---
abstract: 'A numerical investigation of a non-commutative field theory defined via the spectral action principle is conducted. The construction of this triple relies on an 8-dimensional Clifford algebra. Following to the standard procedure of non-commutative geometry, the spectral action is computed for the product of the triple $(\mathcal{A}_4,\mathcal{H}_4,\mathcal{D}_4)$ with a matrix-valued spectral triple. Using Monte Carlo simulation we study various quantities such as the energy density, the specific heat density and some order parameters varying the matrix size and the independent parameters of the model.'
---

\

[**First numerical approach to a Grosse-Wulkenhaar model** ]{}

0.5 cm

[ Bernardino Spisso]{}\
*Mathematisches Institut der Westfälischen Wilhelms-Universität*\
*Einsteinstraße 62, D-48149 Münster, Germany*\
*e-mail:nispisso@tin.it*

1 cm

Introduction 
=============

The main object of this work is a particular non-commutative field theory which is derived using the spectral action principle and then treated numerically. Non-commutativity can be found in many fields of physics like quantum field theories, string theory, condensed matter physics. The first application of non-commutativity into physics is dated from the middle of the last century inspired by the ideas of quantum mechanics, where starting from classical mechanics, the commutative algebra of functions on the phase space is replaced by a non-commutative 