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arXiv:math-ph/0012014 · Journal of Mathematical Physics

A rigorous real time Feynman path integral

Ken Loo

Published Jan 1, 1999• 16 citations
arXivHTMLPDF

Abstract

Using improper Riemann integrals, we will formulate a rigorous version of the real-time, time-sliced Feynman path integral for the L2 transition probability amplitude. We will do this for nonvector potential Hamiltonians with potential which has, at most, a finite number of discontinuities and singularities. We will also provide a Nonstandard Analysis version of our formulation.

DOI
10.1063/1.532760

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Using improper Riemann integrals, we will formulate a rigorous version of the real-time, time-sliced Feynman path integral for the L2 transition probability amplitude. We will do this for nonvector potential Hamiltonians with potential which has, at most, a finite number of discontinuities and singularities. We will also provide a Nonstandard Analysis version of our formulation.
We will derive a rigorous real-time propagator for the non-relativistic quantum mechanical L 2 transition probability amplitude and for the non-relativistic wavefunction. The propagator will be given explicitly in terms of the time evolution operator. The derivation will be for all self-adjoint non-vector potential Hamiltonians. For systems with potentials that carry at most a finite number of singularity and discontinuities, we will show that our propagator can be written in the form of a rigorous real-time, time-sliced Feynman path integral via improper Riemann integrals. We will also derive the Feynman path integral in a non-standard analysis formulation. Finally, we will compute the propagator for the harmonic oscillator using the non-standard analysis Feynman path-integral formulation; we will compute the propagator without using any knowledge of the classical properties of the harmonic oscillator.
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
We will extend Nonrelativistic Quantum Mechanics as a theory in $L\sp2$ to a theory in the space of distributions. We will provide 3 major theories of Nonrelativist Quantum Mechanics. First, we will extend the concept of an integral kernel for the evolution operator to a distribution kernel for the $L\sp2$ transition probability amplitude. Second, we will extend the $L\sp2$ Schrodinger's equation to a distributions Schrodinger's equation. Lastly, we will rigorously prove that; Feynman's original formulation of the real time, time-sliced path integral is well defined when formulated on the $L\sp2$ transition probability amplitude.
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.