Generations: Three Prints, in Colour

Abstract: We point out a somewhat mysterious appearance of $SU_c(3)$ 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, $*$: $i \mapsto -i$. The entire result is achieved using only the eight-dimensional complex octonions as a single ingredient.

• The paper demonstrates that complex octonions form a 64-dimensional Clifford algebra embedding SU(3) generators, which underpin the structure of three fermion generations.
• It employs octonionic chains to model particle-antiparticle transitions via complex conjugation, aligning algebraic transformations with Standard Model symmetries.
• The research introduces a novel algebraic framework that may unify gauge bosons and fermions, opening innovative avenues in unified quantum field theories.

Representation of $SU(3)$ via Complex Octonions

Introduction

The paper "Generations: Three Prints, in Colour" explores the intriguing occurrence of $SU(3)$ representations within the framework of Clifford algebra derived from complex octonions. This formulation suggests a chromodynamic organization of three generations of Standard Model particles, offering a promising structure for further study in quantum chromodynamics (QCD) and beyond.

Construction of Clifford Algebra from Complex Octonions

The paper begins by constructing a 64-complex-dimensional Clifford algebra $C$ using the complex octonions ($\mathbb{C} \otimes \mathbb{O}$). The complex octonions, characterized by their non-associative nature, enable the construction of "octonionic chains" which act on other octonions via left multiplication, forming an associative algebra $\mathbb{C} \otimes \overleftarrow{\mathbb{O}$.

This associative space subsequently manifests as a representation of Clifford algebra $C$, formed by a series of maps represented as sequences of octonion multiplications. The paper explicitly derives how these chains map one octonion to another, thereby defining the algebra's vector structure.

Identification of $SU(3)$ Generators

Within this algebra, the paper identifies the generators of $SU(3)$ as specific elements of the chain algebra. Following the formulations of previous work, these generators exhibit the characteristic commutation relations of $SU(3)$, confirming their validity. Specifically, the eight generators $\Lambda_n\nu$ and their complex conjugates $-\Lambda_n^*\nu^*$ form two representations in the 64-dimensional space.

Chromodynamic Partitioning of the Algebra

The structure of the octonionic chain algebra enables the $SU(3)$ generators to partition the remaining space into subcomponents akin to chromodynamic configurations. It splits into six triplets, six singlets, and corresponding antiparticle counterparts. This categorization reflects the particle structures found in three full generations of Standard Model fermions.

Furthermore, the transition from particle to antiparticle within this framework involves a simple complex conjugate operation, $*$: $i \mapsto -i$, highlighting an elegant symmetry. This property stands in contrast to conventional models where such transitions might require matrix operations.

Implementation and Analysis

In practical terms, the paper details sample calculations to illustrate how $SU(3)$ generators act on elements of the octonionic chain algebra. These calculations verify that the transformations are consistent with known particle physics models, such as converting color states under $SU(3)$ operations.

Implications and Future Research

This research opens avenues in multiple areas of particle physics and unified theories. It provides a basis for reinvestigating division algebra-based models and hints at possible extensions to incorporate the electro-weak sector. The existence of projectors like $\nu$ raises questions about their fundamental significance, where further exploration could link them to Jordan algebras or other structures.

Conclusion

The paper successfully demonstrates how complex octonions and their derived algebraic structures can characterize three generations of fermionic particles efficiently. This approach suggests novel ways to unify gauge bosons with fermions and matter degrees of freedom, potentially simplifying particle physics models without introducing additional particles. This work lays the groundwork for further research into intrinsic algebraic properties that underlie physical symmetries in quantum field theories.