Zitterbewegung in Biquaternionic Form

Introduction

The Zitterbewegung — the "trembling motion" — is the oscillatory part of the position of a Dirac particle. It was found by Schrödinger in 1930, in the course of solving the free Dirac equation. In the single-particle theory the velocity operator has eigenvalues $\pm c$, so a Dirac particle cannot have a sharp velocity; a wave packet assembled from positive- and negative-frequency plane waves has a mean position whose expectation value carries a rapid oscillation. Its angular frequency is $2E/\hbar$, which equals $2mc^2/\hbar$ for a particle at rest, and its amplitude is of the order of half the reduced Compton wavelength $\hbar/(2mc)$.

The biquaternion framework of these articles proposes that the complex structure of $\mathbb{B}$ divides it into two sectors, $\mathbb{B} = \mathbb{M}_+ \oplus \mathbb{M}_-$. The natural suggestion is that this is the same two-component structure the Zitterbewegung exploits: that positive- and negative-energy components are the $\mathbb{M}_-$ and $\mathbb{M}_+$ sectors, and that the trembling is therefore the observable face of the complexification.

This article tests that suggestion rather than assuming it. The finding is mixed, and it is stated plainly from the outset. The oscillation can be derived and its frequency checked, and the mass does set the sector rotation: the rest-energy phase turns the two sectors into one another at $mc^2/\hbar$; but the identification of the $\pm$-energy components with the two sectors does not hold, because the energy-sign split is generated by an object that has no representative in the biquaternion algebra at all, while the sector split is generated by an operation that lives inside it. The two splittings have the same physical scale — the mass — and that is why the suggestion is tempting, but they are splittings of different objects by different operations. Section Does the Sector Split Do Real Work? gives the argument, and Section What Survives separates the claims that hold from the one that does not.

The article is organised as follows. The next section recalls the standard picture and its contested interpretation. The section after fixes the biquaternion notation and the plane-wave branches, following the parent. The following section derives the oscillation, states the exact operator identity, and checks frequency and amplitude on cases chosen here rather than on the case that suggested them. A short section then isolates the genuinely algebraic content — the mass term as a diagonal action with opposite signs on the sectors, and the rest-energy phase as a rotation between them — and shows that the Zitterbewegung phase is the square of that rotation. The central section tests the sector identification and reports its failure. The article closes with a summary and the surviving statements. Nothing here depends on the companion article on the electron; the connection is noted where relevant.

The Standard Picture

The free Dirac equation in the parent's spinor-module form is

$$ (i\hbar\gamma^\mu\partial_\mu - mc)\psi = 0, $$

or, in natural units $\hbar = c = 1$, $(i\gamma^\mu\partial_\mu - m)\psi = 0$. Multiplying by $\gamma^0$ and separating the time derivative puts it in Hamiltonian form,

$$ i\,\partial_t\psi = \hat{H}\psi, \qquad \hat{H} = c\,\boldsymbol{\alpha}\cdot\hat{\mathbf{p}} + \beta mc^2, $$

where $\beta = \gamma^0$ and $\boldsymbol{\alpha}^k = \gamma^0\gamma^k$; we restore $\hbar$ and $c$ when a physical statement is made. The matrices satisfy

$$ \{\alpha^j,\alpha^k\} = 2\delta^{jk}I, \qquad \{\alpha^k,\beta\} = 0, \qquad (\alpha^k)^2 = I, $$

so the velocity operator $c\boldsymbol{\alpha}$ has eigenvalues $\pm c$ and the Hamiltonian has eigenvalues $\pm E$ with $E = \sqrt{\mathbf{p}^2c^2 + m^2c^4}$.

The oscillation follows from the Heisenberg equation. Because $\{\alpha^k,\hat H\} = 2cp_k$ as an operator identity, the velocity in the Heisenberg picture has the exact form (natural units)

$$ \alpha^k(t) \;=\; \frac{cp_k}{\hat H} \;+\; e^{i\hat H t}\Big(\alpha^k(0) - \frac{cp_k}{\hat H}\Big)e^{-i\hat H t}, \qquad \hat H = c\,\boldsymbol{\alpha}\cdot\hat{\mathbf{p}} + \beta mc^2, $$

in which the first term is the constant drift velocity and the second is the trembling term. Its phase evolves as $e^{\pm 2iEt}$ on the two energy branches, so the angular frequency is $2E/\hbar$, and integrating once gives the position

$$ x^k(t) = x^k(0) + \frac{c^2p_k}{\hat H}\,t \;-\; \frac{i\hbar c}{2\hat H}\Big(e^{2i\hat H t/\hbar}-1\Big)\Big(\alpha^k(0) - \frac{cp_k}{\hat H}\Big). $$

At rest, $\hat H = \beta mc^2$ and the oscillating term has frequency $2mc^2/\hbar$ and amplitude of order $\hbar/(2mc)$, half the reduced Compton wavelength. The interference reading is immediate: a positive-frequency component $e^{-iEt/\hbar}$ and a negative-frequency component $e^{+iEt/\hbar}$ beat at $2E/\hbar$, and the trembling term is exactly that beat.

The interpretation of this result is contested, and the article does not settle it. Three positions are in the literature. First, the trembling is an artefact of forcing a single-particle interpretation on a theory whose solutions include both signs of energy: the Foldy–Wouthuysen mean position, the observable position for a positive-energy particle, has no trembling at leading order, so the oscillation is removed rather than seen. Second, the effect is not entirely empty: the same elimination of the small component that produces the mean position leaves the Darwin term at the next order in $1/c$, and the Darwin term and the Zitterbewegung are two faces of the same admixture of the negative-energy component — the exercise on the non-relativistic limit derives that term, and the derivation is not repeated here. Third, in quantum field theory the position operator of a single particle is not fundamental, the free-field Zitterbewegung is not directly observable, and the effect is reinterpreted (in some treatments absorbed, in others associated with virtual pair creation and annihilation). None of these readings denies the algebra above; they disagree about what it describes.

The Biquaternionic Setting

The notation is that of the parent and its read list. The biquaternion algebra is $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$, with basis $e_0 = 1, e_1, e_2, e_3$, $e_k^2 = -e_0$, scalar imaginary $i$, and biquaternionic gradient

$$ \tilde\nabla = e_0\,\partial_{ict} + e_1\,\partial_x + e_2\,\partial_y + e_3\,\partial_z. $$

The Hermitian conjugation is $\tilde Q^{*} = \overline{\tilde Q^{\natural}}$, and it defines the two sectors by their fixed points,

$$ \mathbb{M}_- = \{\tilde Q : \tilde Q^{*} = -\tilde Q\}, \qquad \mathbb{M}_+ = \{\tilde Q : \tilde Q^{*} = \tilde Q\}, \qquad \mathbb{B} = \mathbb{M}_+ \oplus \mathbb{M}_- . $$

The two sectors are exchanged by multiplication by $i$: $i\mathbb{M}_\pm = \mathbb{M}_\mp$. An element of $\mathbb{M}_-$ has imaginary scalar part and real vector part; an element of $\mathbb{M}_+$ has real scalar part and imaginary vector part. The quaternion subspace is $\mathbb{H}_{\mathbb{B}}$, and the trace formula used for expectation values is $\mathrm{Tr}(\tilde P\tilde H) = 2\,\mathrm{Sc}(\tilde P\tilde H)$.

The biquaternionic Dirac equation is the parent's linear, chirality-off-diagonal pair

$$ \tilde\nabla\tilde\Psi_R = m\tilde\Psi_L, \qquad \tilde\nabla^{\natural}\tilde\Psi_L = m\tilde\Psi_R, \qquad \tilde\Psi^\flat = -\tilde\Psi^{*}, $$

with massless limit $\tilde\nabla\tilde\Psi = 0$. The conjugation $\flat$ is the algebra's real structure, not the mass term. Its plane-wave solutions are $\tilde\Psi = \tilde\Psi_0\exp(\tilde k\tilde{Q})$ with wave biquaternion $\tilde k = e_0k_0 + \mathbf{k}$ and the single mass-shell condition

$$ \tilde k\tilde k^{\natural} = -\frac{m^2c^2}{\hbar^2}. $$

The two roots of that condition — positive and negative frequency — are the two branches $u^{(r)}(\mathbf{p})e^{-i(Et-\mathbf{p}\cdot\mathbf{x})}$ and $v^{(r)}(\mathbf{p})e^{+i(Et-\mathbf{p}\cdot\mathbf{x})}$ constructed in the parent, with $E = +\sqrt{\mathbf{p}^2c^2 + m^2c^4}$. The spin sums, normalizations, and the non-relativistic reduction belong to the parent; the present article uses only the branches and the mass shell.

One fact about the field decomposition will be needed throughout. For $\tilde\Psi = \tilde\Psi_+ + \tilde\Psi_-$ with $\tilde\Psi_\pm \in \mathbb{M}_\pm$, Hermitian conjugation acts as $\tilde\Psi_\pm^{*} = \pm\tilde\Psi_\pm$, so

$$ \tilde\Psi^\flat = -\tilde\Psi^{*} = -\tilde\Psi_+ + \tilde\Psi_- , $$

and $\flat$ acts as $-1$ on the $\mathbb{M}_+$ part and $+1$ on the $\mathbb{M}_-$ part: it is the algebra's real structure, and a coupling built on it — a Majorana-type mass — would carry those opposite signs. What the diagonal action of the real structure, the chirality coupling of the mass, and the rest-energy phase do to the sectors is the subject of the section The Complexification and the Rest-Energy Phase.

Recomputing the Oscillation

The matrices of the parent's Dirac basis are

$$ \gamma^0 = \begin{pmatrix} 0 & I_2 \\ I_2 & 0\end{pmatrix}, \qquad \gamma^k = \begin{pmatrix} 0 & \sigma_k \\ -\sigma_k & 0\end{pmatrix}, $$

satisfying $\{\gamma^\mu,\gamma^\nu\} = 2g^{\mu\nu}I_4$ with $g = \mathrm{diag}(+1,-1,-1,-1)$. From these, $\beta = \gamma^0$ and $\alpha^k = \gamma^0\gamma^k = \mathrm{diag}(-\sigma_k,\sigma_k)$, and the relations $\{\alpha^j,\alpha^k\} = 2\delta^{jk}I$, $\{\alpha^k,\beta\} = 0$, and $\hat H^2 = E^2I$ all follow and were recomputed.

The operator identity for the Heisenberg velocity is the exact statement

$$ \alpha^k(t) = \frac{cp_k}{\hat H} + e^{i\hat H t/\hbar}\Big(\alpha^k(0) - \frac{cp_k}{\hat H}\Big)e^{-i\hat H t/\hbar}, $$

in which $cp_k/\hat H$ denotes the operator $cp_k\hat H^{-1} = cp_k\hat H/E^2$ — not a multiple of the identity, since $\hat H$ is a matrix. Because $\hat H$ anticommutes with $\alpha^k - cp_k/\hat H$, the oscillating factor evolves on each energy eigenspace as $e^{\pm 2iEt/\hbar}$.

The identity was checked by direct matrix computation on a case chosen here rather than on the rest case that motivated it: momentum $\mathbf{p}/mc = (0.3,\,0.2,\,-0.1)$, so $|\mathbf{p}| = 0.374\,mc$ and $E = 1.0677\,mc^2$, at the three times $t = 0.017, 0.05, 0.23$ in units of $\hbar/(mc^2)$. The left side $e^{i\hat Ht/\hbar}\alpha^k e^{-i\hat Ht/\hbar}$ and the right side above agreed to a maximum absolute entry of $1.2\times10^{-16}$ over all three components and all times. The rest case is then a consequence, $E = mc^2$, not an input.

The frequency is $2E/\hbar$ on general grounds and $2mc^2/\hbar$ at rest. The numbers for the electron, $m_ec^2 = 0.510998950$ MeV and $\hbar = 6.582119569\times10^{-22}$ MeV s, are

$$ \omega_Z^{\text{rest}} = \frac{2m_ec^2}{\hbar} = 1.55269\times10^{21}\ \text{rad s}^{-1}, \qquad f_Z = \frac{\omega_Z}{2\pi} = 2.47118\times10^{20}\ \text{Hz}, $$

which is the standard value. On a relativistic case chosen independently, $\gamma = 2$ so $E = 2mc^2$, the formula gives $2E/\hbar = 4m_ec^2/\hbar = 3.10538\times10^{21}\ \text{rad s}^{-1}$ — twice the rest value, as the formula requires and as the rest case alone would not reveal. The oscillation amplitude is $\hbar/(2mc)$, which for the electron is $193.1$ fm, half the reduced Compton wavelength $386.2$ fm. Both the frequency and the amplitude are the standard ones; the biquaternion notation reproduces them without alteration.

It should be said clearly that this derivation uses $\beta$ and $\boldsymbol{\alpha}$ and therefore does not stay inside $\mathbb{B}$. That will matter in the test that follows.

The Complexification and the Rest-Energy Phase

There is a genuinely algebraic statement available, and it is the strongest positive result of the article. It has two parts.

The real structure is diagonal on the sectors. The decomposition of the setting section gives $\tilde\Psi^\flat = -\tilde\Psi_+ + \tilde\Psi_-$: the conjugation $\flat$ acts as $-1$ on the $\mathbb{M}_+$ part and $+1$ on the $\mathbb{M}_-$ part, so it preserves each sector and marks it with a sign. This is a property of the algebra's real structure, not of the mass term; the parent's mass is the linear, chirality-off-diagonal pair, which couples the two central ideals (the chiralities) and leaves each element's sector membership untouched. The operation that exchanges the two sectors is multiplication by the central $i$, $i\mathbb{M}_\pm = \mathbb{M}_\mp$.

The rest-energy phase rotates the sectors into one another. Factoring out the rest energy writes the free phase as

$$ e^{-imc^2t/\hbar} = \cos\!\Big(\frac{mc^2t}{\hbar}\Big)e_0 \;-\; \sin\!\Big(\frac{mc^2t}{\hbar}\Big)(ie_0), $$

and the two terms lie in different sectors: $e_0 \in \mathbb{M}_+$ and $ie_0 \in \mathbb{M}_-$ (an imaginary scalar is anti-Hermitian). The rest-energy phase is therefore a rotation between the sectors, at angular frequency $mc^2/\hbar$: it takes the $\mathbb{M}_+$ direction $e_0$ toward the $\mathbb{M}_-$ direction $ie_0$ and back. This is the sense in which the complexification rotates one sector into the other; the rotation is the central phase, whose frequency the mass fixes through the mass shell.

The Zitterbewegung frequency is exactly twice this rate. That is not a coincidence of arithmetic but an algebraic relation: the oscillating factor of the velocity identity is

$$ e^{2i\hat H t/\hbar} = \big(e^{i\hat H t/\hbar}\big)^2, $$

the square of the free-evolution phase, whose rest-frame value is the sector rotation above. So the trembling is not the sector rotation itself; it is the second harmonic of it. The factor of two is the statement that the beat of two branches of opposite frequency is twice the frequency of either branch, and it is the same factor by which the rest-energy phase and the Zitterbewegung frequency differ.

Two consequences deserve emphasis. First, since the rest-energy phase rotates the sectors and its rate is the mass, the scale of the trembling — the mass gap $2mc^2$ — is set by the same quantity that sets the sector rotation. That is the real content of the suggestion that the trembling and the complexification are related. Second, a relation of scale is not a relation of identity: the trembling is a beat of two energy branches, the sector rotation is a rotation in the algebra, and the following section shows these are different decompositions.

Does the Sector Split Do Real Work?

The claim to test is: the positive- and negative-energy components are the $\mathbb{M}_+$ and $\mathbb{M}_-$ sectors. The test fails, and the reason is structural rather than numerical.

The energy-sign split is generated outside $\mathbb{B}$. The projector onto positive energy at momentum $\mathbf{p}$ is

$$ P_+(\mathbf{p}) = \tfrac12\Big(I + \frac{c\,\boldsymbol{\alpha}\cdot\mathbf{p} + \beta mc^2}{E}\Big) = \tfrac12\Big(I + \frac{\hat H}{E}\Big), $$

and it contains $\beta = \gamma^0$. But $\gamma^0$ is an odd element of the Clifford algebra: it is a single generator, and $\mathbb{B}$ is isomorphic to the even subalgebra $\mathrm{Cl}^+_{1,3}$. An odd element has no representative in $\mathbb{B}$ at all, and a fortiori none in either sector. This is not a convention; it is tested by the parity grading. With $\omega = \gamma^0\gamma^1\gamma^2\gamma^3$ the volume element, even elements commute with $\omega$ and odd elements anticommute with it; direct computation gives $\{\beta,\omega\} = 0$ (so $\beta$ is odd) and $[\alpha^k,\omega] = 0$ (so the $\alpha^k$ are even). The energy-sign split and the sector split are therefore generated by different operations: $P_\pm(\mathbf{p})$ by $\beta$, which is outside $\mathbb{B}$, and the sector projectors $\tfrac12(I\pm{}^{*})$ by Hermitian conjugation, which is inside it.

They are also splits of different objects. The sector split is a decomposition of the algebra element $\tilde\Psi \in \mathbb{B}$ into $\tilde\Psi_+ + \tilde\Psi_-$. The energy-sign split is a decomposition of the spinor module — of the four-component Dirac spinor, with positive and negative energy each two-dimensional. The biquaternion algebra $\mathbb{B}\cong M_2(\mathbb{C})$ acts on a two-dimensional complex module; the four-component spinor is the direct sum of that module with its conjugate, and the energy projection mixes the two summands. Identifying a decomposition of the algebra with a decomposition of the module, without a stated dictionary, is a category error, and the parent article explicitly leaves the dictionary as a convention.

The natural dictionary the corpus supplies points the other way. For a normalized two-component spinor the parent identifies the rank-one Hermitian form

$$ \psi\psi^\dagger \;\longmapsto\; \tfrac12\big(e_0 + i\hat{\mathbf{n}}\big) \in \mathbb{M}_+, $$

the idempotent of the informational sector, independent of energy sign. Both rest-frame branches of the parent — $u^{(r)}(0)\propto(\xi^{(r)},0)$ and $v^{(r)}(0)\propto(0,\eta^{(r)})$ in the Dirac basis — reduce to a single two-component spinor and therefore to an $\mathbb{M}_+$ object. The sector $\mathbb{M}_-$ is where the current lives, $\tilde J = ic\,j^0e_0 + \mathbf{j} \in \mathbb{M}_-$, not where the negative-energy component lives. So on the dictionary actually available, the sector split tracks the operator–vector (state–current) distinction, not the energy sign, and both energy signs land in the same sector.

A momentum argument reinforces this. The sector split is pointwise and momentum-independent: whether an element is Hermitian is a property of the element, not of any Fourier mode it carries. The energy projection is momentum-dependent through $\boldsymbol{\alpha}\cdot\mathbf{p}$. A momentum-independent algebraic split cannot be a momentum-dependent spectral projection except in the trivial case $\mathbf{p} = 0$, and even there the two rest branches both land in $\mathbb{M}_+$.

The honest conclusion, then, is that the $\mathbb{M}_\pm$ split renames the Hermitian decomposition of the algebra and not the $\pm$-energy decomposition of the spinor. It is not obviously "doing work": the two splittings share the mass scale, but the interference that produces the trembling is between energy branches, and the energy branches are not the sectors. If a $\mathbb{B}$-internal splitting is wanted to carry the interference, the candidate is not the sector split but chirality: the chirality operator $\gamma_5$ is Clifford-even, hence has a representative in $\mathbb{B}$, and in the massless limit chirality locks to helicity, so that the two branches become the two chiralities. Whether that relabelling is the correct one is not settled here, and the status of the chirality idempotents in the sector decomposition is recorded as an open item. What is settled is that the $\pm$-energy split is not the $\mathbb{M}_\pm$ split, for the reasons above.

What Survives

The following table separates the claims.

Claim	Status
The trembling is an interference of $\pm$-frequency branches, frequency $2E/\hbar$	Established; the standard result, recomputed here
Rest-frame frequency $2mc^2/\hbar$, amplitude $\hbar/(2mc)$	Established; checked numerically above
The real structure $\flat$ acts with opposite signs on $\mathbb{M}_\pm$	Established; diagonal, verified from $\tilde\Psi_\pm^{*} = \pm\tilde\Psi_\pm$
The rest-energy phase rotates the sectors at $mc^2/\hbar$	Established; $e^{-i\theta} = \cos\theta\,e_0 - \sin\theta\,(ie_0)$
The Zitterbewegung phase is the square of the free-evolution phase	Established; hence the frequency is doubled
$\pm$-energy components are the $\mathbb{M}_\mp$ sectors	False as an identity; the generator $\beta$ lies outside $\mathbb{B}$
The $\mathbb{M}_\pm$ split does work beyond relabelling the Hermitian decomposition	Not established; the interference is between energy branches, which are not the sectors

What survives is a relation of scale and phase, not of identity. The mass sets both the sector-rotation rate and the gap whose interference is the trembling; the trembling frequency is twice the sector-rotation frequency because the trembling phase is the square of the evolution phase. This makes the Zitterbewegung a place where the mass term and the two sectors are simultaneously visible, and it is a legitimate point of contact between the Dirac solutions and the sector decomposition. It does not make the trembling the observable face of the complexification in the strong sense proposed.

Zitterbewegung and the Electron

The Zitterbewegung is usually introduced as a property of the electron, and a companion article on the electron in biquaternionic form has since been written in this group. Nothing in the present article depends on it. If the sector decomposition is to acquire physical content, the electron article is the natural place to look: the electron's rest energy is the scale of both the sector rotation and the trembling, so any attempt to identify an observable with the informational sector would have to pass through that scale. The present negative result is a constraint on that attempt, not a substitute for it.

Summary

The Zitterbewegung was derived in the biquaternion notation of the series. The Heisenberg velocity satisfies the exact identity $\alpha^k(t) = cp_k/\hat H + e^{i\hat Ht/\hbar}(\alpha^k(0)-cp_k/\hat H)e^{-i\hat Ht/\hbar}$, checked by matrix computation at momentum $|\mathbf{p}| = 0.374\,mc$ and three times, with maximum error $1.2\times10^{-16}$; the oscillating term has frequency $2E/\hbar$, equal to $2mc^2/\hbar$ at rest, and the electron value $1.55269\times10^{21}\ \text{rad s}^{-1}$ and amplitude $\hbar/(2mc) = 193.1$ fm were confirmed.

The genuinely algebraic content is that the mass is the linear coupling between the two chiralities (the algebra's two central ideals), that the real structure $\flat$ acts with opposite signs on the two sectors, and that the rest-energy phase $e^{-imc^2t/\hbar} = \cos(mc^2t/\hbar)e_0 - \sin(mc^2t/\hbar)(ie_0)$ rotates the two sectors into one another at $mc^2/\hbar$. The Zitterbewegung phase is the square of this evolution phase, so its frequency is exactly twice the sector-rotation frequency.

The structural claim that motivated the article fails. The positive- and negative-energy components are not the $\mathbb{M}_+$ and $\mathbb{M}_-$ sectors. The energy-sign projector is built from $\beta = \gamma^0$, an odd Clifford element with no representative in $\mathbb{B} \cong \mathrm{Cl}^+_{1,3}$, while the sector projectors are built from Hermitian conjugation; the two are splits of different objects (module versus algebra); and on the dictionary the series supplies, both energy branches correspond to $\mathbb{M}_+$ objects while the current lives in $\mathbb{M}_-$. The relation between the trembling and the complexification is therefore one of phase and scale — the trembling is the second harmonic of the sector rotation — and not the identity the strong form of the claim requires.

Summary of Notation

Symbol	Meaning
$\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$	Biquaternion algebra
$e_0 = 1, e_1, e_2, e_3$	Quaternion basis, $e_k^2 = -e_0$
$i$	Scalar imaginary, $i^2 = -1$
$\mathbb{M}_-$, $\mathbb{M}_+$	Anti-Hermitian (material) and Hermitian (informational) subspaces
$\mathbb{H}_{\mathbb{B}}$	Real quaternion subspace
$\tilde\nabla = e_0\partial_{ict} + e_k\partial_k$	Biquaternionic gradient (Dirac operator)
$\tilde\Psi$, $\tilde\Psi^\flat = -\tilde\Psi^{*}$	Biquaternion Dirac field and anti-Hermitian conjugate (the algebra's real structure; not the mass term)
$\tilde k\tilde k^{\natural} = -m^2c^2/\hbar^2$	Biquaternionic mass-shell condition
$u^{(r)}, v^{(r)}$	Positive- and negative-frequency spinors
$\beta = \gamma^0$, $\alpha^k = \gamma^0\gamma^k$	Dirac matrices; $\beta$ odd, $\alpha^k$ even
$\hat H = c\boldsymbol{\alpha}\cdot\hat{\mathbf{p}} + \beta mc^2$	Dirac Hamiltonian, $\hat H^2 = E^2I$
$\omega = \gamma^0\gamma^1\gamma^2\gamma^3$	Volume element (parity grading)
$\omega_Z = 2E/\hbar$	Zitterbewegung angular frequency
$\hbar/(2mc)$	Zitterbewegung amplitude (order)
$\mathrm{Tr}(\tilde P\tilde H) = 2\,\mathrm{Sc}(\tilde P\tilde H)$	Trace formula