A Rigorous Real Time Feynman Path Integral

where φ, ψ ∈ L, H = −~ 2m ∆+V (~x) is essentially self-adjoint, H̄ is the closure ofH , and φ, ψ, V each carries at most a finite number of singularities and discontinuities. In flavor of physics literature, we will formulate the Feynman path integral with improper Riemann integrals. In hope that with further research we can formulate a rigorous polygonal path integral, we will also provide a Nonstandard Analysis version of the Feynman path integral. Using Nonstandard Analysis is not essential to our formulation, and the idea of using Nonstandard Analysis on the Feynman path integral is not a new concept. For readers interested in Nonstandard Analysis, and its applications to Feynman path integrals, see [1], [10], [13], [19], [22], and references within. We will assume that the reader is familiar with Nonstandard Analysis. In physics, the Feynman path integral is formulated on the propagator and it is formally given by (see [11], [14], and [21])