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Journal of the Mathematical Society of Japan

An integration by parts formula for Feynman path integrals

Daisuke Fujiwara

Published Oct 1, 2013• 9 citations
Publisher

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 $S(\gamma)$.

DOI
10.2969/jmsj/06541273

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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: ∫Ωx,yDF(γ)[p(γ)]eiνS(γ)D(γ)\int_{\Omega_{x,y}}DF(\gamma)[p(\gamma)]e^{i\nu S(\gamma)} {\cal D}(\gamma)∫Ωx,y​​DF(γ)[p(γ)]eiνS(γ)D(γ) =−∫Ωx,yF(γ)Div p(γ)eiνS(γ)D(γ)−iν∫Ωx,yF(γ)DS(γ)[p(γ)]eiνS(γ)D(γ).= -\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).=−∫Ωx,y​​F(γ)Divp(γ)eiνS(γ)D(γ)−iν∫Ωx,y​​F(γ)DS(γ)[p(γ)]eiνS(γ)D(γ). (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).
(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
The phase corresponding to a certain path in the Feynman path integral is derived with the help of the de Broglie hypothesis. Zusammenfassung. Die einem bestimmten Pfad entsprechende Phase im Feynman-Pfadintegral wird mit Hilfe der de Broglie-Hypothese abgeleitet. In the beginning quantum mechanics was built up by analogy with the formulation of classical mechanics in terms of the Hamilton function. In 1933, Dirac [2] announced the idea of formulating quantum mechanics on the basis of the Lagrange function. He started from the analogy between the treatment of classical motion as a canonical transformation and the transformational properties of operators with respect to a time translation. This idea of Dirac was developed further by Feynman [3], who regarded a quantum-mechanical motion from one space-time point to another as a superposition of motions along all possible paths connecting the two points. Each path contributes to the total probability amplitude a term proportional to where is the classical action corresponding to this path and is Planck's constant. Feynman [3] adopted the form of the phase in the expression (1) as a postulate. Another way, however, of deriving the phase is to use the de Broglie hypothesis [1], as shown below. According to this hypothesis, it is possible to associate any particle with a certain frequency proportional to its energy (Hamiltonian function), i.e., and a wave vector proportional to the particle's momentum as Now consider, in the sense of Feynman's formulation, a particle moving along a certain path from point 1 to point 2. During the motion, let and be the radius vectors of the particle at the times t, t+ dt respectively. According to the de Broglie hypothesis, we can associate a plane wave with the particle in the neighbourhood of these points. The phase change of this wave along the particle's path during the motion between the points and consists of space and time phase changes: The term in the parentheses is precisely the value of the Lagrange function at the point . The phase change of the wavefunction corresponding to the whole path can be obtained by integration of the term expressing the infinitesimal phase change: In this way, on the basis of the de Broglie hypothesis, we have reached full agreement with Feynman's postulate: the phase of the probability amplitude corresponding to a certain path is equal to , where is the classical action corresponding to this path. References [1] de Broglie L 1923 Comptes rendus 177 548 - 550 [2] Dirac P A M 1933 Phys. Z. der Sowjetunion 3 64 [3] Feynman R 1948 Rev. Mod. Phys. 2 367
Let us recapitulate. Although the form (4.32) for the path integral was motivated by analogy with finite-dimensional integrals, in no way is it a definition. Rather, as we have stressed repeatedly, it is an algorithm for handling the Feynman perturbation series. Strong coupling apart, all the manipulations of the path integral that we have performed (integration by parts, the introduction of auxiliary fields, the inclusion of background fields) stand in one-to-one correspondence to rearrangements of the perturbation series. In fact, path integrals are unnecessary in the derivation of expressions like (4.50). See Fried (1972), for example. The path integrals are as rigorous – or non-rigorous – as the series, irrespective of whether they can be given proper mathematical definition or not. At the same time, the analogy with finite-dimensional integrals suggests approaches that go beyond this purely operational stance. In particular, because of the dominant role played by the classical action in the path integral formalism we would like to use it to establish the relation of quantum dynamics to classical dynamics. This is the most compelling reason for introducing the path integral formalism, requiring a distinct relaxation in our treatment of it. By formal analogy with finite-dimensional saddle-point and stationary-phase methods, elevating the classical action to the phase of the integrand promotes series expansions in ħ as the natural way to proceed.
Efforts to give an improved mathematical meaning to Feynman's path integral formulation of quantum mechanics started soon after its introduction and continue to this day. In the present paper, one common thread of development is followed over many years, with contributions made by various authors. The present version of this line of development involves a continuous-time regularization for a general phase space path integral and provides, in the author's opinion at least, perhaps the optimal formulation of the path integral.