An integration by parts formula for Feynman path integrals

Abstract

We are concerned with rigorously defined, by time slicing approximation method, Feynman path integral $\int_{\Omega_{x,y}} F(\gamma) e^{i\nu S(\gamma)} {\cal D}(\gamma)$ of a functional $F(\gamma)$, cf. [13]. Here $\Omega_{x,y}$ is the set of paths $\gamma(t)$ in R$^d$ starting from a point $y \in$ R$^d$ at time $0$ and arriving at $x\in$ R$^d$ at time $T$, $S(\gamma)$ is the action of $\gamma$ and $\nu=2\pi h^{-1}$, with Planck's constant $h$. Assuming that $p(\gamma)$ is a vector field on the path space with suitable property, we prove the following integration by parts formula for Feynman path integrals:

$ \int_{\Omega_{x,y}}DF(\gamma)[p(\gamma)]e^{i\nu S(\gamma)} {\cal D}(\gamma) $

$ = -\int_{\Omega_{x,y}} F(\gamma) {\rm Div}\, p(\gamma) e^{i\nu S(\gamma)} {\cal D}(\gamma) -i\nu \int_{\Omega_{x,y}} F(\gamma)DS(\gamma)[p(\gamma)]e^{i\nu S(\gamma)}{\cal D}(\gamma). $ (1)

Here $DF(\gamma)[p(\gamma)]$ and $DS(\gamma)[p(\gamma)]$ are differentials of $F(\gamma)$ and $S(\gamma)$ evaluated in the direction of $p(\gamma)$, respectively, and ${\rm Div}\, p(\gamma)$ is divergence of vector field $p(\gamma)$. This formula is an analogy to Elworthy's integration by parts formula for Wiener integrals, cf. [1]. As an application, we prove a semiclassical asymptotic formula of the Feynman path integrals which gives us a sharp information in the case $F(\gamma^*)=0$. Here $\gamma^*$ is the stationary point of the phase $S(\gamma)$.