The 2021 paper "Geometrical interpretation of the photon position operator with commuting components" (Dobrski, Przanowski, Tosiek and Turrubiates; preprint title "The geometrical interpretation of the photon position operator") derived Hawton's photon position operator X⃗^\hat{\vec{X}} with commuting components as a flat connection in momentum space and closed by posing three open problems: (i) a compact distributional form of the configuration-space eigenfunctions, (ii) a detailed understanding of the properties of those eigenfunctions, and (iii) a reformulation of the photon Weyl–Wigner–Moyal (phase-space) formalism in which the operator pair (p⃗^,X⃗^)(\hat{\vec{p}},\hat{\vec{X}}) replaces the interpretationally problematic (p⃗^,x⃗^)(\hat{\vec{p}},\hat{\vec{x}}). Problem (i) has been solved and problem (ii) largely settled by González-López and Martínez Alonso (Phys. Rev. A 114, 022204 (2026)), who derived closed-form elliptic-integral expressions for the configuration-space position eigenfunctions together with their singular-plane structure and asymptotics. The surviving open core is problem (iii): construct, or rigorously exclude, a phase-space formulation of single-photon quantum mechanics whose canonical pair is the momentum and the genuine commuting position operator X⃗^\hat{\vec{X}}, with a Wigner function, star product and Moyal dynamics respecting the transverse two-helicity structure of photon states. The only existing photon Wigner function (Przanowski–Tosiek–Turrubiates 2020) is built on the Fourier-conjugate x⃗^\hat{\vec{x}}, which for a relativistic particle does not represent a position observable; no photon Wigner function built on the position operator X⃗^\hat{\vec{X}} is known.

Background

Photon wave mechanics is the first-quantized quantum mechanics of a single photon, in which the photon is described by a wave function — most commonly the Riemann–Silberstein vector F=E+iB\mathbf{F}=\mathbf{E}+i\mathbf{B}, a complex combination of the electric and magnetic fields that packages the vacuum Maxwell equations into a Dirac-like equation. A long-standing obstruction is the position observable: Newton and Wigner showed in 1949 that massive particles admit localized position states, but massless particles with spin do not, and the early candidate of Pryce (1948) has component operators that do not commute with each other, so it cannot define a sharp photon position. In 1999 Hawton constructed a photon position operator X⃗^\hat{\vec{X}} whose three components do commute, which satisfies the canonical commutation relations [X^j,k^l]=iδjl[\hat{X}_j,\hat{k}_l]=i\delta_{jl} with the wave-vector operator, commutes with the helicity operator (the projection of spin onto the momentum direction, the actual spin degree of freedom of a photon), and is Hermitian with respect to the Białynicki–Birula scalar product ⟨Φ~∣Ψ~⟩BB=∫d3k ((2π)3∣k⃗∣)−1Φ~†(k⃗)Ψ~(k⃗)\langle\widetilde{\Phi}|\widetilde{\Psi}\rangle_{BB}=\int d^3k\,((2\pi)^3|\vec{k}|)^{-1}\widetilde{\Phi}^\dagger(\vec{k})\widetilde{\Psi}(\vec{k}) appropriate to the Riemann–Silberstein wave function. Its momentum-space eigenfunctions are transverse polarized plane waves; in configuration space they are singular distributions, not spherically symmetric. Dobrski, Przanowski, Tosiek and Turrubiates (2021) gave this operator a geometric reading: X⃗^=iD⃗^\hat{\vec{X}}=i\hat{\vec{D}}, where D⃗^\hat{\vec{D}} is a flat connection (a covariant derivative with zero curvature) in the tangent bundle of momentum space with the positive k3k_3 half-axis removed, a structure closely analogous to the Wu–Yang fiber-bundle treatment of the Dirac magnetic monopole; the unavoidable string singularity can be removed by a vector-bundle construction, as those authors showed in a 2023 follow-up. Separately, the Weyl–Wigner–Moyal (WWM) formalism is the phase-space formulation of quantum mechanics: states are represented by Wigner functions on a classical phase space, observables by Weyl symbols, operator composition by the noncommutative star product, and dynamics by the Moyal bracket. Przanowski, Tosiek and Turrubiates (2020) constructed a photon Wigner function on the phase space of the pair (p⃗^,x⃗^)(\hat{\vec{p}},\hat{\vec{x}}), where x⃗^\hat{\vec{x}} is the Fourier conjugate of the momentum; but, as those authors themselves stressed, for a relativistic particle x⃗^\hat{\vec{x}} does not represent a position observable, so the physical interpretation of that phase space is compromised. This tension — a well-defined commuting photon position operator exists, yet the existing photon phase-space formalism cannot use it — is what motivates the open problem.

Problem Statement

Can the Weyl–Wigner–Moyal (phase-space) formalism for the photon be reformulated so that the canonical pair underlying the phase space is (p⃗^,X⃗^)(\hat{\vec{p}},\hat{\vec{X}}) — the photon momentum together with the Hawton-type position operator with commuting components — in place of the conventional (p⃗^,x⃗^)(\hat{\vec{p}},\hat{\vec{x}}), whose position entry is the Fourier conjugate of momentum and, for a relativistic particle, does not represent a position observable? This is the open question posed in the Summary of Dobrski, Przanowski, Tosiek and Turrubiates (Phys. Rev. A 104, 042206 (2021)), reformulated here as the surviving core of that paper's three acknowledged open problems. A complete answer either (a) constructs such a formalism: a Weyl correspondence assigning symbols to observables and a Wigner function W(k⃗,X⃗)W(\vec{k},\vec{X}) for photon states on the (k⃗,X⃗)(\vec{k},\vec{X}) phase space, respecting the transversality constraint k⃗⋅Ψ~(k⃗)=0\vec{k}\cdot\widetilde{\Psi}(\vec{k})=0 and the two helicity sectors of the photon state space and the Białynicki–Birula scalar product, together with the associated star product reproducing operator composition (including the full operator content of X⃗^\hat{\vec{X}}: commuting components, canonical commutation with momentum, helicity compatibility), a Moyal-bracket equation of motion reproducing free photon propagation under H^=cℏ∣k⃗∣\hat{H}=c\hbar|\vec{k}|, and, optionally, an interpretation of the position marginals of WW that reproduces the Białynicki–Birula energy density — the algebraic construction just listed is the mandatory core of (a), while the marginal interpretation is optional, and if it is supplied its localization convention must be stated explicitly — or (b) establishes a rigorous no-go theorem showing that no such reformulation with these mandatory properties exists, identifying exactly which requirement fails. The construction must genuinely use the position operator X⃗^\hat{\vec{X}} with its operator content, not a relabeled Fourier-conjugate variable.

References

  1. Michal Dobrski, Maciej Przanowski, Jaromir Tosiek, Francisco J. Turrubiates, Geometrical interpretation of the photon position operator with commuting components (preprint title: The geometrical interpretation of the photon position operator), Physical Review A 104, 042206 (2021), DOI: 10.1103/PhysRevA.104.042206, arXiv:2104.04351, https://arxiv.org/abs/2104.04351
  2. Artemio González-López, Luis Martínez Alonso, Photon position eigenstates in configuration space, Physical Review A 114, 022204 (2026), DOI: 10.1103/ll57-rj9l, arXiv:2605.25857, https://arxiv.org/abs/2605.25857
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