Symmetrized double quantum stochastic product integrals

A theory is developed of product integrals of the form Pi (a <s <b) Pi (c < t <d)(1+g[h] (ds,dt)). Here [a,b[ and [c,d[ are disjoint finite subinterval s of R+, and g[h] is a formal power series in the indeterminate h whose con stant term is zero and whose coefficients are elements of LxL, where L is t he space of basic differentials of a multidimensional quantum stochastic ca lculus. The product integrals are themselves formal power series in h whose coefficients are finite sums of iterated stochastic integrals against the elements of L. They are symmetrized in such a way that Pi (a <s <b) Pi (c < t <d)(1+g[h](ds,dt)) is the image, obtained by applying the representation J([a,b[)xJ([c,d[) to the coefficients, where J([a,b[) is the representation canonically associated with the interval [a,b[, of a formal power series P i Pi (1+dg[h]) whose coefficients lie in UxU, where U is the universal enve loping algebra of the Lie algebra L. It is shown that the naturally conject ured multiplication rule, analogous to the multiplication rule for simple p roduct integrals, holds in the commutative case. (C) 2000 American Institut e of Physics. [S0022-2488(00)02812-7].