On the Notation of Maxwell's Equations (Renamed On the Notation of the Field Equations of Electrodynamics)

Scientific Paper
Title	On the Notation of Maxwell's Equations (Renamed On the Notation of the Field Equations of Electrodynamics)
Read in full	Link to paper
Author(s)	Andre Waser
Keywords	Maxwell's Equations, electrodynamics, Field
Published	2000
No. of pages	14

Read the full paper here

Abstract

Maxwell's equations are the cornerstone in electrodynamics. Despite the fact that these equations are more than a hundred years old, they are still subject to changes in content or notation. To get an impression over the historical development of Maxwell's equations, the equation systems in different notations are summarized.

Overview

This is a survey rather than a polemic. André Waser, writing in 2000 and revising through 2007, sets out side by side the successive notations in which Maxwell's equations have been written — the original twenty scalar equations of 1865, Maxwell's own quaternion form from the Treatise of 1873, the Heaviside–Gibbs vector notation that displaced it, the Minkowski four-vector and tensor form, and a series of modern proposals that either expand the equation set or complexify it. The point of the exercise is to make visible how much of what is now taught as "Maxwell's equations" is a later editorial choice, and how much content was dropped along the way.

Waser's departure from the textbook account is one of emphasis rather than dissent. He stresses that today's four equations are a subset of Maxwell's original system: the vector potential A, the magnetic potential field Ω and the magnetic mass m that Maxwell introduced in the Treatise have all been eliminated, along with the continuity equation, Ohm's law and in some presentations Faraday's law. His closing judgement is unusually restrained for a dissident venue — after reviewing every symmetrising proposal he concludes that the asymmetry of Maxwell's equations "still are correct", because no magnetic monopoles have been found.

The notations surveyed

Maxwell's original twenty equations

Waser reproduces the 1865 system in Maxwell's own lettering alongside its modern vector equivalent: the total-current equations relating J_tot to j and ∂D/∂t; the definition μH = ∇ × A; the circuital equation ∇ × H = 4πJ; the Faraday force law E = μ(v × H) − ∂A/∂t − ∇φ; the constitutive relations εE = D and σE = j; Coulomb's law ρ + ∇·D = 0; and the continuity equation ∂ρ/∂t + ∇·j = 0. He notes that three of the twenty are immediately recognisable as Ohm's law, the Faraday force and the continuity equation, and that the original set — unlike its modern descendant — carries the vector potential explicitly.

The quaternion form of 1873

Maxwell restated his results in quaternion notation in the Treatise, defining field vectors as quaternions with no scalar part and a differential operator ∇ = id/dx₁ + jd/dx₂ + kd/dx₃, with the prefixes S. and V. extracting scalar and vector parts. Waser observes that Maxwell "has never really calculated with quaternions but only uses either the scalar or the vector part", and that this is precisely why the notation lost out to Heaviside and Gibbs despite Peter Guthrie Tait's advocacy. Two things appear here for the first time: a magnetic potential field Ω with H = −∇Ω, and a magnetic mass m = S.∇M. Waser remarks that Maxwell's first formulation of a magnetic charge density was then forgotten for more than half a century until Paul Dirac revived magnetic monopoles in 1931.

Today's vector notation

Waser derives the modern four from the 1865 set — substituting the total current into the circuital equation to get ∇ × H = j + ∂D/∂t, extracting the potential equation E = −∂A/∂t − ∇φ from the Faraday force law, and taking the curl and divergence of μH = ∇ × A to obtain ∇ × E = −∂B/∂t and ∇·B = 0. He notes that the potentials φ and A were long held to be mathematical conveniences without physical existence, and cites the Aharonov–Bohm experiment as showing this is not so: "Many reasons point out that the potentials φ and A really are the cause of the force fields E and H."

Real expansions

Four proposals are treated as genuine extensions rather than re-lettering. The Hertz ansatz, which Waser credits Thomas E Phipps with recovering, replaces partial with total derivatives, d/dt = ∂/∂t + v·∇, giving ∇ × H = j + dD/dt and −∇ × E = dB/dt. Hertz read v as the absolute motion of aether elements; Phipps reads it as the velocity of a test charge relative to the observer, which makes the equations invariant under a Galilean transformation while reducing to Maxwell's for v = 0. The Dirac ansatz adds a magnetic current density j_m and magnetic charge density ρ_m, requiring complementary potentials φ and C and, as Waser notes, two distinct kinds of photon interacting differently with matter — neither of which has been observed. The Harmuth ansatz, extended by Konstantin Meyl, goes further by removing the source terms altogether and introducing a specific magnetic conductivity s; Harmuth used it to solve signal propagation in lossy media, and Meyl takes the resulting equation as fundamental and reads electric charges as secondary effects of dipoles. The Múnera–Guzmán ansatz of Hector A Munera and Octavio Guzmán works with the combinations N ≡ B − E and P ≡ B + E and ω ≡ ct; Maxwell's four equations are recovered as sums and differences of the four Múnera–Guzmán equations, and the analysis suggests a non-trivial magnetic scalar field alongside the electric one.

Imaginary and higher-dimensional expansions

The Minkowski form is set out in full: the event vector X = x + ict, the four-current, the four-potential A = i φ + A, the d'Alembertian, the field tensor F_μν ≡ ∂A_ν/∂x_μ − ∂A_μ/∂x_ν, and the two tensor equations from which all four Maxwell equations follow. Waser then reviews three complex notations: Inomata's, which puts the "missing" magnetic charge and current on the imaginary axis; Rauscher's eight-dimensional scheme, which gives every field and charge density a real and an imaginary part and so yields two complementary Maxwell sets; and Honig's imaginary-quaternion form, in which — Waser notes with interest — "there does not exist one single real number at all", every quantity being attached either to i or to a Hamilton unit.

Assessment

The paper's value is documentary and it does that job well. Waser has assembled in fourteen pages a comparison that is otherwise scattered across a century of literature, and the side-by-side presentation of Maxwell's 1865 lettering against modern vectors is genuinely useful for anyone who wants to see what the Heaviside–Gibbs compression actually cost. The observation that Maxwell himself introduced a magnetic mass and a magnetic scalar potential in the Treatise, three decades before Dirac, is a real historical point and not a manufactured one. So is the reminder that the continuity equation and Ohm's law are not derivable from the modern four but were part of the original system. Waser is also careful to attribute: Hertz to Phipps, the source-free extension to Harmuth and Meyl, the N/P formulation to Múnera and Guzmán.

The intellectual honesty of the conclusion deserves particular note. Having laid out every proposal for symmetrising the equations, Waser declines to endorse any of them. His argument is a clean disjunction: either the electric field is a relative-motion artefact, as special relativity holds, in which case monopoles cannot exist; or the magnetic force field derives from a scalar potential, in which case they can. Since none has been found in extensive searching, he concludes that "no magnetic potential fields must be postulated and that the non symmetry in Maxwell's equations still are correct" — and warns that imaginary-number symmetrisation "covers the danger that with the simple mathematical tool 'i' a symmetric formulation can be reached vastly, but that the physical models do become nebulous". That is a criticism of much of the dissident literature, delivered from inside it.

Where the paper is weaker is in evaluation. Several of the surveyed proposals are mutually incompatible — the Harmuth–Meyl system denies monopoles of either kind while the Dirac system requires magnetic ones — and Waser records this without adjudicating between them or noting the experimental constraints each faces. The Hertz–Phipps ansatz is presented as achieving Galilean invariance without confronting the results it must then explain away: the Michelson–Morley null result, the Kennedy–Thorndike experiment, and the velocity-independence of the measured speed of light to parts in 10¹⁷ in modern resonator tests. Similarly, the Harmuth equations are described as solving impulse propagation in lossy media, but no indication is given whether their predictions differ measurably from ordinary electrodynamics with finite conductivity — which is the only question that would settle the matter. The claim that the Aharonov–Bohm effect shows the potentials "really are the cause of the force fields" also overstates: the effect establishes that the potentials carry gauge-invariant physical information around a loop, not that they are causally prior to the fields. Read as a catalogue rather than an argument, however, the paper is sound on its own terms and does not claim more than it delivers.

See also

• Andre Waser
• Maxwell's Equations
• Electrodynamics
• Electromagnetism
• James Clerk Maxwell
• Oliver Heaviside
• Paul Dirac
• Thomas E Phipps
• Hector A Munera
• Konstantin Meyl
• Lorentz Force
• Quantum Electrodynamics