Maxwell's Equations: New Light on Old Problems
| Scientific Paper | |
|---|---|
| Title | Maxwell's Equations: New Light on Old Problems |
| Read in full | Link to paper |
| Author(s) | David F Roscoe |
| Keywords | Maxwell's Equations |
| Published | 2006 |
| Journal | Apeiron |
| Volume | 13 |
| Number | 2 |
| No. of pages | 34 |
| Pages | 206-239 |
Read the full paper here
Abstract
Maxwell's equations possess a certain generic structural property which is well-known, but rarely discussed. By considering this property as primary, we are able to derive the complete mathematical structure of Maxwell's equations described in terms of the orthogonality properties defined between certain spaces of linear operators. But, we find that the classical theory, whilst recovered intact here, is incomplete in the sense that the recovered Maxwell field is irreducibly associated with an additional massive vector field. In the overall context, this massive vector field can only be interpreted as a manifestation of a classical massive photon. One immediate consequence is that the Lorentz force law must be generalized and can be trivially made perfectly Newtonian once the massive vector field is accounted for.
Overview
D. F. Roscoe, of the School of Mathematics and Statistics at the University of Sheffield, takes an unusual route to a contested conclusion. He does not set out to improve electrodynamics or to fix a known defect; the work was driven, he says, "by a spirit of curiosity concerning the general structure of Maxwell's equations." That structure — which he labels Property A — is that the canonical covariant Maxwell equations can be expressed as identities arising from mutual orthogonality between certain spaces of linear differential operators. Adding Property B, the centrality of the Poincaré group, he asks the general question: what is the complete class of theories possessing both properties?
The answer is that Maxwell's equations are recovered intact, but only as one half of a larger structure. Alongside the electromagnetic field there appears, unavoidably, an additional massive vector field, irreducibly associated with it — "wherever the Maxwell field exists, then so does the massive vector field and vice versa." Roscoe emphasises the logical form of the claim: if one accepts the Maxwell field at all, one must accept the massive field with it. Since a purely classical theory cannot deliver a particle, he interprets the massive field as the classical representation of a massive photon. This distinguishes his approach from earlier massive-photon proposals — those of de Broglie, Bohm and Vigier — where, he notes, the mass "must be 'put in by hand', usually by constructing some variation of standard electromagnetic theory." Here it is a derived consequence rather than an ad hoc addition.
The argument
Maxwell's equations as identities
Roscoe's starting observation is elementary but pointed. Once Fab is defined from the four-potential, the Jacobi identity is automatically satisfied; and since the current is conserved, ∂aFai = 4πJi/c is mutually equivalent to ∂2Fij/∂xi∂xj = 0, which is itself an identity under the definition. So "the covariant formulation of Maxwell's equations can be reduced to a pair of identities" — the physics entering only when the conserved current is identified with the flow of charge. Writing Fab = Pkabφk for linear differential operators P, both identities become algebraic orthogonality statements between operator sets.
The Lagrangian and the eigensystem
Appendix A argues that a Lagrangian producing identities of this kind must be differentially homogeneous and index-interchange invariant, giving a six-constant general density that reduces, after removing redundancy, to L = −∂Ψij/∂xk ∂Ψji/∂xk + λ ∂Ψkj/∂xi ∂Ψik/∂xj for a single free parameter λ. Expanding the sixteen degrees of freedom of Ψab over undetermined operators Ukab converts the Euler–Lagrange system into a manifestly Poincaré-invariant algebraic eigenvalue problem σijXiXjUk = λ2Uk, whose eigenvalues are simple multiples of the d'Alembertian. Because the σ matrices are symmetric, eigenvectors from distinct eigenvalues are orthogonal — and "it is these orthogonality relations which give rise, amongst other things, to the classical equations of electrodynamics."
There are five distinct eigenvalues, λ = 2, 1, 1, 0, 0, with eigenspaces of dimension one, three, three, three and six, denoted R(1,sy), R(3,sk), R(3,sy), G(3,sk) and G(6,sy). The assignment is the paper's central result:
- R(3,sk) yields classical electromagnetism;
- G(3,sk) yields the electromagnetic dual (indeed Gab = εabmnFmn);
- the field equations follow from the orthogonalities R(1,sy) ⊥ R(3,sk) and G(3,sk) ⊥ R(3,sk);
- R(3,sy) — sharing the eigenvalue λ = 1 with electromagnetism — yields the massive vector field;
- G(6,sy) plays no evident role.
Recovering the canonical formalism
The construction from R(3,sk) is built on a three-vector A = (α2, α3, α4) and no scalar. Roscoe shows that the act of choosing three indices from four "has the effect of making the omitted integer special": the omitted index turns out to be the temporal axis, A is the classical magnetic vector potential, and introducing an arbitrary scalar via a linear transformation reduces the formalism exactly to Fab = XaA′b − XbA′a — the standard four-potential form, with the added scalar identifiable as the electric scalar potential. In the unaugmented R(3,sk) theory the magnetic field takes its usual form but the electric field does not, because the scalar potential is absent; there is no electrostatics.
A stationary longitudinal wave
Section 8 examines free-space waves of the vector potential. The homogeneous condition splits into two cases. If n̂·A0 = 0 one obtains the ordinary transverse wave propagating at c, which Roscoe shows "corresponds exactly to those solutions which arise from the conventional formalism when the Coulomb gauge is chosen." If n̂·A0 ≠ 0 one is forced to n4 = 0 and obtains AL = αn̂ exp(in̂·x̂): a stationary longitudinal wave with no counterpart in the conventional theory. This wave gives E = B = 0, so a non-trivial vector potential accompanies a null electromagnetic field, and no effect propagates at all — "the only way an effect can be observed is that the charged particle must pass through the stationary wave." Roscoe concedes that stationary longitudinal waves in vacuum present "an entirely new level of incomprehensibility since, now, nothing is travelling anywhere," and offers in an appendix a non-wavy solution interpretable as a fluctuating material vacuum of which such waves would be disturbances.
The massive field and why it cannot be absent
From R(3,sy) the most general field Vab = XaVb + XbVa obeys, after using orthogonality with R(1,sy) and the identity ∂jVj = 0, the equation □Va = m2Va + J0a — a massive vector field, whose three components must each satisfy the Klein–Gordon equation.
The irreducibility argument is the paper's pivot. The operator Ptrs generating Va is the same operator that generates Fab from the vector potential. Forming the inner product A′iVi and choosing the basis (r,s) = (1,4),(2,4),(3,4) gives A′iVi = −iE·a, where a = (α5,α6,α7). Since a and E are independent, a can be oriented arbitrarily, so the option a ⊥ E is excluded and Va = 0 forces E = 0. Choosing instead (2,3),(3,1),(1,2) gives the same conclusion for B. Hence "the absence of the massive vector field implies the absence of the electromagnetic field," and Roscoe concludes the massive field must be identified with a classical massive photon.
A Newtonian Lorentz force
Because R(3,sk) and R(3,sy) share λ = 1, the general solution at that eigenvalue is Ψab = Fab + Gab, so the force should generalize to Fa = (e/c)ViFai + (e/c)ViGai. Roscoe invokes the well-known fact that the Lorentz forces two moving charges exert on each other are not equal and opposite even non-relativistically, so that "dynamical reactions, and the freedom to include them, are missing from classical electrodynamics." Identifying Gab with radiated massive photons, the extra term becomes the reaction on a charge of its own action on the source field, and imposing F(1) + F(2) = 0 for a two-particle system becomes a constraint on the reaction fields rather than a violated law.
Reconciliation and wider stakes
Roscoe acknowledges that a massive photon "is radically at variance with classical qed," and offers a reconciliation: since the R(3,sk) formalism has no scalar potential and hence no electrostatics, and since electrostatics is inherently a test-particle idealization (a charge Q assumed unaffected by the charge q it acts on), any formalism requiring the scalar potential embeds test-particle assumptions. The move from his formalism to the canonical one is then "the transition from a 'real world' electromagnetism to its test-particle idealization." He cites Vigier's claim that the Michelson–Morley–Miller interferometer sequence is consistent with a photon mass bounded above by about 10−68 kg, notes that de Broglie–Bohm pilot-wave theory arguably requires massive photons, and closes with the astrophysical stake: since the standard argument against non-expansion redshift mechanisms is that any such mechanism would blur images and broaden spectra, and since "such arguments are predicated directly upon the notion of the massless photon," admitting photon mass reopens the question in principle.
Assessment
The paper's method is its most attractive feature and is genuinely unusual. Rather than modifying Maxwell's equations and examining the consequences, Roscoe identifies a structural property they possess, poses the general classification problem, and finds Maxwell's equations embedded in a strictly larger answer. That is a legitimate and interesting way to look for missing physics, and the intermediate results are real: the reduction of the covariant equations to a pair of identities is correct; the eigenspace decomposition is clean; the identification of G(3,sk) with the dual field via the Levi-Civita tensor is a good internal consistency check, since it was not built in; and the recovery of the canonical four-potential formalism, together with the observation that the omitted index of a three-from-four choice becomes the temporal axis, is an elegant piece of work. The paper is also honest about its own oddities — the "entirely new level of incomprehensibility" of the stationary longitudinal wave is Roscoe's own phrase, not a critic's.
The difficulties concentrate at the joints. The claimed logical necessity is only as strong as the Lagrangian that produces the eigensystem, and that Lagrangian is arrived at in Appendix A by a chain of plausibility arguments — differential homogeneity, second-order field equations, index-interchange invariance — each reasonable but none forced. The reduction from six constants to the single λ is asserted as removal of "redundancy" without a demonstration that no physics is lost. Since everything downstream, including the massive field, follows from this particular density, the phrase "unavoidable conclusion" claims more than the derivation delivers: what is shown is that this generalization contains a massive field, not that any theory with Properties A and B must.
The irreducibility argument, which carries the paper's headline claim, is narrower than it appears. It establishes that Va = 0 implies E = B = 0 — that is, no electromagnetic field without the massive field. But the mass parameter m is nowhere determined; the theory permits m = 0, in which case Va remains a companion field but the "massive photon" evaporates and nothing conflicts with QED. Roscoe does not address why m must be non-zero, which is precisely the point at issue. Relatedly, the identification of a classical vector field with a photon is an interpretive leap the paper itself half-concedes ("the 'photon as particle' can never be recovered from a purely classical theory such as the one considered here").
Empirically the position is difficult. A photon mass modifies Coulomb's law to a Yukawa form and gives the vacuum a dispersion, and both are tightly constrained: Coulomb-law and geomagnetic tests bound the photon mass at roughly 10−54 to 10−52 kg, and the arrival-time dispersion of fast radio bursts across radio frequencies now provides an independent bound of comparable strength. Vigier's 10−68 kg, cited here as an upper limit, is far below all of these, which means it is consistent with them but also means the effect is unobservably small — and a mass that small cannot support the astrophysical hope Roscoe raises in §11.4, since a redshift mechanism weak enough to respect the dispersion bounds cannot produce the observed cosmological redshifts. That section is the weakest in the paper: it gestures at a possibility without proposing a mechanism or an order of magnitude.
The generalized Lorentz force is a fair observation poorly cashed out. It is true that the Lorentz forces between two moving charges are not equal and opposite; but the standard resolution is not that a term is missing — it is that the electromagnetic field itself carries momentum, and momentum conservation holds for the combined particle-plus-field system. Roscoe does not engage with this, so his "reaction field" solves a problem that classical electrodynamics already solves, and does so by introducing an undetermined field constrained only by the requirement that it produce the right answer. Likewise the stationary longitudinal wave, which produces E = B = 0 and no propagating effect, is by construction almost undetectable; the suggestion that a charge must pass through it is at least an observational program, but no experiment is proposed.
Judged as mathematics, the paper is careful and its structural insight into Maxwell's equations is worth having independently of the massive-photon conclusion. Judged as physics, the chain from a plausibility-argued Lagrangian to a classical massive photon to reopened cosmology stretches further at each link than the one before, and the crucial quantity — the mass itself — is never pinned down.