Spectral action in matrix form

TL;DR

This paper develops a matrix-based quantization approach for the spectral action in noncommutative geometry, preserving its structure and relating it to Yang-Mills theories, demonstrated through a toy electroweak model.

Contribution

It introduces a novel matrix form quantization method for the spectral action, maintaining noncommutative geometric features and connecting to Yang-Mills theories.

Findings

Quantization rules derived in matrix form

Spectral action relates to Yang-Mills theory

Application demonstrated on a toy electroweak model

Abstract

Quantization of the noncommutative geometric spectral action has so far been performed on the final component form of the action where all traces over the Dirac matrices and symmetry algebra are carried out. In this work, in order to preserve the noncommutative geometric structure of the formalism, we derive the quantization rules for propagators and vertices in matrix form. We show that the results in the case of a product of a four-dimensional Euclidean manifold by a finite space, could be cast in the form of that of a Yang-Mills theory. We illustrate the procedure for the toy electroweak model.