Space and time dimensions of algebras with applications to Lorentzian noncommutative geometry and quantum electrodynamics

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 assigning an ordered pair (s, t) of space and time dimensions (or equivalently an ordered pair (m, n) of metric and KO dimensions) modulo 8 to any algebraic structure (that we call CPT corepresentation) represented over a Hilbert space by two self- adjoint involutions and an anti-unitary operator having specific commutation relations. It is shown that this assignment is compatible with the tensor product: the space and time dimensions of the tensor product of two CPT corepresentations are the sums of the space and time dimensions of its factors, and the same holds for the metric and KO dimensions. This could provide an interpretation of the presence of such algebras in PT -symmetric Hamiltonians or the description of topological matter. This construction is used to build an indefinite (i.e., pseudo-Riemannian) version of the spectral triple of noncommutative geometry, defined over a Krein space and classified by the pair (m, n) instead of the KO dimension only. Within this framework, we can express the Lagrangian (both bosonic and fermionic) of a Lorentzian almost-commutative spectral triple. We exhibit a space of physical states that solves the fermion-doubling problem. The example of quantum electrodynamics is described.

Mots clés: Pseudo-Riemannian manifolds; noncommutative geometry; quantum electrodynamics; 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 quantum electrodynamics. Journal of Mathematical Physics, 2018, 59 (6), pp.062303. ⟨10.1063/1.4986228⟩. ⟨hal-01398231v2⟩