Semiclassical states in quantum gravity: Curvature associated to a Voronoi graph by Daz-Polo and Garay

This week I’ returning to a much more fundamental level and reviewing a paper on Voronoi graphs. These are a method of dividing up a space into triangles and my very early work on this blog was looking at random triangulations. This paper outlines an attempt to compute the  curvature of a surface that didn’t work – that’s science, but we can build on that to find a method that does work.

The building blocks of a quantum theory of general relativity are
expected to be discrete structures. Loop quantum gravity is formulated using a basis of spin networkswave functions over oriented graphs with coloured edges. Semiclassical states should,
however, reproduce the classical smooth geometry in the appropriate limits. The question of how to recover a continuous geometry from these discrete structures is, therefore, relevant in this context. The authors explore this problem from a rather general mathematical perspective using  properties of Voronoi graphs to search for their compatible continuous geometries. They test the previously proposed methods for computing the curvature associated to such graphs and analyse the framework in detail  in the light of the
results obtained.


General relativity describes the gravitational interaction as a consequence of the curvature of space-time, a 4-dimensional Lorentzian manifold. Given this geometric nature of gravity, it is expected that, when quantizing, a prescription for quantum
geometry would arise based on more fundamental discrete structures, rather than on smooth differential manifolds.

Loop quantum gravity (LQG) is a candidate theory for such a quantization of general relativity. The fundamental objects , thebasis of the kinematical Hilbert space are the so-called spin network states, which are defined as wave functions constructed over oriented coloured graphs. The building blocks of the theory are, therefore, discrete combinatorial structures  – graphs. The theory provides quantum operators with a direct geometric interpretation, areas and volumes, which happen to have discrete eigenvalues, reinforcing the idea of a discrete geometry.

This perspective of considering abstract combinatorial structures as the fundamental objects of the theory is also adopted in other approaches to quantum gravity, such as spin-foam models, causal dynamical triangulations and causal sets . Also, the algebraic quantum gravity approach follows the same spirit of constructing a quantum theory of gravity from an abstract combinatorial structure.

Despite the variety of successful results obtained in LQG, the search for a semiclassical sector of the theory that would connect with the classical description given by general relativity in terms of a smooth manifold is still under research. An interesting question for the description of a semiclassical sector, given the combinatorial nature of the building blocks of the theory, would be whether there
is any correspondence between certain types of graphs and the continuous classical geometries. Tentatively, this would allow for the construction of gravitational coherent states corresponding to solutions of Einstein eld equations. While it is certainly true
that spin networks are a particular basis, and coherent states constructed from them might resemble nothing like a graph, some works seem to indicate that these graphs do actually represent the structure of space-time at the fundamental level.

This raises a very interesting question. How does the transition between a fundamental discrete geometry, encoded in a graph structure, and the continuous geometry we experience in every-day life happen? In particular, how does a one- dimensional structure give rise to 3-dimensional smooth space? A step towards answering these questions could be to think of these graphs as embedded in the
corresponding continuous geometries they represent. However, the situation is rather the opposite, being the smooth continuous structure an effective structure, emerging from the more fundamental discrete one, and not the other way around. Therefore,
a very relevant question to ask would be: Is there any information, contained in the abstract structure of a graph, that determine  the compatible continuous geometries? Can we determine what types of manifolds  a certain graph can be embedded in? One could even go further and ask whether any additional geometric information, like curvature, can be extracted from the very abstract structure of the graph itself. The goal is, therefore, to construct a unique correspondence between the discrete structures given by graphs, which in general do not carry geometric information, and smooth manifolds.

This problem was studied in the context of quantum gravity by Bombelli, Corichi and Winkler, who proposed a statistical method to compute the curvature of the manifold that would be associated to a certain class of graphs, based on Voronoi diagrams, giving a new step towards the semiclassical limit of LQG. Indeed, due to their properties, Voronoi diagrams appear naturally when addressing this kind of problems.They also play an important role in the discrete approach to general relativity provided by Regge calculus.

Voronoi diagrams are generated from a metric manifold and, by construction, contain geometric information from it. What was proposed, however, is to throw away all additional geometric information and to keep only the abstract structure of
the one-dimensional graph that forms the skeleton of the Voronoi diagram. Then, the task is to study if there are any imprints of the original geometry which remain in this abstract graph structure. Although the work is somewhat preliminary and, for the
most part, restricted to 2-dimensional surfacesz, it tackles very interesting questions and explores a novel path towards a semiclassical regime in LQG. The results obtained could also provide a useful tool for the causal dynamical triangulations approach.

 Curvature associated to a graph

A Voronoi diagram is constructed in the following way. For a set of points -seeds, on a metric space, each highest-dimensional cell of the Voronoi diagram contains only one seed, and comprises the region of space closer to that one seed than to any of the others. Then, co-dimension n cells are made by sets of points equidistant to n + 1 seeds, e.g., in 2 dimensions, the edges (1-dimensional cells) of the Voronoi diagram are the lines separating two of these regions, and are therefore equidistant to two seeds. In the same way, vertices (0-dimensional cells) are equidistant to three seeds.
Therefore, except in degenerate situations which are avoided by randomly sprinkling the seed, the valence of all vertices in a D-dimensional Voronoi diagram is D + 1. Another interesting property of Voronoi diagrams is that their dual graph is the so- called Delaunay triangulation, whose vertices are the Voronoi seeds. By construction, for a given set of seeds on a metric space the corresponding Voronoi diagram is uniquely defined.

vor fig1

The starting point is to consider a given surface on which we randomly sprinkle a set of points, that will be the seeds
for the Voronoi construction. A Voronoi cell-complex is constructed, containing zero, one, and two-dimensional cells (vertices, edges and faces). We keep, then, the abstract structure of the one-dimensional graph encoded, for instance, in an adjacency matrix. We are, thus, left with an abstract graph.

vor fig2

All vertices of the Voronoi graph are tri-valent. This gives rise
to the following relation between the total number V of vertices in the graph and the total number of edges E:


since every vertex is shared by three edges and every edge contains two vertices. We will also use the definition of the Euler-Poincare characteristic χ in the two-dimensional case


where F is the total number of faces in the graph (that equals the number of seeds). Finally, one can define the number p of sides of a face (its perimeter in the graph). Taking into account that every edge is shared by two faces, the average p over a set of faces satisfies


The following expression for the Euler-Poincare characteristic χ
can be obtained


in terms of the total number of faces F and the average number of sides of the faces p.

On the other hand, if there is a manifold M associated to the Voronoi diagram, this manifold should have the same topology as the diagram. The Gauss-Bonnet theorem can be used then to relate χ with the integral of the curvature over the manifold. If M is a manifold without boundary (like a sphere), the theorem takes the form


where dA is the area measure and R is the Ricci scalar.

Assume that the region of the graph one is looking at is small enough
so that the curvature can be considered constant. In that case


where As is the total surface area of the sphere.

this formula can also be applied to the sphere patch by defining a density of faces  ρ= F=As = Fp =Ap , where Fp and Ap are respectively the number of faces and area of the patch.


Implementation and results

vor fig3

vor fig8vor fig9

Conclusions and outlook
The problem of reconstructing a continuous geometry starting from a discrete, more fundamental combinatorial structure, like a graph, is interesting for a wide range of research fields. In the case of LQG theory whose Hilbert space is constructed using wave functions defined over graphsthe solution to this problem could provide interesting hints on the construction of semiclassical states, moving toward a connection with classical solutions of the Einstein equations.

In this article the authors discussed and implemented the method proposed  to compute the curvature of a manifold from an abstract Voronoi graph associated to it. By making use of some topological arguments involving the Gauss-Bonnet theorem, a method to statistically compute the curvature in terms of the average number of sides of the faces in the graph is suggested. They tested this
method for the simplest geometries: the unit sphere – constant positive curvature and the plane – zero curvature. They
 found highly unsatisfactory results for the value of the curvature in both cases.


Curvature operator and gravity coupled to a scalar field: the physical Hamiltonian operator by Alesci, Assanioussi and Lewandowski

This week I been reviewing some conference proceedings the FFP14 Conference, Marseilles 2014. One paper I particularly like is this one by Alesci, Assanioussi and Lewandowski. I have been doing a some collaboration with Alesci and Assanoussi on the Hamiltonian Operator.

Gravity (minimally) coupled to a massless scalar field

The theory of 3+1 gravity (Lorentzian) minimally coupled to a free massless scalar field Φ(x) is described by the action




Assuming that


The Hamiltonian constraint is solved for π using the diff. constraint


Φ becomes the emergent time.

In the region (+,+), an equivalent model could be obtained by keeping the Gauss and Diff. constraints and reformulating the scalar constraints




Quantization of the model:Hilbert space and Gauss constraint

The kinematical Hilbert space is defined as



Where its elements are


The gauge invariant subspace



Quantization of the model: Hilbert space of gauge & diff. invariant states

The space of the Gauss & vector constraints is defined as


Quantization of the model: Physical Hamiltonian

Solving the scalar constraint C‘ in the quantum theory is equivalent to finding solutions to


given a quantum observable  ,  the dynamics in this quantum theory is generated by





Quantization of the model: construction of the Hamiltonian operator

Interested in constructing the quantum operator corresponding to the classical quantity




Lorentzian part:


The final quantum operator corresponding to the Lorentzian part:




Properties of this operator:

  • Gauge & Diffeomorphism invariant
  • Cylindrically consistent  if the averaging used in defining the curvature operator is restricted to non zero contributions;
  • Self-adjoint;
  • Discrete spectrum & compact expression for the matrix elements expressed explicitly in terms of the colouring on the spin network basis

Euclidean part:


The resulting operator:


The action of the full H on a spin network state can be expressed as


Introduce the adjoint operator of  H



For two spin network states have

Picture30 Picture31

Define a symmetric operator


Which acts on s-n states as


Define the physical Hamiltonian for this deparametrized model


Properties of the final operator   hphys[N]   :

  • Gauge and Diffeomorphism invariant
  • Cylindrically consistent  if the averaging used in defining the operator is restricted to only non zero contributions
  • Symmetric /Self-adjoint
  • Discrete spectrum  andcompact expression for the matrix elements expressed explicitly in terms of the colouring on the spin network basis – volume operator not involved


The paper presented a way of implementing the Hamiltonian operator in the case of the deparametrized model of gravity with a scalar field using:

  • The simple and well defined curvature operator;
  • A new regularization scheme that allows to define an adjoint operator for the regularized expression and hence construct a symmetric Hamiltonian operator;

This operator verifies the properties of gauge symmetries and cylindrical consistency could be imposed.

Related articles

Special properties of spin network functions

My work on quantum geometric operators makes heavy use of special properties of the spin network functions in particular the fact that the action of the flux operators on spin network basis states can be mapped to the problem of evaluating angular momentum operators on angular momentum eigenstates, which is familiar from ordinary quantum mechanics. This results in a great simplification and provides a convenient way to work in the SU(2)gauge-invariant regime using powerful techniques from the recoupling theory of angular momenta.

This post provide details on the conventions used in the construction of recoupling schemes. The post is organized as follows:

  • Basic properties of matrix representations of SU(2), whose matrix elements are used for the definition of spin network functions.
  • The theory of angular momentum from quantum mechanics and the notion of recoupling of an arbitrary number of angular momenta in terms of recoupling schemes.
  • The definition of 6j-symbols andtheir basic properties. This provides an explicit notion of recoupling schemes in terms of polynomials of quantum numbers.

Representations of SU(2)
Irreducible matrix representations of SU(2) can be constructed in (2j+1)-dimensional linear vector spaces, where j ≥ 0 is a half integer number; j = 0 denotes the trivial representation.

General Conventions — Defining Representation j = 1/2
Generators of SU(2) use the τ-matrices given by τk := − iσk, with σk being the Pauli-matrices:

Additionally use


SU(2) Representations
In the defining representation of SU(2), for a group element h ∈ SU(2)





and we have the additional properties that


Use the following convention for the matrix elements


For the τk’s we additionally have


General Conventions for (2j + 1)-dimensional SU(2) Representation Matrices
General Formula for SU(2) Matrix Element

The (2j +1)-dimensional representation matrix of h ∈ SU(2), given in terms of the parameters of the defining representation can be written as


where ℓ takes all integer values such that none of the factorials in the denominator gets a negative argument. By
construction every representation of SU(2) consists of special unitary matrices and


Generators and ε-Metric
Applying the representation matrix element formula and the ansatz,


where in the defining two dimensional representation the exponential can be explicitly evaluated, one obtains

and finds that


Angular Momentum Theory
Basic Definitions
The angular momentum orthonormal basis u(j,m; n) =|j m ; n〉 of a general (2j + 1) dimensional representation of SU(2). The index n stands for additional quantum numbers, not affected by the action of the angular momentum operators J fulfilling the commutation


Formulate ladder operators as


The |j m ; n〉  simultaneously diagonalize the two operators: the squared total angular momentum (J)² and the
magnetic quantum number J³ :


That is, |j m ; n〉 is a maximal set of simultaneous eigenvectors of (J)² and J³ find the following commutation relations


such that for the (2j + 1)-dimensional matrix representation with arbitrary weight j



Fundamental Recoupling

Clebsch -Gordan theorem on tensorized representations of SU(2):
Theorem b1

If we couple two angular momenta j1, j2, we can get resulting angular momenta j12 varying in the range |j1 − j2| ≤ j12 ≤ j1 + j2. The tensor product space of two representations of SU(2) decomposes into a direct sum of representation spaces, with one space for every possible value of recoupling j12 with the according dimension
2j12 + 1.


Recoupling of n Angular Momenta — 3nj-Symbols
The successive coupling of three angular momenta to a resulting j can be generalized to an n-fold tensor product of representations            πj1 ⊗ πj2 ⊗ . . . ⊗ πjn  by reducing out step by step every pair of representations.

This procedure is carried out until all tensor products are reduced out. One then ends up with a direct sum of representations, each of which has a weight corresponding to an allowed value of the total angular momentum to which the n single angular momenta j1, j2, . . . , jn can couple. However, there is an arbitrariness in how one couples the n angular momenta together, that is the order in which πj1 ⊗πj2 ⊗. . .⊗πjn is reduced out matters.

Consider a system of n angular momenta. First we fix a labelling of these momenta, such that we have j1, j2, . . . , jn. Again the first choice would be a tensor basis |j m〉 of all single angular momentum states |jk mk〉, k = 1 . . . n defined by:


Recoupling Scheme
recoupling b1

Standard Basis

Theorem b2



Properties of Recoupling Schemes

A general standard recoupling scheme is defined as follows:

For the scalar product of two recoupling schemes we have:


Properties of the 6j-Symbols
6j-symbols are the basic structure in recoupling calculations, as every coupling of n angular momenta can be expressed in terms of them.
The 6j-symbol is defined in


The factors in the summation are Clebsch-Gordon coefficients.

Explicit Evaluation of the 6j-Symbols
A general formula for the numerical value of the 6j-symbols has been derived by Racah


Symmetry Properties
The 6j-symbols are invariant under:

  • any permutation of the columns:


• simultaneous interchange of the upper and lower arguments of two columns

Orthogonality and Sum Rules
Orthogonality Relations


Composition Relation


Sum Rule of Elliot and Biedenharn


Properties of the Volume Operator in Loop Quantum Gravity by Brunnemann and Rideout

This week I’ve returned more strongly to my work on the numerical spectra of quantum geometrical operators. This post looks at an analysis of the Ashtekar and Lewandowski version of the volume operator.

The authors analyze the spectral properties of the volume operator of Ashtekar and Lewandowski in Loop Quantum Gravity, which is the quantum analogue of the classical volume expression for regions in three dimensional Riemannian space. The analysis also considers generic graph vertices of valence greater than four. The authors find that the geometry of the underlying vertex characterizes the spectral properties of the volume operator, in particular the presence of a volume gap – a smallest non-zero eigenvalue in the spectrum is found to depend on the vertex embedding.

The authors  compute the set of all non-spatially diffeomorphic non-coplanar vertex embeddings for vertices of valence 5–7 and argue that these sets can be used to label spatial diffeomorphism invariant states. it is seen that  gauge invariance connects vertex geometry and the representation properties of the underlying gauge group in a natural way. Analytical results on the spectrum of 4-valent vertices show the presence of a volume gap.

Loop Quantum Gravity (LQG) is a candidate for a quantum theory of gravity. It is an attempt to canonically quantize General Relativity. The resulting quantum theory is formulated as an SU(2) gauge theory.

General Relativity is put into a Hamiltonian formulation by introducing a foliation of four dimensional spacetime into spatial three dimensional hypersurfaces ∑, with the orthogonal timelike
direction parametrized by t.  first class constraints which have to be imposed on the theory so that it obeys the dynamics of Einstein’s equations and is independent of the particular choice of foliation. These constraints are the three spatial diffeomorphism or vector
constraints which generate diffeomorphisms and the so called Hamilton constraint which generates deformations of the hypersurfaces in the t- foliation direction. In addition there are three Gauss constraints due to the introduction of additional SU(2) gauge degrees of freedom.

The theory is then quantized on the kinematical level in terms of holonomies h and electric fluxes E. Kinematical states are defined over collection of edges of embedded graphs. The physical states have to be constructed by imposing the operator version of the constraints on the thus defined kinematical theory.

There are well defined operators in the kinematical quantum theory which correspond to differential geometric objects, such as the length of curves, the area of surfaces, and the volume of regions in the spatial foliation hypersurfaces. All these quantities have discrete spectra, which can be traced back to the compactness of the gauge group SU(2). A coherent state framework address questions on the correct semiclassical limit of the theory.

A central role in investigations is played by the area and the volume operators.The volume operator is a crucial object not only in order to analyze matter coupling to LQG, but also for evaluating the action of the Hamilton constraint operator in order to construct the physical sector of LQG. By construction the spectral properties of the constraint operators are driven by the spectrum of  the volume operator.

Loop Quantum Gravity

Hamiltonian Formulation
In order to cast General Relativity into the Hamiltonian formalism, one has to perform a foliation M ≅ Rx∑ of the four dimensional spacetime manifold M into three dimensional spacelike hypersurfaces with transverse time direction labelled by a foliation parameter t ∈ R. The four dimensional metric g can then be decomposed into the three metric q(x) on the spatial slices ∑ and its extrinsic curvature K, which serve as canonical variables. Introducing new variables due to Ashtekar  may  take
as canonical variables electric fields E and connections A as used in the canonical formulation of Yang Mills theories. Here a, b = 1 . . . 3 denote spatial (tensor) indices, i, j = 1 . . . 3 denote su(2)-

The occurrence of SU(2) as a gauge group comes from the fact that it is necessary for the coupling of spinorial matter and it is the universal covering group of SO(3) which arises naturally when one

rewrites the three metric q in terms of cotriads as


The pair (A,E) is related to (q,K) as


with Γ being the spin connection.

The pair (A,E) obeys the Poisson bracket:


The theory is subject to constraints which arise due to the background independence of General Relativity. It can be treated as suggested by Dirac . There are:

  • three vector (spatial diffeomorphism) constraints


  •  a scalar – Hamilton constraint


  • three Gauss constraints


The Volume Operator
Definition of the Volume Operator

The operator corresponding to the volume V (R) of a spatial region R ⊂ ∑


Acting on gauge invariant spin network states, is defined as


The sum in has to be taken over all vertices v of the underlying graph γ. At each vertex v one has to sum over all possible ordered triples (ei , ej , ek).

Matrix Formulation



where Q is a totally antisymmetric matrix with purely real elements. Its eigenvalues λ are purely imaginary and come in pairs λQ = ±iλ. Choosing Z = 1 the volume operator V has the same eigenstates as Q and its eigenvalues λ = |λQ|½

We are left with the task of calculating the spectra of totally antisymmetric real matrices of the form:


Analytical Results on the Gauge Invariant 4-Vertex

The spectrum of the volume operator at a given gauge invariant 4-vertex is simple, that is all its eigenvalues, except zero, come in pairs, and there are no further degeneracies or accumulation points in the spectrum. An explicit expression for the eigenstates of the volume operator in terms of polynomials of its matrix elements and its eigenvalues can be given as:




The specific matrix realization of the volume operator plays a crucial role here since it can be written as an antisymmetric purely imaginary matrix having non-zero entries only on the first off-diagonal. This makes it possible to apply  techniques from orthogonal polynomials and Jacobi-matrices.