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CRC Press eBooks

Nonstandard Analytical Construction of a Path Space Measure for the 4-D Dirac Equation

Tõru Nakamura, Keiji Watanabe, Hiroshi Ezawa

Published Nov 27, 2020• 0 citations
Publisher

Abstract

Nonstandard analysis is used to construct a measure over paths in momentum space for the path integral solution to the 4-D Dirac equation. In this approach, it is sufficient simply to assign a 4 × 4 matrix to each path in a discretized momentum space over discretized time. The standard part of our measure is the same as the measure given by B.Gaveau 1 .

DOI
10.1201/9781003078296-17

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Nonstandard analysis is used to construct a measure over paths in momentum space for the path integral solution to the 4-D Dirac equation. In this approach, it is sufficient simply to assign a 4 × 4 matrix to each path in a discretized momentum space over discretized time. The standard part of our measure is the same as the measure given by B.Gaveau 1 .
Nonstandard analysis is used to construct a measure over paths for the path integral solution to the Dirac equation in 3+1 dimension. Paths are considered in momentum space, because the Green function in the configuration space contains a derivative of δ function which keeps us from assigning a measure over paths. The solution is obtained not only as the standard part of a nonstandard path sum with respect to a nonstandard measure, but also as a standard path integral with respect to a standard measure extracted from the nonstandard one. The result is an extension of Gaveau’s work [B. Gaveau, J. Funct. Anal. 58, 310–319 (1984)].
A nonstandard path space * -measure is constructed to justify the path integral formula for the Dirac equation in two-dimensional space–time. A standard measure as well as a standard path integral is obtained from it. We also show that, even for the Schrödinger equation, for which there is no standard measure appropriate for a path integral, there exists a nonstandard measure to define a * -path integral whose standard part agrees with the ordinary path integral as defined by a limit from time-slice approximant.
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.