The Ether and Photons in the Electro-Gravimagnetic Programme

Introduction

The companion articles The Electro-Gravimagnetic Field and the Magnetic-Charge–Mass Hypothesis and Harmonic Elementary Particles and the Periodic System of Atoms record an external programme, due to L. A. Alexeyeva: the reading of the biquaternionic A-field as an electro-gravimagnetic (EGM) field, under the hypothesis that magnetic-charge density is gravitational mass density. The first article follows the programme's field theory and its law structure. The second follows its charge–current field into the monochromatic sector and records the particle and periodic-system reading built on it.

This article records the third slice of the same programme, in which it turns from the source to the field. The programme names the EGM field the ether, and reads the monochromatic solutions of the field equation as photons and as light: an elementary photon emitted by a concentrated charge, free photons as plane waves, and light as a superposition of elementary photons over a spectral band. It closes with the programme's claim about gravity — that no pure gravitational waves exist — which is the one place in the whole programme where a claim can be measured against observation.

The task is the corpus's usual one: record what is derived, mark what is chosen, and state the cost. The conventions are the corpus's. The field is the companion article's A-field, $\boldsymbol{\mathcal A} = \sqrt{\epsilon}\,\mathbf{E} + i\sqrt{\mu}\,\mathbf{H}$, and the paper's biquaternion of tension, $\tilde{A} = i\alpha + \boldsymbol{\mathcal A}$, is that field with the scalar part the paper calls the density of the ether. The paper's conjugation — the vector part negated and the coefficients complex-conjugated, the paper's mutual biquaternion and the conjugation it uses for the energy–momentum — is the corpus's ${}^{*}$ throughout. The symbol $\nabla$ is the pure-vector gradient acting by left quaternion multiplication, so that $\nabla\circ\nabla = -\Delta$; it is the $\nabla$ of the harmonic article, not the corpus's time-carrying gradient.

The Ether and the Two States of the Field

The programme's postulate is the companion article's, and the first thing this paper does with it is to read it as an ontology. The postulate is

$$ \nabla^{+}\tilde{A} = \tilde{\Theta}, \qquad \nabla^{\pm} = \partial_\tau \pm i\nabla , \qquad \tilde{\Theta} = i\rho + \mathbf J , $$

with $\tilde{A}$ the tension biquaternion and $\tilde{\Theta}$ the charge–current biquaternion. On its left is the field, on its right the charges and currents, and the relation says that the sources are the bigradient of the field.

One algebraic remark fixes what the object is. The classical biquaternion form of Maxwell's equations requires the scalar part of the field biquaternion to vanish, and that constraint is what makes the complex three-vector $\boldsymbol{\mathcal A}$ the whole of the field. The paper's own statement of what its EGM field is: the restriction is dropped — the scalar part returns, and the paper names it the density of the ether. That is the whole of the extension, and it is the cleanest statement in the programme of what the programme is; the object is the one the companion article already records. The consistency check the paper offers is that at $\alpha = 0$ the energy–momentum biquaternion collapses to the classical electromagnetic energy density and the Poynting vector, and that check is the companion article's, recorded there. Two corrections to the framing belong with it: the paper credits the complex characteristic of the electromagnetic field to Hamilton, whereas the corpus's history article dates that object to Silberstein in 1907 — the Riemann–Silberstein vector — and keeps the priority there; and the programme presents its model as a generalization of Maxwell and Dirac, while only the Maxwell half appears in this paper.

The paper states the reading in one sentence: the field is primary and the substance is secondary; substance is the physical manifestation of the heterogeneity and motion of the ether. Hence the name: the EGM field is the ether, its scalar part is the density of the ether, and its vector part is the tension of the electric and gravimagnetic fields.

Two things must be said about the name, and the corpus says them rather than letting it pass.

It is a name and a choice, not a derivation. Every equation below is the EGM article's equation. Calling the A-field the ether changes nothing algebraically; what it asserts is that the field is the fundamental object and matter its derivative. That is the same kind of move as the magnetic-charge–mass identification, and it carries the same status: chosen, not forced.
It is not the nineteenth-century ether. No equation of the programme singles out a rest frame; the field equations are the same covariant ones the companion article records. The programme's ether is the field itself under another name, and the relativistic structure of the equations is untouched by it. The name is therefore misleading to a reader who expects a medium, and a reader must be told that it is not one.

The Time-Harmonic Reduction and the Photon Equation

A monochromatic field is one whose time dependence is a single exponential,

$$ \tilde{A}(\tau,\mathbf{x}) = \tilde{\Phi}(\mathbf{x},\omega)\,e^{-i\omega\tau}, \qquad \omega > 0 , $$

so that $\partial_\tau \to -i\omega$. Substituting into the postulate gives the programme's photon equation,

$$ \nabla^{-}_{\omega}\tilde{\Phi} = i\,\tilde{\Theta}(\mathbf{x},\omega), \qquad \nabla^{\pm}_{\omega} = \omega \pm \nabla , $$

in which the frequency of the field plays the part the mass plays in a massive wave equation. The $\omega$-gradients are the corpus's complex gradients with the time dependence divided out, $\nabla^{+}_{\omega} = i\,D_{-}$ and $\nabla^{-}_{\omega} = i\,D_{+}$ at fixed frequency, so the photon equation is the EGM postulate and nothing further; the factor of $i$ is the paper's convention and not a new term. Both the dictionary above and the factorisation below were recomputed, the dictionary exactly and the factorisation to the finite-difference floor.

The second fact is the factorisation. Because $\nabla\circ\nabla = -\Delta$,

$$ \nabla^{-}_{\omega}\circ\,\nabla^{+}_{\omega} = \nabla^{+}_{\omega}\circ\,\nabla^{-}_{\omega} = \omega^{2} + \Delta , $$

the scalar Helmholtz operator. So a first-order operator here has a second-order inverse: the biamplitude of a photon solves a Helmholtz equation, and, conversely, every Helmholtz potential generates a solution of the photon equation by the single application of $\nabla^{+}_{\omega}$. This is the same factorisation the harmonic article records for the charge–current field, applied on the other side of the postulate, and it carries the same cost. The Helmholtz equation is an eigenvalue equation only when a boundary condition is supplied, the programme supplies none, and the frequency is a free continuous parameter of every solution below. The visible band that appears in the last section is put in by hand, exactly as the musical scale was.

The paper observes the type change that comes with the reduction: the biwave equation is hyperbolic, a system of eight first-order equations in the paper's count — the size the complex three-vector form halves — while the stationary equation for the biamplitude is elliptic. An elliptic equation is a boundary-value problem, which is exactly the structure an eigenvalue problem would need. The type change is therefore the sharpest statement of the cost: the programme's own reduction is what makes a boundary condition necessary, and a boundary condition is what the programme does not supply.

The Elementary Photon

The inverse of the photon equation is built from the fundamental solution of the Helmholtz equation that satisfies the radiation condition,

$$ \psi(\mathbf{x},\omega) = -\frac{e^{\,i\omega r}}{4\pi r}, \qquad r = \|\mathbf{x}\| , \qquad \left(\Delta + \omega^{2}\right)\psi = \delta(\mathbf{x}) , $$

the outgoing spherical kernel, whose flux through a small sphere was computed and is exactly $+1$, and which satisfies the Helmholtz equation away from the origin. Applying the conjugate $\omega$-gradient to the source convolved with this kernel gives the photon biquaternion of any monochromatic source,

$$ \tilde{\Phi}(\mathbf{x},\omega) = i\,\nabla^{+}_{\omega}\left\{\tilde{\Theta}(\mathbf{x},\omega) * \psi(\mathbf{x},\omega)\right\}, $$

so that a source distribution is an emitter whose field is the $\omega$-gradient of a Helmholtz potential. The class of solutions the paper works in is the biquaternions whose components are generalized functions of slow growth — tempered distributions — which is what makes the convolved sources here and the layered sources of the light section below legitimate rather than formal. For a concentrated charge, $\tilde{\Theta} = -i\,\delta(\mathbf{x})$, the kernel is evaluated at once and the result is the elementary photon: a spherical EGM wave whose scalar and vector amplitudes are

$$ \left|\alpha_{0}\right| = \frac{\omega}{4\pi r}, \qquad \left\|\boldsymbol{\mathcal A}_{0}\right\| = \frac{\sqrt{\omega^{2} + 1/r^{2}}}{4\pi r} , \qquad \mathbf{e}_{x} = \frac{\mathbf{x}}{r}, $$

both recomputed. The wave is a spherical EGM wave emitted by a point charge, propagating at the speed of the field, decaying as $r^{-1}$; its density and tension amplitudes are proportional to its frequency; its energy density falls as $r^{-2}$ and the vector part of its energy–momentum biquaternion is radial along $\mathbf{e}_{x}$, both recomputed. The paper's energy–momentum display for it, $\tfrac12\tilde{\Phi}^{0}\tilde{\Phi}^{0{}^{*}} = w + i\mathbf{P}$, is $w \propto \omega^{2} + 1/(2r^{2})$ and $i\mathbf{P} \propto i\omega^{2}\mathbf{e}_{x}$ under one shared prefactor, and that bracket is exactly the recomputed one; the overall prefactor is the paper's own and is not self-consistent between its two displays of the elementary photon, so the corpus carries the bracket and not the number. The two powers of $r$ in the tension are the near field and the radiated field, and that structure is the standard one for a spherical wave — the corpus's retarded-potential and layer machinery is in Radiation from Accelerated Charges in Biquaternionic Form and Distributions on Surfaces, Layers, and Jump Conditions. What the programme adds is the packaging in one biquaternion and the reading of the scalar part as an ether density.

Free Photons as Plane Waves

The free photon equation is the homogeneous one,

$$ \nabla^{-}_{\omega}\tilde{\Phi} = 0 , $$

and because of the factorisation every solution of it is $\nabla^{+}_{\omega}$ applied to a Helmholtz potential. Taking the plane potentials $\varphi_{j} = e^{\,i(\mathbf{k},\mathbf{x})}e_{j}$ with $\|\mathbf{k}\| = \omega$ and $e_{0} = 1$, the programme's four modes are, in the corpus's notation and recomputed exactly,

$$ \tilde{\Phi}_{\mathbf{k}0} = \omega\,\varphi\left(1 + i\,\mathbf{e}_{k}\right), \qquad \tilde{\Phi}_{\mathbf{k}j} = \varphi\left(\omega\,e_{j} - i\,k_{j} + i\,[\mathbf{k},e_{j}]\right), \qquad j = 1,2,3 , $$

with $\varphi = e^{\,i(\mathbf{k},\mathbf{x})}$ and $\mathbf{e}_{k} = \mathbf{k}/\omega$. All four satisfy the free equation, and their content differs in a way the programme reads as physics.

The scalar-potential mode is longitudinal and has the two field vectors parallel. For $j = 0$ the mode has $\mathbf{E} = -(\omega\sin\theta/\sqrt{\epsilon})\,\mathbf{e}_{k}$ and $\mathbf{H} = (\omega\cos\theta/\sqrt{\mu})\,\mathbf{e}_{k}$ with $\theta = (\mathbf{k},\mathbf{x})$, so both fields lie along the wave vector and are parallel up to a relative sign set by the phase — recomputed. No other mode of the construction is of this kind: a potential axis along $\mathbf{k}$ reproduces this mode up to a factor, which is the redundancy recorded below. The paper calls it the plane longitudinal harmonic EGM wave.
A vector-potential mode is transverse exactly when its potential is. Its scalar (ether-density) amplitude is $\alpha = -k_{j}\varphi$, recomputed, so it vanishes precisely when $e_{j} \perp \mathbf{k}$. In that case $\mathbf{E} \perp \mathbf{k}$, $\mathbf{H} \perp \mathbf{k}$ and $\mathbf{E} \perp \mathbf{H}$ — the tensional wave of the paper, with zero ether density — while for $k_{j} \neq 0$ the mode carries a longitudinal part. This is where the ether section's restriction returns: the tensional wave is precisely the mode whose scalar part vanishes, that is, the member of the family that survives when the field biquaternion is required to have no scalar part. The classically admissible photons and the ether-density-free photons are the same photons, and the programme's extra content sits entirely in the modes the classical restriction excludes. So the programme's free photons include a longitudinal polarisation, which vacuum electromagnetism excludes. The EGM article has already recorded the same reading from the programme's 2016 paper, where the longitudinal modes were built from scalar potentials; the two constructions are one construction, and the corpus does not count them twice.
The energy density is $\omega^{2}$ and the flux is along the wave vector. With the corpus's ${}^{*}$, the biquaternion $\tfrac12\tilde{\Phi}\tilde{\Phi}^{*}$ has the real scalar part $\omega^{2}$, independent of position, for every mode, and vector part $i\omega^{2}\mathbf{e}_{k}$; both are recomputed. The paper's own display instead places the vector-potential modes' flux along their polarisation axis, $\omega^{2}\cos\gamma_{j}\,e_{j}$, vanishing for the tensional wave, and that form does not reproduce from the paper's own amplitude formula. The corpus carries the recomputed direction; the paper's bilinear pairing is its own and the degraded text layer does not expose it, so the disagreement is recorded rather than resolved.

One structural remark belongs here, because the paper presents the four modes as four independent polarisations. They are not. The free plane equation is an equation of eigenvectors, $\tilde{\Phi} = i\mathbf{e}_{k}\tilde{\Phi}$, whose solution space is two complex dimensions; the four modes span it twice over. The rank of the four amplitudes was computed and is two, so two of the four complex coefficients are redundant: the programme's four polarisations carry four real parameters, not eight.

Light as a Cloud of Photons

Because the source enters linearly, the representations compose. Every emitted monochromatic field is a superposition of elementary photons weighted by the source,

$$ \tilde{\Phi}(\mathbf{x},\omega) = i\,\tilde{\Theta}(\mathbf{x},\omega) * \tilde{\Phi}^{0}(\mathbf{x},\omega), $$

the convolution being over the spatial variables, so the elementary photon is the kernel of the map from a charge–current distribution to its field. Over a band of frequencies the paper sums these contributions into what it calls light,

$$ \Lambda(\mathbf{x},\tau) = \int_{\omega_{1}}^{\omega_{2}} \tilde{\Phi}(\mathbf{x},\omega)\,e^{-i\omega\tau}\,d\omega , \qquad \tilde{\Xi}_{\Lambda} = \Lambda\Lambda^{*} , $$

and the free-photon cloud is the same construction with the four plane modes weighted and summed,

$$ \mathrm{O}(\mathbf{x},\tau) = \sum_{j=0}^{3}\int_{\omega_{1}}^{\omega_{2}} e^{-i\omega\tau} \left\{\Omega_{j}(\mathbf{x},\omega) * \int_{\|\mathbf{k}\|=\omega} b_{j}(\mathbf{k},\omega)\, \tilde{\Phi}_{\mathbf{k}j}(\mathbf{x},\omega)\,dS(\mathbf{k})\right\} d\omega , \qquad \tilde{\Xi}_{\mathrm{O}} = \tfrac12\,\mathrm{O}\,\mathrm{O}^{*} , $$

with $b_{j}$ and $\Omega_{j}$ arbitrary regular functions and biquaternions that admit the convolution. The paper notes that the convolutions may also be taken for singular biquaternions, such as simple and double layers, under the layer convolution rules, and closes the section with the speculation that such clouds "apparently describe ball lightning".

Two structural remarks belong on the free-photon formula, which is the more general of the two. It sums the four polarisation types at each frequency, each with its own arbitrary weight and its own arbitrary biquaternion, and it integrates an arbitrary angular distribution over the sphere $\|\mathbf{k}\| = \omega$. Every ingredient of a beam of light is therefore an input — the band, the angular weights, the polarisation weights and the biquaternions — so the formula describes the set of possible light rather than any light, and that is the boundary-condition gap in its light-theoretic form. Second, the arbitrary biquaternion enters by convolution, which is the structural biquaternion of the harmonic article's crystal construction under another name; the corpus records the construction once and cross-refers rather than counting it twice. And because the four plane modes span a two-dimensional space (the rank computation above), a sum over the four types with four independent weights carries twice the degrees of freedom the free solution space has: two of the four weights are not independent of the other two. That is the same redundancy the plane-wave section found, reappearing where the paper uses the four modes most freely.

The corpus's assessment is short. The superposition is linear and the "cloud of elementary photons" is the spectral integral of the field over a frequency band; that is the standard Fourier and Green-function representation of the field of a source, and a monochromatic field being a continuum of frequencies, the band is an input — the visible range — rather than a result. The layer remark is the corpus's own layer calculus Distributions on Surfaces, Layers, and Jump Conditions under another name. The ball-lightning sentence is a speculation: the paper offers it no model, no equation and no number, and the corpus records it as a remark and not as a claim. The name "photon" is likewise a reading of a monochromatic classical wave; the programme's photon carries no quantum, no spin and no statistics, exactly as its earlier "particles" carried none.

One remark of the paper's is worth keeping for a different reason. It proposes the construction of light phenomena from these formulas as a programming exercise for students, a variety of solutions being free to compute. That is a fair description of what the formulas are — a parametrised family of solutions with the physics read into the parameters — and it is the most candid sentence in the paper, because it states the status of the construction without claiming more for it.

No Pure Gravitational Waves

The paper's physical conclusion is a negative one about gravity. From its formulas for the ether and the photons it states that no pure gravitational waves exist: any change of the gravitational field entails a change of the electromagnetic field, a photon is an EGM wave that contains a gravitational component, and it is that component which determines the light pressure.

Two separate things are being claimed, and they have different standing.

The first — that the two fields cannot be varied independently — is true inside the construction and is a consequence of the construction. The electric and gravimagnetic halves are the real and imaginary parts of one biquaternion, and in the programme's own free modes they are locked together: the longitudinal mode has both non-zero and parallel, the tensional mode has both non-zero and perpendicular, and the scalar-potential mode has both non-zero whatever the phase (all recomputed). No mode of the construction has one half identically zero. But that is a statement about a biquaternion, not about the world; it holds because the two fields were put into one object, and the programme's identification of that object with nature is the hypothesis recorded in the companion article.

The second — that nature contains no pure gravitational waves — is testable and it fails. A pure gravitational wave is precisely what the direct detection of gravitational waves observes: the detectors' events arrive with no electromagnetic counterpart, and the one event with a counterpart, GW170817, is famous because the joint observation was exceptional. The corpus's own Gravitational Waves in Biquaternionic Form has the linearised gravitational wave with its two polarisations, and that wave is not a biquaternion of the EGM kind. The claim is therefore the most exposed statement in the whole programme, and it is the one the corpus cannot leave unremarked.

The light-pressure remark is of a weaker kind. The pressure of light on a surface follows in the standard theory from the momentum the electromagnetic field carries, with no gravitational component required; attributing it to the gravitational half is a re-reading of a known effect, and a re-reading of an effect already explained is not evidence for the reading.

What the Corpus Takes and What It Does Not

The corpus takes four things, and they are small.

The one-line description of what the EGM field is: the classical biquaternion Maxwell form with the zero-scalar-part restriction dropped. Nothing else in the programme states its content as economically, it is exactly true as algebra, and it locates the whole difference from classical electromagnetism in a single constraint — which is also, by the plane-wave section, exactly where the programme's extra content sits.
The exact plane-mode amplitudes, and with them the observation that in this sector the electric and gravimagnetic fields are locked into one biquaternion, so no mode has one half absent. That is a structural fact about the programme's own object, and the tensional mode is its cleanest instance.
The longitudinal mode as the programme's characteristic prediction. The programme's free photons are not all transverse, and this is the sharpest thing it says; it is also the thing vacuum electromagnetism forbids, so it is a prediction that is false if the programme's field is the electromagnetic field. The corpus's shock and front articles already carry the same longitudinal content from the programme's earlier papers, and the reader should see the family as one claim.
The reduction of a mono-chromatic emitted field to a source convolved with one kernel. This is the retarded Helmholtz representation, and it is correct; it is the corpus's own Green-function machinery and it needs no new verification.

The corpus does not take the ether, the photon reading, the cloud, the ball lightning or the gravitational-wave claim. The ether adds a name and an ontology to an unchanged equation. The photon is a monochromatic classical wave under a quantum name. The cloud is a spectral integral under a particle name. The ball-lightning remark has nothing behind it. And the gravitational-wave claim is contradicted by the observation it would have to survive. The programme's own summary of its value is that a biquaternion description "allows determining its characteristics at any point in space-time what is impossible in models of quantum mechanics" — which is true of any classical field description, and is the same claim the harmonic article has already examined.

Summary

The 2020 formulation of the electro-gravimagnetic programme names the EGM field the ether and reads its monochromatic solutions as photons. The programme's own statement of its content is one line: take the classical biquaternion form of Maxwell's equations, whose field biquaternion is required to have no scalar part, and drop that restriction; the returned scalar part is the ether density and the vector part carries the electric and gravimagnetic fields. The algebra is the companion articles': the postulate $\nabla^{+}\tilde{A} = \tilde{\Theta}$, the monochromatic reduction $\nabla^{-}_{\omega}\tilde{\Phi} = i\tilde{\Theta}$ with $\nabla^{\pm}_{\omega} = \omega \pm \nabla$, and the factorisation $\nabla^{-}_{\omega}\circ\nabla^{+}_{\omega} = \omega^{2} + \Delta$, which makes the first-order photon equation invertible by the outgoing Helmholtz kernel $\psi = -e^{\,i\omega r}/(4\pi r)$. The kernel gives the elementary photon of a concentrated charge, a spherical wave with amplitudes $\omega/(4\pi r)$ and $\sqrt{\omega^{2}+1/r^{2}}/(4\pi r)$, an energy density falling as $r^{-2}$ and a radial flux; the homogeneous equation gives free photons as the four plane modes $\omega\varphi(1+i\mathbf{e}_{k})$ and $\varphi(\omega e_{j}-ik_{j}+i[\mathbf{k},e_{j}])$, of which the first is longitudinal with the two field vectors parallel and the others are transverse exactly when their potentials are; and a source's field is the source convolved with the elementary photon, which the paper reads as light, a cloud of elementary photons over a visible band.

Every one of those steps was recomputed, and every one is the corpus's machinery under the programme's names: the Helmholtz factorisation is the harmonic article's, the outgoing kernel is the standard one, the convolution is the Green-function representation, and the "cloud" is a spectral integral. The reduction is also a change of type, hyperbolic to elliptic, which the paper states and which is precisely why a boundary condition would be needed and why its absence is the programme's root gap. What is genuinely the programme's is the packaging of the two fields into one biquaternion and the two readings that follow from it: the longitudinal photon, which vacuum electromagnetism forbids, and the claim that no pure gravitational waves exist, which the direct detection of gravitational waves without electromagnetic counterparts contradicts. Beside those, the ether is a name, the photon is a monochromatic wave, and the light pressure is an effect already explained. The corpus records the formulation because the companion articles raise the question of what the programme does with the field, and this is its answer, and because the field-primary reading is the clearest statement of what the programme assumes.

Summary of Notation

Symbol	Meaning
$\tilde{A} = i\alpha + \boldsymbol{\mathcal A}$	Biquaternion of tension; $\alpha$ is the density of the ether
$\boldsymbol{\mathcal A} = \sqrt{\epsilon}\mathbf{E} + i\sqrt{\mu}\mathbf{H}$	A-field, the companion article's field
$\alpha$	Scalar part of the tension, named the density of the ether
$\tilde{\Theta} = i\rho + \mathbf J$	Charge–current biquaternion (corpus symbol)
$\nabla^{\pm} = \partial_\tau \pm i\nabla$	Mutual complex gradients of the postulate
$\nabla = e_{1}\partial_{x}+e_{2}\partial_{y}+e_{3}\partial_{z}$	Pure-vector gradient, left quaternion multiplication
$\nabla\circ\nabla = -\Delta$	The square of $\nabla$
$\nabla^{\pm}_{\omega} = \omega \pm \nabla$	The paper's $\omega$-gradients; $\nabla^{\pm}_{\omega} = iD_{\mp}$ at fixed frequency
$\tilde{\Phi}(\mathbf{x},\omega)$	Biamplitude, $\tilde{A}(\tau,\mathbf{x}) = \tilde{\Phi}(\mathbf{x},\omega)e^{-i\omega\tau}$
$\nabla^{-}_{\omega}\tilde{\Phi} = i\tilde{\Theta}$	The photon equation
$\nabla^{-}_{\omega}\circ\nabla^{+}_{\omega} = \omega^{2} + \Delta$	Helmholtz factorisation of the photon equation
$\psi = -e^{\,i\omega r}/(4\pi r)$	Outgoing Helmholtz kernel, radiation condition, $(\Delta+\omega^{2})\psi = \delta$
$\tilde{\Phi}^{0}$	Elementary photon, the kernel of the source-to-field map
$\alpha_{0}$, $\boldsymbol{\mathcal A}_{0}$	Scalar and vector amplitudes of the elementary photon (the paper writes $\rho_{0}$ and $\\|\Phi_{0}\\|$)
$\tilde{\Phi}_{\mathbf{k}j}$	Free plane modes, $j = 0,1,2,3$
$\mathbf{e}_{k} = \mathbf{k}/\omega$	Unit wave direction, $\\|\mathbf{k}\\| = \omega$
$\varphi = e^{\,i(\mathbf{k},\mathbf{x})}$	Plane Helmholtz potential
$\alpha = -k_{j}\varphi$	Ether-density amplitude of the $j$-th mode; zero iff $e_{j}\perp\mathbf{k}$
$\mathbf{e}_{x} = \mathbf{x}/r$	Radial unit vector
$\Lambda$, $\tilde{\Xi}_{\Lambda}$	Light and its energy–momentum, a spectral integral over the band
$\mathrm{O}$, $\tilde{\Xi}_{\mathrm{O}}$	Free-photon cloud and its energy–momentum (the paper writes omicron)
$b_{j}$, $\Omega_{j}$	Arbitrary angular weight and structural biquaternion of the $j$-th free-photon type
$\tilde{\Phi}^{*} = \bar{s} - \bar{\mathbf V}$	Vector-negating, coefficient-conjugating conjugation