Bianchi's entropy result-what to ask, what to learn from it

[marcus]

I listened to the rest of the talk, and to the answers to Wen and Gurau's questions a couple of times...
...
Do you have some more illuminating definition of what he's calling intrinsic and extrinsic geometry? It looks like the state $|\Omega\rangle$ that he uses in his density matrix is presumably the state composed of the "intrinsic" degrees of freedom forming the horizon. So how would the statistical mechanics compute some other degrees of freedom?

I'm glad you got to hear the rest of the talk, including the Q&A towards the end around minute 60. You have a really good question to write email to Bianchi about.
I'm sure he would appreciate interest from physics colleagues and would be happy to clarify the distinction.

Relevant links in case anyone else is reading the thread:
http://arxiv.org/abs/1204.5122
and in the hour-long colloquium talk+QA
http://pirsa.org/12050053/ [video]

Physicsmonkey already earlier in this thread had a question that he wrote to Bianchi about and quickly got a reply. It was back near the start of thread, I forget exactly where.

Yeah, it was on the second page of the thread, here:

I reached out to bianchi for clarification about his area formula. In the interest of keeping his privacy, I will just summarize the main points of his brief reply that are apparently common knowledge.

In short, both \sqrt{j(j+1)} and j are acceptable area operators (they differ by an operator ordering ambiguity that vanishes as \hbar \rightarrow 0 (which I guess here means something like j \rightarrow \infty as fzero and others suggested).

The two criteria for an area operator are apparently 1) that its eigenvalues go to j in the large j limit and 2) that its eigenvalue vanish for j=0.

More systematically, bianchi is using a Schwinger oscillator type representation where we have two operators a_i and the spins are \vec{J} = \frac{1}{2} a^+ \vec{\sigma} a. The total spin of the representation can be read off from the total number N = a_1^+ a_1 + a_2^+ a_2 = 2j. On the other hand, you can work out J^2 for yourself to find J^2 = \frac{1}{4}( N^2 + 2N) which one easily verifies gives J^2 = j(j+1). Thus by |\vec{L}| bianchi appears to mean N/2.

It is again interesting to see this kind of representation appearing in a useful way since it is quite important in condensed matter.

Among other things this PhMo post reminds me of the nice point of courtesy that one does not quote someone's email without first asking permission, but one can paraphrase points which are treated as common knowledge. It seems like the right way for someone at advanced academic level to get clarification. I hope, if you write Bianchi about this you will share the main points of his reply with us as PhMo did.

I should probably not interject my own perception of this as it might only cause confusion but, that said, I would like to comment.
Entropy can only be defined with an implied/explicit observer. I believe the idea of an HORIZON is also observer-dependent. If one generalizes and gets away from designating a particular observer, the mathematical language will nevertheless indicate a class or family of observers which share the horizon.
Bianchi develops the Loop BH entropy in a way that seems to me clearly aware of the observer at each stage, although he eventually is able to generalize and cancel out dependence on any particular class or family.

This is in contrast to how I remember the Loop treatment of BH entropy back in the 1990s. I could well be wrong--not having checked back and reviewed those earlier LQG papers. But as I recall it was not so clear, with them, where the observer was and what he was looking at and measuring.

The analysis, as I recall, was done more in a conceptual vacuum. So one was looking at states only of the BH horizon ("intrinsic") without any surrounding geometric or dynamically interacting ("extrinsic") context.
I think Bianchi is going deeper, imagining more, including more in his analysis. I like the fact that he has an actual quantum THERMOMETER with which the observer a little ways outside the horizon can measure the temperature. Stylistically I like the concrete detail in the Colloquium slide where the coffee mug falls in and a new FACET of the quantum state of the horizon is created. The whole treatment AFAICS is deeper, more concrete, more interactive than what I remember from the 1990s papers.

But this is just my personal take. To get a satisfactory answer to your question about the precise meaning of the intrinsic/extrinsic distinction I would guess requires an email to Bianchi. Unless Physicsmonkey or the likes thereof care to explain.

 

[marcus]

It's nice that one of us at PF exchanged an email with Bianchi and got a point in the paper clarified.

I reached out to bianchi for clarification about his area formula. In the interest of keeping his privacy, I will just summarize the main points of his brief reply that are apparently common knowledge.

In short, both \sqrt{j(j+1)} and j are acceptable area operators (they differ by an operator ordering ambiguity that vanishes as \hbar \rightarrow 0 (which I guess here means something like j \rightarrow \infty as fzero and others suggested).
...
...
It is again interesting to see this kind of representation appearing in a useful way since it is quite important in condensed matter.

I see there are signs of fairly wide interest in Bianchi's result. He is scheduled to give two talks next week at a big international conference in Stockholm--the Marcel Grossmann triennial meeting (over 1000 participants have registered for this year's MG13). Eugenio has a 30 minute time slot in the parallel session QG1 (Tuesday 3 July) and another 30 minute in the Thursday session QG4. I just learned the titles of his two tallks and found the abstracts.

http://ntsrvg9-5.icra.it/mg13/FMPro...s&talk_accept=yes&-max=50&-recid=42004&-find=
Session QG4 - Loop quantum gravity cosmology and black holes
Speaker: Bianchi, Eugenio
Entropy of Non-Extremal Black Holes from Loop Gravity
Abstract: We compute the thermodynamic entropy of non-extremal black holes using the quantum dynamics of Loop Gravity. The horizon entropy is finite, scales linearly with the area A, and reproduces the Bekenstein-Hawking expression S = A/4 with the one-fourth coefficient for all values of the Immirzi parameter. The near-horizon geometry of a non-extremal black hole - as seen by a stationary observer - is described by a Rindler horizon. We introduce the notion of a quantum Rindler horizon in the framework of Loop Gravity. The system is described by a quantum surface and the dynamics is generated by the boost Hamiltonion of Lorentzian Spinfoams. We show that the expectation value of the boost Hamiltonian reproduces the local horizon energy of Frodden, Ghosh and Perez. We study the coupling of the geometry of the quantum horizon to a two-level system and show that it thermalizes to the local Unruh temperature. The derived values of the energy and the temperature allow one to compute the thermodynamic entropy of the quantum horizon. The relation with the Spinfoam partition function is discussed.
Talk view--------------------------

http://ntsrvg9-5.icra.it/mg13/FMPro...s&talk_accept=yes&-max=50&-recid=42199&-find=
Session QG1 - Loop Quantum Gravity, Quantum Geometry, Spin Foams
Speaker: Bianchi, Eugenio
Horizons in spin foam gravity
Abstract:Spin foams provide a formulation of loop quantum gravity in which local Lorentz invariance is a manifest symmetry of quantum space-time. I review progress in determining horizon boundary conditions in this approach, and discuss the thermal properties of the quantum horizon.
Talk view: [No link, I suppose that some of the talks will be viewable next week, and this field will be filled in for some of them.]

For an overview of the parallel sessions including links to specific ones, see:
http://www.icra.it/mg/mg13/parallel_sessions.htm
There are 4 specifically Loop sessions each about 4:30 long--each making time for 8 thirty-minute talks and a coffee break. Or more if some talks are limited to 20 minutes.
QG1 A and B ("Loop Quantum Gravity, Quantum Geometry, Spin Foams") chaired by Lewandowski
QG4 A and B ("Loop quantum gravity cosmology and black holes") chaired by Pullin and Singh
Plus there are two more related sessions on devising tests of QG not limited to Loop.
QG2 A and B ("Quantum Gravity Phenomenology") chaired by Amelino-Camelia

 

[marcus]

It would be interesting to see a PERTURBATIVE confirmation of Bianchi's result. A uniformly accelerating observer in Minkowski space has a Rindler horizon (beyond which stuff can't affect him, is out of causal touch with him).

So one can have gravitons as perturbations of Minkowski geometry and look at entropy in that situation.

I should look at Bianchi's ILQGS talk again. He just recently gave a seminar talk, which is online.
Slides: http://relativity.phys.lsu.edu/ilqgs/bianchi101612.pdf
Audio: http://relativity.phys.lsu.edu/ilqgs/bianchi101612.wav
Since this talk was in October, there is sure to be new stuff compared with the May 2012 paper we started this discussion thread with.

 

[francesca]

It would be interesting to see a PERTURBATIVE confirmation of Bianchi's result.

Here it is: http://arxiv.org/abs/1211.0522

Horizon entanglement entropy and universality of the graviton coupling
Eugenio Bianchi
(Submitted on 2 Nov 2012)
We compute the low-energy variation of the horizon entanglement entropy for matter fields and gravitons in Minkowski space. While the entropy is divergent, the variation under a perturbation of the vacuum state is finite and proportional to the energy flux through the Rindler horizon. Due to the universal coupling of gravitons to the energy-momentum tensor, the variation of the entanglement entropy is universal and equal to the change in area of the event horizon divided by 4 times Newton's constant - independently from the number and type of matter fields. The physical mechanism presented provides an explanation of the microscopic origin of the Bekenstein-Hawking entropy in terms of entanglement entropy.
Comments: 7 pages

 

[marcus]

... gravitons in Minkowski space... The physical mechanism presented provides an explanation of the microscopic origin of the Bekenstein-Hawking entropy in terms of entanglement entropy.

Yes! It's significant that the analysis is done in flat 4D space. It does not require a black hole event horizon! No largescale curved geometry is needed. The explanation is quantum field theoretical.

It independently confirms the Bekenstein-Hawking entropy S=A/4 and so gives a reasonable QFT suggestion for where that entropy comes from.

This might be a step towards understanding what the microscopic degrees of freedom are that underlie quantum spacetime and from which GR arises at large scale. Thanks, Francesca!