Gravity and the standard model with neutrino mixing

Abstract

Abstract. We present an effective unified theory based on noncommutative geometry for the standard model with neutrino mixing, minimally coupled to gravity. The unification is based on the symplectic unitary group in Hilbert space and on the spectral action. It yields all the detailed structure of the standard model with several predictions at unification scale. Besides the familiar predictions for the gauge couplings as for GUT theories, it predicts the Higgs scattering parameter and the sum of the squares of Yukawa couplings. From these relations one can extract predictions at low energy, giving in particular a Higgs mass around 170 GeV and a top mass compatible with present experimental value. The geometric picture that emerges is that space-time is the product of an ordinary spin manifold (for which the theory would deliver Einstein gravity) by a finite noncommutative geometry F. The discrete space F is of KO-dimension 6 modulo 8 and of metric dimension 0, and accounts for all the intricacies of the standard model with its spontaneous symmetry breaking Higgs sector.

Quantum Gravity: Unification of Principles and Interactions, and Promises of Spectral Geometry

“…The Closing round table of the International Congress of Mathematicians (Madrid, August [22][23][24][25][26][27][28][29][30]2006) was devoted to the topic Are pure and applied mathematics drifting apart? As panelist, Manin subdivided the mathematization, i.e., the way mathematics can tell us something about the external world, into three modes of functioning (similarly Bohle, Booß and Jensen 1983, [10], see also [13]):…”

“…The Dirac operator on the finite spectral triple (A , H, D) depends on 31 real parameters, which physically correspond to the masses for leptons and quarks etc. The main result of [23] states that the asymptotic formula Λ → ∞ for the spectral action functional (5.2) yields the full Lagrangian of the standard model with neutrino mixing and Majorana mass terms. Note that through the spin manifold M gravity is naturally built into the model.…”

Non-commutative Renormalization

“…In the version of the standard model developped by Alain Connes and followers [76] there is some non commutative geometry but restricted to a very simple internal space. This model when fed with the spectral action principle reproduces in astonishing detail all the standard model terms.…”

Moduli Spaces of Dirac Operators for Finite Spectral Triples

“…Jureit, and Stephan [15,16] have since adopted the new value for the KO-dimension, but further assume orientability and Poincaré duality. As well, Stephan [25] has proposed an alternative finite spectral triple for the current NCG Standard Model with the same physical content but satisfying Poincaré duality; it also just fails to be S 0 -real in the same manner as the finite geometry of [4]; in the same paper, Stephan also discusses non-orientable finite spectral triples.…”

“…However, there were certain issues with the then-current version of the NCG Standard Model, including difficulty with accomodating massive neutrinos and the so-called fermion doubling problem, that were only to be resolved in the 2006 papers by Connes [7] and by Chamseddine, Connes and Marcolli [4], which use the Euclidean signature of earlier papers, and by Barrett [1], which instead uses Lorentzian signature; we restrict our attention to the Euclidean signature approach of [7] and [4], which has more recently been set forth in the monograph [8] of Connes and Marcolli. The finite spectral triple of the current version has KOdimension 6 mod 8 instead of 0 mod 8, fails to be orientable, and only satisfies a certain modified version of Poincaré duality.…”