Algebraic theory of product integrals in quantum stochastic calculus

Motivated by the search for solutions of the quantum Yang-Baxter equation, an algebraic theory of quantum stochastic product integrals is developed. T he product integrators are formal power series in an indeterminate h whose coefficients are elements of the Lie algebra L labelling the usual integrat ors of a many-dimensional quantum stochastic calculus. The product integral s are also formal power series in h, whose coefficients are finite iterated additive stochastic integrals which act on the exponential domain in the F ock space of the calculus and which represent elements of the universal env eloping algebra U of L. They obey a multiplication rule suggested by the qu antum Ito product formula, and are characterized among all such formal powe r series by a grouplike property. (C) 2000 American Institute of Physics. [ S0022-2488(00)02107-1].