  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.
