Tag Archives: Regge calculus

Exact and asymptotic computations of elementary spin networks

This week I have been following up some work which I was introduced to in Dimitri Marinelli’s PhD thesis ‘Single and collective dynamics of discretized geometries’. Essentially this involves the analysis of the volume operator.  This is really exciting for me as it is in my specialist research area –  the numerical analysis of Quantum geometric operators and their spectra. I’ll be following up the literature survey with numerical work in sagemath.

The paper I’ll look at this week is ‘Exact and asymptotic computations of elementary spin networks: classification of the quantum–classical boundaries’ by  Bitencourt, Marzuoli,  Ragni, Anderson and and Aquilanti.

There has been increasing interest to the issues of exact computations and asymptotics of spin networks. The large–entries regimes – semiclassical limits, occur in many areas of physics and in particular in discretization algorithms of applied quantum mechanics.

The authors extend recent work on the basic building block of spin networks, namely the Wigner 6j symbol or Racah coefficient,  by exploiting its self–dual properties and studying it as a function of two discrete variables. This arises from its original definition as an orthogonal angular momentum recoupling matrix. Progress comes
from recognizing its role in the foundation of the modern theory of classical orthogonal polynomials, as extended to include discrete variables. Features of the imaging of various regimes of these orthonormal matrices are made explicit by computational
advances –based on traditional and new recurrence relations– which allow an interpretation of the observed behaviors in terms of an underlying Hamiltonian formulation.

The paper provides a contribution to the understanding of the transition between two extreme modes of the 6j, corresponding to the nearly classical and the fully quantum regimes, by studying the boundary lines – caustics in the plane of the two matrix labels. This analysis marks the evolution of the turning points of relevance for the semiclassical regimes and highlights the key role of the Regge symmetries of the 6j.

Introduction

The diagrammatic tools for spin networks were developed by the Yutsis school and  in connection with applications to discretized models for quantum gravity after Penrose, Ponzano and Regge.

The basic building blocks of all spin networks are the Wigner 6j symbols or Racah coefficients, which are studied here by exploiting their self dual properties and looking at them as functions of two variables. This approach is natural in view of their origin as matrix elements describing recoupling between alternative angular momentum binary coupling schemes, or between alternative hyperspherical harmonics.

Semiclassical and asymptotic views are introduced to describe the dependence on parameters. They originated from the association due to Racah and Wigner to geometrical features, respectively a dihedral angle and the volume of an associated tetrahedron, which is the starting point of the seminal paper by Ponzano and Regge . Their results provided an impressive insight into the functional dependence of angular momentum functions showing a quantum mechanical picture in terms of formulas which describe classical and non–classical discrete wavelike regimes, as well as the transition between them.

The screen: mirror, Piero and Regge symmetries

The 6j symbol becomes the eigenfunction of the Schrodinger–like equation in the variable q, a continuous generalization of j12:

exactequ1

where Ψ(q) is related to

exactequ2

and p² is related with the square of the volume V of the associated tetrahedron.

exactequtetraThe Cayley–Menger determinant permits to calculate the square of the volume of a generic tetrahedron in terms of squares of its edge lengths according to:

exactequ3

The condition for the tetrahedron with fixed edge lengths to exist as a polyhedron in Euclidean 3-space amounts to require V²> 0, while the V²= 0 and V²< 0 cases were associated by Ponzano and Regge to “flat” and nonclassical tetrahedral configurations respectively.

Major insight is provided by plotting both 6js and geometrical functions -volumes, products of face areas – of the associated tetrahedra in a 2-dimensional j12 -􀀀 j23 plane , in whch the square “screen” of allowed ranges of j12 and j23 is used in all the pictures
below.

  •  The mirror symmetry. The appearance of squares of tetrahedron edges entails that the invariance with respect to the exchange J ↔− 􀀀J implies formally j ↔ – 􀀀j 􀀀-1 with respect to the entries of the 6j symbol.
  • Piero line. In general, an exchange of opposite edges of a tetrahedron corresponds to different tetrahedra and different symbols. In Piero formula, there is a term due to this difference that vanishes when any pair of opposite edges are equal.
  • Regge symmetries. The these arises through connection with the projective geometry of the elementary quantum of space, which
    is associated to the polygonal inequalities -triangular and quadrilateral in the 6j case -, which have to be enforced in
    any spin networks.

The basic Regge symmetry can be written in the following form:

exactequ4

The range of both J12 and J23, namely the size of the screen, is given by 2min (J1, J2, J3, J, J1 +ρ , J2 +ρ ,J3 +ρ, J + ρ).

Features of the tetrahedron volume function

Looking at the volume V as a function of  x=J12 and  y=J23 we get the expressions for the xVmaxand yVmax that correspond to the maximum of the volume for a fixed value of x or y:

exactequ5

The plots of these are  called “ridge” curves on the x,y-screen. Each one marks configurations of the associated tetrahedron when two specific pairs of triangular faces are orthogonal. The corresponding values of the volume (xVmax,xand yVmax,y) are

exactequ8F is the area of the triangle with sides a, b and c.Curves corresponding to V = 0, the caustic curves, obey the equations:

exactequ11

exactequfig1

exactequfig1text

 

Symmetric and limiting cases

When some or all the j’s are equal, interesting features appear in the screen. Similarly when some are larger than others.

exactequfig2

Symmetric cases

exactequfig3

 

exactequfig4Limiting cases

We can discuss the caustics of the 3j symbols as the limiting case of the corresponding 6j where three entries are larger than the other ones:

exactequ12

exactequfig5

exactequfig6

exactequfig7

Conclusion

The extensive images of the exactly calculated 6j’s on the square screens illustrate how the caustic curves studied in this paper separate the classical and nonclassical regions, where they show wavelike and evanescent behaviour respectively. Limiting
cases, and in particular those referring to 3j and Wigner’s d matrix elements can be analogously depicted and discussed. Interesting also are the ridge lines, which separate the images in the screen tending to qualitatively different flattening of the quadrilateral,
namely convex in the upper right region, concave in the upper left and lower right ones, and crossed in the lower left region.

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The tetrahedron and its Regge conjugate

This week I have been reading the PhD thesis ‘Single and collective dynamics of discretized geometries’ by Dimitri Marinelli. In this post I’ll look at a small portion about Regge calculus, the  tetrahedron and its Regge conjugate.

Regge Calculus is a dynamical theory of space-time introduced in 1961 by Regge as a discrete approximation for the Einstein theory of gravity. The basic idea is to replace a smooth space-time with a collection of simplices. The collective dynamics of these geometric objects is driven by the Regge action and the dynamical variables are their edge lengths – which play the role of the metric tensor of General Relativity. Simplices are the n-dimensional generalization of triangles and tetrahedra. Regge Calculus inspired and is at the base of almost all the present discretized models for a quantum theory of gravity for at least two reasons:

  • It is a discretized model, so it represents a possible atomistic system typical of quantum systems
  • There is a deep connection between the Regge action, the asymptotic of the 6j symbol and a path integral formulation of gravity.

Let’s see  how the Regge transformation acts on a tetrahedral shape. The formulas

reggeequ4.01

and the association between 6j symbol and an Euclidean tetrahedron tell us that any Regge transformation acts on four edges of a tetrahedron keeping a pair of opposite edges unchanged. The Regge-transformed tetrahedra is called `conjugate’.

Using the Ponzano-Regge formula for the 6j,

reggeequ2.18we can immediately say that the volume of a tetrahedron and that of a Regge transformed one must coincide.

thereom23

tetrahedron

The volume of a tetrahedron is also invariant under the Regge transformation of four consecutive edges.

The volume of a tetrahedron, being a function of six parameters, can be expressed in several ways. For the tetrahedron below:

tetrahedron with dihedral angleThe ‘orientated’ volume reads, 

reggeequ3.2.6

where AABC and AACD are respectively the areas of the triangles ABC and ACD, lAC is the length of the common edge and β is the dihedral angle between these two faces.

The importance of the Regge symmetry is that it constrains the shape dynamics of a single tetrahedron,  it relates different tetrahedra equating their quantum representations and it is the key tool to understand the classical motion of a four-bar linkage mechanical systems and its link to the the quantum dynamics of tetrahedra.

This thesis also contains a section on the Askey scheme which I’ll be following up in future posts:

askey scheme

 

 

 

 

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.

vor fig1

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.

vor fig2

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:

vorequ1

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

vorequ2

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

vorequ3

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

vorequ4

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

vorequ5

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

vorequ6

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.

vorequ7

Implementation and results

vor fig3

vor fig8vor fig9

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.

Learning about Quantum Gravity with a Couple of Nodes by Boria, Garay and Vidotto

This week I have looking at  paper about Quantum gravity on a graph with just two nodes. This has great technical information on performing quantum cosmological calculations which I’ll review in another post. In this post I just want to give a brief qualitative outline of the paper.

Loop Quantum Gravity provides a natural truncation of the infinite degrees of freedom of gravity by setting the theory on a finite graph. In this paper the authors review this procedure  and  present the construction of the canonical theory on a simple graph, formed by only two nodes.

They  review the U(N) framework, which provides a powerful tool for the canonical study of this model, and a formulation of the system based on spinors. They also consider  the covariant theory, which allows the  derivation of the  the model from a more complex formulation.

Why graphs?

Discrete gravity

The essential idea behind the graph truncation can be traced to
Regge calculus, which is based on the idea of approximating spacetime with a triangulation, where the metric is everywhere flat except on the triangles. On a fixed spacelike surface, Regge
calculus induces a discrete 3-geometry defined on a 3d triangulation, where the metric is everywhere flat except on the bones. The two-skeleton of the dual of this 3d cellular decomposition is a graph Γ, obtained by taking a point -a node- inside each cell, and connecting it to the node in an adjacent cell by a link, puncturing the triangle shared by the two cells.

Spin networks

In Loop Quantum Gravity, the spinnetwork basis |Γ, j, ni 〉 is an orthonormal basis that diagonalizes the area and volume operators. The states in this basis are labelled by a graph Γand two quantum numbers coloring it: a spin j at each link  and a volume eigenvalue  at each node n. The  Hilbert space HΓ obtained by considering only the states on the  graph Γis precisely the Hilbert space of an SU(2) Yang-Mills theory on this lattice. Penrose’s spin-geometry theorem connects this Hilbert space with the description of the geometry of the cellular decomposition: states in this Hilbert space admit a geometrical interpretation as a quantum version of the 3-geometry. That is, a Regge 3-geometry defined on a triangulation with dual graph Γ can be approximated by semiclassical state in HΓ.

Holonomies

In the canonical quantization of General Relativity, in order to implement Dirac quantization, it’s convenient to choose the densitized inverse triad Ea –Ashtekar‘s electric field and the Ashtekar-Barbero connection Ai as conjugate variables, and then use the flux of Ei and the the holonomy h as fundamental variables for the quantization. The holonomies can be taken along the links of the graph, and the densitized inverse triad can be smeared over the faces of the triangulation. This connects the holonomy triad variables to the discrete geometry picture.

The common point of these different derivations is 3d coordinate gauge invariance. This invariance is the reason for the use of abstract graphs: it removes the physical meaning of the location of the graph on the manifold. Therefore the graph is just a combinatorial object, that codes the adjacency of the nodes. Each node describe a quantum of space, and the graph describes the relations between different pieces of space. The Hilbert subspaces associated to distinct but topologically equivalent embedded graphs are identified, and each graph space contains the Hilbert spaces of all the subgraphs.

nodefig1

Doing physics with few nodes

The restriction to a fixed triangulation or a fixed graph amounts only to a truncation of the theory, by cutting down the theory to an approximate theory with a finite number of degrees of freedom. Truncations are always needed in quantum field theory, in order to extract numbers from the theory -in appropriate physical
regimes even a low-order approximation can be effective.

Discretizing a continuous geometry by a given graph is nothing but coarse graning the theory. The discreteness introduced by this process is different from the fundamental quantum discreteness
of the theory. The first is the discreteness of the abstract graphs; the later is the discreteness of the spectra of the area and volume operator on each given HΓ.

There are also  two different expansions:

  • The graph expansion obtained by a refinement of the graph, valid at scales smaller that the curvature scale R.
  • The semiclassical expansion,  a large-distance limit on each graph, valid at scales larger that the Planck scale Lp

nodefig2

Interesting physics can arise even by considering a simple graph, with few nodes, and comparing our results with classical discrete gravity. This is true for FLRW cosmologies. It has in fact been proven numerically  that the dynamics of a closed universe, with homogeneous and isotropic geometry, can be captured by 5, 16 and 600 nodes –  the regular triangulations of a 3-sphere.

nodefig3

 On the left, a 4d building block of spacetime and, on the right, the evolution of 5, 16 and 500 of these building block (dashed lines), modelling a closed universe, compared whit the continuous analytic
solution (solid line).

The cosmological interpretation

The most striking example, where this kind of approximation applies, is given by cosmology itself. Modern cosmology is based on the cosmological principle, that says that the dynamics of a homogeneous and isotropic space approximates well our universe. The presence of inhomogeneities can be disregarded at a first order approximation, where we consider the dynamics as described at the scale of the scale factor, namely the size of the universe.

Working with a graph corresponds to choosing how many degrees of freedom to describe. A graph with a single degree of freedom is just one node: in a certain sense, this is the case of usual Loop Quantum Cosmology . To add degrees of freedom, we add nodes and links with a colouring. These further degrees of freedom are a natural way to
describe inhomogeneities and anisotropies , present in our universe.

The easiest thing that can be done is to pass from n = 1 to n = 2 nodes. We choose to connect them by L = 4 links, because in this way the dual graph will be two tetrahedra glued together, and this can be viewed as the triangulation of a 3-sphere .

nodefig4

Choosing a graph corresponding to a triangulation is useful when we want to associate an intuitive interpretation to our model. In order to understand how this can be concretely use to do quantum gravity and quantum cosmology, we need to place on the graph the SU(2) variables.

LQG can describe large semiclassical geometries also over a small number of nodes and links. This paper  reviewed a number of constructions in Loop Quantum Gravity, based on the idea of truncating the Hilbert space of the theory down to the states supported on a simple graph with two nodes.

nodefig5

Below there is a diagram of the solutions found using a 2-node graph. The quantum Hamiltonian of the 2-node model is mathematically analogous to the gravitational part of the Hamiltonian in LQC. Following this analogy, we can interpret the results obtained as the classical analogous of the quantum big bounce found in LQC.

nodefig7

The restriction of the full LQG Hilbert space to a simple graph is a truncation of the degrees of freedom of the full theory. It defines an approximation where concrete calculations can be performed. The approximation is viable in physical situations where only a small number of the degrees of freedom of General Relativity are relevant.

A characteristic example is cosmology. The 2-node graphs with 4 links defines the simplest triangulation of a 3-sphere and can accommodate the anisotropic degrees of freedom of a Bianchi IX model, plus some inhomogeneous degrees of freedom.  In this context, a Bohr-Oppenheimer approximation provides a tool to separate heavy and light degrees of freedom, and extract the
FLRW dynamics. This way of deriving quantum cosmology from LQG is different from the usual one: in standard loop quantum cosmology, the strategy is to start from a symmetry-reduced system, and quantize the single or the few degrees of freedom that survive in the symmetry reduction. In this paper  a truncated version of the full quantum theory of gravity is consided.

A curvature operator for LQG by Alesci, Assanioussi and Lewandowski

In  this paper the authors introduce a new operator in Loop Quantum Gravity – the 3D curvature operator – related to the 3-dimensional scalar curvature. The construction is based on Regge Calculus. It is defined starting from the classical expression of the Regge curvature, they also  derive its properties and discuss  the semiclassical limit.

Introduction

Loop Quantum Gravity is  a theory which aims to give a quantum description of General Relativity. The theory presents two complementary descriptions based on the canonical and the covariant approach – spinfoams.

The canonical approach  implements the Dirac quantization procedure for GR in Ashtekar-Barbero variables formulated in terms of the so called holonomy-flux algebra : it considers smooth manifolds and on those defines a system of paths and dual surfaces over which connection and electric field can be smeared, and then quantizes the system, obtaining the full Hilbert space as the projective limit of the Hilbert space defined on a single graph.

The covariant – spinfoams approach is instead based on the Plebanski formulation of GR, implemented starting from a simplicial decomposition of the manifold, i.e. restricting to piecewise linear flat geometries.

The two formulations share the same kinematics namely the spin-network basis  first introduced by Penrose . In the spinfoam setting then with its piecewise linear nature an interpretation of the spin-networks in terms of quantum polyhedra naturally arise. This interpretation is not needed in the canonical formalism, which deals directly with continuous geometries that in the quantum theory result just in polymeric quantum geometries. However  it has been proven that the discrete classical phase space on a fixed graph of the canonical approach based on the holonomy-flux algebra can be related to the symplectic reduction of the continuous phase space respect to a flatness constraint; this construction allows the  reconciliation of the loop gravity geometrical interpretation in terms of singular geometry, and the spin foam interpretation in terms of piecewise flat geometry, since it can be shown that both geometries belong to the same equivalence class.

The idea developed in this paper is  the following: the Lorentzian term of the Hamiltonian constraint can be seen as the Einstein-Hilbert action in 3d and we know how to write this expression using Regge Calculus in terms of geometrical quantities, i.e. lengths and angles. The length and the angles are available in LQG and Spinfoams as operators.

Regge calculus 

Regge calculus  is a discrete approximation of general relativity which approximates spaces with smooth curvature by piecewise flat spaces: given a n-dimensional Riemannian manifold, considering a simplicial decomposition  approximation  assuming that curvature lies only on the hinges of , namely on its n − 2 simplices. In this context, Regge  derived the simplicial equivalent of the Einstein-Hilbert action:

curveequ1

where the sum extends to all the hinges h with measure Vh and deficit angle ε:

curveequ2

θ is the dihedral angle at the hinge h of the simplex  and the sum extends to all the simplices sharing the hinge h. The coefficient α is the number of simplices sharing the hinge h or twice this number if the hinge is respectively in the bulk, or on the boundary of the triangulation.

The equation: curveequ1

can also be written as:

curveequ3

Which is better adapted to the quantization scheme in this paper:

The purpose of this paper is to define a scalar curvature operator for LQG implementing a regularization of SEH in terms of a simplicial decomposition and  promote this expression to a well defined operator acting on the LQG kinematical Hilbert space. In view of the application to the Lorentzian Hamiltonian constraint of the 4-dimensional theory, we are interested in spaces of dimension n = 3. Therefore the expression we want to quantize is:

curveequ4where L is the length of the hinge h belonging to the simplex s.

To  generalize the classical Regge expression for the integrated scalar curvature.

First introduce the definition of a cellular decomposition: a cellular decomposition C of a space Σ is a disjoint union or partition of open cells of varying dimension satisfying thefollowing conditions:

  1. An n-dimensional open cell is a topological space which is homeomorphic to the n-dimensional open ball.
  2. The boundary of the closure of an n-dimensional cell is contained in a finite union of cells of lower dimension.

In 3-d Regge calculus we consider a simplicial decomposition of a 3-d manifold which is  special cellular decomposition. Using the ∈-cone structure we induce a flat manifold with localized conical defects.Those conical defects lie only on the 1-simplices and encode curvature.The final expression of the integrated scalar curvature in the general case can be written as:curveequ5

where the first sum now is over the 3-cells c and αh is the number of 3-cells sharing the hinge h. This the classical formula that is adopted to express the integrated scalar curvature and it’s the basis of the construction to define a curvature operator.

Start by writing the classical expressions for the length and the dihedral angle in terms of the densitized triad – so called electric field.
Given a curve γ embedded in a 3-manifold Σ:

curveequ10a

the length L(γ) of the curve in terms of the electric field Ei is:

curveequ10

where

curveequ11

where the Ei’s are evaluated at:

curveequ11a

To define the dihedral angle, we consider two surfaces intersecting in the curve γ. The dihedral angle between those two surfaces is then:

curveequ12

Therefore can express Regge action in terms of the densitized triads as follows:

curveequ13

The next step is to match Regge calculus context with LQG framework. This is achieved by invoking the duality between spin-networks and quanta of space that allows us to describe for example spin-networks in terms of quantum polyhedra. The second step is to define a regularization scheme for the classical expressions that we have.

 Spin networks and decomposition of space

In LQG, we define the kinematical Hilbert space H of quantum states as the completion of the linear space of cylindrical functions Ψ(Γ) on all possible graphs Γ. An orthonormal basis in H can be introduced, called the spin-network basis, so that for each graph Γwe can define proper subspace HΓ of H spanned by the spin-network states defined on . Those proper subspaces HΓ are orthogonal to each other and they allow to decompose H as:

curveequ14

A spin-network state is defined as an embedded colored graph denoted |Γ, j, n> where Γ is the graph while the labels j are quantum numbers standing for SU(2) representations (i.e spins) associated to edges, and n are quantum numbers standing for SU(2) intertwiners associated to nodes.

curvefig1

For each spin-network graph define a covering cellular decomposition as follows. A cellular decomposition C of a three-dimensional space ∑ built on a graph Γis said to be a covering cellular decomposition of Γ if:

  1. Each 3-cell of C contains at most one vertex of Γ ;
  2. Each 2-cell or face of C is punctured at most by one edge of  Γand the intersection belongs to the interior of the edge;
  3. Two 3-cells of C are glued such that the identified 2-cells match.
  4. If two 2-cells on the boundary of a 3-cell intersect, then their intersection is a connected 1-cell.

Having such a decomposition we can use it to write the classical expression and promote it to an operator through the quantization of the length and the dihedral angle separately.

The length operator

The  approach used to construct a scalar curvature operator in this paper uses the dual picture, therefore Bianchi’s length operator  which is constructed based on the same dual picture of quantum geometry is chosen for this task. See the post:

Bianchi’s length operator  is:

curveequ20

 

The index ω = (n, e1, e2) stands for a wedge -two edges e1 and e2 intersecting in a node n- in the graph Γ dual to the two faces intersecting in the curve γ. While Y γω) and V are respectively the two-handed operator and the volume operator:

curveequ21The dihedral angle operator

Considering a partition that decomposes a region R delimited by
two surfaces S1 and S2 intersecting in γ, we get the following expression:

curveequ24

On the quantum level, the fluxes are just the SU(2) generators ~ J associated to the edges of the spin-network. Therefore we can write a simple expression for the dihedral angle operator θik in the conventional intertwiner basis:

curveequ26

Where i and k label the two intersecting edges forming the wedge ω dual to the two faces intersecting in the curve γ. The numbers ji, jk and jik are respectively the values of the spins i, j and their coupling.

The curvature operator

We focus on a small region which contains only one hinge of the decomposition. In this paper  the curvature is written as a combination of the length of this hinge and the deficit angle around it.

Define a quantum curvature operator Rc as:

curveequ28

This operator is the quantum analog of the classical expressioncurveequ28a . It is hermitian and depends on the choice of C. We can define an operator  Rc  representing the action  in the region contained in the 3-cell c:

curveequ29

Now evaluate the action of the operator Rc on a  function Ψ, a 3-cell c either contains one node of Γ or no node at all. If it does not contain a node we have;

curveequ30

If the 3-cell does contain a node, say n, then

curveequ31

where ω labels the wedges containing the node n and selected by the 3-cell c. Introducing the coefficient κ(c, ωn) which is equal to 1 when the wedge is selected by the 3-cell c and 0 otherwise. Then

curveequ32

The action of Rc on Ψ is then:

curveequ33

The action of the operator Rc depends on the 3-cells containing the nodes of Γand the cells glued to them . Hence, it can be written:

curveequ34

 

We can express the action of the final curvature operator R which does not depend on the decomposition any more as:

curveequ37

 

Spectrum of the curvature operator

The case of a four valent node with all spins equal, j1 = j2 = j3 = j4 = jo, then for the geometry dual to a loop of three four-valent nodes with equal internal spins labeling the links forming the loop and equal external spins is shown below:

curvefig5

 

Semi-classical properties

In this case the semi-classical limit, large spins limit, does not mean the continuous limit but rather a discrete limit which is classical Regge calculus.Below are shown the expectation values of the curvature operator on Livine-Speziale coherent states  in the case of a regular four-valent node as a function of the spin jo.

curvefig6

Below is shown the expectation values of the curvature operator on Rovelli-Speziale semi-classical tetrahedra as a function of the spin in the case of a regular four-valent node  and for the internal geometry in the case of three four-valent nodes with equal internal spins and equal external spins .

The Rovelli-Speziale semi-classical tetrahedron is a semiclassical quantum state corresponding to the classical geometry of the tetrahedron determined by the areas A1, . . . ,A4 of its faces and
two dihedral angles θ12, θ34 between A1 and A2 respectively A3 and A4. It is defined as a state in the intertwiner basis |j12>

curveequ40

with coefficients cj12 such that:

curveequ41

In the large scale limit, for all ij. The large scale limit considered here is taken when all spins are large. The expression of the coefficients cj12 meeting these the requirements is:

curveequ42

where jo and ko are given real numbers respectively linked to θ12 and θ34 through the following equations:

curveequ43

σj12 is the variance which is appropriately fixed and the phase φ(jo, ko) is the dihedral angle to jo in an auxiliary tetrahedron related to the asymptotic of the 6j symbol performing the change
of coupling in the intertwiner basis.

For a classical regular tetrahedron, using the expression  for Regge action, the integrated classical curvature scales linearly in terms of the length of its hinges because the angles do not change in the equilateral configuration when the length is rescaled, which means that the integrated classical curvature scales as square root function of the area of a face. Below we see that the expected values of R on coherent states and semi-classical tetrahedra for large spins scales as a square root function of the spin, this matches nicely the
semi-classical evolution we expect.

curvefig7

 

 

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A new spinfoam vertex for quantum gravity by Livine and Speziale

In this paper the authors introduce a new spinfoam vertex to be used in models of 4d quantum gravity based on SU(2) and SO(4) BF theory plus constraints. It can be seen as the conventional vertex of SU(2) BF theory, the 15j symbol, in a particular basis constructed using SU(2) coherent states. This basis makes the geometric interpretation of the variables transparent – they are the vectors normal to the triangles within each tetrahedron. They study the condition under which these states can be considered semiclassical, and show that the semiclassical ones dominate the evaluation of quantum correlations. Finally, they describe how the constraints reducing BF to gravity can be directly written in terms of the new variables, and how the semiclassicality of the states might improve understanding the correct way to implement  the constraints.

Introduction
The spinfoam formalism for loop quantum gravity is a covariant approach to  the definition of the dynamics of quantum General Relativity. It provides transition  amplitudes between spin network states. The most studied example in the literature is the
Barrett–Crane model. This model has interesting aspects, such as the inclusion  of Regge calculus in a precise way, but it can not be considered a complete proposal. In  particular, recent developments on the semiclassical limit show that it does not give the full correct dynamics for the free graviton propagator. In this paper the authors introduce a  new model that can be taken as the starting point for the definition of a better behaved  dynamics.

Most spinfoam models, including Barrett-Crane, are based on BF theory, a topological theory whose relevance for 4d quantum gravity has long been conjectured, and has been  exploited in a number of ways. In constructing a specific model for quantum gravity, there
is a key difficulty of quantum BF theory that has to be overcome: not all the variables  describing a classical geometry turn out to commute. This fact has two important  consequences.

The first is that there is in general no classical geometry associated to the spin network in the boundary of the spinfoam. This leads immediately to the problem of finding semiclassical quantum states that approximate a given classical geometry, in the sense in which wave packets or coherent states approximate classical configurations in ordinary quantum theory. This is the problem of defining ‘coherent states’ for LQG.

The second consequence concerns the definition of the dynamics. This is typically obtained by constraining the BF theory, a mechanism well understood classically, but still unsettled at the quantum level. The constraints involve non-commuting variables, and the way to
properly impose them is still an open issue. In particular, it can be argued that the specific procedure leading to the BC model imposes them too strongly, a fact which also complicates a proper match with the states of the canonical theory (LQG) living on the  boundary. These key difficulties are present for both lorentzian and euclidean signatures.

Consider a definition of the partition function of SU(2) BF theory on a Regge triangulation, where the dynamical variables entering the sum have a clear semiclassical interpretation: they are the normal vectors n associated to triangles t within a tetrahedron t.  The classical geometry of this discrete manifold (areas, angles, volumes, etc.) can be  described in a transparent way in these variables, provided they satisfy a constraint for each  tetrahedron of the triangulation. This constraint is the closure condition, that says that the  sum of the four normals associated to each tetrahedron must vanish. If this condition is  satisfied, this new choice of variables positively addresses the issues described above.

First of all, from the boundary point of view, each tetrahedron corresponds to a node of the  boundary spin network. The states associated to the new variables are linear superpositions of the conventional ones, with the property of minimizing the uncertainty of the non commuting operators: they thus provide a solution to the problem of  finding coherent states. Upon satisfying the closure condition, the states are coherent and carry a given classical geometry.

Secondly, as far as the dynamics is concerned, the constraints reducing BF theory to GR are functions of the full bivectorial structure of the B field, this structure is related to the vectors n in a precise way.

A key point of this approach is the closure condition. This is crucial for the semiclassical interpretation of the states. This is not satisfied by all the states entering the partition function. Yet one of the main results of this paper is to show that quantum correlations are
dominated by the semiclassical states in the large spin limit. The key to this mechanism lies in the fact that the coherent states introduced are not normalized, and the configurations maximizing the norm are the ones satisfying the closure condition – the configurations
satisfying the closure condition are exponentially dominating. The results obtained are valid for SU(2) BF theory, but can be straightforwardly extended to Spin(4), and the same logic
applied to any Lie group, including noncompact cases which are of interest for lorentzian signatures.

The new vertex amplitude
The vertex amplitude that characterizes this spinfoam model can be constructed using the conventional procedure for the spinfoam quantization of SU(2) BF theory.

Recall that there is a vector space

spinfoam vertex equ0
associated with each edge of the spinfoam. Here the half-integer (spin) j labels the irreducible representations of SU(2), and F is the number of faces around the  edge. If the spinfoam is defined on a Regge triangulation, F = 4 for every edge. The spinfoam quantization assigns to each edge an integral over the group, that is evaluated by
inserting in Ho the resolution of identity

spinfoam vertex equ1

The labels i are called intertwiners, and the sums run over all half-integer values allowed by the Clebsch-Gordan conditions. Taking into account the combinatorial structure of  the Regge triangulation (four triangles t in every tetrahedron t, five tetrahedra in every
4-simplex s), one ends up with the partition function,

spinfoam vertex equ2

where dj = 2j+ 1 is the dimension of the irrep j, and the vertex amplitude is As(j, i)= {15j}, a well known object from the recoupling theory of SU(2).

This partition function endows each 4-simplex with 15 quantum numbers, the ten irrep labels j and the five intertwiner labels i. Using LQG operators associated with the boundary of the 4-simplex, the ten jtcan be interpreted as areas of the ten triangles in the 4-simplex while the five it give 3d dihedral angles between triangles (one – out of the  possible six – for each tetrahedron). On the other hand, the complete characterization of  a classical geometry on the 3d boundary requires 20 parameters, such as the ten areas plus
two dihedral angles for each tetrahedron. Therefore the quantum numbers of the partition function are not enough to characterize a classical geometry. To overcome these difficulties, this paper writes the same partition function in different variables, which will endow each 4-simplex with enough geometric information.

The key observation is that SU(2) coherent states |j, n>, here n is a unit vector on the two- sphere S2, provide a (overcomplete) basis for the irreps. By group averaging the tensor  product of F coherent states, we obtain a vector

spinfoam vertex equ2a

in Ho. Here j and n denote the collections of all j’s and n’s. The set of all these vectors when varying the n’s forms an overcomplete basis in Ho. Using this basis the  resolution of the identity in Ho can be written as

spinfoam vertex equ3a
This formula can be used in the edge integration. The combinatorics is the same as before.  In particular, it assigns to a triangle t a normal n for each tetrahedron sharing t. Within a  single 4-simplex, there are two tetrahedra sharing a triangle, denoted u(t)and d(t).  Furthermore, it assigns two group integrals to each tetrahedron, one for each 4-simplex  sharing it. Taking also into account the dj factors and using the conventional spinfoam  procedure, gives

spinfoam vertex equ4
where the new vertex is

spinfoam vertex equ5

The new vertex gives a quantization of BF theory in terms of the variables j and n. Together,  these represent the full bivector Bi(x) discretized on triangles belonging to tetrahedra. Taking the B field constant on each tetrahedron, the discretization and quantization procedures can  be schematically summarized as follows,

spinfoam vertex equ6

In the quantum theory, the field Bi (x) is represented by a vector jn associated to each triangle in a given tetrahedron. The set of all these vectors can be used to describe a  classical discrete geometry.

On the other hand, the conventional quantization of BF theory differs in the quantization  step, which reads

spinfoam vertex equ7

where Jt are SU(2) generators associated to each triangle. Consequently the variables entering the partition function are irrep labels j and intertwiner labels i. The first variables  are related to discretization of the modulus of B, and thus to the area of triangles. The  intertwiner labels are related to one angle for each tetrahedron. Therefore in these  variables a classical geometry is hidden, and this in turn makes it hard to implement the  dynamics.

This vertex can be used directly for SU(2) BF theory, and straightforwardly generalized  to the Spin(4) case, exploiting the homomorphism Spin(4) = SU(2) ×SU(2).

spinfoam vertex equ8
The Barrett Crane vertex can be obtained in its integral representation as,

spinfoam vertex equ9

To understand the geometry of this new vertex amplitude, it is crucial to study its asymptotic behaviour in the large spin limit. The large spin limit is dominated by values of the group elements such that the factors of the integrand in

spinfoam vertex equ5

are close to one, namely such that
spinfoam vertex equ10

This has a compelling geometric interpretation. Recall that u(t) and d(t) are the normals to the same triangle as as seen from the two tetrahedra sharing it. They are changed   into one another by a gravitational holonomy. This is in contrast with the BC model, which fixes u(t) = d(t) thus not allowing any gravitational parallel transport between tetrahedra. This is another way of seeing the well known problem that the Barrett Crane model has not enough  degrees of freedom.

Coherent intertwiners
This section focuses on the building blocks of the new vertex, namely the states |j,n>  associated to the tetrahedra. Before studying the details of the mathematical  structure of these states, let us discuss their physical meaning. In the canonical picture, a  tetrahedron is dual to a 4-valent node in a spin network. More in general, when the edge
of the spinfoam is on the boundary, there is a one–to–one correspondence between the number of faces F around it, and the valence V of the node of the boundary spin network. Given a discrete atom of space dual to the node with V faces, its classical geometry can be entirely determined using the V areas and 2(V-3) angles between them. Alternatively, one can use the (non–unit) normal vectors n associated with the faces, constrained to close, namely to satisfy sum(ni) = 0.

On the other hand, the conventional basis used in gives quantum numbers for V areas but only V-3 angles. This is immediate from the fact that one associates SU(2) generators Ji with each face, and only V- 3 of the possible scalar products Ji · Jk commute among each
other. Thus half the classical angles are missing. To solve this problem, it is argued that the states |j,n> carry enough information to describe a classical geometry for the discrete atom of space dual to the node. In particular, we interpret the vectors j, n precisely with the
meaning of normal vectors n to the triangle, and we read the geometry off them.

This requires the vectors to satisfy:

spinfoam vertex equ11
This is a closure condition, if it holds, all classical geometric observables can be parametrized as O({j,n}).

To describe the mathematical details, recall basic properties of the SU(2) coherent states. A coherent state is obtained via the group action on the highest weight state,

spinfoam vertex equ12

where g(n) is a group element rotating the north pole z = (0,0,1) to the unit vector n (and such that its rotation axis is orthogonal to z). These coherent states are semiclassical in the sense that they localize the direction n of the angular momentum.

This localization property is preserved by the tensor product of irreps: in the the large spin limit we have for instance <Ji·Jk> ~ jijk ni·nk for any i and k. More in general any operator O({Ji })on the tensor product of coherent states satisfies:

spinfoam vertex equ13

with vanishing relative uncertainty.

So far, the n’s are completely free parameters. If the closure condition holds, then the states admit a semiclassical interpretation.

The partition function is the tensor product of coherent states projected on the invariant subspace Ho, in order to implement gauge invariance. This is achieved by group averaging. The resulting states written as a linear combination of the conventional basis of intertwiners are

spinfoam vertex equ14
We call these states coherent intertwiners. Projecting onto Ho does not force the V-simplex to close identically, but it does imply that closed simplices dominate the dynamics.
Four-valent case: the coherent tetrahedron

spinfoam vertex equ14

shows the construction of coherent intertwiners for nodes of generic valence. In the 4-valent case, whose dual geometric picture is a tetrahedron, is of particular interest as it enters the construction of vertex amplitudes for spinfoam models. The coherent tetrahedron can be decomposed in the conventional basis of virtual links, which are fixed by choosing to add J1 and J2 first.

Introducing the shorthand notation |i> = |ji,ni>, in the 4-valent case we have,

spinfoam vertex equ31

To study the asymptotics, it is convenient to introduce an auxiliary unit vector n, writing the character in the basis of coherent states,

spinfoam vertex equ32

This gives,

spinfoam vertex equ34

with,

spinfoam vertex equ35
The asymptotics of these coefficients are dominated by g and h close to the identity. Expanding S around p= q = 0 and denoting r = (p,q), we have

spinfoam vertex equ36

where N is the following 6-dimensional vector,

spinfoam vertex equ37

The Hessian matrix has the following structure,

spinfoam vertex equ38

 

where,

spinfoam vertex equ39-40

The asymptotics are given by

spinfoam vertex equ41

where det H can expressed in term of 3 × 3 determinants as,

For fixed ji, ni, this integral i represents the probability of the eigenstate j12 as a function of the ni’s. For closed configurations, we expect this to be a Gaussian peaked on the semiclassical value computed from the ni’s. Let’s consider for simplicity the equilateral case. In this case, we expect j12 to be peaked around j such that

spinfoam vertex equ41a
The diagram below shows that for large spin this is approximated by the Gaussian

spinfoam vertex equ42
where N(jo)is the normalization.

spinfoam vertex fig1

Analogous results hold for arbitrary closed configurations. This shows in a concrete way in which sense |j,n> represents a semiclassical state for a quantum tetrahedron, and more in general for a V-simplex dual to a node of valence V.

Towards the quantum gravity amplitude

How can  the coherent state presented here can be used to define the dynamics of quantum GR, starting from the spinfoam model?Spinfoam quantization of GR usually relies on reformulating GR as a constrained BF theory with an action of the following type

spinfoam vertex equ43

ω is a connection valued on a given Lie algebra (for euclidean signature, su(2) or spin(4)), and Bµ. is a bivector field (or two-form) with values in the same Lie algebra. The term C(B) includes polynomial constraints, reducing topological BF to GR. Typically, it gives the set of second class constraints expressing B in terms of the tetrad field (or vierbein)  and leading to GR in the first order formalism.

The BF theory can be quantized by discretizing the spacetime manifold with a Regge triangulation (or more generally a cellular decomposition), and then evaluating the partition function, where the variables are the representations j and intertwiners i . The natural extension of this procedure to quantize GR with action would be to discretize the constraints C(B) and include them in the computation of the discretized path integral.

 

The partition function for the quantum BF theory in terms of coherent states offers a natural way to impose the constraints on average. The key is that the partition function provides a quantization of BF theory where the B field is represented not through generators of the group, but as representation and
intertwiner labels.

The vertex is dominated in the large spin limit by semiclassical states
satisfying the closure condition for each tetrahedron and the relation between adjacent tetrahedra in the same 4-simplex. On these states the variables j and n give classical values with an (almost minimal) uncertainty decreasing as the jt’s increase.

If we use the j and n to construct a Regge geometry, we expect that the role of the constraints is to generate deficit angles when we glue together various 4-simplices, thus allowing the geometry to be curved.

Conclusions

The standard intertwiner basis which leads to BF theory with the vertex amplitude given by the {15j} symbol does not appear to be the most suitable one to study the semiclassical geometry of BF theory. Furthermore, it also makes it hard to understand the quantum
structure of the constraints reducing BF to GR.

This paper considered a basis constructed out of SU(2) coherent states. It defined nonnormalized coherent intertwiners, and studied their norm as a function of the geometric configuration. For each configuration, the norm is an integral over SU(2) that can be solved exactly.  Evaluation of the leading order of this integral in the large spin limit proves a very accurate approximation even for small spins, and shows very neatly that the norm is exponentially maximized
by the states admitting a semiclassical interpretation, namely the ones whose quantum numbers can be interpreted as vectors j, n describing the classical discrete geometry of a V -simplex. Due to this result, the semiclassical states will dominate the
evaluation of quantum correlations. Using these coherent intertwiners w the BF partition function can be rewritten with a new
vertex amplitude, where the discrete Bt(τ ) variables are interpreted in terms of the vectors j,n. This reformulation of the BF spinfoam amplitudes  improves the geometric interpretation of the theory.

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