10 G.W. JOHNSON and M.L. LAPIDUS
DEFINITION 0.1. Le_t F be a function from C[0,t] to C. Given X 0,
i| L (R ) and K e m , we consider the expression
(0.13) (K.(F)uO(O = / F(A'1/2x+0^(A"1/2x(t)+Odm(x).
C0
The operator-valued function space integral K.(F) exists for X 0 if
(0.13) defines Kx(F) as an element of dt(L2(RN)) . If, in addition, KX(F),
as a function of X, has an extension (necessarily unique) to an analytic
function on C, and a strongly continuous function on C,, we say that
K.(F) exists for X e C,. When X ±s_ purely imaginary, K (F) ijs _ called the
(analytic) operator-valued Feynman integral of F.
REMARK 0.1. The function F in Definition 0.1 (often referred to as a
"functional" in the physics literature), need not be everywhere defined;
however, in order to have K.(F) defined for all X 0, it must be the
-1/2
case that, for every X 0, F(x ' x+O is defined for m * Lebesgue-a.e.
(x,0 CQ x RN.
Given another function G on C[0,t], we say that F is equivalent to
GCF~ G) if, for every X 0, F(x"1/'2x+C) = G(x""1^2x+^) for m * Lebesgue-
a.e. (x,S) £ CQ x KN. [Note that if F - G and KX(F) exists for
X e C+, then KX(G) exists and Kx(F) = Kx(G) for X e C+.] This equiva-
lence, which may appear strange to begin with, is necessitated by the
pathology of Wiener measure under scale change and the fact that infi-
nitely many scale changes (corresponding to all X 0) are involved here.
See [20] for a discussion of this and related matters.
Interest in the "Feynman integral" stems from Feynman1s 1948 paper
[10] which gave a formula for the evolution of a quantum system in terms
of certain heuristically defined path integrals. Making Feynman's ideas
mathematically rigorous in a useful way has proven difficult. There have
been many approaches taken to this problem; that is, many Feynman in-
tegrals. A good introduction to this topic as well as many further
references can be found in the recent book of Exner [9, esp. Chaps. 5
and 6]. For X purely imaginary, Kx(F), as above, provides one way to
make Feynman's definition precise.
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