Spectre (quantum reality is full of phantoms from the unfinished past)

Reality bites irony too

On August 4, 2008 Alain Connes wrote a post entitled Irony in the blog Noncommutative geometry:

In a rather ironical manner the first Higgs mass that is now excluded by the Tevatron latest results is precisely 170 GeV, namely the one that was favored in the NCG interpretation of the Standard Model, from the unification of the quartic Higgs self-coupling with the other gauge couplings and making the "big desert" hypothesis, which assumes that there is no new physics (besides the neutrino mixing) up to the unification scale.

Lubos Motl's commented it in the following way:

Dear Prof Connes, 

thanks for this amazing speed and integrity. I am sure it wouldn't be matched by any other author of unusual and unexpected predictions in physics I know of... 

I wonder whether you appreciate the special role of the value 170 GeV. It's the value for which the Higgs quartic self-interaction is high enough for the RG running to push it to a divergent value - the Landau pole - at an accessible energy scale, namely the GUT scale.

[The Standard Model] has certain couplings and they simply must be allowed to be any real number.

Only the requirement of enhanced symmetries or the absence of anomalies or divergences or ghosts at special points are legitimate reasons to pick preferred values of the masses and other parameters of the Standard Model in any quantum field (-like) theory. 

So even if a particular language makes it harder (or impossible) to write the Standard Model with generic values of the couplings in your (or another) formalism, it can't be viewed as a trusted prediction because it is always numerology that depends on the "language" i.e. the particular NCG reformulation of the Standard Model.

So QFT itself can't answer such questions, e.g. the values of the couplings, not even when one rewrites it with new symbols... 

Now, when you have a sufficient anti-NCG momentum this week, it may be a great idea for someone like me to ask you to learn the rest of the string/M massive tower that you have been neglecting so far...  

So I would like to boldly use the opportunity to invite you to throw away a young-man's maverickness for a while, to learn string/M-theory this and next week (or month) ... and to solve all the remaining open problems of string theory by this Christmas, including a non-perturbative universal definition of the theory applicable across the configuration space (landscape), the vacuum selection problem, and the cosmological constant problem...

Thanks, you can surely do it. I admire you and I have met way too many smart people who admire you even qualitatively more than I do so all this stuff is surely realistic.

All the best

August 5, 2008 at 5:57 PM

Jacques Distler reacted to Lubos' claim about the Landau pole at Planck scale for a 170 GeV prediction:

I am hesitant to wade into this, but I believe that's wrong. 170 GeV came out of an RG analysis, alright. But, rather than diverging at the GUT scale, the Higgs quartic self-coupling "unifies" with the gauge couplings at that scale (the precise formula is in their paper, or in my blog post). 

There are plenty of problems with their scenario, but the Higgs self-coupling hitting its Landau pole at the GUT scale is not one of them.

August 6, 2008 at 6:56 AM

Motl replied:

Your article didn't help me to increase my belief that it should be possible to predict from a new, different, equivalent formulation of a field theory relationships that cannot be extracted from the old-fashioned definitions of the theory.

... in heterotic string theory, all these couplings come from some worldsheet correlators that share a "common ancestor". So there could perhaps be a "unification" of quartic couplings with gauge couplings at the string scale. But these things are not model-independent, not even in big classes of string vacua, so I don't believe that they could be universal in all "good" theories sharing the same low-energy SM limit.

August 6, 2008 at 9:26 AM

 Distler did also:

... At low energies (E≪Λ), everything can be phrased in the language of an effective QFT. What they have is something completely different from a local QFT, for energies above the GUT scale.

August 6, 2008 at 1:01 PM

Fedele Lizzi eventually remarked:

Alain will certainly remember that in various incarnations of the model, starting from his work with Lott, and in particular before the spectral action, the mass of the Higgs went as far up as 250 GeV if I remember well, if not higher. 

The spectral action has been consistent in favouring a more or less "light" Higgs, and some sort of desert. This is probably a consequence of the requirement of the unification of coupling constants, which makes the model akin to SU(5).

The latest version of the model of Chamseddine Connes and Marcolli (ACM)^2 is certainly the most powerful and coherent one, but it assumes an almost commutative geometry all the way to very high scale, and as a consequence that the renormalization group analysis can be done in the usual way. I still find astonishing the fact that a model comes up with an Higgs mass "in the right ball park" from purely geometric considerations.

As first thing we should judge how unchangeable is the Higgs mass prediction in the (ACM)^2 model. The model is complicated, and I cannot say I mastered all of its details, but I think it is fairly solid. It is its strenght, it may become its problem. So its enhancement cannot come from the inside. It probably requires a wider change. My personal bias is to think that probably already at scales which would lie in the big desert some effects of the fact that spacetime is not describable by ordinary geometry should show. Probably the fact that neutrino masses are so small hint at a different mechanism, beyond see-saw.

In 2012, one month exactly after the official announcement of the discovery of a Higgs boson like 125 GeV resonance at LHC, Connes wrote in his blog post :

Since 4 years ago I thought that there was an unavoidable incompatibility between the spectral model and experiment... Now 4 years have passed and we finally know that the Brout-Englert-Higgs particle exists and has a mass of around 125 Gev... this certainly slowed down quite a bit the interest in the spectral model since there seemed to be no easy way out and whatever one would try would not succeed in lowering the Higgs mass {prediction}. The reason for this post today is that this incompatibility has now finally been resolved in a fully satisfactory manner in a joint work with my collaborator Ali Chamseddine, the paper is now on arXiv at 1208.1030 

What is truly remarkable is that there is no need to modify the spectral model in any way, it had already the correct ingredients and our mistake was to have neglected the role of a real scalar field which was already present and whose couplings (with the Higgs field in particular) were already computed in 2010 as one can see in 1004.0464. This completely changes the perspective on the spectral model, all the more because the above scalar field has been independently suggested by several groups as a way for stabilizing the Standard Model in spite of the low experimental Higgs mass. So, after this fruitful interaction with experimental results, it is fair to conclude that there is a real chance that the spectral approach to high energy physics is on the right track for a geometric unification of all known forces including gravity...

A DRESS FOR THE BEGGAR ?

Then two years later on November 9, 2014 came probably the most unexpected and rewarding result of the  spectral noncommutative geometric program. As advertised by his leader:

The purpose of this post is to explain a recent discovery that we did with my two physicists collaborators Ali Chamseddine and Slava Mukhanov. We wrote a long paper Geometry and the Quantum: Basics which we put on the arXiv, but somehow I feel the urge to explain the result in non-technical terms...

In particle physics there is a well accepted notion of particle which is the same as that of irreducible representation of the Poincaré group. It is thus natural to expect that the notion of particle in Quantum Gravity will involve irreducible representations in Hilbert space, and the question is "of what?". What we have found is a candidate answer which is a degree 4 analogue of the Heisenberg canonical commutation relation [p,q]=ih. The degree 4 is related to the dimension of space-time. The role of the operator p is now played by the Dirac operator D. The role of q is played by the Feynman slash of real fields, so that one applies the same recipe to spatial variables as one does to momentum variables. The equation is then of the form E(Z[D,Z]⁴)=γ where γ is the chirality and where the E of an operator is its projection on the commutant of the gamma matrices used to define the Feynman slash. 

Our main results then are that:

1) Every spin 4-manifold M (smooth compact connected) appears as an irreducible representation of our two-sided equation.

2) The algebra generated by the slashed fields is the algebra of functions on M with values in A=M₂(ℍ)⊕M₄(ℂ), which is exactly the slightly noncommutative algebra needed to produce gravity coupled to the Standard Model minimally extended to an asymptotically free theory.

3) The only constraint on the Riemannian metric of the 4-manifold is that its volume is quantized, which means that it is an integer (larger than 4) in Planck units.

The great advantage of 3) is that, since the volume is quantized, the huge cosmological term which dominates the spectral action is now quantized and no longer interferes with the equations of motion which as a result of our many years collaboration with Ali Chamseddine gives back the Einstein equations coupled with the Standard Model.

The big plus of 2) is that we finally understand the meaning of the strange choice of algebras that seems to be privileged by nature: it is the simplest way of replacing a number of coordinates by a single operator. Moreover as the result of our collaboration with Walter van Suijlekom, we found that the slight extension of the Standard Model to a Pati-Salam model given by the algebra M₂(ℍ)⊕M₄(ℂ) greatly improves things from the mathematical standpoint while moreover making the model asymptotically free!

To get a mental picture of the meaning of 1), I will try an image which came gradually while we were working on the problem of realizing all spin 4-manifolds with arbitrarily large quantized volume as a solution to the equation.

"The Euclidean space-time history unfolds to macroscopic dimension from the product of two 4-spheres of Planckian volume as a butterfly unfolds from its chrysalis."

PARTICLES IN QUANTUM GRAVITY

Last year (August 11, 2015) Connes announced his last physical progress obtained with Ali Chamseddine and Walter van Suijlekom:

From the spectral action principle, the dynamics and interactions are described by the spectral action, tr(f(D/Λ)) where Λ is a cutoff scale and f an even and positive function. In the present case, it can be expanded using heat kernel methods, 

tr(f(D/Λ))∼F₄ Λ⁴a₀+F₂ Λ²a₂+F₀ a₄+⋯ 

where F₄,F₂,F₀ are coefficients related to the function f and a_k are Seeley deWitt coefficients, expressed in terms of the curvature of M and (derivatives of) the gauge and scalar fields. This action is interpreted as an effective field theory for energies lower than Λ. 

One important feature of the spectral action is that it gives the usual Pati–Salam action with unification of the gauge couplings...  

Normalizing {the terms from the scale-invariant part F₀ a₄ in the spectral action for the spectral Pati–Salam model} to give the Yang–Mills Lagrangian demands

(F0/2π²)g_L²=(F0/2π²)g_R²=(F0/2π²)g²=1/4, 

which requires gauge coupling unification...  

Since it is well known that the SM gauge couplings do not meet exactly, it is crucial to investigate the running of the Pati–Salam gauge couplings beyond the Standard Model and to find a scale Λ where there is grand unification:  

g_R(Λ)=g_L(Λ)=g(Λ). 

This would then be the scale at which the spectral action is valid as an effective theory. There is a hierarchy of three energy scales: SM, an intermediate mass scale mR where symmetry breaking occurs and which is related to the neutrino Majorana masses (10¹¹−10¹³ GeV), and the GUT scale Λ. 

In the paper, we analyze the running of the gauge couplings according to the usual (one-loop) RG equation. As mentioned before, depending on the assumptions on {the Dirac operator which operates within the noncommutative fine structure of spacetime}, one may vary to a limited extent the scalar particle content ... {nevertheless} we establish grand unification for all of the scenarios with unification scale of the order of 10¹⁶ GeV, thus confirming validity of the spectral action at the corresponding scale. 

Grand Unification in the Spectral Pati-Salam Model

As one can see it took more than one year as wished by Lubos for Connes and his mathematician and physicist colleagues to propose some tentative solutions to the vacuum selection problem and the cosmological constant one (o_~)