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

Quote by fzero 					 				 				... Since the BH is not a pure state, the correct way to do the computation is to compute the energy from (9) in an ensemble.  This will reintroduce the factors of $\mu^*$ and $\gamma$ that were found in the polymer paper.
Why so? I see no reason that combining states of the form (9) to make a mixed state would need to introduce a $\gamma$. Please explain.

Recognitions:      Gold Member        Science  Advisor  In case anyone is new to the discussion the "polymer" paper just referred to is from a years and a half ago and is: http://arxiv.org/abs/1011.5628 Black Hole Entropy, Loop Gravity, and Polymer Physics Eugenio Bianchi (Submitted on 25 Nov 2010) Loop Gravity provides a microscopic derivation of Black Hole entropy. In this paper, I show that the microstates counted admit a semiclassical description in terms of shapes of a tessellated horizon. The counting of microstates and the computation of the entropy can be done via a mapping to an equivalent statistical mechanical problem: the counting of conformations of a closed polymer chain. This correspondence suggests a number of intriguing relations between the thermodynamics of Black Holes and the physics of polymers. 13 pages, 2 figures  The main paper we are discussing is the one Bianchi just posted this week. For convenience, since we just turned a page, I will give the link and abstract again: 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

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Quote by marcus 					 				 				Why so?  I see no reason that combining states of the form (9) to make a mixed state would need to introduce a $\gamma$.  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 $N_i$ in the polymer paper, but I will use the notation of the new paper and call them $N_i$. The area of a given tesselation is

$$A = \sum_f 8\pi G\hbar \gamma N_f j_f$$

and we have a constraint that

$$\sum_f N_f = N.$$

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

$$\langle A \rangle_{\mathrm{ens.}} = A_H$$

where we're summing over the distribution of $N_f$. 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

$$A_H = \sum_f 8\pi G\hbar \gamma j_f .$$

This state alone is not a black hole. This is the step that allowed Bianchi to hide the factor of $\gamma$.

Quote by fzero 					 				 				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 $\gamma$ is not something new introduced by Eugenio Bianchi, but rather a general fact in LQG black holes. The $\gamma$-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 $SU(2)$ instead of $U(1)$ - let me notice here that the original proposal of Rovelli in 1996 was to use $SU(2)$ and the shift to $U(1)$ 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.

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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 by fzero 					 				 				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 by francesca 					 				 				The independence wrt the Immirzi parameter $\gamma$ is not something new introduced by Eugenio Bianchi, but rather a general fact in LQG black holes. The $\gamma$-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 $SU(2)$ instead of $U(1)$ - let me notice here that the original proposal of Rovelli in 1996 was to use $SU(2)$ and the shift to $U(1)$ 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

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Quote by marcus 					 				 				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?

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Quote by atyy 					 				 				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.

Recognitions:      Gold Member        Science  Advisor  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

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Quote by francesca 					 				 				The independence wrt the Immirzi parameter $\gamma$ is not something new introduced by Eugenio Bianchi, but rather a general fact in LQG black holes. The $\gamma$-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 $SU(2)$ instead of $U(1)$ - let me notice here that the original proposal of Rovelli in 1996 was to use $SU(2)$ and the shift to $U(1)$ 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 $\gamma$ 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.

Quote by fzero 					 				 				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 $γ$-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 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.

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Quote by fzero 					 				 				...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.

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Quote by francesca 					 				 				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 $\delta E$. This is the same semiclassical reasoning as Hawking, the quantum computation of the energy $E$ is not done, but $\delta E$ is properly accounted for.

Finally, it's already clear how the "energy ensemble" works. The states that Bianchi uses satisfy $\vec{K} = \gamma \vec{L}$ as well as $|\vec{L}| = |L_z|$. 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 $\mu$ by $\gamma$.

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Quote by marcus 					 				 				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.

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Quote by francesca 					 				 				The independence wrt the Immirzi parameter $\gamma$ is not something new introduced by Eugenio Bianchi, but rather a general fact in LQG black holes. The $\gamma$-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 $SU(2)$ instead of $U(1)$ - let me notice here that the original proposal of Rovelli in 1996 was to use $SU(2)$ and the shift to $U(1)$ 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.

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Quote by atyy 					 				 				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.

Recognitions:      Gold Member        Science  Advisor  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.

Recognitions:      Gold Member        Science  Advisor  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

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