mardi 30 septembre 2014

Epistemology and phenomenology of B-L gauged extension of the Standard Model

Gauging a global symmetry and fixing the electric charge
Many attempts were made to combine space-time symmetries with internal symmetries, e.g., theories based on SU(6) and in the 1970s the emergence of supersymmetry, which from 1980’s became the dominant theme in both theory and experiment. Although experiments ultimately decide which symmetries live and which die, either way they leave a lasting impact on the field. In this article, I will focus on a new symmetry of particle physics, the B−L symmetry, which is a global symmetry of the standard model and appears to be emerging as a local symmetry designed for understanding the physics of neutrino masses...  
A key prediction of the standard model (SM) is that neutrino masses vanish since, unlike other fermions, which have both left and right handed chiralities in the theory, there is no right handed neutrino but just the left handed SU(2)L partner of eL. Because several experiments have confirmed since 1990’s that neutrinos have mass, the simplest extension of SM is to add to it one right handed neutrino per generation to account for this fact. As soon as this is done, one not only has Tr(U(1)B-L[SU(2)L]2)=0 and Tr(U(1)B-L[U(1)Y ]2)=0 but also Tr(B-L)3=0. This allows for the possibility of gauging the U(1)B-L quantum number, which gives B−L a dynamical role... 
... it was pointed out independently by Marshak and me [11] and Davidson [12], that the electric charge formula now becomes
Q = I3L + I3R + (B−L) / 2  ...
This is a considerable improvement over the SM electric charge formula... in the sense that all terms in the formula are determined through physical considerations of weak, left and right isospin, and baryon and lepton number, that reflect independent characteristics of the various elementary particles. No freely floating parameters are needed to fix electric charges as in the standard model. Electric charge is no more a free parameter but is connected to other physical quantum numbers in the theory. As a result, a number of interesting implications follow...
(Submitted on 26 Sep 2014)

Looking for neutron oscillations, Majorana fermions, new gauge bosons...
The question that has to be tackled is whether there exist a theory that combines neutrino mass via the seesaw mechanism which predicts an observable [neutron-antineutron oscillation] and yet keeps the proton ... stable. One example of such a theory was presented in 1980 in [16]. This model presents an embedding of the left-right seesaw model into a quark lepton unified framework using the gauge group ... SU(2)L×SU(2)R×SU(4)c with the symmetry breaking suggested in [the Pati-Salam model] rather than in Ref.13... The sixteen chiral fermions of the SU(2)L×SU(2)R×SU(4)c model fit into the sixteen dimensional spinor representation of the SO(10) group [20] which can be the final grand unification group for left-right symmetry as well as B−L gauge symmetry...
... we have provided a broad brush overview of the history of B-L as a new symmetry in particle physics and how in recent years following the discovery of neutrino masses, interest in this possible new symmetry has grown enormously. In particular, its connection to both neutrino mass and baryon number violation have provided new insights into physics beyond the standard model. All these have to be confirmed experimentally. At the same time, there are phenomenological studies of many different aspects of this symmetry. To summarize the efforts to unravel the degree of freedom corresponding to local B-L symmetry experimentally, I mention only a few topics. Deciphering whether neutrinos are Majorana fermions is a direct confirmation of whether B-L symmetry is broken or not. This does not say whether it is a global or local symmetry. Furthermore, by itself, discovery of ββ decay cannot tell where the scale of B-L symmetry breaking is. Supplemented by a discovery (or non-discovery) of neutron oscillation, one can get an idea about a possible range (or exclude a possible range) of this scale but not the actual scale. The most definitive way to discover the scale of B-L symmetry is to directly search for the gauge boson associated with this in the collider such as the LHC [29]. The same could also be inferred from a discovery of the WR combined with a Majorana right handed neutrino. Such searches are currently under way at the LHC [30a, 30b].

To deepen the historical perspective on our forty year-old Standard Model the interested reader may also benefit from reading an already twenty year old other Mohapatra paper :

Freedom from Adler-Bell-Jackiw anomalies is a primary requirement for the renormalizability of a gauge theory of chiral fermions, which forms the basis of the successful standard model of electroweak interactions and its many extensions. In this article, we explore to what extent, the assumptions behind the standard model as well as the observed quantization of electric charges of quarks and leptons can be understood using the various anomaly constraints and how the situation changes as one tries to incorporate a nonvanishing neutrino mass.

Understanding the Standard Model
R. N. Mohapatra (University of Maryland)
(Submitted on 1 Apr 1994)

lundi 29 septembre 2014

Our (quantum and gravitating) physical universe from a conservative (neither compulsory stringy nor mandatory supersymmetric) viewpoint

Welcome to observationally driven quantum cosmological times!

The Planck measurements have unambiguously confirmed the main predictions of the theory of quantum origin of the universe structure. Namely, the adiabatic nature and the Gaussian origin of primordial perturbations were established beyond any reasonable doubt. Even more amazing, more nontrivial infrared logarithmic tilt of the spectrum, first predicted in [2], was discovered at 6 sigma confidence level. The simplest way to amplify the quantum fluctuations is provided by the stage of inflation. Although nobody doubt the quantum origin of the primordial fluctuations, there are still claims in the literature that basically the same mechanism of amplification of quantum fluctuations can work also either in a bouncing universe on the stage of super slow contraction [18] or in conformal rolling scenario [19]. The generated spectra in the alternative theories are not the predictions of the theory, but rather postdictions which are constructed to be in agreement with observations. Nevertheless, this is not enough to rule out these possibilities at the level of a ”theorem”. Thus, at the moment the only robustly established experimental fact is the quantum origin of the universe structure with a little uncertainty left for the mechanism of amplification of quantum fluctuations. To firmly establish that namely inflation has provided us this mechanism one has to find the primordial gravitational waves the lower bound on which for the spectral index ns= 0.96 corresponds to r about 0.003.
Farewell to some plagues of postmodern inflating speculations?

Production of a closed universe does not cost any energy because the positive energy of the matter is entirely compensated by the negative energy of the gravitational self-interaction of this matter. Therefore the closed universe can be produced as a result of quantum fluctuations [7]. One can expect that quantum fluctuations are essential only at Planck scale and only quantum universes with internal mass of order 10−5g can easily emerge. If gravity is an attractive force then such universes will immediately recollapse. However, as it was pointed out in [8],[9],[10], this does not happen if for some reason the equation of state within the Planckian universe corresponds to the cosmological constant p ≈−ε, where p is the pressure and ε is the energy density. In this case the gravitational field, determined by ε+ 3p ≈−2ε, is repulsive and instead of collapsing the universe starts to expand with acceleration. As a result the size of the closed universe grows exponentially and its total mass (as well as the number degrees of freedom) also increases exponentially fast. The energy needed to produce the matter comes from the gravitational reservoir with unbounded from below energy. This is a rough picture of the emergence of the causal universe in Minkowski space [8] or from nothing [9],[10]1, which was proposed in 80th. This picture is also well supported by quantization of space-time in noncommutative geometry [11]. The stage of accelerated expansion is useful for amplifying quantum fluctuations which later serve as the seeds for galaxy formation [1], [2] and, moreover, it also amplifies the quantum fluctuations of transverse degrees of freedom of the gravitational field (gravitational waves) [12]. 
There are many inflationary scenarios in the literature the only purpose of which is to provide us with the stage of the quasi-exponential expansion. These scenarios mainly differ by the choice of a slow-roll scalar field potential “justified” by a “fundamental theory”. However such theory is not yet known and hence any particular potential is merely based on the prejudices of the authors. I believe that under such circumstances the more plausible approach is an effective description of inflation in terms of the effective equation of state  1+p/ε=1+w(N) parametrized by the number of e-folds left until the end of inflation  N. As it was shown in [13] even the simplest choice for the equation of state allows us to cover nearly all scenarios and to prove that the most valuable predictions of the theory of quantum origin of the universe structure are robust. [...] 
If we want to avoid the selfreproduction and do not face the problem of initial conditions the required equation of state must simultaneously satisfy the following requirements: 
  1. 1 + w(N) ≃ 1 at N ≃ 1 (to have graceful exit), 
  2. 1 + w (N) ≤ 2/3 at N ≃ Nm (to solve initial condition problem), 
  3. 1 + w (N) ≪ 1 for 1 < N < Nm (inflation), 
  4. 1 + w (N) > ε(N) for 1 < N < Nm (no selfreproduction). 


//work in progress.

mardi 23 septembre 2014

How to save (or rather softly break) supersymmetry with the help of noncommutative geometry?

SUSY  bashing soft breaking ;-)

We describe how a soft supersymmetry breaking Lagrangian arises naturally in the context of almost-commutative geometries that fall within the classification of those having a supersymmetric particle content as well as a supersymmetric spectral action. All contributions to such a Lagrangian are seen to either be generated automatically after introducing gaugino masses to the theory or coming from the second Seeley-DeWitt coefficient that is already part of the spectral action. In noncommutative geometry, a supersymmetric particle content and the appearance of a soft breaking Lagrangian thus appear to be intimately connected to each other.

(Submitted on 21 Sep 2014)

Parallel but opposite or dual and complementary histories of two different ideas?

Supersymmetry,  a decade ago

Supersymmetry is now 30 years old. The first supersymmetric field theory in four dimensions - a version of supersymmetric quantum electrodynamics (QED) - was found by Golfand and Likhtman in 1970 and published in 1971. At that time the use of graded algebras in the extension of the Poincaré group was far outside the mainstream of high-energy physics. 
Three decades later, it would not be an exaggeration to say that supersymmetry dominates high-energy physics theoretically and has the potential to dominate experimentally as well. In fact, many people believe that it will play the same revolutionary role in the physics of the 21st century as special and general relativity did in the physics of the 20th century. 
This belief is based on the aesthetic appeal of the theory, on some indirect evidence and on the fact that there is no theoretical alternative in sight. Since the discovery of supersymmetry, immense theoretical effort has been invested in this field. More than 30,000 theoretical papers have been published and we are about to enter a new stage of direct experimental searches. 
The largest-scale experiments in fundamental science are those that are being prepared now at the LHC at CERN, of which one of the primary targets is the experimental discovery of supersymmetry. 
The history of supersymmetry is exceptional. In the past, virtually all major conceptual breakthroughs have occurred because physicists were trying to understand some established aspect of nature. In contrast, the discovery of supersymmetry in the early 1970s was a purely intellectual achievement, driven by the logic of theoretical development rather than by the pressure of existing data.

Keith Olive and Misha Shifman, Minnesota

CERN COURIER, Feb 26, 2001

Spectral noncommutative geometry, right now
Spectral noncommutative geometry is approximately 25 years old. The first noncommutative model of spacetime - offering a geometric understanding of the Higgs term in the Glashow-Weinberg-Salam Lagrangian - was proposed by A. Connes in 1988 and was further developed and expanded to the Standard Model with J. Lott in 1990. At that time the use of operator algebraic tools to characterize spectrally (and extend much further) Riemannian manifolds was far outside the mainstream of high-energy physics.  
Two decades later, it would not be an understatement to say that the now well established tools from spectral noncommutative geometry are still largely ignored by high-energy physics theoreticians more accustomed to highly sophisticated but classical differential geometry of String Theory mathematics. Nevertheless constant progress in the understanding of the spectral version of the Standard Model and its possible extensions give hope to the rise of a genuine spectral noncommutative phenomenology. In fact, one can reasonably expect that it will help to realize in the 21st century a synthesis of quantum physics and general relativity from the 20th century. 
This hope is based on the epistemological logic and potential heuristics of the theory, on the direct detection of one Standard Model-like Higgs boson and on the fact that there is no experimental sign of other new fundamental particle at the highest accessible energies from colliders. Since the beginning of spectral noncommutative geometry, important breakthroughs have been done in this field but progress has been slow and difficult. Only 300 theoretical papers have been published but we have in some sense already entered in the first stage of direct experimental investigations. 
Among the largest-scale experiments in fundamental science are those that are planned to restart at the LHC at CERN, of which one of the target could be the search for a Z' boson associated to a U(1)B-L extension of the Standard Model gauge group compatible with the parsimonious spectral predictions
The history of spectral noncommutative geometry is quite original. In the recent past, the more publicized theories have implied lot of physicists trying to extrapolate the quantum knowledge gained from strong interaction to explain the weakness of classical gravitation. In contrast, the development of spectral noncommutative geometry for physics started with the work of a lonely mathematician looking for a conceptual understanding of the Brout-Englert-Higgs mechanism that provides mass to the bosons of  the weak interaction. 

dimanche 21 septembre 2014

There are more things in theory and phenomenology, my friend, than are covered in more popular blogs.

More things in theory
  • Is there a theory that properly combines quantum field theory with gravity?  
  • Why do we observe precisely the SM-particles, and why not more? Or less?  
  • What accounts for the dark matter that astrophysicists observe? Why do all fermions appear in three copies that are identical,save for their masses?  
  • What keeps the Higgs boson mass stable when considering loop corrections?  
  • Why is the mass of the neutrinos so small compared to that of—say— the top quark? 
In the past decades the academic community has witnessed the birth of a plethora of theories that address one or more of the above questions. Some of them entail only minor modifications to the SM, others require us to radically reconsider the origin of the laws of nature. The hope is that there will scientific progress in the upcoming years via the falsification of many such theories by the results of the Large Hadron Collider. We live in fascinating times indeed! 
In this thesis we focus on an alternative way to obtain particle theories such as the SM on the one hand, and on a particular extension of the SM on the other. In fact, it is the combination of both that we are after. The first of these comes from the field of noncommutative geometry (NCG). Historically, this is a branch of mathematics, but it has applications in physics. From the latter point of view it can be considered as a generalization of Einstein’s theory of General Relativity in the sense that it admits spaces to exhibit some notion of noncommutativity. particular class of noncommutative geometries —called almost-commutative geometries (ACGs)— does a marvelous job at describing gauge theories, of which the SM is an example. These ACGs are constructed by combining a commutative geometry, consisting of a curved space(time) on which there ‘live’ fermions, with a so-called finite noncommutative geometry... The constraints that are imposed on these ACGs by the axioms of NCG translate to properties of particle theories that are actually observed in experiments. NCG thus provides us new ways, of geometrical nature, to understand theories such as the SM. The latter in fact comes out as very natural in this context. 
The aforementioned extension of the Standard Model encompasses supersymmetry. This line of thought was once devised to ‘get the most’ out of quantum field theories, by using all its possible symmetries. Applying it to the SM in particular requires extending it with a set of new particles, one for each particle that we have currently observed. This leads to a theory that provides an answer to some of the fundamental open questions, raised above. The theory is called the Minimal Supersymmetric Standard Model (MSSM). At the LHC, experimenters vigorously look for signals that hint at its validity. To date, however, these have not been observed. Despite this lack of experimental success, the MSSM remains one of the prime candidates for a ‘beyond the Standard Model’ theory. 
A natural question to ask is then if noncommutative geometry and supersymmetry go well together,i.e. if the framework of NCG admits models that exhibit supersymmetry. This question has already been around for some time, but despite several previous attempts by others, its answer was still inconclusive.  
This PhD thesis is devoted mainly to address this subject and combining NCG and supersymmetry. We have restricted ourselves to the class of almost-commutative geometries,in combination with the spectral action principle, a combination that was of immense value in obtaining the SM from NCG. In addition, we have restricted our analysis  to finite KO-dimension 6, that allows us to solve the fermion doubling problem in 4 space-time dimensions.  
We have first turned to general extensions of the SM in NCG, supersymmetric or not. We have translated several physical demands (anomaly-freedom, correct hypercharges) and properties (the existence of a GUT-point) into constraints on the multiplicities of particles. The SM only satisfies these constraints when three right-handed neutrinos are added to the particle content. Although the MSSM particle content is anomaly free, it does not yield a GUT-point nor does it give the correct hypercharges. This last problem can be solved by introducing the notion of R-parity —one that is characteristic for supersymmetry— in the context of almost-commutative geometries and modifying some expressions accordingly.  
In order to answer the question of whether a certain ACG exhibits supersymmetry or not, a distinction must be made between the almost-commutative geometry itself and its associated action functional. Necessary for supersymmetry is the equality of fermionic and bosonic degrees of freedom. At the level of the ACGs, this leads us to the identification of supersymmetric building blocks and a diagrammatic approach to manage calculations. These are additions to the ACG (consisting of components of the finite Hilbert space and Dirac operator) that yield degrees of freedom eligible for supersymmetry. Since this demand must hold both on shell and off shell, we are forced to introduce (non-physical) auxiliary fields by hand, since the spectral action is interpreted as the on shell action. The requirement for the total action to actually be supersymmetric (i.e. its variation under the supersymmetry transformations vanishes) then depends on the value of the components of the finite Dirac operator. In total,we have identified five such building blocks. For each of them, the action that results corresponds in form to a term in the superfield method, in which supersymmetry is most often phrased... In the process, all formal properties of and demands on almost-commutative geometries are respected.  
Characteristic for this approach is that each new addition to an ACG provides extra contributions to the prefactors of terms that were previously already present in the action. This requires reassessing all interactions with each newly added building block. To manage this, we have set up a list with all possible terms that occur in the action and all possible contributions to them from each building block. The action is then supersymmetric if for a particular set of building blocks all the pre-factors of the terms that occur in the action can be equated to the value required for supersymmetry. At least for the most straightforward situations (a single building block of the second type, a single building block of the third type) this is not the case; the set up turns out to be over-constrained. An interesting phenomenon occurring is that in some cases the demand of a supersymmetric action puts constraints on the number of particle generations.  
Inseparable from supersymmetry is its breaking, required to give (realistic) masses to the particles appearing in the theory. We observe that soft supersymmetry breaking interactions appear automatically in the spectral action. Hence, NCG provides a new soft supersymmetry breaking mechanism. There are in fact only two supersymmetry breaking sources: the trace of the finite Dirac operators squared, which yields mass-like terms for the scalars, and gaugino masses. The second is the most prominent one. Interestingly, the gaugino masses provide a cascade of other soft breaking interactions, each of them associated to one of the five building blocks. In particular, they also give contributions to the scalar mass terms. These are of opposite sign with respect to those from the trace of the finite Dirac operator squared. This is required for the scalar mass terms to have the right sign needed to prevent them from maximally breaking the gauge group.  
This sets the stage for answering the central question, concerning a noncommutative version of the MSSM. There exists a set of building blocks whose particle content corresponds to that of the MSSM and whose fermionic interactions coincide with those of the MSSM. However, the relevant constraints on the four-scalar interactions that were mentioned above can only be satisfied for a non-integer number of particle generations. Thus, the almost-commutative geometry whose particle content is equal to that of the MSSM has a spectral action that is not supersymmetric. 
Properties of this theory hint at possible extensions of the MSSM that do satisfy all constraints, but to find it (or any other positive example of a supersymmetric NCG for that matter) requires a more constructive, and possibly automated, approach. If such a search would yield one or more positive results, these will —due to the stringency of this approach— at least enjoy a very special status.
Thesis defended on 5 September 2014 (p 121)

More things in phenomenology
Although the Standard Model takes a prominent place [14,16,7] within the possible models of almost-commutative geometries one can to go further and construct models beyond the Standard Model. The techniques from the classification scheme developed in [14] were used to enlarge the Standard Model [24,25,23,26], but most of these models [24,25,23] suffer from a similar shortcoming as the Standard Model: The mass of the SMS is in general too high compared to the experimental value. Here the model in 26 will be of central interest, since it predicted approximately the correct SMS mass. In the case of finite spectral triples of KO-dimension six [2,5] a different classification leads to more general versions of the Standard Model algebra [7], under some extra assumptions. Considering the first order axiom as being dynamically imposed on the spectral triple one finds a Pati-Salam type model [9] . From the same geometrical basis one can promote the Majorana mass of the neutrinos to a scalar field [8,11] which allows to lower the SMS mass to its experimental value.
The model we are investigating extends the Standard Model [6] by N generations of chiral X1-and X2-particles and vectorlike Vc/w-particles. It is a variation of the model in [26] and the model in [27] . For details of the following calculations as well as the construction of the spectral triple we refer the reader to [27]. In particular its Krajewski diagram is depicted in figure 4 ? The necessary computational adaptations to the model in this publication are straightforward. The gauge group of the Standard Model is enlarged by an extra U(1)subgroup, so the total group is G=U(1)Y×SU(2)W×SU(3)c×U(1)X. The Standard Model particles are neutral with respect to the U(1) subgroup while the X-particles are neutral with respect to the Standard model subgroup GSM=U(1)Y×SU(2)W×SU(3)c. Furthermore the model contains two scalar fields: a scalar field in the SMS representation and a new scalar field carrying only a U(1)X charge. They induce a symmetry breaking mechanism G→U(1)em×SU(3)c... 
From the experimental point of view the question is of course the detectability of the new particles (at the LHC?) and whether the dark sector contains suitable dark matter candidates. One would expect more kinetic mixing of U(1) to the hypercharge group U(1)compared to the model [27] due to the fact that here the V-particles may have a different masses and therefore mix the abelian groups. This poses the question how Z0-like the ZX-boson are and whether they are already excluded (at least for the mass range explored in this publication). The model certainly has a rich phenomenology and is one of the few known models to be consistent with (most) experiments, with the axioms of noncommutative geometry and with the boundary conditions imposed by the Spectral Action.
 Example for the mass eigenvalues mH (bottom) and mϕ (top) of the scalar fields H and ϕ with respect to v2. Here the top quark mass is taken to be mt(mZ ) = 173.5 GeV and the experimental SMS mass mexp. = 125.5 GeV. We obtain a heavy scalar with mϕ = 320 GeV and a U(1)X-scalar boson mass mZx = 172 GeV.
(Submitted on 14 May 2013)


samedi 20 septembre 2014

Successive slidings of SUSY looking for a better place...

... in the sun of phenomenological reality
Already shortly after the advent of supersymmetry it was realized [98] that if it is a real symmetry of nature, then the superpartners should be of equal mass. This, however, is very much not the case. If it were, we should have seen all the sfermions and gauginos that feature in the Minimal supersymmetric Standard Model (MSSM, e.g. [39]) in particle accelerators by now. In the context of the MSSM we need [55] a supersymmetry breaking Higgs potential to get electroweak symmetry breaking and give mass to the SM particles. Somehow there should be a mechanism at play that breaks supersymmetry. Over the years many mechanisms have been suggested that break supersymmetry and explain why the masses of superpartners should be different at low scales. Ideally this should be mediated by a spontaneous symmetry breaking mechanism, such as D-term [80] or F-term [45] supersymmetry breaking. But phenomenologically such schemes are disfavoured, for they require that ‘in each family at least one slepton/squark is lighter than the corresponding fermion’ [39, §9.1].
Alternatively, supersymmetry can be broken explicitly by means of a supersymmetry breaking Lagrangian. In order for the solution to the hierarchy problem that supersymmetry provides to remain useful, the terms in this supersymmetry breaking Lagrangian should be soft [51]. This means that such terms have couplings of positive mass dimension, not yielding the quadratically divergent loop corrections that would spoil the solution to the hierarchy problem (the enormous sensitivity of the Higgs boson mass to perturbative corrections) that supersymmetry provides.

... in a spectral noncommutative frame

For long, the [top] figure that was presented in [87] was believed to be correctly describing the (empty) intersection between noncommutative geometry and supersymmetry. In our view this should be slightly modified, to yield [the bottom] figure... The question that is still open is how big that intersection actually is, and exactly how ‘far’ the MSSM is away from it, i.e. if there are models resembling the MSSM that do exhibit a supersymmetric action. This calls for a constructive approach generating a list of (experimentally viable) supersymmetric spectral triples.

lundi 15 septembre 2014

A spectral non commutative shot in the darkness of quantum physics and Einstein's gravity

Pointing to a way out of the cosmological constant problem... 

 ... in noncommutative geometry the volume of [a] compact manifold is quantized in terms of Planck units. This solves a basic difficulty of the spectral action [1] whose huge cosmological term is now quantized and no longer contributes to the field equations... 
Let us study consequences of the four volume quantization for Einstein gravity... First we consider Euclidian compact spacetime and implement ... [two] kinematic constraints [associated with quantization of 4d spacetime in the spectral non commutative framework] in the action for gravity through Lagrange multipliers. This action then becomes
where 8πG=1. Notice that the last term is a four-form and represents the volume element of a unit four-sphere and can be written in differential forms as -1/(4!8)*Tr(YdY∧dY∧dY∧dY) and is independent of variation of the metric. Taking into account the [two] kinematic constraints, variation of the action with respect to the metric gives  
Gµν+1/2gµνλ = 0,          
where Gµν=Rµν−1/2Rgµν is the Einstein tensor. Tracing this equation gives 
and as a result equations for the gravitational field become traceless 
Gµν−1/4gµνG= 0.           (15) 
Using the Bianchi identity these equations imply that ∂µG=0, and hence 
where Λ is the cosmological constant arising as a constant of integration... 
One immediate application is that, in the path integration formulation of gravity, and in light of having only the traceless Einstein equation (15), integration over the scale factor is now replaced by a sum of the winding numbers with an appropriate weight factor. We note that for the present universe, the winding number equal to the number of Planck quanta is of the order of 1061[7].
(last revised 9 Sep 2014 (this version, v2))

Providing an Ariadne's thread to go through the dark matter and dark energy maze 
... In reality spacetime is Lorentzian and generically has one noncompact dimension corresponding to time. Therefore, the condition for the volume quantization is literally non-applicable there. However being implemented in the Euclidian action it leads nevertheless to the appearance of the cosmological constant as an integration constant even in the Lorentzian spacetime... The Lorentzian action for the gravity is 

[where Ya... depend on the coordinates xµ and the variable X is necessary to implement a third constraint expressed in the last term - namely the quantization of the volume of compact 3d hypersurfaces in 4d spacetime. This term] corresponds to mimetic dark matter [10],[11]. Thus the resulting action describes both dark matter and dark energy. Both substances arise automatically when the kinematic 4d and 3d compact volume quantization in noncommutative geometry is incorporated in the gravity action.

Proposing a conformal extension of Einstein’s general theory of relativity
Recently a new interesting model of mimetic dark matter was suggested in [1] and was further elaborated in [2, 3]. The basic idea is remarkably simple. The physical metric gphysµν is considered to be a function of a scalar field φ and a fundamental metric gµν, where the physical metric is defined as 
The physical metric gphysµν is invariant with respect to the Weyl transformation of the metric gµν,

Then it was shown in [1] and in [2] that the ordinary Einstein-Hilbert action con- structed using the physical metric gphysµν possesses many interesting properties. In fact, the model analyzed below is a conformal extension of Einstein’s general theory of relativity. The local Weyl invariance is ensured by introducing an extra degree of freedom that as was shown in [1] has the form of pressureless perfect fluid that, according to [1], can mimic the behavior of a real cold dark matter. 
Historically, Gunnar Nordström was the first to construct a relativistic theory of gravity as a scalar field theory [4] whose geometric reformulation [5] was the first metric theory of gravity. The physical metric of this gravitational theory was defined as a conformal transformation of the flat Minkowski metric, gµν2ηµν where φ is the scalar field of Nordstr¨om’s theory. In other words, it was a theory of conformally flat spacetimes. The structure of the field equation, R=24πGT, where R and T are the traces of the Ricci tensor and the energy-momentum tensor respectively, closely resembled the field equation of the general theory of relativity formulated by Einstein in the following year. Intriguingly, the idea of mimetic matter [1] is to introduce additional fields in the conformal factor that relates the physical and auxiliary metrics (1) in such a way that the physical metric remains invariant under the conformal transformation (2).
(Submitted on 15 Apr 2014 (v1), last revised 6 Jul 2014 (this version, v3))

Opening a promising playground for new cosmological and astrophysical model
We discuss ghost free models of the recently suggested mimetic dark matter theory. This theory is shown to be a conformal extension of Einstein general relativity. Dark matter originates from gauging out its local Weyl invariance as an extra degree of freedom which describes a potential flow of the pressureless perfect fluid. For a positive energy density of this fluid the theory is free of ghost instabilities, which gives strong preference to stable configurations with a positive scalar curvature and trace of the matter stress tensor. Instabilities caused by caustics of the geodesic flow, inherent in this model, serve as a motivation for an alternative conformal extension of Einstein theory, based on the generalized Proca vector field. A potential part of this field modifies the inflationary stage in cosmology, whereas its rotational part at the post inflationary epoch might simulate rotating flows of dark matter.
(Submitted on 13 Nov 2013)

... we have extended the Mimetic Dark Matter to mimic any background cosmology. This can be achieved by adding an appropriate potential V (φ) to the original metric. As simple examples we have discussed quintessence, inflation and bouncing universe with vanishing speed of sound for perturbations.
Further, we have found another interesting novel extension which allows for the nontrivial speed of sound. This can be achieved by adding higher-order derivative terms to the action without increasing the number of degrees of freedom in the system. This allows one to quantize the inflationary scalar perturbations using standard techniques. It is demonstrated that these perturbations can have novel observational features absent in the case of k-inflation models. In particular it is possible to strongly suppress the gravitational waves from inflation, seemingly without any non-Gaussianity. It would be very interesting to analyze whether one can observationally distinguish Mimetic Inflation from other models.
Finally, such a modification opens up a new interesting playground for modeling Dark Matter, where the speed of sound can be very small but not exactly vanishing and the behavior of the mimetic matter can deviate from the usual perfect-fluid-like dust.
(Submitted on 16 Mar 2014 (v1), last revised 14 Apr 2014 (this version, v2))

dimanche 14 septembre 2014

Heisenberg relation(s) for spectral non commutative spacetime?

The first Heisenberg-like commutation relation for a spectral non commutative spacetime / La première relation de commutation de type Heisenberg pour un espacetemps spectral non commutatif

In this letter we shall take equation (2), and its two sided refinement (4) below using the real structure, as a geometric analogue of the Heisenberg commutation relations [p,q]=i where D plays the role of p (momentum) and Y the role of q (coordinate) and use it as a starting point of quantization of geometry with quanta corresponding to irreducible representations of the operator relations. The above integrality result on the volume is a hint of quantization of geometry. We first use the one-sided (2) as the equations of motion of some field theory on M and describe the solutions as follows. (For details and proofs see [])... 
Each geometric quantum is a sphere of arbitrary shape and unit volume (in Planck units).
It would seem at this point that only disconnected geometries fit in this framework but this is ignoring an es-sential piece of structure of the NCG framework, which allows one to refine (2). It is the real structure J, an antilinear isometry in the Hilbert space H which is the algebraic counterpart of charge conjugation. This leads to refine the quantization condition by taking J into account as the two-sided equation

where E is the spectral projection for {1,i}⊂C of the double slash Y=Y+⊕Y-∈ C(M,C+⊕C-). It is the classification of finite geometries of [4] which suggested to use the direct sum C+⊕C-  of two Clifford algebras and the algebra C(M,C+⊕C-). It turns out moreover that in dimension 4 one has C+=M2(H) and C-=M2(C) which is in perfect agreement with the algebraic constituents of the Standard Model
(last revised 9 Sep 2014 (this version, v2))

Verstehen Sie nur Bahnhof? Vergessen Sie die Bahn und halten nur den Hof! / Vous n'y pigez que dalle? Oubliez la trajectoire et ne gardez que le halo 

Über den anschaulichen Inhalt der quantentheoretischen Kinematik und MechanikHeisenberg, 23. März 1927
//ajout d'une référence le 9 novembre 2014

samedi 13 septembre 2014

1814-2014 : 200 years of spectroscopy / 200 ans de spectroscopie

From the Fraunhofer spectral lines of Sun light... / Du spectre de raies solaires de Fraunhöfer 

[In 1814 Joseph von Fraunhofer invented the spectroscope,] looking for ways to check (and improve) the quality of telescopes he was making. He rediscovered the dark lines in the sun's spectrum while measuring the dispersive powers of various kinds of glass for light of different colors. As he worked on this project, he noticed that a bright 'orange' line (due to sodium, but he didn't know that) in the spectrum of the flame he was using was in the same position as the dark D-line (see below). This same line had been observed in flames from alcohol and sulfur as well as from candles. Fraunhofer mapped out the 574 thin black lines that he observed in the sun's spectrum. Eight of the most prominent lines were labeled A to G. Today, these lines are known as the Fraunhofer lines 

... to the Bekenstein-Mukhanov discrete spectrum of  Black Holes Hawking radiation? ... au spectre de Bekenstein-Mukhanov discret pour le rayonnement de Hawking des trous noirs

[The area quantization of 2d manifolds] can have far-reaching consequences for black holes and de Sitter space. In particular, the area of the black hole horizon must be quantized in integers of the Planck area (see also [8]). Because the area of a black hole of mass M is equal to 
A = 16πM2, 
this implies mass quantization
Mn = √n / 2 
As it was shown in [9] Hawking radiation in this case can be considered as a result of quantum transitions from the level n to the nearby levels n−1,n−2,... As a result even for large black holes Hawking radiation is emitted in discrete lines and the spectrum with the thermal envelope is not continuous. The distance between the nearby lines for large black holes is of order
ω = Mn −Mn-1 ≃ 1 / 4√n =1 / 8M 
and proportional to Hawking temperature, while the width of the line is expected to be at least ten times less than the distance between the lines [9]. Note that taking the minimal area to be α larger than the Planck area changes the distance between the lines by a factor α2. Thus area quantization can be experimentally verified if evaporating black hole will be discovered.
(last revised 9 Sep 2014 (this version, v2))

We develop the idea that, in quantum gravity where the horizon fluctuates, a black hole should have a discrete mass spectrum with concomitant line emission. Simple arguments fix the spacing of the lines, which should be broad but unblended. Assuming uniformity of the matrix elements for quantum transitions between near levels, we work out the probabilities for the emission of a specified series of quanta and the intensities of the spectral lines. The thermal character of the radiation is entirely due to the degeneracy of the levels, the same degeneracy that becomes manifest as black hole entropy. One prediction is that there should be no lines with wavelength of order the black hole size or larger. This makes it possible to test quantum gravity with black holes well above Planck scale.

(Submitted on 10 May 1995)

A spectral dream : quantum gravitation phenomenology  

last revised 8 Dec 2011 (this version, v2)

mardi 9 septembre 2014

From quantum black hole "atoms" of gravitation to quanta of spectral noncommutative spacetime / Des trous noirs quantiques, "atomes" de la gravitation, aux quanta d'espacetemps spectral non commutatif

Celebrating 40 years of quantum black hole physics... / Célébrons le quarantième anniversaire de la physique quantique des trous noirs...
Expressions are derived for the mass of a stationary axisymmetric solution of the Einstein equations containing a black hole surrounded by matter and for the difference in mass between two neighboring such solutions. Two of the quantities which appear in these expressions, namely the area A of the event horizon and the "surface gravity"; of the black hole, have a close analogy with entropy and temperature respectively. This analogy suggests the formulation of four laws of black hole mechanics which corre- spond to and in some ways transcend the four laws of thermodynamics.
J. M. Bardeen, B. Carter and S. W. Hawking 
24 Jan, 1973
Using well understood arguments from quantum theory we discuss the nature of the quantum spectra of the mass, charge, and angular momentum of the Kerr black hole. We argue that the mass spectrum is discrete, and infer a for the allowed mass levels by first pointing out the analogy between the squared irreducible mass of the Kerr hole and the action integral of mechanics, then quantizing the squared irreducible mass by the Bohr-Sommerfeld quantization rule... The result is consistent with the correspondence principle...
Jacob D. Bekenstein
QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that ... the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 1017s which is very long compared to the Planck time ≈ 10−43s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10−6(M/M)K where κ is the surface gravity of the black hole. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (M/M)−3s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe. Any such black hole of mass less than 1015g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs
S. W. Hawking,
01 March 1974

... the spectral noncommutative way! / ... à la mode spectrale non commutative !
In the construction of spectral manifolds in noncommutative geometry, a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of real scalar fields naturally appears and implies, by equality with the index formula, the quantization of the volume. We first show that this condition implies that the manifold decomposes into disconnected spheres which will represent quanta of geometry. We then refine the condition by involving the real structure and two types of geometric quanta, and show that connected manifolds with large quantized volume are then obtained as solutions. When this condition is adopted in the gravitational action it leads to the quantization of the four volume with the cosmological constant obtained as an integration constant. Restricting the condition to a three dimensional hypersurface implies quantization of the three volume and the possible appearance of mimetic dark matter. When restricting to a two dimensional hypersurface, under appropriate boundary conditions, this results in the quantization of area and has many interesting applications to black hole physics.
Ali H. Chamseddine, Alain Connes, and Viatcheslav Mukhanov,
Mon, 8 Sep. 2014

//retouches éditoriales le  13/09/14