Derivation-based Noncommutative Field Theories on $AF$ algebras

TL;DR

This paper introduces a new approach to noncommutative gauge field theories using AF $C^*$-algebras, focusing on derivation-based calculus and physical applications like mass spectra from symmetry breaking.

Contribution

It develops a derivation-based framework for noncommutative gauge theories on AF algebras, integrating mathematical structures and physical applications.

Findings

Constructed a derivation-based differential calculus on AF algebras.

Analyzed modules, connections, and metrics within this framework.

Numerical results on mass spectra from spontaneous symmetry breaking.

Abstract

In this paper, we start the investigation of a new natural approach to "unifying" noncommutative gauge field theories (NCGFT) based on approximately finite-dimensional ($AF$) $C^*$-algebras. The defining inductive sequence of an $AF$ $C^*$-algebra is lifted to enable the construction of a sequence of NCGFT of Yang-Mills-Higgs types. The present paper focus on derivation-based noncommutative field theories. A mathematical study of the ingredients involved in the construction of a NCGFT is given in the framework of $AF$ $C^*$-algebras: derivation-based differential calculus, modules, connections, metrics and Hodge $\star$-operators, Lagrangians... Some physical applications concerning mass spectra generated by Spontaneous Symmetry Breaking Mechanisms (SSBM) are proposed using numerical computations for specific situations.