Gravity as Entanglement Thermodynamics

[S.Daedalus]

Gravity as "Entanglement Thermodynamics"

The recent paper by Lashkari, McDermott and Van Raamsdonk, Gravitational Dynamics from Entanglement "Thermodynamics", has prompted me to consider this approach (which I think I've posted about once or twice on here) once again, and to gather some opinions on this approach to quantum gravity.

Briefly, I see it essentially as originating from Sakharov's ideas on induced gravity, mainly because that was what Ted Jacobson was working on before writing his seminal paper, on the Einstein equation of state. There were, however, some earlier bids to get spacetime and gravity starting from the quantum, e.g. von Weizsäcker's ur-theory, Penrose's spin networks (which in their original formulation referred more directly to 'exchanges' of units of spin between larger systems, from which he was able to derive the structure of three dimensional space), and Finkelstein's space-time code; the former two (in a sense) have found a modern information theoretic successor in the arguments by Müller and Masanes, who show that exchanging qubits in order to accumulate direction information necessarily leads to the appearance of three dimensional space.

More generally, extensions of quantum mechanics based on the quaternion and octonion algebras have been investigated, leading to 5+1 and 9+1 dimensional spacetimes respectively, which seems to point to the fact that at the heart of this relationship is the simple fact that the SLOCC group of single qubits in these cases are just $SL(2,\mathbb{C})$ for standard QM, and $SL(2,\mathbb{H})$ and $SL(2,\mathbb{O})$ for the quaternionic and octonionic cases respectively, which are in turn isomorphic to the Lorentz groups $SO(3,1)$, $SO(5,1)$ and $SO(9,1)$.

Thus, getting spacetime from the quantum is an old idea, that seems to be cashed out now thanks to the termodynamic wrinkle introduced by Jacobson, and refined by (among others) Padmanabhan and most intriguingly Van Raamsdonk (there's also, of course, Verlinde's 'entropic gravity', but I tend to see this more as a toy model of the more developed ideas). The basic idea of this is that if the Bekenstein-Hawking area-entropy relation holds, Einstein's equations can be deduced from simple thermodynamics, making gravity effectively an emergent rather than fundamental force (which is only natural, since spacetime itself is not fundamental in this approach).

The added wrinkle here is that the origin of BH entropy is supposed to lie in quantum mechanical entanglement. One of the first to realize that entanglement entropy, like BH entropy, follows an area law was Srednicki; however, unless you impose a cutoff, the entanglement entropy is divergent. Last year, though, Jacobson has argued that the emergence of gravity effectively renders the entropy finite.

Of late, this picture has become important in the discussion of the black hole firewall problem, with the Maldacena/Susskind ("ER=EPR") conjecture that entangled particles should be connected by a wormhole in the gravitational dual; the recent 'fuzz or fire'-conference even featured a special session on spacetime from entanglement.

All this seems like it should have connections to holography as it is more usually understood, i.e. in the AdS/CFT context, per e.g. Swingle's work on conceptualizing entanglement renormalization as a discrete version of the correspondence (I'm not clear on the details here, and would love some pointers, though). At some point, the words 'Ryu-Takayanagi formula' should probably be used.

The picture that's developing, to my eyes, is roughly the following: spacetime is a fundamentally quantum mechanical object, with separate quantum states yielding separate spacetime pieces, which can be connected by entanglement ('entanglement as glue', Lubos has called it somewhere). Gravity is nothing but the dynamics of this entanglement, governed by thermodynamics. It's then not a fundamental force; rather, you get it extra, if you start with the right quantum (field) theory.

This obviously raises some intriguing questions. First of all, as already Jacobson remarked in his '95 paper, quantizing gravity may then be just kind of a category error, unable to yield the true microscopic degrees of freedom, like quantizing water waves does not yield H₂O atoms. That might alleviate some worries about the nonrenormalizability of quantum gravity (if it's useful as a theory at all, it's certainly an effective theory, so there's no real need for it to be renormalizable), and the areas of conflict between QM and GR might just be those where the effective theory no longer describes the situation well---i.e. the 'out of equilibrium'-situations (singularities in black holes, the big bang etc.).

Another interesting question is precisely what is needed for quantum theory to yield gravity in this way. The arguments pointing towards 3+1 dimensional spacetime from quantum theory seem to be quite generic, as you really only need two level quantum systems for that. But when does gravity fall out as entanglement thermodynamics? Do you need a QFT, or even a CFT?

Of course (and the main reason for my starting this thread), it's also possible that I've gotten this whole thing wrong, and am thinking about it in a completely muddle-headed way---because frankly, I'm a bit surprised at the relative lack of discussion regarding what seems (to me, anyway) to be a real shot at getting around the problems of combining quantum theory and relativity in a consistent manner. So if all of this is just wrong (or 'not even'), I'd humbly ask to be educated.

Otherwise, what are your thoughts on the matter? Merely a theoretical curiosity, or some genuine new (I don't want to say 'paradigm changing') development? What does it mean in relation to established quantum gravity proposals, be they stringy, loopy, or something else-y? (One thing I remember from Penrose is the remark that essentially, quantum systems come with their own three-geometry, regardless of what other geometry they may be embedded in; I've sometimes thought that maybe this could be a good alternative way of thinking about dimensional reduction in strings.)


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(Apologies for the length, and thanks if you've stuck it out 'till here...)

 

[atyy]

I think it's a major theme. We've been trying to follow the ideas in https://www.physicsforums.com/showthread.php?t=376399.

Off the top of my head, the major old papers about entanglement and spacetime are:
Maldacena Eternal Black Holes in AdS
Ryu and Takayanagi Holographic Derivation of Entanglement Entropy from AdS/CFT
Swingle Entanglement Renormalization and Holography
van Raamsdonk Comments on quantum gravity and entanglement

Th old papers didn't have much dynamics. There have been many papers eg. http://arxiv.org/abs/1006.4090, http://arxiv.org/abs/1008.3439, http://arxiv.org/abs/1305.7244, http://arxiv.org/abs/1303.1080 looking at the time evolution of the entanglement entropy using the Ryu-Takayanagi formula - and these have provided support that the ideas are in the right direction - but I think the new articles are now really trying to get the Einstein equation out.

In addition to Lashkiri, McDermott and Raamsdonk's Gravitational Dynamics From Entanglement "Thermodynamics" that you mentioned, I think it's interesting to also look at Bhattacharya and Takayanagi's Entropic Counterpart of Perturbative Einstein Equation.

Related papers from the LQG side:
Bianchi Black hole entropy from graviton entanglement
Pranzetti Black hole entropy from KMS-states of quantum isolated horizons (The terminology here is a bit unusual, but by von Neumann entropy, he does mean entropy from entanglemement)

 

[marcus]

...
From the LQG side:
Bianchi Black hole entropy from graviton entanglement
Pranzetti Black hole entropy from KMS-states of quantum isolated horizons (The terminology here is a bit unusual, but by von Neumann entropy, he does mean entropy from entanglemement)

In the paper you cite http://arxiv.org/abs/1211.0522 LQG does not enter (although the connection with a nonperturbative result using LQG in another Bianchi paper is mentioned at the end).
The paper of Bianchi that you cite is a low-energy perturbative, essentially (semi)classical, analysis.

The LQG-based paper, on the other hand, is:
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

Bianchi's plenary talk at the recent (July 2013) Loops conference covers BOTH results. Here's the video:
http://pirsa.org/13070048/
http://pirsa.org/displayFlash.php?id=13070048
Eugenio Bianchi, Perimeter Institute
Entanglement, Bekenstein-Hawking Entropy, and Spinfoams
I review recent developments on vacuum entanglement perturbations in perturbative quantum gravity and spinfoams, and discuss their relevance for understanding the nature of black hole entropy.

The talk+discussion lasted a bit over 45 minutes. I think it's good he has shown how to get the same result (BH entropy = entanglement entropy) using different approaches---both perturbative and nonperturbative (spin foam). Makes the whole thing more solid and reliable.

For completeness, here is the abstract of the paper that Atyy cited (in which LQG does not enter in the analysis but is mentioned at the end):
http://arxiv.org/abs/1211.0522
Black hole entropy from graviton entanglement
Eugenio Bianchi
(Submitted on 2 Nov 2012 (v1), last revised 7 Jan 2013 (this version, v2))
We argue that the entropy of a black hole is due to the entanglement of matter fields and gravitons across the horizon. While the entanglement entropy of the vacuum is divergent because of UV correlations, we show that low-energy perturbations of the vacuum result in a finite change in the entanglement entropy. The change is proportional to the energy flux through the horizon, and equals the change in area of the event horizon divided by 4 times Newton's constant - independently from the number and type of matter fields. The phenomenon is local in nature and applies both to black hole horizons and to cosmological horizons, thus providing a microscopic derivation of the Bekenstein-Hawking area law. The physical mechanism presented relies on the universal coupling of gravitons to the energy-momentum tensor, i.e. on the equivalence principle.
4 pages

 

[marcus]

Bianchi's first paper (the nonperturbative LQG/spinfoam approach) is especially significant because it gets the 1/4 coefficient right.
Recent comment by Lee Smolin:
http://www.math.columbia.edu/~woit/wordpress/?p=6208&cpage=2#comment-159323

==quote==
Moreover, recent work deriving black hole thermodynamics from quantum gravity by Bianchi, both perturbative (arXiv:1211.0522) and non-perturbative (arXiv:1204.5122) shows that the black hole entropy is best understood as an entanglement entropy. I would suggest that this be taken seriously as it is the only calculation of the BH entropy that gets the 1/4 right without any parameter fixing for a generic non-extremal black hole.
==endquote==

 

[marcus]

Steven D., it looks to me as if your opener post raises two distinct questions:
(1) gravity as statistical mechanics of some kind of "geo-molecule" we haven't yet identified

(2) (BH-related and some other geometric-type) entropy = entanglement

You can conveniently merge the questions because if (1) could be definitely and affirmatively answered then (2) could presumably follow. But it's also useful to distinguish two "levels" of question. Eugenio Bianchi and others now seem to have partially proven (2) both perturbatively at the level of effective QG, and nonperturbativley in LQG, proven at least in interesting cases like Rindler flat space, and BH.

Given the progress in just the past 2 years, I would expect this result to continue being extended.

But that doesn't identify what the "geo-molecules" are---the geometric units that participate in the dynamic geometry called gravity. What are they? Bits of angle, flecks of area, chunks of volume?
A shimmering foam of indefinite polyhedra that have not yet decided how many faces they want to have?

A lot of LQG/spinfoam research has been aimed at trying to find out what these geometric units might be, and when there is a guess as to what they are then trying out calcullations with them.

for any given idea of "geo-molecules", should we think of them INFORMATIONALLY as the results of geometric MEASUREMENTS we might ideally make? A finite number of interconnected measurments of angle, area, volume etc.? Or should we think of them as really THERE in nature?

At least naively, I would suppose that if the plan is to do statistical mechanics with them, and talk about the temperature and entropy of geometry, then maybe they better be real. But reserve judgment and won't worry about that now.

Anyway, it's interesting that if you look at Carlo Rovelli's papers you see that up thru 2011 he was intently focused on developing the spinfoam formalism for LQG and proving various things about it like Lorentz covariance etc etc...

But since 2012 he and collaborators have been primarily focused on the problem of formulating general covariant quantum statistical mechanics and on thermodynamics in the GR or general covariant context. A bunch of younger people seem to have taken over responsibility for the Loop gravity spin foam formalism (I'm thinking of Jon Engle, of Wolgang Wieland, and many others) so plenty of progress is occurring on that front.

The point, though, is that a bunch of LQG researchers seem to have split off and gotten focused on exactly the sort of stuff you suggested in your opener post.
How do you do general covariant (diffeomorphism invariant) QSM? To do quantum statistical mechanics (without a prior selected time) you presumably need to have "geo-atoms" and you need some way for a global time to arise.

I hope you have been thinking along these lines too, and find these questions interesting.

At the minimum, it would seem, there has to be an observable algebra. Probably a C* algebra. And the STATE of the system (or perhaps universe) has to be definable as a positive functional on the algebra. That really is the bare minimum, for a quantum system, more basic even that the Hilbert space ordinarily used to define the observables as operators. So then given a state (which in this sense is good for all time, except we don't have time defined yet) you can GET a state-dependent time defined as Tomita-flow on the observables. One parameter group of automorphisms. Alain Connes and Rovelli wrote a paper about this in the 1990s. My guess is that gen. cov. QSM is going to take shape in that math context.

The "particles" of geometry may then come to light as truncations of the algebra, or so I vaguely imagine. But that's too far ahead to be speculating.

 

[marcus]

Straws-in-the-wind (indicating directions of future research):

http://arxiv.org/abs/1306.5206
The boundary is mixed
Eugenio Bianchi, Hal M. Haggard, Carlo Rovelli
8 pages, 2 figures

http://arxiv.org/abs/1302.0724
Death and resurrection of the zeroth principle of thermodynamics
Hal M. Haggard, Carlo Rovelli
5 pages, 2 figures

http://arxiv.org/abs/1211.2166
The spin connection of twisted geometry
Hal M. Haggard, Carlo Rovelli, Francesca Vidotto, Wolfgang Wieland
5 pages, 2 figures

http://arxiv.org/abs/1209.0065
General relativistic statistical mechanics
Carlo Rovelli
A tentative second step in the thermal time direction, 10 years after the paper with Connes. The aim is the full thermodynamics of gravity. The language of the paper is a bit technical: look at the Appendix first (expanded in version 2)
10 pages

 

[S.Daedalus]

I think it's a major theme. We've been trying to follow the ideas in https://www.physicsforums.com/showthread.php?t=376399.

Hi atyy, thanks for the pointers; I've been sort of keeping an eye on the thread, and while it's all clearly related, I think there's some work needed to untangle what relates to what, and how. For instance, there's the condensed matter angle: you can get sort of an emergent metric in an appropriate condensed matter system, such that excitations 'feel' a particular kind of vacuum spacetime. An interesting angle here is that you don't have to start with a special relativistic system to get the excitations to behave as if they were in a Minkowski space; I think Olaf Dreyer ('internal relativity') has thought about that. And of course, the precise structure of the entanglement in a condensed system is an indicator of its characteristics, phase, criticality, things like that. But it's not clear to me how that relates to the 'spacetime-from-entanglement' angle precisely; it's one of these things I thing someone (other than me ) should really sit down and make clear for good.

Steven D., it looks to me as if your opener post raises two distinct questions:
(1) gravity as statistical mechanics of some kind of "geo-molecule" we haven't yet identified

(2) (BH-related and some other geometric-type) entropy = entanglement

You can conveniently merge the questions because if (1) could be definitely and affirmatively answered then (2) could presumably follow. But it's also useful to distinguish two "levels" of question. Eugenio Bianchi and others now seem to have partially proven (2) both perturbatively at the level of effective QG, and nonperturbativley in LQG, proven at least in interesting cases like Rindler flat space, and BH.

Given the progress in just the past 2 years, I would expect this result to continue being extended.

But that doesn't identify what the "geo-molecules" are---the geometric units that participate in the dynamic geometry called gravity. What are they? Bits of angle, flecks of area, chunks of volume?
A shimmering foam of indefinite polyhedra that have not yet decided how many faces they want to have?

I think that part of the idea is that it's not all that significant what the precise lower level details are, at least if you think about gravity only: like universal phenomena in condensed matter and effective field theory, you can to a certain extend just 'forget' about the low-level kinematics. As long as you have entanglement and an area law, you can use the Jacobson-style thermodynamic arguments and come up with gravity.

This is part of the reason I'm not sure that the loop angle isn't too specific at this point: ultimately, its motivation is to find a consistent quantum theory of gravity, and I think the promise of this approach is that you might not have to. So even if you can quantize gravity, and if you can do it using loop variables, then it's rather more like a quantum field theory of phonons, for example, as the spacetime itself emerges from some ('pre-geometric') quantum considerations.

Now it might turn out that these pre-geometric quantum objects are in some sense related to loops or spinfoams, or can at least be usefully described as such, but I think one needs to establish a clear nomological hierarchy here; for instance, deriving black hole entropy from graviton entanglement seems, at least to me, to be kind of putting the cart before the horse: it's ultimately the black hole entropy (derived from the entanglement of whatever the underlying degrees of freedom may be) that necessitates gravity, and thus (provided it makes sense to then quantize gravity) leads to the existence of gravitons.

The point, though, is that a bunch of LQG researchers seem to have split off and gotten focused on exactly the sort of stuff you suggested in your opener post.
How do you do general covariant (diffeomorphism invariant) QSM? To do quantum statistical mechanics (without a prior selected time) you presumably need to have "geo-atoms" and you need some way for a global time to arise.

I hope you have been thinking along these lines too, and find these questions interesting.

Well, as I said, I come at this from the different angle of really viewing spacetime, and gravity, as ultimately emergent, the result of a coarse-grained description of the entanglement properties of some underlying quantum theory. But then, also things like diffeo invariance and so on aren't really fundamental anymore; as I see it, this isn't about a thermodynamics of spacetime so much as a thermodynamic description of something giving rise to spacetime.

 

[atyy]

Hi atyy, thanks for the pointers; I've been sort of keeping an eye on the thread, and while it's all clearly related, I think there's some work needed to untangle what relates to what, and how. For instance, there's the condensed matter angle: you can get sort of an emergent metric in an appropriate condensed matter system, such that excitations 'feel' a particular kind of vacuum spacetime. An interesting angle here is that you don't have to start with a special relativistic system to get the excitations to behave as if they were in a Minkowski space; I think Olaf Dreyer ('internal relativity') has thought about that. And of course, the precise structure of the entanglement in a condensed system is an indicator of its characteristics, phase, criticality, things like that. But it's not clear to me how that relates to the 'spacetime-from-entanglement' angle precisely; it's one of these things I thing someone (other than me ) should really sit down and make clear for good.

One shouldn't take my use of "condensed matter" too seriously. After all, it is a field which was "saved" by an HEP outsider (Wilson). OTOH, that outsider working on a insightful hint from a condensed matter guy (Kadanoff) is why anyone understands quantum field theory.

Anyway, the main idea is to try to understand why AdS/CFT works intuitively. One of the old pictures was that the extra dimension in the bulk emerges as the scale of renormalization applied to the boundary. Then Ryu and Takayanagi gave their formula relating boundary entanglement and bulk geometry. What Swingle (http://www.phys.washington.edu/~karch/Conferences/Aspen/brianswingle.pdf) did was to relate bulk geometry and boundary entanglement with bulk entanglement and renormalization by noting the connection between AdS/CFT and MERA. MERA, invented by Guifre Vidal, is a variational wave function used to approximate condensed matter systems and relates entanglement and renormalization. The way boundary entanglement entropy is calculated in MERA is highly analogous to the Ryu-Takayanagi formula using bulk minimal surfaces. MERA is an example of a tensor network, which are variational wave functions for tractable simulation of strongly interacting condensed matter systems. Tensor networks grew out of trying to understand and generalize DMRG, which worked well only in strongly interacting 1D systems. The DMRG itself was invented by Steven White by applying Wilsonian renormalization to numerical simulation.

A side idea is that in AdS/CFT the gauge theory defines the gravitational theory. In condensed matter, one can have emergent relativistic gauge theory from lattice systems. So can we have double emergence here: nonrelativistic lattice → lattice gauge theory → bulk gravity?

Another side idea is that the tensor network formalism is related to the spin networks and spin foams of LQG. Can LQG help provide dynamics for tensor networks?

 

[marcus]

...
Now it might turn out that these pre-geometric quantum objects are in some sense related to loops or spinfoams,...

The main goal of the Loop program as I see it---put in the simplest terms---has always been pre-geometry. The basic objects of the theory are seen as something which (if the theory is successful) gives rise to spacetime. When I say spacetime I'm thinking "geometry" IOW a web of geometric relationships.

So you are pretty much right on target here except I guess the ideal outcome would be BE rather than "related to". I can only give you the best account I can as a watcher from the sidelines. I follow the literature.
The pre-geometric objects being studied in the LQG program are mostly labeled GRAPHS called spin networks and labeled 2-complexes called spin foams. (The "loop" in the name LQG goes back to the early 1990s before the focus shifted from loops to graphs, and thence to foams.)
Because the approach has always been primarily non-perturbative it has not been primarily about GRAVITONS or with quantizing gravity as a FORCE. The idea is that gravity=geometry and one wants to discover underlying pre-geometric objects that give rise to space-time.

That said, there are some recent trends to point out. Some recent directions in LQG research are illustrated by those "straws-in-the-wind" papers I mentioned. I see two main directions, one is easy to talk about and one is hard to talk about because the terminology is not firmly established as yet.

1. Easy to talk about: naively put, I see people using different types of labels. Instead of spins, they can be using holonomies, group elements, twistors (pairs of spinors)... So it is to some extent a misnomer to say that the pre-geometric objects being studied, or the quantum states of geometry being studied, are "spin" networks, or "spin" foams. Simply because the edges of the graph are carrying other stuff.

2. Hard to talk about: a number of recent papers are more abstract and I would say "top-down" in that they don't specify any particular pre-geometry, don't talk about networks/foams and the like.
And of course they don't assume a differential MANIFOLD either!
So what do I mean when I say "general relativistic" QFT, or "general relativistic" QSM? Or "general covariant" QSM (quantum statistical mechanics)?
What does general covariance mean if there is no smooth manifold?
I think it means you build the pregeometric theory in such a way that when it gives rise to spacetime the quantum theory of spacetime geometry and matter it gives rise to will be diffeomorphism invariant.

It means, for instance, that you don't put a preferred time-progression in by hand, and you make locality be entirely relational from the start. I'm not clear about what all it means for a theory to be "general covariant" before there is any differential manifold. Maybe we can get back to these issues and I should stop here.

Well, as I said, I come at this from the different angle of really viewing spacetime, and gravity, as ultimately emergent, the result of a coarse-grained description of the entanglement properties of some underlying quantum theory.

YES!

But then, also things like diffeo invariance and so on aren't really fundamental anymore...

I think they need to be re-defined/re-named to apply to pregeometric context.

 

[S.Daedalus]

Sorry for being so slow to respond to my own thread, and many thanks for the answers so far given, I'm reading attentively; I just keep putting off replying because I keep feeling that I should put some more thought into these things, and I don't really have much time to think these days...

Anyway, I did manage to watch van Raamsdonk's 'fuzz or fire'-seminar today, and one question I'm not entirely clear about emerged right in the beginning, when he said 'start with some state in a holographic field theory'. The problem is, I'm unclear about exactly what a holographic field theory is---of course, I know examples of such theories, and have some general idea of how the holographic dictionary works in such cases, but what I'm missing is something that needs to be true of a theory for it to be a 'holographic field theory' (obviously, it needs to have a holographic dual, but that's a bit circular for a definition). I know that the theories appear to be quite varied---occasionally, one reads about things like the gravity dual of the Ising model, etc. (Which, by the way, is somewhat interesting: after all, (the continuum limit of) the (3D) Ising model can be described by a noncritical string theory, so in this case, it looks like we have something like a gravity dual of a string theory--!?)

In any case, what's known about theories exhibiting some form of holography? I can think of at least two possibilities: 1) it has to be a CFT (although I think that probably not all CFTs have geometric duals, at least such that they aren't highly singular), and 2) it needs to permit an area law for the entanglement entropy. OK, there's a third possibility, that anything quantum has some form of spacetime dual, though perhaps not necessarily a classical one, or one in which the Einstein equations hold---this would sort of be the ideal fulfillment of the ideas of v. Weizsäcker and Penrose I mentioned in my OP. Then, you could basically just start with qubits, have them interact---leading to a spin network of sorts in Penrose's original formulation---giving you 3-space, and (at least in the case you have an area law) the entropy of the resulting entanglement gives you Einstein gravity via the usual arguments. At least, this would be a nice story to tell, but I'm less certain about whether it's also a true one.

Be that as it may, if somebody knows of research attacking/answering the above question, i.e. ideally completing the sentence 'you know your theory has a spacetime dual if...', then I'd be grateful to hear it.

Marcus, the reason I'm not sure about making this about loops is that essentially, there seem to be theories more general, that don't admit a useful description in loop variables, which nevertheless lead to emergent gravitational theories; it may be that then these theories can be described in terms of loop variables, but in a sense, you'd be blinded to what's going on below. Though of course the other possibility is that what's going on in the theories with gravity dual that gives rise to that dual can always be described in a loopy way, for instance, in the down-to-earth spin-network way originally proposed by Penrose: there, you had quite concrete physical system exchanging units of spin, leading to the $SU(2)$ rep labelled graphs we all know and love. Such a picture might apply in a quite literal sense to a theory giving rise to some form of spacetime.

And also, there's the angle atyy mentioned, about the relationship between tensor network states and spin networks; and again, typically, in the ground states described by a tensor net, you will have an area law. So it seems as if it's all connected somehow, and this thread is an attempt to provide some discussion about how that might be, i.e. about what we start out with, and where we end up, following the yarn until it untangles, so to speak.

So in this sense, maybe a few concrete questions might help. What's the simplest system to which we can meaningfully associate a spacetime geometry? What conditions must be satisfied to make this association? How does this relate to the different realizations of holography---AdS/CFT, MERA, the general idea of 'spacetime from entanglement'?

Ideally, I would like to be able to tell a story like the following: take this physical system (Ising model, qubits interacting,...). You can cast its ground state in the form of a tensor network like this, leading to a geometry like that. Or, the entanglement follows an area law, and variations in the entanglement entropy produce variations in the associated geometry that satisfy Einstein's equation. Or, the continuum limit of this is described by a CFT, with this AdS dual. Most importantly, perhaps, would be some elucidation on how these are ultimately saying the same thing---if they do, that is.

So, is there such a story, or anything that comes close (and if there is, can someone explain it to me :tongue:)? I realize I'm asking much, and I'm thankful for any opinion; but to me, this whole cluster of ideas seems like it should be intimately and exquisitely interrelated, but I at least am fundamentally confused on what, precisely, these interrelations are (and how, if at all, these things relate to the earlier ideas I've mentioned in my OP).

 

[marcus]

Hi Stephen D,
I think you raised the issue of pre-geometry. Loop is one of a number of ideas people have in that direction so I was arguing for its INCLUSION in what you were considering. Not for "making this about loops" exclusively.

You might be interested in watching the first 10 minutes or so of TOM BANKS' talk, given the 28th. I had to run some errands so did not get to listen further. He reported on an email exchange with Polchinski where as I understand it they agreed that "AdS/CFT can't say anything about what's inside a black hole".

And Tom Banks promised to show, in his talk, how in the matrix theory context he was working in, "unitarity is compatible with information loss".

On the one hand that seems like an attention-getting shocker. On the other hand it could be just a way of summarizing what Smolin was saying. Namely evolution is unitary but if a BH forms then the Hilbertspace is direct sum of states we can access and states that are down the hole.

Banks could ALSO have come to that conclusion, particularly since he and Polchinski reportedly had agreed not to expect AdS/CFT to apply---the boundary can't be expected to recover all the information that went down the hole.

As time permits I'll go back and listen to Banks again. It sounded intriguing and (unless I misunderstood) heretical.

 

[S.Daedalus]

Hi Stephen D,
I think you raised the issue of pre-geometry. Loop is one of a number of ideas people have in that direction so I was arguing for its INCLUSION in what you were considering. Not for "making this about loops" exclusively.

Sorry if what I said sounded overly harsh, I didn't mean it that way; I'm grateful for getting your perspective in this thread. But in a way, I have a feeling that this whole issue is something that goes beyond loops. LQG is a specific quantum theory, and I think we're here perhaps rather talking about a feature of quantum theory in general or as such, or at least something associated with a large class of quantum theories, and it's that what I'd like to get to. In any case, as I said, it may be that this whatever-it-is can always be described in a loopy way, so it's good to keep an eye open in that direction.

 

[marcus]

Sorry if what I said sounded overly harsh, I didn't mean it that way; I'm grateful for getting your perspective in this thread. But in a way, I have a feeling that this whole issue is something that goes beyond loops. LQG is a specific quantum theory, and I think we're here perhaps rather talking about a feature of quantum theory in general or as such, or at least something associated with a large class of quantum theories, and it's that what I'd like to get to. In any case, as I said, it may be that this whatever-it-is can always be described in a loopy way, so it's good to keep an eye open in that direction.

But that's fine! You were not one iota harsh. I think you should go with your feeling that LQG is not the sort of pre-geometry (maybe even pre-quantum theory) you have in mind. I presented my viewpoint (namely that Loop has horses in the pre-geometry race) emphatically enough that I don't need to and shouldn't re-iterate it in this thread.

So maybe you could lay out the questions and conceptual initiatives you have in mind at greater length and we can stay closer to topic from here on. You mentioned watching van Raamsdonk's talk at the KITP workshop.
It's clear there is a crisis around the firewall and people are venturing out in surprising new directions looking for ways to resolve it. The KITP talks I've watched from this past week or so have been unusually interesting. I didn't get to van Raamsdonk but I went back and watched quite a lot of Tom Banks. The talk is 72 minutes and he only gets to BH at minute 59. He says at minute 69 that he is now ready to address JOE POLCHINSKI'S point and it is only at minute 71 that he says he has reconciled exact unitarity of the S matrix with information loss.

I don't want to say anything negative--Tom Banks has my full respect. But I found the most interesting part of the talk was the first 3 minutes where he recounted an email exchange with Polchinski and quoted Joe as saying: "I'm an agnostic about the firewall, but I'm convinced that no model like AdS/CFT or matrix theory tells you anything about the BH interior."

That from Polchinski is something I think calls for explanation. It came as a bit of a shock.

 

[Haelfix]

Real quick, b/c I am pressed for time.

Regarding the AdS/CFT interior 'blindness'. I feel that Ted Jacobsen explains this point pretty well in his talk, as does Susskind in the ER-EPR talk. It has to do with this notion of precursor states, and the fact that the interior state's CFT mirror involves physics not just from a single region (say Region I), but also from scattering from region 2 and 3, which necessarily involve physics in the causal future of the ADS mirror (again it helps to look at the Penrose diagram Jacobsen draws).

The point is the following, AdS/CFT is a bit like a blackbox. You ask a certain class of general intuition question, and you get a very specific answer. For instance, does unitarity hold, to which it answers unambigously yes (and I believe almost everyone at the conference believes that the AdS/CFT conjecture is true).

What it is terrible at, is answering exactly why or how various properties that seem self evident in the dual holds in microscopic detail in the bulk; and the more you attempt to peer into the blackbox, the more you get a sort of fuzziness. This is why it hasn't helped as much as we'd like with the Firewall question. It doesn't really appear to be suited for that type of question.

 

[marcus]

Just to have for convenient reference, I'll copy in the abstract of the article linked in the opening post. I expect this must have been the basis for at least part of van Raamsdonk's talk at the KITP BH workshop that's been going on.
http://arxiv.org/abs/1308.3716
Gravitational Dynamics From Entanglement "Thermodynamics"
Nima Lashkari, Michael B. McDermott, Mark Van Raamsdonk
(Submitted on 16 Aug 2013)

In a general conformal field theory, perturbations to the vacuum state obey the relation δS = δE, where δS is the change in entanglement entropy of an arbitrary ball-shaped region, and δE is the change in “hyperbolic” energy of this region. In this note, we show that for holographic conformal field theories, this relation, together with the holographic connection between entanglement entropies and areas of extremal surfaces and the standard connection between the field theory stress tensor and the boundary behavior of the metric, implies that geometry dual to the perturbed state satisfies Einstein’s equations expanded to linear order about pure AdS.
15 pages.

EDIT: Haelfix, didn't see yours until I posted this. Wise observation about fuzzy:

...
What [AdS/CFT] is terrible at, is answering exactly why or how various properties that seem self evident in the dual holds in microscopic detail in the bulk; and the more you attempt to peer into the blackbox, the more you get a sort of fuzziness. This is why it hasn't helped as much as we'd like with the Firewall question. It doesn't really appear to be suited for that type of question.

 

[S.Daedalus]

Regarding the AdS/CFT interior 'blindness'. I feel that Ted Jacobsen explains this point pretty well in his talk, as does Susskind in the ER-EPR talk. It has to do with this notion of precursor states, and the fact that the interior state's CFT mirror involves physics not just from a single region (say Region I), but also from scattering from region 2 and 3, which necessarily involve physics in the causal future of the ADS mirror (again it helps to look at the Penrose diagram Jacobsen draws).

Somewhat off-topic, but it was fun watching Susskind explain things like quantum error correction, having a background in quantum information (well, starting to build a background there more like). They don't tell you it relates to black holes when they teach you QEC codes! But it's really intriguing how many quantum info concepts crop up in HEP physics these days; but then again, it's in a sense just a very general frame to talk about quantum stuff, I suppose.

Some things Susskind said related to the possibility of influencing states in different regions of the AdS geometry, which is possible only if you have a non-vanishing commutator for some reason. It's interesting (though perhaps unrelated) that the mandatory vanishing of commutators at spacelike separation in QFT in a sense is in conflict with the Bekenstein-Hawking bound, because it means that you can in a sense 'store' arbitrarily much information in an arbitrarily small piece of spacetime, and retrieve it perfectly later. I'm wondering if a modification of this would allow the firewall problem to be circumvented by introducing an explicit backaction of the measurements performed by Alice on the degrees of freedom in the black hole. But then again I suppose repealing the locality of field theory would be quite a drastic move.

But that's fine! You were not one iota harsh. I think you should go with your feeling that LQG is not the sort of pre-geometry (maybe even pre-quantum theory) you have in mind. I presented my viewpoint (namely that Loop has horses in the pre-geometry race) emphatically enough that I don't need to and shouldn't re-iterate it in this thread.

Please, bring it up whenever you deem it appropriate; I don't want to be responsible for any sort of censorship in this thread.

So maybe you could lay out the questions and conceptual initiatives you have in mind at greater length and we can stay closer to topic from here on.

I'll try to do that. In a sense, what I have in mind is getting clear the relationship between quantum theory and general relativity. The received view, as far as I can tell, is that general relativity is a particular kind of theory, like for instance electrodynamics, that awaits its quantization in order to enable at least a conceptual, and ideally even nomological (as in the case of string theory) unification with the other theories we know of, which constitute the standard model. But I see the developments in the cluster 'entanglement thermodynamics' and maybe in the wider field of holography as challenging this idea: general relativity is not something external to quantum theory, but rather, at least under certain circumstances, comes part and parcel with the latter.

The aim of this thread is, at least in part, to suss out if, and when, this is true, that is, in which cases you can construct a spacetime picture from the quantum description of some system. Of course, that's still quite broad, and I don't expect there to emerge a cut-and-dried answer (that may for all I know not even exist), but I tend to try and grasp things by looking at them from different perspectives, and considering what connects those perspectives; in the concrete case, what makes things like holography, MERA, ER=EPR (somebody really needs to come up with a dual 'P=1' conjecture, ideally usable to prove ER=EPR) and entropic area laws all be aspects of the same kind of phenomenon.

In light of this, perhaps it's useful to hark back to atyy's comment:

A side idea is that in AdS/CFT the gauge theory defines the gravitational theory. In condensed matter, one can have emergent relativistic gauge theory from lattice systems. So can we have double emergence here: nonrelativistic lattice → lattice gauge theory → bulk gravity?

This would, I think, be an interesting picture to develop further. There are some very simple systems that give rise to area laws---1D gapped spin chains spring to mind (which, IIRC, in their ground state always satisfy an area law).

Susskind in his talk took a few steps in the direction I have in mind, with considering geometries made up by entangled qubits; in general, there probably won't be any nice classical geometry associated with something like that (in general, there are very few states satisfying an area law*), but, he argued, at least his intuition was that if you scramble things up a bit, then you should get something geometric at least in an appropriate limit. Getting to the heart of what might be meant by such statements, and collecting the evidence produced in the literature, is what I'd like to do here (though atyy's thread already contains quite a nice collection, I'm grateful for all references thrown my way).



*Incidentally, there's a concrete bit of confusion I'd appreciate somebody clearing up for me---there's a heuristic argument for entanglement area laws that's sometimes floated around that goes somewhat like this: start out with a quantum system in a pure state; trace out the degrees of freedom in a given volume. Then, the entropy of both regions, inside and outside of the hole, must match; but since the area of the traced-out volume are the only quantity both regions have in common, that means the entropy must be proportional to this area (I actually walk past this argument on a poster every day when I go into the office, but I didn't notice it until a few days ago). But actually, it's not hard to see that most states of a given quantum system don't follow an area law. So what gives?

 

[atyy]

In any case, what's known about theories exhibiting some form of holography? I can think of at least two possibilities: 1) it has to be a CFT (although I think that probably not all CFTs have geometric duals, at least such that they aren't highly singular), and 2) it needs to permit an area law for the entanglement entropy. OK, there's a third possibility, that anything quantum has some form of spacetime dual, though perhaps not necessarily a classical one, or one in which the Einstein equations hold---this would sort of be the ideal fulfillment of the ideas of v. Weizsäcker and Penrose I mentioned in my OP. Then, you could basically just start with qubits, have them interact---leading to a spin network of sorts in Penrose's original formulation---giving you 3-space, and (at least in the case you have an area law) the entropy of the resulting entanglement gives you Einstein gravity via the usual arguments. At least, this would be a nice story to tell, but I'm less certain about whether it's also a true one.

"Extra dimension is renormalization scale" alone is too rough to talk about Einstein gravity, since it can be used for any QFT that flows to a UV fixed point. Same with the AdS/MERA "entanglement renomalization is spacetime", since it applies to any CFT. Heemskerk, Penedones, Polchinski and Sully http://arxiv.org/abs/0907.0151 term the geometry generated by renormalization "coarse holography", as it generates locality only above the scale of AdS. They go on to specify some additional conditions needed for sharp locality. Swingle's second AdS/MERA paper http://arxiv.org/abs/1209.3304 sketches what the tensor network equivalent of those conditions might be. Maldacena acknowledges a bunch of problems with the tensor network picture in his talk http://online.kitp.ucsb.edu/online/fuzzorfire-m13/maldacena/, and says it's mainly because of analogies that he thinks it's in the right direction.

Take a look at Physics Monkey's comment https://www.physicsforums.com/showpost.php?p=4042674&postcount=60. Also the intermediate value theorem around 4-5 min of his talk http://www.perimeterinstitute.ca/videos/asymmetry-protected-emergent-e8-symmetry.

The other line of work that's relevant is Takayanagi's http://arxiv.org/abs/1304.7100 and http://arxiv.org/abs/1308.3792 to see how Einstein equations in the bulk constrain boundary entanglement entropy. This line of work is complementary to the Lashkari, McDermott and van Raamsdonk work of the OP http://arxiv.org/abs/1308.3716 of trying to get the Einstein equations in the bulk from a theory in which the Ryu-Takayanagi formula holds. In a prequel to those papers, Takayanagi begins to address the questions with a continuous version of the MERA http://arxiv.org/abs/1208.3469. I think these look like good developments to try to make the picture more quantitative. I also like that with "thermodynamics" coming in, maybe we'll get more understanding of Jacobson's derivation, as you mention in the OP.

A side idea is that in AdS/CFT the gauge theory defines the gravitational theory. In condensed matter, one can have emergent relativistic gauge theory from lattice systems. So can we have double emergence here: nonrelativistic lattice → lattice gauge theory → bulk gravity?

This would, I think, be an interesting picture to develop further. There are some very simple systems that give rise to area laws---1D gapped spin chains spring to mind (which, IIRC, in their ground state always satisfy an area law).

Raman Sundrum's http://arxiv.org/abs/1106.4501 discusses emergent relativity in section 9. He also talks about the coarse and sharp locality issue based on the Heemskerk et al paper.

*Incidentally, there's a concrete bit of confusion I'd appreciate somebody clearing up for me---there's a heuristic argument for entanglement area laws that's sometimes floated around that goes somewhat like this: start out with a quantum system in a pure state; trace out the degrees of freedom in a given volume. Then, the entropy of both regions, inside and outside of the hole, must match; but since the area of the traced-out volume are the only quantity both regions have in common, that means the entropy must be proportional to this area (I actually walk past this argument on a poster every day when I go into the office, but I didn't notice it until a few days ago). But actually, it's not hard to see that most states of a given quantum system don't follow an area law. So what gives?

I'm not sure. Would it work to say a function of the area - then volume = A^3/2 ? Actually, the equality of the entropy for both subsystems holds even if you don't define "inside" versus "boundary" spins, so maybe the answer is that in general the inside bits are entangled with the outside bits and that is what guarantees the equality. For example in systems with a form of "long-range entanglement" http://arxiv.org/abs/1004.3835, there is an additive violation of the area law http://arxiv.org/abs/hep-th/0510092.

 

[atyy]

S. Daedulus, since you seem to be a quantum information person, would you like to comment on:

Rohrlich, Thermodynamical analogues in quantum information theory (2001)?

It's interesting to type "entanglement thermodynamics" as a search term in the arxiv. Among the things that come up are:

Mukohyama, Seriu, Kodama Entanglement thermodynamics (1998).

Takayanagi does mention Muohyama et al's paper in http://arxiv.org/abs/1212.1164 , as well as

Fursaev, 'Thermodynamics' of Minimal Surfaces and Entropic Origin of Gravity (2010).

Fursaev does cite Verlinde as inspiration. He also wrote a paper http://arxiv.org/abs/hep-th/9703178 on black hole entropy in induced gravity, which cites Bombelli, Koul, Lee, Sorkin and Srednicki for their entanglement entropy = BH entropy idea.

The key result used in the Lashkari, McDermott and van Raamsdonk paper is Blanco, Casini, Hung, Myer's Relative Entropy and Holography. They do cite introductory quantum information work by Vedral and Sagawa for the second law. Incidentally, Vidal, who invented MERA, is a quantum information guy.

 

[S.Daedalus]

You've got a nice collection of references at your fingertips, atyy! Especially these:

"Extra dimension is renormalization scale" alone is too rough to talk about Einstein gravity, since it can be used for any QFT that flows to a UV fixed point. Same with the AdS/MERA "entanglement renomalization is spacetime", since it applies to any CFT. Heemskerk, Penedones, Polchinski and Sully http://arxiv.org/abs/0907.0151 term the geometry generated by renormalization "coarse holography", as it generates locality only above the scale of AdS. They go on to specify some additional conditions needed for sharp locality. Swingle's second AdS/MERA paper http://arxiv.org/abs/1209.3304 sketches what the tensor network equivalent of those conditions might be. Maldacena acknowledges a bunch of problems with the tensor network picture in his talk http://online.kitp.ucsb.edu/online/fuzzorfire-m13/maldacena/, and says it's mainly because of analogies that he thinks it's in the right direction.

Seem to be very much what I'm looking for. Also, I agree that the Takayanagi/van Raamsdonk stuff is complementary, as you put it: it seems to me that essentially, Takayanagi et al. showed the implication 'Einstein equations -> constraints on entanglement entropy', and van Raamsdonk et al. supplemented the other direction (if I don't misunderstand things). I'll probably watch Maldacena's talk either today or tomorrow.

I'm not sure. Would it work to say a function of the area - then volume = A^3/2 ?

But is it true that the entropy of the outer volume is extensive with the volume of the 'hole'? Seems strange to me...

S. Daedulus, since you seem to be a quantum information person, would you like to comment on:

Rohrlich, Thermodynamical analogues in quantum information theory (2001)?

I'll have a look at that paper. Is there anything specific you'd like to discuss?

Fursaev, 'Thermodynamics' of Minimal Surfaces and Entropic Origin of Gravity (2010).

Fursaev does cite Verlinde as inspiration. He also wrote a paper http://arxiv.org/abs/hep-th/9703178 on black hole entropy in induced gravity, which cites Bombelli, Koul, Lee, Sorkin and Srednicki for their entanglement entropy = BH entropy idea.

Yes, I think I mentioned induced gravity in my OP; it's also what Jacobson worked on before he came out with his Einstein equation of state. The idea that entanglement entropy = BH entropy is particularly natural in induced gravity; if all you've got is minimally coupled fields, then the conjecture can be made precise, basically because the induced Newton's constant is inversely proportional to the number of fields, see the review by Solodukhin, sect. 8.2. One of the things I wonder is in what sense induced gravity is just the same wine in different bottles, so to speak.

 

[S.Daedalus]

S. Daedulus, since you seem to be a quantum information person, would you like to comment on:

Rohrlich, Thermodynamical analogues in quantum information theory (2001)?

The first thing I noticed about the paper was that it contains the brilliant sentence:

To explain teleportation, we approximate Captain Kirk as a single spin-1/2 particle in an unknown state $|K\rangle$.

:tongue2:
(One could hold that this just about does justice to the depth of William Shatner's acting.)

In any case, you probably know that in QIT, entanglement is considered a resource: using entanglement---in the simplest case, in the form of shared singlet states---tasks can be accomplished that can't be performed classically. A simple example of such a task is of course the violation of a Bell inequality, which is classically impossible; but this by itself isn't very interesting as far as applications go (though the degree of violation of a Bell inequality can in some cases be used to quantify how much better you can do using quantum resources than using classical resources, for example in communication complexity tasks).

Now, one thing Rohrlich considers is how to extract usable entanglement, i.e. singlet pairs, from some shared quantum state. The result here is that the fraction of singlet pairs that can be obtained from some shared states is bounded by the entanglement entropy. He then uses this to explain an analogue to the second law of thermodynamics concerning entanglement, which is that you can't produce entanglement using local operations (and classical communication, that is, transfer of classical information). The argument is that if there were a more efficient way of producing singlet states, then you could also create entanglement locally, analogously to how you could violate the second law if there were a process more efficient than a Carnot process.

Ultimately, however, I'm not sure these analogues are very deep: after all, entanglement entropy has its name for a reason---it's just the von Neumann entropy of the reduced density matrix, which itself is a straightforward generalization of Shannon entropy from probability distributions to density matrices. And of course, the relations between Shannon and thermodynamical entropy are well known; so I'm not sure that things could have been any different. But I haven't really thought much about it.

 

[atyy]

I'll have a look at that paper. Is there anything specific you'd like to discuss?

Ultimately, however, I'm not sure these analogues are very deep: after all, entanglement entropy has its name for a reason---it's just the von Neumann entropy of the reduced density matrix, which itself is a straightforward generalization of Shannon entropy from probability distributions to density matrices. And of course, the relations between Shannon and thermodynamical entropy are well known; so I'm not sure that things could have been any different. But I haven't really thought much about it.

Yes, I was wondering whether there was any consensus among quantum information folks about whether this analogy was deep or useful. At any rate, it looks mainly like an analogy with the second law, whereas the first law is what Jacobson and the new papers use.

At first glance I thought the earliest form of the first law that's used by Lashkari, McDermott and Van Raamsdonk was the Fursaev paper. However Bhattacharya, Nozaki, Takayanagi and Ugajin cite that and Verlinde's entropic gravity to say that it is not the same, because the sign in their dE ~ dS relation is opposite! Blanco, Casini, Hung and Myers do cite the Bhattacharya paper as an earlier form of their dE ~ dS relation, but caution that dE is "in general a different 'type' of energy, the modular 'energy'. So I wonder if the Bhattacharya and Blanco papers agree completely.

Yes, I think I mentioned induced gravity in my OP; it's also what Jacobson worked on before he came out with his Einstein equation of state. The idea that entanglement entropy = BH entropy is particularly natural in induced gravity; if all you've got is minimally coupled fields, then the conjecture can be made precise, basically because the induced Newton's constant is inversely proportional to the number of fields, see the review by Solodukhin, sect. 8.2. One of the things I wonder is in what sense induced gravity is just the same wine in different bottles, so to speak.

Yes, that's why I mentioned the Frolov and Fursaev paper. Unfortunately, the Solodukhin review concludes that in their model (or at least that of Frolov, Fursaev and Zelnikov) the entanglement entropy isn't the BH entropy. Induced gravity is such a nice idea that many hope it will work. Weinberg and Witten ruled out many models of emergent gravity by their Weinberg-Witten theorem, but explicitly noted in their paper that their theorem did not apply to Sakharov's induced gravity. Strominger did hopefully comment "If gravity is induced [9], which means that Newton’s constant is zero at tree level and arises as a one loop correction, then the entanglement entropy is responsible for all of the entropy, and reproduces the area law with the correct coefficient [7,10]. This might in fact be the case in string theory, where the Einstein action is induced at one loop from open strings, but this notion has yet to be made precise. Recent progress [11] has revealed a rich holographic relation between entanglement entropy and minimal surfaces including horizons. Related observations appear in [12]."

But is it true that the entropy of the outer volume is extensive with the volume of the 'hole'? Seems strange to me...

http://arxiv.org/abs/0808.3773 state this in the last paragraph of their section II. One of their references is http://arxiv.org/abs/hep-th/9601132. Also, for a finite dimensional system in a pure state, if you split it in two, the entanglement entropy of each subsystem is equal http://e3.physik.uni-dortmund.de/~suter/Vorlesung/QIV_WS11/Entanglement.pdf, http://www-thphys.physics.ox.ac.uk/people/JohnCardy/seminars/statphys.pdf (I don't know if this is true for infinite dimensional systems).

 

[S.Daedalus]

Induced gravity is such a nice idea that many hope it will work. Weinberg and Witten ruled out many models of emergent gravity by their Weinberg-Witten theorem, but explicitly noted in their paper that their theorem did not apply to Sakharov's induced gravity. Strominger did hopefully comment "If gravity is induced [9], which means that Newton’s constant is zero at tree level and arises as a one loop correction, then the entanglement entropy is responsible for all of the entropy, and reproduces the area law with the correct coefficient [7,10]. This might in fact be the case in string theory, where the Einstein action is induced at one loop from open strings, but this notion has yet to be made precise. Recent progress [11] has revealed a rich holographic relation between entanglement entropy and minimal surfaces including horizons. Related observations appear in [12]."

That's a perspective that I think I haven't encountered before. String theory obviously has a fundamental spin-2 excitation, which in general is sufficient for getting Einstein gravity; I don't know what it means in this context that the Einstein-Hilbert action is induced at one loop. After all, the latter can be true in theories that lack a spin-2 mode. On the other hand, I've thought that the emergence of gravity and spacetime in matrix models seems to be similar to induced gravity models.

--Ah, I just noticed he's talking about open strings, whereas the graviton is a closed string; but of course, if you have open strings, you also get closed strings (no?).

Another thing I'd like to get more clear about is the preference of AdS space in this sort of approach, the problem here being that our universe doesn't really look AdS-y, so we might end up talking about something that doesn't really have any real-world applicability. On the other hand, AdS-space is just one solution of Einstein's equations, and if they come up wholesale from this approach, maybe the problem is just something like we're doing perturbation around one particular solution, and can't see the full space of possibilities from here. Also, I don't think induced gravity has a similar limitation; you start with a manifold 'flapping in the breeze', and then get out that it must follow the Einstein equations. I know there's been some steps towards dS/CFT, but still, this fixation on AdS has always struck me as somewhat quaint.

http://arxiv.org/abs/0808.3773 state this in the last paragraph of their section II. One of their references is http://arxiv.org/abs/hep-th/9601132. Also, for a finite dimensional system in a pure state, if you split it in two, the entanglement entropy of each subsystem is equal http://e3.physik.uni-dortmund.de/~suter/Vorlesung/QIV_WS11/Entanglement.pdf, http://www-thphys.physics.ox.ac.uk/people/JohnCardy/seminars/statphys.pdf (I don't know if this is true for infinite dimensional systems).

It ought to be true generically, but you'd expect the EE to diverge in general. In a sense, it's intuitive: you can't do more than entangle each spin (finite dimensional system) maximally, so a lower bound on the entanglement should be given by the singlet fraction of the system, which goes with the total number of spins. Not sure why this seemed so counterintuitive to me yesterday...

 

[atyy]

In the paper you cite http://arxiv.org/abs/1211.0522 LQG does not enter (although the connection with a nonperturbative result using LQG in another Bianchi paper is mentioned at the end).
The paper of Bianchi that you cite is a low-energy perturbative, essentially (semi)classical, analysis.

The LQG-based paper, on the other hand, is:
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

Bianchi's plenary talk at the recent (July 2013) Loops conference covers BOTH results. Here's the video:
http://pirsa.org/13070048/
http://pirsa.org/displayFlash.php?id=13070048
Eugenio Bianchi, Perimeter Institute
Entanglement, Bekenstein-Hawking Entropy, and Spinfoams
I review recent developments on vacuum entanglement perturbations in perturbative quantum gravity and spinfoams, and discuss their relevance for understanding the nature of black hole entropy.

The talk+discussion lasted a bit over 45 minutes. I think it's good he has shown how to get the same result (BH entropy = entanglement entropy) using different approaches---both perturbative and nonperturbative (spin foam). Makes the whole thing more solid and reliable.

For completeness, here is the abstract of the paper that Atyy cited (in which LQG does not enter in the analysis but is mentioned at the end):
http://arxiv.org/abs/1211.0522
Black hole entropy from graviton entanglement
Eugenio Bianchi
(Submitted on 2 Nov 2012 (v1), last revised 7 Jan 2013 (this version, v2))
We argue that the entropy of a black hole is due to the entanglement of matter fields and gravitons across the horizon. While the entanglement entropy of the vacuum is divergent because of UV correlations, we show that low-energy perturbations of the vacuum result in a finite change in the entanglement entropy. The change is proportional to the energy flux through the horizon, and equals the change in area of the event horizon divided by 4 times Newton's constant - independently from the number and type of matter fields. The phenomenon is local in nature and applies both to black hole horizons and to cosmological horizons, thus providing a microscopic derivation of the Bekenstein-Hawking area law. The physical mechanism presented relies on the universal coupling of gravitons to the energy-momentum tensor, i.e. on the equivalence principle.
4 pages

Bianchi-Myers http://arxiv.org/abs/1212.5183 does refer to the second Bianchi paper you listed http://arxiv.org/abs/1211.0522. The first Bianchi paper you listed http://arxiv.org/abs/1204.5122 has nothing explicit about entanglement. However it is built on the Frodden-Ghosh-Perez http://arxiv.org/abs/1110.4055 E=TdS analogue in LQG. The Lashkari-McDermott-van Raamsdonk http://arxiv.org/abs/1308.3716 is based on Blanco-Casini-Hung-Myers's http://arxiv.org/abs/1305.3182 E=TdS in which E is the modular Hamiltonian on the boundary, which as you pointed out in the Tomita flow thread is the thermal Hamiltonian of thermal time and the entanglement Hamiltonian in condensed matter, as mentioned by the Chirco-Haggard-Rovelli http://arxiv.org/abs/1309.0777.

It would be interesting to know if the Frodden-Ghosh-Perez E=TdS is in fact also a modular Hamiltonian, or related to it. Blanco-Casini-Hung-Myers do comment "Some recent references [38, 63] also consider relations similar to the first law of thermodynamics, i.e., dE = T dS, for entanglement entropy". [38] is Bhattacharya-Nozaki-Takayanagi-Ugajin http://arxiv.org/abs/1212.1164 and [63] is the second Bianchi paper you listed. Bianchi-Myers has a whole section on the modular Hamiltonian, so I assume Bianchi should be telling us soon.

 

[atyy]

I don't think the second part of Bianchi's Loops talk http://pirsa.org/13070048/ corresponds to his Entropy of Non-Extremal Black Holes from Loop Gravity

I think the talk corresponds to these two papers:
Bianchi, Black hole entropy from graviton entanglement
Bianchi and Myers, On the Architecture of Spacetime Geometry

The reason is that the final equations on slide 42 look like Bianchi and Myers's Eq 13-15. The "Non-Extremal" paper does not calculate the entropy from entanglement. Bianchi and Myers's Eq 15 looks different from the equations in the "Non-Extremal" paper. I wonder whether that'd be sorted out if Bianchi used the relative entropy instead of the entanglement entropy.

 

[Jim Kata]

Leonard Susskind has a good survey of these ideas on youtube, .
The gist of the idea, as my feeble brain could understand it, is that you can use the correspondence between the near horizon geometry of a black hole to Rindler space to associate an entropy to spacetime via Bekenstein Hawking formula. The fields that bend spacetime can then be interpreted as changing the entropy, and vice versa. So a changing density matrix affects spacetime.

 

[Physics Monkey]

One thing I don't understand with all these various papers attempting to argue that entanglement controls the geometry is that the Hilbert space is always not quite well defined. It seems clear that for fluctuations on top of a smooth non-black hole background, there is effectively a local Hilbert space for matter and graviton excitations. On the other hand, we know this just can't ultimately be true.

So, for example, what stuff's entropy is the entropy that Bianchi and Meyers equate with the area? Somehow this looks innocuous enough, but something subtle is happening when one has an event horizon. And so one is left with the concern that without being completely precise about what the Hilbert space is, e.g. in some versions of the firewall argument, one cannot really formulate a sharp contradiction.

One thing one can do with tensor networks is associated a state space to every "opening up" of the links of the network. In other words, we just take part of the network with some physical labels and some auxilliary labels. This defines a map from auxilliary labels to physical labels. And when we choose some subregion in the bulk, I feel that we are doing something like this. Curiously, the state space associated with nearby regions may not be orthogonal and may be embedded in a larger Hilbert space in a non-trivial way.

 

[MichaelManthey]

Marcus wrote

But that doesn't identify what the "geo-molecules" are---the geometric units that participate in the dynamic geometry called gravity. What are they? Bits of angle, flecks of area, chunks of volume? A shimmering foam of indefinite polyhedra that have not yet decided how many faces they want to have?

My answer to this question is a set of quaternion isomorphs - dubbed TauQuernions - that are also irreversible, and also are entanglement operators. [Notation: geometric (Clifford) algebra, 1-vectors ${a,b,c...}$, $ab=-ba$, over $Z_3 = {0,1,-1}$.] The quaternion triple $(ab,ac,bc)$ maps to the tauquernion triple $(ab-cd,ac+bd,ad-bc)$. That is, a single quaternion, say $ab$, performs reversible space-like rotations, and likewise a single tauquernion $ab-cd$ rotates through isomorphic states. But $ab-cd$ has no inverse, and so is also time-like. Furthermore, it turns out that $ab-cd$ is a quantum entanglement operator.

Rather than filling this space with more too-brief explanation, please see TauQuernions.org, where you will find distributed computational systems and standard quantum mechanics described in the same formalism. That paper is called "TauQuernions $τ_i, τ_j, τ_k$: 3+1 Dissipative Space out of Quantum Mechanics".

Because the finite, discrete, computational context is foreign, you might though want to begin with the synchonization paper, "Synchronization - the Font of Physical Structure", which establishes the basic connection between computation and quantum physics, which is that the computational synchronization operators (Wait, Signal) correspond respectively to (nilpotent, idempotent), ie. (boson, fermion). "Riemann Fever" was for fun. The thesis by Matzke is a straight-forward read.

 

[marcus]

Hi Michael, thanks for the reply. You are replying to a rather speculative post #5 of this thread which I wrote back in August and (I'm embarrassed to say) forgot about.
Here's a link in case anyone wants to look back:
https://www.physicsforums.com/showthread.php?p=4484815#post4484815

I actually don't have any further ideas on that. When I get a moment I'll go see what your ideas are like. the idea that gravity (geometry) could be based on the statistical mechanics of "geo-atoms" of some as-yet-undetermined nature is fascinating, but so remote. I'm inclined to adopt a wait-and-see attitude for now.