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

Abstract

This paper investigates how the spinor space of the electroweak gauge group $\text{SU}_{I}(2) \times \text{U}_{Y}(1)$ 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, without complexification or matrix representation. In fact, the defining projector of the spinor space corresponds to the zero matrix in the defining representation of $\text{SU}(2)$, and hence this construction has no $2 \times 2$ complex matrix counterpart. We subsequently show that the spinor space of $\mathbb{R}_4$ naturally accommodates irreducible representations (irreps) for a single generation of chiral Standard Model leptons, including a sterile right-chiral neutrino. Left- and right-chiral particles arise due to the grade-parity of the irreps, explaining geometrically why weak isospin acts exclusively on left-chiral states. This manuscript shows how the real Clifford algebra $\mathbb{R}_4$ is the smallest algebra to house irreps for both the $\text{SU}_{I}(2) \times \text{U}_{Y}(1)$ gauge bosons and the Standard Model leptons. Simultaneously the geometric approach presents a way to compute interactions that does not depend on irreps, but which instead allows computations on the higher level of bosons and leptons. The emergence of the correct interactions directly from first principles highlights the promise of this framework not only for the geometric foundations of Electroweak Theory, but also for the Standard Model and 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.