Journal of the Mathematical Society of Japan

An integration by parts formula for Feynman path integrals


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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

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

First available in Project Euclid: 24 October 2013

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Zentralblatt MATH identifier

Primary: 81S40: Path integrals [See also 58D30]
Secondary: 35A08: Fundamental solutions 46T12: Measure (Gaussian, cylindrical, etc.) and integrals (Feynman, path, Fresnel, etc.) on manifolds [See also 28Cxx, 46G12, 60-XX] 58D30: Applications (in quantum mechanics (Feynman path integrals), relativity, fluid dynamics, etc.) 81Q20: Semiclassical techniques, including WKB and Maslov methods

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


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.

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