A numerical approach to harmonic non-commutative spectral field theory

TL;DR

This paper presents a numerical method to investigate a complex non-commutative spectral field theory based on spectral triples and matrix discretization, overcoming analytical solution challenges.

Contribution

It introduces a numerical approach to study harmonic non-commutative spectral field theories where analytical solutions are infeasible.

Findings

Numerical results demonstrate the feasibility of analyzing complex spectral actions.

Discretization enables investigation of correlation functions in non-commutative models.

The approach shows promise for exploring spectral field theories in the Moyal matrix basis.

Abstract

The object of this work is the numerical investigation of a non-commutative field theory defined via the spectral action principle. The Starting point is a spectral triple (A,H,D) referred to as harmonic. The construction of these data relies on an 8-dimensional Clifford algebra. The spectral action is computed for the product of the triple (A,H,D) with a matrix-valued spectral triple. Renormalization theory associates to the spectral action a probability measure. Its associated correlation functions define then a field theory. In the perturbative approach this measure is constructed as a formal power series. This requires explicit knowledge of the solutions of the Euler-Lagrange equations. For the model under consideration, it turns out impossible to obtain these solutions. An alternative approach consists in a discretization of all variables and a numerical investigation of the behavior of the correlation functions when the discretization becomes finer. Despite the complexity of the approximated spectral action, some reliable numerical results are obtained, showing that a numerical treatment of this kind of models in the Moyal matrix basis is possible.