Listening to the little quantum music of the spectral model
*The author is Einstein. Here are his words in a letter to Arnold Sommerfeld dated October 29, 1912:
At the moment I am working solely on the problem of gravitation and believe 1 will be able to overcome all difficulties with the help of a local, friendly mathematician. But one thing is certain, that I have never worked so hard in my life, and that I have been injected with a great awe of mathematics, which in my naiveté until now I only viewed as a pure luxury in its subtler forms!'** I have particularly in mind Ali Chamseddine and Sacha Mukhanov but one should add Walter van Suijlekom and Matilde Marcolli.
We make no apology for this relatively high-brow approach to such an apparently simple matter.
“The origin of the quark and lepton masses is shrouded in mystery” . Some thirty years ago, attempts to solve the enigma based on textures of the quark mass matrices, purposedly reflecting mass hierarchies and “nearest-neighbour” interactions, were very popular. Now, in the late eighties, Branco, Lavoura and Mota  showed that, within the SM, the zero pattern ..., a central ingredient of Fritzsch’s well-known Ansatz for the mass matrices, is devoid of any particular physical meaning...
Although perhaps this was not immediately clear at the time, paper  marked a watershed in the theory of flavour mixing. In algebraic terms, it establishes that the linear subspace of matrices ...[with the former zero pattern] is universal for the group action of unitaries effecting chiral basis transformations, that respect the charged-current term of the Lagrangian. That is, any mass matrix can be transformed to that form without modifying the corresponding CKM matrix.
Fast-forwarding to the present time, notwithstanding steady experimental progress  and a huge amount of theoretical work by many authors, we cannot be sure of being any closer to solving the “Meroitic” problem  of divining the spectrum behind the known data. Disappointingly, textures are still going strong in some quarters [6, 7]. However, it seems fair to state that the focus of attention in the search for underlying symmetry and/or dynamics has turned to the mixing matrix VCKM itself —or the lepton mixing matrix VPMNS, for that matter. Models seeking to discern finite groups of “horizontal symmetry” behind the mixing patterns [8–10] and studies such as  of empirical mass relations do appear to respond to that type of investigation. There are of course many other ideas hawked in the market.
Still, “the Higgs boson must know something we do not know”, and we would dearly like to know it. Perhaps it is time again that we bend the stick again towards the issue of the mass matrices. A perennial question in flavour-mixing theory is the following. Suppose the mass eigenvalues and the empirical mixing matrix, or equivalent data for fermion multiplets, are given: what is the space of mass matrices compatible with these data? A possible path towards the answer to that question involves a detour through the realm of noncommutative geometry (NCG). In , when grappling with the classification problem for Riemannian manifolds, Connes dubbed the CKM construction a “toy model” for geometrical placement problems in general. His abstract formulation of the latter helps to inject some fresh thinking into the subject. We make no apology for this relatively high-brow approach to such an apparently simple matter.
Noncommutative geometry and flavour mixing prepared by Jose M. Gracia-BondiaOctober 22, 2013