We study generalized Dyson series and their representation by
generalized Feynman diagrams as well as the closely related topic of
Feynman's time-ordered operational calculus for noncommuting operators.
These perturbation series are obtained by replacing ordinary Lebesgue
measure in the time integration involved in the Feynman-Kac functional by
an arbitrary Lebesgue-Stieltjes measure; we then calculate the Wiener
and Feynman path integrals of the corresponding functional. Our Dyson
series provide a means of carrying out the "disentangling" which is a
crucial element of Feynman's operational calculus. We are also able to
treat far more general functionals than the traditional exponential func-
tional; in fact, the class of functionals dealt with forms a rather large
commutative Banach algebra.
An intriguing aspect of the present theory is that it builds bridges
between several areas of mathematical physics, operator theory and path
integration. Combinatorial considerations permeate all facets of this
1980 Mathematics Subject ClassIfteation. Primary 05A99, 28C20,
47A55, 58D30, 81C12, 81C20; Secondary 05C99, 26A42, 28A33, 44A99, 46J99,
47D05, 47D45, 81C30, 81C35, 82A99.
Key words and phrases. Wiener measure, Wiener functional, path
integration, Feynman path integral, Feynman diagrams, Dyson series,
perturbation series, time-ordering, disentangling of noncommuting
operators Feynman's operational calculus, Banach algebras, functional