Abstract
We are concerned with rigorously defined, by time slicing approximation method, Feynman path integral ∫Ωx,y F(γ) eiν S(γ) $¥cal D$(γ) of a functional F(γ), cf. [13]. Here Ωx,y is the set of paths γ(t) in Rd starting from a point y ∈ Rd at time 0 and arriving at x ∈ Rd at time T, S(γ) is the action of γ and ν = 2πh−1, with Planck's constant h. Assuming that p(γ) is a vector field on the path space with suitable property, we prove the following integration by parts formula for Feynman path integrals:
$¥int$Ωx,yDF(γ)[p(γ)]eiνS(γ) $¥cal D$(γ)
= −$¥int$Ωx,y F(γ) Div p(γ) eiν S(γ) $¥cal D$(γ) − iν $¥int$Ωx,y F(γ)DS(γ)[p(γ)]eiν S(γ) $¥cal D$(γ).
(1)
Here DF(γ)[p(γ)] and DS(γ)[p(γ)] are differentials of F(γ) and S(γ) evaluated in the direction of p(γ), respectively, and Div p(γ) is divergence of vector field p(γ). 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(γ*) = 0. Here γ* is the stationary point of the phase S(γ).
