A new algebraic structure in the standard model of particle physics

TL;DR

This paper presents a novel algebraic reformulation of non-commutative geometry that captures the standard model's structure more precisely and offers new geometric insights into electroweak symmetry breaking.

Contribution

It introduces a simple super-algebra framework for real-spectral-triples, recovering axioms and constraints of NCG and applying it successfully to the standard model.

Findings

New geometric constraints are physically meaningful and phenomenologically correct.

Provides a geometric interpretation of electroweak symmetry breaking.

Offers more restrictive and insightful structure than traditional effective field theories.

Abstract

We introduce a new formulation of the real-spectral-triple formalism in non-commutative geometry (NCG): we explain its mathematical advantages and its success in capturing the structure of the standard model of particle physics. The idea, in brief, is to represent $A$ (the algebra of differential forms on some possibly-noncommutative space) on $H$ (the Hilbert space of spinors on that space), and to reinterpret this representation as a simple super-algebra $B=A\oplus H$ with even part $A$ and odd part $H$. $B$ is the fundamental object in our approach: we show that (nearly) all of the basic axioms and assumptions of the traditional real-spectral-triple formalism of NCG are elegantly recovered from the simple requirement that $B$ should be a differential graded $\ast$-algebra (or "$\ast$-DGA"). Moreover, this requirement also yields other, new, geometrical constraints. When we apply our formalism to the NCG traditionally used to describe the standard model of particle physics, we find that these new constraints are physically meaningful and phenomenologically correct. In particular, these new constraints provide a novel interpretation of electroweak symmetry breaking that is geometric rather than dynamical. This formalism is more restrictive than effective field theory, and so explains more about the observed structure of the standard model, and offers more guidance about physics beyond the standard model.