Bianchi's entropy result--what to ask, what to learn from it I think most if not all here are familiar with the idea that entropy, by definition, is not an absolute but depends on the observer. (Padmanabhan loves to make that point. :-D) There may also be an explicit scale-dependence. And in the Loop context one expects the Immirzi parameter to run with scale. Likewise black hole horizon temperature is highly dependent on how far away the observer is hovering. So there is this interesting and suggestive nexus of ideas that we need to pick apart and learn something from. Bianchi has just made a significant contribution to this. 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
Re: Bianchi's entropy result--what to ask, what to learn from it Bianchi uses the classic Clausius definition of entropy ∂S = ∂E/T and makes it very clear where the observer is hovering, at what distance from horizon. So observer's measurement of E and T depends on that, but the effects cancel and to first order he gets S = A/4. Earlier treatments of BH entropy did not use the Clausius relation. Instead, they employed state counting. One assumes the observer is low-resolution and can make only the coarsest distinctions. So the more states he confuses with each other, the more entropy. You take the log of the number of states and that's it. Or if it is a Hilbertspace of quantum states, you take the log of the dimension of the Hilbertspace. Ted Jacobson made some critical comments about this in a 2007 paper, which is Bianchi's reference [20] at the end. I would be really interested to know Jacobson's reaction to Bianchi's paper. http://arxiv.org/abs/0707.4026 Renormalization and black hole entropy in Loop Quantum Gravity Ted Jacobson 7 pages (Submitted on 26 Jul 2007) "Microscopic state counting for a black hole in Loop Quantum Gravity yields a result proportional to horizon area, and inversely proportional to Newton's constant and the Immirzi parameter. It is argued here that before this result can be compared to the Bekenstein-Hawking entropy of a macroscopic black hole, the scale dependence of both Newton's constant and the area must be accounted for. The two entropies could then agree for any value of the Immirzi parameter, if a certain renormalization property holds." Bianchi also introduces the concept of "quantum Rindler horizon" which I don't recall being used in earlier Loop BH entropy papers. If you know of an instance, please let me know--I could have simply missed it. Mathematically the idea of "γ-simple" unitary representations of SL(2,C) is intriguing and could turn out to be a fertile useful concept. It was already there, he just found a good terminology, I think, and occasionally in math that can be important. I wonder if one might conclude that the bare value of the Immirzi is 0.2375. In many papers that study the long distance limit they let gamma go to zero----meaning that the region stays the same size but its geometry gets less fuzzy. Less "rumpled" like an unmade bed is rumpled. Then gamma=0.2375 would represent the maximal rumpling of nature. Just speculating It's classy to use the Clausius definition of entropy. In my humble opinion if you ever want a beard this is the kind to have: http://en.wikipedia.org/wiki/Rudolf_Clausius (1822-1866) Under no circumstances do you want one like this http://en.wikipedia.org/wiki/Ludwig_Boltzmann
Re: Bianchi's entropy result--what to ask, what to learn from it If there's no state counting, isn't this just a semiclassical calculation, like Hawking's?
Re: Bianchi's entropy result--what to ask, what to learn from it What intrigues me is the appearance of the boost generator. I would like to understand this operator better in the loop gravity context. The connection between rindler horizons and the boost generator is long known, but I have a lot of interest in this topic because we have recently been able to put this connection to good use in condensed matter. Of course, all this spin network stuff reminds me of my old pal tensor networks, and I wonder if there is some grand synthesis (involving tensor networks, entanglement, holography, ...) possible here.
Re: Bianchi's entropy result--what to ask, what to learn from it No, it's not semiclassical. In fact in the paper all the ingredients are derived from the full quantum theory. The relation energy-area was found by Frodden Gosh Perez using Einstein equations; here it is found using the boost generator given by Spinfoam Theory. The calculation in order to find Unruh temperature is done here again using the boost generator, it's completely new. And finally there is the remarkable demonstration that the Spinfoam amplitude implies the right distribution, that yields Hawking entropy.
Re: Bianchi's entropy result--what to ask, what to learn from it Having read the paper a little more closely, I have some basic confusion about what is going on: 1. Although Bianchi claims that E and A don't commute, it looks like on the image under [itex] Y_\gamma [/itex] of the spatial spin network states they are essentially identical. This seems to be so because of the [itex]\gamma[/itex]-simple constraint Eq. 6 2. Related to 1, in what sense can an eigenstate of energy and area possibly have an entropy? 3. Everything looks like a product state over facets, but I would expect entropy and thermalization to be associated with some interactions between facets. 4. Is there a [itex] \rho [/itex] for which [itex] S = - \text{tr}(\rho \log{\rho})[/itex]? 5. What is the physical state space? Is it the finite spin network basis (given a set of punctures)? Surely the continuous space of SL(2,C) representations are not the physical states? I should say that I haven't yet processed the temperature derivation section, although it looks like a standard unruh-type setup. Perhaps some of the answers can be found there, but many of these issues seem more basic as if they should be understood before tackling the issue of temperature.
Re: Bianchi's entropy result--what to ask, what to learn from it I am interested in the spectrum of the reduced density matrix of spatial regions inside bulk materials. This spectrum knows a lot about entanglement e.g. the entanglement entropy is computable from it. If [itex] \rho_R = \exp{(-H_R)}[/itex] is the reduced density matrix of region R in the ground state, then the spectrum of [itex] H_R [/itex] (defined by this equation) is the entanglement spectrum. It is an old result in Lorentz invariant field theory that when R is the half space, say x>0, then [itex] H_R = 2 \pi K [/itex] with [itex] K [/itex] the boost generator mixing x and t. Thus for LI field theory we know the entanglement spectrum for a special subregion, the half space. The form of the operator [itex] K [/itex] is [itex] K = \int_{x>0} dx\, dx^2 ... dx^d \left( x T^{tt} \right)[/itex] (at t=0) and hence it looks like the physical Hamiltonian with an edge. We used this to show that in many cases the entanglement spectrum shares many universal features with the energy spectrum of a physical edge. In other words, the imaginary entanglement cut becomes a real physical cut in the system. A simple example is provided by the fractional quantum Hall effect. In that case a physical edge always has a chiral edge mode circulating around the sample. Using the technology above we were able to show that the entanglement spectrum also has this chiral edge mode. So even on a system with no boundary you can, by looking at entanglement, detect the existence of protected chiral edge states.
Re: Bianchi's entropy result--what to ask, what to learn from it But isn't the action at the end a semiclassical one?
Re: Bianchi's entropy result--what to ask, what to learn from it The main result(s) of the paper are proved in the first 4 pages up thru the section called *Entropy of the Quantum Horizon*. You must be talking about some action that appears in pages 1-4, but I can't figure out which. There is the section on page 5 which I see as kind of a postscript. It contains some interesting reflections and points to some future work (a paper which Wieland and Bianchi have in the works.) But that is not essential to the main work of the paper, it's more interpretive afterthought, and it does mention something that occurs in the "semiclassical limt of the Spinfoam path integral..." But that hardly means that the whole paper is proving things only at the semiclassical level (this is what some of your earlier comments seemed to be suggesting.)
Re: Bianchi's entropy result--what to ask, what to learn from it A simpler reason to worry about semiclassicality is found in the early pages, especially after Eq. 8 and Eq. 9. There Bianchi makes heavy use of the classical results to identify the right operator to call the "energy" of the horizon. One could worry in the usual way that this identification is semiclassical. For example, will the quantum hair proposal of Ghosh-Perez be captured by these identifications?
Re: Bianchi's entropy result--what to ask, what to learn from it The Clausius relation is classical. The calculation done here is semiclassical because [itex]\delta E[/itex] corresponds to the addition of a single quantum of energy. In this regard, it's not that different from Hawking's approach. A fully quantum treatment must involve the counting of microstates. In fact, the derivation of the energy of the black-hole is quite confusing from entropy considerations. Bianchi says that the Rindler surface is described by the state [tex]|s\rangle = \otimes_f | j_f \rangle,[/tex] which results from a tesselation into the facets [itex]f[/itex]. But this is a pure state and should have zero information-theoretic entropy. I'm not sure if it even makes sense to talk about other tesselations in this framework, but from the statistical point of view, one would want a mixed state obtained by summing over tesselations. The black hole should then turn out to be a maximal entropy configuration.
Re: Bianchi's entropy result--what to ask, what to learn from it I've been appreciating your comments, since you know a lot about this. I'm glad you took an interest and read the paper. Part of the confusion could be due to problems with notation. I think of what we have now as a draft to which more explanation could be added. I could be wrong but I don't think it says "E and A don't commute". The OPERATORS for energy and area are denoted H and A, are they not? The letter E seems to denote a quantity. At one point he says E = <s|H|s>, so as a quantity it would commute with everything I suppose. The energy operator H is defined by eqn (8) and seems to be composed of boost pieces. The area operator seems to be composed of rotation pieces. Correct me if I'm wrong these don't commute as operators, do they? Equation (6) just says they have the same matrix element form. Let me know if I'm saying something really stupid. So anyway I think on page 3, middle of first column, where he says "the energy does not commute with the area of the quantum horizon" what he means is "H and A don't commute." Is this right? You are by far the expert in this context.
Re: Bianchi's entropy result--what to ask, what to learn from it How is the operator [itex]|\vec{L}_f|[/itex] defined? I understand the subscript, but is it supposed to be the operator whose eigenvalue is the square root of that of [itex]|\vec{L}_f|^2[/itex]? How do we find [itex]j_f[/itex] instead of [itex]\sqrt{j_f(j_f+1)}[/itex] (or has the limit of large [itex]j_f[/itex] been taken?) In any case, the states [itex]|E\rangle[/itex] defined below (9) are simultaneous eigenstates of [itex]|L|, L_z[/itex] and [itex]K_z[/itex], so [itex]H[/itex] and [itex]A[/itex] commute on them as operators.
Re: Bianchi's entropy result--what to ask, what to learn from it Just to be clear, what I said in post #14 was in reply to this of Physics Monkey: Physics Monkey also asked about the physical state space. It may help us to better understand the paper and even some of the notation if we read the first paragraph, where he refers to an earlier paper of his about black hole entropy. This is his reference [4] ==quote first paragraph== 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== Here is [4]: 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 This was a year and a half ago and employed an entirely different method, namely (semiclassical) state-counting. But some of the notation and definitions undoubtably overlap, so this paper might be of use.
Re: Bianchi's entropy result--what to ask, what to learn from it In that paper, he uses [itex]A_f \sim \sqrt{j_f(j_f+1)}[/itex] and the entropy still has the coefficient of [itex]\gamma[/itex]. The troublesome thing is that, in the new paper [tex] E \sim \sum_f j_f [/tex] is not proportional to [tex] A \sim \sum_f \sqrt{j_f(j_f+1)}.[/tex] Using the Clausius relation gives a correction to the area law. At first order, the correction is proportional to [itex]N[/itex], the number of facets. The 2010 paper, if it applies here, suggests in eq (19) that [itex] N \sim A[/itex], so this corrects the coefficient of the leading term (away from 1/4). Edit: there is a mistake in the estimate above, see https://www.physicsforums.com/showpost.php?p=3884283&postcount=22
Re: Bianchi's entropy result--what to ask, what to learn from it That's fascinating. In the original case of a Lorentz invariant field theory on the half space, are there also "edge states"?
Re: Bianchi's entropy result--what to ask, what to learn from it I didn't even notice this at first, but it looks like bianchi is either doing the large j limit or made an important mistake?
Re: Bianchi's entropy result--what to ask, what to learn from it There certainly can be. Not all Lorentz invariant theories have protected physical edge states on a half space, but we showed that if they do then the half space entanglement spectrum (with no physical edge) also has the universal aspects of these physical edge states.