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arXiv:2203.14555

A Comment on the "Photon position operator with commuting components" by Margaret Hawton

Arkadiusz Jadczyk, Agnieszka Matylda Schlichtinger

Published Mar 28, 2022• 1 citation
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Abstract

We show that the position operator with commuting components proposed by M. Hawton [M. Hawton, Phys. Rev. A {\bf 59}, 954 (1999)] and developed in subsequent papers, including the recent ones, does not have the properties required for a photon position operator. Depending on the exact interpretation of the results it either concerns a triplet of massless spin zero particles rather than the photon, or does not have the required covariance properties under space rotations.

DOI
10.48550/arxiv.2203.14555

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Reply to arXiv:2203.14555

Margaret Hawton

This comment refutes the claim made by A. Jadczyk and A.M. Schlichtinger in arXiv:2203.14555 that the photon position operator with commuting components proposed in 1999 does not have the properties required for a photon position operator.

arXiv:2205.01020•May 2, 2022•1 citation

Photon position operators and localized bases

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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.

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We show that the position operator with commuting components proposed by M. Hawton [M. Hawton, Phys. Rev. A {\bf 59}, 954 (1999)] and developed in subsequent papers, including the recent ones, does not have the properties required for a photon position operator. Depending on the exact interpretation of the results it either concerns a triplet of massless spin zero particles rather than the photon, or does not have the required covariance properties under space rotations.
It is shown that the photon position operator X⃗^\hat{\vec{X}}X^ with commuting components introduced by Margaret Hawton can be written in the momentum representation as X⃗^=iD⃗^\hat{\vec{X}}=i \hat{\vec{D}}X^=iD^, where D⃗^\hat{\vec{D}}D^ is a flat connection in the tangent bundle T(R3∖{(0,0,k3)∈R3:k3≥0})T(\mathbb{R}^3 \setminus \{ (0,0,k_3) \in \mathbb{R}^3 : k_3 \geq 0\})T(R3∖{(0,0,k3​)∈R3:k3​≥0}) over R3∖{(0,0,k3)∈R3:k3≥0}\mathbb{R}^3 \setminus \{ (0,0,k_3) \in \mathbb{R}^3 : k_3 \geq 0\}R3∖{(0,0,k3​)∈R3:k3​≥0} equipped with the Cartesian structure. Moreover, D⃗^\hat{\vec{D}}D^ is such that the tangent 222-planes orthogonal to the momentum are parallelly propagated with respect to D⃗^\hat{\vec{D}}D^ and, also, D⃗^\hat{\vec{D}}D^ is an anti-Hermitian operator with respect to the scalar product ⟨Ψ∣H^−2s∣Φ⟩\langle \mathbf{\Psi} | \hat{H}^{-2s} |\mathbf{\Phi} \rangle⟨Ψ∣H^−2s∣Φ⟩. The eigenfunctions ΨX⃗(x⃗)\mathbf{\Psi}_{\vec{X}} (\vec{x})ΨX​(x) of the position operator X⃗^\hat{\vec{X}}X^ are found.
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.
We propose a position operator for the photon. The components commute and commute with helicity, and the two operators form a complete set of commuting observables. However, the simultaneous eigenvectors of this position operator and helicity do not rotate locally. For a normalized wavepacket state vector, the position-helicity amplitudes give a complete characterization of the physical properties of the state vector, yet their moduli-squared have rotation properties that do not have a simple interpretation. Thus we call these pseudo-amplitudes. We show that there is a regime of wavepacket state vectors, those with a small fractional spread around an average momentum, where the position-helicity probability pseudo-densities rotate very nearly locally as scalar functions. This is just the regime that is needed to describe a scattering experiment involving photons. We consider other measures of localization for the photon, including those proposed by other authors. It is important to note that exactly the same problems arise if we try to characterize the state vector of a massive particle with spin by position and helicity. The problems are not the result of the masslessness of the photon but arise because of its limited helicity spectrum.
In order to discuss localization experiments and also to extend the consistent history interpretation of quantum mechanics to relativistic properties, the techniques introduced in a previous paper [J. Math. Phys. 38, 697 (1997)] are applied to the localization of a photon in a given region of space. An essential requirement is to exclude arbitrarily large wavelengths. The method is valid for a particle with any mass and spin. Though there is no proper position operator for a photon, one never needs one in practice. Causality is valid up to exponentially small corrections.