Subluminal and Superluminal Electromagnetic Waves and the Lepton Mass Spectrum

Introduction

This article records an external programme, that of Waldyr Alves Rodrigues Jr. and Jayme Vaz Jr. as presented in Subluminal and Superluminal Electromagnetic Waves and the Lepton Mass Spectrum (arXiv:hep-th/9607231, 1996). It works in the Clifford bundle of differential forms $C\ell(M)$ and studies the single equation

$$ \partial F = 0 $$

for a 2-form field $F$, together with the spinor sector of the same algebra. From that starting point it claims three things. The first is that the free Maxwell equation has subluminal and superluminal solutions that are undistorted progressive waves and carry non-null field invariants and longitudinal field components — a family larger than the transverse plane waves of the textbook account. The second is that, writing the field as a spinor bilinear $F=\psi\gamma_{21}\widetilde{\psi}$, the free Maxwell equation is equivalent to a non-linear Dirac–Hestenes equation for $\psi$, and that this non-linear equation linearises exactly when the Takabayasi angle $\beta$ is constant; the same section shows the massless Dirac equation to be a Maxwell equation with an axial (magnetic) current. The third is a lepton mass spectrum: at the Takabayasi angles $\pi/2$ and $3\pi/2$ the equation yields the muon mass, and with one further hypothesis the tau mass.

The corpus records the programme and does not endorse it. Its value to this corpus is twofold. The structural part — the spinor representation of the field, the criterion that selects the linear equation, and the axial-current form of the massless equation — is rigorous and checkable, and it bears directly on questions the corpus states elsewhere: the invariant classification of The Field-Strength Biquaternion and Its Invariants, which classifies the field by its two invariants and notes that they are not conserved by the free evolution, but does not exhibit the family of free solutions that carry non-null invariants; the Dirac–Hestenes formulation of The Dirac–Hestenes Equation and Spacetime Algebra in Biquaternionic Form, which already carries the Yvon–Takabayasi angle as an "angle zero for a pure electron state" but not the criterion that makes the equation linear; and the axial current of The Magnetic Monopole in Biquaternionic Form, which the massless equation here reproduces. The mass claim is of a different kind — an algebraic mass-generation proposal — and is recorded with that standing, next to the other two the corpus already holds.

One convention differs from the corpus and is flagged once, so that formulas are not misread between the two. The source works in the Clifford bundle $C\ell(M)$ with the metric $g=\mathrm{diag}(+1,-1,-1,-1)$, the convention the corpus itself adopts for its gamma matrices. Its volume element $\gamma_5=\gamma_0\gamma_1\gamma_2\gamma_3$ satisfies $\gamma_5^2=-1$, and it sets $i\equiv\gamma_5$: the source's "$i$" in the electric–magnetic notation $F=\mathbf{E}+i\mathbf{B}$ is therefore the duality operator, the corpus's central imaginary unit, and not a second imaginary number. The source's fixed bivector $\gamma_{21}=\gamma_2\gamma_1$ is the spin plane that multiplies the spinor on the right, the role played in the corpus's Dirac–Hestenes article by the fixed element $I\sigma_3$. The even subalgebra $C\ell^+(M)$, in which the spinor $\psi$ lives, is the biquaternion algebra $\mathbb{B}$ under the corpus's dictionary; the source's route is nevertheless the Clifford-bundle one, and the corpus records it as a different technical road to the same structural equivalences, cross-referencing rather than merging.

Maxwell's Equations in the Clifford Bundle

The plane wave and the duality operator

The free equation $\partial F=0$ is solved by the plane wave

$$ F = f\,e^{\gamma_5 kx}, $$

with $f$ a constant 2-form and $k$ the propagation vector. Applying $k$ on the left gives $kF=0$, and applying $k$ again gives $k^2F=0$, so the propagation vector is null,

$$ k^2=0 \quad\Longleftrightarrow\quad k_0=\pm|\mathbf{k}| . $$

Multiplying $kF=0$ on the right by $F$ gives $F^2=0$: the plane wave has null invariants. Both facts are the standard ones, and the source's point in restating them is the contrast with what follows — the plane waves of the free equation sit on the special null case, and the source then exhibits solutions that do not.

The role of the volume element is worth one paragraph, because the source insists on it. Since $\gamma_5^2=-1$, the exponential splits as $e^{\gamma_5 kx}=\cos kx+\gamma_5\sin kx$, so

$$ F = f\cos kx + \gamma_5 f\sin kx . $$

Writing $F=\mathbf{E}+\gamma_5\mathbf{B}$ and $f=\mathbf{e}$, this is $\mathbf{E}+\gamma_5\mathbf{B}=\mathbf{e}\cos kx+\gamma_5\mathbf{e}\sin kx$: the electric and magnetic parts are out of phase by a quarter period, and the "$i$" of the standard complex formulation is precisely the $\gamma_5$ that produces that phase, not a formal device. The source calls this the "secret" that the complex notation hides, and it is the observation on which the rest of its plane-wave discussion is built. That $\gamma_5$ is the duality operator is also why the duality rotation $F\mapsto e^{\gamma_5\beta}F$ of Exercise: Duality Rotation and the Riemann–Silberstein Vector appears below as the parameter that governs the non-linear equation.

The Hertz potential

The solutions of the next two sections are generated by a Hertz potential. The source's theorem is that if a 2-form $\pi$ satisfies the wave equation $\partial^2\pi=0$ and the vector potential is taken as $A=-\delta\pi$ with $\delta$ the coderivative, then $F=\partial A$ solves the Maxwell equation $\partial F=0$. The proof is short — $\delta A=-\delta^2\pi=0$, so $F=dA$ and $\partial F=(d-\delta)(d-\delta)A=-(d\delta+\delta d)A=0$ since $d\delta\pi=-\delta d\pi$ follows from $\partial^2\pi=0$ — and it is the technical device that converts a scalar solution of the wave equation into an electromagnetic field with the desired properties.

Subluminal Solutions

The stationary solution

The source takes the Hertz potential to be $\pi=\Phi\,\gamma_1\gamma_2$ with the scalar part separated as

$$ \Phi(t,\mathbf{x}) = \varphi(\mathbf{x})\,e^{\gamma_5\Omega t}, $$

so that the wave equation becomes the Helmholtz equation $\nabla^2\varphi+\Omega^2\varphi=0$. The simplest solution is the spherical one $\varphi(\mathbf{x})=C\,\sin(\Omega r)/r$, and the resulting stationary field $F_0$ is regular at the origin and vanishes at infinity — a localised field configuration at rest in the adapted frame. Written in the electric–magnetic split $F_0=\mathbf{E}_0+\gamma_5\mathbf{B}_0$ it is

$$ \mathbf{E}_0=-\mathbf{W}\sin\Omega t, \qquad \mathbf{B}_0=\mathbf{W}\cos\Omega t, $$

with $\mathbf{W}(\mathbf{x})$ a fixed vector field built from spherical harmonics and powers of $1/r$.

Three of its properties are the point of the construction. First, the invariant $I_1=\mathbf{E}_0^2-c^2\mathbf{B}_0^2$ and the pseudo-invariant $I_2=\mathbf{E}_0\cdot\mathbf{B}_0$ are not zero: $\mathbf{E}_0$ and $\mathbf{B}_0$ are both parallel to $\mathbf{W}$ and oscillate a quarter period apart, so $I_2=\mathbf{E}_0\cdot\mathbf{B}_0$ is generally non-zero while $I_1=-|\mathbf{W}|^2\cos 2\Omega t$ oscillates between the two signs. The field is of the generic electric–magnetic-parallel type, not the null type, and it is a free solution of the source-free Maxwell equation that carries a non-null field. Second, the field is not transverse: the source shows by boosting the stationary solution to a moving subluminal one that the propagation vector is no longer orthogonal to the field, in contrast to the plane wave. That is why its construction matters to claims about longitudinal electromagnetic waves, for which the source cites Evans. Third, $\mathbf{W}$ satisfies the Beltrami (force-free) equation

$$ \mathrm{rot}\,\mathbf{W} = -\Omega\,\mathbf{W}, $$

so that the configuration is a helical force-free field, the object that appears in plasma physics and in the earlier "force-free" literature the source cites. The energy integral of $F_0$ alone diverges, and the source notes that finite-energy wave packets over a distribution of frequencies can be formed instead.

Purely electromagnetic particles

The subluminal solutions bear on a question older than the formalism. Einstein, following Poincaré and Ehrenfest, asked whether a purely electromagnetic particle can exist: a field configuration $F_p$ with current $J_p$ obeying $\partial F_p=J_p$ and the subsidiary condition $J_p\cdot F_p=0$. In vector form the condition reads $\rho_p\mathbf{E}_p=0$, $\mathbf{j}_p\cdot\mathbf{E}_p=0$, $\mathbf{j}_p\times\mathbf{B}_p=0$, and Einstein concluded that the only solution is $J_p=0$ — but, the source observes, only if $J_p$ is time-like. If $J_p$ is allowed to be space-like there is a frame in which $\rho_p=0$, and a solution with $\mathbf{E}_p\cdot\mathbf{B}_p=0$ and $\mathbf{j}_p=kC\,\mathbf{B}_p$ exists, with $k=\pm1$ a chirality. The connection with the section above is that the free stationary field $F_0$ can be split as a field plus a current, $\partial F_0=0$ being equivalent to $\partial F_p=J_p$ for a suitable space-like $J_p$; the free equation then models a standing purely electromagnetic particle. This is the source's route into the older programme of Waite, Barut and Zeni, which it cites; it is recorded here as a programme item, not as an established particle model.

Superluminal Solutions: the X-Wave

The same Hertz-potential device produces superluminal solutions. The scalar seed is taken from the family of X-waves,

$$ \Phi_{X_n}(t,\mathbf{x}) = e^{in\theta}\int_0^\infty B(k)\,J_n(k\rho\sin\eta)\,e^{-k[a_0-i(z\cos\eta-t)]}\,dk, $$

with $\eta$ the axicon angle and $J_n$ the Bessel function. With $B(k)=a_0$ the integral is elementary and gives a closed-form superluminal electromagnetic X-wave (SEXW), a localised beam-like solution of the wave equation in which $z\cos\eta-t$ replaces the plane-wave phase — the field pattern travels faster than $c$ along the axis while the individual wave fronts do not. The source treats these as valid solutions of the free Maxwell equation and points to Rodrigues and Lu for the physical devices that would generate them.

The superluminality must be read with the corpus's usual care, and the source's own phrasing invites the mistake. The X-wave is a localised pattern whose amplitude profile advances superluminally; this is the pattern (or phase) speed of a finite wave packet and is not a speed of energy, information or signal. A free-field solution with a superluminal pattern is not in conflict with causality, and the corpus records the claim in that sense only. The same construction yields superluminal solutions of the Weyl equation, which the source offers as a possible reading of neutrino puzzles; that part is speculative and is not carried.

The Field as a Spinor Bilinear

Two constraints, six degrees of freedom

The structural heart of the paper is the representation of the field as a spinor bilinear. With $\psi$ a Dirac–Hestenes spinor, an element of the even subalgebra $C\ell^+(M)$, the field is written

$$ F = \psi\,\gamma_{21}\,\widetilde{\psi}, $$

where $\widetilde{\psi}$ is the reverse of $\psi$. The counting matters: a general even spinor has eight real degrees of freedom, while the field $F$ has six, so two constraints must be imposed on $\psi$ for the correspondence to be one-to-one up to a normalisation. The source chooses them as $\partial\cdot j=0$ and $\partial\cdot g=0$, where $j$ and $g$ are the vector and axial-vector currents read off the derivatives of $\psi$. This is the same eight-versus-six counting the corpus records in its Dirac–Hestenes article, which reaches it through the parameters of the rotor decomposition; the constraints are the source's extra data.

Substituting $F=\psi\gamma_{21}\widetilde{\psi}$ into the generalised Maxwell equation $\partial F=J$ and using the canonical decomposition $\psi=\rho^{1/2}e^{\gamma_5\beta/2}R$ — the rotor $R$, the density $\rho$, and the Takabayasi angle $\beta$ — turns the field equation into a spinorial equation for $\psi$:

$$ \partial\psi\gamma_{21} = \frac{e^{\gamma_5\beta}}{2\rho}\,J\,\psi + \lambda\,\psi\gamma_0 + \gamma_5\kappa\,\psi\gamma_0 , $$

with $\lambda$ and $\kappa$ the two scalars built from the bivector $\Omega=v^\mu\Omega_\mu$. Its free case $J=0$ is

$$ \partial\psi\gamma_{21} = \lambda\,\psi\gamma_0 + \gamma_5\kappa\,\psi\gamma_0 , $$

which has the shape of the Dirac–Hestenes equation but with two scalar coefficients instead of one mass. The two-constraint condition forces their ratio to a constant, and after a duality rotation by that constant the relation becomes $\kappa/\Lambda=\tan\beta$, i.e. the two coefficients are the single mass times the cosine and sine of the Takabayasi angle.

The non-linear Dirac–Hestenes equation

Written in full, the equation for $\psi$ is a non-linear Dirac–Hestenes equation

$$ \partial\psi\gamma_{21} = \Lambda\,\psi\gamma_0 e^{\beta\gamma_5} + \gamma_5 K\,\psi\gamma_0 e^{\beta\gamma_5} + \frac{1}{2\rho}\,e^{\beta\gamma_5} J\,\psi . $$

The non-linearity is exactly the factor $e^{\beta\gamma_5}$: when $\beta$ is constant the factor is a constant duality rotation and the equation becomes linear. Concretely, with $\psi=\varphi\,e^{\gamma_5\beta/2}$ and $\varphi=\sqrt{\rho}\,R$, the phase factor on the right is cancelled by the phase factor of the decomposition — $e^{\gamma_5\beta/2}\gamma_0=\gamma_0e^{-\gamma_5\beta/2}$ because $\gamma_5$ anticommutes with $\gamma_0$ — and the free equation reduces to the linear Dirac–Hestenes equation

$$ \partial\varphi\gamma_{21} = m\,\varphi\gamma_0 , $$

with $\Lambda=m$ identified as the mass. The criterion is therefore sharp, and it is the source's key structural statement: the Maxwell equation for the field is equivalent to the linear Dirac equation for the spinor exactly when the Takabayasi angle is constant, and the particular values $\beta=0$ and $\beta=\pi$ select the electron and the positron. The plane-wave solutions of the corpus's own Dirac exercise have constant $\beta$ — zero for the positive-energy ones — so they sit on precisely this linearisable class, and the source supplies the general reason why.

Two consequences the source draws from the same algebra are worth recording. The first is a superselection rule: the superposition principle survives only for spinors of the same $\beta$, because $e^{\beta\gamma_5}$ is not linear across different angles; the source reads this as particles and antiparticles forming separate superselection sectors. The second is the momentum relation. Multiplying the non-linear equation by $\psi^{-1}$ gives $p=mv$ for every angle $\beta$, whereas the linear Dirac–Hestenes equation gives $p=e^{\beta\gamma_5}mv$, which for $p$ and $v$ real forces $\beta=0$ or $\beta=\pi$ and thus the choice between positive and negative energy. In the non-linear reading there is no negative-energy problem to interpret: the equation gives $p=mv$ throughout, and the source proposes to reinterpret the energy projector as a particle projector instead.

The projection operators

The framework of projectors is what carries the mass spectrum, so it is stated precisely. The energy projector of the linear theory is

$$ \Lambda_\pm(\psi) = \tfrac12\left(\psi \pm \gamma_0\psi\gamma_0\right), $$

and the source introduces the one-parameter family

$$ \Lambda_\beta(\psi) = \tfrac12\left(\psi + e^{\gamma_5\beta}\gamma_0\psi\gamma_0\right), $$

which reduces to $\Lambda_+$ at $\beta=0$ and to $\Lambda_-$ at $\beta=\pi$. It is a projector for every $\beta$. The reason is a single identity: the map $\psi\mapsto e^{\gamma_5\beta}\gamma_0\psi\gamma_0$ squares to the identity, because $\gamma_5$ anticommutes with $\gamma_0$ and $\gamma_0^2=1$, so that $\tfrac12(1+T)$ is a projector whenever $T^2=1$. The shift $\beta\mapsto\beta+\pi$ negates $T$ and therefore exchanges the projector with its complement:

$$ \Lambda_\beta + \Lambda_{\beta+\pi} = 1, \qquad \Lambda_\beta\,\Lambda_{\beta+\pi} = 0 . $$

This is the projector statement the next section uses, and it is the corrected form of the relation printed in the source (which reads $\Lambda_\beta=\Lambda_{\beta+\pi}$; see the companion .context for the check).

Maxwell and the Massless Dirac Equation

The equivalence runs in the other direction as well, and that direction carries the axial current. The massless Dirac equation $\partial\psi=0$ for $\psi$ in the even subalgebra is equivalent to a generalised Maxwell equation

$$ \partial F = J_e - \gamma_5 J_m = J , $$

with both an electric current $J_e$ and an axial (magnetic) current $-\gamma_5J_m$. For the positive-parity eigenstate $\psi^\uparrow$ the electric current vanishes, $j_e=0$, and the equivalent Maxwell equation reads

$$ \partial F^\uparrow = -\gamma_5 J_m , $$

that is, a massless Dirac field is a Maxwell field with a magnetic current and no electric current. The mechanism is the parity split: the source shows that $\psi^\uparrow$ decouples into a pair of Weyl fields of opposite charge, $\partial\psi^\uparrow_+ + g\gamma_5B\psi^\uparrow_+=0$ and $\partial\psi^\uparrow_- - g\gamma_5B\psi^\uparrow_-=0$, so that a single $\psi^\uparrow$ coupled to a potential describes a particle of charge $+g$ and a particle of charge $-g$ — which the source identifies with the pair of magnetic monopoles of Lochak's construction. This is the exact counterpart of The Magnetic Monopole in Biquaternionic Form, and it is a further route in the corpus by which a magnetic current appears in a spinorial or quaternionic reformulation of Maxwell's equations. The source adds one structural remark that the corpus records as a curiosity: the Tetrode energy–momentum one-form of the non-linear equation is a kind of square root of the Maxwell energy–momentum one-form $S_\mu=\tfrac12 F\gamma_\mu F$.

The Lepton Mass Spectrum

The mass formula

The final section of the paper uses the projector family to propose a mass spectrum. The electron is the free configuration $F_e$ with $\beta=0$ (the positron with $\beta=\pi$); the muon is modelled as the electromagnetic configuration $F_e+F^\uparrow$, and the magnetic current $J_m=c\,q_m\,\varphi\gamma_0\widetilde{\varphi}$ with magnetic charge $q_m$ is inserted into the non-linear equation. In Gaussian units, with the field normalised as $F=k\,(e\hbar/mc)\,\psi\gamma_{21}\widetilde{\psi}$ and the constant fixed at $k=2\pi/3$, the equation for $\varphi$ becomes a linear Dirac–Hestenes equation precisely when the $\gamma_5$-coefficient vanishes,

$$ N=-\frac{3m}{e}q_m\cos\beta = 0 \quad\Longleftrightarrow\quad \beta=\frac{\pi}{2}\ \text{or}\ \frac{3\pi}{2}, $$

and the effective mass is then

$$ M = m + \frac{3m}{e}\,q_m . $$

Dirac's quantisation condition for the monopole, $eq_m/\hbar c=n/2$, gives $q_m=n\,e/(2\alpha)$ with $\alpha=e^2/\hbar c$ the fine-structure constant, and hence

$$ M = m\left(1+\frac{3n}{2\alpha}\right). $$

At $n=1$ this is the muon, $M=206.55\,m_e$: the source reports the muon mass to better than a tenth of a percent from an equation in which the only inputs are the electron mass, the fine-structure constant and a monopole quantum number. An extra hypothesis (the tau as an excited state with $n=p^4$) extends this to the spectrum

$$ M_p = m\left(1+\frac{3}{2\alpha}\sum_{l=0}^{p} l^4\right), $$

which is a formula found earlier by Barut (1980) by quite different arguments. It gives $M_0=m_e$, $M_1=m_\mu$, $M_2=m_\tau$, and a fourth state $M_3\approx2.0\times10^4\,m_e\approx10.3$ GeV that has not been observed. The projector values $\beta=0,\pi,\pi/2,3\pi/2$ are read as the electron, positron, muon and antimuon respectively — the discrete phase of the spinor-to-field map becomes a discrete mass.

The numbers the formula gives

The formula is elementary, and the corpus has recomputed it (see the companion .context for the script). With $\alpha=1/137.036$ and the electron mass as unit:

State	$\sum_{l=0}^{p}l^4$	$M_p/m_e$	$M_p$ (MeV)	Nearest measured lepton
$p=0$	$0$	$1.000$	$0.51$	electron, $0.511$
$p=1$	$1$	$206.55$	$105.5$	muon, $105.7$
$p=2$	$17$	$3495.4$	$1786$	tau, $1777$
$p=3$	$98$	$20145$	$10294$	none known

The muon reproduction is good to about $0.1\%$ and the tau to about $0.5\%$, which is the sense in which the source calls the agreement excellent. Two printed slips in the source were found in the same check and are recorded in the .context: the tau ratio is printed as $3845$ where the formula and the source's own printed energy give $3495$, and the projector relation of the previous section is printed without its complement. Neither affects the formula the source proposes; both would mislead a reader who transcribes the paper's printed numbers.

The honest standing of the mass claim is that it is a third algebraic mass-generation proposal in the corpus, alongside the eigenvalue sector of the electro-gravimagnetic programme and the eigenvalue equations of Gsponer–Hurni. Like the others it is an algebraic relation that reproduces one or two charged-lepton masses, not a spectrum derived from first principles, and the single formula that extends it to a "spectrum" rests on the ad hoc identification $n=p^4$. The test the corpus applies to all three is the same: whether the full charged-lepton and quark spectrum follows with no further input. Until then the formula is recorded as a striking numerical coincidence with a structural origin, not as a derivation.

Status, Contrasts, and Boundaries

What the programme establishes is the structural half. That the free Maxwell equation has free solutions with non-null invariants is true and is a genuine correction to the reflex that "free electromagnetic waves are null transverse plane waves"; the classification of The Field-Strength Biquaternion and Its Invariants admits such fields but does not exhibit them, and the subluminal construction here supplies the missing example. That the field is a spinor bilinear and that the Maxwell equation for the field is a non-linear Dirac–Hestenes equation which linearises precisely at constant Takabayasi angle is likewise a clean structural statement, and it explains from a second direction why the corpus's plane-wave Dirac solutions are simple. That the massless Dirac equation is a Maxwell equation with an axial current is the third, and it reproduces the corpus's monopole sector.

The boundaries are these. The Clifford-bundle route is an external formalism; the corpus's own treatment is the biquaternion one, and the two are related through the even-subalgebra dictionary rather than identified. The "superluminal" of the X-wave is a pattern speed of a localised solution, not a signalling speed, and must not be read otherwise. The purely electromagnetic particle is a programme item revived from Einstein and Waite, not a constructed particle. And the mass spectrum is an algebraic proposal whose extension to a full spectrum is a hypothesis, tested above against the two masses it does reproduce.

Summary

The programme of Rodrigues and Vaz works in the Clifford bundle and presses the free Maxwell equation $\partial F=0$ in two directions. It exhibits subluminal and superluminal undistorted progressive wave solutions with non-null invariants and longitudinal components, built from a Hertz potential and a spherical Helmholtz solution, and connects the free stationary solution to the old question of purely electromagnetic particles. It represents the field as the spinor bilinear $F=\psi\gamma_{21}\widetilde{\psi}$ and shows the Maxwell equation to be a non-linear Dirac–Hestenes equation, with the Takabayasi angle as the knob: constant $\beta$ makes it linear, $0$ and $\pi$ make it the electron and positron, $\pi/2$ and $3\pi/2$ make it the muon. In the other direction the massless Dirac equation is a Maxwell equation with an axial (magnetic) current, which is the route by which a monopole pair appears. The lepton mass formula $M_p=m(1+\frac{3}{2\alpha}\sum_{l=0}^{p}l^4)$ reproduces the muon and tau masses to within a percent and predicts a fourth charged lepton near $10.3$ GeV; it is Barut's formula recovered from a geometric phase, and it is a numerical coincidence with a structural origin until a full spectrum follows.

Summary of Notation

Symbol	Meaning
$C\ell(M)$	Clifford bundle of differential forms over Minkowski spacetime
$C\ell^+(M)$	Its even subalgebra; the biquaternion algebra $\mathbb{B}$ under the corpus's dictionary
$g=\mathrm{diag}(+1,-1,-1,-1)$	Metric; the corpus's gamma-matrix convention
$\gamma_\mu$	Clifford generators, $\{\gamma_\mu,\gamma_\nu\}=2g_{\mu\nu}$
$\gamma_5=\gamma_0\gamma_1\gamma_2\gamma_3$	Volume element (pseudoscalar); $\gamma_5^2=-1$; the source's "$i$" and the corpus's central imaginary unit
$\gamma_{21}=\gamma_2\gamma_1$	Fixed spin-plane bivector multiplying the spinor on the right
$\partial=\gamma^\mu\partial_\mu$	Dirac operator; $\delta$ the coderivative, $d$ the exterior derivative
$F$, $F_0$, $F_e$, $F^\uparrow$	Field 2-form; the stationary subluminal solution; the electron field; the positive-parity field
$\pi$, $A$	Hertz potential; vector potential, $A=-\delta\pi$
$f$, $k$, $x$	Constant 2-form; propagation vector ($k^2=0$); position
$\mathbf{E}_0,\mathbf{B}_0,\mathbf{W}$	Fields of the stationary solution and their common vector, $\mathrm{rot}\,\mathbf{W}=-\Omega\mathbf{W}$
$\Omega$	Intrinsic frequency of the subluminal solution ($\Omega$ in the Helmholtz equation $\nabla^2\varphi+\Omega^2\varphi=0$)
$\Phi_{X_n}$, $\eta$	Scalar X-wave seed; axicon angle
$\psi$, $\widetilde{\psi}$	Dirac–Hestenes spinor (even element) and its reverse
$\psi=\rho^{1/2}e^{\gamma_5\beta/2}R$	Canonical decomposition: rotor $R$, density $\rho$, angle $\beta$
$\beta$	Takabayasi (Yvon–Takabayasi) angle; $0,\pi$ electron/positron, $\pi/2,3\pi/2$ muon/antimuon
$j$, $g$, $J=J_e-\gamma_5J_m$	Vector current, axial current, total current with electric and magnetic parts
$\Lambda_\pm,\ \Lambda_\beta$	Energy projector $\tfrac12(\psi\pm\gamma_0\psi\gamma_0)$; one-parameter projector family
$S=\tfrac12\psi\gamma_{21}\psi^{-1}$	Bivector of the canonical analysis; $S_\mu=\tfrac12 F\gamma_\mu F$ the Maxwell energy–momentum
$\lambda,\kappa,\Lambda,K$	Scalar coefficients of the spinorial Maxwell and non-linear equations
$q_m$, $n$, $\alpha$	Magnetic charge, monopole quantum number, fine-structure constant $e^2/\hbar c$
$M_p=m(1+\frac{3}{2\alpha}\sum_{l=0}^{p}l^4)$	The lepton mass formula; $p=0,1,2$ electron, muon, tau