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. v

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