Noncommutative geometry as a framework for unification of all fundamental interactions including gravity. Part I.

Abstract

We examine the hypothesis that space-time is a product of a continuous four-dimensional manifold times a finite space. A new tensorial notation is developed to present the various constructs of noncommutative geometry. In particular, this notation is used to determine the spectral data of the standard model. The particle spectrum with all of its symmetries is derived, almost uniquely, under the assumption of irreducibility and of dimension 6 modulo 8 for the finite space. The reduction from the natural symmetry group SU (2) × SU (2) × SU (4) to U (1) × SU (2) × SU (3) is a consequence of the hypothesis that the two layers of space-time are finite distance apart but is non-dynamical. The square of the Dirac operator, and all geometrical invariants that appear in the calculation of the heat kernel expansion are evaluated. We re-derive the leading order terms in the spectral action. The geometrical action yields unification of all fundamental interactions including gravity at very high energies. We make the following predictions: (i) The number of fermions per family is 16. There is a right-handed neutrino with a see-saw mechanism. Moreover, the zeroth order spectral action obtained with a cut-off function is consistent with experimental data up to few percent. We discuss a number of open issues. We prepare the ground for computing higher order corrections since the predicted mass of the Higgs field is quite sensitive to the higher order corrections. We speculate on the nature of the noncommutative space at Planckian energies and the possible role of the fundamental group for the problem of generations.

Resilience of the spectral standard model

“…It turned out that the RG equations of the combined Higgs-singlet system solves the stabilization problem faced with a light Higgs field of the order of 125 Gev avoiding making the Higgs quartic coupling negative at very high energies. Remarkably, the form of the Higgs-singlet potential proposed recently agree with the one we derived before from the spectral action [2]. The quartic couplings are determined at unification scale in terms of the gauge and Yukawa couplings.…”

“…Their result shows that, adding this new scalar field, everything would be fine, with a Higgs of 125 Gev, for SM at high energies. It turns out that by miracle their proposed couplings are exactly the same as the ones which were delivered before by the spectral action in [2].…”

“…The main result of this paper is that the full spectral action as computed in our previous paper [2], published in 2010, in fact already contains the solution to this second difficulty and at the same time shows the compatibility of the spectral model with the above low experimental Higgs mass. The fact is that in our previous prediction of the Higgs mass, we assumed, incorrectly, that the additional real singlet field σ responsible for the neutrino Majorana masses [2] would not interfere much with the Higgs mass prediction and could be ignored.…”

“…The fact is that in our previous prediction of the Higgs mass, we assumed, incorrectly, that the additional real singlet field σ responsible for the neutrino Majorana masses [2] would not interfere much with the Higgs mass prediction and could be ignored. We became aware of the non-trivial role played by such a scalar field when we realized that it has the same structure of couplings as in the papers by various authors [3][4][5][6][7] who proposed to solve the above instability problem of the Standard Model precisely by…”

“…In fact the singlet responsible for the right-handed neutrino mass gets mixed in a non-trivial way with the Higgs field. The potential was derived and given in full detail in our earlier work [2]. Recently and in more than one work [3][4][5][6], it was shown that adding a singlet (real or complex) scalar field, whose potential mixes with the Higgs field, has important consequences.…”