Path space measures for Dirac and Schrödinger equations: Nonstandard analytical approach

Abstract

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.