Probing Clifford Algebras Through Spin Groups: A Standard Model Perspective

TL;DR

This paper investigates the relationship between complex and real Clifford algebras using spin groups, revealing how Standard Model gauge groups can be derived from higher-dimensional Clifford algebra structures.

Contribution

It introduces a method to connect complex Clifford algebras with their real counterparts via spin groups, providing geometric insights into Standard Model gauge symmetries.

Findings

Standard Model gauge groups emerge from higher-dimensional Euclidean Clifford algebras.

Gauge symmetries can be translated from complex to real Clifford algebras using Witt decomposition.

Bivector structures preserve gauge symmetry, linking algebraic and geometric properties.

Abstract

Division algebras have demonstrated their utility in studying non-associative algebras and their connection to the Standard Model through complex Clifford algebras. This article focuses on exploring the connection between these complex Clifford algebras and their corresponding real Clifford algebras providing insight into geometric properties of bivector gauge symmetries. We first generate gauge symmetries in the complex Clifford algebra through a general Witt decomposition. Gauge symmetries act as a constraint on the underlying real Clifford algebra, where they're then translated from their complex form to their bivector counterpart. Spin group arguments allow the identification of bivector structures which preserve the gauge symmetry yielding the corresponding real Clifford algebra. We conclude that Standard Model gauge groups emerge from higher-dimensional Clifford algebras carrying Euclidean signatures, where particle states are recognized as a combination of basis elements corresponding to complex Euclidean Clifford algebras.