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Trends in mathematics

Canonical Photon Position Operator with Commuting Components

Michał Dobrski, Maciej Przanowski, Jaromir Tosiek, Francisco J. Turrubiates

Published Jan 1, 2023• 0 citations
Publisher
DOI
10.1007/978-3-031-30284-8_11

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An examination is made of the consequences for the quantum mechanics of spinning particles of equations characteristic of Lorentz-covariant position variables. These equations are commutator analogs of the Poisson bracket equations that express the familiar transformation properties of space-time events in classical mechanics. For a particle of zero spin it is found that the usual canonical coordinate is the unique solution of these equations. For a particle with positive spin there is no position operator which satisfies these equations and has commuting components. For a particle and antiparticle there is a unique solution with commuting components which is valid for all values of the spin and reduces for zero spin to the canonical coordinate. For spin \textonehalf{} this is the Foldy-Wouthuysen transform of the position operator of the Dirac equation. A generalization of the inverse Foldy-Wouthuysen transformation, valid for any value of the spin, appears as a unique unitary transformation which takes this generalized Dirac position to the canonical coordinate. The application of this transformation to the canonical form of the Hamiltonian gives a generalization of the Dirac equation Hamiltonian. This is developed and compared with the literature for spin 1. It gives a nonlocal equation as the spin 1 analog of the Dirac equation.
We use a new limiting procedure, developed to study quantum-optical phase, to examine canonically conjugate operators in general. We find that Dirac's assumption that photon number and phase should be canonically conjugate variables, similar to momentum and position, is essentially correct. The difficulties with Dirac's approach are shown to arise through use of a form of the canonical commutator which, although the only possible form in the usual infinite Hilbert space approach, is not sufficiently general to be used as a model for a number-phase commutator. The approach in this paper unifies the theory of conjugate operators, which include photon number and phase, angular momentum and angle, and momentum and position as particular cases. The usual position-momentum commutator is regained from a more generally applicable expression by means of a domain restriction which cannot be used for the phase-number commutator.
We extend a procedure for the construction of photon position operators with transverse eigenvectors and commuting components [Phys. Rev. A 59, 954 (1999)] to body rotations described by three Euler angles. The axial angle can be made a function of the two polar angles, and different choices of the functional dependence are analogous to different gauges of a magnetic field. Symmetries broken by a choice of gauge are reestablished by transformations within the gauge group. The approach allows several previous proposals to be related. Because of the coupling of the photon momentum and spin, our position operator, like that proposed by Pryce, is a matrix that does not commute with the spin operator. Unlike the Pryce operator, however, our operator has commuting components, but the commutators of these components with the total angular momentum require an extra term to rotate the matrices for each vector component around the momentum direction. Several proofs of the nonexistence of a photon position operator with commuting components are based on overly restrictive premises that do not apply here.