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Electroweak Structure and Three Fermion Generations in Clifford Algebra with S3 Family Symmetry

Niels Gresnigt

TL;DR

The paper addresses the origin of three fermion generations within a Clifford-algebraic framework by embedding the full SM gauge group into $\mathbb{C}\ell(10)$ and introducing an intrinsic $S_3$ family symmetry that permutes three gauge-equivalent fermion sectors without duplicating gauge bosons. Fermions are realized as minimal left ideals of $\mathbb{C}\ell(10)$, while gauge transformations act via the adjoint commutator and must commute with $S_3$, ensuring a unique gauge sector across generations. The construction proceeds by first building a $\mathbb{C}\ell(8)$ backbone with a Witt-basis and primitive idempotents, then extending to $\mathbb{C}\ell(10)$ to incorporate $SU(2)_L\times U(1)_Y$; the three generations arise from the $S_3$ action and remain linearly independent. The framework yields generation-aware charge assignments and a structurally constrained spectrum, suggesting novel links between discrete family symmetry, tri-hypercharge structures, and observable flavour physics with potential phenomenological implications.

Abstract

We construct an explicit algebraic realisation of three fermion generations within a single Clifford algebra, transforming under the full Standard Model $SU(3)_C\times SU(2)_L\times U(1)_Y$ gauge group, in which an intrinsic $S_3$ family symmetry permutes three algebraically distinguished but gauge-equivalent fermion sectors without replicating the gauge bosons. Fermionic states are represented by minimal left ideals of the complex Clifford algebra $\mathbb{C}\ell(10)$, while the three-generation structure arises from an embedded discrete $S_3$ symmetry acting on the space of algebraic spinors. The Standard Model gauge generators are identified as elements commuting with this $S_3$ action and act on physical states via the adjoint (commutator) action. The resulting spectrum reproduces the correct Standard Model quantum numbers for three linearly independent generations of fermions.

Electroweak Structure and Three Fermion Generations in Clifford Algebra with S3 Family Symmetry

TL;DR

The paper addresses the origin of three fermion generations within a Clifford-algebraic framework by embedding the full SM gauge group into and introducing an intrinsic family symmetry that permutes three gauge-equivalent fermion sectors without duplicating gauge bosons. Fermions are realized as minimal left ideals of , while gauge transformations act via the adjoint commutator and must commute with , ensuring a unique gauge sector across generations. The construction proceeds by first building a backbone with a Witt-basis and primitive idempotents, then extending to to incorporate ; the three generations arise from the action and remain linearly independent. The framework yields generation-aware charge assignments and a structurally constrained spectrum, suggesting novel links between discrete family symmetry, tri-hypercharge structures, and observable flavour physics with potential phenomenological implications.

Abstract

We construct an explicit algebraic realisation of three fermion generations within a single Clifford algebra, transforming under the full Standard Model gauge group, in which an intrinsic family symmetry permutes three algebraically distinguished but gauge-equivalent fermion sectors without replicating the gauge bosons. Fermionic states are represented by minimal left ideals of the complex Clifford algebra , while the three-generation structure arises from an embedded discrete symmetry acting on the space of algebraic spinors. The Standard Model gauge generators are identified as elements commuting with this action and act on physical states via the adjoint (commutator) action. The resulting spectrum reproduces the correct Standard Model quantum numbers for three linearly independent generations of fermions.
Paper Structure (20 sections, 95 equations, 2 tables)

This paper contains 20 sections, 95 equations, 2 tables.

Table of Contents

  1. Introduction
  2. $\mathbb{C}\ell(8)$ basis and its minimal left ideals
  3. Primitive idempotents in $\mathbb{C}\ell(8)$
  4. Witt-basis preserving symmetries
  5. The discrete $S_3$ symmetry in $\mathbb{C}\ell(8)$
  6. The action of $\epsilon$ and $\psi_3$ on primitive idempotents
  7. Three generations of fermions with $SU(3)_C\times U(1)_{em}$ gauge symmetry in $\mathbb{C}\ell(8)$
  8. Spinor transformations and localisation of the gauge action
  9. Enlarging $\mathbb{C}\ell(8)$ to $\mathbb{C}\ell(10)$
  10. Embedding $S_3$ into $\mathbb{C}\ell(10)$
  11. Three generations of fermion states in $\mathbb{C}\ell(10)$
  12. The colour symmetry $SU(3)_C$
  13. The chiral weak symmetry $SU(2)_L$
  14. An $S_3$ invariant projector
  15. $S_3$ invariant $SU(2)_L$ generators
  16. ...and 5 more sections