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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Spin-Foams for All Loop Quantum Gravity by Kaminski ,Kisielowski and Lewandowski

This week I return more directly to work connected to the Hamiltonian constraint operator and look at an important paper on the spin-foam formalism.

In this paper the simplicial framework of Engle-Pereira-Rovelli-Livine spin-foam models is generalized to match the diffeomorphism invariant framework of loop quantum gravity. The simplicial
spin-foams are generalized to arbitrary linear 2-cell spin-foams. The resulting framework admits all the spin-network states of loop quantum gravity, not only those defined by triangulations or cubulations.

In particular the notion of embedded spin-foam  allows the consideration of  knotting or linking spin-foam histories. The vertex structure and the vertex amplitude are naturally generalized to arbitrary valency case. The correspondence between all the SU(2) intertwiners and the SU(2)xSU(2) EPRL intertwiners is shown to be 1-1 in the case of the Barbero-Immirzinparameter γ≥ 1.

Recent spin-foam models of the gravitational field

Spin-foams were introduced as histories of quantum spin-network states of loop quantum gravity (LQG) . That idea gave rise to spin foam models  (SFM). The spin-foam model of 3-dimensional gravity is derived from a discretization of the BF action.

For the 4-dimensional gravity there were several approaches, including the Barrett-Crane spinfoam model.  The BC model is mathematically elegant, but  it was shown not to have sufficiently many degrees of freedom to ensure the correct classical limit.

A suitable modification was found by  two teams: (i) Engle, Pereira, Rovelli and Livine and (ii) Freidel and Krasnov who found systematic derivations of a spin-foam model of gravitational field using as the starting point a discretization of the Holst action.  This led to a spin-foam model valid for the Barbero-Immirzi parameter belonging to the interval −1 ≤γ ≤ 1 which is  a promising candidate for a path integral formulation of LQG, The values of the Barbero-Immirzi parameter predicted by various black hole models also belong to this range.

The  aim of the authors of this paper is to  develop  spin-foam models and most importantly the EPRL model so that they can be used to define spin-foam histories of an arbitrary spin network state of LQG. The notion of embedded spin-foam we use, allows to consider knotting or linking spin-foam histories. Since the knots and links may play a role in LQG, it is important to keep these topological degrees of freedom in a spin-foam approach.
The authors also characterize the structure of a general spin-foam vertex. They  encode all the information about the vertex structure in the spin-network induced on the boundary of the neighborhood of
a given vertex. They also define the generalize the EPRL spin-foams  and the Engle-Pereira-Rovelli-Livine intertwiner to general spin foams.

Abstract spin-networks

Given a compact group G, a G-spin-network is a triple ( γ,ρ,ι  ), an oriented, piecewise linear 1-complex (a graph) equipped with two colourings: ρ and  ι  defined below

The colouring  ρ, maps the set (γ) of the 1-cells (edges) in γ into the set of  irreducible representations of G. That is, to every edge e we assign an irreducible representation e defined in the Hilbert space He.

The colouring  ι  maps each vertex v ∈γ(o) (where (γ0) denotes the set of 0-cells of γ) into the subspace

Spin-foams of spin-networks and of spin-network functions

The motivation of the spin-foam approach to LQG is to develop an analog of the Feynman path integral,  the paths, should be suitably defined histories of the spin-network states.

Spin-foams

A foam is an oriented linear 2-cell complex with  boundary. Briefly, each foam  consists of 2-cells (faces), 1-cells (edges), and 0-cells (vertices).

The faces are polygons, their sides are edges, the ends of the edges are vertices. Faces and edges are oriented, and the orientation of an edge is independent of the orientation of the face it is contained in. Each edge is contained in several faces, each vertex is contained in several edges.

Given a foam , a spin-foam κ is defined by introducing two colourings:

To every internal vertex of a spin-foam ( κ,ρ,ι  ) we can naturally assign a number by contracting the invariants  which colour the incoming and outgoing edges and we call it the spin-foam vertex trace at a vertex v.

The spin-foam vertex trace has a clear interpretation in terms of the spin-networks.

Example of  spin-foam evolution

The recent spin-foam models of Barrett-Crane , Engle-Pereira-Rovelli-Livine  and Freidel-Krasnov  fall into a common scheme. Combine it with the assumption that the spin-foams are histories of the embedded spin-network functions defined within the framework
of the diffeomorphism invariant quantization of the theory of connections. The resulting scheme reads as follows:

• First, ignoring the simplicity constraints, one quantizes the theory using the Hilbert space
• Next, the simplicity constraints are suitably quantized to become linear quantum constraints imposed on the elements.
• The amplitude of each spin-foam ( κ,ρ,ι  ) is defined by using the spin-foam trace.

Simple intertwiners

There are three main proposals for the simple intertwiners:

1.  That of Barrett-Crane (BC) corresponding to the Palatini action
2. That of Engle-Pereira-Rovelli-Livine (EPRL) corresponding the Holst action with the value of the Barbero-Immirzi parameter γ= ±1
3. That of Freidel-Krasnov (FK) also corresponding to the Holst action with the value of the Barbero-Immirzi parameter γ= ±1,

Summary

The authors conclusion is that we do not have to reformulate LQG in terms of thebpiecewise linear category and triangulations to match it with the EPRL  SFM. The generalized spinfoam framework is compatible with the original LQG framework and accommodates all the spinnetwork states and the diffeomorphism covariance.

This papers generalization goes in two directions:

The first one is from spin-foams defined on simplicial complexes to spin-foams defined on the arbitrary linear 2-cell complexes. The main tools of the SF models needed to describe 4- dimensional spacetime are available: spin-foam, boundary spin-network, characterization of vertex, vertex amplitude, the scheme of the SF models of 4-dim gravity, the EPRL intertwiners.

The second direction is from abstract spin-foams to embedded spin-foams, histories of the embedded spin-network.2 For example the notion of knots and links is again available in that framework.
One may for example consider a spin-foam history which turns an unknotted embedded circle into a knotted circle.

The most important result is the characterization of a vertex of a generalized spin-foam. The structure of each vertex can be completely encoded in a spin-network induced locally on the boundary of the vertex neighbourhood. The spin-network is used for the natural generalization of the vertex amplitude used in the simplicial spin-foam models. The characterization of the vertices
leads to a general construction of a general spin-foam. The set of all the possible vertices is given by the set of the spin-networks and and set of all spin-foams can be obtained by gluing the vertices
and the “static” spin-foams.

In the literature, the 3j, 6j, 10j and 15j symbols are used extensively in the context of the BC and EPRL intertwiners and amplitudes. From the  point of view of this paper, they are just related to a specific
choice of the basis in the space of intertwiners valid in the case of the simplicial spin-foams and spin-networks.

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Quantum geometry from phase space reduction by Conrady and Freidel

In this paper the authors give an explicit isomorphism between the usual spin network basis and the direct quantization of the reduced phase space of tetrahedra. The main outcome is a formula that describes the space of SU(2) invariant states by an integral over coherent states satisfying the closure constraint exactly, or equivalently, as an integral over the space of classical tetrahedra. This provides an explicit realization of theorems by Guillemin–Sternberg and Hall that describe the commutation of quantization and reduction.This resulted to express the FK spin foam model as an integral over classical tetrahedra and the asymptotics of the vertex amplitude is determined.

Introduction
Recently following the proposal of new spin foam models for gravity a number of important developments have occurred. These models allow the inclusion of a nontrivial Immirzi parameter and have been shown to satisfy two very important consistency requirements:
firstly, in the semiclassical limit they are asymptotically equivalent to the usual Regge discretisation of gravity, independently of the complexity of the underlying cell complex. Secondly, they have been shown to possess SU(2) spin network states as boundary states.

These new developments link with the evolution of the understanding of the simplicity constraints. These simplicity constraints state in any dimension that the bivector used to
contract the curvature tensor in the Einstein action comes from a frame field. In a simplicial context this constraint splits into three classes: there are the face simplicity constraints (among bivectors associated to the same face), the cross–simplicity constraints (among
bivectors associated to faces of the same tetrahedron) and the volume constraints (among opposite bivectors). All these constraints are quadratic, but the volume constraint, because it depends on different tetrahedra, also involves the connection and is therefore extremely hard to quantise. The first major advance came from the work of Baez, Barrett and Crane who showed that it is possible to linearise the volume constraint and replace it by a constraint living only at one tetrahedron. This new constraint is the closure
constraint and the discrete analogue of the Gauss law generating gauge transformations. The second key insight in this direction came from the work of Engle, Pereira and Rovelli who showed that one can linearise the simplicity and cross simplicity constraints, opening
the path toward a new way of constructing spin foam models and allowing the incorporation of the Immirzi parameter.
The purpose of the paper is to show that the new spin foam model can be written explicitly in terms of a sum of amplitudes, where all simplicity constraints are imposed strongly in the path integral, and where all the boundary spin networks have a geometrical interpretation even before taking the semiclassical limit.

Along with the construction of new models there is also a merging of two lines of thought on spin foam models that developed in parallel for a long time and are now  intersecting.

One line of thought, can be traced back to the seminal work of Barbieri – see post: ‘Quantum tetrahedra and simplicial spin networks, who realized that spin network states of loop quantum gravity can be understood by applying the quantization procedure to a collection of geometric tetrahedra in 3 spatial dimensions. This important work suggested that spin network states and quantum gravity may be about quantizing geometric structures. This idea was then quickly applied to the problem of quantization of a geometric simplex in R4, with the result being the construction of the Barrett–Crane model. This line of thought was also developed by Engle, Livine, Pereira, Rovelli. The main problem in this approach is that the geometry associated with the quantum tetrahedra is fuzzy and cannot be resolved sharply due to the noncommutativity of the geometric ‘flux’ operators. This problem prevents a priori a sharp coupling between neighboring 4d building blocks, and that was the main issue with the Barrett-Crane model.

The second line of thought can be traced back to the work of Reisenberger who proposed to think about quantum gravity directly in terms of a path integral approach in which we integrate over classical configurations living on a 2–dimensional spine and where
spin foam models arise from a type of bulk discretization. This line of reasoning is related to the Plebanski reformulation of gravity as a constraint SO(4) BF theory. In this approach it is quite easy to get a consistent gluing condition, however, it is much less trivial to find models which have a natural and simple algebraic expression in terms
of recoupling coefficients and it is even more of a challenge to get SU(2) boundary spin networks as boundary states.

The merging between these two approaches starts with a work of Livine and Speziale who proposed to label spin network states not with the usual intertwiner basis, but with ‘coherent intertwiners’ that are labelled by four vectors whose norm is fixed to be the area
of the faces of the tetrahedron. This has led to a very efficient and geometrical way of deriving the new spin foam models
There is however a caveat – the coherent intertwiner resolves the fuzzyness of the quantum space of a tetrahedron by construction, but the classical configuration no longer satisfies the closure constraint unless one takes the semiclassical limit and it is therefore not geometrical.

The resolution of these problems proposed by the authors in ths paper here is to use coherent states that are associated to the geometrical configuration – see post:’ A semiclassical tetrahedron‘ and relate these ‘geometrical’ states to the usual spin network
states or the coherent intertwiner states.

This paper gives a construction of such states and shows that they form an over-complete basis of the space of four–valent intertwiners and gives the explicit isomorphism between this new basis of states, where the closure constraint is imposed strongly, and the coherent intertwiner basis. The proof of this isomorphism amounts essentially to showing  that ‘quantisation commutes with reduction’ and utilises heavily the work of Guillemin and  Sternberg on geometrical quantisation.

The main formula of the paper expresses the decomposition of the identity in the space of four–valent intertwiners Hj= (Vj1 .· · · .Vj4)as an integral over coherent states satisfying the closure constraint, and so as an integral over the space of classical tetrahedra:

This identity can be applied it to the spin foam models: it allows the formulation the path integrals in terms of variables on which the closure constraints are imposed strongly. These variables are well–suited to analyze the asymptotic large spin behaviour of amplitudes, and are used in this paper to derive the asymptotics of the FK vertex amplitude.

The paper describes in detail the phase space of the classical tetrahedron and shows that it can be obtained as a symplectic quotient. It states the classical part of the Guillemin–Sternberg isomorphism, which is a isomorphism between a constrained phase space divided by the action of the gauge group and the unconstrained phase space divided by the complexification of the group. It also gives a detailed discussion of coherent states and their link with geometrical quantisation and a construction of the isomorphism between the new geometrical basis and the coherent intertwiner basis.This enables the FK. spin foam model to be written in terms of the new tetrahedral states and use this to derive the asymptotics of the vertex amplitude.