  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.
