marcus said:
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 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
\log \mathscr{Z} = - \log ( 1 - z \sum_j (2j+1) e^{-\beta E_j} ), ~~z = e^{\beta\mu}.
We use the Schwinger basis and G\hbar =\ell_p^2, then
E_j = \bar{\kappa} \hbar \gamma j .
Using \beta\bar{\kappa} = 2\pi/\hbar, we can write
\beta E_j = 2\pi \gamma j.
The ensemble energy is
\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}}.
Let's get a neater expression from this by noting that
\sum_j (2j+1) E_j e^{-\beta E_j} = -\frac{ \gamma f'(\gamma) }{\beta}<br />
,
where
f(\gamma) = \sum_j (2j+1) e^{-2\pi\gamma j}.
Now the energy constraint is
\frac{\bar{\kappa} A}{8\pi G} = \bar{E} =-\frac{1}{\beta} \frac{\gamma f'(\gamma)}{1-z f(\gamma)},
so we can write
A =- 4G\hbar \frac{\gamma f'(\gamma)}{1-z f(\gamma)}.
The right-hand side of this expression is what we mean by F(\gamma,\mu). The area A is a fixed input, so it is a transcendental equation that relates \gamma and \mu.
We can also note immediately that the leading contribution to the entropy is
S =- \frac{\gamma f'(\gamma)}{1-z f(\gamma)} +\cdots.
In terms of microscopic quantities, this looks \gamma dependent, but the area constraint sets it to a macroscopic constant.
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.
As for the other terms, there are a variety of ways to express them using the expressions
\bar{N} = \frac{zf}{1-zf}, ~~~ zf = \frac{\bar{N}}{\bar{N}+1}.
In particular
S = \beta \bar{E} - \beta \mu \bar{N} + \log\mathscr{Z} ,
=- \frac{\gamma f'(\gamma)}{1-z f(\gamma)} -\beta\mu \frac{zf}{1-zf} - \log (1-zf),
=-(\bar{N}+1) \gamma f' - \beta\mu \bar{N} + \log(\bar{N}+1).
To try to examine these terms, it's useful to work at large \bar{N}, for which
e^{-\beta\mu} \approx f(\gamma) \approx \frac{2\pi\gamma + 1}{\pi^2\gamma^2} e^{-\pi\gamma}.
One thing to note about this expression is that there doesn't seem to be any limiting value of \mu for which \gamma\rightarrow 0. In any case, we can use this to write
S\approx -\bar{N}\gamma f' +\bar{N} \log f + \log\bar{N}.
The first two terms are roughly of the same order for \gamma = O(1). The relation between \gamma and \mu is too unwieldy to do much analytically, but maybe some rough numerics could prove insightful.
Edit: Actually, when \mu =0, f\approx 1, so \log f\approx 0. From the 1st term, it turns out that
\bar{N} \approx 0.4227 \frac{A}{4G},
so the 3rd term goes like \log A. 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.
and i don't see anything in your posts that implies it. So a simple "no" answer would suffice.