ABSTRACT

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

work.

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

calculus.

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