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Progress of Theoretical Physics

Remarks on the Time Transformation of Path-Integral Measures

Reijiro Kubo

Published Feb 1, 1987• 0 citations
Publisher

Abstract

By means of a time transformation of path-integral measures we rewrite the usual Feynman path integral as a path integral of determinants with the Wiener measure. We derive a formula, which is useful in evaluating path integrals.
DOI
10.1143/ptp.77.201

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(and its generalisations that include magnetic fields) can be roughly divided into two classes. In the aproaches of the first class that we shall not discuss here in detail the Feynman integral is not supposed to be a genuine integral, but is specified as some generalised functional on an appropriate space of functions, which can be defined, for example, as the limit of certain discrete approximations (see e.g. [16], [44] and more recent papers [24], [25], [34]), by means of analytical continuation (see e.g. [26], [23] and references therein), by extensions of Parceval’s identity and by related axiomatic definitions (see [2], [5], [30], [11], [42] and references therein) or by means of the white noise analysis (see [22]), see also [46] for the discussion of path integral applied to the Dirac equation. These approaches still cover only a very restrictive class of potentials, for example, singular potentials were considered only by white noise analysis approach but only in one-dimensional case (see e.g. [6] and references therein). In the approaches of the second class, one tries to define the infinite-dimensional Feynmann integral as a genuine integral over a bona fide σ-additive measure on an appropriate space of trajectories. The first attempt made in [19] to construct such a measure was erronious and led to understanding that there is no direct generalisation of Wiener measure that can give an analog of FeynmanKac formula for the case of Schrodinger operators. A correct construction of the Feynman integral in terms of the Wiener measure was proposed in [15] and was based on the idea of the rotation of the classical trajectories in complex
We are concerned with rigorously defined, by time slicing approximation method, Feynman path integral ∫Ωx,yF(γ)eiνS(γ)D(γ)\int_{\Omega_{x,y}} F(\gamma) e^{i\nu S(\gamma)} {\cal D}(\gamma)∫Ωx,y​​F(γ)eiνS(γ)D(γ) of a functional F(γ)F(\gamma)F(γ), cf. [13]. Here Ωx,y\Omega_{x,y}Ωx,y​ is the set of paths γ(t)\gamma(t)γ(t) in Rd^dd starting from a point y∈y \iny∈ Rd^dd at time 000 and arriving at x∈x\inx∈ Rd^dd at time TTT, S(γ)S(\gamma)S(γ) is the action of γ\gammaγ and ν=2πh−1\nu=2\pi h^{-1}ν=2πh−1, with Planck's constant hhh. Assuming that p(γ)p(\gamma)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_{\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(γ)[p(γ)]DF(\gamma)[p(\gamma)]DF(γ)[p(γ)] and DS(γ)[p(γ)]DS(\gamma)[p(\gamma)]DS(γ)[p(γ)] are differentials of F(γ)F(\gamma)F(γ) and S(γ)S(\gamma)S(γ) evaluated in the direction of p(γ)p(\gamma)p(γ), respectively, and Div p(γ){\rm Div}\, p(\gamma)Divp(γ) is divergence of vector field p(γ)p(\gamma)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∗)=0F(\gamma^*)=0F(γ∗)=0. Here γ∗\gamma^*γ∗ is the stationary point of the phase S(γ)S(\gamma)S(γ).