Tag Archives: Special unitary group

U(N) Coherent States for Loop Quantum Gravity by Freidel and Livine

This week I have been reading a paper by Freidel and Livine which investigates the geometry of the space of N-valent SU(2)intertwiners. In this paper, the authors propose a new set of holomorphic operators acting on this space and a new set of coherent states which are covariant under U(N) transformations. These states are labeled by elements of the Grassmannian GrN,2, and have  a geometrical interpretation in terms of framed polyhedra and are related to  coherent intertwiners – see the post

Loop quantum gravity is a  canonical quantization of general relativity where the quantum states of geometry are the so-called spin network states. A spin network is based on a graph 􀀀 dressed up with half-integer spins je on its edges and intertwiners iv on its vertices. The spins define quanta of area while the intertwiners describe chunks of space volume. The dynamics then acts on the spins je and intertwiners iv, and can also deform the underlying graph 􀀀.

In this paper, the authors focus on the structure of the space of intertwiners describing the chunk of space. They focus their study on a region associated with a single vertex of a graph and arbitrary high valency. Associated with this setting there is a classical geometrical description. To each edge going out of this vertex there is associated a dual surface element or face whose area is given by the spin label. The collection of these faces encloses a 3-dimensional volume whose boundary forms a 2-dimensional polygon with the topology of a sphere. This 2d-polygon is such that each vertex is trivalent. At the quantum level the choice of intertwiner iv attached to the vertex describes the shape of the full dual surface and gives the volume contained in that surface.

The space of N-valent intertwiners carries an irreducible representation of the unitary group U(N). These irreducible representations of U(N) are labeled by one integer: the total area of the dual surface – defined as the sum of the spins coming
through this surface. The U(N) transformations deform of the shape of the intertwiner at fixed area. This provides a clean geometric interpretation to the space of intertwiners as wavefunctions over the space of classical N-faced polyhedron. It also leads to a clearer picture of what the discrete surface dual to the intertwiner should look like in the semi-classical regime.

In this work the authors present the explicit construction for new coherent states which are covariant under U(N), then compute their norm, scalar product and show that they provide an overcomplete
basis. They also compute their semi-classical expectation values and uncertainties and show that they are simply related to the Livine-Speziale coherent intertwiners used in the construction of the Engle-Pereira-Rovelli-Livine (EPRL) and Freidel-Krasnov (FK) spinfoam models and their corresponding semi-classical boundary states. These new coherent states confirm the polyhedron interpretation of the intertwiner space and show the relevance of the U(1) phase/frame attached to each face, which appears very similar to the extra phase entering the definition of the discrete twisted geometries for loop gravity.

Considering the space of all N-valent intertwiners, they  decompose it separating the intertwiners with different total area u1conequ1:


Each space H(J)N at fixed total area carries an irreducible representation of U(N). The u(N) generators Eij are quadratic operators in the harmonic oscillators of the Schwinger representation of the su(2)-algebra. the full space HN as a Fock space by introducing annihilation and creation operators

The full space HN is established as a Fock space by introducing annihilation and creation operators  Fij , Fij , which allow transitions between intertwiners with different total areas.  These creation operators can be used to define U(N) coherent states |J,  zi 〉 ∝ Fz| 0〉labeled by the total area J and a set of N spinors zi . These states turn out to have very interesting properties.

  • They transform simply under U(N)-transformations                             u |J,  zi 〉 = |J(u z)i 〉
  • They get simply rescaled under global GL(2,C) transformation acting on all the spinors: u1conequ3In particular, they are invariant under global SL(2,C)  transformations.
  • They are coherent states  and are obtained by the action of U(N) on highest weight states. These highest weight vectors correspond to bivalent intertwiners such as the state defined by u1conequ4
  • For large areas J, they are semi-classical states peaked around the expectation values for the u(N) generators: u1conequ5
  • The scalar product between two coherent states is easily computed:u1conequ6
  • They are related to the coherent and holomorphic intertwiners, writing |j,zi〉 for the usual group-averaged tensor product of SU(2) coherent states defining the coherent intertwiners, we have: u1conequ7

The authors believe that this U(N) framework opens the door to many applications in loop quantum gravity and spinfoam models.


Related Posts


Deformations of Polyhedra and Polygons by the Unitary Group by Livine

This week whilst preparing some calculations I have been looking at the paper ‘Deformations of Polyhedra and Polygons by the Unitary Group’. In the paper the author inspired by loop quantum gravity, the spinorial formalism and the structures of twisted geometry discusses the phase space of polyhedra in three dimensions and its quantization, which serve as basic building of the kinematical states of discrete geometry.

They show that the Grassmannian space U(N)/(U(N − 2) × SU(2)) is the space of framed convex polyhedra with N faces up to 3d rotations. The framing consists in the additional information of a U(1) phase per face. This provides an extension of the Kapovich-Milson phase space  for polyhedra with fixed number of faces and fixed areas for each face – see the post:

Polyhedra in loop quantum gravity

They describe the Grassmannian as the symplectic quotient C2N//SU(2), which provides canonical complex variables for the Poisson bracket. This construction allows a natural U(N) action on the space of polyhedra, which has two main features. First, U(N) transformations act non-trivially on polyhedra and change the area and shape of each individual face. Second, this action is cyclic: it allows us to go between any two polyhedra with fixed total area  – sum of the areas of the faces.

On quantization, the framed polyhedron phase space leads to the Hilbert space of SU(2) intertwiners, which is interpreted as the space of quantum polyhedra. By performing a canonical quantization from the complex variables of C2N//SU(2) all the classical features are automatically exported to the quantum level. Each face carries now a irreducible representation of SU(2) – a half-integer spin j, which defines the area of the face. Intertwiners are then SU(2)-invariant states in the tensor product of these irreducible representations. These intertwiners are the basic
building block of the spin network states of quantum geometry in loop quantum gravity.

The U(N) action on the space of intertwiners changes the spins of the faces and each Hilbert space for fixed total area defines an irreducible representation of the unitary group U(N). The U(N) action is cyclic and allows us to generate the whole Hilbert space from the action of U(N) transformation on the highest weight
vector. This construction provides coherent intertwiner states peaked on classical polyhedra.

At the classical level, we can use the U(N) structure of the space of polyhedra to compute the averages of polynomial observables over the ensemble of polyhedra and  to use the Itzykson-Zuber formula from matrix models  as a generating functional for these averages. It computes the integral over U(N) of the exponential of the matrix elements of a unitary matrix tensor its complex conjugate.

At the quantum level the character formula, giving the trace of unitary transformations either over the standard basis or the coherent intertwiner basis, provides an extension of the Itzykson-Zuber formula. It allows us in principle to generate the expectation values of all polynomial observables and so their spectrum.

This paper defines and describe the phase space of framed polyhedra, its parameterization in terms of spinor variables and the action of U(N) transformations. Then it shows  how to compute the averages and correlations of polynomial observables using group integrals over U(N) and the Itzykson-Zuber integral as a generating function. It discusses the quantum case, with the Hilbert space
of SU(2) intertwiners, coherent states and the character formula.

The paper also investigates polygons in two dimensions and shows that the unitary group is replaced by the orthogonal group and that
the Grassmannian Ø(N)/(Ø(N −2)×SO(2)) defines the phase space for framed polygons. It then discusses the issue of gluing such polygons together into a consistent 2d cellular decomposition, as a toy model for the gluing of framed polyhedra into 3d discrete manifolds. These constructions are relevant to quantum gravity in 2+1 and 3+1 dimensions, especially to discrete approaches based on a description of the geometry using glued polygons and polyhedra such as loop quantum gravity  and dynamical

The paper’s goal is to clarify how to parametrize the set of polygons or polyhedra and their deformations, and to introduce mathematical tools to compute the average and correlations of observables over the ensemble of polygons or polyhedra at the classical level and then the spectrum and expectation values of geometrical operators on
the space of quantum polygons or polyhedra at the quantum level.

Related posts


Polyhedra in spacetime from null vectors by Neiman

This week I have been studying a nice paper about Polyhedra in spacetime.

The paper considers convex spacelike polyhedra oriented in Minkowski space. These are classical analogues of spinfoam intertwiners. There is  a parametrization of these shapes using null face normals. This construction is dimension-independent and in 3+1d, it provides the spacetime picture behind the property of the loop quantum gravity intertwiner space in spinor form that the closure constraint is always satisfied after some  SL(2,C) rotation.These  variables can be incorporated in a 4-simplex action that reproduces the large-spin behaviour of the Barrett–Crane vertex amplitude.

In loop quantum gravity and in spinfoam models, convex polyhedra are fundamental objects. Specifically, the intertwiners between rotation-group representations that feature in these theories can be viewed as the quantum versions of convex polyhedra. This makes the parametrization of such shapes a subject of interest for  LQG.
In kinematical LQG, one deals with the SU(2) intertwiners, which correspond to 3d polyhedra in a local 3d Euclidean frame. These polyhedra are naturally parametrized in terms of area-normal vectors: each face i is associated with a vector xi, such that its norm
equals the face area Ai, and its direction is orthogonal to the face. The area normals must satisfy a ‘closure constraint’:


Minkowski’s reconstruction theorem guarantees a one-to-one correspondence between space-spanning sets of vectors xi that satisfy (1) and convex polyhedra with a spatial orientation. In
LQG, the vectors xi correspond to the SU(2) fluxes. The closure condition  then encodes the Gauss constraint, which also generates spatial rotations of the polyhedron.

In the EPRL/FK spinfoam, the SU(2) intertwiners get lifted into SL(2,C) and are acted on by SL(2,C) ,Lorentz, rotations. Geometrically, this endows the polyhedra with an orientation in the local 3+1d Minkowski frame of a spinfoam vertex. The polyhedron’s
orientation is now correlated with those of the other polyhedra surrounding the vertex, so that together they define a generalized 4-polytope. In analogy with the spatial case, a polyhedron with spacetime orientation can be parametrized by a set of area-normal
simple bivectors Bi. In addition to closure, these bivectors must also satisfy a cross-simplicity


In this paper, the author presents a different parametrization of convex spacelike polyhedra with spacetime orientation. Instead of bivectors Bi, they associate null vectors i to the polyhedron’s
faces. This parametrization does not require any constraints between the variables on different faces. It is unusual in that both the area and the full orientation of each face are functions of the data on all the faces. This construction, like the area-vector and
area-bivector constructions above, is dimension-independent. So we can parametrize d-dimensional convex spacelike polytopes with (d − 1)-dimensional faces, oriented in a (d + 1)-dimensional Minkowski spacetime.  These variables can be to construct an action principle for a Lorentzian 4-simplex. The action principle reproduces the large spin behaviour of the Barrett–Crane spinfoam vertex. In particular, it recovers the Regge action for the classical simplicial gravity, up to a possible sign and the existence of additional,degenerate solutions.

In d = 2, 3 spatial dimensions, the parametrization is  contained in the spinor-based description of the LQG intertwiners. There, the face normals are constructed as squares of spinors. It was observed that the closure constraint in these variables can always be satisfied by acting on the spinors with an SL(2,C) boost.  The simple spacetime picture presented in this paper is new. Hopefully, it will contribute to the geometric interpretation of the modern spinor and twistor variables in LQG.

The parametrization
Consider a set of N null vectors liμ in the (d + 1)-dimensional Minkowski space Rd,1, where i = 1, 2, . . . ,N and d ≥2.  Assume the following conditions on the null vectors liμ.

  • The liμ span the Minkowski space and  N ≥  d + 1.
  • The  liμ  are either all future-pointing or all past-pointing.

The central observation in this paper is that such sets of null vectors are in one-to-one correspondence with convex d-dimensional spacelike polytopes oriented in Rd,1.

Constructing the polytrope

Consider a set {liμ} ,take the sum of the liμ normalized
to unit length:


The unit vector nμ is timelike, with the same time orientation as the liμ. Now take nμ to be the unit normal to the spacelike polytope. To construct the polytope in the spacelike hyperplane ∑ orthogonal to nμ define the projections of the null vectors liμinto this hyperplane:


The spacelike vectors siμ  automatically sum up to zero. Also, since the liμ span the spacetime, the siμ must span the hyperplane ∑ . By the Minkowski reconstruction theorem, it follows that the siμ are the (d − 1)-area normals of a unique convex d-dimensional polytope in . In this way, the null vectors li define a d-polytope oriented in spacetime.

Basic features of the parametrization.

The vectors  are liμ  associated to the polytope’s (d −1)-dimensional faces and are null normals to these faces. The orientation of a spacelike (d − 1)-plane in Rd,1 is in one-to-one correspondence with the directions of its two null normals. So each liμ carries partial information about the orientation of the ith face. The second null normal to the face is a function of all the liμ. It can be expressed as:


where  nμ is given by


Similarly, the area Ai of each face is a function of the
null normals liμ to all the faces:


The total area of the faces has the simple expression:


A (d+1)-simplex action

To construct a (d + 1)-simplex action that reproduces in the d = 3 case the large-spin behaviour of the Barrett–Crane spinfoam vertex.

At the level of degree-of-freedom counting, the shape of a (d +1)-simplex is determined by the (d + 1)(d + 2)/2 areas Aab of its (d − 1)-faces. These areas are directly analogous to the spins that appear in the Barrett–Crane spinfoam. Let us fix a set of values for Aab and consider the action:


Then restrict to the variations where:


The stationary points of the action  have the following properties. For each a, the vectors  labμ define a d-simplex with unit normal naμ
and (d − 1)-face areas Aab.


A (d − 1)-face in a (d + 1)-simplex, shared by two d-simplices a and b. The diagram depicts the 1+1d plane orthogonal to the face. The dashed lines are the two null rays in this normal plane.


The d-simplices automatically agree on the areas of their shared (d −1)-faces. The two d-simplices agree not only on the area of their shared (d − 1)-face, but also on the orientation of its (d − 1)-plane in spacetime. In other words, they agree on the face’s area-normal bivector:


The area bivectors defined  automatically satisfy closure and cross-simplicity:


We conclude that the stationary points are in one-to-one correspondence with the bivector geometries of the Barrett-Crane model with an action of the form:



Numerical work with sagemath 25: The Wigner D matrix

This week I have been studying the paper ‘A New Realisation of Quantum Geometry’. I’ll review the paper in the next post and follow up that with an analysis of the area operator in  the SU(2) case.

What I’m posting at the moment is some exploratory work looking at the behaviour of the Wigner D matrix elements and U(1)area operator using python and sagemath.

Below I look at the general behaviour of the Wigner D matrix elements:



Below  Here I  look at the behaviour of the spectrum of the area operator with U(1).



Below I look at the how the area varies with μ for the values 0.1, 0.3, 0.5:



The geometry of the tetrahedron and asymptotics of the 6j-symbol

This week I have been studying a useful PhD thesis ‘Asymptotics of quantum spin networks‘ by van der Veen. In this post I’ll look a the section dealing with the geometry of the tetrahedron.

Spin networks and their evaluations

In the special case of the tetrahedron graph the author presents a complete solution to the rationality property of the generating series of all evaluations with a fixed underlying graph using the combinatorics of the chromatic evaluation of a spin network, and a complete study of the asymptotics of the  6j-symbols in all cases: Euclidean, Plane or Minkowskian ,using the theory of Borel transform.


 The three realizations of the dual metric tetrahedron depending on the sign of the Cayley-Menger determinant det(C). Also shown are the two exponential growth factors Λ ± of the 6j-symbol.

The author also computes the asymptotic expansions for all possible colorings,  including the degenerate and non-physical cases. They find that  quantities in the asymptotic expansion can be expressed as geometric properties of the dual tetrahedron graph interpreted as a metric polyhedron.

letting ( Γ,γ ) denote the tetrahedral spin network colored by an admissible coloring . Consider the planar dual graph which is also a tetrahedron with the same labelling on the edges. Regarding the labels of as edge lengths one may ask whether ( Γ,γ ) can be realized as a metric Euclidean tetrahedron with these edge lengths. The admissibility of ( Γ,γ ) implies that all faces of  the dual tetrahedron satisfy the triangle inequality. A well-known theorem of metric geometry implies that ( Γ,γ ) can be realized in exactly one of three flat geometries

(a) Euclidean 3-dimensional space R³
(b) Minkowskian space R²’¹
(c) Plane Euclidean R².

Which of the above applies  is decided by the sign of the Cayley-Menger determinant det(C) which is a degree six polynomial in the six edge labels.

Its assumed that ( Γ,γ ) is nondegenerate in the sense that all faces of are two dimensional. In the degenerate case, the evaluation of 6-symbol is a ratio of factorials, whose asymptotics are easily obtained from Stirling’s formula.

The geometry of the tetrahedron

The asymptotics of the 6j-symbols are related to the geometry of the planar dual tetrahedron. The 6j-symbol is a tetrahedral spin network ( Γ,γ ) admissibly labeled as shown below:


with  γ= (a, b, c, d, e, f). Its dual tetrahedron  ( Γ,γ ) is also labelled by  γ. The tetrahedron and its planar dual, together with an ordering of the vertices and a colouring of the edges of the dual is depicted below. When a more systematic notation for the edge labels is needed  they will be denoted them by dij it follows that

(a, b, c, d, e, f) = (d12, d23, d14, d34, d13, d24)


We can interpret the labels of the dual tetrahedron as edge lengths in a suitable flat geometry. A condition that allows one to realize ( Γ,γ )  in a flat metric space such that the edge lengths equal the edge labels. Labeling the vertices of ( Γ,γ ). We can formulate such a condition in terms of the Cayley-Menger determinant. This is
a homogeneous polynomial of degree 3 in the six variables a2, . . . , f2. A  definition of the determinant is:

Given numbers dij we can define the  Cayley-Menger matrix by

Cij = 1−d²ij/2 for i, j ≥1


Cij = sgn(i−j) for when i = 0 or j = 0.

In terms of the coloring = (a, b, c, d, e, f) of a tetrahedron, we have:


The sign of the Cayley-Menger determinant determines in what space the tetrahedron can be realized such that the edge labels equal the edge lengths

  • If det(C) > 0 then the tetrahedron is realized in Euclidean space R³.
  •  If det(C) = 0 then the tetrahedron is realized in the Euclidean plane R².
  • If det(C) < 0 then the tetrahedron is realized in Minkowski space R²’¹.

In each case the volume of the tetrahedron is given by


The 6 dimensional space of non-degenerate tetrahedra consists of regions of Minkowskian and regions of Euclidean tetrahedra. It turns out to be a cone that is made up from one connected component of three dimensional Euclidean tetrahedra and two connected components of Minkowskian tetrahedra. The three dimensional Euclidean and Minkowskian tetrahedra are separated by Plane tetrahedra. The Plane tetrahedra also form two connected components, representatives of which are depicted below:


The tetrahedra in the Plane component that look like a triangle with an interior point are called triangular and the Plane tetrahedra from that other component that look like a quadrangle together with its diagonals are called quadrangular.  The same names are used for the corresponding Minkowskian components.

An integer representative of the triangular Plane tetrahedra is not easy to find as the smallest example is (37, 37, 13, 13, 24, 30).

Let’s look at the dihedral angles of a tetrahedron realized in either of the three above spaces. The cosine and sine of these angles can be expressed in terms of certain minors of the Cayley-Menger
matrix. Define the adjugate matrix ad(C) whose ij entry is (−1)i+j times the determinant of the matrix obtained from C by deleting the i-th row and the j-th column. Define Ciijj to be the matrix obtained from C by deleting both the i-th row and column and the j-th row and column.

The Law of Sines and the Law of Cosines are well-known formulae for a triangle in the Euclidean plane. let’s look at the Law of Sines and the Law of Cosines for a tetrahedron in all three flat geometries.

If we let θkl be the exterior dihedral angle at the opposite edge ij. The following formula is valid for all non-degenerate tetrahedra:


Related articles

GFT Condensates and Cosmology

This week I have been studying some papers and a seminar by Lorenzo Sindoni and Daniele Oriti on Spacetime as a Bose-Einstein Condensate. I have also been reading a great book ‘The Universe in a Helium Droplet’ by Volovik and a really good PhD Thesis, ‘Appearing Out of Nowhere: The Emergence of Spacetime in Quantum Gravity‘ by Karen Crowther – I be posting about these next time.

Spacetime as a Bose-Einstein Condensate has been discussed in a number of other posts including:

Simple condensates

Within the context of  Group Field Theory (GFT), which is a field theory on an auxiliary group manifold. It incorporates many ideas and structures from LQG and spinfoam models in a second quantized language. Spacetime should emerge from the collective dynamics of the microscopic degrees of freedom. Within Condensates all the quanta are in the same state. These simple quantum states of the full theory, can be put in correspondence with Bianchi cosmologies via symmetry reduction at the quantum level. This leads to an effective dynamics for cosmology which makes  contact with LQC and Friedmann  equations.

Group Field Theories the second quantization language for discrete geometry

Group field theories are quantum field theories over a group manifold. The basic defintiion of a GFT is


which can denoted as:


The theory is formulated in terms of a Fock space and Bosonic statistics is used.


Gauge invariance on the right is required, that is:


GFT quanta: spin network vertices  and quantum tetrahedra

Considering D=4 with group G=SU(2). These quanta have a natural interpretation in terms of 4-valent spin-network vertices.



Via a noncommutative Fourier transform it can be formulated in group variables. Considering  SU(2), we have:



We now  have a second quantized theory that creates quantum tetrahedra

gftfig3 represented as gftequ6.

 Correlation functions of GFT and spinfoams

When computing the correlation functions between boundary states the Feynman rules glue tetrahedra into 4-simplices. This is controlled by the combinatorics of the interaction term. This amplitude is designed to match spinfoam amplitudes. For example,
the interaction kernel can be chosen to be the EPRL vertex in a group representation.


The dynamics can be designed to give rise to the transition amplitudes with sum over 4d geometries included using a discrete path integral for gravity.

By  proceeding as in condensed matter physics and we can design
trial states, parametrised by relatively few variables, and deduce from the dynamics of the fundamental model the optimal induced dynamics.

Now we select some trial states to getthe  effective continuum dynamics. We choose trial states that contain the relevant information about the regime that we want to explore. Fock space suggests several interesting possibilities such as field coherent states;


This is a simple state, but not a state with an exact finite number of particles. It is  inspired by the idea that spacetime is a sort of condensate and can be generalized to other states  such assqueezed, and multimode.

The condensates can be naturally interpreted as homogeneous cosmologies:


Elementary quanta possessing the same wavefunction so that  the metric tensor in the frame of the tetrahedron is the same everywhere. This Vertex or wavefunction homogeneity can be interpreted in terms of homogeneous cosmologies, once a
reconstruction procedure into a 3D group manifold has been specified.The reconstruction procedure is based on the idea that each of these tetrahedra is embedded into a background manifold: the edges are aligned with a basis of left invariant vector fields.

Quanta in a Quantum Spacetime

The Frontiers of Fundamental Physics 14 conference again captures my interest this week and I’ve been looking at a paper, ‘ How Many Quanta are there in a Quantum Spacetime?

In this paper the authors develop a technique for describing quantum states of the gravitational field in terms of coarse grained spin networks. They show that the number of nodes and links and the values of the spin depend on the observables chosen for the description of the state. So in order to say how many quanta are in a quantum spacetime further information about what has been measured has to be given.


The electromagnetic field can be viewed as formed by individual photons. This is a consequence of quantum theory. Similarly, quantum theory is likely to imply a granularity of the gravitational field, and therefore a granularity of space.

How many quanta form a macroscopic region of space? This question has implications for the quantum physics of black holes, scattering calculations in non perturbative quantum gravity and quantum cosmology. It is related to the question of the number of nodes representing a macroscopic geometry in a spin network state in loop gravity. In this context, it takes the following form: what is the relation between a state with many nodes and small spins, and a state with few nodes but large spins?

Quanta of space

A quanta of space may be  a quanta of energy from the excitation of the gravitational field. In loop quantum gravity, each quanta is a  quantum polyhedron. The geometry of quantum polyhedron defined by graph. We associate a state or element of Hilbert space for each quanta of space. The basis which spanned this Hilbert space is the spin network basis.

A quantum tetrahedron and its dual space geometry: the graph 

quantum tetrahedron

A graph γ is a finite set N of element n called nodes and a set of L of oriented couples called links l = (n, n’). Each node corresponds to one quantum tetrahedron. Four links pointing out from the node correspond to each triangle of the tetrahedron.

How many quanta in a field?

Consider a free scalar field in a finite box, in a classical configuration φ(x,t). The standard quantum-field-theoretical number operator, which sums the number the quanta on each mode, has a well defined classical limit. The number operator is

quantaequ1where an and an are the annihilation and creation operators for the mode n of the field and the sum is over the modes, namely the Fourier components, of the field. Since the energy can be expressed as a sum over modes as


where ωn is the angular frequency and En its energy of the mode n, it follows that the number of particles is given


which is a well defined classical expression that can be directly obtained from φ(x,t) by computing the energy in each mode. Therefore each classical configuration defines a total particle-number N and a distribution of these particles over the modes


Subset graphs

The state space of loop quantum gravity contains subspaces Hγ associated to abstract graphs γ. A graph γ is defined by a finite set N of |N| elements n called nodes and a set L of |L| oriented couples l = (n, n’) called links.

A pure state |ψ〉 determines the density matrix ργ= |ψ〉〈ψ|. A generic state can be written in the form


In the loop gravity the operators defined on Hγ can be interpreted as the description of the geometry of |N| quantum polyhedra connected to one another when there is a link between the corresponding nodes.

Given a graph , define a subset graph Γ which partitions N into subsets N such that each N is a set of nodes connected among themselves by sequences of links entirely formed by nodes in N.

quantafig1 We define the area of the big link by


and the volume of the big node by


where we recall that v is the expression for the classical volume of a polyhedron. The operators AL and VN commute, so they can be diagonalized together. The quantum numbers of the big areas are half integers JL and the quantum numbers of the volume are VN.

Course graining spin networks

Coarse-graining the entire graph into a graph Γ formed by a single node N with legs b

quantafig2A  set of small links l that are contained in a single large link L.

quantafig3Any general coarse-graining is a combination of collecting nodes and summing links


The geometry of the subset graph

The geometrical interpretation of the coarse grained states in HΓ is that these describe the geometry of connected polyhedra. The partition that defines the subset graph Γ is a coarse-graining of the polyhedra into larger chunks of space. The surfaces that separate these larger chunks of space are labelled by the big links L and are formed by joining the individual faces labelled by the links l in L.

In general, it is clearly not the case that the area AL is equal to the sum of the areas Al of all l in L. However, this is the case if all these faces are parallel and have the same orientation. Similarly, in general, it is clearly not the case that the volume VN is equal to the sum of the volumes Vn for the n in N. However, this is true if in gluing n polyhedra one obtains a at polyhedron with flat faces.

The two operators,


provide a good measure of the failure of the geometry that the state associates to Γto be flat.

To have a good visualization of the coarse-grained geometries, it is helpful to consider the classical picture. In the 4-dimensional theory, the graph is defined at the boundary of a 3-dimensional hypersurface, the spin operator on the links is related to the area operator byquantaequ48a

Given a 3-valent graph with spins operators Jla, Jlb , and Jlc on each link, the dihedral angle between Jlb and Jlc can be obtained from the angle operator, defined by


Applying this operator to the spin network state gives the dihedral angle between Jland Jlc on the internal links lb and lc.


The Regge intrinsic curvature of a discretized manifold is given by the deficit angle on the hinges, the (n -2) dimensional simplices of the n-dimensional simplex. Thus, given a loopgraph with n-external links, the deficit angle for a general n-polytope (n-valent loop graph) is:


Coarse grained area

The boundary of spacetime is a 3-dimensional space. Triangulation on the boundary is defined using flat polyhedra. Every closed, flat, n polyhedron satisfies the closure relation on the node given by


Consider the net of a polyhedron:


Since the interior of polyhedron is at, the closure relation can be written as


Then the area operator on the base is


and we can define the coarse-grained area as:


Thus, for a 2-dimensional surface, we can always think the coarse grained area AL as the area of the base of a polyhedron, while the total sum of area Al is the area around the hat – the area of n triangle which form the net of the polyhedron.

The differences between the coarse-grained and the fine-grained area gives a good measurement on how the space deviates from being flat. It is possible to obtain the explicit relation between the Regge curvature with these area differences in some special cases.

The Regge curvature for a 2-dimensional surface is defined as 2 minus the sum of all dihedral angle surrounding a point of the triangulation, which is n. The Regge curvature as a function of the coarse-grained and the fine-grained area is:


In the classical limit, it is clear that there can be states where ε = 0 or AL = 0. These correspond to geometries where the normals to the facets forming the large surface L are parallel. However, this is only true in the classical limit, namely disregarding Planck scale effects. If we take Planck-scale effects into account, we have the  result that


quantaequ63where n + 1 is the number of facets. Therefore the fine grained area is always strictly larger than the coarse grained area. There is a Planck length square contribution for each additional facet. It is as if there was an irreducible Planck-scale fluctuation in the orientation of the facets.

 Coarse-grained volume


In the same manner as the surface’s coarse-graining, we triangulate a 3-dimensional chunk of space using n symmetric tetrahedra. The Regge curvature is defined by the dihedral angle on the bones of the tetrahedra. Using the volume of one tetrahedron,



we obtain the fine-grained volume, which is


The coarse-grained volume is the volume of the 3-dimensional base, which is the volume of the n-diamond:


so the Regge curvature is


Notice that this is just a classical example. In the quantum picture, adding two quantum tetrahedra does not gives only a triangular bipyramid, it could give other possible geometries which have 6 facets, i.e., a parallelepiped, or a pentagonal-pyramid.


The number of quanta is not an absolute property of a quantum state: it depends on the basis on which the state is expanded. In turn, this depends on the way we are interacting with the system. The quanta of the gravitational field we interact with, are those described by the quantum numbers of coarse-grained operators like AL and V, not the maximally fine-grained ones.

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