samedi 5 novembre 2016

The dream of a direct experimental test of Connes or rather "causal cones" theory is trembling

From mathematical concepts to physical models and "Euclidean" almost commutative geometries to "Lorentzian" almost commutative spacetimes

The set-up of the Dirac equation [1] was a prodigious step towards the unification of quantum and relativistic principles. Surprisingly enough, the differential operator engaged in the Dirac equation turned out to play a pivotal role in differential geometry [2]. Moreover, it lies at the heart of Connes’ theory of noncommutative geometry [3, 4], which extends such classical notions as differentiation, distance [3] and causality [5] to an abstract algebraic setting. Nowadays, noncommutative geometry centred around the concept of the Dirac operator provides a compelling framework for the study of fundamental interactions [6], yielding concrete testable predictions in the domain of elementary particles [7, 8] and gravitational physics [9, 10]. 

Building upon Connes’ ideas [11] we model a single massive Dirac fermion with the help of an almost commutative spacetime. The latter turns out to provide a geometric description of one of the peculiarities of the Dirac theory – the Zitterbewegung [12]. We show that the causal structure of the almost commutative spacetime at hand puts an explicit bound on the frequency of the ‘trembling motion’ of a single Dirac fermion propagating in a possibly curved spacetime and interacting with the electromagnetic and a Higgs-like scalar field. Next, we explain how the concept of quantum simulation [13] can be promoted to emulate almost commutative spacetimes, thus opening the door to a direct experimental test of Connes’ theory.


The original explanation of the mechanism behind Zitterbewegung given by Schrödinger [12], and refined by several authors [14, 16, 17], relates it to the interference between the positive and negative energy parts of the Dirac wave packet... 

The tangibility of Zitterbewegung for actual fermions ... is generally questioned [18] (see however [19]). On the other hand, the realness of this effect has been confirmed in various Dirac-like systems [15, 20, 21] and was found responsible for the appearance of a minimal conductivity and a sub-Poissonian shot noise in graphene [22, 23].  

There exists an alternative viewpoint on Zitterbewegung (known also under the name of ‘chiral oscillations’ [24, 25]) relating it to the spin components of the fermionic wave function [26–30]. Any Dirac spinor can be uniquely decomposed as a sum of two Weyl spinors ψ=ψ+... The Weyl spinors ψ±, being eigenstates of the chirality operator γ5...   have opposite chirality. By acting with the projector Pa± on Equation (1) we obtain iγµµψ± = mcψ∓ ... which can be seen as two coupled equations for Weyl spinors ψ±, one acting as a source of the other [30]. Since Weyl spinors are massless, they move with the speed of light and one obtains a ‘zigzag picture’ of a massive fermion [29, Figure 25.1]. In the fermion’s rest frame, the period of oscillations between the two eigenstates of chirality precisely equals TZB = π/(mc2). [28].

The two points of view on Zitterbewegung are closely related in the Dirac wave packet formalism [25]. In particular, purely positive/negative energy solutions to the Dirac equation also do not exhibit chiral oscillations [24].  

Hestenes contended [26–28] that the ‘chiral’ interpretation of Zitterbewegung is more natural, as the origin of the effect resides in the geometry of spacetime. In the present Letter we support this claim, although we argue that the very notion of geometry needs to be refined... 
The basic objects of noncommutative geometry à la Connes [3] are spectral triples (A, H, D) consisting of a (dense subalgebra of a) C* -algebra A, a Hilbert space H with a faithful representation of A and an unbounded self-adjoint operator D acting on H. The original framework was designed to describe spaces of Euclidean signature and has recently been extended to encompass the Lorentzian ones [3136]. In the latter case, the main conceptual change consists in endowed the Hilbert space H with an indefinite inner product, turning it into a Krein space K [37] — a vector space equipped with an indefinite non-degenerate inner product — and in requiring D to be self-adjoint with respect to the indefinite inner product.

"Connes theory" could mean "theory of causal cones" in spectral noncommutative geometry
Causality is one of the most fundamental principles underlying physical theories. Within Einstein’s theory it is defined as a partial order relation on the set of events: pq means that q lies in the future of p. However, noncommutative spaces (not only in the framework of spectral triples) typically admit only a global description and the very notion of an event does not make sense. This raises the question: what is the scene for causal relations and what is an operational meaning of a ‘noncommutative spacetime’? 
In [5] we put forward the idea that a ‘noncommutative spacetime’ ought to be understood as the space of (pure) states of a, possibly noncommutative, C* -algebra. The motivation behind this step is twofold: Firstly, if the algebra at hand is of the form AM=Cc(M) ..., then its pure states P(AM) are in one-to-one correspondence with the events in M[5]. Secondly, C*-algebras provide a unified framework for an operational formulation of both classical and quantum physics [39–41]. In this context, states on an abstract C*- algebra of observables can be understood as the actual states of a given physical system. For instance, if AF= Mn(C) ..., then pure states in P(AF) are precisely the n-qubits, whereas the mixed states S(AF) correspond to the density matrices [41]. In [5] we have shown that such a ‘noncommutative spacetime’ admits a sensible notion of a causal structure associated with a Lorentzian spectral triple. To this end, one has to identify a specific subset C of the algebra A named the ‘causal cone’... 
The space of pure states of an almost commutative geometry has a particularly pellucid physical interpretation: Let A =AMAF ..., then P(A)≃M×F for some finite dimensional space F. In other words, all pure states on A are separable [43]. Hence, the space of pure states of an almost commutative spacetime is the Cartesian product of the spacetime M and an ‘inner’ space of states of the model.
... in almost commutative spacetimes, Einstein’s causality in the {commutative} spacetime component is not violated. On the other hand, in [38] and [44] we discovered that the extended causal structure imposes highly non-trivial restrictions on the evolution also in the ‘inner’ space of the model. We shall now apply the mathematical results obtained in [44] to lift the veil on the nature of Zitterbewegung...
Connes first observed [11] that the (Euclidean version of the) above almost commutative model provides a geometric viewpoint on Zitterbewegung. However, Connes remark was only qualitative and focused on regarding the Higgs field as a gauge boson operating on the finite space {−,+} (see also [50]). We discovered that taking into account the Lorentzian aspects of this model reveals a deeper, quantitative, connection between geometry and the ‘trembling motion’ of fermions (cf. [44, Theorem 9]):
Theorem 6.  Let τ(γ) be the proper time along a causal curve in M. Two states (p,−),(q,+) ∈ P(A) are causally related with (p,−)(q,+) if and only if there exists a causal curve γ giving pq in M, such that τ(γ)≥π/(2|µ|c2).  
Restoring the physical dimensions in the model (which is unambiguous as slashed D has the dimension L-1 , thus DF must have so), we arrive at: 
(p,−)(q,+)   ⇔     pq and τ (γ) ≥ π/(2|µ|c2). (6)
The number on the RHS of (6) is precisely half of the Zitterbewegung period of a Dirac fermion of mass |µ| = m. It is striking to realise that the possibility of Zitterbewegung is encoded in the geometry of a purely classical model. The bound on the frequency of the fermion’s quivering is of kinematic origin – we have invoked {a fermionic} action... only to identify the relevant degrees of freedom. The apparent abrupt change of state implied by Theorem 6 becomes more transparent when one considers the subspace of mixed states of indefinite chirality. In the space M×[−1,+1] ⊂ S(A) the boundary of the causal cone becomes a continuous surface [44, Fig. 2] permitting a smooth evolution of the expectation value of chirality.  
The advantage of the presented model is its general covariance, which guarantees that Theorem 6 applies in any globally hyperbolic spacetime M. But the framework of noncommutative geometry is even more flexible and allows one to accommodate, via the ‘fluctuations’ of the Dirac operator, other fields interacting with the fermion [8]...

Emulating almost commutative spacetimes ?
In an (analogue) quantum simulation some aspects of the dynamics of a complicated quantum system are mimicked in a simpler one, which is under control [13]. Given an almost commutative spectral triple... together with {a} fermionic action... one can extend this concept to probe almost commutative spacetimes in table-top experiments... 
The quantum simulation of a single free Dirac fermion in a flat 2-dimensional spacetime has been successfully accomplished with cold atoms [15], trapped ions [20] and photonic systems [21]. Furthermore, a suitable experimental setup has been proposed using superconductors [54], semiconductors [55, 56] and graphene [57]. In the trapped-ion setting [20, 58], the mass of the simulated fermion can be introduced dynamically, what enables a simulation of a Dirac fermion coupled to a (real) scalar field. Moreover, a possibility of studying the impact of an external electromagnetic field on Zitterbewegung using the framework of [20] was suggested in [59]. 
In the presented almost commutative model... Zitterbewegung in the wave packet formalism is not a single-frequency oscillation [16]. Nevertheless, given concrete initial and final states of a simulator system one can  unravel the consequences of {the Connes theory} drawing from the fact that the causal cone is completely characterised in [44] and suitable computational tools to handle mixed states were devised in [60]... 
For a free electron the period of Zitterbewegung... is of the order of 10-20s, which is well beyond the currently available experimental time resolution. Moreover, to model a genuine electron one would need to employ the quantum field theoretic description, which seems to exclude Zitterbewegung [18], at least in flat spacetimes [19]. The scheme presented in this Letter suggests, however, that the very foundations of quantum gauge theories might need to be refined. The principle of micro-causality — requiring the observables in space-like separated regions to commute — is at the core of all axiomatic approaches to quantum field theory [39, 61]. If the fields have additional degrees of freedom then their background geometry is that of an almost commutative spacetime. Consequently, when constructing a quantum theory of fields, one should take into account its inherent causal structure. 
The concept of causality in the space of states is at the core of the presented model. When applied to other almost commutative spacetimes (see [38] for another gauge model), it might cast a new light on such perplexing phenomena as neutrino or quark mixing, which also involve a ‘motion’ in the fermion’s internal space. Finally, one can reach beyond the almost commutative setting and study the causal structure of genuinely noncommutative spacetimes [62]. This perspective suggests that, contrary to the common belief [63, 64], causal structure need not breakdown at the Planck scale, but the very notion of spacetime geometry needs to be refined.

jeudi 1 septembre 2016

(Spacetime models) For your eyes (but not) only

//This post has been slightly edited on 17 September 2016

Defending another point of view on Solid Theoretical Research in Natural Geometric Structures
Here is a selection of videos from 2014 to 2016 (plus a bonus older audio file at the end) gathered as interesting representatives of most recent progress in spectral (mostly noncommutative) geometry to improve our models in spacetime with potential applications in high energy physics beyond the minimal Standard Model. Click on the title of each video to watch it.

Speaker: Achim Kempf (09/2015)

Speaker: Fedele Lizzi (05/2014)

Speaker: John W. Barrett (12/2015)

The standard model from non-commutative geometry: what? why? what's new?
Speakers: Latham Boyle and Shane Farnsworth (05/2015)

Spectral action and cosmology
Speaker: Mairi Sakellariadou (04/2016)

Spectral Geometric Unification
Speaker: Ali Chamseddine (06/2014)

Noncommutative geometry and physics (123)
Speaker : Alain Connes (03/2016)

And last but not least here is a 10 year-old "golden oldie" conference this post aims at celebrating but for your ears only since it is the proper tool to hear the music of spheres (click on the title below to listen to the audio file)

Noncommutative geometry and the standard model with neutrino mixing

Summarizing a ten year old achievement...
... namely the complete derivation of the Lagrangian density of the minimal Standard Model exactly as given in Veltman’s book Diagrammatica through the thorough study of the spectral action on a specific almost-commutative manifold to model spacetime. From the following compact expression that synthesizes the spectral noncommutative geometric understanding of the Standard Model of particle physics

one recovers after a patient computation the long equation below compiled by generations of physicists...

Full details are available here (starting from p 217).

mercredi 31 août 2016

A view to a Kill (a farewell to continuous spacetime?)

Light rays to illuminate (the geometrodynamics behind) the entangled dark body radiation of black holes  
This post will be the penultimate of my August-series with titles constantly borrowed from the James Bond films list. 
More seriously it is also the follow up of the last one as announced in its ending picture caption. I try to convey once more the physical insight of the 't Hooft vision on black holes with his own worlds but using texts I have not quoted yet and which are more synthetic. 

what happens to the quantum information carried by particles that enter the black hole, soon to be absorbed by the central singularity? Do they wiggle their way out again, or at least the quantum information carried along with them? When you take the existing theory and equations literally, there is no chance that they can wiggle out. Should we ignore this?  
First of all, we claim that this problem is a much more elementary and important one than what the literature suggests, and that most of the suggested cures are not well enough thought through. What should be done, is to stretch as much as is possible what one can do using standard, mainstream physics. Then, finally, impose the demand that black holes should behave, as much as possible, as ordinary forms of matter. When doing this properly, this leads to amazing observations.
Only few people noted that the application of existing physical knowledge can bring us much further than any wild speculation. Only recently we discovered a way to calculate things that could have been applied decades earlier. Our observation was [22] that we can use a partial wave expansion to describe energy and momentum entering the black hole, and that, applying Einstein’s equations, the information carried in by each partial wave is carried out again, in a partial wave with the same values of the quantum numbers ℓ and m 
This calculation seems to be as elementary as the calculation of the spectrum of a hydrogen atom using quantum mechanics. As in the hydrogen atom, the boundary conditions are of crucial importance, but in a black hole, these boundary conditions are quite counter intuitive 
What this calculation does, is to expose our problem more clearly than ever before, and now we can see what the answers must be. The general coordinate transformation relating the “inside” of a black hole with what an outside observer sees, must be a topologically nontrivial one. [23,24,25] Only then do the equations make sense.  
This does remarkable things with the fabric of space and time itself, not noticed before. For one thing, space and time must be discrete.
Gerard ’t Hooft (10 June 2016)

For the quantization of gravity, it is crucial to understand the role of (real as well as virtual) black holes. The region in the immediate vicinity of the horizon, is characterised by the fact that time translations are substituted by Lorentz transformations. This means that observables near the horizon are undergoing unlimited processes of Lorentz contraction. 
Light rays are essential for defining the exact location of a black hole horizon: it is the boundary that separates regions from which escape to the outside universe by light rays is still possible, from the domain where all light rays are trapped. We shall now show how to use light rays to describe the backreaction of Hawking particles upon the presence of matter entering the black hole. 
The density matrix, that is, the probability distribution, of Hawking particles basically only depends on mass, charge and angular momentum of the black hole, but the actual configuration of the out-going particles characterises the microstate of the black hole, of which we have a large number (≈ e4πMin natural units) of distinct elements [The number 4πM2, corresponding to the black hole entropy, also represents the total number of Hawking particles emitted by a black hole in its life time [14]]. Whenever a particle enters the hole from outside, transitions to different microstates take place. This happens because a particle entering a black hole interacts with what comes out. The most disruptive interaction is the gravitational one, in spite of its apparent weakness... 
The gravity field surrounding a particle causes a very slight Shapiro effect: a passing light beam is slowed down and dragged towards the particle [15]. For a particle at rest, this effect is very small, but when the particle is Lorentz boosted, its gravity field increases in strength. At the black hole horizon, this Lorentz boost enhances its energy exponentially in time. 
Let pµin(Ω) be the momentum distribution of the in-going particles at the spot Ω=(θ,φ) on the horizon, and δxµout(Ω) the Shapiro displacement for the Hawking particles going out. An elementary calculation shows that 
δxµout(Ω)=8πG ∫d2Ω′f(Ω,Ω′)pµin(Ω'),     (4.1)
where f(Ω, Ω′) is a Green function, defined by ∆f(Ω,Ω′)=−δ2(Ω,Ω′) , and here, the angular operator ∆S is defined by
S=∆−1 = −ℓ(ℓ+1)−1 .                          (4.2) 
Often, we ignore the −1. It was further assumed that the momentum pµin is dominated by the radial component pin, so that, also, its derivatives w.r.t. θ and φ are small. Thus, whatever the positions xµout(Ω) of the Hawking particles in the original microstate were, they are now replaced by the displaced positions, and since the particles emerge with a trajectory that separates itself from the horizon exponentially in Schwarzschild time, the significance of this displacement also increases exponentially in time. This is how information regarding ingoing particles is imprinted in the out-going ones. In Ref [17], a formal expression for the ensuing unitary evolution matrix is derived, to be summarised in the next section. One important feature is seen to emerge: the effect is purely geometrical. Only the momentum distribution pµin(Ω) (mainly the minus component) enters. Consequently, unitarity of this evolution (scattering) matrix implies that these in-going particles are to be entirely characterised by their momenta...  
The question is now, how to deduce the black hole microstates from this evolution matrix. Let us simplify the situation a bit by replacing the Schwarzschild black hole metric by Rindler space. This implies that the angular coordinates Ω=(θ,φ) are replaced by two transverse, flat coordinates x=(x,y).... The Green function f(Ω,Ω′) becomes 
f(x,x′) = −1/2π log(|xx′|).                    (5.1) 
Reserving the symbol z for the radial component of the positions, we write the displacement (4.1) due to the Shapiro shift as 
δzout(x) = −4G δp(x′) log(|xx′|/C).   (5.2) 
The constant C will soon drop out. Assume now that a black hole produced by one given initial state |in0⟩, upon its final explosion leads to a given final state |out0We then calculate the final state when a slight modification is brought about to the state |in⟩. Let the modification consist of adding one light particle with momentum δp− entering the Rindler horizon at the transverse position x. All particles at the transverse position x' in the final state |out⟩ are then dragged along such that their out-coordinate - is modified by an amount given by Eq. (5.2). 
We write this modification as a property of the black hole scattering matrix:  
|in0⟩ |out0⟩ → S |in0 + δp(x)⟩ = exp(−i ∫d2x p+out(x'z|out0⟩,     (5.3)
... now we can reach any other initial state |in⟩, when described by the distribution of the total momentum going in, ptot(x) (as compared to the original initial state |in0), to find the new final state |out⟩ as a displacement of the original finite state |out0⟩: 
⟨out|S| in⟩ = ⟨out0|S|in0⟩ exp[4iG ∫d2x′ log(|x'x/C) p+out(x'pin(x)].        (5.4) 
Note that, here, the operators pin(xand p+out(x') both describe the total momenta of all in- and out going particles as distributions on the Rindler horizon. The important step to be taken now is to postulate that the entire Hilbert space of the in-particles is spanned by the function pin(x), and the black hole scattering matrix maps that Hilbert space onto the space of all particles going out, spanned by the function p+out(x'). We arrive at the unitary scattering matrix S : 
p+out|S|pin⟩ = N exp[ 4iG ∫d2x′ log(|x'x/C) p+out(x'pin(x)].              (5.5) 
This procedure leads to one common factor N , which now can be fixed (apart from an insignificant overall phase) by demanding unitarity... From Eq. (5.3) and the expression (5.5) for the scattering matrix, we can now deduce the relations and commutation rules between the operators pin(x), p+out(x'), zin(x) and z+out(x'), where the latter can be regarded as the coordinates of in- and out-going particles relative to the Rindler horizon... The algebra ... is quite different from the usual Fock space algebra for the elementary particles. In fact, it resembles a bit more the algebra of excited states of a closed string theory, but even that is not the same...
{As the operators z± and  p±  form a linear set one can decompose these new physical degrees of freedom into eigenmodes through decoupled k-waves in Rindler space. This diagonalisation provides in particular a pair of equations that can be seen as boundary conditions on the horizon implying the waves moving in are transformed into waves going out (these waves are not to be interpreted as particles as they may consist of many ones). This can be interpreted} as a real bounce against the horizon. The information is passed on from the in-going to the out-going particles. We do emphasise that in- and out-going particles were not assumed to affect the metric of the horizon, which is fine as long as they do not pass by one another at distances comparable to the Planck length or shorter; in that case, the gravitational effect of the transverse momenta must be taken into account. For the rest, no other assumptions have been made than that the longitudinal components of the gravitational fields of in- and out-going particles should not be ignored. This must be accurate as long as we keep the transverse distances on the horizon large compared to the Planck length. 
It is also important to emphasise that, even though we describe modes of in-going 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 [20], an observer falling in 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 nocloning theorems, we believe that this situation is asking for a more delicate quantum formalism...
One could try to compute the black hole entropy from the contributions of these reflecting modes. For each mode, the result is finite... Using the thermodynamical equations ... one can derive the contribution of each mode with transverse wave number k to the total entropy... The expression we obtained must now be summed over the values k. If we take these to describe a finite part of the black hole horizon area, we see that the summed expression will be proportional to the area, as expected, but the sum diverges quadratically for large k... The explanation for this divergence is that... our expressions are inaccurate at very large k, where transverse gravitational forces should be taken into account 
It is not easy to correct for this shortcoming, but we can guess how one ought to proceed. It was remarked already in Refs. [18], that the algebraic expressions we obtain on the 2-dimensional horizon, take the form of functional integrals very much resembling those of string theory. We did treat the transverse position variables x and wave number variables k very differently from the longitudinal variables z± and  p± , but it is clear that we are dealing with the full expressions of an S2 sphere. This sphere should be given two arbitrary coordinates σ=(σ12), after which these should be fixed by a gauge condition relating them to the transverse coordinates x. We took σ=x, but apparently this fails when the longitudinal variables fluctuate too wildly. As long as a more precise procedure has not been found, we can simply insert a cut-off for the transverse wave numbers k, or equivalently, the angular momentum quantum numbers ℓ for a finite-size black hole. The divergence is quadratic. The expected expression, Hawking’s entropy S=4πM2, comes about of we introduce a sharp cut-off at |k|≈MPlanck. Such wave number cut-offs impy that the conjugate variables, x, or equivalently, Ω=(θ,φ), form a discrete lattice [21]... 
Note, that we did not apply second quantization, such as in Ref. [17], since now we are not dealing with a quantum field theory. At every value of k, there are exactly two wave functions Ψ...(one at each side of the horizon, which mix). Second quantization would fail here, since the microstates appear to be determined by a single function such as p...
It was observed that the in-going and out-going particles with which we started, produce vertex insertions as in a string world sheet, as if all particles considered should be regarded as closed string loops. It all takes the form of a string theory. Strings were not put in, however, rather, they come out as inevitable objects! But beware, these are not “ordinary” strings. The black hole horizon is the string world sheet [18]. If ordinary strings were to be Wick rotated to form space-like string world sheets, all factors i would disappear from the action, whereas our expressions are still in the complex plane, as if the string slope parameter α′ should have the purely imaginary value 4Gi. In most string treatments of black holes, the string world sheets are assumed to be in the longitudinal direction, that is, the world sheets are taken to be orthogonal or dual to the horizon.  
Our analysis appears to be closely related to ideas using the BMS approach [22], although there, the emphasis seems to be specially on the in- and out-going gravitational waves, while we focus on all particle types entering or leaving the black hole. Secondly, ... we attribute the black hole properties to the immediate surroundings of the future and past event horizon. Also, one may note that both approaches now focus on light-like geodesics, which justifies attempts to employ conformally invariant (or covariant) descriptions of quantum gravity

What is the required Nature's Book keeping system?
A (last) mid-summer night's dream (or day's homework) could (have) be(en) to (start to) harness its potential for the construction of the proper geometric framework for quantum gravity. 
I am amazed by the physical intuition of 't Hooft that shines through his prospective quantum modelisation of black hole dynamics. I must confess I am also intrigued by the vague but pushy analogy one might tentatively establish between the physicist computing scheme outlined in bold characters in the text above with the recent new degrees of freedom devised by spectral noncommutative geometers to describe tentative quanta of geometry. Since the quantum information of incoming matter is supposed to impinge on outgoing one only through a momentum operator while two position operators are required to describe in a unitary way outgoing matter, could it be that a set made of a Dirac operator D and two Feynman-slashed position operators Y and Y' as defined by noncommutative geometry is the Nature's book keeping system to track the entangled spacetime-matter dynamics of black hole horizons?

Laws of gravity, as they are known today, then suggest that all forms of matter will be geometric: the way they affect the curvature of space and time is the only form of information that is conserved [5]–[7]. We think we drew the important conclusion that observations of the sort mentioned here, will be the only way to reconcile finiteness of the degrees of freedom with an unbounded group of local Lorentz transformations. This is extremely important, if true. It means that Fock space will not be the appropriate language; rather, we get something that resembles a bit more string theory, which is also basically geometric. String theories known today, however, seem not yet to be based on very sound book keeping... 
The demand from string theory that space and time themselves must feature either 10 or 26 dimensions, however, seems to be too restrictive. If indeed, as we suspect, physical degrees of freedom form discrete sets, then dimensionality of space and time may be less well-defined notions, so that such ‘predictions’ from string theory may well be due to some mathematical idealisation having little to do with reality. All in all, we are badly in need of a more orderly listing of all conceivable configurations of physical variables in a small region of space and time... 
String theory was an interesting guess, but may well have been a too wild one. We are guessing the mathematical structures that are likely to play a role in the future, but we fall short on grasping their internal physical coherence and meaning. For this, more patience is needed.
(Submitted on 29 Apr 2016)

Comment: the scrutinizing reader will notice that in the first text of 't Hooft I quote today I choose not to select his commentary about noncommutative geometry. This is because I think it addresses specifically the noncommutative field theory program, or to say more geometrically it refers to the construction of noncommutative spaces via deformation of commutative algebra while I am interested in this post (and in this blog) to chronicle advances in the isospectral deformation of classical manifolds, what I use to call spectral noncommutative geometric program (see chap 19  in this review for details).

To (almost) conclude this 2016 August-series
There is real adventure to be had at a time in which pure mathematics, theoretical physics, astronomy, philosophy and experiment are all coming together in a manner unseen for almost a hundred years. You probably have to go back still further to the seventeenth-century Scientific Revolution to get a full sense of what I mean, to the era of ‘natural philosophy’, Copernicus, Galileo, Newton. I do sometimes hear some of my physicist colleagues lamenting the old days when physics was young, when there was not a vast mountain of theory or technique that a young researcher had to climb before even starting out... actually we are at the birth of a new such era right now.
On Space and Time, Shahn Majid, 2007

Now it seems that the empirical notions on which the metric determinations of Space are based, the concept of a solid body and that of a light ray, lose their validity in the infinitely small; it is therefore quite definitely conceivable that the metric relations of Space in the infinitely small do not conform to the hypotheses of geometry; and in fact one ought to assume this as soon as it permits a simpler way of explaining phenomena. The question of the validity of the hypotheses of geometry in the infinitely small is connected with the question of the basis for the metric relations of space. In connection with this question, which may indeed still be ranked as part of the study of Space, the above remark is applicable, that in a discrete manifold the principle of metric relations is already contained in the concept of the manifold, but in a continuous one it must come from something else. Therefore, either the reality underlying Space must form a discrete manifold, or the basis for the metric relations must be sought outside it, in binding forces acting upon it. An answer to these questions can be found only by starting from that conception of phenomena which has hitherto been approved by experience, for which Newton laid the foundation, and gradually modifying it under the compulsion of facts which cannot be explained by it. Investigations like the one just made, which begin from general concepts, can serve only to insure that this work is not hindered by too restricted concepts, and that progress in comprehending the connection of things is not obstructed by traditional prejudices. This leads us away into the domain of another science, the realm of physics, into which the nature of the present occasion does not allow us to enter”.
Bernhard Riemann (10th June 1853)

mardi 30 août 2016

(How not to be afraid by wormhole) Octopussy (or Medusa)

//The title of this post has been slightly edited on 1st September 2016

Octopus wormhole (or wormhole octopussy?) 
(Olena Shmahalo/Quanta Magazine)
The ER = EPR idea posits that entangled particles inside and outside of a black hole’s event horizon are connected via wormholes.

"The story of Medusa tells you she destroys those who contemplate only her reflection, which is of course horrible, instead of contemplating her real face, which is the face of wholeness" 
Patrick MoranSwimming in stone

Contemplating quantum gravity in its full nonlocal glory... without fearing for local burns at the black hole horizon?

The modern representation of Karl Schwarzschild’s spherically symmetric solution of Einstein’s equations reads

ds2 = −(1 −2M/r) dt2 + 1/(1 − 2M/r) dr2 + r2(dθ2 + sin2θ dϕ2 ) . (1.1) 
[Note: In Schwarzschild’s original work [1], the coordinate r in Eq. (1.1) was called R, while he chose an other radial coordinate r such that the point R=2M corresponds to r=0 , since it seemed to be obvious to expect a singular mass distribution at the origin of the coordinate frame. Today, we know that this was unnecessary, for two reasons: first, one is free to choose the most convenient coordinate system anyway, and secondly, the surface r=2M does not represent a physical singularity at all, but just a coordinate singularity, much like the north pole of the Earth. It is the black hole horizon.]
As we now know very well, matter can enter the black hole through the horizon, defined by the surface r=2M , while in the standard, unquantised theory, nothing can emerge out of it. The horizon is a one way door. [Note: On some web pages, these facts are still being disputed, which we can only attribute to ignorance. Schwarzschild, who wrote his paper in less than two months after Einstein’s discovery, could be excused for not immediately realising the rather subtle features of black hole horizons, which required several years to be cleared up, but today’s experts cannot afford to make such mistakes.] In the coordinates of Eq. (1.1), the point r = 0 is a real physical singularity. 
Even though the horizon appears to be a regular region of space-time, we do have a problem with it. According to Hawking’s well-known result [2], it is due to vacuum fluctuations that a distant observer will observe particles leaving the black hole: Hawking radiation. These particles have a thermal spectrum, independent of the black hole formation process.
Hawking’s original conclusion was that this result must imply that a black hole as a physical object violates the laws of quantum mechanics: even if it originates from matter in a single quantum state, it ends up in a thermal, that is, a quantum mechanically mixed state. How could it be that a derivation that uses quantum mechanics can yield a result violating the laws of this theory? Hawking particles are now understood to be formed at the horizon, not, as was originally thought, somewhere near the r=0 singularity in its past...
(Submitted on 13 Nov 2015)

Hawking particles emitted by a black hole are usually found to have thermal spectra, if not exactly, then by a very good approximation. Here, we argue differently. It was discovered that spherical partial waves of in-going and out-going matter can be described by unitary evolution operators independently, which allows for studies of space-time properties that were not possible before. Unitarity dictates space-time, as seen by a distant observer, to be topologically non-trivial. Consequently, Hawking particles are only locally thermal, but globally not: we explain why Hawking particles emerging from one hemisphere of a black hole must be 100 % entangled with the Hawking particles emerging from the other hemisphere. This produces exclusively pure quantum states evolving in a unitary manner, and removes the interior region for the outside observer, while it still completely agrees locally with the laws of general relativity. Unitarity is a starting point; no other assumptions are made. Region I and the diametrically opposite region II of the Penrose diagram represent antipodal points in a PT or CPT relation, as was suggested before. On the horizon itself, antipodal points are identified.
(Submitted on 14 Jan 2016 (v1), last revised 14 Apr 2016 (this version, v4))

To make a long story short: 't Hooft proposes to solve the  information loss paradox of quantum evaporating black holes arguing that a pure state of collapsing matter forming a black hole evaporates indeed through the radiation of Hawking particles out of the horizon but globally this radiation is not thermal but made of entangled pairs of Hawking particles emitted from antipodal points on the two hemispheres of the horizon (thus restoring the unitary evolution of black hole evaporation).

Another speculative way to "feel" quantum gravity with just a snow ball's chance in hell against quantum fluctuations
The following work I am going to highlight now might be based on less solid grounds than (and potentially be incompatible with) the 't Hooft work but I find it a very evocative outline of tentative ideas about quantum gravity to be worth a quote here:

Here we use a thought experiment, based on a comparison of Hawking radiation with Unruh radiation, to show that these two quantum phenomenon imply a small violation of the equivalence principle. The manner in which the equivalence principle is violated by the comparison of these two effects might point toward a resolution of some of the short comings of general relativity such as the existence of singularities for certain space-times and the difficulty in formulating a quantum version of general relativity...
Hawking radiation [8] is the thermal radiation emitted by a black hole of mass, M. It occurs as a consequence of placing quantum fields in the gravitational background of a black hole. An observer who stays at a fixed distance, R, from a black hole of mass, M, will measure a temperature given by [9
Normally, the Hawking temperature THawking is quoted for an observer a large distance from the black hole (i.e. R → ∞) so the blue shift factor √(1−2GM/c2R) is not written down. It is kept here since it is crucial for seeing how the equivalence principle is violated. 
By the equivalence principle an observer, accelerating through flat, Minkowski spacetime, should also measure thermal radiation. Otherwise the observer could immediately tell the difference between a gravitational field and an accelerating frame – the accelerating frame would be the one in which no thermal radiation is detected. Soon after Hawking’s original paper on black hole radiation it was shown that an accelerating observer (with an acceleration of a = |a|) does detect thermal radiation with a temperature given by 
TUnruh =  a/2πckB

This radiation and TUnruh are know as Unruh radiation and the Unruh temperature respectively [10]. Thus, at least qualitatively, there is no violation of the equivalence principle – an observer in an Einstein elevator fixed at a distance, R, from a black hole will measure both a downward acceleration toward the floor of the elevator and thermal radiation at a temperature THawking ; an observer in an Einstein elevator which is accelerating through flat, Minkowski space-time will also measure both a downward acceleration and thermal radiation. However, looking at this situation quantitatively uncovers a violation of the equivalence principle except in the limit as the observer approaches the event horizon... 
We now ask “What are the possible implications, for gravity, of this violation of the equivalence principle from the above thought experiment?”. In addressing this question we will assume that the strength of the gravitational effects are proportional to or connected with the Hawking temperature and the inertial effects are proportional to or connected with the Unruh temperature. In particular we assume the ratio of the gravitational and inertial masses are connected with the ratio of the Hawking temperature to the Unruh temperature. This assumption is not trivial since the violation of the equivalence principle discussed above deals with the Einstein elevator formulation of the equivalence principle while the distinction between gravitational and inertial masses is a different formulation of the equivalence principle. 
First we look n the near horizon limit R → 2GM/c2 a. Here the gravitational effects dominate the inertial effects since 
THawking → c3/[8πGMkB] > TUnruh  → /2πckB   × (GM/R2)
For example, taking M = MSun ... and R = 1AU ... yields THawking ≈6.2×10−8K and TUnruh≈2.4×10−23K. While THawking is 15 orders of magnitude larger than TUnruh both temperatures are effectively zero when compared to something like the 2.7K cosmic microwave background. We take this as an indication that the violation of equivalence principle, implied by the difference in Hawking and Unruh temperatures for the same accelerations, is a very small effect under normal conditions i.e. low energy density, low gravitational field strength, non-relativistic velocity. Nevertheless, the implication would be that the variation in gravitational mass is slight larger than the variation in inertial mass. This might have some bearing on the rotation curves of galaxies. The anomalous velocity profile of outlying stars orbiting the galactic center is usually explained by an enhancement of the gravitational force coming from the presence of dark matter. Here, the enhancement of gravity over inertia would come from the slight dominance of gravitational mass over inertial mass at large distances from the galactic center. 
 Second we look in the near horizon limit R→2GM/c2 and find
 THawking →... TUnruh
Thus, in the near horizon region the equivalence principle is restored. (Note that exactly at the horizon both temperatures diverge to the same infinite value due to the blue shift factor. This is as expected since for an observer fixed just above the horizon the local acceleration and Hawking temperature both diverge). This is surprising. One might have guessed that in a region of stronger gravitational field strength, such as near the horizon versus far from the horizon, the violation of the equivalence principle would be more pronounced; that the divergence between quantum mechanics (as represented by Hawking and Unruh radiation) and general relativity would be larger. The fact that this is not the case might be taken as an indication that general relativity and quantum mechanics are more compatible, not less, as the strength of the gravitational field increases...
In the near horizon limit R→2GM/c2 our arguments pick out the event horizon as special – it is the surface where the (local) equivalence principle on which general relativity is based, becomes compatible with (non-local) quantum mechanics as represented by Hawking and Unruh radiation. 
Finally, behind the horizon (i.e. R<2GM/c2  ) the expressions for Hawking temperature and Unruh temperature break down, and we continue our journey inward based on the following conjecture: Outside the horizon (R>2GM/c2  ) gravitational effects dominate inertial effects; near the horizon ... gravitational effects and inertial effects are equivalent; thus we postulate that inside the horizon ... inertial effects dominate gravitational effects. As R→0 inertial effects will become ever more dominant over gravitational effects... one can say that ... any ... material that has fallen to this point inside the horizon, is “frozen” and non-dynamical, since the inertial mass of the material will have increased to the point where further motion is impossible. This postulated transition of gravity, under conditions of high mass-energy density, to a non-dynamical theory also might have relevance for the difficulty of consistently calculating quantum corrections to the gravitational field i.e. the long standing and unresolved problem of formulating a quantum theory of gravity...
Black holes are often described as a scientific version of Hell – a place of extreme conditions which is inhospitable to any person who falls inside. Taking into account the above postulated picture, if a black hole is to be compared to Hell then it would be the Hell of Dante’s Inferno, the center of which is frozen, rather than the traditional hot Hell of fire and brimstone

A radiolarian (skeleton) with a bilateral symmetry may mimic the radiating horizon of a black hole with its antipodal entangled Hawking particles better than an octopussy ;-) Its lattice structure may also fit pretty well for the discreteness of spacetime that emerges from the quantum gravity effects at the black hole horizon but this is another story for a later post...