Four Components of One Product

The Poynting theorem is a scalar equation. Energy density changes in time, energy flows across a boundary, work is done on the charges; three terms, one relation, and the whole of electromagnetic energy accounting in classical field theory.

Nothing about it is wrong. What is worth asking is whether it is the whole of what the construction contains, or one projection of something larger — because the object it is built from, in the quaternion formulation, is not a scalar at all.

Where the Poynting vector comes from

Recall the product rule for two quaternions. Multiplying a pair of them generates several distinct pieces: a product of scalars, minus a dot product, two scalar-times-vector terms, and a cross product.

The classical Poynting vector, E × B over μ₀, is exactly one of those pieces. It is the cross-product term and nothing else.

The objects involved are these. Each field biquaternion carries a scalar part built from the gauge quantity b, and a vector part in which the electric and magnetic fields appear together:

and the energy-flux object is formed in strict analogy with the classical construction, using the complex conjugate of the magnetic biquaternion:

Note which conjugation is meant. Complexifying the coefficients made two inequivalent conjugations available — one flipping the Hamiltonian units, one flipping the complexifying unit — and it is the second that appears here. The choice is not cosmetic: with the other one the scalar part of the product acquires a real residue proportional to E·B, and the clean separation below does not occur.

Note also the coefficient β sitting in the scalar parts. It is not a typographical decoration and it is not determined by anything written so far. What it is doing there is the subject of the next section.

So the question of what else the product contains is not speculative. Build the fields into biquaternions carrying their own scalar parts, form the energy-flux object in strict analogy with the classical construction, and the same product rule that generates the cross product also generates everything else it is capable of generating. Four algebraically independent components come out: real scalar, imaginary scalar, real vector, imaginary vector.

Each must be satisfied separately. That is the structural difference from the vector formulation, and it is worth stating plainly: writing the theory in terms of E × B does not merely re-notate a four-component relation, it retains one component and has no expression for the other three.

What follows is what those four components evaluate to. The results are not uniform, and the non-uniformity is the substance of the exercise.

A coefficient nobody had to choose

Before any of the four components can be evaluated, there is a question about the construction itself that is easy to walk past.

The scalar part of the potential derivative is the Lorenz condition, and the Lorenz condition vanishes. To distribute a vanishing quantity between the two field biquaternions one writes it as a term minus itself — and the coefficient attached to that pair is fixed by nothing. The two contributions cancel whatever value it takes, so the defining requirement places no constraint on it at all.

The asymmetry is visible in the construction. The vector sector is split evenly between the two objects, each receiving half. The scalar sector, as conventionally written, assigns the full amount to each.

One might expect this to be harmless, since it was introduced by rewriting zero. It is not. The gauge scalar enters the scalar part of the flux object quadratically and the vector part linearly, so their ratio depends on the coefficient and does not cancel. A quantity introduced by splitting a vanishing identity can become significant once its two halves are separated into distinct objects; cancellation at the point of introduction guarantees nothing thereafter.

Something else has to fix it. Expanding the scalar-sector relation and using Gauss’s law, the source term cancels exactly against a term in the divergence, and what remains is a residual condition on the gauge scalar and the field:

In the regime of a purely longitudinal wavefront — no magnetic field, and the kinematic identity relating the gauge scalar to the wavefront field — that condition is satisfied by one value, and only one:

Exactly, for a plane wavefront. Asymptotically for the retarded spherical one, where the required value carries a correction that decays as one over distance. The result was checked by symbolic differentiation from the potentials and independently by finite differences on a numerical grid, across five different source waveforms.

The conventional choice — the one that comes out of writing the construction the obvious way, without noticing there was a choice — satisfies the condition in no regime at all. Its residual saturates near unity and does not decay with distance.

This is worth pausing on, because it is the kind of thing a compact formalism can hide. A free parameter entered through an identity that vanishes, survived into a relation that does not, and had to be fixed by a physical regime rather than by algebra. Writing the theory compactly did not make the choice; it made the choice invisible until the components were separated and evaluated one at a time.

The imaginary vector component: momentum

This one reduces exactly to the standard electromagnetic momentum-conservation law:

with the field momentum density, the Maxwell stress tensor and the Lorentz force density in their usual roles. The directional-derivative terms that appear on evaluation are the field-tension components of the stress tensor; the contribution from the charge density cancels algebraically against a term in the divergence of the stress tensor.

Two features of this are worth noticing. The first is that momentum conservation and energy conservation are not two separate results derived by two separate arguments — they are two components of one product, and neither had to be sought.

The second is that the gauge scalar drops out of this channel entirely. Its terms appear on both sides of the evaluation and cancel identically, whatever value the coefficient of the previous section is given. Whatever that quantity is doing elsewhere, it does nothing here.

This is a reconstruction of known physics, not a new relation. It is also the strongest evidence that the construction is doing something legitimate: a formalism that produced a novel result in every channel would be more suspicious, not less.

The imaginary scalar component: a constraint in disguise

This channel yields

and it is easy to misread. It resembles the familiar statement that the magnetic part of the Lorentz force does no work — but that concerns the force on an individual charge moving with some velocity, and holds automatically. This says something else: that no current flows along the local magnetic field direction, anywhere.

That is a substantive restriction, and it is false in general. The counterexample is immediate and standard. Take a force-free field, the canonical model for solar coronal and tokamak equilibria:

a legitimate Maxwell solution, in which the current is everywhere parallel to the field and the quantity above is non-zero by construction.

So this component is not a law recovered by the algebra. It is a restriction delimiting the class of configurations for which the full source coupling can be imposed at all — and it excludes magnetised plasmas, where parallel conduction dominates. It contains no dependence on the gauge scalar, and holds unchanged in gauges where that scalar vanishes identically.

Reporting this as a constraint rather than as a result is the difference between a decomposition and a claim.

The real vector component: something already named

The classical part of this channel is the curl of the Poynting vector — a quantity that already has a name and a literature. It is the electromagnetic vorticity, called Poynting spin in work on optical angular momentum, and it is known to be non-zero for optical vortices independently of polarisation.

So the decomposition arrives at a quantity other people are already studying, by a route that had no knowledge of them. That is a reasonable indication that the channels are picking out real structure rather than algebraic debris.

This channel also carries a term depending on the gauge scalar, and that term is not mysterious. It is the curl of the same flux that appears in the scalar channel — gauge-dependent by construction, and inheriting exactly the diagnostic status that quantity carries there. It is not a second physical effect sitting alongside the electromagnetic vorticity; it is the rotational part of a bookkeeping quantity.

In the spherically symmetric geometry where the gauge scalar is actually put to use, it vanishes identically. And where the geometry does not annihilate it, its size can be computed rather than guessed: for a sphere a few centimetres across charged over some ten nanoseconds, observed a few radii out, the whole channel comes to around a hundredth of a watt per cubic metre. Identifiable in principle, negligible in practice at those parameters.

Over-determined

There is a further result that only appears once all four channels are evaluated together, and it works against the construction rather than for it.

The natural expectation, having built a single compact equation, is that its components will be satisfied automatically by Maxwell’s equations — that the object is a repackaging rather than a set of additional demands. Tested directly, substituting Maxwell-consistent sources and comparing both symbolically and at numerically evaluated points, that expectation does not hold. Not for one channel, and not for another with a different sign convention: not for any of them.

None of the four components is an unconditional identity. The quaternionic equation is over-determined, and each component has to be assessed on its own terms rather than inherited as a consequence of the others.

What that assessment produces is not uniform, and the non-uniformity is the substance of the exercise. One channel imposes a restriction on admissible configurations. Two evaluate in closed form to expressions that reconstruct known physics but do not coincide with the source pairing they were supposed to match. One yields a relation with no counterpart in the vector formulation, and holds in a stated regime rather than everywhere.

This is a limitation of the formalism and it belongs in any account of it. A compact object with four components is an advantage only where the four components mean the same kind of thing; here each means something different, and no amount of algebraic elegance collapses them into one.

A structural result: what survives operand reversal

The most interesting outcome of the decomposition has nothing to do with sources, and follows from one line of algebra.

Quaternion multiplication is non-commutative. The energy-flux object was built with a particular operand order; build it with the order reversed, and compare.

The scalar part is unchanged. The vector part changes sign in its Poynting contribution and leaves the other flux term untouched.

The reason is visible in the product rule. The cross product is the only piece that is antisymmetric under exchange of the two factors; the scalar-times-vector terms are symmetric. So separating the product into its symmetric and antisymmetric parts separates the two fluxes exactly:

The classical Poynting vector is precisely the antisymmetric part of the biquaternionic product. It exists only because quaternion multiplication does not commute, and it would not survive in a hypothetical commutative reduction of the same algebra. The other flux term is the symmetric part, independent of operand order, and owes its existence instead to the field objects each carrying a scalar component that an ordinary pairing of E and B does not have.

These are not two terms of the same kind, appearing side by side in a sum. They have different algebraic origins, and the vector formulation retains one and has no slot for the other.

What the exercise establishes

None of the four channels comes for free. One imposes a restriction that excludes an important class of physical configurations. Two reconstruct things already known — momentum conservation, and a quantity the optical angular-momentum literature studies under another name — without matching the source pairing that was supposed to produce them. One yields a relation with no counterpart in the vector formulation, valid in a stated regime. And the construction itself turned out to carry a coefficient that had to be fixed from outside.

Stated that way it sounds like a poor return, and it is worth saying why it is not. The point of a systematic decomposition is that it is systematic: the four components were not selected, they are what the product rule generates, and evaluating all of them — including the ones that produce nothing new, and the one that produces an unwanted constraint — is what makes the exercise a measurement of the formalism’s reach rather than an advertisement for it.

The result of that measurement: the quaternionic form of the energy-flux relation contains the classical Poynting theorem, standard momentum conservation, a quantity already studied elsewhere, a constraint that rules out magnetised plasmas, a normalisation that algebra alone cannot fix, and one relation that the vector formulation cannot write down.

What that last one says, and what it is a statement about, is a separate matter. It is the subject of a further article.

Sources

• J. D. Jackson, Classical Electrodynamics, 3rd ed. New York: Wiley, 1998, Ch. 6 and 8.
• V. Epp, U. Guselnikova, and I. Kamenskaya, “Angular momentum transferred by the field of a moving point charge,” arXiv:2111.00321, 2021.
• M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt., vol. 11, 094001, 2009.
• P. Shi et al., “A decomposition of light’s spin angular momentum density,” Light: Science & Applications, vol. 13, 160, 2024.
• S. Filippi, “A Systematic Algebraic Decomposition of the Electromagnetic Energy-Flux Biquaternion: The Lorenz Gauge b-Channel and the Structure of Its Companion Terms,” preprint, 2026.