Krein spectral triples and the fermionic action

TL;DR

This paper introduces Krein spectral triples, extending spectral triples to Krein spaces, to better formulate the fermionic action in almost-commutative geometries, successfully recovering key physical theories without relying on a real structure.

Contribution

It defines Krein spectral triples and demonstrates their effectiveness in formulating fermionic actions for fundamental physical models, improving upon previous Hilbert space approaches.

Findings

Correct Lagrangians for electrodynamics, electro-weak theory, and the Standard Model

Krein spectral triples recover physical actions without real structures

Explicit calculations validate the approach

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.