The photon position operator (non-commuting) and its string-like eigenstates

TL;DR

This paper defines and analyzes the non-commuting photon position operator, introduces its string-like eigenstates called photon strings, and explores their properties and representations in different geometries.

Contribution

It presents three consistent definitions of the photon position operator, introduces photon string eigenstates, and studies their properties in straight line and circular geometries.

Findings

Photon position operator is non-commuting but has eigenstates called photon strings.

Photon strings include electric and magnetic types, with exact solutions on lines and circles.

Radial photon position operators are non-equivalent but transform into each other in the helical representation.

Abstract

The paper provides three main definitions of the Cartesian photon position operator based on: boost generator, the transversality condition and the helicity operator. In each case, the correctness of the definition and Hermitianness of the operator in the domain of physical states are proven. All considered definitions lead to the same form of the Cartesian position operator in the domain of physical states. Radial photon position operators were also defined, but they turned out to be non-equivalent. Nevertheless, the most useful two radial operators turned out to be twin counterparts in the sense of the transformation to the helical representation, which is an alternative positional representation. The components of the photon position operator do not commute, but in analogy to the problem of eigenangular momentum, its eigenstates do exist. The eigenstates of the two components of the position operator (including the radial component) are called photon strings. Photon strings on a straight line and on a circle have been studied. The usual, previously known photon string states were named electric strings, but new magnetic strings were also introduced. Exact helical photon strings on a straight line, as well as hybrid helical photon strings on a circle, were also considered. On the other hand, infinitely short photon magnetic strings turned out to be flat photon vortices on planes with a normalization formula looks like as point-localized particles.