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Quantum mechanical Wiener processes
JOURNAL
OF MULTIVARIATE
ANALYSIS
Quantum A. M. Mathematics
7, 107-124
(1977)
Mechanical
Wiener
COCKROFT
Communicated
R. L. HUDSON
AND
Department, University Nottingham NG7
Processes
of Nottingham, ZRD, England by K.
University
Park,
Urbanik
Quantum mechanical analogues of Wiener processes are defined and their existence proved, in terms of which the field operators of extremal universally invariant representations of the canonical commutation relations are expressible as stochastic integrals. A noncommutative analogue of the Wiener transformation is constructed and shown to have properties analogous to the classical Wiener transformation.
1. INTRODUCTION In [4] a quantum mechanical analogue of the central limit theorem was formulated, in which the r6le of a random variable 5 was played by a pair (p, Q) of quantum mechanical canonical momentum and position observables satisfying the Heisenberg commutation relation
pq - qp = -4.
(l-1)
In this analogue the limit distributions, analogues of the normal distribution, are, modulo a linear canonical transformation, characterised by ‘the fact that the corresponding characteristic functions are of form
(1.2) where u is a real parameter with u 3 1 and E denotes expectation. In [8] a class of representations of the canonical commutation relations over a complex Hilbert space ‘W, the so-called extremul uniwersallyinvariant representations,were introduced. These can be characterized by characteristic functionals of form
Received
June
22, 1976.
AMS Subject Classification: 28-A-40, 60-00, Key words: Canonical, quantum mechanics,
60-H-05, Wiener
81-A-20. process,
Wiener
transform.
107 Copyright All rights
c:, 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
0047-259X
108
COCKROFT
AND
HUDSON
where CJ3 1, ‘% is a real-linear subspace of ‘%’ such that %’ == $9 -;- i(31 and, for f, g E ‘%, n(f), +(g) are the canonical field observables associated with the representation. It is clear that the functionals (1.2) are special cases of (1.3) obtained by taking %’ to be the complex plane and % to be the real line. We may thus regard the latter in general as characterising canonical analogues of isonormal distributions over the real Hilbert space R Now if ‘$I is the space L,“[O, I] of real-valued Lesbegue square-integrable functions on the unit interval, the isonormal distribution with variance parameter u can be realised in terms of a Wiener process X with the same variance by mapping the vector f E % to the stochastic integral sf dX. This suggests that, if ‘%’ = L2[0, 11, ‘% = LV2[0, 11, the field operators rr(f ), 4(g) of the universally invariant representation (2) are expressible in the form
x(f) = j-f@>
4(g) = j-gdQ
(1.4)
in terms of a pair of elements (P, Q) f orming a canonical analogue of the Wiener process with variance parameter o. Our purpose in this work is to formulate a precise notion of such an analogue, to verify the validity of the stochastic integral expressions (1.4) for field operators of the universally invariant representations, and to construct a canonical analogue of the Wiener tranform [2, 61 which is an isometry in a certain space consisting essentially of “functions” of the noncommuting field operators T( f ) and $(g). In a sequel [3] we shall show that the construction of a Wiener process as a limit of random walks has its canonical analogue, giving another approach to canonical Wiener processes, and we shall establish an analogue of the samplepath continuity property. In Section 2 we define canonical Wiener processes and in Section 3 their existence and essential uniqueness are demonstrated using the anticipated connection with universally invariant representations. Section 4 establishes the existence and properties of the canonical Wiener transformation for the case of one degree of freedom, analogous to the Wiener transformation over !R related to the Fourier-Plancherel transformation through the mutual equivalence of Gauss and Lebesgue measure, in this case. In Section 5 this special case is used to set up the general Wiener transformation. We shall frequently have occasion to refer to pairs of self-adjoint operators S, Tin a Hilbert space as satisfying a commutation relation of form [S, T] = ic, where c is a real number. By this is always meant that the unitary operators eizs, eiuT satisfy the corresponding Weyl relation eiaSeir/T
=
e-ix!/(aeigTeixS
6-G y
E q.
QUANTUM MECHANICAL WIENER PROCESSES
109
In particular, commutativity of S and T meanscommutativity of the unitary operators eiti, eigTfor all X, y E R, and if this holds then the operator S + T can be defined as the infinitesimal generator of the unitary group x --f eixTeisS.
2.
CANONICAL WIENER PROCESSES
By a Wiener processwith variance a2 > 0, we shall understand a stochastic process8 = {3((t): t E [0, I]} indexed by [0, I] such that
(1) E(0) = 0 almost surely, (2) B is Gaussianwith meanzero and lE(B(s)S(t)) = 4% A t, where s A t denotesthe minimum of s and t. (2) is equivalent to: (2)’ If, for each subinterval A = (a, b] of [0, 11, LJ~denotes the random variable (b - ~)-l/~(s(b’(b) - E(a)), then for arbitrary disjoint A, , A, ,... the random variables fdl , sdB,... are independent and identically normally distributed with mean zero and variance u2. It is well known that there exists a probability measurew, , Wiener measure u2, on the Bore1 field of the spaceV of real-valued continuous functions on [O, l] vanishing at 0, equipped with the uniform metric topology, which realisesthe processin the probability space(W, wJ as
with variance
qt,
X) = X(t)
(X E U).
Moreover if, for f EL,~[O, 11, [(f ) denotesthe stochasticintegral
in W, w,), then 5(f 1 is normally distributed with mean0 and variance a2l/f 1j2, linear in f and continuous in the sensethat if fn convergesstrongly to f, t(f,,) converges in probability to [(f ). In the terminology of [7l, f + t(f) is a realisation of the isonormal distribution of variance a2over Lr2[0, 11. Modifying to more concise equivalent forms the definitions of [4], we shall say that physically independent canonical pairs (p, , qr), (p2 , q2),..., that is pairs of self-adjoint operators in a Hilbert space5 satisfying
[Pj 7P7cl= [%, !7kl = 0, [Pf, 4kl = --i&k,
COCKROFT AND HUDSON
110
are (stochastically) independentin the state determined by the unit vector # E 8 if, denoting by lE(A) the quantum mechanicalexpectation (A#, #}, we have
IEfi A, = fj E(A,J t ) I=1
j=l
for arbitrary n = 2, 3,... and elementsA, ,..., A, of the von Neumann algebras 4, ,**a,4, generated by spectral projections of distinct pairs (pi, , qi,),..., (pi, 9a,,). The pairs (PI , ql), (~2 , q&. are identically distributed if, for each j # k and arbitrary A E J& , [E(L&) = lE(A), where ~~~is the normal isomorphism from 4 to Mk carrying each exp(i(xpj + yqj)) to exp(i(xp, + yq,J), whoseexistence follows from the von Neumann uniquenesstheorem. We shall say that the canonical pair (p, q) is normally distributed with mean zero and variance ~2 if (1) holds. We note that an arbitrarily normally distributed (in the senseof [4]) canonical pair is equivalent under a linear canonicaltransformation to a normally distributed canonical pair with mean zero and variance o2 3 1. DEFINITION. A canonical Wiener processwith variance u2 is a pair (P, Q) consisting of functions from the unit interval [0, l] to self-adjoint operators acting in a Hilbert spacesj equipped with a unit statevector $ which determines expectations, such that
(0) V’(s),p(t)1 = [Q(s),SW1 = 0, [P(s), Q(t)] = -is A t, s,t E[O,11; (1) P(0) = Q(0) = 0; (2) if, for each subinterval d = (a, b] of [0, I], (pd , q4) denotes the canonicalpair (for such it is readily seento be, using (0)) ((b - a)-‘12(W4 - P(a)>, (b - a)-““(Q(b)
- Q(a))>,
then for arbitrary disjoint d, , d, ,..., the canonicalpairs (pdl , q&,), (pdp , qd2),... are independentand identically normally distributed with meanzero and variance u2. We shall say that (P, Q) is cyclic if Z/ is a cyclic vector for the set of unitary operators constituting the one-parameter unitary groups generated by P(t) and Q(t), t E [0, 11. It is clear from the Heisenberguncertainty principle that a canonical Wiener processwith variance a2can exist only if a2 > 1.
3. EXISTENCE AND UNIQUENESS OF CANONICAL WIENER PROCESSES Let !JI be a real Hilbert space.By a representationof the canonicalcommutation relations over % we shall mean a pair (U, I’) of unitary representationsof the additive group ‘9I in a (complex) Hilbert space& whose restrictions to each
QUANTUM
MECHANICAL
WIENER
PROCESSES
111
finite-dimensional subspace of ‘% are strongly continuous in the natural vector topology, and which satisfy U(f)
V(g) = e”(r*g)V(g) U(f)
(f, g E St)*
Given such a representation,we may write U(f)
= exp(in(f
V( f ) = exP(+(&!)),
)),
(3.1)
where n, 4 are self-adjoint operator-valued linear functions on ‘B satisfying Mf
)8 4dl
=
Ed(f )S &?>I
Mf
), d(g)1= -i(f,
= 0,
(3.2)
g>.
Two cyclic representationsof the canonical commutation relations over % (U, V), (U’, V’) in Hilbert spaces8, $‘, equipped with specified unit cyclic vectors tj, $’ are equivalent if there is a unitary transformation from & to 5’ carrying # into a scalar multiple of $’ under conjugation by which each U(f) is carried into U’(f) and V(f) into V’( f ). It is then well known that cyclic representations are characterised to within equivalence by the characteristic functional F(f, g) = e-(llzi(f*g)(
U( f ) V(g)+,
4).
Using our terminology, the extremal universally invariant representationof [S] over a complex Hilbert space %’ may be regarded as representationsof the canonical commutation relations over a real-linear subspaceR of !R’ for which %’ = ‘94+ i%, characterisedby the characteristic functionals Fn( f, g) = e-~~~/~~~ilfll~+11sll*~
(0 >, 1).
(3.3)
The following theorem establishesthe existence and essentialuniqueness of the cyclic canonicalWiener processwith variance a2. THEOREM 1. Let ‘9I be the real Hilbert space of real-valued Lebesgue squareintegrable functions on [0, l] and for (T >, 1 let (U, V) be a cyclic representation of the canonical commutation relations over 5R with unit cyclic vector #for whizh the characteristic functiotd is F, , given by (3.3). Denote by v( f ), +( f ), (f E ‘8) the correspondingfield operators defined by (3.1). Then if x[~,~I denotes the indicator function of the interval [0, t], the operators
w
683/7/1-8
=
4xkd
8(t)
=
(b(xr0.d
(t ED 11)
(3.4)
112
COCKROFT
AND
HUDSON
constitute a cyclic canonical Wiener process with variance 9 for which # is the state vector. Conversely if (P, Q) is a cyclic canonical Wiener process with variance 2, then there exists a representation of the canonical commutation relations (U, V) over % for which the state vector is cyclic with characteristic functionaljri, , such that (P, Q) is given by (3.4). Proof. Given the representation (U, V), it is clear from (3.2) that P, Q defined by (3.4) satisfy the defining properties (0) and (1) for a canonicalWiener process.To verify (2), observe first that for an arbitrary subinterval d = (a, 61 of [0, 11,p, , qAare given by
pA = ~((b - a>-1’2xA),
qA
=
$((b
-
a)-1’2xA)!
and so, with expectations determined by the cyclic vector (6, lE(ei(opA+~*A))
=
FJx(b
-
a)-1/2xA
, y(b
-
a)-lj2xA)
Hence (PA , qA) is normally distributed with mean zero and variance u*. More generally for disjoint subintervals A, , A, ,..., A, , the joint characteristic function
Of
(pAI
, qdl)~~~~~
= exp ( -(02/4) i
(PA,,+,
qAN)
is
65” + Y,B)),
using the orthonormality of the vectors (bj - aj)-1/2xd, in ‘8. Thus the joint characteristic functional factorises, whence using the analysisof [4] it is easily Seen that (PAN, 4+-, (PAN, PAN) are independent. Thus (P, Q) is a canonical Wiener processwrth variance us; we must show that it is cyclic. To do this, we first showthat # is cyclic for the operators(U(fJ, V(fO): fO E !R,}, where here and below !I$, denotes the classof all step functions (finite linear
QUANTUM
MECHANICAL
combinations of indicator f~L?~~~fcl~&~%~
WIENER
functions
PROCESSES
of subintervals)
113
on [0, I]. For arbitrary
IIU(f)W)#- U(fcJ Ir(&#II2 = <(V-g>Y-f) - ~(-&I)U(-foNU(f)w> - wh) %5J>~~ 4) = ((1- v-g> V-f +fo>mJo> - V-go)Y-f0 +f) Vg)+ 1)A9) = 2 - e-i<-f+fo*g)( U( -f - e-i<-fo+f~g>< U( -f. =
2 _ _
+ fo) V( -g + go)+, 4) + f) V( -go + g)b 4)
e-i<-f+fo.!?>e’l/2,ic-f+f,.
-~+~,>e-~0~/4~~ll-f+f~l/~+ll-~+B~Il*~
e-i<-f,+f.~~e~l/2~i<-fO+f’-8~+8~e-~u~/4~~Il-fo+fll*+ll-9o+~ll~~~
For fixed f, g, since 5&, is dense in % it is clear that by appropriate choice off-, , g, , this expression can be made arbitrarily small. Since + is cyclic for { U( f ), V( f ): f E ‘St) it follows that it is cyclic for {U(f), V( f ): f E %,J. We now show that forjo , g, E R, , U(f,) V(gO) is a polynomial in the elements of the one-parameter unitary groups generated by P(t), Q(t), t E [0, I]. It is clearly sufficient to show this for U(fJ and V(g,,) separately; we do so for U(fJ, the proof for V(g,,) being similar. Writing f,, E %,, as fo = f
%Aj
9
j=l
where d, ,..., .4, are subintervals of [0, I], we have df0)
= i:
cim’(xA,)
j=l
where A, = (a, , bj]. Hence U(f0) =
eim’fo’
=
e~B~-a,cj(P(b,)-P(aj))
=
fi
(eiciP(bj)e-icjP(aj))
+1
as required. Thus (P, Q) is cyclic. Conversely, let there be given a cyclic canonical Wiener process (P, Q) with variance 02 having state vector Z/LWe construct an associated representation of
114
COCKROFT
AND
HUDSOS
the canonical commutation relations over ‘8, defining U(f,), instance only forfa E !RO. For
V(f,)
in the first
fo ==j=l T cjXAj , 4 = Caj7bjl, we set .qf,)
=
fi
(eiejP(b,)e-icjP(ai));
j=l
equivalently - P(Uj)) = i
4fo> = i 4Wi> j=l
Cj(bj - Uj)l+$
,
j=l
(3.5)
d(h) = i c,(Q(V - Q(d) = i cj(b - Q#‘~cL,~ . j=l
i=l
If also g, E ‘$I,-,, we may assume without loss of generality that g, can be expressed in the form Bo = f d&A, j=l
and that the intervals A, ,..., A, are disjoint. Then using the defining property (0) of the canonical Wiener process,
l?(foh Qo)l = kul>9 54go)l = 0 and b(fo),
‘#(go)]
=
[f:
Cj(bj
-
aj)lptbj
i=l
= -i
, i
dj(bj
-
“1)“‘4Aj]
14
i
cjdi(bj - q)
j=l
= -i
QUANTUM MECHANICAL WIENER PROCESSES
115
(~2/4)(cj’(bj-ai)+dj~(bj-al))
=ge=e -(aZ/4)~;,1(cj*(b~--aj)+~j2(bj-~j)) = e-l~2/4)lIlf~lla+ll~~I12)
It follows from (3.5) that P(t), Q(t) are expressiblein the form (3.4). Since + is cyclic for {&Plt), eiZOlt); x E R, t E [0, l]} it is cyclic for { U(fs), V((fJ; f0 E !&}. Thus the proof is completed if it can be shown that the domain of definition of U, V can be extended from ‘%,,to its completion % to obtain a representation of the canonical commutation relation with characteristic functional F, . That such an extension can be performed follows from [I, Lemma 2.31,together with the observation that if lh,,+m [/fm - fn I/ = 0, then limn,n+ooE(f, - fn , 0) = lE(O,fnL -fn) = l,where limm,n+co
E(f, g) =
e”12)i(f*g)F,,(f,
g)
is the characteristic functional, as defined by Araki and Woods in [l], corresponding to F, . 1 It is instructive to observe that the infinitesimal form of the extension concluding the proof of Theorem 1 amountsto extending the domainsof definition of the integrals 4f)
= /f(t)
dfw
b(f)
= j-f(t)
dQ(t),
which are initially meaningful for step functions f, to arbitrary f E%. Thus we may regard the pair (n( f ), $( f )) for generalf E ‘8 as a stochastic integral off with respect to the canonical Wiener process(P, Q). 4. THE CANONICAL WIENER TRANSFORMATION CASE OF ONE DEGREE OF FREEDOM
IN THE
Let dn/dx = (~T~cJ~)+~ exp( -+(x2/u2))
denote the normal density with mean zero and variance u2 on R. The Wiener
116
COCKROFT
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HUDSON
transform of a complex polynomial A in a single indeterminate is the polynomial WA(~) = IzER .4(2112x $ iy) &J(X). It is well known that w is a bijective map from the space of such polynomials to itself, with inverse given by w+A(y)
= 1 A(21Rv - iy) h(x), xeP
and that w is isometric for the inner product
(A,B)=s,,,44B(x) W4, and thus extends uniquely to a unitary operator, the Wiener transformation, in the completion with respect to this inner product. The latter space may be identified with the Hilbert space La@, dn), and the Wiener transformation is then the product of three isometries M-lflM, where M maps L2(R, dn) to L2(R) by multiplication by (dn/d~)l/~ and S is the Fourier-Plancherel operator on L2(R). The significance of the Wiener transformation is that, unlike the Fourier-Plancherel transformation, it generalizes to the case where [w is replaced by an infinite-dimensional real Hilbert space and n by an isonormal distribution on this space. In this section we construct an analogue of the Wiener transformation, defined in the first instance on complex polynomials in two indeterminates satisfying the Heisenberg commutation relation (I. 1). That such a generalisation is possible rests on the following observation. Let p, 4 be indeterminates satisfying the Heisenberg relations pp - qp = -i and denote by ~3 the complex polynomial algebra generated byp and q, equipped with the involution * which makesp and q self-adjoint. Then the elements of the algebraic tensor product 93 @ B 21/2p1 F ip2 ,
21%1 F iq2 ,
(4.1)
where p, =p @ 1, q1 = q @ 1, p, = 1 @p, q2 = 1 @ q also satisfy (1.1). From this it follows that there are algebra isomorphisms (but not *-isomorphisms) rlr from J% into g @ .98under which the images of p and q are given by (4.1). These provide unambiguous analogues of the 42 complexifications under which a polynomial A(x) in x is mapped into A(2112x F iy). We compare this observation with the “addition theorem” underlying the canonical central limit theorem of [4], that there is for each n == 2, 3,... an
QUANTUM
MECHANICAL
WIENER
PROCESSES
from (in this case a *-isomorphism) an obvious notation, p and q map to
isomorphism which, with
n-yp,
+ --* + PA
117
9 into @y=r G? under
n-+71 + *** + qn),
since the latter alsosatisfy (1.1). Let w be a state, that is a positive linear functional, taking the value 1 on the identity, on the *-algebra B. Then we define a linear map w(l) from 9 @ @ to @ by w(l) c Aj @ B, = c W(Aj)Bj . (4.2) j i The normal state with meanzero and variance u2 > 1 can be defined by w(A) = tr L(A)P
(4.3)
where L is the Schr6dinger representation of g, that is the *-representation [5] by operators in the Hilbert spaceL2(W) with common domain the space Y(R) of infinitely differentiable rapidly decreasingfunctions, under which p and q map to the Schrijdinger operatorsp,, and qO , respectively, where p, = -i(d/dx),
ww
= M4,
and p is the density operator (positive operator of unit trace) corresponding to the normal distribution with mean zero and variance u2, i.e., such that tr P
exP(+%l+
Y%)) = exP(-(ae/4)(x2 + Y”))
(4.4)
(it can be shown [9] that p mapsL2(R)into the domain Y(R) of the operator c(A), and that the product c(A)p is a trace-class operator in L2(R)). Alternatively w can be defined by w(A(p, n)) = 4-@/W,
-iWW)
ev(-(u2/4)(~2 + y2) + Bier) lr=y=o, (4.5)
where the polynomial A@, q) is arrangedso that each constituent monomial is ordered as pmqn. The equivalence of this to the preceding definition follows from (4.3) (4.4) and the relation exp(i(% + Yqd = expWhd eMk-J
exd-tier).
We introduce the mappingsfrom B to itself w* = WI+, where q* are the “ 2/Z complexifications” defined above, w is the normal state on @ with meanzero and variance u2, and w1is given by (4.2), and prove LEMMA
1.
W-W+A
= W+W-A
= A for all A E B.
cocKRo~T
118
Proof. For A = A@, 4) arranged successively
?+A = 42%
AND HUDSON so as to ensure validity
of (4.3,
we have
+ ipZ, 2’l”Ql + iQ2), r
W+A = A
-211% g
i
X exp (-
+ ip, -.-21/2i-&
$ (x2 + ~2) + 4iv)
W-W+A = A (-2lizi-$
+ i (-2ijzi-$
- 21i2i$
x
= A
x
exp
(
i
+ iQ)
+ i (--21i2i$
lx w o , -- ip),
- i*))
- G (x2 + yz) + $ky - f
*W-=0 d-d-0
p + 21J2 &
(
exp (-
(x’2 + y’“) + &k/y’)
a
-i;),Q+2q-
aY’ -‘ar
.a
(4.4)
)I
% (X2 + y2) + +ixy - % (d2 + Y’~) + &Yy ‘) 1 2-y-0 d-f/'-O
Now for m, n = 1, 2,...
a T-z% (
.am a .a% I( -J-j-T-"ay 1 X exp
(
- f
(x2 + y2) + $ixy - -$ (d2 + y’2) + &Yy’)
=(
- 4 (x’ - ix) + +i(y’
x
exp
(
- z$))”
(-
- $ (x2 + y2) + $ixy - ;
$ (y’ - iy) + $i(d (x’2 +y’2)
- ix))”
+ gdy’)
(to see this note that the operators (a/&‘) - i(a/&), (ajay’) - i(a/ay) annihilate the coefficient of the exponential) and this expression vanishes when x, y, x’, y’ are set equal to zero. Hence, using a formal Taylor expansion about (p, Q) in (4.6), we may conclude that W, W-A = A. A similar argument shows that W+W-A = A. 1
LEMMA 2. For A, &EL%, w(( W-A) B*) = w(A( W+B)*).
.
QUANTUM
MECHANICAL
WIENER
119
PROCESSES
Proof. We note first that since the elements 2112p1 + ;P2, 2r12q1+ iq2, 2=/2p2 - ip, , 211Zq2 - z$r satisfy the same commutation relations as p, , ql , p2 , qz , there exists an algebra automorphism 77of B @ B mapping the latter
elementsinto the former. We showthat the functional w @ w is invariant under 7; w @ w(7JC)= w @ w(C) for all C E .% @ 9. It suffices to consider the case when C is a monomial p;“lppqpqia. Then, using (4.5) w 0 f-&C) = w @ W((2’l”Pl
=
(
-Q/2
+ ip2)m’(21/2p2 - ipl)m~(21/2q~ + iq2)n’(21/2q2 - iql)n8)
a + 2t)“’ ax1 2
x ( 421/2
a
x exp
$
(-
+
(41/a
q1
8Yl
&
(42112
2
- g*
&
-
q2
aY1
?Y2
(xl2
+
y12)
1
+
Qiwl
-
z
(x2$
+
yz2)
+
4ix2y2)
1 o,-Ypo +*pr,-0
.
Making the formal changeof variable (under which the exponential is invariant) xl = 21j2x+ ix’,
yl = 2112y + iy’,
x2 = 2112*'
- ix,
y2 I
2V2y
-
iy
this expressionbecomes
x exp - $ (x2 + y2) + ?jxy - G (x’2 + y’2) + @x5’) (
1
,
it-it-0 CC’=&-0
which is w @ w(C) asrequired, using (4.5). Now we can write w((W-A) B*) = w @ w(7+4(1 @ B)*) = w 0 47(7-4I 0 q*)) = w 0 w(A @ l(T)+B)*) = w(A( W+B)*) using the definition of W, , the fact that 7 is an algebra isomorphism, and the easily verified relations. 77)-A =A
@ 1,
7u 0 4”
= (rl+B)*.
I
120
COCKROFT
AND
HUDSON
Lemmas 1 and 2 imply that W+ is a preisometric operator in the pre-inner product space(in fact a true inner product spaceexcept in the casewhen a2 = 1) obtained by equipping 9 with the pre-inner product (A, B)
= w(AB*)
(4.7)
and that W- is its inverse. We may therefore construct a unitary operator W which is the unique unitary extension to the completion of the isometric operator induced by W, in the quotient inner product spaceof @ by the null spaceof the pre-inner product (4.7). We call W the canonical Wiener transformation for one degreeof freedom.
5. THE CANONICAL WIENER TRANSFORMATION
IN THE GENERAL CASE
Let 93 be a real Hilbert space.By the polynomial algebra over ‘8 we mean the complex associativealgebra with identity Q@R)generated by commuting elementsE(f ), f E % satisfying the linearity relation
&f + 49 =
at(f)
+ hXg)
(f,gE%a,bER),
equipped with the involution * which makeseach t(f) self-adjoint. The map where &(g) = (g, f) extends uniquely to an isomorphismr from 5(f)-&, O?(R)to the algebraof polynomial functions on R in the senseof [6]. Let n be the isonormal distribution with variance parameter a2 on !R Then w(A) = s T(A) dn determinesa state w of OZ(!K);equivalently w may be characterised by its values on monomialsof form [(fJl ... [(fn>r, in which fi ,...,fn are orthonormal vectors in %, viz.,
46(fi>” ... S(fJn) = fi ((-i(a/ax,))‘~ e-(02’2)q 1q”. .=r,+ . j=l The Wiener transformation may be defined as follows. Let 8;t be the isomorphisms (not *-isomorphisms) from Cn(%) into the algebraic tensor product GZ(!R)@ @(%) which map f(f) into 2112[(f) @ 1 F il @ .$(f ), and let w1 be the linear map F 4 0 4 --t C 44Pj j from U@I) @ O@R)into OQI). Define wf = wrB* . Then it can be shown that the operators w+ are inversesof each other, and that they are mutually adjoint for the inner product in @(!K) (A,
Bj
= w(AB*).
(5.1)
QUANTUM
MECHANICAL
WIENER
121
PROCESSES
The Wiener transformation w is the unique unitary extension to the completion of the isometric operator induced by w+ in the quotient inner product spaceof rZ!(%)by the null spaceof the inner product space(5.1) (in fact (5.1) is always a true inner product in the classicalcase). The isomorphism7 extends to a Hilbert spaceisomorphismfrom this completion onto the Hilbert spaceof square-integrablerandom variablesof a realisation of the distribution n which are measurablewith respect to the u field generated by {n( f ): f E R}. The unitary operator in the latter spaceconjugate to w under this extension is the usual Wiener transform, for instancethat of [6]. We proceed to construct a canonical analogueof the foregoing. The canonical polynomial algebra over the real Hilbert space ‘% is the complex associative algebra with identity 9Y(%) generatedby elements&, f ), $( f ), f E 9I satisfying daf
+ bg) = an(f) m(f)
p(g)
+ Wg),
= r(g) 4f
C(af + bg) = a+(f ),
$(f ) Hg)
df)4(g)-Ng)df)
I+
W(g),
= $(g) 4(f )T
(5.2)
= -i
for f, g E 3, a, b E R, equipped with the involution * which makeseach n( f ) and 4(g) self-adjoint. Since~~(f)=21~2.rr(f)~1~il~~(f),~~(f)=21~2~(f)~1fil~(b(f) satisfy the relations (5.2), there exist unique isomorphismsqh (not *-isomorphisms) from &‘(‘%) into the algebraic tensor product B(!JI) @ @(%) under which the imagesof r( f ), 4( f ) are v*( f ), +*( f ), respectively. For given variance parameter us > 1, we define a state OJ%on L?@R)by its values on monomialsof form r(fi)‘l *.* r(fJ+(f$1 .*. +(fia)sm, where fi ,..., fn are orthonormal vectors in ‘8, viz.,
That wsris indeed a state can be seenfrom the following equivalent form of this definition: Denote by B(%r) the subalgebraof 9Y(‘B)consistingof polynomials in elements r( f ), u( f ) for f ranging over some finite-dimensional subspace‘RI of %I; let {fi ,..., fN} be an orthonormal basisof !RI . Then the correspondence r(cj
%fj) -+ Xj XjPj , $(x:i xjfj> -+ Cr %qi , where PI ,...,
PN
, 41 ,..a9 qN are the
Schrodinger operatorsacting in L2(RN) by pj = -;(a/&,), qj = X~extends to a unique *-representation [5] rst, of .%9(‘9&) inL2(RN) by operatorshaving domain 9&P’). Then JV)
= tr m,(4,
(5.3)
122
COCKROFT
AND
HUDSON
where p is the density operator in L2(BP’) such that
(It can be shown that IT%, extends uniquely to a traceable operator with domain L2(KC”)). As in the classicalcase,we let wrw be the linear map
from 9(%) @ 9Y(%) to &Y(s), and we set W*” Lemmas1 and 2 we have
= ~~~7%. Then, gene&sing
THEOREM2. adjoint
fw
The mappings Wkg are inverses of each other and are mutually the pre-inner product determined by u; that is, for A, B E S?(%),
(1)
W-“W+%A
= W+“WeRA
(2)
w%(( W-%A)B*)
= A,
= wR(A( W+“B)*).
Proof. Every pair A, B E@(%) belongsto 9(%r) for somefinite-dimensional subspace‘$ of S?,and if L is the identity injection of @(%J into 9?(s) we have
lW3 = wp1, wwL=w wx. It is therefore sufficient to prove the theorem under the additional hypothesis that % is finite dimensional. We fix an orthonormal basis(fi ,..., fN) for !%.The correspondence
7r( C xjfj 1 *
Cx5P$ 9
(b (1 xLh) + C xi4j P
where p, = 1 @ ..a @ $ @ ... @ 1, qj = 1 @ ..* @ ‘G @ ... @ 1 extends uniquely to a *-isomorphism 5 from a(%) onto the algebraic tensor product @,tl c@,where 9 is the canonicalpolynomial algebraof Section 4 for one degree of freedom, with generatorsp, q satisfyingpq - qp = -i. Using (5.3) and (5.4), it is easily seenthat (5.5)
QUANTUM
MECHANICAL
WIENER
PROCESSES
123
where w is the normal state with mean zero and variance u2 of g of Section 4, and that
W*R= 5-l & wi 5, ( j=l 1
(5.6)
where W* are the mappings from 99 to itself of Section 4. Using (5.6) we can now write, for arbitrary A E g’(s) W-3 W+RA = 5-l & (W-W,)
[A = A,
j=l
making use of Lemma 1. A similar argument shows that W+‘RW-RA = A. Thus (1) is established. If A, B E a’(%) are product elements, of form 5-l &, Aj , 5-l @Tsl Bj , respectively, we have, using (5.5) and (5.6), wx(( W_‘RA)B*)
= n w(( W-A,)&.*) j=l = fi w(A,(W+B,)*) j=l
= w%(A( W+%B)*) making use of Lemma 2. Since arbitrary A, BE B(%) can be expressed as 1 linear combinations of product elements, (2) is proved. Thus, as in the case of one degree of freedom, W&s is a pre-isometric operator in the pre-inner product space obtained by equipping g!(g) with the pre-inner product {A, B) = wR(AB*) (5.7) and lV-% is its inverse. The canonical Wiener transformation over R can now be defined as the unitary extension fl to the completion of the isometric operator induced by W+W in the quotient inner product space of .%Y(%)by the null space of pre-inner product (5.7).
REFERENCES [l]
ARAKI, H. AND WOODS, E. J. (1963). Representations of the canonical commutation relations describing the equilibrium states of the free Bose gas. J. Math. Phys. 4 637-662. [2] CAMERON, R. H. AND MARTIN, W. T. (1947). Fourier-Wiener transforms of functionals belonging to L, over the space C. Duke Math. J. 14 99-107. [3] COCKROFT, A.M., GUDDER, S. P., AND HUDSON, R. L. (1977). A quantum-mechanical functional central limit theorem. J. Multiaariate Anal. 7 12.5-148.
124 [4]
[5] [6] [7] [8]
[9]
COCKROFT
AND
HUDSON
CUSHEN, C. D. AND HUDSON, R. L. (1971). A quantum-mechanical central limit theorem. J. Appl. Prob. 8 454-469. POWERS, R. T. (1971). Self-adjoint algebras of unbounded operators. C’ovrv~,lr~rz. Math. Phys. 21 85-124. SEGAL, I. E. (1956). Tensor algebras over Hilbert spaces I. Trans. Amer. Math. Sot. 81 106-l 34. SEGAL, I. E. (1958). Distributions in Hilbert space and canonical systems of operators. Trans. Amer. Math. Sot. 88 12-40. SEGAL, I. E. (1962). Mathematical characterization of the physical vacuum. Ill. J. Math. 6 500-523. WORONOWICZ, S. L. (1970). The quantum problem of moments. Rep. Math. Phys. 1 135-145.
OF MULTIVARIATE
ANALYSIS
Quantum A. M. Mathematics
7, 107-124
(1977)
Mechanical
Wiener
COCKROFT
Communicated
R. L. HUDSON
AND
Department, University Nottingham NG7
Processes
of Nottingham, ZRD, England by K.
University
Park,
Urbanik
Quantum mechanical analogues of Wiener processes are defined and their existence proved, in terms of which the field operators of extremal universally invariant representations of the canonical commutation relations are expressible as stochastic integrals. A noncommutative analogue of the Wiener transformation is constructed and shown to have properties analogous to the classical Wiener transformation.
1. INTRODUCTION In [4] a quantum mechanical analogue of the central limit theorem was formulated, in which the r6le of a random variable 5 was played by a pair (p, Q) of quantum mechanical canonical momentum and position observables satisfying the Heisenberg commutation relation
pq - qp = -4.
(l-1)
In this analogue the limit distributions, analogues of the normal distribution, are, modulo a linear canonical transformation, characterised by ‘the fact that the corresponding characteristic functions are of form
(1.2) where u is a real parameter with u 3 1 and E denotes expectation. In [8] a class of representations of the canonical commutation relations over a complex Hilbert space ‘W, the so-called extremul uniwersallyinvariant representations,were introduced. These can be characterized by characteristic functionals of form
Received
June
22, 1976.
AMS Subject Classification: 28-A-40, 60-00, Key words: Canonical, quantum mechanics,
60-H-05, Wiener
81-A-20. process,
Wiener
transform.
107 Copyright All rights
c:, 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
0047-259X
108
COCKROFT
AND
HUDSON
where CJ3 1, ‘% is a real-linear subspace of ‘%’ such that %’ == $9 -;- i(31 and, for f, g E ‘%, n(f), +(g) are the canonical field observables associated with the representation. It is clear that the functionals (1.2) are special cases of (1.3) obtained by taking %’ to be the complex plane and % to be the real line. We may thus regard the latter in general as characterising canonical analogues of isonormal distributions over the real Hilbert space R Now if ‘$I is the space L,“[O, I] of real-valued Lesbegue square-integrable functions on the unit interval, the isonormal distribution with variance parameter u can be realised in terms of a Wiener process X with the same variance by mapping the vector f E % to the stochastic integral sf dX. This suggests that, if ‘%’ = L2[0, 11, ‘% = LV2[0, 11, the field operators rr(f ), 4(g) of the universally invariant representation (2) are expressible in the form
x(f) = j-f@>
4(g) = j-gdQ
(1.4)
in terms of a pair of elements (P, Q) f orming a canonical analogue of the Wiener process with variance parameter o. Our purpose in this work is to formulate a precise notion of such an analogue, to verify the validity of the stochastic integral expressions (1.4) for field operators of the universally invariant representations, and to construct a canonical analogue of the Wiener tranform [2, 61 which is an isometry in a certain space consisting essentially of “functions” of the noncommuting field operators T( f ) and $(g). In a sequel [3] we shall show that the construction of a Wiener process as a limit of random walks has its canonical analogue, giving another approach to canonical Wiener processes, and we shall establish an analogue of the samplepath continuity property. In Section 2 we define canonical Wiener processes and in Section 3 their existence and essential uniqueness are demonstrated using the anticipated connection with universally invariant representations. Section 4 establishes the existence and properties of the canonical Wiener transformation for the case of one degree of freedom, analogous to the Wiener transformation over !R related to the Fourier-Plancherel transformation through the mutual equivalence of Gauss and Lebesgue measure, in this case. In Section 5 this special case is used to set up the general Wiener transformation. We shall frequently have occasion to refer to pairs of self-adjoint operators S, Tin a Hilbert space as satisfying a commutation relation of form [S, T] = ic, where c is a real number. By this is always meant that the unitary operators eizs, eiuT satisfy the corresponding Weyl relation eiaSeir/T
=
e-ix!/(aeigTeixS
6-G y
E q.
QUANTUM MECHANICAL WIENER PROCESSES
109
In particular, commutativity of S and T meanscommutativity of the unitary operators eiti, eigTfor all X, y E R, and if this holds then the operator S + T can be defined as the infinitesimal generator of the unitary group x --f eixTeisS.
2.
CANONICAL WIENER PROCESSES
By a Wiener processwith variance a2 > 0, we shall understand a stochastic process8 = {3((t): t E [0, I]} indexed by [0, I] such that
(1) E(0) = 0 almost surely, (2) B is Gaussianwith meanzero and lE(B(s)S(t)) = 4% A t, where s A t denotesthe minimum of s and t. (2) is equivalent to: (2)’ If, for each subinterval A = (a, b] of [0, 11, LJ~denotes the random variable (b - ~)-l/~(s(b’(b) - E(a)), then for arbitrary disjoint A, , A, ,... the random variables fdl , sdB,... are independent and identically normally distributed with mean zero and variance u2. It is well known that there exists a probability measurew, , Wiener measure u2, on the Bore1 field of the spaceV of real-valued continuous functions on [O, l] vanishing at 0, equipped with the uniform metric topology, which realisesthe processin the probability space(W, wJ as
with variance
qt,
X) = X(t)
(X E U).
Moreover if, for f EL,~[O, 11, [(f ) denotesthe stochasticintegral
in W, w,), then 5(f 1 is normally distributed with mean0 and variance a2l/f 1j2, linear in f and continuous in the sensethat if fn convergesstrongly to f, t(f,,) converges in probability to [(f ). In the terminology of [7l, f + t(f) is a realisation of the isonormal distribution of variance a2over Lr2[0, 11. Modifying to more concise equivalent forms the definitions of [4], we shall say that physically independent canonical pairs (p, , qr), (p2 , q2),..., that is pairs of self-adjoint operators in a Hilbert space5 satisfying
[Pj 7P7cl= [%, !7kl = 0, [Pf, 4kl = --i&k,
COCKROFT AND HUDSON
110
are (stochastically) independentin the state determined by the unit vector # E 8 if, denoting by lE(A) the quantum mechanicalexpectation (A#, #}, we have
IEfi A, = fj E(A,J t ) I=1
j=l
for arbitrary n = 2, 3,... and elementsA, ,..., A, of the von Neumann algebras 4, ,**a,4, generated by spectral projections of distinct pairs (pi, , qi,),..., (pi, 9a,,). The pairs (PI , ql), (~2 , q&. are identically distributed if, for each j # k and arbitrary A E J& , [E(L&) = lE(A), where ~~~is the normal isomorphism from 4 to Mk carrying each exp(i(xpj + yqj)) to exp(i(xp, + yq,J), whoseexistence follows from the von Neumann uniquenesstheorem. We shall say that the canonical pair (p, q) is normally distributed with mean zero and variance ~2 if (1) holds. We note that an arbitrarily normally distributed (in the senseof [4]) canonical pair is equivalent under a linear canonicaltransformation to a normally distributed canonical pair with mean zero and variance o2 3 1. DEFINITION. A canonical Wiener processwith variance u2 is a pair (P, Q) consisting of functions from the unit interval [0, l] to self-adjoint operators acting in a Hilbert spacesj equipped with a unit statevector $ which determines expectations, such that
(0) V’(s),p(t)1 = [Q(s),SW1 = 0, [P(s), Q(t)] = -is A t, s,t E[O,11; (1) P(0) = Q(0) = 0; (2) if, for each subinterval d = (a, b] of [0, I], (pd , q4) denotes the canonicalpair (for such it is readily seento be, using (0)) ((b - a)-‘12(W4 - P(a)>, (b - a)-““(Q(b)
- Q(a))>,
then for arbitrary disjoint d, , d, ,..., the canonicalpairs (pdl , q&,), (pdp , qd2),... are independentand identically normally distributed with meanzero and variance u2. We shall say that (P, Q) is cyclic if Z/ is a cyclic vector for the set of unitary operators constituting the one-parameter unitary groups generated by P(t) and Q(t), t E [0, 11. It is clear from the Heisenberguncertainty principle that a canonical Wiener processwith variance a2can exist only if a2 > 1.
3. EXISTENCE AND UNIQUENESS OF CANONICAL WIENER PROCESSES Let !JI be a real Hilbert space.By a representationof the canonicalcommutation relations over % we shall mean a pair (U, I’) of unitary representationsof the additive group ‘9I in a (complex) Hilbert space& whose restrictions to each
QUANTUM
MECHANICAL
WIENER
PROCESSES
111
finite-dimensional subspace of ‘% are strongly continuous in the natural vector topology, and which satisfy U(f)
V(g) = e”(r*g)V(g) U(f)
(f, g E St)*
Given such a representation,we may write U(f)
= exp(in(f
V( f ) = exP(+(&!)),
)),
(3.1)
where n, 4 are self-adjoint operator-valued linear functions on ‘B satisfying Mf
)8 4dl
=
Ed(f )S &?>I
Mf
), d(g)1= -i(f,
= 0,
(3.2)
g>.
Two cyclic representationsof the canonical commutation relations over % (U, V), (U’, V’) in Hilbert spaces8, $‘, equipped with specified unit cyclic vectors tj, $’ are equivalent if there is a unitary transformation from & to 5’ carrying # into a scalar multiple of $’ under conjugation by which each U(f) is carried into U’(f) and V(f) into V’( f ). It is then well known that cyclic representations are characterised to within equivalence by the characteristic functional F(f, g) = e-(llzi(f*g)(
U( f ) V(g)+,
4).
Using our terminology, the extremal universally invariant representationof [S] over a complex Hilbert space %’ may be regarded as representationsof the canonical commutation relations over a real-linear subspaceR of !R’ for which %’ = ‘94+ i%, characterisedby the characteristic functionals Fn( f, g) = e-~~~/~~~ilfll~+11sll*~
(0 >, 1).
(3.3)
The following theorem establishesthe existence and essentialuniqueness of the cyclic canonicalWiener processwith variance a2. THEOREM 1. Let ‘9I be the real Hilbert space of real-valued Lebesgue squareintegrable functions on [0, l] and for (T >, 1 let (U, V) be a cyclic representation of the canonical commutation relations over 5R with unit cyclic vector #for whizh the characteristic functiotd is F, , given by (3.3). Denote by v( f ), +( f ), (f E ‘8) the correspondingfield operators defined by (3.1). Then if x[~,~I denotes the indicator function of the interval [0, t], the operators
w
683/7/1-8
=
4xkd
8(t)
=
(b(xr0.d
(t ED 11)
(3.4)
112
COCKROFT
AND
HUDSON
constitute a cyclic canonical Wiener process with variance 9 for which # is the state vector. Conversely if (P, Q) is a cyclic canonical Wiener process with variance 2, then there exists a representation of the canonical commutation relations (U, V) over % for which the state vector is cyclic with characteristic functionaljri, , such that (P, Q) is given by (3.4). Proof. Given the representation (U, V), it is clear from (3.2) that P, Q defined by (3.4) satisfy the defining properties (0) and (1) for a canonicalWiener process.To verify (2), observe first that for an arbitrary subinterval d = (a, 61 of [0, 11,p, , qAare given by
pA = ~((b - a>-1’2xA),
qA
=
$((b
-
a)-1’2xA)!
and so, with expectations determined by the cyclic vector (6, lE(ei(opA+~*A))
=
FJx(b
-
a)-1/2xA
, y(b
-
a)-lj2xA)
Hence (PA , qA) is normally distributed with mean zero and variance u*. More generally for disjoint subintervals A, , A, ,..., A, , the joint characteristic function
Of
(pAI
, qdl)~~~~~
= exp ( -(02/4) i
(PA,,+,
qAN)
is
65” + Y,B)),
using the orthonormality of the vectors (bj - aj)-1/2xd, in ‘8. Thus the joint characteristic functional factorises, whence using the analysisof [4] it is easily Seen that (PAN, 4+-, (PAN, PAN) are independent. Thus (P, Q) is a canonical Wiener processwrth variance us; we must show that it is cyclic. To do this, we first showthat # is cyclic for the operators(U(fJ, V(fO): fO E !R,}, where here and below !I$, denotes the classof all step functions (finite linear
QUANTUM
MECHANICAL
combinations of indicator f~L?~~~fcl~&~%~
WIENER
functions
PROCESSES
of subintervals)
113
on [0, I]. For arbitrary
IIU(f)W)#- U(fcJ Ir(&#II2 = <(V-g>Y-f) - ~(-&I)U(-foNU(f)w> - wh) %5J>~~ 4) = ((1- v-g> V-f +fo>mJo> - V-go)Y-f0 +f) Vg)+ 1)A9) = 2 - e-i<-f+fo*g)( U( -f - e-i<-fo+f~g>< U( -f. =
2 _ _
+ fo) V( -g + go)+, 4) + f) V( -go + g)b 4)
e-i<-f+fo.!?>e’l/2,ic-f+f,.
-~+~,>e-~0~/4~~ll-f+f~l/~+ll-~+B~Il*~
e-i<-f,+f.~~e~l/2~i<-fO+f’-8~+8~e-~u~/4~~Il-fo+fll*+ll-9o+~ll~~~
For fixed f, g, since 5&, is dense in % it is clear that by appropriate choice off-, , g, , this expression can be made arbitrarily small. Since + is cyclic for { U( f ), V( f ): f E ‘St) it follows that it is cyclic for {U(f), V( f ): f E %,J. We now show that forjo , g, E R, , U(f,) V(gO) is a polynomial in the elements of the one-parameter unitary groups generated by P(t), Q(t), t E [0, I]. It is clearly sufficient to show this for U(fJ and V(g,,) separately; we do so for U(fJ, the proof for V(g,,) being similar. Writing f,, E %,, as fo = f
%Aj
9
j=l
where d, ,..., .4, are subintervals of [0, I], we have df0)
= i:
cim’(xA,)
j=l
where A, = (a, , bj]. Hence U(f0) =
eim’fo’
=
e~B~-a,cj(P(b,)-P(aj))
=
fi
(eiciP(bj)e-icjP(aj))
+1
as required. Thus (P, Q) is cyclic. Conversely, let there be given a cyclic canonical Wiener process (P, Q) with variance 02 having state vector Z/LWe construct an associated representation of
114
COCKROFT
AND
HUDSOS
the canonical commutation relations over ‘8, defining U(f,), instance only forfa E !RO. For
V(f,)
in the first
fo ==j=l T cjXAj , 4 = Caj7bjl, we set .qf,)
=
fi
(eiejP(b,)e-icjP(ai));
j=l
equivalently - P(Uj)) = i
4fo> = i 4Wi> j=l
Cj(bj - Uj)l+$
,
j=l
(3.5)
d(h) = i c,(Q(V - Q(d) = i cj(b - Q#‘~cL,~ . j=l
i=l
If also g, E ‘$I,-,, we may assume without loss of generality that g, can be expressed in the form Bo = f d&A, j=l
and that the intervals A, ,..., A, are disjoint. Then using the defining property (0) of the canonical Wiener process,
l?(foh Qo)l = kul>9 54go)l = 0 and b(fo),
‘#(go)]
=
[f:
Cj(bj
-
aj)lptbj
i=l
= -i
, i
dj(bj
-
“1)“‘4Aj]
14
i
cjdi(bj - q)
j=l
= -i
QUANTUM MECHANICAL WIENER PROCESSES
115
(~2/4)(cj’(bj-ai)+dj~(bj-al))
=ge=e -(aZ/4)~;,1(cj*(b~--aj)+~j2(bj-~j)) = e-l~2/4)lIlf~lla+ll~~I12)
It follows from (3.5) that P(t), Q(t) are expressiblein the form (3.4). Since + is cyclic for {&Plt), eiZOlt); x E R, t E [0, l]} it is cyclic for { U(fs), V((fJ; f0 E !&}. Thus the proof is completed if it can be shown that the domain of definition of U, V can be extended from ‘%,,to its completion % to obtain a representation of the canonical commutation relation with characteristic functional F, . That such an extension can be performed follows from [I, Lemma 2.31,together with the observation that if lh,,+m [/fm - fn I/ = 0, then limn,n+ooE(f, - fn , 0) = lE(O,fnL -fn) = l,where limm,n+co
E(f, g) =
e”12)i(f*g)F,,(f,
g)
is the characteristic functional, as defined by Araki and Woods in [l], corresponding to F, . 1 It is instructive to observe that the infinitesimal form of the extension concluding the proof of Theorem 1 amountsto extending the domainsof definition of the integrals 4f)
= /f(t)
dfw
b(f)
= j-f(t)
dQ(t),
which are initially meaningful for step functions f, to arbitrary f E%. Thus we may regard the pair (n( f ), $( f )) for generalf E ‘8 as a stochastic integral off with respect to the canonical Wiener process(P, Q). 4. THE CANONICAL WIENER TRANSFORMATION CASE OF ONE DEGREE OF FREEDOM
IN THE
Let dn/dx = (~T~cJ~)+~ exp( -+(x2/u2))
denote the normal density with mean zero and variance u2 on R. The Wiener
116
COCKROFT
AND
HUDSON
transform of a complex polynomial A in a single indeterminate is the polynomial WA(~) = IzER .4(2112x $ iy) &J(X). It is well known that w is a bijective map from the space of such polynomials to itself, with inverse given by w+A(y)
= 1 A(21Rv - iy) h(x), xeP
and that w is isometric for the inner product
(A,B)=s,,,44B(x) W4, and thus extends uniquely to a unitary operator, the Wiener transformation, in the completion with respect to this inner product. The latter space may be identified with the Hilbert space La@, dn), and the Wiener transformation is then the product of three isometries M-lflM, where M maps L2(R, dn) to L2(R) by multiplication by (dn/d~)l/~ and S is the Fourier-Plancherel operator on L2(R). The significance of the Wiener transformation is that, unlike the Fourier-Plancherel transformation, it generalizes to the case where [w is replaced by an infinite-dimensional real Hilbert space and n by an isonormal distribution on this space. In this section we construct an analogue of the Wiener transformation, defined in the first instance on complex polynomials in two indeterminates satisfying the Heisenberg commutation relation (I. 1). That such a generalisation is possible rests on the following observation. Let p, 4 be indeterminates satisfying the Heisenberg relations pp - qp = -i and denote by ~3 the complex polynomial algebra generated byp and q, equipped with the involution * which makesp and q self-adjoint. Then the elements of the algebraic tensor product 93 @ B 21/2p1 F ip2 ,
21%1 F iq2 ,
(4.1)
where p, =p @ 1, q1 = q @ 1, p, = 1 @p, q2 = 1 @ q also satisfy (1.1). From this it follows that there are algebra isomorphisms (but not *-isomorphisms) rlr from J% into g @ .98under which the images of p and q are given by (4.1). These provide unambiguous analogues of the 42 complexifications under which a polynomial A(x) in x is mapped into A(2112x F iy). We compare this observation with the “addition theorem” underlying the canonical central limit theorem of [4], that there is for each n == 2, 3,... an
QUANTUM
MECHANICAL
WIENER
PROCESSES
from (in this case a *-isomorphism) an obvious notation, p and q map to
isomorphism which, with
n-yp,
+ --* + PA
117
9 into @y=r G? under
n-+71 + *** + qn),
since the latter alsosatisfy (1.1). Let w be a state, that is a positive linear functional, taking the value 1 on the identity, on the *-algebra B. Then we define a linear map w(l) from 9 @ @ to @ by w(l) c Aj @ B, = c W(Aj)Bj . (4.2) j i The normal state with meanzero and variance u2 > 1 can be defined by w(A) = tr L(A)P
(4.3)
where L is the Schr6dinger representation of g, that is the *-representation [5] by operators in the Hilbert spaceL2(W) with common domain the space Y(R) of infinitely differentiable rapidly decreasingfunctions, under which p and q map to the Schrijdinger operatorsp,, and qO , respectively, where p, = -i(d/dx),
ww
= M4,
and p is the density operator (positive operator of unit trace) corresponding to the normal distribution with mean zero and variance u2, i.e., such that tr P
exP(+%l+
Y%)) = exP(-(ae/4)(x2 + Y”))
(4.4)
(it can be shown [9] that p mapsL2(R)into the domain Y(R) of the operator c(A), and that the product c(A)p is a trace-class operator in L2(R)). Alternatively w can be defined by w(A(p, n)) = 4-@/W,
-iWW)
ev(-(u2/4)(~2 + y2) + Bier) lr=y=o, (4.5)
where the polynomial A@, q) is arrangedso that each constituent monomial is ordered as pmqn. The equivalence of this to the preceding definition follows from (4.3) (4.4) and the relation exp(i(% + Yqd = expWhd eMk-J
exd-tier).
We introduce the mappingsfrom B to itself w* = WI+, where q* are the “ 2/Z complexifications” defined above, w is the normal state on @ with meanzero and variance u2, and w1is given by (4.2), and prove LEMMA
1.
W-W+A
= W+W-A
= A for all A E B.
cocKRo~T
118
Proof. For A = A@, 4) arranged successively
?+A = 42%
AND HUDSON so as to ensure validity
of (4.3,
we have
+ ipZ, 2’l”Ql + iQ2), r
W+A = A
-211% g
i
X exp (-
+ ip, -.-21/2i-&
$ (x2 + ~2) + 4iv)
W-W+A = A (-2lizi-$
+ i (-2ijzi-$
- 21i2i$
x
= A
x
exp
(
i
+ iQ)
+ i (--21i2i$
lx w o , -- ip),
- i*))
- G (x2 + yz) + $ky - f
*W-=0 d-d-0
p + 21J2 &
(
exp (-
(x’2 + y’“) + &k/y’)
a
-i;),Q+2q-
aY’ -‘ar
.a
(4.4)
)I
% (X2 + y2) + +ixy - % (d2 + Y’~) + &Yy ‘) 1 2-y-0 d-f/'-O
Now for m, n = 1, 2,...
a T-z% (
.am a .a% I( -J-j-T-"ay 1 X exp
(
- f
(x2 + y2) + $ixy - -$ (d2 + y’2) + &Yy’)
=(
- 4 (x’ - ix) + +i(y’
x
exp
(
- z$))”
(-
- $ (x2 + y2) + $ixy - ;
$ (y’ - iy) + $i(d (x’2 +y’2)
- ix))”
+ gdy’)
(to see this note that the operators (a/&‘) - i(a/&), (ajay’) - i(a/ay) annihilate the coefficient of the exponential) and this expression vanishes when x, y, x’, y’ are set equal to zero. Hence, using a formal Taylor expansion about (p, Q) in (4.6), we may conclude that W, W-A = A. A similar argument shows that W+W-A = A. 1
LEMMA 2. For A, &EL%, w(( W-A) B*) = w(A( W+B)*).
.
QUANTUM
MECHANICAL
WIENER
119
PROCESSES
Proof. We note first that since the elements 2112p1 + ;P2, 2r12q1+ iq2, 2=/2p2 - ip, , 211Zq2 - z$r satisfy the same commutation relations as p, , ql , p2 , qz , there exists an algebra automorphism 77of B @ B mapping the latter
elementsinto the former. We showthat the functional w @ w is invariant under 7; w @ w(7JC)= w @ w(C) for all C E .% @ 9. It suffices to consider the case when C is a monomial p;“lppqpqia. Then, using (4.5) w 0 f-&C) = w @ W((2’l”Pl
=
(
-Q/2
+ ip2)m’(21/2p2 - ipl)m~(21/2q~ + iq2)n’(21/2q2 - iql)n8)
a + 2t)“’ ax1 2
x ( 421/2
a
x exp
$
(-
+
(41/a
q1
8Yl
&
(42112
2
- g*
&
-
q2
aY1
?Y2
(xl2
+
y12)
1
+
Qiwl
-
z
(x2$
+
yz2)
+
4ix2y2)
1 o,-Ypo +*pr,-0
.
Making the formal changeof variable (under which the exponential is invariant) xl = 21j2x+ ix’,
yl = 2112y + iy’,
x2 = 2112*'
- ix,
y2 I
2V2y
-
iy
this expressionbecomes
x exp - $ (x2 + y2) + ?jxy - G (x’2 + y’2) + @x5’) (
1
,
it-it-0 CC’=&-0
which is w @ w(C) asrequired, using (4.5). Now we can write w((W-A) B*) = w @ w(7+4(1 @ B)*) = w 0 47(7-4I 0 q*)) = w 0 w(A @ l(T)+B)*) = w(A( W+B)*) using the definition of W, , the fact that 7 is an algebra isomorphism, and the easily verified relations. 77)-A =A
@ 1,
7u 0 4”
= (rl+B)*.
I
120
COCKROFT
AND
HUDSON
Lemmas 1 and 2 imply that W+ is a preisometric operator in the pre-inner product space(in fact a true inner product spaceexcept in the casewhen a2 = 1) obtained by equipping 9 with the pre-inner product (A, B)
= w(AB*)
(4.7)
and that W- is its inverse. We may therefore construct a unitary operator W which is the unique unitary extension to the completion of the isometric operator induced by W, in the quotient inner product spaceof @ by the null spaceof the pre-inner product (4.7). We call W the canonical Wiener transformation for one degreeof freedom.
5. THE CANONICAL WIENER TRANSFORMATION
IN THE GENERAL CASE
Let 93 be a real Hilbert space.By the polynomial algebra over ‘8 we mean the complex associativealgebra with identity Q@R)generated by commuting elementsE(f ), f E % satisfying the linearity relation
&f + 49 =
at(f)
+ hXg)
(f,gE%a,bER),
equipped with the involution * which makeseach t(f) self-adjoint. The map where &(g) = (g, f) extends uniquely to an isomorphismr from 5(f)-&, O?(R)to the algebraof polynomial functions on R in the senseof [6]. Let n be the isonormal distribution with variance parameter a2 on !R Then w(A) = s T(A) dn determinesa state w of OZ(!K);equivalently w may be characterised by its values on monomialsof form [(fJl ... [(fn>r, in which fi ,...,fn are orthonormal vectors in %, viz.,
46(fi>” ... S(fJn) = fi ((-i(a/ax,))‘~ e-(02’2)q 1q”. .=r,+ . j=l The Wiener transformation may be defined as follows. Let 8;t be the isomorphisms (not *-isomorphisms) from Cn(%) into the algebraic tensor product GZ(!R)@ @(%) which map f(f) into 2112[(f) @ 1 F il @ .$(f ), and let w1 be the linear map F 4 0 4 --t C 44Pj j from U@I) @ O@R)into OQI). Define wf = wrB* . Then it can be shown that the operators w+ are inversesof each other, and that they are mutually adjoint for the inner product in @(!K) (A,
Bj
= w(AB*).
(5.1)
QUANTUM
MECHANICAL
WIENER
121
PROCESSES
The Wiener transformation w is the unique unitary extension to the completion of the isometric operator induced by w+ in the quotient inner product spaceof rZ!(%)by the null spaceof the inner product space(5.1) (in fact (5.1) is always a true inner product in the classicalcase). The isomorphism7 extends to a Hilbert spaceisomorphismfrom this completion onto the Hilbert spaceof square-integrablerandom variablesof a realisation of the distribution n which are measurablewith respect to the u field generated by {n( f ): f E R}. The unitary operator in the latter spaceconjugate to w under this extension is the usual Wiener transform, for instancethat of [6]. We proceed to construct a canonical analogueof the foregoing. The canonical polynomial algebra over the real Hilbert space ‘% is the complex associative algebra with identity 9Y(%) generatedby elements&, f ), $( f ), f E 9I satisfying daf
+ bg) = an(f) m(f)
p(g)
+ Wg),
= r(g) 4f
C(af + bg) = a+(f ),
$(f ) Hg)
df)4(g)-Ng)df)
I+
W(g),
= $(g) 4(f )T
(5.2)
= -i
for f, g E 3, a, b E R, equipped with the involution * which makeseach n( f ) and 4(g) self-adjoint. Since~~(f)=21~2.rr(f)~1~il~~(f),~~(f)=21~2~(f)~1fil~(b(f) satisfy the relations (5.2), there exist unique isomorphismsqh (not *-isomorphisms) from &‘(‘%) into the algebraic tensor product B(!JI) @ @(%) under which the imagesof r( f ), 4( f ) are v*( f ), +*( f ), respectively. For given variance parameter us > 1, we define a state OJ%on L?@R)by its values on monomialsof form r(fi)‘l *.* r(fJ+(f$1 .*. +(fia)sm, where fi ,..., fn are orthonormal vectors in ‘8, viz.,
That wsris indeed a state can be seenfrom the following equivalent form of this definition: Denote by B(%r) the subalgebraof 9Y(‘B)consistingof polynomials in elements r( f ), u( f ) for f ranging over some finite-dimensional subspace‘RI of %I; let {fi ,..., fN} be an orthonormal basisof !RI . Then the correspondence r(cj
%fj) -+ Xj XjPj , $(x:i xjfj> -+ Cr %qi , where PI ,...,
PN
, 41 ,..a9 qN are the
Schrodinger operatorsacting in L2(RN) by pj = -;(a/&,), qj = X~extends to a unique *-representation [5] rst, of .%9(‘9&) inL2(RN) by operatorshaving domain 9&P’). Then JV)
= tr m,(4,
(5.3)
122
COCKROFT
AND
HUDSON
where p is the density operator in L2(BP’) such that
(It can be shown that IT%, extends uniquely to a traceable operator with domain L2(KC”)). As in the classicalcase,we let wrw be the linear map
from 9(%) @ 9Y(%) to &Y(s), and we set W*” Lemmas1 and 2 we have
= ~~~7%. Then, gene&sing
THEOREM2. adjoint
fw
The mappings Wkg are inverses of each other and are mutually the pre-inner product determined by u; that is, for A, B E S?(%),
(1)
W-“W+%A
= W+“WeRA
(2)
w%(( W-%A)B*)
= A,
= wR(A( W+“B)*).
Proof. Every pair A, B E@(%) belongsto 9(%r) for somefinite-dimensional subspace‘$ of S?,and if L is the identity injection of @(%J into 9?(s) we have
lW3 = wp1, wwL=w wx. It is therefore sufficient to prove the theorem under the additional hypothesis that % is finite dimensional. We fix an orthonormal basis(fi ,..., fN) for !%.The correspondence
7r( C xjfj 1 *
Cx5P$ 9
(b (1 xLh) + C xi4j P
where p, = 1 @ ..a @ $ @ ... @ 1, qj = 1 @ ..* @ ‘G @ ... @ 1 extends uniquely to a *-isomorphism 5 from a(%) onto the algebraic tensor product @,tl c@,where 9 is the canonicalpolynomial algebraof Section 4 for one degree of freedom, with generatorsp, q satisfyingpq - qp = -i. Using (5.3) and (5.4), it is easily seenthat (5.5)
QUANTUM
MECHANICAL
WIENER
PROCESSES
123
where w is the normal state with mean zero and variance u2 of g of Section 4, and that
W*R= 5-l & wi 5, ( j=l 1
(5.6)
where W* are the mappings from 99 to itself of Section 4. Using (5.6) we can now write, for arbitrary A E g’(s) W-3 W+RA = 5-l & (W-W,)
[A = A,
j=l
making use of Lemma 1. A similar argument shows that W+‘RW-RA = A. Thus (1) is established. If A, B E a’(%) are product elements, of form 5-l &, Aj , 5-l @Tsl Bj , respectively, we have, using (5.5) and (5.6), wx(( W_‘RA)B*)
= n w(( W-A,)&.*) j=l = fi w(A,(W+B,)*) j=l
= w%(A( W+%B)*) making use of Lemma 2. Since arbitrary A, BE B(%) can be expressed as 1 linear combinations of product elements, (2) is proved. Thus, as in the case of one degree of freedom, W&s is a pre-isometric operator in the pre-inner product space obtained by equipping g!(g) with the pre-inner product {A, B) = wR(AB*) (5.7) and lV-% is its inverse. The canonical Wiener transformation over R can now be defined as the unitary extension fl to the completion of the isometric operator induced by W+W in the quotient inner product space of .%Y(%)by the null space of pre-inner product (5.7).
REFERENCES [l]
ARAKI, H. AND WOODS, E. J. (1963). Representations of the canonical commutation relations describing the equilibrium states of the free Bose gas. J. Math. Phys. 4 637-662. [2] CAMERON, R. H. AND MARTIN, W. T. (1947). Fourier-Wiener transforms of functionals belonging to L, over the space C. Duke Math. J. 14 99-107. [3] COCKROFT, A.M., GUDDER, S. P., AND HUDSON, R. L. (1977). A quantum-mechanical functional central limit theorem. J. Multiaariate Anal. 7 12.5-148.
124 [4]
[5] [6] [7] [8]
[9]
COCKROFT
AND
HUDSON
CUSHEN, C. D. AND HUDSON, R. L. (1971). A quantum-mechanical central limit theorem. J. Appl. Prob. 8 454-469. POWERS, R. T. (1971). Self-adjoint algebras of unbounded operators. C’ovrv~,lr~rz. Math. Phys. 21 85-124. SEGAL, I. E. (1956). Tensor algebras over Hilbert spaces I. Trans. Amer. Math. Sot. 81 106-l 34. SEGAL, I. E. (1958). Distributions in Hilbert space and canonical systems of operators. Trans. Amer. Math. Sot. 88 12-40. SEGAL, I. E. (1962). Mathematical characterization of the physical vacuum. Ill. J. Math. 6 500-523. WORONOWICZ, S. L. (1970). The quantum problem of moments. Rep. Math. Phys. 1 135-145.