# Maxwell quantum mechanics

## TL;DR

This paper develops a first quantized, Lorentz-covariant quantum theory of the photon based on Maxwell's classical fields, defining a positive-definite photon number density and a Hermitian position operator.

## Contribution

It introduces a novel photon wave function with a conserved four-current, localized eigenvectors, and maintains covariance and causality within a quantum framework.

## Key findings

- Photon position operator is Hermitian with localized eigenvectors.
- Photon probability amplitude is the real part of the projection onto position eigenvectors.
- The theory ensures causal propagation and zero net energy absorption without charged matter.

## 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.