Historical Note on Relativistic Theories of Electromagnetism
| Scientific Paper | |
|---|---|
| Title | Historical Note on Relativistic Theories of Electromagnetism |
| Read in full | Link to paper |
| Author(s) | Valeri V Dvoeglazov |
| Keywords | Quantum electrodynamics, light, electromagnetism, quantum theory of light |
| Published | 1998 |
| Journal | Apeiron |
| Volume | 5 |
| Number | 1-2 |
| No. of pages | 19 |
| Pages | 69-88 |
Read the full paper here
Abstract
Quantum electrodynamics is the well-accepted theory. However, we feel it is useful to look at formalisms that provide alternative ways to describe light, because in the recent years the development of quantum field theories based primarily on the gauge principle has encountered considerable difficulties. There is a wide variety of generalized theories, and they are characterized mainly by the introduction of additional parameters and/or longitudinal modes of electromagnetism. The Majorana-Oppenheimer form of electrodynamics, the Sachs theory of Elementary Matter, the analysis of the action-at-a-distance concept, presented recently by Chubykalo and Smirnov-Rueda, and the analysis of the claimed "longitudinality" of the antisymmetric tensor field after quantization are reviewed in this essay. We also list recent advances in the Weinberg 2 (2J+1)(2J+1) formalism (which is built on First Principles) and in the Majorana theory of neutral particles. These may serve as starting points for constructing a quantum theory of light.
Overview
This is a survey essay, not a theory paper: Valeri Dvoeglazov of the Universidad Autónoma de Zacatecas walks through roughly a dozen alternative relativistic formulations of Electromagnetism, from J. R. Oppenheimer's 1931 matrix electrodynamics to the Evans–Vigier B(3) field of the 1990s, and asks which of them might serve as a starting point for a quantum theory of Light that does not rest on the gauge principle. He opens by conceding that "the accuracy of the predictions of quantum electrodynamics is without precedent," which sets the tone: this is not an attack on Quantum Electrodynamics as a calculational instrument but a survey of what has been left unexplored.
The motivating claim is that gauge-principle field theory had by 1998 run into "serious difficulties," and Dvoeglazov itemises the anomalies he thinks the standard model does not satisfactorily explain: the LANL neutrino oscillation experiment, the atmospheric neutrino anomaly and the solar neutrino puzzle (all implying Neutrino mass), tensor couplings in π− and K+ decays, the Dark Matter problem, the observed periodicity in the number distribution of galaxies, the QCD "spin crisis", and a group of claimed superluminal phenomena — negative mass-squared neutrinos, tunnelling photons, X-shaped waves and apparent superluminal expansion in quasars. The unifying thread running through the alternatives he reviews is the appearance of longitudinal and scalar modes of the electromagnetic field, which orthodox electrodynamics either excludes or gauges away.
The formalisms reviewed
E = 0 solutions
The essay begins with what Dvoeglazov treats as the key anomaly: the first-order massless equations (J·p ± p0)φR,L = 0 admit acausal dispersion relations, and for Spin j = 1 an E = 0 solution. He notes that this is not new — Oppenheimer (1931), Weinberg (1964) and Gianetto had all encountered it. Weinberg's observation is quoted at length: for j = ½ these are the Weyl equations, for j = 1 "just Maxwell's free-space equations for left- and right-circularly polarized radiation", and their first-order character "seems to me to be of no great significance." Oppenheimer connected the E = 0 mode to the electrostatic solutions of Maxwell's equations — puzzling, since the free-space equations contain no charge densities yet the dispersion relation produces the mode anyway. Recami, in private communication, is reported as identifying E = 0 solutions with a tachyon of infinite velocity.
The 'baroque' formalism and Majorana–Oppenheimer form
Imaeda (1950) and Ohmura (1956), attempting to solve the electron stability problem, added scalar and pseudoscalar fields to Maxwell's theory, with monopoles and magnetic currents; the equations acquire gradient terms, curl H − ∂E/∂t = −i + grad e0, and so on. Ohmura noted the resulting longitudinal photons and asked whether γ-rays keep their transverse character at high energy. Written in matrix form with the Majorana–Oppenheimer matrices, these become the (0,0)⊕(1,0) representation of the Poincaré group — ρμ∂μψ = f, where the field function packs Ek ± iBk together with a zero component. Dvoeglazov notes that with sources switched off the dispersion relations reduce to E = ±|p| only, and cites Moses's rule that a field with non-zero divergence must have that part subtracted before it counts as a "final field" — a convention he thinks needs firmer justification. He also reviews the Lyttleton–Bondi modification, in which additional terms of order l−2 (with l of order the radius of the universe) encode a photon mass and were used to explain cosmic expansion; Chambers showed the Watson and Lyttleton–Bondi generalisations equivalent below that scale, and both able to describe local creation of charge.
Sachs's theory of elementary matter
Mendel Sachs's formalism builds spinorial functions from the combinations Gk = Hk + iEk, with the dynamical equation σμ∂μφ = ϒ. Sachs's own claim, quoted here, is that this is "not merely a rewriting of the vector form of the field equations" but "a true generalization in the sense of transcending the predictions of the older form", equivalent to eight real conservation laws. Dvoeglazov lists the reported consequences — small but non-zero neutrino masses and an infinite neutrino spectrum, the Planck blackbody distribution, the hydrogen spectrum including the Lamb shift, a basis for charge quantisation, the muon lifetime, and an electron–muon mass splitting depending on the Fine Structure Constant through the geometry of the physical vacuum. His interjection into Sachs's quotation is telling: the "physical vacuum" as a degenerate gas of spin-zero objects is, he writes, "longitudinal and scalar photons, in fact!" He calls the results "impressive."
Phase, evolution parameter, action-at-a-distance
Staruszkiewicz's action adds a longitudinal piece ∂μS∂μS with S a scalar phase field, which Dvoeglazov identifies as a development of the Dirac–Fock–Podolsky model of a gradient current. Staruszkiewicz's questions — whether a system can be governed by charge conservation alone, whether there can be "a pure charge not attached to a nonelectromagnetic piece of matter" — are answered by showing that the electrodynamics of a gradient current is a closed dynamical system, with total charge related to the change in phase between past and future timelike infinity. Dvoeglazov thinks this research "can help to understand the nature of the charge and of the fine structure constant." Horwitz's theory with an invariant evolution parameter τ, developing Stueckelberg's worldline mechanics, yields a five-potential electrodynamics whose Maxwell limit is recovered by integrating over τ; Tanimura's generalisation of the Feynman–Dyson proof independently produces a scalar field Gμ = ∂μf alongside the usual tensor, which Horwitz identified with Fμ5. Chubykalo and Smirnov-Rueda's argument for reviving instantaneous action-at-a-distance is reviewed with its convection displacement current jdisp = −(1/4π)(v·∇)E — described as "a resurrection of the Hertz' ideas", later defended by T. E. Phipps Jr., of replacing the partial by the total derivative in Maxwell's equations. Belinfante's much earlier work, in which scalar and longitudinal photons appear in pairs in the zeroth-order approximation, is noted as anticipating Sachs, and as having concluded that signals can be transmitted faster than c — which Dvoeglazov connects to Nimtz's claim of transmitting Mozart's Symphony No. 40 through a 114 mm barrier at 4.7c.
The B(3) field and the notoph
Myron W Evans and Jean-Pierre Vigier's longitudinal B(3) field, defined through the cyclic relations B(1)×B(2) = iB(0)B(3)* and permutations, replaces O(2) gauge geometry by non-Abelian O(3). Dvoeglazov reports having proven the relativistic covariance of the B-cyclic relations himself, and notes Ahluwalia and Sawicki's result that in the light-front (1,0)⊕(0,1) formulation the longitudinal "bispinor" is directly proportional to the particle's Mass, so the massless limit deserves further study. He is careful to footnote his position: "Although I frequently disagree with Dr. M. W. Evans, his main idea is reasonable." He then traces the longitudinal idea back to Whittaker at the start of the century, whose general solution of the d'Alembert equation expresses any electrodynamic field in terms of two scalar potential functions, which in modern language relate to the Hertz potentials.
The antisymmetric tensor field ("notoph") is where his own technical contribution lies. Ogievetskii and Polubarinov claimed such a field must be longitudinal in the quantum theory, owing to a new gauge invariance plus supplementary conditions, and the claim became widely accepted. Dvoeglazov objects: if the antisymmetric tensor field were purely longitudinal, it becomes impossible to understand why in classical electromagnetism it is transverse — "this induces speculations about the incorrectness of the Correspondence Principle" — and it contradicts Weinberg's theorem B − A = λ. The resolution he reports is that the alleged longitudinality is an artefact of imposing the generalised Lorentz condition ∂μFμν|Ψ⟩ = 0 on the states, which brings with it the Gupta–Bleuler indefinite-metric problem.
The Weinberg 2(2j+1) formalism
The final and longest technical section reviews Weinberg's (j,0)⊕(0,j) theory, built on just two postulates — relativistic invariance and causality (vanishing of the field (anti)commutator at spacelike separation). Dvoeglazov reproduces the Barut–Muzinich–Williams matrices, the dynamical equations, and the covariant propagator, which is not the Wick-theorem propagator because of extra equal-time δ-function terms. Weinberg's massless theorem is quoted in full: a helicity-λ operator "can only be used to construct fields which transform according to representations (A,B) such that B − A = λ" — so a left-circularly polarised photon goes with (1,0) or (3/2,½), "but not with the vector potential, (½,½)". Dvoeglazov's own results are then listed: Sankaranarayanan's alternative (1,0)⊕(0,1) equation, in which a boson and its antiboson carry opposite intrinsic parities; the first explicit example of a Bargmann–Wightman–Wigner-type theory; self/anti-self charge conjugate spinors that are eigenstates neither of parity nor of ordinary helicity, requiring a new "chiral helicity" operator −γ5h; automatically parity-violating gauge currents; and possible oscillations between them.
Assessment
As a piece of scholarship the essay's real value is bibliographic and historical. Dvoeglazov's genuine contribution is showing that ideas usually presented as fringe innovations — longitudinal photon modes, scalar companions to the electromagnetic field, action-at-a-distance, the failure of purely transverse descriptions — have a continuous and largely forgotten pedigree in the orthodox literature: Whittaker's two scalar potentials, Oppenheimer in Physical Review in 1931, Imaeda and Ohmura in Progress of Theoretical Physics in the 1950s, Lyttleton and Bondi and Belinfante, Ogievetskii and Polubarinov, and Weinberg's own remarks about the significance of first-order equations. Recovering that lineage is worth doing, and it is done with a care that many polemical papers lack; the footnote distancing himself from Evans while endorsing the core idea is characteristic of the essay's willingness to separate a formalism from its advocate. His specific technical criticism of the "longitudinality" of the antisymmetric tensor field is the strongest argument in the paper, because it is a properly internal one: the alleged result follows from a supplementary condition on the states, it conflicts with Weinberg's own B − A = λ theorem, and it fails the correspondence limit. That is a real objection, not a preference.
The weaknesses follow from the genre. This is a catalogue, and its unifying claim — that longitudinal and scalar modes are the missing ingredient — is asserted through juxtaposition rather than argued. Formalisms with quite different content are grouped because each yields some extra scalar or longitudinal degree of freedom: Sachs's spinor theory, Horwitz's five-dimensional off-shell electrodynamics, Chubykalo's instantaneous action-at-a-distance and the B(3) field are not variants of one another, and several are mutually inconsistent. No criterion is offered for choosing among them, and none of the reviewed theories is subjected to the comparison that would matter — a prediction differing from QED at a level presently measurable. Sachs's impressive-sounding list (Lamb shift, muon lifetime, blackbody spectrum, e–μ mass ratio) is transmitted at face value with the exclamation "That was impressive work and these are impressive results!" and no number is checked; given that the electron anomalous magnetic moment agrees with QED to better than one part in 1012, any rival account of the same phenomena needs quantitative comparison at that level, which the essay never asks for.
The list of standard-model anomalies has also not aged uniformly. Neutrino oscillation — presented in 1998 as an embarrassment for the standard model — was confirmed by Super-Kamiokande and SNO and is now accommodated by neutrino masses without abandoning gauge theory; the LANL (LSND) result specifically was not confirmed by KARMEN and was excluded over most of its parameter space by MiniBooNE. Superluminal quasar expansion has a standard relativistic-beaming explanation, and the Nimtz tunnelling experiments are widely understood as pulse-reshaping without superluminal signalling — a point the essay does not engage, though it is precisely the question at issue. Using such a list as motivation is legitimate; treating each item as evidence for longitudinal electromagnetic modes specifically is a leap the paper does not attempt to justify.
Read for what it is — a working physicist's annotated map of the roads not taken in relativistic electrodynamics, with an honest statement of which ones he thinks are worth walking — the essay is useful and largely fair. Read as an argument that quantum electrodynamics should be replaced, it does not make the case, and mostly does not try to.