# Calculations on Quantum Cuboids and the EPRL-FK path integral for quantum gravity

This week I have been studying a really great paper looking at Quantum Cuboids and path-integral calculations for the EPRL vertex in LQG and also beginning to write some calculational software tools for performing these calculations using Sagemath.

In this work the authors investigate the 4d path integral for Euclidean quantum gravity on a hypercubic lattice, as given by the EPRL-FK model. To tackle the problem, they restrict the path to a set of quantum geometries that reflects the lattice symmetries. In particular, the sum over intertwiners is restricted to quantum cuboids, that is,  coherent intertwiners which describe a cuboidal
geometry in the large-j limit.

Using asymptotic expressions for the vertex amplitude, several interesting properties of the state sum are found.

• The value of coupling constants in the amplitude functions determines whether geometric or non-geometric configurations dominate the path integral.
• There is a critical value of the coupling constant α, which separates two phases.  In one phase the main contribution
comes from very irregular and crumpled states. In the other phase, the dominant contribution comes from a highly regular configuration, which can be interpreted as flat Euclidean space, with small non-geometric perturbations around it.
• States which describe boundary geometry with high
torsion have exponentially suppressed physical norm.

The symmetry-restricted state sum

Will work on a regular hypercubic lattice in 4d. On this lattice consider only states which conform to the lattice symmetry. This is a condition on the intertwiners, which  corresponds to cuboids.
A cuboid is completely determined by its three edge lengths, or equivalently by its three areas.

All internal angles are π/2 , and the condition of regular cuboids on all dual edges of the lattice result in a high degree of symmetries on the labels: The area and hence the spin on each two parallel squares of the lattice which are translations perpendicular to the squares, have to be equal.

The high degree of symmetry will make all quantum geometries flat. The analysis carried out here is therefore not suited for describing local curvature.

Introduction

The plan of the paper is as follows:

• Review of the EPRL-FK spin foam model
• Semiclassical regime of the path integral
• Construction of the quantum cuboid intertwiner
• Full vertex amplitude, in particular describe its asymptotic expression for large spins
• Numerical investigation of the quantum path integral

The spin foam state sum  employed is the Euclidean EPRL-FK model with Barbero-Immirzi parameter γ < 1. The EPRL-FK model is defined on an arbitrary 2-complexes. A 2-complex 􀀀 is determined by its vertices v, its edges e connecting two vertices, and faces f which are bounded by the edges.

The path integral is formulated as a sum over states. A state in this context is given by a collection of spins –  irreducible representations
jf ∈ 1/2 N of SU(2) to the faces, as well as a collection of intertwiners ιe on edges.

The actual sum is given by

where Af , Ae and Av are the face-, edge- and vertex- amplitude functions, depending on the state. The sum has to be carried out over all spins, and over an orthonormal orthonormal basis in the intertwiner space at each edge.

The allowed spins jf in the EPRL-FK model are such
that  are both also half-integer spins.

The face amplitudes are either

The edge amplitudes Ae are usually taken to be equal to 1.

In Sagemath code this looks like:

Coherent intertwiners

In this paper, the space-time manifold used is  M∼ T³×[0, 1] is the product of the 3-torus T3 and a closed interval. The space is compactified toroidally. M is covered by 4d hypercubes, which
form a regular hypercubic lattice H.There is a vertex for each hypercube, and two vertices are connected by an edge whenever two hypercubes intersect ina 3d cube. The faces of 􀀀 are dual to squares in H, on which four hypercubes meet.The geometry will be encoded in the state, by specification of spins jf
and intertwiners ιe.

Intertwiners ιe can be given a geometric interpretation in terms of polyhedra in R³. Given a collection of spins j1, . . . jn and vectors n1, . . . nn which close . Can define the coherent polyhedron

The geometric interpretation is that of a polyhedron, with face areas jf and face normals ni. The closure condition ensures that such a polyhedron exists.

We are interested in the large j-regime of the quantum cuboids. In this limit, these become classical cuboids  which are completely specified by their three areas. Therefore, a
semiclassical configuration is given by an assignment of
areas a = lp² to the squares of the hypercubic lattice.

Denote the four directions in the lattice by x, y, z, t. The areas satisfy

The two constraints which reduce the twisted geometric
configurations to geometric configurations are given by:

For a non-geometric configuration, define the 4-volume of a hypercube as:

Define the four diameters to be:

then we have, V4 = dxdydzdt

We also define the non- geometricity as:

as a measure of the deviation from the constraints.

In sagemath code this looks like:

Quantum Cuboids

We let’s look at  the quantum theory. In the 2-complex, every edge has six faces attached to it, corresponding to the six faces of the cubes. So any intertwiner in the state-sum will be six-valent, and therefore can be described by a coherent polyhedron with six faces. In our setup, we restrict the state-sum to coherent cuboids, or quantum cuboids. A cuboid is characterized by areas on opposite sides of the cuboid being equal, and the respective normals being negatives of one another

The state ιj1,j2,j3 is given by:

The vertex amplitude for a Barbero-Immirzi parameter γ < 1 factorizes as Av = A+vAv with

with the complex action

where, a is the source node of the link l, while b is its target node.

Large j asymptotics
The amplitudes A±v possess an asymptotic expression for large jl. There are two distinct stationary and critical points, satisfying the equations.

for all links ab . Using the convention shown below

having fixed g0 = 1, the two solutions Σ1 and Σ2 are

The amplitudes A±satisfy, in the large j limit,

In the large j-limit, the norm squared of the quantum cuboid states is given by:

For the state sum, in the large-j limit on a regular hypercubic lattice:

In sagemath code this looks like:

Related articles

# 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 :

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: In 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
• For large areas J, they are semi-classical states peaked around the expectation values for the u(N) generators:
• The scalar product between two coherent states is easily computed:
• 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:

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
triangulations.

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

# 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:

# Generating Functionals for Spin Foam Amplitudes by Hnybida

This week I have been reading the PhD thesis ‘Generating Functionals for Spin Foam Amplitudes’ by Jeff Hnybida. This is a few useful topic because the exact evaluation could be useful to study
recursion relations, coarse graining moves, or to perform numerical calculations.

In the various approaches to Quantum Gravity such as Loop Quantum Gravity, Spin Foam Models and Tensor-Group Field theories use invariant tensors on a group, called intertwiners, as the basic building block of transition amplitudes. For the group SU(2) the contraction of these intertwiners in the pattern of a graph produces spin network amplitudes.

In this paper a generating functional for the exact evaluation of a coherent representation of these spin network amplitudes is constructed. This generating functional is defined for arbitrary graphs and depends only on a pair of spinors for each edge. The generating functional is a meromorphic polynomial in the spinor invariants which is determined by the cycle structure of the graph.
The expansion of the spin network generating function is given in terms of a basis of SU(2) intertwiners consisting of the monomials of the holomorphic spinor invariants. This basis, the discrete-coherent basis, is labelled by the degrees of the monomials and is  discrete. It also contains the precise amount of data needed to specify points in the classical space of closed polyhedra.

The focus the paper is on the 4-valent basis, which is the case of interest for Quantum Gravity. Simple relations between the discrete-coherent basis, the orthonormal basis, and the coherent basis are found.

The 4-simplex amplitude in this basis depends on 20 spins and is referred to as the 20j symbol. The 20j symbol is the exact evaluation of the coherent 4-simplex amplitude.

The asymptotic limit of the 20j symbol is found to give a generalization of the Regge action to Twisted Geometry.

3d quantum gravity

A triple of edge vectors meeting at a node must be invariant under the local rotational gauge transformations

There is only one invariant rank three tensor on SU(2) up to normalization: The Wigner 3j symbol or Clebsch-Gordan coefficient. The 3j symbol has the interpretation as a quantum triangle and its three spins correspond to the lengths of its three edges, which close to form a triangle due to the SU(2) invariance. Contracting four 3j symbols in the pattern of a tetrahedron gives the  well-known 6j symbol which is the amplitude for each tetrahedron.

Coherent BF Theory
The coherent intertwiners are  a coherent state representation of the space of invariant tensors on SU(2). The exact evaluations computed later are a result of a special exponentiating property of
coherent states. Each SU(2) coherent state is labelled by a spinor |z 〉,  |z] denotes its contragradient version. Using a bra-ket notation for the spinors

such that given two spinors z and w the two invariants which can be formed by contracting with either epsilon or delta are denoted

The exponentiating property of the coherent states corresponds to the fact that the spin j representation is simply the tensor product of 2j copies of the spinor |z〉⊗2j   . A coherent rank n tensor on SU(2) is therefore the tensor product of n exponentiated spinors.

To make the coherent tensor invariant we group average using the Haar measure

which is the denition of the Livine-Speziale intertwiner.

The coherent 6j symbol is constructed by contracting 4 coherent intertwiners in the pattern of a tetrahedron. Labeling each vertex by i = 1,..,4 and edges by pairs (ij) this amplitude depends on 6 spins jij = jji and 12 spinors |zij 〉≠ |zji〉 where the upper index denotes the vertex and the lower index the connected vertex. The coherent
amplitude in 3d is given by

The asymptotics of the coherent amplitude have been studied extensively, however the actual evaluation of these amplitudes was not known. While the asymptotic analysis is important to check the semi-classical limit, the exact evaluation could be useful to study
recursion relations, coarse graining moves, or to perform numerical calculations.

To obtain the exact evaluation we use a special property of the Haar measure on SU(2) to express the group integrals above as Gaussian integrals. The generating functional is defined as

we are able to compute the Gaussian integrals in above, not just for the tetrahedral graph but for any arbitrary graph. Performing the Gaussian integrals produces a determinant depending purely on the spinors. The determinant can be evaluated in general and can be expressed in terms of loops of the spin network graph.

For example, after integration and evaluating the determinant, the generating functional of the 3-simplex takes the form:

4d quantum gravity

General Relativity in four dimensions is not topological, but it can  be formulated by a constrained four dimensional BF theory. That is if B is constrained to be of the form

for a real tetrad 1-form e then the BF action becomes the Hilbert Palatini action for General Relativity. The aim of the spin foam program is to formulate a discretized version of these constraints that can break the topological invariance of BF theory and give rise
to the local degrees of freedom of gravity.

The advantage of formulating GR as a constrained BF theory is that, instead of quantizing Plebanski’s action, we can instead use the topological nature of BF theory to quantize  the discretized BF action and impose the  discretized constraints at the quantum level.
The first model of this type was proposed by Barret and Crane.
While this is not a quantization of a constrained system in the sense of Dirac it is a quantization of the Gupta-Bleuler type which was realised by Livine and Speziale  and led to corrected versions of the Barret-Crane model by Engle, Livine, Pereira, Rovelli  and by Freidel, Krasnov.

The behaviour of our spin network generating functional under
general coarse graining moves is a simple transformation of the coarse grained action in terms of lattice paths. For a square lattice, the generating functional expressed as sums over loops similar to gives precisely the partition function for the 2d Ising model.

Since the Ising model and its renormalization are very well understood this example could provide a toy model for which one could base a study of the more complicated spin foam renormalization.

Related articles

# The Spinfoam Framework for Quantum Gravity by Livine

This post looks at  the part of Livine’s  great PhD thesis about the quantum tetrahedron.

Below is a table from this thesis looking at the types of structures found in quantum gravity:

I’ll be looking at the quantum tetrahedron, intertwiner and group field structures.

Loop quantum gravity provides a mathematically  rigorous quantisation of  operators for geometric observables such as area and volume and also shows that the spectra of these operators are discrete in Planck units. This leads to a clear picture of discrete quantum geometry.

There is  a straightforward geometric interpretation of spin network states: the edges e of the graph are dual to elementary surfaces whose area is given by the spin j carried to the edge, and the vertices v are dual to elementary chunks of 3d space bounded by those elementary surfaces and whose volume is determined by the intertwiner  living at the vertex.

Spin network states

This interpretation points towards the reconstruction of a discrete geometry dual to the spin network state, with classical polyhedra reconstructed around each vertex whose faces are dual to the edges attached to the vertex and whose exact shape would depend on the explicit intertwiner living at the vertex. This point of view has been particularly developed from the perspective of geometric quantization. It is possible to see intertwiners as quantum polyhedra.

See the post: Polyhedra in loop quantum gravity

In particular, a lot of  work has focused on the interpretation of 4-valent intertwiners as quantum tetrahedron. This point of view has been particularly useful to build spinfoam models as quantized 4-dimensional triangulations

The quantum tetrahedron

In order to  identify intertwiners as quantum polyhedra and spin network states as discrete geometries, we need to be able to build semi-classical intertwiner states whose shape would be peaked on classical polyhedra and then to glue them together in order to build semi-classical spin network states peaked on classical discrete geometries.

There has  been a lot of research work done on developing concepts such as complexifier coherent states introduced by Thiemann, the related holomorphic spin network states  and coherent intertwiner states introduced by Livine Speziale.

Particular recent lines of research which seems to re-unify these works and viewpoints are the twisted geometry framework  and the U(N) framework for intertwiners which actually converge themselves to a unified picture of coherent spin network states as semiclassical discrete geometries . These frameworks are partly inspired from the picture of coherent intertwiners and allow to define explicit variables which control the shape of intertwiners and also parameterize classical polyhedra, thus creating an explicit bridge between the two.

Related articles

# Semiclassical analysis of Loop Quantum Gravity by Claudio Perini

This week I have been reading the PhD thesis Semiclassical analysis of Loop Quantum Gravity by Claudio Perini .

Semiclassical states for quantum gravity

The concept of semiclassical state of geometry is a key ingredient in the semiclassical analysis of LQG. Semiclassical states are kinematical states peaked on a prescribed intrinsic and extrinsic geometry of space. The simplest semiclassical geometry one can consider is the one associated to a single node of a spin-network with given spin labels. The node is labeled by an intertwiner, i.e. an invariant tensor in the tensor product of the representations meeting at the node. However a generic intertwiner does not admit a semiclassical interpretation because expectation values of non-commuting geometric operators acting on the node do not give the correct classical result in the large spin limit. For example, the 4-valent intertwiners defined with the virtual spin do not have the right semiclassical behavior; one has to take a superposition of them with a specific weight in order to construct semiclassical intertwiners.

The Rovelli-Speziale quantum tetrahedron  is an example of semiclassical geometry; there the weight in the linear superposition of virtual links is taken as a Gaussian with phase. The Rovelli-Speziale quantum tetrahedron is actually equivalent to the Livine-Speziale coherent intertwiner with valence 4 more precisely, the former constitutes the asymptotic expansion of the latter for large spins. Coherent intertwiners are as the geometric quantization of the classical phase space associated to the degrees of freedom of a tetrahedron.

In  recent graviton propagator calculations of semiclassical states
associated to a spin-network graph Γ have been  considered. The states used in the definition of semiclassical n-point functions ) are labeled by a spin jo and an angle ξe per link e of the
graph, and for each node a set of unit vectors n, one for each link surrounding that node. Such variables are suggested by the simplicial interpretation of these states: the graph Γ is in fact assumed to be dual to a simplicial decomposition of the spatial manifold, the vectors n are associated to unit-normals to faces of tetrahedra, and the spin jo is the average of the area of a face. Moreover, the simplicial extrinsic curvature is an angle associated to faces shared by tetrahedra and is identified with the label ξe. Therefore, these
states are labeled by an intrinsic and extrinsic simplicial 3-geometry. They are obtained via a superposition over spins of spin-networks having nodes labeled by Livine-Speziale coherent intertwiners.
The coefficients cj of the superposition over spins are given by a Gaussian times a phase as originally proposed by Rovelli.

Such proposal is motivated by the need of having a state peaked both on the area and on the extrinsic angle. The dispersion is chosen to be given by

so that, in the large jo limit, both variables have vanishing relative dispersions. Moreover, a recent result of Freidel and Speziale strengthens the status of these classical labels: they show that the
phase space associated to a graph in LQG can actually be described in terms of the labels (jo , ξe, ne, n′) associated to links of the graph. The states have good semiclassical properties and a clear geometrical
interpretation.

Within the canonical framework, Thiemann and collaborators have strongly advocated the use of complexifier coherent states . Such states are labeled by a graph Γ and by an assignment of a SL(2,C) group element to each of its links. The state is obtained from the gauge-invariant projection of a product over links of modified heat-kernels for the complexification of SU(2). Their peakedness properties have been studied in detail.

Perini’s thesis presents the  proposal of coherent spin-network states: the proposal is to consider the gauge invariant projection of
a product over links of Hall’s heat-kernels for the cotangent bundle of SU(2). The labels of the state are the ones used in Spin Foams: two normals, a spin and an angle for each link of the graph. This set of labels can be written as an element of SL(2,C) per link of the graph. Therefore, these states coincide with Thiemann’s coherent states with the area operator chosen as complexifier, the SL(2,C) labels written in terms of the phase space variables (jo , ξe, ne, n′e) and the heat-kernel time given as a function of jo.

The author shows that, for large jo , coherent spin-networks reduce to the semiclassical states used in the spinfoam framework. In particular that they reproduce a superposition over spins of spin-networks with nodes labeled by Livine-Speziale coherent intertwiners and coefficients cj given by a Gaussian times a phase as originally proposed by Rovelli. This provides a clear interpretation of the geometry these states are peaked on.

Livine-Speziale coherent intertwiners
In ordinary Quantum Mechanics, SU(2) coherent states are defined as the states that minimize the dispersion

of the angular momentum operator J, acting as a generator of rotations on the representation space Hj ≃C2j+1 of the spin j representation of SU(2). On the usual basis |j,m> formed by simultaneous eigenstates of J2 and J3 we have

so the maximal and minimal weight vectors |j,±j> are coherent states. Starting from |j, j〉, the whole set of coherent states is constructed through the group action

One can take a subset of them labelled by unit vectors on the sphere S²:

where n is a unit vector defining a direction on the sphere S² and g(n) a SU(2) group element rotating the direction z ≡ (0, 0, 1) into the direction n. In other words, a coherent states is a state satisfying

For each n there is a U(1) family of coherent states and they are related one another by a phase factor. The choice of this arbitrary phase is equivalent to a section of the Hopf fiber bundles :

S2 ≃ SU(2)/U(1) → SU(2).

Explicitly, denoting n = (sin θ cos φ, sin θ sin φ, cos θ), a possible section is

where m ≡ (sin φ,−cos φ, 0) is a unit vector orthogonal both to z andn. A coherent state can be expanded in the usual basis as

Coherent states are normalized but not orthogonal, and their scalar product is

where A is the area of the geodesic triangle on the sphere S² with vertices z, n`₁` and n`₂`. Furthermore they provide an overcomplete basis for the Hilbert space Hj of the spin j irreducible representation of SU(2), and the resolution of the identity can be written as

with d²n the normalized Lebesgue measure on the sphere S².

The Livine-Speziale coherent intertwiners are naturally defined taking the tensor product of V coherent states (V stands for valence of the node) and projecting onto the gauge-invariant subspace:

Here the projection is implemented by group averaging. They are labeled by V spins j and V unit vectors n. The states |j, n〉o carry enough information to describe a classical geometry associated to the node. Interpret the vectors j,n as normal vectors to triangles, normalized to the areas j of the triangles.

Coherent tetrahedron

The case of 4-valent coherent intertwiners is of particular importance for LQG and especially for Spin Foam Models. In fact it is the lowest valence carrying a non-zero volume and most SFM’s are build over a simplicial triangulation, so that the boundary state space has only 4-valent nodes, dual to tetrahedra. A 4-valent coherent intertwiner with normals satisfying the closure condition can be interpreted as a semiclassical tetrahedron. In fact expectation values of geometric operators associated to a node give the correct classical quantities in the semiclassical regime. This regime is identified with the large spin (large areas) asymptotics. In the following we give some details.

Define n = jn as the normals normalized to the area. In terms of them, the volume (squared) of the tetrahedron is given by the simple relation:

The geometric quantization of these degrees of freedom is based on the identification of generators Ji of SU(2) as quantum operators corresponding to the  ni . This construction gives directly
the same quantum geometry that one finds via a much longer path by quantising the phase space of General Relativity, that is via Loop Quantum Gravity. The squared lengths |ni|² are the SU(2) Casimir
operators C²(j), as in LQG. A quantum tetrahedron with fixed areas lives in the tensor product ⊗Hj .The closure constraint reads:

and imposes that the state of the quantum tetrahedron is invariant under global rotations. The state space of the quantum tetrahedron with given areas is thus the Hilbert space of intertwiners

The operators Ji · Jj are well defined on this space, and so is the operator

Its absolute value |U| can immediately be identified with the quantization of the classical squared volume 36V²,  in agreement with standard LQG results.

To find the angle operators,  introduce the quantities  Jij := Ji+  Jj. Given these quantities, the angle operators θij can be recovered from

The quantum geometry of a tetrahedron is encoded in the operators Ji ², Jij² and U, acting on Ij₁…j`₄`. It is a fact that out of the six independent classical variables parametrizing a tetrahedron, only
five commute in the quantum theory. Indeed while we have

[Jk², Ji · Jj] = 0,

it is easy to see that:

A complete set of commuting operators, in the sense of Dirac, is given by the operators Ji² , J`₁``₂`². In other words, a basis for Ij`₁`…j4 is provided by the eigenvectors of any one of the operators Jij² . We write the corresponding eigenbasis as |j〉ij . These are virtual links here we are introducing them via a geometric quantization of the classical tetrahedron. For instance, the basis |j〉`₁``₂` diagonalises the four triangle areas and the dihedral angle θ`₁``₂` or, equivalently, the area A`₁``₂` of one internal parallelogram. The relation between different basis is  obtained from SU(2) recoupling theory: the matrix describing the change of basis in the space of intertwiners is given by the usual Wigner 6j-symbol,

so that

Notice that from the orthogonality relation of the 6j-symbol,

we have

The states |j〉`₁``₂` are eigenvectors of the five commuting geometrical operators Ji² , J²`₁``₂`    so the average value of the operator corresponding to the sixth classical observable, say J²`₁``₃` is on these states maximally spread. This means that a basis state has undetermined classical geometry or, in other words, is not an eigenstate of the geometry.  Then led to consider superpositions of states to be able to study the semiclassical limit of the geometry. Suitable superpositions could be constructed for instance requiring that they minimise the uncertainty relations between non–commuting observables, such as

States minimising the uncertainty above are usually called coherent states. Coherent intertwiners seem not to verify exactly , but they are such that all relative uncertainties h〈Δ²〉Jij/〈J²ij〉, or equivalently 〈Δθij〉/〈θij〉, vanish in the large scale limit. The limit is defined by taking the limit when all spins involved go uniformly to infinity, namely ji = λki with λ → ∞. Because of these good semiclassical properties we can associate to a coherent intertwiner the geometrical interpretation of a semiclassical tetrahedron; an analogous interpretation should be also valid nodes of higher valence.

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