La cℏorégraphie d'Alice et BoB sur l'horizon d'un trou noir

En attendant l'hypothétique rendez-vous des quanta d'espacetemps avec les paires intriquées de particules de Hawkings  ...

Physiciens écoutez cette phrase est pour vous
Le trou noir est un astre à la taille volage
Qui ne se laisse pas quantifier par vous
Ses quanta se cachent sur l'horizon face à nous
Quand on le croit plein son coeur vide pour nous
vibre, vibre tandis que voyagent
 et dansent appariés ses quanta en un ballet flou
Cherchez-les ils sont un peu partout...

sur un air célèbre du fou chantant 

Comme le Soleil a rendez-vous avec la Lune d'après la chanson française fameuse de Charles Trenet, la relativité générale et la mécanique quantique doivent bien se rencontrer quelque part, par exemple sur l'horizon des trous noirs. Les modalités de cette réunion n'ont pas encore été observées ni même probablement complètement comprises malgré la théorie "thermodynamique" initiée dans les années 70 puis popularisées dans les années 90 entre autre par Stephen Hawking. Certes elle apporte des éléments d'informations qui convainquent la majorité des physiciens en postulant l'existence d'un processus d'émission de particules qui émergent du trou noir via des fluctuations du vide quantique et conduisent à une lente évaporation de l'astre sombre à cause de son couplage gravitationnel au reste de l'univers ... Or il existe une théorie sinon dissidente du moins nettement moins vulgarisée, développée avec patience et obstination par Gerard 't Hooft, théoricien hollandais et héros quantique reconnu de ses pairs mais méconnu du grand public, qui ne renonce pas à une description quantique unitaire de l'évolution des trous noirs. Ses réflexions initiées dans les années quatre-vingt se sont enrichies de nouvelles idées et d'une hypothèse sinon originale du moins presque oubliée et récemment remise en avant permettant une avancée significative déjà discutée dans ce blog ici et là. 

Le billet d'aujourd'hui se penche sur le travail d'un trio de jeunes physicien-ne-s éduqué-e-s comme l'immense majorité de leurs contemporain-e-s au sein généreux de la théorie des cordes et qui explorent la voie ouverte par leur illustre aîné avec leurs propres outils conceptuels, ce qui permet d'apprécier sous un angle différent le travail de 't Hooft et nous offrent l'occasion de présenter un tableau imagé assez clair du ballet chorégraphié par la mécanique quantique et l'interaction gravitationnelle entre Alice et BoB ou plus explicitement entre un paquet d'onde de matière "traversant" l'horizon d'un trou noir et une paire de particules de Hawking quantiquement intriquées de façon antipodale sur l'horizon en question :

We revisit the old black hole S-Matrix construction and its new partial wave expansion of ’t Hooft. Inspired by old ideas from non-critical string theory & Matrix Quantum Mechanics, we reformulate the scattering in terms of a quantum mechanical model—of waves scattering off inverted harmonic oscillator potentials—that exactly reproduces the unitary black hole S-Matrix for all spherical harmonics; each partial wave corresponds to an inverted harmonic oscillator with ground state energy that is shifted relative to the s-wave oscillator. Identifying a connection to 2d string theory allows us to show that there is an exponential degeneracy in how a given total initial energy may be distributed among many partial waves of the 4d black hole...

Antipodal entanglement 

Unitarity of the S-Matrix demands that both the left and right exteriors in the two-sided Penrose diagram need to be accounted for; they capture the transmitted and reflected pieces of the wave-function, respectively. In the quantum mechanics model, there appears to be an ambiguity of how to associate the two regions I and II of the scattering diagram in Fig. 1 to the two exteriors of the Penrose diagram. We saw, in the previous section, that the quantum mechanical model appears to support the creation of physical black holes by exciting appropriate oscillators. Therefore, in this picture there is necessarily only one physical exterior. To resolve the issue of two exteriors, it was proposed that one must make an antipodal identification on the Penrose diagram [19]; see figure 3. Unitarity is arguably a better physical consistency condition than a demand of the maximal analytic extension. The precise identification is given by x → Jx with  

J : (u +, u−, θ, φ) ←→ (−u +, −u −, π − θ, π + φ).                  (5.1)

[... the simpler mapping of identifying points in I, II via (u⁺, u−, θ, φ)↔(−u⁺,−u−,θ,φ) is singular on the axis u⁺, u−=0]. Note that J has no fixed points and is also an involution, in that J²=1. Such an identification implies that spheres on antipodal points in the Penrose diagram are identified with each other. In particular, this means 

u^±(θ, φ) = − u^±(π − θ, π + φ) and p^±(θ, φ) = − p^±(π − θ, π + φ).     (5.2) 

Therefore, noting that the spherical harmonics then obey Y_l,m(π-θ,π+φ)= (−1)^l Y_l,m(θ,φ), we see that only those modes with an l that is odd contribute. However, owing to the validity of the S-Matrix only in the region of space-time that is near the horizon, this identification is presumably valid only in this region. 

Global identifications of the two exteriors have been considered in the past [56–58]. The physics of the scattering, with this identification is now clear. In-going wave-packets move towards the horizon where gravitational back-reaction is strongest according to an asymptotic observer. Most of the information then passes through the antipodal region and a small fraction is reflected back. Turning on quantum mechanics implies that ingoing position is imprinted on outgoing momenta and consequently, an highly localised ingoing wave-packet transforms into two outgoing pieces—transmitted and reflected ones—but both having highly localised momenta. Their positions, however, are highly delocalised. This is how large wavelength Hawking particles are produced out of short wavelength wave-packets and an IR-UV connection seems to be at play. Interestingly, the maximal entanglement between the antipodal out-going modes suggests a wormhole connecting each pair [59]; the geometric wormhole connects the reflected and transmitted Hilbert spaces. Furthermore, as the study of the Wigner time-delay showed, the reflected and transmitted pieces arrive with a time-delay that scales logarithmically in the energy of the in-going wave. This behaviour appears to be very closely related to scrambling time (not the lifetime of the black hole) and we leave a more detailed investigation of this feature to the future. One may also wonder why transmitted pieces dominate the reflected ones. It may be that the attractive nature of gravity is the actor behind the scene. 

Approximate thermality

We now turn to the issue of thermality of the radiated spectrum. Given a number density, say Nin(k) as a function of the energy k, we know that there is a unitary matrix that relates it to radiated spectrum. This unitary matrix is precisely the S-Matrix of the theory...

In our context, since we do not yet have a first principles construction of the appropriate second quantised theory, this in-state may be chosen. For instance, a simple pulse with a wide-rectangular shape would suffice. One may hope to create such a pulse microscopically, by going to the second quantised description and creating a coherent state. Alternatively, one may hope to realize a matrix quantum mechanics model that realizes a field theory in the limit of large number of particles. After all, we know that each oscillator in our model really corresponds to a partial wave and not a single particle in the four dimensional black hole picture.

Second Quantization v/s Matrix Quantum Mechanics

Given the quantum mechanical model we have studied in this article, we may naively promote the wave-functions ψlm into fields to obtain a second quantized Lagrangian: ...

The form of the Lagrangian being first order in derivatives indicates that the Rindler fields are naturally fermionic. In this description we have a collection of different species of fermionic fields labelled by the {l, m} indices. And the interaction between different harmonics would correspond to interacting fermions of the kind above. The conceptual trouble with this approach is that each “particle” to be promoted to a field is in reality a partial wave as can be seen from the four-dimensional picture. Therefore, second quantizing this model may not be straight-forward [20]. It appears to be more appealing to think of each partial wave as actually arising from an N-particle matrix quantum mechanics model which in the large-N limit yields a second quantized description. Since N counts the number of degrees of freedom, it is naturally related to c via 

1/N²∼ c = 8πG R²∼ l²_P /R².                   (5.8) 

Therefore, N appears to count the truly microscopic Planckian degrees of freedom that the black hole is composed of. The collection of partial waves describing the Schwarzschild black hole would then be a collection of such N-particle matrix quantum mechanics models. Another possibility is to describe the total system in terms of a single matrix model but including higher representations/non-singlet states to describe the higher l modes. This seems promising because if one fixes the ground state energy of the lowest l=0 (or l=1 after antipodal) oscillator, the higher l oscillators have missing poles in their density of states compared to the l=0, much similar to what was found for the adjoint and higher representations in [60]...

To sharpen any microscopic statements about the S-matrix, one might first need to derive an MQM model that regulates Planckian effects.

The Black Hole S-Matrix from Quantum Mechanics

Panagiotis Betzios, Nava Gaddam, Olga Papadoulaki

(Submitted on 26 Jul 2016 (v1), last revised 23 Nov 2016 (this version, v2))

Puissent d'autres physiciennes et physiciens à l'oreille cette fois familière au hiératique chant spectral noncommutatif suivre l'exemple précédent et enrichir à leur tour de leur répertoir propre la mystérieuse musique du cℏoeur quantique des trous noirs révélée par G. 't Hooft en trouvant comment accorder ses paires de particules antipodales intriquées sur l'horizon du trou noir avec les deux types de quanta de volume à la base des solutions d'espacetemps spinoriels à quatre dimensions pour l'équation de Chamseddine-Connes-Mukhanov.
//Modification du titre du billet et du corps du texte en français le 25 Mars 2017

Alice and Bob in Wonderland (Video here)