The Magnetic Monopole in Biquaternionic Form

Introduction

In the standard formulation, Maxwell's equations treat electricity and magnetism asymmetrically. Electric charge and current source the field through

$$ \mathrm{div}\,\mathbf{D} = \rho_e, \qquad \mathrm{rot}\,\mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} + \mathbf{J}_e, $$

while the remaining two equations,

$$ \mathrm{div}\,\mathbf{B} = 0, \qquad \mathrm{rot}\,\mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, $$

are homogeneous: they hold identically once the field is written in terms of a potential. There is no magnetic charge, and the symmetry that would exchange the electric and magnetic halves of the field is broken by the absence of magnetic sources.

The preceding articles established two facts that bear directly on this asymmetry. First, the field-strength biquaternion

$$ \tilde{F} = i\sqrt{\epsilon}\,\mathbf{E} - \sqrt{\mu}\,\mathbf{H} $$

carries the electric field in its imaginary half, in the Hermitian subspace $\mathbb{M}_+$, and the magnetic field in its real half, in the anti-Hermitian subspace $\mathbb{M}_-$: the two halves of the electromagnetic field occupy the two complementary sectors of the algebra. Second, the electric–magnetic duality rotation acts on $\tilde{F}$ by multiplication by a scalar phase, so it is a rotation inside the algebra rather than a Lorentz transformation. This article asks what the biquaternion algebra says about the object that would restore the symmetry on the source side: the magnetic monopole.

The fit is real, and it should be shown rather than asserted. Magnetic sources extend the single biquaternionic Maxwell equation cleanly, with the electric and magnetic source terms assembling into one complex biquaternion; the duality rotation then becomes an exact internal symmetry of the sourced equations, not only of the source-free ones; and the Dirac quantisation condition appears as a condition on the holonomy of the biquaternionic potential. The monopole is, in this precise sense, the object for which the second, magnetic half of the electromagnetic structure is needed.

The article is equally concerned with what the framework does not do. The biquaternion algebra is classical; it contains no $\hbar$, and it neither predicts that a monopole exists nor fixes its mass or charge. No magnetic monopole has ever been observed. The algebra accepts a monopole; it does not manufacture one.

Maxwell's Equations with Magnetic Sources

Let the medium have permittivity $\epsilon$ and permeability $\mu$, with $\mathbf{D} = \epsilon\mathbf{E}$, $\mathbf{B} = \mu\mathbf{H}$ and $c = 1/\sqrt{\epsilon\mu}$. Admit a magnetic charge density $\rho_m$ and a magnetic current density $\mathbf{J}_m$, and write the symmetric Maxwell system

$$ \mathrm{div}\,\mathbf{D} = \rho_e, \qquad \mathrm{rot}\,\mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} + \mathbf{J}_e, $$

$$ \mathrm{div}\,\mathbf{B} = \rho_m, \qquad \mathrm{rot}\,\mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} - \mathbf{J}_m . $$

With $\rho_m = \mathbf{J}_m = 0$ this is the system of the parent article. The electric source is the four-current biquaternion of that article, now written with a subscript,

$$ \tilde{R}_e = \frac{i\rho_e}{\sqrt{\epsilon}} + \sqrt{\mu}\,\mathbf{J}_e \in \mathbb{M}_-, $$

and the magnetic source is the analogous biquaternion

$$ \tilde{R}_m = \frac{i\rho_m}{\sqrt{\mu}} + \sqrt{\epsilon}\,\mathbf{J}_m \in \mathbb{M}_- . $$

Both sources are four-currents and therefore lie in the material subspace $\mathbb{M}_-$; the factors of $1/\sqrt{\mu}$ and $\sqrt{\epsilon}$ are the magnetic counterparts of the electric normalizations, chosen so that the combined equation below carries no explicit $\epsilon$ or $\mu$. With these definitions the four symmetric Maxwell equations collapse into the single biquaternionic equation

$$ \boxed{\;\tilde{\nabla}\tilde{F} = -\tilde{R}_e - i\,\tilde{R}_m\;} $$

with the gradient $\tilde{\nabla} = e_0\partial_{ict} + e_1\partial_x + e_2\partial_y + e_3\partial_z$ of the parent article. The factor of $-i$ in front of the magnetic source is not decoration: it is the same scalar phase that implements duality, and it is what places the magnetic contribution in the complementary sector of the algebra.

The reduction is best seen by taking scalar and vector parts. For a pure-vector field strength, $\tilde{\nabla}\tilde{F}$ has scalar part $-\mathrm{div}\,\mathbf{F}$ and vector part $\partial_{ict}\mathbf{F} + \mathrm{rot}\,\mathbf{F}$. The scalar part of the equation gives

$$ \mathrm{div}\,\mathbf{F} = \frac{i\rho_e}{\sqrt{\epsilon}} - \frac{\rho_m}{\sqrt{\mu}}, $$

and the vector part gives

$$ \partial_{ict}\mathbf{F} + \mathrm{rot}\,\mathbf{F} = -\sqrt{\mu}\,\mathbf{J}_e - i\sqrt{\epsilon}\,\mathbf{J}_m . $$

Substituting $\mathbf{F} = i\sqrt{\epsilon}\mathbf{E} - \sqrt{\mu}\mathbf{H}$ and separating real and imaginary parts returns the four equations above. Two cases verify the signs independently of the case that suggested them. For a magnetostatic configuration with $\rho_e = \mathbf{J}_e = \mathbf{J}_m = 0$ and $\partial_t = 0$, the scalar equation reads $\mathrm{div}\,\mathbf{F} = -\rho_m/\sqrt{\mu}$; since $\mathbf{F} = -\mathbf{B}/\sqrt{\mu}$ and $\mathrm{div}\,\mathbf{B} = \rho_m$, both sides equal $-\rho_m/\sqrt{\mu}$, and the vector part is zero on both sides because $\mathrm{rot}\,\mathbf{H} = 0$. For a static magnetic current with $\rho_m = 0$, $\mathrm{rot}\,\mathbf{E} = -\mathbf{J}_m$ and $\mathrm{rot}\,\mathbf{H} = 0$, the vector equation gives $\mathrm{rot}\,\mathbf{F} = i\sqrt{\epsilon}\,\mathrm{rot}\,\mathbf{E} = -i\sqrt{\epsilon}\,\mathbf{J}_m$, matching the right-hand side. The two checks use different components and different source types.

Charge conservation has a magnetic counterpart. Just as the electric source is constrained by $\mathrm{Sc}(\tilde{\nabla}^{\natural}\tilde{R}_e) = 0$, which is $\mathrm{div}\,\mathbf{J}_e + \partial_t\rho_e = 0$, the magnetic source satisfies $\mathrm{Sc}(\tilde{\nabla}^{\natural}\tilde{R}_m) = 0$, which is $\mathrm{div}\,\mathbf{J}_m + \partial_t\rho_m = 0$. Magnetic charge is conserved for exactly the algebraic reason electric charge is: the conservation law is the integrability condition of the field equation, and the two sources are subject to it separately because they occupy the two complementary sectors under $\mathrm{Sc}$.

The Dual Field-Strength Biquaternion

The symmetric system invites a second field-strength biquaternion. Define the dual field configuration by the duality transformation of the field-strength article at $\theta = \pi/2$,

$$ \mathbf{E} \mapsto c\,\mathbf{B}, \qquad \mathbf{B} \mapsto -\frac{1}{c}\,\mathbf{E}, $$

and let $\tilde{F}_\star$ be the field-strength biquaternion built from the dual fields. Using $\mathbf{B} = \mu\mathbf{H}$ and $c = 1/\sqrt{\epsilon\mu}$,

$$ \tilde{F}_\star = i\sqrt{\epsilon}\,(c\mathbf{B}) - \sqrt{\mu}\left(-\frac{1}{c\mu}\mathbf{E}\right) = \frac{i}{\sqrt{\mu}}\,\mathbf{B} + \sqrt{\epsilon}\,\mathbf{E} = -i\,\tilde{F}, $$

the last equality following from $\tilde{F} = i\sqrt{\epsilon}\mathbf{E} - \mathbf{B}/\sqrt{\mu}$. The dual field-strength biquaternion is therefore not an independent object: Hodge duality is multiplication by the scalar phase $-i$. This is the algebraic fact behind the Riemann–Silberstein linearization, and it is why the dual carries the electric and magnetic parts with their roles exchanged — the real and imaginary halves of $-i\tilde{F}$ are exactly the magnetic and electric halves of $\tilde{F}$, in the opposite slots.

A reader arriving from a tensor formulation might expect the combination $\tilde{F} + i\tilde{F}_\star$ to double the field content. It does not: $\tilde{F} + i\tilde{F}_\star = \tilde{F} + i(-i\tilde{F}) = 2\tilde{F}$. The field strength already is the complexified object, its electric and magnetic halves being its Hermitian and anti-Hermitian parts. Where the complexification genuinely adds content is in the sources. They combine into

$$ \tilde{\mathcal{R}} = \tilde{R}_e + i\tilde{R}_m, $$

whose $\mathbb{M}_-$ part is electric and whose $\mathbb{M}_+$ part is magnetic, and the field equation becomes $\tilde{\nabla}\tilde{F} = -\tilde{\mathcal{R}}$. This is the object on which duality acts nontrivially, and it is the reason a magnetic source is not a mere relabelling of the electric one: it enters through the other sector.

Duality as an Internal Rotation

The field-strength article established that for a real angle $\theta$ the duality transformation rotates the Riemann–Silberstein vector, $\mathbf{V} \mapsto e^{-i\theta}\mathbf{V}$. In biquaternion terms, and since $\tilde{F} = i\sqrt{\epsilon}\,\mathbf{V}$, this is

$$ \tilde{F} \;\longmapsto\; e^{-i\theta}\,\tilde{F}. $$

The verification is direct and does not rely on the special value $\theta = \pi/2$: substituting the general rotation $\mathbf{E} \mapsto \mathbf{E}\cos\theta + c\mathbf{B}\sin\theta$, $\mathbf{B} \mapsto \mathbf{B}\cos\theta - c^{-1}\mathbf{E}\sin\theta$ into the definition of $\tilde{F}$, the coefficient of the electric term is $i\sqrt{\epsilon}\cos\theta + \sqrt{\epsilon}\sin\theta$ and the coefficient of the magnetic term is $i(\sin\theta)/\sqrt{\mu} - (\cos\theta)/\sqrt{\mu}$, which are precisely the coefficients in $e^{-i\theta}\tilde{F} = (\cos\theta - i\sin\theta)(i\sqrt{\epsilon}\mathbf{E} - \mathbf{B}/\sqrt{\mu})$. The identity holds for every $\theta$, and the special case $\theta = \pi/2$ reproduces the dual field of the previous section.

The phase $e^{-i\theta}$ is a complex scalar of the algebra, an element of the scalar subspace $\mathbb{C}_{\mathbb{B}}$; it commutes with the quaternion units $e_1, e_2, e_3$ and hence with every element of $\mathbb{B}$, including every Lorentz rotor. Consequently:

Duality commutes with the Lorentz group. It is not a Lorentz transformation and cannot be absorbed into one; it is an internal $U(1)$ symmetry generated by the scalar imaginary $i$.
Duality mixes the two sectors. Since $\tilde{F} = i\sqrt{\epsilon}\mathbf{E} - \sqrt{\mu}\mathbf{H}$, the rotated field $e^{-i\theta}\tilde{F}$ has both a Hermitian and an anti-Hermitian part for generic $\theta$, and the electric and magnetic halves are exchanged continuously.
Duality rotates the biquaternion norm. Because $N$ is bilinear, $N(e^{-i\theta}\tilde{F}) = e^{-2i\theta}N(\tilde{F})$, in agreement with the doubled-angle rotation of the pair $(I_1, 2cI_2)$ recorded in the field-strength article.

The new content of the sourced theory is that duality is a symmetry of the equations with sources. If $(\tilde{F}, \tilde{\mathcal{R}})$ satisfies $\tilde{\nabla}\tilde{F} = -\tilde{\mathcal{R}}$, then so does $(e^{-i\theta}\tilde{F}, e^{-i\theta}\tilde{\mathcal{R}})$, because $\tilde{\nabla}$ commutes with the scalar phase. At $\theta = \pi/2$ the rotation sends $\tilde{\mathcal{R}} = \tilde{R}_e + i\tilde{R}_m$ to $-i\tilde{R}_e + \tilde{R}_m$: the electric source is replaced by a magnetic one and vice versa, up to the scalar phase. Without magnetic charge there is no partner for the electric source, and duality is a symmetry only of the source-free equations — the observation recorded in the field-strength article. With magnetic charge the rotation closes on a symmetry of the full sourced theory, and it exchanges electric and magnetic charge. This is the precise sense in which the monopole is the object the second half of the electromagnetic structure is needed for.

The Monopole Field

The static sector of the parent article carries over verbatim. With $\partial_t = 0$ the gradient reduces to $\tilde{\nabla}_{\mathrm{stat}} = e_1\partial_x + e_2\partial_y + e_3\partial_z$, and the sourced equation becomes

$$ \tilde{\nabla}_{\mathrm{stat}}\tilde{F} = -\tilde{R}_e - i\tilde{R}_m . $$

For a point magnetic charge $g$ at the origin, $\rho_m = g\,\delta(\mathbf{x})$ and $\mathbf{J}_m = 0$, so $\tilde{R}_m = ig\,\delta(\mathbf{x})e_0/\sqrt{\mu}$ and the right-hand side is the real scalar source $g\,\delta(\mathbf{x})e_0/\sqrt{\mu}$. The parent article's static Green's function then gives

$$ \tilde{F}(\mathbf{x}) = -\frac{g}{4\pi\sqrt{\mu}}\,\frac{\hat{\mathbf{x}}}{\|\mathbf{x}\|^2}, \qquad \mathbf{B}(\mathbf{x}) = \frac{g}{4\pi}\,\frac{\hat{\mathbf{x}}}{\|\mathbf{x}\|^2}, \qquad \mathbf{E} = 0, $$

the biquaternionic form of the Dirac monopole field $\mathbf{B} = g\hat{\mathbf{x}}/(4\pi\|\mathbf{x}\|^2)$. Substituting back confirms the equation, and the sign is fixed by the scalar part: $\mathrm{div}\,\mathbf{F} = -\rho_m/\sqrt{\mu}$ while $\mathrm{div}\,\mathbf{B} = \rho_m$.

The monopole field has a structural feature worth isolating. The electric Coulomb field of a point charge $q$ is, from the parent article, $\tilde{F} = iq\hat{\mathbf{x}}/(4\pi\sqrt{\epsilon}\|\mathbf{x}\|^2)$: a purely imaginary vector, hence an element of the Hermitian subspace $\mathbb{M}_+$. The magnetic Coulomb field above is a purely real vector, hence an element of the anti-Hermitian subspace $\mathbb{M}_-$. A static electric charge is thus a source whose field lies in the informational half of the algebra; a static magnetic charge is a source whose field lies in the material half. The duality rotation, which multiplies by $-i$, interchanges the two. This is the sharpest statement the framework makes about the monopole: it is the source whose field populates the sector complementary to the electric one.

The field is also classically singular in the familiar way. Its energy density $W = \tfrac{1}{2}\mu\mathbf{H}^2 = \mathbf{B}^2/(2\mu) \propto g^2/r^4$ integrates to a linearly divergent total energy for a point source, so the classical point monopole has infinite self-energy. Its mass is therefore not a prediction of the algebra but a model-dependent quantity: finite-mass monopoles arise only when the point singularity is resolved by additional structure, as in the 't Hooft–Polyakov solitons. The biquaternion algebra is silent on that structure.

The Two Invariants and Magnetic Charge

The biquaternion norm of the field strength was computed in the previous article as

$$ N(\tilde{F}) = -\epsilon\left(I_1 + 2ic\,I_2\right), \qquad I_1 = \mathbf{E}^2 - c^2\mathbf{B}^2, \qquad I_2 = \mathbf{E}\cdot\mathbf{B}. $$

The first invariant $I_1$ is an ordinary scalar; the second, $I_2$, is a pseudoscalar, odd under parity, and it is the imaginary part of the complex biquaternion norm. Because $N(\tilde{F}) \mapsto e^{-2i\theta}N(\tilde{F})$ under duality, the second invariant is exactly the one that duality rotates into the first: the pair $(I_1, 2cI_2)$ turns by the doubled angle.

Two different objects in this framework have a claim to the name "second invariant", and the monopole is responsible for only one of them. The pointwise second invariant is $I_2 = \mathbf{E}\cdot\mathbf{B}$, proportional to the Pontryagin density $F_{\mu\nu}\star F^{\mu\nu}$ of the field tensor. On a static pure monopole it vanishes: $\mathbf{E} = 0$ implies $I_2 = 0$, while $I_1 = -c^2\mathbf{B}^2 < 0$, so the monopole is a purely magnetic-type field in the invariant classification, on the negative-$I_1$ branch. The second invariant is nonzero only for a dyon, a source carrying both electric and magnetic charge, where the electric and magnetic fields are both present and generally not orthogonal. The monopole does not make $I_2$ nonzero; it is the dyon that does.

The topological second invariant is the magnetic charge itself. The electric charge is measured by the flux of $\mathbf{D}$ through a closed surface, $Q_e \propto \oint_{S^2}\mathbf{D}\cdot d\mathbf{A}$, and the magnetic charge by the flux of $\mathbf{B}$,

$$ Q_m \propto \oint_{S^2}\mathbf{B}\cdot d\mathbf{A}, $$

which is why $\rho_m$ appears in the scalar equation. This charge is conserved by the magnetic integrability condition of the second section, and it is a topological characteristic of the field configuration rather than the integral of a local density. If the brief phrase "the second invariant" is read as the second conserved charge of the theory, it is apt: electric charge is the first, magnetic charge is the second, and without monopoles the second is identically zero for every configuration. If it is read as the pointwise invariant $I_2$, it is not: that quantity vanishes on every static monopole. The framework supports the first reading and corrects the second, and the two should not be conflated.

The Dirac Quantisation Condition

A point monopole cannot be described by a single smooth potential, and this is where the algebra meets quantum mechanics. If $\tilde{F}$ were the vector part of $\tilde{\nabla}^{\natural}\tilde{A}$ for one biquaternionic potential $\tilde{A}$ defined everywhere, then the Bianchi identity would hold and no magnetic source could appear. A monopole forces a patchwise potential, with the two patches related on their overlap by a gauge transformation $\tilde{A}' = \tilde{A} - \tilde{\nabla}\Gamma$ with a multivalued $\Gamma$ — the biquaternionic form of the Dirac string. A charged particle transported around the string accumulates the Aharonov–Bohm phase

$$ \exp\left(\frac{i e}{\hbar c}\oint \mathbf{A}\cdot d\boldsymbol{\ell}\right) = \exp\left(\frac{i e g}{\hbar c}\right), $$

using that the flux through a surface bounded by the loop is the monopole charge $g$ in the normalization of the previous section. The phase must be trivial, $\exp(ieg/\hbar c) = 1$, so the product of the two charges is quantised:

$$ \boxed{\;e\,g = 2\pi n\,\hbar c\;}, \qquad n \in \mathbb{Z}. $$

This is the Dirac quantisation condition, written in Heaviside–Lorentz units (the numerical prefactor absorbs factors of $4\pi$ and $\epsilon_0$ in other systems, and $n$ is the winding number of the multivalued gauge parameter). Because $e$ and $g$ appear only as a product, the condition relates them: given the electric charge, it fixes the magnetic charge to be $g = 2\pi n\hbar c/e$. Equivalently, the magnetic fine-structure constant is

$$ \alpha_m = \frac{g^2}{4\pi\hbar c} = \frac{n^2}{4\alpha}, $$

where $\alpha = e^2/(4\pi\hbar c)$ is the electric one. With $\alpha \approx 1/137$, this gives $\alpha_m \approx 34.25\,n^2$: the monopole couples strongly, at a strength of order the inverse of the electric coupling. That is the physical content the biquaternion formulation gathers into a single statement — the phase of the biquaternionic gauge freedom around the string must be trivial, and the charge product is quantised in units of $\hbar c$ accordingly.

The condition is a re-expression, not a prediction. It is a statement about the consistency of quantum mechanics in the presence of a magnetic charge; it constrains the product $eg$ given that a monopole exists. It does not assert that any monopole exists, it does not fix $e$ or $g$ individually, and it says nothing about the monopole's mass. It also imports $\hbar$ from outside the algebra: the biquaternion framework is a complexified classical structure, and every step of the derivation above is classical until the single-valuedness of the phase is imposed. The algebra supplies the home for the gauge freedom; quantum mechanics supplies the quantum.

A Neighbouring Reading: The Monopole as Torsion

The monopole of this article is an abelian magnetic source, coupled through the dual field strength. A neighbouring programme reaches a magnetic charge from a different direction, and because it is the sharpest contrast available it is worth stating here. In the algebrodynamical programme of Kassandrov (recorded in The Algebrodynamical Programme) the electromagnetic potential is complex, $A_\mu=a_\mu+ib_\mu$ with $a_\mu$ and $b_\mu$ real, and the same connection carries the two parts as two geometric fields of a Weyl–Cartan space: the real part $a_\mu$ is the Weyl non-metricity, and the imaginary part $b_\mu$ is the pseudotrace of the torsion. The Coulomb field is the Weyl part and the magnetic monopole is the torsion, so that here magnetic charge is not a source added to Maxwell's equations but a property of the affine connection. The consequence the source draws runs against this article: because torsion does not appear in the equations of geodesics, a monopole built this way would exert no force on a test particle and would be "entirely unobservable". Where the monopole here is an observable charge in Maxwell's equation, the programme's is a geometric field the equations of motion do not see.

The reading also carries an existence statement the corpus's framework lacks. In the programme the connection, its Weyl part and its torsion are all fixed by the single primary equation, and the same connection was shown by Stepanov to be the only space-time connection compatible with a spinor-bundle structure carrying the conventional covariant spinor derivative. A magnetic charge therefore comes out of an algebraic condition instead of being inserted, which is the closest any source in this corpus comes to the "requires, not merely accepts" statement that the section below records as missing. The visible sector is exactly the one this article calls observable: in the programme's 1998 statement of the static solution the electric field is real — $\vec E$ carries the Coulomb field, the magnetic dipole moment and the electric quadrupole moment — while the magnetic field is pure imaginary, and the source groups the magnetic charge together with the electric dipole moment as "phantom", carried by the torsion and entering no equation of motion, so that neither would be observable. Two cautions keep it from settling the matter: the connection, its torsion and the interpretation are the programme's, obtained from a nonlinear primary equation this corpus does not use; and the programme's own conclusion is that the monopole so obtained would be unobservable, which is the opposite of a prediction to be tested. The two readings are recorded side by side, and neither is adopted.

What the Framework Does Not Say

The temptation with a symmetry this clean is to read a prediction out of it. It is not there, and the gap is worth stating plainly.

Existence. Nothing in the algebra privileges a configuration with $\rho_m \neq 0$ over one with $\rho_m = 0$. The sourced equation accepts magnetic charge; it does not require it. The symmetric system is a larger theory of which the physical one — with $\rho_m \equiv 0$ — is a consistent special case.
Mass. The point monopole has divergent classical self-energy, so the algebra cannot fix a mass. Finite masses come from the mechanism that resolves the core and are model-dependent: the 't Hooft–Polyakov soliton mass is of order $4\pi M_W/g^2$, and grand-unified monopoles are enormous. None of this is in the algebra.
Observation. No magnetic monopole has been observed; the null results bound the monopole flux, not the algebra. The framework is consistent with them because it predicts nothing.
Quantum structure. The qubit operator algebra of $\mathbb{M}_+$ is available, but nothing here couples it to the integer $n$ of the Dirac condition. Whether that integer can be tied to the discrete structure of the informational sector is not addressed and is recorded as open.

A structural argument from the Clifford literature. The gaps above are about what the algebra does and does not require. There is also an argument within the Clifford formulation against magnetic sources, and it is worth stating because it is not empirical. Gsponer and Hurni observe that in the biquaternion form of Maxwell's equations the source current and the potential must be reversion symmetric, and that the equation that follows — their $\nabla\tilde{F} - \tilde{F}^{\sim}\nabla = 0$ — is stronger than the tensor Bianchi identity: it excludes any kind of magnetic source, not merely smooth ones. Reversion invariance thus enforces the electric–magnetic asymmetry as a structural feature of the formulation rather than as an experimental input. The corpus records this as the one structural (rather than observational) counter-argument to magnetic charge in the Clifford literature, and notes that it bears directly on programmes that add magnetic charge by hand, such as the electro-gravimagnetic programme of The Electro-Gravimagnetic Field and the Magnetic-Charge–Mass Hypothesis. The argument is conditional: it applies to formulations whose fields are reversion-covariant, and it is not a proof that magnetic charge is impossible in every biquaternionic setting.

The relation as a reversal parity. The same relation has a second reading, and it places the monopole-free statement in the framework's classical layer. The companion article The Proca Equation: Massive Spin 1 in Biquaternionic Form records that Maxwell's equation with a mass term is one of the descendants of Lanczos's coupled system $\tilde{\nabla}\tilde{A}=m\tilde{B}$, $\tilde{\nabla}\tilde{B}=m\tilde{A}$, and that the descendant has two reversal parities: the reversal-even combination is Proca's massive equation (the Maxwell equation in its massless limit), and the reversal-odd combination is exactly the relation $\nabla\tilde{F} - \tilde{F}^{\sim}\nabla = 0$ written above. The absence of magnetic charge is therefore, in this formulation, the vanishing of the reversal-odd half of a single equation whose reversal-even half is the massive or massless vector equation the corpus already writes — a parity statement about one equation, not a law added to the field equations from outside. It is the same content as the opening observation of this article — that a single smooth potential for the field strength makes the Bianchi identity hold — seen from the other side: a magnetic source is what makes the reversal-odd half nonzero, and it does so precisely by forcing the potential to be patchwise rather than one ordinal-invariant biquaternion.

The honest summary is that the biquaternion framework is an excellent language for the monopole, and that language is a language of reformulation. Whether it can be made to say something new — a mass, a coupling, a selection rule — is the open question.

Summary

Maxwell's equations with magnetic sources extend the biquaternionic Maxwell equation to

$$ \tilde{\nabla}\tilde{F} = -\tilde{R}_e - i\tilde{R}_m, \qquad \tilde{R}_m = \frac{i\rho_m}{\sqrt{\mu}} + \sqrt{\epsilon}\,\mathbf{J}_m, $$

with the factor of $-i$ placing the magnetic source in the sector complementary to the electric one. The two sources assemble into the single complex source biquaternion $\tilde{\mathcal{R}} = \tilde{R}_e + i\tilde{R}_m$, and the dual field-strength biquaternion is $\tilde{F}_\star = -i\tilde{F}$: Hodge duality is internal multiplication by a scalar phase, not an independent field.

The duality rotation $\tilde{F} \mapsto e^{-i\theta}\tilde{F}$ is generated by a scalar in $\mathbb{C}_{\mathbb{B}}$, commutes with the quaternion units and with every Lorentz rotor, and is therefore an internal $U(1)$ symmetry, not a spacetime one. It rotates the two sectors of the field into one another, and with magnetic sources present it becomes a symmetry of the sourced equations, exchanging electric and magnetic charge. That is the structural role of the monopole in this framework.

The static monopole field is $\tilde{F} = -g\hat{\mathbf{x}}/(4\pi\sqrt{\mu}\|\mathbf{x}\|^2)$, a purely real vector in $\mathbb{M}_-$, in contrast with the electric Coulomb field, a purely imaginary vector in $\mathbb{M}_+$. A static monopole has $I_2 = \mathbf{E}\cdot\mathbf{B} = 0$ and $I_1 < 0$; the second pointwise invariant is nonzero only for a dyon. The genuinely "second" quantity is the topological magnetic charge $Q_m \propto \oint_{S^2}\mathbf{B}\cdot d\mathbf{A}$, conserved because it is the integrability condition of the magnetic equation.

The Dirac quantisation condition $eg = 2\pi n\hbar c$ (Heaviside–Lorentz units) emerges from the single-valuedness of the biquaternionic gauge phase around the Dirac string, equivalently $\alpha_m = n^2/(4\alpha)$. It constrains the product of the charges; it does not predict the monopole's existence or mass, and it imports $\hbar$ from quantum mechanics. The framework re-expresses the monopole and its quantisation faithfully, and it is silent where the standard theory is silent.

A neighbouring programme reads the monopole differently, and the contrast is sharp enough to keep beside this account. In Kassandrov's algebrodynamics the potential is complex, its real part is the Weyl non-metricity of a Weyl–Cartan connection and its imaginary part is the torsion pseudotrace; the Coulomb field is the Weyl part and the magnetic monopole is the torsion, forced by the programme's primary equation rather than inserted. Because torsion does not enter the equations of geodesics, the source concludes that the monopole so obtained would be unobservable. That is the opposite of the reading recorded here — an observable charge in Maxwell's equation — and the two are recorded side by side, neither adopted.

Summary of Notation

Symbol	Meaning
$\mathbb{B}$	Biquaternion algebra
$e_0 = 1, e_1, e_2, e_3$	Quaternion basis, $e_k^2 = -e_0$
$i$	Scalar imaginary, $i^2 = -1$
$\mathbb{C}_{\mathbb{B}}, \mathbb{H}_{\mathbb{B}}$	Scalar subspace, real-quaternion subspace
$\mathbb{M}_+, \mathbb{M}_-$	Hermitian and anti-Hermitian subspaces
$\tilde{\nabla}$	Biquaternionic gradient
$\tilde{F} = i\sqrt{\epsilon}\,\mathbf{E} - \sqrt{\mu}\,\mathbf{H}$	Field-strength biquaternion (pure vector)
$\mathbf{E}, \mathbf{H}, \mathbf{B} = \mu\mathbf{H}$	Electric field, magnetic field, magnetic induction
$\epsilon, \mu$	Permittivity and permeability of the medium
$c = 1/\sqrt{\epsilon\mu}$	Speed of light in the medium
$\tilde{F}_\star = -i\tilde{F}$	Dual field-strength biquaternion
$\tilde{R}_e = i\rho_e/\sqrt{\epsilon} + \sqrt{\mu}\,\mathbf{J}_e$	Electric source biquaternion
$\tilde{R}_m = i\rho_m/\sqrt{\mu} + \sqrt{\epsilon}\,\mathbf{J}_m$	Magnetic source biquaternion
$\tilde{\mathcal{R}} = \tilde{R}_e + i\tilde{R}_m$	Complex source biquaternion
$\rho_e, \mathbf{J}_e; \rho_m, \mathbf{J}_m$	Electric and magnetic charge and current densities
$N(\tilde{F}) = -\epsilon(I_1 + 2icI_2)$	Biquaternion norm of the field strength
$I_1 = \mathbf{E}^2 - c^2\mathbf{B}^2$	First invariant (scalar)
$I_2 = \mathbf{E}\cdot\mathbf{B}$	Second invariant (pseudoscalar)
$\mathbf{V} = \mathbf{E} + ic\mathbf{B}$	Riemann–Silberstein vector, $\tilde{F} = i\sqrt{\epsilon}\mathbf{V}$
$g$	Magnetic charge
$Q_m \propto \oint_{S^2}\mathbf{B}\cdot d\mathbf{A}$	Magnetic (topological) charge
$e$	Electric charge
$n \in \mathbb{Z}$	Winding number in the Dirac condition
$\alpha, \alpha_m$	Electric and magnetic fine-structure constants