GENERALIZED DYSON SERIES AND FEYNMAN'S OPERATIONAL CALCULUS 3

generalized Feynman graphs, and the reader may find it helpful, after a

brief look at Section 5, to draw such graphs while following the proofs

and examples in Sections 1-4.

Different situations may lead to generalized Dyson series and Feynman

graphs with diverse combinatorial structures as is illustrated in Section

3 where we present miscellaneous examples. We remark that, up to this

point, we have discussed only cases involving a single measure n and a

single potential 6. However, we can, for example, treat functionals formed

by composing an analytic function of several complex variables with func-

tions of the form (0.2). We mention in particular Example 3.6 which we

use to make explicit some of the ties with Feynman's operational calculus.

Feynman's time-ordered operational calculus, introduced in [11],is

based on the interesting observation that noncommuting operators A and B

can be treated as though they commuted; a time index is attached to them

to indicate the order of operation. More specifically, Feynman suggests

writing

r

CO.4) A(Sl)B(s2)

BA if s- , s«

AB if s2 s-.

^undefined if s. , = s2.

One then performs the desired calculations just as if A and B were com-

muting. Eventually one wants to restore the conventional ordering of the

operators; Feynman refers to this as "disentangling". He says [11, p.110]:

"The process is not always easy to perform and, in fact, is the central

problem of this operator calculus".

Our generalized Dyson series provide a means of carrying out this

disentangling process for a rather large class of operators. It is the

use of path integration that enables us to accomplish this. Some possible

relations with path integration were already suggested in Feynman's

paper [11, p. 108 and Appendices A-C, pp. 124-127] and in the book of

Feynman and Hibbs [12, pp. 355-356].

We note that we will, for example, be integrating expressions similar

to the left-hand side of (0.4) over a square (0,t) * (0,t), and, when this

is done with respect to measures with nonzero discrete part, one cannot