Mathematical Remarks on the Feynman Path Integral for Nonrelativistic Quantum Electrodynamics

TL;DR

This paper rigorously defines the Feynman path integral for nonrelativistic quantum electrodynamics, including cases without constraints, and demonstrates how Coulomb potentials naturally emerge from photon interactions.

Contribution

It introduces a new rigorous mathematical formulation of the Feynman path integral in quantum electrodynamics, extending previous formal definitions and clarifying the role of Coulomb potentials.

Findings

Rigorous definition of the Feynman path integral with constraints.

Extension to unconstrained Feynman path integral.

Demonstration of Coulomb potentials arising from photon interactions.

Abstract

The Feynman path integral for nonrelativistic quantum electrodynamics is studied mathematically of a standard model in physics, where the electromagnetic potential is assumed to be periodic with respect to a large box and quantized thorough its Fourier coefficients. In physics, the Feynman path integral for nonrelativistic quantum electrodynamics is defined very formally. For example, as is often seen, even independent variables are not so clear. First, the Feynman path integral is defined rigorously under the constraints familiar in physics. Secondly, the Feynman path integral is also defined rigorously without the constraints, which is stated in Feynman and Hibbs' book without any comments. So, our definition may be completely new. Thirdly, the vacuum and the state of photons of momentums and polarization states are expressed by means of concrete functions of variables consisting of the Fourier coefficients of the electromagnetic potential. Our results above have many applications as is seen in Feynman and Hibbs' book, though the applications are not rigorous so far. It is also proved rigorously by means of the distribution theory that the Coulomb potentials between charged particles naturally appear in the Feynman path integral above. As is well known, this shows that photons give the Coulomb forth.