A geometric basis for the standard-model gauge group

TL;DR

This paper presents a geometric model of the standard model using Clifford algebra Cl_7, revealing how gauge groups and particle properties emerge from seven-dimensional rotations and spinor transformations.

Contribution

It introduces a novel geometric framework that derives the standard model gauge groups and particle charge assignments from Clifford algebra-based rotations.

Findings

Reproduces the fundamental fermion charge assignments.

Identifies geometric origins of SU(2)_L, SU(3)_C, and U(1)_Y groups.

Provides a basis for the Higgs field within a seven-dimensional space.

Abstract

A geometric approach to the standard model in terms of the Clifford algebra Cl_7 is advanced. A key feature of the model is its use of an algebraic spinor for one generation of leptons and quarks. Spinor transformations separate into left-sided ("exterior") and right-sided ("interior") types. By definition, Poincare transformations are exterior ones. We consider all rotations in the seven-dimensional space that (1) conserve the spacetime components of the particle and antiparticle currents and (2) do not couple the right-chiral neutrino. These rotations comprise additional exterior transformations that commute with the Poincare group and form the group SU(2)_L, interior ones that constitute SU(3)_C, and a unique group of coupled double-sided rotations with U(1)_Y symmetry. The spinor mediates a physical coupling of Poincare and isotopic symmetries within the restrictions of the Coleman--Mandula theorem. The four extra spacelike dimensions in the model form a basis for the Higgs isodoublet field, whose symmetry requires the chirality of SU(2). The charge assignments of both the fundamental fermions and the Higgs boson are produced exactly.