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

Michał Dobrski ^1,*, Maciej Przanowski^1,†, Jaromir Tosiek ^1,‡, and Francisco J. Turrubiates ^2,§

¹Institute of Physics, Łódź University of Technology, Wólczańska 217/221, 93-005 Łódź, Poland
²Departamento de Física, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Unidad Adolfo López Mateos, Edificio 9, 07738 Ciudad de México, Mexico

^*michal.dobrski@p.lodz.pl
^†maciej.przanowski@p.lodz.pl
^‡jaromir.tosiek@p.lodz.pl
^§fturrubiatess@ipn.mx

Phys. Rev. A 107, 042208 – Published 11 April, 2023

DOI: https://doi.org/10.1103/PhysRevA.107.042208

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 Białynicki–Birula scalar product, and is 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 $R^{3} ∖ {(0, 0, 0)}$ . This observation enables us to reformulate the quantum mechanics of a single photon on $(R^{3} ∖ {(0, 0, 0)}) \times C^{2}$ .

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

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