lundi 30 juin 2014

The spectral noncommutative programme = STRINGS from a conservative General Relativistic and QFT inspired heuristic viewpoint

What are STRINGS?
Here is a very nice answer by the famous (string)physicist Juan Maldacena, reported in the recent post from Peter Woit: String Theory Visions
Solid Theoretical Research INatural Geometric Structures
Juan Maldacena , Geometry and Quantum Mechanics Vision Talks at Strings 2014, 27 juin 2014
What is the spectral noncommutative programme for physics?
A written answer in 2007:
[Our] approach to physics can be summarized as a strategy to interpret the complicated input of the phenomenological Lagrangian of gravity coupled with matter as coming from a fine structure (of the form [continuous ]M[anifold]×F[inite space]) in the geometry of space-time. Extrapolating this to unification scale (i.e. assuming the big desert) gives predictions which can be compared with experiment. Of course we do not believe that the big desert is there and a key test when “new physics” will be observed is to decide whether it will be possible to interpret the new terms of the Lagrangian in the same manner from noncommutative spaces and the spectral action. This type of test already occurred with the new neutrino physics coming from the Kamiokande experiment and for quite some time I believed that the new terms would simply not fit with the spectral action principle. It is only thanks to the simple idea of decoupling the KO-dimension from the metric dimension that the problem was resolved (this was also done independently by John Barrett [1] with a similar solution). At a more fundamental level the fact that the action functional can be obtained from spectral data suggests that instead of just looking at the inner fluctuations of a product metric on M×F, one should view that as a special case of a fully unified theory at the operator theoretic level i.e. a kind of spectral random matrix theory where the [Dirac] operator varies in the symplectic ensemble...
Alain Connes, Noncommutative geometry and the spectral model of space-time, 2007

An informal oral answer in 2014, after having removed a several year (Higgs mass prediction gl)itch:
Probably the true geometry [to describe physics] is entirely finite...
There is no no-go theorem known in quantum gravity that you cannot replace a [continuum] space by a discrete space...
The idea is that whereas it is not possible to "cut" the riemannian space into small points by respecting the symmetries [like the Lorentz one] it is perfectly sensible to cut it [following the spectral action principle] to do it by cutting [the spectrum of the Dirac operator acting] in the Hilbert space of [spin 1/2] fermions [at high energy] and then you have all the symmetries that you want and this has been done in few examples but it's a general principle * [in the operator theoretic framework of non commutative geometry].
The general principle is that probably the true physical theory is a theory of random matrices. The random matrices are the Dirac operators and they are finite dimensional, very high dimensional but finite... This is why you probably don't have any problem with the functional integral... What we are trying to do is we are trying to guess that gradually, from the shadow that we have which is partly riemannian and now from this strory you can see it is only partly Riemannian ... it's a little bit non commutative and probably when you go further in energy it will become more and more non commutative.
(transcription of) Alain Connes (' answer to a question raised by the blogger at séminaire Algèbres d'opérateurs) during a presentation by Ali Chamseddine entitled: Spectral geometric unification, 06/26/2014

The spectral noncommutative programme belongs to STRINGS in the Maldacena vision! I wonder if this could make some researcher involved in spectral non commutative geometry eligible to get a nomination for a Breakthrough Prize in Fundamental Physics or Mathematics (and one could also ask  if Lubos Motl would agree ;-)

* Addendum
There is one important advantage of the spectral point of view when compared to the old idea of a discrete space-time, which is that continuous symmetry groups survive the operation of truncating the Hilbert space H to the finite dimensional subspace H(Λ) corresponding to eigenvectors of the Dirac operator for eigenvalues ≤ Λ where Λ is a cutoff scale. Indeed, any unitary operator U commuting with D will automatically restrict to H(Λ). It could well be that the coherence of the spectral action principle indicates that our continuum picture of space-time is only an approximation to a completely finite spectral geometry whose underlying Hilbert space is finite dimensional. Of course the basic ingredients such as J and γ will still be present, but the algebra A itself will have no reason to remain commutative. In this scenario, once we go up in energy towards the unification scale, the small amount of noncommutativity encoded in the finite geometry F to model the present scale, will gradually creep in and invade the whole algebra of coordinates which will become a huge matrix algebra at Planck scale. The noncommutativity of the algebra of coordinates means that the “internal” degrees of freedom have gradually replaced the external ones and that the notion of “point” has disappeared since a matrix algebra admits only one irreducible representation.
Ali H. Chamseddine, Alain Connes, Submitted on 5 Aug 2010
//last edition: July the 11th 2014

jeudi 26 juin 2014

Bossons sur le (grand frère du) boson scalaire (de Higgs à 125GeV)

Naturalité du boson de Higgs associée à un pseudo-dilaton dans un contexte de théorique quantique des champs 
En ce temps estival, le blogueur se plonge dans le flot des prépublications scientifiques récentes qui tournent autour de l'extension du secteur scalaire des théories quantiques des champs, toujours guidé par son outil méta-heuristique favori (la géométrie non commutative et le principe d'action spectrale).
On commence aujourd'hui avec un article qui fait singulièrement écho à la problématique évoquée dans un précédent billet intitulé Naturalité des extensions minimales du secteur scalaire du Modèle Standard
The Higgs boson presents several well-known puzzles associated with the problem of the naturalness of the existence of a low mass fundamental 0+ field in quantum field theory. The naturalness issue is associated with how scale symmetry is implemented (or not) for the Higgs boson, and there has been a recent upsurge of interest in models that attempt to maintain a classical scale invariance which is broken only by scale anomalies [14]. Here we explore this idea in the context of an extension of the Standard Model (SM) that includes a new gauge singlet scalar field σ coupled to the Higgs sector via ultra-weak couplings. In particular, we assume that the Higgs couples to the singlet field σ through a portal interaction ζ1σ2HH.. Electroweak breaking is induced when σ acquires a VEV by quantum loops, i.e., through Coleman- Weinberg (CW) symmetry breaking [5], and thus yields a mass for σ and for the Higgs boson. We consider the case that the σ field VEV f is much larger than the weak scale, f ≫ vweak, in which case the coupling ζ1 must be ultra-weak, 1| = m2/ f2 ≪ 1.
At first sight, constructing a model with ultra-weak scalar couplings would seem to be a foolish thing to do since most SM couplings are either technically naturally small (e.g., the electron or up and down quark Higgs-Yukawa couplings) or are of order the gauge couplings, such as gtop ∼ g3. For example, the Higgs quartic coupling λ receives additive contributions from the large O(1) couplings gtop, g2 and g1, and thus λ is not ultra-weak. Therefore we must ask if ζ1 can be technically naturally small. The answer is yes: there exists a custodial symmetry for ultra-weak couplings amongst singlet fields. This is a “shift symmetry” and it has a Noether current whose divergence is small, ∝ ζi. This is the reason why ultra-weak couplings can remain ultra-weak in the renormalization group (RG) evolution; the ’t Hooft naturalness of ultra-weak couplings is the exact shift symmetry in the limit ζi→ 0. We have seen shift symmetry in another guise before. Shift symmetry naturally casts σ as a pseudo-dilaton...
Given that the scale of gauge couplings in the SM is O(1), the shift symmetry limit can exist only if the σi are gauge singlet fields. Indeed, it is not meaningful to talk about shift symmetries for fields that carry gauge charges such as the Higgs boson (unless one is interested in the consequences of dynamics in the limit that gauge couplings can be ignored). The couplings λi of fields such as the Higgs boson will receive additive corrections from gauge couplings and will not be multiplicatively renormalized. They will run according to the RG and become comparable in size to the gauge couplings. Of course, our argument is subject to gravitational effects. All fields including σ couple to gravity, which is a gauge theory, so the condition of ultra-weak ζi couplings is subject to whether or not the shift symmetry can be maintained in the context of gravity. This can be done if the contributions to the RG equations from conformal couplings ξi, which appear in terms like 1/2ξσ2R, can remain ultra-weak. These, in turn, will involve effective gravitational couplings, an example of which is the recent “Agravity” model of Salvio and Strumia [10]...
Hence, the shift symmetry may be a powerful constraint that admits a natural sector of ultra-weakly coupled physics...
Up to now we have assumed that the theory obeys classical scale invariance in the sense that scale invariance is broken only through the trace anomaly. This assumes, as is the case in dimensional regularization, that the radiative corrections to scalar masses that are quadratically dependent on the cut-off scale are cancelled by the bare mass terms, leaving the scalars massless before spontaneous symmetry breaking. This makes sense in a pure field theory because only the renormalized masses are physical. However, new physics at a high scale can spoil this by introducing contributions to the scalar masses that are proportional to the high scale. This is the case if there is a stage of Grand Unification, for which the contributions are proportional to the mass scale of the heavy GUT states, but can also happen even if there are no massive states, for example when the new scale is generated by the CW mechanism. In the model presented here, such corrections would affect the Higgs mass and give rise to the usual hierarchy problem, but they also affect the singlet state, despite its ultra-weak couplings, because a contribution to the σ mass squared of O(ζiΛ2) will dominate over the CW potential for   Λ > O(TeV). To avoid this we envisage two possibilities.
The first is that there are no high scales of the type discussed above. Of course this cannot be true if gravity is included, but, as discussed above, it may be that gravity respects the shift symmetry and the gravitational corrections to the dilaton mass are small. However, one would still expect an unacceptably large contribution to the Higgs mass, thereby reintroducing the hierarchy problem. Alternatively, if the model is UV complete so that it does not have Landau poles, gravity may not contribute to the scalar masses at all [11]. This case is analogous to that of a pure field theory with classical scale invariance and guarantees that the scalar sector remains massless in the absence of spontaneous symmetry breaking.

The second possibility is to super-symmetrize the model so that the quadratic mass terms have a low SUSY scale cut-off. In this case, one can have a stage of Grand Unification without introducing unacceptably large scalar mass contributions. A supersymmetric version of the model requires an additional Higgs doublet that somewhat complicates the model.
Kyle Allison, Christopher T. Hill, et Graham G. Ross, Ultra-weak sector, Higgs boson mass, and the dilaton, 24 avril 2014

A la recherche d'une symétrie cachée qui rendrait naturel le (couplage ultra-faible du) boson de Higgs (à un scalaire réel singulet?)
The field σ with ultra-weak couplings is formally analogous to a dilaton, as occurs in a spontaneous breaking of scale symmetry. Let us examine this relationship. Spontaneous scale symmetry breaking can be viewed in two ways. The conventional description is to start with a scale invariant theory, containing a dilaton with a shift-invariant potential, and matter fields. The dilaton’s shift symmetry is broken by the coupling to matter, e.g., as in Yukawa couplings. The stress-tensor is traceless. The dilaton can then acquire a nonzero VEV, and the matter fields then acquire mass, but the stress tensor remains traceless. Hence, we end up with a scale invariant theory, massive matter, and a massless dilaton as the Nambu-Goldstone boson. Alternatively, we can start with massive matter fields, and we include a dilaton with a shift-invariant potential, but with couplings to matter that again break the shift symmetry. Now we compute the stress tensor and find that it is not traceless, i.e., the scale current is not conserved. However, we can find a linear combination of the scale current and the dilaton shift current that is conserved; the theory has a hidden symmetry after all.

mercredi 25 juin 2014

Avertissement (au lecteur et encouragement) à l'étudiant

Un conseil en forme d'avertissement pour le futur
Voilà ce que disait Abdus Salam,  (cité par Remo Ruffini dans un recueil d'articles intitulé Matter Particled), à son étudiant Yuri Ne'eman, alors que ce dernier s'était mis en tête de découvrir une structure mathématique dans le zoo des centaines de résonances hadroniques et autres particules du même nom identifiées par les accélérateurs de particules de l'époque (qui atteignaient alors une échelle d'énergie de l'ordre de la dizaine de GeV).
"You are embarking on a highly  speculative venture and your one year fellowship may be over with nothing to show!"
Abdus Salam s'adressant à son étudiant Yuri Ne'eman, c.1960

Heureusement pour le jeune physicien à qui s'adressait cet avertissement, son programme de recherche s'avéra rapidement fructueux:
Taking an as yet untried algebraic route, [Ne'eman] mastered the theory of Lie algebras and studied Cartan's 1894 classification of the simplest ones, noting that what he was after was an algebra of rank r = 2 to accommodate Isospin and Strangeness. By October 1960 he had identified SU(3) as the classifying symmetry of the hadrons ...
 Remo Ruffini, Matter Particled, 1997

Ici le blogueur ne peut s'empêcher de penser à tous les courageux thésards engagés actuellement dans des travaux théoriques pour comprendre la physique des particules à des échelles d'énergie 1000 fois plus élevées qu'il y a cinquante ans, tâche d'autant plus difficile qu'à l'heure qu'il est la situation phénoménologique est exactement l'inverse de celle qui prévalait au début des années soixante, à savoir l'absence de nouvelles particules fondamentales au delà du boson de Higgs et du quark top! 

Peut-être faut-il alors encourager les quelques audacieux thésards qui se coltinent les algèbres d'opérateurs de la  géométrie spectrale non commutative pour comprendre la structure mathématique si particulière du Modèle Standard? On peut en particulier citer et soutenir comme ils le méritent: l'intrépide Agostino Devastato qui explore (sous la direction de Fedele Lizzi et Pierre Martinetti) les potentialités d'un modèle de grande unification algébrique jusqu'à l'échelle de Planck et le courageux Shane Farnsworth qui propose (avec son directeur de thèse Latham Boyle) d'étendre le formalisme non commutatif à des algèbres non associatives (parvenant semble-t-il à réduire les hypothèses qui sous-tendent l'unification géométrique spectrale de toutes les interactions fondamentales connues).

Encouragement(s) à aller toujours plus loin rétrospectivement

The first tentative steps towards the idea of a deeper layer of particles within the hadrons was made in 1962 by Ne'eman ... and his colleague Haim Goldbeg-Ophir. They wrote a paper suggesting that baryons might each be made up to three more fundamental particles... The paper attracted little attention, partly..., as Ne'eman has aknowledged, 'because it did not go far enough'. The authors had not yet decided whether to regard the fundamental components as proper particles or as abstract fields that did not materialize as particles.
John et Mary Gribbin, Richard Feynman, a Life in Science, 1998

Noncommutative geometry was shown to provide a promising framework for unification of all fundamental interactions including gravity [3], [5], [6], [12], [10]. Historically, the search to identify the structure of the noncommutative space followed the bottom-up approach where the known spectrum of the fermionic particles was used to determine the geometric data that defines the space. This bottom-up approach involved an interesting interplay with experiments. While at first the experimental evidence of neutrino oscillations contradicted the first attempt [6], it was realized several years later in 2006 ([12]) that the obstruction to get neutrino oscillations was naturally eliminated by dropping the equality between the metric dimension of space-time (which is equal to 4 as far as we know) and its KO-dimension which is only defined modulo 8. When the latter is set equal to 2 modulo 8 [2], [4] (using the freedom to adjust the geometry of the finite space encoding the fine structure of space-time) everything works fine, the neutrino oscillations are there as well as the see-saw mechanism which appears for free as an unexpected bonus. Incidentally, this also solved the fermionic doubling problem by allowing a simultaneous Weyl-Majorana condition on the fermions to halve the degrees of freedom. The second interplay with experiments occurred a bit later when it became clear that the mass of the Brout-Englert-Higgs boson would not comply with the restriction (that mH=170GeV) imposed by the validity of the Standard Model up to the unification scale. This obstruction to lower mH was overcome in [11] simply by taking into account a scalar field which was already present in the full model which we had computed previously in [10]. One lesson which we learned on that occasion is that we have to take all the fields of the noncommutative spectral model seriously, without making assumptions not backed up by valid analysis, especially because of the almost uniqueness of the Standard Model (SM) in the noncommutative setting.
 Ali H. Chamseddine, Alain Connes et Walter D. van Suijlekom, Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification, 30 octobre 2013

Last musings (about why studying non commutative spectral models) / Dernières réflexions (autour de l'intérêt d'étudier des modèles spectraux non commutatifs)
... it would be fitting for scientists to think of themselves as members of an expedition sent to explore an unfamiliar but navigable ocean whose size and shapes are dimly understood. However profitable it may be to make cabotage along the rich coastal cities of the Supersymmetric Commonwealth or exciting to participate in M-theory regatta, it would be tragic to neglect support to the spectral noncommutative parties already working their way across the strait of the TeV (scale), following the Higgs (co)m(p)ass towards the Cape of Grand Unification, dreaming to see a little further the Pillars of Planck otherwise reaching the Ocean of all Geometries at least back to the sources of Space-Time.
... Les scientifiques pourraient se voir comme les membres d'une expédition envoyée pour explorer une mer inconnue mais navigable dont la taille et la forme ne seraient que vaguement comprises. Aussi rentable soit le cabotage le long des riches villes côtières de la République Supersymétrique ou aussi excitant soit la participation à des régates M-théoriques, il serait dommageable de négliger pour autant le soutien aux explorateurs spectraux non commutatifs déjà engagés dans la traversée du détroit (de l'échelle) du Te(ra-électron)V(olt) en suivant la direction du compas de Higgs et qui font route vers le cap de Grande Unification rêvant de voir un peu plus loin les colonnes de Planck pour sinon atteindre l'Océan de toutes les Géométries du moins remonter jusqu'aux sources de l'Espace-Temps.
 à la manière de Steven Weinberg dans Why build accelerators? (p73)

//Dernières retouches éditoriales 30/06/14 03/06/15

jeudi 19 juin 2014

(L'appel) pour un programme d'unification géométrique spectrale (non commutatif)

Le prochain et dernier cours d'Ali Chamseddine sur un modèle d'unification géométrique spectral aura lieu à l'IHES non pas le 25 juin comme prévu initialement mais le mercredi 2 juillet 2014 afin de ne pas interférer avec la conférence en l'honneur de Maxim Kontsevich : Algèbre, Géométrie et Physique. Signalons aussi que le séminaire d'Algèbres d'Opérateurs accueillera Chamseddine pour un exposé sur le même sujet ce jeudi 26 à 15h à l'UPMC au batiment Sophie Germain (de l'université Paris Diderot).

Appel (spectral non commutatif) du 1819 juin
Les théoriciens qui, depuis de nombreuses années, sont à la tête de la physique des hautes énergies, ont formé un magistère.
Ce magistère, alléguant un problème de naturalité de la masse du boson de Higgs, a pour ainsi dire renoncé à diffuser et enseigner autre chose que la supersymétrie comme solution viable au problème.
Certes, le programme d’unification géométrique spectral basé sur un modèle d’espacetemps presque commutatif a failli dans sa prévision de masse du boson scalaire responsable de la brisure de symétrie électrofaible.
Infiniment plus que la renormalisabilité, ce sont les arguments de naturalité, les symétries, les dualités qui ont ébloui les théoriciens. Ce sont les arguments de naturalité, les symétries, les dualités qui ont fait miroiter aux expérimentateurs l'existence de nouvelles particules à l'échelle du TeV.
Mais le dernier mot est-il dit? L'espérance d’un autre type de physique au-delà du modèle standard doit-elle disparaître ? La défaite est-elle définitive? Non!
Ecoutez Ali Chamseddine et Alain Connes, eux qui nous parlent en connaissance de cause et nous disent que rien n'est perdu pour un modèle d’unification géométrique spectral non commutatif. Les mêmes moyens qui ont invalidé le précédent modèle peuvent faire venir la victoire du prochain.
Car le modèle d’unification spectral basé sur la géométrie non commutative n'est pas ad-hoc! Il repose sur les principes de la physique quantique et de la relativité générale. Il peut faire bloc avec le modèle renormalisable de Yang-Mills-Higgs et la théorie d’Einstein qui tiennent la mer et continuent la lutte.
La recherche n'est pas limitée au territoire actuellement désert de la physique des grands collisionneurs. La question de savoir quelle est la physique au delà du modèle standard n'est pas tranchée par la mesure de la masse du Higgs. Cette quête est une quête à toutes les échelles d’énergie. Toutes les pannes et les retards, toutes les erreurs, tous les bruits de fond, n'empêchent pas qu'il y a, quelque part une expérience en cours ou à venir, avec les moyens nécessaires pour découvrir un jour le signal d'une nouvelle physique. Frappés aujourd'hui par l'absence d'un tel signe empirique fiable, nous pourrons le débusquer dans l'avenir grâce à de nouvelles prédictions. Le destin de la physique est là.
Moi, blogueur, ponctuellement auditeur libre à l’IHES (ou au Collège de France), j'invite les scientifiques et les doctorants de tous pays connectés à Internet ou qui se trouvent en île de France ou qui viendraient à s'y trouver, avec leurs bourses de recherche ou de thèse ou sans financement, j'invite les passionnés de sciences et les étudiants de physique et de mathématiques connectés à Internet ou qui se trouvent en île de France ou qui viendraient à s'y trouver, à découvrir le modèle d’unification géométrique spectral non commutatif et à suivre le dernier cours d’Ali Chamseddine à l’IHES.
Quoiqu'il arrive, la flamme de la résistance ne doit pas s'éteindre et ne s'éteindra pas ;-)
Demain, comme aujourd'hui, je serai à mon poste de blogueur pour essayer d'en apprendre davantage sur le programme d’unification géométrique spectral et diffuser l'information.
à la manière de qui vous savez...

mercredi 4 juin 2014

Un après-midi pluvieux avec Ali Chamseddine, Thibault Damour et Pierre Cartier

Le soleil d'une unification géométrique spectrale des interactions fondamentales (se lèvera-t-il demain?) 
Ce mercredi 4 juin 2014, alors que la pluie arrosait généreusement toute l'île de France, avait lieu à l'IHES le premier d'une série de quatre cours d'Ali Chamseddine, intitulée Spectral Geometric Unification. La séance ayant été enregistrée, la vidéo sera peut-être mise en ligne comme c’est le cas de certains cours et séminaires récents de l’IHES (à moins que les ronflements sporadiques d'une personne qui avait oublié l’une des 12 règles de l’art de dormir pendant les séminaires soient jugés trop irrévérencieux ;-). Parmi les auditeurs plus attentifs, se trouvait Thibault Damour, posant des questions pour éclaircir la signification de certaines notes au tableau (souvent pas assez visibles), connaître le statut logique de certaines égalités algébriques (axiomes ou résultats calculatoires), préciser la nature mathématique ou physique de certains objets clés du formalisme mis en jeux (espace-temps euclidien compact ou non, opérateur de Dirac à spectre continu et/ou discret, espace de Hilbert à une ou plusieurs particules?). Pierre Cartier était là aussi, actif en mathématicien bienveillant secondant le physicien sur quelques points.
Le professeur Chamseddine s'est d'abord efforcé dans ce premier cours de présenter rapidement les données de base de la géométrie spectrale non commutative conçue par Alain Connes (depuis bientôt 20 ans) à savoir celle de triplet spectral avec une structure réelle. Rappelons pour l'occasion comment ce dernier défendait son paradigme géométrique dans l'article originel:
Thanks to the recent experimental confirmations of general relativity from the data given by binary pulsars [4] there is little doubt that Riemannian geometry provides the right framework to understand the large scale structure of space-time.
The situation is quite different if one wants to consider the short scale structure of space-time... In particular there is no good reason to presume that the texture of space-time will still be the 4-dimensional continuum at such scales.
In this paper we shall propose a new paradigm of geometric space which allows us to incorporate completely different small scale structures. It will be clear from the start that our framework is general enough. It will of course include ordinary Riemannian spaces but it will treat the discrete spaces on the same footing as the continuum, thus allowing for a mixture of the two. It also will allow for the possibility of noncommuting coordinates [6]. Finally it is quite different from the geometry arising in string theory but is not incompatible with the latter since supersymmetric conformal field theory gives a geometric structure in our sense whose low energy part can be defined in our framework and compared to the target space geometry [7].
It will require the most work to show that our new paradigm still deserves the name of geometry. We shall need for that purpose to adapt the tools of the differential and integral calculus to our new framework. This will be done by building a long dictionary which relates the usual calculus (done with local differentiation of functions) with the new calculus which will be done with operators in Hilbert space and spectral analysis, commutators... The first two lines of the dictionary give the usual interpretation of variable quantities in quantum mechanics as operators in Hilbert space. For this reason and many others (which include integrality results) the new calculus can be called the quantized calculus’ but the reader who has seen the word “quantized” overused so many times may as well drop it and use “spectral calculus” instead.
 A. Connes, Noncommutative geometry and reality, 4 avril 1995 
...à suivre // rédaction encore en cours

Mise à jour du 25 juillet 2014
Vidéos des quatre cours à l'IHES accessibles ici. 
Transparents des deux dernières conférences d'Ali Chamseddine à Frontiers of Fundamental Physics Marseille juillet 2014:
conférence plénièreconférence spécialisée