Quaternionic Charges

• Quaternionic charges are a unifying framework where quaternion-valued quantities encode multiple physical, gauge, and topological invariants.
• They are applied in diverse contexts, including dyon theories, superconductivity, and band topology, to analyze transformation laws and conserved currents.
• The non-commutative nature of quaternions organizes complex interaction terms and invariants, offering new insights into electromagnetic, gravitational, and quantum systems.

“Quaternionic charges” denotes a family of constructions in which charge-like data are encoded by quaternion-valued variables or by invariants naturally attached to quaternionic geometry. In the literature, the term does not refer to a single universal conserved quantity. It may mean a literal quaternion $Q=e-ig-jm-kh$ unifying electric, magnetic, gravitational, and Heavisidian charges; a gauge charge assignment for quaternion-valued superconducting order parameters, where $q\mapsto e^{2i\phi}q$ and $Q\propto \mathrm{Sc}(q^2)\mapsto e^{4i\phi}Q$; a non-abelian topological charge valued in the quaternion group $Q_8$; or a system of quaternionic coordinates from which electric charge and related quantum numbers are extracted as linear functionals (Negi et al., 2010, Tantardini et al., 18 Jan 2026, Finck et al., 25 Jul 2025, Jansson, 2024).

1. Terminological range and recurrent structures

Across the cited literature, quaternionic charges arise whenever quaternions are used to package several real or complex components into one algebraic object and the physical “charge” is then read from its transformation law, bilinear invariants, or homotopy class. The common algebraic background is the quaternion basis

$$q=a_0\mathbf{1}+a_x\boldsymbol{i}+a_y\boldsymbol{j}+a_z\boldsymbol{k},$$

with $$\boldsymbol{i}^2=\boldsymbol{j}^2=\boldsymbol{k}^2=\boldsymbol{i}\boldsymbol{j}\boldsymbol{k}=-1$$, but the physical meaning attached to the quaternion varies sharply by context (Negi et al., 2010, Tantardini et al., 18 Jan 2026).

Setting	Quaternionic object	Meaning of “charge”
Gravi-electromagnetic dyon theory	$Q=e-ig-jm-kh$	Four physical charges in one quaternion
Quaternionic superconductivity	$q$, $Q\propto\mathrm{Sc}(q^2)$	Gauge charges $2e$ and $q\mapsto e^{2i\phi}q$0
Three-band real band topology	$q\mapsto e^{2i\phi}q$1	Non-abelian topological loop charge
$q\mapsto e^{2i\phi}q$2 electroweak model	$q\mapsto e^{2i\phi}q$3	Electric charge as a linear functional

A recurrent misconception is that quaternionic charge must be a new conserved quantum number. That is not generally the case. In the quaternionic superconductivity framework, the paper states explicitly that “quaternionic charges” are “not a new conserved quantum number,” but a unifying language for electromagnetic charge, symmetry transformation properties, and topological indices (Tantardini et al., 18 Jan 2026). In quaternionic scalar field theory, by contrast, the relevant conserved quantities remain sums or weighted sums of ordinary $q\mapsto e^{2i\phi}q$4 charges carried by complex components, with no additional charge triplet directly associated with $q\mapsto e^{2i\phi}q$5 (Giardino, 2022). This suggests that the expression should be read contextually rather than ontologically.

2. Unified charge algebras in dyonic and gravi-electromagnetic theories

One major use of the term is literal and algebraic: the charge itself is taken to be a quaternion. In the quaternionic generalization of the Schwinger–Zwanziger dyon, the unified charge is

$q\mapsto e^{2i\phi}q$6

where $q\mapsto e^{2i\phi}q$7 is electric charge, $q\mapsto e^{2i\phi}q$8 magnetic charge, $q\mapsto e^{2i\phi}q$9 gravitational charge, and $Q\propto \mathrm{Sc}(q^2)\mapsto e^{4i\phi}Q$0 Heavisidian charge. Quaternion conjugation gives $Q\propto \mathrm{Sc}(q^2)\mapsto e^{4i\phi}Q$1, and the product $Q\propto \mathrm{Sc}(q^2)\mapsto e^{4i\phi}Q$2 decomposes into four real couplings $Q\propto \mathrm{Sc}(q^2)\mapsto e^{4i\phi}Q$3, which reproduce the coefficients entering the force law and the conserved angular momentum. The generalized Schwinger–Zwanziger condition becomes

$Q\propto \mathrm{Sc}(q^2)\mapsto e^{4i\phi}Q$4

with $Q\propto \mathrm{Sc}(q^2)\mapsto e^{4i\phi}Q$5 the nontrivial quaternionic components of the two-body coupling (Negi et al., 2010).

In this formulation, quaternionic charge does not merely bundle four numbers. The non-commutative product organizes which bilinear combinations enter radial forces, topological couplings, and quantization rules. The usual electromagnetic dyon condition $Q\propto \mathrm{Sc}(q^2)\mapsto e^{4i\phi}Q$6 and its gravito-dyon analogue appear as lower-dimensional projections obtained by setting subsets of components to zero. The theory therefore treats electromagnetic dyons, gravito-dyons, and mixed two-charge sectors as different slices of the same quaternionic algebraic object (Negi et al., 2010).

A related but structurally distinct construction appears in quaternion gravi-electromagnetism, where the generalized charge is first written as a complex quantity

$Q\propto \mathrm{Sc}(q^2)\mapsto e^{4i\phi}Q$7

with $Q\propto \mathrm{Sc}(q^2)\mapsto e^{4i\phi}Q$8 and $Q\propto \mathrm{Sc}(q^2)\mapsto e^{4i\phi}Q$9, and then embedded into quaternionic currents, potentials, and differential operators. There the complex part organizes the internal gravity–electromagnetism doublet, while the quaternionic basis organizes spacetime structure. The field equations

$Q_8$0

are quaternionically covariant, but the gravity–electromagnetism mixing itself is carried by a complex duality rotation rather than by the quaternion units $Q_8$1 (Rawat et al., 2011). This comparison is useful because it shows two incompatible but instructive meanings of “quaternionic charge”: either the charge itself is quaternion-valued, or it is a complex or multi-component charge embedded into a quaternionic formalism.

3. Internal quantum numbers, color, and supersymmetric central charges

In particle-theoretic model building, quaternionic charges often mean quaternionic coordinates from which standard quantum numbers are reconstructed. In the $Q_8$2 root-system model, each fermion flavor is identified with a quaternion

$Q_8$3

with $Q_8$4 designated as fundamental quantum numbers. Electric charge is then defined by the linear functional

$Q_8$5

The same quaternionic coordinates also re-express $Q_8$6, while the binary tetrahedral group $Q_8$7 acts by quaternion multiplication on the $Q_8$8-cell of $Q_8$9 roots and serves as the rule for electroweak interactions (Jansson, 2024). In that setting, quaternionic charge means both a coordinate assignment and a discrete interaction mechanism.

A different use appears in quaternionic Dirac theory for colored particles. There, the three imaginary units $$q=a_0\mathbf{1}+a_x\boldsymbol{i}+a_y\boldsymbol{j}+a_z\boldsymbol{k},$$0 are associated with the three colors, single and double quaternionic conjugations define additional conserved currents, and electromagnetic coupling with a quaternionic operator $$q=a_0\mathbf{1}+a_x\boldsymbol{i}+a_y\boldsymbol{j}+a_z\boldsymbol{k},$$1 yields composite quasi-particles carrying effective charges $$q=a_0\mathbf{1}+a_x\boldsymbol{i}+a_y\boldsymbol{j}+a_z\boldsymbol{k},$$2 and $$q=a_0\mathbf{1}+a_x\boldsymbol{i}+a_y\boldsymbol{j}+a_z\boldsymbol{k},$$3. The resulting proton- and neutron-like composites are color-neutral because the allowed two- and three-body quaternionic charge combinations must satisfy permutation invariance under the $$q=a_0\mathbf{1}+a_x\boldsymbol{i}+a_y\boldsymbol{j}+a_z\boldsymbol{k},$$4 basis exchange (Welch, 2008). Here quaternionic charge is not a single scalar observable but an internal algebraic content controlling color assignments, conserved currents, and effective fractional electric charge.

Quaternionic superspace introduces yet another meaning. In quaternionic $$q=a_0\mathbf{1}+a_x\boldsymbol{i}+a_y\boldsymbol{j}+a_z\boldsymbol{k},$$5-extended supertwistor constructions, the superspace has $$q=a_0\mathbf{1}+a_x\boldsymbol{i}+a_y\boldsymbol{j}+a_z\boldsymbol{k},$$6 even-quaternionic coordinates and $$q=a_0\mathbf{1}+a_x\boldsymbol{i}+a_y\boldsymbol{j}+a_z\boldsymbol{k},$$7 odd-quaternionic coordinates, each quaternion carrying four complex fields. The paper identifies tensorial central charges with generators associated to the tensorial representation parameters $$q=a_0\mathbf{1}+a_x\boldsymbol{i}+a_y\boldsymbol{j}+a_z\boldsymbol{k},$$8 inside the superspace matrix $$q=a_0\mathbf{1}+a_x\boldsymbol{i}+a_y\boldsymbol{j}+a_z\boldsymbol{k},$$9, and states that for even $$\boldsymbol{i}^2=\boldsymbol{j}^2=\boldsymbol{k}^2=\boldsymbol{i}\boldsymbol{j}\boldsymbol{k}=-1$$0 the maximal possible internal group is

$$\boldsymbol{i}^2=\boldsymbol{j}^2=\boldsymbol{k}^2=\boldsymbol{i}\boldsymbol{j}\boldsymbol{k}=-1$$1

In this language, quaternionic charges are tensorial central charges realized geometrically as internal quaternionic coordinates and algebraically as $$\boldsymbol{i}^2=\boldsymbol{j}^2=\boldsymbol{k}^2=\boldsymbol{i}\boldsymbol{j}\boldsymbol{k}=-1$$2-compatible generators (Cirilo-Lombardo et al., 2016).

4. Gauge charge in quaternionic superconductivity

In quaternionic superconductivity, the spinful gap is rewritten as a single quaternion field

$$\boldsymbol{i}^2=\boldsymbol{j}^2=\boldsymbol{k}^2=\boldsymbol{i}\boldsymbol{j}\boldsymbol{k}=-1$$3

which packages the spin-singlet and spin-triplet pairing amplitudes into one order parameter. Under a global electromagnetic $$\boldsymbol{i}^2=\boldsymbol{j}^2=\boldsymbol{k}^2=\boldsymbol{i}\boldsymbol{j}\boldsymbol{k}=-1$$4 gauge rotation, $$\boldsymbol{i}^2=\boldsymbol{j}^2=\boldsymbol{k}^2=\boldsymbol{i}\boldsymbol{j}\boldsymbol{k}=-1$$5, so the quaternion field carries charge $$\boldsymbol{i}^2=\boldsymbol{j}^2=\boldsymbol{k}^2=\boldsymbol{i}\boldsymbol{j}\boldsymbol{k}=-1$$6. The scalar part of its square,

$$\boldsymbol{i}^2=\boldsymbol{j}^2=\boldsymbol{k}^2=\boldsymbol{i}\boldsymbol{j}\boldsymbol{k}=-1$$7

transforms as $$\boldsymbol{i}^2=\boldsymbol{j}^2=\boldsymbol{k}^2=\boldsymbol{i}\boldsymbol{j}\boldsymbol{k}=-1$$8 and therefore carries charge $$\boldsymbol{i}^2=\boldsymbol{j}^2=\boldsymbol{k}^2=\boldsymbol{i}\boldsymbol{j}\boldsymbol{k}=-1$$9. The corresponding Ginzburg–Landau functional uses the covariant derivatives

$Q=e-ig-jm-kh$0

and includes a trilinear coupling $Q=e-ig-jm-kh$1 that locks the quartet phase to twice the Cooper-pair phase, $Q=e-ig-jm-kh$2 (Tantardini et al., 18 Jan 2026).

This framework makes the charge content of the theory algebraically transparent. The same quaternion field generates both the primary $Q=e-ig-jm-kh$3 condensate and a composite $Q=e-ig-jm-kh$4 quarteting field. Above the usual $Q=e-ig-jm-kh$5 transition, integrating out Gaussian $Q=e-ig-jm-kh$6-fluctuations renormalizes the quarteting mass to

$Q=e-ig-jm-kh$7

so vestigial charge-$Q=e-ig-jm-kh$8 order appears when $Q=e-ig-jm-kh$9 while $q$0. In $q$1D, $q$2; in $q$3D, $q$4. Device-level consequences then follow from the $q$5 charge assignment: a pure-$q$6 vortex carries $q$7 flux, the current-phase relation can enter a regime with $q$8, and the Josephson response displays doubled ac emission and even-only Shapiro steps (Tantardini et al., 18 Jan 2026).

The same paper also ties gauge charge to topology. In $q$9D class DIII, the $Q\propto\mathrm{Sc}(q^2)$0 invariant is computed directly from $Q\propto\mathrm{Sc}(q^2)$1 via a Pfaffian formula at TRIM, while in $Q\propto\mathrm{Sc}(q^2)$2D the normalized quaternion $Q\propto\mathrm{Sc}(q^2)$3 supports an integer winding. This is a particularly dense use of the term: electromagnetic charges, symmetry covariance, and topological charges are all encoded in the same quaternionic variables (Tantardini et al., 18 Jan 2026).

5. Topological, homotopical, and geometric charges

In real three-band band theory, quaternionic charge appears as a genuinely non-abelian topological invariant. For a real symmetric Bloch Hamiltonian with three bands, the space of orthonormal eigenframes modulo sign choices is

$Q\propto\mathrm{Sc}(q^2)$4

so

$Q\propto\mathrm{Sc}(q^2)$5

A closed loop $Q\propto\mathrm{Sc}(q^2)$6 in the Brillouin zone avoiding degeneracies therefore carries a quaternionic charge

$Q\propto\mathrm{Sc}(q^2)$7

Small loops around isolated Dirac points acquire charges $Q\propto\mathrm{Sc}(q^2)$8 or $Q\propto\mathrm{Sc}(q^2)$9 depending on which pair of bands crosses, and non-abelian braiding conjugates these charges. The product of loop charges determines whether two Dirac points can annihilate: total charge $2e$0 allows annihilation, while nontrivial total charge obstructs it. The same obstruction is tracked by the patch Euler number on a subregion without additional degeneracies (Finck et al., 25 Jul 2025).

A broader homotopical use appears in orbi-$2e$1-theory. There, microscopic topological phases and brane charges are first placed in unstable Cohomotopy and then measured in stable cohomology by maps that factor through the quaternionic Hopf fibration $2e$2. The relevant stable datum is a quaternionic orientation

$2e$3

restricting to the unit on $2e$4. In that formulation, quaternionic charges are not ordinary source charges but Cohomotopy classes or their images in twisted, $2e$5-equivariant orbi-$2e$6-theory; the paper applies this to gapped nodal lines in the Brillouin torus and to M-string charges in M5-brane probes (Sati et al., 16 Nov 2025).

Quaternionic pluripotential theory provides a geometric analogue. For a quaternionic $2e$7-subharmonic function $2e$8, the measure

$2e$9

acts as a nonnegative quaternionic $q\mapsto e^{2i\phi}q$00-Hessian charge density, and the total charge on a set $q\mapsto e^{2i\phi}q$01 is

$q\mapsto e^{2i\phi}q$02

The theory then defines $q\mapsto e^{2i\phi}q$03-capacity, equilibrium potentials, and an $q\mapsto e^{2i\phi}q$04-Lelong number $q\mapsto e^{2i\phi}q$05 measuring local charge concentration at a point. Its fundamental solution satisfies

$q\mapsto e^{2i\phi}q$06

so point charges are represented by Dirac masses for the quaternionic Hessian operator (Liu et al., 2022). This extends the term “quaternionic charge” from particle and band theory to a nonlinear geometric measure theory.

6. Charge operators, induced sources, and conceptual limits

In free quaternionic scalar field theory on a real Hilbert space, the charge structure is multi-component but still abelian. The field can be quantized in either a four-component or a two-component scheme, and the total conserved charge is

$q\mapsto e^{2i\phi}q$07

in the four-component case, or

$q\mapsto e^{2i\phi}q$08

in the two-component case. The corresponding charge operators are sums or weighted sums of ordinary Klein–Gordon $q\mapsto e^{2i\phi}q$09 charges, and the paper states that there is no additional triplet of charges directly associated with the quaternionic units $q\mapsto e^{2i\phi}q$10 (Giardino, 2022). This is an important limiting case: quaternionic structure need not imply new non-abelian charge quantum numbers.

Quaternionic electrodynamics and dyonic plasma theory move in the opposite direction by treating electric and magnetic source sectors as quaternionic from the outset. In quaternionic electrodynamics, the electric and magnetic fields, charge densities, and currents are all taken to be pure quaternionic, the magnetic field contains a self-interaction term

$q\mapsto e^{2i\phi}q$11

and magnetic charge density arises dynamically from $q\mapsto e^{2i\phi}q$12. The resulting field angular momentum gives the generalized Dirac quantization condition

$q\mapsto e^{2i\phi}q$13

In dyonic cold plasma, quaternionic hydro-electromagnetic fields encode electric and magnetic charge densities and currents, and the quaternionic field equation yields generalized Dirac–Maxwell equations, electric and magnetic continuity equations, Bernoulli-like energy conservation, Navier–Stokes-like momentum balance, and two wave families: Langmuir-like electron waves and ’t Hooft–Polyakov-like monopole waves (Giardino, 2020, Chanyal, 2019).

A more speculative use appears in electrodynamics with quaternionic mass, where mass is promoted to

$q\mapsto e^{2i\phi}q$14

There, the vector part $q\mapsto e^{2i\phi}q$15 induces effective electric charge and current densities such as

$q\mapsto e^{2i\phi}q$16

and the paper maps $q\mapsto e^{2i\phi}q$17 and $q\mapsto e^{2i\phi}q$18 to the spatial and temporal derivatives of an axion field, thereby relating quaternionic-mass electrodynamics to axion electrodynamics and to gauge-invariant Maxwell–Proca equations (Arbab, 2022). This suggests a further extension of the term from intrinsic charges to induced source terms generated by quaternionic material parameters.

Taken together, these usages show that “quaternionic charges” is best understood as a polysemous technical expression. Depending on the setting, it may denote quaternion-valued physical charges, gauge charges carried by quaternionic condensates, internal or central charges in superspace, non-abelian topological charges, Cohomotopy or $q\mapsto e^{2i\phi}q$19-theoretic brane charges, Hessian charge measures, or effective source terms induced by quaternionic constitutive structure. This suggests that the unifying content is not a common conserved observable but a common algebraic mechanism: quaternions organize several charge sectors, symmetry actions, or invariants into one non-commutative object whose scalar, vector, and conjugation properties carry the physically relevant information (Tantardini et al., 18 Jan 2026).