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 ,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
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