  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.
