### My own private ostinato

**A tentative articulate answer to some recent Lubos Motl comments that appeared here and there about physics on non-commutative spaces**

In a recent post* Lubos Motl appears to me as reducing the noncommutative geometrization of physics by Alain Connes and his collaborators as a fancy formalism to rewrite quantum field theory in an abstract way but it is much more than that. I am going to skip through his most easily proved wrong assertion

just providing the two following informative examples:Connes' "fix" that reduced the prediction to 125 GeV was largely ignored by the later pro-Connes literature that kept on insisting that 170 GeV is indeed what the theory predicts.

NCG made a prediction for the SM Higgs at approximately 170 GeV [CIS99,KS06,CCM07], which is now ruled out by experiment [A+12, C+12b]. Since then a number of solutions to the ‘Higgs mass problem’ have been proposed. Notably by Estrada and Marcolli who include gravitation corrections to obtain a 125 GeV Higgs [EM13a], while Chamseddine and Connes propose an alternative solution in which they add an extra scalar field σ into the model by hand [CC12]. Later papers by Devastato et al. [DLM14a] and Chamseddine et al. [CCvS13] construct standard model extensions which naturally include the σ field as an output. Our fused algebra formulation of the NCG SM {FB15b} offers a natural solution to the problem which I will outline in full detail...

The recent discovery of the Higgs boson with a massm_{H}≃126 Gev suggests the big desert hypothesis should be questioned. There is indeed an instability in the electroweak vacuum which is meta-stable rather than stable (see [5] for the most recent update). There does not seem to be a consensus in the community whether this is an important problem or not: on the one hand the mean time of this meta-stable state is longer than the age of the universe, on the other hand in some cosmological scenario the meta-stability may be problematic [26, 27]. Still, the fact thatm_{H}is almost at the boundary value between the stable and meta-stable phases of the electroweak vacuum suggests that “something may be going on”. In particular, particle physicists have shown how a new scalar field suitably coupled to the Higgs - usually denoted σ - can cure the instability (e.g. [13, 25]).

Taking into account this extra field in the NCG description of the SM induces a modification of the flow of the Higgs mass, governed by the parameter r = k_{ν }/k_{t }which is the ratio of the Dirac mass of the neutrino and of the Yukawa coupling of the quark top. Remarkably, for any value of Λ between10^{12}and10^{17}GeV, there exists a realistic value r ≃ 1 which brings back the computed value ofm_{H}to 126 Gev [8].

The question is then to generate the extra field σ in agreement with the tools of noncommutative geometry. Early attempts in this direction have been done in [34], but they require the adjunction of new fermions (see [35] for a recent state of the art). In [8], a scalar σ correctly coupled to the Higgs is obtained without touching the fermionic content of the model, simply by turning the Majorana massk_{R}of the neutrino into a field k_{R}→k_{R}σ.

Twisted spectral triple for the Standard Model and spontaneous breaking of the Grand SymmetryAgostino Devastato, Pierre Martinetti(Submitted on 5 Nov 2014 (v1), last revised 29 Jan 2015 (this version, v2))

I prefer to insist first on a problem of terminology that Lubos skips completely in his post and comments:

We shall make the distinction between noncommutative spacetimes intended as spaces whose coordinates no longer commute, and spectral geometries intended as a space whose algebra of functions is nonnecessarily commutative.

Noncommutative spacetimes, ... , are obtained as a deformation of a usual space by trading the (commutative) coordinate functions x_{µ}, x_{ν}of a manifold with coordinate operatorsq_{µ}, q_{ν}satisfying non-trivial commutation relations. Besides the seminal quantum spacetime model of Doplicher, Fredenhagen, Roberts [2] treated in L. Tomassini talk [3], such spaces are present in many - if not all - approaches to quantum gravity, including loop quantum gravity, string theory... as well as in more phenomenology-oriented models like doubly special relativity...

Noncommutative spacetimes also emerged very early as a possible solution to ultraviolet divergencies in quantum field theory, especially in the work of Snyder [6]. Quantum field and gauge theories on noncommutative spacetimes have thus developed as a theory on their own, independently of any consideration on quantum gravity...

.Spectral geometries [11]... consists in a generalization of Gelfand duality between locally compact spaces and C∗-algebras, so that to encompass all the aspects of Riemannian geometry [12] beyond topology. It furnishes a geometrical interpretation of the Lagrangian of the standard model of elementary particles [13, 14], as well as some possibilities to go beyond [15, 16]..

Beyond the Standard Model with noncommutative geometry, strolling towards quantum gravityPierre Martinetti (Submitted on 28 Oct 2015)

There is a second aspect completely skipped by the author of

*the Reference Frame*blog. It is the following. To quote the physicist Thomas Schucker :

... noncommutative {spectral geometries} are close enough to Riemannian spaces such that Einstein’s derivation of gravity from Riemannian geometry carries over to noncommutative{spectral geometries}. In Connes’ derivation, the entire Yang-Mills-Higgs action pops up as a companion to the Einstein-Hilbert action, just like the magnetic field pops up as a companion to the electric field, when the latter is generalized to Minkowskian geometry, i.e. special relativity.

I think Lubos completely misses or eludes this point in his paragraph (and the followings) starting with :(Submitted on 29 Mar 2010)

There are many detailed questions that Connes can't quite answer that show that he doesn't really know what he's doing. One of these questions is really elementary: Is gravity supposed to be a part of his picture?

The understanding (checking by computation) of what Schucker claims requires time and skill (that I do not claim to have). Some accomplished physicists did. Lubos and all string theorists have skill. The current paucity of empirical evidence for TeV scale susy/wim-particles may give time to some young bold ones to appreciate the fact that Connes noncommutative geometry has been extended since 2006 and might have already uncovered at least one really nontrivial fact in our quest to quantum gravity:

...any connected Riemannian Spin 4-manifold with quantized volume >4 (in suitable units) appears as an irreducible representation of {some proposed} two-sided commutation relations in dimension 4 and ... th{is} representation give{s} a seductive model of the "particle picture" for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics

(Submitted on 4 Nov 2014)

In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with the index formula, the quantization of the volume. We first show that this condition implies that the manifold decomposes into disconnected spheres which will represent quanta of geometry. We then refine the condition by involving the real structure and two types of geometric quanta, and show that connected spin-manifolds with large quantized volume are then obtained as solutions. The two algebrasM_{2}(ℍ)andM_{4}(ℂ)are obtained which are the exact constituents of the Standard Model. Using the two maps fromM^{4}to S^{2}the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. We give several physical applications of this scheme such as quantization of the cosmological constant, mimetic dark matter and area quantization of black holes.

(Submitted on 8 Sep 2014 (v1), last revised 11 Feb 2015 (this version, v4))

These higher degree Heisenberg-like new commutation relations with the help of the spectral action principle might select the very specific effective quantum field theory beyond the Standard Model viable up to the seesaw scale and further to grand unification. Moreover the proposed quanta of geometry might provide non trivial insights into the black hole microscopic degrees of freedom and the dark sector of the lambda-CDM cosmological standard model. This is my tentative answer to Lubos, in particular when he writes:

Lubos finished his post with the following sentence:

Now, Connes and collaborators claim to have something clearly different from the usual rules of quantum field theory (or string theory). The discovery of a new framework that would be "on par" with quantum field theory or string theory would surely be a huge one, just like the discovery of additional dimensions of the spacetime of any kind. Except that we have never been shown what the Connes' framework actually is, how to decide whether a paper describing a model of this kind belongs to Connes' framework or not. And we haven't been given any genuine evidence that the additional dimensions of Connes' type exist...

Connes et al. basically want to have a non-singular compactification without branes and they still want to claim that they may decouple some ordinary standard-model-like physics from everything else – like the excited strings or (even if you decided that those don't exist) the black hole microstates which surely have to exist. But that's almost certainly not possible. I don't have a totally rock-solid proof but it seems to follow from what we know from many lines of research and it's a good enough reason to ignore Connes' research direction as a wrong one unless he finds something that is really nontrivial, which he hasn't done yet...

Lubos finished his post with the following sentence:

I could not say it better indeed!"The principles producing theories that seem to work should be taken very seriously"

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