Vol. 36 (1995)
REPORTS
TOWARDS
ON MATHEMATICAL
No. I
PHYSlCS
CHARACTERIZING QUANTUM FERMIONIC EVOLUTIONS
STOCHASTIC
V. R. STRULECKAJA Department
of Mathematics, University of Nottingham, (Received December
28, 1994 
University Park, Nottingham NG7 2RD, UK Revised March 20, 1995)
The generalization to the case of &graded von Neumann algebras of evolutions satisfying a quantum stochastic differential equation with bounded coefficients and driven by fermionic noise is presented. It is proved that the correspondence between fermionic evolutions and stochastic cocycles in this case is parallel to the one in the boson case [ll]. Use of the unification technique [14] allows us to benefit from working in the boson Fock space.
1. Introduction
In this paper we are concerned with a cocycle characterization of an evolution on a &graded von Neumann algebra satisfying a quantum stochastic differential equation (QSDE) driven by fermionic noise. In [ll], assuming norm continuity of the reduced semigroup, evolutions of this type but driven by bosonic noise are characterized by a stochastic cocycle property with respect to second quantized shifts. This result has been generalized, both to flows on algebras by Bradshaw [4] and to the case of unbounded driving coefficients by Accardi and Mohari [l] in the framework of bosonic stochastic calculus. Our aim is to show how this theory can be generalized to the case of a &graded system algebra, when the evolution is the solution of a QSDE with bounded coefficients driven by fermionic noise. We show that the classes of stochastic evolutions with bounded coefficients and of corresponding cocycles with uniformly continuous reduced semigroup are coextensive. Although we use the technique of unification of fermionic and bosonic quantum stochastic calculus, the fermionic result is by no means an immediate corollary of the bosonic one. The outline of this paper is as follows. In Section 2 we give the basis of bosonic quantum stochastic calculus, deferring some of the definitions to Section 4 for our convenience. Here we also introduce our notation. In Section 3 the technique of unification of bosonic and fermionic quantum stochastic calculus [14] is introduced. In Section 4 we give an account of &graded von Neumann algebras. In Section 5 we define a fermionic cocycle and show that the solution of the corresponding QSDE is such a cocycle. Finally, in Section 6 we show that all such cocycles for which the reduced semigroup is uniformly continuous arise as solutions of fermionic QSDE. F31
64
V. R. STRULECKAJA
2. Preliminaries
and elements
of quantum
stochastic
calculus
Here we give a brief account of bosonic quantum stochastic calculus. Let h be a complex Hilbert space, and let r(h) denote the boson Fock space over h. For , each element f E h let 4(f) d enote a coherent (exponential) vector (1, f, l/(n!): 8 f . . .), wh ere @f denotes the symmetric tensor product of rr copies of f. The 1ine”ar span of {$(F) : f E h} is total in r(h). We denote it by &. Let h = ht @ ht be the direct sum decomposition of the Hilbert space h (E L2(!R+) = L2[0,t ] @ L2[t, CQ) here). We identify the boson Fock space T(h) with the tensor product T(ht) I%T(h’), so that the exponential domain E factorizes as an algebraic tensor product & = Et @Et. We denote by .? the algebraic tensor product ho @I&. Let r, denote the Fock space over h = L2(1), where I may denote R, R+, R, or an interval [a,b] C R+. We write r+ for r(L2(R+)), and r for r(L2(lR)). The three basic processes of bosonic stochastic calculus ([12, 171) are the gauge K’, the creation Kg and the annihilation Ki processes defined by their action on elements u @ g(f), u E ho @ E, 4(f) E E, f E L*(R+): (Kl(t)u
@$(f))
= f
Ircou @+(exP J wJ4f)~ (2.1)
Kt is symmetric and Ki and Ki are mutually adjoint on
[email protected]&. Given a contraction 5’ on h, the second quantized contraction r(S) is defined by its action on elements of I as ~(OJJ(~) = Q(S.f). The It6 multiplication rules of the basic differentials dKi, i = 1,4, of the bosonic quantum stochastic calculus [12, 171, where dK1 = CM’, dK2 = dKg, dK3 = dKi and dK4 being a time differential dt, are given by the following identities: dK’dK’
= dK1,
dK’dK2
= dK2,
dK3dK’
= dK3,
dK3dK2 = dK4,
(2.3)
with all the others being zero. A bounded process X = (X(t) : t E IR+) defined together with its adjoint Xt on the common domain in 2 is adapted if for each t E II?,+we can write X(t) = Xt @ 1” (xqt)
= x,t @ It) [17]. Given the adapted process Ei, i = 1,4, the stochastic integral M(t) = $ EidKi (repeated suffix summation) is adapted. In [6] it has been shown that the bosonic creation and annihilation field operators at and a can be expressed as stochastic integrals against the basic integrators Ki and K& respectively,
4f) = j ?dK;,
&f) = j fdK:,,
QUANTUM
in the sense
The
FERMIONIC
EVOLUTIONS
65
that
operators
[a(f),
STOCHASTIC
a#(f)
are unbounded
and satisfy
the canonical
&)I = CatLOa+(g)1= 0, [a(f)! a+(g)1= (.f,.9)1
3. Unification
for
commutation
relations
f. 9 E L”(R+).
technique
Now we shall introduce are
the
fermion
creation
j RdK;, II
K;(t)
and
annihilation
processes
follow
ing [14]. These
I&t)
where
Since
R = (I)K’
=
is the parity
with K’
process,
=
j RdK:,, 0
defined
(3.1)
by (2.1)
so that
dK;
= RdK:,,
dK;
= RdK:,.
(3.2)
dK;
= RdK;,
dK:, = RdK;.
(3.3)
R2 = 1, we have
The fermion field operators b(f), b+(f) basic differentials of bosonic stochastic b(f) = The operators
[
jdK; be(f)
=
s
fRdK;,
are bounded
b+(f) =
and satisfy
(CAR) P(f), b(g)l+ [b+(f),b+(g)]+= vacuum q!,(O) is annihilated by b(f) and representation of the CAR. As is seen from (3.1) and (3.4) we Fock space I’(h). The basic differentials of fermionic dK4 = dt [15, 91. Ito’s multiplication given by the identities (2.2) with (dK’, differentials dKb, i = 2,3, and dKb, i 4. &graded
can be expressed calculus: s
as stochastic
f dK;
the canonical
=
s
integrals
fRdK;.
against
(3.4)
anticommutation
relations
0, [Kf),bt(g)l+ = (.f,dL f,g E L*@+).The is cyclic for b+(f). Thus are to remain
we obtain
in the framework
the Fock
of the boson
stochastic calculus are dK’, dK$, dKi and rules of fermionic stochastic calculus are dK*, dK”, dK4) = (dK’, dKg, dKg, dK4). The = 2,3, are related via R as in (3.2) (3.3).
*algebras
A Qgraded *algebra M E B(h) is a pair (M, 7) comprising of a *algebra M and a *automorphism y of M, such that y* = id. We write M = Mf CBM, where M* are the tixed point spaces of y and 7, respectively, obtained by conjugation by the parity operator R E B(h), XT = RXR for X E M, and M * = {X E M : X = X if X E X X if X E M ’ }. We shall always use
V. R. STRULECKAJA
66
R(t)$(f) = $~(fXla,~l + f~(~,~)) to &grade M10,~l. We notice that conjugation by this leaves invariant each of the subalgebras,MI, I C [O,T], 0 I T < CCL We denote the &graded tensor product by &. Observe that if I = [s, t] then consists of i&graded ampliations. The MI C hIo,,l% {WI)+ @ B(~I)) @,_) &graded tensor product operator ZG T, z E MO, T E MI, is defined by the action on 2: tX
[email protected](t))(‘l~ @ Kf))
= FX~ @ T+(f)>
(4.1)
where the sign is minus only if both u and T (each of definite parity) are odd. We identify the process K$ i = 2,3, of Section 3 with &graded ampliations to the domain 2. From now on let MI denote the &graded von Neumann subalgebra of M = B(P+) generated by the fermion field operator J?(f), f E h and suppf C 1 defined by (3.4): MI := {t?(f) : suppf 5 1)“. Given an initial Hilbert space ho carrying the &graded initial (system) algebra MO obtained by conjugating by the parity operator RQ, we define the &graded van Neumann algebra: NI := {Mr, & Mr : I c .+}“. The algebra NI is equipped with the product &graded structure inherited from the algebras MO and Mr. The NI, I = [s,t c R+, 0 5 s 5 t < co, form a double filtration of B(/Q @ r+) 1 c N12,,t,I, whenever 0 5 s’ 5 s < t 5 t’ < o;), and UICR Nr in the sense that Nr,,tI = B(h0 @T+). The vacuum state conditional expectation ES exists onto each NI. It is characterized by the property: IE~[T~i~‘(f,)...b#ffi)] = (q(O),
@‘(fnx~c) . . * @VixrMO))T
% b#‘(fnxd. . . @“(fixr),
where T E MO, b# (fn). . . b#l (f~) is a polynomial in the creation and annihilation operators b#(fi), fi E Lz(R+), and 1’ denotes the complement of I in R+. E1 satisfies the projective property IE~,.IEI, = EI, if 11 c 12 C R +. Here lEt denotes the conditional expectation IEI, I = [O,t]. An adapted process X = (X(t) : t E R+) is a martingale if it satisfies the identity E,[X(t)]
= X(s)
vs < t.
We shall recall also the fundamental result of quantum stochastic calculus by Parthasarathy and Sinha [18] set originally in a framework of boson stochastic calculus. The martingale is regular if there is a Radon measure p on I c R+ such that for s < t, u E ro,s 65ti(O)r,,, > hII[X(9

[email protected])1412+ KII[W(4
 x*m412 I P([SIal1412.
QUANTUM
STOCHASTIC FERMIONIC EVOLUTIONS
67
It can be shown that, in this case, X can be represented as a stochastic integral against the fundamental martingales of quantum stochastic calculus, which are the gauge, creation and annihilation processes. We shall conclude this section by giving a brief view of references to original works on &graded algebras and their applications for the purposes of quantum stochastic calculus. &graded algebras were constructed and studied by C. Chevalley [S]. E. B. Davies [7] studied a skew tensor product of von Neumann algebras in the framework of involuntary automorphisms of IV*algebras. In [2, 3, 15, 161 one can find applications to quantum stochastic calculus problems.
5. Stochastic
evolutions
and fermionic
cocycles
Let the process U(t) be a solution of the fermionic dU = UlidKi;
QSDE
uo = 1,
(5.1)
where the coefficients li, i = 1,4, are bounded operators in ha identified with their &graded ampliations to ho @ r+ with 11 and 14 even and 12 and l3 odd. Suppose further that the coefficients l,, i = 1,4, satisfy the unitarity conditions which are found, under boundedness assumption, by equating to zero the stochastic differentials of UtlJ and lJ& according to Ito’s multiplication rules (2.2). These are [15]: 11 +
lf + 1Ilf = 1; + 11+ $11 = 0,
12+ 1p + 1112= lf? + 12+
[email protected] 1p
+ 13 +
141 + 14 +
= 0,
$71, = 13 + 1p + 131i = 0,
(5.2)
1p1; = 14 + la + 13lf = 0.
Remark 5.1: Notice, that the unitarity conditions (5.2) differ from the ones in [8] since all differentials are now applied on the right. These conditions were proved in [14] to be sufficient for the existence of a unique unitary process satisfying (5.1) in the case of trivial initial grading. Corresponding results were obtained in [3] in the absence of a gauge term and working in a different formulation of fermionic stochastic calculus, and in a more general case in [4] still in the framework of the latter formulation.
We define the twoparameter family of operators on b @ r+ to be Ii,,, := {Us:‘U, : 0 C s I t < m}. Then for any 0 < r 5 s i t < 00 we have that Ur,JJs,t = u,iuJJ,‘G We regard
(5.3) as describing
a twoparameter
= UT.,.
evolution. The family {Uqs.,: 0 < s with the unitary process t H U,
I t < LX} is called the stochastic evolution associated
on ha @ r+, that is the solution of (5.1).
(5.3)
68
V. R. STRULECKAJA
Denote
by I(,!&) the second quantization
{of’” >  % ZtEekise
St(f)(z) = Like S,, the operator r(S,)r(S,) = T(Ss+,). to the initial space.
I($) F(St)
of the right shift S, on L*(iR+):
is isometric and, furthermore, it satisfies the equality is an even operator; we identify it with its ampliation
DEFINITON 5.2. A femionic cocycle on ho @ r+ is a family {U,,t :0 < s 5 t < cm} of even unitary operators satisfying the following conditions for all 0 < T I s 2 t < 00
(i)
u?,.JJ%t = Ur.t,
(ii)
us,, E Nst,,
(iii)
r(ss)*u,,tr(ss)
where N& Section 4.
(5.4) = Uo,ts,
is the even subalgebra
of the Z2graded von Neumann
PROPOSITION 5.3. Zf {US,, : 0 c s 5 t < m}
H I&$&]
is a fermionic
algebra N,,,
of
cocycle, then t
is a semigroujx
Proof By the properties
of conditional
expectations:
Eo[Us+tl= ~o[~s[~o,s~;,J?~+tl = ~o[~o,s~s[~s,s+tll = ~o[~o,,~o[~s,s+tl1
~o[~o,,~o[~(s,)*~,.,+t~(s,)1 = ~o[~o,~l~o[~o,tl. =
Denoting semigroup.
iE#JS+,]
by PS+t, we see that PS+t = PSPt. Thus t H E,[U,,t]
is a n
We say that (US,,) is uniformly continuous if t H lE,[US,t] is norm continuous. Henceforth we only consider uniformly continuous cocycles (cf. [ll]). THEOREM
5.4.Let U be a solution
of the QSDE (5.1), with 1, E NO, i = 1,4,
constant, and 11 =w1,
12 = 1,
1s = I*wY,
l4 = ih  $*I,
(5.5)
where w* = wl is unitary, h is selfadjoint and 1 is arbitrary in B(ho), and y is the parity automolphism. Then {US,, :0 < s 5 t < m} is a fermionic cocycle. Note that conditions (5.5) are equivalent to conditions (5.2). If U is required to be even, which seems to be a natural assumption, then w and h are necessatily even and 1 is odd. Proof: The uniqueness of the solution to equation (5.1), which is unitaryvalued when its coefficients satisfy conditions (5.5), but with an arbitrary unitary initial value, is obtained by straightforward generalization to the &graded case of Theorem 7.1
QUANTUM
STOCHASTIC
FERMIONIC
6’)
EVOLUTIONS
of [12] in the boson case, and proven for the initial value 1 in Theorem 6.2 in [2]. From this, (5.4i) follows. While fixing s E R+, denote Us(z) = U,*U,+s (notice that UO,, = U(s)) and observe that Us(z) satisfies the integral equation (5.6)
U,(z) = I + 1 U&)li CMj(r). 0
where Iii(.) := K”(s+.)  K”(s), i = 1,3, are the new basic martingales. Applying the Picard iteration procedure, we can obtain the unique solution of (5.6) [12], and since each of the iterates belongs to the algebra i2rs.s+z, the (strong) limit belongs to this algebra as well, Us(z) E hrs.s+z. This gives us (5.4ii). By arguments similar to the ones in [ll], we show that (5.4iii) is fulfilled. Indeed, since r(s,)*~V,.,I’(s,~) = J&_,, r + T(S,)*U(s)U(s + T)T(S,) is an adapted process. We denote it by V(T). Applying the first fundamental formula of quantum stochastic calculus [12, 171 for f,g E L*(R+), U, 2’ E ho, q,(f) E E, one gets
+
by C;t~((ss(fx[o.tJ)>(~:)),u(.~)*u(~)((S,(fx[o,~])~~(~)~~~ @TNGm9)~+
+ (u
[email protected]((s&f)(~)),U(s)*~(~)((S,(SX[o.t]))(~)~~3Ro~ + (u c?G~?J((&f)(~)),u(s)*u(+w
@~((s~(gx[o,~])~(~~)))t
@~((Ssg>(~)>~l dz;
putting s  s = y we finally obtain = (u E ti(f),
j &%)*U(s)*u(Y
+ s)r(S)L
dK”(p)v @ :r(g)),
0
where dh”, i = l3 4, are the fundamental differentials of fermionic quantum stochastic calculus. Note that the second quantization r(Ss) commutes with the overall Fock space parity operator r(I) and so is even. Thus V also satisfies (5.6). By the uniqueness H argument, V = U. This completes the proof. 6. From
cocycles
to unitary
evolutions
To prove the converse of Theorem 5.4, we shall use the remarkable result, which is the representability of regular martingales as stochastic integrals against the fundamental martingales of stochastic calculus [18]. This result may easily be adapted to the case of fermionic stochastic calculus.
70
V. R. STRULECKAJA
THEOREM 6.1.Let Us,t be a cocycle. Then there exists li E JI&, i = 1,4, satisfying the unitarity conditions (6.2) with 11 and 14 even, 12 and 13 odd, such that U(t) = Uo,, (= Ut) is a solution of the QSDE U(0) = I.
U(t) = UlidK”,
(6.1)
Remark 6.2: In the nonunit variance boson case the statement of Theorem 5.1 was established in [lo] and in the unit variance boson case in [ll]. We shall apply arguments similar to those employed in the latter. Proof of Theorem 6.1: By Proposition 5.3 Pt s IEo[Uo,,] satisfies the semigroup condition P,+t = PSPt. Recall that we are assuming that Pt is norm continuous. Then there exists a bounded operator N such that Pt = etN. Define a bounded adapted process X by X(t)
= Ut 
j
U,Nds.
0
Then X(t)
is a martingale. Ea[X(t)]
Indeed, for 0 5 s < t < CQ,
= IEo[U,j,,]  IEo[ j Uo,sN ds] = Pt 
j
P(s)N
ds = I
0
0
and j U(r)N dr]
K[
[email protected])  X(s)] = K [Uo,t  uo,, = I& [Uo,. [U.,,  I
= lE,
=
j U,;,Ndr]] s
I[U(s)Eo [us,,
j&,,Ndr]] S
E, [U(s)Eo [Wa)*{u,,t I
= JE, [U(s)IEo [U(t  s)  I 
j Ua,rN
ts s ,(r
dr]W)}]
 s)N dr]]
0
= lE,[U(s)Eo[X(t
 s)  I]] = 0.
Thus, X is an Na,svalued martingale as claimed. Next we show that X is a regular martingale, so that we can apply the martingale representation theorem [16]. Since P is contractive for 0 5 a 5 b < T < CO, and u E To,, @ +(Or_), one obtains by Schwarz’s inequality Il[
[email protected]) 
xw1412 =
Ua,b

I
) a
Ua,,N
W))*
QUANTUM
STOCHASTIC FERMIONIC EVOLUTIONS
71
 11~112 + I([ j ua.sNds]?L~~*}. a
By definition
of a semigroup
it is easy to see that
=
1 J
(UT mw~~~
F (b 
a>llNII11~112
0
and (Ua,b%
(Ua,b

+)
=
((1

Ua,b)%
u).
Thus, for both X and Xt the following estimate holds: Il[X%)

x%)1~l125 4(b  a)llNII ll42 + w  ~~*11~11*114l*. (6.2)
For 0 5 a 5 b < co the r.h.s. of (6.2) is bounded and positive. As in the boson case, it may easily be shown that the r.h.s. is dominated by a multiple of Lebesgue measure for each finite interval [a, b], 0 5 a 5 b < 03. Thus, X is regular and being a bounded operatorvalued regular Fock martingale it can be represented as a sum of stochastic integrals of bounded operatorvalued processes, say E,, i = 1,4, against the basic martingales dKi, i = v. Setting I%(s) := U(s)*Ei, we obtain V(t) = I + j U(S)li(S)dKi(S)
(6.3)
0
with Z4 G N. Note that the generator N = /4 of the norm continuous semigroup t ++ IEt[Us,t], and that 11 is even, and 12 and Zs are odd. We define new operators on an enlarged Fock space hi 6 r = ho G r_ 8 r+ by their ampliations to ho63 r as follows: c,,, = IF_ &?J U+ ii = Ir_ G&(t), h” = Ir_ c%P?(t), E = Ir_ 8 Iho % R(t) the parity operator, and the new right shift operator Ts on L*(R) and its second quantization ?;, = Ih,, 6 r(T,). Note that T, and r(T,) are unitary. Then (6.3) can be thought of as an equation for operators on h&r+, ho = hoBr_: i?(t) = I +
j
iT(S)7~(S)K”(S).
0
Notice that, as before, have
~~*~~‘,,& = 6(t  s)(:= fi~,~,~) for 0 < s 5 t < co, and we
72
V. R. STRULECKAJA
ts = I +
J iT(T){F,$(?+ s&s>dKi(T) 0
for 0 5 s < t < co, since r, commutes with &?, 7&.) = F$&
for any i = 1,4. Thus
+ s)T9.
Putting T + s = t we have (6.4) Let us, formally, set s = t. We get
that is rl(t) = F&O)?;,* = c,(O),
(6.5)
since Ft is unitary and commutes with Ti. Thus, up to’ planar set of zero Lebesgue measure (6.5) holds, as (6.4) is satisfied almost everywhere in {(s, t); s 2 t}. For T > 0 and 0 5 s < t < T < co there exists Ti, and therefore li, which is N constant on [0, T]. Since T can be arbitrarily large, this completes the proof. Remark 6.3: The proof of Theorem 6.1 is yet one more result in the theory of fermionic Fock space evolution based on the ‘exponentialvectorsspan’ formulation of the theory, rather than the ‘multiparticlevector’ formulation, mentioned in Remark 5.1. Once again, the unification technique enabled us to proceed with this generalization in a rather natural way. Acknowledgement
I wish to express my gratitude to Professor R. L. Hudson, who drew my attention to this problem and for the hours of discussions he has generously granted me, and to the referee for a number of comments which helped me to improve the readability of these notes. REFERENCES Accardi, L. and Mohari, A.: On the structure of classical and quantum flows, Universita degli Studi di Roma II, Italy. Preprint, 1994. D. B.: Publ. RIMS, Kyoto Univ. 23, 1 (1987) 17. PI Applebaum, D. B. and Hudson, R. L.: Commun. Malh. Phys. 96 (1984) 473. [31 Applebaum, [41 Bradshaw, W. S.: Bull. Math. 2nd series, 116 (1992) 1. Chevalley, C.: Publ. Math. Sot. Japan I. (1955). PI F51 Cockroft, A. and Hudson, R. L.: J. Multivar. Anal. 7 (1978), 107. [71 Davies, E. B.: Trans. Amer. Math. Sot. 158 (1971) 115. Hudson, R. L.: ht. J. Theor. Phys. 32 (1993), 2413. PI 191 Hudson, R. L. and Lindsay, J. M.: J. Funct. Anal. 61 (1985), 202. Hudson, R. L. and Lindsay, .I. M.: Uses of nonFock quantum brownian motion and a quantum WI martingale representation theorem. Quantum Probability and Applications II, Proceedings, ed. Accardi, L. et al., Springer, Berlin, Lecture Notes in Mathematics, 1104 (1985).
PI
QUANTUM STOCHASTIC FERMIONIC
EVOLUTIONS
73
L. and Lindsay, .I. M.: Mu&. Proc. Cumb. Phil. Sot. 102 (1987) 363. L. and Parthasarathy, K. R.: Commun. Mar/r. Phys. 93 (1984) 301. L. and Parthasarathy. K. R.: Acta Appl. Math. 2 (1984), 353. L. and Parthasarathy, K. R.: Commun. Math. Phys. 104 (1986). 457. L. and Struleckaja, V. R.: On nonabelian cohomology over Zsgraded*algebras. Len. Murh. Phys. (1995) (to appear). ]lh] Hudson, R. L. and Struleckaja, V. R.: Fermionic quantum stochastic flows, Letf. Math. Phys. (19Y5), to appear. [ 171 Parthasarathy, K. R.: An Introduction to Quantum Stochastic Calculus. Birkhauser, Base1 1992. [18] Parthasarathy, K. R. and Sinha, K. B.: J. Funct. Anal. 67 (1986). 126. [ 1l] 1121 [13] [I41 [lS]
Hudson, Hudson, Hudson, Hudson, Hudson,
R. R. R. R. R.