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

TL;DR

This paper introduces a hyper-complex quaternionic framework for Dirac, electrodynamics, and electroweak theories, revealing new algebraic distinctions and sign disagreements with the standard model.

Contribution

It develops a novel quaternionic formulation of fundamental fields, highlighting differences in weak neutral current signs and algebraic structures.

Findings

Correct magnetic moment for charged spin-1/2 particles from coupling to electromagnetism.

Algebraic distinction between leptonic and Higgs fields in extended electroweak theory.

Disagreement with the standard model on signs of weak neutral currents.

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.