Photon position eigenvectors, Wigner's little group and Berry's phase

TL;DR

This paper links photon position eigenvectors with the E(2) symmetry of the little group, showing their role in localized states and Berry phase effects in twisted light.

Contribution

It demonstrates the E(2) symmetry in photon position eigenvectors and their basis for localized states with angular momentum, connecting to Berry phase phenomena.

Findings

Photon position eigenvectors exhibit E(2) symmetry.

Eigenvectors form a basis for localized twisted light states.

Rotation of symmetry axis induces Berry phase displacement.

Abstract

We show that the cylindrical symmetry of the eigenvectors of the photon position operator with commuting components, x, reflects the E(2) symmetry of the photon little group. The eigenvectors of x form a basis of localized states that have definite angular momentum, J, parallel to their common axis of symmetry. This basis is well suited to the description of "twisted light" that has been the subject of many recent experiments and calculations. Rotation of the axis of symmetry of this basis results in the observed Berry phase displacement. We prove that {x1,x2,J3} is a realization of the two dimensional Euclidean e(2) algebra that effects genuine infinitesimal displacements in configuration space.