How Does Spin Foam Model Relate to Regge Calculus in Quantum Gravity?

[marcus]

See what you think of this:


http://arxiv.org/abs/1105.0216
Regge gravity from spinfoams
Elena Magliaro, Claudio Perini
8 pages
(Submitted on 1 May 2011)
We consider spinfoam quantum gravity in the double scaling limit $$\gamma\rightarrow 0$$, $$j\rightarrow\infty$$ with $$\gamma j$$ constant, where $$\gamma$$ is the Immirzi parameter, j is the spin and $$\gamma j$$ 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 $$\hbar\rightarrow 0$$, we find new corrections due to the discreteness of geometric spectra which is controlled by $$\gamma$$.

 

[marcus]

This is based on the version of Loop presented in the
Zakopane lectures. It would seem as if gradual progress is being made towards recovering GR in the appropriate limit.

This paper removes some restrictive assumptions that were made in the authors' earlier work, and they explicitly mention what needs further attention.

My guess is that Rovelli will discuss the progress on this front in his plenary talk on 25 May at the Loops conference.

Perini and Magliaro are also talking about other stuff in the parallel sessions. I looked up their talks to see if there was any connection shedding light on this paper:

Renormalization of spinfoam graviton propagator.
Claudio Perini
M. Ambientales Room. Monday, May, 23rd, 17:00 - 17:20.
Abstract:
We study the correlation functions of metric operators in spinfoam quantum gravity, when more than one spinfoam vertex is considered. Specifically, we compare the renormalized propagator (1->5 move) with the previous single-vertex calculations and find that they coincide to first order, in the flat regime. The result strongly supports the general idea that the spinfoam truncation on a finite 2-complex with few vertices is a good and physically motivated approximation for the study of quantum amplitudes over a semiclassical, flat background.

Curvature in spinfoams.
Claudio Perini
M. Ambientales Room. Tuesday, May, 24th, 15:05 - 15:25.
Abstract:
We consider spinfoam quantum gravity. We show in a simple case that the amplitude projects over a nontrivial (curved) classical geometry. This suggests that, at least for spinfoams without bubbles and for large values of the boundary spins, the amplitude takes the form of a path integral over Regge metrics, thus enforcing discrete Einstein equations in the classical limit.

Coherent states for FLRW space-times in loop quantum gravity.
Elena Magliaro
M. Ambientales Room. Friday, May, 27th, 18:30 - 18:50.
Abstract:
We construct a class of coherent spin-network states that capture proprieties of curved space-times of the Friedmann-Lamaitre-Robertson-Walker type on which they are peaked. The data coded by a coherent state are associated to a cellular decomposition of a spatial t=const. section with dual graph given by the complete five-vertex graph, though the construction can be easily generalized to other graphs. The labels of coherent states are complex SL(2,C) variables, one for each link of the graph and are computed through a smearing process starting from a continuum extrinsic and intrinsic geometry of the canonical surface. The construction covers both Euclidean and Lorentzian signatures; in the Euclidean case and in the limit of flat space we reproduce the simplicial 4-simplex semiclassical states used in Spin Foams.

Magliaro and Perini are postdocs in Ashtekar's QG group at Penn State. They are recent Marseille
PhDs.
In case anyone would like to see the abstracts in context of the full conference program here is a link:
http://loops11.iem.csic.es/loops11/...spinfoams&catid=36:parallel-sesions&Itemid=73

 

[Sardano]

However gamma should have a fixed value in order to properly get the entropy of a BH...

 

[atyy]

However gamma should have a fixed value in order to properly get the entropy of a BH...

Also, they talk about gamma running due to renormalization. http://arxiv.org/abs/1103.6264 seems to indicate that for some forms of coarse graining, the resulting spin foam is different from the type considered here.

I do wonder whether the LQG black hole calculation really calculated the Bekenstein-Hawking entropy, or if it is more like the entanglement entropy, in which case it'd be a correction to the Bekenstein-Hawking term.

 

[marcus]

However gamma should have a fixed value in order to properly get the entropy of a BH...

That's debatable. It was proposed already quite a while ago that gamma could run. In fact should run. A number of researchers have written on that.

 

[atyy]

That's debatable. It was proposed already quite a while ago that gamma could run. In fact should run. A number of researchers have written on that.

But in Jacobson's argument, the value that the Immirzi parameter runs to isn't zero.

OTOH, here they take the large spin limit. I believe Ashtekar in his ILQGS talk with Rovelli and Freidel said, following an argument of Perez, that a black hole doesn't exist in the large spin limit, and should be formed from lots of small spins (which this paper does acknowledge as a different sort of limit that they don't treat).

 

[fzero]

It is a weird limit. In the Holst action, we see that the topological term dominates and we are in the maximally parity-violating phase of the theory. The classical equations of motion are not the Einstein equations, but vanish if the Einstein equation is imposed.

Also, Benedetti and Speziale did not find any physical running of the Immirzi parameter at one-loop in pure gravity, consistent with the fact that divergences start at 2-loops. However, when they coupled the Holst action to fermions, they found that no finite $$\gamma$$ ever runs to $$\gamma \rightarrow 0$$.

This also doesn't seem to be a new result, since it's attributed as equation (101) in Rovelli's Zakopane lectures to the Nottingham group. Perhaps there's a better way to look at the continuum limit with a more direct relation to $$\hbar\rightarrow 0$$.

 

[marcus]

Interesting observations, all! I'll add one bit of intuition, which is not meant as argument or to engage any of the details that people have brought up. It's just a way of looking at it:

The immirzi "gamma" is our handle on the limitations of measurement. Because it controls the area gap---the discreteness of the spectra of the geometric operators that measure areas and volumes etc. Intuitively think of gamma as representing an unavoidable discreteness of geometric measurement.

You can't have a continuum limit unless the immirzi--> 0.

On the other hand you can't have a large scale limit unless j's go to infinity. Because the j labels determine the volumes (node measures) and the areas (linking adjacent nodes).

In that sense it seems intuitively reasonable to want to hold j gamma constant while taking gamma to zero. It captures the idea of both a continuum and a large scale limiting process.

 

[fzero]

You can't have a continuum limit unless the immirzi--> 0.

On the other hand you can't have a large scale limit unless j's go to infinity. Because the j labels determine the volumes (node measures) and the areas (linking adjacent nodes).

In that sense it seems intuitively reasonable to want to hold j gamma constant while taking gamma to zero. It captures the idea of both a continuum and a large scale limiting process.

In the "First 25 years" paper, Rovelli argues the opposite. In section IIC, a few paragraphs after eq (18), he says that

It follows that the continuum limit is intrinsically different, and in fact much simpler, than in lattice QCD. There is no lattice spacing to take to zero, but only the refinement of the two-complex to take to infinity. Therefore the continuum limit is entirely captured by (16).

I can't say much more without it being speculation, since there are many details that I haven't ironed out for myself.

 

[claude.perin]

I agree there is something more to understand in this kind of continuum limit. However, this limit is also considered in the symmetry reduced theory (loop quantum cosmology: LQC) and in the last graviton propagator calculation (arXiv:0905.4082), and it yelds the correct low-energy physics! One can also look at the beautiful paper by M. Bojowald arXiv:gr-qc/0105113 to realize that:

1) the limit gamma->0 (and j->infty) is NOT equivalent to keeping only the Holst parity violating, topological term in the action, but actually is exactly the other way around: it is the limit in which the theory reduces to standard quantum cosmology with only the Einstein-Hilbert term. However, there are parity-braking gamma-corrections, a pure LQG predicion (is it observable? we should look at the forthcoming precision observations of CMB... ;-) Intuetively, this happens because the theory with small finite gamma is very different from the theory with gamma exactly zero (this is a well known discontinuity of classical GR)

2) the classical equations of motion of LQC are obtained in the simultaneous continuum gamma->0 and semiclassical hbar->0 limit, just as in the full LQG theory considered in the new paper arXiv:1105.0216

Here I post some comments to different questions that have appeared

- The equation (101) in Rovelli's Zakopane lectures refers to the asymptotics of a single vertex amplitude. A single vertex can only describe a flat chunk of spacetime, a trivial spacetime endowed with Minkowski metric. But curvature is concentrated on spinfoam dual faces in the 2-complex, where many 4-simplices are glued together. The main purpose of arXiv:1105.0216 is to generalize the previous classical limits to the physically interesting cases of curved spacetimes and general 2-complexes.

- I think that we still cannot say that the gamma->0 limit is a 'running' in the conventional sense. If it is a sort of running, it should be linked to the coarse-graining of the 2-complex. But the spinfoam cut-off on the number of vertices in the 2-complex is neither UV nor IR, but rather on the ratio between the largest and lowest wavelengths allowed (the spinfoam 'lattice' does not possesses an intrinsic scale, the scale is determined by the dynamical field itself, the gravitational field), hence it looks very different from the Wilsonian coarse-graining of standard QFT on a background.

There are 2 kind of continuum limit, and this may generate some confusion. The full continuum limit is defined as the infinite refinement limit (on an infinite 2-complex), as in Rovelli's last reviews. However, in order to compute some physics in the approximation given by the truncation to a finite 2-complex, as done in most semiclassical calculations, the only way to take the continuum and semiclassical limit is to look at gamma->0 and j->infty. The relation between the two is yet to be discovered...

 

[marcus]

Claude,
Enchanté! Vous êtes le bienvenu! Fzero,
You mentioned that section of Rovelli's December review---the "25 years" paper. He's right in the sense that there is no lattice spacing to take to zero. And he's right that what he has there is the best you do for a continuum limit unless you allow immirzi gamma to go to zero.

But I don't necessarily agree with the terms within which Rovelli is working there---the context of assumptions and limitations in the passage you quote. I recall Atyy and I discussed that part of the December paper earlier.

To me it is intuitively reasonable that to get a continuum limit you may have to relax the assumption that gamma is constant and let it go to zero. Then it is a new game, and what he said in December does not apply.

We'll see. The authors are both Rovelli PhDs and have worked closely with him. They acknowledge discussions in this paper. So he knows about this, and will almost certainly refer to it in his overview on Wednesday 25 May.

Don't worry about our speculating just a little bit in this case! It is a fast moving field and we have to look ahead a little. How it looks could change between December and May.

 

[atyy]

There are 2 kind of continuum limit, and this may generate some confusion. The full continuum limit is defined as the infinite refinement limit (on an infinite 2-complex), as in Rovelli's last reviews. However, in order to compute some physics in the approximation given by the truncation to a finite 2-complex, as done in most semiclassical calculations, the only way to take the continuum and semiclassical limit is to look at gamma->0 and j->infty. The relation between the two is yet to be discovered...

Conceptually, is it right that one should take the full continuum limit first then the hbar to zero limit? So present work on the semiclassical limit hope that these will commute?

 

[atyy]

Back in http://arxiv.org/abs/0905.4082 , the requirement for the Immirzi parameter to go to zero to get the correct LQG propagator is dependent on the choice of boundary state. Maybe the boundary state chosen is not correct? OTOH, the boundary state they chose is important, because "The feature of this boundary state is that it selects only one of the critical points, extracting exp iSRegge from the asymptotics of the EPRL spin foam vertex." Also, I don't really understand why large j is the relevant limit (as opposed to hbar going to zero).

 

[marcus]

More about GR from Loop appeared on arxiv today.
Continued work on the graviton propagator. This seems to use the same limit as the opening post paper. Namely gamma --> 0 and j --> infinity. Thought you might be interested.http://arxiv.org/abs/1105.0566
Euclidean three-point function in loop and perturbative gravity
Carlo Rovelli, Mingyi Zhang
16 pages
(Submitted on 3 May 2011)
"We compute the leading order of the three-point function in loop quantum gravity, using the vertex expansion of the Euclidean version of the new spin foam dynamics, in the region of gamma<1. We find results consistent with Regge calculus in the limit gamma->0 and j->infinity. We also compute the tree-level three-point function of perturbative quantum general relativity in position space, and discuss the possibility of directly comparing the two results."

Now it's clear these results will come out at the conference in three weeks.

 

[atyy]

! "We are not persuaded by this intuition (in spite of the fact that one of the authors is quite responsible for propagandizing it" !

Hmmm, very interesting. If this really pans out, I will be surprised. My intuition from CDT was that it would work only in the Lorentzian case. Yet here it seems ok so far for the Euclidean case.

 

[marcus]

Humorous quote! Here is the context on page 3 out of which it is lifted:
==quote==
The second idea for computing n-point functions is the vertex expansion [4]. This is the idea of studying the approximation to Eq.(4) given by the lowest order in the σ → ∞ limit, namely using small graphs and small two-complexes...

The vertex expansion has appeared counterintuitive to some, on the base of the intuition that the large distance limit of quantum gravity could be reached only by states defined on very fine graphs, and with very fine two-complexes. We are not persuaded by this intuition (in spite of the fact that one of the authors is quite responsible for propagandizing it [26–28]) for a number of reasons. The main one is the following. It has been shown that under appropriate conditions Eq.(9) can approximates a Regge path integral for large spins [25, 29, 30]. Regge calculus is an approximation to general relativity that is good up to order O(l2/ρ2), where l is the typical Regge discretization length and ρ is the typical curvature ra- dius. This implies that Regge theory on a coarse lattice is good as long as we look at small curvatures scale. In particular, it is obviously perfectly good on flat space, where in fact it is exact, because the Regge simplices are themselves flat, and is good as long as we look at weak field perturbations of long wavelength. This is precisely the limit in which we want to study the theory here. In this limit, it is therefore reasonable to explore whether the vertex expansion give any sensible result.
==endquote==

At one time in the past, Rovelli argued against using the vertex expansion. Now he is explaining the reasons for using it. References 26-28 are from 1992-1994. That is what the droll mention of "propagandizing" is a about (joint work with Ashtekar and Smolin from the early days). I think the statute of limitations has run out on that one "biggrin: Here is more, from the end:

==quote==
...Because of these various technical complications a direct comparison with the weak field expansion in gμν requires more work. On the other hand, it is not clear that this work is of real interest, since the key result of the consistency of the loop dynamics with the Regge one is already established.
IV. CONCLUSION
We have computed the three-point function of loop quantum gravity, starting from the background independent spinfoam dynamics, at the lowest order in the vertex expansion. We have shown that this is equivalent to the one of perturbative Regge calculus in the limit γ → 0, j→∞ and γj=A.

Given the good indications on the large distance limit of the n-point functions for Euclidean quantum gravity, we think the most urgent open problem is to extend these results to the Lorentzian case, and to the theory with matter [46, 47] and cosmological constant [48–50].
Among the problem that we leave open, are the followingl...

==endquote==

 

[marcus]

I see that the new Rovelli Zhang paper on the graviton 3-point function cites the March 2011 one of Magliaro Perini. (This is one that he will be presenting at conference.) The new one is too recent to have been cited yet or scheduled for presentation, but perhaps it will find its way into discussion.

 

[atyy]

I guess this is still all perturbative (Rovelli and Zhang's comments before their Eq 14), whereas CDT is "non-perturbative". So we still don't know if the theory taken to all orders is defined.

 

[marcus]

Let's get the key passage laid out in front of us so we can examine it and quote from adequate context:

==quote Rovelli Zhang page 1==
..A strategy to address the problem has been developing in recent years, based on two ideas.

The first is to define n-point functions over a background by storing the information about the background in the boundary state [1]. In covariant loop gravity [2, 3], this technique yields a definite expression for the theory’s n-point functions.

The second is to explore the expansion of this expression order by order in the number of interaction vertices [4]. Although perhaps counter-intuitive, this expansion has proven effective in certain regimes; for details see [5, 6].

In particular, the low-energy limit of the two-point function (the “graviton propagator”) obtained in this way from the improved-Barrett-Crane spin foam dynamics [7–12] (sometime denoted the EPRL/FK model) correctly matches the graviton propagator of pure gravity in a transverse radial gauge (harmonic gauge) [13, 14]. This result has been possible thanks to the introduction of the coherent intertwiner basis [15] and the asymptotic analysis of vertex amplitude [16, 17].

The obvious next step is to compute the three-point function. In this paper we begin the three-point function analysis. We compute the three-point function from the non-perturbative theory. As in [14], we work in the Euclidean regime and with the Barbero-Immirzi parameter 0 < γ < 1 where the amplitude defined in [11] and that defined in [12] coincide.

Our main result is the following. We consider the limit, introduced in [14], where the Barbero-Immirzi parameter is taken to zero γ → 0, and the spin of the boundary state is taken to infinity j → ∞, keeping the size of the quantum geometry A ∼ γj finite and fixed.

This limit corresponds to neglecting Planck scale discreteness effects, at large finite distances. In this limit, the three-point function we obtain exactly matches the one obtained from Regge calculus [18].

This implies that the spin foam dynamics is consistent with a discretization of general relativity, not just in the quadratic approximation, but also to the first order in the interaction terms.

The relation between the Regge and Loop three-point function [on the one hand] and the three-point function of the weak field perturbation expansion of general relativity around flat space, on the other hand, remains elusive. We compute explicitly the perturbative three-point function in position space in the transverse gauge (harmonic gauge), and we discuss the technical difficulty of comparing this with the Regge/Loop one.
==endquote==

 

[marcus]

I guess this is still all perturbative (Rovelli and Zhang's comments before their Eq 14), whereas CDT is "non-perturbative". So we still don't know if the theory taken to all orders is defined.

Calling it it "still all perturbative" could be misleading since there is a clear difference between a perturbation series and an ordinary infinite series.

The vertex expansion is an ordinary infinite series and one can, for instance, take the first term of it as a first order approximation.

In case anyone else is reading, the paper discussed here is:

http://arxiv.org/abs/1105.0566
Euclidean three-point function in loop and perturbative gravity
Carlo Rovelli, Mingyi Zhang
16 pages
(Submitted on 3 May 2011)
"We compute the leading order of the three-point function in loop quantum gravity, using the vertex expansion of the Euclidean version of the new spin foam dynamics, in the region of gamma<1. We find results consistent with Regge calculus in the limit gamma->0 and j->infinity. We also compute the tree-level three-point function of perturbative quantum general relativity in position space, and discuss the possibility of directly comparing the two results."

They explicitly contrast what they are doing with the perturbative approach. They compute the 3-point function by their non-perturbative method (to first order) and then ALSO compute the 3-point function in perturbative gravity, to see what comparison can be made.
It turns out to be difficult.

So one has the Loop/Regge result on one hand, and the perturbation series result on the other. The point of the paper is somewhat obscured if one calls Rovelli Zhang "still all perturbative".

It is still first order however! Meaning that they only treat the first term in the vertex expansion. (That is what they are telling you in the discussion you mentioned prior to equation 14.)