Photon position eigenvectors lead to complete photon wave mechanics

TL;DR

This paper constructs a photon position operator with commuting components, leading to a complete photon wave mechanics framework that includes vortex-structured eigenvectors and satisfies Maxwell's equations.

Contribution

It introduces a novel photon position operator with vortex eigenvectors, enabling a consistent wave mechanics for single photons that aligns with quantum mechanics principles.

Findings

Photon position eigenvectors have vortex structures.

Two classes of position eigenvectors lead to different wave functions.

Photon wave functions satisfy Maxwell's equations.

Abstract

We have recently constructed a photon position operator with commuting components. This was long thought to be impossible, but our position eigenvectors have a vortex structure like twisted light. Thus they are not spherically symmetric and the position operator does not transform as a vector, so that previous non-existence arguments do not apply. We find two classes of position eigenvectors and obtain photon wave functions by projection onto the bases of position eigenkets that they define, following the usual rules of quantum mechanics. The hermitian position operator, r0, leads to a Landau-Peierls wave function, while field-like eigenvectors of the nonhermitian position operator and its adjoint lead to a biorthonormal basis. These two bases are equivalent in the sense that they are related by a similarity transformation. The eigenvectors of the nonhermitian position operators lead to a field-potential wave function pair. These field-like positive frequency wave functions satisfy Maxwell's equations, and thus justify the supposition that MEs describe single photon wave mechanics. The expectation value of the number operator is photon density with undetected photons integrated over, consistent with Feynman's conclusion that the density of non-interacting particles can be interpreted as probability density.