vendredi 14 novembre 2014

Grothendieck and the theory of everything (/grand unified theory) / Grothendieck et la théorie unitaire

La disparition hier du légendaire mathématicien Alexandre Grothendieck m'a fait découvrir un peu par hasard l'extrait que voici de ses secrètement célèbres mémoires d'outre-tombe :

« On a appelé “théorie unitaire” une théorie hypothétique qui arriverait à “unifier” et à concilier [une] multitude de théories partielles […]. J'ai le sentiment que la réflexion fondamentale qui attend d'être entreprise aura à se placer sur deux niveaux différents :
1). Une réflexion de nature “philosophique”, sur la notion même de “modèle mathématique” pour une portion de la réalité. Depuis les succès de la théorie newtonienne, c'est devenu un axiome tacite du physicien qu'il existe un modèle mathématique (voire même, un modèle unique, ou “le” modèle) pour exprimer la réalité physique de façon parfaite, sans “décollement” ni bavure. Ce consensus, qui fait loi depuis plus de deux siècles, est comme une sorte de vestige fossile de la vivante vision d'un Pythagore que “Tout est nombre”. Peut-être est-ce là le nouveau “cercle invisible”, qui a remplacé les anciens cercles métaphysiques pour limiter l'Univers du physicien (alors que la race des “philosophes de la nature” semble définitivement éteinte, supplantée haut la main par celle des ordinateurs…). […] Ce serait le moment ou jamais de soumettre cet axiome à une critique serrée, et peut-être même, de “démontrer”, au-delà de tout doute possible, qu'il n'est pas fondé : qu'il n'existe pas de modèle mathématique rigoureux unique, rendant compte de l'ensemble des phénomènes dits “physiques” répertoriés jusqu'à présent. 
[…]
2). C'est après une telle réflexion seulement, il me semble, que la question “technique” de dégager un modèle explicite, plus satisfaisant que ses devanciers, prend tout son sens. Ce serait le moment alors, peut-être, de se dégager d'un deuxième axiome tacite du physicien, remontant à l'antiquité, lui, et profondément ancré dans notre mode de perception même de l'espace : c'est celui de la nature continue de l'espace et du temps (ou de l'espace-temps), du “lieu” donc où se déroulent les “phénomènes physiques”. Il doit y avoir déjà quinze ou vingt ans, en feuilletant le modeste volume constituant l'oeuvre complète de Riemann, j'avais été frappé par une remarque de lui “en passant”. Il y fait observer qu'il se pourrait bien que la structure ultime de l'espace soit “discrète”, et que les représentations “continues” que nous nous en faisons constituent peut-être une simplification (excessive peut-être, à la longue…) d'une réalité plus complexe ; que pour l'esprit humain, “le continu” était plus aisé à saisir que “le discontinu”, et qu'il nous sert, par suite, comme une “approximation” pour appréhender le discontinu […]. [A]u sens strictement logique, c'est plutôt le discontinu qui, traditionnellement, a servi comme mode d'approche technique vers le continu. 
[…] Toujours est-il que de trouver un modèle “satisfaisant” (ou, au besoin, un ensemble de tels modèles, se “raccordant” de façon aussi satisfaisante que possible…), que celui-ci soit “continu”, “discret” ou de nature “mixte” — un tel travail mettra en jeu sûrement une grande imagination conceptuelle, et un art consommé pour appréhender et mettre à jour des structures mathématiques de type nouveau. Ce genre d'imagination ou de “flair” me semble chose rare, non seulement parmi les physiciens (où Einstein et Schrödinger semblent avoir été parmi les rares exceptions), mais même parmi les mathématiciens (et là je parle en pleine connaissance de cause). Pour résumer, je prévois que le renouvellement attendu (s'il doit encore venir…) viendra plutôt d'un mathématicien dans l'âme ; bien informé des grands problèmes de la physique, que d'un physicien. Mais surtout, il y faudra un homme ayant “l'ouverture philosophique” pour saisir le nœud du problème. Celui-ci n'est nullement de nature technique, mais bien un problème fondamental de “philosophie de la nature” »
Chapitre 2. Promenade à travers une œuvre ou l’Enfant et la Mère. § 2.20. Coup d’œil chez les voisins d’en face, p. 80 (transcription d’Yves Pocchiola), 
Alexandre GROTHENDIECK 1928-2014

English translation* :
The hypothetical theory that would ‘unify’ and reconcile a multitude of partial theories has been called a ‘unitary theory’. I have the feeling that the kind of fundamental thinking that needs to be undertaken will have to take two different forms: 
1.A kind of ‘philosophical’ reflection on the very ‘mathematical model’ for a part of reality. Ever since the success of Newtonian theory, it has been a tacit axiom of the physicist’s that there must be a mathematical model (perhaps even a unique model or ‘the’ model) to express reality in a perfect manner, without any ‘detachment’ or blurring. This consensus, which has lasted for two centuries, is a kind of fossil-like remains of Pythagoras’ living vision of a world in which ‘Everything is a number’. Perhaps this is the new ‘invisible circle’, which has replaced the ancient metaphysical circles, and limits the World of the physicist (now that the race of ‘natural philosophers’ seems to be definitely dead, overtaken high-handedly by the computer…) […] Now is the best possible moment to submit this axiom to a concentrated critique, and perhaps even to ‘demonstrate’, beyond all possible doubt, that it has no foundation: that there is no unique and rigorous mathematical model under which all phenomena so far identified as ‘physical’ can be made to fall. 
[…] 
2.It is only after such a process of reflection that, it seems to me, the ‘technical’ question of finding an explicit model, more satisfying than its predecessors, begins to have real meaning. Perhaps then would be the moment to enfranchise ourselves from a second tacit axiom of the physicist, this time going right back to the ancients, and deeply anchored in our perception of space: that is the continuity of time and space (or the space-time continuum), the ‘place’, then, where ‘physical phenomena’ occur. 
It must be fifteen or twenty years ago when, flicking through the modest volume that holds Riemann’s complete works, I was struck by a passing comment that he makes. He remarks that it could well be that the ultimate structure of space is ‘discrete’, and that the ‘continuous’representations that we observe are perhaps simplified (excessively so, perhaps, in the long run…) versions of a more complex reality; that for the human mind, ‘continuity’ is easier to grasp than ‘discontinuity’ […]. In a strictly logical sense, it has traditionally been the discontinuous that has served as a technical method for approaching the continuous. It nevertheless remains true that the finding of a ‘satisfactory’ model (or, if need be, several such models, which would ‘connect’ in as satisfactory a manner as possible…) – whether this model was ‘continuous’, ‘discrete’ or of a ‘mixed’ nature – would require a great conceptual imagination, and a consummate art for apprehending and updating mathematical structures of a new type. This kind of imagination or ‘flair’ is rare indeed, not only amongst physicists (Einstein and Schrödinger seem to be notable exceptions), but even amongst mathematicians (and there I am speaking in full knowledge of the facts). 
To sum up, I predict that the long-awaited renewal (if it is still coming…) will come from a born mathematician well-informed about the big questions of physics rather than from a physicist. But above all, we will need a man with the kind of ‘philosophical openness’ necessary to take hold of the heart of the problem. This problem is by no means a technical one, but is rather a fundamental question of ‘natural philosophy’. 
Harvests and Seeds
Chapter 2. A Walk through my work or The Child and the Mother. 2.20 A glance over the road, p.80 (transcription by Yves Pocchiola)
Alexandre GROTHENDIECK 1928-2014

jeudi 13 novembre 2014

One or two things I know about a (spectral noncommutative) dynamical symmetry breaking of a Pati-Salam symmetry down to the Standard Model one

A nice recap
In a recent article submitted on 5 Nov 2014 on arxiv and entitled : Twisted spectral triple for the Standard Model and spontaneous breaking of the Grand SymmetryAgostino Devastato and Pierre Martinetti have written a nice (clear,  explicit) summary of the state of the art of the spectral non-commutative geometric based physics beyond the Standard Model in general and report also on important progress in their own work :
Noncommutative geometry [NCG] provides a description of the standard model of elementary particles [SM] in which the mass of the Higgs −at unification scale Λ−is a function of the other parameters of the theory, especially the Yukawa coupling of fermions [7]. Assuming there is no new physics between the electroweak and the unification scales (the “big desert hypothesis”), the flow of this mass under the renormalization group yields a prediction for the Higgs observable mass mH. It is well known that in the absence of new physics the three constants of interaction fail to meet at a single unification scale, but form a triangle which lays between 1013 and 1017 GeV. The situation can be improved by taking into account higher order term in the NCG action [19], or gravitational effects [18]. Nevertheless, the prediction of mH is not much sensible on the choice of the unification scale... 
The recent discovery of the Higgs boson with a mass mH≃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 [3] 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-stabililty may be problematic [23, 24]. Still, the fact that mH 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. [1122]) 
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=kν/kt , 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 between 1012 and 1017 Gev, there exists a realistic value r≃1 which brings back the computed value of mH to 126 Gev [6]. 
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 [29], but they require the adjunction of new fermions (see [30] for a recent state of the art). In [6], a scalar σ correctly coupled to the Higgs is obtained without touching the fermionic content of the model, simply by turning the Majorana mass kR of the neutrino into a field kR → kR σ... Usually the bosonic fields in NCG are generated by inner fluctuations of the geometry. However this does not work for the field σ because of the first-order condition [[D,a],JbJ -1] = 0 ∀a,b ∈A ... where A and D are the algebra and the Dirac operator of the spectral triple of the standard model, and J the real structure. 
In [9, 10] it was shown how to obtain σ by a inner fluctuation that does not satisfy the first-order condition, but in such a way that the latter is retrieved dynamically, as a minimum of the spectral action. The field σ is then interpreted as an excitation around this minimum


A new twist(ted spectral triple) 


... in [20] another way had been investigated to generate σ in agreement with the first-order condition, taking advantage of the fermion doubling in the Hilbert space H of the spectral triple of the SM [26, 27, 28].  
More specifically, under natural assumptions on the representation of the algebra and an ad-hoc symplectic hypothesis, it is shown in [5] that the algebra in the spectral triple of the SM should be a sub-algebra of C(M)⊗AF, where M is a Riemannian compact spin manifold (usually of dimension 4) while 
 AF=Ma(H)⊕M2a(C)    a∈N  ... 
The algebra of the standard model  
Asm:=C⊕H⊕M3(C)  ... 
is obtained from AF for a=2, by the grading and the first-order conditions. Starting instead with the “grand algebra” (a=4) 
 AG:= M4(H)⊕M8(C)  ... 
one generates the field σ by a inner fluctuation which respects the first-order condition imposed by the part Dv of the Dirac operator that contains the Majorana mass kR [20].  The breaking to Asm is then obtained by the first-order condition imposed by the free Dirac operator \(\begin{equation} D\!\!\!/ \end{equation}\):= \(\begin{equation} \partial\!\!\!/ \end{equation}\)⊗ I.  
Unfortunately, before this breaking not only is the first-order condition not satisfied, but the commutator 
[\(\begin{equation} D\!\!\!/ \end{equation}\),A] A∈C(M)⊗AG  ... 
is never bounded. This is problematic both for physics, because the connection 1-form containing the gauge bosons is unbounded; and from a mathematical point of view, because the construction of a Fredholm module over A and Hochschild character cocycle depends on the boundedness of the [later] commutator...  

In this paper, we solve this problem by using instead a twisted spectral triple (A,H,D,ρ) [14]. Rather than requiring the boundedness of the commutator, one asks that there exists a automorphism ρ of A such that the twisted commutator 
[D,a]ρ  := Da−ρ(a)D  ... 
is bounded for any a ∈A. Accordingly, we introduce... a twisted first-order condition 
[[D,a]ρ,JbJ-1]ρ :=[D,a]ρJbJ-1 −Jρ(b)J-1[D,a]ρ=0 ∀a,b ∈A  ... 
We then show that a for a suitable choice of a subalgebra B of C(M)⊗AG, a twisted fluctuation of \(\begin{equation} D\!\!\!/ \end{equation}\)+Dv that satisfies [the twisted first-order condition] generates a field σ - slightly different from the one of [6] - together with an additional vector field Xµ. 

Furthermore, the breaking to the standard model is now spontaneous, as conjectured by Lizzi in [20]. Namely the reduction of the grand algebra AG to Asm is obtained dynamically, as a minimum of the spectral action. The scalar and the vector fields then play a role similar as the one of the Higgs in the electroweak symmetry breaking.  

Mathematically, twists make sense as explained in [14], for the Chern character of finitely summable spectral triples extends to the twisted case, and lands in ordinary (untwisted) cyclic cohomology. Twisted spectral triples have been introduced to deal with type III examples, such as those arising from transverse geometry of codimension one foliation. It is quite surprising that the same tool allows a rigorous implementation in NCG of the idea of a “bigger symmetry beyond the SM”. 
The main results of the paper are summarized in the following theorem. 
Theorem 1.1. Let H be the Hilbert space of the standard model described in §2.1. There exists a sub-algebra B of the grand algebra AG containing Asm together with an automorphism ρ of C(M)⊗B such that 
  • i) (C(M)⊗B,H, \(\begin{equation} D\!\!\!/ \end{equation}\)+Dv; ρ) is a twisted spectral triple satisfying the twisted 1st-order condition (1.8); 
  • ii) a twisted fluctuation of \(\begin{equation} D\!\!\!/ \end{equation}\)+Dv by B generates an extra scalar field σ, together with an additional vector field Xµ; 
  • iii) the spectral triple of the standard model is obtained as the minimum of the spectral action induced by a twisted fluctuation of / D. The same result is obtained from a twisted fluctuation of \(\begin{equation} D\!\!\!/ \end{equation}\)+Dv, neglecting the interaction term between σ and Xµ. ...
Starting with the “not so grand algebra” B, one builds a twisted spectral triple whose fluctuations generate both an extra scalar field σ and an additional vector field Xµ. This is a Pati-Salam like model - the unitary of B yields both an SU(2)R and an SU(2)L - but in a pre-geometric phase since the Lorentz symmetry (in our case: the Euclidean SO(n) symmetry) is not explicit. The spectral action spontaneously breaks this model to the standard model, with both a scalar and a vector field playing a role similar as the one of Higgs field. We thus have a dynamical model of emergent geometry.

The last word

Finally, let us mention a very recent work of Chamseddine, Connes and Mukhanov [8] where the algebra AF for a=2 is obtained without the ad-hoc symplectic hypothesis, but from an higher degree Heisenberg relation for the space-time coordinates. It would be interesting to understand whether the case a=4 enters this framework.

mercredi 5 novembre 2014

Déplier l'espace-temps euclidien depuis l'échelle électrofaible pour mieux le quantifier / Unfold the euclidean spacetime from the electroweak scale the better to quantize it

Un petit pas vers l'union de la gravitation et de la physique quantique, un grand pas pour le modèle spectral non commutatif de l'espacetemps / one small step for unification of gravitation and quantum physics,  one giant leap for the spectral noncommutative model of spacetime
In this paper we have uncovered a higher analogue of the Heisenberg commutation relation whose irreducible representations provide a tentative picture for quanta of geometry. We have shown that 4-dimensional Spin geometries with quantized volume give such irreducible representations of [a] two-sided [Heisenberg-like commutation] relation involving the Dirac operator and the Feynman slash of scalar fields and the two possibilities for the Clifford algebras which provide the gamma matrices with which the scalar fields are contracted. These instantonic fields provide maps Y,Y' from the four-dimensional manifold Mto S4. The intuitive picture using the two maps from Mto Sis that the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. The volume of space-time is quantized in terms of the sum of the two winding numbers of the two maps. More suggestively 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. Moreover, amazingly, in dimension 4 the algebras of Clifford valued functions which appear naturally from the Feynman slash of scalar fields coincide exactly with the algebras that were singled out in our algebraic understanding of the standard model using noncommutative geometry thus yielding the natural guess that the spectral action will give the unification of gravity with the Standard Model (more precisely of its asymptotically free extension as a Pati-Salam model as explained in [5]).
(traduction du blogueur)
Dans cet article, nous avons mis au jour un analogue généralisé de la relation de commutation de Heisenberg dont les représentations irréductibles pourraient s'interpréter comme les quanta de la géométrie. Nous avons montré que les géométries spinorielles à 4 dimensions avec un volume quantifié donnent de telles représentations irréductibles pour une relation de commutation de Heisenberg impliquant l'opérateur de Dirac et l’opérateur dit « slash » (ou coupure) de Feynman qui associe aux champs scalaires les matrices gamma représentant deux algèbres de Clifford avec lesquelles les champs sont contractés. Ces champs instantoniques permettent de construire deux applications Y, Y ' d’une variété à 4 dimensions M4 vers la 4-sphère S4. L'image intuitive utilisant les deux applications de M4 à S4 est que la variété à 4 dimensions est construite à partir d'un grand nombre de répliques de deux types de 4-sphères de volume unité dans l’échelle de Planck. Le volume de l'espace-temps est quantifié comme une somme de deux termes correspondants aux indices des deux applications. De façon plus suggestive l'histoire de l’espace-temps euclidien se déploie à l'échelle macroscopique à partir du produit de deux 4-sphères de volume élémentaire égal au volume de Planck comme un papillon se déploie hors de sa chrysalide. En outre, et de façon étonnante, les fonctions à valeur dans les algèbres de  Clifford qui apparaissent naturellement dans l’opérateur de coupure de Feynman agissant sur les scalaires, coïncident exactement avec les algèbres mises en évidence dans notre compréhension algébrique du modèle standard à l'aide de la géométrie non commutative ; cela nous conduit à supposer naturellement que l’action spectrale devrait permettre l'unification de la gravité avec le modèle standard (plus précisément avec son extension asymptotiquement libre à savoir le modèle de Pati-Salam explicité dans [5]).
(Submitted on 4 Nov 2014)

Souhaitons que ces trois navigateurs spectraux qui viennent tout juste de commencer à déplier l'espace-temps euclidien non-commutatif depuis l'échelle électrofaible du boson de Higgs puissent aussi un jour aider les explorateurs du réél qui vont bientôt replonger dans le zeptoespace avec une carte lorentzienne commutative.

Ajout du 13/11/2014 :
Pour en savoir plus voir lire le billet pulié par Alain Connes sur le blog Noncommutative geometry.