The Unified Standard Model

Abstract

The aim of this work is to find a simple mathematical framework for our established description of particle physics. We demonstrate that the particular gauge structure, group representations and charge assignments of the Standard Model particles are all captured by the algebra M(8,$\mathbb{C})$ of complex 8$\times$8 matrices. This algebra is well motivated by its close relation to the normed division algebra of octonions. (Anti-)particle states are identified with basis elements of the vector space M(8,$\mathbb{C})$. Gauge transformations are simply described by the algebra acting on itself. Our result shows that all particles and gauge structures of the Standard Model are contained in the tensor product of all four normed division algebras, with the quaternions providing the Lorentz representations. Interestingly, the space M(8,$\mathbb{C})$ contains two additional elements independent of the Standard Model particles, hinting at a minimal amount of new physics.

arXiv comment29 pages; Minor revisions and clarifications added

References in corpus (17)

• Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC
• Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC
• Gravity coupled with matter and foundation of non-commutative geometry
• The Spectral Action Principle
• Resilience of the Spectral Standard Model
• Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra
• Octonions, E6, and Particle Physics
• Generations: Three Prints, in Colour
• Geometry and the Quantum: Basics
• Standard model physics from an algebra?
• Charge quantization from a number operator
• Octonions, exceptional Jordan algebra and the role of the group F_4 in particle physics
• Three fermion generations with two unbroken gauge symmetries from the complex sedenions
• The Standard Model Algebra - Leptons, Quarks, and Gauge from the Complex Clifford Algebra Cl6
• Non-Commutative Geometry, Non-Associative Geometry and the Standard Model of Particle Physics
• Towards a unified theory of ideals
• An octonionic formulation of the M-theory algebra

Cited by in corpus (12)

• Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra
• One generation of standard model Weyl representations as a single copy of $\mathbb{R}\otimes \mathbb{C}\otimes \mathbb{H}\otimes \mathbb{O}$
• An Exceptional G(2) Extension of the Standard Model from the Correspondence with Cayley-Dickson Algebras Automorphism Groups
• Clifford algebra Cl(0,6) approach to beyond the standard model and naturalness problems
• Algebraic realisation of three fermion generations with $S_3$ family and unbroken gauge symmetry from $\mathbb{C}\ell (8)$
• Listening to Celestial Algebras
• Three Generations and a Trio of Trialities
• A Superalgebra Within: representations of lightest standard model particles form a ${\mathbb{Z}}_2^5$-graded algebra
• A Novel Approach To Particle Representations
• An Algebraic Roadmap of Particle Theories, Part II: Theoretical checkpoints
• Quantum number conservation: a tool in the design and analysis of high energy experiments
• An Algebraic Roadmap of Particle Theories, Part I: General construction