October, 2013 An integration by parts formula for Feynman path integrals
Daisuke FUJIWARA
J. Math. Soc. Japan 65(4): 1273-1318 (October, 2013). DOI: 10.2969/jmsj/06541273

## 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 $S(\gamma)$.

## Citation

Daisuke FUJIWARA. "An integration by parts formula for Feynman path integrals." J. Math. Soc. Japan 65 (4) 1273 - 1318, October, 2013. https://doi.org/10.2969/jmsj/06541273

## Information

Published: October, 2013
First available in Project Euclid: 24 October 2013

zbMATH: 1286.81132
MathSciNet: MR3127824
Digital Object Identifier: 10.2969/jmsj/06541273

Subjects:
Primary: 81S40
Secondary: 35A08 , 46T12 , 58D30 , 81Q20

Keywords: Feynman path integrals , Feynman propagator , integration by parts , quantum mechanics , Schrödinger equation , semiclassical techniques , Wiener integrals