We extend a classification of irreducible, almost commutative geometries whose spectral action is dynamically non-degenerate to internal algebras that have four simple summands. PACS-92: 11.15 Gauge field theories MSC-91: 81T13 Yang-Mills and other gauge theories

In this paper we will classify the finite spectral triples with KO-dimension six, following the classification found in [1,2,3,4], with up to four summands in the matrix algebra. Again, heavy use is made of Krajewski diagrams [5]. Furthermore we will show that any real finite spectral triple in KO-dimension 6 is automatically S 0 -real. This work has been inspired by the recent paper by Alain Connes [6] and John Barrett [7].In the classification we find that the standard model of particle physics in its minimal version fits the axioms of noncommutative geometry in the case of KO-dimension six. By minimal version it is meant that at least one neutrino has to be massless and mass-terms mixing particles and antiparticles are prohibited.CPT-P75-2006 PACS-92: 11.15 Gauge field theories MSC-91: 81T13 Yang-Mills and other gauge theories

We extend a classification of irreducible, almost-commutative geometries whose spectral action is dynamically non-degenerate, to internal algebras that have six simple summands. We find essentially four particle models: An extension of the standard model by a new species of fermions with vectorlike coupling to the gauge group and gauge invariant masses, two versions of the electro-strong model and a variety of the electro-strong model with Higgs mechanism. -92: 11.15 Gauge field theories MSC-91: 81T13 Yang-Mills and other gauge theories 1 Privatgelehrter, Kiel (Germany) 2 christophstephan@gmx.de PACS