Space and time dimensions of algebras with applications to Lorentzian noncommutative geometry and the standard model

Nadir Bizi (1) , Christian Brouder (1) , Fabien Besnard (2)

1 IMPMC - Institut de minéralogie, de physique des matériaux et de cosmochimie
2 (EPF) - Ecole Polytechnique Féminine

An analogy with real Clifford algebras on even-dimensional vector spaces suggests to assign a space dimension and a time dimension (modulo 8) to an algebra (represented over a complex Hilbert space) containing two self-adjoint involutions and an anti-unitary operator with specific commutation relations. It is shown that this assignment is compatible with the tensor product, in the sense that a tensor product of such algebras corresponds to the addition of the space and time dimensions. This could provide an interpretation of the presence of such algebras in P T-symmetric Hamiltonians or the description of topological matter. This construction is used to build the tensor product of Lorentzian (and more generally pseudo-Riemannian) spectral triples, defined over a Krein space. The application to the standard model of particles suggests the identity of the time and space dimensions of the total (manifold+finite algebra) spectral triple. It also suggests the emergence of the pseudo-orthogonal group SO(4, 6) in a grand unified theory.

Mots clés: Pseudo-Riemannian manifolds; noncommutative geometry; standard model unification; Clifford algebras

Domaines: Physique des Hautes Energies - Théorie [hep-th]; Systèmes mésoscopiques et effet Hall quantique [cond-mat.mes-hall]

Nadir Bizi, Christian Brouder, Fabien Besnard. Space and time dimensions of algebras with applications to Lorentzian noncommutative geometry and the standard model: Space and time dimensions of algebras. 2016. ⟨hal-01398231v1⟩