Abstract
Starting from the formulation of pseudo-Riemannian generalisation of real spectral triples we develop the data of geometries over finite-dimensional algebras with indefinite metric and their Riemannian parts. We then discuss the Standard Model spectral triple in this formalism and interpret the physical symmetry preserving the lepton number as a shadow of a finite pseudo-Riemannian structure.
Highlights
Noncommutative geometry offers an intriguing possibility of a new insight into the structure of all fundamental interactions, linking purely geometric gravity with the electroweak and strong interactions of elementary fermions [1]
We apply the discussion to the standard model spectral triple, and classify possible time orientations leading to restrictions on the physical parameters and symmetries
We propose that the consistent model-building for the physical interactions and possible extensions of the standard model within the noncommutative geometry framework should use possibly the pseudo-Riemannian extension of finite spectral triples, for which we present a consistent and clear framework
Summary
Noncommutative geometry offers an intriguing possibility of a new insight into the structure of all fundamental interactions, linking purely geometric gravity with the electroweak and strong interactions of elementary fermions [1]. Though the quest for the better understanding of the structure has already brought new results, some issues still remain unsolved, like the consistent Lorentzian framework for standard model description [8,9,10], the fermion doubling problem [11,12] or the classification of possible Dirac operators [13,14].
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