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

fzero

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

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/(4l_p²), 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

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.

marcus

Thanks for the comment on Pranzetti's paper! Not sure exactly what you mean by "semiclassical" though.

You are right that the derivation of S = A/4 (without Immirzi dependence) happens in the first 3 pages.

Indeed the grand canonical ensemble of quantum states of horizon's geometry is set up in the first few equations on page 2, where he gives the "canonical partition function" of quantum states, e.g. equation (3).

There is no hint of "QFT on a curved spacetime" there. Punctures are simply where spin network states go through the horizon. Their edges carry area quantum numbers j.
A class of spin network states is specified by {s_j} in his notation. There s_j is the number of spin network edges with spin label j, which pass thru the horizon.
So the approach is fully background independent. There is no prior geometry. All the geometry is in the spin networks which are quantum states of geometry. Standard in Loop gravity.


So he sets up to derive S = A/4 with those equations (2) (3) (4) .....(9)

The fully quantum conclusion is equation (9) where you see the quantum corrections as well as the leading term which has the 1/4 coefficient.

Then to recover the Bek Hawk. result he of course takes a limit so that the quantum corrections go away. That is equation (10) in the next paragraph.

But already before that in equation (9), which is not semiclassical, you see there is no dependence on Immirzi.

Anyway that is how I read it. Do you see equations (2 - 9) as in any way semiclassical?
For me they come entirely within ordinary spin network Loop gravity.

fzero

I was a bit too fast to comment on the Pranzetti paper. This is for two reasons: 1. I was swayed by his own reference to a semiclassical analysis to fix [itex]\beta \bar{\kappa} = 2\pi/\hbar[/itex], and 2. My own thermodynamics is quite rusty.

It appears to me that Pranzetti's calculation in the grand canonical ensemble (GCE) is equivalent (there is a question of the role of the large [itex]N[/itex] limit that I comment on below) to the calculation Bianchi made in the microcanonical ensemble (MCE) with the area/energy constraint. In both cases you find the correct leading term in the entropy, but for a fixed value of the Immirzi parameter. In the Bianchi calculation, the fixing of the Immirzi parameter is explicit. In the Pranzetti calculation it is hidden, but he refers to it below his equation (11). I will discuss this more in a bit.

Now, using the MCE is rather transparent. We're aiming to count microstates and not worry about dynamical processes like emission or absorption, so we can fix the number of quanta and energy. To ensure that we're using the correct mixed state, we impose the area constraint and extremize the entropy.

In the GCE, we allow the number of quanta and energy to fluctuate but we use a heat bath to fix the temperature of the system state. We end up getting the same answer for the entropy as before when we fix the average energy to be the appropriate multiple of the energy. The setup is a bit unphysical for a real BH, but for our purposes we can always imagine feeding the right amount of matter in to balance the radiation coming out. It also seems to me to be nicer to impose the area constraint dynamically, rather than by hand, but this is more opinion than a serious objection.

However, what the GCE also seems to do for us is let us avoid the large [itex]N[/itex] limit. For Bianchi, the large [itex]N[/itex] limit was not just important to allow us to use the Stirling approximation, but it was also important in obtaining a manageable form for the number of states with the same occupation numbers ([itex]\Omega[/itex] in B10). Pranzetti's use of the GCE seems to remove the need for us to take this limit, at the expense of an extra free parameter, the chemical potential. This leads us naturally to the new conundrum surrounding the Immirzi parameter.

We will work with an area operator that is a mix of the ones that Pranzetti and Bianchi use:

[tex] \hat{H} | \{ s_j\}\rangle = \hbar \bar{\kappa} \gamma \sum_j s_j j | \{ s_j\}\rangle .[/tex]

I am using Pranzetti's notation, but I have set [itex]\hbar G = \ell_p^2[/itex] for convenience, as well as chosen the Schwinger-type basis of Bianchi to simplify some calculations later.

Now, it could be that I don't understand the state space well enough and there is some inequivalence between the Bianchi and Pranzetti pictures. This would presumably address the large [itex]N[/itex] questions above. Somehow the difference would go away in the large [itex]N[/itex] limit, explaining why they agree. In any case, I will keep going under the assumption that I understand the mechanics of the states, if not their complete motivation.

It will be convenient to define a parameter

[itex]\nu = \hbar \beta \bar{\kappa} \gamma.[/itex]

If we set [itex]\beta \bar{\kappa} = 2\pi/\hbar[/itex], then we can write the Immrizi parameter as

[tex] \gamma = \frac{\nu}{2\pi}.[/tex]

Now, we can write all thermodynamic quantities in terms of the function

[tex] f(\nu) = \sum_j (2j+1) e^{-\nu j} \longrightarrow \frac{4}{\nu^2} (\nu+1) e^{-\nu/2} ~\mathrm{as}~N\rightarrow \infty.[/tex]

This is the same expression that turned up in the result for the occupation numbers in B10, so I've included the value for the sum that we find in the large [itex]N[/itex] limit.

The relevant equation from Pranzetti is (10), which we write as

[tex] f(\nu) = \frac{\bar{N}}{\bar{N}+1} e^{-\beta \mu} . [/tex]

In general, this is a transcendental equation that determines [itex]\nu[/itex] (equivalently [itex]\gamma[/itex]) in terms of [itex]\mu[/itex] and [itex]\bar{N}[/itex]. In the large [itex]\bar{N}[/itex] limit, the explicit [itex]\bar{N}[/itex] dependence drops out. If we did not wish to take the limit, we could just eliminate [itex]\bar{N}[/itex] using the energy constraint, which is

[itex] \frac{\bar{\kappa}}{8\pi G} A = \bar{E} = - \hbar \bar{\kappa} \gamma (\bar{N}+1) \frac{d}{d\nu} \log f.[/itex]

Now the chemical potential represents the cost in energy to take a particle from the heat bath and place it into the system (the isolated horizon or BH). Physically, we might think that this is related to the surface gravity of the horizon. Perhaps we might think that it is zero, since this is precisely the notion of energy that is ambiguous in a gravitational system. It might be possible to address this by thinking more carefully about how we have to define the GCE. I will probably think a bit more about it, but in any case, the Immrizi parameter now depends implicitly on the chemical potential.

If we set the chemical potential to zero and take the large [itex]\bar{N}[/itex] limit, then we will recover the same numerical solution as in the modified Bianchi calculation

[itex]\nu \sim 2.086,~~~\gamma \sim 0.322.[/itex]

For other values of [itex]\mu[/itex], we will obtain some other value of [itex]\gamma[/itex].

This is either strange or expected. On the one hand, it might seem strange that we need to change the quantum of area to accommodate a change in chemical potential. On the other hand, we might think that whatever change we made to the system represents some sort of change in the natural energy scale of the problem and the Immirzi parameter must run.

Anyway, the upshot of all of this is that B10 and Pranzetti look like correct computations for the state spaces they are using. Their results seem to agree qualitatively and quantitatively in the common regime of validity. I still think that B12 is pulling a bit of a fast one by using a pure state, but I understand it to be correct as a semiclassical computation, rather than a purely quantum one.

marcus

I was a bit hasty myself (we had company yesterday evening ). What I meant and should have said is that already in equation (9), which is not semiclassical, you can see there is no dependence on Immirzi in the leading term, which is the area term. The other two terms can be considered corrections, that appear in the full quantum version of the entropy equation, namely equation (9).

I would expect there to be some dependence on Immirzi in the second term. Perhaps both, I haven't thought much about it. But it is the leading term that is the area term, where you see the proportionality of the entropy with the area, and that is where you don't get dependence on Immirzi. I should have made that clearer.

BTW just for clarification the μ that appears in the second term of (9) is called the chemical potential. Doubtless familiar to you, Fzero, but others might be reading. The N-bar that appears in the second term is the average number of punctures. I write it Ñ to avoid having to resort to LaTex.

I really like this Pranzetti paper! For convenience here's the link:
http://arxiv.org/abs/1204.0702
The second term in equation (9) is quite interesting. I think I first saw it in the Ghosh Perez paper last year, but I'm not sure. It is -μβÑ. The β, as he says, can be interpreted as a "generic temperature β for the preferred local observer O hovering outside the horizon at proper distance l, as result..." It seems to be this second term which you are scrutinizing in your above post #56. More power to you .

fzero

As it turns out, the leading term proportional to area comes entirely from the [itex]\beta \bar{E}[/itex] term. For the subleading terms we find

[tex] -\mu \beta \bar{N} =-\bar{N} \log z = \bar{N} \log f + \bar{N} \log\frac{\bar{N}+1}{\bar{N}},[/tex]

[tex]\log \mathscr{Z}=\log(\bar{N}+1).[/tex]

So we can write

[tex] S = \frac{A}{4G\hbar} + \bar{N} \log f + \log \frac{(\bar{N}+1)^{\bar{N}+1}}{\bar{N}^\bar{N}}.[/tex]

Now, remember that

[tex](\log f)' = c \frac{A}{\bar{N}+1}[/tex]

by the energy constraint. We can't explicitly integrate this because [itex]\bar{N}[/itex] is (defined transcendentally as) a function of [itex]\nu[/itex].

It is interesting to try to compare this to the large N result in B10:

[tex] S = \frac{A}{4G\hbar} - \frac{3}{2} \log \frac{A}{G\hbar}. [/tex]

Naively, it doesn't appear possible to reproduce the [itex]\log A[/itex] correction, since [itex]\log f\rightarrow 0[/itex]in the [itex]\mu=0[/itex], large [itex]\bar{N}[/itex] limit. Perhaps there is some subtlety in taking these limits.

marcus

I don't think people are entirely sure what the correction terms are. They all seem to agree that the MAIN term for the entropy is A/4.
And that obviously does not depend on the Immirzi.

And from what i see they expect the correction terms, which arise when you do a full quantum treatment of the BH entropy, to involve the Immirzi.

Bianchi says as much in his recent paper. And if I remember right this is explictly the case both with Ghosh Perez and with Pranzetti.

So this is the picture that is emerging more or less across the board with these young researchers' work on Loop BH.

Supposing they are right and the work is born out, then I think this allows for the Immirzi to run. Run with what? With scale? with energy density?

Then the size of the BH would affect the slight corrections in the formula by which the entropy was calculated.

Maybe the "bare" UV value of Immirzi is, say, 0.274. And she runs to zero when you go to larger and larger scale.

Just speculating
========================

I should copy Pranzetti's equation (9) since that seems to be the main equation we are discussing.===quote http://arxiv.org/abs/1204.0702 page 3 equation (9)===

S = (βκ/8πG)A -μβÑ + log curlyZ
==endquote==

curlyZ is defined in equation (2) as a function of the local observer's temperature β and is the grand canonical partition function for the gas of punctures. These are the links of spin networks sticking out thru the BH horizon.

The term (βκ/8πG) turns out to be 1/4, in the appropriate units, with G = hbar = c = 1.
This is when β is seen to be the Unruh temperature associated with the acceleration which the observer must maintain in order to continue hovering at a fixed distance above the horizon, which has surface gravity κ.

I have no particular reason to copy in equation (9) at this point. We have been discussing it for the past I-don't-know-how-many posts. But I just wanted to finally write it in for completeness, for the record so to speak.

fzero

You can't quite say that the leading contribution to the entropy doesn't depend on the Immirzi parameter. Either it does, or a new parameter is introduced into the model that serves to fix the Immirzi parameter mid-calculation. In particular, we have roughly 4 different modern methods of state counting that have appeared in the literature:

1. SU(2) Chern-Simons with level [itex]k\rightarrow \infty[/itex], microcanonical ensemble: entropy depends on [itex]\gamma[/itex] as

[tex]S = \frac{c_1}{\gamma} \frac{A}{4G}.[/tex]

2. Microcanonical ensemble with area/energy constraint: entropy depends on [itex]\gamma[/itex] as

[tex]S = \frac{c_2}{\gamma} \frac{A}{4G},[/tex]

where [itex]c_2[/itex] is set at a critical value via the constraint. Value agrees with approach 1 above.

3. SU(2) CS with finite level, MCE: entropy does not depend explicitly on [itex]\gamma[/itex], but after extremizing [itex]\gamma[/itex] depends on the level (see, for example, the discussion around fig 6 in http://arxiv.org/abs/1103.2723v1). Value quickly converges to [itex]k=\infty[/itex] result for [itex]k\geq 4[/itex], so consistent with 1 and 2.

4. Grand canonical ensemble with area/energy constraint, chemical potential [itex]\mu[/itex] introduced: entropy does not depend on [itex]\gamma[/itex].
Thermodynamics, including energy constraint, fix [itex]\gamma[/itex] in terms of [itex]\mu[/itex]. In [itex]\mu =0[/itex], large [itex]A[/itex] limit, recover value of [itex]\gamma[/itex] consistent with approaches 1 and 2.

Engle et al, in 1103.2723 linked above, claim that

"As we have just seen, k does modify the leading but does not modify the subleading corrections of the entropy. In that sense, the logarithmic corrections seems to be universal and independent of the Immirzi parameter"

But I think they're making the mistake of forgetting that their critical exponent [itex]\alpha[/itex] is defined in terms of [itex]\gamma[/itex]. Essentially the ratio of [itex]\alpha[/itex] and [itex]\gamma[/itex] must take a critical value when computing the entropy.

Running of the Immirzi parameter already has consequences at lowest orders in the approaches outlined below. Extending the observations of Larsen and Wilzcek and Jacobsen, the space of running parameters is [itex](G,\gamma, \mu, k, \ldots)[/itex]. The fact that critical parameters appear in most approaches also gives another effective quantity that will be a function of scale. I certainly don't know enough to speculate on the consequences.

marcus

Let's suppose Pranzetti's equation is right. Here's his equation (9)
===quote http://arxiv.org/abs/1204.0702 page 3 equation (9)===

S = (βκ/8πG)A -μβÑ + log curlyZ
==endquote==

Here is the first term,
(βκ/8πG)A

Are you saying that this term depends on the Immirzi?

marcus

I don't see how you could possibly be saying that and i don't see anything in your posts that implies it. So a simple "no" answer would suffice.

Just to be clear, I'd like to be sure of that. So that I know we both agree that the first term in Pranzetti's eqn (9) does not depend on the Immirzi.

And in that case we can look at the other two terms, try to estimate their size etc, if you are so inclined. But first let's be sure we understand each other about the leading term.

fzero

The leading term doesn't depend on [itex]\gamma[/itex] in the following sense. It gives

[tex]S = \frac{A}{4G},[/tex]

where [itex]A[/itex] is a macroscopic parameter. In terms of the microscopic parameters,

[tex] A = F(\gamma,\mu).[/tex]

So [itex]A[/itex] is the macroscopic area at a specific value of [itex]\gamma[/itex], much the same way that the microcanonical result [itex]c A/\gamma[/itex] is the area at a specific [itex]\gamma[/itex].

I've been looking at whether or not there's some way to derive an expression for the entropy that makes sense without appealing to the area constraint. I haven't found anything useful so far.

I wrote down expressions for the other two terms in post #58. They are also implicitly functions of [itex]\gamma[/itex], but they vanish in the large N limit.

I worked out what was confusing me about the [itex]\log A[/itex] term. What had happened was B10 partially reproduces the "quantum" corrections from the CS theory (they at least agree at large N). These corrections have been ignored in Pranzetti, so there's no point in looking for them.

marcus

I'm not sure you understand. In the Loop papers I've seen where γ → 0 all the areas remain constant. Spin network labels are increased precisely in accordance with this requirement. So jγ = const. Having gamma, the Immirzi parameter, run does not necessarily introduce any variation in the area. That holds for any area, not only for the areas of BH horizons.

So I would say your first statement is right. The leading term coefficient has no Immirzi dependence.
S = A/4
But your second statement A = F(gamma, mu) does not connect with how I've seen things done in Loop gravity.

I think it's pretty clear that the leading term in (9) need not change as gamma runs, as, for example, γ → 0. It would be interesting, though, to learn something about the dependence of the other two terms, and their sizes relative to the leading term.

Various papers by Bianchi, Magliaro, Perini exemplify this so-called "double scaling limit" it makes sense to keep the overall region of space the same size as you vary parameters. I suspect that the proven usefulness of this type of limit is one of the motivations here: i.e. reasons for interest in the new work giving Immirzi parameter greater freedom.

fzero

I can be more specific using the expressions in post #56. There are some missing factors in those expressions, so let me give some more detail here and clear up the mistakes.

We start with

[tex]\log \mathscr{Z} = - \log ( 1 - z \sum_j (2j+1) e^{-\beta E_j} ), ~~z = e^{\beta\mu}.[/tex]

We use the Schwinger basis and [itex]G\hbar =\ell_p^2[/itex], then

[tex]E_j = \bar{\kappa} \hbar \gamma j .[/tex]

Using [itex]\beta\bar{\kappa} = 2\pi/\hbar[/itex], we can write

[tex]\beta E_j = 2\pi \gamma j.[/tex]

The ensemble energy is

[tex] \bar{E} = - \frac{\partial}{\partial \beta} \log \mathscr{Z} = \frac{z\sum_j (2j+1)E_j e^{-\beta E_j}}{1 - z \sum_j (2j+1) e^{-\beta E_j}}.[/tex]

Let's get a neater expression from this by noting that

[tex] \sum_j (2j+1) E_j e^{-\beta E_j} = -\frac{ \gamma f'(\gamma) }{\beta}
,[/tex]

where

[tex] f(\gamma) = \sum_j (2j+1) e^{-2\pi\gamma j}.[/tex]

Now the energy constraint is

[tex] \frac{\bar{\kappa} A}{8\pi G} = \bar{E} =-\frac{1}{\beta} \frac{\gamma f'(\gamma)}{1-z f(\gamma)}, [/tex]

so we can write

[tex] A =- 4G\hbar \frac{\gamma f'(\gamma)}{1-z f(\gamma)}. [/tex]

The right-hand side of this expression is what we mean by [itex]F(\gamma,\mu)[/itex]. The area [itex]A[/itex] is a fixed input, so it is a transcendental equation that relates [itex]\gamma[/itex] and [itex]\mu[/itex].

We can also note immediately that the leading contribution to the entropy is

[tex] S =- \frac{\gamma f'(\gamma)}{1-z f(\gamma)} +\cdots. [/tex]

In terms of microscopic quantities, this looks [itex]\gamma[/itex] dependent, but the area constraint sets it to a macroscopic constant.



As for the other terms, there are a variety of ways to express them using the expressions

[tex] \bar{N} = \frac{zf}{1-zf}, ~~~ zf = \frac{\bar{N}}{\bar{N}+1}.[/tex]

In particular

[tex] S = \beta \bar{E} - \beta \mu \bar{N} + \log\mathscr{Z} ,[/tex]
[tex] =- \frac{\gamma f'(\gamma)}{1-z f(\gamma)} -\beta\mu \frac{zf}{1-zf} - \log (1-zf),[/tex]
[tex]=-(\bar{N}+1) \gamma f' - \beta\mu \bar{N} + \log(\bar{N}+1).[/tex]

To try to examine these terms, it's useful to work at large [itex]\bar{N}[/itex], for which

[tex] e^{-\beta\mu} \approx f(\gamma) \approx \frac{2\pi\gamma + 1}{\pi^2\gamma^2} e^{-\pi\gamma}.[/tex]

One thing to note about this expression is that there doesn't seem to be any limiting value of [itex]\mu[/itex] for which [itex]\gamma\rightarrow 0[/itex]. In any case, we can use this to write

[tex]S\approx -\bar{N}\gamma f' +\bar{N} \log f + \log\bar{N}.[/tex]

The first two terms are roughly of the same order for [itex]\gamma = O(1)[/itex]. The relation between [itex]\gamma[/itex] and [itex]\mu[/itex] is too unwieldy to do much analytically, but maybe some rough numerics could prove insightful.

Edit: Actually, when [itex]\mu =0[/itex], [itex]f\approx 1[/itex], so [itex]\log f\approx 0[/itex]. From the 1st term, it turns out that

[tex]\bar{N} \approx 0.4227 \frac{A}{4G},[/tex]

so the 3rd term goes like [itex]\log A[/itex]. However, as I mentioned earlier, there are other corrections to the partition function that have not been taken into account that contribute to the log.

marcus

Since we're on a new page I should probably recap what the main topic is. Haven't done that for a while.

http://arxiv.org/abs/1204.5122
Entropy of Non-Extremal Black Holes from Loop Gravity
Eugenio Bianchi
(Submitted on 23 Apr 2012)
We compute the 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.
6 pages, 1 figure

==quote first paragraph==
There is strong theoretical evidence that Black Holes have a finite thermodynamic entropy equal to one quarter the area A of the horizon [1]. Providing a microscopic derivation of the Bekenstein-Hawking entropy

S_BH = A/(4G hbar)

is a major task for a candidate theory of quantum gravity. Loop Gravity [2] has been shown to provide a geometric explanation of the finiteness of the entropy and of the proportionality to the area of the horizon [3]. The microstates are quantum geometries of the horizon [4]. What has been missing until recently is the identification of the near-horizon quantum dynamics and a derivation of the universal form of the Bekenstein-Hawking entropy with its 1/4 prefactor. This is achieved in this letter.
==endquote==

Over the past year or so there have been several Loop gravity papers by various authors (Ghosh, Perez, Pranzetti, Frodden, Engle, Noui...) supporting this general conclusion. If it is sustained (and I think Bianchi's treatment of it will be, possibly among others) this will constitute a landmark. AFAIK no other approach to Quantum Gravity has achieved such a result at the equivalent level of generality. In stringy context the 1/4 prefactor was derived only for highly special extreme cases not expected to be observed in nature.
So it would be natural if Bianchi's paper were to occasion an incredulous outcry from some quarters. We'll have to see if that happens. Anyway the story isn't over, Bianchi and Wieland have a followup paper in the works. Others I mentioned (or forgot to mention) may have as well.

A nice choice of units is made in this paper. c = k_B = 1, so that at all times one sees the dependence on G and hbar and can immediately see what the effect of varying them would be. IOW time is measured in meters and temperature is measured in joules. In such units the planck area is Ghbar so A/Ghbar, as a ratio of areas, is dimensionless (a unitless number) and also, since k_B=1, entropy, which might otherwise be expressed as energy/temperature, turns out to be dimensionless. So the above equation is simply an equality of pure numbers.

marcus

Additional perspective on the significance of the LQG entropy result can be gleaned from this excerpt at the close of Bianchi's conclusion section.
==quote conclusions, page 5==
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.
==endquote==

marcus

Now that we have the "Discrete Symmetries" paper (May 2012 Rov+Wilson-Ewing) a natural question to ask about Bianchi's entropy paper is what if any changes would follow from changing over to the proposed S' action?

In "Discrete Symmetries" RWE consider the effect of time and parity reversal on the conventional Holst action S[e,ω] that has so far been the basis of covariant LQG, i.e. of spinfoam dynamic geometry.

They propose two alternative actions, since these are closely related I will just consider one (S') for simplicity. You can look up the other (S") in their paper if you wish.

The classical basis for spinfoam QG is the Holst action. A 4D manifold M equipped with internal Minkowski space M at each point together with a tetrad e (one-forms valued in M) and a connection ω. The conventional Holst action is:
S[e,ω]=∫e^IΛe^JΛ(∗ + 1/γ) F_{I J}

Here the ∗ denotes the Hodge dual. Rovelli and Wilson-Ewing propose a new action S' that uses the signum of det e: s = sign(det e) defined to be zero if det e = 0 and otherwise ±1.

S'[e,ω]=∫e^IΛe^JΛ(s ∗ + 1/γ) F_{I J}

===========
http://arxiv.org/abs/1205.0733
Discrete Symmetries in Covariant LQG
Carlo Rovelli, Edward Wilson-Ewing
(Submitted on 3 May 2012)
We study time-reversal and parity ---on the physical manifold and in internal space--- in covariant loop gravity. We consider a minor modification of the Holst action which makes it transform coherently under such transformations. The classical theory is not affected but the quantum theory is slightly different. In particular, the simplicity constraints are slightly modified and this restricts orientation flips in a spinfoam to occur only across degenerate regions, thus reducing the sources of potential divergences.
8 pages

So what if any effect would the modified simplicity constraints have on the BH entropy results of Bianchi and others? Here is the modified simplicity constraint for S':
K+sγL=0
This seems to conflict with the idea in the Bianchi paper of a γ-simple representation for which K-γL=0
Section IV "Quantum Theory" starting on page 3 of the RWE paper is specifically about this kind of question:

"Let us now study the effect of using the modified simplicity condition on the quantum theory. We refer the readers to [1, 9, 11, 12] for the general construction. In the quantum theory, π_f^IJ is promoted to a quantum operator which is identified as the generator of SL(2, C) over a suitable space formed by SL(2,C) unitary representations. K_f and L_f are then the generators of boosts and rotations respectively...
...Therefore the key effect of the introduction of the sign s is that the quantum theory now includes both positive and negative k representations..."
This seems very interesting--I'm just now trying to understand it.