1. Introduction We begin with a brief background of the basic ideas behind the Feynman integral. Our task is made simpler by the recent appearance of the excellent monograph  by G.W. Johnson and M. Lapidus. A discussion of the Feynman integral in greater depth than is given here is presented in Chapter 7 of  and this isa basic reference at the appropriate points below. As much as possible, the notation of  is used when it is needed. For a quantum particle in one dimension, the evolution of its state is expressed in terms of the continuous unitary group e−itH/ , t ∈ R of operators. The dynamics is speciﬁed by the Hamiltonian operator H. The integral kernel K(x, y, t), x, y ∈ R, of the operator e−itH/ isknown as the propagator of the quantum system, although this may only be a distributional expression. Knowledge of the propagator therefore speciﬁes the evolution of quantum states. The quantization of the dynamics of a classical mechanical system means expressing the quantum propagator in terms of classical quantities. Usually this is done by replacing classical observables such as position andmomentum by their corresponding operator counterparts. By contrast, R. Feynman  tried to write the quantum propagator K(x, y, t) ˙ directly in terms of the classical action S(q, q, t) by means of the formula (1.1) K(x, y, t) = N −1
0,t Here Cx,y is the set of all continuous paths ω such that ω(0) = x and ω(t) = y, N is a 0,t ‘normalization factor’ and Dqis ‘uniform measure’ on Cx,y . For a single particle of mass m on the line with a real valued potential V , the action of a classical path q : [0, t] → R is the expression t
S(q, q, t) = ˙
1 m q(s)2 − V (q(s)) ds. ˙ 2
An attempt can be made to interpret formula (1.1) by taking a positive integer n, dividing the interval [0, t] into n equal parts and setting Kn (x, y, t) equalto (1.3) Cn exp
t m (xj − xj−1 )2 − V (xj ) 2(t/n) n
dx1 · · · dxn−1 ,
where x0 = x and xn = y and the normalization constant Cn is given by −im 2π (t/n)
Date: February 3, 2004. 2000 Mathematics Subject Classiﬁcation. Primary 81S40, 58D30; Secondary 46G10, 28B05.
The exponent in the integrand of expression (1.3) is i/times an approximation to the action (1.2) in the case that q is a polygonal path [33, Equation (7.4.1)]. One might hope that the approximations Kn (x, y, t), n = 1, 2, . . . , converge in an appropriate sense to the propagator K(x, y, t). One diﬃculty is that the integrand of expression (1.3) has absolute value one, so the integral is not absolutely convergent. Neverthless, Kn (x, y, t) can be viewedas the kernel of an operator which converges in the strong operator topology as n → ∞ for a large class of potentials V [33, Theorem 7.5.1]. The limiting operator is e−itH/ for the associated quantum Hamiltonian operator H. The reinterpretation of formula (1.1) in terms of limits of operator products was ﬁrst given by E. Nelson  and is actually part of a more general theory of theapproximation of semigroups of operators [33, Chapter 11]. A mathematical physicist may simply declare that the story ended with the paper , but formula (1.1) has continued to inspire a great deal more mathematics, a part of which has a detailed exposition in the monograph . Moreover, the heuristic Feynman integral, taking the essential ideas of formula (1.1), has produced useful formulae in lowdimensional topology and knot theory, see [33, Section 20.2] and  for a discussion of these developments. The geometric nature of these applications mean that there is no fundamental time parameter with which to provide subdivisions and form operator products (but see the discussion of the multiplicative property of the heuristic Feynman path integral on [33, p. 648]). In the viewpoint of E....