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

j_{f} ∈ 1/2 N of SU(2) to the faces, as well as a collection of intertwiners ι_{e} on edges.

The actual sum is given by

where A_{f} , A_{e} and A_{v} 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 j_{f} in the EPRL-FK model are such

that are both also half-integer spins.

The face amplitudes are either

The edge amplitudes A_{e} 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 j_{f}

and intertwiners ι_{e}.

Intertwiners ι_{e} can be given a geometric interpretation in terms of polyhedra in R³. Given a collection of spins j_{1}, . . . j_{n} and vectors n_{1}, . . . n_{n} which close . Can define the coherent polyhedron

The geometric interpretation is that of a polyhedron, with face areas j_{f} and face normals n_{i}. 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 = l_{p}² 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, V_{4} = d_{x}d_{y}d_{z}d_{t}

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 ιj_{1},j_{2},j_{3} is given by:

The vertex amplitude for a Barbero-Immirzi parameter γ < 1 factorizes as A_{v} = A^{+}_{v}A^{–}_{v} 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 j_{l}. There are two distinct stationary and critical points, satisfying the equations.

for all links ab . Using the convention shown below

having fixed g_{0} = 1, the two solutions Σ_{1} and Σ_{2} are

The amplitudes A^{±}_{v }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**

- Discrete
**harmonic functions (johndcook.com)**