Why we may need a new spacetime model...

... to understand the "dark far side" of the Higgs (that stares at unification)?

This post is sort of a personal addendum to the recent Why We Need A New Collider by Tommaso Dorigo and older Higgs on the Moon by Adam Falkowski. The subtitle is a wink to the current Chinese landing on the 'Dark Side' of the Moon.

Late 2018 marked Hundred Years of Gauge Theory. For this occasion Gerard 't Hooft was invited to make a lecture where he argued the following :

Gauge invariance was introduced by Hermann Weyl as an attempt to unify Einstein’s General Relativity with electromagnetism, by adding local scale transformations, affecting the gauge of a weighing scale... This did not quite work. 

But you do get electromagnetism if charged fields ψ transform as ψ→e ^iω(x)ψ... Quantizing the fields with gauge invariance, leads to the correct quantum theory for electrically charged particles...Infinities in the procedure could be made to cancel one an other, if done with sufficient care. Renormalizable theories contained scalar fields ϕ(x), fermionic fields ψ(x) and the electromagnetic fields Fµν(x) and Aµ(x)... 

But the weak interaction also appeared to be mediated by a vector particle, just like the photon. There should be at least three types of weak photons, W +, W −, and Z, besides the photon, γ. In the 1960s, M. Veltman was convinced by the experimental evidence: The weak interactions had to be some modification of a Yang-Mills theory. The Yang-Mills photons had to have mass, and the neutral component Z, if that existed at all, would couple differently to charged and neutral currents. Veltman attempted to formulate the renormalization procedure for the modified theory. But his modified theory was not gauge-invariant. 

We learned that gauge invariance was not allowed to be messed with...  

The Brout-Englert-Higgs mechanism (not a symmetry breaking!) must be invoked to represent the masses of the photons. Moreover, by allowing the Higgs field also to couple to the fermions, we could allow masses for charged leptons and neutral leptons to differ from one another. So we solved not one, but two mass problems for the electro-weak theory... 

To be renormalizable, the short distance structure of the theory must be exactly that of a pure gauge theory. We need exact local gauge invariance... 

Our original proofs of renormalizability were based on various techniques... Faddeev-Popov ghost Lagrangian... η and η' ... anti-commuting, scalar fields... By combining the Feynman rules, physical particles were found to obey the non-Abelian generalisation if the Ward-Takahashi identities, a symmetry relating... diagrams. However, we did not recognise this as a symmetry between the fields... However, it is a super-symmetry between fields: Becchi Rouet Stora Tyutin [symmetry]...

With the Large Hadron Collider, we are now seeing a glimpse of the future of quantum field theory ! The mass of the Higgs particle – finally found – is very close to the value that flattens off the curves of the running coupling constants as if we are approaching a domain with scale invariance, more precisely: local conformal symmetry. This may mean that we can approach quantum gravity more quickly than expected. 

Past and Future of Gauge Theory, 

WE-Heraeus-Seminar on “Hundred Years of Gauge Theory” Opening talk, July 30, 2018

Gerard 't Hooft

I stop the long quote here as the rest is no more history but a specific speculation (the curious reader can follow in the footsteps of 't Hooft along the conformal symmetry track in Singularities, horizons, firewalls, and local conformal symmetry). Staying nevertheless with 't Hooft for an outlook on physics I'd like to focus on the following:

When trying to describe nature’s most general state, we have to ask what the ultimate state will look like when we squeeze as much matter as is possible into the tiniest possible volume. One usually expects one or more black holes to form. All presently known laws of fundamental physics make use of quantum mechanics, so, even if one is skeptical about this, it is natural first to search within the quantum formalism...
Laws of gravity, as they are known today, then suggest that all forms of matter will be geometric: the way they affect the curvature of space and time is the only form of information that is conserved [5]–[7]...
The demand from string theory that space and time themselves must feature either 10 or 26 dimensions, however, seems to be too restrictive. If indeed, as we suspect, physical degrees of freedom form discrete sets, then dimensionality of space and time may be less well-defined notions, so that such ‘predictions’ from string theory may well be due to some mathematical idealization having little to do with reality. All in all, we are badly in need of a more orderly listing of all conceivable configurations of physical variables in a small region of space and time. 
As the reader may notice, this demand by itself is also not formulated very sharply. This is exactly the point we wish to bring forward in this paper. Part of our problem is the precise formulation of our question or questions. Nature’s book keeping system must be outlined together with the answers to the question how the variables will evolve in time. It is important to observe that the big revolutions in science often came with improvements in our way to phrase the physical degrees of freedom, together with new proposals as to how these evolve. The grand total is what we call physical law. 
Smolin complains that science today is slowing down considerably, and blames this to the rise of string theory and its staggering promises, at the expense of other approaches in the basic sciences. But Smolin forgets that the real revolutions in science are often only recognised in hindsight. To our judgement, it is quite conceivable that many or all of our questions concerning nature’s book keeping system will be solved in the not so distant future. However, the road towards these solutions will consist of very small but mathematically precisely formulated steps in our way of thinking. String theory was an interesting guess, but may well have been a too wild one. We are guessing the mathematical structures that are likely to play a role in the future, but we fall short on grasping their internal physical coherence and meaning. For this, more patience is needed.

Nature's Book Keeping System

Gerard t Hooft (Submitted on 29 Apr 2016)

As the reader knows 't Hooft was a PhD student in 1970 when he achieved  to prove with Veltman the renormalizability of the Glashow Weinberg Salam electroweak unification model completed with the Brout-Englert-Higgs mechanism, then convincing high energy physicists about the merits of Yang-Mills quantum gauge field theory. And he was still a post-doc when he was the first to understand that chromodynamics could fit with the same framework to describe strong interactions, paving the way to the experimental exploration of the Standard Model physical spectrum programmatically described by Jean Iliopoulos in 1974. 

Let's repeat symbolically this handing over from 't Hooft to Iliopoulos now:

In the past decades we have learned that all fundamental interactions are described by gauge theories. Gauge transformations are of two sorts: 
• Diffeomorphisms in space-time. 
• Internal symmetry transformations which apply on some internal space. However, the parameters of these transformations are functions of the space-time point, but not of the point of the internal space. The latter is fixed and does not participate in the dynamics. 

Question: Is there a space on which internal symmetry transformations act as diffeomorphisms? 

Answer: Yes, but it is a space with non-commutative geometry. A space defined by an algebra of matrix-valued functions 

This approach has been developed by Alain Connes and collaborators. Some recent references are ... [0706.3688, 1411.0977, 1409.2471]. The quantum mechanical non-commutativity involves the introduction of a deformation parameter, to wit ℏ, which defines the phase space commutation relations among position and momentum operators for every degree of freedom. The goal here is to extend this notion of non commutativity using mathematical ideas from spectral theory. The construction involves a fundamental spectral triplet: Given a spin manifold M, the triplet consists of:
• a Hilbert space H; 
• an algebra of functions A which are C∞(M) [A generalises the notion of space];

• the Dirac operator D, which plays the role of the inverse of distance... 

To this basic triplet we can add decorations, such as chirality, CPT, etc. 

Although these assumptions sound rather weak, the conditions imposed by spectral theory are quite powerful and restrict the possible choices of A. In particular, the gauge algebras of the Standard Model appear together with General Relativity. Although, to my opinion, the technical part of this construction is not yet fully developed, several physical results may emerge naturally. An incomplete list includes: 
• I mentioned already the special role of the Brout-Englert-Higgs scalar as the distance between different branes in the phase with spontaneously broken symmetry.
• This approach may provide the way we are all looking for to unify gauge theories and Gravity. 
• It may give some predictions for the Standard Model parameters. An example was a relation between the masses of the vector and the scalar bosons of the electroweak theory. In the traditional framework of renormalisable perturbation theory such a prediction is impossible because there is no zero in the corresponding β-function[22]. 

• It may offer a new insight for the mysteries of dark matter and dark energy.

... non-commutative geometry has come to stay! It is part of gauge theories. Whether it will turn out to be convenient for us to use, is still questionable. It will depend on our ability to simplify the mathematics sufficiently, or to master them deeply, in order to get new insights. Quite independently, let me point out that the spectacular accuracy reached by experiments, as well as theoretical calculations, has made particle physics a precision science. Therefore, "approximate" theories are no more sufficient. A discrepancy by a few percent implies that we do not have the right theory! This strongly restricts the possible ways to go beyond the Standard Model. On the other hand, the completion of the Standard Model strongly indicates that new and exciting Physics is around the corner. But for the moment, we see no corner! Could non-commutative geometry show the way out of this dilemma?

Gauge Theories and non-Commutative Geometry: A review

John Iliopoulos 2018

This review is more or less a written version of a lecture given  by Iliopoulos on different occasions and at various locations, the one above was for David Gross' birthday! It's too bad the same lecture delivered at Ludwig-Maximilians-University of Munich (in the presence of the cosmologist Viatcheslav Mukhanov) requires a login to be watched as is available online now(!) and its questions & answers part is interesting. Indeed Iliopoulos had the opportunity to motivate his personal doubts about the relevance of spectral noncommutative geometry for current high energy physics because of its lack of a definite quantum version (particularly of the proper operators to define BRST symmetry). Nevertheless he also expressed his hopes regarding the connection established recently in the spectral noncommutative geometric framework between the mimetic dark matter and dark energy models developped by Mukhanov with Ali Chamseddine and the hypothesis of 4D/3D quantized volumes of Lorentzian space-time. 

To get a feeling of the potential opportunities offered by this deep geometric book keeping system which happened to meet Yang Mills and Higgs physics 30 years ago now, I propose to the imaginative graduate student and [her, his] bold enough PhD supervisor to engage with Einstein-Hilbert action under the spell of a quantum rapture following Chamseddine:

...a Lorentzian space-time volume quantization is possible, provided that the field X [which maps into ℝ the geodesics normal to the moving 3D hypersurfaces that generate the 4D manifold]... satisfies a length preserving condition... In the synchronous gauge, [the field X] is identified with the time coordinate and modifies Einstein equations [only in the longitudinal sector] by giving an energy-momentum tensor in the absence of matter, giving rise to mimetic cold matter. We have shown that this field, which arises naturally from the three space quantization condition can be used to construct realistic cosmological models such as inflation without the need to introduce additional scalar fields. By including terms in the action of the form f (☐X) which do occur in the spectral action as can be seen from considerations of the scale invariance, it is possible to avoid singularities in Friedmann, Kasner [35] or Black hole solutions [36]. This is possible because the contributions of the field X to the energy-momentum tensor would allow, for special functions f(☐X) to limit the curvature, preventing the singularities from occurring.

Quanta of Space-Time and Axiomatization of Physics... celebrating the one century anniversary of the program announced in 1916 by Hilbert entitled " Foundations of Mathematics and Physics"

Ali H. Chamseddine (Submitted on 27 Feb 2017)