Krein spectral triples and the fermionic action

Abstract

Motivated by the space of spinors on a Lorentzian manifold, we define Krein spectral triples, which generalise spectral triples from Hilbert spaces to Krein spaces. This Krein space approach allows for an improved formulation of the fermionic action for almost-commutative manifolds. We show by explicit calculation that this action functional recovers the correct Lagrangians for the cases of electrodynamics, the electro-weak theory, and the Standard Model. The description of these examples does not require a real structure, unless one includes Majorana masses, in which case the internal spaces also exhibit a Krein space structure.

17 pages

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Cited by in corpus (13)

• Lorentz signature and twisted spectral triples
• Wick Rotation and Fermion Doubling in Noncommutative Geometry
• The noncommutative geometry of Zitterbewegung
• Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces
• Spectral Noncommutative Geometry, Standard Model and all that
• Clifford Structures in Noncommutative Geometry and the Extended Scalar Sector
• A $U(1{)}_{B-L}$-extension of the Standard Model from Noncommutative Geometry
• Lorentzian fermionic action by twisting euclidean spectral triples
• Noncommutative geometrical origin of the energy-momentum dispersion relation
• Families of spectral triples and foliations of space(time)
• Doppler shift in semi-Riemannian signature and the non-uniqueness of the Krein space of spinors
• Algebraic backgrounds: a framework for noncommutative Kaluza-Klein theory
• A critical survey of twisted spectral triples beyond the Standard Model