The Minimal Coupling of the Biquaternion Dirac Field to Electromagnetism

Introduction

Minimal coupling is the prescription by which a charged field is coupled to the electromagnetic field: the ordinary derivative is replaced by the covariant derivative. In the biquaternion framework both objects are already fixed by the gauge principle. The connection is the potential biquaternion $\tilde{A}$ of Maxwell's Equations in the Biquaternionic Form, its gauge transformation is

$$ \tilde{A}' = \tilde{A} - \tilde{\nabla}\Gamma, $$

and the covariant derivative is

$$ D = \tilde{\nabla} + \frac{iq}{\hbar}\tilde{A}, \qquad D_\mu = \partial_\mu + \frac{iq}{\hbar}A_\mu . $$

The gauge principle article derived $D$ from the localization of the algebra's central phase and verified that $D\tilde{\Psi}$ transforms homogeneously. It applied the result to a general biquaternion field, not specifically to the Dirac field. The Dirac Equation in Biquaternionic Form names the minimal coupling of the Dirac field as the subject of the companion treatment (its open question 7); the precise biquaternion form is representation-dependent. This article supplies that statement, and reports what the recomputation gives.

Three results are established here. The first is the coupled equation itself: replacing the gradient in the free biquaternion Dirac equation by the covariant derivative gives $D\tilde{\Psi}_R = m\tilde{\Psi}_L$, $\bar{D}\tilde{\Psi}_L = m\tilde{\Psi}_R$, whose components are $\sum_\mu e_\mu(\partial_\mu + \frac{iq}{\hbar}A_\mu)\tilde{\Psi}_R = m\tilde{\Psi}_L$ and its conjugate partner. The second is that this coupling is exactly gauge covariant, in the massive case as well as the massless one — $D\tilde{\Psi} = 0$ and the massive chiral pair each hold in every gauge when they hold in one — because the parent's mass term is the linear chiral pair and the central phase passes through it; the axial symmetry, not the phase symmetry, is what the mass breaks. The third is a negative result about the interaction current: the naive gauge-invariant bilinear $i\tilde{\Psi}\tilde{\Psi}^{*}$ is a material-sector object but is not conserved on solutions of the massless equation, so the electromagnetic current of the coupled system is not a pure biquaternion product of this kind.

The division between what is established and what is interpretation is stated at the outset and kept explicit.

• Established, and recomputed below. The minimal-coupling prescription $D = \tilde{\nabla} + \frac{iq}{\hbar}\tilde{A}$ acting on the biquaternion Dirac field by left multiplication; the coupled equation and its component form; the exact gauge covariance of the coupled equation, in the massive case as well as the massless one; the gauge invariance and material-sector membership of $i\tilde{\Psi}\tilde{\Psi}^{*}$; and the non-conservation of that bilinear.

• Interpretation. Reading the coupled equation as "the Dirac field in a background electromagnetic connection" is a geometric reading of an algebraic construction. The algebra supplies the transformation law and the covariance; the connection/curvature picture is a consistent reading, as in the gauge principle article.

• Gaps, left visible. The biquaternion form of the conserved interaction current is not the naive bilinear, and is not settled here. Whether the left or the right action on the algebra is the physical matter representation is a choice the parent left open. The axial symmetry broken by the mass — and whether it can be gauged — is not treated here. These are stated as gaps in the sections where they arise and collected in the open questions.

The article is organized as follows. The next section recalls the Dirac field and the phase symmetry that is to be localized. The section after that states the minimal-coupling prescription. The following section proves the covariance in the massless case and fixes the sign conventions. The next section treats the massive case, where the linear mass term preserves the covariance and the axial symmetry is what the mass breaks. A section examines the interaction current and the Maxwell source, and reports the negative result. A section relates the construction to the standard spinor-module minimal coupling. A section separates what the algebra supplies from what it only transcribes. The article closes with open questions.

Conventions. We use those of the companion articles throughout. The biquaternion algebra is $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$, the quaternion basis is $e_0 = 1, e_1, e_2, e_3$ with $e_k^2 = -e_0$, and $i$ is the scalar imaginary, $i^2 = -1$, commuting with every $e_k$. The fixed-point subspaces are $\mathbb{M}_-$ (anti-Hermitian, the material sector) and $\mathbb{M}_+$ (Hermitian, the informational sector); $\mathbb{H}_{\mathbb{B}}$ is the real-quaternion subspace and $\mathbb{C}_{\mathbb{B}} = \mathrm{span}_{\mathbb{R}}\{e_0, ie_0\}$ is the complex scalar subspace, which is the center of the algebra. The biquaternionic gradient is $\tilde{\nabla} = e_0\partial_{ict} + e_1\partial_x + e_2\partial_y + e_3\partial_z$, its quaternion conjugate is $\tilde{\nabla}^{\natural} = e_0\partial_{ict} - e_1\partial_x - e_2\partial_y - e_3\partial_z$, and $\Box = \tilde{\nabla}\tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural}\tilde{\nabla}$. The potential and field strength are $\tilde{A} = i\phi/c\,e_0 + \mathbf{A}$ and $\tilde{F} = i\sqrt{\epsilon}\,\mathbf{E} - \sqrt{\mu}\,\mathbf{H}$, the source is $\tilde{R} = \frac{i\rho}{\sqrt{\epsilon}}e_0 + \sqrt{\mu}\,\mathbf{J}$, and the Maxwell equation is $\tilde{\nabla}\tilde{F} = -\tilde{R}$. Throughout, $c = 1/\sqrt{\epsilon\mu}$ is the speed of light in the medium and $c_0$ the vacuum speed of light. The Dirac field is $\tilde{\Psi}\in\mathbb{B}$ with $\tilde{\Psi}^\flat = -\tilde{\Psi}^{*}$ and $\tilde{\Psi}^{*} = \overline{\tilde{\Psi}^{\natural}}$. The mass term is the linear chiral pair $\tilde{\nabla}\tilde{\Psi}_R = m\tilde{\Psi}_L$, $\tilde{\nabla}^{\natural}\tilde{\Psi}_L = m\tilde{\Psi}_R$; the conjugation $\tilde{\Psi}^\flat$ is the algebra's real structure and is not the mass. The trace formula $\mathrm{Tr}(\tilde{P}\tilde{H}) = 2\,\mathrm{Sc}(\tilde{P}\tilde{H})$ is inherited unchanged.

The Dirac Field and Its Phase Symmetry

The biquaternion Dirac field is a biquaternion-valued field $\tilde{\Psi}(\tilde{Q}) \in \mathbb{B}$ satisfying the free equation

$$ \tilde{\nabla}\tilde{\Psi}_R = m\,\tilde{\Psi}_L, \qquad \tilde{\nabla}^{\natural}\tilde{\Psi}_L = m\,\tilde{\Psi}_R, \qquad \tilde{\Psi}^\flat = -\tilde{\Psi}^{*} = -\overline{\tilde{\Psi}^{\natural}}, $$

which reduces to

$$ \tilde{\nabla}\tilde{\Psi} = 0 $$

when $m = 0$. The massless equation is identical in form to the source-free biquaternion Maxwell equation obeyed by the field-strength biquaternion $\tilde{F}$; the difference is the representation, the Maxwell field living in the vector part of the algebra and the Dirac field in the full algebra. The two are the two representations of the same Clifford algebra, as the parent article records.

The symmetry that minimal coupling localizes is the algebra's central $U(1)$, established in the gauge principle article. The center of $\mathbb{B}$ is the complex scalar subspace

$$ \mathbb{C}_{\mathbb{B}} = \mathrm{span}_{\mathbb{R}}\{e_0,\, ie_0\}, $$

and its unitary elements are the phases $\lambda = e^{i\theta}$ with $\theta \in \mathbb{R}$. A constant central phase

$$ \tilde{\Psi} \;\longmapsto\; e^{i\theta}\tilde{\Psi}, \qquad \theta \ \text{constant}, $$

is a symmetry of the equation, massive or massless, because a central constant commutes with the basis and with the derivatives and therefore passes through the gradient and through the linear mass term:

$$ \tilde{\nabla}\!\left(e^{i\theta}\tilde{\Psi}\right) = e^{i\theta}\,\tilde{\nabla}\tilde{\Psi}. $$

The massive equation is invariant as well: the parent's mass term is the linear chiral pair, so a central phase passes through it, and the massive equation carries the same continuous symmetry as the massless one. What the mass term breaks is the axial symmetry, not the phase. The algebra's anti-Hermitian conjugation $\flat$ remains antilinear,

$$ \left(\lambda\tilde{\Psi}\right)^\flat = \lambda^{*}\,\tilde{\Psi}^\flat $$

for a central phase $\lambda$, so a mass term that paired the field with its conjugate would require $\lambda = \lambda^{*}$ and would retain only $\lambda = \pm 1$; but that is a property of the real structure $\flat$, not of the parent's mass term, and it is developed in the companion articles on the neutrino and on chirality. The continuous coupling therefore belongs to the massive field as well as to the massless one.

The charge $q$ is the coupling constant of the field, introduced by the gauge principle. It is a parameter: the algebra does not fix its value, its sign, or its quantization. For the electron $q = -e$; the value and the sign are inserted, as The Electron in Biquaternionic Form records.

The Minimal Coupling Prescription

The prescription replaces the gradient in the free equation by the covariant derivative:

$$ \tilde{\nabla} \;\longrightarrow\; D = \tilde{\nabla} + \frac{iq}{\hbar}\tilde{A}, \qquad D\tilde{\Psi} = \tilde{\nabla}\tilde{\Psi} + \frac{iq}{\hbar}\,\tilde{A}\,\tilde{\Psi}. $$

The connection acts by left multiplication on the field. Applied to the free Dirac equation in its linear, chirality-off-diagonal form, the prescription gives the minimally coupled biquaternion Dirac equation

$$ \boxed{\ D\tilde{\Psi}_R = m\,\tilde{\Psi}_L, \qquad \bar{D}\tilde{\Psi}_L = m\,\tilde{\Psi}_R\ } $$

that is,

$$ \tilde{\nabla}\tilde{\Psi}_R + \frac{iq}{\hbar}\,\tilde{A}\,\tilde{\Psi}_R = m\,\tilde{\Psi}_L, \qquad \tilde{\nabla}^{\natural}\tilde{\Psi}_L + \frac{iq}{\hbar}\,\tilde{A}^{\natural}\,\tilde{\Psi}_L = m\,\tilde{\Psi}_R, $$

and in the massless case

$$ D\tilde{\Psi} = 0, \qquad \text{i.e.} \qquad \tilde{\nabla}\tilde{\Psi} + \frac{iq}{\hbar}\,\tilde{A}\,\tilde{\Psi} = 0 . $$

The term $\frac{iq}{\hbar}\tilde{A}\tilde{\Psi}$ is the interaction term. It is exactly the term that the localization of the phase forces, and it is the biquaternion representative of the familiar $\frac{iq}{\hbar}\gamma^\mu A_\mu$ coupling of the spinor-module Dirac equation.

Component form. Write the potential as $\tilde{A} = \sum_\mu A_\mu e_\mu$ with complex scalar coefficients $A_\mu$ — the abelian case, in which the connection coefficients are central scalars and commute with one another. Since each $A_\mu$ is a scalar and commutes with the basis, the covariant derivative separates into components:

$$ D = \sum_{\mu=0}^{3} e_\mu\left(\partial_\mu + \frac{iq}{\hbar}A_\mu\right), \qquad D_\mu = \partial_\mu + \frac{iq}{\hbar}A_\mu, $$

with $\partial_0 = \partial_{ict}$ and $\partial_k = \partial_{x_k}$. This is the identical component form derived in the gauge principle article, now read as an operator on the Dirac field. Setting $\hbar = 1$ gives $D_\mu = \partial_\mu + iqA_\mu$, the convention used for the minimal substitution in the companion article on the Dirac equation's solutions and the non-relativistic limit.

The connection is a material-sector object. The gauge principle article shows that a real gauge function $\Gamma$ keeps $\tilde{A}' = \tilde{A} - \tilde{\nabla}\Gamma$ in $\mathbb{M}_-$. The coupling is therefore the coupling of the Dirac field to a material-sector connection, and the ingredient $\frac{iq}{\hbar}\tilde{A}$ in $D$ is Hermitian. This is inherited, not rederived.

Sign convention. The gauge principle article fixes the pair of signs $\tilde{A}' = \tilde{A} - \tilde{\nabla}\Gamma$ and $D = \tilde{\nabla} + \frac{iq}{\hbar}\tilde{A}$ with the phase $\lambda = e^{iq\Gamma/\hbar}$, and its companion records that the two must be kept together. The open question in the parent Dirac article writes the standard physics convention $\partial_\mu \to \partial_\mu - iqA_\mu/\hbar$, with the opposite sign. That is the same coupling with the opposite sign of $q$ (or of $\tilde{A}$); this article inherits the gauge principle's sign and does not silently flip it. The covariance checks below use the inherited pair.

Left action and right action. The prescription above acts on the left. The algebra also admits a mirror construction, with the derivative taken on the right and the connection inserted on the right:

$$ \tilde{\Psi}\overleftarrow{\tilde{\nabla}} + \frac{iq}{\hbar}\,\tilde{\Psi}\,\tilde{A}, \qquad \tilde{\Psi}\overleftarrow{\tilde{\nabla}} := \sum_\mu (\partial_\mu\tilde{\Psi})\,e_\mu . $$

Both constructions are gauge covariant (verified below). What is not covariant is the mixed combination — the left gradient with the connection inserted on the right. The two consistent constructions correspond to the two ways the algebra can act on the field, and the choice between them is a choice of matter representation: on which module, and from which side, the Dirac field sits. The gauge principle article records this as an open question, and this article does not resolve it; it states the left-action form, which is the one the gauge principle produces, and records the right-action mirror as the alternative.

Gauge Covariance of the Coupled Equation

The gauge transformation is

$$ \tilde{\Psi}' = \lambda\,\tilde{\Psi}, \qquad \lambda = e^{iq\Gamma(\tilde{Q})/\hbar}, \qquad \Gamma \ \text{real}, $$

together with $\tilde{A}' = \tilde{A} - \tilde{\nabla}\Gamma$. The covariance is the same computation as in the gauge principle article, repeated here because it is the justification of the prescription. The Leibniz rule for the left action of the gradient,

$$ \tilde{\nabla}\!\left(\lambda\tilde{\Psi}\right) = \lambda\,\tilde{\nabla}\tilde{\Psi} + \left(\tilde{\nabla}\lambda\right)\tilde{\Psi}, $$

and the chain rule for the central phase,

$$ \tilde{\nabla}\lambda = \frac{iq}{\hbar}\left(\tilde{\nabla}\Gamma\right)\lambda , $$

combine into

$$ \tilde{\nabla}\!\left(\lambda\tilde{\Psi}\right) = \lambda\left[\tilde{\nabla}\tilde{\Psi} + \frac{iq}{\hbar}\left(\tilde{\nabla}\Gamma\right)\tilde{\Psi}\right]. $$

The covariant derivative built from $\tilde{A}'$ then gives

$$ \begin{aligned} D'(\lambda\tilde{\Psi}) &= \tilde{\nabla}(\lambda\tilde{\Psi}) + \frac{iq}{\hbar}\left(\tilde{A} - \tilde{\nabla}\Gamma\right)(\lambda\tilde{\Psi}) \\ &= \lambda\,\tilde{\nabla}\tilde{\Psi} + \frac{iq}{\hbar}\left(\tilde{\nabla}\Gamma\right)\lambda\tilde{\Psi} + \frac{iq}{\hbar}\lambda\,\tilde{A}\,\tilde{\Psi} - \frac{iq}{\hbar}\left(\tilde{\nabla}\Gamma\right)\lambda\tilde{\Psi} \\ &= \lambda\left[\tilde{\nabla}\tilde{\Psi} + \frac{iq}{\hbar}\tilde{A}\tilde{\Psi}\right] = \lambda\,D\tilde{\Psi}. \end{aligned} $$

The two $\tilde{\nabla}\Gamma$ terms cancel and the central $\lambda$ moves to the left. Consequently

$$ D\tilde{\Psi} = 0 \qquad \Longleftrightarrow \qquad D'\tilde{\Psi}' = 0 : $$

the massless minimally coupled biquaternion Dirac equation is exactly gauge covariant. It holds in every gauge when it holds in one, and the phase of the field is not observable on its own. This is the charged massless fermion — the biquaternion form of the gauged Weyl field.

The same computation gives the mirror statement for the right action. With the derivative on the right and the connection inserted on the right, and with the same transformation laws for $\tilde{\Psi}$ and $\tilde{A}$, the right covariant derivative is also homogeneous, so the right-action coupled equation is also covariant. The mixed insertion is not: the two gradient terms involving $\tilde{\nabla}\Gamma$ no longer cancel, because one acts on the field from the left and the other multiplies it from the right. This is the algebraic content of the statement that the matter representation must be chosen consistently.

The Massive Case: Gauge Covariance and the Axial Symmetry

The massive equation is where the mass term's role is decided, and with the linear chiral pair it is benign. Suppose the minimally coupled massive equation holds in one gauge:

$$ D\tilde{\Psi}_R = m\tilde{\Psi}_L, \qquad \bar{D}\tilde{\Psi}_L = m\tilde{\Psi}_R . $$

Because $D$ is covariant, the left-hand side of the transformed first equation is

$$ D'\tilde{\Psi}_R' = \lambda\,D\tilde{\Psi}_R , $$

and the right-hand side is the transformed mass term, which for a linear mass term carries the same factor:

$$ m\tilde{\Psi}_L' = m\,\lambda\,\tilde{\Psi}_L . $$

The two sides transform identically, because the central phase commutes with the mass. The same holds for the second equation, with $\bar{\lambda}$. The massive coupled equation is therefore form-invariant under the local phase, and it holds in every gauge when it holds in one — exactly as in the massless case.

This is the biquaternion form of the standard fact that a mass term linear in the field is compatible with a gauged continuous $U(1)$ (fermion number). A mass term that instead pairs the field with its conjugate — a Majorana-type mass, built on the algebra's anti-Hermitian conjugation $\flat$, which is antilinear — would acquire the factor $e^{2iq\Gamma/\hbar}$ and would be form-invariant only for $\lambda = \pm 1$. That obstruction is a property of the real structure $\flat$, not of the parent's mass term, and it belongs to the companion articles on the neutrino and on chirality rather than here.

What this is. The computation is elementary and exact, and it is the biquaternion form of a familiar fact: a mass term linear in the field is compatible with a gauged continuous $U(1)$ (fermion number), whereas a mass term that pairs a field with its conjugate — a Majorana-type mass — is not. The complex scalar realization of the phase symmetry in the gauge principle article has a linear mass term, and so does the biquaternion Dirac field in its parent's present form; both keep their $U(1)$ even when massive. The standard four-component Dirac equation has the linear mass term $m\psi$ and is gauge covariant under minimal coupling, and the biquaternion chiral pair is its algebra-level transcription. The object the phase cannot pass through is the antilinear conjugation $\flat$, which is retained as the algebra's real structure and is no longer the mass term.

What it is not. The article does not declare the standard minimal coupling deficient, and with the linear mass term it does not need to: the massive sector is gauge covariant on the same footing as the massless one. What the biquaternion form adds is the algebra-level location of the coupling — the covariant derivative $D = \tilde{\nabla} + \frac{iq}{\hbar}\tilde{A}$ acting by left multiplication, with the central phase passing through the linear mass and the axial symmetry broken instead. The residual question of the left-versus-right matter representation is the parent's, and is collected in the open questions below.

The reading that is taken. Of the two readings previously left open, the parent's linear mass term selects the second: the algebra-level equation is the real-form expression of an ordinary Dirac mass, the physical coupling is the standard spinor-module one, and the massive sector carries the charge. The first reading — that the framework's massive equation is a Majorana-type equation, with no continuously charged massive field — remains available, but for a different equation, one genuinely built on $\flat$; it is now a statement about the algebra's real structure rather than about the parent, and it is where the companion articles on the neutrino and on chirality continue the discussion.

The Interaction Current and the Maxwell Source

Minimal coupling has a second half: the coupled field must source the electromagnetic field. In the biquaternion framework the sourced Maxwell equation is $\tilde{\nabla}\tilde{F} = -\tilde{R}$, and the source $\tilde{R}$ satisfies the integrability condition $\mathrm{Sc}(\tilde{\nabla}^{\natural}\tilde{R}) = 0$, the biquaternionic form of charge conservation.

The natural candidate for the current of the Dirac field is the bilinear

$$ \tilde{J} = i\,\tilde{\Psi}\,\tilde{\Psi}^{*} . $$

Two of its properties are immediate and exact. It is gauge invariant, because for a unimodular central phase

$$ \left(\lambda\tilde{\Psi}\right)\left(\lambda\tilde{\Psi}\right)^\dagger = \lambda\,\tilde{\Psi}\tilde{\Psi}^{*}\,\lambda^{*} = |\lambda|^2\,\tilde{\Psi}\tilde{\Psi}^{*} = \tilde{\Psi}\tilde{\Psi}^{*} , $$

and it is anti-Hermitian, $\tilde{J}^{*} = -\tilde{J}$, hence an element of the material sector $\mathbb{M}_-$ — the same sector as the source $\tilde{R}' = ic\rho + \mathbf{J}$ of the Maxwell article. Its scalar part is $i\,\mathrm{Sc}(\tilde{\Psi}\tilde{\Psi}^{*}) = i\,\|\tilde{\Psi}\|_E^2$, a positive imaginary number, matching the $ic\rho$ form of a positive density. It is, by these tests, a plausible current.

It is not conserved. The divergence of $\tilde{J}$ in the $ict$ convention is the scalar part of $\tilde{\nabla}^{\natural}\tilde{J}$,

$$ \mathrm{Sc}\!\left(\tilde{\nabla}^{\natural}\tilde{J}\right) = \partial_{ict}J_0 + \mathrm{div}\,\mathbf{J}, $$

the quantity that would have to vanish on solutions. It does not. To check this without relying on the case that suggested the candidate, take a field generated from a wave-equation potential: if $\phi$ is a biquaternion whose every component satisfies the scalar wave equation $\Box\phi = 0$, then

$$ \tilde{\Psi} = \tilde{\nabla}^{\natural}\phi $$

satisfies $\tilde{\nabla}\tilde{\Psi} = \tilde{\nabla}\tilde{\nabla}^{\natural}\phi = \Box\phi = 0$, so it is a genuine solution of the massless equation. On such a solution, with $\phi$ built from superposed plane waves so that no accidental cancellation is available, the divergence $\mathrm{Sc}(\tilde{\nabla}^{\natural}\tilde{J})$ is not zero; the computation gives values of order $10^2$ at generic points, for either sign of the bilinear. The naive biquaternion product is therefore not the conserved electromagnetic current of the Dirac field.

Why, and what the current is instead. The reason is representation-theoretic. The conserved current of the Dirac field is the spinor-module bilinear $j^\mu = \bar{\psi}\gamma^\mu\psi$ with the Dirac adjoint $\bar{\psi} = \psi^\dagger\gamma^0$. The factor $\gamma^0$ is a Clifford-odd element; it is not in the even subalgebra $\mathbb{C}\ell_{1,3}^+ \cong \mathbb{B}$, and it is exactly the extra ingredient that the biquaternion formulation needs to recover the four-component Dirac spinor from its pair of Weyl spinors. A bilinear formed only from $\tilde{\Psi}$ and $\tilde{\Psi}^{*}$ inside $\mathbb{B}$ cannot supply it. The physical current therefore lies partly outside the algebra, and the pure-biquaternion form of the interaction current is not given by the naive product. That the electron's current is nevertheless an $\mathbb{M}_-$ four-vector — $\tilde{J} = ic\,j^0e_0 + \mathbf{j}$ with $j^0 = \psi^\dagger\psi$ — is a statement about the spinor-module transcription, not about a product in $\mathbb{B}$.

This is the same representation question as the left/right action of the previous sections, met from the current side. The parent Dirac article's open question flags the representation conventions; this article records that they are not idle. The biquaternion form of the conserved current, and the sense in which the sourced Maxwell equation $\tilde{\nabla}\tilde{F} = -\tilde{R}$ is fed by the Dirac field, remain open.

Recovering the Standard Minimal Coupling

The construction meets the standard theory in the spinor-module transcription. The biquaternion field, viewed as an element of $\mathbb{B}\cong M_2(\mathbb{C})$, is a pair of two-component Weyl spinors, and the biquaternion gradient plays the role of the Dirac operator $\not\partial = \gamma^\mu\partial_\mu$ on the spinor module. Under that transcription, left multiplication by the connection $\tilde{A} = \sum_\mu A_\mu e_\mu$ is the representative of the matrix $\gamma^\mu A_\mu$, and the coupled biquaternion equation becomes the standard minimally coupled Dirac equation

$$ \left(i\hbar\gamma^\mu D_\mu - mc\right)\psi = 0, \qquad D_\mu = \partial_\mu + \frac{iq}{\hbar}A_\mu , $$

provided the mass term is the linear Dirac mass $m\psi$. With the linear mass the equation is gauge covariant in the standard way, the conserved current is $j^\mu = \bar{\psi}\gamma^\mu\psi$, and the standard consequences — the tree-level gyromagnetic factor $g = 2$, the correct non-relativistic limit, the magnetic moment — follow by the computations of the companion articles on the electron and on the solutions and non-relativistic limit. Those results are not repeated here; the point for this article is that they are obtained with a linear mass term, which is exactly the form the parent's biquaternion equation carries, so the transcription and the standard treatment agree on the massive sector as well as on the massless one.

The transcription carries the representation conventions that the parent Dirac article explicitly left open — the identification of the biquaternion units with the gamma matrices, the placement of the two chiralities in the algebra, and the role of the Dirac adjoint. This article does not fix them. What it fixes is the biquaternion form of the coupling itself: the covariant derivative $D = \tilde{\nabla} + \frac{iq}{\hbar}\tilde{A}$ acting by left multiplication, the component operators $D_\mu = \partial_\mu + \frac{iq}{\hbar}A_\mu$, and the coupled equation $D\tilde{\Psi}_R = m\tilde{\Psi}_L$, $\bar{D}\tilde{\Psi}_L = m\tilde{\Psi}_R$.

What the Algebra Supplies and What It Only Transcribes

It is worth separating the two, because they are easily conflated.

What the algebra supplies.

• A canonical abelian coupling. The gauge group is the unitary part of the center, and the center is the only place a central phase can live. The $U(1)$ that minimal coupling gauges is attached to the algebra without a choice of representation, as the gauge principle article establishes.
• The covariant derivative as the localization of the phase. The interaction term $\frac{iq}{\hbar}\tilde{A}\tilde{\Psi}$ is not added by hand: it is the term the localized phase forces, and the connection it forces is exactly the potential biquaternion of the Maxwell article.
• Exact covariance of the massless coupled equation, with the component operators $D_\mu$ and the phase law $\tilde{A}' = \tilde{A} - \tilde{\nabla}\Gamma$ inherited unchanged.
• A material-sector connection. For a real gauge function the connection stays in $\mathbb{M}_-$, so the coupling respects the sector decomposition the framework is built on.

What it only transcribes.

• The minimal-coupling principle itself. The derivation is the standard global-to-local argument carried into the biquaternion algebra. The algebra supplies a home for it and makes the component structure transparent; it does not supply a reason for the gauge field to exist.
• The value of the coupling. The charge $q$ is a parameter; the algebra does not fix it.
• The standard spinor-module results. The conserved current, the non-relativistic limit, and $g=2$ are properties of the Dirac equation the framework contains, obtained with the linear mass term on the spinor module; the framework transcribes them.

The commutativity that makes this unambiguous — and the sector where it fails. The whole prescription rests on a property of internal gauge forces that this article has used without naming: for an internal gauge group the substitution $\partial \to D$ gives the same equation whether it is applied to the action or to the Euler–Lagrange equation of motion, so it does not matter which is gauged. The reason is that the internal generators commute with $\gamma^a$, and the abelian $U(1)$ of this article is the simplest case. The property is not universal. For a Dirac fermion coupled to gravity in the standard vierbein formulation, the gauge generators are the Lorentz generators $S^{cd}$, which do not commute with $\gamma^a$ — they satisfy $[\gamma^c,S^{ab}] = V^{ab}{}_d\gamma^d$ — and the two procedures then differ by an anticommutator term; J. Fredsted (arXiv:1906.12200v3 [physics.gen-ph], 2019) proves the mismatch and constructs a world-index formalism in which it is removed, recorded in Curved Spacetime and the Biquaternion Framework. The consequence for this article is a boundary rather than a defect: the minimal-coupling prescription derived here is unambiguous because its gauge group is internal, and it must not be transferred to the gravitational sector unchanged, where the same prescription is a different and inequivalent construction.

What is interpretation. Reading the coupled equation as a Dirac field propagating in a background connection is a geometric interpretation. The algebraic content is the transformation law, the covariance of the massless equation, and the explicit component form; the bundle picture is a consistent reading of that content, as in the gauge principle article.

Open Questions

1. The massive sector and the axial symmetry. With the linear mass term the vector $U(1)$ survives and the massive sector is gauge covariant, so the minimal coupling derived here applies to it. What the mass breaks is the axial symmetry between the two central ideals. Can that symmetry be gauged, and does the framework's central-element structure supply anything analogous to the would-be Goldstone statement for the breaking? The former question — whether the parent's mass term is a Majorana-type coupling — is answered in its second branch by the linear mass term.

2. The matter representation. The gauge principle produces a left-action covariant derivative; a right-action mirror is equally covariant, and the mixed insertion is not. Which action — which spinor module — is the physical Dirac field in? What fixes it inside the algebra?

3. The interaction current. The naive gauge-invariant bilinear $i\tilde{\Psi}\tilde{\Psi}^{*}$ is in $\mathbb{M}_-$ but is not conserved. The physical current uses the Clifford-odd $\gamma^0$, outside $\mathbb{B}$. Is there a biquaternion-natural current, perhaps on a restricted module, or is the current irreducibly a representation-level object?

4. The sourced system. How does the Dirac current feed the biquaternionic Maxwell equation $\tilde{\nabla}\tilde{F} = -\tilde{R}$, and does the coupled system close on the algebra or require the spinor module? The integrability condition $\mathrm{Sc}(\tilde{\nabla}^{\natural}\tilde{R}) = 0$ must be recovered from the current.

5. The shared generator. The internal phase $e^{iq\Gamma/\hbar}$ and the complex time coordinate $ict$ both use the single element $i$ of $\mathbb{B}$. Is this identification content-bearing, or vacuous because the phase is central? (The gauge principle article's open question.)

6. The non-abelian extension. When the connection becomes $\mathbb{B}$-valued, the field must be placed in a representation of the gauge group and the left/right question becomes sharp. The planned companions on the covariant derivative, non-abelian fields, and Yang–Mills must settle it.

7. Empirical content. As everywhere in the framework, the unresolved question is whether any of this yields a prediction distinguishing it from standard gauge theory. The minimal coupling as presented is a reformulation; the framework-level question of empirical contact remains open.

Summary

The minimal coupling of the biquaternion Dirac field to electromagnetism is the replacement of the gradient by the covariant derivative of the gauge principle,

$$ D = \tilde{\nabla} + \frac{iq}{\hbar}\tilde{A}, \qquad D_\mu = \partial_\mu + \frac{iq}{\hbar}A_\mu, $$

acting by left multiplication on the field. The coupled equation is

$$ D\tilde{\Psi}_R = m\tilde{\Psi}_L, \qquad \bar{D}\tilde{\Psi}_L = m\tilde{\Psi}_R, \qquad \tilde{\nabla}\tilde{\Psi}_R + \frac{iq}{\hbar}\,\tilde{A}\,\tilde{\Psi}_R = m\,\tilde{\Psi}_L, $$

with the interaction term $\frac{iq}{\hbar}\tilde{A}\tilde{\Psi}$ that the localized central phase forces, and the massless case $D\tilde{\Psi} = 0$.

The massless and the massive coupled equations are both exactly gauge covariant: with $\tilde{\Psi}' = e^{iq\Gamma/\hbar}\tilde{\Psi}$ and $\tilde{A}' = \tilde{A} - \tilde{\nabla}\Gamma$, one has $D'\tilde{\Psi}' = \lambda D\tilde{\Psi}$, and the parent's linear mass term acquires the same factor $\lambda$, so $D\tilde{\Psi}_R = m\tilde{\Psi}_L$ holds in every gauge when it holds in one. The continuous $U(1)$ of the massless theory survives the mass. The biquaternion form of the statement that a Majorana-type mass term — built on the antilinear real structure $\flat$ — is incompatible with a gauged continuous $U(1)$ is given in the companion articles on the neutrino and on chirality; the standard linear Dirac mass, which is the parent's form, does not have that obstruction, and on the spinor module the standard minimal coupling is recovered.

The interaction current is not a pure biquaternion product. The gauge-invariant, material-sector bilinear $i\tilde{\Psi}\tilde{\Psi}^{*}$ is not conserved on solutions of the massless equation; the physical current is the spinor-module $j^\mu = \bar{\psi}\gamma^\mu\psi$, whose Dirac adjoint uses the Clifford-odd $\gamma^0$ outside $\mathbb{B}$. The pure-biquaternion form of the current, and the closure of the sourced system, are left as open questions, together with the choice of the left or right matter representation and the axial symmetry of the massive sector.

Summary of Notation

Further Reading

• P. A. M. Dirac, "The quantum theory of the electron," Proceedings of the Royal Society A 117 (1928) 610–624, for the original equation and the minimal coupling.
• Hermann Weyl, "Elektron und Gravitation," Zeitschrift für Physik 56 (1929) 330–352, for the origin of the local phase and the gauge principle.
• J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, 1964), for the standard treatment of minimal coupling, the conserved current, and the non-relativistic limit.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, 1995), and S. Weinberg, The Quantum Theory of Fields, Vol. 1 (Cambridge, 1995), for the standard abelian and non-abelian gauge couplings.
• Claude Itzykson and Jean-Bernard Zuber, Quantum Field Theory (McGraw-Hill, 1980), for the Majorana and Dirac mass terms and the fermion-number symmetry.
• Pertti Lounesto, Clifford Algebras and Spinors (Cambridge, 2001), for the identification of the biquaternion algebra with the even part of $\mathrm{Cl}_{1,3}$ and the relation of $\gamma^0$ to the even subalgebra.
• Chris Doran and Anthony Lasenby, Geometric Algebra for Physicists (Cambridge, 2003), for the gauge-theoretic reading of the covariant derivative in geometric algebra.
• David Hestenes, Space-Time Algebra (Gordon and Breach, 1966), for the original formulation of the Dirac equation in spacetime algebra.
• Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, Vol. 1 (Cambridge, 1984), for the two-component spinor calculus and the Dirac adjoint.
• J. Fredsted, "Obtaining consistent Lorentz gauging for a gravitationally coupled fermion," arXiv:1906.12200v3 [physics.gen-ph] (2019), for the external result recorded in What the Algebra Supplies and What It Only Transcribes: that localization commutes with the Euler–Lagrange variation for an internal gauge force (the property this article's abelian prescription silently relies on) and fails for the Lorentz gauge group of the standard vierbein coupling. Cited as an external claim; the gravitational construction is recorded in Curved Spacetime and the Biquaternion Framework.
• Companion articles: The Gauge Principle in Biquaternionic Form; Maxwell's Equations in the Biquaternionic Form; The Dirac Equation in Biquaternionic Form; The Dirac Equation in Biquaternionic Form — Solutions and the Non-Relativistic Limit; The Electron in Biquaternionic Form; Canonical Quantization of the Biquaternion Dirac Field; Canonical Quantization of the Biquaternion Maxwell Field; The Anti-Hermitian Subspace $\mathbb{M}_-$ as the Material Sector; The Hermitian Subspace $\mathbb{M}_+$ as the Informational Sector.