Generations: Three Prints, in Colour

Abstract

We point out a somewhat mysterious appearance of $S{U}_c(3)$ representations, which exhibit the behaviour of three full generations of standard model particles. These representations are found in the Clifford algebra $\mathbb{C}l(6)$, arising from the complex octonions. In this paper, we explain how this 64-complex-dimensional space comes about. With the algebra in place, we then identify generators of $SU(3)$ within it. These $SU(3)$ generators then act to partition the remaining part of the 64-dimensional Clifford algebra into six triplets, six singlets, and their antiparticles. That is, the algebra mirrors the chromodynamic structure of exactly three generations of the standard model's fermions. Passing from particle to antiparticle, or vice versa, requires nothing more than effecting the complex conjugate, $\ast$: $i\mapsto -i$. The entire result is achieved using only the eight-dimensional complex octonions as a single ingredient.

5 pages, 3 figures

References in corpus (12)

• Exceptional Deconfinement in G(2) Gauge Theory
• Exceptional Confinement in G(2) Gauge Theory
• Color Screening, Casimir Scaling, and Domain Structure in G(2) and SU(N) Gauge Theories
• Octonions, E6, and Particle Physics
• G_2 gauge theory at finite temperature
• Casimir scaling in G(2) lattice gauge theory
• Chiral symmetry and spectral properties of the Dirac operator in G2 Yang-Mills Theory
• Towards a unified theory of ideals
• An octonionic formulation of the M-theory algebra
• Octonionic Representations of GL(8,R) and GL(4,C)
• Division Algebras; Spinors; Idempotents; The Algebraic Structure of Reality
• An Exceptional Electroweak Model

Cited by in corpus (36)

• Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra
• $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y\left(\times U(1{)}_X\right)$ as a symmetry of division algebraic ladder operators
• Charge quantization from a number operator
• Three fermion generations with two unbroken gauge symmetries from the complex sedenions
• Braids, normed division algebras, and Standard Model symmetries
• Division algebraic symmetry breaking
• Superconducting currents and charge gradients in the octonion spaces
• The Unified Standard Model
• A non-Abelian twist to integer quantum Hall states
• Subgroups of Clifford algebras
• One generation of standard model Weyl representations as a single copy of $\mathbb{R}\otimes \mathbb{C}\otimes \mathbb{H}\otimes \mathbb{O}$
• Braids, 3-manifolds, elementary particles: number theory and symmetry in particle physics
• Octonion Internal Space Algebra for the Standard Model
• Quaternionic Wavefunction
• The $C\ell (8)$ algebra of three fermion generations with spin and full internal symmetries
• Three generations of colored fermions with $S_3$ family symmetry from Cayley-Dickson sedenions
• Torques and angular momenta of fluid elements in the octonion spaces
• Some spin textures relevant to magnetic moments in the octonion spaces
• Braided fermions from Hurwitz algebras
• Lessons from Einstein's 1915 discovery of general relativity
• Dixon-Rosenfeld Lines and the Standard Model
• A group-theorist's perspective on symmetry groups in physics
• Algebraic realisation of three fermion generations with $S_3$ family and unbroken gauge symmetry from $\mathbb{C}\ell (8)$
• Three Generations and a Trio of Trialities
• Two incompatible types of invariants in the octonion spaces
• Potential applications of modular representation theory to quantum mechanics
• Finite symmetry groups in physics
• Options for a finite group model of quantum mechanics
• Eight Equilibrium and continuity equations within the material media
• A Superalgebra Within: representations of lightest standard model particles form a ${\mathbb{Z}}_2^5$-graded algebra
• Some field equations with composite operators in octonion spaces
• Basic postulates of some coordinate transformations within material media
• Coherent States and Particle Scattering in Loop Quantum Gravity
• Comparison of group-theoretical strategies for unification of 3+1 structures in physics
• An Algebraic Roadmap of Particle Theories, Part I: General construction
• An Algebraic Roadmap of Particle Theories, Part II: Theoretical checkpoints