Which assumption must fail: reconciling a non-zero photon mass with observed TEM propagation in coaxial cables

Steininger, Mieling and Chruściel (Phys. Rev. D 107, 056013 (2023)) proved that the Proca equation — the standard classical model of a massive photon — is incompatible with the transverse-electromagnetic (TEM) modes observed to propagate in coaxial cables, for any non-zero photon mass, under three auxiliary assumptions: charge conservation, a field strength free of Dirac-delta distributions, and ideal (perfectly conducting) conductors. The four statements are mutually incompatible, and the paper shows that Lorentz-covariant mass terms built only from the metric and the medium four-velocity cannot restore TEM modes, while a term involving a preferred spacelike vector along the cable axis can — at the cost of extra structure. The open problem is to settle which assumption must be abandoned: (1) construct, or rule out, a Lorentz-covariant, experimentally viable massive-photon electrodynamics that reproduces observed guided-wave propagation without preferred structures; (2) determine whether relaxing the regularity of the field strength (allowing surface-delta contributions) can coherently restore TEM modes while retaining a well-defined energy-momentum density; or (3) determine whether realistic finite-conductivity conductors evade the spectral obstruction. No resolution of any of these routes is known in the literature to date: the later works either move to boundary-free settings or to dielectric fibres, where the no-go does not arise.

Background

Whether the photon has a non-zero rest mass is a long-standing question of classical electrodynamics and particle physics. The standard classical model of a massive photon replaces Maxwell's equations with Proca's equation,

\eta^{\alpha\beta}\partial_\alpha F_{\beta\gamma}-\mu^2 A_\gamma=-\eta_{\gamma\alpha}j^\alpha,\qquad F_{\alpha\beta}=\partial_\alpha A_\beta-\partial_\beta A_\alpha,

where $\eta_{\alpha\beta}=\mathrm{diag}(-1,+1,+1,+1)$ is the Minkowski metric, $A_\alpha$ the vector potential, $j^\alpha$ the conserved electric current, and $\mu$ the photon mass (the Particle Data Group bound $\mu<10^{-18}\,\mathrm{eV}/c^2$ is quoted in the source paper). Because $F_{\alpha\beta}$ is antisymmetric, charge conservation $\partial_\alpha j^\alpha=0$ implies the Lorenz-type constraint $\eta^{\alpha\beta}\partial_\alpha A_\beta=0$ .

A coaxial cable consists of an inner conductor, a dielectric (often near-vacuum) annulus, and an outer conducting shield; its principal propagating mode is the transverse-electromagnetic (TEM) mode, in which both electric and magnetic fields are transverse to the direction of propagation and the observed dispersion relation is $\omega=\pm\beta$ in vacuum ( $\beta$ the propagation constant, $\omega$ the angular frequency), or $n\omega=\pm\beta$ in a dielectric of refractive index $n$ . In 1971, Goldhaber and Nieto proposed a program to bound the photon mass using transmission-line methods, and Kroll argued in the same issue that the TEM-mode dispersion in such experiments is insensitive to a small photon mass; the claim that guided-wave tests probe $\mu$ entered the photon-mass literature reviewed by Goldhaber and Nieto in 2010.

Steininger, Mieling and Chruściel (2023) overturned this. Modeling the conductors as ideal ( $F_{\alpha\beta}$ vanishing inside, with induced surface currents), the no-Dirac-delta regularity of $F_{\alpha\beta}$ — motivated by the requirement of an everywhere well-defined energy-momentum density, since the energy-momentum tensor involves squares of $F_{\alpha\beta}$ and squares of Dirac deltas are ill-defined — forces the potential to satisfy Dirichlet boundary conditions $A_\alpha|_{\partial U}=0$ on the annular cross-section $U$ . Fourier decomposition then reduces the problem to the eigenvalue problem of the transverse Laplacian on $U$ with Dirichlet data, whose smallest eigenvalue $\xi_0>0$ is of order $r_1^{-2}$ (e.g. $\xi_0\approx 9.75\,r_1^{-2}$ for outer radius $r_2=2r_1$ ), yielding $\omega^2>\beta^2+\xi_0$ , which contradicts the observed $\omega=\pm\beta$ no matter how small $\mu$ is. The argument extends to linear isotropic dielectrics via Lagrangians built on Gordon's optical metric $\gamma_{\alpha\beta}=\eta_{\alpha\beta}+(1-n^2)u_\alpha u_\beta$ (the effective metric encoding light propagation in a medium with refractive index $n$ and four-velocity $u^\alpha$ ), for both natural choices of the mass matrix $m_{\alpha\beta}\in\{\eta_{\alpha\beta},\gamma_{\alpha\beta}\}$ .

The paper's conclusion is that four statements are mutually incompatible: (i) Proca's equation holds, (ii) charge conservation holds, (iii) the electromagnetic field has an everywhere well-defined energy-momentum density, and (iv) ideal conductors are admissible idealizations — while TEM modes are observed. The authors then ask "whether there exist an equation which sidesteps the problem raised above". They analyze the most general Lorentz-covariant mass matrix compatible with time-reversal symmetry and the medium, $m^{\alpha\beta}=\eta^{\alpha\beta}+p\,u^\alpha u^\beta-q\,\ell^\alpha\ell^\beta$ with $\ell^\alpha$ a unit spacelike vector along the cable's symmetry axis: TEM modes survive only for the singular choice $p=q=1$ (only the transverse field components acquire mass, with a residual gauge freedom), which requires the preferred direction $\ell^\alpha$ ; no modification without such extra structure is available. Follow-up work by the same group sharpens the boundary structure of the obstruction: the no-go theorem does not arise in step-index optical fibres (dielectric interfaces, no conductors), where all Proca modes are $O(\mu^2 a^2)$ perturbations of the Maxwell modes, and the Proca Cauchy problem in gravitating dielectric media is well posed away from conducting boundaries. None of this, however, settles which assumption must fail in the experimentally relevant coaxial geometry.

Problem Statement

Given the no-go theorem of Steininger–Mieling–Chruściel, determine which of the mutually incompatible assumptions behind it must be abandoned for a non-zero photon mass to be compatible with the observed TEM propagation in coaxial cables. The common scientific objective is to settle whether any physically acceptable massive-photon theory reproduces observed guided-wave propagation, along one of the three alternative escape routes afforded by the source's list of mutually incompatible assumptions (the source itself explicitly raises the question underlying route (1); routes (2) and (3) relax its assumptions (iii) and (iv), respectively):

(1) Construct a Lorentz-covariant, experimentally viable massive-photon theory (field equations or Lagrangian, together with the induced interface conditions) that reproduces the observed TEM dispersion $\omega=\pm\beta$ in vacuum and $n\omega=\pm\beta$ in dielectrics, to experimental precision, without introducing a preferred spacelike vector field or other extra structure beyond the medium's four-velocity; or prove that no such theory exists.

(2) Determine whether the regularity requirement that $F_{\alpha\beta}$ contain no Dirac-delta distributions can be relaxed — allowing surface-delta contributions at conductor interfaces — while retaining an everywhere well-defined (possibly renormalized) energy-momentum density and thereby coherently restoring TEM modes with $\mu\neq 0$ .

(3) Determine whether replacing the ideal-conductor boundary conditions by a realistic finite-conductivity model (finite conductivity, penetration effects, boundary currents) changes the transverse spectral problem sufficiently to permit TEM-like propagation with $\mu\neq 0$ , for example by deriving and solving the resulting two-region transmission/eigenvalue problem and its $\mu\to 0$ limit.

The three routes are alternatives, not separate deliverables: a complete answer either exhibits one route that provably works — including the resulting theory's consistency with existing photon-mass bounds and with the observed dispersion within experimental precision — or establishes that none of the three routes yields a physically acceptable theory, thereby settling which assumption must be abandoned. Definitively closing only a proper subset of the routes, or an impossibility theorem restricted to a class of theories that leaves open whether a viable theory exists outside it, does not settle which assumption must fail and does not complete the objective.

References

F. Steininger, T. B. Mieling, P. T. Chruściel, 'No Proca photons', Physical Review D 107, 056013 (2023), DOI: 10.1103/physrevd.107.056013, arXiv:2212.12408, https://arxiv.org/abs/2212.12408
F. Steininger, P. T. Chruściel, 'The Cauchy problem for the Proca equation in gravitating dielectric media', Journal of Mathematical Physics 64, 072501 (2023), DOI: 10.1063/5.0156319, arXiv:2305.01563, https://arxiv.org/abs/2305.01563
F. Steininger, 'Proca Fields in Step-index Optical Fibres', arXiv:2302.12729 (2023), https://arxiv.org/abs/2302.12729
A. S. Goldhaber, M. M. Nieto, 'How to Catch a Photon and Measure its Mass', Physical Review Letters 26, 1390–1392 (1971), DOI: 10.1103/PhysRevLett.26.1390, https://doi.org/10.1103/PhysRevLett.26.1390
N. M. Kroll, 'Theoretical Interpretation of a Recent Experimental Investigation of the Photon Rest Mass', Physical Review Letters 26, 1395–1398 (1971), DOI: 10.1103/PhysRevLett.26.1395, https://doi.org/10.1103/PhysRevLett.26.1395
A. S. Goldhaber, M. M. Nieto, 'Photon and graviton mass limits', Reviews of Modern Physics 82, 939–979 (2010), DOI: 10.1103/RevModPhys.82.939, https://doi.org/10.1103/RevModPhys.82.939
S. Saito, 'Dirac equation for photons in a fibre: Origin of polarisation', Heliyon 10, e28367 (2024), DOI: 10.1016/j.heliyon.2024.e28367, https://doi.org/10.1016/j.heliyon.2024.e28367