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Reviews in Mathematical Physics

THE PHASE SPACE FEYNMAN PATH INTEGRAL WITH GAUGE INVARIANCE AND ITS CONVERGENCE

Wataru Ichinose

Published Nov 1, 2000• 10 citations
Publisher

Abstract

Convergence of a time-slicing approximation is studied in a general way of the Feynman path integral in phase space with an electromagnetic potential. In the present paper a new approximation of the Feynman path integral in phase space, different from the familiar one in physics, is proposed so that its convergence can be shown independently of the choice of electromagnetic potentials and that gauge invariance of the Feynman path integral defined by its limit can be shown directly. It is also shown that the Feynman path integral in phase space defined above can be expressed in the form of the Feynman path integral in configuration space defined in the familiar way in physics, which Feynman himself stated heuristically.

DOI
10.1142/s0129055x00000630

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There are many ways to give a rigorous meaning to the Feynman path integral. In the present paper especially the method of the time-slicing approximation determined through broken line paths is studied. It was proved that these time-slicing approximate integrals of the Feynman path integral in configuration space and also in phase space converge in L2 space as the discretization parameter tends to zero. In the present paper it is shown that these time-slicing approximate integrals converge in some weighted Sobolev spaces as well. Next as an application of this convergence result in the weighted Sobolev spaces, the path integral representation of correlation functions is studied of the position and the momentum operators. We note that their path integral representation is given in phase space. It is shown that the approximate integrals of correlation functions converge or diverge as the discretization parameter tends to zero. We note that the divergence of the approximate integrals reflects the uncertainty principle in quantum mechanics.
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Abstract. In this paper, we will show the existence and uniqueness of a real-time, time-sliced Feynman path integral for quantum systems with vector potential. Our formulation of the path integral will be derived on the L 2 transition probability amplitude via improper Riemann integrals. Our formulation will hold for vector potential Hamiltonian for which its potential and vector potential each carries at most a finite number of singularities and discontinuities. 1. Introduction. In his paper(see [22] footnote 13), Feynman observed that by using wave functions, ill-defined oscillatory integrals can be given rigorous meaning. We will use Feynman’s observation and use wave functions to provide a convergence factor in the derivation of a real time propagator that takes the form of an L 2 transition probability amplitude and the derivation of a real time, time sliced Feynman
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.