0. INTRODUCTION AND PRELIMINARIES

0.1. Introduction.

Let C = C[0,t] denote the space of continuous functions on [0,t] with

N

values in ]R . In the study of the Feynman-Kac formula and of the Feynman

integral, a particular class of functionals on C[0,t] has been of para-

mount importance:

(0.1) F(y) := exp { / e(s,y(s))ds},

(0,t)

where the "potential" 6 is a complex-valued function on [0,t] x IR . In

this paper, we consider analytic functions f(z) of the functional

(0.2) F,(y) := / e(s,y(s))dn(s),

1 (0,t)

where n belongs to M = M(0,t), the space of complex Borel measures on

(0,t) [3, Chap. 4; 42, pp. 19-23]. We calculate the associated Wiener

integral and, after analytic continuation, obtain the corresponding

Feynman integral. In carrying out the Wiener path integral, it is

advantageous to use the unique decomposition of the measure n, n = y + v,

into its continuous part y and its discrete part v [3, p. 12; 42, p.22].

This decomposition, with appropriate care taken with the time-ordering

and the combinatorics involved, leads to a "generalized Dyson series".

If f(z) = exp(z) and n = y =: &, where i is ordinary Lebesgue measure

on (0,t), the perturbation series, in "real time", is just the classical

Dyson series [8; 45, Chap. 11.f].

The additional flexibility provided by the use of Lebesgue-Stieltjes

measures in this context has many implications, allowing us to broaden

and unify known concepts and to introduce new ones having an interest in

their own right. When n = y is a continuous measure, the generalized

Dyson series has the same formal appearance as in the classical case.

However, even when y is absolutely continuous, very different interpre-

tations are suggested; for example, all the mass could be concentrated

Received by the editors October 10, 1985.

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