Some Properties of Quantum Lévy Area in Fock and Non-fock Quantum Stochastic Calculus.

We consider the analogue of Lévy area, de…ned as an iterated stochastic integral, obtained by replacing the two independent component onedimensional Brownian motions by the mutually non-commuting momentum and position Brownian motions P and Q of either Fock or non-Fock quantum stochastic calculus, which are also stochastically independent in a certain sense. We show that the resulting quantum Lévy area is trivially distributed in the Fock case, but has non-trivial distribution in non-Fock quantum stochastic calculus. We also show that it behaves di¤erently from the classical Lévy area under a kind of time reversal, in both the Fock and non-Fock cases. 1. Introduction. Lévy’s stochastic area [7] can be de…ned in terms of iterated stochastic integration as 1 2 Z 0 x 1 called the variance. In the tensor product H H, where H denotes the Hilbert dual space of H, de…ne operators W (f) =W ( p 2+1 2 f) W p 2 1 2 f ; f 2 h, where for a bounded operator K on H, K denotes the operator K = (K ) and denotes the element 7! h ; i of H. Then the family W = (W (f); f 2 h) satis…es the ccr (2.1). Moreover in the state = ; the processes P and Q got by replacing W in (2.2) by W are Brownian motions of variance 2 in the sense that instead of (2.3) we have (2.5) E [exp(ix(P (t)] = E [exp(ixQ (t)] = exp t 2 2 x : We shall …nd another realisation [1] of the non-Fock quantum Brownian motions P and Q useful. Denote by C;F ;P the standard Wiener space realisation of two dimensional standard classical Brownian motion, so that C is the space of continuous R-valued functions ! = (!1; !2) on R with !(0) = 0; F is the -…eld of subsets of C generated by the evaluations X(t; !) = !1(t); Y (t; !) = !2(t); ! = (!1; !2) 2 C; and P is two-dimensional Wiener measure, which makes the processes X and Y into independent standard unit-variance Brownian motions. In the Hilbert space tensor product H = H L(C;F ;P); equipped with the unit vector 1 where 1 is the constant function 1(!) = 1 on C we may de…ne (2.6) P (t) = P (t) I + p ( 2 1)I mult X(t) (2.7) Q (t) = Q(t) I + p ( 2 1)I mult Y (t) wheremultF denotes the operator of multiplication by the function F on L(C;F ;P): More rigorously we may de…ne the corresponding Weyl operators as exp (ix(P (t)) = exp (ix(P (t)) exp ix p ( 2 1)mult X(t) exp(iyQ (t)) = exp(iyQ(t)) exp ix p ( 2 1)mult Y (t) : (2.8) Then it can be veri…ed that both P and Q again satisfy (2.5) where now is the unit vector 1C2 ; as well as the commutation relations (2.4). In the rest of the paper we shall denote by the same symbol (P;Q) the pair consisting either of the Fock momentum and position Brownian motions or the pair (P ; Q ) as de…ned in either of the alternative ways above, making clear which is intended where necessary. The same symbol E will similarly denote either E or E : Because of their mutual noncommutativity, in orthodox quantum theory it is not possible to measure P and Q simultaneously and it is therefore meaningless from the point of view of quantum physics to speak of their stochastic independence. Nevertheless they retain a property that in classical probability is tantamount to 4 SHANG CHEN AND ROBIN HUDSON independence, namely factorization of joint characteristic functions, in the sense of Theorem 1 which follows below. We must …rst give a rigorous de…nition, for arbitrary real numbers 1; 2; :::; m and 1; 2; :::; n; and nonnegative mumbers s1; s2; :::; sm and t1; t2; :::; tn; of the unitary operator exp n i Pm j=1 jP (sj) + Pn k=1 kQ(tk) o : It is de…ned to be the value W (1) of the one parameter unitary group (Wx)x2R ; where