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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