# 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.

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