Bianchi's PI talk on quantum polyhedra in LQG

[marcus]

http://pirsa.org/10110052/
Quantum polyhedra in loop quantum gravity
Eugenio Bianchi
"Interwiners describe quanta of space in loop quantum gravity. In this talk I show that the Hilbert space of SU(2) intertwiners has as semiclassical limit the phase space of a classical system originally considered by Minkowski: convex polyhedra with N facets of given areas and normals. This result sharpens Penrose spin-geometry theorem. The knowledge of the classical system associated to intertwiner space can be fruitfully used: I show that many properties of the spectrum of the volume operator can be derived via Bohr-Sommerfeld quantization of the volume of a classical polyhedron. Moreover, a recent derivation of the entropy of a Black Hole involves the calculation of the dimension of the associated SU(2) intertwiner space. I describe a semiclassical version of this calculation: the microstates counted are shapes of a tessellated horizon having facets of given areas and normals. The calculation reproduces the area law, together with the logarithmic corrections to the entropy."

For various reasons this talk has a special interest for me---and perhaps others of us.

First the Penrose theorem that says spin-network = geometry has always been hard to understand. Why would a graph with links labeled by spins (that is in effect by dimensionalities of SU2 representations) describe a quantum state of metric geometry? How does the Penrose "metric operator" work? How does it just happen that the LQG Hilbertspace of quantum states of geometry just happens to have a basis consisting of spin-labeled graphs (i.e. spin networks)? Bianchi explaining a sharpening of Penrose' theorem might open some new insight.

Second the intertwiner labels on the NODES of the graph have also been a little hard to understand. How should one think of intertwiners? Why should they be associated with VOLUMES? Recently there appeared this paper which identifies intertwiners with "quantum polyhedra." The talk should make definite what is meant by quantum polyhedra and might give some new understanding of how to think about intertwiners.

Third, it's always good to get new insight about Hawking BH entropy law. How to think about the degrees of freedom of the BH horizon, the microstates that carry the BH entropy? One doesn't want to hope for too much, but maybe the talk could also give some additional perspective on that.

I will try to think what papers one might want to look at before watching the online video of the talk.

 

[marcus]

First for some overview relating to Penrose' theorem there is page 3 of
http://arxiv.org/abs/1010.1939

Section IV Penrose Metric Operator

Section IV-A Spin networks as quantum 3-geometries

==a few excerpts from page 3 ==
In the next section I show (following [29]) that the states in this boundary space have a natural interpretation as 3-geometries, thanks to a beautiful theorem by Penrose.
...

The boundary Hilbert space (8) has a natural interpretation as a space of quantum metrics, that was early recognized by Roger Penrose. The natural “momentum” operator on L₂[SU2] is the derivative operator...
...
FIG. 2: The angle defined by the Penrose metric operator on the graph.
...
Penrose spin-geometry theorem then gives states in H_Γ a consistent interpretation as quantized 3-geometries.
The metric operator G_ll′ determines the angle between the links l and l at the node n [19, 30, 31] (see Fig.2).
... A volume element associated to the node n can be defined in terms of Penrose metric operator, using standard relations between metric and volume element [32].
...
The Area and Volume operators A_l and V_n form a complete set of commuting observables in H_Γ, in the sense of Dirac. The spectrum of both operators can be computed [32]; it is discrete and it has a minimum step between zero and the lowest non-vanishing eigenvalue.
...
[My comment: note that the definition is Lorentz invariant.]
...
The results above equip the boundary states of the model (4) with a geometrical interpretation: the spin network state ψ_{Γ,jl ,in} is interpreted as representing a granular space. Each node is a quantized “chunk”, or “quantum” of space (see Fig.3); the graph gives the connectivity relations between these quanta; [the intertwiner i_n at node n] is the quantum number of the volume of the n’th quantum of space; and [the spin j_l on the link l] is the quantum number of the area of the elementary surface separating the adjacent nodes...
==endquote==

 

[marcus]

Second, I mentioned another reason to be interested in the talk was to get a better understanding of intertwiners. They seem to be the natural quantum numbers labeling chunks of volume (in a quantum geometry), and also to be associated with quantum polyhedra.So another paper to look at, for pre-talk prep, might be this September one:

http://arxiv.org/abs/1009.3402
Polyhedra in loop quantum gravity
Eugenio Bianchi, Pietro Donà, Simone Speziale
32 pages, many figures
(Submitted on 17 Sep 2010)
"Interwiners are the building blocks of spin-network states. The space of intertwiners is the quantization of a classical symplectic manifold introduced by Kapovich and Millson. Here we show that a theorem by Minkowski allows us to interpret generic configurations in this space as bounded convex polyhedra in Euclidean space: a polyhedron is uniquely described by the areas and normals to its faces. We provide a reconstruction of the geometry of the polyhedron: we give formulas for the edge lengths, the volume and the adjacency of its faces. At the quantum level, this correspondence allows us to identify an intertwiner with the state of a quantum polyhedron, thus generalizing the notion of quantum tetrahedron familiar in the loop quantum gravity literature. Moreover, coherent intertwiners result to be peaked on the classical geometry of a polyhedron. We discuss the relevance of this result for loop quantum gravity. In particular, coherent spin-network states with nodes of arbitrary valence represent a collection of semiclassical polyhedra. Furthermore, we introduce an operator that measures the volume of a quantum polyhedron and examine its relation with the standard volume operator of loop quantum gravity. We also comment on the semiclassical limit of spinfoams with non-simplicial graphs."

And then there is the third source of interest (at least for me personally)---black hole entropy. Maybe someone else can suggest what might be a good paper to read that would relate to what Bianchi says about BH entropy in the abstract of his talk.
==quote==

...Moreover, a recent derivation of the entropy of a Black Hole involves the calculation of the dimension of the associated SU(2) intertwiner space. I describe a semiclassical version of this calculation: the microstates counted are shapes of a tessellated horizon having facets of given areas and normals. The calculation reproduces the area law, together with the logarithmic corrections to the entropy."
==endquote==

 

[marcus]

About black hole entropy and intertwiners, here is a May 2009 paper
http://arxiv.org/abs/0905.3168 (Engle, Noui, Perez)
and here are the (so far) 19 papers that cite that:
http://arxiv.org/cits/0905.3168

The first paper to cite that one was also May 2009
http://arxiv.org/abs/0905.4916 (Krasnov Rovelli)
and here are the (so far) 13 papers that cite that one:
http://arxiv.org/cits/0905.4916

Citing both of these two 2009 papers is an October 2010 paper published in Physical Review D that has no arxiv entry, curiously enough. There is only a Spires entry for it:

Detailed black hole state counting in loop quantum gravity.
Ivan Agullo, (Wisconsin U., Milwaukee), J. Fernando Barbero, (Madrid, Inst. Estructura Materia), Enrique F. Borja, (Erlangen - Nuremberg U., Theorie III), Jacobo Diaz-Polo, (Penn State U.), Eduardo J.S. Villasenor, (Carlos III U., Madrid & Madrid, Inst. Estructura Materia) . Oct 15, 2010. (Published Oct 15, 2010) . 31pp..

However there are two earlier papers by the same authors which were posted on arxiv:
http://arXiv.org/abs/0906.4529 (publ. Physical Review D)
http://arXiv.org/abs/0802.4077 (publ. Physical Review Letters)

This work involves the use of the space of intertwiners (equivalent in their case to a Chern-Simon Hilbert space) in calculating the BH entropy.
==quote from 0906.4529 ==
''Each of these lists gives a contribution to the entropy equal to the dimension of the Hilbert
space H^CS (j₁ , . . . , j_n )of the Chern-Simons theory associated with the fixed choice of spins j_p at each puncture p of the horizon. When the Immirzi parameter satisfies |γ | ≤ √3 the space H^CS (j₁ , . . . , j_n ) coincides with the invariant subspace of the tensor product of the irreducible SU(2) representations [j_p ] labeled by those spins and hence dim[ H ^CS (j₁ , . . . , j_n )= dim[Inv(⊗_p [j_p ])] .

==endquote==
The space of intertwiners is by definition the "invariant subspace of the tensor product of the irreducible SU(2) representations" given by the list of spins. So that is what they are using, though they do not employ the intertwiner terminology.

 

[atyy]

I'm not sure if the polyhedra in Bianchi's talk is the same as this, but here's some other LQG work discussing polyhedra:

http://arxiv.org/abs/0902.0351
Quantum geometry from phase space reduction
Florian Conrady, Laurent Freidel
"It is also interesting to note that along with the construction of new models we are also witnessing a merging of two lines of thoughts on spin foam models that developed in parallel for a long time and are now finally intersecting. One line of thought, which is more canonical in spirit, can be traced back to the seminal work of Barbieri [13], who realized that spin network states of loop quantum gravity can be understood (and heuristically derived) by applying the quantization procedure to a collection of geometric tetrahedra in 3 spatial dimensions. ...

The second line of thought can be traced back to the work of Reisenberger [14] who proposed to think about quantum gravity directly in terms of a path integral approach in which we integrate over classical configurations living on a 2–dimensional spine and where spin foam models arise from a type of bulk discretization. ...

The merging between these two approaches starts with a work of Livine and Speziale [17] who proposed to label spin network states not with the usual intertwiner basis, but with “coherent intertwiners” that are labelled by four vectors whose norm is fixed to be the area of the faces of the tetrahedron. This is almost the missing link between the two approaches and it has led to a very efficient and geometrical way of deriving the new spin foam models [2, 3].

http://arxiv.org/abs/0905.3627
Holomorphic Factorization for a Quantum Tetrahedron
Laurent Freidel, Kirill Krasnov, Etera R. Livine
(Submitted on 22 May 2009 (v1), last revised 27 Mar 2010 (this version, v2))
We provide a holomorphic description of the Hilbert space H(j_1,..,j_n) of SU(2)-invariant tensors (intertwiners) and establish a holomorphically factorized formula for the decomposition of identity in H(j_1,..,j_n). 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. Our results provide a new interpretation for this quantity as being, in the limit of large conformal dimensions, the exponential of the Kahler potential of the symplectic manifold whose quantization gives H(j_1,..,j_n). 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. We describe how the holomorphic intertwiners are related to the usual real ones by computing their overlap. The semi-classical analysis of these overlap coefficients in the case of large spins allows us to obtain an explicit relation between the real and holomorphic description of the space of shapes of the tetrahedron. Our results are of direct relevance for the subjects of loop quantum gravity and spin foams, but also add an interesting new twist to the story of the bulk/boundary correspondence.

"The integration kernel here turns out to be just the n-point function of the AdS/CFT duality [10], given by an integral over the 3-dimensional hyperbolic space of a product of n so-called bulk-to-boundary propagators ...

We would like to finish this section by pointing out that our results imply that the n-point function of the bulk-boundary correspondence of string theory has the interpretation of the (exponential of the) Kahler potential on the space of shapes Sj . This is surprising, at least to the present authors, and appears to be a new result."

 

[marcus]

Additional context re PI's QG seminar.

Wednesday 3 November
Maïté Dupuis (ENS Lyon, where Livine has a QG group)
http://pirsa.org/10110072
U(N) framework and simplicity constraints for spin foam models
"In the context of loop quantum gravity and spin foam models, the simplicity constraints are essential in that they allow to write general relativity as a constrained topological theory. I will first recall the spin foam quantization procedure and focus more particularly on the step consisting in implementing the simplicity constraints. Then, I will present the U(N) framework initially developed for SU(2) intertwiners. Finally, I will show how we can apply this new framework to impose the simplicity constraints in the context of 4d Euclidean gravity and present new solutions defined in term of U(N) coherent states."
(some possible refs: 1005.2090, 1006.5666, 1010.5451 ?)

Tuesday 9 November
Eugenio Bianchi (Marseille)
http://pirsa.org/10110052/

Tuesday 16 November
Matteo Smerlak (Marseille)
http://pirsa.org/10110071

 

[marcus]

The earlier papers Atyy mentioned seem to be about the tetrahedron. Much recent work, I seem to recall, has focused on the quantum tetrahedron (rather than the general polyhedron) and on vertices with a limited valence.

That makes sense as an exploratory strategy---look at simple cases first, then generalize later. One can see the current work on quantum polyhedra as part of a coherent program, which carries forward research such as that by Freidel et al in the papers cited.

Freidel is at PI and I imagine he will make some comment during the seminar presentation, four days from now.

 

[marcus]

Today is Bianchi's talk. They have posted the video and the talk is a good one.

http://pirsa.org/10110052/
Quantum polyhedra in loop quantum gravity
Eugenio Bianchi
"Interwiners describe quanta of space in loop quantum gravity. In this talk I show that the Hilbert space of SU(2) intertwiners has as semiclassical limit the phase space of a classical system originally considered by Minkowski: convex polyhedra with N facets of given areas and normals. This result sharpens Penrose spin-geometry theorem. The knowledge of the classical system associated to intertwiner space can be fruitfully used: I show that many properties of the spectrum of the volume operator can be derived via Bohr-Sommerfeld quantization of the volume of a classical polyhedron. Moreover, a recent derivation of the entropy of a Black Hole involves the calculation of the dimension of the associated SU(2) intertwiner space. I describe a semiclassical version of this calculation: the microstates counted are shapes of a tessellated horizon having facets of given areas and normals. The calculation reproduces the area law, together with the logarithmic corrections to the entropy."

Bianchi covers a range of topics. A paper related to the talk (among several) that one might wish to review before watching the video is
http://arxiv.org/abs/1009.3402
Polyhedra in loop quantum gravity
Eugenio Bianchi, Pietro Donà, Simone Speziale

Also one week from today there will be a second interesting talk:
Tuesday 16 November
Matteo Smerlak (Marseille)
http://pirsa.org/10110071
Is temperature the speed of time? Thermal time and the Tolman effect

There was a paper posted recently at arxiv, on that topic. One could review the paper before watching the talk. http://arxiv.org/abs/1005.2985 Here is the abstract:
"The thermal time hypothesis has been introduced as a possible basis for a fully general-relativistic thermodynamics. Here we use the notion of thermal time to study thermal equilibrium on stationary spacetimes. Notably, we show that the Tolman-Ehrenfest effect (the variation of temperature in space so that $$T\sqrt{g_{00}}$$ remains constant) can be reappraised as a manifestation of this fact: at thermal equilibrium, temperature is locally the rate of flow of thermal time with respect to proper time - pictorially, "the speed of (thermal) time". Our derivation of the Tolman-Ehrenfest effect makes no reference to the physical mechanisms underlying thermalization, thus illustrating the import of the notion of thermal time."