Gauge-Independent Quantum Electrodynamics

Abstract

We discuss a covariant, but not manifestly covariant, form of quantum electrodynamics. No gauge-dependent potentials are introduced as independent (canonical) variables. Only the transverse electromagnetic field is quantized as a photon field. We formulate the theory first in interaction representation, although only flat space-like surfaces $\sigma$ are considered. The interaction operator given by Eq. (10) is then used for describing Lorentz transformations (rotations of $\sigma$) as well as time dependence (parallel progress of $\sigma$). The integrability of the generalized Schrödinger equation is then proved. As we transform to Heisenberg representation the electron wave function $\psi$ loses its spinor character and the transverse photon field $\mathfrak{E}$, $\mathfrak{B}$ its tensor character, but by adding the coulomb field ${\mathbf{E}}_{11}$ to $\mathfrak{E}$ we restore the tensor character of the electromagnetic field. The gauge-independent quantum electrodynamics of Pauli's Handbuch article is a special form of the result thus obtained for the particular case that the number of electrons is known and finite. Our theory has a more general form allowing use of position (hole) theory.