Construction of a photon position operator with commuting components from natural axioms

TL;DR

This paper constructs a photon position operator with commuting components based on natural axioms, linking it to geometric structures and reformulating single-photon quantum mechanics.

Contribution

It introduces a new photon position operator satisfying natural axioms and relates it to flat connections and vector bundles, advancing photon quantum mechanics theory.

Findings

Photon position operator with commuting components derived

Operator commutes with photon helicity and is Hermitian

Reformulation of single-photon quantum mechanics on a geometric manifold

Abstract

A general form of the photon position operator with commuting components fulfilling some natural axioms is obtained. This operator commutes with the photon helicity operator, is Hermitian with respect to the Bialynicki-Birula scalar product and defined up to a unitary transformation preserving the transversality condition. It is shown that, using the procedure analogous to the one introduced by T. T. Wu and C. N. Yang for the case of the Dirac magnetic monopole, the photon position operator can be defined by a flat connection in some trivial vector bundle over $\mathbb{R}^3 \setminus \{(0,0,0)\}$. This observation enables us to reformulate quantum mechanics of a~single photon on $(\mathbb{R}^{3} \setminus \{(0,0,0)\}) \times \mathbb{C}^2$.