The Immirzi limit-1108.0832, 1105.0216, 1107.1320

[marcus]

The Immirzi limit--1108.0832, 1105.0216, 1107.1320

If the Ghosh Perez result (http://arxiv.org/abs/1107.1320 ) on BH entropy is sustained the Immirzi gamma is free to be used in in defining limits such as the classical, or alternatively the continuum limit in loop geometry.

Or I suppose it might conceivably be used in renormalization procedures, coarse-graining, and so forth.

I had not thought much about this, so can't offer an opinion as to what applications are likely to make sense. But if you have a look at the new treatment of BH entropy by Ghosh Perez you will see the Immirzi is not pinned down. It does not enter as a multiplicative factor in the main expression for the entropy--only in a correction term.

In 1108.0832 Rovelli describes a way of taking the classical limit by letting gamma go to zero. See note 7 at the bottom of page 5. http://arxiv.org/abs/1108.0832
He refers there to three other papers, by Magliaro Perini, and also by You Ding and Eugenio Bianchi. I listed one of them 1105.0216 in the headline.

I will fetch the page 5 quote:

There is a formal way to take the classical limit without sending the dimensions of the individual polyhedra to infinity. Since the eigenvalues of the geometrical quantities are proportional to (powers of) the Immirzi parameter γ, one can formally take γ to zero in order to explore the classical limit at fixed boundary triangulation and at fixed boundary size. The γ → 0 limit has been studied in [28–30].

In ref. [28] http://arxiv.org/abs/1105.0216 Magliaro and Perini have already considered this γ → 0 limit. As I recall it is how they prove that loop geometry achieves the Regge GR limit. I have to review this to be sure. Hopefully someone else will be interested enough to take a look.

 

[marcus]

The immirzi γ is basically the size of the characteristic area of LQG expressed in Planck units.

Ghosh Perez give a number theory reasoning that there is a natural upper bound above which the theory doesn't make sense---as I recall this upper bound (defined number theoretically or combinatorially) is about 0.274. Have to check their paper about that.

But they allow γ to be any positive number between 0 and that upper bound. Anything in the interval [0, .274].

Say we are calculating a transition amplitude for some spinnetwork, which serves as a boundary of a spin foam. I'm imagining letting spins j → ∞ and Immirzi γ → 0 simultaneously so that the size of the setup stays the same. Is that the classical limit that Magliaro Perini are talking about?

Company is coming. Have to go. Back later.

 

[marcus]

Magliaro and Perini posted another paper about this limit, just this week. They are calling it the double scaling limit. Their paper was published in the August 2011 issue of European Physical Letters (EPL), so the arxiv posting is the published version, put up after the journal itself appeared.

http://arxiv.org/abs/1108.2258
Emergence of gravity from spinfoams
Elena Magliaro, Claudio Perini
(Submitted on 10 Aug 2011)
We find a nontrivial regime of spinfoam quantum gravity that reproduces classical Einstein equations. This is the double scaling limit of small Immirzi parameter (gamma), large spins (j) with physical area (gamma times j) constant. In addition to quantum corrections in the Planck constant, we find new corrections in the Immirzi parameter due to the quantum discreteness of spacetime. The result is a strong evidence that the spinfoam covariant quantization of general relativity possesses the correct classical limit.
9 pages, shorter version of "Regge gravity from spinfoams"
Published in European Physical Letters, vol 95, n 3 (August 2011)

I already talked about the earlier paper, the May 2011 "Regge gravity from spinfoams."
But I didnt give the abstract. I'll do that, since it spells out some details and gives notation.

Abstract of 1105.0216:
We consider spinfoam quantum gravity in the double scaling limit γ→0, j→∞ with γj=const., where γ is the Immirzi parameter, j is the spin and γj gives the physical area in Planck units. We show how in this regime the partition function for a 2-complex takes the form of a path integral over continuous Regge metrics and enforces Einstein equations in the semiclassical regime. The Immirzi parameter must be considered as dynamical in the sense that it runs towards zero when the small wavelengths are integrated out. In addition to quantum corrections which vanish for ℏ→0, we find new corrections due to the discreteness of geometric spectra which is controlled by γ.
===

I see now that "double scaling limit" is better terminology. I should have used that term instead of saying "Immirzi limit." What interested me about it to start with was the fact that they let the Immirzi parameter run.

Letting gamma go to zero seems to be morally similar to "integrating out the small wavelengths." Or a sort of coarse-graining? You airbrush out the discrete quantum nature of geometry? Shave off the quantum hair?

===============

Magliaro Perini don't really need the Ghosh Perez result, which I mentioned at the beginning of the thread. They explain why it is legitimate to let Immirzi gamma go to zero, even without BH entropy being independently A/4. And the GP result has not been completely reviewed yet, we don't know if it will be sustained. But all the same for me the GP result sheds light on this whole business. In the GP context the gamma enters into a purely quantum-hair term, the "chemical potential" saying how important you are going to consider one additional puncture to be. Taking gamma to zero says you are seeing the BH in a completely hairless way, as classical as it can possibly be. Ignoring the punctures or defects that are part of the quantum BH state. So Ghosh and Perez separate stuff out for you so you can think about it from a different perspective.

Still, the Magliaro Perini result does not actually depend on it.

 

[Physics Monkey]

Regarding the black hole paper, the logic of Ghosh-Perez is not clear to me. They introduce a new term in the entropy which they refer to as quantum hair.

Now this may be a useful decomposition, but won't it be true in a thermodynamic limit that their N is proportional to A and hence the area law is still dependent on gamma?

Put another way, if we are to believe that the N term can be ignored, how do we know that the semiclassical result only captures the explicit area term?

They also make a comment about "natural" values of the chemical potential and reintroduce the tuning that sets gamma to a fixed value.

Have they really fixed the gamma dependence of the area law and freed up gamma for other jobs? How does their result interface with string calculations where the exact semiclassical answer can be accounted for via counting microscopic states i.e. without adding any quantum hair?

 

[marcus]

I'm glad you read the Ghosh Perez paper! It is a preliminary opening to a new direction and potentially important if confirmed. It would be nice if enough people know the paper for us to talk about it.

I should make clear that would not completely free up the Immirzi. The best way is to read what they say on page 4:

==quote page 4 first column==
The hair N has its origin in the underlying quantum geometry and hence, the first law of classical isolated horizons does not possesses this term. Classically, the only natural value of the chemical potential is zero, which implies
1 = ∑(2j + 1) exp(−2πγ√j(j + 1)). This fixes the value of the Immirzi parameter reported earlier and from (20) the entropy S = A/4l²_P . This result (with some mild differences depending up on the IH model) was obtained in all previous counting [8]. Our present result can clearly reproduce these earlier results. However, it differs in many important ways from the existing view- point. First of all, in (20) the Immirzi parameter does not appear as a multiplicative constant. It appears in an additive correction to the semiclassical expression. This additive term is the quantum correction to the semiclas- sical entropy induced by the quantum hair N . This result is more robust in the sense that the semiclassical results are reproduced even when γ does not exactly obey the constraint and the chemical potential is not exactly zero. Even to reproduce all earlier results one only requires the chemical potential to be only close to zero; more precisely N → ∞ and σ ∼ O(1/N), so that the quantum correction to the entropy σN ∼ O(1).
==endquote==
In other words at a classical level the only natural value, as G&P say, is for gamma to satisfy the equation
1 = ∑(2j + 1) exp(−2πγ√j(j + 1)). This means gamma must be 0.274.

But that still leaves some wiggle-room. To recover semiclassical results, G&P say, they just need σN to be of order unity. They do not need gamma to exactly satisfy the constraint, so it does not need to be exactly 0.274 (or whatever the solution of that equation is, I know it is close to 0.274).

Have they really fixed the gamma dependence of the area law and freed up gamma for other jobs?...

Well, perhaps not! The Ghosh Perez result might not sustain scrutiny (other people I gather are checking it out) and even if confirmed it might conceivably turn out to be as significant as I think!

The main job where I see increased gamma freedom helping is with the continuum limit as Magliaro Perini approach it. There you want to take gamma to zero.

The way Magliaro Perini currently present this is that their double-scaling procedure is formal.

They let j → ∞ (the area and volume quantum numbers get large) which if you don't do anything else would make the lattice grow in size without bound. But then they also require γj = constant. The effect is in some sense to make the cell size shrink. So it captures the idea of a continuum limit.

So formally they need to send γ → 0. Now maybe I am wrong and the Ghosh Perez result has no bearing on this. But it seems to me that Ghosh Perez are ALSO getting involved with sending gamma to zero. At this point I just have a dim intuition that there is a possible physical connection between the MP continuum limit and the GP black hole.

 

[Physics Monkey]

I'm glad you read the Ghosh Perez paper! It is a preliminary opening to a new direction and potentially important if confirmed. It would be nice if enough people know the paper for us to talk about it.

I should make clear that would not completely free up the Immirzi. The best way is to read what they say on page 4:

==quote page 4 first column==
The hair N has its origin in the underlying quantum geometry and hence, the first law of classical isolated horizons does not possesses this term. Classically, the only natural value of the chemical potential is zero, which implies
1 = ∑(2j + 1) exp(−2πγ√j(j + 1)). This fixes the value of the Immirzi parameter reported earlier and from (20) the entropy S = A/4l²_P . This result (with some mild differences depending up on the IH model) was obtained in all previous counting [8]. Our present result can clearly reproduce these earlier results. However, it differs in many important ways from the existing view- point. First of all, in (20) the Immirzi parameter does not appear as a multiplicative constant. It appears in an additive correction to the semiclassical expression. This additive term is the quantum correction to the semiclas- sical entropy induced by the quantum hair N . This result is more robust in the sense that the semiclassical results are reproduced even when γ does not exactly obey the constraint and the chemical potential is not exactly zero. Even to reproduce all earlier results one only requires the chemical potential to be only close to zero; more precisely N → ∞ and σ ∼ O(1/N), so that the quantum correction to the entropy σN ∼ O(1).
==endquote==

I wouldn't really call this result more robust. It's completely standard to have subleading (in system size) corrections to various quantities in thermodynamic systems. It is also well known that black hole entropy can have all kinds of subleading corrections.

In other words at a classical level the only natural value, as G&P say, is for gamma to satisfy the equation
1 = ∑(2j + 1) exp(−2πγ√j(j + 1)). This means gamma must be 0.274.

But that still leaves some wiggle-room. To recover semiclassical results, G&P say, they just need σN to be of order unity. They do not need gamma to exactly satisfy the constraint, so it does not need to be exactly 0.274 (or whatever the solution of that equation is, I know it is close to 0.274).
Well, perhaps not! The Ghosh Perez result might not sustain scrutiny (other people I gather are checking it out) and even if confirmed it might conceivably turn out to be as significant as I think!

The main job where I see increased gamma freedom helping is with the continuum limit as Magliaro Perini approach it. There you want to take gamma to zero.

The way Magliaro Perini currently present this is that their double-scaling procedure is formal.

They let j → ∞ (the area and volume quantum numbers get large) which if you don't do anything else would make the lattice grow in size without bound. But then they also require γj = constant. The effect is in some sense to make the cell size shrink. So it captures the idea of a continuum limit.

So formally they need to send γ → 0. Now maybe I am wrong and the Ghosh Perez result has no bearing on this. But it seems to me that Ghosh Perez are ALSO getting involved with sending gamma to zero. At this point I just have a dim intuition that there is a possible physical connection between the MP continuum limit and the GP black hole.

What one needs apparently is order 1 freedom in gamma, not order 1/N freedom. In other words, 0 is not .274 plus order 1/N, so I don't see at all why you think the black hole paper has anything to do taking the gamma goes to zero limit. I see the black hole paper as helping only if it really frees up gamma for order 1 changes.