# 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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# Group field theories generating polyhedral complexes by Thürigen

This week I have been studying recent developments in  Group Field Theories. Group field theories are a generalization of matrix models which provide both a second quantized reformulation of loop quantum gravity as well as generating functions for spin foam models. Other posts looking at this include:

While states in canonical loop quantum gravity are based on graphs with vertices of arbitrary valence, group field theories have been defined so far in a simplicial setting such that states have support only on graphs of fixed valency. This has led to the question whether group field theory can indeed cover the whole state space of loop quantum gravity.

The paper discusses  the combinatorial structure of the complexes generated by the group field theory partition function. These new group field theories strengthen the links between the various quantum gravity approaches and  might also prove useful in the investigation of renormalizability.

The combinatorial structure of group field theory
The common notion of GFT is that of a quantum field theory on group manifolds with a particular kind of non-local interaction vertices. A group field is a function of a Lie group G and the GFT is defined by a partition function

the action is of the form:

The evaluation of expectation values of quantum observables O[φ], leads to a series of Gaussian integrals evaluated
through Wick contraction which are catalogued by Feynman diagrams Γ,

where sym(Γ) are the combinatorial factors related to the automorphism group of the Feynman diagram Γ:

The specific non-locality of each vertex is captured by a boundary graph. In the interaction term in each group field term  can be represented by a graph consisting of a k-valent vertex connected to k univalent vertices. One may further understand the graph as
the boundary  of a two-dimensional complex a with a single internal vertex v. Such a one-vertex two-complex a is called a spin foam atom.

The GFT Feynman diagrams in the perturbative sum have the structure of two complexes because Wick contractions effect bondings of such atoms along patches. The combinatorial
structure of a term in the perturbative sum is then a collection of spin foam atoms, one for each vertex kernel, quotiented by a set of bonding maps, one for each Wick contraction. Because of this construction such a two-complex will be called a spin foam molecule.

The crucial idea to create arbitrary boundary graphs in a more efficient way is to distinguish between virtual and real edges and obtain arbitrary graphs from regular ones by contraction of the
virtual edges.

In terms of these contractions, any spin foam molecule can be obtained from a molecule constructed from labelled regular graphs.

Conclusions
This paper has aimed to  generalization of GFT to be compatible with LQG. It has clarified the combinatorial structure underlying the amplitudes of perturbative GFT using the notion of spin foam atoms and molecules and discussed their possible spacetime interpretation.

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# Group field theory as the 2nd quantization of Loop Quantum Gravity by Daniele Oriti

This week I have been reviewing Daniele Oriti’s work, reading his Frontiers of Fundamental Physics 14 conference  paper – Group field theory: A quantum field theory for the atoms of space  and making notes on an earlier paper, Group field theory as the 2nd quantization of Loop Quantum Gravity. I’m quite interested in Oriti’s work as can be seen in the posts:

Introduction

We know that there exist a one-to-one correspondence between spin foam models and group field theories, in the sense that for any assignment of a spin foam amplitude for a given cellular complex,
there exist a group field theory, specified by a choice of field and action, that reproduces the same amplitude for the GFT Feynman diagram dual to the given cellular complex. Conversely, any given group field theory is also a definition of a spin foam model in that it specifies uniquely the Feynman amplitudes associated to the cellular complexes appearing in its perturbative expansion. Thus group field theories encode the same information and thus
define the same dynamics of quantum geometry as spin foam models.

That group field theories are a second quantized version of loop quantum gravity is shown to be  the result of a straightforward second quantization of spin networks kinematics and dynamics, which allows to map any definition of a canonical
dynamics of spin networks, thus of loop quantum gravity, to a specific group field theory encoding the same content in field-theoretic language. This map is very general and exact, on top of being rather simple. It puts in one-to-one correspondence the Hilbert space of the canonical theory and its associated algebra of quantum observables, including any operator defining the quantum dynamics, with a GFT Fock space of states and algebra of operators  and its dynamics, defined in terms of a classical action and quantum equations for its n-point functions.

GFT is often presented as the 2nd quantized version of LQG. This is true in a precise sense: reformulation of LQG as GFT very general correspondence both kinematical and dynamical. Do not need to pass through Spin Foams . The LQG Spinfoam correspondence is  obtained via GFT. This reformulation provides powerful new tools to address open issues in LQG, including GFT renormalization  and Effective quantum cosmology from GFT condensates.

Group field theory from the Loop Quantum Gravity perspective:a QFT of spin networks

Lets look at the second quantization of spin networks states and the correspondence between loop quantum gravity and group field theory. LQG states or spin network states can be understood as many-particle states analogously to those found in particle physics and condensed matter theory.

As an example consider the tetrahedral graph formed by four vertices and six links joining them pairwise

The group elements Gij are assigned to each link of the graph, with Gij=Gij-1. Assume  gauge invariance at each vertex i of the graph. The basic point is that any loop quantum gravity state can be seen as a linear combination of states describing disconnected open spin network vertices, of arbitrary number, with additional conditions enforcing gluing conditions and encoding the connectivity of the graph.

Spin networks in 2nd quantization

A Fock vacuum is the no-space” (“emptiest”) state |0〉 , this is the LQG vacuum –  the natural background independent, diffeo-invariant vacuum state.

The  2nd quantization of LQG kinematics leads to a definition of quantum fields that is very close to the standard non-relativistic one used in condensed matter theory, and that is fully compatible
with the kinematical scalar product of the canonical theory. In turn, this can be seen as coming directly from the definition of the Hilbert space of a single tetrahedron or more generally a quantum polyhedron.

The single field  quantum is the spin network vertex or tetrahedron – the so called building block of space.

A generic quantum state is anarbitrary collection of spin network vertices including glued ones or tetrahedra including glued ones.

The natural quanta of space in the 2nd quantized language are open spin network vertices. We know from the canonical theory that they carry area and volume information, and know their pre geometric properties  from results in quantum simplicial geometry.

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# Semiclassical states in quantum gravity: Curvature associated to a Voronoi graph by Daz-Polo and Garay

This week I’ returning to a much more fundamental level and reviewing a paper on Voronoi graphs. These are a method of dividing up a space into triangles and my very early work on this blog was looking at random triangulations. This paper outlines an attempt to compute the  curvature of a surface that didn’t work – that’s science, but we can build on that to find a method that does work.

The building blocks of a quantum theory of general relativity are
expected to be discrete structures. Loop quantum gravity is formulated using a basis of spin networkswave functions over oriented graphs with coloured edges. Semiclassical states should,
however, reproduce the classical smooth geometry in the appropriate limits. The question of how to recover a continuous geometry from these discrete structures is, therefore, relevant in this context. The authors explore this problem from a rather general mathematical perspective using  properties of Voronoi graphs to search for their compatible continuous geometries. They test the previously proposed methods for computing the curvature associated to such graphs and analyse the framework in detail  in the light of the
results obtained.

Introduction

General relativity describes the gravitational interaction as a consequence of the curvature of space-time, a 4-dimensional Lorentzian manifold. Given this geometric nature of gravity, it is expected that, when quantizing, a prescription for quantum
geometry would arise based on more fundamental discrete structures, rather than on smooth differential manifolds.

Loop quantum gravity (LQG) is a candidate theory for such a quantization of general relativity. The fundamental objects , thebasis of the kinematical Hilbert space are the so-called spin network states, which are defined as wave functions constructed over oriented coloured graphs. The building blocks of the theory are, therefore, discrete combinatorial structures  – graphs. The theory provides quantum operators with a direct geometric interpretation, areas and volumes, which happen to have discrete eigenvalues, reinforcing the idea of a discrete geometry.

This perspective of considering abstract combinatorial structures as the fundamental objects of the theory is also adopted in other approaches to quantum gravity, such as spin-foam models, causal dynamical triangulations and causal sets . Also, the algebraic quantum gravity approach follows the same spirit of constructing a quantum theory of gravity from an abstract combinatorial structure.

Despite the variety of successful results obtained in LQG, the search for a semiclassical sector of the theory that would connect with the classical description given by general relativity in terms of a smooth manifold is still under research. An interesting question for the description of a semiclassical sector, given the combinatorial nature of the building blocks of the theory, would be whether there
is any correspondence between certain types of graphs and the continuous classical geometries. Tentatively, this would allow for the construction of gravitational coherent states corresponding to solutions of Einstein eld equations. While it is certainly true
that spin networks are a particular basis, and coherent states constructed from them might resemble nothing like a graph, some works seem to indicate that these graphs do actually represent the structure of space-time at the fundamental level.

This raises a very interesting question. How does the transition between a fundamental discrete geometry, encoded in a graph structure, and the continuous geometry we experience in every-day life happen? In particular, how does a one- dimensional structure give rise to 3-dimensional smooth space? A step towards answering these questions could be to think of these graphs as embedded in the
corresponding continuous geometries they represent. However, the situation is rather the opposite, being the smooth continuous structure an effective structure, emerging from the more fundamental discrete one, and not the other way around. Therefore,
a very relevant question to ask would be: Is there any information, contained in the abstract structure of a graph, that determine  the compatible continuous geometries? Can we determine what types of manifolds  a certain graph can be embedded in? One could even go further and ask whether any additional geometric information, like curvature, can be extracted from the very abstract structure of the graph itself. The goal is, therefore, to construct a unique correspondence between the discrete structures given by graphs, which in general do not carry geometric information, and smooth manifolds.

This problem was studied in the context of quantum gravity by Bombelli, Corichi and Winkler, who proposed a statistical method to compute the curvature of the manifold that would be associated to a certain class of graphs, based on Voronoi diagrams, giving a new step towards the semiclassical limit of LQG. Indeed, due to their properties, Voronoi diagrams appear naturally when addressing this kind of problems.They also play an important role in the discrete approach to general relativity provided by Regge calculus.

Voronoi diagrams are generated from a metric manifold and, by construction, contain geometric information from it. What was proposed, however, is to throw away all additional geometric information and to keep only the abstract structure of
the one-dimensional graph that forms the skeleton of the Voronoi diagram. Then, the task is to study if there are any imprints of the original geometry which remain in this abstract graph structure. Although the work is somewhat preliminary and, for the
most part, restricted to 2-dimensional surfacesz, it tackles very interesting questions and explores a novel path towards a semiclassical regime in LQG. The results obtained could also provide a useful tool for the causal dynamical triangulations approach.

Curvature associated to a graph

A Voronoi diagram is constructed in the following way. For a set of points -seeds, on a metric space, each highest-dimensional cell of the Voronoi diagram contains only one seed, and comprises the region of space closer to that one seed than to any of the others. Then, co-dimension n cells are made by sets of points equidistant to n + 1 seeds, e.g., in 2 dimensions, the edges (1-dimensional cells) of the Voronoi diagram are the lines separating two of these regions, and are therefore equidistant to two seeds. In the same way, vertices (0-dimensional cells) are equidistant to three seeds.
Therefore, except in degenerate situations which are avoided by randomly sprinkling the seed, the valence of all vertices in a D-dimensional Voronoi diagram is D + 1. Another interesting property of Voronoi diagrams is that their dual graph is the so- called Delaunay triangulation, whose vertices are the Voronoi seeds. By construction, for a given set of seeds on a metric space the corresponding Voronoi diagram is uniquely defined.

The starting point is to consider a given surface on which we randomly sprinkle a set of points, that will be the seeds
for the Voronoi construction. A Voronoi cell-complex is constructed, containing zero, one, and two-dimensional cells (vertices, edges and faces). We keep, then, the abstract structure of the one-dimensional graph encoded, for instance, in an adjacency matrix. We are, thus, left with an abstract graph.

All vertices of the Voronoi graph are tri-valent. This gives rise
to the following relation between the total number V of vertices in the graph and the total number of edges E:

since every vertex is shared by three edges and every edge contains two vertices. We will also use the definition of the Euler-Poincare characteristic χ in the two-dimensional case

where F is the total number of faces in the graph (that equals the number of seeds). Finally, one can define the number p of sides of a face (its perimeter in the graph). Taking into account that every edge is shared by two faces, the average p over a set of faces satisfies

The following expression for the Euler-Poincare characteristic χ
can be obtained

in terms of the total number of faces F and the average number of sides of the faces p.

On the other hand, if there is a manifold M associated to the Voronoi diagram, this manifold should have the same topology as the diagram. The Gauss-Bonnet theorem can be used then to relate χ with the integral of the curvature over the manifold. If M is a manifold without boundary (like a sphere), the theorem takes the form

where dA is the area measure and R is the Ricci scalar.

Assume that the region of the graph one is looking at is small enough
so that the curvature can be considered constant. In that case

where As is the total surface area of the sphere.

this formula can also be applied to the sphere patch by defining a density of faces  ρ= F=As = Fp =Ap , where Fp and Ap are respectively the number of faces and area of the patch.

Implementation and results

Conclusions and outlook
The problem of reconstructing a continuous geometry starting from a discrete, more fundamental combinatorial structure, like a graph, is interesting for a wide range of research fields. In the case of LQG theory whose Hilbert space is constructed using wave functions defined over graphsthe solution to this problem could provide interesting hints on the construction of semiclassical states, moving toward a connection with classical solutions of the Einstein equations.

In this article the authors discussed and implemented the method proposed  to compute the curvature of a manifold from an abstract Voronoi graph associated to it. By making use of some topological arguments involving the Gauss-Bonnet theorem, a method to statistically compute the curvature in terms of the average number of sides of the faces in the graph is suggested. They tested this
method for the simplest geometries: the unit sphere – constant positive curvature and the plane – zero curvature. They
found highly unsatisfactory results for the value of the curvature in both cases.

# Properties of the Volume Operator in Loop Quantum Gravity by Brunnemann and Rideout

This week I’ve returned more strongly to my work on the numerical spectra of quantum geometrical operators. This post looks at an analysis of the Ashtekar and Lewandowski version of the volume operator.

The authors analyze the spectral properties of the volume operator of Ashtekar and Lewandowski in Loop Quantum Gravity, which is the quantum analogue of the classical volume expression for regions in three dimensional Riemannian space. The analysis also considers generic graph vertices of valence greater than four. The authors find that the geometry of the underlying vertex characterizes the spectral properties of the volume operator, in particular the presence of a volume gap – a smallest non-zero eigenvalue in the spectrum is found to depend on the vertex embedding.

The authors  compute the set of all non-spatially diffeomorphic non-coplanar vertex embeddings for vertices of valence 5–7 and argue that these sets can be used to label spatial diffeomorphism invariant states. it is seen that  gauge invariance connects vertex geometry and the representation properties of the underlying gauge group in a natural way. Analytical results on the spectrum of 4-valent vertices show the presence of a volume gap.

Loop Quantum Gravity (LQG) is a candidate for a quantum theory of gravity. It is an attempt to canonically quantize General Relativity. The resulting quantum theory is formulated as an SU(2) gauge theory.

General Relativity is put into a Hamiltonian formulation by introducing a foliation of four dimensional spacetime into spatial three dimensional hypersurfaces ∑, with the orthogonal timelike
direction parametrized by t.  first class constraints which have to be imposed on the theory so that it obeys the dynamics of Einstein’s equations and is independent of the particular choice of foliation. These constraints are the three spatial diffeomorphism or vector
constraints which generate diffeomorphisms and the so called Hamilton constraint which generates deformations of the hypersurfaces in the t- foliation direction. In addition there are three Gauss constraints due to the introduction of additional SU(2) gauge degrees of freedom.

The theory is then quantized on the kinematical level in terms of holonomies h and electric fluxes E. Kinematical states are defined over collection of edges of embedded graphs. The physical states have to be constructed by imposing the operator version of the constraints on the thus defined kinematical theory.

There are well defined operators in the kinematical quantum theory which correspond to differential geometric objects, such as the length of curves, the area of surfaces, and the volume of regions in the spatial foliation hypersurfaces. All these quantities have discrete spectra, which can be traced back to the compactness of the gauge group SU(2). A coherent state framework address questions on the correct semiclassical limit of the theory.

A central role in investigations is played by the area and the volume operators.The volume operator is a crucial object not only in order to analyze matter coupling to LQG, but also for evaluating the action of the Hamilton constraint operator in order to construct the physical sector of LQG. By construction the spectral properties of the constraint operators are driven by the spectrum of  the volume operator.

Loop Quantum Gravity

Hamiltonian Formulation
In order to cast General Relativity into the Hamiltonian formalism, one has to perform a foliation M ≅ Rx∑ of the four dimensional spacetime manifold M into three dimensional spacelike hypersurfaces with transverse time direction labelled by a foliation parameter t ∈ R. The four dimensional metric g can then be decomposed into the three metric q(x) on the spatial slices ∑ and its extrinsic curvature K, which serve as canonical variables. Introducing new variables due to Ashtekar  may  take
as canonical variables electric fields E and connections A as used in the canonical formulation of Yang Mills theories. Here a, b = 1 . . . 3 denote spatial (tensor) indices, i, j = 1 . . . 3 denote su(2)-
indices.

The occurrence of SU(2) as a gauge group comes from the fact that it is necessary for the coupling of spinorial matter and it is the universal covering group of SO(3) which arises naturally when one

rewrites the three metric q in terms of cotriads as

The pair (A,E) is related to (q,K) as

with Γ being the spin connection.

The pair (A,E) obeys the Poisson bracket:

Constraints
The theory is subject to constraints which arise due to the background independence of General Relativity. It can be treated as suggested by Dirac . There are:

• three vector (spatial diffeomorphism) constraints

•  a scalar – Hamilton constraint

• three Gauss constraints

The Volume Operator
Definition of the Volume Operator

The operator corresponding to the volume V (R) of a spatial region R ⊂ ∑

Acting on gauge invariant spin network states, is defined as

The sum in has to be taken over all vertices v of the underlying graph γ. At each vertex v one has to sum over all possible ordered triples (ei , ej , ek).

Matrix Formulation

Consider,

where Q is a totally antisymmetric matrix with purely real elements. Its eigenvalues λ are purely imaginary and come in pairs λQ = ±iλ. Choosing Z = 1 the volume operator V has the same eigenstates as Q and its eigenvalues λ = |λQ|½

We are left with the task of calculating the spectra of totally antisymmetric real matrices of the form:

Analytical Results on the Gauge Invariant 4-Vertex

The spectrum of the volume operator at a given gauge invariant 4-vertex is simple, that is all its eigenvalues, except zero, come in pairs, and there are no further degeneracies or accumulation points in the spectrum. An explicit expression for the eigenstates of the volume operator in terms of polynomials of its matrix elements and its eigenvalues can be given as:

where,

The specific matrix realization of the volume operator plays a crucial role here since it can be written as an antisymmetric purely imaginary matrix having non-zero entries only on the first off-diagonal. This makes it possible to apply  techniques from orthogonal polynomials and Jacobi-matrices.

# Quantum cosmology of loop quantum gravity condensates: An example by Gielen

This week I have mainly been studying the work done during the Google Summer of Code workshops, in particular that on sagemath knot theory at:

This work looks great and I’ll be using the results in some of my calculations later in the summer.

Another topic I’ve been reviewing is the idea of spacetime as a Bose -Einstein condensate. This together with emergent, entropic and  thermodynamic gravitation seem to be an area into which the quantum tetrahedron approach could naturally fit via statistical mechanics.

In the paper, Quantum cosmology of loop quantum gravity condensates, the author reviews the idea that spatially homogeneous universes can be described in loop quantum gravity as condensates of elementary excitations of space. Their treatment by second-quantised group field theory formalism allows the adaptation of techniques from the description of Bose–Einstein condensates in condensed matter physics. Dynamical equations for the states can be derived directly from the underlying quantum gravity dynamics. The analogue of the Gross–Pitaevskii equation defines an anisotropic  quantum cosmology model, in which the condensate wavefunction becomes a quantum cosmology wavefunction on minisuperspace.

Introduction
The spacetimes relevant for cosmology are to a very good approximation spatially homogeneous. One can use this fact and perform a symmetry reduction of the classical theory – general relativity coupled to a scalar field or other matter – assuming spatial
homogeneity, followed by a quantisation of the reduced system. Inhomogeneities are usually added perturbatively. This leads to models of quantum cosmology which can be studied  without the need for a full theory of quantum gravity.

Loop quantum gravity (LQG) has some of the structures one would expect in a full theory of quantum gravity: kinematical states corresponding to functionals of the Ashtekar–Barbero connection can be rigorously defined, and geometric observables such
as areas and volumes exist as well-defined operators, typically with discrete spectrum. The use of the LQG formalism in quantising symmetry-reduced gravity leads to loop quantum cosmology (LQC).

Because of the well-defined structures of LQG, LQC allows a rigorous analysis of issues that could not be addressed within the Wheeler– DeWitt quantisation used in conventional quantum cosmology, such as a definition of the physical inner product. More recently, LQC has made closer contact with CMB observations, and the usual inflationary scenario is now discussed within LQC.

A new approach towards addressing the issue of how to describe cosmologically relevant universes in loop quantum gravity uses the group field theory (GFT) formalism, itself a second quantisation formulation of the kinematics and dynamics of LQG: one has a Fock space of LQG spin network vertices or tetrahedra, as building blocks of a simplicial complex, annihilated and created by the field operator ϕ and its Hermitian conjugate ϕ†, respectively. The advantage of using this reformulation is that field-theoretic techniques are available, as a GFT is a standard quantum field theory on a curved group manifold. In particular, one can define coherent or squeezed states for the GFT field, analogous to states used in the physics of Bose– Einstein condensates or in quantum optics; these represent quantum gravity condensates. They describe a large number of degrees of freedom of quantum geometry in the same microscopic quantum state, which is the analogue of homogeneity for a differentiable metric geometry. After embedding a condensate of tetrahedra into a smooth manifold representing a spatial hypersurface, one shows that the spatial metric in a fixed frame reconstructed from the quantum state is compatible with spatial homogeneity. As the number of tetrahedra is taken to infinity, a continuum homogeneous metric can be approximated to a better and better degree.

At this stage, the condensate states defined in this way are kinematical. They are gauge-invariant by construction, and represent geometric data invariant under spatial diffeomorphisms. The strategy followed for extracting information about the dynamics of these states is the use of Schwinger–Dyson equations of a given GFT model. These give constraints on the n point functions of the theory evaluated in a given condensate state – approximating a non-perturbative vacuum, which can be translated into differential equations for the condensate wavefunction used in the definition of the state. This is analogous to condensate states in many-body quantum physics, where such an expectation value gives, in the simplest case, the Gross–Pitaevskii equation for the condensate
wavefunction. The truncation of the infinite tower of such equations to the simplest ones is part of the approximations made. The effective dynamical equations obtained can be viewed as defining a quantum cosmology model, with the condensate wavefunction interpreted as a quantum cosmology wavefunction. This provides a general procedure for deriving an effective cosmological dynamics directly from the underlying theory of quantum gravity. It canbe shown that  a particular quantum cosmology equation of this type, in a semiclassical WKB limit and for isotropic universes, reduces to the classical Friedmann equation of homogeneous,
isotropic universes in general relativity.

See posts:

Let’s  analyse more carefully the quantum cosmological models derived from quantum gravity condensate states in GFT. In particular, the formalism identifies the gauge-invariant configuration space of a tetrahedron with the minisuperspace of homogeneous generally anisotropic geometries.

Using a convenient set of variables the gauge-invariant geometric data, can be mapped to the variables of a general anisotropic Bianchi model it is possible to  find simple solutions to the full quantum equation, corresponding to isotropic universes.

They can only satisfy the condition of rapid oscillation of the WKB approximation for large positive values of the coupling μ in the GFT model. For μ < 0, states are sharply peaked on small values for the curvature, describing a condensate of near-flat building blocks, but these do not oscillate. This supports the view that rather than requiring semiclassical behaviour at the Planck scale, semiclassicality should be imposed only on large-scale observables.

From quantum gravity condensates to quantum cosmology

Review the relevant steps in the construction of effective quantum cosmology equations for quantum gravity condensates. Use group field theory (GFT) formalism, which is a second quantisation formulation of loop quantum gravity spin networks of fixed valency, or their dual interpretation as simplicial geometries.

The basic structures of the GFT formalism in four dimensions are a complex-valued field ϕ : G⁴ → C, satisfying a gauge invariance property

and the basic non-relativistic commutation relations imposed in the quantum theory

These relations  are analogous to those of non-relativistic scalar field theory, where the mode expansion of the field operator defines annihilation operators.

In GFT, the domain of the field is four copies of a Lie group G, interpreted as the local gauge group of gravity, which can be taken to be G = Spin(4) for Riemannian and G = SL(2,C) for Lorentzian models. In loop quantum gravity, the gauge group is the one given by the classical Ashtekar–Barbero formulation, G = SU(2). This property encodes invariance under gauge transformations acting on spin network vertices.

The Fock vacuum |Ø〉 is analogous to the diffeomorphism-invariant Ashtekar–Lewandowski vacuum of LQG, with zero expectation value for all area or volume operators. The conjugate  ϕ acting on the Fock vacuum |Ø〉  creates a GFT particle, interpreted as a 4-valent spin network vertex or a dual tetrahedron:

The geometric data attached to this tetrahedron, four group elements gI ∈ G, is interpreted as parallel transports of a gravitational connection along links dual to the four faces. The LQG interpretation of this is that of a state that fixes the parallel transports of the Ashtekar–Barbero connection to be gI along the four links given by the spin network, while they are undetermined everywhere else.

In the canonical formalism of Ashtekar and Barbero, the canonically conjugate variable to the connection is a densitised inverse triad, with dimensions of area, that encodes the spatial metric. The GFT formalism can be translated into this momentum space formulation by use of a non-commutative Fourier transform

The geometric interpretation of the variables B ∈ g is as geometric bivectors associated to a spatial triad e, defined by the integral over a face △ of the tetrahedron. Hence, the one-particle state

Defines a tetrahedron with minimal uncertainty in the fluxes, that is the oriented area elements given by B . In the LQG interpretation this state completely determines the metric variables for one tetrahedron, while being independent of all other degrees of freedom of geometry in a spatial hypersurface.

The idea of quantum gravity condensates is to use many excitations over the Fock space vacuum all in the same microscopic configuration, to better and better approximate a smooth homogeneous metric or connection, as a many-particle state can contain information about the connection and the metric at many different points in space. Choosing this information such that it is compatible with a spatially homogeneous metric while leaving the particle number N free, the limit N → ∞ corresponds to a continuum limit in which a homogeneous metric geometry is recovered.

In the simplest case, the definition for GFT condensate states is

where N(σ) is a normalisation factor. The exponential creates a coherent configuration of many building blocks of geometry. At fixed particle number N, a state of the form σⁿ|Ø〉 would be interpreted as defining a metric (or connection) that looks spatially homogeneous when measured at the N positions of the tetrahedra, given an embedding into space usually there is a sum over all possible particle numbers. The condensate picture does not use a fixed graph or discretisation of space.

The GFT condensate is defined in terms of a wavefunction on G⁴
invariant under separate left and right actions of G on G⁴ . The strategy is then to demand that the condensate solves the GFT quantum dynamics, expressed in terms of the Schwinger–Dyson equations which relate different n-point functions for the condensate. An important approximation is to only consider the simplest Schwinger– Dyson equations, which will give equations of the form

This is analogous to the case of the Bose–Einstein condensate where the simplest equation of this typegives the Gross–Pitaevskii equation.

In the case of a real condensate, the condensate wavefunction Ψ (x), corresponding to a nonzero expectation value of the field operator, has a direct physical interpretation: expressing it in terms of amplitude and phase,  one can rewrite the
Gross–Pitaevskii equation to discover that ρ(x) and v(x) = ∇θ(x) satisfy hydrodynamic equations in which they correspond to the density and the velocity of the quantum fluid defined by the condensate. Microscopic quantum variables and macroscopic classical variables are directly related.

The wavefunction σ or ξ of the GFT condensate should play a similar role. It is not just a function of the geometric data for a single tetrahedron, but equivalently a function on a minisuperspace of spatially homogeneous universes. The effective dynamics for it, extracted from the fundamental quantum gravity dynamics given by a GFT model, can then be interpreted as a quantum cosmology model.

Minisuperspace – gauge-invariant configuration space of a tetrahedron

Condensate states are determined by a wavefunction σ, which is
a complex-valued function on the space of four group elements for given gauge group G which is invariant under

is a function on G\G⁴/G. This quotient space is a smooth manifold
with boundary, without a group structure. It is the gauge-invariant configuration space of the geometric data associated to a tetrahedron. When the effective quantum dynamics of GFT condensate states is reinterpreted as quantum cosmology equations, G\G⁴/G becomes a minisuperspace of spatially homogeneous geometries.

Conclusion
Condensate states in group field theory can be used to derive effective quantum cosmology models directly from the dynamics of a quantum theory of discrete geometries. This can be illustrated by the interpretation of the configuration space of gauge-invariant geometric data of a tetrahedron, the domain of the condensate
wavefunction, as a minisuperspace of spatially homogeneous 3-metrics.

I’ll also looking at the calculations behind this paper in more detail in a later post.

# Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint by Alesci, Thiemann, and Zipfel

This week I have been continuing my work on the Hamiltonian constraint in Loop Quantum Gravity,  The main paper I’ve been studying this week is ‘Linking covariant and canonical LQG: new solutions to the Euclidean Scalar Constraint’. Fortunately enough linking   covariant and canonical LQG was also the topic of a recent seminar by Zipfel in the ilqgs spring program.

The authors of this paper emphasize that spin-foam models could realize a projection on the physical Hilbert space of canonical Loop Quantum Gravity (LQG). As a test  the authors analyze the one-vertex expansion of a simple Euclidean spin-foam. They find that for fixed Barbero-Immirzi parameter γ= 1 the one vertex-amplitude in the KKL prescription annihilates the Euclidean Hamiltonian constraint of LQG. Since for γ = 1 the Lorentzian part of the Hamiltonian constraint does not contribute this gives rise to new solutions of the Euclidean theory. Furthermore, they fi nd that the new states only depend on the diagonal matrix elements of the volume. This seems to be a generic property when applying the spin-foam projector.

To circumvent the problems of the canonical theory, a
covariant formulation of Quantum Gravity, the so-called spin-foam model was introduced. This model is mainly based on the observation that the Holst action for GR  de fines  a constrained BF-theory. The strategy is first to quantize discrete BF-theory and then to implement the so called simplicity constraints. The main building block of the model is a linear two-complex  κ embedded into 4-dimensional space-time M whose boundary is given by an initial and final gauge invariant spin-network, Ψi respectively Ψf , living on the initial respectively final spatial hyper surface of a
foliation of M. The physical information is encoded in the spin-foam amplitude.

where Af , Ae and Av are the amplitudes associated to the internal faces, edges and vertices of  κ and B contains the boundary amplitudes.

Each spin-foam can be thought of as generalized
Feynman diagram contributing to the transition amplitude from an ingoing spin-network to an outgoing spin-network. By summing over all possible two-complexes one obtains the complete transition amplitude between ψi and ψf .

The main idea in this paper is that if f spin-foams provide a rigging map  the physical inner product would be given by

and the rigging map would correspond  to

Since all constraints are satis ed in Hphys the physical scalar product must obey

for all ψout, ψn ∈ Hkin.

As a test  the authors consider an easy spin-foam amplitude and show that

where  κ is a two-complex with only one internal vertex such that  Φ is a spin-network induced on the boundary of  κ and Hn is the Hamiltonian constraint acting on the node n.

Hamiltonian constraint

The classical Hamiltonian constraint is

where,

The constraint can be split into its Euclidean part H = Tr[F∧e] and Lorentzian part HL = C- H.

Using,

where V is the volume of an arbitrary region  ∑ containing the point x. Smearing the constraints with lapse function N(x) gives

This expression requires a regularization in order to obtain a well-de fined operator on Hkin. Using a triangulation T of the manifold  into elementary tetrahedra with analytic links adapted to the graph Γ of an arbitrary spin-network.

Three non-planar links de fine a tetrahedron . Now  decompose H[N] into a sum of one term per each tetrahedron of the triangulation,

To define  the classical regularized Hamiltonian constraint as,

The connection A  and the curvature are regularized  by the holonomy h in SU (2), where in the fundamental representation m = ½. This gives,

which converges to the Hamiltonian constraint if the triangulation is sufficiently fine.

As seen in the post

This can be generalized with a trace in an arbitrary irreducible representation m leading to

this converges to H[N] as well.

Properties

The important properties of the Euclidean Hamiltonian constraint are;

when acting on a spin-network state, the operator reduces to a sum over terms each acting on individual nodes. Acting on nodes of valence n the operator gives

The Hamiltonian constraint on di ffeomorphism invariant states is independent from the refi nement of the triangulation.

Since the Ashtekar-Lewandowsk volume operator annihilates coplanar nodes and gauge invariant nodes of valence three H does not act on the new nodes – the so called extraordinary nodes.

Action on a trivalent node

To ompute the action of the operator on a trivalent node where all links are outgoing, denote a trivalent node by |n(ji,jj , jk)> ≡ |n3>, whereas ji, jj ,jk are the spins of the adjacent links ei, ej , ek:

To quantize  [N] the holonomies and the volume are replaced by their corresponding operators and the Poisson bracket is replaced by a commutator. Since the volume operator vanishes on a gauge invariant trivalent node  only need to compute;

so, h(m) creates a free index in the m-representation located at the node , making it non-gauge invariant and a new node on the link ek:

so we get,

where the range of the sums over a, b is determined by the Clebsch-Gordan conditions and

The complete action of the operator on a trivalent state |n(ji, jj , jk)> can be obtained by contracting the trace part with εijk. So, H projects on a linear combination of three spin networks which differ by exactly one new link labelled by m between each couple of the oldlinks at the node.

Action on a 4-valent node

The computation for a 4-valent node |n4> is

where i labels the intertwiner – inner link.

The holonomy h(m) changes the valency of the node and the Volume therefore acts on the 5-valent non-gauge invariant node. Graphically this corresponds to

and finally we get:

This can be simpli ed to;

SPIN-FOAM

Using  the de finition of an Euclidean spin-foam models as suggested by Kaminski, Kisielowski and Lewandowski (KKL) and since we,re   only interested in the evaluation of a spin-foam amplitude we choose a combinatorial de finition of the model:

Consider an oriented two-complex  de fined as the union of the set of faces (2-cells) F, edges (1-cells) E and vertices (0-cells) V such that every edge e is a 1-face of at least one face f (e ∈ ∂f) and every vertex V is a 0-face of at least one edge e (v ∈∂e).

We call edges which are contained in more than one face f internal and denote the set of all internal edges by Eint.  All vertices adjacent to more than one internal edge are also called internal and denote the set of these vertices by Vint. The boundary ∂κ is the union of all external vertices (-nodes) n ∉ Vint and external edges (links) l ∉Eint.

A spin-foam is a triple (κ, ρ; I) consisting of a proper foam whose faces are labelled by irreducible representations of a Lie-group G, in this case here SO(4) and whose internal edges are labeled by intertwiners I. This induces a spin-network structure ∂(κ, ρ; I) on the boundary of κ.

The BF partition function can be rewritten as

Av de fines an SO (4) invariant function on the graph Γ induced on the boundary of the vertex v

In the EPRL model  the simplicity constraint is imposed weakly so,

It follows immediately that

defi nes the EPRL vertex amplitude with

Expanding the delta function in terms of spin-network function and integrating over the group elements gives

In order to evaluate the fusion coefficients by graphical calculus it is convenient to work with 3j-symbols instead of Clebsch-Gordan coefficients. When replacing the Clebsch-Gordan coefficients we have to multiply by an overall factor .

Spin-foam projector

Given any couple of ingoing and outgoing kinematical states ψout, ψin, the Physical scalar product can be
formally de fined by

where η is a projector – Rigging map onto the Kernel of the Hamiltonian constraint.

Suppose that the transition amplitude Z

can be expressed in terms of a sum of spin-foams, then

this can be interpreted as a function on the boundary graph ∂κ;

in the EPRL sector.

Restricting the boundary elements h ∈ SU (2)  ⊂ SO (4) then;

where |S>N is a normalized spin-network function on SU(2). This fi nally implies;

NEW SOLUTIONS TO THE EUCLIDEAN HAMILTONIAN CONSTRAINT

Compute new solutions to the Euclidean Hamiltonian constraint by employing spin-foam methods. Show that

in the Euclidean sector with γ= 1 and s = 1, where κ is a 2-complex with only one internal vertex.

Trivalent nodes

Consider the simplest possible case given by an initial and final state  |Θ>, characterized by two trivalent nodes joined by three links:

the only states produced by the Hamiltonian acting on a node, are given by a linear combination of spin-networks that diff er from the original one by the presence of an extraordinary (new) link. In particular the term will be non vanishing only if |s> is of the kind:

The simplest two-complex κ(Θ,s) with only one internal vertex de fining a cobordism between   |Θ> and |s> is a tube  Θ x[0,1] with an additional face between the internal vertex and the new link m,

Since the space of three-valent intertwiners is one-dimensional and all labelings jf are fixed by the states   |Θ> , |s>  the fi rst sum  is trivial.

Γv= s and therefore we have,

where the sign factor is due to the orientation of s, The fusion coefficients contribute four 9j symbols since,

The full amplitude is

and

This yields,

The last two terms are equivalent to the first term when exchanging jk↔ jj. The EPRL spin-foam reduces just to the SU(2) BF amplitude that is just the single 6j  in the first line.

Now using the defi nition of a 9j in terms of three 6j’s,

The 9j’s involved in this expression can be reordered using the permutation symmetries  giving,

the statesare solutions of the Euclidean Hamiltonian constraint

The spin-foam amplitude selects only those terms which depend on the diagonal elements on the volume. This  simplifies the calculation since we do not have to evaluate the volume explicitly.

Four valent nodes

The case with ψ in = ψout = |n4>where

The matrix element <s| |n4>  is non-vanishing if |s>is of the form

Choose again a complex  of the form

The vertex trace  can be evaluated by graphical calculus

The fusion coefficients give two 9j symbols for the two trivalent edges and two 15j- symbols for the two four-valent edges.The fusion coefficients reduce to 1 when γ=1.Taking the scalar product  we obtain,

Taking the scalar product with the Hamiltonian gives,

Summing over a and using the orthogonality relation gives,

The three 6j’s in the two terms defi ne a 9j ,summing over the indexes and b respectively gives:

the final result is,

As for the trivalent vertex the spin-foam amplitude just takes those elements into account which depend on the diagonal Volume elements.

CONCLUSIONS

LQG is grounded on two parallel constructions; the canonical and the covariant ones. In this paper the authors  construct a simple spin-foam amplitude which annihilates the Hamiltonian constraint .They found that in the euclidean sector with signature s = 1 and Barbero-Immirzi parameter γ = 1 the Euclidean Hamiltonian constraint is annihilated by a spin-foam amplitude Z for a simple two-complex with only one internal vertex. The one vertex amplitudes of BF theory are explicit analytic solutions of the Hamiltonian theory.

Also the 6j symbol associated to every face is annihilated by the Euclidean scalar constraint. This is a generalization of the work by Bonzom-Freidel in the context of 3d gravity where they found that the 6jis annihilated by a suitable quantization of the 3d scalar constraint F = 0 . The spin-foam amplitude diagonalizes the Volume.

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