On the Quantum-Mechanics of a Single Photon

TL;DR

This paper presents a Lorentz-covariant quantum-mechanical wave equation for a single photon based on a Dirac-type equation, overcoming previous obstacles and ensuring proper probability interpretation and transversality of photon modes.

Contribution

It introduces a novel Dirac-type equation for photon wave functions that is Lorentz-covariant, probabilistically consistent, and automatically transversal, advancing the quantum description of photons.

Findings

Photon wave function yields conserved non-negative probabilities.

Photon momentum and energy satisfy Einstein relations.

Photon modes are automatically transversal without extra constraints.

Abstract

It is shown that a Dirac(-type) equation for a rank-two bi-spinor field on Minkowski (configuration) spacetime furnishes a Lorentz-covariant quantum-mechanical wave equation in position-space representation for a single free photon. This equation does not encounter any of the roadblocks that have obstructed previous attempts (by various authors) to formulate a {quantum-mechanical} photon wave equation. In particular, it implies that the photon wave function yields conserved non-negative Born-rule-type quantum probabilities, and that its probability current density four-vector transforms properly under Lorentz transformations. Moreover, the eigenvalues of the pertinent photon Dirac Hamiltonian and the vector eigenvalues of the photon momentum operator yield the familiar Einstein relations $E=\hbar\omega$ and ${\bf p}=\hbar{\bf k}$, respectively. Furthermore, these spin-1 wave modes are automatically transversal without the need of an additional constraint on the initial data. Some comments on other proposals to set up a photon wave equation are supplied as well.