An integration by parts formula for Feynman path integrals. (English) Zbl 1286.81132

Fujiwara, Daisuke

J. Math. Soc. Japan 65, No. 4, 1273-1318 (2013).

This paper gives a proof of the integration-by-parts relation appearing in the abstract \[ \begin{split} \int\limits_{\Omega_{x,y}} DF\left(\gamma\right)\left[p\left(\gamma\right)\right] e^{i\nu S\left(\gamma\right)} {\mathcal D}\left(\gamma\right)\\ = - \int\limits_{\Omega_{x,y}} F\left(\gamma\right) \mathrm{Div}p\left(\gamma\right) e^{i\nu S\left(\gamma\right)} {\mathcal D}\left(\gamma\right) - i \nu \int\limits_{\Omega_{x,y}} F\left(\gamma\right) DS\left(\gamma\right)\left[p\left(\gamma\right)\right] e^{i\nu S\left(\gamma\right)} {\mathcal D}\left(\gamma\right).\end{split} \] The value of this paper is the careful definition of all quantities and the rigorous proof of the above formula. The authors define the path integral by a time-slicing procedure in eq.(9). The functional \(F(\gamma)\) has to be a \(m\)-smooth, where \(m\)-smoothness is defined in definition (1.1). \(p(\gamma)\) is the tangent vector field to the path \(\gamma\) and \(p(\gamma)\) has to be an \(m'\)-admissible vector field, where \(m'\)-admissibility is defined in definition (3.1). Furthermore the authors require that two out of the three quantities \(DF(\gamma)[p(\gamma)]\), \(F(\gamma) \mathrm{Div}p(\gamma)\) or \(F(\gamma) DS(\gamma)[p(\gamma)]\) are \(F\)-integrable, where the concept of \(F\)-integrability is defined by requiring that the corresponding time-slicing limit of the path-integral exists. In addition, there is the condition \(T \leq \mu\), which relates the path intervall \([0,T]\) and the potential \(V(t,x)\) through eq.(5) and eq.(6). Under these conditions the authors prove the integration-by-parts relation.
The last section of the paper gives an application of the integration-by-parts relation towards the semiclassical asymptotic behaviour of Feynman path integrals.

Reviewer: Stefan Weinzierl (Mainz)

MSC:

81S40	Path integrals in quantum mechanics
35A08	Fundamental solutions to PDEs
46T12	Measure (Gaussian, cylindrical, etc.) and integrals (Feynman, path, Fresnel, etc.) on manifolds
58D30	Applications of manifolds of mappings to the sciences
81Q20	Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics

Keywords:

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

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