AN ALGEBRAIC FRAMEWORK FOR DESCRIBING THREE FERMION GENERATIONS USING A DISCRETE FAMILY SYMMETRY

Gourlay, Liam (2025) AN ALGEBRAIC FRAMEWORK FOR DESCRIBING THREE FERMION GENERATIONS USING A DISCRETE FAMILY SYMMETRY PhD thesis, University of Liverpool.

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Abstract

The Standard Model of particle physics successfully describes known particles and interactions, but fails to explain why there are three generations of fermions or why the gauge group takes the specific form $SU(3)_C \times SU(2)_L \times U(1)_Y$. It also offers no explanation as to why particular gauge groups correspond to certain particle multiplets or singlets, and it lacks a reason for the quantisation of electric charge. This thesis develops an algebraic framework to model the three fermion generations using symmetries derived from Cayley-Dickson and Clifford algebras. The model is constructed using minimal left and right ideals in $\mathbb{C}\ell(8)$ and $\mathbb{C}\ell(10)$, with gauge symmetries motivated from the internal structure of the Cayley-Dickson and Clifford algebras. Basis vectors of the minimal ideals are associated with particles, while the symmetries that act via commutators on these basis vectors are interpreted as gauge group generators. The sedenion algebra features an $S_3$ automorphism group, the permutation group on three elements, which cyclically permutes three sets of minimal ideals. These ideals are associated with the three generations of fermions, with $S_3$ acting as a family symmetry. Triality is the automorphism of $\text{Spin}(8)$ that permutes its three 8-dimensional representations, two spinors and one vector. By forming appropriate linear combinations of the minimal ideals permuted by $S_3$, one finds two generations of fermions transforming as left/right-handed spinors and one generation transforming as a vector space. The family symmetry $S_3$ is then seen to permute linear combinations of these triality states. Building on earlier models that use division algebras to describe the Standard Model algebraically, this thesis proposes that the $S_3$ automorphisms of the sedenion algebra provide a natural algebraic origin for the existence of three generations. This work additionally offers a novel link between the discrete symmetry $S_3$ and triality, suggesting a unified algebraic explanation for the three fermion generations.

Item Type:	Thesis (PhD)
Uncontrolled Keywords:	Discrete Symmetry, Three Generations, Sedenions, Division Algebras, Standard Model, Minimal Ideals, Triality
Divisions:	Faculty of Science & Engineering Faculty of Science & Engineering > School of Physical Sciences Faculty of Science & Engineering > School of Physical Sciences > Mathematical Sciences
Depositing User:	Symplectic Admin
Date Deposited:	11 Feb 2026 14:05
Last Modified:	11 Feb 2026 14:06
DOI:	10.17638/03196062
Supervisors:	Gresnigt, Niels Kouwenhoven, Mattheus Gorbahn, Martin
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3196062
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