Noncommutative geometry and structure of space–time
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I give a summary review of the research program using noncommutative geometry as a framework to determine the structure of space–time. Classification of finite noncommutative spaces under few assumptions reveals why nature chose the Standard Model and the reasons behind the particular form of gauge fields, Higgs fields and fermions as well as the origin of symmetry breaking. It also points that at high energies the Standard Model is a truncation of Pati–Salam unified model of leptons and quarks. The same conclusions are arrived at uniquely without making any assumptions except for an axiom which is a higher form of Heisenberg commutation relations quantizing the volume of space–time. We establish the existence of two kinds of quanta of geometry in the form of unit spheres of Planck length. We provide answers to many of the questions which are not answered by other approaches, however, more research is needed to answer the remaining challenging questions.
KeywordsNoncommutative Geometry Unification
Mathematics Subject Classification58B34 81T75 83C65
I would like to thank Alain Connes for a fruitful collaboration on the topic of noncommutative geometry since 1996. I would also like to thank Walter van Suijlekom and Slava Mukhanov for essential contributions to this program of research. This research is supported in part by the National Science Foundation under Grant No. Phys-1518371.
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