Mathematical Constraints on Gauge in Maxwellian Electrodynamics

Scientific Paper
Title	Mathematical Constraints on Gauge in Maxwellian Electrodynamics
Read in full	Link to paper
Author(s)	E Comay
Keywords	Classical Electrodynamics, Gauge, Variational Principle, Quantum Mechanics
Published	2008
Journal	Apeiron
Volume	15
Number	2
No. of pages	16
Pages	16

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Abstract

The structure of classical electrodynamics based on the variational principle together with causality and spacetime homogeneity is analyzed. It is proved that in this case the 4-potentials are defined uniquely. On the other hand, the approach where Maxwell equations and the Lorentz law of force are regarded as cornerstones of the theory allows gauge transformations. For this reason, the two theories are not equivalent. A simple example substantiates this conclusion. Quantum physics is linked to the variational principle and it is proved that the same result holds for it. The compatibility of this conclusion with gauge invariance of the Lagrangian density is explained. Several alternative possibilities that may follow this work are pointed out.

Overview

This Apeiron paper argues that two formulations of electrodynamics which physicists normally treat as interchangeable are in fact not equivalent theories. Comay names them explicitly: Maxwell–Lorentz electrodynamics (MLE), in which Maxwell's equations and the Lorentz force law are taken as the axioms; and variational electrodynamics (VE), in which the variational principle — together with causality and the homogeneity of space-time — is taken as the foundation. Maxwell's equations and the Lorentz law can be derived from the variational principle, but the two theories rest on different sets of axioms, so, Comay insists, "the validity of their equivalence is not a priori clear."

The specific point at issue is the gauge degree of freedom. In MLE the equations of motion involve only the fields F^μν, which are untouched by a gauge transformation, so any gauge function whatever may be used. Comay's claim is that VE is far more restrictive: once dimensional consistency, Lorentz covariance, causality and space-time homogeneity are imposed, the gauge function Φ(x^μ) is forced to be a constant, its 4-gradient vanishes identically, and the retarded Liénard–Wiechert 4-potentials are unique. Because quantum mechanics is built on the variational principle, the same conclusion is then argued to hold there too: "gauge transformations are forbidden in quantum physics."

The paper is deliberately modest about what it establishes. Comay closes by saying it "is not intended to present a comprehensive solution of a physical problem but to draw the attention of the physical community to a problem which deserves a further analysis," and lists several mutually exclusive ways the matter might be resolved. Units with c = 1 and ħ = 1 are used throughout, so length [L] is the only dimension, and the metric is diagonal (1, −1, −1, −1).

The argument

The motivating observation

Comay opens with an elementary dimensional argument. Every term of a physical expression must have the same dimensions, and if an analytic function F(q) has a power series with more than one term, then q must be dimensionless — as with the e^−E/KT factor of the Maxwell–Boltzmann distribution, where KT has the dimensions of energy. If the expression is relativistic, covariance further requires q to be a Lorentz scalar. The wave function's phase e^{i(k·x−ωt)} satisfies both requirements.

But in quantum mechanics the charged particle's sector acquires the gauge-dependent factor e^{ieΦ(x^μ)}, and in the units used here the electric charge is a pure number (e² ≈ 1/137). The same two requirements therefore demand that Φ be a dimensionless Lorentz scalar — yet, Comay notes, the standard textbook treatment allows Φ to be an arbitrary function of the space-time coordinates. He adds that other gauge-related oddities are already in the literature: in the Coulomb gauge, as Jackson observes, a transverse electric current is found throughout the entire space in spite of actual charge localization.

Deriving the constraint in classical physics

The standard Lagrangian density is taken as

L = −(1/16π) F^μνF_μν − j^μA_μ

For a single charge of given motion, Maxwell's equations plus causality yield the retarded Liénard–Wiechert 4-potentials A^μ = e v^μ/(R^αv_α), with v^μ the 4-velocity at the retarded time and R^μ the 4-vector from the retarded space-time point to the field point. A gauge transformation is A′_μ = A_μ − Φ_,μ.

The dimensional counting then runs: the action is a dimensionless Lorentz scalar, so every term of the Lagrangian density has dimension [L⁻⁴]; the 4-current components are charge and current densities of dimension [L⁻³]; hence A^μ has dimension [L⁻¹], as the Liénard–Wiechert potentials indeed do; hence Φ_,μ also has dimension [L⁻¹] and Φ must be a dimensionless Lorentz scalar.

Comay then asks what such a function of the coordinates can look like. Any scalar built from coordinates must be a fully contracted tensorial expression, so it can be written as a sum of products of powers of quantities of the form f_a,b(x^μ) = (x^μ − x^μ_a)(x_μ − x_bμ). Causality and space-time homogeneity leave only one distinguished point available — the retarded position of the charge — so this collapses to R^μR_μ, which the retardation condition makes vanish identically. Therefore Φ is a constant, the gauge 4-vector Φ_,μ vanishes, and the Liénard–Wiechert 4-potential is unique.

The counterexample: a free motionless charge

Section 3 gives the concrete case that carries the paper. Take a single motionless particle of mass m and charge e sitting in a region where the external fields vanish. In MLE the Lorentz force is zero, the particle stays at rest, and its energy is the constant E = m. In VE one sets the external 4-potentials to zero, A_(ext)μ = 0, writes the Hamiltonian

H = [m² + (P − eA)²]^1/2 + eφ

and with P = 0 recovers the same E = m. So far the two agree exactly.

Now apply the gauge function Φ = t², giving A′_(ext)μ = (−2t, 0, 0, 0). Comay lists four objections:

Φ = t² has the dimensions [L²], whereas in VE it must be dimensionless.
It is the U⁰⁰ entry of the second-rank tensor U^μν = x^μx^ν, not a Lorentz scalar.
Substituted into the Hamiltonian it gives H′ = m − 2et, so the energy of a closed system is no longer a constant of the motion.
The physical state — one motionless particle in null fields — is manifestly time-independent, yet the gauge freedom has converted a trivially time-independent Hamiltonian into a time-dependent one, and a time-dependent Hamiltonian means energy is not conserved.

In MLE nothing whatever changes, since F′^μν = F^μν = 0 and the equations of motion depend only on the fields. Comay's verdict: "the gauge degree of freedom destroys VE," and MLE and VE are therefore not equivalent theories.

The quantum version

Section 4 repeats the exercise for the Dirac field. From the Lagrangian density L = ψ̄[γ^μ(i∂_μ − eA_μ) − m]ψ comes the Dirac Hamiltonian H = α·(P − eA) + βm + eφ. The quantum gauge transformation is the pair A′_μ = A_μ − Φ_,μ together with ψ′(x^μ) = e^ieΦψ(x^μ), and the Lagrangian density is indeed invariant under it.

For the spin-up motionless Dirac particle, ψ = e^−imt(1, 0, 0, 0) gives E = m. Under the same gauge choice Φ = t² the wave function becomes ψ′ = e^iet²e^−imt(1, 0, 0, 0), and

H′ψ′ = i ∂ψ′/∂t = (m − 2et)ψ′ → ⟨H′⟩ = m − 2et

— "precisely the same discrepancy" found classically.

Comay is careful to explain why this does not contradict the textbook gauge invariance of the Lagrangian density. In the Dirac Lagrangian the two parts of the gauge transformation cancel each other, so the action, the associated phase and any interference pattern are formally unaffected. The Hamiltonian is different: it does not contain the time derivative of the gauge-transformed wave function, so "one term has no counterpart and the Hamiltonian varies." That, on his account, is the asymmetry that has been overlooked.

What might follow

Section 5 lists possible outcomes rather than choosing among them:

All gauge applications survive within MLE untouched, since a gauge transformation there merely adds a zero to the fields, and "a zero is consistent with all dimensions and with all tensorial quantities." Gauge transformations also remain an important tool for solving Maxwell's equations, since a solution of the homogeneous equation may always be added to a particular solution of the inhomogeneous one.
Gauge freedom in VE might yet be vindicated by some further theoretical structure — in which case, Comay notes, MLE and VE are still not exactly equivalent, because VE would need that extra (as yet unknown) structure and MLE would not.
Gauge operations might survive in VE while gauge invariance ceases to be a mandatory criterion for accepting an electrodynamic expression. Comay says this "minimal theoretical change" was his original motivation for the work.
Most drastically, some or all gauge-based operations might be forbidden in VE, in which case results now obtained by gauge arguments — especially experimentally confirmed ones — would have to be re-derived by other means.

He also notes that the outcome is "probably linked" to Jackson's analysis of the Coulomb gauge and its ghost-like transverse current in empty space.

Assessment

The paper's strength is its economy. It does not attack Maxwell's equations, quantum electrodynamics or any measured result; it isolates a single narrow question — what class of function a gauge function is allowed to be, once one insists on dimensional consistency and Lorentz covariance in the action — and pushes it with a worked example a reader can check in a few lines. The distinction it draws between the gauge invariance of the Lagrangian density and the non-invariance of the Hamiltonian is a real and often glossed-over asymmetry, and the observation that Φ appears in the exponent e^ieΦ of the wave function, where a dimensionful argument would be meaningless, is genuinely pointed. Comay's framing is also unusually restrained: he offers four alternative resolutions and asks for further analysis rather than announcing a refutation.

The difficulties are equally clear. Most importantly, the counterexample Φ = t² is chosen to be dimensionally illegal — Comay's own objection 1 says so — so the demonstration that it wrecks the Hamiltonian may show only that a badly chosen gauge function misbehaves, not that gauge freedom as such is forbidden. The standard reply, which the paper does not engage in detail, is that the canonical momentum P is itself gauge-dependent, so the Hamiltonian's numerical value is not the physical energy and its change under a gauge transformation is expected rather than fatal; the gauge-invariant quantity is the kinetic momentum P − eA. Comay's response is essentially his fourth item — that a physically time-independent state ought to have a time-independent Hamiltonian — but this is asserted from physical intuition rather than derived.

The step from "causality and homogeneity leave only the retarded point available" to "therefore Φ must be a constant" also carries a good deal of weight for a single paragraph, and it is developed only for the field of a single charge whose motion is prescribed, with the linearity of Maxwell's equations and superposition invoked to extend it to a general system.

Finally, the conclusion is very strong. Gauge invariance is one of the organising principles of the Standard Model, and a proof that gauge transformations are inconsistent with quantum physics would touch an enormous body of successfully tested calculation — quantum electrodynamics being the most precisely verified theory in physics. Comay acknowledges this obliquely in his fourth scenario, where he grants that "electromagnetic relations that have been confirmed in experiments are expected to be proved successfully by other methods" — an expectation, not a demonstration. The paper is best read as it presents itself: a sharply posed consistency question about the foundations, not a completed replacement theory.