Consequences of volume quantisation of space-time

Here is the fourth fragment of my Lover's Dictionary of Spectral Physics:

Quantisation of spacetime in the spectral noncommutative framework based on the Heisenberg-like Chamseddine-Connes-Mukhanov equation(s)

... when the intellectual despotism of the Church, which had been maintained through the Middle Ages, had crumbled, and a wave of scepticism threatened to sweep away all that had seemed most fixed, those who believed in Truth clung to Geometry as to a rock, and it was the highest ideal of every scientist to carry on his science "more geometrico".

Space-time-matter, 

Hermann Weyl, 1917

La nature est localement quantique et spectralement noncommutative

Folklore 

As the last echo to the quantisation of matter-radiation interaction started with Planck in 1900, the recent completion of the standard model of unification of the strong and electroweak interactions has advanced our ideas of the subatomic world a step further. It is even not unreasonable to conceive that the next scale of particle physics lies beyond 10¹⁰ GeV! Now wider expanses and greater depths are thus exposed to the searching eye of knowledge (to quote Hermann Weyl in a 100 year-old reflexive text about the foundation of differential geometry and general relativity). The more so as tremendous progress in astrophysics have seen the emergence of a testable cosmological standard model that offers a window on energy scales of which we had hardly a hope to probe a decade ago. Both standard models have already brought us much nearer to grasping the plan that underlies all physical happening.
May be it is time now to repeat the saga initiated by Planck and Einstein (from the experimental data collected by an army of spectroscopists raised by Newton) to envision a quantisation of space-time. This is what is reported below through a spectral ride on the loop where the micro and macro worlds meet to uncover regions of which we had not even a presentiment. To say shortly the consequence of the quantisation of spacetime based on a Heisenberg-like equation found by Chamseddine, Connes and Mukhanov might be a pretty unique unified  noncommutative framework for space-time-matter-radiation as we know it here (13 TeV) and now (2,7 K).

... by starting from a quantization condition on the volume of the noncommutative space, all fields and their interactions are predicted and given by a Pati-Salam model which has three special cases one of which is the Standard Model with neutrino masses and a singlet field. The spectral Standard Model predicts unification of gauge couplings and the correct mass for the top quark and is consistent with a low Higgs mass of 125 Gev. The unification model is assumed to hold at the unification scale and when the gauge, Yukawa and Higgs couplings relations are taken as initial conditions on the RGE, one finds complete agreement with experiment, except for the meeting of the gauge couplings which are off by 4%. This suggests that a Pati-Salam model defines the physics beyond the Standard Model, and where we have shown [16] that it allows for unification of gauge couplings, consistent with experimental data. 

The assumption of volume quantization has consequences on the structure of General Relativity. Equations of motion agree with Einstein equations except for the trace condition, which now determines the Lagrange multiplier enforcing volume quantization. The cosmological constant, although not included in the action, is now an integration constant... To have a physical picture of time we have also considered a four-manifold formed with the topology of R × Σ3, where Σ3 is a three dimensional hypersurface, to allow for space-times with Lorentzian signature. The quantization condition is modified to have two mappings from Σ3 → S 3 and a mapping X : R → R. The resulting algebra of the noncommutative space is unchanged, and the three dimensional volume is quantized provided that the mapping field X is constrained to have unit gradient. This field X modifies only the longitudinal part of the graviton and plays the role of mimetic dust. It thus solves, without extra cost, the dark matter problem [33]. Recently, we have shown that this field X can be used to build realistic cosmological models [34]. In addition, and under certain conditions, could be used to avoid singularities in General relativity for Friedmann, Kasner [35] and Black hole solutions [36]. This is possible because this scalar field modifies the longitudinal sector in GR...   

We have presented enough evidence that a framework where space-time assumed to be governed by noncommutative geometry results in a unified picture of all particles and their interactions. The axioms could be minimized by starting with a volume quantization condition, which is the Chern character formula of the noncommutative space and a special case of the orientability condition. This condition determines uniquely the structure of the noncommutative space. Remarkably, the same structure was also derived, in slightly less unique way, by classifying all finite noncommutative spaces [10].

The picture is very compelling, in contrast to other constructions, such as grand unification, supersymmetry or string theory, where there is no limit on the number of possible models that could be constructed. The picture, however, is ... incomplete as there are still many unanswered questions and we now list few of them. Further studies are needed to determine the structure and hierarchy of the Yukawa couplings, the number of generations, the form of the spectral function and the physics at unification scale, quantizing the fields appearing in the spectral action and in particular the gravitational field. To conclude, noncommutative geometry as a basis for unification, is a predictive and exciting field with very appealing features and many promising new directions for research.

Quanta of Space-Time and Axiomatization of Physics

Ali H. Chamseddine 

Contribution to the special issue of IJGMMP celebrating the one century anniversary of the program announced in 1916 by Hilbert entitled " Foundations of Mathematics and Physics", (Submitted on 27 Feb 2017)

Remark: the reader is warmly invited to have a careful look at the article above and particularly on the pages starting from section 9 Consequences of volume quantization (for astrophysics and cosmology) where the technical details of how the geometric invariants of a field derived from the covariantly isolated scale factor of general relativity can mimic dark matter without the need for new interactions, in the perfectly classical differential geometric language.