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

by marcus
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fzero
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Apr27-12, 10:18 PM
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Quote Quote by marcus View Post
Why so? I see no reason that combining states of the form (9) to make a mixed state would need to introduce a [itex]\gamma[/itex]. Please explain.
Each pure state can be thought of as a set of occupation numbers associated with which facets we use to tesselate the surface. These are the [itex]N_i[/itex] in the polymer paper, but I will use the notation of the new paper and call them [itex]N_i[/itex]. The area of a given tesselation is

[tex]A = \sum_f 8\pi G\hbar \gamma N_f j_f [/tex]

and we have a constraint that

[tex] \sum_f N_f = N.[/tex]

Furthermore, we have to require that our mixed state matches the data of the black hole. For whatever the appropriate distribution, this can be written as an expectation value

[tex] \langle A \rangle_{\mathrm{ens.}} = A_H[/tex]

where we're summing over the distribution of [itex]N_f[/itex]. I put the subscript on the ket to note that this isn't just the expectation value in the pure state.

It is logical in this program to use Bianchi's polymer distribution and demand that the BH state maximizes the entropy. This will result in the same steepest descent condition as in (16) in the polymer paper. The computation of the energy should follow similar steps as those following that equation, leading to the factors I'm referring to.

Basically, if both papers are correct (and they already have many important methods in common), the final answers for the entropy have to agree. Because the mixed state will have an occupation number associated with which faces are used to must satisfy the same constraint (16) as in the polymer paper. It is not enough to just pick a pure state and demand that

[tex]A_H = \sum_f 8\pi G\hbar \gamma j_f .[/tex]

This state alone is not a black hole. This is the step that allowed Bianchi to hide the factor of [itex]\gamma[/itex].
francesca
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Apr28-12, 08:11 AM
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Quote Quote by fzero View Post
The polymer microstate calculation had an explicit dependence on the Immirzi parameter. The only reason the present calculation does not have this dependence is because Bianchi uses a single pure state to do the calculation.
The independence wrt the Immirzi parameter [itex]\gamma[/itex] is not something new introduced by Eugenio Bianchi, but rather a general fact in LQG black holes. The [itex]\gamma[/itex]-dependence was present in the old treatment of LQG black holes, indeed. But in 2009 Engle, Noui and Perez presented a new treatment (based on [itex]SU(2)[/itex] instead of [itex]U(1)[/itex] - let me notice here that the original proposal of Rovelli in 1996 was to use [itex]SU(2)[/itex] and the shift to [itex]U(1)[/itex] appeared in the paper by Krasnov and others) so that the entropy is correctly achieved without fixing the Immirzi parameter.

References:
1. Black hole entropy and SU(2) Chern-Simons theory.
2. Black hole entropy from the SU(2)-invariant formulation of Type I isolated horizons
3. Static isolated horizons: SU(2) invariant phase space, quantization, and black hole entropy
4. Radiation from quantum weakly dynamical horizons in LQG.
marcus
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Apr28-12, 09:48 AM
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Fzero, thanks for your careful detailed response to my question! It is very helpful to see spelled out why you found the paper flawed, and the conclusion (in the Loop context) that entropy is independent of the Immirzi parameter to be invalid. Everybody benefits from this kind of careful study (although I disagree with you.)
Quote Quote by fzero View Post
The polymer microstate calculation had an explicit dependence on the Immirzi parameter. The only reason the present calculation does not have this dependence is because Bianchi uses a single pure state to do the calculation...
Quote Quote by francesca View Post
The independence wrt the Immirzi parameter [itex]\gamma[/itex] is not something new introduced by Eugenio Bianchi, but rather a general fact in LQG black holes. The [itex]\gamma[/itex]-dependence was present in the old treatment of LQG black holes, indeed. But in 2009 Engle, Noui and Perez presented a new treatment (based on [itex]SU(2)[/itex] instead of [itex]U(1)[/itex] - let me notice here that the original proposal of Rovelli in 1996 was to use [itex]SU(2)[/itex] and the shift to [itex]U(1)[/itex] appeared in the paper by Krasnov and others) so that the entropy is correctly achieved without fixing the Immirzi parameter.

References:
1. Black hole entropy and SU(2) Chern-Simons theory.
2. Black hole entropy from the SU(2)-invariant formulation of Type I isolated horizons
3. Static isolated horizons: SU(2) invariant phase space, quantization, and black hole entropy
4. Radiation from quantum weakly dynamical horizons in LQG.
I'm beginning to get a better sense of the historical development. The key reference seems to be #3. The first two lead up to it, but they don't seem to explicitly break free from dependence on the Immirzi parameter. They lay the groundwork, if I am not mistaken. I'll quote the abstract of your reference #3. The November 2010 paper of Perez and Pranzetti.

http://inspirehep.net/record/877359?ln=en
http://arxiv.org/abs/1011.2961
Static isolated horizons: SU(2) invariant phase space, quantization, and black hole entropy
Alejandro Perez, Daniele Pranzetti
(Submitted on 12 Nov 2010)
We study the classical field theoretical formulation of static generic isolated horizons in a manifestly SU(2) invariant formulation. We show that the usual classical description requires revision in the non-static case due to the breaking of diffeomorphism invariance at the horizon leading to the non conservation of the usual pre-symplectic structure. We argue how this difficulty could be avoided by a simple enlargement of the field content at the horizon that restores diffeomorphism invariance. Restricting our attention to static isolated horizons we study the effective theories describing the boundary degrees of freedom. A quantization of the horizon degrees of freedom is proposed. By defining a statistical mechanical ensemble where only the area A of the horizon is fixed macroscopically-states with fluctuations away from spherical symmetry are allowed-we show that it is possible to obtain agreement with the Hawking's area law---S = A/4 (in Planck Units)---without fixing the Immirzi parameter to any particular value: consistency with the area law only imposes a relationship between the Immirzi parameter and the level of the Chern-Simons theory involved in the effective description of the horizon degrees of freedom.
26 pages, published in Entropy 13 (2011) 744-777
atyy
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Apr28-12, 10:13 AM
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Quote Quote by marcus View Post
http://arxiv.org/abs/1011.2961
Static isolated horizons: SU(2) invariant phase space, quantization, and black hole entropy
Alejandro Perez, Daniele Pranzetti
(Submitted on 12 Nov 2010)
We study the classical field theoretical formulation of static generic isolated horizons in a manifestly SU(2) invariant formulation. We show that the usual classical description requires revision in the non-static case due to the breaking of diffeomorphism invariance at the horizon leading to the non conservation of the usual pre-symplectic structure. We argue how this difficulty could be avoided by a simple enlargement of the field content at the horizon that restores diffeomorphism invariance. Restricting our attention to static isolated horizons we study the effective theories describing the boundary degrees of freedom. A quantization of the horizon degrees of freedom is proposed. By defining a statistical mechanical ensemble where only the area A of the horizon is fixed macroscopically-states with fluctuations away from spherical symmetry are allowed-we show that it is possible to obtain agreement with the Hawking's area law---S = A/4 (in Planck Units)---without fixing the Immirzi parameter to any particular value: consistency with the area law only imposes a relationship between the Immirzi parameter and the level of the Chern-Simons theory involved in the effective description of the horizon degrees of freedom.
26 pages, published in Entropy 13 (2011) 744-777
So what is the relationship between the Immizi parameter and the level of the Chern-Simons theory in Bianchi's new calculation?
marcus
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Apr28-12, 10:37 AM
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Quote Quote by atyy View Post
So what is the relationship between the Immizi parameter and the level of the Chern-Simons theory in Bianchi's new calculation?
Why should there be any at all? I don't see in Bianchi's paper any reference to the 2010 Perez Pranzetti paper. What you ask sounds to me like a good research topic. There might or might not be some interesting connection. I don't think one can determine that simply based on the research papers already available. I could be wrong of course. Maybe Francesca will correct me, and answer your question. She has her own research to do though.

This breaking free from dependence of entropy on the Immirzi looks to me like a gradual historical process that has been happening by various routes on different fronts. I think of it as a kind of blind tectonic shift. Perhaps the earliest sign being Jacobson's 2007 paper.

Wait. Bianchi's reference [3] cites (in addition to papers by Rovelli 1996 and by Ashtekar et al 1998) the 2010 ENP paper Engle Noui Perez. That was the first one Francesca listed. So there is an indirect reference to Chern Simons level. Maybe we can glimpse some connection by looking at the ENP paper.
marcus
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Apr28-12, 11:37 AM
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This is fascinating, I thought the 2009 ENP paper still inextricably involved Immirzi dependence, but I may have missed something. Bianchi cites it and it was the first one on Francesca's list. I need to take a closer look.
http://arxiv.org/abs/0905.3168
Black hole entropy and SU(2) Chern-Simons theory
Jonathan Engle, Karim Noui, Alejandro Perez
(Submitted on 19 May 2009)
Black holes in equilibrium can be defined locally in terms of the so-called isolated horizon boundary condition given on a null surface representing the event horizon. We show that this boundary condition can be treated in a manifestly SU(2) invariant manner. Upon quantization, state counting is expressed in terms of the dimension of Chern-Simons Hilbert spaces on a sphere with marked points. Moreover, the counting can be mapped to counting the number of SU(2) intertwiners compatible with the spins that label the defects. The resulting BH entropy is proportional to aH with logarithmic corrections Δ S=-3/2 log aH. Our treatment from first principles completely settles previous controversies concerning the counting of states.
4 pages, published in in Physical Review Letters 2010
fzero
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Apr28-12, 12:02 PM
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Quote Quote by francesca View Post
The independence wrt the Immirzi parameter [itex]\gamma[/itex] is not something new introduced by Eugenio Bianchi, but rather a general fact in LQG black holes. The [itex]\gamma[/itex]-dependence was present in the old treatment of LQG black holes, indeed. But in 2009 Engle, Noui and Perez presented a new treatment (based on [itex]SU(2)[/itex] instead of [itex]U(1)[/itex] - let me notice here that the original proposal of Rovelli in 1996 was to use [itex]SU(2)[/itex] and the shift to [itex]U(1)[/itex] appeared in the paper by Krasnov and others) so that the entropy is correctly achieved without fixing the Immirzi parameter.

References:
1. Black hole entropy and SU(2) Chern-Simons theory.
2. Black hole entropy from the SU(2)-invariant formulation of Type I isolated horizons
3. Static isolated horizons: SU(2) invariant phase space, quantization, and black hole entropy
4. Radiation from quantum weakly dynamical horizons in LQG.
I am not claiming that the dependence on [itex]\gamma[/itex] is new or that there weren't earlier papers that claimed that they could avoid it. The simple fact is that Bianchi's polymer result had this dependence. His new result does not. I have explained the reason for the discrepancy, and it has nothing to do with any gauge fixing. In the earlier paper he uses the proper mixed state for the black hole, while in the new paper he uses a pure state. The new paper does not compute the entropy of a black hole.

If you disagree, please explain which Bianchi paper is wrong and why.
francesca
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Apr28-12, 01:13 PM
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Quote Quote by fzero View Post
If you disagree, please explain which Bianchi paper is wrong and why.
I don't disagree with you :-)
because the papers by Bianchi are both right,
but the two calculations are done using different ensembles.

This is a tricky point that could have been overlooked. All the previous calculations used the area ensemble, namely one counts how many spin states there are for a given area. You are right to say that a [itex]γ[/itex]-dependence is unavoidable. This is also written in the paper (even if in a very compact manner, it would be nice to have a more extended comment on this issue):
Quote Quote by arXiv:1204.5122
The result obtained directly addresses some of the difficulties found in the original Loop Gravity derivation of Black-Hole entropy where the area-ensemble is used [3] and the Immirzi parameter shows up as an ambiguity in the expression of the entropy [20]. Introducing the notion of horizon energy in the quantum theory, we find that the entropy of large black holes is independent from the Immirzi parameter. Quantum gravity corrections to the entropy and the temperature of small black holes are expected to depend on the Immirzi parameter.
So a central point in the paper is the introduction of the energy ensemble, where the energy of the black hole is fixed. This choice is guided by the physical intuition that the energy is the key object being interested in the heat exchanges between the black hole and its neighborhood. This is a thermodynamical reasoning. Of course one can also look at the statistics of the energy ensemble: this is not what has been done in this paper, but I think that people are already working on this for a follow-up paper.
marcus
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Apr28-12, 01:53 PM
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Quote Quote by fzero View Post
...while in the new paper he uses a pure state. The new paper does not compute the entropy of a black hole.
If you disagree, please explain which Bianchi paper is wrong and why.
Hi Fzero, it's fun having you take such an interest in Bianchi's new entropy paper. Perhaps I should wait for F. to reply since you were addressing her, but she may have more urgent things to do. So I'll tell you my hunch.
I think probably all or most theory papers by creative people are in some respect wrong. They open up and develop new paths. The important papers are never the final word, they shine a light ahead into the dark.

My hunch is that the new Bianchi paper (which I think is basically a draft) probably has places where the reasoning could be improved or clarified. I also think that his conclusion is probably right and will stand! That's just a guess but it seems to be the way a lot of recent Loop BH work is going. Quite a lot of the younger-generation people are beginning to see reasons why BH entropy is independent from Immirzi. I'm just now realizing how many, and how many of them are still postdoc or have recently taken their first faculty appointment (e.g. Engle, Noui, Durka, Pranzetti, Bianchi..). It has the makings of a little revolution--we'll have to see how it goes.

I don't think I need to argue with you. You have decided to disbelieve the result because the argument is based on considering a pure quantum state. I think it's fine for you to say this whenever the occasion arises I do not think the reasoning actually rests on that singlestate basis, but that's MY perception not yours.

As I see it, he's really considering a PROCESS which adds or subtracts a little facet of area and bit of energy from each one of a huge swarm of pure states.
In the case of each pure state he verifies that ∂A/4 = ∂E/T
So "by superposition" he reasons that for the whole swarm it is always true that ∂A/4 = ∂E/T. So, in effect, QED.

But I think it's fine for you to remain unalterably opposed to Bianchi's paper and to firmly declare things like "The new paper does not compute the entropy of a black hole." I don't especially want you to agree with me. And I could be wrong! I'm basically going to wait and see until the next paper on this, by Bianchi and Wolfgang Wieland, comes out. It's in prep. And the last thing I want to do is argue with you. I won't know what I really think about this until I see the followup paper(s).

OOPS! I didn't realize F. had already replied! So this is superfluous, but I think I nevertheless won't erase it.
Hi Francesca, I didn't think you would reply, so wanted to pay Fzero the courtesy of saying something in response to his interesting post.
fzero
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Apr28-12, 02:14 PM
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Quote Quote by francesca View Post
So a central point in the paper is the introduction of the energy ensemble, where the energy of the black hole is fixed. This choice is guided by the physical intuition that the energy is the key object being interested in the heat exchanges between the black hole and its neighborhood. This is a thermodynamical reasoning. Of course one can also look at the statistics of the energy ensemble: this is not what has been done in this paper, but I think that people are already working on this for a follow-up paper.
There are several problems here.

First, there is no "ensemble" in the latest paper. As you mention, no statistics are addressed, but left to future work. So, as I've been saying, the state being considered is not that of a black hole (pure state vs mixed state). The role of the area ensemble in the polymer paper was not just to count microstates, but was a cruicial part of selecting the correct black hole state.

As you say, what is left is a thermodynamic calculation. What is being treated quantum mechanically is the change in energy [itex]\delta E[/itex]. This is the same semiclassical reasoning as Hawking, the quantum computation of the energy [itex]E[/itex] is not done, but [itex]\delta E[/itex] is properly accounted for.

Finally, it's already clear how the "energy ensemble" works. The states that Bianchi uses satisfy [itex]\vec{K} = \gamma \vec{L}[/itex] as well as [itex]|\vec{L}| = |L_z|[/itex]. This is something that PhysicsMonkey was asking about a couple of days ago. Therefore the energy is directly proportional to the area. If we were to count microstates subject to the energy constraint, we'd find essentially the same result as he did in the polymer paper. It looks like the only change amounts to a rescaling of the Lagrange multiplier [itex]\mu[/itex] by [itex]\gamma[/itex].
fzero
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Apr28-12, 02:23 PM
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Quote Quote by marcus View Post
As I see it, he's really considering a PROCESS which adds or subtracts a little facet of area and bit of energy from each one of a huge swarm of pure states.
In the case of each pure state he verifies that ∂A/4 = ∂E/T
So "by superposition" he reasons that for the whole swarm it is always true that ∂A/4 = ∂E/T. So, in effect, QED.
I think I agree with this, but I've argued on a couple of occasions that this is a semiclassical computation. It is not the fully quantum mechanical treatment that I thought was being advertised. It's also not clear whether we have learned much since Hawking was already able to do this calculation and didn't find a dependence on the Immirzi parameter either!

But I think it's fine for you to remain unalterably opposed to Bianchi's paper and to firmly declare things like "The new paper does not compute the entropy of a black hole." I don't especially want you to agree with me. And I could be wrong! I'm basically going to wait and see until the next paper on this, by Bianchi and Wolfgang Wieland, comes out. It's in prep. And the last thing I want to do is argue with you. I won't know what I really think about this until I see the followup paper(s).
I could amend my statement to refer to the fully quantum computation of a BH entropy. I don't have any major objections against the computation when viewed in the spirit of the original semiclassical computations.

I've already explained how the statistical treatment should work. There is no reason to expect that the result from the polymer paper is going to change since the area contraint is equivalent to the energy constraint for the subspace of states that Bianchi is using.
atyy
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Apr28-12, 03:50 PM
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Quote Quote by francesca View Post
The independence wrt the Immirzi parameter [itex]\gamma[/itex] is not something new introduced by Eugenio Bianchi, but rather a general fact in LQG black holes. The [itex]\gamma[/itex]-dependence was present in the old treatment of LQG black holes, indeed. But in 2009 Engle, Noui and Perez presented a new treatment (based on [itex]SU(2)[/itex] instead of [itex]U(1)[/itex] - let me notice here that the original proposal of Rovelli in 1996 was to use [itex]SU(2)[/itex] and the shift to [itex]U(1)[/itex] appeared in the paper by Krasnov and others) so that the entropy is correctly achieved without fixing the Immirzi parameter.

References:
1. Black hole entropy and SU(2) Chern-Simons theory.
2. Black hole entropy from the SU(2)-invariant formulation of Type I isolated horizons
3. Static isolated horizons: SU(2) invariant phase space, quantization, and black hole entropy
4. Radiation from quantum weakly dynamical horizons in LQG.
I don't think they are independent of the Immirzi parameter. Basically, the SU(2) introduces one more parameter k, the level of the Chern-Simons theory. So you have two parameters, and if you fix one, say the Immirzi, you have another to adjust to match the semiclassical calculation of Hawking.
fzero
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Apr28-12, 04:22 PM
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Quote Quote by atyy View Post
I don't think they are independent of the Immirzi parameter. Basically, the SU(2) introduces one more parameter k, the level of the Chern-Simons theory. So you have two parameters, and if ypu fix one, say the Immirzi, you have another to adjust to match the semiclassical calculation of Hawking.
Also, the level must be an integer, so only discrete values of the Immirzi parameter are allowed in those models. This is at odds with the arguments that the Immirzi parameter might be thought of as a running coupling. So on the one hand, if the BH calculations are to be trusted, at least some of the techniques/conclusions of the asymptotic safety programs are not.
marcus
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Apr29-12, 12:22 PM
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For convenience, here's the link to Bianchi's paper:
http://arxiv.org/abs/1204.5122
In his conclusions section on page 5, Bianchi cites a 2003 paper of Jacobson and Parentani which we also might want to keep handy:
http://arxiv.org/abs/gr-qc/0302099
Horizon Entropy
Ted Jacobson, Renaud Parentani
(Submitted on 25 Feb 2003)
Although the laws of thermodynamics are well established for black hole horizons, much less has been said in the literature to support the extension of these laws to more general settings such as an asymptotic de Sitter horizon or a Rindler horizon (the event horizon of an asymptotic uniformly accelerated observer). In the present paper we review the results that have been previously established and argue that the laws of black hole thermodynamics, as well as their underlying statistical mechanical content, extend quite generally to what we call here "causal horizons". The root of this generalization is the local notion of horizon entropy density.
21 pages, one figure, to appear in a special issue of Foundations of Physics in honor of Jacob Bekenstein

Conceptually, Bianchi's paper seems in part to derive from this J&P paper. A Rindler horizon is a type of causal horizon. Bianchi makes central use of the ideas of a quantum Rindler horizon and entropy density.
His derivation of the entropy density, to first order, comes tantalizingly close to a tautology.
He shows that for all pure states of the quantum Rindler horizon it is identically true that
∂A/4 = ∂E/T
The argument that this extends by linearity to superpositions---to mixed states of the quantum Rindler horizon, and large assemblies thereof---is not made explicitly. But a relevant observation is made immediately after equation (20) on page 4:
"Notice that the entropy density is independent of the acceleration a, or equivalently from the distance from the horizon."
This opens the way to our concluding that ∂A/4 = ∂E/T applies as well to mixed states and collections thereof.
Thus any process that increases the BH energy slightly (such as small object like an icecream cone or ukelele falling into the hole) will make the two quantities change in tandem, so that Rindler horizon entropy and area will remain in the same ratio S = A/4.
marcus
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Apr29-12, 05:19 PM
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I mentioned that I'm beginning to see this paper in the context of a small revolution in Loop gravity.
A number of young researchers are posting Loop BH papers which break from the earlier work (1990s) and often find the entropy independed of Immirzi to first order. (Besides Bianchi, some names are Ghosh, Perez, Engle, Noui, Pranzetti, Durka. And there are groundbreaking Loop BH papers by Modesto, Premont-Schwarz, Hossenfelder. I'm probably forgetting some. )
So part of understanding Bianchi's paper, for me, is catching up on the context of other recent Loop BH papers. Here is one that came out earlier this month. You can see there is significant conceptual overlap.

http://arxiv.org/abs/1204.0702
Radiation from quantum weakly dynamical horizons in LQG
Daniele Pranzetti
(Submitted on 3 Apr 2012)
Using the recent thermodynamical study of isolated horizons by Ghosh and Perez, we provide a statistical mechanical analysis of isolated horizons near equilibrium in the grand canonical ensemble. By matching the description of the dynamical phase in terms of weakly dynamical horizons with this local statistical framework, we introduce a notion of temperature in terms of the local surface gravity. This provides further support to the recovering of the semiclassical area law just by means of thermodynamical considerations. Moreover, it allows us to study the radiation process generated by the LQG dynamics near the horizon, providing a quantum gravity description of the horizon evaporation. For large black holes, the spectrum we derive presents a discrete structure which could be potentially observable and might be preserved even after the inclusion of all the relevant transition lines.
Comments: 9 pages, 2 figures
fzero
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Apr29-12, 05:29 PM
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Quote Quote by marcus View Post
He shows that for all pure states of the quantum Rindler horizon it is identically true that
∂A/4 = ∂E/T
The argument that this extends by linearity to superpositions---to mixed states of the quantum Rindler horizon, and large assemblies thereof---is not made explicitly. But a relevant observation is made immediately after equation (20) on page 4:
"Notice that the entropy density is independent of the acceleration a, or equivalently from the distance from the horizon."
This opens the way to our concluding that ∂A/4 = ∂E/T applies as well to mixed states and collections thereof.
I went through the statistical computation of the entropy in Bianchi's polymer model using the energy constraint. I was wrong when I said that there would be a rescaling of the Lagrange multiplier by a factor of [itex]\gamma[/itex]. The important point is that the energy constraint is not merely equivalent to the area constraint, they are in fact exactly the same. From equation (9) of the present paper (let's call it B12 for Bianchi-2012),

[tex]E = \sum_f \hbar\gamma j_f a = \frac{a}{8\pi G} \sum_f A_f = \frac{aA_H}{8\pi G}.[/tex]

We can rewrite this constraint as

[tex] \sum_f j_f = \frac{A_H}{8\pi \gamma G\hbar}.[/tex]

This is the same as the area constraint used in equation (16) of the polymer paper (B10 for Bianchi-2010), except, as previously discussed, here we are using a slightly different basis where the eigenvalues of [itex]|\vec{L}|[/itex] are [itex] j[/itex] rather than [itex]\sqrt{j(j+1)}[/itex]. You can check that the distribution of states depends only on the degeneracy and not on the precise eigenvalue, so the rest of equ (16) is unchanged.

I'd already gone through all of the math in that section of the paper before realizing that there weren't any numerical differences between the constraints, so I might as well report on the result. The derivation of the entropy only differs in the numerical value of the constants derived there. This is the effect of the different eigenvalue spectrum. For example, the occupation numbers at equilibrium satisfy

[tex] p_j \equiv \frac{N_j^*}{N^*} \approx (2j+1) e^{-\mu^* j}.[/tex]

Imposing the normalization requirement

[tex] \sum_j p_j =1[/tex]

can be done by approximating the sums by an integral. I find that [itex]\mu^*[/itex] is the solution to

[tex]\int_0^\infty dx(x+2) e^{-\mu^* (x+1)/2} =1.[/tex]

This leads to the equation

[tex] 4(\mu^*+1) e^{-\mu^*/2} = (\mu^*)^2,[/tex]

which has a numerical solution at

[tex]\mu^* \sim 2.086.[/tex]

This is a little bit different from the value obtained in B10, but in a reasonable neighborhood given the similarity of the normalization constraints.

Similarly, the constant

[tex]\alpha^* = \sum_j j p_j = \frac{1}{4} \int_0^\infty dx(x+1)(x+2) e^{-\mu^* (x+1)/2}[/tex]
[tex]= \frac{e^{-\mu^*/2} }{(\mu^*)^3} ( (\mu^*)^2 + 2 \mu^* + 4) \sim 0.486.[/tex]

The leading term in the entropy is once again

[tex] S = \frac{\kappa}{4G\hbar} \frac{\mu^*}{2\pi\gamma} A_H,[/tex]

so that we require

[tex]\gamma =\frac{\mu^*}{2\pi} \sim 0.322.[/tex]

Presumably the difference between this value of the Immirzi parameter and earlier results is due to the difference in the spacing between the area eigenvalues [itex]\sqrt{j(j+1)}[/itex] vs [itex] j[/itex]. We are effectively using slightly different scales to quantize the area operator.
marcus
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Apr29-12, 06:19 PM
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Compliments on giving these papers a close reading, and thanks for sharing what you are finding out.
It occurs to me that Pranzetti's April 2012 paper may actually be to your liking. At least I hope so! Canonical ensemble etc.
==quote page 3 middle of second column==
Namely, if we assume a stationary near-horizon geometry and we use the Unruh temperature βκ = 2π/hbar for our local accelerated observer O, the entropy expression (9) gives exactly S = A/(4lp2), at the leading order.
==endquote==
http://arxiv.org/abs/1204.0702
here kappa is the local surface gravity, so beta is the Unruh temperature. As one sees, no dependence on Immirzi.
The derivation may conform with your standards. Not sure, but it might.

==quote Pranzetti beginning of page 2==
In this letter, we want to investigate further and more in detail the analogy between a quantum horizon with its punctures and a gas of particles by introducing the main ingredients for a grand canonical ensemble analysis. The basic idea is to regard the bulk and the horizon as forming together an isolated system. The two subsystems are considered separately in thermal equilibrium; then, at some point, a weakly dynamical phase takes place and they interact with each other. This local interaction allows for the possibility of exchange of energy and particles between the two. After such a change of thermodynamic state has taken place, the two subsystems go back to a situation of thermal equilibrium. This picture will be made more precise in the following, where we will concentrate only on the spherically symmetric case. However, let us at this point clarify the framework we are working in: no background structure is introduced at any point, we will work in the quantum gravity regime; no matter is going to be coupled to gravity; the radiation spectrum we will derive is related entirely to emission of quanta of the gravitational field due to dynamical processes described by the LQG approach...
==endquote==

==quote page 2 start of "Entropy" section==
Entropy. Let us now first concentrate on the derivation of the entropy of the gas of punctures (see [18] for the original microcanonical derivation and [19] for a recent review). By working in the grand canonical ensemble— which represents the physically most suitable framework to describe the horizon+bulk system—, it can be shown how the Bekenstein-Hawking semiclassical entropy can be recovered only through thermodynamical considerations. Moreover, the description of the radiation process in the second part of the paper justifies the interpretation of the local notion of surface gravity introduced above as a temperature, which is a fundamental ingredient to recover the semiclassical area law (see below). In this sense, the result of the second part of the paper puts on more solid ground the recent derivation of [16]. This section simply presents a more detailed derivation of the IH entropy in the grand canonical ensemble already performed in [16]. The original part of the paper is contained in the next section.
The grand canonical partition function for the gas of punctures is given by...
===endquote===

Pranzetti is at the Max Planck Institute for Gravitation in a little place called Golm outside Berlin. I wonder whose group he's in. Bianca Dittrich or Dan Oriti probably. I checked. Oriti's group. Pranzetti was a Marseille PhD student before that.
fzero
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Apr29-12, 06:54 PM
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Quote Quote by marcus View Post
Compliments on giving these papers a close reading, and thanks for sharing what you are finding out.
It occurs to me that Pranzetti's April 2012 paper may actually be to your liking. At least I hope so! Canonical ensemble etc.
==quote page 3 second column==
Namely, if we assume a stationary near-horizon geometry and we use the Unruh temperature βκ = 2π/hbar for our local accelerated observer O, the entropy expression (9) gives exactly S = A/(4lp2), at the leading order.
==endquote==
http://arxiv.org/abs/1204.0702
here kappa is the local surface gravity, so beta is the Unruh temperature.
The derivation may conform with your standards. Not sure, but it might.
The calculation he's referring to in that particular paragraph is a semiclassical one, so it's on par with the one in B12 and Hawking's original one. Someone should probably be able to explain why the semiclassical results seem somewhat universal. I have a feeling that it's just that adding quantum bits of area are the same as what Bekenstein and Hawking were doing back in the 70s. As long as you cook up the right relationship between the energy of the state (being added) and the corresponding area, you will find the same result when you use the Clausius formula.

This is not to say that you don't learn something from these approaches. But don't be confused that the semiclassical computations are as exciting as a quantum treatment that accounts for the right microstates.

The rest of the Pranzetti paper seems to be about developing the grand canonical ensemble for the spin microstates. This is interesting, and something like it is needed to properly treat radiation (he addresses this of course). However, as you can see from his equation (9), the grand canonical formalism is a specific type of correction to the leading term computed by the microcanonical ensemble. So it's probably not immediately relevant to the specific issues I've been discussing.

Addendum: I saw your last edit and believe that the above comments continue to apply. The "thermodynamic" calculation is the semiclassical one. The corrections from the grand canonical ensemble do not affect the leading order entropy computed in the microcanonical formalism at large occupation numbers. The corrections represent quantum corrections due to particle exchange between the horizon and the exterior, so represent subleading corrections. These are likely of smaller order than terms that we've already dropped in the [itex]N_j\gg 1[/itex] limit.


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