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An integration by parts formula for Feynman path integrals

机译：Feynman路径积分的零件积分公式

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We are concerned with rigorously defined, by time slicing approximation method, Feynman path integral ∫_(Ω_(x,y)) F_(γ)e~(ivS_(γ))D_(γ) of a functional F_(γ), cf. [13]. Here Ω_(x,y) is the set of paths γ(t) in R~d starting from a point y ∈ R~d at time 0 and arriving at x ∈ R~d at time T, S_(γ) is the action of γ and v = 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: ∫_(Ω_(x,y))DF_(γ)[p_(γ)]e~(ivS_(γ))D_(γ)= -∫_(Ω_(x,y))~F_(γ)Div p_(γ)e~(ivS_(γ))D_(γ)- iv∫_(Ω_(x,y))F_(γ)DS_(γ)[p_(γ)]e~(ivS_(γ))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).
机译：我们关注的是通过时间切片逼近方法严格定义功能F_（γ）的Feynman路径积分∫_（Ω_（x，y））F_（γ）e〜（ivS_（γ））D_（γ）， cf. [13]。此处Ω_（x，y）是R〜d中从时间y点y∈R〜d开始到时间T到达x∈R〜d的路径γ（t）的集合，S_（γ）是γ和v =2πh〜（-1）的作用，普朗克常数h。假设p_（γ）是路径空间上具有适当性质的矢量场，我们通过分部公式证明Feynman路径积分的以下积分：∫_（Ω_（x，y））DF_（γ）[p_（γ） ] e〜（ivS_（γ））D_（γ）=-∫_（Ω_（x，y））〜F_（γ）Div p_（γ）e〜（ivS_（γ））D_（γ）-iv∫ _（Ω_（x，y））F_（γ）DS_（γ）[p_（γ）] e〜（ivS_（γ））D_（γ）（1）此处DF_（γ）[p_（γ）]和DS_（γ）[p_（γ）]分别是在p_（γ）方向上求得的F_（γ）和S_（γ）的微分，而Div p_（γ）是矢量场p_（γ）的散度。这个公式类似于Elworthy对Wiener积分的零件积分公式，参见。 [1]。作为一个应用，我们证明了Feynman路径积分的半经典渐近公式，在F_（γ〜*）= 0的情况下，它可以为我们提供清晰的信息。此处，γ〜*是S_（γ）相的固定点。

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