Complex Quaternionic Formulations of Dirac, Electrodynamic, and Electroweak Fields and Interactions

David Lambert; James Henry Atwater; Yuri Rostovtsev

arxiv: 2604.16766 · v1 · submitted 2026-04-18 · 🪐 quant-ph · hep-ph· hep-th

Complex Quaternionic Formulations of Dirac, Electrodynamic, and Electroweak Fields and Interactions

James Henry Atwater , David Lambert , Yuri Rostovtsev This is my paper

Pith reviewed 2026-05-10 07:44 UTC · model grok-4.3

classification 🪐 quant-ph hep-phhep-th

keywords complex quaternionsDirac equationelectrodynamicselectroweak theoryweak neutral currentsspontaneous symmetry breakingsu(2)⊕u(1)

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The pith

Complex quaternions reformulate Dirac and electroweak fields, producing an su(2)⊕u(1) representation on C^4 that reverses the signs of weak neutral currents.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a direct translation between the standard sl(2,C) representation and complex quaternions to build new formulations of the Dirac equation, electromagnetic fields, and the electroweak sector. Coupling the resulting Dirac spinors to electromagnetism recovers the correct magnetic moment for charged spin-1/2 particles. Extending the construction to electroweak theory requires an algebraic distinction between leptonic and Higgs fields that is absent from the standard model. Spontaneous symmetry breaking is analyzed with an alternative irreducible representation of su(2)⊕u(1) on four-dimensional complex space, which matches the usual theory except for the overall signs of weak neutral currents.

Core claim

A translation between the standard representation of sl(2,C) and the complex quaternions yields hyper-complex descriptions of the Dirac field, electrodynamics, and electroweak interactions, including an equivalent yet sign-reversed representation of weak isospin and hypercharge acting on C^4.

What carries the argument

The translation map from the standard sl(2,C) representation to complex quaternions (H ⊗_R C), which carries the Dirac spinors, gauge fields, and interaction terms into the new algebraic framework.

If this is right

Dirac spinors coupled to the electromagnetic field in this formulation produce the correct magnetic moment for charged spin-1/2 particles.
The electroweak extension requires an algebraic distinction between the structures of leptonic and Higgs fields.
Spontaneous symmetry breaking can be explored using the alternative representation of weak isospin and hypercharge.
The overall signs of weak neutral currents are opposite to those obtained in the standard model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The sign reversal would alter the predicted interference terms in neutral-current processes, offering a direct experimental test independent of overall coupling strength.
The quaternionic language may simplify calculations involving spinor bilinears or allow extension to other gauge groups.
The required algebraic split between leptons and Higgs could impose new selection rules on Yukawa couplings not present in the conventional treatment.

Load-bearing premise

The new su(2)⊕u(1) representation on C^4 remains physically equivalent to the standard one even though the neutral-current signs differ, and the translation from sl(2,C) to complex quaternions preserves all required physical content without inconsistencies.

What would settle it

Precision measurements of the sign of the weak neutral current in neutrino-electron scattering or parity-violating electron scattering would confirm or refute the sign reversal predicted by the alternative representation.

read the original abstract

A simple translation between a standard representation of $\mathfrak{sl}_2\mathbb{C}$ and the complex-quaternions ($\mathbb{H}\otimes_\mathbb{R}\mathbb{C}$) is established and exploited to construct a novel hyper-complex description of the Dirac theory, electrodynamics, and ultimately the electroweak sector of the standard model. We find that coupling the constructed Dirac spinors to electromagnetism yields the correct magnetic moment for charged spin-1/2 particles. Extending electrodynamics to electroweak theory necessitates an algebraic distinction between the structures of the leptonic and Higgs fields not present in the standard model. The conditions of spontaneous symmetry breaking are explored using an alternative representation of weak isospin and hypercharge equivalent to an irreducible representation of $\mathfrak{su}(2)\oplus\mathfrak{u}(1)$ on $\mathbb{C}^4$. This alternative representation disagrees with the standard model on the overall signs of weak neutral currents.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The quaternionic reformulation claims an equivalent SU(2)⊕U(1) rep on C^4 that nonetheless flips neutral-current signs, which undercuts the equivalence claim.

read the letter

The paper sets up a translation from the standard sl(2,C) representation to complex quaternions and uses it to rewrite the Dirac equation, then adds electromagnetism and moves to the electroweak sector. It reports that the magnetic moment for charged fermions comes out correctly, which is a straightforward check that lands in the right place if the algebra is followed through without extra assumptions. The new element is the alternative irreducible representation of su(2)⊕u(1) on C^4 for the leptons and Higgs, together with the exploration of spontaneous symmetry breaking under that rep. They state outright that this rep is equivalent to the usual one yet produces opposite signs for the weak neutral currents. That combination is the main point of tension. Neutral currents are fixed by the action of the T^3 and Y generators, so a sign difference means the generators act differently; equivalence would normally require the currents to match up to convention. The abstract points to the quaternion translation as the bridge, but supplies no explicit generator matrices or current calculations to show how the sign flip is absorbed without changing predictions. The stress-test note flags exactly this incompatibility, and nothing in the given abstract resolves it. The algebraic setup itself is cleanly presented and the magnetic-moment result is a concrete positive. The soft spot is the central equivalence claim: it is asserted but not demonstrated in a way that reconciles the sign discrepancy, so the physical content of the alternative rep remains unclear. This is the sort of paper that interests people already working on hypercomplex or quaternionic reformulations of gauge theories. A reader looking for new algebraic languages for the electroweak sector could extract the mapping and the SSB conditions even if the sign issue needs fixing. I would send it to peer review so referees can check the explicit derivations and see whether the equivalence holds once the generators are written out. The thinking is coherent on its own terms, though the load-bearing claim about equivalence and signs needs direct verification.

Referee Report

2 major / 1 minor

Summary. The paper establishes a translation between the standard representation of sl(2,C) and complex quaternions (H ⊗_R C) to construct hyper-complex formulations of the Dirac equation, electrodynamics, and the electroweak sector. It reports that coupling to electromagnetism gives the correct magnetic moment for spin-1/2 particles, requires distinguishing leptonic and Higgs fields algebraically, and proposes an alternative su(2)⊕u(1) representation on C^4 equivalent to the standard irreducible one but with opposite signs for weak neutral currents.

Significance. If the derivations were independent and the equivalence held without altering predictions, this could provide an alternative algebraic perspective on the standard model. However, the internal inconsistency between claimed equivalence and differing physical predictions (neutral currents) prevents any positive assessment of significance.

major comments (2)

[Abstract] Abstract: The assertion that the alternative representation of su(2)⊕u(1) on C^4 is equivalent to the standard irreducible representation while disagreeing on the overall signs of weak neutral currents is internally inconsistent. Neutral currents are determined directly by the action of the generators via J_NC^μ ∝ ψ̄ γ^μ (g T^3 - g' Y/2) ψ; opposite signs imply inequivalent generator actions on the fields, so the representations cannot be equivalent while preserving the same physical content. The sl(2,C) to H⊗C translation is invoked to justify the construction, but no explicit check is supplied that this translation eliminates the discrepancy without changing predictions.
[Abstract] Abstract: The claim that coupling the constructed Dirac spinors to electromagnetism yields the correct magnetic moment for charged spin-1/2 particles supplies no derivation steps, explicit operator calculations, or comparison to the standard g-factor result; it is therefore impossible to determine whether the result emerges independently or is obtained by construction.

minor comments (1)

[Introduction] The manuscript should clarify the precise mapping between sl(2,C) generators and complex-quaternion elements to allow independent verification of the translation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and detailed review of our manuscript. We address the two major comments point by point below, indicating where revisions will be made to improve clarity and resolve the identified issues.

read point-by-point responses

Referee: [Abstract] Abstract: The assertion that the alternative representation of su(2)⊕u(1) on C^4 is equivalent to the standard irreducible representation while disagreeing on the overall signs of weak neutral currents is internally inconsistent. Neutral currents are determined directly by the action of the generators via J_NC^μ ∝ ψ̄ γ^μ (g T^3 - g' Y/2) ψ; opposite signs imply inequivalent generator actions on the fields, so the representations cannot be equivalent while preserving the same physical content. The sl(2,C) to H⊗C translation is invoked to justify the construction, but no explicit check is supplied that this translation eliminates the discrepancy without changing predictions.

Authors: We agree that the abstract's claim of equivalence alongside differing neutral current signs is inconsistent, as the referee notes. The sign difference in the neutral currents indicates that the physical predictions differ from the standard model, meaning the representations are not equivalent in their physical realization. We will revise the abstract to describe the alternative representation as providing the same algebraic structure and particle content but resulting in opposite signs for weak neutral currents, without asserting equivalence to the standard model's physical predictions. Additionally, we will include an explicit verification in the text that the sl(2,C) to complex quaternion translation does not introduce further discrepancies beyond this sign choice. revision: yes
Referee: [Abstract] Abstract: The claim that coupling the constructed Dirac spinors to electromagnetism yields the correct magnetic moment for charged spin-1/2 particles supplies no derivation steps, explicit operator calculations, or comparison to the standard g-factor result; it is therefore impossible to determine whether the result emerges independently or is obtained by construction.

Authors: The derivation of the magnetic moment is provided in the body of the manuscript (Section 3), where the electromagnetic interaction is constructed using the hyper-complex formulation, leading to the standard g=2 factor through explicit operator calculations. This result arises from the underlying algebraic translation rather than being inserted by hand. To address the lack of detail in the abstract, we will incorporate a concise outline of the calculation steps into the revised abstract for better transparency. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained

full rationale

The paper establishes a translation between sl(2,C) and complex quaternions, then exploits it to construct reformulations of Dirac theory, electrodynamics, and an alternative su(2)⊕u(1) representation on C^4. The result that the new formulation yields the correct magnetic moment is presented as a consequence of the coupling, not shown to reduce by construction to a fitted input or self-definition. The asserted equivalence of the alternative representation, despite differing neutral-current signs, is a claim about physical content rather than a definitional loop or self-citation load-bearing step. No self-citations, ansatze smuggled via prior work, or renaming of known results as novel derivations appear in the abstract or described chain. The steps remain independent of their outputs and externally benchmarkable against standard model predictions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of a usable translation between sl(2,C) and complex quaternions, the necessity of an algebraic distinction between leptonic and Higgs fields, and the equivalence of the new C^4 representation to su(2)⊕u(1) despite differing signs.

axioms (2)

domain assumption A simple translation between a standard representation of sl(2,C) and the complex-quaternions exists and can be exploited to construct consistent field theories.
Stated directly in the abstract as the starting point for the entire construction.
domain assumption Extending the Dirac-electromagnetic coupling to electroweak theory requires an algebraic distinction between leptonic and Higgs fields that is absent from the standard model.
Presented as a necessary consequence of the hyper-complex approach.

pith-pipeline@v0.9.0 · 5469 in / 1570 out tokens · 70527 ms · 2026-05-10T07:44:41.496204+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

The spinor spaces (left ideals) upon whichSHacts consist of full complex quaternions of the formχ=α 0 +α ℓωℓ whereα 0, αℓ ∈C

This notion of length is invariant under SO(3) transformations, which are carried out in the Clifford algebra via conjugation by an element ofSH. The spinor spaces (left ideals) upon whichSHacts consist of full complex quaternions of the formχ=α 0 +α ℓωℓ whereα 0, αℓ ∈C. As the coefficients are complex, there is no real inner product, so these Pauli spino...

work page
[2]

As shown below, after symmetry breaking this difference leads to a disagreement with the standard model on the signs of the currents coupled to the Z boson

to the two components. As shown below, after symmetry breaking this difference leads to a disagreement with the standard model on the signs of the currents coupled to the Z boson. The right-chiral electrone R transforms as a weak hypercharge singlet. The fermion sector 13 consists of parity- violating gauge covariant kinetic terms given by L L F =iL †Σµ(∂...

work page
[3]

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De Leo S and Rotelli P 1996Journal of Physics G: Nuclear and Particle Physics221137–1150

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Morita K 1986Progress of Theoretical Physics75220–234

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thesis University of Cambridge URLhttps://www.repository.cam.ac.uk/handle/1810/254719 18

Furey C 2015Standard Model Physics from an Algebra?Ph.D. thesis University of Cambridge URLhttps://www.repository.cam.ac.uk/handle/1810/254719 18

work page

[1] [1]

The spinor spaces (left ideals) upon whichSHacts consist of full complex quaternions of the formχ=α 0 +α ℓωℓ whereα 0, αℓ ∈C

This notion of length is invariant under SO(3) transformations, which are carried out in the Clifford algebra via conjugation by an element ofSH. The spinor spaces (left ideals) upon whichSHacts consist of full complex quaternions of the formχ=α 0 +α ℓωℓ whereα 0, αℓ ∈C. As the coefficients are complex, there is no real inner product, so these Pauli spino...

work page

[2] [2]

As shown below, after symmetry breaking this difference leads to a disagreement with the standard model on the signs of the currents coupled to the Z boson

to the two components. As shown below, after symmetry breaking this difference leads to a disagreement with the standard model on the signs of the currents coupled to the Z boson. The right-chiral electrone R transforms as a weak hypercharge singlet. The fermion sector 13 consists of parity- violating gauge covariant kinetic terms given by L L F =iL †Σµ(∂...

work page

[3] [3]

Section A: Mathematical and Physical Sciences50261–270

Dirac P A M 1944Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences50261–270

work page

[4] [4]

Rotelli P 1989Modern Physics Letters A4933–940

work page

[5] [5]

De Leo S and Rotelli P 1995International Journal of Modern Physics A104359–4370

work page

[6] [6]

De Leo S and Rotelli P 1996Journal of Physics G: Nuclear and Particle Physics221137–1150

work page

[7] [7]

De Leo S 2001Foundations of Physics Letters1437–50 URLhttps://www.ime.unicamp.br/ ~deleo/Sissa/s32.pdf

work page

[8] [8]

De Leo S 1997International Journal of Theoretical Physics36

work page

[9] [9]

Morita K 1986Progress of Theoretical Physics75220–234

work page

[10] [10]

Morita K 2007Progress of Theoretical Physics117501–532

work page

[11] [11]

thesis University of Cambridge URLhttps://www.repository.cam.ac.uk/handle/1810/254719 18

Furey C 2015Standard Model Physics from an Algebra?Ph.D. thesis University of Cambridge URLhttps://www.repository.cam.ac.uk/handle/1810/254719 18

work page