Stefano Filippi

The Fifth Equation

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The previous article left one domain out. This is it.

Applying the same operator to the electromagnetic potentials returns Maxwell’s equations, as one would expect from the pattern established so far. It also returns something the other domains did not produce: a fifth relation, appearing on exactly the same footing as the four field equations, which the standard treatment classifies as a choice rather than a law.

Two quaternions

Build the potentials into a single object with the same structure used throughout — scalar part imaginary, vector part real — and do the same for the sources:

A~=𝐢cφ+𝐀,J~=𝐢cρ+𝐉\tilde{A} = \frac{\mathbf{i}}{c}\varphi + \mathbf{A}, \qquad \tilde{J} = \mathbf{i}c\rho + \mathbf{J}

with φ the scalar potential, A the vector potential, ρ the charge density and J the current density. Nothing has been assumed about how these are related; that is what the operator is for.

One application

Apply the operator once, expand using the product rule, and identify the fields in the usual way — the electric field as minus the gradient of the scalar potential minus the time derivative of the vector potential, the magnetic field as the curl of the vector potential. The result separates into its natural parts:

∇~A~=−(1c2∂φ∂t+∇⋅𝐀)−𝐢c𝐄+𝐁\tilde{\nabla}\tilde{A} = -\left(\frac{1}{c^2}\frac{\partial \varphi}{\partial t} + \nabla\cdot\mathbf{A}\right) – \frac{\mathbf{i}}{c}\mathbf{E} + \mathbf{B}

The vector part carries E and B, packaged together in one object. That is unremarkable; several nineteenth and twentieth century treatments do the same.

The scalar part is the interesting one. Written out, it is

1c2∂φ∂t+∇⋅𝐀=0\frac{1}{c^2}\frac{\partial \varphi}{\partial t} + \nabla\cdot\mathbf{A} = 0

which is the Lorenz condition — the relation published in 1867 and misattributed ever since, discussed in an earlier article.

It has not been imposed here. It has not been chosen for convenience, or adopted to simplify anything. It came out of the same single application of the same operator that produced the fields, occupying the scalar slot that the fields do not occupy. Setting it to zero is one natural way of reading the equation; but the quantity itself is there whether or not it vanishes.

Two applications

Apply the operator twice — that is, apply the d’Alembertian established two articles ago — and set the result equal to the current quaternion:

∇~2A~=μ0J~\tilde{\nabla}^{2}\tilde{A} = \mu_0 \tilde{J}

Scalar and vector parts separate directly into the two wave equations for the potentials:

□φ=ρε0,□𝐀=μ0𝐉\Box\,\varphi = \frac{\rho}{\varepsilon_0}, \qquad \Box\,\mathbf{A} = \mu_0\mathbf{J}

From here the recovery of Maxwell’s equations is standard vector calculus and needs no quaternions at all. Expanding the vector wave equation with the identity relating the Laplacian of a vector field to the gradient of its divergence and the curl of its curl, and substituting the scalar relation from the single application above, gives the Ampère–Maxwell law. Expanding the scalar wave equation using the definition of E gives Gauss’s law.

The fourth Maxwell equation requires no work at all. Since the magnetic field is defined as a curl, and the divergence of a curl vanishes identically, the statement that B has zero divergence holds automatically — independently of any gauge, in any formulation whatever.

What is actually being claimed

Nothing above is new physics, and none of it should be read as such. Maxwell’s equations came out of the construction because they were put in, encoded in the definitions of the fields in terms of the potentials. A formalism that failed to return them would be wrong.

The claim is narrower, and concerns where the Lorenz condition sits.

In the standard vector treatment it is a gauge choice. The potentials are underdetermined; one fixes the freedom by imposing an extra condition; the Lorenz condition is one such, adopted because it makes the wave equations come out symmetrically and because it respects relativistic causality. It is a good choice, and it is universally described as a choice. It has no claim to stand alongside the divergence of E or the divergence of B as a field equation.

In the quaternion formulation it is not in the same category. It appears as the scalar component of an object whose vector component is the fields, produced by one application of one operator. Structurally it is a fifth equation among five, not an auxiliary constraint appended to four.

This is a statement about formalism, and it is worth being exact about what it does and does not establish. It does not show that the Lorenz condition is physically privileged over other gauge choices — gauge freedom is real, and the physics is unchanged under a change of gauge. What it shows is that the vector reduction of the 1880s, which split the four components of the potential into a scalar and a vector treated asymmetrically ever since, also demoted a relation that the original algebra had produced on equal terms with the rest.

An arrival from two directions

There is a coincidence here worth noticing, because it is the kind of thing that is easy to overinterpret and also easy to dismiss.

Lorenz reached his condition in 1867 by a physical route: he was constructing solutions of the field equations that incorporate the finite speed of light, and the condition falls out of requiring the potentials to propagate causally. He was not choosing a gauge; he was writing down retarded potentials and finding that they satisfied a relation.

The route above is purely algebraic. No causality argument was made, no retardation was assumed, nothing physical was required beyond the definitions of the fields. The condition appears because the quaternion product has a scalar slot and something has to go in it.

Two independent arrivals at the same relation do not prove anything on their own. Convergences of that sort happen, and sometimes they are coincidence. But it is at least a reason to look more carefully at the quantity sitting in that scalar slot, rather than treating it as a bookkeeping condition that happens to be convenient.

What that quantity does, when the same formalism is applied to the transport of energy rather than to the field equations, is a separate question and a longer one.

Sources

  • L. V. Lorenz, “On the Identity of the Vibrations of Light with Electrical Currents,” Phil. Mag., series 4, vol. 34, pp. 287–301, 1867.
  • A. W. Conway, “On the applications of quaternions to some recent developments of electrical theory,” Proc. R. Ir. Acad., vol. 29, pp. 1–9, 1911.
  • L. Silberstein, “Quaternionic form of relativity,” Phil. Mag., vol. 23, pp. 790–809, 1912.
  • J. D. Jackson and L. B. Okun, “Historical Roots of Gauge Invariance,” Rev. Mod. Phys., vol. 73, pp. 663–680, 2001.

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