Lepton Triptych I: Geometric Foundations of Electroweak Symmetry in the Real Clifford Algebra $\text{Cl}_4(\mathbb{R})$

TL;DR

The paper addresses a geometrically grounded derivation of the electroweak gauge group by constructing a real spinor space within $R^4$ that does not rely on complexification. It introduces a forgetful idempotent and a spine-based isoclinic framework to realize two copies of SU(2) and the hypercharge U(1) as intrinsic parts of the Clifford algebra, yielding left-handed lepton doublets and right-handed singlets with correct quantum numbers. Key contributions include a complete real-spinor construction with 8 real DOF, a consistent innor and outor spinor product formalism, and a direct geometric explanation for handedness as grade-parity, along with a natural mapping to SM leptons and gauge bosons. The approach promises a robust geometric foundation for Electroweak Theory and broader GUT extensions, setting the stage for future work on SU(3) and spacetime Yang-Mills in subsequent panels.

Abstract

This paper investigates how the gauge group $\text{SU}_{I}(2) \times \text{U}_{Y}(1)$ of the electroweak interactions can be derived using recent geometric techniques within the real Clifford Algebra $\mathbb{R}_4 = \text{Cl}_4(\mathbb{R})$. Central to this approach is a novel procedure for constructing the spinor space of $\mathbb{R}_4$ directly, \emph{without} complexification. We show that $\mathbb{R}_4$ naturally accommodates representations for the $\text{SU}_{I}(2) \times \text{U}_{Y}(1)$ gauge bosons and a single generation of chiral Standard Model leptons, with weak isospin acting exclusively on left-chiral states. Specifically, under hypercharge and isospin $(Y, I_3)$, $\mathbb{R}_4$ contains $(-1, \mp \tfrac{1}{2})$ irreps for left-chiral electrons and neutrinos, a $(-2, 0)$ irrep for a right-chiral electron, and a $(0, 0)$ irrep for a sterile right-chiral neutrino. The distinction between left- and right-chiral particles arises from the grade parity of the irreps, providing a natural geometric explanation for why only left-chiral particles couple to $\text{SU}_{I}(2)$. The emergence of the correct eigenvalues directly from first principles highlights the promise of this framework for the geometric foundations of Electroweak Theory and the Standard Model, as well as for Grand Unified Theories more broadly. This paper is the first panel of the Lepton Triptych, which will ultimately present the full Yang-Mills theory of the electroweak model based on these principles.

Figures (1)

• Figure 1: Fock space structure of the particle states. Note how the arrows are bi-directional, since the vectors $v_i = e^{\theta_i b_i} \mathbf {e}_{i}$ serve as both raising and lowering operators, captured algebraically by the fact that $v_{i}^2 = 1$. Moreover, the $\nu_L$ and $\varepsilon_L$ states couple to $\text{SU}_{I}({2})$.