mardi 15 mars 2016

"Black holes should be entirely consistent with the laws of quantum mechanics"

...the most promising alley towards further understanding of nature’s book keeping system near the Planck scale (?)

According to the classical picture of a black hole, it appears to be a sink that absorbs all matter aimed at it, without leaving a trace. The earliest descriptions of the quantum effects near a black hole, which lead to the marvellous conclusion that, due to vacuum polarisation, particles are emitted by a black hole [1], still suggested that quantum mechanics cannot prevent information to disappear. This, however, was quickly put in doubt [2][3]. Precisely because the laws of thermodynamics apply to black holes [4][1] , black holes must also be constrained to form quantum states with orthonormality and unitarity conditions. 
When this was realised, the author came with a possible scenario [5]. Particles going in, do have an effect on the Hawking particles coming out, by modifying their quantum states, in spite of the fact that their thermodynamical distribution remains unaffected. The explanation of this effect is that particles going in interact with the particles going out. Ordinary, Standard Model interactions are too weak to determine the quantum states of the out-going particles in such a way that a unitary evolution operator could emerge, but the gravitational interactions, paradoxically, are so strong here that they dominate completely. Indeed, their effects can cause the relation between in- and out-going states to be unitary... 
Recently, it was found that the black hole back reaction can be calculated in a more systematic fashion. We did have a problem, which is that the gravitational force attributes too much hair to a black hole. This is because the effects of the transverse gravitational fields could not yet be taken into account. Particles with too high angular momenta would contribute without bound, and this is obviously not correct. The search was for a method to give a transverse cut-off. A cut-off was proposed by Hawking et al [7], which is a good starting point, while, being qualitative, it does not yet provide us with the exact expression for Hawking’s entropy.
There is however an other question that can be answered: how can we separate the physical degrees of freedom near the horizon, so that each can be followed separately? This question was not posed until recently, and we found an astonishing answer [8]: we can diagonalise the information retrieval process completely, so that we can visualise how information literally bounces{*} against the horizon. Where, in previous calculations, a brick wall had been postulated [2], a brick wall emerges naturally and inevitably when the diagonalised variables are used. And there is more. We found that the degrees of freedom in region II of the Penrose diagram get mixed with the degrees of freedom in region I. This is an inevitable element of the theory: gravitational deformations of space-time due to the gravitational fields of the particles going in and out, cause transitions from one region into the other. The question how this can be understood physically was not answered in our last paper. Here, we answer it, with a caveat to be mentioned in the end...
Penrose diagram for Schwarzschild black hole, showing regions I and II , a particle going in in region # I , and particles going out in region I and region II . The shift caused by the in-particle is the same in both cases, but in region II the particle seems to go backwards in time. Since, in region II , the particle is shifted away from the event horizon, region II experiences the same particle as a negative energy one, or an annihilated particle.
The Hawking particles emerging from one hemisphere of the black holes, are maximally entangled with the particles emerging from the other hemisphere...
The central issue in our report is that the {scattering matrix used to describe the evolution between in and out going states} ... must be unitary, while it mixes the two regions of the Penrose diagram, so we have to conclude that region II also represents part of physical space-time. The only reasonable choice appears to be to identify region II with the antipodes...  
In our theory, something happens that has never been explicitly noticed: The arrow of time, in both regions, at all points near the horizon, must be taken to be the same as the external, Schwarzschild time, τ = t/4GM . This is contrary to standard practice [15] and also contrary to the author’s own earlier expectation. However, in order to keep unitarity, this is exactly what was done in the calculations reported about in this paper, and now we claim that calculations have to be done this way. It implies that, in terms of the Penrose coordinates, the identification involves a PT transformation [Presumably, this should be PCT, but in our formalism the notion of antiparticles was not yet introduced; including electromagnetism in our formalism may well clarify this point.]: time in region II goes backwards. The gravitational shift effects that we incorporate, bring us from region I into region II and back, and we would not have unitarity if we used the time coordinate employed by local observers. Indeed, using the τ coordinate, we get the desired feature that our identification procedure commutes with time translations.  
Note that, within our formalism, the ‘interior region’ of a black hole disappears altogether, so that no problems with firewalls can arise...
An elegant way to phrase the new proposed theory, is to say that, when the first trapped region opens up, we can regard it as a very tiny black hole, coming into existence via a very tiny gravitational instanton. The fact that this tiny instanton has antipodal identifications is a minute modification of space-time structure inside the trapped region; then, when the region opens up wide, the new configuration grows together with it. A local observer near the horizon, sees both Penrose’s regions I and II , not realising that region II is a (C)PT image of the antipodal part of the hole, since the same laws of physics apply there. This is why we say we do not violate general relativity with our identifications. 
We observed that antipodal identification of points on the horizon is inevitable if we want a unitary evolution operator. Formally, from the moment that a trapped space-time region forms, we must already identify antipodal points on the crossing point of future and past event horizons. Our point is that this remains invisible for ‘experiments’, until one waits to see the quantum effects of a decaying and vanishing black hole. Clearly, our familiar notions of space and time will have to be thoroughly revised. The advantage of our procedure of splitting things up in partial wave expansions, is that different partial waves are completely uncoupled, so that we are left with very simple, finite-dimensional quantum mechanics for each wave, where one can exactly see what is going on.  
The partial wave decomposition employed here should be distinguished from the usual partial wave decompositions in first- or second-quantized particle theories. We are forced to treat particles not as being point-like, but as forming a finite set of membranes that each take the shape of a partial wave. So it is not true that this would restrict total angular momenta of all particles any way. Although these partial wave have a classical appearance, we insist that they form legitimate representations of our operator algebra. They can be interpreted as a reformulation of the coordinates of all particles entering and leaving the black hole, a number that is roughly equal to R2 (in Planck units) [14]. The partial waves are then nothing but a band-limited mode decomposition as was done if Ref [15]. 
Also, one should not expect a majority of Hawking particles to emerge with l values close to the Planck limit. To the contrary, as was emphasised by Dvali [16], Hawking radiation in practice is dominated by S -waves, with small tails in higher l modes, which are strongly suppressed by their Boltzmann factors. It is the micro-states that we arrange according to their (l, m) values. 
Let us emphasise, once again, that each partial wave decouples from all other partial waves, and this fact should be seen as a major discovery. It enables us to form a very simple picture of the structure of space-time at or near the Planck scale, without having to take our refuge in functional variables and integrals, which often obscure things. One finds that space and time have exciting features. The most important problem has always been that, at the black hole horizon, the local observer must allow for unlimited Lorentz boosts. These cause gravitational back reactions that are also unlimited. We now have a handle to cope with that situation : it was discovered that in-going particles exchange position operators with momentum operators, to turn into out-going particles.
Let us also emphasise that hardly any ‘approximation’ has been made. Authors of other publications often belittle our results by claiming that it is merely a ‘classical approximation’ or something like that. To the contrary, the algebra on which it is based is very compelling, having been derived from impeccable physical arguments. Indeed, the physics is very accurate as soon as we look at l values well below the maximal limit (at the Planck scale). How exactly to perform the cut-off at the maximal values of l is not precisely understood today, but the Planckian regime has not yet been well understood by anybody. 
(Submitted on 14 Jan 2016 (v1), last revised 28 Mar 2016 (this version, v3))

* It is also important to emphasise that, even though we describe modes of infalling matter that “bounce back against the horizon”, these bounces only refer to the information our particles are carrying, while the particles will continue their way falling inwards as seen by a co-moving observer. In accordance with the notion of Black Hole Complementarity, an infalling observer only sees matter going in all the way, and nothing of the Hawking matter being re-emitted, since that is seen as pure vacuum by this observer. Rather than stating that this would violate no-cloning theorems, we believe that this situation is asking for a more delicate quantum formalism.
(Submitted on 5 Sep 2015)

//last edition 01 April 2016
//last edit August 24, 2016
Note that the bold typing of the text is mine.

mardi 8 mars 2016

Nous n'irons pas sans vous, Mesdames, à travers les espaces non commutatifs

Mairi Sakellariadou or the contemporary cosmologist as a math-physics boundary spanner 
To construct a quantum theory of gravity coupled to matter, one can either neglect matter altogether (as for instance in Loop Quantum Gravity), or consider instead that the interaction between gravity and matter is the most important aspect of the dynamics. The latter is indeed the philosophy followed in the context of Noncommutative Geometry, aiming at obtaining matter and gravity from (noncommutative) geometry [26]. The geometry is considered to be the tensor product of a continuous 4-dimensional geometry for space-time and an internal 0-dimensional geometry for the gauge content of the theory, namely the Standard Model of particle physics. The fruitful outcome of this approach is that considering gravity alone on the product space one obtains gravity and matter on the ordinary 4-dimensional space-time.  
The conventional way to identify the geometry of a given object is by measuring distances between its points. There is however a different approach proposed more than a century ago by Weyl, who suggested to identify a membrane’s shape 4 through the way it vibrates as one bangs it. Studying the frequencies of the resulting fundamental tone and overtones, one may get information about the membrane’s geometry. Following Weyl’s law, the largest frequencies of the sound of a membrane are basically determined by the area of the membrane and not by its shape, thus verifying Lorentz’s conjecture.  
In a paper under the intriguing title “Can one hear the shape of a drum?”, Kac has formulated the puzzle of whether one can reconstruct the geometry of a n-dimensional manifold (possibly with boundary) from the eigenvalues of the Laplacian on that manifold [27]. In other words, the relevant question is whether one (assuming he/she had perfect pitch) can deduce the precise shape of a drum just from hearing the fundamental tone and all overtones, even though one cannot really see the drum itself. Kac had stated that one most probably cannot hear the shape of a drum, nevertheless he investigated how much about the tambourine’s shape one can infer from the knowledge of all eigenvalues of the wave equation that describes the vibrating object (the membrane).  
From the mathematical point of view, the question posed by Kac can be formulated as “How well do we understand the wave equation?”. For ordinary Riemannian geometry, the shape of the Laplacian does not fully determine the metric, so the shape of the drum cannot be heard, implying that we do not have a complete understanding of the (rather simple) wave equation. But what about the product space considered within the Noncommutative Geometry context?  
Every space vibrates at certain frequencies, hence one may consider the universe as a vibrating membrane, a tambourine. Let us also consider that space is identified with the product space of the 4-dimensional space-time and the internal space, defined by a mathematical object, called the spectral triple given by the algebra of coordinates, the Hilbert space and the Dirac operator corresponding to the inverse of the Euclidean propagator of fermions. The dynamics of the spectral triple are governed by a spectral action [28] summing up all frequencies of vibration of the product space. One may thus ask whether he/she can hear the shape of this product space. The spectrum of the Dirac operator together with the unitary equivalence class of its noncommutative spin geometry, fully determine the metric and its spin structure. Hence, one may indeed hear the shape of a spinorial drum.  
Considering the spectral action and gravity coupled to matter, it was shown that one can obtain gravity and the Standard Model of elementary particles [29]. Thus, noncommutative spectral geometry offers a purely geometrical explanation for the Standard Model, the most successful particle physics model we still have at hand. 
Let us hence consider General Relativity as an effective theory and build a cosmological model based on Noncommutative Geometry along the lines of the spectral action. Given that this model lives by construction at very high energy scales, it provides a natural framework to construct early universe cosmological models. Studying these models one can test the validity of the Noncommutative Geometry proposal based on the spectral action, and in addition address some early universe open issues [30].
(Submitted on 15 Jul 2014)

Matilde Marcolli : the ronin of noncommutative geometry 
Cosmology is currently undergoing one of the most exciting phases of rapid development, with sophisticated theoretical modeling being tested against very accurate observational data for both the Cosmic Microwave Background (CMB) and the matter distribution in the Universe. This is therefore a highly appropriate time for a broad range of mathematical models of particle physics and cosmology to formulate testable predictions that can be confronted with the data 
While model building within the framework of string and brane scenarios and their possible implications for particle physics and cosmology have been widely developed in recent years, less attention has been devoted to other sources of theoretical high energy physics models that are capable of producing a range of predictions, both in the particle physics and cosmology context. It is especially interesting to look for alternative models, which deliver predictions that are distinguishable from those obtained within the framework of string theory. Particle physics models derived within the framework of Noncommutative Geometry recently emerged as a source for new cosmological models, [11], [45], [46], [51], [52], [53], [54].  
Among the most interesting features of these models of particle physics based on noncommutative geometry is the fact that the physical Lagrangian of the model is completely computed from a simple geometric input (the choice of a finite dimensional algebra), so that the physics is very tightly constrained by the underlying geometry.  
The features that link the noncommutative geometry models to areas of current interest to theoretical cosmologists are the fact that the action functional of these models, the spectral action, behaves in the large energy asymptotic expansion like a modified gravity model, with additional coupling to matter. Various models of modified gravity have been extensively studied by theoretical cosmologists in recent years. Another feature, which is particular to the noncommutative geometry models we consider here, is the fact that the nonperturbative form of the spectral action determines a slow-roll inflation potential, which shows a coupling of spatial geometry (cosmic topology) and the possible inflation scenarios. The model also exhibits couplings of matter and gravity, which provide early universe models with variable effective gravitational and cosmological constants.  
The results described in this paper are mostly based on recent joint work with Elena Pierpaoli [45], with Elena Pierpaoli and Kevin Teh [46], and with Daniel Kolodrubetz [38], as well as on earlier joint work with Ali Chamseddine and Alain Connes [21].