6 G.W. JOHNSON and M.L. LAPIDUS
In Section 6, we show that the general class of functionals treated
in Section 2 forms a commutative Banach algebra, and we discuss the
related functional calculus. We finish with a discussion of some
connections with Feynman's operational calculus.
Possible physical interpretations are provided in various places
throughout the paper.
A great variety of Feynman diagrams and perturbation expansions
appear in the physics literature. We should make it clear that we do
not claim here to be generalizing all of those.
Parts of the present paper were announced in .
0.2. Notation and Preliminaries.
In A through I below, we recall some facts and introduce most of the
notation which we will require. With the possible exception of G and I,
we suggest that the reader go over the material quickly and then return
to it if and when it is necessary.
First we mention some general references: For the theory of the
Wiener process and applications of path integration, the reader may wish
to consult [13,14,24,25,46,50]. For semigroup theory, we mention [6,
Chap. 8; 15; 26]; for the theory of the Bochner integral, we refer to the
treatise of Hille and Phillips [15, Chap. III]. Finally, the basic facts
of measure theory used in this paper can be found in [42, §§1.3 and 1.4,
pp. 12-26] and [3,43,49].
A. C, C,, C,: These denote, respectively, the complex numbers, the
complex numbers with positive real part, and the nonzero complex numbers
with nonnegative real part.
B. L (B ) : The space of Borel measurable, C-valued functions \J on
B such that \i\ is integrable with respect to Lebesgue measure on B .
C. L (B ) : The space of Borel measurable, C-valued functions on
B which are essentially bounded.
2 N ° ° N
More formally, the elements of L (B ) and L (B ) are equivalence
classes of functions, with I|J- . and i|)
said to be equivalent if they are
equal almost everywhere (a.e.) with respect to Lebesgue measure.