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Quantum stochastic differential equations in terms of quantum white noise
Nonlinear
Analysis,
Theory,
Methods
PII: SO362-546X(!W)OO157-0
& Applications, Vol. 30, No. 1, pp. 279-290.1997 Proc. 2nd World Congress of Nonlinear Analysts @ 1997 Elsevier Science Ltd printed in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00
QUANTUM STOCHASTIC DIFFERENTIAL IN TERMS OF QUANTUM WHITE NOBUAKI Graduate
School of Polymathematics,
EQUATIONS NOISE
OBATA Nagoya University,
Nagoya 464-01, Japan
words and phrases: White noise distribution theory, Fock space, Operator symbol, Wick symbol, Quantum stochastic process, Quantum white noise, Wick product, Quantum Key
stochastic differential equation.
1 INTRODUCTION Quantum stochastic differential equations have been discussedconsiderably in mathematics and physics. From the mathematical aspect in 1984 Hudson and Parthasarathy [6] established a quantum analogue of It8 theory with in mind a quantum stochastic differential equation of the form z -1 , dZ = (LIdA + LzdA + LsdA’ + L,dt)Z, -o(1.1) where Li are operators acting on an initial (or system) Hilbert space ‘X; {A,}, {A,} and {A,*} are respectively the number process, the annihilation process and the creation process on the Fock space r( L2(R)); and therefore, the solution {Et} will be a process of operators acting on F(L2(IR)) @7-L In fact, equation (1.1) is understood as a formal representation of the integral equation (1.2) -t” - I + 0*(LlZ8dAs + L2EsdA, + L,&dA: + L&ds), J where the integrals are Itb type quantum stochastic integrals of adapted processes. In short, the role of an infinitesimal increment of the Brownian motion dB, in the classical It6 theory is played by dAl, dAt and d&. As a result, the solution should be an adapted process. For comprehensive account see [9], [17]. In respect of various applications it is important to remove the adaptedness condition (in this connection seealso [2]) and to include quantum stochastic differential equations with more general (or singular) coefficients. The white noise distribution theory (WNDT for short; see e.g., [8] for the recent progress) has brought new aspects to quantum stochastic processesand quantum stochastic integrals [13], [16], quantum It6 formula [5], quantum martingales [14] and quantum stochastic differential equations [15]. The idea of WNDT, originally tracing back to Hida [4], is to introduce a particular Gelfand triple called the Kondmtiev-Shit space: (Jqp c L2(E*,P) g W2W)
c (Jq,
OlP
(1.3)
so that the white noise process is a (E)i-valued smooth function obtained by differentiating the Brownian motion with respect to the time parameter. 219
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It is also noticeable that the pointwisely defined annihilation operator at and creation operator at become continuous operators for themselves on the Kondratiev-Streit space (1.3). Moreover, we have dAt = at dt, dAf = at dt, dAt = a,‘at dt and thereby equation
(1.1) is brought into a usual differential dZ dt
where o is the Wick product.
equation for operators:
+ Ls.Fa, + L3at*z + L&s
=
ba,*Zat
=
(Lla,*at + L2a, + L3at + L4) 0 Z,
(14
From our aspect (1.4) should be regarded as a special case of dE -& = L,oc”+M,,
(1.5)
where { Lt } and { &ft} are quantum stochastic processes in r( L2(R)) @ X. The previous paper [15] studied equations of the form (1.5) without an initial space ‘Pt. In this paper, formulating a quantum stochastic process on r(L2(lR)) 8 N, we shall discuss existence and uniqueness of a solution of (1.5). A difficulty caused by a non-trivial ‘H is that the Wick product becomes non-commutative. The functional analytic background of our discussion relies upon [ll], 1121. 2
WHITE
NOISE
DISTRIBUTION
THEORY
Let H = L’(IR, dt) be the Hilbert space of R-valued L2-functions denoted by ] . 1s. We then consider the real Gelfand triple E = S(R) c H = L2(R,dt)
on lR of which
norm is
c E’ = S’(R).
The real inner product of H and the canonical bilinear form on E* x E are compatible and so denoted by the same symbol (., +). Let p be the standard Gaussian measure on E’ and L2(E*,p) the Hilbert space of C-valued L2-functions on E*. There is a unitary isomorphism between L2( E’, p) and the Boson Fock space r(Hc) determined uniquely by the correspondence
(In general, for a real vector space X the complexification is denoted by Xc.) celebrated Wiener-It&Segal isomorphism. If 4 E L2(E*, CL) and (fn)rzO E F(Hc) in this manner, we write 4 N (fm) for simplicity. It is then noted that
This is the are related
(2-l) where ]I 4 )]e is the L2-norm of 4 E L2(E*, p). Taking (2.1) into account, we introduce a family of norms with p E 0%by setting m
indices -1
< p < 1 and
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where PP)B
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1f,, IP = 1(A@*)f’f,, lo, A = 1 + t2 - dz/dt2. Suppose p 2 0 and 0 5 p < 1. Then = (4; II d Ilp,p< oo} becomes a Hilbert space and
a countable Hilbert nuclear space. On the other hand, II . IIPp + is a Hilbertian L2(E*,~) and we denote by (J!?-~)-P the completion. It then holds that (E);
55 inpd_lm(E-P)-~
norm on
= U (EPp))p. PLO
The Kondratiev-Streitspace (1.3) is thus obtained. will be denoted by ((v, .)). Then we have
The canonical bilinear form on (E);
x (E)p
where (., +) is the canonical C-bilinear form on (E$“)* x Ep. The case of /3 = 0 in (1.3) is also called the Hida-K&o-Tulcenaka spaceand denoted by (E) C L2(E*, /J) C (E)‘. 3
QUANTUM
WHITE
NOISE
In general, for two locally convex spacesX, 9 let L(X, 9) denote the space of continuous linear operators from X into 9. When otherwise stated, .C(X, 9) is endowed with the topology of bounded convergence. For simplicity we also write L(X) = C(X, X). The annihilation operator at a point t E R, denoted by at, is an operator in C((E)p, (E)p) uniquely determined by a& = <(t)&, < E EC. Its adjoint ai E C((E&,(E);) is called the creation operator at a point t. These are not operator-valued distributions but continuous operators for themselves. Moreover, the map t I+ at E L((E)p, (E)p) is infinitely many differentiable; hence so is t H ai E C(( E);, (E);). Since the natural inclusions w%3~
(EM
= w%
(J%
WE);7
(E&) c w%~
(E&)
are continuous, t I+ IV, = at + ai E L(( E)a, (E);) is also infinitely many differentiable. The pair {(at, 4)) or {Wt} is called the quantum white noise. It is known [8, $9.31that the pointwise product of (E)p is extended to a product of a test function $J E (E)p and a generalized function 9 E (E&, which is denoted by @4 E (E);. In other words, each @ E (E); gives rise to a multiplication operator; moreover, (E); + w%, (E&) b ecomesa continuous injection. The quantum white noise {Wt} is nothing but the one-parameter family of multiplication operators by the classical white noise processwhich lives in (E);, see e.g., [ll, $4.11. 4
OPERATOR
SYMBOLS
Let 7-l be another Hilbert space which is refered to as an initial Hilbert space. For our purpose we need X-valued white noise functions, more precisely, our discussion will be based upon (E)B @‘Ii c L2(E*, p) 63‘8 c ((E),g 8 7-1)’ = (E); ~3X,
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We note that
w% @x7c-q @7-0csw% @m37W))
as vector spaces.
Moreover, for a sequence the convergences in C( (E)p @ X, (E); @ X) and in C( (E)p @ (E)p, L(X)) are equivalent. On the othe hand, we have topological isomorphisms:
which follow from fundamental properties of the z-tensor product and the kernel theorem. For Z E L((E)p @ 7-& (E); @I‘FI) an C(E)-valued function 2 on EC x EC defined by (%,rlh
u) = ((G$
@U)> c&J@v>),
5>v~Ec,
‘~L,uE%
is called the symbol of 5’. Since every continuous operator c” E L((E)p @ 7-& (E)l; @ ‘lf) is determined uniquely by its symbol, many questions about an operator can be discussed in terms of its symbol. The idea of symbol is due to Berezin [3] and has been developed by Kree and Raszka [7]. A detailed study of operator symbols in terms of WNDT has been done in (111, [12] where only the case of Hida-Kubo-Takenaka space (p = 0) is discussed, however, the extension to the case of Kondratiev-Streit space (arbitrary 0 5 p < 1) is straightforward in general. Theorem 4.1 An L(X)-valued f unction 0 defined on EC x EC is the symbol of an operator C( ( E)P @I 7-l, (E); @ 7-l) if and only ij (i) for any <,
Theorem 4.2 Let {&} be a sequence of operators in L((E)p @ 3-1, (E); @ 7-l). Then {E,,} converges to 0 in C( ( E)P 8 7-l, (E); @ZI3-1) or equivalently in C( ( E)P @J(E)p, C(X)) if and only if there exist a sequence {en} of positive numbers converging to 0, constant numbers K 2 0 and p 2 0 such that
II m, 4 lb I 5
enexpK(]~jP*+]g]$),
QUANTUM
STOCHASTIC
<,~EEc,
n=1,2,....
PROCESSES
Let T c Iw be an interval. According to [13] a one-parameter family of operators { Et}lE~ C w% @XT w; @Et> is called a quantum stochastic process whenever t H Et is continuous. By definition {al}, {a:} and {Wt} are quantum stochastic processes.If {Zt} c L((E)p, (E);) is a quantum stochastic process, so is the amplification {Et 8 I}. We often write Et for Zt 8 I
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for brevity. In general, the continuity oft H Et E L((E)p @%, (E& @Ja) is verified with the help of Theorem 4.2. We begin with the following general result. Lemma 5.1 Let X be a Fre’chet space with defining seminorms {II . II,}. For a continuous maptHLtEC(X,X*) dfie ned on a closedfinite intemral [a, b] there exists a continuous map tl+ Zt E C(X, X’) such that
Moreover,
$ Et = Lt holds in L(X, X’).
PROOF. For fixed t E T and 4, $J E X the right hand side of (5.1) is just the Riemann integral of a continuous function s I+ ((L&J, +)). We denote it by &(<,v). Obviously Bt is a bilinear form on X x X. For the existence of Zt E L(X,X*) it suffices to show that Bt is separately continuous. By assumption { Lt ; t E [a, b]} is a compact subset of C(X, x’) and hence bounded. Then by the Banach-Steinhaus theorem [18, Theorem 33.11 it is an equicontinuous subset; namely, for any bounded subset B C X there exist Q:and C 2 0 such that II Lt4 IIB = g; I GA 111)) I I c II 4 IL 7 4 E x, a
P&A $115 lt I ((LA SDIds 5 W - 4 II d IL,
4 Ex,
for some (Y and C 2 0. Hence 4 H B,($, $J) is continuous; and by duality $J I+ B,(q5, $J) is also continuous. Consequently, Bt is separately continuous. The rest of assertion (continuity and differentiability oft H St) is verified in a standard manner. qed Lemma 5.2 If{L,} and {a;L,}.
C
C((E)~@X,(E&C3K)
is a quantum stochastic process, so are {L,a,}
PROOF. (The proof given in [13] for ‘FI = c can not be applied to this case.) Let BI, B2 c (Q c37-t be bounded subsets. First note that
II Las -
bat
IIBI,BzI II L(h - 4 IIB1,Bz + II CL - Lbt llB,,B,.
As for the second term we seethat
II CL -
Lth
llB,,B,= II La-
Lt
Ilat(B1),Bz + 0 as s + t>
where we note that at(B 1) is again a bounded subset. Since s I+ L,q5 E (E);3 @‘?t is continuous for a fixed 4 E (E)p @IX, the image of a closed finite interval [a, b] is compact and hence bounded in (E); @ 3-1;namely,
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In other words, {L:$ ; $ E B 2, a 5 s 5 b} is a weakly bounded subset of ((E) @X)‘; it is equicontinuous, that is, one may find C 2 0 and p > 0
hence
Then for a 5 s 5 b,
=
sup 401
sup I&B,
(5.3) Since t H at 8 I E L((E)p 8 7-&(E)p @ Z) is continuous, (5.3) goes to 0 as s + t.
qed
Let {Lt} c L((E)p @‘H,(E); 8 Et> b e a quantum stochastic process. Then, by virtue of Lemmas 5.1 and 5.2, we define new quantum stochastic processesby
tLds, tLas t ds, J J J a
a
a
a:L, ds.
(5.4)
The second and the third are called the quantum stochastic integrals of {L,} annihilation and creation processes,respectively. In particular, At =
J0
ta,ds,
A; =
t J0
t a; ds,
A, =
J0
against the
a;a, ds,
are respectively the annihilation process, the creation process and the number process of Hudson-Parthasarathy [6]. Contrary to their theory we consider the number process as being not independent of {At} and {AZ}. M oreover, we proved in (131that any quantum stochastic process in L((E), (E)*) can be decomposed into a sum of “generalized” quantum stochastic integrals against the annihilation process, against the creation process and against the time. We are thus convinced that {at} and {at} are the “primary” quantum noises. The situation for a quantum stochastic process in L((E)p 8 ‘FI, (E);3 8 3-1)is expected to be similar. 6
WICK
PRODUCT
We first discuss the case of .C((E)p, (E);). It follows from the characterization of operator symbols (Theorem 4.1) that for two operators c”i, z”2 E C((E)p, (E);) there exists a unique operator E E C((E)p, (E);) such that
(6.1) The above E is . called the Wick product and is denoted by Z = Zi properties: IO, =:- -ZoI=c”,
Moreover, we note that
(~10~~)o~~=~~o(~20~~),
o
Es. Note some algebraic
(EIoE2)*=S;oz;.
(6.2)
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Namely, equipped with the Wick product L((E)p, (E)f3) becomes a commutative As for the annihilation and creation operators we have a, 0 at = asat,
* asoat *- - ata,,
a: 0 at = a:at,
algebra.
a: 0 ai = atar.
(6.3)
More generally, it holds that a:l...a:,Zat,.-.atm
=zo(a:l.-.
aIlat
. . . atm),
E E Jwqp,
(qj).
It is known [15] that the Wick product is a unique bilinear map from C( (E)p, (E);) x L((E)p, (E)l;) into C((E)p, (E);) which is (i) separately continuous; (ii) associative; and (iii) satisfies (6.3). For an operator E E L((E)p, (E);) the function g(<,q) Remark = e-(tg’r) ((E&, &)), <,q E EC, is sometimes called the Wick symbol [3]. Th is is more like the standard notion of a symbol of a pseudo-differential operator (see e.g., [lo]) an d is slightly convenient for the Wick product since (El o E’s)“(<, 7) = gi(& v)gz(<, 77). H owever, the difference between the operator symbol and the Wick symbol being only the exponential factor e-(tlq), it is a routine work to translate statements for operator symbols into ones for Wick symbols. In this paper we use only operator symbols just to avoid confusion. We next consider the case of L(( E)a @ ti:, (E); @ 3-t). Again the Wick symbol is defined by (6.1) though sr(r, r])gs(t, 7) in the right hand side is the usual product (composition) of two operators in C(X). Th is is also called the Wick product and again denoted by El o .Yz. The algebraic properties in (6.2) are again valid; however, contrary to the case of ‘,Y = @ the Wick product in the vector-valued case is not commutative. In fact, by definition we have
‘7 WICK Any operator E’ E C((E)@ 63 74 (E); integral kernel operators:
EXPONENTIAL ~3 3-1) ad mi t s an infinite
series expansion
in terms of
(7.1) where the series converges in L((E)p has a formal integral expression: ‘b(K) If an operator
= L,+-
8 X, (E)i; QDX). Recall that an integral kernel operator
Kc%, . . . , St, h,. . . , tm)azl . . . a:,at, . . - at,dsl
. . . ds&
. . . dt,.
E E ,C( (E)p 8 X, (E);3 @ X) is expressed as in (7. l), we put degc” =
sup{l+m;
qm
#O}.
It can happen that deg E = 00. If Z E C(( E)a 8 Z, (E); @Jtit> has a finite degree deg Z < 00, i.e., if E is* a finite sum of integral kernel operators, then E E C( (E) @I7-l:,(E)* @ 7f). Namely, for an operator of finite degree p is not important.
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7.1 For E E C( (E) 8 ti, (E)’ @X) we consider @CC, 7) = exp (CS, rl) + e-(t’9)s(5, 7)) .
Then 0 is the symbol of an operator
in L((E)a@E,
(E)f,@X)
(7.2)
i;f and only i,fdegZ
5 2/(1-p).
PROOF. The “if” part is proved by straightforward norm estimates. For the “only if” part we employ Hadamard’s factorization theorem for entire holomorphic functions. qed The operator of which symbol is given by (7.2) is called the Wick exponential of Z and denoted by wexp 5. In view of the argument in [15] we see that wexp Z = go;zy
~%+O.~.O~,
pJ=I,
n times
where the series converges in L((E)p @X, (E& @ tit> if and only if deg E 5 2/(1 - p). Lemma 7.2 Let Ei E C((E)@Z,(E)*@‘,V)
withdegz;
< 00, i = 1,2. IfZro$
= ZzoZi,
we have
(wexp Zi) 0 (wexp Zz) = wexp (Zr + Zs). In particular,
for Z E L((I.3)
8 T-l,(E)*
63 FL) with degZ < 00,
wexp E owexp(-Z)
= I.
The proof is straightforward. Note that 3 H wexp z- is not continuous. In fact, the Wick exponential is defined only for E of finite degree and such operators do not constitute an open set in C((E) @ 3-1,(E)* @7-r!). Nevertheless we have map defined on an interval 8 7&(E)* 8 7-L) b e a continuous p) fort E T. Then t H wexp Et E C((E)p 63‘FI, (E); 63X) t H Et is differentiable and (dZt/dt) o ,-,t ” - Et o (dc”,/dt), then
Lemma 7.3 Let t H Et E L((E) T c JR such that deg Zt 5 2/(1-
If
is continuous.
in addition
dz
d
z wexp Zt = ~owexpZ”t=wexp.Z~Oo dt holds in C((E)p
8
@ ?l, (I?);
QUANTUM
dC dt
~3 7-l).
STOCHASTIC
DIFFERENTIAL
EQUATIONS
We begin with a differential equation in Wick product form. Theorem and consider
and {A&} be quantum stochastic processes in L((E) @ 3-1,(E)’ 8 7-l) ualue problem: ., dS = LtoZ+A& z = Eel E C((E) 63 7-l, (lx)’ c3 7-l). (8.1) t=o dt
8.1 Let {L,} the initial
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If deg Lt 5 2/(1-
Congress
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p) and if 0, =
O~OLL, = LtoOt, then there
287
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exists a unique
solution
to (8.1)
-t-
” - wexp J&0
t (J
in L((E)p
t L,ds, J0
~3 7-i, (E)l;
(8.2) @ Ii)
which
is given
by
. (8.3) > PROOF. It follows immediately from the expansion of Lt (see (7.1)) that deg Rt 5 deg Lt. Therefore wexp (&Q,) E L((E)~@‘Z-I, (E);;3@7i). I n general, the Wick product of two quantum stochastic processesis again a quantum stochastic process. Hence by a step-by-step argument we seethat {Et} defined as in (8.3) is a quantum stochastic processin L( (E)p @ ?I!, (E)l; @ 7i). Obviously, it is differentiable; and by a direct computation with Lemmas 7.2 and 7.3 we can check that (8.3) is a solution of (8.1). Th e uniqueness follows by observing the differential equation for operator symbols corresponding to (8.1). qed The commutativity condition (8.2) is automatically satisfied when ?l = @. Even in case of a non-trivial ?I!, with the help of time-ordered (Wick) exponential (see e.g., [lo]) one can discuss without assuming the commutativity condition. Detailed study in this connection will appear elsewhere. Before illustrating someexamples, we note relation between the Wick product and the usual product (composition) of operators. Let U be the space of all Z E L((E)p @ Pi, (E& @ X) which admit expansions of the form: Z = Cz=, Zo,m(~O,m),i.e., contain no creation operators. Lemma 8.2 For any R E 2i f~ L((E)p
0
wexp(-QS)OMSds+Ze
@X)
and Z E L((E)p
~3 7d) we have
R’OS”f-lR*E.
EoR=c”R, PROOF.
63 N, (E)l;
By assumption we put R = CzXo ZO,m(~0,~). Then the symbol is given by fi((,q)
= e(c,o) 2 K.o,,((@~), m=O
and hence (E 0 f2
jg, 7) = e- (BOWL,
77)fi(E, rl) = 5 qt, m=O
77)~o,m(Prn).
(8.4)
On the other hand, since
which is verified easily, we have (wx,
du, V)
=
((wk
8 4, 4, B 4)
(8.5) From (8.4) and (8.5) we see that E o 52 = ZR. duality.
The rest of the assertion is obtained by qed
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Lemma 8.3 For any H E L((E)p,
(E);)
and Z E ,!Z((E)p @ 31, (E); @I7i) we have
(H@I)OZ=~O((H@I).
(f-4.6)
PROOF. By a direct computation we seethat (H QDI)-([, 7) = H^(s, 7)1, which is a scalar operator on X. Then (8.6) follows immediately by computing symbols. qed
By virtue of Lemmas 8.2 and 8.3 a wide class of quantum stochastic differential equations comes into our discussion. Example
1 Let Li E C(X), i = 1,2,3,4, and consider the initial value problem:
where L; stands for I @ Li, at for at @ I, and so forth. It follows from Lemmas 8.2 and 8.3 that
Llu,*Zut
+ L2Eut + L&E
for any s” E L((E)p @ti, (E); @Z). (8.1) where
Lt = Llu;ut+
+ L4Z = (Lluiut
+ Last + L&
+ L4) 0 2,
Th en equation (8.7) is brought into a particular case of
Lzut + Lxu,’ + Lq,
A!& = 0.
Moreover, the commutativity condition (8.2) is obviously satisfied, and degLt = 2 implies p = 0. Consequently, (8.7) has a unique solution in C((E) @3t, (E)* @ ‘X). We shall remark at the end of this section that equation (8.7) is equivalent to a typical quantum stochastic differential equation of Hudson-Parthasarathy, see (1.1). Example 2 A similar argument as in Example 1 can be applied to an equation involving higher powers of quantum white noisessuch as
The unique solution lies in L((E)p @ X, (E); @ ti), w here 0 5 p < 1 should be chosen as sup{m + 72; L,,n # 0) 5 z/(1 -8. Quant urn stochastic differential quations involving higher powers of quantum white noiseshave recently discussedby Accardi [l]. Note that such equations are fairly singular from the usual aspect and are considered as Schrodinger equations with singular Hamiltonians in interacting representations. Finally we observe that our approach is one of the possible generalizations of quantum stochastic differential equations of Hudson-Parthasarathy type. Start with
dE = (L:‘)dAt + Li2)dAt + Li3)dA; + Ly)dt)E,
zltco = zoo,
where {Li”)} c .fZ((E) @J‘H, (E)* @ X) is a quantum stochastic process. According to the original idea of Hudson-Parthasarathy [6], assuming that {Lii)} is an adapted process, we
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bring (8.9) into an integral equation (see (1.2) in $1) and then obtain a usual differential equation for operator symbols:
+5(t)(L/2kj(O
r]) + q(t)(L~%j(~,
77)+ (Li4ktj(1,
77).
(8.10)
Thus (8.9) is understood as a formal expression and its precise meaning is given by (8.10). On the other hand, we can consider the initial value problem: (8.11) which is a particular form of (8.1). Contrary to (8.9), equation (8.11) is a readily well-posed differential equation for operators. Obviously, in terms of operator symbols (8.11) becomes $Z(h)
=
e-((l?){~(t)ll(t)~j(E, +E(#k17)
17)+
+ &i%7)
(8.12)
+ ~(~,1))}%7).
Then equations (8.10) and (8.12) coincide if (Lps”, jg, or equivalently if
rl) = 6 ((,9)LT((, &(&?J), L(“) - = L(“) 0 ” t Et -t>
i = 1,2,3,4,
i = 1,2,3,4.
t
(8.13)
As is seen in Lemma 8.2, a sufficient condition for (8.13) is that (Li’))’ E U n C((E)p @ 7l). Consider a particular case where Lr) = 1 @ L;, L.I E L(X). Then {L{“)} is an adapted (constant) process and Lj”)E = Lr) o Z for any E E C((E)p @ X,(E); @ ‘FI). In that case equations (8.9) and (8.11) become equivalent; and the latter has been discussedin Example 1. In other words, a typical quantum stochastic differential equation of Hudson-Parthasarathy (see also (1.1)) is included in our framework. REFERENCES 1. ACCARDI L., Applications Nagoya University (1996). 2. BELAVKIN finct.
Anal.
3. BEREZIN (1971).
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theory,
and stochastic
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Lectures
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at
analysis in Fock scale, J.
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H.-H.,
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12. OBATA 185-232
N., Operator (1994).
13. OBATA (1995).
N., Generalized
K. R., Q uantum (1984).
Distribution
probability
noise calculus
of Nonlinear
Ito’s formula and stochastic in quantum
for probabihsts,
Lect. Notes
V. E. & STERNIN
field theory,
white
in Math.
Ann. Inst.
Vol. 1538, Springer-
B. YU., Methods
and Fock space, Lect. Notes
stochastic
evolutions,
CRC Press (1996).
Theory,
caIculus on vector-valued quantum
Analysts
and symbols of operators
10. NAZAIKINSKII V. E., SHATALOV Analysis, de Gruyter (1995). 11. OBATA N., White Verlag (1994).
Congress
of Noncommutative
in Math. Vol. 1577, Springer-
noise functionals,
J. Funct.
processes on Fock space, Publ. RIMS
Anal.
121,
31, 667-702
14. OBATA N., White noise approach to quantum martingales, in Probability Theory and Mathematical Statistics (Edited by S. WATANABE, M. FUKUSHIMA, YU. V. PROHOROV and A. N. SHIRYAEV), World Scientific, 379-386 (1996). 15. OBATA N., Wick product of white noise operators and its application differential equations, RIMS Kokyuroku 957, 167-185 (1996). 16. OBATA N., Integral kernel operators on Fock space - Generalizations tum dynamics, to appear in Acta Appl. Math. 17. PARTHASARATHY (1992). 18. TREVES
F., Topological
K. R., An Vector
Introduction
Spaces,
to
Distributions
Quantum and
and applications
Stochastic
Kernels,
to quantum
CaZcuZus,
Academic
stochastic to quanBirkhsuser
Press (1967).
Analysis,
Theory,
Methods
PII: SO362-546X(!W)OO157-0
& Applications, Vol. 30, No. 1, pp. 279-290.1997 Proc. 2nd World Congress of Nonlinear Analysts @ 1997 Elsevier Science Ltd printed in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00
QUANTUM STOCHASTIC DIFFERENTIAL IN TERMS OF QUANTUM WHITE NOBUAKI Graduate
School of Polymathematics,
EQUATIONS NOISE
OBATA Nagoya University,
Nagoya 464-01, Japan
words and phrases: White noise distribution theory, Fock space, Operator symbol, Wick symbol, Quantum stochastic process, Quantum white noise, Wick product, Quantum Key
stochastic differential equation.
1 INTRODUCTION Quantum stochastic differential equations have been discussedconsiderably in mathematics and physics. From the mathematical aspect in 1984 Hudson and Parthasarathy [6] established a quantum analogue of It8 theory with in mind a quantum stochastic differential equation of the form z -1 , dZ = (LIdA + LzdA + LsdA’ + L,dt)Z, -o(1.1) where Li are operators acting on an initial (or system) Hilbert space ‘X; {A,}, {A,} and {A,*} are respectively the number process, the annihilation process and the creation process on the Fock space r( L2(R)); and therefore, the solution {Et} will be a process of operators acting on F(L2(IR)) @7-L In fact, equation (1.1) is understood as a formal representation of the integral equation (1.2) -t” - I + 0*(LlZ8dAs + L2EsdA, + L,&dA: + L&ds), J where the integrals are Itb type quantum stochastic integrals of adapted processes. In short, the role of an infinitesimal increment of the Brownian motion dB, in the classical It6 theory is played by dAl, dAt and d&. As a result, the solution should be an adapted process. For comprehensive account see [9], [17]. In respect of various applications it is important to remove the adaptedness condition (in this connection seealso [2]) and to include quantum stochastic differential equations with more general (or singular) coefficients. The white noise distribution theory (WNDT for short; see e.g., [8] for the recent progress) has brought new aspects to quantum stochastic processesand quantum stochastic integrals [13], [16], quantum It6 formula [5], quantum martingales [14] and quantum stochastic differential equations [15]. The idea of WNDT, originally tracing back to Hida [4], is to introduce a particular Gelfand triple called the Kondmtiev-Shit space: (Jqp c L2(E*,P) g W2W)
c (Jq,
OlP
(1.3)
so that the white noise process is a (E)i-valued smooth function obtained by differentiating the Brownian motion with respect to the time parameter. 219
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It is also noticeable that the pointwisely defined annihilation operator at and creation operator at become continuous operators for themselves on the Kondratiev-Streit space (1.3). Moreover, we have dAt = at dt, dAf = at dt, dAt = a,‘at dt and thereby equation
(1.1) is brought into a usual differential dZ dt
where o is the Wick product.
equation for operators:
+ Ls.Fa, + L3at*z + L&s
=
ba,*Zat
=
(Lla,*at + L2a, + L3at + L4) 0 Z,
(14
From our aspect (1.4) should be regarded as a special case of dE -& = L,oc”+M,,
(1.5)
where { Lt } and { &ft} are quantum stochastic processes in r( L2(R)) @ X. The previous paper [15] studied equations of the form (1.5) without an initial space ‘Pt. In this paper, formulating a quantum stochastic process on r(L2(lR)) 8 N, we shall discuss existence and uniqueness of a solution of (1.5). A difficulty caused by a non-trivial ‘H is that the Wick product becomes non-commutative. The functional analytic background of our discussion relies upon [ll], 1121. 2
WHITE
NOISE
DISTRIBUTION
THEORY
Let H = L’(IR, dt) be the Hilbert space of R-valued L2-functions denoted by ] . 1s. We then consider the real Gelfand triple E = S(R) c H = L2(R,dt)
on lR of which
norm is
c E’ = S’(R).
The real inner product of H and the canonical bilinear form on E* x E are compatible and so denoted by the same symbol (., +). Let p be the standard Gaussian measure on E’ and L2(E*,p) the Hilbert space of C-valued L2-functions on E*. There is a unitary isomorphism between L2( E’, p) and the Boson Fock space r(Hc) determined uniquely by the correspondence
(In general, for a real vector space X the complexification is denoted by Xc.) celebrated Wiener-It&Segal isomorphism. If 4 E L2(E*, CL) and (fn)rzO E F(Hc) in this manner, we write 4 N (fm) for simplicity. It is then noted that
This is the are related
(2-l) where ]I 4 )]e is the L2-norm of 4 E L2(E*, p). Taking (2.1) into account, we introduce a family of norms with p E 0%by setting m
indices -1
< p < 1 and
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where PP)B
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1f,, IP = 1(A@*)f’f,, lo, A = 1 + t2 - dz/dt2. Suppose p 2 0 and 0 5 p < 1. Then = (4; II d Ilp,p< oo} becomes a Hilbert space and
a countable Hilbert nuclear space. On the other hand, II . IIPp + is a Hilbertian L2(E*,~) and we denote by (J!?-~)-P the completion. It then holds that (E);
55 inpd_lm(E-P)-~
norm on
= U (EPp))p. PLO
The Kondratiev-Streitspace (1.3) is thus obtained. will be denoted by ((v, .)). Then we have
The canonical bilinear form on (E);
x (E)p
where (., +) is the canonical C-bilinear form on (E$“)* x Ep. The case of /3 = 0 in (1.3) is also called the Hida-K&o-Tulcenaka spaceand denoted by (E) C L2(E*, /J) C (E)‘. 3
QUANTUM
WHITE
NOISE
In general, for two locally convex spacesX, 9 let L(X, 9) denote the space of continuous linear operators from X into 9. When otherwise stated, .C(X, 9) is endowed with the topology of bounded convergence. For simplicity we also write L(X) = C(X, X). The annihilation operator at a point t E R, denoted by at, is an operator in C((E)p, (E)p) uniquely determined by a& = <(t)&, < E EC. Its adjoint ai E C((E&,(E);) is called the creation operator at a point t. These are not operator-valued distributions but continuous operators for themselves. Moreover, the map t I+ at E L((E)p, (E)p) is infinitely many differentiable; hence so is t H ai E C(( E);, (E);). Since the natural inclusions w%3~
(EM
= w%
(J%
WE);7
(E&) c w%~
(E&)
are continuous, t I+ IV, = at + ai E L(( E)a, (E);) is also infinitely many differentiable. The pair {(at, 4)) or {Wt} is called the quantum white noise. It is known [8, $9.31that the pointwise product of (E)p is extended to a product of a test function $J E (E)p and a generalized function 9 E (E&, which is denoted by @4 E (E);. In other words, each @ E (E); gives rise to a multiplication operator; moreover, (E); + w%, (E&) b ecomesa continuous injection. The quantum white noise {Wt} is nothing but the one-parameter family of multiplication operators by the classical white noise processwhich lives in (E);, see e.g., [ll, $4.11. 4
OPERATOR
SYMBOLS
Let 7-l be another Hilbert space which is refered to as an initial Hilbert space. For our purpose we need X-valued white noise functions, more precisely, our discussion will be based upon (E)B @‘Ii c L2(E*, p) 63‘8 c ((E),g 8 7-1)’ = (E); ~3X,
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We note that
w% @x7c-q @7-0csw% @m37W))
as vector spaces.
Moreover, for a sequence the convergences in C( (E)p @ X, (E); @ X) and in C( (E)p @ (E)p, L(X)) are equivalent. On the othe hand, we have topological isomorphisms:
which follow from fundamental properties of the z-tensor product and the kernel theorem. For Z E L((E)p @ 7-& (E); @I‘FI) an C(E)-valued function 2 on EC x EC defined by (%,rlh
u) = ((G$
@U)> c&J@v>),
5>v~Ec,
‘~L,uE%
is called the symbol of 5’. Since every continuous operator c” E L((E)p @ 7-& (E)l; @ ‘lf) is determined uniquely by its symbol, many questions about an operator can be discussed in terms of its symbol. The idea of symbol is due to Berezin [3] and has been developed by Kree and Raszka [7]. A detailed study of operator symbols in terms of WNDT has been done in (111, [12] where only the case of Hida-Kubo-Takenaka space (p = 0) is discussed, however, the extension to the case of Kondratiev-Streit space (arbitrary 0 5 p < 1) is straightforward in general. Theorem 4.1 An L(X)-valued f unction 0 defined on EC x EC is the symbol of an operator C( ( E)P @I 7-l, (E); @ 7-l) if and only ij (i) for any <,
Theorem 4.2 Let {&} be a sequence of operators in L((E)p @ 3-1, (E); @ 7-l). Then {E,,} converges to 0 in C( ( E)P 8 7-l, (E); @ZI3-1) or equivalently in C( ( E)P @J(E)p, C(X)) if and only if there exist a sequence {en} of positive numbers converging to 0, constant numbers K 2 0 and p 2 0 such that
II m, 4 lb I 5
enexpK(]~jP*+]g]$),
QUANTUM
STOCHASTIC
<,~EEc,
n=1,2,....
PROCESSES
Let T c Iw be an interval. According to [13] a one-parameter family of operators { Et}lE~ C w% @XT w; @Et> is called a quantum stochastic process whenever t H Et is continuous. By definition {al}, {a:} and {Wt} are quantum stochastic processes.If {Zt} c L((E)p, (E);) is a quantum stochastic process, so is the amplification {Et 8 I}. We often write Et for Zt 8 I
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for brevity. In general, the continuity oft H Et E L((E)p @%, (E& @Ja) is verified with the help of Theorem 4.2. We begin with the following general result. Lemma 5.1 Let X be a Fre’chet space with defining seminorms {II . II,}. For a continuous maptHLtEC(X,X*) dfie ned on a closedfinite intemral [a, b] there exists a continuous map tl+ Zt E C(X, X’) such that
Moreover,
$ Et = Lt holds in L(X, X’).
PROOF. For fixed t E T and 4, $J E X the right hand side of (5.1) is just the Riemann integral of a continuous function s I+ ((L&J, +)). We denote it by &(<,v). Obviously Bt is a bilinear form on X x X. For the existence of Zt E L(X,X*) it suffices to show that Bt is separately continuous. By assumption { Lt ; t E [a, b]} is a compact subset of C(X, x’) and hence bounded. Then by the Banach-Steinhaus theorem [18, Theorem 33.11 it is an equicontinuous subset; namely, for any bounded subset B C X there exist Q:and C 2 0 such that II Lt4 IIB = g; I GA 111)) I I c II 4 IL 7 4 E x, a
P&A $115 lt I ((LA SDIds 5 W - 4 II d IL,
4 Ex,
for some (Y and C 2 0. Hence 4 H B,($, $J) is continuous; and by duality $J I+ B,(q5, $J) is also continuous. Consequently, Bt is separately continuous. The rest of assertion (continuity and differentiability oft H St) is verified in a standard manner. qed Lemma 5.2 If{L,} and {a;L,}.
C
C((E)~@X,(E&C3K)
is a quantum stochastic process, so are {L,a,}
PROOF. (The proof given in [13] for ‘FI = c can not be applied to this case.) Let BI, B2 c (Q c37-t be bounded subsets. First note that
II Las -
bat
IIBI,BzI II L(h - 4 IIB1,Bz + II CL - Lbt llB,,B,.
As for the second term we seethat
II CL -
Lth
llB,,B,= II La-
Lt
Ilat(B1),Bz + 0 as s + t>
where we note that at(B 1) is again a bounded subset. Since s I+ L,q5 E (E);3 @‘?t is continuous for a fixed 4 E (E)p @IX, the image of a closed finite interval [a, b] is compact and hence bounded in (E); @ 3-1;namely,
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In other words, {L:$ ; $ E B 2, a 5 s 5 b} is a weakly bounded subset of ((E) @X)‘; it is equicontinuous, that is, one may find C 2 0 and p > 0
hence
Then for a 5 s 5 b,
=
sup 401
sup I&B,
(5.3) Since t H at 8 I E L((E)p 8 7-&(E)p @ Z) is continuous, (5.3) goes to 0 as s + t.
qed
Let {Lt} c L((E)p @‘H,(E); 8 Et> b e a quantum stochastic process. Then, by virtue of Lemmas 5.1 and 5.2, we define new quantum stochastic processesby
tLds, tLas t ds, J J J a
a
a
a:L, ds.
(5.4)
The second and the third are called the quantum stochastic integrals of {L,} annihilation and creation processes,respectively. In particular, At =
J0
ta,ds,
A; =
t J0
t a; ds,
A, =
J0
against the
a;a, ds,
are respectively the annihilation process, the creation process and the number process of Hudson-Parthasarathy [6]. Contrary to their theory we consider the number process as being not independent of {At} and {AZ}. M oreover, we proved in (131that any quantum stochastic process in L((E), (E)*) can be decomposed into a sum of “generalized” quantum stochastic integrals against the annihilation process, against the creation process and against the time. We are thus convinced that {at} and {at} are the “primary” quantum noises. The situation for a quantum stochastic process in L((E)p 8 ‘FI, (E);3 8 3-1)is expected to be similar. 6
WICK
PRODUCT
We first discuss the case of .C((E)p, (E);). It follows from the characterization of operator symbols (Theorem 4.1) that for two operators c”i, z”2 E C((E)p, (E);) there exists a unique operator E E C((E)p, (E);) such that
(6.1) The above E is . called the Wick product and is denoted by Z = Zi properties: IO, =:- -ZoI=c”,
Moreover, we note that
(~10~~)o~~=~~o(~20~~),
o
Es. Note some algebraic
(EIoE2)*=S;oz;.
(6.2)
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Namely, equipped with the Wick product L((E)p, (E)f3) becomes a commutative As for the annihilation and creation operators we have a, 0 at = asat,
* asoat *- - ata,,
a: 0 at = a:at,
algebra.
a: 0 ai = atar.
(6.3)
More generally, it holds that a:l...a:,Zat,.-.atm
=zo(a:l.-.
aIlat
. . . atm),
E E Jwqp,
(qj).
It is known [15] that the Wick product is a unique bilinear map from C( (E)p, (E);) x L((E)p, (E)l;) into C((E)p, (E);) which is (i) separately continuous; (ii) associative; and (iii) satisfies (6.3). For an operator E E L((E)p, (E);) the function g(<,q) Remark = e-(tg’r) ((E&, &)), <,q E EC, is sometimes called the Wick symbol [3]. Th is is more like the standard notion of a symbol of a pseudo-differential operator (see e.g., [lo]) an d is slightly convenient for the Wick product since (El o E’s)“(<, 7) = gi(& v)gz(<, 77). H owever, the difference between the operator symbol and the Wick symbol being only the exponential factor e-(tlq), it is a routine work to translate statements for operator symbols into ones for Wick symbols. In this paper we use only operator symbols just to avoid confusion. We next consider the case of L(( E)a @ ti:, (E); @ 3-t). Again the Wick symbol is defined by (6.1) though sr(r, r])gs(t, 7) in the right hand side is the usual product (composition) of two operators in C(X). Th is is also called the Wick product and again denoted by El o .Yz. The algebraic properties in (6.2) are again valid; however, contrary to the case of ‘,Y = @ the Wick product in the vector-valued case is not commutative. In fact, by definition we have
‘7 WICK Any operator E’ E C((E)@ 63 74 (E); integral kernel operators:
EXPONENTIAL ~3 3-1) ad mi t s an infinite
series expansion
in terms of
(7.1) where the series converges in L((E)p has a formal integral expression: ‘b(K) If an operator
= L,+-
8 X, (E)i; QDX). Recall that an integral kernel operator
Kc%, . . . , St, h,. . . , tm)azl . . . a:,at, . . - at,dsl
. . . ds&
. . . dt,.
E E ,C( (E)p 8 X, (E);3 @ X) is expressed as in (7. l), we put degc” =
sup{l+m;
qm
#O}.
It can happen that deg E = 00. If Z E C(( E)a 8 Z, (E); @Jtit> has a finite degree deg Z < 00, i.e., if E is* a finite sum of integral kernel operators, then E E C( (E) @I7-l:,(E)* @ 7f). Namely, for an operator of finite degree p is not important.
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7.1 For E E C( (E) 8 ti, (E)’ @X) we consider @CC, 7) = exp (CS, rl) + e-(t’9)s(5, 7)) .
Then 0 is the symbol of an operator
in L((E)a@E,
(E)f,@X)
(7.2)
i;f and only i,fdegZ
5 2/(1-p).
PROOF. The “if” part is proved by straightforward norm estimates. For the “only if” part we employ Hadamard’s factorization theorem for entire holomorphic functions. qed The operator of which symbol is given by (7.2) is called the Wick exponential of Z and denoted by wexp 5. In view of the argument in [15] we see that wexp Z = go;zy
~%+O.~.O~,
pJ=I,
n times
where the series converges in L((E)p @X, (E& @ tit> if and only if deg E 5 2/(1 - p). Lemma 7.2 Let Ei E C((E)@Z,(E)*@‘,V)
withdegz;
< 00, i = 1,2. IfZro$
= ZzoZi,
we have
(wexp Zi) 0 (wexp Zz) = wexp (Zr + Zs). In particular,
for Z E L((I.3)
8 T-l,(E)*
63 FL) with degZ < 00,
wexp E owexp(-Z)
= I.
The proof is straightforward. Note that 3 H wexp z- is not continuous. In fact, the Wick exponential is defined only for E of finite degree and such operators do not constitute an open set in C((E) @ 3-1,(E)* @7-r!). Nevertheless we have map defined on an interval 8 7&(E)* 8 7-L) b e a continuous p) fort E T. Then t H wexp Et E C((E)p 63‘FI, (E); 63X) t H Et is differentiable and (dZt/dt) o ,-,t ” - Et o (dc”,/dt), then
Lemma 7.3 Let t H Et E L((E) T c JR such that deg Zt 5 2/(1-
If
is continuous.
in addition
dz
d
z wexp Zt = ~owexpZ”t=wexp.Z~Oo dt holds in C((E)p
8
@ ?l, (I?);
QUANTUM
dC dt
~3 7-l).
STOCHASTIC
DIFFERENTIAL
EQUATIONS
We begin with a differential equation in Wick product form. Theorem and consider
and {A&} be quantum stochastic processes in L((E) @ 3-1,(E)’ 8 7-l) ualue problem: ., dS = LtoZ+A& z = Eel E C((E) 63 7-l, (lx)’ c3 7-l). (8.1) t=o dt
8.1 Let {L,} the initial
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Congress
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p) and if 0, =
O~OLL, = LtoOt, then there
287
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exists a unique
solution
to (8.1)
-t-
” - wexp J&0
t (J
in L((E)p
t L,ds, J0
~3 7-i, (E)l;
(8.2) @ Ii)
which
is given
by
. (8.3) > PROOF. It follows immediately from the expansion of Lt (see (7.1)) that deg Rt 5 deg Lt. Therefore wexp (&Q,) E L((E)~@‘Z-I, (E);;3@7i). I n general, the Wick product of two quantum stochastic processesis again a quantum stochastic process. Hence by a step-by-step argument we seethat {Et} defined as in (8.3) is a quantum stochastic processin L( (E)p @ ?I!, (E)l; @ 7i). Obviously, it is differentiable; and by a direct computation with Lemmas 7.2 and 7.3 we can check that (8.3) is a solution of (8.1). Th e uniqueness follows by observing the differential equation for operator symbols corresponding to (8.1). qed The commutativity condition (8.2) is automatically satisfied when ?l = @. Even in case of a non-trivial ?I!, with the help of time-ordered (Wick) exponential (see e.g., [lo]) one can discuss without assuming the commutativity condition. Detailed study in this connection will appear elsewhere. Before illustrating someexamples, we note relation between the Wick product and the usual product (composition) of operators. Let U be the space of all Z E L((E)p @ Pi, (E& @ X) which admit expansions of the form: Z = Cz=, Zo,m(~O,m),i.e., contain no creation operators. Lemma 8.2 For any R E 2i f~ L((E)p
0
wexp(-QS)OMSds+Ze
@X)
and Z E L((E)p
~3 7d) we have
R’OS”f-lR*E.
EoR=c”R, PROOF.
63 N, (E)l;
By assumption we put R = CzXo ZO,m(~0,~). Then the symbol is given by fi((,q)
= e(c,o) 2 K.o,,((@~), m=O
and hence (E 0 f2
jg, 7) = e- (BOWL,
77)fi(E, rl) = 5 qt, m=O
77)~o,m(Prn).
(8.4)
On the other hand, since
which is verified easily, we have (wx,
du, V)
=
((wk
8 4, 4, B 4)
(8.5) From (8.4) and (8.5) we see that E o 52 = ZR. duality.
The rest of the assertion is obtained by qed
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Lemma 8.3 For any H E L((E)p,
(E);)
and Z E ,!Z((E)p @ 31, (E); @I7i) we have
(H@I)OZ=~O((H@I).
(f-4.6)
PROOF. By a direct computation we seethat (H QDI)-([, 7) = H^(s, 7)1, which is a scalar operator on X. Then (8.6) follows immediately by computing symbols. qed
By virtue of Lemmas 8.2 and 8.3 a wide class of quantum stochastic differential equations comes into our discussion. Example
1 Let Li E C(X), i = 1,2,3,4, and consider the initial value problem:
where L; stands for I @ Li, at for at @ I, and so forth. It follows from Lemmas 8.2 and 8.3 that
Llu,*Zut
+ L2Eut + L&E
for any s” E L((E)p @ti, (E); @Z). (8.1) where
Lt = Llu;ut+
+ L4Z = (Lluiut
+ Last + L&
+ L4) 0 2,
Th en equation (8.7) is brought into a particular case of
Lzut + Lxu,’ + Lq,
A!& = 0.
Moreover, the commutativity condition (8.2) is obviously satisfied, and degLt = 2 implies p = 0. Consequently, (8.7) has a unique solution in C((E) @3t, (E)* @ ‘X). We shall remark at the end of this section that equation (8.7) is equivalent to a typical quantum stochastic differential equation of Hudson-Parthasarathy, see (1.1). Example 2 A similar argument as in Example 1 can be applied to an equation involving higher powers of quantum white noisessuch as
The unique solution lies in L((E)p @ X, (E); @ ti), w here 0 5 p < 1 should be chosen as sup{m + 72; L,,n # 0) 5 z/(1 -8. Quant urn stochastic differential quations involving higher powers of quantum white noiseshave recently discussedby Accardi [l]. Note that such equations are fairly singular from the usual aspect and are considered as Schrodinger equations with singular Hamiltonians in interacting representations. Finally we observe that our approach is one of the possible generalizations of quantum stochastic differential equations of Hudson-Parthasarathy type. Start with
dE = (L:‘)dAt + Li2)dAt + Li3)dA; + Ly)dt)E,
zltco = zoo,
where {Li”)} c .fZ((E) @J‘H, (E)* @ X) is a quantum stochastic process. According to the original idea of Hudson-Parthasarathy [6], assuming that {Lii)} is an adapted process, we
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bring (8.9) into an integral equation (see (1.2) in $1) and then obtain a usual differential equation for operator symbols:
+5(t)(L/2kj(O
r]) + q(t)(L~%j(~,
77)+ (Li4ktj(1,
77).
(8.10)
Thus (8.9) is understood as a formal expression and its precise meaning is given by (8.10). On the other hand, we can consider the initial value problem: (8.11) which is a particular form of (8.1). Contrary to (8.9), equation (8.11) is a readily well-posed differential equation for operators. Obviously, in terms of operator symbols (8.11) becomes $Z(h)
=
e-((l?){~(t)ll(t)~j(E, +E(#k17)
17)+
+ &i%7)
(8.12)
+ ~(~,1))}%7).
Then equations (8.10) and (8.12) coincide if (Lps”, jg, or equivalently if
rl) = 6 ((,9)LT((, &(&?J), L(“) - = L(“) 0 ” t Et -t>
i = 1,2,3,4,
i = 1,2,3,4.
t
(8.13)
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