Journal of the Mathematical Society of Japan

An integration by parts formula for Feynman path integrals

Daisuke FUJIWARA

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 $S(\gamma)$.

Article information

Source
J. Math. Soc. Japan, Volume 65, Number 4 (2013), 1273-1318.

Dates
First available in Project Euclid: 24 October 2013

https://projecteuclid.org/euclid.jmsj/1382620193

Digital Object Identifier
doi:10.2969/jmsj/06541273

Mathematical Reviews number (MathSciNet)
MR3127824

Zentralblatt MATH identifier
1286.81132

Citation

FUJIWARA, Daisuke. An integration by parts formula for Feynman path integrals. J. Math. Soc. Japan 65 (2013), no. 4, 1273--1318. doi:10.2969/jmsj/06541273. https://projecteuclid.org/euclid.jmsj/1382620193

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