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.