1. INTRODUCTION: DEFINITION OF THE SEQUENTIAL FEYNMAN INTEGRAL
It is the purpose of this paper to give a simple sequential definition of the
Feynman integral which is applicable to a rather large class of functionals and which
can be conveniently manipulated. We shall show that our integral has good translation
and rotation properties and permutes with other integrals and sums in a reasonable way.
It is defined as the limit of a sequence of finite dimensional Lebesgue integrals, (or
rather as the common limit of a set of such sequences.) The definition involves no
statistics, functional analysis, analytic continuation, repeated limits, or integrals
in function space.
In order to introduce our definition, we first present some necessary notation.
NOTATION: Let C = C[a,b] be the space of continuous functions x(t) on [a,b] such
v
that x(a)=0 and let C [a,b] = X C[a,b] .
Let a subdivision a of [a,b] be given:
a:[a = T T , T
0
. . . T , . . . T = bJ] .
u o 1 2 k m
Let X = X(t) be a polygonal curve in C based on a subdivision a and the
matrix of real numbers § = {§. . } , and defined by
J k
(1.1) X(t) ^X(t,7,?) = [X^t,a,?),...,X
v
(t,cr,?)]
where
X.(t,a,§) = 3* X K ii£ s-i-
0 T k " T k - l
when T
H
t T
k
, k = l,2,...,m , §
Q
s o .
(We note that as § ranges over all of vm dimensional real space, the polygonal
functions X((,)5°"5) range over all polygonal approximations to the functions inC [a,b]
based on the subdivision a . Specifically if x is a particular element of C [a,b]
and we set §. =X.(T.) , the function X((*)3°"5) is the polygonal approximation of
x based on the subdivision CT .)
Received by the editors June 2U,1981.
Presented to the Society April 16,1982.
1
Previous Page Next Page