jeudi 22 janvier 2015

Wer's nicht glaubt (in der quantengravitation), zahlt einen Taler / Whosoever does not believe (in quantum gravity), must pay a euro

A tentative historical introduction for a book that it is currently possible to continue now:

Here, as [Matveï] Bronstein wrote, "solid lines correspond to existing theories, dotted ones to problems not yet solved" (Bronstein 1933b, p. 15)[*]. Relativistic quantum theory, that is, ch theory, was then the center of attention. But, as a rule, gravity was ignored. 
Bronstein also wrote about the quantum limits of general relativity in other papers, in the context of astrophysics and cosmology (see, for example, Bronstein 1933a). One phrase from Bronstein's popular article of 1929 (which concerned Einstein's attempt to unify gravitation and electromagnetism) reveals his general attitude to the problem of the quantum generalization of general relativity: "The construction of a space-time geometry that could result not only in laws of gravitation and electromagnetism but also in quantum laws is the greatest task ever to confront physics" (Bronstein 1929, p. 25). Thus, it is no surprise that Bronstein chose the quantization of gravitation as the topic of his dissertation (when the system of scientific degrees was introduced in the USSR). However, this choice was rather surprising in the contemporary scientific environment, because in the 1930s, fundamental theorists were concentrating on quantum electrodynamics and nuclear physics. 
Bronstein defended his thesis in November 1935. His examiners, the prominent Soviet theorists Vladimir Fock and Igor Tamm, praised the thesis highly as "the first work on the quantization of gravitation resulting in a physical outcome" (Gorelik and Frenkel 1985, p. 317). Bronstein's dissertation and two corresponding publications (Bronstein 1936a, 1936b) were mainly devoted to the quantization of weak gravitational fields, but one of the most important results is an analysis of the compatibility of quantum concepts and classical general relativity in the general case... 
Bronstein considered gravitation in the weak-field approximation (where it is described by a tensor field in Minkowski space) in accordance with Heisenberg and Pauli's general scheme of field quantization (Heisenberg and Pauli 1929). From quantized weak gravitation, he deduced two consequences: (1) a quantum formula for the intensity of gravitational radiation coinciding in the classical limit with Einstein's formula, and (2) the Newtonian law of gravitation as a consequence of quantum-gravitational interactions. 
... Bronstein's results had real significance, because the peculiar character of the gravitational field (namely, its identification with the space-time metric) gave rise to doubts about the possibility of synthesizing quantum concepts and general relativity. For example, Yakov I. Frenkel (the head of the theoretical department at the Leningrad Physical-Technical Institute, where Bronstein worked) was very sceptical about the possibility of quantizing gravitation, because he, like some other physicists, considered gravitation as a macroscopic, effective property of matter. It should be mentioned that even in the 1960s, Leon Rosenfeld supposed that the quantization of gravitation might be meaningless since the gravitational field probably has only a classical, macroscopic nature (Rosenfeld 1963)—and Rosenfeld was the first to consider a formula for the expression of quantized gravitation. On the other hand, Einstein's attitude was also well known. He believed that the correct full theory was separated from general relativity by a much smaller distance, so to speak, than from quantum theory.
Before obtaining the two mentioned results and just after deducing commutation relations, Bronstein analyzed the measurability of the gravitational field, taking into account quantum restrictions. The question of measurability had occupied an important place in fundamental physics since Heisenberg had discovered the restrictions on measurability resulting from the indeterminacy relations (restrictions on the measurability of conjugate parameters). As physicists looked toward the development of ch-theory (relativistic quantum theory), the question of measurability attracted great attention, especially after Landau and Peierls rejected in their 1931 paper the concept of an "electromagnetic field at a point," based upon its immeasurability within the ch-framework. This paper led to the detailed and careful analysis of the situation that was undertaken in 1933 by Bohr and Rosenfeld. They saved a local field description in quantum electrodynamics, but at the high price of assuming arbitrarily high densities of mass and electrical charge. 
Bronstein had an eye for this subject, and just after Bohr and Rosenfeld's paper, he summed up the situation in a short but clear note (Bronstein 1934). So it was quite natural that in considering quantum gravity Bronstein decided to analyze the problem of the measurability of the gravitational field within the cGh-framework. [...]

Bronstein wrote that the preceding considerations are analogous to those in quantum electrodynamics. But there arises here the essential difference between quantum electrodynamics and quantum gravity, because
in formal quantum electrodynamics, which does not take into consideration the structure of the elementary charge, there is no consideration limiting the increase of density ρ. With sufficiently high charge density in the test body, the measurement of the electrical field may be arbitrarily precise. In nature, there are probably limits to the density of the electrical charge... but formal quantum electrodynamics does not take these limits into account... The quantum theory of gravitation represents a quite different case: it has to take into account the fact that the gravitational radius of the test body must be less than its linear dimensions ... (Bronstein 1936b, p. 217)
Bronstein understood that "the absolute limit is calculated roughly" (in the weak-field framework), but he believed that "an analogous result will be valid also in a more exact theory." He formulated the fundamental conclusion as follows:
The elimination of the logical inconsistencies connected with this requires a radical reconstruction of the theory, and in particular, the rejection of a Riemannian geometry dealing, as we have seen here, with values unobservable in principle, and perhaps also the rejection of our ordinary concepts of space and time, replacing them by some much deeper and non evident concepts. Wer's nicht glaubt, bezahlt einen Taler. (Bronstein 1936b, p. 218) (The same German phrase concludes one** of the Grimm brothers' very improbable fairy tales).

Theoretical physicists are now confident that the role of the Planck values in quantum gravity, cosmology, and elementary particle theory will emerge from a unified theory of all fundamental interactions and that the Planck scales characterize the region in which the intensities of all fundamental interactions become comparable. If these expectations come true, the present report might become useful as the historical introduction for the book that it is currently impossible to write, The Small-Scale Structure of Space-Time. 
by Gennady Gorelik (1992)
Studies in the history of general relativity. [Einstein Studies. Vol.3]

* blogger's comment : in the Bronstein's diagram at the beginning of this post it is hard to miss the lack of a block labeled "Thermodynamics / Statistical physics" on the first line but this is another story for another post with the following question : is there any universal constant one should associate to the missing block, like a large integer N ...? 
(**addendum, 1st March 2015, the tale is titled "Vom klugen Schneiderlein" or The cunning little tailor, vielen Dank an meine Kollege Elizabeth M. ;-)
A possible first chapter of the book
Our knowledge of spacetime is described by two existing theories:
  • General Relativity
  • The Standard Model of particle physics
General relativity describes spacetime as far as large scales are concerned and is based on the geometric paradigm discovered by Riemann. It replaces the flat (pseudo) metric of Poincaré, Einstein, and Minkowski, by a curved spacetime metric whose components form the gravitational potential. The basic equations are Einstein equations... which have a clear geometric meaning and are derived from a simple action principle. Many processes in physics can be understood in terms of an action principle, which says, roughly speaking, that the actual observed process minimizes some functional, the action, over the space of possible processes...

This Einstein-Hilbert action SEH, from which the Einstein equations are derived in empty space, is replaced in the presence of matter by the combination (1) S = SEH + SSM where the second term SSM is the standard model action which encapsulates our knowledge of all the different kinds of elementary particles to be found in nature. While the Einstein-Hilbert action SEH has a clear geometric meaning, the additional term SSM ... is quite complicated (it takes about four hours to typeset the formula) and is begging for a better understanding... 

Our goal in this expository text is to explain that a conceptual understanding of the full action functional is now available (joint work with A. Chamseddine and M. Marcolli [6], [8]) and shows that the additional term SSM exhibits the fine texture of the geometry of spacetime. This fine texture appears as the product of the ordinary 4-dimensional continuum by a very specific finite discrete space F. Just to get a mental picture one may, in first approximation,think of F as a space consisting of two points. The product space then appears as a 4-dimensional continuum with “two sides”. As we shall see, after a judicious choice of F, one obtains the full action functional (1) as describing pure gravity on the product space M × F. It is crucial, of course, to understand the “raison d’être” of the space F and to explain why crossing the ordinary continuum with such a space is necessary from first principles. As we shall see below such an explanation is now available...
Alain Connes

A summary of the second chapter under way

... 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 the two-sided 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 M4to S4. The intuitive picture using the two maps from M4 to S4 is 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]).
(Submitted on 4 Nov 2014)

mardi 20 janvier 2015

The hitchhiker's guide to the whole quantum galaxy

Walking in the footsteps of Gravitation and the Standard Model...

The road to the resolution of the grand problem of theoretical physics – the search for a unified theory of all fundamental forces – does not come with many road signs. The work by Connes and coworkers on the standard model of particle physics, where the standard model coupled to general relativity is reformulated as a single gravitational model written in the language of noncommutative geometry, appears to be such a road sign. From the road sign, which we believe this formulation of the standard model is, we read of three travel advices for the road ahead: 
1. It is a formulation of fundamental physics in terms of pure geometry. Thus, it suggests that one should look for a unified theory which is gravitational in its origin.
2. The unifying principle in Connes formulation of the standard model hinges completely on the noncommutativity of the algebra of observables. Thus, it suggests that one should search for a suitable noncommutative algebra. 
We pick up the third travel advice from the fact that Connes work on the standard model coupled to general relativity is essentially classical. With its gravitational origin this is hardly a surprise: if the opposite was the case it would presumably involve quantum gravity and the problem of finding a unified theory would be solved. This, however, suggests: 
3. That we look for a theory which is quantum in its origin.
If we combine these three points we find that they suggest to look for a model of quantum gravity that involves an algebra of observables which is sufficiently noncommutative and subsequently arrive at a principle of unification by applying the machinery of noncommutative geometry. The aim of this review paper is to report on efforts made in this direction. In particular we shall report on efforts made to combine noncommutative geometry with canonical quantum gravity.
(Submitted on 28 Mar 2012)

... saying goodbye to our classical space-time relationship ?
Despite the realization that important conceptual problems are still affecting quantum mechanics as a fundamental theory of nature (the problem of measurement, the problem of time for covariant quantum theories, the conflicts with classical determinism) and despite the recurrent claims of a need for a theory that supersedes, modifies or extends quantum theory (hidden variables, several alternative interpretations, collapse of wave function, deterministic derivations of quantum theory), very few people have pointed out (perhaps because of a misinterpretation of N. Bohr’s correspondence principle) that quantum theory still now is essentially an incomplete theory, incapable of “standing on its own feet”. 
In all the current formulations of quantum theory, the basic degrees of freedom of a theory are specifically introduced “by hand” and make always reference to a classical underlying geometry. 
The choice of the degrees of freedom is usually done in elementary quantum mechanics through “quantization procedures” via the imposition of Weyl (or Heisenberg) commutation relations for conjugated observables starting from a classical pair of position and momentum as in Dirac canonical quantization and, more generally, as in Weyl quantization, associating quantum observables starting from functions living on a classical phase-space. 
In quantum field theory of free fields again, the local Weyl algebras are obtained by second quantization from symplectic spaces that originate from propagators of hyperbolic operators living on a classical Lorentzian manifold.  
Even in algebraic quantum field theory, the most abstract axiomatization available, we have seen that there is always an underlying classical space-time as an indexing base of the net of quantum observables or a category of classical space-time geometries as a domain of the C∗-algebra-valued functor that defines the theory. 
What is even more incredible is that, although it is widely recognized that many of the problems encountered in quantum field theory can be traced back to “unhealthy usage” of classical notions of space-time manifolds (divergences, convergence of Feynman n integrals), the previous problem of intrinsic characterization of degrees of freedom “inside quantum theory” has rarely, if ever, been seriously considered (with some notable exceptions) and that most of the attempts to cure the problem are simply trying to substitute “by hand” commutative classical geometries with non-commutative counterparts without addressing the issue from within quantum physics. 
Among those really notable exceptions in this direction (see the companion review [30, Section 5.4] for more details), we have to mention the few attempts to spectrally reconstruct spacetime from operationally defined data of observables and states in algebraic quantum field theory starting from U. Bannier [18] and culminating in S.J. Summers, R. White [243] and, at least form the ideological point view, the efforts in information theoretical approaches for an operational definition of quantum theory as, for example, in A. Grinbaum [127, 128, 129]. 
A very interesting passage, asserting that in a fundamental theory the notion of space-time must be derived a posteriori and understood through relations between “quantum events”, can be found in R. Haag [132, Section VII, Concluding remarks]. Our main ideological motivation for the work on “modular algebraic quantum gravity” presented here comes from the view [25, 27, 28, 30] that:
space-time should be spectrally reconstructed a posteriori from a basic operational theory of observables and states; A. Connes’ non-commutative geometry provides the natural environment where to attempt an implementation of the spectral reconstruction of space-time; Tomita–Takesaki modular theory should be the main tool to achieve the previous goals, associating to operational data, spectral non-commutative geometries.
A very interesting recent work by C. Rovelli, F. Vidotto [218, see Section III B] is making some clear progress in the direction of the first two claims above obtaining relations between entropy of graphs describing the quantum geometry of loop gravity and the “spectral geometry” given by the non-relativistic Hamiltonian of a single particle interacting with the gravitational field. Still the deepest and at the same time most visionary proposal for a program aiming at the reconstruction of space-time via interacting events from purely quantum theoretical constructs is the one that has been given by C. Rovelli [214, Section 5.6.4] inside the framework of his relational theory of quantum mechanics [215, 237]. 
The modest ideas suggested here might be seen as just a very partial attempt to set up a mathematical apparatus capable of implementing C. Rovelli’s intuition, merging it with noncommutative geometry and utilizing instruments from category theory, higher C∗-categories and Fell bundles, in order to formulate a theory of relational quantum mechanics. [...] 
Finally we must stress that, contrary to most of the proposals for fundamental theories in
physics that are usually of an ontological character, postulating basic microscopic degrees of freedom and their dynamics with the goal to explain known macroscopic behaviour, our approach (if ever successful) will only provide an absolutely general operational formalism to model information acquisition and communication/interaction between quantum observers (described via certain categories of algebras of operators) and to extract from that some geometrical data in the form of a non-commutative geometry of the system. Possible connections of these ideas to “quantum information theory” and “quantum computation” are also under consideration [25].
(Submitted on 23 Jul 2010 (v1), last revised 19 Aug 2010 (this version, v2))

lundi 19 janvier 2015

Quo vadis dark matter theoretical research ?

The stuff some dreams of quantum gravity are made of ...
The origin of Dark Matter (DM) is one of the oldest and biggest puzzles in cosmology and particle physics. Surprisingly, even basic macroscopic nature of DM is not yet understoodindeed, we do not know whether DM is a gas of some particles beyond the Standard Model (SM) {... composed from the so-called Weakly Interacting Massive Particles (WIMPs) [1], sterile right-handed neutrinos [2] and even different and some times numerous copies of the SM, see e.g. [3, 4] just to mention few options. Whether the axion DM [5, 6] represents a condensate or not is still debated, for the most recent discussion see [7]. DM can also be} a gas of primordial black holes see e.g. [8, 9], or some other macroscopic objects, see e.g. [10], or some fluid e.g. [11] or Bose-Einstein condensates – some classical scalar fields e.g. [12], [13, 14] or even some effective solid [15]. In the latter approach where one assumes high occupation numbers of some new fields, we can also incorporate the relativistic version of MOND [16] - TeVeS [17] and numerous others modifications of general relativity (GR), e.g. [18]. The simplest modification of GR can be achieved by promoting it to a scalar-tensor theory. 
GR enjoys a very powerful symmetry – diffeomorphism invariance. One of the manifestations of its power is that one can parametrize the metric gµν by a scalar field ϕ and an auxiliary metric lµν in a general disformal way [19 

gµν = C(ϕ,X)lµν +D(ϕ,X)ϕ,µϕ,ν         (1.1)

where X = 1/2*lµνϕ,µϕ,ν and C(ϕ,X) and D(ϕ,X) are free functions, and obtain the Einstein equations (for gµν) by variation of the action with respect to ϕ and lµν instead of gµν, see [25]. The only exception from this rule corresponds to a singular parameterisation when [25] 

D(ϕ,X)= f (ϕ)− C(ϕ,X)/2X .           (1.2) 

When the transformation is singular, there are new degrees of freedom and new physics modifying GR. Mimetic Dark Matter [26] is one of the theories of this type and makes use of the transformation (1.1) with C =2X and D =0. It is important that in this case the system is Weyl invariant with respect to the transformations of the auxiliary metric lµν. Soon it was realised that Mimetic Dark Matter is equivalent to the fluid description of irrotational dust [27, 28] with the mimetic field ϕ playing the role of the velocity potential. Models of this type also appear in the IR limit of the projectable version of Hořava-Lifshitz gravity [29, 30, 31] and correspond to a scalar version of the so-called Einstein Aether [32]. Surprisingly these models can also emerge in the non-commutative geometry [33]. In [34] this class of systems was further extended by i) adding a potential V(ϕ) which allows to obtain an arbitrary equation of state for this dust-like matter with zero sound speed, as it was done earlier in [35]; ii) by introducing the higher derivatives (HD) which provide a nonvanishing sound speed. The latter modification allowed one to study inflationary models with the creation of quantum cosmological perturbations. Moreover, this finite sound speed can suppress the structure on small scales [36] and have other interesting phenomenological consequences. 
(Submitted on 22 Dec 2014)

samedi 17 janvier 2015

Finite naturalness of a scalar singlet extension of the standard model ?

Exploring an "ugly" model with very small  couplings to the Standard Model which does not care about quadratically divergent quantum corrections...
... like the "good" supersymmetric universe but evades the "bad" multiverse solution:
The naturalness principle strongly influenced high-energy physics in the past decades [1], leading to the belief that physics beyond the Standard Model (SM) must exist at a scale ΛNP  such that quadratically divergent quantum corrections to the Higgs squared mass are made finite (presumably up to a log divergence) and not much larger than the Higgs mass Mh itself. This ideology started to conflict with data after TeVatron measured the top mass (which implies a sizeable order-one top Yukawa coupling λt) and after LEP excluded new charged particles below 100 GeV [2]. Indeed, imposing that the SM one loop correction to M2h... is smaller than M2h×∆, where ∆ is the amount of allowed fine-tuning (ideally ∆<∼1), implies (computing the fine tuning  with respect to high-scale parameters) ΛNP <∼√∆× 50GeV . 
The most plausible new physics motivated by naturalness is supersymmetry. [...] 
However, no new physics has been so far seen at LHC with √s = 8 TeV, such that, in models that aim to be valid up to high energies, the unit of measure for ∆ presently is the kilo-fine-tuning. While this is not conclusive evidence, while special models that minimise fine-tuning are being considered, while naturalness arguments can be weakened by allowing for a finer tuning, while various searches have not yet been performed, while LHC will run at higher energy, etc, it is fair to say that the most straightforward interpretation of present data is that the naturalness ideology is wrong. 
This situation leads to consider the opposite extremum: the Higgs is light due to huge cancellations [4] because ‘anthropic selection’ destroyed naturalness. 
Here we explore an intermediate possibility, that sometimes appeared in the literature, more or less implicitly. We name it ‘finite naturalness’. The idea is that we should ignore the uncomputable quadratic divergences, so that the Higgs mass is naturally small as long as there are no heavier particles that give large finite contributions to the Higgs mass.[...] 
Anyhow, the goal of this paper is not advocating for the ‘finite naturalness’ scenario. Instead, we want to explore how experiments can test if it satisfied in nature. [...] 
Another experimental result which might suggest the presence of physics beyond the SM is the fact that the SM potential (for the currently favored values of the Higgs and top masses) develops an instability at field values above about 108 GeV, leading to vacuum decay with a rate much longer than the age of the universe [7]. 
There are many ways to avoid this instability, which employ loop corrections from new particles with sizeable couplings to the Higgs [...]. Thereby, in the context of finite naturalness, this kind of new physics is expected to be around the weak scale. 
This is however not a general conclusion. Indeed there is one special model where the instability is avoided by a tree level effect with small couplings. Adding to the SM a scalar singlet S with interactions to the Higgs described by the potential [38] 
 V = λH(HH −v2)2 + λS(SS −w2)2 + 2λHS(HH −v2)( SS −w2)   
the low-energy Higgs quartic coupling is given by λ = λH −HS2/λS at tree level. This model allows to stabilize the SM vacuum compatibly with ‘finite naturalness’ even if the singlet is much above the weak scale, provided that the couplings λHS and λS are small. A singlet with this kind of couplings is present within an attempt of deriving the SM from the framework of non commutative geometry [39]. 
Finally, observations of cosmological inhomogeneities suggest that the full theory incorporates some mechanism for inflation. At the moment the connection with the SM is unknown, even at a speculative level. A successful inflaton must have a flat potential, which is difficult to achieve in models; at quantum level flatness usually demands small couplings of the inflaton to SM particles. An inflaton decoupled from the SM would satisfy ‘finite naturalness’. A free scalar S with mass M≈1013 GeV is the simplest inflaton candidate; it satisfies finite naturalness provided that its couplings to the Higgs λHS is smaller than about 10-10. It is interesting to notice that this roughly is the maximal mass compatible with ‘finite naturalness’: at three loops gravity gives a finite correction to the Higgs mass, δm2 ∼y2M6/M4PL(4π)6 [we thank A. Arvinataki, S. Dimopoulos and S. Dubovsky for having noticed and pointed to us such effect]. 
(Submitted on 28 Mar 2013 (v1), last revised 29 Apr 2014 (this version, v3))

mardi 13 janvier 2015

New hopes for a noncommutative dual UV/IR phenomenological manifestation ?

From noncommutative quantum field theory ...
In this paper we have proven that the real φ4-model on (Euclidean) noncommutative R4 is renormalisable to all orders in perturbation theory. The bare action of relevant and marginal couplings of the model is parametrised by four (divergent) quantities which require normalisation to the experimental data at a physical renormalisation scale. The corresponding physical parameters which determine the model are the mass, the field amplitude (to be normalised to 1), the coupling constant and (in addition to the commutative version) the frequency of an harmonic oscillator potential. The appearance of the oscillator potential is not a bad trick but a true physical effect. It is the self-consistent solution of the UV/IR-mixing problem found in the traditional noncommutative φ4-model in momentum space. It implements the duality (see also [4]) that noncommutativity relevant at short distances goes hand in hand with a modified structure of space relevant at large distances. Such a modified structure of space at very large distances seems to be in contradiction with experimental data. But this is not true. Neither position space nor momentum space are the adapted frames to interpret the model. An invariant characterisation of the model is the spectrum of the Laplace-like operator which defines the free theory. Due to the link to Meixner polynomials, the spectrum is discrete. ... we see that the spectrum of the squared momentum variable has an equidistant spacing of 4Ω/θ . Thus, √(4Ω/θ) is the minimal (non-vanishing) momentum of the scalar field which is allowed in the noncommutative universe. We can thus identify the parameter √Ω with the ratio of the Planck length to the size of the (finite!) universe. Thus, for typical momenta on earth, the discretisation is not visible. 
(Submitted on 20 Jan 2004 (v1), last revised 4 Oct 2004 (this version, v2))

... to spectral noncommutative geometry
The spectral action delivers a huge mass term [of the Higgs] and one can check that it is consistent with the sign and order of magnitude of the quadratic divergence of the self-energy of this scalar field. However though this shows compatibility with a small low energy value it does by no means allow one to justify such a small value. Giving the term − 2H2 at unification scale and hoping to get a small value when running the theory down to low energies by applying the renormalization group, one is facing a huge fine tuning  problem. Thus one should rather try to find a physical principle to explain why one obtains such a small value at low scale. In the noncommutative geometry model M × F of space-time the size of the finite space F is governed by the inverse of the Higgs mass. Thus the above problem has a simple geometric interpretation: Why is the space F so large (by a factor of 1016) in Planck units? There is a striking similarity between this problem and the problem of the large size of space in Planck units. This suggests that it would be very worthwhile to develop cosmology in the context of the noncommutative geometry model of space-time, with in particular the preliminary step of the Lorentzian formulation of the spectral action.
This also brings us to the important role played by the dilaton field which determines the scale in the theory. The spectral action is taken to be a function of the twisted Dirac operator so that D2 is replaced with e−φD2e−φ. In [10] we have shown that the spectral action is scale invariant, except for the dilaton kinetic energy. Moreover, one can show that after rescaling the physical fields, the scalar potential of the theory will be independent of the dilaton at the classical level. At the quantum level, the dilaton acquires a Coleman-Weinberg potential [21] and will have a vev of the order of the Planck mass [22]. The fact that the Higgs mass is damped by a factor of e−2φ, can be the basis of an explanation of the hierarchy problem.
(Submitted on 30 Nov 2008)

"It's very important to have a mental picture of what it's going on now ... it's very surprising why spacetime is macroscopic because when you write any type of equation in [quantum] physics you find something with a Planck size... and the idea is that eventually when we do cosmology  the picture that is emerging [from our most recent work] is that the ...[spacetime] manifold M is like a butterfly ... folded in the product of two  spheres of [Planckian] size but itself it has a macroscopic size and it unfolds... and one amazing thing is when you write down these maps [from M] to the spheres with their degrees [or winding numbers, related to the number of quanta of  geometry] you find out there are natural operations in mathematics which are coming from self maps of the sphere. Now what is the effect of these operations on the butterfly ? They multiply the volume [of spacetime] by a constant ... and of course this is extremely reminiscent of e-folding in inflation..."
Alain Connes (17.12.2014) (oral transcript by the blogger)

vendredi 2 janvier 2015

2015 : Ready to unravel the maze of dark matter and cosmological inflation...

... following the Ariane thread of a spectral noncommutative "quantum gravity" ?
In the following video (from 17 december 2014) Alain Connes explains how a quantized 4D euclidean spacetime dressed with (a Pati Salam asymptotically free extension of) the Standard Model emerges from an operator algebraic Heisenberg like equation first proposed in this article from september 2014.

The video is a lively exposition of the mathematical work that appeared in the following paper with hopefully inspiring ideas for physics in the last quarter of the talk (specifically from 1h37'48'' when Connes gives literally a hand-waving argument how a macroscopic spin manifold can be folded inside the products of two Planck sized 4D spheres). "Having established the mathematical foundation for the quantization of geometry, we shall present consequences and physical applications of these results in a forthcoming publication" is the ending sentence of the aforementioned paper : needless to say expectations are high for this new year that the blogger wishes rich in delight and discoveries for all his readers!