Noncommutative operations on Wiener functionals and Feynman's operational calculus

Abstract

The authors recently introduced a family { A t t⩾0} of Banach algebras of functionals on Wiener space and showed that, for every F in A t the functional integrals K λ t ( F) exist and are given by a perturbation expansion which serves to “disentangle,” in the sense of Feynman's operational calculus, the operator K λ t ( F). When λ is purely imaginary, K λ t ( F) is the analytic operator-valued Feynman integral of F. In this paper, noncommutative operations ∗ and + on Wiener functionals are defined which enable the authors to provide a precise and rigorous interpretation of certain aspects of Feynman's operational calculus for noncommuting operators. In particular, the authors show that if FϵA t1 and G ϵ A t2 then F ∗G ϵA t1+t2 and K λ t 1 + t 2 (F ∗ G) = K λ t 1 (F)K λ t 2 (G) . Hence, the product of operators which can be disentangled using their recent results can itself be disentangled. The noncommutative operations, the disentangling algebras { A t}, and the functional integrals possess a rich interlocking structure.