samedi 21 janvier 2017

S'il vous plaît M. 't Hooft... chante nous un cℏoeur quantique d'espace-temps

(Dreaming of) A portrait of a black hole horizon (as a quantum cℏoir of spacetime)...
... reading 't Hooft last preprint on his proposed new constraints on the topology and the boundary conditions of general coordinate transformations to solve both the firewall and information problems in black hole physics :
In this paper, we shall primarily make use of a partial answer that we claim to have arrived at recently [10]: the necessity of revising the boundary conditions for Nature’s degrees of freedom at the horizon of a black hole. Since our analysis started out with our desire for consistent descriptions of stationary (or approximately stationary) black holes, it was not immediately clear how the revised boundary conditions should have been enforced during the formation of a black hole, but, in a somewhat formal fashion, one may well argue that, during black hole formation, the horizon starts out stretching over an infinitesimally tiny region; it opens up at a single point [Upon close inspection, one might conclude that the horizon first forms on a fractal subspace of spacetime, but since the scale at which this fractal extends may end up to be small even in Planck units, we ignore this complication in this paper.] in space and time. At that single point, it now appears to be necessary to revise the structure of this infinitesimal horizon to obey the new boundary condition, but since all this should happen at Planckian dimensions, the revision needed in our laws of Nature here can easily be argued to have escaped our notice up to today. 

After the horizon opens up, a black hole can grow quite big; the black hole horizon area grows rapidly towards macroscopic sizes during collapse, and as our modified boundary condition keeps track, it turns space and time into a non-trivial topological manifold. As our new boundary condition must end up as being indestructible, its implications will be sizeable. We emphasise that, nevertheless, our modified boundary condition will not affect the visible properties of a black hole in the classical limit. Also, we shall ensure that the modified boundary condition is of a kind that is not directly observable for a local observer. 

The boundary condition that we shall arrive at is characterised as an antipodal identification. In short, what it means is that the region of space-time inside the horizon is removed completely, as if by surgery, after which the edges are glued together by identifying the antipodes. This is continued throughout the lifetime of the black hole [Do keep in mind that, strictly speaking, the horizon is entirely timeless.] It is important, subsequently to insist that, locally, space and time remain smooth across the seams, while particles, including the information they carry, can cross. The seams must be locally invisible—only global observers notice this boundary condition. We argue that the antipodal mapping is the only way to attach the edges together such that strict geometrical conditions are obeyed. 

It boils down to a single “new physics” ingredient in black hole physics as soon as quantum effects are being considered
  • from now on, when quantised particles and fields are considered, only those general coordinate transformations are permitted that map space and time continuously, and they must be one-to-one,
a condition not obeyed by the standard, classical Schwarzschild metric: every space-time point in the physically observable part of the universe is mapped onto two points in the Kruskal-Szekeres coordinates. 

As will be demonstrated (subsection 3.1), mapping the Schwarzschild metric onto the space-time metric of a local observer forces us to glue together regions in such a way that time-inversion takes place. Inverting the time direction is associated with an interchange of creation operators and annihilation operators. This implies that a region almost devoid of particles for one observer, is mapped onto a region almost filled with particles for the other observer. At first sight, this seems to be an unfamiliar and unwanted complication, but it is unavoidable...

At first sight, it may seem that our way of handling space and time near a black hole, will make a decent quantum field theoretic description of the elementary particles in there, hopelessly inadequate but the contrary is true. [... in earlier reports, the author had expressed his opinion that the causal order of events has to be respected by the coordinate transformations used; we now found compelling reasons to abandon that demand]. The apparently drastic rearrangement of the space-time continuum is exactly what is needed to arrive at pure quantum states for the black hole, and to obtain a unitary scattering matrix, so as to eradicate both the black hole information problem and the firewall problem, while accurately respecting the laws of general relativity.

There is a number of points that we should keep in mind. One is that wild guesses concerning possible answers, such as ‘novel uncertainty relations’, will be almost fruitless, as history shows*. The best thing to do is to split our problems into small pieces, and try to address each of these small fragments of questions in turn. Every now and then, such fragmented questions will lead to surprises. It helps enormously if we can convince ourselves of the correctness of our partial answers, and it is these that we should be able to use as new starting points for our next steps.  
On the other hand, we do not claim that all mysteries are resolved now. A systematic procedure must be found for a one-to-one mapping of the states generated by the spherical waves of momentum distributions and positions, onto states of the Fock space of a quantum field theory (some grand unified version of the standard model, relevant in the vicinity of the Planck scale, simply referred to as “standard model” elsewhere in this paper). It is here that the machinery of string theory might be of much help. 
Another point where our theory becomes vague is where the Planckian dimensions are reached. Usually it is assumed that string theory will provide all the answers, but string theory did not tell us about gravitational back reaction or antipodal identification, so we respectfully conclude that string theory is not fool-proof. 

* Personal comment or wishful(l) thinking : more recent history might prove this 't Hooft statement is not exactly right...

Credit: X-ray: NASA/CXC/Penn State/B.Luo et al.
Press Image and Caption
This is the deepest X-ray image ever obtained... collected in eleven and a half weeks, of Chandra spatial telescope observing time. The image comes from what is known as the Chandra Deep Field-South. The central region of the image contains the highest concentration of supermassive black holes ever seen, equivalent to about 5,000 objects that would fit into the area on the sky covered by the full Moon and about a billion over the entire sky...

About 70% of the objects in the ... image are supermassive black holes, which may range in mass from about 100,000 to ten billion times the mass of the Sun. Gas falling towards these black holes becomes much hotter as it approaches the event horizon, or point of no return, producing bright X-ray emission.
For Release: January 5, 2017

This post is dedicated to a hiker friend, Emmanuel ;-)

mardi 17 janvier 2017

Physics has one quantum (or rather two quanta?) to pick with mathematics / Les mathématiques ont maille à partir avec la physique quantique (ou plutôt deux quanta à partager?)

The videos of the second Alain Connes' lecture at Collège de France for 2017 (entitled "La géométrie et le quantique") that took place last week are available now (same site as last ones mentioned in the former post).
Some slides to illustrate the title of this post:

As a teaser to the third lecture (on next Thursday 19 January) let's note that in the last six minutes of the second lecture (video "12 janvier 15h45-17h") Connes provides his general philosophy to address "quantum gravity" problems with functional integral calculation on 4D Euclidean geometries with 3D+1 Minkowski space-time "boundaries" (fitting with Hawking and Gibbons results)...

jeudi 12 janvier 2017

A quantum leap beyond classical Riemannian geometry brought up in the realm of experimental physics

If I understand correctly, it seems for the first time Alain Connes accepted his 2017 (possibly last?) lectures at College de France be filmed. I remember a few years ago he explained to his audience he was reluctant to accept a broadcasting of these particular lectures because he was afraid it could impart his freedom to think aloud. 

It is a great opportunity to all ingenious undergraduate seniors at a university anywhere in the world (see former post) and the luckiest students in Paris area to engage in a fascinating but challenging* conceptual trip (*these Connes lectures are in French ;-) 

The first "episode" (two videos) is available here
The second one is today (I don't know but I wish it will broadcasted too!)

Here are a few slides from the first videos as an invitation to a journey with a mathematician who unfolded spacetime from X to Higgs.
Voici quelques captures d'écran du premier cours en guise d'invitation au voyage avec un mathématicien qui a déplié l'espacetemps des rayons X au boson de Higgs.

From electroweak energy scale to the top and down back (2017 Challenge for youngsters and others)

 Reader information: if you are looking for exotica you are on the quite wrong blog, keep up with "M. Trump goes to Washington, D.C." and his twitter feed instead ;-)

A reasonably simple informative 2016 paper (not cited yet*) on high energy physics...
...for the education of my best gifted high school student *with a node to a recent post of another gifted but former high school student, Lubos Motl, who complained about the disappointing composition of top-cited 2016 HEP papers.

An extra U(1)' gauge symmetry is a common presence in many attempts to go beyond the Standard Model (SM). It represents, from a low-energy perspective, the simplest extension that can be attached to the SM gauge group, at the same time, from the opposite high-energy point of view, an extra abelian factor is almost an unavoidable leftover from the breaking of many GUT scenarios [1]. 
If we adopt a (grand) unification paradigm, it is therefore feasible that a regime ruled by the gauge structure SU(3)CSU(2)LU(1)YU(1)' could populate the sequence of effective descriptions scaling from the GUT energy, before breaking into the SM one. The last step may be triggered by the nontrivial vacuum expectation of a scalar field χ, that is, consequently, required to be SM-singlet. If such U(1)' breaking is realised at the TeV scale, then there are realistic prospects of an interesting interplay with the current LHC probe, the precise traits of such phenomenological characterisation being dictated by the extended matter content ([2–6]). Beyond the scalar sector, where a SM-singlet accounts for the extra U(1)' breaking, and the neutral vector Z' , to accomplish gauge invariance, one extra fermion per generation is needed to cancel gauge and gravitational anomalies in a minimal way. This scenario has been the subject of a recent up-to-date investigation [7] where we exploited the bounds and the discovery potential of current and forthcoming collider searches. The more promising regions of the allowed parameter space have supplied the boundary conditions for a Next-to-Leading-Order (NLO) vacuum stability analysis, that we performed extrapolating the model to higher energies with two-loop β functions. As a result, the explored regions have been labeled with the maximal energy scale up to which they would provide a coherent (stable and perturbative) extrapolation of the model. 
The role of the Renormalisation Group (RG) extrapolation does not exhaust its insight power with the stability analysis. As we will illustrate, for the particular case of our minimal SM⊗U(1)' regime, the RG may draw clear indications also about the high-energy regime that is expected to take place. In a combined effort, all the phenomenological and formal aspects of this analysis will contribute to unveil a consistent link between the low-energy model characterisation, and a stable, perturbative, ultraviolet (UV) completion... 
The class of models encompassed in [7] adds, to the SM field content, a massive neutral gauge boson plus a scalar and three extra fermions, all transforming trivially under the SM gauge group. This extended spectrum is naturally introduced to minimally account for gauge invariance, anomaly freedom and the requirement of a massive Z' . We notice, as a valuable consequence of the previous setting, that the presence of a new abelian factor has forced the introduction of states that complete the 16-dimensional representation of SO(10) which is a further motivation to explore the possible UV fate of such model. Anomaly cancellation also rules the possible U(1)’ charges, leaving to the ratio of just two parameters the definition of the allowed charge assignments. In a low-energy investigation the common choice is to highlight the Hypercharge operator Y, so that the generator of the extra U(1)' is constrained to the form Y' = (B − L) + (g/g'1)Y, where B and L are, respectively, the Baryon and the Lepton number of the fields. The overall gauge strength g'1 and the mixing g rule, therefore, the content of the B − L and the Hypercharge operators in Y'. The renormalisable interactions that arise from the extended field content can promptly realise a Type I seesaw mechanism to account for neutrino masses.
... the past investigations of the Z' have been strongly affected by bounds coming from EW Precision Tests (EWPTs) from LEP2. The first Run of LHC at 8 TeV and L = 20 fb−1 has generated even more stringent bounds at the TeV scale. These can be extracted using a signal-to-background analysis for the Drell-Yan channel... Also the extended scalar sector creates numerous chances to reveal and characterise the class of models under study. We have limited the related new parameter space, that we parameterised with the new scalar mass mH2 and the mixing angle α, considering the bounds from the direct detection probes, and comparing the signal produced with the one measured of the discovered Higgs at 125.09 GeV...
The presence of multiple abelian factors is a peculiar trait of this class of models. The induced kinetic mixing, absorbed in the contribution of the coupling g to the covariant derivative 
Dµ = ∂µ + ig1YBµ + i(gY + g'1 YB-L)B'µ + . . . , (3) 
may shed light, supported by a precise RG inspection, on the UV embedding that precedes the U(1)’ regime [10–12].
The key for this analysis is in the matching of the low and the high-energy generators basis, used to describe the abelian sector. In our phenomenological survey, adherence with the SM regime suggested the use of the Hypercharge for one of the two U(1). In turn, within the constraints of anomaly cancellation, we chose B-L for the other. The mixing would then provide the component of the Y 0 generator in the Hypercharge direction. From a high-energy perspective is more appropriate to work with the basis that is naturally provided by the embedding of the abelian factors in the unifying group. For example, a Left-Right (LR) symmetric regime SU(2)RU(1)B-L, which includes U(1)RU(1)B-L, would select the corresponding YR and YB-L set of generators. Close to the energy scale of the LeftRight symmetry breaking, the mixing between the YR and YB-L is zero, being protected by the overall non-abelian gauge symmetry of SU(2)R. It is possible, with the appropriate normalisation, to match our SM-oriented parameters (g1, g'1g) with the ones corresponding to the (candidate) high-energy basis (gR, gB-LgR/B-L). Therefore, following the RG evolution of  gR/B-L in terms of g1, g'1 and g, we can recognise, by its zeroing at a given energy, the restoration of a Left-Right symmetry.
This analysis can be promptly extended to include thresholds of SO(10) that represent a realistic UV embedding. These involve, in addition to the Left-Right case discussed, a direct breaking of a Pati-Salam (PS) group into our model, and the flipped SU(5) case (fig. 2). Choosing as boundary conditions, for the RG extrapolation, benchmarks points inspired by our phenomenological analysis, the parameter space offers regions that, in case of discovery, would clearly reveal the presence of one of the previous embeddings (fig. 3).

Figure 2. The SO(10) breaking chart illustrating the chains investigated in this work. The cases for the Flipped SU(5) and Pati-Salam required also additional unification conditions involving the non-abelian gauge sector

Figure 3. (a) When we consider the stability and perturbativity analysis, the colours refer to the different regions defined by the maximum energy up to which the model is stable and perturbative. The same energy/colour relation is also used for the unification study, referring to regions that fulfil the given unification requirement. (b) Regions with Flipped SU(5) restoration. (c) Regions with LR and PS restoration.

... The minimal character of this U(1)’ extension of the SM makes particularly efficient the use of vacuum stability and perturbativity as constraining requirements to shape the viable parameter space [13–15]. The vacuum stability is addressed asking for the extended scalar potential to be bounded from below, λ1 > 0 , λ2 > 0 , 4λ1λ2 − λ 2 3 > 0 . (4) Together with the perturbativity requirement it is also challenges the viability of a given unification scenario. If we accept the minimal content of the model, then a coherent extrapolation asks for the maximum scale of stability and perturbativity to be greater than the one realising a successful embedding. By relying on the analysis presented in [7], we may exploit this further constrains. Our final results (fig. 4) give an illustration of how the interplay of the tools presented may enrich the forthcoming collider profiling of specific regions of the parameter space. Moreover, in a possible post-discovery phase, frictions with the measured scenarios would help in outlining the degrees of freedom necessary to recover stability, when a promising unification of the gauge sector is at hand.
Figure 4. (a) The effect of the stability requirement to the LR and PS restoration. The similar analysis for the Flipped SU(5) case would result trivially in the surviving only of the case with α = 0.3. Explicit matching of the stability and perturbativity map with the unification regions. Case α = 0.1 (b) and α = 0.3 (c).

A final remark
In Lubos Motl's post to which I referred in the beginning one can read:
I think that if there are some ingenious undergraduate seniors at a university anywhere in the world, they have a much harder time to turn into stars than in other periods of the history of physics.
I wish a young student should prove him wrong! In the meantime I propose to the most ingenious graduate ones to challenge their mathematical skills and physical intuition reading the two following fascinating 2016 review articles which seemed to have escaped Lubos radar and most popular science outlets. I encourage them to check the proof or first build a “mental picture” oftheir own understanding of these research works and construct more and more penetrating mental and practical tools to explore previously hidden aspects of our reality.

The firewall transformation for black holes and some of its implications
Gerard 't Hooft  (Submitted on 27 Dec 2016) 
A promising strategy for better understanding space and time at the Planck scale, is outlined and further pursued. It is explained in detail, how black hole unitarity demands the existence of transformations that can remove firewalls. This must then be combined with a continuity condition on the horizon, with antipodal identification as an inevitable consequence. The antipodal identification comes with a CPT inversion. We claim to have arrived at 'new physics', but rather than string theory, our 'new physics' concerns new constraints on the topology and the boundary conditions of general coordinate transformations. The resulting theory is conceptually quite non trivial, and more analysis is needed. A strong entanglement between Hawking particles at opposite sides of the black hole is suspected, but questions remain. A few misconceptions concerning black holes, originating from older investigations, are discussed.

(Submitted on 3 Jun 2016) 
This is a tribute to Abdus Salam's memory whose insight and creative thinking set for me a role model to follow. In this contribution I show that the simple requirement of volume quantization in space-time (with Euclidean signature) uniquely determines the geometry to be that of a noncommutative space whose finite part is based on an algebra that leads to Pati-Salam grand unified models. The Standard Model corresponds to a special case where a mathematical constraint (order one condition) is satisfied. This provides evidence that Salam was a visionary who was generations ahead of his time.

mercredi 11 janvier 2017

How far do we understand the 125 GeV Higgs boson (and its consequences)? A 2017 perspective

From local...
Run 1 accumulated striking evidence that the Higgs field is the cause of the screening of the weak interaction at long distances. Indeed, the observation and measurement of the H → ZZ → 4 channel indicate that the Higgs field develops a vacuum expectation value (vev) that is not invariant under the SU(2)L × U(1)Y gauge symmetry of the SM. Furthermore, this vev seems to be the common source of the Z-boson mass and the coupling between the Higgs boson and the Z boson. However, this evidence only addresses the question of how the symmetry of the weak interaction is broken. It does not address the question of why the symmetry is broken or why the Higgs field acquires an expectation value.  The situation is simply summarized in the following tautology :
Why is electroweak symmetry broken?
 Because the Higgs potential is unstable at the origin. 

Why is the Higgs potential unstable at the origin? 
Because otherwise EW symmetry would not be broken.  

The discovery of a Higgs boson allowed first glimpses into a new sector of the microscopic world. Now comes the time of the detailed exploration of this new Higgs sector. And some key questions about the Higgs boson emerge:  

1. Is it the SM Higgs?
2. Is it an elementary or a composite particle?
3. Is it unique and solitary? Or are there additional states populating the Higgs sector?
4. Is it eternal or only temporarily living in a metastable vacuum?
5. Is its mass natural following the criteria of Dirac, Wilson or ’t Hooft?
6. Is it the first superparticle ever observed?
7. Is it really responsible for the masses of all the elementary particles?
8. Is it mainly produced by top quarks or by new heavy vector-like particles?
9. Is it a portal to a hidden world forming the dark matter component of the Universe?
10. Is it at the origin of the matter-antimatter asymmetry?
11. Has it driven the primordial inflationary expansion of the Universe?

The answers to these questions will have profound implications on our understanding of the fundamental laws of physics... 

... to global point of view
The recent discovery of the Higgs boson [1, 2] and the ongoing measurements of its properties [3] are in good agreement with the hypothesis that this particle is a remnant of the Brout-Englert-Higgs mechanism, i.e. the spontaneous breaking of SU(2)L×U(1)Y → U(1)QED.

While the precise determination of the Higgs and gauge boson masses, as well as the interactions of the Higgs boson with elementary particles, including itself, will continue to improve our understanding of the scalar potential’s local structure in the vicinity of the vacuum, its global structure, which can possibly explain the nature of electroweak symmetry breaking, is very difficult to probe experimentally.

For example, the nature of the Higgs, whether elementary or composite, is still an open question. Even if the Higgs is assumed to be elementary, the shape of its potential remains unknown. It could be of mexican-hat shape as in the Standard Model (SM), or it could be deformed by strong quantum corrections due to virtual effects of additional fields. Were the Higgs boson to be a composite pseudo-Nambu-Goldstone boson of a strongly-coupled sector, one would expect a periodic potential involving trigonometric functions. In all cases, the Higgs mass is fixed by the curvature of the potential at its minimum, and so in the vicinity of the latter the shape of the potential will be similar in all possible models. Nevertheless, deviations are allowed away from the minimum. For example, one could have a barrier at zero temperature between the vacuum and the origin of field-space. Moreover, in composite Higgs models the relation between the Higgs field’s vacuum expectation value (VEV) and the gauge boson masses differs from its SM counterpart, and thus the location of the minimum in field-space may vary.

Discriminating between the different possibilities is of fundamental importance for our understanding of nature and, hence, the embedding of the effective Standard Model in an underlying UV theory. This motivates to consider possible observables which could be sensitive to the Higgs potential beyond its minimum. A possible candidate is the energy scale of baryon-number-violating processes. If baryon number is only violated by the anomaly under the weak interactions, then it follows that processes that violate baryon-number are associated with transitions between vacua classified by their weak topological charge. The minimum energy barrier between these vacua thus sets the expected scale of baryon-violating processes, which is an observable that could potentially be probed by experiments, either at colliders [4–9] or cosmic ray and neutrino detectors [10–15]. Getting accurate predictions for the rates of baryon-number-violating interactions is a difficult problem, due to a possible breakdown of the semiclassical expansion used to compute vacuum transitions. After extensive discussion in the literature (see for example [16–24]) the latest estimates point towards rates that could be probed by future experiments [25, 26]; for recent analyses of measurement prospects at colliders, cosmic ray and neutrino detectors, see for example [27–29].
(Submitted on 16 Nov 2016)

A very personal (thus naive) spectral perspective go beyond the above tautology

Why is electroweak symmetry broken?
 Because spacetime geometry has a fine structure or more crudely a "discrete" dimension at the zeptometer scale that the discovery of the Higgs boson makes it possible to uncover provided one understands it through the spectral noncommutative point of view.

Why is the Higgs potential unstable at the origin? 
This is a consequence of the spectral action principle applied to the proper almost commutative 4D manifold. The latter is a small (but topologically highly nontrivial) extension of our ordinary continuous and commutative geometric model of spacetime while the former is a stronger hypothesis than the usual diffeomorphism invariance of the action of general relativity.

...and propose tentative answers or rather educated guess
1. Is it the SM Higgs? Yes 
2. Is it an elementary or a composite particle? It is elementary in the spectral model of particle physics compatible with current experiments and observations.  
3. Is it unique and solitary? Or are there additional states populating the Higgs sector? There should be more scalars responsible for several gauge symmetry breaking but they may be at partial or grand unification energy scales inacessible to terrestrial particle accelerators. The 125 GeV Higgs boson could be very weakly mixed with (but strongly coupled to) a big brother ("big broson")  responsible for the type I seesaw mechanism that gives very low masses to left-handed neutrinos. 
4. Is it eternal or only temporarily living in a metastable vacuum? The coupling with the above very massive scalar "big broson" should stabilise the vacuum.  

5. Is its mass natural following the criteria of Dirac, Wilson or ’t Hooft? This question could be settled once the proper fine structure of spacetime is established and the quantum dynamics of scalars in this new arena is better understood.  

6. Is it the first superparticle ever observed? There might be no need for that hypothesis to paraphrase a famous Laplace quote

7. Is it really responsible for the masses of all the elementary particles? There could exist a dilaton scalar ruling all the masses so to speak and responsible for a spontaneous symmetry breaking of Weyl invariance

8. Is it mainly produced by top quarks or by new heavy vector-like particles? Noncommutative geometry and the spectral action principle provide a conceptual explanation for the standard model algebra and the number of fundamental fermions by generation so there should be no need for heavy vector-like particles to explain the production rate of Higgs boson at the LHC. 

9. Is it a portal to a hidden world forming the dark matter component of the Universe? In a metaphorical way the answer could be yes. Indeed, understanding how to accommodate the measured Higgs mass in the spectral noncommutative framework has helped to uncover new mathematical structures to build 4D spin manifolds from a higher degree Heisenberg commutation relation. This provides new perspectives on the dark component as not composed of unknown particles or fields but mimicking some kind of new quanta of geometry.
10. Is it at the origin of the matter-antimatter asymmetry? In an indirect way one cold say yes. The phenomenological consequences of the spectral standard model have not been extensively probed but they might be close to the predictions of a minimal nonsupersymmetric SO(10) model which values of the parameters obtained from the low energy observables yield a baryon assymetry in agreement with observations.

11. Has it driven the primordial inflationary expansion of the Universe? This question has not been investigated serioulsy in the most advanced spectral modelisation of spacetime and matter at my knowledge but I think some people responsible for an interesting extension of the standard model not that far from the spectral one work on this...

The above picture could look like a pretty bleak one for the youngest ingenious physicists engaged for instance in ATLAS and CMS collaborations who cope with data from LHC run 1 and 2. Nevertheless I have no doubt they will make their way exploring unchartered territories with different compasses to guide them and they will improve our knowledge of interactions in the zeptouniverse. Moreover I have only drawn very rough lines and it is not impossible to imagine discovering in less orthodox or apparently more contrieved spectral models new right-handed gauge bosons at the LHC ... Subtle is the Lord! Besides  most details are not understandable by myself but I have worked on it here up until now with you, dear reader!

Fortune favors the prepared mind.
La chance ne sourit qu'aux esprits bien préparés.
 Louis Pasteur

dimanche 8 janvier 2017

Non commutativity : the scientific term that ought to be more widely known in 2017!

An unofficial contribution to the annual challenge proposed by

The word "noncommutativity" will certainly have been one of the most representative of the fertility of mathematics in the 20th century. If the commutative / noncommutative dichotomy is present in the nineteenth century, the advent of group theory, what is now called non-commutative mathematics, has taken off after the new mechanics adapted to the world Atomic scale was born. Quantum Mechanics has drawn an ontology of non-commutative mathematics, just as it has created a new physical paradigm for our perception of the world. From a philosophical point of view it is strange that the negation of a concept, of a formula, also becomes positively anchored in an almost universal conceptual aspect: in principle, since the "commutative" is one, the non-commutative should be multiple. We often speak of the commutative and the non-commutative, as if there were only one occurrence of the latter. This, it seems to me, reflects the profound paradigm shift that abandonment represents, in a physical theory or field of mathematics, the postulate [A, B]: = AB - BA = 0. 
Should we see [A, B] = 0 or [A, B] ≠ 0 as a constraint? 
On the other hand, if two given matrices A and B commute or do not commute, it turns out that there are families of matrices which almost commute, thus providing the possibility of a transition from non-commutative to commutative. This transition is difficult and offers an extreme richness, trace according to us of the depth of the paradigmatic change between the commutative and the non-commutative. We will try to give some examples ...
 English translation from the following original text in French by Thierry Paul :

Le mot “noncommutatif” aura certainement été l’un des plus représentatifs de la fécondité des mathématiques au XXème siècle. Si la dichotomie commutatif / noncommutatif est présente au XIXème siècle, dès l’avènement de la théorie des groupes, ce que l’on appelle de nos jours les mathématiques non commutatives ont pris tout leur essor après que la nouvelle mécanique adaptée au monde à l’échelle atomique ait vu le jour. La Mécanique Quantique a dessiné une ontologie du non commutatif en mathématique, tout comme elle a créé un nouveau paradigme physique pour notre perception du monde. D’un point de vue philosophique il est étrange que la négation d’un concept, d’une formule, devienne aussi positivement ancré dans un aspect conceptuel presque universel : en principe, puisque le “commutatif” est un, le non commutatif devrait être multiple. Or on parle souvent du commutatif et du non commutatif, comme s’il n’y avait qu’une seule occurrence de ce dernier. Cela traduit, il me semble, le changement de paradigme profond que représente l’abandon, dans une théorie physique ou domaine des mathématiques le postulat [A, B] := AB − BA = 0.  
Faut-il voit [A, B] = 0 ou bien [A, B] ≠ 0 comme une contrainte ?  
D’autre part, si deux matrices données A et B commutent ou bien ne commutent pas entre elles, il s’avère qu’il existe des familles de matrices qui commutent presque, offrant ainsi la possibilité d’une transition du non commutatif vers le commutatif. Cette transition est difficile et offre une richesse extrême, trace selon nous de la profondeur du changement paradigmatique entre le commutatif et le non commutatif. Nous allons essayer d’en donner quelques exemples...

Addendum du 12/01/17

Voici une manière figurée de sentir la richesse de la vision noncommutative du monde qui est parfaitement incarnée dans la puissance du langage ordinaire avec l'anagramme (tirée de Anagrammes pour lire dans les pensées) que voici:
Ondes gravitationnelles
Le vent d'orages lointains
Dans une vision strictement commutative, les deux lignes suivantes perdent littéralement leur sens pour ne plus être que le magma suivant : 
Cet exemple est tiré de l'avantun des derniers transparents du premier cours de 2017 d'Alain Connes au collège de France dont les deux vidéos sont disponibles ici.

samedi 7 janvier 2017

Task list and best wishes for 2017

Rather than relying on a breakthrough discovery in high energy physics in 2017 which is really difficult to envision, I am quite convinced that with what we know of the complete standard model of particle physics and what we don't know from the dark energy and dark matter phenomenological parametrisation in the cosmological concordance model there are dramatic improvement possible : in model building for grand unification of gauge coupling constants at and beyond the zeptometer and also in inflation, baryogenesis and seesaw mechanisms on the zetta-eV side. 
I think with the last progress in the spectral noncommutative geometric modelisation of spacetime, fundamental fermionic matter and bosonic gauge and scalar interaction mediators, theoretical physicists could build a pretty consistent view of the dreamed micro-macro connection epitomised by the famous Glashow snake. Another task for bold and gifted experimenters will be to imagine and then design the proper tools to probe the realm of some potential new quanta of spacetime waiting for us to be uncovered.
Best wishes for 2017 to them, to the readers of this blog and to all others!

Deux anagrammes à clef* pour commencer la nouvelle année tirées d'Anagrammes renversantes et combinés par le blogueur à la façon d'un haïku :

Et les particules élémentaires  tissèrent l'espace et la lumière 1, 
tout en faisant résonner en moi la disparition de Majorana ...3  
j'adorai la dimension à part 2.

*Parcours de découverte à la recherche des clefs ou simplement pour apprécier le chemin déjà parcouru dans la modélisation spectrale de notre espace-temps physique "à la mode" noncommutative:

Une classification des facteurs de type III (inception)
L'interaction entre physique théorique et mathématiques pure sous l'angle des algèbres d'opérateurs (pro domo advocacy for a quantum perspective on spacetime)
Essay on Physics and Non-commutative Geometry (a roadmap?)
Classical Bosons in a Noncommutative Geometry (the Higgs mechanism pops up from
noncommutative geometric extension of abelian gauge field theory)

Lessons from Quantum Field Theory - Hopf Algebras and Spacetime Geometries 
On the foundation of noncommutative geometry (historical perspective)
Noncommutative geometry and the standard model (very pedagogical introduction)
Renormalization, the Riemann–Hilbert correspondence, and motivic Galois theory 
Can we predict the value of the Higgs mass? (a critique about a wrong interpretation)

Noncommutative geometry and physics (a mathematical physics memento)
Non-compact spectral triples with finite volume
3Resilience of the Spectral Standard Model (hint of a canonical completion of the second model)
3Asymptotic safety, hypergeometric functions, and the Higgs mass in spectral action models (another more versatile completion)
Gauge networks in noncommutative geometry
Inner Fluctuations in Noncommutative Geometry without the first order condition
(breakthrough : a former axiom is removed)
Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification 
("natural" connection between the electroweak and seesaw scales) 
Renormalizability Conditions for Almost-Commutative Manifolds 
Quanta of Geometry: Noncommutative Aspects (breakthrough : mimetic dark matter  and dark energy derivable from noncommutative geometry) 
Grand Unification in the Spectral Pati-Salam Model (a threefold third canonical model)  
Pati-Salam Unification from Non-commutative Geometry and the TeV-scale W_R boson (the first truly testable but noncanonical third-like model?)
The noncommutative geometry of Zitterbewegung  (a table-top experiment for quantum simulation?)
Resolving Cosmological Singularities and Nonsingular Black Hole (not necessarily connected to the spectral noncommutative framework but naturally embeddable into?)
2017 update!
Geometry and the quantum 
Quanta of Space-Time ...

That's all for this brief tapestry of an Odyssey through the building of spectral noncommutative geometry. Let's be prepared to engage sooner or later in the Iliad singing the confrontation of spectral Pati-Salam model and mimetic extension of general relativity predictions with  the phenomenological world.

//last edit 08/01/17
//last edit 09/01/17
//last edit 22/02/17
//last edit 08/03/17 (link to Geometry and the quantum was changed)

jeudi 5 janvier 2017

A 2016 biased review

Three slides from a talk by Cern theorist Alessandro Strumia 
at Madrid, September 29, 2016

Dans la mythologie grecque, la sibylle est une prêtresse d'Apollon qui personnalise la divination et prophétise. Elle le faisait dans un langage énigmatique permettant de nombreuses interprétations, ce qui la mettait à l'abri de toute contestation ultérieure. Fameuse est sa prophétie orale pour un soldat « Ibis redibis non morieris in bello ». Si une virgule est placée avant le « non », la phrase devient «Tu iras, tu reviendras, tu ne mourras pas en guerre », mais si la virgule était placée après le « non », la phrase est « Tu iras, tu ne reviendras pas, tu mourras en guerre ».
Cette pratique ... a donné le qualificatif de « sibyllin » qu'on attribue à des écrits ou des paroles obscures, énigmatiques, mystérieuses ou à double sens. La sibylle figure l'être humain élevé à une dimension surnaturelle, lui permettant de communiquer avec le divin et d'en livrer les messages, tels le possédé, le prophète, l'écho des oracles, l'instrument de la révélation.

adapté de source :

Anagramme personnelle en guise de conclusion :

Oh belle supersymétrie ! 
Hère ? Sibylle prometteuse trompeuse ...

Ajout du 10/01/17 (légèrement amendé le 11/01/17)
Commentaire de l'anagramme en forme d'avertissement au lecteur 
Jouer avec les mots est évidemment beaucoup plus simple que d'émettre une hypothèse scientifique sérieuse et d'explorer toutes ses conséquences pour en tester la profondeur et la pertinence, que ce soit dans le monde réel ou celui des idées. L'anagramme précédente n'a pas pour vocation de se moquer à peu de frais des générations de physiciens qui se sont succédées depuis les années soixante dix pour concevoir des théories et réaliser des expériences visant à tester les prédictions des précédentes, sans parvenir jusqu'à présent à valider la détection d'une particule supersymétrique malgré l'énergie déployée (voire par exemple cette série de conférences datées de décembre 2016 en l'honneur de Pierre Fayet, l'un des pères des modèles supersymétriques en France)... Elle est seulement là pour alerter sur le danger d'un programme scientifique de grande envergure (big science) mobilisant des moyens économiques mais surtout humains considérables qui serait guidé par une vision borgne de l'ensemble des imaginaires possibles (ou mieux plausibles) à explorer. 

Je pense que la part de l'effort de recherche consacrée à l'exploration de la physique des particules au delà du modèle standard sans l'hypothèse supersymétrique a été insuffisante dans le passé à cause peut-être d'un excès de confiance dans cette hypothèse qui transparait encore aujourd'hui dans la sphère médiatique même si une évolution sensible est visible dans celle plus spécialisée des publications scientifiques comme le montre le graphique mis en exergue par Alessandro Strumia ci-dessus. Évidemment la critique a posteriori est plus facile à faire que la prospective à long terme ou la prévision de l'avenir...

Sur le plan des idées, il me semble - comme essayent de l'illustrer explicitement certains billets voire implicitement l'ensemble de ce blog - que nous n'avons pas encore tiré toutes les leçons heuristiques de la découverte il y a quatre ans maintenant du boson scalaire de 125 GeV. Évidemment les spécialistes de la question le savent bien (voir futur billet) mais je regrette que cette information ne soit pas diffusée et rendue accessible à un public plus large. D'autre part il me semble aussi dommageable que le public plus restreint des physiciens ne soit pas davantage sensible au parfum que l'alambic de la géométrie spectrale noncommutative (inspirée de la relativité générale et de la physique quantique) distille à partir des trois fruits mûrs de la physique des interactions fondamentales : je veux parler du modèle standard, des changements de saveur des neutrinos et du secteur sombre du modèle de concordance cosmologique. Peut-être vaudrait-il d'ailleurs mieux parler pour ce dernier point de "paramétrisation de l'inconnu" dans la dynamique gravitationnelle aux échelles galactique et cosmologique (je détourne ici une formule que Jean Iliopoulos appliquait au boson de Higgs comme description effective du mécanisme de brisure spontanée de la symétrie électrofaible). Quoiqu'il en soit je suis heureux de constater que les organisateurs de la série de conférences célébrant le travail scientifique de Pierre Fayet (parmi lesquels J. Iliopoulos justement) avaient jugé bon d'inviter un cosmologiste de renom, en l'occurrence Viatcheslav Mukhanov, parler du lien phénoménologique qu'il pourrait y avoir entre énergie noire et matière noire d'une part et "quantification du volume", respectivement pour l'espace-temps quadridimensionnel et l'espace tridimensionnel. En style télégraphique et en anglais dans le texte :
Last slide of a conference Dec 9 2016 : Non-commutative Geometry and Mimetic Dark Matter by S. Mukhanov

 //dernière retouche le 123/012/17