Maxwell quantum mechanics

Abstract

We extend classical Maxwell field theory to a first quantized theory of the photon by deriving a conserved Lorentz four-current whose zero component is a positive definite number density. Fields are real and their positive (negative) frequency parts are interpreted as absorption (emission) of a positive energy photon. With invariant plane wave normalization, the photon position operator is Hermitian with instantaneously localized eigenvectors that transform as Lorentz four-vectors. Reality of the fields and wave function ensure causal propagation and zero net absorption of energy in the absence of charged matter. The photon probability amplitude is the real part of the projection of the photon's state vector onto a basis of position eigenvectors and its square implements the Born rule. Manifest covariance and consistency with quantum field theory is maintained through use of the electromagnetic four-potential and the Lorenz gauge.