A theory is developed of product integrals of the form ∏a<s<b ∏c<t<d(1+g[h] (ds,dt)).
are disjoint finite subintervals of +,
is a formal power series in the indeterminate h
whose constant term is zero and whose coefficients are elements of L⊗L,
is the space of basic differentials of a multidimensional quantum stochastic calculus. 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 ∏a<s<b ∏c<t<d(1+g[h](ds,dt))
is the image, obtained by applying the representation J[a,b[⊗J[c,d[
to the coefficients, where J[a,b[
is the representation canonically associated with the interval [a,b[,
of a formal power series ∏ ∏(1+dg[h])
whose coefficients lie in U⊗U,
is the universal enveloping algebra of the Lie algebra L
. It is shown that the naturally conjectured multiplication rule, analogous to the multiplication rule for simple product integrals, holds in the commutative case. © 2000 American Institute of Physics.