Lorentzian version of the noncommutative geometry of the standard model of particle physics

A formulation of the noncommutative geometry for the standard model of particle physics with a Lorentzian signature metric is presented. The elimination of the fermion doubling in the Lorentzian case is achieved by a modification of Connes' internal space geometry ['Gravity coupled with matter and the foundation of non-commutative geometry', Commun. Math. Phys. 182, 155-176 (1996)] so that it has signature 6 (mod 8) rather than 0. The fermionic part of the Connes-Chamseddine spectral action can be formulated, and it is shown that it allows an extension with right-handed neutrinos and the correct mass terms for the seesaw mechanism of neutrino mass generation