Condensed matter physics, area laws & LQG?

[atyy]

Indeed, in my original paper I tried to emphasize that the identification of the minimal curve in the MERA graph really only gives a bound on the entanglement, although in numerical practice the bonds tend to all give the same contribution.

This issue has been a bit of a puzzle actually, since MERA seems to work too well i.e. is too like classical gravity even when we expect that the putative dual theory shouldn't be. On the other hand, maybe the gravity theory is always quasi-local on some scale. After all, the RG equations are local in any QFT. In general, it seems that there are many different length scales at which we can discuss locality, including the Planck length, the string length, and the AdS radius (approaching from the holographic side).

I naively think of renormalization as usually acting on the Hamiltonian or action of the theory, and it's not very obvious what entanglement is doing, which I think of more as a property of a wave function. OTOH, MERA seems something like "wave function renormalization"? Are these two sorts of renormalization related? Off the top of my head, I can think of this paper by Balasubramanian and colleagues, who mention at the end that their work might be related to the connection you drew.

 

[audioloop]

I think Markopoulou and Oriti have been sniffing this out a long time. Note that Wen has heuristically linked tensor networks and AdS/CFT (strings!) in the final slide of http://dao.mit.edu/~wen//09QHtop.pdf.

not LQG but related

http://arxiv.org/pdf/1203.5367.pdf

...Condensed matter physics provides a third conceptual framework...
...In this context, QG can be treated analogously to crystallographic models of condensed matter physics...
...Quantum graphity is a background independent model that provides an alternative viewpoint on the notion and structure of space, based on condensed matter concepts but extended to a dynamic quantum lattice...

http://arxiv.org/pdf/hep-th/0611197v1.pdf

...As the system cools and the temperature drops, however, one or more phase transitions may occur in which the j degrees of freedom will become frozen. How the system cools depends on the relations between different coupling constants...

 

[atyy]

The Gravity Dual of a Density Matrix by Bartlomiej Czech, Joanna L. Karczmarek, Fernando Nogueira and Mark Van Raamsdonk seems to address a similar question as Nozaki, Ryu and Takayanagi. They cite Ryu and Takayanagi's earlier work. On p11, they have a section about reconstructing bulk metrics from extremal surface areas. On p16, they discuss how spacetime could emerge from entanglement, and in footnote 25 explicitly cite Swingle's AdS/CFT/MERA paper as advocating a similar picture.

 

[atyy]

Spin networks in LQG have an interpretation as geometry.
http://arxiv.org/abs/1007.0402
http://arxiv.org/abs/1102.3660
"The mathematics needed to describe such quanta of space is provided by the theory of spin networks ... The other way around, the Hilbert space of SU(2) Yang-Mills lattice theory admits an interpretation as a description of quantized geometries, formed by quanta of space, as we shall see in a moment. This interpretation forms the content of the "spin-geometry" theorem by Roger Penrose, and an earlier related theorem by Hermann Minkowski. These two theorems ground the kinematics of LQG."

Some tensor networks also have an interpretation as geometry.
http://arxiv.org/abs/0905.1317
http://arxiv.org/abs/1106.1082
http://arxiv.org/abs/1208.3469
"By making a close contact with the holographic formula of the entanglement entropy, we propose a general definition of the metric in the MERA in the extra holographic direction, which is formulated purely in terms of quantum field theoretical data."

Are these two forms of geometry related?

 

[atyy]

Rovelli says "A generic state of the geometry is not a spin network state: it is a linear superposition of spin networks." And "Classically, each node represents a polyhedron, thanks to Minkowski's theorem, but the polyhedra picture holds only in the classical limit and cannot be taken literally in the quantum theory. ... In other words, these are "polyhedra" in the same sense in which a particle with spin is a "rotating body"."

Singh and Vidal have a new paper, in which they say "Consequently, a tensor network made of SU(2)-invariant tensors decomposes as a linear superposition of spin networks ... As a practical demonstration we describe the SU(2)-invariant version of the MERA ..."

 

[Physics Monkey]

I naively think of renormalization as usually acting on the Hamiltonian or action of the theory, and it's not very obvious what entanglement is doing, which I think of more as a property of a wave function. OTOH, MERA seems something like "wave function renormalization"? Are these two sorts of renormalization related? Off the top of my head, I can think of this paper by Balasubramanian and colleagues, who mention at the end that their work might be related to the connection you drew.

There is definitely a relationship, not that I have a super clear idea what it is precisely. I'm actually struggling a lot with this at the moment, since as you point out there is a bit of a gulf between the very action oriented setup of holography and traditional field theory and the very state oriented setup of these modern quantum info methods. I would like to bridge this gulf a bit to bring the two closer together.

At a practical level, they seem to contain much of the same information. Certain scaling dimensions, operator product coefficients, central charges, and so on can be obtained from either method. In so far as these data define a conformal field theory, say, then the methods appear to contain the same information.

I think one very useful approach is to think about representing the operator e^{-\beta H}. When thinking about this operator maybe its not so mysterious why Hamiltonian RG and wavefunction RG convey the same information.

 

[marcus]

Rovelli says "A generic state of the geometry is not a spin network state: it is a linear superposition of spin networks." And "Classically, each node represents a polyhedron, thanks to Minkowski's theorem, but the polyhedra picture holds only in the classical limit and cannot be taken literally in the quantum theory. ... In other words, these are "polyhedra" in the same sense in which a particle with spin is a "rotating body"."

Singh and Vidal have a new paper, in which they say "Consequently, a tensor network made of SU(2)-invariant tensors decomposes as a linear superposition of spin networks ... As a practical demonstration we describe the SU(2)-invariant version of the MERA ..."

Since you are exploring the relation between LQG and condensed matter physics, you might like to look over what I think in a sense replaces the 2011 paper of Rovelli you quote here (http://arxiv.org/abs/1102.3660 ).

This is a set of 26 slides from a talk given in Stockholm in July at the MG13 meeting.

http://www.cpt.univ-mrs.fr/~rovelli/RovelliStockholmSpinFoam.pdf

It's an up-to-date condensed but fairly complete overview of Loop gravity, main results and open problems, as Rovelli sees it.

 

[atyy]

There is definitely a relationship, not that I have a super clear idea what it is precisely. I'm actually struggling a lot with this at the moment, since as you point out there is a bit of a gulf between the very action oriented setup of holography and traditional field theory and the very state oriented setup of these modern quantum info methods. I would like to bridge this gulf a bit to bring the two closer together.

At a practical level, they seem to contain much of the same information. Certain scaling dimensions, operator product coefficients, central charges, and so on can be obtained from either method. In so far as these data define a conformal field theory, say, then the methods appear to contain the same information.

I think one very useful approach is to think about representing the operator e^{-\beta H}. When thinking about this operator maybe its not so mysterious why Hamiltonian RG and wavefunction RG convey the same information.

Hmm, apparently the string theorists also have more than one sense of renormalization. I was looking at the discussion in your paper, and you mentioned that Lawrence and Sever emphasize that the renormalization flow depends on the state, which was a surprise to me. I looked up citations to their work, and found that Heemskerk and Polchinski say that de Boer et al's renormalization is non-Wilsonian! In Heemskerk and Polchinski's terminology "Wilsonian couplings and evolution depend only on scales above the cutoff and are independent of the state."

 

[atyy]

There is an interesting comment in McGreevy's notes that the locality in Wilsonian renormalization is not as local as the locality in AdS/CFT. He refers to a paper by Heemskerk and colleagues who say "Thus, energy-radius holography nicely explains part of the emergence of the bulk spacetime, but also misses a critical aspect. The existence of locality down to a fixed physical scale that can be parametrically smaller than the AdS length remains a mystery in the CFT. Thus, we refer to coarse holography and sharp holography, and it is the latter that we seek to explain".

Also, LQG doesn't seem to assign geometries to generic spin network states, except in the large spin limit. More generally the states seem to be twisted geometries. But I don't know if LQG's conception of geometry of a tensor network state is the same as that as proposed for condensed matter.

 

[atyy]

Regarding action and wave function renormalization, the other place where I've seen a statement about the state during action renormalization is in Tom Banks's QFT text. IIRC, I believe he says renormalization assumes all the high energy degrees of freedom are in their ground state. I don't know what exactly that means, maybe the Balasubramanian paper is explaining the same idea?

Edit: I looked up Banks's comment (p138): "In QFT we will always assume that the high frequency degrees of freedom are in their ground state, so that the Green function is the one defined by Feynman ..."

 

[Physics Monkey]

It's not completely clear to me what these sorts of statements mean since the high energy dof don't really exist on their own. From a condensed matter perspective the high energy modes are changing all the time and the type of low energy theory you have determines the type of RG you do e.g. scaling to a point in critical theories or scaling to a surface in fermi surface systems.

I don't know how much this really differs from the standard wilsonian picture (which we all certainly use regularly). Morally speaking it seems that the rg flow ofter wanders into difficult to navigate terrain where the effective degrees of freedom change dramatically e.g. electrons at high energy and anyons at low energy.

 

[Physics Monkey]

Area laws, condensed matter, and a little bit of holography/gravity make an appearance in an essay I wrote for FQXi http://fqxi.org/community/forum/topic/1559

Check it out and give me a vote if you like it.

 

[atyy]

I see your classic paper finally got accepted by PRD!

Here's another addition to the AdS/MERA literature:

Refined Holographic Entanglement Entropy for the AdS Solitons and AdS black Holes
Masafumi Ishihara, Feng-Li Lin, Bo Ning
"We consider the refinement of the holographic entanglement entropy on a disk region for the holographic dual theories to the AdS solitons ... based on AdS/MERA conjecture, we postulate that the IR fixed-point state for the non-extremal AdS soliton is a trivial product state."

I don't immediately see a relation between the geometrical interpretations of AdS/MERA and LQG's spin networks, but let me list the one LQG paper I know that makes an explicit but bizarre connection to AdS/CFT:

Holomorphic Factorization for a Quantum Tetrahedron
Laurent Freidel, Kirill Krasnov, Etera R. Livine
"Interestingly, the integration kernel that appears in the decomposition formula turns out to be the n-point function of bulk/boundary dualities of string theory. ... For the case n=4, the symplectic manifold in question has the interpretation of the space of “shapes” of a geometric tetrahedron with fixed face areas, and our results provide a description for the quantum tetrahedron in terms of holomorphic coherent states."

 

[atyy]

http://arxiv.org/abs/1209.3304
Constructing holographic spacetimes using entanglement renormalization
Brian Swingle
(Submitted on 14 Sep 2012)
We elaborate on our earlier proposal connecting entanglement renormalization and holographic duality in which we argued that a tensor network can be reinterpreted as a kind of skeleton for an emergent holographic space. Here we address the question of the large N limit where on the holographic side the gravity theory becomes classical and a non-fluctuating smooth spacetime description emerges. We show how a number of features of holographic duality in the large N limit emerge naturally from entanglement renormalization, including a classical spacetime generated by entanglement, a sparse spectrum of operator dimensions, and phase transitions in mutual information. We also address questions related to bulk locality below the AdS radius, holographic duals of weakly coupled large N theories, Fermi surfaces in holography, and the holographic interpretation of branching MERA. Some of our considerations are inspired by the idea of quantum expanders which are generalized quantum transformations that add a definite amount of entropy to most states. Since we identify entanglement with geometry, we thus argue that classical spacetime may be built from quantum expanders (or something like them).

Goes beyond the original AdS/MERA paper by using "we" - not sure whether that's royal or not Snow monkeys are Japanese, so it's probably the latter.

 

[Physics Monkey]

I never heard of Sun Wukong before, but I like him.

 

[atyy]

Me too! http://www.youtube.com/watch?v=mOV4JUOb9j0&feature=fvwrel

 

[atyy]

@Physics Monkey, I'm still reading your latest paper slowly, but just wanted to say that it's very nicely written that even a lay person like me can understand it! It formulates more sharply all the vague questions I've been having, and begins to answer them.

 

[Physics Monkey]

@atyy, thanks a lot for your kind comment. I'm glad you found it vaguely comprehensible.

 

[atyy]

http://arxiv.org/abs/1210.6759
Holographic Entanglement Entropy of AdS Solitons and Tensor Network States
Javier Molina-Vilaplana
(Submitted on 25 Oct 2012)
The recent proposal connecting the AdS/CFT correspondence and entanglement renormalization tensor network states (MERA) is investigated by showing that the entanglement entropy and the two point functions in a type of hybrid tensor network state composed by a finite number of MERA layers and a matrix product state (MPS) acting as a cap layer, imitate the behaviour of the holographic entanglement entropy and the two point functions in the AdS soliton geometry. Within the context of AdS/CFT, AdS solitons represent theories with a mass gap, i.e gapped systems. From these observations, an explicit connection between the entanglement structure of the tensor network and those parameters which define the AdS soliton geometry is provided.

 

[atyy]

http://arxiv.org/abs/1210.7244
Entanglement entropy in de Sitter space
Juan Maldacena, Guilherme L. Pimentel
"We then study the entanglement entropy of field theories with a gravity dual. When the dual is known, we use the proposal of [10,11] to calculate the entropy. It boils down to an extremal area problem. The answer for the entanglement entropy depends drasticallyon the properties of the gravity dual. In particular, if the gravity dual has a hyperbolic Friedman-Robertson-Walker spacetime inside, then there is a non-zero contribution at order N2 for the “interesting” piece of the entanglement entropy. Otherwise, the order N2 contribution vanishes."

 

[atyy]

It does seem that the relationship between renormalization flow and holography is not well understood. Here is an interesting article about scheme dependence.

http://arxiv.org/abs/1211.1729
Holographic interpretations of the renormalization group
Vijay Balasubramanian, Monica Guica, Albion Lawrence
(Submitted on 7 Nov 2012 (v1), last revised 27 Nov 2012 (this version, v2))
In semiclassical holographic duality, the running couplings of a field theory are conventionally identified with the classical solutions of field equations in the dual gravitational theory. However, this identification is unclear when the bulk fields fluctuate. Recent work has used a Wilsonian framework to propose an alternative identification of the running couplings in terms of non-fluctuating data; in the classical limit, these new couplings do not satisfy the bulk equations of motion. We study renormalization scheme dependence in the latter formalism, and show that a scheme exists in which couplings to single trace operators realize particular solutions to the bulk equations of motion, in the semiclassical limit. This occurs for operators with dimension \Delta \notin \frac{d}{2} + \mathbb{Z}, for sufficiently low momenta. We then clarify the relation between the saddle point approximation to the Wilsonian effective action (S_W) and boundary conditions at a cutoff surface in AdS space. In particular, we interpret non-local multi-trace operators in S_W as arising in Lorentzian AdS space from the temporary passage of excitations through the UV region that has been integrated out. Coarse-graining these operators makes the action effectively local.

Not directly related, but MERA fans may like to see how other people use the word "disentangle": http://techtalks.tv/talks/opening-remarks/57645/ (13:45)

 

[atyy]

marcus has listed an interesting new paper in his bibliography. It shows the LQG people are thinking about AdS/CFT and using MERA as a tool to understand it. Swingle's original paper is cited. Bianchi needs to read the new paper too, and link it up with Friedel, Krasnov, and Livine's mysterious observation I'm also glad they are thinking about induced gravity. Weinberg and Witten explicitly say it evades their no-go theorem.

http://arxiv.org/abs/1212.5183
On the Architecture of Spacetime Geometry
Eugenio Bianchi, Robert C. Myers
(Submitted on 20 Dec 2012)
We propose entanglement entropy as a probe of the architecture of spacetime in quantum gravity. We argue that the leading contribution to this entropy satisfies an area law for any sufficiently large region in a smooth spacetime, which, in fact, is given by the Bekenstein-Hawking formula. This conjecture is supported by various lines of evidence from perturbative quantum gravity, simplified models of induced gravity and loop quantum gravity, as well as the AdS/CFT correspondence.

 

[marcus]

As I recall, Physicsmonkey indicated he was Brian Swingle earlier in this thread, so there is a PF connection! Not only does Bianchi cite Brian's paper but he and coauthor thank him in the acknowledgments, for discussions.

 

[John86]

http://arxiv.org/abs/1212.5121
Modular transformation and bosonic/fermionic topological orders in Abelian fractional quantum Hall states
Xiao-Gang Wen
(Submitted on 20 Dec 2012)
The non-Abelian geometric phases of the degenerate ground states was proposed as a physically measurable defining properties of topological order in 1990. In this paper we discuss in detail such a quantitative characterization of topological order, using generic Abelian fractional quantum Hall states as examples. We show that the non-Abelian geometric phases not only contain information about the quasi-particle statistics, they also contain information about the Hall viscosity and the chiral central charge of the edge states. The chiral central charge appears as the universal 1/A correction to the Hall viscosity (where A is the area of the space). Thus, the non-Abelian geometric phases (both the Abelian part and the non-Abelian part) may provide a way to completely characterize 2D topological order. Also the non-Abelian part of the geometric phases gives rise to a projective representation of the modular group (or SL(2,Z)).

http://arxiv.org/abs/1212.4863
Boundary Degeneracy of Topological Order
Juven Wang, Xiao-Gang Wen
(Submitted on 19 Dec 2012)
We introduce the notion of boundary degeneracy of topologically ordered states on a compact orientable spatial manifold with boundaries, and emphasize that it provides richer information than the bulk degeneracy. Beyond the bulk-edge correspondence, we find the ground state degeneracy of fully gapped edge states depends on boundary gapping conditions. We develop a quantitative description of different types of boundary gapping conditions by viewing them as different ways of non-fractionalized particle condensation on the boundary. This allows us to derive the ground state degeneracy formula in terms of boundary gapping conditions, which reveals the fusion algebra of fractionalized quasiparticles. We apply our results to Toric code and Levin-Wen string-net models. By measuring the boundary degeneracy on a cylinder, we predict Z_k gauge theory and U(1)_k x U(1)_k non-chiral fractional quantum hall state at even integer k can be experimentally distinguished. Our works refine definitions of symmetry protected topological order and intrinsic topological order.

http://arxiv.org/abs/1212.2121
2D Lattice Model Construction of Symmetry-Protected Topological Phases
Peng Ye, Xiao-Gang Wen
(Submitted on 10 Dec 2012)
We propose a general approach to construct symmetry protected topological (SPT) states (ie the short-range entangled states with symmetry) in 2D spin/boson systems on lattice. In our approach, we fractionalize spins/bosons into different fermions, which occupy nontrivial Chern bands. After the Gutzwiller projection of the free fermion state obtained by filling the Chern bands, we can obtain SPT states on lattice. In particular, we constructed a U(1) SPT state, a SO(3) SPT state, and a SU(2) SPT state on lattice.

http://arxiv.org/abs/1212.1827
Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases
Ling-Yan Hung, Xiao-Gang Wen
(Submitted on 8 Dec 2012)
We study the quantized topological terms in a weak-coupling gauge theory with gauge group $G_g$ and a global symmetry $G_s$ in $d$ space-time dimensions. We show that the quantized topological terms are classified by a pair $(G,\nu_d)$, where $G$ is an extension of $G_s$ by $G_g$ and $\nu_d$ an element in group cohomology $\mathcal{H}^d(G,\R/\Z)$. When $d=3$ and/or when $G_s$ is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (ie gapped long-range entangled phases with symmetry). Thus those SET phases are classified by $\mathcal{H}^d(G,\R/\Z)$, where $G/G_g=G_s$. We also apply our theory to a simple case $G_s=G_g=Z_2$, which leads to 12 different SET phases where quasiparticles have different patterns of fractional $G_s=Z_2$ quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry $G_s$, which may lead to different fractionalizations of $G_s$ quantum numbers and different fractional statistics (if in 2+1D).

 

[atyy]

Not directly related, but MERA fans may like to see how other people use the word "disentangle": http://techtalks.tv/talks/opening-remarks/57645/ (13:45)

I was only kidding there - but it turns out that Jason Morton works on both tensor networks and deep learning - apparently with the same mathematics!

Andrew Critch, Jason Morton. Polynomial constraints on representing entangled qubits as matrix product states

Jason Morton, Jacob Biamonte. Undecidability in Tensor Network States

Jason Morton. https://www.ipam.ucla.edu/publications/gss2012/gss2012_10605.pdf

 

[jasonmorton]

Hey, Jason here, thanks for the mention! I absolutely do think there is a connection between MERA and Deep Learning, as mentioned in the paper with Critch. I am trying to work out the details and hope to have more news this spring.

 

[atyy]

Hey, Jason here, thanks for the mention! I absolutely do think there is a connection between MERA and Deep Learning, as mentioned in the paper with Critch. I am trying to work out the details and hope to have more news this spring.

That is very cool! I see I gave the wrong link for your paper with Critch above, so let me correct that.

http://arxiv.org/abs/1210.2812
Polynomial constraints on representing entangled qubits as matrix product states
Andrew Critch, Jason Morton
"A conjectured dictionary between tensor network state models and classical probabilistic graphical models was presented in [11]. In this dictionary, matrix product states correspond to hidden Markov models, the density matrix renormalization group (DMRG) algorithm to the forward-backward algorithm, tree tensor networks to general Markov models, projected entangled pair states (PEPS) to Markov or conditional random fields, and the multi-scale entanglement renormalization ansatz (MERA) loosely to deep belief networks. In this work we formalize the first of these correspondences and use it to algebraically characterize quantum states representable by MPS and study their identifiability."

Incidentally, I came across your work via http://keck.ucsf.edu/~surya/ thesis. He's a fellow neurobiologist who had Sturmfels on his thesis committee. His website says "Although during my graduate work I played around with black holes, eleven dimensions, and little loops of string, I am now more fascinated by the world of biology which is full of incredible amounts of data but a relative paucity of theoretical frameworks within which to interpret and understand this data." But perhaps there are black holes in the brain after all To be honest, the deep architectures are genuinely inspired by biology, and although most of the learning rules seem unphysiological, experimental neurobiologists like me do find the DNN work informative. I have to admit I find DNNs more intuituitive, and I do wonder why it's ok to transfer the weights from a DBN to a DNN.

 

[atyy]

A new paper which says that the mutual information describes entanglement at finite temperature better than the entanglement entropy. Both of Physics Monkey's AdS/MERA papers are cited.

http://arxiv.org/abs/1212.4764
Holographic Mutual Information at Finite Temperature
Willy Fischler, Arnab Kundu, Sandipan Kundu
(Submitted on 19 Dec 2012)
Using the Ryu-Takayanagi conjectured formula for entanglement entropy in the context of gauge-gravity duality, we investigate properties of mutual information between two disjoint rectangular sub-systems in finite temperature relativistic conformal field theories in d-spacetime dimensions and non-relativistic scale-invariant theories in some generic examples. In all these cases mutual information undergoes a transition beyond which it is identically zero. We study this transition in details and find universal qualitative features for the above class of theories which has holographic dual descriptions. We also obtain analytical results for mutual information in specific regime of the parameter space. This demonstrates that mutual information contains the quantum entanglement part of the entanglement entropy, which is otherwise dominated by the thermal entropy at large temperatures.

Incidentally, there was an interesting result that despite correlations diverging at criticality, the mutual information in a classical stat mech Ising model peaks away from criticality. The result seems to have been confirmed. Hoefully this means that the brain isn't critical;)

http://arxiv.org/abs/1011.4421
Mutual information in classical spin models
Johannes Wilms, Matthias Troyer, Frank Verstraete
"The total many-body correlations present in finite temperature classical spin systems are studied using the concept of mutual information. As opposed to zero-temperature quantum phase transitions, the total correlations are not maximal at the phase transition, but reach a maximum in the high temperature paramagnetic phase."

http://arxiv.org/abs/1210.5707
Information theoretic aspects of the two-dimensional Ising model
Hon Wai Lau, Peter Grassberger
"All this suggests strongly that it is the slope of the mutual information, not the mutual information itself, that diverge at the critical point."

 

[atyy]

http://arxiv.org/abs/1302.5703
Holographic Local Quenches and Entanglement Density
Masahiro Nozaki, Tokiro Numasawa, Tadashi Takayanagi
(Submitted on 22 Feb 2013)
We propose a free falling particle in an AdS space as a holographic model of local quench. Local quenches are triggered by local excitations in a given quantum system. We calculate the time-evolution of holographic entanglement entropy. We confirm a logarithmic time-evolution, which is known to be typical in two dimensional local quenches. To study the structure of quantum entanglement in general quantum systems, we introduce a new quantity which we call entanglement density and apply this analysis to quantum quenches. We show that this quantity is directly related to the energy density in a small size limit. Moreover, we find a simple relationship between the amount of quantum information possessed by a massive object and its total energy based on the AdS/CFT.

"Now we would like to consider how to describe local quenches by using tensor networks. ...

Finally we would like to ask what is the holographic origin of gravitational force."

 

[atyy]

marcus highlighted this beautiful talk by Rivasseau in his bibliography.

http://pirsa.org/13020132/
Quantum Gravity as Random Geometry
Vincent Rivasseau
Abstract: Matrix models, random maps and Liouville field theory are prime tools which connect random geometry and quantum gravity in two dimensions. The tensor track is a new program to extend this connection to higher dimensions through the corresponding notions of tensor models, colored triangulations and tensor group field theories.
27/02/2013