Photon Quantum Mechanics

TL;DR

Addresses the problem of localizing photons by promoting EM fields to a covariant quantum theory via second quantization in the Lorenz gauge. It derives a real, normalized single-photon density $ρ_p(x)$ with $∫ ρ_p(x) d^3x = 1$ and defines Schrödinger-like operators for position, momentum, energy, and angular momentum. The approach yields a consistent photon current and four-momentum densities in a manifestly covariant framework and clarifies the role of positive and negative frequencies. This work provides a practical, CPT-consistent route to single-photon quantum mechanics suitable for first-pass context in AI and search tooling.

Abstract

We second quantize the standard electromagnetic Lagrangian with a subsiduary Lorenz gauge constraint to obtain a covariant theory of the discrete excitations of the classical EM field that can properly be called photons. The longitudinal photon number is zero due to cancellation of Gupta-Bleuler like terms. Physical photons are described by a real number density whose spatial integral is unity so it can be interpreted as the probability density to find a photon at position x'. Energy density is not separated into is positive and negative frequency parts so the nonlocal frequency operator is not required.