf. [5, Appendix] and see [23]), via the following simple construction: Even for a general (non-commutative) C'-algebra ^ 3 11, any such map 7 £ V\(A) with 7 : B —> A (and hence B Abelian!) as in (2.4.,i) uniquely determines a partition of the unit ou, = {Ai £ A+\ £ ; A; = B}, simply defining Ai = 7(e,) with the minimal projectors e; £ B (i = 1 , . . . , dimB); and this is the natural choice of QU, for the calculation of the entropy /ta,(#T,7) in (i) (or of the "related", but much better calculable "fuzzy-entropy" [5, App.], [23]), as the latter depends (in the final effect, besides on w and 0T) only on the set {7(ej) = Ai}, even when "ignoring" that 7 is actually linear. - In turn, again generally, any such partition of the unit a = {Ai £ , 4 + | i = 1 , . . . ,n : £"=1 ^ = 11} gives a positive unital map 7„ : Ba —> A with 23a = (Dffi
243 entropy (2.1,iii) generally as h(T) = s u p a g 0 + ( / 4 ) h(T,U(a)), again by (2.2.,d), also for X not necessarily metric. By the same argument (2.2,d), we know that for 7 £ V\{A), 7 : B -> A = C(X) with "associated" a , £ 0 ^ " ( ^ ) as before, and U(cu,) £ O ( X ) as just above: h(T,U(ch))=
sup
h(T,U(l(P)j),
which is clear from the fact that for S = {e; £ B|i = 1 , . . . , d i m B } £ 0 + ( B ) with the minimal projectors e ; 6 B w e have 7(5) = <x, and then obviously U(eu,) >- U(f(B)) V/? £ Of(B); or "symbolically" Z ^ o , ) = V^ 6 o+( B ) W (7(/3)) as a "supremum" (note that for the finite-dimensional Ahelian preimage-algebra B there exist only finitely many different U(y(fi)) £ O(X) as 8 varies over Of(B), such that the supremum over 0 + ( B ) is actually attained as a maximum). In this sense, cty is again the "natural" choice for computing the entropy (2.1,ii) of T w.r.t. an open cover "associated" with 7 £ Vi(A); and by analogy with (2.4,i & ii) this makes it tempting to define the topological entropy (2.1,iii) of (X,T) formally "algebraically" for A = C(X) and BT £ * - Aut(.A) as before, using the above expression as a definition for "K(9T,I)" and again defining K(9T) = sup g7> (A)h(6T,f), such that (obviously from the above discussion) again ft(#r) = h(T) would be fulfilled (corresponding to (2.4,ii)). However, for the desired generalization of this still only formally algebraic "definition" of h(8) to 8 £ * — Aut(,4.) for non-Abelian C*-algebras A (see Sect. 3), where in addition for 7 £ CVi(A) with 7 : B —t A and also non-commutative preimage-algebra B there is no uniquely determined at, £ Of(A) any more (as there is for 7 £ V\{A) above), it seems to be necessary (see Sect. 3) to "mimick" the definition [1] of ^ ( # , 7 ) as in (2.4,i) even further in the algebraic definition of "h{8,7)" (although still equivalent to the above first attempt in the Ahelian case A = C(X) and 7 £ Vi(A))\ and consequently to consider the maps 7 £ {C)'Pi(A) as the "proper" (in Section 3, non-Abelian) algebraic generalization of the open covers U £ O(X), instead of the more directly related partitions a £ Of (A) of 1 € A (as discussed before), using those maps 7 already in the C*-algebraic reformulation (and then non-Abelian generalization) of the entropy (2.1,i) for open covers U £ 0(X), as it then enters in the definition (2.1,ii) of h{T,U). First, we know from the above discussion of the latter entropy when restricted to all U = W(a) with a £ Ot(A) "associated" to 7 £ Vi(A), -y:B->A = C(X) and a = 7 (/3) for some 3 £ 0 ^ ( 5 ) , that a fortiori also for the former .ff-entropy (2.1,i): H(U{ch))=
max
H(U{I{B)))
is valid, with a-, = 7(0") as before (follows from (2.2,a)). Thus the right hand side offers itself as the (again first only formally - ) "algebraic" definition of an entropy # ( 7 ) for 7 £ {C)V\(A); and it turns out that this definition can be made "purely" algebraic rather easily, but since in the definition (2.1,i) of H{U(a)) = log N(U(a)) we possibly omit sets Ui £ U{a) to get a minimal subcover U(a)' £ O(X), corresponding to omitting certain elements A{ £ a £ Ot(A), we first have to weaken the requirement that a be a partition of unity B £ A
244
2.3
C*-algebraic reformulation of t h e classical t h e o r y
We do this using the notion of strict positivity in a (general, not necessarily Abelian) unital C*-algebra: Definition (2.5): Let A 9 11 be a C*-algebra, then A £ A+ is said to be strictly positive (denoted A > 0 in contradistinction to A > 0), if the following (rather obviously) equivalent conditions (i)-(iii) are fulfilled:
(i)
(ii) 3A-1 £ A : AA-1 = A~lA = 11 (iii) 3e > 0 : A > e ■ 1. Note that in particular for an Abelian C - a l g e b r a A, again jsomorphic to C(X) for some X as before via Gelfand's isomorphism A : A —> C(X), we have that A > 0 in A iff A(z) > 0 Vx € X. Again for a general C'-algebra A B H, clearly A+ B A> 0 •<=>■ A2 > 0, but as an easy exercise (in elementary C*-theory) one sees that for a set a = {A^ £ A+\i = 1 , . . . ,n} also even
(EA)>0^(EA ! )>0; and we call such a set a fulfilling these equivalent conditions a "positive operator cover'' for A, and 0 + ( . 4 ) D Ot(A) denotes the set of all these a. Now in turn again for Abelian A = C(X), we can directly extend our "definition" (in Sect. (2.2) above) of the map 0 ^ - 4 ) 9 a H* U(a)[e O(X)] also to a £ 0+(A) but even to a = {Ai £ .4+} £ 0 + ( . 4 ) , and obviously from above a £ C+(.4) <S=> W(a) £ O(X). Furthermore, for A = C(X) and a = {A{},P = {Bj} £ 0+(A) we can define their "refinement" a V /3 = { 4 ; • Bj £ .4+}, such that obviously a V /3 £ 0+(A) and W(a V /3) = W(a) V U{0) with the join of Def. (2.1) in O(X). And for T : X -» X with induced fly £ * - Aut(.4) as in (2.4), we have again obviously: T-lU{a) = U{9T{a)) Va £ 0+{A). Now, we can first "algebraically translate" the classical definition of topological entropy in Def. (2.1) as follows: Definition (2.6): Let A be an Abelian C - a l g e b r a with 8 £ * - Aut(.4). For a,/? £ 0+(A) we define their common refinement a V j3 £ 0+(A) as just above. (i) N(a) — min{card a'\a' C a, a ' £ 0+(A)} "operator subcover1' a' of a £ 0 + ( . 4 ) .
£ IN denotes the cardinality of a minimal
For 7 i , . . . , 7 „ £ Vi(A) with 7* : Bk -> A {Bk hence a C*-algebra; k = 1 , . . . , n ) , their entropy is defined by ^.■■•,7n)-
max {(PHTl)
A>(7"))}
finite-dimensional
logAr(7l(/91)V...V7„(/3„)),
Abelian
245 where the maximum is taken (first as supremum) over the set of n-tuples with el ements 0k(rik) 6 Oi(Bk) determined by the resp. map 74 (Vfc = l , . . . , n ; i.e. for Ik = 1i also Pkilk) = Pli.1l)), and the maximum rather obviously exists (cf. the discussion in Sect. (2.2), in particular before Sect. (2.3), and remember Prop.(2.2,b); see also Sect. 3 below). (ii) The entropy of 8 w.r.t. 7 £ V\{A) is defined as h(8,y) = limn-,,*, ^H{^,8 0 7 , . . . ,8"^ o 7), where again the limit obviously exists (cf. (2.2,b & c)). (iii) The "topological" entropy of 8 is h(8) = sup 7 € T , , «
h(8,y).
Summarizing our findings thus far, we have the T h e o r e m (2.7):(cf. [5]) Let (X,T) ated" C*-dynamical system A = C(X) and (2.6) coincide: h(T) = K{8T).
be a topological dynamical system with "associ and 8T £ * — Aut(.A), then the definitions (2.1)
Corollary ( 2 . 8 ) : Let A be a (separable) Abelian C*-algebra with 8 € * — Aut(^4), and denote by SA the set of 0-invariant states w = u 0 8 £ SA, then obviously by (2.3,iv), (2,4,ii) and (2.7): h(8) = sup„ €S » hu(8), with the CNT entropy on the right hand side. In view of the definition (2.6,iii) of h(8) = sup^ e 7 , l / / t j/i(ff,7), resp. the result hw(8) = P-,s/Pi(A) h*(fii 7) °f (2.4,i) for Abelian A, it is tempting to think (cf. [5]) of a direct "purely algebraic" proof of (2.8), simply by showing that K{8,i) = s\rpw&ss hu(8,f), or at least ^(0i7) & l 0 ( f i , 7 ) Vui £ SA. But this can certainly not be true for all 7 £ Vi(A), if ever, for the following reason: As shown in [1], A.„(#,7) is norm-continuous in 7 even for non-Abelian A and 7 £ CVi(A) with our notation from Sect. (2.2), what then implies the ''non-Abelian KS theorem" for nuclear A (recently generalized for quasi-local C*-algebras by Park and Shin [2]). Just to remind the reader and again fix the notation, we state su
P r o p o s i t i o n (2.9)[1]: Let A 3 11 be a (general) unital C*-algebra with 8 £ * Aut(,A), u £ SA. (i) V 7 £ CVM), 7 ■ & -► A, and Ve > 0 3 5 ( d i m S , e ) > 0 : V7' £ CVl{A), A with ||7 - 7'|| < S, the CNT entropies \hu{8,f) - K{8,i)\ < e.
7 ' : B ->
(ii) If A is nuclear [25], there exists a net of completely positive unital maps <7„ : A —» B„, T„ : Bv —» A with finite-dimensional C"-algebras (or full matrix algebras) B„ (u £ N, a directed set), such that lim^gjv ||T„ 0 a„{A) — A\\ = 0, VA £ A Then as a corollary of (i), quite obviously the full CNT entropy hw(8) = lim„ejv b.^(8,Tv), although {T„\V £ N} C CVi(A). As is well known (and rather easily seen), there exists an "approximating" sequence o-n : A —> B n , r„ : Bn —> A (n £ IN) iff A is separable; and in this case A.„(0) = lim„_ 0 0 fc„(0,T„).
246 But for our "topological" entropy h{e,y) of (2.6,ii) with A Abelian and 7 g Vi(A) before, we have unfortunately the following continuity-"no go" result:
as
P r o p o s i t i o n ( 2 . 1 0 ) : Let A 3 11 be an Abelian C*-algebra with 6 g * - Aut(.A). (i) For 7 g Vi(A), -y:B->A{B finite-dimensional Abelian), and VS > 0, 3 7 ' g ■y'-B-^A such that ||7 - 7 ' | | < S, but fc(0,7') = 0.
Vi(A),
(ii) For any 7 ' g Vi{A) with h{6,7') = 0 as above, Be(-f') > 0 such that ft(0,7") = 0 V 7 " g Vi{A), Y'-B^A with || 7 ' - 7 " | | < e. We leave the proof to the reader as an easy but tedious exercise (going through the above definitions (2.6,i & ii) and using the correspondence V\(A) 3 7 >-» a-, g Of (A) as discussed in Sect. (2.2) for computing # ( 7 1 , . . . ,7„) = log ^(ct^, V . . . V 04.) = 0 whenever 3Ak g o^h : Ah > 0 Vfc = 1 , . . . , n ) . - Thus the 7 ' : B -» .4 with ft(0,7#) = 0 are an open dense subset of {7 g Pi(A)\f ■ B —► A} 3 7' in norm; and hence, also in view of the proof [1] of the "non-Abelian KS theorem" (2.9,ii) for hw(6), it is clear that the mere existence of an approximating sequence of maps r„ : An —> A,
247 In addition, it is evident by (2.2,g) and definition (2.6,ii) of h(6T,Tn) homeomorphism T : X -> X, that h(0T) - l i m , , ^ K(6T, r„).
= h(T,U(/3n))
for any
This detailed simple proof is an exception of our rule to skip all of them in this paper; but actually (2.11) paves the way out of the "no-go" impasse (2.10) via the following more general sublemma (with essentially the same proof): S u b l e m m a ( 2 . 1 1 , 1 ) : Let A 3 1 be a non—separable Abelian (7*—algebra, and note that Of (A) with the partial pre-order y as introduced before is a (pre-)directed set with the common refinement (as in (2.6)) aV /3 >- a,fi G Of (A). Considering V\(A) 3 fa as a net indexed by a G Of (A) with our notation from Sect. (2.2), there exists a net {
3 3.1
Q u a n t u m Topological Entropy for Systems
C*—dynamical
T h e general theory for nuclear C*-algebras
Definition ( 3 . 1 ) : Let A 3 D be a (first general) non-Abelian C*-algebra. Using the general definition (2.5), we introduce the following notations: O(A) denotes the set of all finite ''operator covers" a = {A,- E A\ £ i A\A{ > 0}, 02{A) is the set of (not necessarily positive) "operator partitions of unity" (OPUs) /3 = {Bj € A\ Ej B)Bj = 11}, and 0}(A) denotes t h e set of positive "square root" OPUs /3 = {Bj € A+\EjB] = H}. Note that obviously Ot{A) C 02(A) C O(A) resp. 0}(A) C 0+(A) C 0{A) with 0+ as in Sect. (2.3), and "symbolically" for /? £ Of (A) element-wise (02) G Of(A), resp. for a G Of{A) in turn ( o s ) G O ^ ^ l ) . For a,f3 £ O(A), a = {At} and /3 = {Bj}, we can well define their "ordered refinement" aSJB = {BjAi} G 0(A), as we have for Y.j B'jBj > e • H (for some e > 0, using (2.5,iii)) that Y,ij AlB'jBjAi > eZi A'iAi > 0. Obviously, V* : O x O —* O is associative (because A is so), but not commutative for non-Abelian A, and V • 02 x C 2 -» 02 (but V : 0+ x 0+ -> O, with image more than 0+\). On the other hand, for Abelian A and a,/3 G 0 + ( - 4 ) of course aV/3 = a V /3 with the "classical" refinement as used in Def. (2.6).
248 Now we can first generalize for non-Abelian A only the first part of Def. (2.6,i), and also the partial pre-order >- on 0+(A) as in (2.11 & ,1), but necessarily also extended to all of O(A): Definition (3.2): Let again A 3 11 be a general C*-algebra. (i) For a € O(A), N(a) = m i n { c a r d a ' | a ' C a, a ' £ O(A)} £ IN denotes t h e cardinality of a minimal "operator subcover'' a' of a . (ii) We define a partial pre-order y on 0{A) by a y 0 for a , 0 £ 0 ( . 4 ) , if V.A; £ a 3 B i W 6 /? and A; € (0,1] : A ^ . M ; < B^Bm. Note that these are really generalizations of the "classical" definitions, although for mally different for Ahelian A; remember that generally a £ 0+(A) ■$=> a2 £ 0+(A) (element-wise!), resp. for A, B £ A+ : XA < B = > \* A* < B' (being equivalent for Abelian A). P r o p o s i t i o n ( 3 . 3 ) : For these preceding definitions, we have the following rather ob vious properties: Va,/3,7 £ O(A), ad (ii) (a) a -< ay0, but not necessarily 0 -< aSJ0 (i.e. the "directed join" V does not render (O, y) a directed set any more). (b) 0 y 7 = > aV/3 >- oSlf, but no< necessarily also 0va >- 7Va! a d (i)
(c) N{a) < N(aV0) < N{a) ■ N(0), but not necessarily also N(0) < N(aV0), as follows from (a) together with (f) below (this generalizes (2.2,a & b) for N partly). (d) N(e(a))
= N(a) V0 £ * - Aut(.A) (generalizes (2.2,c) for N).
(e) JV(aVaV . . . \/a) = N(a) Va £ 0+(.A) and for arbitrary V-"powers" of a (generalizes N{U) = N(U V U) in (2.1,i), as follows from U V U y U and U V U -< U for U £ O ( X ) by (2.2,a)), but not necessarily also N(aVa) < N(a)
totae0(A)\O+(A), (f) a -< 0 = > # ( a ) < ]V(/3) (generalizes (2.2,a) for N). L e m m a ( 3 . 4 ) : Let B be a /mzte-dimensional C*-algebra, i.e. *-isomorphic to ®2-i Mdk = B with full dk x dk-matrix algebras Mdtt and define D(B) to be the dimension of a maximal Abelian subalgebra C C B: D(B) = £ J = 1 d*. Then for any a £ O(B) there exists a 0 £ 0 + ( B ) , /3 = {e; £ B+\i = 1 , . . . , D{B) ■ Y,?=i d > 0} with minimal projectors ei £ B, such that 0 y a as defined in (3.2,ii). Corollary (3.5): Let B again be a finite-dimensional C*-algebra with D(B) defined as in (3.4), then Va £ O(B) : JV(a) < D(B), as follows from (3.3,f) & (3.4). Obvi ously, there exists a 0 £ Of(B) ("I O j ( S ) with again minimal projectors e ; , y3 = {e< £ B+|i = 1 , . . . , D ( B ) : E ^ i ^ i = A}, such that N(0) = D(B); and together we have that m a x „ e 0 { B ) N(a) = D{B) = max f l £ 0 + ( B ) N(0).
249 D e f i n i t i o n ( 3 . 6 ) : Let 7 : B —> A be a positive linear map between C*-algebras B and A, then we define the '^-application" 7[»]j : B —» «4+ (as a map, also denoted by "absolute value" I7I) by -y[B]2 = [y(B*B)]'. Obviously, |-y| : B -» A+ is still positive but no* linear any more (only still positive homogeneous: 7[AB] 2 = A • 7[B] 2 , VA > 0, Be B); and if 7 is a *-homomorphism, then |7|B+ = 7B+- (Less trivially, this latter statement has a partial converse: If 7 is even completely positive, then 7 [ £ ] 2 = f(B) for a particular B £ B implies that B is in the "left multiplicative domain" of "y, i.e. ■'y(C) • ^(.Z?) = *y(CZ?) VC 6 $J see [26].) Note that generally f[B]2 = -f(B) at least whenever B = B2 £ B+ and y(B) = 7 ( B ) 2 £ A+ are projections. L e m m a (3.7): (i) Let 7 : B —* A\ and 8 : A\ —> At be positive linear maps between C*-algebras B , AX,A2, then %[B]a]a = (5 o 7 ) [ £ ] 2 Vfl £ B (or alternatively, symbolically: |0| o
M = |0°7l)(ii) If 0 is even a *-homomorphism, then also 0(7[.B]2) = (8of)[B]2 VB € B (or 0 o | 7 | = \8 o 7I; note that also for 7 a *-homomorphism and 8 general: \8 o 7I = \8\ o 7). Definition ( 3 . 8 ) : Let 7 1 , . . . , 7 „ be positive unital (linear) maps jk : Bk —> A from finite-dimensional C"*-algebras Bk into the C'-algebra A 3 11, then their entropy is defined #(7i.---.7n)=
,
m a x
,
mlog^(7i[ai]2V...V7„[o„]2),
where the maximum is taken (first as supremum) over the set of ra-tuples with elements <*k £ 02(Bk) determined by the respective map 74 (i.e. for 74 = 7/ also a/,(-yk) = <**(7*)), and the maximum exists because of (3.11,ii) and (3.10) below; and where of course 7i[ajt] 2 £ 02(A) is understood element-wise. For Abelian A and 7 1 , . . . ,7„ £ Vi(A), i.e. also Bk Abelian Vfc, this definition rather obviously coincides again with (2.6,i); and on the other hand for non-Abelian A but the inclusion *-homomorphism 7* = i s t : Bk —* A of finite-dimensional *-suialgebras Bk C A, we see from (3.6) that z B t [a*] 2 = iB t (a*) V a t £ 02(Bk), and thus we can denote H(B1,...,Bn)
= H(vBl,...,tBn)=
max
log # ( a , \ 7 . . .tfa,,).
L e m m a ( 3 . 9 ) : Let 7 : B —> A be a positive unital (linear) map from a dimensional C*-algebra B into the C*-algebra A B 1, then Va £ 0 + ( S ) :
finite-
(i) iV(7[a] 2 ) < JV(a), where we should emphasize at this occasion that we understand it as denned by the arguments of JV (or H) to which algebra they refer (here A on l.h.s., B on r.h.s.), and (ii) also 7 ( a ) £ 0+(A) with N(-y[a}2) < N(f(a)). In particular, for a positive operator cover B of B, 3 = {Pi = p} £ S + | £,-?,• > 0} with projections p{ even N(-y[B]2) = JV( 7 (/3)).
250 Proof: The inequalities (i) resp. (ii) follow from the implications (i) E . A ? > ° = * £ < 7 ( A ? ) > ° (obvious by (2.5,iii)), resp. (ii) Y,if(A)2 > 0 = » T,i~t(A{) > 0 (for which Schwarz positivity of 7 is not even necessary, as 7 ( J 4 2 ) > ~f(A)2 VA = A' by Kadison's ancient result [27]); which obviously both cannot be reversed in general. But in (ii), somewhat ''paradoxically'', always 7 ( a ) € 0+(A), as follows from our well-known equivalence (cf. after (2.5)): a ; A ? > 0 <=> E i At > 0 =► T,i-r(Ai) > 0 <=* ZirtAi)2 > 0. And if even A\ = -A; Vi, we can use the latter equivalence to reverse the inequality (ii). Corollary ( 3 . 1 0 ) : Let 7 : B -> A be as in (3.9), then £ ( 7 ) < l o g D ( S ) = H(B), as follows from (3.9,i),(3.5) and the definitions. Thus H(f) = max.aeo+,BAogN(f[a]2) is really a maximum; but even H(f) = max ae o 2 (B) log ^(7(0)2) does not give more, because for a = {Ai G B\^2iA'iAi = 1} e 02(B) we have that (symbolically) | o | = {{A$Ai}' £ B+} £ 0}{B), and obviously by Def. (3.6): 7(0)2 = 7[|o|] 2 . By the same argument, we can take a„(7 n ) 6 0 2 ( S n ) in Def. (3.8), but not for k = 1 , . . . , (n — 1) if we want to keep to the equivalent expression for suialgebras Bk C A in (3.8). P r o p o s i t i o n ( 3 . 1 1 ) : Let A 3 H be a (first general) C*-algebra, then the entropy functional (3.8) has the following general properties: (i) For 8/, : Bk —» Ak and 7* : Ak —► A positive unital (linear) maps with finitedimensional C*-algebras Ak,Bk (Vfc = 1 , . . . ,n), such that (cf. Def. (3.8)!) for 7,- = jj also 7; 0 B{ = 7 i o 8j (Vi,j = 1 , . . . , n ) : # ( 7 l o 9U... ,fn o Sn) < # ( 7 l , . . . , 7 n ) , which follows from 8h[0${Bk)]% C 0${Ak) using (3.7,i) (and generalizes (2.2,a) for H)-
'
(ii) For fk'Ak—*A B{ll,
'
positive unital maps with finite-dimensional (7"-algebras Ak, . . . ,7m) < # ( 7 l , • • ■ , 7 » ) < H(fl
. . . ,7m) +
fffrm+l,
...,
7n
)
Vm < n (where literally the order of ( 1 , . . . , n ) has to be respected), which follows from (3.3,c) and generalizes (2.2,a &: b) for H partly. If in addition 7; f 7,- Vi ^ j £ { l , . . . , n } , then even 5{yh>. ..,iit) < #(7i> ■ ■ • >7») V£ < 71 and 1 < ij < i 2 < . . . < it < 71, as follows directly from Def. (3.8). (iii) For 7! : A\ -» A as before, # ( 7 1 , . . . ,71) = # ( 7 1 ) for arbitrarily many repetitions of only one single 7 J (as follows from (3.3,e) because of the definition (3.8) for H, and generalizes H(UWU) = H{U) in (2.1,i) as noted in (3.3,e)). (iv) For 9 6 * - Aut(.4) and 7fc : Ak -» A as before, H{6ofu . . . ,
,7,,),
251 (v) If there exists a finite-dimensional C*-subalgebra B C A such that Im(7fc) C B Vfc = l,...,n, then # ( 7 l , . . . ) 7 „ ) < logD(B) = H(B) with the "Abelian di mension" D{B) as introduced in (3.4); which follows again from (3.4) applied to 7i[<*i] 2 \7... fynKl* £
||B*|| < 1} C 7 ( B + ) ,
( < H(B) by (3.10) above); which follows directly from
Corollary ( 3 . 1 2 ) : ad (i) If j k : A -» A and At C Bk with conditional expectations Ek . Bk —» .4* (such t h a t t h e inclusions u t . At —> Bk have Ek as left o-inverse: Ej, 0 nt = I d ^ t ; and assuming that for Ai = Aj also iJ; = Ej, Vi,j = l , . . . , n ) , then # ( 7 1 o Ex,... ,7„ o £ „ ) = # ( 7 1 , . . . ,7„). Thus we may again assume (as in [1, (111.5,1)]) for the computation of H(fi,... ,jn) that the 7* are maps from resp. full matrix algebras Mit into A; or 7* £ Ki(A) (with our notation at the beginning of Sect. (2.2)) if we restrict to completely positive unital maps 74 6 CVi(A). Definition ( 3 . 1 3 ) : We define n positive unital (linear) maps 7* : Bk —» A (fc = 1 , . . . ,n) to be independently covering (for the C*-algebra A B H), if for Bk £ B% but -fk(Bk) not invertible (7*(B fe ) ^ 0, Vfc) 3u € SA : w o lk{Bk) = 0 Vfc = 1 , . . . ,71. By (2.5,i), this condition is equivalent to the following reformulations: BkeB+
(k = l,...,n):~(k(Bk)?0
Vfc = ^
£>fc(i?0*0 k=\
<»: BkeB£
(k = l,...,n):Yl7k{Bk)>0
=>•
3k : fk(Bk)
> 0.
We say that two maps 71 and 7 2 as before are "commuting" (denoted symbolically [71,72] = 0) if [ll(B1),l2(B2)} = 0 VB* £ Bk (fc = 1,2). P r o p o s i t i o n ( 3 . 1 4 ) : If n positive unital (linear) maps 7* : Bk —> A from finitedimensional C*-algebras Bk into A 3 11 (fc = 1, ■ ■ ■ ,n) are independently covering and pairwise commuting, [7;,7j] = 0 Vi ^ j £ { l , . . . , n } , then # ( 7 1 , . . . , 7 , , ) = E]t=i # ( 7 * ) ; i.e. t h e optimal upper bound of (3.11,ii) is attained. In particular, this is the case if A = «4j ® . . . ® A . with (e.g. nuclear) unital C*-algebras A 3 Bt and 7* : Bk -» flj ® . . . ® A ® . . . ® 11„ (Vfc = 1,.._., n ) . S k e t c h of Proof: Let ak 6 0$(Bk) be such that H(jk) = log N(lk[ak]2) Vfc = 1 , . . . , n , by (3.10). Then, for the corresponding ' ^ - m i n i m a l " subcovers 7*[o4]2 £ 0+(A) (a'k C ctk with cardajj. = expH(fk)), clearly 71[o4]2V . . . V7n.[o4]2 is always a subcover of the "unprimed" 71-fold V-refinement of all 7t[afc]2; but it is even minimalhy the assumption
252 (3.13) for {*fk\k = l , . . . , n } , as otherwise at least one a'k could not be 7t-minimal (an H elementary C'-exercise for the reader). Thus by Def. (3.8), # ( 7 ! , . . . ,7„) > E L i ilk)\ and (3.11,ii) gives the converse. The rest is obvious. L e m m a ( 3 . 1 5 ) : Let Ai,Ai C A be unital C - s u i a l g e b r a s of A 3 11 which are com muting ([Ai, ^2] = 0), then i^t, and i^ a (the *-homomorphic inclusions) are independently covering iffAi and A2 are (7*-independent in the sense of [28]; with proof of the nontrivial implication similar to that of (3.2,(1)=>(2)) in [28]. R e m a r k ( 3 . 1 6 ) : For an re-tuple of positive unital (linear) maps 74 : Bk —» A (k = 1 , . . . , re), an "Abelian model" is determined by an n-tuple (0:1(71),..., a„(7„)) of 0^(7*) € Oj"(Bfc) as in Def. (3.8), and is defined as follows: For a* = {fif*' e 8£\i* = l,...,Nk: E ^ i ^ i ^ ) 2 = M . choose the Abelian C*< algebra C = ®J = i(©i 1 =i( C); t ), with corresponding minimal projectors e(i,,...,y £ C. Using the notations I„ = ( i i , . . . ,i„) £ IN" and Ein = 7„[B,„] 2 • . . . • 7i[Bi 1 ]2, define a positive unital linear map Pu : A—* C (for fixed state w £ Sj) hy
PU(A) = J2v(El
■ A ■ Eu) ■ w{ElEK)-'
■ ein,
and a state /iw £ Sc by /J u (e/„) = u(Ej„El*)Then, for fixed ( a x , . . . ,arl), w i-> P„ is a map from the state space SA of A into the set V\{A, C) of positive unital maps from A to C; and a; i-» /i„ is a weak-* continuous affine map M : SA —► Sc, whose adjoint Af * : C —> .A is given by the positive unital (linear) map M*(ejn) = Ej Ei„. Now, it is not true in general that /i„ o P„ = u Vw £ SA (only if u> is "tracial" at least for •£/„, V7„); but C/ n : u i-» /io,(e/„) = w 0 M*(e/„) is a weak-* continuous (affine) function on SA, Cj^ £ C(SA), and for /? M = { C / J / „ } £ Oi + (C(S,i)) we have ff(7l>... , 7 n ) = m a x { M = M ( a i , . . " „ . ) } l o g JV(/3M), by definitions (3.8), (3.2,i) resp. (2.6,i), and (2.5,i). Note, finally, that for the pure states ej^ £ Sc C C*, defined by ej„(ej„) = 5/„j K) we get the "reduced" states e'K o P„(A) = U ; ( E £ • A ■ Eu) ■ / t „ ( e / , J - \ VA"G A, D e f i n i t i o n ( 3 . 1 7 ) : For a positive unital map of finite rank 7 • B —> A into a C*algebra A 3 31, and for 8 £ * - Aut(.A), the "entropy" of 8 w.r.t 7 is defined by h(8,-y) = lim,,-,,,, ^H("1,80 7 , . . . , 8"-1 0 7 ) , where the limit exists by (3.11,ii & iv), and actually is the infimum. P r o p o s i t i o n ( 3 . 1 8 ) : The "entropy" (3.17) has the following easily provable properties (as corollaries of (3.11)): (i) h(8,f)
< H(-y), generalizes (2.2,e).
(ii) For 72 : B2 —» A and 71 : # i -> B2 with finite-dimensional Bk (k = 1,2): H0,ti o 7i) < ^(^,72); generalizes (2.2,d).
(iii) h(uo8oo-'l,o-of)
= h(8,j) W £ * - Aut(„4).
253 (iv) R ( 0 n ) 7 ) < \n\ -h(e,-t) is not provable here.
Vra 6 TL (generalizes (2.2,f), if n = - 1 ) ; but K(9,y) <
ft(0n,7)
(v) For A = Ax ® At, 7* : Bk -> A (e.g. with nuclear Ak, k = 1,2), and 9 = 6x ® 02 with 0* £ * - A u t ( A ) : A(»,7i ® 72) > Keut\) + M ^ J ) . as follows from Lemma (3.19) below; but the converse inequality is unfortunately improvable thus far (cf. [29, Lemma(3.4)] for the C N T entropy). (vi) If there exists a sequence of positive unital injective maps f„ : Bn —> A of finite rank, such that for given 7 : B —» A: {7(Bx)'> ■ 9 o 7 ( B 2 ) * . . . . . 0 - 1 o 7 ( S „ ) . . . . . 7 ( ^ 0 " | V B t £ B+, ||B*|| < 1} C 7 „ ( S + ) , Vn 6 IN, then h(9,j)
< liminf^jtf i # ( 7 „ ) .
L e m m a ( 3 . 1 9 ) : Actually, we need for the proof of (v) the following corollary of the proof of (3.14): For A = Ax ® A2, let fik £ 02(Ak) (e.g. with nuclear Ak k = 1,2), and <*i = 0x ® Bj, 0*2 = Hi ® £2 £ 02(A), then ^ ( 0 ^ 0 2 ) = ^ ( a i ) • N(a2) (again by much the classical argument as implicitely used in [16], using (3.13) for Ak). Definition ( 3 . 2 0 ) : Let A 9 11 be a nuclear C'-algebra and choose an approximating net of completely positive unital maps r = {T„ £ CV\(A)\v £ N}, r„ : A„ —> A (with corresponding av : A —* Av) as defined in (2.9,ii). We define the "r-topological" entropy of 9 £ * — Aut(«4) by repeating (2.12): ftT(0) = limsup„ € A r ft(5,r„). For IT £ * — Aut(_/4), we use the notation cr(r) = {
O9O a - 1 ) = hT(9), W £ * - Aut(.4); generalizes (2.2,h).
(ii) hT(9n) < \n\ • hT(9), Vn £ TL (unfortunately, the converse inequality is unprovable thus far; cf. [30, Corollary(2.9)] for the CNT entropy); generalizes (2.3,i) partly. (iii) For A - Ax ® A2 with nuclear Ak and 9k £ * - A u t ( A ) , f> = h ® 92 £ * - Aut(.4); and for approximating nets Tk = {rVk . Avt —> Ak\vk £ Nk} of Ak, also r = {r^ ® T ^ : A ® Av, —» ^|("i> "2) € Nx * N2] is an approximating net of A Then by (3.18,v): ftT(0) > fcT1(0i) + ft„(02); generalizes (2.3,ii) partly. P r o b l e m ( 3 . 2 2 ) : Characterize those approximating nets r of A as in (3.20), for which the (first "formal") supremum h(9) = sup 7 € C P l ( y t ) ft(«,7) = s u p , ^ ^ fc(0,7) (cf. (3.12)) is attained: hT(9) = h(0) W9 £ * - Aut(.4); cf. (2.12) for Abelian A. Is there any relation still to the partial pre-order (3.2,ii) for non-Abelian Al Note that it is possibly not true in general that K(r){9) = fc(0), contrasting (3.21,i) above.
254
3.2
AF algebra theory and first applications
Let A = U n £ IN-4n be a (separable) unital AF algebra, i.e. An C A.+1 (Vn e IN) with unital inclusions of finite—dimensional C*-algebras An, and A = A^ is the norm comple tion of the inductive limit A » = U„ 6 IN A n which is a locally semisimple *-algebra (with pre-C*-norm). A is nuclear with approximating maps (2.9,ii) given by r„ = ij^ . An —» A (*-homomorphic inclusions) and with any chosen norm-one projections (conditional ex pectations)
\n\-K{8,B)
<
h{6,B2).
Vne7L.
(iii) If there exists an increasing sequence B„ C Bn+i of finite-dimensional subalgebras Bn C A such that B, 8(B),..., 8"~1(B) C B„ (as not necessarily commuting subalge bras) Vn e IN, then h{8, B) < H m i n f ^ ^ ^H(Bn). S u b c o r o l l a r y ( 3 . 2 5 ) : For an AF algebra A = A^ limn-.oo k(8, A„); follows from (i).
as in Def. (3.23), hAa,(8)
=
Again generally, we can concretize also (3.11) by C o r o l l a r y ( 3 . 2 6 ) : The entropy H{Bu...,Bn) with (3.8) for finite-dimensional su&algebras Bk C A (h = 1,... ,n) has the following general properties (from (3.11)): (i) For Bk H(B,
C Ak
C A, such that for At Bn)
= Aj also Bt = Bj (Vi,j
=
l,...,n),
(ii) H(B1,...,Bm)
<
,B) = H(B) with arbitrarily many repetitions of only one single B.
(iv) For6£*-Aut(A),H(8(B1),...,8(Bn)) (v) If there exists a H(Blt...,Bn)
finite-dimensional
= B
H(B1,...,Bn).
C A such that Bx,...,Bn
C B
then
255 (vi) If B%,... , B„ are pairwise commuting ([Bi,Bj] = 0 Vi ^ j) and independently cover ing in the sense of (3.14),(3.15), then H(BU ...,B„) = £ L i B{Bk). Note that in (vi) the condition (3.13) for {Bk\k = 1 , . . . , n } is seemingly stronger than only pairwise C*-independence of (0,-, Bj) Vt ^ j in the sense of [28], as it means that also the "multiple correlations" between Bi,...,Bn vanish in the sense of [28, (3.2,4)]. P r o p o s i t i o n ( 3 . 2 7 ) : In addition, the entropy (3.8) has the following special properties referring to resp. only one (particular) finite-dimensional su&algebra of A:
(i) For Si,.. -, B„ C A as in (3.26), H{BU..., Bn-UB„, ...,Bn) = B(BU..., B„) with arbitrarily many repetitions of the last entry B„ in H (but not provable for the other entries B\,. ■. ,B„_i analogously!); generalizes (3.26,iii) a tiny bit further. (ii) For Bi,B2
C A, we define Bi C B 2 <=» VBi e Bx 3B2 G B 2 : \\Bt - B , | | < 8\\Bt\\.
Then if Bi C B 2 C A with S < 10" 4 , always H{B{) < H(B2) more than one argument analogously!).
(but not provable for
C o r o l l a r y ( 3 . 2 8 ) : Defining in (ii): | | B i - B , | | = inf{5 > 0|B, CB2,B2 C B j } , we have obviously: ||Bi - B 2 || = S with 6 < 10" 4 = > H{Bi) = H{B2). Although this looks like a "first step" towards a norm-continuity of %{6,B) analogous to (2.9,i), it seems to be already the "last" step (into an impasse); not to speak about the generalization for h(8,f) (for 7 6 CVi(A)), in view of our "no-go" Proposition (2.10) already for Abelian A. Note that the premise of (3.28) is actually nonsense for Abelian A 3 Bi,B2, as then rather obviously ||Bi — B s || = 1 VBi ^ B 2 ; which could be viewed as rendering any non-commutative analogue of (2.9,i) "unnecessary" even for h(6,B) (cf. however [31, Prop.l] and its discussion in (3.35,2) below). P r o o f of ( 3 . 2 7 ) : (i) By Def. (3.8), we have to consider for the l.h.s. JV(a : Va 2 V . . . Vo^VovV . . . Va„) with ak G Ot{Bh). Now o V ? ^ . . . tfon € 0 2 ( B n ) only, but as in (3.10) B n = | a „ \ 7 a „ \ 7 . . . \ 7 a n | e 0 £ ( B „ ) , and obviously N(ax<j... Vo^-W/S,,) is the same as the above expression. Thus H{B\,... ,B„,B„,... ,B„) < ff(Bj,... , B „ _ i , B n ) , and (3.26,ii) gives the converse. (ii) By Thm. (5.3) of [32], there exists a unitary U £ .A (such that ||n - U\\ < with UBiU' C B 2 . Thus by (3.26,i & iv): H(B2) > HiUBiU*) = H{BX).
120\/S)
Now let again A = U„ e IN -^n D e AP, then A is uniquely determined up to * isomorphism by the norm-dense, locally semisimple *-subalgebra Ax = U n e IN "^"1 o r equivalently by a corresponding Bratteli diagram [33], which is in turn equivalent to the sequence of inclusion matrices { [ A . -> A.+i]|Vn G IN} (cf. [34] for the notation and ter minology in the following). D e f i n i t i o n ( 3 . 2 9 ) : We say with Choda [34] that the sequence {A»} n e ]N a s above is periodic with period p, if 3n 0 £ IN such that V? > n0:
256 (i) [Aj -» Aj+i] = [Aj+P -»
Aj+P+1]
(ii) The (hence necessarily square-) matrix Tj = [Aj —> Aj+P] is primitive (■<=>■ 3< £ IN : (Tf)ik > 0 Vi, /s •*==>• by the inclusion Aj C A;+
=
T
(^ii(n))/<%Ti)i
wnere
(%£\) corresponding to the trace T £
5 ^ by
fljb(„) a 1 6 *k e minimal central projections of An (Vfc(ra) =
1 , . . . , N„, Vra £ IN). As indicated in [34], we have tj = /? • tj+p
Vj > n0. Fix j > n0 and
let Tj = minjfcjy) tW.> and Sj = m a x ^ j t B ' - y Then since
i = Y, 4 i+ " p) 4 i+np) = £/3~M i) 4 :,+ " p) *W+np)
Mi)
(note that Nj = Ni+np Vn £ IN), we get / S ^ J 1 < E ^ " 1 " " 0 < fi^j1, Vn £ IN. By Def. (3.4), E * 4* + " , ' ) = ^KA-t-p). so that by (3.5) and Def. (3.8): ralog/3 - loga,- < H(Aj+np) < nlog/3 - logr,, Vn 6 IN, and H m . ^ . \B(A.) = l i m ^ iff(.A,+np) = ilog/3D e f i n i t i o n ( 3 . 3 1 ) : For an AF algebra .4 = U„ e ]N A . = A ^ , « £ » - Aut(.A) is said to be "^loo-shifty" if the following conditions are fulfilled (cf. [34]): .,6m-\Aj)
(i) V ] > e M 3
as (not
(ii) there is a sequence {n,- £ IN|nj+i > rij}.^ such that Aj,8n'(Aj),.. . ,0*"»'(Aj) are pairwise commuting and independently covering as in (3.26,v), VJfc € IN; and such that lim^oo ^ ^ = 0. T h e o r e m ( 3 . 3 2 ) : Let A = U„ e IN A be an AF algebra with an "A=o-shifty" auto morphism Be*Aut(A), then hAJ{8) = l i m , ^ , ^H(An)Proof:[34] By (3.25), (3.26;ii, iv & v) (cf. also (3.24,iii)!), hAJ6)
= lim h(8,Aj)
= lim Urn -H(Ai,0{Aj),...,8n-1(Aj))
j—too
* J ™ i f e bB{A»
<
j-»oo n-too 71
■ ■ •en'i(A^
—
.,8n-\Aj))]
+ H(8-*\Aj),..
< Urn lira j i n ^ -[H(a(An))
+ H(8"-i+1
= lim lim inf - [ # ( A . ) +
ff(Ay-i)]
o a(Aii-1))] = lim inf
= S M ,
<
257 On the other hand, by (3.24,ii) resp. (3.21,ii): n , • KA_{8)
> KA„(8"i)
> ft(On',A) =
H(AS)
by assumption (ii) and (3.26,vi). Hence Tit
h**.(') A„(e) >
v n
i
A \
=
TTf A \
ff(-4,)
_ n
• TTi
A \
" = y^ - ' 3 j"^» 3 nj
which implies hAao{8) > limsup„_ 0 0 ±H{A„)
i
(again using assumption (ii) on {n,}).
C o r o l l a r y ( 3 . 3 3 ) : Let A = U „ € I N A , be an A F algebra with a periodic sequence {A»}„ S ]N (with period p and Perron-Frobenius eigenvalue /3 of the inclusion matrix [Aj —» Aj+P] Vj > n 0 ) , then for any " . ^ - s h i f t y " 6 E*- Aut(.A): hAac(8) = i l o g / 3 . E x a m p l e s (3.34): (a) Let A(n) = <S)ke7i{Mn)k be the ra°°-UHF algebra and AN(n) = ®£ = _ JV (iW n )* I A » ( i ) = Ujve]N An(n); then for 9n £ * — Aut(.4(n)) determined by the unit shift ("n-shift"?) 8n : ( M . ) 4 -» (M B )i+i (V* £ IN), we get I U . W ( « L ) = logn (foUows from (3.33) with p = 1 and [-4jv(n) —► -4j\r+i(Ti)] = n £ IN). (b) As{g) = C"({e ; |i = -JV,. ..,N}) with the relations e\ = eit e\ = 11 and e.-ey = e,-ei(-l) a ( l '- , ' l ) (i f j) with S : IN -» {0,1}. Again A ^ s ) = U„ € IN ^AT(S) and Ag = A * , ^ ) ; then for 6g £ * — Aut(^4,) determined by 0,(ei) = ei+\ Vi € ffi, we get hA„(g){9g) = j l o g 2 if g(n) = 1 Vra £ IN or if 3n 0 : j ( n ) = 0 Vra > n0 (but no< with j = 0, in which case we would have the classical "2-shift" with h(8) = log 2, corresponding to example (a) with 8 restricted to a maximally Abelian (7*-subalgebra C({0, l } 2 2 ) C .4(2)); see [14] for the proof. (c) -4i\r(A) = C*({pi\i = —N,..., N}) with the relations p,- = p*{ = p], PiPi±\Pi = Ap; and \pi,pi\ = 0 Vi, j : |t - j \ > 2; for A £ {(4cos 2 i ) - » | m £ IN \ {1,2}}. Again -4<„(A) = U^glN Av(A) and A = A»(A). Then, for 0* £ * - Aut(Ax) determined by S\(Pi) = Pi+i Vi £ TL, we get hAooW(8x) = - ^ l o g A . (Note that (c) for A = \ is identical with (b) for g(n) = S„i by pi = | ( e ; + H); and for A ^ | , (c) follows again from (3.33) by the same argument as in [34]). (d) Let (AA,@A) be the shift 9A on an AF algebra AA associated with a topological Markov chain as treated by Evans [31] with his "AF-imitation" of the topological entropy (via the Connes-St0rmer entropy [6]); but without repeating this lengthy example here, we just compare with Evans' notation: Our (AA,8A) = (CA,o-0) of Evans, where A is an aperiodic (n x n ) - m a t r i x with entries in {0,1}, and our A, = N. (with a £ IN) resp. Aoo(A) = U, £ IN N, of Evans. Then, repeating the proof of the main theorem in [31] with our H instead of Evans' H and using properties (3.26,i & iii) and (3.10) of H (resp. its definition (3.8)), we immediately get the same result:
258 'MootAjCM = 1°6 ^i where A is the spectral radius of A (note that we do not need at all even the ingredients of Evans' Prop. 2 for the second part of the rewritten proof, as it amounts in our case to the same estimate by "log |.M,,,+jt|" in Evans' notation as the first part, only from below instead from above). C o n c l u d i n g R e m a r k s (3.35): 1. Possibly, the main problem (3.22) of our thus ''pedestrian" approach could be solved more easily in this special AF case as above, i.e. the problem to characterize (or identify) the norm-generating locally semisimple *-subalgebras Ax. of the A F algebra A such that the (first "formal") supremum h(0) = sup 76C7 , l (_ 4) ft(6,7) is attained:
W ) = *(*). W e * - AutC/4). 2. The advantage of Evans' approach [31] cited in (d) is to "circumvent" this problem (1.) by using the norm continuity (2.9,i) of the Connes-St0rmer entropy ([6], for finite-dimensional subalgebras B C A); but on the other hand, this "AF-imitation" of the topological entropy has the serious drawback of an "AF-limitation" in the fol lowing sense: To extend Evans' definition further for n o n - A F (nuclear) C""-algebras, one would have to repeat his construction with the CNT entropy as in (2.9,i) for 7 € CPi(A) instead of the Connes-St0rmer entropy; and then it seems to be impos sible (to show?) that this analogously extended definition of Evans' entropy }IB(8) coincides again with the "classical" topological entropy (2.1,iii) for a homeomorphism T : X —> X of a connected compact space X: h.E(8T) = h(T)? (Cf. [5, App.], and contrast with Thm. (2.7) here.) 3. Naively, one could think of a direct "classical" definition of a "quantum" topological entropy hci(6) for a. C*-dynamical system (A,9), simply defining hci(6) = h(Tg) with the topological entropy (2.1,iii) for the adjoint (affine) weak-* homeomorphism Te : SA -» SA (defined by Te(u>) = u> o 8, Vu £ SA). But by Thm. (2.3,iii) this is no generalization of the topological entropy h(T) ^ h{Tia), and one could only think to regard it as a non-Abelian analogue; but also that is definitely ruled out by the following most simple "counter-example": In (3.34,a) already with n = 2, hci{02) = oo; which we leave to the reader as a not quite trivial exercise (see [10]). 4. In all the AF examples (3.34), we always have the desirable generalization of (2.8): '"■AoW — suPu€S» hu(6), with the CNT entropy on the r.h.s.; but without having computed hT{6) for any non-AF nuclear C'-algebra with ("optimal") approximating net r (and also not yet the CNT entropy, for non-trivial u> and S, by the way!), we are far from a conjecture (not to speak about a general proof, cf. already [5] for Abelian
A). A c k n o w l e d g e m e n t s : I have to thank the organizers of "QP VIII", Profs. L. Accardi and W. von Waldenfels, for inviting me to this meeting in Oberwolfach with its included financial support by the institute; and thanks go to the referee for pointing out a (terrible) mistake in the original proof of the nevertheless correct Lemma (3.30).
259 Financial support by "Fonds zur Forderung der wissenschaftUchen Forschung in Osterreich" (Proj. P7101-Phy) is gratefully acknowledged. R e m a r k : Two earlier versions of this contribution have been previously distributed as preprints UWThPh-1991-62 (Vienna) resp. NTZ-37/1991 (Leipzig).
References [1] Connes, A., Namhofer, H. and Thirring, W.: "Dynamical Entropy for C* Algebras and von Neumann Algebras", Commun. Math. Phys. 112, 691-719 (1987). [2] Park, Y. M. and Shin, H. H.: "Dynamical Entropy of Quasi-local Algebras in Quan t u m Statistical Mechanics", Commun. Math. Phys. 144, 149-161 (1992). [3] Effros, E. G.: "Why the Circle is Connected: An Introduction to Quantized Topol ogy", Mathematical Intelligencer 11, 27-34 (1989). [4] Blackadar, B. E.: "A simple unital projectionless C*-algebra", J. Operator Theory 5, 63-71 (1981). [5] Hudetz, T.: "Algebraic Topological Entropy", in Nonlinear Dynamics and Quantum Dynamical Systems, G. A. Leonov et al. (eds.), Akademie-Verlag Berlin (1990), p. 27-41. [6] Connes, A. and St0rmer, E.: "Entropy for Automorphisms of Hi von Neumann Alge bras", Acta Math. 134, 289-306 (1975). [7] Connes, A.: "Entropie de Kolmogoroff—Sinai et mecanique statistique quantique", C. R. Acad. Sci. Paris Ser. I Math. 3 0 1 , 1-6 (1985). [8] Hudetz, T.: "Non-linear Entropy Functionals and a Characteristic Invariant of Sym metry Group Actions on Infinite Quantum Systems", in Selected Topics in QFT and Mathematical Physics, J. Niederle and J. Fischer (eds.), World Scientific, Singapore (1990), p. 110-124. [9] Lindblad, G.: "Dynamical Entropy for Quantum Systems", in Quantum Probability and Applications III, L. Accardi and W. von Waldenfels (eds.), Springer LNM 1303, Berlin (1988), p. 183-191. [10] Hudetz, T.: "Quantum Topological Entropy: The 'Canonical' Approach", in prepa ration (1992). [11] Hudetz, T.: Ph.D. Thesis, Univ. Vienna (in German), unpublished (1992). [12] Emch, G. G.: "Kolmogorov Flows, Dynamical Entropies and Mechanics", to appear in Quantum Probability VII (New Delhi), L. Accardi et al. (eds.), World Scientific, Singapore (1992).
260 260 260 K-Systems , Commun. Math. Phys. 125, [13] ]Narnhofer, H. and Thirring, W.: "Quantum 565-577 (1989).
[13] ] Narnhofer, H. and Thirring, W.: "Chaotic [13] ] Properties of the Noncommutative 2-shift", [14] 1 to appear in From Phase Transitions to Chaos, Topics in Modern Statistical Physics, [14] 1 G. Gyorgyi et al. (eds.), World Scientific, [14] 1 Singapore (1992). [15] !Sauvageot, J.-L. and Thouvenot, J.-P.: "Une nouvelle definition de l'entropie dynamique des systemes non commutatifs", Commun. Math. Phys. 145, 411-423 (1992). [15] ! Adler, R. L., Konheim, A. G. and McAndrew, [15] ! M. H.: "Topological Entropy", Trans. [16] , Am. Math. Soc. 114, 309-319 (1965). [16] , Bauer, W. and Sigmund, K.: "Topological [16] , Dynamics of Transformations Induced on [17]1 the Space of Probability Measures", Monatshefte Math. 79, 81-92 (1975). [17]1 [17]1 [18] !Sigmund, K.: "Affine Transformations on the Space of Probability Measures", Societe Mathematique de Prance, Asterisque 5 1 , 415-427 (1978). [18] ! [18] ! [19] 1Dinaburg, E. I.: "On the Relations Among Various Entropy Characteristics of Dynam ical Systems", Mathematics of the USSR - Izvestiya 5, 337-378 (1971); reprinted in Scientific, Singapore (1991), p. 97-138. [19] 1 Dynamical Systems, Ya. G. Sinai (ed.), [19]World 1 [20] ( Goodman, T. N. T.:"Relating Topological Entropy and Measure Entropy", Bull. Lon don Math. Soc. 3, 176-180 (1971). [20] ( [20] ( [21] ]Kuriyama, K.: "Entropy of a Finite Partition of Fuzzy Sets", J. Math. Anal. Appl. 94, 38-43 (1983). [21] ] [22] ]Malicky, P. and Riecan, B.: "On Theory and Related Topics II, H. 135-138. [22] ]
[21] ] the Entropy of Dynamical Systems", in Ergodic Michel et al. (eds.), Teubner, Leipzig (1987), p. [22] ]
[23] 1Hudetz, T.: "Entropy and Dynamical Entropy for Continuous Fuzzy Partitions", in preparation (1992). [23] 1 [23] 1 [24] .Gillmann, L. and Jerison, M.: Rings of Continuous Functions, Springer GTM 43, New York (1960). [24] . [24] . [25] I Choi, M.-D. and Effros, E. G.: "Nuclear C - a l g e b r a s and the Approximation Prop erty", Am. J. Math. 100, 61-79 (1978).
[25] I [25] I [26] I Choi, M.-D.: "A Schwarz Inequality for Positive Linear Maps on C*-algebras", Illinois J. Math. 18, 565-574 (1974). [26] I [26] I [27] ]Kadison, R. V.: "A Generalized Schwarz Inequality and Algebraic Invariants for Op <erator Algebras", Ann. Math. 56, 494-503 (1952). [27] ]
[27] ]
<
<
261 [28] Summers, S. J.: "On the Independence of Local Algebras in Quantum Field Theory", Rev. Math. Phys. 2, 201-247 (1990). [29] St0rmer, E. and Voiculescu, D.: "Entropy of Bogoliubov Automorphisms of the Canonical Anticommutation Relations", Commun. Math. Phys. 133, 521-542 (1990). [30] Hudetz, T.: "Spacetime Dynamical Entropy of Quantum Systems", Lett. Math. Phys. 16, 151-161 (1988). [31] Evans, D. E.: "Entropy of Automorphisms of AF Algebras", Publ. RIMS Kyoto Univ. 18, 1045-1051 (1982). [32] Christensen, E.: "Near Inclusions of <7*-algebras", Acta Math. 144, 249-265 (1980). [33] Bratteli, 0.: "Inductive Limits of Finite Dimensional C""-algebras", Trans. Am. Math. Soc. 171, 195-234 (1972). [34] Choda, M.: "Entropy for *-Endomorphisms and Relative Entropy for Subalgebras", J. Operator Theory (to appear).
Quantum Probability and Related Topics Vol. VIII (pp. 263-270) ©1993 World Scientific Publishing Company
ON FEYNMAN-KAC COCYCLES R L Hudson Mathematics Department University of Nottingham Nottingham NG7 2RD UK
Abstract. We consider the operator Feynman-Kac formula in which the unperturbed semigroup is got by contracting a quantum stochastic evolution, and ask when is the perturbation cocycle of exponential integral form.
§1. Statement of the problem An analysis of the Feynman-Kac formula [HIP] at operator rather than super-operator [Ac] level, reveals that the unperturbed semigroup is the expectation of a random unitary evolution and the per turbed semigroup is the expectation of a perturbation of this evolution which is effected by a cocycle possessing covariance properties for the one-dimensional Euclidean group of translations and reflections of the real line. Such random evolutions are readily constructed using quantum stochastic calculus. Let Us, denote the solution at time s of the quantum stochastic differential equation dU = U(ldAf-l*dA + (ih-it*[)ds),
U(l) = 1
(1.1)
in which / and h = h* are ampliations to the Fock space T\L2(R)) of operators in an initial space K0. Then (t/ (i , : i S t) is a covariantly adapted evolution in the sense of [HIP] in which the filtration of von Neumann algebras (N,(NSJ : s 2 t)) is got by taking N = B(K0) ®B(r(L2(R))),
N,j = B(3C0) ® 1 ® B(r(L 2 [/, s])) <8 I
and the Euclidean group acts by second quantising shifts and reflections on L2(H). If h = 0 (Us,) is reflectively covariantly adapted. The vacuum conditional expectation onto BQC0) is a reflectively covariant reducing map. The reduced evolution T, = E 0 [ ^ + , . J is the contraction semigroup on H0 generated by ih-\l*l.
Given VeB(K0)
identified with its
264
ampliation to Fock space we may now construct the cocycle Mv = (M^,, s 5 () defined by the ordi nary differential equation
a,<( = - < X , w - , \
M,r, = i,
the perturbed evolution <,=<,£'../ and the corresponding semigroup T , v = E 0 [Z// + ,, S ]
whose generator is i/i-j/*/+V. The formula e'<*-^'-"'=E0[M/+/,s{/st,J
(1.2)
is die abstract form of Feynman-Kac formula of [HIP]. The standard Feynman-Kac formula of Brownian motion is obtained by taking h = 0, I = I* = p, the momentum operator in Ka = L2(R), identifying i(A1-A) as Brownian motion, and taking the perturbation potential V to be a function of q. The 'oscillator process' Feynman-Kac formula of die Omstein-Uhlenbeck process [Si] is got by tak ing / = a, die annihilation operator q + ip [HIP]. In bodi rnese cases, and also [HPl] in die case of die 'ami Ornstein-Uhlenbeck process' got by taking / = a1', provided mat V is a function of q (or more generally, of a fixed real linear combination V = v(xp+yq) of p and q), the process (Us ,VU~} : j 5 l ) is commutative. (Note that in all diree cases / is unbounded.) Consequently [HIP] die cocycle A/v can be expressed in exponential integral form Ml, = e x p l - j " U^VU'l
dz
The purpose of diis paper is to investigate conditions on V, h and / under which this is me case. By covariance, die problem amounts to the following. When is the process (V(t): t 5 0) commutative, where V(t) = U(t)VU(t)~l, and V(t) = U, 0? Acknowledgement I am grateful to J M Lindsay for pointing out an error in an earlier version of mis work and for suggesting numerous other improvements.
265 §2. The bounded case Assuming that V, h and / are bounded operators on the initial space Ha we denote by j the flow jt(x) = U(i)xU(t)-\
xeB(Xa)
on B{3C0) and by a, a t and t its structure maps a(x) = [l,x], r(x) =
a\x)=[x,l*),
Hk,x)-i(l*lx-2I*xl+xl*0,
so that dj,(x) = x + ! Jo
(j,(a(x))dA*+j,(.a\x))dA+j,(r(x))ds).
Theorem 2.1. Under these assumptions, the process V(f) is commutative if and only if, for arbitrary n e N, [V,<-„...r,(V)] = 0,
(2.1)
where Kl,...,Kn are arbitrary elements of the set of structure maps if = [a, af,x\. Proof. Assuming that [VJ,(V)] = 0 ,
I H
we have, since dV = 0 and V commutes with dA*. dA and dt 0 = [V,dj,(V)] = I
[VJ,«V))] dK,
Keif
where dK is the stochastic differential corresponding to f e / By independence of the stochastic dif ferentials [Li] we must have [V,i(n(V»] = 0 , ( 5 0 for each Keif. Repeating the argument, differentiating n times in total, and finally setting t = 0, we obtain (2.1). For the converse we use an adapted form of the argument of Theorem 7.1 of [FS]. We have [V,;,(V)] = 2
[ [V,/,(JC
266 Since (2.1) holds, we may iterate, obtaining
[VJ,(V))=
I
\
[V,j,.(Kn...K1(V))]dK„(sn)...dKl(s1).
Using the basic estimate of quantum stochastic calculus in me form II f EdKu®y(f)\\2 Jo
=S ctf.T)
[ Jo
\\E(s)u®v(f)\\'1ds
for 1 =S T and locally bounded / , we find that for arbitrary such / and u e X0, using the contractivity of
u ||[V,/,(V)1«®IK/)I 2 «^2^^II^1II"®^/)II 2 -T* 0 where M = max( \a\, B(B(X0)).
\\af\\ [T|| } and a , a f and T are regarded as elements of the Banach space
Hence [VJt(V)]
= 0 as required. D
We consider the case when X0 = L 2 (R) and seek evolutions U, for which the Feynman-Kac cocycle Mv will have exponential integral form for every potential of the form V = v(q) of a bounded measurable function of the position operator q. By Theorem 2.1 we must have, for all such V, [V,[/,V]] = 0.
(2.2)
If u is a continuous bijective function, for example if v(x) = tanhAx for fixed i e R , operators which commute widi V are themselves multiplications so we can write U.V]=V,
(2.3)
where V is multiplication by a function 0. Since a d , : V t-> [/, V] is a derivation, die same is true for products of such functions v and hence, by the Stone-Weierstrass theorem and boundedness of the map adi, for arbitrary V belonging to the commutative C*-aIgebra of multiplications by continuous functions on R possessing limits at =F~. Passing to weak operator limits using the weak operator continuity of ad, on bounded subsets of the von Neumann algebra B(X0) we conclude that (2.3) holds for multiplications V by arbitrary bounded measurable functions and hence that ad, is a derivation of die von Neumann algebra M of such multiplications. Since Abelian von Neumann algebras have no non-trivial derivations we see diat a = ad, vanishes on M and hence, since M is maximal Abelian, that l e M . In particular, for all V e M, a(V) = 0 and
T(V) = i[h,V}-i(l*lV-21*Vl+VI*l)
= i[h,V].
We may now repeat die argument with / replaced by ih to conclude that h e M and T{V) = 0 for all V e M. Thus, for V e M , j,(V) = V for all / and the Feynman-Kac formula (1.2) takes the trivial form
267 Ki*-i;«j
- ^ E o t e - ' ^ J = «-' «-'vvE E00[C/J+,.J = e -'^'< ; '-i'*')
of a product of commuting semigroups of elements of M. We have proved Theorem 2.2. It is not possible to find a unitary process, driven by a stochastic differential equation (1.1) with bounded coefficients ( and h, for which there is a non-trivial perturbation cocycle of exponential integral form for arbitrary V of the form v(q) where v is a bounded measurable function.
§3. Some remarks about Feynman-Kac cocycles for the Weyl algebra We denoted by W the Weyl algebra of polynomials in the indeterminate p and q satisfying [p,q] = - iJ , [p,q] equipped with the involution t which makes p and q self-adjoint. Theorem 3.1. Let a, a* and T be linear maps W —> W satisfying the structure relations of a quantum flow a(xy) = a(x)y+xa(y),
af(x) = a(;c a(xf)+f )t
t z(xy) = T(x)y+xr(y) (x)a(y), r(x)y+xt(y) + a' af(x)a(y),
T t == rTf.f.
(3.1) (3-D (3.2) (3.2)
In order that for every element V e W of form V= = v(xp v(xp+yq), i , j ee R +■>«). x,y that is a polynomial in a real linear combination of p and q, the condition (2.1) holds, it is necessary and sufficient that a and r take the form o(r) a « = [/,i], [/,*],
f T(AT) = = i[h,x]-i(.l i(x) lx-2lfxl+xlf0, it*.*] -£(/'&-2/^/+*/*/),
(3.3)
where / and h are, respectively, complex and real linear polynomials in p and q. Proof. To prove the necessity, we note first that, since every derivation of If is inner [Se], [HP1], a is of form a(x) = [!,x] for some / e W, and furthermore since r0 defined by t t *oM = - K ' t t - 2 / * x / + ^ / / )
satisfies (3.2), T must be of form
268 i[h,x]-l(lflx-2[*xl+xlt0,
T(x) =
where h = h? e W. To see that / is linear in p and q we first take V = q. From (2.1) we have [?.[',
Similarly, taking V=p,
we find that / must be of the form
qh(p) + k(p) where h and k are polynomials. Combining these statements we conclude that / must be of the form / = bpq + cp + dq + e, where b.cd.e
E C. NOW take V = p + q. Then a(V) = [l,V] = ib(-q+p)-ic
+ id.
It follows from [V, a(V)] = 0 that 6 = 0 and hence that / is linear in p and q as claimed. It can now be verified that each V = v{xp+yq) commutes with zQ(V) = -j(lflV-2lfVl and must thus commute with the difference i[h,V].
+ Vlf!) as well as with T(V)
Repetition of the argument for / shows that h
must also be linear in p and q; it is a real linear combination since h — h^ Conversely suppose that a and % have the forms (3.3). Fix x,y e W. An element w of W com mutes with all polynomials v(xp+yq),
in particular with xp+yq
itself, if and only if it is itself such a
polynomial. The proof will therefore be completed by showing that, if w is such a polynomial then so too are a(w), a^(w), r(w), since then repeated action of the maps a, af and T on the polynomial V can only yield further polynomials, which will commute with V. Assuming that I = cp + dq + e we have C = cp + dq + e and hence a(w) = i(dx-cy)w',
a^(w) = -i(dx-cy)w',
where w' is the derivative of the polynomial w. Writing T(W) in the form T(W) = - j / t a ( v v ) - 5 a t ( i v ) ; we have, using (3.4), that [xp+yq, T(w)] =
-ii(yc-xd)(dx-cy)w+
= 0. Hence r(w) is also a polynomial as required. □
ii(dx-cy)(yc-xd)w'
(3.4)
269 §4. Conclusion Theorem 3.2 indicates that the only Feynman-Kac cocycles constructed by means of quantum stochastic calculus for one-dimensional quantum particle mechanics for which the perturbation cocycle has exponential integral form correspond to the linear quantum flows of [HPl], in which creation and annihilation operators satisfy
linear quantum stochastic differential
equations with constant
coefficients. (The possibility of including the term i[h,x] in (3.3) was not considered in [HPl], but the new freedom generated thereby amounts only to transforming to new canonical coordinates related to the old by a phase-space translation.) In [HPl] three canonical forms for such flows were found under the actions of linear canonical transformations and gauge transformations in Fock space, for which the corresponding Feynman-Kac formulae are, respectively, the usual one for Brownian motion, the 'oscillator process' [Si] one corresponding to the Omstein-Uhlenbeck process, and that for the 'anti-Ornstein-Uhlenbeck process' of [HPl]. Thus it may be asserted that these three are the only 'useful' Feynman-Kac formulae. However, to reach this conclusion we have had to assume exponen tial integral form for the cocycle whenever V is a function of a fixed linear combination xp+yq, and our analysis evidently exploits the resulting canonical symmetry. The question of classifying Feynman-Kac cocycles of exponential-integral type only for potentials which are functions of q remains open.
References [Ac]
Accardi L, A quantum formulation of the Feynman-Kac formula, Random Fields, Vol I, Proceedings Esztergom 1979, ed J Fritz et al, North Holland (1981).
[FS]
Fagnola F and Sinha K B, On quantum flows with unbounded structure maps, to appear in J London Math Soc.
[HIP]
Hudson R L, Ion P D F and Parthasarathy K R, Time-orthogonal unitary dilations and noncommutative Feynman-Kac formulae, Commun Math Phys 83 761-80 (1982).
[HPl]
Hudson R L and Parthasarathy K R, Construction of quantum diffusions, pp 173-198, in Quantum Probability, proceedings, Rome 1982, ed L Accardi et al. Springer LNM 1055.
[HP2]
Hudson R L and Parthasarathy K R, Quantum Ito's formula and stochastic evolutions, Com mun Math Phys 93 301-323 (1984).
[Li]
Lindsay J M, Independence for quantum stochastic integrators, pp 325-332 in Quantum Pro bability VI, eds L Accardi et al. World Scientific (1991).
270 [Se]
Segal I E, Quantized differential forms. Topology 7 147-172 (1968).
[Si]
Simon B, Functional integration and quantum physics. Academic Press, New York (1979).
Quantum Probability and Related Topics Vol. VIII (pp. 271-280) ©1993 World Scientific Publishing Company
THE KERNEL OF A FOCK SPACE OPERATOR I J M Lindsay §0. Introduction Symmetric Fock space over L (Af) is usually viewed as consisting of squaresummable sequences (/„) of symmetric square-integrable functions (/„ on M"). When M is non-atomic however, it is often convenient to replace the sequence (/„) by a single function / on F, the finite power set of M. In this way Fock space becomes an L -space itself — the measure on F being composed from the product measures on M" in a natural fashion ([Gui]). Exploiting this viewpoint, a class of integral-sum operators has been introduced for describing solutions of quantum stochastic differential equations ([Maa], [Mel] p.305, [Me2] p.39, [LI], [LP]). The operators have a formal similarity with the Hilbert-Schmidt class, they are defined however in terms of the union and partition operations which are closely connected with the gradient operator and HitsudaSkorohod integral of Malliavin calculus (see [L2]). Here we are concerned with the 3-argument integral-sum operators introduced by Meyer. The operator X with kernel x is formally the multiple quantum stochastic integral HI where dAfr _
r
x(p,o,T)dA*dAadAT,
(0.1)
) = dA* ...dA*^ etc and A*,A,A are respectively the creation, preser
vation and annihilation processes ([HuP], [Par]). Indeed the action of X on k e L2(r) may be deduced by using the representation k = l k((D)dA%80 and applying the formal quantum Ito relations (see [Mel] p.81, [LM]). In this note the uniqueness of the kernel of such operators is demonstrated. More specifically, it is shown that if an integral-sum operator annihilates a finite particle
272
domain or, equally, an exponential domain, then the kernel must be zero. This in turn indicates a limitation of this class: if an integral-sum operator has domain including either of these domains, then it is determined by its action on that domain. Precisely how the kernel is determined is a little delicate — this will be described in a compan ion paper ([BL]). Operators of the form (0.1) but with a fourth variable, integrated over in the ordinary sense, have also been considered ([LI]). They have the attractive property that the kernel of a product of two such operators is given simply by a sum, over partitions, of products of the two kernels evaluated at appropriate sets. Thus, unlike the convolution of 3-argument kernels, no integration is involved. However, uniqueness clearly fails here. In contrast, if one allows distributions as kernels, then 2-arguments suffice since (0.1) coincides with jj 22K(.a,y)dA*dA K(a,y)dA%dAr r = IT II
i
K{a, K(a, y) d*dYy da dy
where K is the distribution
"lb
,y)(p(aua, couy)dadady
^S
be the map which takes 0 to 0, and reduces s e 5 (n) to the collection of its coordi nates: (ji,...,«„}.
273
r$ inherits a measurable structure from S as follows. Let & be a cr-algebra on 5, then V c r is measurable if, for each n,
(n\rlm-n(0-\Unrn)),
where i0(£7) = 1 if 0 e L7, and 0 otherwise, and m" is the completion of the product measure m". One consequence of non-atomicity that is used all the time is that, for d22, (<» e (rs)d: cOjCCOj *
>0
L^m (Mn).
For a function q>: S—>C, let n^\ Fg —> C denote the product function
:oi continuous and The map q> e L2(M) i-> K® e L2(FM) is
{x9:q>6L2(M), H-plU ^ 1} is total in L2(FM). The fundamental property of the symmetric measure is: it-Lemma: For d > 2 let g : Fd —> C be non-negative, or integrable, then ^...^(cr1,...,crd)dcr1...da,=^|a|
= £T g(a 1 ,.., 0 r d )dcr
where the sum is over partitions of a into d parts: (ai,...,£fy), and dFm(<7) is abbrevi ated to da. A proof may be found in [LP]. As noted by Meyer ([Me 3]), the ^-relation is linear in g, and is trivial for functions of the form 7^® • • • ® ^ ,
274
§2. Integral-sum operators Fix a cr-finite, non-atomic, separable measure space M in which each singleton set is measurable, and let F = rM be its symmetric measure space. For a function x:T3 -> C, x' will denote the following transform of x: x-: (a,p,y) >-> V
x(a,a>,y).
(2.1)
x may be recovered from x' by the Mobius-type inversion: x(a,p,y) = Y
a
(-l)mx'(a,a>,y),
(2.1)'
where the sum is over partitions of p into two parts: (a),a>). Measurability of x (in the product cr-algebra of r ) is equivalent to measurability of x' Definition 2.1: Let x : F 3 —» C and fe -.T -¥ C be measurable, and consider the fol lowing condition on the pair (x, k): \x(a,p,co)k(a)
I
(2.2)
If (x,k) satisfies (2.2), then write X^k for the a.e. defined function (X3A:)(cr) = [ Y
x(ai,a2,o))k(coua2^Jai)dQ)
(2.3)
the sum being over all partitions of a into three: (ori,a2>a3)The condition (2.2) on x, for fixed k, is equivalent to the following condition on its transform x'\ \
\x'(a,p,(o)k(coKjP)\dco < ~
for a.a. (a,p")er 2 ,
(2.2)'
moreover, when satisfied, (X3k)k(a) = f £
x'(a,a,ffl)/t(fi>uS) da
(2.3)'
for a.a. cr. One advantage of this alternative representation is the identity HI
h(avP)x'(a,p,r)k{pvy)dadpdy=
f h(<j)(X3k)(cj)da
(2.4)
275
— valid when either side is defined — which is an immediate consequence of the ^-Lemma. Some other routine consequences of the iLemma are collected next. Lemma 2.2: Let x : f -> C and k : r -» C be measurable. Then (i) (ii)
x=0 a.e. if and only if x'=0 a.e.; if either x=0 a.e. or fc=0 a.e. then X$k is well-defined and vanishes a.e.
Thus each kernel x determines a partially defined linear operator X3 on L0(I~) — the linear space of measurable functions modulo null functions. Sufficient conditions for X3 to be a densely defined operator on L2(F) are given in [Me 2], [L1] and [BL]. The following classes of functions are relevant: 3ta :=[*eL°(D:||*| ( a ) := { j
a#cr|fc(cr)|2dcrl
< ~|
(a>0)
(2.5)
* : = n f l > 0 x* x\ := \xeL°(.n:
3K > 0 s.t. jj ^ sup|a#ac#J'A-#^jc(a,y3,7)|2dady < ~
Va,c > o|.
Kernels from XI determine an algebra of operators each leaving X invariant, and the kernel of the product is given by a convolution-like product of the corresponding ker nels ([Mel] p309, [LI]). X3 is bounded as an operator from Xa to K\, for values of (a,b) depending on x, moreover the bound is given by an L xLTxL norm on the kernel — see [Bel] and [BL] Proposition 2.2. §3. Uniqueness of the kernel Injectivity of the map from kernel to integral-sum operator will be demonstrated in two senses: the operator is first viewed on finite particle vectors, and secondly on an exponential domain. Let M and F be as in §2 and fix a countable generating ring %, consisting of sets with finite measure. Let ## and 8% denote respectively the linear spans of ( j f r , ( ^ f » e N , Ee3t, q> E 3 }
276
K:
reT,
% , u) ■= {
for any finite set F, where the measure on rF
is the counting measure:
//({«})= 1 Va<=/>. Proposition 3.1 Let x: rM —> C. Suppose that x is locally integrable in the third variable, in the sense that for each
neN,Ee9l,
oo for j \x(a,/5,o))\ \x(a,P,m)\ da for a.a. a.a. (a,j5) e r^ . da < < °° (M)e/i. 'r„(E)
Then (i) X^k is defined for each
ke.9^;
(ii) if X3 annihilates &%, then x = 0 a.e. Proof: If & € &x then for some n € H, K > 0 and
Ee9l,
1*1 «*&/•«,#). so (JC, A) satisfies (2.2), and X$k is defined. Suppose X3 annihilates $% . Fix j,k,le and let xj t
IM, put n = y' +fc,m = & + / and /? = min{«,m},
denote x' times the indicator function of rjxrkxri.
;
countable, there is a null set N such that for at j y } r 2ja c for each
I
Then, since 5) is
Nnrm,
x'{a,a,co){x == Q0 K(p)((0^ JCC) da = rnq))(,(Qua)<\cD x'(a,a,o))(zr a
n
ue [Q + iQ]CT and .$ e Q . Thus
^ X ^ I a c c ^ I l o *
1
* ^
,,-,„_,(a, a, co) = 0
for
su)]
in L2(r),
277
for each (s,u); the totality of ( i „ : u e [Q + iQ]a] in L\r(a))\ and the totality of {(l.-s sp): i s Q} in C p+1 ; and Fubini's Theorem, we see that for a.a. (cr,o>)e.T2, x'm-ijin-i(.a,a,co) = 0 Va c a, i = 0,...,p. Putting i = k and applying the -^-Lemma gives 0 =
Hji X a c a \xUi(">a,a))\ dadco
= jjj
\xj,kt^a,p,a»\dadpdtu.
Since j , k and / were arbitrary, x' = 0 a.e., and so by Lemma 2.2 x = 0 a.e. also.
■ Proposition 3.2: Let jc: rM —> C be locally integrable in its third variable in the sense that for each E e %., j \x(a,p,co)\ da < ~ nE)
J
for a.a. (a,/3)er^
.
Then (i) X$k is defined for each ke8%; (ii) if Xj, annihilates 8%, then x = 0 a.e. Proof: If ke8%, then | k | ^ Kxr(E) for some X' > 0 and £ e #, so (x, k) satisfies (2.2) and X3fc is defined. Suppose that Xj, annihilates <S#. Since 3) is countable there is a null set N such that for otN, j da>^(fl))Y
^(cf)jc'(a,a,©) = 0
for each q>e33 for which |MU « 1. Letting q> run through {(pe3>((TiU): \\
278 In the context of distributions, the map (jc9®Kw®n¥,x') = (x
(p,y) i->
is called the symbol of the operator X3. We have shown that 3-argument integral-sum operators are determined by their symbols. Another consequence which may be of use in Fock space analysis is: Corollary 3.3: The following collection is total in L (T ): « := {"V,®*^®*^: { j r f t « i c f l f c ® « ^ : ftft-eS, ||
1 = 1,2}
is orthogonal to S. Let x be the inverse transform of
y, given by (2.1)'. Then for each ;ry e 8%, ( y real-valued), {K9®Kww®n: ®K¥v,y) 0 = (^®n ,y)
{a^XjKy,) == (ltp,X$Xy)
V
and so Xjfiy = 0 a.e. Since X3 annihilates
■
The proof of Proposition 3.1 should have appeared in [LI]. Unfortunately it did not survive the transition from preprint to publication. An indication of the result was given by Meyer, who examined the matrix elements of X3 with respect to 3- and 2particle functions ([Me 2] p.46). Attal is considering the multidimensional case, and has results on the uniqueness of (n + 2n)-argument kernels ([At 2]).
279
References [At 1]
S Attal: Characterisations of some operators on Fock space, Quantum Proba bility & Related Topics VIII (this volume).
[At2]
S Attal: Problemes d'unicite dans les representations d'operateurs sur l'espace de Fock. Sim. de Prob. XXVI (to appear).
[Bel]
V P Belavkin: A quantum non-adapted Ito formula and stochastic analysis in Fock scale, J. Fund. Anal. 102 no. 2 (1991) 414-447.
[BL]
V P Belavkin and J M Lindsay: The kernel of a Fock space operator II, Quantum Probability & Related Topics VIII (this volume).
[Gui]
A Guichardet: "Symmetric Hilbert spaces and related topics". Springer LNM 261 (1972).
[HOS]
T Hida, N Obata and K Saito: Infinite dimensional rotations and Laplacians in terms of white noise calculus, Nagoya Math J. 128 (1992) 65-93.
[Hua]
Z-Y Huang: Quantum white noises — white noise approach to quantum sto chastic calculus, Nagoya Math. J. 129 (1993) 1-20.
[HuP]
R L Hudson and K R Parthasarathy: Quantum Ito's formula and stochastic evolutions, Commun. Math. Phys. 93 (1984) 301-323.
[LI]
J M Lindsay: On set convolutions and integral-sum kernel operators, in "Proceedings of Vth International Vilnius Conference on Probability and Mathematical Statistics, 1989" (ed. B Grigelionis et al) Volume II, 105-123, VSP, Utrecht 1990.
[L2]
J M Lindsay: Quantum and non-causal stochastic calculus, Probab. Th. Rel. Fields (to appear).
[LM]
J M Lindsay and H Maassen: An integral kernel approach to noise, in "Quantum Probability and Applications III: Proceedings, Oberwolfach 1987" (ed. L Accardi and W von Waldenfels). Springer LNM 1303 (1988) 192-208.
280
[LP]
J M Lindsay and K R Parthasarathy: Cohomology of power sets with appli cations in quantum probability, Commun. Math. Phys. 124 (1989) 337-364.
[Maa]
H Maassen: Quantum Markov processes in Fock space described by integral kernels, in "Quantum Probability and Applications II: Proceedings, Heidel berg 1984", (ed. L Accardi and W von Waldenfels). Springer LNM 1136 (1985) 361-374.
[Mel]
P-A Meyer: Elements de probabilites quantiques I-V, in "Seminaire de Probabilites XX" (ed. J Az6ma and M Yor). Springer LNM 1204 (1986) 186-312.
[Me 2]
P-A Meyer: Elements de probabilites quantiques VI-VIII, in "Seminaire de Probabilites XXI" (ed. J Azema, P-A Meyer and M Yor). Springer LNM 1247 (1987) 34-80.
[Me 3]
P-A Meyer: Fock spaces in classical and non-commutative probability, Ch IV, Strasbourg (1989).
[Par]
K R Parthasarathy: "An Introduction to Quantum Stochastic Calculus", Birkhaiiser Verlag, Basel/Boston/Berlin, (1992). Mathematics Department University Park Nottingham NG7 2RD England
Quantum Probability and Related Topics Vol. VIII (pp. 281-295) ©1993 World Scientific Publishing Company
BRAIDED GROUPS A N D BRAID STATISTICS Shahn Majid1 Department of Applied Mathematics & Theoretical Physics University of Cambridge Cambridge CB3 9EW, U.K. ABSTRACT We report on a generalization of groups and supergroups relevant to particles of braid statistics. This is based on the notion of a braided group, for which the ring of co-ordinate functions is acted upon by a braid group action 9, generalizing * = ±1 familiar for super-groups. We explain the physical motivation, the current status of the programme, and some problems that must be solved for a systematic approach to braided increment processes.
1
INTRODUCTION
The physical motivation for the present work comes from a recent proof of a bose-fermi statis tics theorem of Doplicher and Roberts in the framework of algebraic quantum field theory, as explained in [5]. The interesting thing about this proof is that it works only in four space-time dimensions. In three or two dimensions the superselection structure of the theory need not cor respond to that of bose or fermi particles but rather to particles of braid statistics[8][7][12]. It is well known that the processes that arise as central limits of baths of particles depend only on the statistics of the particles, rather than on their internal symmetries. In general, this means that it is hard to obtain new and yet physical kinds of processes. Because the situation in two and three dimensions is different, with new more general braid statistics allowed, it suggests that we should look there for new physical kinds of noise processes. Briefly, the results of [5] can be outlined as follows. Let 0 denote the "diamond-shaped" intersection between the forward light cone of a point in space-time and the rear light cone of a future point (in two space-time dimensions these are rectangles). Let .4(0) denote the algebra of observables of the quantum theory that are localized in 0. Thus the entire C algebra of observables is the completion A = UoA(O). If a region is not of the type 0 we define its algebra of localized observables in the same way as the completion over all O in the region. For an algebraic quantum field theory one makes various postulates. Among the most important is Haag duality
A(01)CA(02)',
WiCO'i
where the first prime denotes commutant and the second denotes spacelike complement. Of course, there should be an action a of the Poincare group P such that ai(A(0)) 'SERC Fellow and Drapers Fellow of Pembroke College, Cambridge
=
A(L{0))
282 for all I € P. Finally, there should be a vacuum representation ir0 of A such that no{A{0)) =
MAO'))'. A representation ff of A is localized at 0 if it coincides with the vacuum when restricted to .4(0'). Such representations are "close" to the vacuum. One of the results in [5] is that such localized representations are of the form IT = ■Kg°P where p : A —► A is an endomorphism and is the identity when restricted to .4(0') (an endomorphism localized at 0). If 7ri,7r2 correspond to local endomorphisms pi,P2, then we can obtain a new representation iri ® iti corresponding to a local endomorphism obtained from p\ o pi. In constructing the latter as a local endomorphism, the local region of p\ has to be transported round to that of p2. Note that this fails in spacetime dimension 2 where the causal complement of a point is not connected. There are also some more subtle problems in dimension 3. See [5] for more details. Theorem 1.1 [4][5] In space-time dimension n = 4 the endomorphisms p form a symmetric monoidal category C with duals. Moreover, C = Rep{G), the representations of a compact group G, which is reconstructed. This G acts on a larger algebra T with fixed points A = T . In this theorem, A is the algebra of observables while T is the bigger "field algebra" and G is the global internal symmetry group. Only the G-invariant combinations of the fields in T are physically observable. For example, is should be possible to identify quantities like a spinor field ipa as lying in T but only G-invariant combinations such as ipip would be among the observables in A. Such identification of T with physical fields remains to be proven in practice, but it can be expected. Also, the local endomorphisms describe super-selection sectors forming a general symmetric monoidal category i.e., there are isomorphisms $ : -K^ ® 7T2—fl"2 ® TVI giving an action of the symmetric group. However, in terms of T the fields can be chosen such that the action is that of bosonic or fermionic statistics. By contrast in n = 2 or n = 3 we have for C a braided monoidal category[7][12]. This is a generalization for which $ 2 / id. Briefly, a braided monoidal category is a collection of objects and morphisms, a functor ®, unit object 1, associativity isomorphisms $ and quasisymmetry isomorphisms (or braiding) *[10), see also [16, Sec. 7]. In the present case the objects are *i,*i,ITS, •■ •. The isomorphisms * , l l T j , T 3 - ffi ®(T 2 ® T 3 ) - ( T I $^2)® *3 obey Machine's well-known pentagon coherence condition[14]. This asserts that the different ways to bracket expressions can be consistently identified, i.e. we can suppress the brackets and * . The quasisymmetry isomorphisms *»-,,»-2 : it\ ® n2-T2 ® Ti obey the conditions (suppressing $) *
*m,lr,8ir3 = *»,,iis 0 *»».«»i
*ir,l = id = * l t » .
(1)
This asserts that $ gives an action of the braid group on tensor products of objects. The details were reviewed already in [15, Sec. 3] (in the context of quantum groups) so we do not do so again here. An example is the category of super-spaces where $ = ± l r according to the degrees (r is the usual twist map). More general examples where * 2 / id will be described explicitly below.
283 Hence in n = 2 or n = 3 we cannot identify C with the representations of any group. Many authors have therefore conjectured as to what should be the object playing the role of G. For example, it is known that the representations of quantum groups (in the strict sense of quasitriangular Hopf algebras) form such categories. This was explained in [15, Sec. 3] and indeed quantum groups could furnish examples of generalized G. For a general algebraic quantum field however, there seems to be no theorem that the role of G has to be played by a quantum group. Rather, we have argued in [17][18] that in general the role of G should be played not by a quantum group, but something more general, which we call a braided group[18][20]. It is this notion that we wish to explain in the present paper. A braided group A is a generalization of the ring of functions on a group or supergroup. In the familiar language of Hopf algebras, it is a braided-commutative Hopf algebra in a braided monoidal category (just as a supergroup is, for our purposes, a supercommutative Hopf algebra in the category of superspaces.) Note that because braided groups are commutative (albeit in a non-commutative category), they are geometric objects rather than non-commutative-geometric or ''quantum" objects. This is in keeping with the situation in n = 4 where G is the classical symmetry group of the quantum theory. On the other hand every ordinary quantum group can be viewed by a procedure of transmutation[18] as an braided-commutative Hopf algebra in a braided category. Hence the point of view of braided groups is a very general one. It represents a shift from the philosophy of non-commutative geometry, now familiar in quantum probability, to the philosophy of super-geometry and its generalizations. T h e o r e m 1.2 [19][18] Let C be a braided monoidal category with duals. Then there exists a braided group Aut(C) in a cocompletion ofC and a functor C — ► Rep(Aut(C)) compatible with the forgetful functor. Here Rep denotes Aut(C)-comodules in C and indeed, Aut(C) can be characterized as the uni versal object with such properties[20]. The proof of the theorem is based on a Tannaka-Krein or Fourier Transform theorem well known in harmonic analysis (where a group is reconstructed from its representations), but now generalized to the braided setting. Some similar Hopf algebras have also been found in another context in [13]. Much work remains to be done in order to actually identify such Aut(C) as the physical "internal symmetries" of actual algebraic quantum field theories in n = 2, n = 3. On the other hand this need not deter us from going ahead to construct increment processes on purely mathematical examples. It is such mathematical examples of categories C and braided groups resulting from them that we shall describe in detail here. They are perhaps toy models of real physical systems along the lines above. We turn in Section 2 to a more detailed explanation of what a braided group is. This is based on [18][20] but includes pedagogical details omitted there. In Section 3 we describe some examples of matrix braided groups, such as the braided group BSL(2). This is based on [21]. Finally, in Section 4 we conclude with a brief discussion of what would be needed to actually develop stationary independent increment processed based on braided groups (or quantum braided groups), and give another example of a braided category which could be useful for this. The formalism that we propose is along the lines introduced for Hopf algebras and super-groups or super-Hopf algebras in [1]. Many results in the context of Hopf algebras have
284 been obtained already in pioneering works such as [9][25][26][24] and their sequels. Moreover, it has become clear that there are still more general examples of processes, and some have definite braid-like aspects. See for example the talk of R. Speicher at this conference[3]. Hence it seems likely that at least some of these examples can be put into the general framework of processes built on braided groups, and perhaps given a systematic physical interpretation in terms of algebraic quantum field theory or otherwise. While such results will certainly not be achieved here, it is hoped that the present introduction to braided groups may at least be useful for experts in quantum probability wishing to undertake this line of development.
2
H O P F ALGEBRAS IN BRAIDED CATEGORIES
A braided group is a certain type of Hopf algebra in a braided category. We explain the latter notion first. In doing this, we give some of the pedagogical details omitted in the existing works by the author on this topic. Recall first that a Hopf algebra in the ordinary sense (over a field, say C) means (A, A, t, S) where A is a unital algebra, A • A —> A® A (the coproduct) is coassociative in the sense (A ®id) o A = (id® A) o A and an algebra homomorphism. Here A® A has the tensor product algebra structure, t : A —> C (the counit) obeys (c®id) o A = id = (id®e) o A. The antipode S : A —* A obeys ■ o (5®id) o A = l^e = • o (id® 5) o A and plays the role of group inverse. For an introduction, see [15]. A well known example is A = Z°°(G), the functions on a discrete group (more generally, for locally compact groups, this is a Hopf-von Neumann or Kac algebra). The coproduct is (Af)(x,y)
= f(xy)
(2)
for / € X°°(G) and x,y € G. The right hand side defines an element of i°°(G) ® L°°(G). Of course, examples of this type, corresponding to functions on groups, are commutative. A super-Hopf algebra is similar except that A = Ao(B A\ is a Z2-graded algebra (the product map respects the total degree) and A is an algebra homomorphism provided A ® A is endowed with the super tensor product algebra structure, (a®b)(c®d) = (-lf^(ac®bd).
(3)
Here |a| is the degree of a homogeneous element a etc. More generally, fix a braided monoidal category C with quasisymmetry * . We suppress 3>. An algebra in C just means an object A of C and a morphism 4 ® A —» A, along with n: 1 — ► A (the unit morphism). This is a sb'ght abuse of notation (the term "monoid" would be more precise) - in practice our categories are fc-linear over a field and we require all morphisms to respect this so that we really have algebras. Proposition 2.1 Let A be an algebra inC. Then ■A®A ■ (A® A)®(A®Af^A(A®A)®(A® defines an associative product on A® A.
A)%'A® A.
285
Figure 1: Proof of Proposition 2.1
Proof The proof is not entirely trivial, and is indicated in Figure 1. We have adopted here a diagrammatic notation essential to any serious work with braided categories, as for example in [19]. As explained in [15, Sec. 3], we write $ = ^ and $ _ 1 =yM. We will write all morphisms pointing downwards. The product is written as a U-vertex (and later on we will write the coproduct as a fl-vertex). Functoriality of
(4) (4)
286
Figure 2: (a) Hopf algebra axiom, (b) Commutativity Axiom
We mean to first apply * to b ® c and then multiply on the left by a and on the right by d. Clearly the super tensor product algebra structure in (3) is an example where *(6®c) = (-iyb"c'c® b. The axioms of coassociativity, the counit, and the antipode do not directly involve *, being just the usual ones (as morphisms). Nevertheless, the appearance of * in the axiom for A means that they do involve
287 This novel notion asserts that A behaves braided-commutatively as far as any representation in the class O is concerned. If * 2 = id and • = • o $ then this braided-commutativity condition holds for all comodules, but in general one cannot expect this as explained above. In the example Aut(C) explained in the introduction, O is the image of C under the functor C —> Rep (Aut(C)). Typically, as in the examples in Section 3, the braided-commutativity is reflected directly in the algebraic structure of A itself. What should be the class O is in general for us to define. If A is not very commutative, O will not be very big. It can, however, always be taken to be closed under ®, as the next proposition shows. Proposition 2.5 Let A be a Hopf algebra inC. Let O(A) denote the collection of all comodules of A for which A is braided-commutative in the sense of Definition 2.1,. Then O(A) is a braided monoidal category. It can be called the category of central comodules of A. Proof The tensor product and quasisymmetry are given in the usual way as for groups (albeit in a dual form with comodules rather than modules): the role of the twist map for groups representations is played now by $ in C, viewed now as an intertwiner in O(A). These results are proven in detail and in greater generality in [22, Sec. 3]. Explicitly, the tensor product of two comodules V,W is V®W with ftvigW = (idgiid®-) ° *$A,W ° [Pv®Pw) and * can be seen, by diagrammatic methods, to be an intertwiner. This completes our introduction to the central notions behind working with Hopf algebras and group-like objects in braided monoidal categories.
3
EXAMPLES OF MATRIX BRAIDED GROUPS
In this section we review some examples of matrix braided groups. In the setting of the Intro duction, the internal symmetry group in 4 space-time dimensions is typically a matrix groups like SU(n) (e.g. particle colour or isospin). So it should be braided group analogues of just such matrix groups that arise in 2 or 3 spacetime dimensions. To explain our notation, recall first our treatment of ordinary groups G in terms of the algebra of functions. We saw in (2) that the group structure is expressed in terms of the coproduct A. For matrix groups the algebra of functions is generated by the tautological ones u'j defined by u'j(x) = x'j for any matrix x 6 G C Mn{C). According to (2) the coproduct of these is (AU'JXZ,!,)
= u'j(xy) = {xyjj = £ > V i = £ K i ® A'X*^)k
k
The counit and antipode can similarly be expressed in this way. Of course these u'j are com mutative as functions on G. Matrix braided groups are generated by similar u'j with the same matrix form of A but now non-commutative. Instead, the u'j are braided-commutative. This can be contrasted with matrix quantum groups as in [9][1]: the difference is that the generators live in a braided category and the apparent non-commutativity is of a very specific form, namely commutativity in that category. For example, if C is the category of super-spaces then a matrix braided group means a matrix super-group. Theorem 3.1 [21] There are braided group analogs of all the classical Lie groups.
288 This is proven in [21] by borrowing some ideas from the theory of quantum groups. Indeed, it is known that all the classical Lie groups have quantum group analogs. The matrix versions of these were introduced in [6] by means of bialgebras A{R) associated to R 6 Mn(C) ® M„(C) solving the quantum Yang-Baxter equations (QYBE). The latter is the equation R12R13R23 = R23R13R12 in M „ ® M „ ® M „ , where JJ12 = # ® i d etc. The construction was already reviewed in [15, Sec. 2]. Suitable R are known for all the classical Lie groups as well as ways to quotient the bialgebra by "determinant" type relations to obtain Hopf algebras with antipode. In fact, the construction works more generally for any invertible R obeying the QYBE for which R'2 is invertible. Here ' 2 denotes transposition in the second Mn factor. Define the matrices $, $ ' in M„2 ® M„2 by
V'jKL=
tfVjo^'V'^V.^V0.*
E
(5)
a,b,c,d
V'jKL=
"£, ^Uic,kaaRi,k\R'lAR\kd-
(6)
a,b,c,d
Here R = ((.ft' 2 ) -1 )' 2 and i" = {ia,h),J = (io.ii) etc are multi-indices running from (1,1),(1,2), ..., (n,n). W is a matrix obeying the QYBE and * ' is a variant of *P. $ ' =
(
u V = J2 uLuJy'KL'j
(7)
0 1 q-q-1 J,L 0 Q YBE associated to the Jones knot 0 polynomial. 0 1 Writing 0 " u = ^ " * j , the braided-commutativity where u = K' 0 ;, etc. The action of the braid group on the {u1} (extending to an action on all relationss (7) f7j become 0 0 0 q) of B{R)) is
CD-
*(M 7 g uK) = ^
L M
® uJ9KL'j.
(8)
J,L
With this action, the matrix coproduct and counit Ati'j = J2 «'* ® «*i.
*(«'i) = Fj
(9)
k
make B(R) into a braided-commutative bialgebra in u braided category. For nice R we can quotient B(R) further by suitable determinant-type relations to obtain a braided group
((
Example 3.3 [21] Let R = I Example
qq 00 Q
00 q
*
0\ 0\ I.
This is the well-known solution of the
9 00 10 q-q00 ' TH'S iS ihe well-known ~11 solution of the Q YBE associated to the Jones knot polynomial. Writing 0 0 10 0q)" u = ^ " * j , the braided-commutativity relations s (7) f7j become 0 polynomial. 0 0 q) Q YBE associated to the Jones knot Writing u = ^ ° ^ , the braided-commutativity
CD-
relations (7) become ba = q2ab,
ca = q~2ac,
da = ad, 2
db = bd + (1 - q~ )ab,
be = cb + (1 - q~2)a(d - a) 2
cd = dc + (1 - q~ )ca
(10) (11)
This describes a bialgebra BM(2) in a braided category. The additional relation od - q*cb = 1
(12)
289
gives the braided group BSL(2). The quasisymmetry or "braiding" is given by (8). A sample of this is (the others are similar), V(a®a)
= a®a + (l-g2)6®c
tf(a®6)
= 6® a
* ( c ® 6 ) = g _ 2 6® c
*(6®a) *(t®6)
2
= a®6 + ( l - i ) 6 ® ( c i - a ) = q2b®b
*(rf®o) = a® d + (1 - q~2)b®c *(d®6) = 6®d.
Note that this is not a Hopf algebra in the ordinary sense. The matrix coproduct (9) only extends consistently to products of generators if we give the generators braid statistics as shown. For example Aba
Aq2ab
=
( A 6 ) ( A a ) = (a ® f> + 6 ® d)(a ® a + 6 ® c)
=
aV(b®a)a + o*(6® b)c + 6*(d®a)a + 6*(d® 6)c
= q2(Aa)(Ab) = q2(a®a + 6®c)(a®6 + 6® d) = q2a^(a ® a)b + q2a
That these expressions coincide can be seen after substituting $(6® a) etc as shown and using the relations in BM(2). That $ 2 ^ id for generic q ^ 1 is evident from $(6®6) = q2b®b, for example. 0\ We see in this example that qas 0q —> 01 the algebra becomes commutative and the braiding
(
q case the commutative algebra of functions on SL(2) in <Example P becomes3.3 trivial. We R recover Example [21] Let = I in this * I. This is the well-known solution of the Q the form explained above, while for generic q the algebra relations in this example are roughly 1 q-q-1 quantum 0 as complicated as those for the 0well-known group S£,(2)[28]. However, the concep Q YBE associated to the Jones knot polynomial. Writing , the braided-commutativity 0 0 1 non-commutative 0 " u = ^ " and* j therefore tual improvement is that whereas SLg(2) is far from admitting relationss (7) f7j become 0 0 0 q) a classical geometrical interpretation, the relations of BSL{2) are simply those of braidedcommutativity from (7) denning a braided 4-plane, SM(2), plus one further quadratic con straint (the determinant condition (12)). This is exactly parallel to the description of SU(2) as the space S3 C Jffi4, or in our case SL(2) C C* Thus working with braided groups rather than quantum groups means that our algebras are "commutative" (albeit in a braided sense) and hence retain their geometric interpretation as the "ring of functions" on a manifold (in the present case a complexified braided-5 3 ).
CD-
(1
q9 00
Example 3.3 [21] Let R = I Example
Q
q
00 *
0\ 0 \ I. This is the well-known solution of the
l 9 ~ 1' 0° 1 q-q0 jj1 . This is the solution of the QYBE Q YBE associated to the Jones knot polynomial. 00 00 10 Writing 0 -q" u/ = ^ " * j , the braided-commutativity known to be become associated to the Alexander-Conway knot relations s (7) f7j 0 0 0 q) polynomial, see for example [11]. Writing u= I I the braided-commutativity relations (7) become b2 = 0, c2 = 0, d - a central, (13)
ab = q~2ba, This describes a bialgebra BM^ and
CD-
ac = q2ca,
be = -q2cb + (1 - q2)(d - a)a.
(14)
in a braided category. The braiding is given by d - a bosonic
290 9(a®a) *(a®6) ¥(6® a) <$(a®c) y(c
= = = = =
a ® a + (1 - q2)b®c 6® a a®fc + ( l - g 2 ) & ® ( d - a ) c®a + (l - q2){d- a)®c a®c.
$(6® 6) $(6®c) *(c®6) if!(c®c)
= = = =
-6® 6 - c ® 6 - (1 --q2){d- - a ) ® ( d - a) -6®c -c®c
We see in this example that a,d are almost bosonic while 6, c are almost fermionic. For example, *(6®6) = -b®b, *(c®6) = - 6 ® c etc. This is reflected also in the algebra relations 62 = 0 etc (because * ' is a variant of * ) . Indeed, as q -> 1 this example becomes the algebra of super-matrices Mxu = {( ° ,\ \ a,d bosonic, b,c fermionic}. This demonstrates the sense in which braided groups and matrices really generalize super groups and matrices. Finally in this section we want to describe a notation for working with braided groups and algebras that more closely resembles the notation used by physicists in describing fermions. In this notation, * and ® are suppressed. Instead, the exchange properties of independent particles are described by the statistics relations (such as commuting or anticommuting statistics). To be concrete we demonstrate the notation on BSL{2). Firstly, distinguish elements of the two copies of BSL(2) in BSL(2) ® BSL(2) by labelling those in the second copy with a prime. Note that this is a, property of where the element lies not of the element itself. Thus for any / in BSL(2), we write / = / ® 1 and / ' = 1 ® / . Then from (4) we have / S ' = / * ( l ® l ) f f = /®ff,
/'ff = * ( / ® s ) .
(15)
The extension of $ from the generators of BSL(2) to arbitrary elements, as well as the functoriality properties of
a'b = ba',
b'a = ab' + (1 - q2)b{d' - a'), b'b = q2bb\
b'c = q~2cb' + (1 + ? 2 )(1 - q^fbc1
b'd = db' + (l-q-2)b(d'-a'),
d'a = ad' + (l-
2
a'c = ca' + (1 - q2)(d- a)c',
c'a = ac',
a''d = da' + (1 - q~2)bc' - (1 - q~2)(d - a)(d' - a')
c'b = q-2bc',
2
c'c = q2cc',
c'd = dc'
2
q~ )bc\ d'b = bd', d'c = cd' + (1 - q~ )(d - a)c', d'd = dd' - q~ {\ - q~2)bc'.
These statistics relations together with the algebra relations (10)-(12) of BSL(2) (and an iden tical set for the primed variables) generate the associative algebra J55Z.(2)® BSL(2). This algebra contains all the information of the braided group. The notation can be extended to higher tensor powers by a labelling each copy of BSL(2) by an integer for its place in the tensor power. Also in this notation, the coproduct takes the form
»(:!)-(: i)ff S) and similarly for higher coproducts. It means that we can work with matrix braided groups in many ways just as for ordinary matrix groups but now with braided-commutativity and braid statistics for their matrix entries.
291
4
TOWARDS BRAIDED I N C R E M E N T PROCESSES
In this section we want to conclude with a, (fairly obvious) proposal for braided increment processes. Our results here are limited to some modest category-theoretic considerations relevant to the general formulation. Of course, the abstract formulation has to be followed by some concrete examples and their realization on Fock spaces. We do not attempt this here but we do indicate some examples of the necessary data related to anyons, and briefly discuss the problems that remain. The definition of a braided increment process that we propose is based on [1], There a quantum independent increment process consists of *-algebra maps j , t : B —> A where A is a (super )-*-algebra equipped with state
= -°{j®j')°
A,
j,f
€
Moi(B,A).
The unit in Moi(B, A) is n o e : B — ► A. Proof
Given j,j',j" j*(j'*j")
in Mor(B, A) we compute =
■o(id®-)°0'®(J'®/'))o(id®A)°A
=
■ o (id ® •) o $~£AA a ((j ® f) ®j") o §B,B,B
=
■ o (• ® id) o ((j® j') ® j") o (A® id) o A =
o (id ® A) o A
{j*j')*j"
Here the first equality is the definition of • applied twice. The second is the functoriality of $ € Nat(®( ® ), ( ® ) ®)[14]. The third is an expression of associativity of A and coassociativity of B in C. The associativity map $ was suppressed in Section 2. After checking the last lemma, we continue to suppress writing it explicitly. We are now in a position to formulate our proposal. Definition 4.2 Let C be a braided monoidal category with unit 1 and quasisymmetry *, A an algebra in C and B a bialgebra in C. Let <j>: A -* 1 be a morphism in C. A braided independent increment process over B is a collection of algebra homomorphisms {jat : B ^
A\ (s,t) €
-H+i * < *} in C such that (i) jr, *jst = jrt for allr < s
292 A ® • ■ • ® A —> A. In axiom (iii) we have used that the set of (s, t) with s < t is ordered. This means that it is quite natural to impose (iii) only for half the cases as shown rather than for all disjoint (s,t),(s',t') as in [1]. Lemma 4.3 Axiom (iii) above is equivalent to ■ o (i, t ® j'y,<) = ■ ° ®~A]A o U't ® is't') for all t' < s. Proof
Assuming (iii) holds we compute for /' < s,
■o{j,t®j,n>)=
■ » ( i . t ® j . v ) o * f l , B ° * B ) B = ■°VA,AU-'t'®i°t)°yB1,B
= ■ ° U"t'® j si) °^B1,B-
For the second equality we used functoriality of * € Nat(®,® op ). This was already used extensively in Section 2. Likewise, the last expression equals the right hand side in the lemma by functoriality of
V v,w(v ® u>) = e
"
w®v
(17)
on homogeneous elements of degree \v\, \w\. Here the degree is defined by g>v — e n « for the action of the generator g of Zn. This is well known to physicists in the context of anyons, e.g.[27][8]. The quantities e n ~ can be called fractional or anyonic statistics. The category was studied systematically in [23] where we showed that it can be identified with the category of representations of a certain quantum group with non-trivial H. We also showed there that any ordinary quantum group containing a group-like element of order n can be turned into an anyonic one by a procedure of transmutation[23]. Likewise, any algebra A containing an element g of order n can be viewed as an anyonic algebra by making A into a Z„-module by the adjoint action. The following anyonic quantum group is obtained by transmutation. One could consider its use, or the use of its dual, for B in Definition 4.2.
293 Example 4.5 [23]. Let n = 4r and q = e^. The anyonic quantum group B(Z'n, Uq(sl(2))') in An has generators g of degree 0 and E,F of degrees \E\ = 2, \F\ = - 2 and relations S" = l,
ET = 0 = F;
gEg-^e^E,
gFg-1 = e " 4 ? F,
9
qEF-FE=
-^±.
9-1
The anyonic quantum group structure is AE = E®g4+l®E, A°E_E = E®1 + 1®E,
AF = F®l + l®F,
eE = eF = 0,
A°E_F = F®l + g4®F,
R=
Y ^=o
SE = -Eg-4, Em®Fm-
S.F = -F (q
- ',
-
(g™ - 1) ■ • • (« - 1)
where the m = 0 term is defined to be 1 <2> 1. The coproduct etc on g are as for the group algebra of Zn. Here A o p is a second coproduct structure and 2 intertwines A and A op according to the axioms of an anyonic quantum group. The problem of actually realizing braided processes in the systematic form above remains an open one, even for ones based on such simple anyonic examples. A problem for the systematic theory is as follows. Firstly, recall that stationary processes are ones where
294 Finally, from this point of view there should be yet more general braided process associated to (quantum) braided groups and braided categories that do not come in any way from quan tum groups. Nevertheless they should still have realizations in terms of ordinary algebras and ordinary analysis in the context of algebraic quantum field theory. This setting was explained in the introduction and provides a third direction for further work. ACKNOWLEDGEMENTS This is based on my talk at the conference. I want to thank the organizers K.R. Parthasarathy and K.B. Sinha for a most stimulating conference and for their very kind hospitality at the Indian Statistical Institute.
REFERENCES [1] L. Accardi, M. Schiirmann, and M. von Waldenfels. Quantum independent increment processes on superalgebras. Math. Zeit., 198:451-477,1988. [2] F.A. Berezin. Analysis with Anticommuting Variables. Kluwer, Dordrecht, 1987. [3] M. Bozejko and R. Speicher. An example of a generalized brownian motion, I, II. Preprints, 1990. [4] S. Doplicher and J.E. Roberts. A new duality theory for compact groups. Inv. Math., 98:157-218,1989. [5] S. Doplicher and J.E. Roberts. Why is there a field algebra with a, compact gauge group describing the superselection structure in particle physics. Comm. Math. Phys., 131:51-107, 1990. [6] L.D. Faddeev, N.Yu. Reshetikhin, and L.A. Takhtajan. Quantization of Lie groups and Lie algebras. In M. Kashiwara and T. Kawai, editors, Algebraic Analysis, Vol. I, pages 129-139. Academic Press, 1988. [7] K. Fredenhagen, K.H. Rehren, and B. Schroer. Superselection sectors with braid statistics and exchange algebras. Comm. Math. Phys., 125:201-226, 1989. [8] J. Frohlich. Statistics of fields, the Yang-Baxter equation and the theory of knots and links. In Proc. Cargese (1987). [9] P. Glockner and W. von Waldenfels. The relations of the non-commutative coefficient algebra of the unitary group. In L. Accardi and W. von Waldenfels, editors, Quantum Probability and Applications IV, number 1396 in Lee. Notes Math., pages 182-220. Springer, 1989. [10] A. Joyal and R. Street. Braided monoidal categories. Mathematics Reports 86008, Macquarie University, 1986. [11] L. Kauffman. Knot polynomials and Yang-Baxter models. In Proc. ICMP, Swansea. Adam Hilger, 1989.
295 [12] R. Longo. Index of subfactors and statistics of quantum fields. 126:217, 1989.
Comm. Math. Phys.,
[13] V.V. Lyubashenko. Tensor categories and RCFT 1,11. Preprint, 1990. [14] S. Mac Lane. Categories for the Working Mathematician. Springer, 1974. GTM vol. 5. [15] S. Majid. Quantum groups and quantum probability. In L. Accardi et al, editor, Quantum Probability and Applications, Trento, 1989. [16] S. Majid. Quasitriangular Hopf algebras and Yang-Baxter equations. Physics A, 5(1):1-91, 1990.
Int. J. Modern
[17] S. Majid. Some physical applications of category theory. In C. Bartocci, U. Bruzzo, and R. Cianci, editors, XlXth DGM, Rapallo (1990), volume 375 of Lee. Notes in Phys., pages 131-142. Springer, 1991. [18] S. Majid. Braided groups and algebraic quantum field theories, 1990. To appear in Lett. Math. Phys. [19] S. Majid. Reconstruction theorems and rational conformal field theories, 1989. To appear in Int. J. Mod. Phys. [20] S. Majid. Braided groups. Preprint, DAMTP/90-42, 1990. [21] S. Majid. Examples of braided groups and braided matrices. Preprint, DAMTP/90-43, 1990. [22] S. Majid.
Transmutation theory and rank for qauntum braided groups.
Preprint,
DAMTP/91-10, 1991. [23] S. Majid. Anyonic quantum groups. Preprint, DAMTP/91-16, 1991. [24] M. Schiirmann. White noise on involutive bialgebras. In Quantum Probability and Appli cations, Trento, 1989. [25] M. Schiirmann. A class of representations of involutive bialgebras. Math. Proc. Camb. Phil. Soc, 107:149-175, 1989. [26] M. Schiirmann. Infiniteley divisible states on cocommutative bialgebras. Heidel. math preprint, 1989. [27] F. Wilczek. Fractional Statistics and Anyon Superconductivity. World. Sci., 1990. Ed. [28] S.L. Woronowicz. Twisted 5T7(2)-group, an example of » non-commutative differential calculus. Publ. RIMS (Kyoto), 23:117-181, 1987.
Quantum Probability and Related Topics Vol. VIII (pp. 297-311) ©1993 World Scientific Publishing Company
A Q U A N T U M P R O B A B I L I T Y T H E O R Y FOR T H E N-LEVEL A T O M by Lamberto Rondoni* Center for Transport Theory and Mathematical Virginia Polytechnic Institute and State Blacksburg, VA 24061-0435
Physics,
University,
(U.S.A.)
rondoni @ vtccl.bitnet
ABSTRACT We define a class of discrete dynamical systems, called Quantum Boltzmann Maps, and we introduce an appropriate perturbative expansion of such maps, in order to construct a model for the dynamics of N-level atoms, N > 2, in a confined radiation field. In particular, we use the increase of the entropy under the maps to prove the convergence of the dynamics to the appropriate equilibrium positions, in the relevant phase spaces.
1. INTRODUCTION In this paper, we construct a theory of TV-level systems using a class of nonlinear dynamical systems, called "Quantum Boltzmann Maps'' (QBM), which were first intro duced in '87 by R.F. Streater [1]. Such maps represent a suitable mathematical tool for the description of certain nonequilibrium physical systems, as they approach their sta tionary states. It is known that the problem of convergence to equilibrium for different classes of physical systems is outstanding, and different theoretical approaches are under consideration. Among these, the method based on QBM belongs to a class of stochastic models which neglect the details of the microscopic interactions among particles, and such that irreversibility emerges as a consequence of the randomness which is put in the process by hand. This approach may not appear as satisfactory as a Hamiltonian theory would be, as the macroscopic behaviour of a physical system is not derived from its microscopic dynamics. Nonetheless, it has good physical justification, since it attempts to describe the effects of the myriad of unknown degrees of freedom, and of noise, which are always present in the real physical situations, and which are not likely to be included in a Hamiltonian
298 formalism. The study of TV-level systems is precisely the subject of a vast literature. For instance, a very incomplete list of the research in the field includes [2 - 5], where the more mathe matical aspects of the theory were investigated, and [6 - 10], where physical applications, with particular enphasis on the theory of LASER'S, were considered. Our goal, here, is to show the versatility of the quantum probabilistic approach based on the QBM, in the construction of new models of physical interest. We will derive a description of JV-level atoms in a finite radiation field, as an approximation of an appropriate QBM, and we will discuss some of its properties. In particular, our focus will be on the asymptotic behaviour of the dynamics. Our approach deals with discrete time models, differently from the most of the existing literature, as we think that discrete schemes are very interesting both from a mathematical and from a physical point of view, and thus worth being further developed [11]. However, our set up allows us to draw results also on the continuous time limit, as it will be explained in a future paper. This work differs from others in the literature also because the radiation field is assumed to be finite, that is to say it contains just a finite number of photons, and so its temperature is not constant. It follows that the result ing "isolated" system is properly described only by nonlinear processes, as explained by Streater in [12]. This paper is organized as follows. Section 2. summarizes some of the most important results on the QBM, from Streater's work [1,12]. In Section 3. a model for the two - level atom is constructed, and its properties are analyzed. Section 4. contains a generalization of the theory developed in Section 3. to the case of /V-level systems.
2. THE QUANTUM BOLTZMANN MAP In this section we briefly review for completeness some notions concerning the QBM, as defined by Streater [1,12]. This map is to be regarded as a model for a set of N < oo quantum harmonic oscillators which constitute a dilute gas whose particles seldom collide, so that memory of such events is lost before other collisions take place. Then, the Hamiltonian of the system is well approximated by the free Hamiltonian, and the interactions can be described by a scattering matrix computed as if there were an infinite time between successive collisions. This can be formalized as follows. Let K be the 1particle space for a set of N harmonic oscillators with frequencies wi, ...,WN, and consider the symmetrized Fock space over K : VS(K) =
299 of observables is taken to be the set of bounded operators on Ta(IC), i.e. A = B(T,(IC)), which is generated by bounded functions of the creation and annihilation operators a*and a, , j = 1,..., JV.
Let E be the set of (normal) states on A -i.e. determined by
a density operator p- and furnish A with the inner product < A,B > = TT(A*B)
(for
A and B of Hilbert-Schmidt class). The entropy of the state p is denned as S(p) = - T r ( p l o g p ) , and the effect of the scattering among particles is represented by a doubly stochastic map T*, adjoint of a doubly stochastic map on A, which is also completely positive in all the practical cases. In particular, the doubly stochastic maps we deal with, in this paper, are obtained as mixtures of conjugations by unitaries, which also preserve the spectral projections, P, say, of the Hamiltonian. Then, the average energy of the state p is preserved under the application of T* to p. In fact, denning the Hamiltonian as Ho = T,uiajaj
= T,EiP>> w e S e t
Ti((T'p)H0)
= Tr(U*pUH0)
= Tr(pUH0U')
= Ti(pTH0)
= Ti{pH0) = E,
where U is a unitary operator, V* is its adjoint, and T is defined by TA = UAU* for every A in A. As discussed in Ref.s[l,12-14], a linear map like T* cannot properly describe the time evolution of many systems of interest in Physics, and extra stochasticity must be introduced. Here, as we do not have a formalism of particles, we cannot resort to the classical Boltzmann's stosszahlansatz, in order to provide such extra stochasticity. However, by analogy, we can make use of the "quasi - free" projection, which replaces a state with finite first and second moments with the state whose third and higher moments vanish, and whose first and second moments equal those of the original state. Such a map was introduced long ago by Bogoliubov, and it has been considered also in more recent works by a number of authors; see Ref.s [15-17] for instance. It was proved by Streater [18] that the quasi free projection, which we denote by Q does not decrease the entropy, and that S(Qp) > S(p) unless Qp = p. Combining the actions of the maps T* and Q we get a new map with many satisfactory features [1,12-14] which we call QBM: DEFINITION. A QBM is a map r : E -► E defined by r = QT", where Q is the quasi - free projection on S, and T* is a completely positive and doubly stochastic map on E. Thus, the dynamics determined by r takes place on the subset of quasi - free states in E. It turns out that the limit of every convergent subsequence of {7-"/>}i° is a quasi free state poo such that TT*Poo = Poo, [!]■
300 3. THE TWO-LEVEL ATOM In this section, we describe how a perturbation expansion of a QBM, <, can be used to construct a theory for a dilute gas of 2-level atoms in a particular heath bath. Naturally, such a physical system has a large number of degrees of freedom, but we do not attempt to explicitly include all of them in our framework. Instead, our approach aims at providing a reduced description of the whole system, by predicting the values that the average populations of the atoms in the first and second energy levels -as well as the average population of the photons- will take at different instants of time. Note that our model may also be interpreted as a theory of a single atom in a radiation field. In order to do that, it suffices to normalize as appropriate the average populations, and view them as probabilities of finding the particle in a certain energy level. Let w\ and u>2 be the energies of the ground and of the excited states, respectively, and assume photons of energy ui = u>i — u>2 can be exchanged with a finite radiation field.
Here, the terms energy and frequency are interchangebly used, as we take the
multiplicative constant between the two quantities to be 1. Let a* and a,, i = 1,2 be the creation and annihilation operators of the atomic levels, and let b' and b be those for the photonic field. The Hamiltonian of the system is Ho = wiJVj + U/2N2 + uiN, where Ni,N2,N
are the number operators of level 1, level 2 and of the photons, and the space
we must consider is ^((E 3 ), the Fock space over (E3- The completely positive and doubly stochastic map that we adopt is defined by T\A = S\AS%, for every A 6 A = ^(^((E 3 )), where S\ = exp{iA(aJ6*a2 + a^bax)} = eixK,
(3)
which is a unitary operator on .4, and K is defined by K = aJ6*a2 + a^bai. We choose this form for S\ because it intuitively describes the fact that one atom of level 2 is created when one atom of level 1 and one photon disappear by absorbtion, and viceversa. Moreover, we write T\ to stress that there is a dependence on a real parameter, A, which could also be a random variable on some sample space, so that averages with probability measures could also be considered. Something similar will be done, indeed, in the next section, in order to model the more general TV-level atom. Clearly [H0, Sx] = 0, and thus the average energy turns out to be a constant of the motion under the action of r = QTt.
For every p in
E, let us introduce a linear functional Wp defined by WP(A) = Ti(pA), for every A in A. (This is what is more apropriately called a normal state). Then, if we define n ; =
WJNi),
for t = 1,2, and no index, we can write
E = wini + ui2n2 + u>n = constant.
(4)
301 Moreover, if we consider the normal state relative to rp , WTp, we get < = WTp(Ni) = Tr((Qr A »iV,) = Tr(( J\V)JV;) = W,(TxNi),
i = 1,2, no index, (5)
because of the definition of the map Q. However, this should not mislead the reader, thinking that Q has no effect. In fact, Q shows up in the computation of higher order moments, setting t o zero all the truncated functions of order 3 and more. It is Q that makes r into a nonlinear dynamical system. If we now perform a perturbative expansion in the variable A of the one point functions, we find that, to the second order, they are left unchanged by the map, provided that their initial values and the initial values of the "mixed" two - point functions -those involving other operators than Ni , N2 and TV, like Wp(aia2),
for instance- all vanish. As such moments do not correspond to physically
measurable quantities, we can impose Wp(ai) = ... = Wp(b*) = 0, without any harm, assuming that no off-diagonal long range order (ODLRO) occurs [19]. In certain cases, we may easily see that also the mixed second moments are left unchanged by the second order map, if the same conditions are satisfied. However, this may not be the case in the most general situation, therefore, in order to avoid unnecessary complications, we change the definition of the QBM introducing Pauli's "forget" map D . £ —* E, which makes such mixed moments vanish, and which was proposed by Pauli in 1928. Thus, we take r = DQT% as the new definition of a QBM, similarly to what was done in some example in Ref.[12], and the dynamics is now contained in the subset of diagonal states; but it should be noted that this doesn't alter the basic properties of r described in Section 2. Moreover, we show that the resulting time evolution is still consistent with the appropriate quantum statistics. If we assume that the first and the mixed second moments all vanish at the initial time, we find that the perturbative expansion of the map defined by Eq.(5) produces the following expressions „; = WT„{Ni) =
WDQ^TXNX)
= nx + WDQp (iX[K,Nt] n'2 = WTP{N2) =
- \\2[K,
WDQp(TxN2)
= n2 + WDQp (iX[K, N2] - \\2[K, „ ' = WTP(N) =
[JSr.JVJ] + ...) = m + \2G
[K,N2]] + ...) = n 2 - A2G
WDQp(TxN)
= n + WDQp (\i[K, N] - \\2[K, where we have [K,Ni] = [K,N] = -[K,N2]
[K, IV]] + . . . ) = » + A2G,
= a%bax - a\b*a2, from which the equalities
[tf, [ # , . . . , [A'.JVY]...]] = {K,[K,...,[I(,N]...)}
=
-IK,[K,...,[K,N2].„]},
302 for terms of the same order, follow. That's why Eq.(6) has a " + G " for n\ and n', while it has a "—G" for n'2. Moreover, it is easy to see that the term G introduced in Eq.(6) depends on the order of the expansion, but it contains even terms only, so that the second order expansion, for instance, is indeed a third order one, similarly to what happens with the diffusion approximation of the classical Boltzmann equation (see Ref.[20] for instance). This is a, consequence of the fact that the evaluation of the terms of odd order -those of the form WDQP([K,
[K, ...,[K, JVJ...]]) with an odd number of J T s - always involves either
one-point functions, which are assumed to vanish, and/or t-point truncated functions, with t > 3, which are set to zero by the application of Q. Also, we see that there are two independent conserved quantites, n\ + n2 = Ci (the total number of atoms), and n 2 + n = C2, so that the resulting dynamical system is 1-dimensional. It follows that the the total energy E, defined by Eq.(4), is conserved also under the approximate maps of any order. Turning to the second order map (exact to third order), we find [K, [K, 7V2]] = 2(a\a\bb*a2'a.2 — a\aib*ba2a2),
and, using the appropriate CCR or CAR relations, Eq.(6)
reduces to
I
n'j = c\ — n2 n 2 = n 2 + A2 [m(l ± n 2 )n - (1 ± n i ) n j ( l + n)] n' = c2 -
(7)
n'2,
where the " + " signs inside the brackets are for bosonic atoms, while the "—" signs are for the fermionic ones. Here, use was made of the fact that the mixed moments are initially vanishing, and so the action of the diagonalizing operator, D, turns to be necessary, to ensure that the form of Eq.(7) does not change at later times. As the map defined by Eq.(7), r 3 say, is only an approximation of r, we don't know a priori whether it shares all the properties of the QBM. In fact, we are now going to prove that some care must be used, and that certain constraints must be satisfied for the approximate theory to be physically meaningful. The first aspect to check is whether r 3 preserves the positivity of the average populations ni , n2 and n, or not. To find out, we use the fact that the dynamics is 1-dimensional, and we parametrize it by n2. Then, with a small abuse of notation, we can write r 3 (n 2 ) = ri2 = n2 + A2 [Zn\ - (1 + 2c)n 2 + e t c 2 ] = n 2 + X2Gb(n2),
(8)
for bosonic atoms, where c = Ci + c 2 ; from which it is easy to see that n'2 is nonnegative if the initial populations are nonnegative and A2 < 1/(1 + 2c) = A^. Similarly, for fermionic atoms we get r 3 (re 2 ) = n'2 = n 2 + A2 [n\ - (1 + 2c2) +
Cj c 2 ]
= n2 + A 2 G/(n 2 ),
(9)
303 where n 2 > 0 if nun2
€ [0,1] , n > 0, and |A| < |A/| with Xj = 1/(1 + 2c 2 ). In both
cases, the condition |A| < |A;| , i = b,f, implies r^(n 2 ) = (
,
for
n s £ [0,c/2],
(10)
0 < r 3 (0) < r 3 (n 2 ) < r 3 (l) < 1 ,
for
n , e [0,1],
(11)
for bosons, and
for fermions. That is to say, Ib = [0, c/2] and / / = [0,1] are trivially compact and convex, invariant sets for r 3 , respectively for the bosonic and for the fermionic systems. Then, the continuity of r 3 together with Brouwer fixed point theorem [21] ensures the existence of a fixed point for r 3 ) in the appropriate invariant set. Moreover, the strict positivity of 7-3(712) in the interior of I\, and Ij -provided h,If
are not just one point- implies the uniqueness
of such a fixed point, i.e.: there is a unique n 2 € /,-, i = b,f, such that G,(n 2 ) = 0. In fact, for 712 € li, it is easily seen that (?j(n 2 ) is negative for n 2 > n 2 , and G,(n 2 ) is positive for 7i2 < fi2- This observation, coupled with Eq.s (10) or (11), and with the nonnegativity of r 3 , allows us to conclude that 7i2 is a global attractor for the dynamics determined by r 3 in I{, and that the convergence is monotonic. Furthermore, using very simple inequalities, we can prove that there exists an invariant closed interval Jj C li such that ^ ( T ^ ) < 1 for n 2 € Ji, so that r 3 is a contraction', and the convergence is exponential, in ■/,. We find that Jf contains If, so r 3 is a global contraction for fermionic systems, while Ji, may or may not contain It,, depending on the initial conditions 7ii,7i2,7i. With these results at hand, and using Eq.(7), it is possible to show that also the positivity of nj and n is preserved under the map r 3 , and that n[ < 1, for fermions, if « i , n 2 € [0,1], n > 0. Also, performing some algebraic manipulation of the equations, we realize that the values of the three populations at the fixed point can be expressed as "! = ^ K - r t _ 1 '
" 2 = e«<*-M) - 1 '
"
=
l^^l
'
(12)
for bosonic atoms, and 1 711
=
e
«„-,) + 1 '
1 2
.
" - e/X-,-1.) + 1 '
"
_
1
,„.
eP» - 1 '
(U)
for fermionic atoms, where the inverse temperature, /?, and the chemical potential, \i, are determined by the value of the energy, E in Eq.(4), and by the total number of atoms d = n\ + 7i2. Summarizing, we we can state the following result. LEMMA 5. Let T3 be the map defined by Eq.(7).
Assume the initial populations
P =
(ni,n 2 ,7i) are nonnegat/ve, and that 7ii,n 2 < 1 in the fermionic case. Then, there exists a real number A' > |Aj| > 0 , i = b, f, such that for |A| < A' the following holds:
304 a) The iterations under r 3 are positivity preserving and { ^ ( P ) } ^ is contained in a compact subset Q of the positive octant of IR3 b) linn^oo T%(P) = P, where P contains the Plank distribution for the bosonic popula tions, and the Fermi distribution for the fermionic ones, relative to the initial values of the energy E and of the total number of atoms c\ in the system.
Moreover, the
convergence is monotonic, and it is exponentially fast in a closed subset of Q. As a corollary of this lemma, we get a result on the increase of the entropy under the map r 3 . Note, that the entropy can be expressed in terms of ni , n 2 and re, using the well known formulas for fermions [22] (and similarly for bosons), from which we get 2
5 6 (re 1 ,n 2 ,re) = ^
[(n,- + l)log(re,-+ 1) - n.logn;] + (re + l ) l o g ( n + l ) - r e l o g r e ,
(14)
i=i
for bosonic atoms, and 2
5 / ( n i , re2, n) = - ^
[»• l o S ». + (1 - n,-) log(l - n,)] + (n + 1) log(n + 1) - n log n, (15)
v=i
in the fermionic case. If we express n\ and re in terms of n2, it is easy to see that - — = 0 if and only if
« i ( l -f n2)re = (rei + l)n2(n + 1),
(16)
UTl2
and dSt —— = 0 if and only if rej(l - n 2 )n = (1 — ni)re(l + n).
(17)
U7l2
Moreover, (d2Si/dn\)
is negative (possibly equal to —oo) in the whole domain of 5,,
i = 6 , / . Therefore, the entropy of both the bosonic and the fermionic cases have a unique maximum, the unique fixed point n 2 in /., and it is strictly convex. Using this fact, introducing the notation a,(n 2 ) = (dSi/dn2)\n:i,
and recalling that the iterations converge
monotonically to the fixed point hi (lemma 5.), we can write a;(n 2 ) > a,(r 3 (n2)) > 0 if
n2 <
re2,
(18)
because n 2 < T3(re2) in this case, and OLi(n2) < Qj(r 3 (n 2 )) < 0
if re2 > n 2 ,
(19)
as n2 > T3(n2) in this second case. Then, integrating these expressions with respect to the variable n 2 , we get the following result: COROLLARY 6. The entropy of the system (with two-level bosonic or fermionic atoms) is a strict Liapunov function for the map r 3 . Precisely, Si(r(n2))
> Si(n2) + \2 \ai(T3(n2))Gi(n2)\
for
i = b,f,
(20)
305 where Q,(n 2 ) and Gi(n2) are different from zero unless n 2 equals n2, the unique fixed point in the domain of the map. We complete the analysis of the two-level systems observing that the fixed points of r 3 are stable with respect to small perturbations of the initial conditions. This is proved by the observation that the fixed points of r 3 depend continuously on the initial conditions, and with an argument similar to the one for the proof of the stability of the fixed points of the classical Boltzmann maps examined in Ref.[14]. Let us turn, now, to the study of the TV-level atom in a finite radiation field.
4. THE MULTILEVEL SYSTEM IN A FINITE RADIATION FIELD The theory outlined in the previous section can be generalized to the case of bosonic or fermionic N-le\el systems in a number of different ways. The approach we adopt here is to form convex combinations of the (JV2 — N)/2 two-level maps relative to all the pairs of energy levels of the system. Let w; be the frequency of the i-th level, and w;; = Uj - w; , u>j > ui{, be the frequency of the photon 7^ exchanged in a transition between the levels "«" and " j " . Then, three different cases are possible. 1) Uij ^ Uki if (i,j) u>ij = uiiti for every two couples (i,j),{k,l),
^ (&,')• 2)
such that direct transitions from i to j and
from k to I are allowed. 3) u>,j = UIM for some of the couples (i,j) and (kj)
only, which
is an intermediate situation between (1) and (2). The reason for considering these three different cases is technical, and has to do with the number of different transitions which are feeded with the same kind of photons. However, situations (2) and (3) can be treated with similar techniques to those of (1). Thus, we will concentrate on case (1) only. Note that these models correspond to systems whose transitions involve precisely one photon at a time. The case in which more photons are exchanged in one time step, in a transition from level i to level j , is more complicated and will be the subject of a future paper. The total number of degrees of freedom of our model is d = N + (N2 - N)/2 = (N2 + /V)/2 but, as we will see, some of them are not independent from the others, and some others might be "inhibited", when certain transitions are forbidden. The space we consider is T(<S,d), A = 6(J r ((C'')) is the algebra of observables, and we denote by S the set of density operators on A. The Hamiltonian is given by N
t'=l
N
j'-l
j = 2 .=1
where JV; and Nij, respectively, are the number operators of the atomic energy levels and of the photonic fields. Hereafter, the levels are ordered according to the growing values of the energy, so that in W;J the first index, i, is always less than the second one, j . A
306 two-level map on £, which regards the energy levels i and j , is a map r tJ that acts like the two-level maps we have studied in the previous section, on the subspace relative to the levels i, j , and to the photons 7,j carrying the energy difference, and it acts as the identity on the rest of the space. Such a map can be written as Ty(/>) = DQT\t>(p), the adjoint action of S\tj = exp(i\{jKij)
where Txti is
® / , and K{j = a'b'jaj + a'bijat, with obvious
notation. Then, T{j takes E into itself, and any convex combination of maps of this kind, like N
3-1
v r *- ==Yl Y,Yl Yu "&*}, a v> Vij "yS^0Q , • J2 Vii = x>
((21) 21 )
T
3=2 3=2 i. == l1
>,3 i,j
is another map of £ into itself, because of the convexity of S. As the number of terms in the convex combination is finite, the perturbative second - order expansion of r is just the convex combination, with same coefficients, of the second order expansions of each r , j . Note that the variable A with respect to which the expansion is performed can be taken from the set {Ay} of the coupling constants of each ry, and all the remaining Ay's can be expressed by Ajj = A ■ constant(i,j).
If the constants are different from 1, a little care
must be used, because the ranges over which the A,j's can vary will be either enlarged or shrunk by a constant. Thus, for simplicity we assume constant(i,j)
= 1 for all couples
(i, j) with i < j . If we take the expectations of the 7V;'s and of the iV,'/s, and we expand to the second order, we end up with maps ry and r -with a small abuse of notation- of a convex subset, Q, of TR,d into itself, which can be written as faj(n)]; ni- - A faj(n)]; = = ra; A yy (( nn )) rij + A y; j ( n ) h i ( n ) ] j = ry [nj{n)]ij = nnu0- -- A Ayy((nn)) faj(n)]y = [n,(n)]k = nk T T
n
n
0'( )]* = = nklki [t ij(n)]k for each two-level contribution, and
for
(22) k^i,j
for (fc,/)(k,l)yt(i,j), for #(«>».
f
[r(n)] [r(n)Ut = = nnkt ++ E £ *^"1> ,«*» AA a ( .n f) c- H £ &- »E+ ^, ^■ *A* ^« (n» )
1
A,tA,fct (n), [[Tri(nn)]ki )]« = »ii - A/tA,
(23)
for the total map r. Here, ng Q is the vector whose components are the average popula tions n,- and B;J , [r,j(n)]^ is the fc-th element of r y ( n ) , and Aij(n) A?- [»;(! ± ,y)ny iy)ny - (1 ± nn;)n.,(l A y ( n ) = A?. ny)]. 4 )n,-(l + By)],
(24)
where the " + " signs in the brackets refer to bosonic atoms, while the " - " signs are for the fermionic ones. A first observation is that there is a conserved quantity for the dynamics, which is W N
JV N
S k-1 r-1
L = J2knk + ^2Y,(k~l)n'kThen, the following holds.
fc=l k=\
fc=2 k=2 1=1 1=1
(25) (25)
307 LEMMA 7. Let the constants Ay relative to the 2-level maps ry, which define r in eq.(21), obey A ?? < A
l
(26) (26)
"^- 22ZTT I +1 '
Then, the map T defined by Eq.(23) preserves the positivity
of the populations of the
energy levels and of the photons. PROOF: We know that each map ry preserves the positivity of the populations of the energy levels as well as of the photons, provided that the relative constant |Ay| is smaller than a positve constant A^- > 1/^/1 + 2cy, where cy = Ci(i,j) + c2(i,j)
=re,-+ 2nj + n,j.
Thus, a convex combination of maps like ry, with small enough constants, is going to be positivity preserving. However, as we have seen in Section 3., the AJ^-'s depend on the cy's, which are determined by the initial conditions, and so they cannot be used in the present context, because of the action of more than one map on the energy levels i and j . On the other hand, each cy is less or equal than L, and positive, at the initial time. Therefore, if we let A?- < 1/(1+ 2£) for every i and j , all the constants of the two-level maps are in their positivity preserving ranges. As a consequence, after one time step, all the populations and the constants C{j are nonnegative, and such that the new bounds on the Ay's, for the positivity preserving property, still contain the range [ - l / i / l + 2L, l / \ / l + 2L]. Thus, the total map r preserves the positivity at all times.
■
Note that the quantity L is just a special member of the family of conserved quantities given by L{n,y) Y^Vknk++^2(yk 'Ys(yk-yi)nik £( n .y) = 5^»»* - ydmk , , k
(27) (27)
k>l
where n is the initial condition and y = {yk} is a vector in TS,N Thus, there are at least N linearly independent conserved quantities for a given initial condition. In addition to these, we occasionally have other independent constants of motion -e.g. when some of the V{j 's vanish- and so we conclude that the dimension of the space where the dynamics takes place is at most N(N - l ) / 2 . We can now prove that every initial condition, n = («i, ■•-,"#> n i2,re 1 3 , ...,rew-i,jv)> is driven by the iterations of r to a fixed point uniquely determined by n itself. The proof is based on a simple convexity argument, involving the uniqueness of the fixed point of r in the convex set in which the iterations lie. THEOREM 8. Consider an N-level system described by the map r defined in Eq.(23). Let n be an initial condition with all nonnegative entries, and with nj < 1 for fermionic atoms. Then, n determines a compact and convex set Q„, through Eq.(25), which is invariant under T. The dimension d' < (N2 - N)/2 of Q„ equals the number of positive coefficients Uij in the definition of r.
Moreover, there is a unique fixed point for r in Qn, n say,
308 to wich the iterations converge, i.e. lim*:_00 Tk(n) = h. If the atoms are fermionic,
then
t
[r (n)]j < 1 for all j's and k's. PROOF: The fact the iterations will remain inside a convex set Q n follows from the fact that positivity is preserved and from Eq.(25). The dimension d! of the space 1R''', where Q n is contained, is easily computed by the observation that each map r y determines one direction in Md, in the sense that r»j(n)—n is proportional to a given element, e;j say, of a base of TR.d, for every n g TR.d. Thus, r ( n ) - n is a linear combination of a fixed sub-base, {e,y} of WLd for every n g JR,d Then, we have d! < (number of 2-level maps in r ) . But each map r,j lives in a subspace which is not contained in the one determined by all the other maps, as the variable n t J appears in the dynamics of r,j only. Thus d! > (number of 2-level maps in r ) . In particular, the map lives in a space of dimension (TV2 — N)/2, if all the coefficients i/,-y are positive. The fermionic variables are confined inside [0,1] simply because convex combinations of numbers in [0,1] are numbers in [0,1], and because each fermionic r y takes [0,1] into itself. Concerning the fixed points of r , Brouwer's fixed point theorem ensures the existence of at least one of such points in <2n. Also, the entropy of the system, in a state with populations given by n = J ^ j J ' y P t j , obeys S(n) = = SS(( £ j > £ VijSiPn), 5(n) E ^UijPi, f t J >j2^s(p,j), i,j V« J
(28) (28)
and so n is a fixed point for r if, and only if, it is fixed for each r,j separately. In fact, assume that n is not fixed for ry, and write s nH *("*) ( ki), =E E> '("*)++ ^2 E S (" )> k k,l k,l
s
k
where s(?i/t) is the contribution to S of the A:-th atomic degree of freedom, and s(nki) is the contribution of the photons yy. Then, using inequality (20), we get
5(B)) << 5(r S(r„-(n)) = S(n u (n)) = JV N
= =
s s nw E l( w * )+ + E X E (("*<)) s
*=i.
n
5
+ s([n>)];) s([n3(n)I,-) + + s([r, 3(tni(a)],-i), + si(Wn)]i) (h>)]0 + J(n)]IJ),
(29)
(29)
(M)*(M)
and, letting JFV, = r,j(n) in Eq.(27), we obtain S(r(n)) > 5(n), i.e. n is not fixed for r . Thus, S is a strict Liapunov function for r and, being continuous, it is bounded in Q n . It follows that the nonnegative quantity F(r*(n)) = S(r* + 1 (n)) - S(r*(n)) converges to zero as k tends to infinity and, because of equations (20) and (29), rk(n) must converge
309 to the manifold of fixed points of r, i.e. to the set of the joint fixed points in Qn of all the TVJ'S. The proof of the theorem will be completed if we can show that there is a unique n € Q n which is fixed for r . Indeed, this is the case, as every fixed point of r is a local maximum for 5 , and there can be no more than one point with that property in Q„. In fact, ft is a maximum for S along the direction of motion of Tij, for every (i,j),
and the
strict convexity of s implies that S is also strictly convex, thus making it impossible for two local maxima to coexist in <2„.
•
We have thus shown that the discrete models defined by Eq.(23) admit a range for their parameters, within which convergence to an equilibrium point always occurs. From this, an appropriate limiting procedure similar to that described in Ref.[ll] would show that the continuous time schemes associated with our QBM behave in the same way. This will be the subject of a future paper.
5. CONCLUSIONS We have introduced the nonlinear dynamical systems called Quantum Boltzmann maps, and we have discussed some of their basic properties and an application to the theory of iV-level systems. In particular, we have discussed the positivity preserving properties of the approximate theory, and the increase of the entropy under the relative map for the populations of the different energy levels and of the heat reservoirs. The global and the asymptotic behaviour of the system, under the approximate QBM has been investigated, and we have found that every initial condition is attracted to the uniquely determined fixed point.
ACKNOWLEDGEMENTS The author wants to thank Prof. L. Accardi, Director of the "Centro Matematico V. Volterra", University of Rome II, and Prof. A. Frigerio for enlightening discussions. The author also thanks the anonymuos referees for pointing out several weaknesses of the first version of this work. Thanks are in order to Prof. R.F. Streater, for reading the manuscript and for constructive criticism. Support from GNFM-CNR is gratefully acknowledged.
310 REFERENCES [1] R.F. Streater A Boltzmann Map for Quantum Oscillators J. Stat. Phys. 48, 753 (1987) [2] R. Diimcke The Low Density Limit for an N-Level System Interacting with a Free Bose or Fermi Gas Comm. Math. Phys. 97 331 (1985) [3] V. Gorini , A. Kossakowski and E.C.G. Sudarshan Completely Positive
Dynamical
Semigroups of N-Level Systems J. Math. Phys. 17(5) 821 (1976) [4] V. Gorini and A. Kosskowski N-Level System in Contact with a Singular Reservoir J. Math. Phys. 17(7) 1298 (1976) [5] A. Frigerio and V. Gorini N-Level Systems in Contact with a Singular Reservoir.
II
J. Math. Phys. 17(12) 2123 (1976) [6] M. H. Mittleman Introduction to the Theory of Laser - Atom Interactions
Plenum
Press, New York (1982) [7] C. Cohen-Tannoudji Light Shifts and Multiple Quantum Transitions in Physics of the One- and Two-Electron Atoms, F. Bopp and H. Kleinpoppen Editors, Wiley Interscience, Amsterdam (1969) [8] P.J. Ungar, D.S. Weiss, E. Riis and S. Chu Optical Molasses and Multilevel
Atoms:
Theory J. Opt. Soc. Am. B 6(11) 2058 (1989) [9] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste and C. Cohen Tannoudji Laser Cooling Below the one-photon Recoil Energy by Velocity - Selective Coherent Popula tion Trapping: Theoretical Analysis J. Opt. Soc. Am. B 6(11) 2112 (1989) [10] C.C. Gerry and R.F. Welch Interactions of a Two-Level Atom with One Mode of Correlated Two-Mode Field States J. Opt. Soc. Am. B 8(4) 868 (1991) [11] L. Rondoni A Stochastic treatment of Reaction and Diffusion, Ph.D. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, August 1991 [12] R.F. Streater The Boltzmann Equation for Discrete Systems in Statistical
Mechanics,
A. Solomon Editor, World Scientific (1988) [13] R.F. Streater Convergence of the Quantum Boltzmann Map Comm. Math. Phys. 98, 177(1985) [14] L. Rondoni Nonlinear Boltzmann Maps in Classical and Quantum Probability (in this volume) [15] O.E. Lanford and D.W. Robinson Approach to Equilibrium of Free Quantum
Systems
Comm. Math. Phys. 24, 193 (1972) [16] E.H. Wichmann Density Matrices Arising from Incomplete Measurements J. Math. Phys. 4, 884 (1963) [17] A.G. Shuhov and Yu. M. Suhov Ergodic Properties of Groups of Bogoliubov Transfor mation of CAR C-algebras Ann. of Phys. 175, 231 (1987)
311 [18] R.F. Streater Entropy and the Central Limit Theorem in Quantum Mechanics J. Phys. A. Math. Gen. 20, 4321 (1987) [19] L.E. Reichl A Modern Course in Statistical Physics University of Texas Press, Austin (1984) [20] V. Boffi Fisica del Reattore Nucleare Vol. 1, Patron Editore, Bologna (1974) [21] V. Hutson and J.S. Pym Applications of Functional Analysis and Operator Theory Academic Press, New York (1980) [22] M. Fannes, Ann. Inst. H. Poincare A, 28, 187 (1978). See also N.M. Hugenholtz Derivation of the Boltzmann
Equation for a Fermi Gas J. Stat. Phys. 32(2) 231
(1983)
* Current address: Department of Theoretical Physics, School of Physics, The University of New South Wales, Kensington, NSW 2033, Australia
Quantum Probability and Related Topics Vol. VIII (pp. 313-328) ©1993 World Scientific Publishing Company
N O N L I N E A R B O L T Z M A N N M A P S IN CLASSICAL A N D Q U A N T U M P R O B A B I L I T Y by Lamberto Rondoni* Center for Transport Theory and Mathematical Virginia Polytechnic Institute and State Blacksburg, VA 24061-0435
Physics,
University,
(U.S.A.)
ABSTRACT We consider a. class of discrete dynamical systems, called Boltzmann Maps, both in classical and quantum probability.
We show that many physical models used in
nonequilibrium statistical physics can be expressed in terms of Boltzmann Maps, and we discuss how new models can be constructed in this framework. In particular, the increase of the entropy under the Boltzmann Maps is used to prove the convergence of the dynamics to the appropriate equilibrium positions, in the relevant phase spaces.
1. INTRODUCTION In this paper, we consider a class of nonlinear dynamical systems, called "Boltzmann Maps" (BM), which were first introduced in the 80's by R.F. Streater [1 - 3]. Such maps represent a suitable mathematical tool for the description of nonequilibrium physical systems, as they approach their stationary states. Several applications of the BM have already been investigated, with the result that certain known features of classical and non
classical systems have been reproduced in the new framework, and new results
have been derived for other systems. In particular, certain problems that arise in sta tistical, atomic and nuclear physics, in chemistry, and in biology have been studied in terms of BM. It is known that the problem of convergence to equilibrium for different classes of physical systems is outstanding, and different theoretical approaches are under consideration. Among these, the method based on BM belongs to a class of stochastic models which neglect the details of the microscopic interactions among particles, and
314 such that irreversibility emerges as a consequence of the randomness which is put in the process by hand. This model may not appear as satisfactory as a Hamiltonian theory would be, as the macroscopic behaviour of a physical system is not derived from its microscopic dynamics. Nonetheless, it has good physical justification, since it attempts to describe the effects of the myriad of unknown degrees of freedom, and of noise, which are always present in the real physical systems, and which are not easily included in a Hamiltonian formalism. Moreover, we will show how some physical insight into the microscopic structure of certain processes can be drawn from the use of BM, and we will give an idea of their potentially very broad applicability. The aim of this paper is to familiarize the reader with the subject of BM, and to stimulate further research. Thus, the discussion will be rather informal and sketchy, as technical details can be found in the quoted references. Our approach is not the most commonly used, and, in fact, the Hamiltonian formal ism is more often found in the literature. In particular, we call the reader's attention to Alicki and Messer's work [25], where a model for a quantum Boltzmann equation is de rived within a Hamiltonian formalism. Also, more recent results on mean-field dynamical semigroups have been presented by Duffield and Werner in Ref.[26].
2. THE CLASSICAL BOLTZMANN MAP The concept of nonlinear "classical" Boltzmann maps (CBM) was first introduced by Stretaer in Ref.[l]. Here, we use the term classical to stress the fact that quantum mechanics doesn't play a role in this theory. The notion of CBM can be summarized as follows. Let fi = {Ai, A2,A3,...}
be a finite or countable set, to be regarded as our theory's
sample space, and consider fim = fi x ■ ■ ■ x fi, the cartesian product of m copies of fi. Denote by Q the simplex of probability measures on fi (real sequences {p,} in /i(fi), such that pi > 0 and £ , p, = 1). and let Qm represent the set of probability measures on fim (real sequences {pi,...im} in h(Um),
such that £>,,...,„, = 1). Define the sampling
map a : Q -» Qm, as a(P) = ®™P, which forms a probabilty in Qm by taking the product of m copies of P = (pi,p 2 ,P3, •■•), a probability in Q. We will interpret a as the map that implements the Boltzmann's Stosszahlansatz, by noting that each component, Pi, of P could represent the probability of sorting one element, Ai, out of fi, and thus each component of a(P) would be the probability of fishing out m independent elements before they interact. Let T : Q m -+ Qm be a doubly stochastic map; i.e. T and its adjoint T* are stochastic
315 maps, which is to say: a > 0 implies Ta. > 0 , ||Ta|| = ||a|| (^-norm) for all a > 0, and the same holds if T is replaced by Tm- The matrix elements of T, under the inner product of l2, will be interpreted as the probability that a set of m particles interact and produce another set of m particles. Let E : Qm —> Q be the expectation onto the first factor, i.e. the summation over all but one index of an element of Q m . This takes probabilities on (lm to probabilities on fi. Then we define a Boltzmann map of order m as follows. DEFINITION 1. A Boltzmann map of order m is a map of the form r(P) =
E(T(c(P))),
i.e. P ^
® r P i-1* T(®?P)
3 + P' = T(P),
for every P 6 Q. Just from the definition, one can easily deduce that THEOREM 1. (a) A Boltzmann map of order m maps Q into itself, (b) The entropy S = — ^2 jPj log pj of the system does not decrease under the action ofr; i.e. S(T(P))
> S(P) for every P G Q.
These are desirable properties, for a theory that describes the approach to equilib rium of a physical system, although they are not shared by all the discrete models. In particular, the preservation of probabilities has a very important interpretation in terms of nonnegativity of densities of particles, or concentrations and soon. Furthermore, the choice of a doubly stochastic map for the description of the interaction of particles, is made necessary, in a sense, by a theorem of Alberti and Uhlmann [4], concerning "chaos - enhancement" due to linear operators. In fact, it is shown in Ref.[4] that the only chaos - enhancing linear operators are doubly stochastic, where the term chaos - enhancing is a generalization of the concept of entropy non-decrease. Although it is possible to proceed with the analysis on countable sample spaces, we now turn to the special case of finite spaces, in order to focus on some applications of particular interest. Therefore, let us assume that fi contains n elements only, then we can prove the following. THEOREM 2. Ifl is a simple eigenvalue ofTT* and P € Q, then Tk(P) -* (1,1,..., l ) / n as k —* oo. (Note that we are in a finite dimensional space, so TT' is well defined). The proof of Theorem 2. relies on the fact that the crucial step for the increase of the entropy determined by r is the application of T, and it is based on the following fundamental result from Ref.[l]
316 THEOREM 3. Let T be a, bistochastic map on MN and let P = {pj}^ be a i.e. 1 > p,>
0 , j = l,...,iV, and Y-iPi ^ *-
Let
the
semiprobability,
eigenvalue 1 ofTT'
be simple,
with a. gap A to the next largest eigenvalue. Let qi = 5DfcTti pk- Then A | | p-- p | | ^ ^53Pk P f c lloo ePk gpi > > ^2 ^Ij^gqj ij lQ g?j ++ 2- A H P Pill
k
(1)
ij
2 where p =--- \\P\\i/N, where Iwvl2 \\P\\i/N, and \\P\\2 = = VIPil VM22 + •-■■++\PN\
Therefore, whenever TT* has a spectral gap A, the entropy of the system increases a. positive amount at every application of r, unless P is the uniform distribution. It follows that the entropy converges to a finite limit as it is a continuous function on a compact domain. At the same time, the sequence { r m P } lies in the same compact set, and so it has a subsequence that converges to P, say. This observation allows the author to prove that S{TP) = S(P), and that P is the limit of all the convergent subsequences of {rmP}.
Then P must be the uniform probability distribution, otherwise Theorem 3.
would be violated. Having introduced the CBM, we now take a look at its applications.
We start
with the observation that it is possible to find appropriate sample spaces and doubly stochastic maps, such that the resulting form of r constitutes a discrete scheme for the time evolution of the concentrations of the different chemicals in a chemical reaction. For instance, consider the following reaction "iAi AN nxAi + +.... .. ++ U nNNA N
= = m,\B\ miB1
+ +.... .. + +
mITIMBM, MBM,
(2)
where the small letters are integers representing the stoichiometric coefficients of the reaction, and the capital letters are symbols of chemicals. Let us denote by p, and by q}, respectively, the concentrations of the A;'s and of the Bj's. Then, assuming that the forward and the backward reaction rates are equal, we have that the following system of ordinary differential equations describes the evolution of the concentrations. ^ dd J-^£
= n,\( q?'...qZ" nJA ( C - < ? M M-ft...iff), -P?'-P7).
..N ij ==i , - h-N
= -m \(q?'...qZ"-mr-PN"), kX(q^...q^-p^...p^\ = k P
*fe== i ,...M h-M
(3) W
where A > 0 is the rate constant. These generalize the equations of [5],[6]. If n\ + ... + njv = mi + ■■■ + m-M we say the system is "balanced". In that case, and normalizing the sum of the concentrations to 1, we can express the discrete form of Eq.(l) Pi ";/*(??" p] = Pi + » j K C -■■■I7"-PV9 M " - Pi" - -P7), PAT).
i3== 1. i. ...N -N
m it = ft - m "»*/*( q'k = * M ry, ( C ' ...«£" - P? -PnNN"), ), -1MM~PT
k = l,l,....M ..M
(4)
317 as a Boltzmann map on the probability space relative to £2 = {Ax, ...,AN,BX,
...,BM}-
In fact, our reaction describes a process in which a particular selection of n particles of kind "A" interact and disappear in order to produce another particular selection of n particles of kind " 5 " , and viceversa. If the set of n reacting /1-particles is different from a permutation of the n-tuple, Rx
x
W,...,N
= {Ai,...,A1,...,AN,...,AN),
with rei
particles of kind J4I, ... , UN of kind AN, then no reaction takes place, i.e. the reaction products are represented by a permutation of the reacting n-tuple itself. If, instead, the reacting n-tuple is RI,..,,I,„,,N,..,,N, °f ■5'i,...,i
M
then the reaction products form a permutation
M = (B\,...,BI,...,BM,...,BM),
with m,\ particles of kind B\, ..., m-M of
kind BM- A similar notation can be developed for a set of n reacting B-particles. Then, the following can be proven [7,8] LEMMA 4. Let R and S be generic n-tuples of elements ofCl, and assume that all the permutations,
TTJ(R) i—► itj(S), are equally probable processes. Then Eq.(4) represents
a CBM of order n, if and only if (i 6 [ 0 , Z B ] , where
. I 1( n - 1 ) !
LB = mm
- -
(n-1)!
'— , — \
'—
\
.
In the case that Eq.(4) is a CBM, the form of the interaction operator is
T =
T'®Ml®...®Mk,
where T' is doubly stochastic, M; = (l/ti)Jt,, are all equal to 1. Moreover, T'(T')'
and Jti is a t,- x U matrix whose entries
has a spectral gap A > 0 if and only if /i e (0,ifl).
The proof of this result is relatively complex in terms of algebraic manipulations of sets of equations, which invlove some combinatorial calculus. For the rest, the result is based on Perron - Frobenius theorem and on the observation that all the entries of X" are positive if and only if fi lies in (0, LB)To make all these concepts clear, let us consider a simple example that shows the main general features: | 2A = B + C |. In this case, the map r acts on a probability space {P = (PA,PB,PC)}
= Q C Ht3, and it transforms P into P* acording to the following
rules: PA = PA ~ I* (P2A - PBPc) =PAP*B=PB+
nD
Pc =PC+
liD.
Ip-D
318 The bistochastic matrix associated with this reaction is given by AA BC CB AB BA AC CA BB (\-1jX AA AA (1-ty
BC CB
ii »
nP-
0
A* M
^2 ^kiR 0 2 1=R ^ ^ 2 0 0 0
0 00
0° °0 0 0 0 00
CC CC 00 >\
°0 00
0
00
00
0
0
0 0
AB AB
p. h 0
0
0
.5
.5
0
0
0
BA BA
0
0
0
.5
.5
0
0
0
0
AC
0
0
0
0
0
.5
.5
0
0
CA
0
0
0
0
0
.5
.5
0
0
BB BB
0
0
00
00
11
0
cCCc V V oo
0
2
0 0 0 0 0 00
o0 o
0
o
0o
0o
0o
o0
= =TT
(5) (5)
i 1 )/
Here the capital letters have been used to better identify which process is represented by which matrix element, and each such matrix element T;J ; M can be interpeted as the probability that the process (i,j) i-> (k,l) takes place. Clearly, T acts like the identity on a part of the vector that is given as input. In fact, T is applied to vectors of the form P® P only, and so the part relative to the components PAB,PBA,PAC,PCA,PBB
an<
i Pec
is left unchanged. On the other hand, the invariant subspace relative to the components PAA-IPBC PCB is "mixed" by the action of T. In fact, the submatrix T'(T')*
a
nd
of TT*, which is
relevant to such a, subspace, has positive entries, and so it has a spectral gap. As a consequence, the application of r increases the entropy until the input vector for T has PAA = PBC = PCB, i-e- p\ ~ VBPC-, which is exactly the equation for the stationary points of r. This, in general, is not enough to prove convergence to the fixed points of T. In fact, 7 has a smooth, connected and compact manifold of fixed points, and so the iterations could wander around, closer and closer to such a manifold, without converging to any one of its points. However, it is easily seen that single reactions give rise to one - dimensional dynamical systems, due to the presence of constants of motion. This observation, coupled with Brouwer's fixed point theorem, allows us to prove that, for every initial condition P 6 Q, there is a unique fixed point P to which the iterations can converge, and thus they do indeed converge. This result, although quite crude, is somewhat interesting, because the map r is not a global contraction in Q, neither it is locally contracting. Note that Q is a subset of IR^ for some K 6 IN, and so we prove that there is no norm or metric of IR
under which T is contracting, simply by checking
that r is not a contraction in the ||.|| 2 -norm (cf. theorem 88.1 in Ref.[24]). Nonetheless, knowing that the system is 1-dimensional, we can restrict our analysis to the appropriate subspace, and draw some more refined conclusions. Thus, parametrizing the motion with a unique variable, x say, we get the following result [7-9]
319 THEOREM 5. For every reaction described by a CBM r, and for every initial condition P € Q, there is a "line of reaction" I > C Q, such that the dynamics {r J (P)}g° is contained in I > , and there is a unique fixed point P e I > to which the iterations converge provided that p. € {0,LB).
Also, there is a p0 > 0, such that the iterations
converge in a monotonic way, if 0 < p < ^o- Moreover, there exists an invariant compact subset I ofTp, exponentially
which attracts the iterations, and there is a p.' > 0, such that T'(P) is convergent to P, if P € I and if0
p!.
The proof of this theorem is based on very standard arguments from the analysis of dynamical systems on the interval, and its consequence is this corollary [7-9] COROLLARY 6. (a) The entropy is a strict Liapunov function for r.
(b) All the fixed
points in the interior of Q are stable. So far, we have considered single reactions only; but it is possible to construct "Generalized CBM", which share the same basic properties of the CBM. In particular, it is possible to take convex combinations of different CBM's -each one of which acts on the simplex relative to a particular sample space- by redefining every CBM on the probability space of the union of the different sample spaces. Doing this, we construct a new map which can be used to model complex chemical reactions, i.e. reactions which are combinations of many single reactions, and which better represent most of the "real life" ones [9]. Consider one of this generalized CBM, r = X], A;r,-, where r, acts on Qi, and the A;'s obey 2 ; A, = 1 , A; > 0. Then, using the convexity properties of the entropy, we prove that a point P is fixed for r if and only if it is fixed for each of the TV'S separately. The analysis of this new kind of dynamical systems is necessarily more complex than that of the CBM, because the notation becomes a little heavier, and especially because the dynamics is not 1-dimensional any more. Therefore, we leave out the details, and we simply state the main result [8,10]. THEOREM 7. Consider a set of N reactions, each described by a Boltzmann order rn,i=l,
map r; of
...,N. Let r be a convex combination of the r, 's. Then
(a) S is a strict Liapunov function for r. (b) Let M be the manifold of the fixed points ofr, and let d(x,M)
= m i n y g x \\x - y\\2.
Then, for every P € Q and every c > 0, there exists a k € IN such that
320 fixed point for r. As we can also prove that the iterations cannot converge to the fixed points on the boundary of the set determined by P, we have got the result that also the systems of convex combinations converge to an equilibrium point. It is now quite interesting to note that the discrete diffusion operator, which we get by taking the central difference approximation to the Laplacian in a. finite dimensional space, is a CBM of order 1. Therefore, if we study a reaction in an m-dimensional container of finite volume, we can subdivide such a, volume in an arbitrary, but finite, number of cells, and then consider the space inhomogeneus problem with the discrete Laplacian, as described by a generalized CBM. We cannot easily take the limit for the number of cells that goes to infinity. However, from a physical point of view the situation is satisfactory, as the space discretization of a finite volume can be made as fine as desired. To make this clear, let us consider the example of a 1-dimensional medium made of L cells, in which a single non-autocatalyitc reaction takes place: n-iAi H
h riff AN = miAjv + i H
(- VHMAN+M-
(6)
For simplicity, assume that the diffusion coefficients of each chemical do not vary from cell to cell, and call d, > 0 the cell relative to the chemical A,-, Let p,j be the concentration of A{ in the j'-th cell, and assume that N+M
L
££w
= i-
(7)
1=1 j = l
Then, the diffusion operator has the form <5 = Si ® ... © <W+M> where , describes the diffusion of the chemical A{. On the other hand, the full reaction operator takes the form L
rr = £A,r„
(8)
i=i
where r,- describes (6) in the z'-th cell, and so the combination of the reaction term with S can be written as r = r r + ( l - A)<5, for A e (0,1), if we have A, > 0 and J \ A; = A. Then, given a P 6 Q, the simplex of probabilities on fi = {An,...,
AN+M,i}f=1,
the action of r
on P produces a new probability P", whose components p*,- can be computed as follows: ph
= m
- * m
(Pii ■ ■ -PN" - ?n
■ • -
- (i - > K ( P , I -
p&)
P'i = Pii - A . W {Pu ■ ■ -PN", - Vu> ■ ■ •<&") + (1 " > K ( P , , . - i - 2 P j , - p J | 1 + i), n
P)L = P,L - \uMj (PTL ■ ■ -p NL - sffl ■ ~q$tL) + (i - AKbto-i - m), for i = 2,..., L - 1, for the chemicals on the left hand side of (6), and 3/1 = VI + Ai/im, (p»J ■ • .pjft - qR ■ • .qgfi) - (1 - \)dN+J(qn
- qj2)
q*, = 9H + Aj/my {ft ■ ■ ■?$ - C ■ -qS/f) + (1 - X)dN+j(qiii_1 - 29ll 1'L = VL + *Lm {PTL ■ ■ -pnNNL ~ ?1L - ■<$&) + (1 - VdN+AUL-l ~ qjL),
Ui+1),
(10)
(9)
321 for i = 2,..., L - 1, for the chemicals on the right, where we have used
L and q] = q3■+ fim,r ] T A;£;(P),
(11)
where Di{P) = pft ■ • -p%N. - g™' •••ififf, from which it M o w s that njpi - n-ipj = Kj,
j = 2,..., N,
and miPl
+ mq, = Ljt
j = 1,..., M, (12)
are constants of motion. We can now find the fixed points of r , given some initial condition P(0) G Q. Recall that r ( P ) = P if and only if 6(P) = P and r;(P) = P , for every i, because P is fixed for r if and only if it is separately fixed for each of its components. Then, if we call pji the components of a fixed point P of r , we must have pji = ... = pji = pj/L for every j = 1,..., TV + M, and £>,'(P) = 0 for every i, i.e.: A(P) = K
(#'■■•&"-«;"'•
where n = J ] n, = 53 m >'
anc
•?£M) = 7 ^ ) = 0,
(13)
^ ^(^*) ' s *^ e disequilibrium parameter of the stirred
reaction, evaluated at the point whose components are the pj's. We know from [7] that there exists a unique solution to (13) in Q, given the particular choice of the constants of motion due to P(0), and we conclude that for every P(0) € Q there exists a unique P € Q such that r*(P(0)) -> P as k —» oo. Moreover, P is stable, as the fixed points of the stirred reaction are. Concerning the explicit form of the diffusion operator, i, we observe that each one of its components, 6i, has the form 1
(^ - *
2
3
4
L-
1
L
di
0
0
0
0
di
l-2dt
di
0
0
0
0
di
1 - 2d;
d,
0
0
0
0
di
1 - 2d, '
0
0
0
0
0
0
l-2d;
di
0
0
0
0
■■■di
\
(14)
1-dJ
in a 1-dimensional medium with equal diffusion coefficients at every cell. So far, we have seen how the study of chemical reactions, under certain hypothesis, can be cast into the form of a CBM. The results we have presented are not the only ones that can be derived in this framework, and we don't know of any other equally general treatment for discrete models. We should note that most of these results were known for the continuous time systems. However, we believe that the discrete systems are
322 worth being investigated of their own merits, and not just because they approximate the continuous models. In fact, besides the physical motivation of this statement [8], there certainly is a strong mathematical motivation, which arises because of the use of discrete schemes in numerical analysis and especially because discrete models are substantially different mathematical objects from their continuous counterparts. In particular, prop erties like the positivity of the solutions, which are trivially verified for the systems of ordinary differential equations, are not shared by all the discrete schemes, so that only particular classes of maps, like the CBM's, can be used. It is also interesting to note that the known results for continuous systems, can be rederived in the framework provided by CBM's, if the rate constant p is interpreted as AAi, where A is the rate constant of Eq.(3). In fact, conditions can be given such that the discrete scheme converges to the solution of the continuous model when At —► 0, [9,10]. The study of chemical reactions is not the only application of the CBM that has been investigated. For instance, Streater has recently shown how the Ising model with Glauber dynamics, and cotransport through membranes can be treated via generalized CBM's of particular kind [11,12].
3. THE FIRST QUANTIZED BOLTZMANN MAP The first quantized Boltzmann map (FQBM) was first introduced by Stretaer in Ref.[2]. It is a direct generalization to quantum mechanics of the CBM, which has the feature of treating quantum particles as distinguishable. However, this model is based on very clear physical intuitions, and many of the results drawn within its framework are used also in the second quantized formalism. Moreover, we may argue that Boltzmann's stosszahlansatz really makes sense only in a scheme with distinguishable particles. It is therefore worth taking a look at the FQBM as a discrete version of the quantum Boltzmann equation. The set up for the FQBM is this: H is a finite dimensional Hilbert space, with < x, y >= £3; S7 yi and a state is obtained from a density operator p which is positive and has unit trace. If the set of linear operators is denoted by A = B(H), and the set of density matrices by E, we can define a doubly stochastic map, T : A —t A, as in the previous section. Define the entropy of a state p as S(p) = - Tr(plogp).
Then,
r maps S into E, and S(rp) > S(p), where equality holds if and only if r(p) = p. For a doubly stochastic map T : A -> A, a sharp estimate on the entropy increase under its action can be given. Furnishing A with the inner product < A, B > = TT(A*B),
we get
THEOREM 8. Let PL be the orthogonal projection of B onto Kei(I - TT*)-*-, and assume that the spectrum u(TT*)
of TT* is contained in [0,1 - A] U {1}, for some
number
323 0 < A < 1. Then the entropy obeys S(T'p)-S(p)>j\\P±p\\\
(15)
where I is the identity map, and p is a positive operator of unit trace. (Note that the domain of T* is invariant under the action of T" and it is contained in the domain of T. Thus TT* is well denned). Let us introduce K., a Hilbert space representing the one-particle space, and assume that our ri is the tensor product ri = K ® ■ ■ ■ ® K, which will be regarded as the n-particle space, if n is the number of factors. Then we define a FQBM, r , as follows. DEFINITION 2. A FQBM is the composition of maps
p„®lp„
T(®lp) P- Tr2...„ [T(®y)} = r(p),
where T is doubly stochastic on A and commutes with the permutations
of the factors
in the tensor products, and Tr2...„ is the trace over all but one of the n factors K. The assumption on the commutativity of T with permutations is technical, and has to do with the fact that the particles are treated as distinguishable, in this theory. The physical picture behind the construction of the FQBM is that of a rarefied gas, in which the time between two collisions among a particle and other particles is very long, so that the single particle Hamiltonian is approximately equal to the free Hamiltonian. Let us assume that ho, the single particle hamiltonian, have eigenvalues e\, e2,...., e^. We denote by Ho the Hamiltonian for n independent particles of the gas. Then Ho takes the form Ho = h0® I ® ■ ■ -®I+ ■■■ + I ® / ® • • • ® /io- Let us assume that H0 has a spectral resolution Ho = J2jEjPj,
where the Pj's are the corresponding spectral projections,
and the Ej's are the eigenvalues. Then we can prove that the FQBM, r , preserves the average 1-particle energy, i.e. Tr((r/))/i 0 ) = Ti(ph0), provided that the doubly stochastic map T preserves the spectral projections of ifoWe now introduce the notion of ergodicity for doubly stochastic maps. DEFINITION 3. A doubly stochastic map which preserves the spectral projections, Pj, of H0 is called ERGODIC ifTA
= A = T*A implies A = £ ; - djPj, where aj is a function
of Ej for every j . Then, the following can be proved: THEOREM 9. Assume that {0,1,2,...} is the spectrum of h0 on K., and let the finite multiplicity
m(j) of the eigenvalue j obey m(j) < cjT, with c and r fixed
constants.
Let H = K, ® ■ ■ ■ ® K. and let T : A -> A be doubly stochastic and commute with the
324 permutations
of tensor products. Assume T is ergodic on every eigenspace of Ho- Then
m
T (p) converges in the trace norm to a Gibbs state -j8/i„
Pos :
(16)
Tr(e-Ph-)
where /? is determined by Tr[h e-0h")
W=^W- Trie-o 01")
d7)
Cearly, the condition on the multiplicity of the eigenvalues is always satisfied for a finite dimensional AC, but it should be noted that this theorem holds also if K, is an infinite dimensional separable Hilbert space [2], A possible application of the FQBM would be to investigate the properties of certain continuous time models, similarly to what was done with the CBM. For instance, Aratari and Frigerio [13], gave an explicit solution in terms of unitary dilations of the quantum Boltzmann equation introduced by Snider [14]. However, the form of the solution is complicated enough that it is difficult to analyze it in a direct way. This could be avoided, for instance, with the aid of the FQBM as, once again, such a map is nothing but a discrete form of the continuous system.
4. THE SECOND QUANTIZED BOLTZMANN MAP The second quantized Boltzmann map (SQBM) was introduced by Streaterin Ref.[3], as a model for a set of M < oo quantum harmonic oscillators. Let K, be the 1-particle space for a set of M harmonic oscillators with frequencies Wi, ...,UIM, and consider the symmetrized Fock space over /C : TS(K) = (E © K © (K, ® K)s © ■ • ■, for bosons, or the antisymmetrized Fock space Ta(K.) = (C © K © A 2 (£) © • ■ •, for fermions, where A*(/C) is the subspace of antisymmetric tensors of gijK. For simplicity, we will mainly concentrate on bosonic systems. Therefore, the algebra of observables of the system is taken to be the set of bounded operators on T s (£),i.e. A = B{TS(IC)), which is generated by the creation and annihilation operators a* and a,j , j = 1, ...,M. Let £ be the set of (normal) states on A, and furnish A with the inner product < A, B >= Tr(A*B).
If
we define the entropy of the state p as in the previous section, we can make use of the same results about the entropy gain under a doubly stochastic map. In particular, the doubly stochastic maps, T, we deal with, in this section, are obtained as mixtures of conjugations of unitaries, thus being completely positive, and we assume that all such unitaries commute with the spectral projections of the Hamiltonian. Then, the average energy of the state p is preserved under the application of T* to p. In fact, defining the
325 Hamiltonian as H0 = ^Uja^aj Tr((T'p)H0)
- £ E\P{, we get
= Tx(U'pUH0)
= Ti{pUH0U*)
= Tr(pTH0)
= Tr(ptf 0 ),
where U is a unitary operator and U* is its adjoint, and T is denned by TA = UAU" for every A in A We can also prove this result: THEOREM 10. l e t Hj = P-H, with H = F(<£N), the Fock space over <EN.
Then, the
restriction Tj ofT on BCHj) maps the identity operator J, into itself, and
S(T'p)-S(P)>
> ^
where Aj is the spectral gap ofTjTj.
P-
,
dim Hi
pzBCHj),
Then the iterations (T')np
(18)
for p 6 S(W) converge
to a density operator which is constant on each B(Hj). Therefore, a map like T" cannot mix the different "energy shells" of a system, and the dynamics it induces can be interpreted as the evolution of a system of thermally isolated parts, each one of which reaches an independent microcanonical state. To achieve mixing of the different shells more stochasticity is needed than that provided by a doubly stochastic map. In the previous two sections, relative to the CBM and to the FQBM, this extra stochasticity is provided by the stosszahlansatz.
Here, we do not have a
formalism of particles any more, and so we cannot resort to such a tool. However, we can argue by analogy, and introduce a map Q which, like the stosszahlansatz, destroys the correlations by replacing a given state with the Gaussian state which has same first and second moments. Such a nonlinear map was already used in previous works [15 - 17], and is called "quasi- free" projection, as it replaces a. state with finite first and second moments with the state whose third and higher moments vanish, and whose first and second moments equal those of the original state. It was proved by Streater [18] that such a map does not decrease the entropy, and that S(Qp) > S(p) unless Qp = p. DEFINITION 4. A SQBM is a map r : E -» E defined by r = QT*, where Q is the quasifree projection on E, and T* is a completely positive and doubly stochastic map on E. Thus, the dynamics determined by r takes place on the subset of quasi - free states in E. It turns out that the limit of every convergent subsequence of {rnp}f free state p^ such that TT*p<x>
=
is a quasi -
P<xf
DEFINITION 5. We say that a state is ERGODICALLY
MIXED if it can be expressed as
p = "£,VjPj, where Pj are the spectral projections of H0, and we say that the doubly stochastic map T is ERGODIC if the only fixed points ofTT' Then the following theorem holds.
are ergodically mixed.
326 THEOREM 11. Suppose the frequencies o>i,... t uw are such that the only quasi-free, ergodically mixed states are canonical, i.e. of the form p , = e~^H"/ Tr(e-" H °) for some (3 > 0. Assume T* conserves energy and is ergodic; then, for any state p of finite energy E , r"p —> p p, where 3 is determined by Tr(p s Ho) = E. From this, it follows that THEOREM 12. Consider a bosonic system with Hamiltonian UII,..,,UIM
be positive and mutually rational.
H0 = $3> a ; j a j a j-
Let
Let T be ergodic and p a state of en
ergy E. Then rnp converges to the canonical state of energy E. A similar result holds for fermionic systems. For some applications of this theory see Refs.[3] and [19]. In another paper [20] we show how a perturbation expansion of the map defined by r can be used to construct a theory for the M-level atom. Finally, we mention Streater and Koseki's works who have recently studied the problem of photon exchange with an infinite heat bath, deriving the relative /'-theorems (F = free energy) both for the discrete, [11], and for the continuous, [22], cases. One application of the SQBM to nuclear physics can be found in [23].
5. CONCLUSIONS We have introduced the nonlinear dynamical systems called Boltzmann maps, in classical and quantum probability, and we have discussed some of their basic properties and applications to physical problems. The stochastic treatment of the approach to equilibrium for isolated, closed and open systems have been outlined, thus showing the versatility of the BM's in describing a wide variety of physical phenomena. In particular, we have discussed how the asymptotic behaviour of the systems described by BM's can be inferred from the properties of the map, how the results about continuous systems can be recovered from the study of their discrete counterparts, and, in some cases, we have given a detailed analysis of the global dynamics.
ACKNOWLEDGEMENTS The author wants to thank Prof. L. Accardi, Director of the "Centro Matematico V. Volterra", University of Rome II, for having suggested the writing of this paper and for the hospitality of the Center.
The author is indebted to Prof.
A. Frigerio, for
enlightening discussions on the topic of this work, and to Prof. R.F. Streater, for many inspiring discussions and for critical reading of the manuscript. Thanks are in order to
327 the anonymous referees for useful comments, and to Prof. P.F. Zweifel, Director of the "Center for Transport Theory and Mathematical Physics", Virginia Polytechnic Institute and State University, where this work has been partly carried out. Partial support from GNFM-CNR is gratefully acknowledged.
REFERENCES [1] R.F. Streater Convergence of the Iterated Boltzmann Map Publ. RIMS, Kyoto Uni versity, 20, 913 (1984) [2] R.F. Streater Convergence of the Quantum Boltzmann Map Comm. Math. Phys. 98, 177 (1985) [3] R.F. Streater A Boltzmann
Map for Quantum Oscillators J.Stat. Phys. 48, 753
(1987) [4] P.M. Alberti and A. Uhlmann Stochasticity and Partial Order D. Reidel Pub. co., Dordrecht (1982) [5] I. Prigogine Evolution Criteria, Variational Properties and Fluctuations. In Nonequilibrium Thermodynamics,
Variational Techniques and Stability R.J. Donnelly, R.
Herman and I. Prigogine (Eds.), University of Chicago Press (1965 ) [6] B. Crell and A. Uhlmann An Example of a Non-Linear Evolution Equation Showing "Chaos - Enhancement"
Letters Math. Phys. 3 463 (1979)
[7] L. Rondoni and R.F. Streater Chemical Reactions as Dynamical Systems on the Interval}.
Stat. Phys. 66(5/6), 1557 (1992)
[8] L. Rondoni A Stochastic Treatment of Reaction and Diffusion Ph.D. Thesis, Virginia Tech, August 1991 [9] L. Rondoni Autocatalytic Reactions as Dynamical Systems on the Interval (submit ted) [10] L. Rondoni Complex Chemical Reactions: a Probabilistic Approach (to appear in NUc. Sci. Eng.) [11] R.F. Streater The F-Theorem for Stochastic models (to appear in Annals of Physics) [12] R.F. Streater Stochastic Models of Cotransport (to appear in Transport Theory and Stat. Phys.) [13] A. Frigerio, C. Aratari Unitary Dilation of a Nonlinear Boltzmann Equationm
Quan
tum Probability and Applications IV, L. Accardi and W. Waldenfels (Eds.), Sringer Verlag, New York (1989) [14] R.F. Snider J. Chem. Phys. 3211 (1960) 1051
328 [15] O.E. Lanford and D.W. Robinson Approach to Equilibrium of Free Quantum
Systems
Comm. Math. Phys. 24, 193 (1972) [16] E.H. Wichmann Density Matrices Arising from Incomplete Measurements J. Math. Phys. 4, 884 (1963) [17] A.G. Shuhov and Yu. M. Suhov Ergodic Properties of Groups of Bogoliubov Trans formation of CAR C-algebras Ann. of Phys. 175, 231 (1987) [18] R.F. Streater Entropy and the Central Limit Theorem in Quantum Mechanics J. Phys. A. Math. Gen. 20, 4321 (1987) [19] R.F. Streater The Boltzmann Equation for Discrete Systems in Statistical
Mechanics
A. Solomon (Ed.), World Scientific [20] L. Rondoni A Quantum Probability Theory for the N-Level Atom (in this volume) [21] N.M. Hugenholtz Derivation of the Boltzmann Equation for n Fermi Gas J. Stat. Phys. 32(2), 231 (1983) [22] S. Koseki, Ph.D. Thesis, King's CoUege, London (1992) [23] R.F. Streater, Cold Fusion King's College, preprint [24] H.G. Heuser Functional Analysis, John Wiley and Sons, New York (1982) [25] R. Alicki and J. Messer Nonlinear Quantum Dynamical Semigroups for
Many-Body
Open Systems J. Stat. Phys. 32(2), 299 (1983) [26] N.G. Dufneld and R.F. Werner Mean Field Dynamical Semigroups on
C-Algebras,
DIAS-STP-90-13
* Current address: Department of Theoretical Physics, School of Physics, The University of New South Wales, Kensington, NSW 2033, Australia
Quantum Probability and Related Topics Vol. VIII (pp. 329-346) ©1993 World Scientific Publishing Company
W E A K COUPLING AND LOW DENSITY LIMITS IN TERMS OF SQUEEZED VECTORS Slawomir Rudnicki, Slawomir Sadowski and R o b e r t Alicki Institute of Theoretical Physics and Astrophysics University of Gdansk Wita Stwosza 57, PL-80-952 Gdansk, Poland
Abstract T h e weak coupling limit for a quantum system interacting with a free Bose or Fermi gas as well as the low density limit for the similar models with bilinear interactions are considered. In both cases the choice of collective squeezed vectors as reference states leads to the proper unitary limit dynamics driven by the quantum Brownian or Poisson processes respectively. The proof of the associated limit theorem is given for the case of weak coupling limit while the low density limit is treated in a separate paper. 1. I N T R O D U C T I O N T h e investigation of limit theorems for quantum evolution initiated by Accardi, Lu and Frigerio [1, 2] and continued by the authors of the present paper also provide a rigorous description of the influence of large quantum reservoirs on small quantum systems in terms of quantum noises. The main idea is to consider the unitary dynamics of the composite system (small system + reservoir) in the interaction picture WtA which depends on the constant A > 0 describing the magnitude of a reservoir's influence on the system. The limit theorems state the convergence Z/AA2 —> lit in the sense of matrix elements constructed in terms of certain reference vectors from the Hilbert space of the reservoir. The limit dynamics Ut should be a unitary solution of the suitable quantum stochastic differential equation in the sense of Hudson and Parthasarathy [3, 4] driven by quantum Brownian or Poisson processes. After the first successful implementation of this scheme for the case of weak coupling limit for a system linearly coupled to Bose field and coherent states as reference vectors [1] one has realized [5, 6] that the existence of unitary limit lit depends essentially on the choice of reference vectors. Namely for the interactions bilinear in fields the use of coherent vectors leads to an unphysical limit dynamics [5]. The aim of this paper is to show that the collective squeezed vectors are proper reference vectors for bilinear interaction both for the weak coupling and low density hmits. We give the proof of the limit theorem for the former case while the later is studied in details in ref. [6]. It is worthwhile to notice that the different and independent approach by Accardi and Lu who used collective number vectors as reference states in the case of weak coupling
330 limit [7] as well as in low density limit [2] (in the Bose case only) gives the same physical solution, i.e. the shapes of the quantum stochastic differential equations describing the limit dynamics Ut are the same. Moreover it gives the same reduced dynamics for small system. Nevertheless the collective squeezed vectors have more clear physical interpretation, furthermore they allowed us to extend our results on the Fermi case too. T h e combined results suggest the following hypothesis: if for two different sets of reference states the limit dynamics are unitary then they are essentially the same. Finally one should mention that the limit theorems discussed here are essential extensions of the Markovian limit procedures [8, 9, 10] performed for the reduced dynamics of the open system. One assumed there as an initial state the tensor product of the fixed state of the reservoir and an arbitrary state of the small system while the description based on the notion of quantum noise admits more complicated and physically interesting initial conditions too. This formalism has found already applications in quantum optics and measurement theory [11, 12]. 2. M A I N I D E A In this chapter we present very briefly the main idea of the problem which is considered in this paper. Let Hs and HR denotes Hilbert spaces for the system S and the reservoir R respectively. Let Uf denotes one-parameter family of unitary operators describing the time evolution of the whole system S + R in the interaction picture. Our aim is to find a certain set of vectors {\t} C HR and a certain nonlinear averaging map \t —> P\\P and for each P\$> to assign explicitly one vector $ € Tg(C2(R.,dt) ® HN) (for a certain Hilbert noise space HN ) such that:
,
Ut)x,
u'®PA*'>
^°
,
Ut
u'®*'>
,
(2.1)
where u,u'£Hs and Ut is a solution of a certain quantum stochastic differential equation (QSDE). It was already mentioned that in the limit X —► 0 the time evolution of the system and reservoir S + R is governed by QSDE. Those equations are driven by quantum Brownian motion or quantum Poisson process in a weak coupling limit or a low density limit respectively. Now we briefly present the ideas of these two types of quantum equations. Let Ut is an operator on W S ® F B ( £ 2 ( R , dr)®W w ) where Hs and HN are two Hilbert spaces and T B ( - ) denotes boson-Fock space. Definition 2.1 Let {Da} be a family of bounded operators on Hs {ga} be a set of vectors ga e HN- Then the following equation is called QSDE by Brownian motion:
dUt = Y.{Do®
dM9a)
- Da ® dA't(ga) - D*a Da \\ga\\l dt } Ut
and driven
, (2.2)
U0 = 1
,
where At(g) = A(x[o,t) ® s) is an annihilation operator and the factor || f f ||l has the property: 2&(|| • | | i ) = < ■ , ■ > (cf. (3.18), (3.26) ). Solution of this equation exists [3] and one can easily check that it is unitary.
331 Definition 2.2 Let a £ 8{Hs) . b 6 B(WAT) are bounded operators, 77 £ HN d X[o,<) is a characteristic function of the interval [0,t) which can be treated either as an operator on C2(K, dt) or a function in £ 2 ( R , dt) . Then the quantum Poisson noise is a family of operators on Ws® ra(£ 2 (R,cfi)®Wjv) denned as follows: an
= a®A*t(br}) + a ® A , ( 6 ) +
ATt(a®b,t})
+ a ® At(b*T)) + a < 77,677 > t
(2.3)
,
where A<(6) = A(x[o,t) ® 6) is a preservation operator [3]. By linearity be extended on all bounded operators on B(Hs) ® B(HN) ■ The following equation is called QSDE driven by quantum Poisson noise: dUt = d / V , ( S - l , 7 j ) « i #0 =
,
Mt
can
(2.4)
1 ,
where S is an operator on B(Hs) ® B(HN) Solution of this equation exists [13] and one can easily check that for unitary S (e.g. scattering matrix) £/t is also unitary. 3. W E A K C O U P L I N G L I M I T Let us assume that reservoir R consists of noninteracting bosons or fermions in a quasi-free equilibrium state wpt, ( ft is an inverse temperature, z is a fugacity ). Oneparticle is described by a bounded from below Hamiltonian Hi on the one-particle Hilbert space 7i\ . ~Hs is a Hilbert space of the system S and HR is a second quantized Hi H$ is a Hamiltonian of the system S on the Hilbert space Hs ■ Let us assume that the interaction between system S and reservoir R is given by: V = A B ® A ( f f ° ) A ( g 1 ) + h.c. where D <E B(Hs) , J 0 , } 1 £ Hi and also assume rotating wave approximation: e«Hs De-i'Hs
and
A(-),A*(-)
_ e-i"'D
< g° , e i ( H l g1 > = 0
for
all
(3.1)
,
satisfy CCR 01• CAR. We
(3.2)
t
(€R
,
(3.3)
i.e. g° and g have disjoint energy spectra. The reservoir can be represented in terms of the Fock space [14]: (3.4) r(WiffiWi) = r(«x)®r(?ii) 1
and the correspondence: A(f)
—> a(y/l±Tf)
wf>,z(A#(f1)---A#(fn))
+ b*(jVTf)
= A„,,{f)
=
(3.5)
, ,
(3.6)
332 where A*, Af 2 denote A, A', A^, z , Ap respectively, state on F(Hi ®Hi). J is an involution,operator commuting with self-adjoint operator such J can be chosen ). Here: T :H : Hi -^ x-^Hi T =
ZZ ee-
Hi
Q is a vacuum Hi ( for each
, 3 7 ((3.7) - )
"- "^- (' l( lT T2 2e e- -^^) )_ _1 1
We have denoted by a(f), a*(f) and b(g), b*(g) the annihilation and creation operators on r ( W i © Hi) labelled by vectors / ffi 0 and 0 ffi g respectively. In all formulae with double sign ± or =p the upper one corresponds to the Bose and the lower one to the Fermi statistics. Now we define reference vectors. Let M; i = 1, 2 are Hilbert-Schmidt operators on Hi . Without lost of generality one can assume the symmetry property M* = ±JMiJ Then the following operator exists and is unitary [15]: =s
B(Mi,M B(Mi,M2)2)
e x p [ - i YH<
hi
'
M l h
, ) > a'{h a*(.h (Jh])+< i)ai)a'{Jhj)+<
- h.c. J ,
h,, M2h, hj > > b'^b^Jh,)} b'^b^Jh,)}
■- h.c. | ,
U
(3.8)
where {h{} is an orthonormal basis in Hi . Using these operators one can define the Bogoliubov transformation
:
B(Mi,M2)a(f)B*(M B(Mi,M ) 2) 2)a{f)B*(M uMu2M
= a(c(Mi) a(c(Mi) f) + a'(Js(Mi)f) a*{J${Mt)f)
,
(3.9) (3.9)
B(MuuM B(M M2)bU)B*{Mi,M 2)b(f)B'(MuM 2)2)
= b(c(M b(c(M22)f) )f)
+ b'{Js{M b'{Js{M2)f) 2)f)
,
(3.10)
where:
'(")^E(*D- ( 2 r + v
M*(MM*)n
^
v v
n=0
* *
.
(3.11) (3.n)
«<«>. =D| > D r« ^ < > ± Definition 3.1
(3.12)
The following vectors are called squeezed vectors : V(M Sl{Mi,M UM22))
(3.13)
B*{Mi,M = B'(M uM2)a 2)U
Using above properties one can easily check that the following state: # <*(M UM w M ul Mi{A*{fl)A # ( / „n)) )) = ~<*(M l M ! ( A ( / i ) - - - ■A*(f UM12,) M 2 )
is a quasi-free state being a perturbation of state choose squeezed vectors as reference vectors.
,,
A* (fi)---A* (f )V(M )V(M M )> M )> 4AM- ■■A#,(f x
u^ | Z
n
ltz n2
u
2
, hence it gives us motivation to
333 By analogy to refs. [1, 2, 5] we can define the averaging map P A *(Afi,A/ 2 ) = *(M 1 \M 2 A )
P\
,
(3.14)
where M? = A /
•>Q./A!
dt St Mi St
,
Ri,QieR
(3.15)
and 5) =
e«(w»+«/»)
(3.16)
is a modified one-particle evolution in the reservoir; u is defined in (3.2). One can easily check that: _ eitHR ye-UHR ^ (3.17) pit(Hs + HR)y e-it(Hs+HR) where HR, is a second quantization of the operator H\ + OJ/2 . It is obvious that both operators Hs + HR and HR give the same evolution in the interaction picture, so we shall use the second one. The vectors P\ *(Afi,M 2 ) we shall name the collective squeezed vectors. According to the scheme presented at the beginning of this chapter we should find a noise space Wjv and then assign each P\^{M\,M2) with the one vector from r f l ( £ 2 ( R , dt) ® WJV) ■ In order to do this we introduce a subspace £ which consists of operators satisfying the following condition: (M,M')=
I Tr(Su M* 5„ M')du
< oo
for
all
M,M'
e C
(3.18)
Definition 3.2 The noise space HN is equal to H'N © W » , where H'N is a completion of the quotient of C with respect to (-, )-norm defined by (3.18). Now in order to formulate the final theorem we need a few technical assumption. Let Co C C is a set of finite-range operators: N
( 3 - 19 )
M = £l/i>
for
/ ; , gj £ fC C Hi
V/,96JC
/
\
,
K
is defined as follows:
>\du
< oo
and
R
We also assume that
VT K C K
,
y/l±TK
C K
.
(3.20) g",gl € K.
(cf. (3.1) ).
334 Let us introduce the notation: De, e = 0,1 where D° = Theorem 3.3 With the notation as above for Mi, M[ e Co e = 1,2 (which nical assumption described in Proposition 5.5) there exists t0 > 0 u,u' £ Hs and for all t e [0,*o) or t e R + in the Bose or tively and finite Qi, Q[, R{, R[ 6 R i = 1,2 : lim < u ® P A * ( M i , M 2 )
,
Uhki
A—0
D,
D1 = D* .
satisfy some tech such that for each Fermi case respec
u'®Px^{M[,M'2)>
(3.21)
'
exists and is equal to
,
u'®$(x[Q'1,K'1]Mi©Xl-H|,-OyMi)>
Ut
,
(3.22) where
Ut
is the unitary solution of QSDE:
dUt = J2 { D<®dA*t(Ge) - D'-'ttdAtiG1)
- \\Gt\\2_DtDl-t®ldt}Ut
c=o,i Uo = 1
, (3.23)
,
where: $(•)
denotes a coherent state normalized to V(C2(R,dt)
in the Fock space G° = -liOeljVTg1 1
G \\M®M'\\2_
1
= -((\VlTfg
>
/
,
,
(3.24)
>< JN/1 ± Tg°\ 0 0 )
= \\Mf_ + \\M'\\2_
\\M\\2_ =
1
® Hjv)
for
,
M,M' € H'N
dt Tr{5 ( M" S* M)
(3.25) , (3.26)
J — oo
In the above theorem we have used the same symbols for elements of Co © Co and HN It is obvious that any element £ 0ffiCo defines an element in the Hilbert space HN J SO such notation should not be confusing. For the proof see the chapter Main Calculus. 4. L O W D E N S I T Y LIMIT The low density limit was described in all details in ref. [6], so now we present only an outline of this problem. We assume that the interaction between system S and reservoir R is given by: V = D®A'(g°)A(g1)
+ D* ® A'(g1) A(g°)
,
(4.1)
where A(-) and A*() satisfy CCR or CAR and D £ B(HS) ff0,?1 € "Hi As in the weak coupling limit case, we also assume the rotating wave approximation (3.2)
335 as well as that g0 and g\ have disjoint energy spectra (3.3). We use the same representation for this system as in the weak coupling limit case. Let M is a Hilbert-Schmidt operator on H\ then the following unitary oper ator exists: 8(M)
= exp{ -J2
< hi,Mhj
> a*{hi)b*{Jhj)
- h.c. }
,
(4.2)
■ ,j
where {hi} is an orthonormal basis in Hi Such operators also define the Bogoliubov transformation (cf. [15]). By analogy to the previous case the following vectors we shall call squeezed vectors: * ( M ) = B*{M)il , (4.3) where
fi
is a vacuum state on
r(Hi ®Hi)
The states define by the above vectors:
u)M(•) = < * ( M ) , • * ( M ) >
(4.4)
are quasi-free and gauge invariant and are perturbations of the equilibrium state (3.6). The averaging map Pz is defined as follows: PZ<S>(M) = ^ ( v ^ / v JQh
dtStMS;)
.
ujp 2
(4.5),
'
One should notice that in the low density limit we choose as a magnitude of reservoir's influence on the system S the fugacity z (3.6) which is proportional (for small z) to the density of the reservoir. In the above definition St is a modified one-particle reservoir's evolution: St = e i ' ( H l + w f t > , (4.6) where Pa is a projector commuting with Wi such that PQ g° = g° and Po g1 — 0 . Existence of such projector is a consequence of the assumption about disjoint energy spectra of g° and g1 Let C is a linear subspace of the space of Hilbert-Schmidt operators consists of operators which satisfy the following condition: (M , M')
=
f Tr( St M* St* M')
< co
for all M , M ' e £
(4.7)
Definition 4.1 The noise space "HN we define as a completion of the quotient C with respect to (-, -)-norm defined above. The final theorem is proved for a certain subset of the set of squeezed vectors, indexed by a subset £0 of the space C, . £ 0 ' consists of finite-range operators: of
N
M » £
1/iXjil
f,geHi
,
336 where /, g satisfy the condition (3.20). We have to assume also that two families of t-functions parameterized by fugacity z: | < ■s/Tfzf
, Stg > |
and
| < s/l ± Tf , Stg > \
for all z small enough are majorized by a function which belongs to also assume that:
C1 ( R )
We should
dt\ < ge, Stg' > | < 1 for e = 0 , l
4||D|| j
In order to make the formulae in the following theorem simpler we introduce the notation: IT(M) = /
duSuMS'
One can easily check that the next formulae in which the above notation is used are well defined. T h e o r e m 4.2 // M,M' £ Co
and t satisfy the following C\ \\D\\t + C2 < 1 C3 < 1
condition: for bosons, for fermions,
(4.8)
whe d
=
max 4||IT(M* -M")gt\\
\\U(i + M'*)gu\\ +
+ ^max^ 4||n(M* -M'*)II(£ + M")g'\\ + mzx4\m
+ M'*)g
\\g"\\ +
, (4.9)
C2 = 4||£>||max / '
dt\
>|
,
J — OO
c 3 = 5 ( c 1 + c2) , where
£ £ H/v
and is defined as follows: £= £
IjJXJjl
.
e=0,l
^
y ««PPll*« 11(E)
ww)p{dE)ge
(4.10)
337 where
P(dE)
is the spectral measure of w em f
[Ils ||(-B)]2 ==
H\
and
is a Radon-Nikodym derivative which exists due to (3.20). finite Q, Q>, R, R' e R : lim < u ® P 2 tf * ((M) M)
,,
2— 0
U* Wt/2/t
Then for all
u,u'eHs
u1 ®P®P z
and
(4.11) (4-11)
exists and is equal to
Ut
W
i,
is a unitary solution of QSDE driven by quantum Poisson dU < £V
(4.12) (4-12)
,. noise:
dAf dAf t(S-l,Tj)U t(S-l,n)Ut
Uo = 1
(4.13)J
(
,
where S ij o unitary operator on Hs ® HN associated with a scattering Si on Hs ® Wj defined in terms of the operators r( #HSs®1 ® i + i1 ® F# i , \ H Hs ® ® 11 ++ 11 ® ®H Hii ++ Vi Vi Namely, let
M, M' 6 C0
1 X
(cf. (3.19)), n
< uu®® M , Su'®M'
matrix
u, u' e Hs i
m
,
>n >-H s®n s®'H N N== YY YY
//
dtdt
<9j <9j , , Stgi>
/«fi , , SjSiuu'® ' ® / jf'j >> . .
i = l i = l ■' R i=l j = l •'R
4>
Mfficoherent state normalized to 1 in tte Fock space
T(£2(R,
dt) ® 1i.fi)
5. M A I N C A L C U L U S This chapter refers to the case of the weak coupling limit. L e m m a 5.1 M„ M< , i/ien. then: lLet ei M\ €&, £ CD , i = 1,2 lim < Uu®®P P A x**( (MMI i, ,M lim < M22)) A->0 A->0
,
«'®P > == U '®PA A*(M;, *(M;,M M ^2 )) >
= «®*(x[Q11,fl ffix[- l j - 2Q]Af 1 H,]Afi , ] ^2 )) = <
.,
IM; ' © X [-«' uu '' ® ® ** (( xX[[o0;i, R.*'. ' , ] A ' ' i ©XI- R2,2-Oil ,-Q2]Mi)> . (5. 1) (5.1)
338 Proof after it):
One can check that (cf. ref. [15] ; §4 th. 3 and §5 th. 2 and discussion
,
,
PxV{M[,M!i)>
=
B(Mx,Mx)B*{MlX,M'2x)U> A
A
A
= <9,
A
, Bfi>=
A
x
(5.2) x
= [det(c(M 1 ,M; )c*(M 1 ,M{ )) d e t ( c ( M , M ^ ) c*(M ,M'2 )) where B and *(■)
]
T1/2
,
is a Bogoliubov transformation for which the corresponding functions (cf. (3.9), (3.10)) are equal: c{Mx,M'x) x
x
s{M ,M' )
c(-)
= c(Mx)c(M'x)^s(Mx*)s(M'x)
,
(5.3)
= s(Mx)c(MlX)
,
(5.4)
-c{Mx*)s(M'x)
det( ■) denotes the Fredholm determinant ( see eg. [16] or any other handbook on integral equations). One can easily check that: c(Mx,M'x)c*(Mx,M'x)
=
= exp{±[MxMx* F
+ M'x M'x* - Mx M'x* - MlX Mx* ]} + 6(MX,M[X)
=
(5.5)
x
= e< + S and
Fx
,
Sx
are self-adjoint, trace-class operators. Hence:
tet{c(Mx,M[x)c*(Mx,M'x))
= eiT'F"
det[l + e~iF*
6xe~iF?]
e
iTrF*
(5.6)
One can check, using the same method as in the Lemma 5.2 that: f?
limsup ||e
lim Tr|A"A| = 0 Therefore there exists
Ao > 0
0 < Tv\e-iF"
.
such that for all 6xe-iF*\
< \\e~i
A < A0 F
' ||2 Tr|<5A| < 1
Then the following inequalities are satisfied: 1 - Trle-f'tfe-itfl
< | d e t [ l + e " i F" Sx e~i <
F
?} | <
expJTrle-i^i^e-T^I}
(5.7)
So we have: lim det[l + e-^F"
Sxe-iF.x]
= l
.
(5 8)
339 From the fact that Mi i = 1, 2 are symmetric or antisymmetric operators for the Bose or Fermi case respectively it results: Tr [M,A M?m] = Tr [ J M?* J J M'{x J) = Tr [M/1* M'f]
(5.9)
Now after straightforward calculations using the method from Lemma 5.2, properties of coherent vectors and the shape of scalar product (3.18) we complete the proof.
■ Let us notice that the initial condition of the equation in the main Theorem 3.3 results from the above Lemma. Let us introduce the notation f, = Stf
for
/ 6 Wi
Ti = Vl±T
,
(5.10)
,
T2 = VT , yu\
_ eit{Hs + HR)y
L e m m a 5.2 Let set AA C {!,•••,K] , be u series of operators such that:
e-it(Hs
K < oo
,
N = card.4
«"*>-{$*?> Kt:,K}\A and
f,g£fc
+ HR)
,
and let
{tk(M£)}k=1
Mk&c
(5 12)
° >
'
, then the following limit exists and is equal to: K
Km \~N
= I T (X[SP,TP)(S)) If
A = 0 Proof
j
< gs/X2+t
,
dai-'-iss
Y[ **( M *) f°l*+t'
,
>
=
Y[(JS3kMkS,t)f>
(5.13)
as e ua , then we mean the product 11*6.4 (') Q ^ *° !• The proof is exactly the same as Proof 7.1 in ref. [6].
L e m m a 5.3 Let Mi, M't £ Co , (5 > 0 , suc/i 2/iffli /or aH
i = 1,2 A< S
< « ® P A * ( M 1 , M 2 ) , (-*)"*" / Jo
, :
n £ N
dsi--- / Jo
, then for all
A > 0
ttere
eiista
dsnV{si)---V(sn)u'®Px9(Mi,Mi)>\<
340 n fCB(Ci\\D\\t + C, -+ A ) D ( \Coell ll (C3 + A)"
for bosons, for fermions,
(5.14)
where:
( 5 - 15 )
c0 = HI KM , d
2
s (24) ||D|| max max (|( M - M' v
'
\(M'
,
C2 = f
,
' i=l,2 e=0,l UV
"
IT.ff'xJT^'-'l)!
dsf(s)
e
,
1
\Tk g >< JT{ g '^ ||r,«?<|| 2 }
)|
,
, (5.16)
,
(5.17)
J — oo
C3 = 2 C , + C 2
,
(5-18)
/ ( a ) = 2 ■ (24) 2 | | £ | | max max
| < T,- S e , T, |
.
(5.19)
Proof Because the proof of this lemma is long and rather technical we perform here an outline of it only. The left-hand side of the equation (5.14) can be estimated by the sum of the following type: ft/*2
I
/ Jo
/•«—i
dst ■ ■ ■ / Jo
dsn < u , D* ■ ■ ■ D* u' >
,
B*{M[x,M'2x)Sl>
c*{-)---c*(-) 2 fr t / A "
t/\
HI ll«'|| ||.D|P
/
«.„_,
/-S
,
ds„
dSl---
Jo
| <
Jo
B{M[X,M'2x)c#{■)■■■
c#{■) B*{M[x,M'2X)
Q.>\
,
or where c*(-) denotes a(-)i a * ( ' ) i K') &*(') operator. The proof can be done in the following steps: 1) We apply the Bogoliubov transformation given by the unitary operator B(M[X, M'2X) (see the marked part above). 2) We bring our expression to the normally ordered form. Because an annihilation op erator acting on the vacuum state gives zero, the final formula is equal to the sum of expressions:
<S(M;\M2A)B*(MJ\M2A) n 3)
,
e*(-)---c*(-)o>
multiplied by the product of scalar products < /, g > where / , g £ 'H\ We move creation operators c*() to the left-hand side of the scalar product and we repeat the procedure described in steps 1) and 2). Now the Bogoliubov transformation is given by: B(Afj\Mj A ) B*{M[X,M'2X)
341 4)
The remaining terms: <
5)
B(M1A,M2i)B'(M;A,M;*)fi>
(5.20)
we estimate by X3k Ck k\ or A3* Ck (2k is the number of creation operators) in the Bose or Fermi case respectively. Now we have to make the most difficult step. We are going to apply the Pule inequality [8] in the following form: rt/\2
,»„_,
/
< where
m
dsn 22 I I h(si*u> ~ SPS) ^
dsi-\2(m-n)
(n — m)\
2 < fi < ■ • • < qm < n
,
f Kt) dt J — OO
l < P i < - < P m < « —1
and
pj < qj
,
5^, is a set of permutation a of numbers l , - - - , m such that 3 =!,■•• ,m. % = 1, • • ■, m , and /i is a non-negative, integrable and symmetric Pi < ?
6)
,
a*(-)---a*(-)Q>
|
where 0 and C is a constant) in the Bose or Fermi case respectively. Now we apply Pule inequality. At the end we sum up all terms and we make simple estimations. The final result is of the form (5.14), where the constants depend on A , but they tend to C\ , C2 or Cz respectively in the limit A —» 0 .
L e m m a 5.4 Let Mi,M<e£a
,
i = 1,2
,
n e N
, then:
,
lim < u ® P A * ( A f 1 , M 2 ) , ( - 0 n A " /
A *-><>
exists.
J0
*>!••- /
Jo
dsnV{s1)---V(sn)u'®Px^{M[,M'2)>
342 Proof In the detailed proof of the Lemma 5.3 one can notice that the terms containing creation operators (5.20) are of order A and vanish in the limit A —» 0 Hence the non-zero contribution may give the pure scalar products of elements from the Hilbert space Hi only. The convergence of a very similar expressions has been shown by Accardi, Lu and Frigerio (ref. [1] theorem 5.1).
P r o p o s i t i o n 5.5 If the constants C\ condition:
, C2
, C3
defined in Lemma 5.3 and time
Ci ||JD|| t + C2 < 1
for bosons,
C3 < 1
for
t
satisfy the (5.21)
fermions,
then the following limit exists: lim < u ® P A * ( MTi/T1 , M\X 2 )\
1 ., U t)x, t/x
,
u' ®Px
J.
Proof Using the Dyson expansion of proof is immediate.
^A A 2
.
(5.22)
and Lemata 5.1, 5.3, 5.4 the
■ Now we shall find a QSDE which describes the limit evolution of the system. Let us introduce the notation:
V°R(t) =
>= lim < u ® P A * ( A / 1 , M 2 )
,
W(%2 u'®Px
>
,
(5.23)
l®{a(Tlg°t)a(T1g1t)+
+ a ( T 1 ? ? ) 6 ( c ( M 2 A ) 7 , ( M 2 A ) T 2 S ( 1 ) + 6 ( c ( M 2 A ) J , ( M 2 A ) T 2 S ? ) « ( r i S ( 1 ) } , (5.24) V&t)
=
l®{b(,T2gl)b(T2g°)+
+ b(T2 g])a{c{M?) Vl(t)
J s{M^)T,
x
= D®B(M1\M2 ){a(T1g°)a(T1gj)
g°t) + a(c{M?) J »(M*) r , g\) b{T2 gf)} +
+ a{T, g°)b'(T2 g\) - a'(c(M$)
Js(M?)Ti
1
x
j ? ) b'{T2 g}) + 1
]B'(M^M2X)
+ b*(T2g°)a(T1g t)-b*(T2g°)a*{c(M?)Js(M1 )T1g t) 0
+ D*®B{M$,M}){b{T2g])b(T2g t) + b(T2 g^a'iT,
(5.25)
+
g\) - b*(c(M})Js(M})T2
+
( 5 . 2 6) g\) a'{T, j ? ) +
a'{T, g])b{T2 gt) - a*{Tx g\)6*,(c(M2A) Js{M$)T2
g°t)
}B*(M*,M}
343 After straightforward calculations and using (5.23)-(5.26) we get: < u , V{t) > - < u, V(0) > = ft/X
i r/x = -— / ds
,
V{s)U$u'®Px<S>(M[,M'2)>
= H m y / ds{
=
, Ux/X2 u> ®B*(M[\M2X)U
,
Dl ® 1 Ux/X, Vfa/X2)
,
D<®1 [Vfc/X2)
u' ® B'(M[X,M2X)Q
>+
> +
1=0,1
< u ® B*(Mx,Mx)a
J2
, Ux/X,}
u' ® B\M'X,M2X)U
>} =
=0,1
= / ds {h(S) + /,(«) + I3(S)} Jo
.
(5.27)
Lemma 5.6 With the same assumption as in Proposition 5.5 h{s) = -*x I ( 9 l ,fl t ] 00(Mi©0 -iX[-R„-Q2]is)(0®M2 Proof mediate.
, ,
| \ / l ± 7 y >< J v T ^ T ? ° | e o )
/
+
o
0 9|jyTg >
(5.28) After straightforward calculations and using Lemma 5.2 the proof is im
Lemma 5.7 With the same assumption as in Proposition 5.5 his) = -ix[Q'1,K<1](s)(|VT±~T<71 X JVT±Tg°\@0 1
- iX[-R'„-Q's](s) ( 0 e \JVTg
a
>< Vfg \
, M[®0))
+
Vis) > (5.29)
Proof
The proof is similar to the proof of the Lemma 5.6.
Lemma 5.8 With the same assumption as in Proposition 5.5 his)
= -\\\s/Y±Tgl
X / V l i l V l e 0|| 1
+ (5.30)
344 Proof
Expanding
WAA2
in the Dyson series we have: .,'/X2
oo
/,(3)= K m - i £
An+1
£H)" ,
/-s„-i
^.••■/
I> e ® [ V K ( S / A 2 ) , V(Sl)---V(sn)} q 0 0 0\
(
Examplethat: 3.3 [21] Let R = I It is Example obvious
[vfa/x*),
/
q
Q
v(Sl)---v(sn)}
*
«'®PA*(M{,M^)> . (5.31) This is the well-known solution of the
I.
=
0 1 q-q-1 0 Q YBE associated to the Jones knot 0 polynomial. 0 1 Writing 0 " u = ^ " * j , the braided-commutativity relationss (7) f7j become 0 0 0 q) An analogous method to Accardi, Frigerio and Lu [1] lead us to the conclusion that the non-zero contribution in (5.31) is given by the term with k = 1 in (5.32) only. Let us consider the first term of V&(s/\2) (5.24) : a(Ti g°a/x2)a(Ti g\/x2) Using the condition (3.3) we obtain:
CD-
lim y
< u ® B"(M*, Mf)(l
= lim-/
, D ® [ofT, g\,^)a{T%
dsi < u ® S * ( M A M A ) n T
®{<2i»J/*i • rirf,>
,
g\,„)
, « A / A 2 ] u' ® B'(M[\M'2X)Q,
DD'®
i < > ±
, T1g°Sl>a(T1gl/x,)b(T1glJ±
+ < T, rf/A, , T, ffit > a(T, 9 ° / A 2 ) KTi <&) }
<
a*{T1g0,/xl)a(T1g°.l)
, Kg)^ A
A
u' ® 0*(M{ , M 2 ) fi >
(5.33) The terms of type a*(-)a(-) in the above formula vanish in the limit A —» 0 after acting on the left-hand side of the scalar product. The terms of a(-)6(-) - type we commute once again with U, and we see that they vanish in the limit A —► 0 because of the same reasons as we discussed in case of formula (5.32). Therefore only the first term in the above formula has a non-zero contribution in the limit A —> 0 . Performing the change of variable s\ —> S\ — s/A 2 lead us to the expression: /•o lim - / dSl < T, g° , T, <£ > < r , ff1 ■, Ti ffs\ >
,
W A /A2+S1
rfsi < Ti ff° , Tj ff°t > < 7\ g1 , 2i
5s\
u'®£*(M;A,M2A)ft> =
x D D ' u ,
Z>(s) >
*/ —C
The rest term of proof.
V^(s/\2)
>=
(5.34) can be calculated in the same way, what completes the
+
345 P r o o f of t h e T h e o r e m 3.3 The following formula results from the Lemata 5.1, 5.6, 5.7, and 5.8:
< u , V(t) > - < u , V(0) >= I ds Jo
{ - | | 1 7 V > < JTig°\
e 0||i
-«X[Qi ,*i](«) (Mi 9 0
x
,
\Tl9
>< JTig°\
-iXlQ>ltBv(*)(\Tigl
>< JTig"\ ffi 0
|| 0 © \JT2gl
> < r 2 ? ° | | | 2 _
-
-iX[-R2,-Q2)(s)(0
© M2
, x
- i X [ - « i , - Q i ] W ( 0 © \JT2g
>+
,
0)
+ > +
> +
0 ffi \JT2g* >
ffi
M[ ffi 0)
>
< D'u,
V(s) > +
0 ffi M£) < D * « , 2 ? ( s ) > }
.
(5.35) From the theory of quantum stochastic processes [3] it results that the following matrix element: <«®*(xw1,R,]Miffix[-RJ,-QJ]M2)
,
U,
U'®$(X[Q[,RII]M[®X[-R2,-Q2]M2)>
(5.36) satisfies t h e integral equation which has exactly the same form as in the formula (5.35), where Ut is the unitary solution of the QSDE (3.23). Using Lemma 5.1 we find the initial condition for Ut . The vectors Mi, M[ 6 HR , i = 1,2 in the formula (5.36) are natural inclusions into HR of the vectors Mi,M[eH'R , 8 = 1,2 in the formula (5.35). Hence the proof of the theorem is completed.
■ Acknowledgement: This work is supported by The Ministry of Education Grants B W / 5 - 4 0 0 - 4 - 0 4 8 - 1 , P B 1436/2/91. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
L. Accardi, A. Frigerio, Y.G. Lu, The weak coupling limit as a quantum functional central limit. Commun. Math. Phys. 1 3 1 , 537-570 (1990). L. Accardi, Y.G. Lu, The low density limit of quantum systems. 3. Phys. A Math. Gen. 2 4 , 3483-3512 (1991). R.L. Hudson, K.R. Parthasarathy, Quantum Ito's formula and stochastic evolutions. Commun. Math. Phys. 9 3 , 301-323 (1984). K.R. Parthasarathy, Quantum Ito's formula Rev. Math. Phys. 1, 89-112 (1989). L. Accardi, Y.G. Lu, R. Alicki, A. Frigerio, An invitation to the weak coupling and low density limits. Quantum Probability vol. VI. World Scientific, Singapore (1991). S. Rudnicki, R. Alicki, S. Sadowski, The low density limit in terms of collective squeezed vectors. To appear in J. Math. Phys. (1992). Y.G. Lu, The weak coupling limit for quadratic interactions. Preprint (1990). J.V. Pule, The Bloch equation. Commun. Math. Phys. 3 8 , 241-256 (1974). E.B. Davies, Markovian master equations. Commun. Math. Phys. 3 9 , 91-110 (1974).
346 [10] [11] [12] [13] [14] [15] [16]
R. Diimke, The low density limit for an N-level system interacting with u, free Bose or Fermi gas. Commun. Math. Phys. 97, 331-359 (1985). A. Barchelli, Quantum stochastic differential equations: an application to the electron shelving effect. J. Phys. A 20 6341-6355 (1987). P. Robinson and H. Maasen Quantum Stochastic Calculus and the Dynamical Stark Effect . Preprint (1990). A. Frigerio, H. Maasen, Quantum Poisson processes and dilations of dynamical semi groups. Prob. Th. Rel. Fields 8 3 , 489-508 (1989). O. Bratelli, D.W. Robinson, Operator algebras and quantum statistical mechanics, vol. II. Springer-Verlag, New York (1981). F.A. Berezin, The method of second quantization. Academic Press, New York (1966). F. Smithies, Integral equations. Cambridge University Press, New York (1958).
Q u a n t u m Probability a n d R e l a t e d Topics Vol. VIII (pp. 3 4 7 - 3 5 2 ) © 1 9 9 3 World Scientific Publishing C o m p a n y
T H E LATTICE OF ADMISSIBLE PARTITIONS Roland Speicher INSTITUT FUR ANGEWANDTE UNIVERSITAT
MATHEMATIK
HEIDELBERG
IM NEUENHEIMER
FELD
294
W-6900 HEIDELBERG FEDERAL REPUBLIC OF GERMANY Abstract. We introduce the notion of 'admissible' partitions, outline its importance for non-commutative probability theory and start the examination of the lattice of admissible partitions by calculating the corresponding Mobius function.
In the following we examine the lattice of admissible partitions, which determi nes the combinatorial structure of the 'free' convolution of Voiculescu [11]. After finishing this work we found out that our lattice of admissible partitions was consi dered before in literature, namely it appeared as 'lattice of non-crossing partitions' in a paper of G. Kreweras: Sur les partitions non croisees d'un cycle. Discr. Math. 1 (1972), 333-350. So our contribution should be regarded as a remind of the ba sic facts about the lattice of non-crossing partitions from a quantum probabilistic point of view. In particular, this point of view allows a proof of Theorem 2 which is much simpler than the one in the paper of Kreweras. The combinatorial aspects of the convolution of probability measures are in a nice way connected with the lattice £„ = V(l,...,n) of all partitions of the set { 1 , . . ., n} (or better with the union of all such lattices for all n): For a measure /i with moments m; and cumulants (semi-invariants) C; (i 6 IN) we can define on the lattice £„ the multiplicative functions tp^ and Q^ by (V = {Vi,... ,V,} e Cn) 5
*v(v) ~ n*^)
and
i=i
^ M : = n
<=i
where for some finite V C IN
and
Q„(V) = Q M (1 # V ) :=
Then the connection between these two functions is given by
= E
V
C#v-
348 and by Mobius inversion
QM=
£
¥V(VMV,V0),
V
where fi(-,-)
is the Mobius function of the lattice £ „ . T h e usual convolution of
probability measures which are determined by their m o m e n t s is easily described in t e r m s of the Q-functions since
Q ^ ^ l n ) = f,,(ln). One should also note t h a t there is another subset of the lattice of all partitions, namely the lattice of interval partitions (which is isomorphic to the Boolean lattice of all subsets of a given set), which has gained some interest in the literature ([1,13]) and also gives rise to some form of convolution. Although the lattice point of view shows in a nice way the combinatorial features of t h e involved forms of convolution, one should, however, keep in mind t h a t it veils all positivity aspects. T h u s from this point of view it is a mystery why the convolution of two positive functionals is positive again. T h a t this is indeed the case for t h e above mentioned three lattices seems to be accidentally, and it would be an interesting task to determine all subsets of the lattice of all partitions having this feature. Let us now start our examination of the lattice of admissible partitions. D E F I N I T I O N : Let V = {Vu ..., V,} be a partition of the ordered set S, i.e. the V{ are ordered and disjoint sets, whose union is 5 . T h e n V is called admissible if for all i,j = l,...,s with V{ = (vx,...,vn) (vt < ••■ < Vn) and V, = (wu...,wm) (w-i < ■ ■ ■ < wm) we have wk
< wk+l
«• wk < vn < wk+1
(fc = 1 , . . . , m - 1).
349 We will denote the set of all admissible partitions of S by Va(S). We can reformulate the definition of 'admissible' in a recursive way: The par tition V = { V i , . . . , V , } is admissible if at least one VJ is a segment of S, i.e. it contains exactly all points of S lying between two points, and V\{Vi} is an admissible partition of S\Vj. In a more pictorial language: If we build bridges by connecting in S the numbers belonging to the same V*, then a. partition is admissible if it is possible to build the corresponding bridge in such a way that the lines do not cross. EXAMPLE: E.g. {(1, 3,5), (2), (4)} and {(1,5), (2,4), (3)} are admissible partitions of {1, 2, 3,4, 5}, not admissible are {(1, 3), (2,4,5)} and {(1,4), (2), (3,5)}. The Vi are called the blocks of a given partition V = {Vi,...,V,}, and the order structure on Va(S) is defined as usual: Vi < V2 exactly if each block of Vi is contained in one block of V2. Of course, the lattice structure of Va(S) depends only on the number of elements of S, i.e. we can restrict for each n to the consideration of some fixed Sn with # 5 n = n, let's say S„ = ( l , . . . , n ) . By 0 n := {(i) I i € Sn} and l n := {Sn} we denote the minimal and the maximal element of Pa(>5n), consisting of n and one block, respectively. One sees easily that the Mobius function fi(-,-) of V^S) is multiplicative in the following weak sense: If T is the disjoint union of Tj and T2 and Vi, Wi are admissible partitions of TJ with Vi < Wi (i = 1,2), then
/*(Vi u Vt, Wi u W2) = n(Vu W1MV2, Wj). This shows that the determination of p. reduces to the calculation of fi(V, 1„) for all n and V € Va(Sn). One notices that for each V € Va(S„) there exists a unique set {fci,..., kT} of integers kt > 2 such that the segment [V, 1„] is isomorphic to ^ ( S * , ) x ••• x V*{Skr)- We shall call this set the class of V: class(V) := {ku...,kT}. The existence of this set can be seen as following: Let us consider [V, 1„] with V = { V i , . . . , V,} € Va(Sn) and put T := ( 1 , . . . , n). If possible, choose a V and two neighbouring elements k < I in Vt, i.e. Vi = ( . . . & , / . . . ) , such that h + l ^ l . Put T0 := T\[k +1,1-
1],
Tx := {&} U [k + 1,1 - 1].
Then
Vo~VnTo is an admissible partition of To and
Vl:={k}U(Vn[k
+1,1-1])
is an admissible partition of Ti and we have [V,l.]srVo,lT.]x[Vi,lrJ].
350 This decomposition reflects t h e admissible character of our partitions, which im plies that the block V; separates the points lying between k a n d I from the points lying outside this interval. If it should happen t h a t one of t h e two factors is isomorphic to Ta(Si), then we will forget this factor. Now we can repeat the above procedure (for each of the factors) until we end up with some [W, l m ] , where we do not find some V, with the above properties any more. But in this case [W, l m ] — "Pa(S\w\)- O n e should note t h a t t h e resulting set {ki,..., k,} is independent of the order of choosing the V{ or the neighbouring k, I. Let us also give some examples for the notion of class: class({(l, 3, 4), (2)}) = {2}, class({(l, 6), ( 2 , 4 , 5), (3)}) = {2, 2 } , class({(l, 6,11), (2, 5), ( 3 , 4 ) , (7, 8), (9,10), (12), (13,14), (15)}) = { 2 , 2 , 3 , 4 } . T h e class of V contains all information for the calculation of /x(V, 1 „ ) , because it allows to reduce /i(V, l n ) to the knowledge of some /i(0;, 1;). But this can be given explicitely with the help of the Catalan numbers Cj, which may be denned as t h e number of lattice paths from (0, —1) to (i, i — 1) lying entirely below the diagonal and which appear in a lot of contextes (see e.g. [2]). For example, CQ = 1, C\ = 1, c 2 = 2, c 3 = 5, c 4 = 14, c 5 = 42. THEOREM 1. 1) Consider V € Va(Sn)
with cJass(V) = {klt...
,kr}.
Then
rT
1 Mv,i»)=n^°^' ^)=n p ^.x»i). *«(V t l.) =
J=I
2) For all k € IN we hare fj.(Ok, lk) = ( —l) f e - 1 c f e _ x . OUTLINE O F P R O O F : T h e first part follows from the fact t h a t the segment [V, l n ] is isomorphic to Ta(Skl) x • ■ ■ X Ta(Skr). For t h e second part one checks that the same argument as for the lattice of all partitions (c.f. [4], Proposition 3) gives 7 1l - 1 T
//z(0„,l i ( 0 „ , l nn)) = =
-J>(V.,l„), i= l
where V; is the atom consisting of n — 2 blocks which contain only one element and one block which contains the first and t h e (i + l ) - t h element of 5 „ . Since class(Vi) = {n- iti} we have / / ( V , , l „ ) = / J ( 0 „ _ , , l „ _ j ) / i ( 0 j , 1,), hence n-l
/ . ( O nn__„ „ l „u-M^i, _ 1 ) / i ( 0 1 , li.),). v(on, in) == --^£>(o *«(0„,1») 1=1
W i t h t h e definition dk = (-l) f c /x(0*+i, l * + t ) this gives dn-1 = X ^ 1 d ^ ^ d , - ^ , which is the recursion formula for the Catalan numbers ck. Because of d0 = di = 1
351 we get the assertion dk = c^. The connection between the lattice of admissible partitions and the Catalan numbers is even more intriguing. The next theorem shows that also the number of all admissible partitions of 5„, which corresponds in the case of the lattice of all partitions to the Bell number Bn, is given by c„. Although this shows that the set of all paths below the diagonal and the set of all admissible partitions can be identified, one should note that the partial order on the partitions does not correspond to the canonical partial order on the paths. Thus, the language of paths does not seem to be the right way of thinking about our lattice structure. In the next theorem we shall also consider such admissible partitions V = { V i , . . . , V , } , where each block V^ contains exactly two elements. Let us denote the set of all admissible partitions of this form by V^{Sn) (where, of course, n should be even). THEOREM 2. We .have /or all n G M
# P a ( 5 „ ) = #7>a2(S2n) = c . OUTLINE OF PROOF: The essential part of the proof for the first equality is already contained in [8]. We only recall the idea: Let {e; | i G ffl0} be an orthonormal basis of a separable Hilbert space and define the operators Z with adjoint I* and A by (i G iZV"0) Pe, = e i + 1 , le0 = 0, Ae 0 = 0,
lei+l = e,, A e I + l = ei+i.
Then one checks that for all n G M <e 0 ,(Z + Z*) 2 "e 0 > = #7> 2 (S 2n ) < eo, (A + Z + r + l ) n e 0 > =
#Va(Sn).
The equation (Z + Z*)2 = A + ZZ+ /*** + 1 and the fact that 11,1*1* are as good as Z, Z* shows the first equality. (From a probabilistic point of view, the operators Z + I* and A + Z + Z* + 1 give a free Gaussian and a free Poisson distribution, respectively.) For the second equality, let us count the elements of Tl(S2n) in the way that we first fix the first block of V, let's say Vi = (l,*)i a n d t n e n s u m o v e r a11 possi bilities for completing this block to an admissible V = {Vi,...,Vn}. The nature of admissible partitions implies that Vi separates the points 2 , . . . , i — 1 from the points i + 1 , . . . , In and thus V G 7> 2 (1,..., In) is built up (if Vi is fixed) from Vx,
352 an element of ^ ( 2 , . . . , i - 1), and an element of V\(i + 1 , . . . , 2n). In p a r t i c u l a r , i has to be even, let's say i = 2k. This implies nn
=E
2
2 #^ (52„)== £ ##rl(s ^ ( ^2lt--22 ) •afT )-#r (s2n -2k2n),-2k), #-Pl(S2n) a(S 2
ik=i
which is t h e recursion formulai for for tt h e C a t a l a n n u m b e r s .
0 References
[1]
[2] [3] [4] [5] [6] [7] [8] !9] [10]
[11] [12] [13]
G.C. Hegerfeld and S. Schulze, Noncommutative C u m u l a n t s for Stochastic Differential Equations and for Generalized Dyson Series, J. S t a t . P h y s . 51 (1988) 691-710 P. Hilton and J. Pederson, Catalan N u m b e r s , Their Generalization, a n d Their Uses, M a t h . Intelligencer 13, No. 2 (1991) 64-75 F . Radulescu, T h e fundamental group of t h e von N e u m a n n algebra of a free group with infinitely many generators, P r e p r i n t G.-C. Rota, On the Foundations of Combinatorial Theory. I T h e o r y of Mobius Functions, Z. Wahrscheinlichkeitstheorie verw. G e b . 2 (1964) 340368 T . P . Speed, C u m u l a n t s and partition lattices, Austral. J. Statist. 25 (1983) 378-388 T . P . Speed, C u m u l a n t s and partition lattices II: generalized fc-statistics, J. Austral. M a t h . Soc. A 40 (1986) 34-53 T . P . Speed, C u m u l a n t s and partition lattices III: multiply-indexed a r r a y s , J. Austral. M a t h . Soc. A 40 (1986) 161-182 R. Speicher, A New Example of ' I n d e p e n d e n c e ' and ' W h i t e Noise', P r o b a b . T h . Rel. Fields 84 (1990) 141-159 R. Speicher, Free convolution and the r a n d o m s u m of matrices, P r e p r i n t , Heidelberg, 1991 D. Voiculescu, Symmetries of some reduced free p r o d u c t C*-algebras, in: H. Araki, C.C. Moore, S. Stratila and D. Voiculescu, eds., O p e r a t o r Algebras and their Connection with Topology and Ergodic Theory (Lect. Notes in Math., vol 1132, Springer, Heidelberg, 1985) 556-588 D. Voiculescu, Addition of certain non-commuting r a n d o m variables, J. Funct. Anal. 66 (1986) 323-346 D. Voiculescu, Limit laws for r a n d o m matrices a n d free p r o d u c t s , Invent. m a t h . 104 (1991) 201-220 W . von Waldenfels, Integral Partitions and Pair Interactions, in: P.A. Meyer, ed., Seminaire de Probabilites IX (Lect. Notes in M a t h . , vol 465, Springer, Heidelberg, 1975) 565-588
Quantum Probability and Related Topics Vol. VIII (pp. 353-369) ©1993 World Scientific Publishing Company
Phonion Limits and Macroscopic quasi-particle spectrum for t h e BCS-model M. Broidioi, B. Momont
' and A.
Verbeure
K.U.Leuven Instituut
voor Theoretische
Fysica.
Celestijnenlaa.il 200 D B-3001 LEUVEN
(Belgium)
Abstract
On the basis of non-commutative central limit theorems, starting from a dynamical sys tem in equilibrium (A,at,u>) fluctuations
the corresponding macroscopic system (A(Lit),dt,uj)
of field
at momentum k is constructed. A mathematical mechanism for the appear
ance of a macroscopic quasi particle structure is developed in the BCS-model. Due to the spontaneous breaking of symmetry below the critical temperature the resulting dispersion relation shows an energy gap at zero momentum. The BCS-model is used as a toy model for explaining the general theory. 'Onderzoeker IIKW Belgium
354
1
Introduction
The role played by phon(i)ons in condensed matter physics is well known.
Phon(i)ons
mediate various interactions between quasi-particles and phon(i)ons interact with exter nal fields, phon(i)ons carry an energy and a quasi-momentum. As essential ingredients we learned from the literature that phon(i)ons are linked to collective or macroscopic observables, and that their dynamics is induced by the microdynamics of the system. Here we illustrate our ideas of how to obtain phonions from first principles in a quan tum mechanical context. Our general mathematical theory for the mechanism of the appearance of phonons is worked out for lattice systems in ref. [9]. We take the opportunity of these lecture notes to work out an example of the continu ous BCS-model. This is a model for long range interactions. It is also completely soluble and in fact well known. So what is new in our treatment is the approach to the model illustrating the mechanism of the appearance of phonions as bona fide quasi-particles. It is also used to illustrate the phenomenon of coarse graining, i.e. at macroscopic level the number of degrees of freedom comes over as being much smaller than at the microscopic level. For long range forces, the spontaneous breaking of a symmetry is accompanied with an energy gap t 0 in the density spectrum at momentum k = 0. Recently it was proved that the energy gap £0 can be found back as a discrete point of the spectrum of zeromomentum fluctuations in models with very long range interactions [1][2][3]. In this paper we consider the continuous BCS-model [4] and extend the above results in the following
355 sense. We develop the theory of field fluctuations at momentum k =£ 0. We show that for any fixed k 6 HI3 these field fluctuations generate a finite dimensional Clifford algebra. The macroscopic dynamics leaves this algebra invariant and yields a discrete spectrum coinciding exactly with the quasi spectrum. However, the main result of this paper is the mathematical mechanism for the understanding of the quasi particle structure in solid state physics.
2
T h e e q u i l i b r i u m s t a t e s of t h e B C S - m o d e l
In this section we determine the translation-invariant equilibrium states of the BCS-model. The model describes a dynamics of a system of Fermions (electrons) with two possible spin states labeled by the index i = 1,2. Therefore the algebra of observables, denoted by A, is generated by the following set of smeared out creation and annihilation operators
W),cr(/);/ei 2 (iR 3 ) satisfying the usual anti-commutation relations (CAR) {C:{f),Cf(g)}
=
(f,g)6,3
{(?-(/), C-(<,)}=0
CtUY = CM) The formal Cf{x)
are related to these by the expression
CM) = ldxf(x)C?(x).
356 The gauge groups {7jv|A 6 [0,2ir)}]=u2
act as automorphisms on A as follows
i}(C-,(f)) C;,(f)e^ e'V* 7*(C£(/)) = C-,{j)
(2.1)
The translation group {ra\a £ IR3} acts as automorphisms r a on A in in the following way
ra(C-(/)) = c;(/ a ) ra(C-(/))
(2.2)
with /„(.r) = /(.r - a). The BCS-Hamiltonian is given by [4] dxVCt{x)-VC-{x) •vc-(i) Hv == Yj :, \\ l 1 dxVCt(x) i=i
-
J v
d s
y')C;(x') (2-3) (x ' ++ y')cr(*') dx Cl ( i ) C + ( x ++ y)y) Irfx'c -^J dx'C22-(*' (2-3) -T, vdyS(y)J I v (y) vdy'S(y' f dy's(y') ) f/ dxct(x)c+(^ Jv V Jv Jv Jv w ithS a real-valu*ed function such that
2
J
e L :(1R ).
Jv Notice that the model-H;imiltonian
with 5 a real-valued function such that ^f^ € L 2 (1R 3 ). Notice that the model-Hamiltonian is invai 'iant under both gauge groups is invariant under both gauge groups ^{Hv) Hv;j ] = l,2. ~f*(Hv) = Hv; The model-Hamiltonian is also space translation invariant TaaH Hy+a. HvT= Hv+aVT- a■a —
As the algebra of observables is the separable CAR-algebra A, which is space translation asymptotically abelian, the translation invariant equilibrium states LJ can be uniquely decomposed into extremal invariant equilibrium states [5]. Now we determine the extremal space translation invariant states. One of the main properties of extremal space translation invariant states is that the average over the translations of a local observable exists i.e. for all A 6 A weak 77 / darJA) darAA) = u[A)t UJ(A)1 w e a k - lim — = v v—x, ' v—» V V Jv Jv
(9 4) (2.4)
v
;
357 and for all B e A weak - lim [— / dara(A),
B] = 0
(2.5)
V—nx> y Jv
These averages are observables at infinity [6].
Let Hv be the BCS-model and u> an
extremal space translation invariant state, then for all local X £ A, using (2.4) and (2.5)
lim
LJ(X'[HV,X])
V —too
lim {UJ(X'[TV,X})
-cluj(X"[Av,X})
v—
-c6w(A'*[A^,A'l)}
(2.6)
where Ay = Jy dy S(y) jf dx r« (Cj-(w)Cf(O)) and b = ^lirn Jv dy S(y) w ( £ _£
Av
= weak - lim / dy S(y) f dx rx (C^(f^)C^(n) n->oo Jy
Jv
\
»
■ J
The above computation yields that for each extremal space translation invariant state u>, there exists an effective Hamiltonian
H% = TV -c(bA'v
+ bAv)
(2.7)
such that for all local A' 6 A
weak — lim
[Hv,X]
= w e a k - \\m[H^,X) V —'oo
=
6w(X).
358 One computes explicitly
+ L(Cn*))
- ^ C , (.,:) W (*» == ~^C+(x) +
- el j dy S(y)C;{x cbjdyS(y)C-{x
C+ + (( . r ) + cbjdyS(y)C;(x Sw{C+(x)) = - y C cbjdyS(y)C;(x ^(C+(.r))
+ y)
(2.8) (2.8)
- y)
(2.9) (2.9)
In order to determine the two point function and the gap-equation in a convenient way we introduce the CAR-two-componenl version of A. For I R 33))eL 0 L 2 2( I(R R 33)) ==-- LL22((IR
(fnh)€l f = (fnh)€L define the creation operators C+(f),
f 6 L by :
c ++(f)={c+(h),C;(h))(/) = (c 1 + (/ 1 ),c 2 -(/ 2 )). C
(2.10) (2.10)
4 /7+(/) = = C C++(h„f) (/^/) 8„C+(f)
((2.H) 2.H)
Then
where hw is the following operator ■ / A„ =
- fA2 '
—cbS -cbS*-*■ ^
-cT>S * ■ I -cbS*K
fA2 '
f/ 2
which can be defined as a self-adjoint operator on L = L (IR 3 )©L 2 (IR 3 ). T h e convolution products * and * are defined as follows : (S * f^x)^ (5 /,)(*) = / dy S(xfdyS(x-y)f,(y) -y)h(y) ((5 5 ••// , ) ( * )) ==
fdyS(y-x)f fdyS(yx)h[y). 2(y).
Define the functions ek± on IR3, for all k £ 1R3, by ckk, , Ix)\ -
X 1
1
!1
t ^ V + i"±wi'
I'
l1
^
I , i-k*l
^ M±(*o j
359 where /j±(fc) =
J b V 2 T \ A V 4 + <=W(2*) 3 |S(fc)|' cb{27c)3fiS(~k)
Then one computes that fcu,4 = e ± ( A ) 4
(2.12)
with e±(fc) = ±\Zfc 4 /4 + c 2 | i | 2 ( » 3 | S ( * : ) l 2 These are called the quasi particle energies at momentum k. Clearly the spectrum of hu is absolutely continuous, because using Fourier analysis, and the property fi+(k)fj._(k)
= —1
one checks t h a t any / € L can be written as
f=Jdk
( ( 4 , / ) 4 + (ei,/)ei).
(2.13)
The extremal space translation invariant KMS-states are determined in the following theorem. P r o p o s i t i o n 2 . 1 . (KMS-solutions) T h e extremal space translation invariant equilibrium states are the quasi-free states OJ on A determined by the two-point function
-(C+(/)C-(5)) = ( , ,
r T
^
:
/ )
(2.14)
where the value b in hw is determined as a solution of the gap-equation 2e (k) 2 ^-/^^TN+
proof the proof is completely analogous to the proof of theorem 2.2. in [7]
<**>
360 The gap-equation always admits a solution 6 = 0. This state is invariant for the gauge group automorphisms. In the low temperature region (fi > /3C) there are solutions to the gap-equation with b = f dyS(y)u
(C 2 "(i/)Cf (0)) =f 0. Hence the broken symmetry in the
BCS-model is the compact gauge group [4][S]. Introducing the Fourier transforms of the creation and annihilation operators ihx bf{k) = 4= 4= £E e" e-'Ht(fc) C,+(.T) =
b~(k) = b+(k)the order parameter 6 can be written as
dk ■S(k)b+(- -k)bt(k)) b=^j-^{jdks(k)bi(-k)bt(k)y
il
1 b= " (27r) 3 /2 u
.
Hence there is a nonvanishing correlation between electron states with opposite spins and impulses in the superconducting phase 6 ^ 0 .
One speaks of the appearance of Cooper
pairs. Notice also that the gap-equation fixes only the absolute value |6|, but not the phase. Therefore each non-zero solution | i | =^ 0 yields an infinite set of extremal invariant states parametrized by the phase of b. The physical spectrum of the micro-dynamics in any state OJ is as usual given by the spectrum of the Hamiltonian H^, implementing the time evolution in the GNSrepresentation of u e'70C-(/)e-'"-< = e"s"C-(/) where £„ is given by formula (2.11) in terms of the one particle Hamiltonian h^. If w is an equilibrium state of the type described in propostion 2.1., it is necessarily time invariant. This implies that zero belongs to the spectrum of Hm.
Furthermore as
361 any such s t a t e is quasi free and the spectrum of h^ is absolutely continuous given by the set ] - oo, +oo[, the spectrum of the physical Hamiltonian H^ in the state u is given by 1R and a discrete spectrum consisting of the single point zero.
3
Macroscopic quasi particle spectrum for the BCSmodel
It is easily checked that the space translation invariant KMS-states of the BCS-model are: even, quasi free and /^-clustering; i.e.
|c/a1...|
for all n > 2 and for all /,- having compact support in L, and where the iJ^
(3.16)
are the
truncated functions defined in the usual way [5] and C * ' = C + or C~ From now on we take a space translation invariant equilibrium state of the BCS-model determined by (2.14) and (2.15) with 6 ^ 0 (T < Tc). In this state w, we consider the local field fluctuations at momentum k in the volume A
FfM) = ^xLdae*'k°T°c*w'f
e L
We are interested in the macroscopic observables lirriA—oo ^t,A(/)> ca-lled the observables of the Fermi fields C#{f)
fluctuation
at momentum k. The limit should be understood
in the sense of a central limit. Notice t h a t one can also consider fluctuations of more complicated local operators, like higher order monomials in the creation and annihilation operators, e.g. energy density,
362 particle density, orderparameter density, etc. [3]. Here we limit ourselves to the simplest situation of the field operators. Because of the cluster condition (3.1G) and the translation invariance of u>, one has the following theorem whose proof is now immediate. T h e o r e m 3 . 1 . (Central limit theorem) Under the above conditions
# ++ lim l i m u. M '( . ) )) ) ==0 o ' (//2 „n++ ■ W ( i ^ ( / . ) - - - ■^■ A " v 1
(
A—coo
*
&
(
/
.
)
■
'
#
(hn)) umw (*&(/0{F*xih )...n rc/^)) /7#2n •■ * M
lim u. A—*oo A—coo
*
J
da i*7,'ka k°u>u(C*->{f^)T (C*11-' (J*?l-I )T 0C*> = E ^:f't-n If /d*e*"' C*"U*j) '(/**)) 0
7T *
i=\J
where the sum is over all permutations
' l 1 .. . .. 7T IT
2n In 7T2I- 1 << ""2/w i t h 7T2/-1 T2/-
== , n""1 K
■■ •■• T2n " 2 n Jj M ■
It is obvious that different micro observables can lead to the same macroscopic fluctu ation observable, e.g. C+(f)
and e~'kaTaC+(f).
This is the property of coarse graining.
As in [9] and [10,p. 6S] we define an equivalence relation on the microscopic algebra with the property that all the micro observables in one equivalence class lead to the same macroscopic fluctuation. We define on A the equivalence relation ~j,. by:
C # ( / ) ~* C*(f) ~* C*{g) C*{g) w F * AA 'l™ li*■ »Uf —oo
2 3 /f,g, g eeLL = LL22(IR (Hl3 )3 )©L © L 2(lR (IR )
) ) >F* *A ) ] ==0 0. . ~- S3))' A(f( / -- Sg)]
363 It follows from a straightforward computation that e.g. Vu^ (F+A(f-g)F-,(f-g)) rn^{FUl-a)Fk-,(f-9))
- ! +*Jlm i + \ ^
m
!(/»(*)-&(*)) + M*) (AW -feW)|a
+ ^ S i ) i + j—^ji |(A(*) -g.(t)) + ^ W (A(fc) +^Si) (AW -ga(fc))|2 So we have the following Lemma 3.2. C'(f)
~* C~{g) C-(j) «*
AW = hit) h{k) h{k) /*(*) = &(*)
It is clear that ~j.. is indeed an equivalence relation.
It is possible to construct a CAR-algebra of macroscopic field fluctuations.
■
We first
consider the limit of the anti-commutators of the local field fluctuations Lemma 3.3.
!$««{*£(/), *&(*)} = fdae>k°(f,gQ) = o ) 3 (W)si(k) +Mk)Mk)) = (Lot m
We define the algebra of Fermion field fluctuations of the system at momentum k as the CAR-algebra A(L).) generated by the new Fermion creation and annihilation operators
/^f# ([/]); ([/]); [/] €e h U = LU L(~, = C c2
364 satisfying the CAR:
{^((/]).^+([
T h e o r e m 3.4. There exists a quasi free state Co on A(Lk)
defined by
* ( n # 1 ([/>]) - - . i f " ( L M ) ) =
(3-17)
^{F*X{h)...F*l{fn))
(3.18)
with two point function
^f'([/l])^#2([/2]))
= Jdae**ikau> (C#'(/i)^C* 2 (/ 2 )) for all [/i], [/2] € L. The microscopic time evolution at = expitS^, in the state LO in duces a dynamics 5; on A(Lk) in the macrostateaj through o*tF*([f]) = i* 1 *^'""'!/]) leaving the state Co invariant and where ft* is determined by Afc[/] = [h^f]
p r o o f Determine the linear functional Co on A{Lk)
by formula (3.17). It follows from
proposition 2.1 and theorem 3.1 that the functional is quasi-free with the right twopoint function. The positivity of tj follows from the positivity of the microstate LO. Using (2.14) one has
0<^.(C+(/)C-(/)) < ( / , / ) ;
feL
365
Since the functional is gauge-invariant in the two-component version a necessary and sufficient condition for the positivity is [5, vol.11, p.44]
o<^(n + (t/])^([/]))
0 < W([/])^([/])) =
/dae-'kMC+(f)raC-(f))
< Jdae-'k°(fJa) = (f,ft proving the positivity of the quasi-free functional. The existence of the dynamics on A(Lk) induced by a, follows from the fact that (eiih"f,eith-gt i.e. (e^lfU^lg})
= {f,gt;
f,g e L
= ([/],[
This is an immediate consequence of the translation invariance of the dynamics : Taot = cttTa. It is well known that each unitary e'"1* map on Lk yields a *-automorphism at of the corresponding CAR-algebra A(Lk) via the formula 5 ( F*([/]) = F # ( e ' ^ [ / ] ) Again the translation invarianceof a, and the time invariance of u> : u>at = u> yield: u> (a, =
fdae-ik°w(ai(C+(f))TaC-(g))
=
[dae-k°u>(C+(fKC-(g))
=
^{F^[f])Fk-([g)))
366 implying the a ( -invariance of Co.
■
We obtained a new dynamical system, namely that of the Fermion field fluctuations of the BCS-model at momentum k. It is given by the triplet (A(L):),^', A(Lk)
»i) with Coat = Co;
is the CAR-algebra of the macroscopic field fluctuations, w is the macro state
induced by the micro equilibrium state w as given by theorem 2.1., a, is the macro evolution induced by the micro evolution a ( . The central limit theorem and therorem 3.4. suggest the identification F*(lf])
= limF*A(/);/£
L
A
i.e. the limit of the Fermion field fluctuations can be identified to a new and well defined Fermion field representation determined by the quasi free state Co. Furthermore the a r time invariance of the state Co, yields as a consequence of the GNS-theorem, that the macro dynamics o ( has a Hamiltonian representation in the state Co:
«<jf ([/]) =
e^F*([f))e-'^
Where
(
Ft
Hk
V K°)
\
( {2-Kf2
lS(k)F+
\
V Ft
+ Fk
i°]
n
V
\
1 %] n 'A + l>S(-lc)F+ (A v°y
K1)
v1 I
v°,
Kl>
V0/
(A V1/
What has to be observed on the level of the dynamics of the fluctuations is the spectrum of the macroscopic Hamiltonian Hk. Our main result of this section is the determination of the spectrum of Hk-
367 T h e o r e m 3 . 5 . The spectrum of Hk is completely discrete and given by the set { ± ^ / 4 + c2|6|2|5(fc)|2(27r)3}
proof
&tF*([f\)
F*(c"k[f})
= =
F*([c'^f})
or equivalently we determine <5, the infinitesimal generator of (oOiplR
6lf([f\)
=
#F*(hk[f})
=
#F*([hwf})
We already showed that the equivalence class [g] is characterized by the two complex numbers gi(k) and gi(k).
Applying this to the equivalence class [hwf], we find for
hk the following matrix: /t 2 /2
-cbS{-k){2-Kfl2
-c65'(i)(27r) 3 / 2
-P/2
y
with eigenvalues 4
= ±\/k*/4
+
c*\b\2\S(k)\*(2ir)3
The corresponding normalized eigenvectors are
1
1
(1
2
\
V*F' JTTW* V^ / with ±
/'it
_
^ 2 / 2 - e* C6(2TT) 3 / 2 S(-A:)
I
368 For it / 0 we obtain a result analogous to the case k = 0 worked out for the Overhausermodel [7], Remark that on the level of the fluctuations the different k ^ 0 modes cor respond to discrete eigenvalues, and hence macroscopically give rise to bound states, macroscopic quasi-states. Notice that
l i m e t = £0 T^O if S(fc = 0) ^ 0
corresponding to a macroscopic quasi paricle of zero momentum. This is the situation of an energy gap, due to the Cooper pairing. The k / 0 field fluctuations can be worked out rigorously in an analogous way for the Overhauser-model, and in principle for any mean field model. We showed that on the level of the field fluctuations in equilibrium a new dynamical system of macroscopic Fermions with a discrete spectrum appears. As such this might be understood as the mathematically rigorous understanding of the so-called selfconsistent field of macroscopic quasi-particle structure. At the basis of our work is a central limit theorem, our theory should also give an understanding of this phenomenon for more realistic systems (i.e. beyond mean field systems), at least in the situations that there is enough clustering; e.g. at high enough temperatures.
369
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4. R. Haag; II Nuovo Cimento 19, 154 (1961).
5. 0 . Bratelli, D.W. Robinson; Operator Algebras and Quantum Statistical Mechanics, Springer, Berlin, 1979, 1981, vol. I, II.
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dynamics; Preprint KUL-TF-90/28; to appear in J. Math. Phys. 1991.
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