Daisuke Fujiwara

Recently N. Kumano-go [15] succeeded in proving that piecewise linear time slicing approximation to Feynman path integral ∫ F(γ)e ivS(γ) script D[γ] actually converges to the limit as the mesh of division of time goes to 0 if the functional F(γ) of paths γ belongs to a certain class of functionals, which includes, as a typical example, Stieltjes integral of the following form F(γ) = ∫0T f(t, γ(t))ρ(dt), (1) where ρ(t) is a function of bounded variation and f(t,x) is a sufficiently smooth function with polynomial growth as |x| → ∞. Moreover, he rigorously showed that the limit, which we call the Feynman path integral, has rich properties (see also [10]). The present paper has two aims. The first aim is to show that a large part of discussion in [15] becomes much simpler and clearer if one uses piecewise classical paths in place of piecewise linear paths. The second aim is to explain that the use of piecewise classical paths naturally leads us to an analytic formula for the second term of the semi-classical asymptotic expansion of the Feynman path integrals under a little stronger assumptions than that in [15]. If F(γ) ≡ 1, this second term coincides with the one given by G. D. Birkhoff [1].

Recently N. Kumano-go [15] succeeded in proving that piecewise linear time slicing approximation to Feynman path integral integral F(gamma)e(i nu(gamma))D[gamma] actually converges to the limit as the mesh of division of time goes to 0 if the functional F(gamma) of paths gamma belongs to a certain class of functionals, which includes, as a typical example, Stieltjes integral of the following form; F(gamma) = integral(T)(o) f (t, gamma(t))rho(dt), (1) where rho(t) is a function of bounded variation and f (t, x) is a sufficiently smooth function with polynomial growth as \x\ --> infinity. Moreover, he rigorously showed that the limit, which we call the Feynman path integral, has rich properties (see also (10]). The present paper has two aims. The first aim is to show that a large part of discussion in [15] becomes much simpler and clearer if. one uses piecewise classical paths in place of piecewise linear paths. The second aim is to explain that the use of piecewise classical paths naturally leads us to an analytic formula for the second term of the semi-classical asymptotic expansion of the Feynman path integrals under a little stronger assumptions than that in [15]. If F(gamma) = 1, this second term coincides with the one given by G. D. Birkhoff [1].

In discussing time slicing approximations of Feymnan's path integrals, one can use stationary phase method for some oscillatory integrals over a space of large dimension. Previously estimates of remainder term of such stationary phase method were given in [2] and [6]. In the present note an improved remainder estimate is given for such stationary phase method. The estimate is sharp enough to give us the second term of semiclassical asymptotic expansion if it is applied to Feynman path integrals of some kind discussed in [6] and [4]. See our forthcoming paper [3].

We give a fairly general class of functionals on a path space so that Feynman path integral has a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of Feynman path integral converges uniformly on compact subsets of the configuration space. Our class of functionals is closed under addition, multiplication, functional differentiation, translation and real linear transformation. The integration by parts and Taylor's expansion formula with respect to functional differentiation holds in Feymnan path integral. Feynman path integral is invariant under translation and orthogonal transformation. The interchange of the order with Riemann-Stieltjes integrals, the interchange of the order with a limit, the semiclassical approximation and the fundamental theorem of calculus in Feynman path integral stay valid as well as N. Kumano-go [Bull. Sci. Math. 128 (3) (2004) 197-251]. (C) 2004 Elsevier SAS. All rights reserved.