On a classification of irreducible almost-commutative geometries IV

Abstract

In this paper, we will classify the finite spectral triples with KO-dimension 6, following the classification found in Iochum, B., Schücker, T., and Stephan, C. A., J. Math. Phys. 45, 5003 (2004); Jureit, J.-H. and Stephan, C. A., J. Math. Phys. 46, 043512 (2005); Schücker, T. (unpublished); Jureit, J.-H., Schücker, T., and Stephan, C. A., J. Math. Phys. 46, 072302 (2005). with up to four summands in the matrix algebra. Again, heavy use is made of Krajewski diagrams [Krajewski, T., J. Geom. Phys. 28, 1 (1998).] This work has been inspired by the recent paper by Connes (unpublished) and Barrett (unpublished). 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 6. By minimal version, it is meant that at least one neutrino has to be massless and mass-terms mixing particles and antiparticles are prohibited.