Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Phase space Feynman path integrals

Sergio A Albeverio, Giuseppina Guatteri, Sonia Mazzucchi

Published Jun 1, 2002• 36 citations
Publisher

Abstract

A rigorous mathematical formulation of the “phase space Feynman path integral” is given in a general setting. This is then applied to yield a representation of solutions of the Schrödinger equation with potential depending both on the position and momentum variables.
DOI
10.1063/1.1470705

Related papers

A mathematical theory of the Feynman path integral for the generalized Pauli equations

Wataru Ichinose

The definitions of the Feynman path integral for the Pauli equation and more general equations in configuration space and in phase space are proposed, probably for the first time. Then it is proved rigorously that the Feynman path integrals are well-defined and are the solutions to the corresponding equations. These Feynman path integrals are defined by the time-slicing method through broken line paths, which is familiar in physics. Our definitions of these Feynman path integrals and our results give the extension of ones for the Schrödinger equation.

Journal of the Mathematical Society of Japan•Jul 1, 2007•4 citations

Hamiltonian Feynman path integrals via the Chernoff formula

Oleg Georgievich Smolyanov, A. G. Tokarev, Aubrey Truman

The main aim of the present paper is using a Chernoff theorem (i.e., the Chernoff formula) to formulate and to prove some rigorous results on representations for solutions of Schrödinger equations by the Hamiltonian Feynman path integrals (=Feynman integrals over trajectories in the phase space). The corresponding theorem is related to the original (Feynman) approach to Feynman path integrals over trajectories in the phase space in much the same way as the famous theorem of Nelson is related to the Feynman approach to the Feynman path integral over trajectories in the configuration space. We also give a representation for solutions of some Schrödinger equations by a series which represents an integral with respect to the complex Poisson measure on trajectories in the phase space.

Journal of Mathematical Physics•Sep 26, 2002•121 citations

A nonstandard representation of Feynman’s path integrals

Touru Nakamura

A nonstandard path space is constructed that gives the mathematically rigorous formulation for the path integral representation of the fundamental solution to the Cauchy problem for the Dirac equation in (1+1)-dimensional space-time. Nonstandard analysis makes the mathematical concepts elementary, consequently, the procedures to prove theorems are considerably simplified. A difference scheme with infinitesimal spacing is available in determining the probability distribution over the path-space and this method is also available for the heat equation and for the free Schrödinger equation.

Journal of Mathematical Physics•Feb 1, 1991•22 citations

The Feynman Path Integral: An Historical Slice

John R. Klauder

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.

arXiv:quant-ph/0303034•WORLD SCIENTIFIC eBooks•Jul 1, 2003•20 citations

A white noise approach to phase space Feynman path integrals

Wolfgang Böck, M. Grothaus

The concepts of phase space Feynman integrals in White Noise Analysis are established. As an example the harmonic oscillator is treated. The approach perfectly reproduces the right physics. I.e., solutions to the Schrödinger equation are obtained and the canonical commutation relations are satisfied. The latter can be shown, since we not only construct the integral but rather the Feynman integrand and the corresponding generating functional.

arXiv:1012.1125•Theory of Probability and Mathematical Statistics•Jan 11, 2013•9 citations

A White Noise Approach to Phase Space Feynman Path Integrals

Wolfgang Böck, Martin Grothaus

The concepts of phase space Feynman integrals in White Noise Analysis are established. As an example the harmonic oscillator is treated. The approach perfectly reproduces the right physics. I.e., solutions to the Schrödinger equation are obtained and the canonical commutation relations are satisfied. The latter can be shown, since we not only construct the integral but rather the Feynman integrand and the corresponding generating functional.

•Aug 17, 2016•18 citations
The definitions of the Feynman path integral for the Pauli equation and more general equations in configuration space and in phase space are proposed, probably for the first time. Then it is proved rigorously that the Feynman path integrals are well-defined and are the solutions to the corresponding equations. These Feynman path integrals are defined by the time-slicing method through broken line paths, which is familiar in physics. Our definitions of these Feynman path integrals and our results give the extension of ones for the Schrödinger equation.
The main aim of the present paper is using a Chernoff theorem (i.e., the Chernoff formula) to formulate and to prove some rigorous results on representations for solutions of Schrödinger equations by the Hamiltonian Feynman path integrals (=Feynman integrals over trajectories in the phase space). The corresponding theorem is related to the original (Feynman) approach to Feynman path integrals over trajectories in the phase space in much the same way as the famous theorem of Nelson is related to the Feynman approach to the Feynman path integral over trajectories in the configuration space. We also give a representation for solutions of some Schrödinger equations by a series which represents an integral with respect to the complex Poisson measure on trajectories in the phase space.
A nonstandard path space is constructed that gives the mathematically rigorous formulation for the path integral representation of the fundamental solution to the Cauchy problem for the Dirac equation in (1+1)-dimensional space-time. Nonstandard analysis makes the mathematical concepts elementary, consequently, the procedures to prove theorems are considerably simplified. A difference scheme with infinitesimal spacing is available in determining the probability distribution over the path-space and this method is also available for the heat equation and for the free Schrödinger equation.
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.