Noncommutative geometry and the standard model of particle physics

Stephan, Christoph Alexander

Alain Connes discovered an algebraic approach to geometry where he replaced ordinary Riemannian spin geometry by spectral triples. A spectral triple is a set of three entities: An algebra, a Dirac operator and a Hilbert space. They encode all the information of the geometry. It is now tempting to render a commutative algebra, i.e. space-time, a little bit noncommutative, by tensorising it with a sum of matrix algebras. One obtains for a certain choice of matrix algebras general relativity and the classical field theory of the standard model of particle physics. These half commutative, half noncommutative spectral triples are called almost-commutative. Every almost-commutative spectral triple represents a Yang-Mills-Higgs model and is a potential candidate for a physical theory. On these spectral triples further physical requirements where imposed, such as non-degenerate fermion masses and renormalisability. From these first principles all almost-commutative spectral triples with up to three algebras, and most with up to four algebras where classified. Surprisingly the standard model took a most prominent position in this classification.

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Stephan, Christoph Alexander: Noncommutative geometry and the standard model of particle physics.

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