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

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.

• The paper introduces a complex-quaternionic framework that reconstructs Dirac, Pauli, and Weyl spinors, preserving standard electromagnetic and electroweak phenomena.
• It establishes explicit isomorphisms between Clifford algebras, Lie algebras, and complex quaternions to derive gauge invariance and conserved currents.
• The alternative electroweak representation predicts distinct effects like weak neutral current sign reversal, suggesting new avenues for beyond-Standard-Model physics.

Complex Quaternionic Formulations of Dirac, Electrodynamic, and Electroweak Fields

Introduction and Motivation

This work develops a hyper-complex, complex-quaternionic ($\mathbb{H}\otimes_\mathbb{R}\mathbb{C}$) formalism encapsulating the Dirac equation, Maxwell's equations, and the electroweak sector of the Standard Model. The motivation lies in seeking a mathematically natural and structurally unifying framework for particle physics, drawing on the algebraic richness of the complex quaternions. The approach is grounded in explicit isomorphisms between Clifford algebras, Lie algebras commonly used in particle physics (notably $\mathfrak{su}(2)$ and its variants), and complex-quaternionic structures. This translation is leveraged to revisit both standard and alternative representations of the symmetries and interactions of the Standard Model.

Complex Quaternionic Structure of Spinors

A central result is the complete reconstruction of Pauli, Weyl, and Dirac spinors within the language of complex quaternions. The algebra $\mathbb{H}\otimes_\mathbb{R}\mathbb{C}$ naturally provides a framework for representing Euclidean and Minkowski Clifford algebras via explicit identifications of quaternionic and complex units with Clifford generators. Both right- and left-chiral Lorentz transformations, spin groups, and their actions on embedded real Minkowski vectors are constructed, recovering the standard invariances and transformation properties of spacetime intervals and spinor bilinears in the new formalism.

Dirac spinors are represented as elements of $\mathbb{H}^2\otimes_\mathbb{R}\mathbb{C}$, and the explicit construction of Dirac gamma matrices in this context leads to an alternative but equivalent realization of the Dirac equation. The formalism preserves conventional discrete symmetries (C, P, T) and their composition, as well as Hermitian norm structures relevant for probabilistic interpretations.

Physical Quantities: Spin, Helicity, and Magnetic Moment

The complex-quaternionic formalism supports the definition and analysis of physical spin, helicity, and the electromagnetic coupling of spin-1/2 particles. The spin operator is formulated in terms of the quaternionic basis, and the complex subalgebra chosen for motion in a fixed direction ($\mathbb{C}\{1, k\}$ for propagation along $z$) is directly connected to helicity eigensolutions.

A detailed computation of the minimally coupled Dirac equation to an electromagnetic field yields the correct form of the Pauli equation in the non-relativistic limit, reproducing the precise magnetic moment for charged fermions, $\vec{\mu} = \frac{e}{2m} \Sigma_\ell$, directly analogous to the standard $\mathbb{C}^4$ formalism. The representation thus retains predictive agreement with the Standard Model at the electrodynamics level.

Reformulating Electrodynamics

Maxwell's equations and the associated Lagrangian are reformulated using the full Clifford algebraic structure over complex quaternions. The embedding of the four-potential, field strength, and spacetime derivative within matrix-valued complex-quaternionic objects reproduces both the homogeneous and sourced Maxwell equations. Conserved electromagnetic current $J=\bar{\psi} \Gamma^\mu \psi$ emerges as a direct consequence of Noether's theorem under $\text{U}(1)$ transformations, preserving the gauge invariance structure of standard electrodynamics.

Electroweak Theory: Standard and Alternative Structures

The complex-quaternionic Dirac formalism accommodates the standard $\mathfrak{su}(2)$0 gauge structure of the electroweak theory. Both the conventional (Pauli matrix-based) and a distinct, irreducible $\mathfrak{su}(2)$1-based representation of weak isospin and hypercharge are realized as matrix subalgebras within $\mathfrak{su}(2)$2.

A key nontrivial result concerns the alternative representation: after spontaneous symmetry breaking (SSB), this choice leads to physical predictions that deviate from the Standard Model. Specifically, the alternative yields:

• An overall sign flip for weak neutral current interactions: The neutral current associated with the $\mathfrak{su}(2)$3 boson couples with a sign opposite to that in the Standard Model. This implies, for example, that the weak neutral force mediated by the $\mathfrak{su}(2)$4 would be repulsive between certain pairs where it is attractive in the Standard Model.
• A negative-definite inner product for the constructed charged vector bosons ($\mathfrak{su}(2)$5), distinguishing their norm structure from the standard theory.
• An algebraic distinction between the electroweak transformation properties of the Higgs and leptonic fields, which is absent in the standard construction.

While the standard choice recovers all usual phenomenology, the alternative highlights the critical dependence of model predictions on the underlying algebraic embeddings of the gauge groups and their representations. This result suggests that subtle structural shifts in the foundational algebra can produce physically distinguishable signatures, both at the Lagrangian and interaction level.

Implications and Theoretical Outlook

The framework establishes a mathematically coherent unification of the Standard Model's spinor and gauge structure within the complex-quaternionic algebra. The structure faithfully recapitulates known results for the Dirac and Maxwell sectors while revealing the possible existence of non-equivalent electroweak structures within the same algebraic envelope.

Practically, the agreement on the magnetic moment and the complete reconstruction of spinor bilinears supports the viability of this formalism for QED and SM-level calculations. The alternative representation with its sign reversals in weak neutral currents suggests the formalism may be employed for investigating putative beyond-Standard-Model physics, extensions of the Higgs sector, or novel symmetry breakings.

The paper concludes by noting a natural pathway for further unification: the usual $\mathfrak{su}(2)$6 gauge group can be realized as a subalgebra of $\mathfrak{su}(2)$7, suggesting compatibility with Quantum Chromodynamics. The extension of this approach to all Standard Model fields, and to possible new gauge or Higgs structures, remains for future work.

Conclusion

The complex-quaternionic approach expounded here generates a complete, mathematically elegant embedding of the Dirac, Maxwell, and electroweak sectors, with explicit algebraic isomorphisms carrying over all standard representations and calculations. The identification of a physically inequivalent alternative representation for weak isospin within the same algebra crucially demonstrates the dependence of physical phenomenology on algebraic structure. This formalism offers a robust platform for both foundational study and potential model-building within and beyond the Standard Model, pending further exploration of chromodynamic and beyond-electroweak sectors.

Reference: "Complex Quaternionic Formulations of Dirac, Electrodynamic, and Electroweak Fields and Interactions" (2604.16766)