The second term of the semi-classical asymptotic expansion for Feynman path integrals with integrand of polynomial growth

Abstract

Recently N. Kumano-go [15] succeeded in proving that piecewise linear time slicing approximation to Feynman path integral $$\int F\left(\gamma \right)e^{i\nu S\left(\gamma \right)}𝒟\left[\gamma \right]$$actually converges to the limit as the mesh of division of time goes to $0$ if the functional $F\left(\gamma \right)$ of paths $\gamma$ belongs to a certain class of functionals, which includes, as a typical example, Stieltjes integral of the following form; $$F\left(\gamma \right)={\int }_0^{T}f(t,\gamma (t\left)\right)\rho \left(dt\right),\left(1\right)$$ where $\rho \left(t\right)$ is a function of bounded variation and $f(t,x)$ is a sufficiently smooth function with polynomial growth as $\left|x\right|\to \mathrm{\infty }$. 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 $F\left(\gamma \right)\equiv 1$, this second term coincides with the one given by G. D. Birkhoff [1].