# 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:

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# Curved Polyhedra

This week I have been studying a great paper:  SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry by  Haggard, Han, Kaminski, and Riello. This post looks at the section of the paper on curved tetrahedra. This carries on the literature review started in the following posts:

Curved Tetrahedra

Using critical point equations, together with the interpretation
of the holonomies Hab we can reconstruct a curved-tetrahedral geometry at every vertex of the graph. This post shows  how to recover the tetrahedral geometry from the holonomies in a constructive way.

The key equation is the closure condition

we will focus on the derived equation

where Ob ∈ SO(3) is the vectorial (spin 1) representation of Hb ∈ SU(2). The Ob are interpreted as parallel transports along specific, simple paths on the tetrahedron 1-skeleton.

The ordered composition of all the paths associated to a tetrahedron is equivalent to the trivial path, hence the identity on the right-hand side of the closure equation.

Flatly Embedded Surfaces

Consider a 4-dimensional spacetime (M4,  gαβ), with no torsion, constant curvature λ , and tetrad eαI . In this spacetime, consider a bounded 2-surface s that is flatly embedded in M4, i.e. such that the wedge product of its space- and time-like normal fields, nα and uβ respectively, is preserved by parallel transport on the surface s.45 Then, the holonomy around s of the torsionless spin connection   is given in the spinor representation by:

where the subscript + indicates the self-dual part of an object, as is the area of s, and we have defined uI and nI to be the internal spacelike and timelike normals to the surface. In the future pointing time gauge  and therefore,

where the last two equalities hold in time-gauge.

Finally we obtain in the future pointing time gauge,

and  in the vectorial representation,

Beyond the properties of area, curvature, and orientation, the shape of s is not defined for the moment. We can  further constrain its geometric degrees of freedom  by requiring each vertex of the graph to be identified with the simplest curved geometrical object with four faces, a homogeneously curved tetrahedron.
The cosmological constant or equivalently the curvature is totally free at each face. So this model cannot be considered a quantization of gravity with a fixed-sign cosmological constant: it is
rather a quantization of gravity with a cosmological constant, the sign of which is determined dynamically, and only semiclassically, by the imposed boundary conditions  the external jab and ξab.

Constant Curvature Tetrahedra

The faces of the curved tetrahedron are spherical or hyperbolic triangles, with a radius of curvature equal to . This means that their areas must lie in the interval orrespectively.

The spherical case is no problem, since SU(2) group elements have the right periodicity in their argument. By looking at the deformed SU(2)q representations with  and  , one only finds spins up to |k|/2, which translates into γj ≤ 6π/|Λ|.

The hyperbolic case, on the other hand, is more subtle.  These subtleties can give rise—in certain cases determined by the choice of the spins—to non-standard geometries that extend across the two sheets of the two-sheeted hyperboloid.

Consider the reconstruction at the vertex 5 of 􀀀Γ5. The closure
equation in the vector representation is

and the special side is 24. We will take the base point to be vertex 4. Because all the holonomies are based at vertex 4, all the nb are defined there, which we notate nb(4). However the property of being flatly embedded means having vanishing extrinsic curvature, and so this makes the normal to a face well-defined at any of its points. The faces 1, 2 and 3  contain vertex 4, and this means that n1(4), n2(4), n3(4) can be directly interpreted as normals to their respective faces, while n4(4) is the vector obtained after parallel-transporting n4 from its face to vertex 4, via the edge 24. That is,

where ocb is the vector representation of the holonomy from vertex b to vertex c, along the side cb.

It is possible to give an expression of the cosines of the dihedral angles directly in terms of the holonomies,

which are a sort of normalized, connected two-point functions of the holonomies.

Once we have unambiguously fixed the cosines of the dihedral angles, these can be used to construct the Gram matrix of the tetrahedron

The determinant of ³Gram determines whether the tetrahedron is hyperbolic or spherical. detG > 0 gives spherical geometry whilst
detG < 0 gives hyperbolic geometry, this therefore provides the crucial information

This fixes the sign of the cosmological constant at a given vertex. Consequently, there is no freedom, within a vertex, to change this sign, and a unique correspondence between the spinfoam and geometric data can finally be established. Note that flipping the sign of the cosmological constant does not change the Gram matrix, since it corresponds to flipping all the ±b. This fact is crucial, since it means that sgn () can actually be calculated.

Finally, from the Gram matrix one can fully reconstruct the curved tetrahedron. In practice this amounts to repeatedly applying the spherical (and/or hyperbolic) law of cosines to first calculate the face angles of the tetrahedron and then its side lengths.

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