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.


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:


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


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:


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:


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




where the Ei’s are evaluated at:


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


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


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:


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.


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:



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:


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:


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:


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:


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;


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


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


The action of Rc on Ψ is then:


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:



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



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:



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.


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>


with coefficients cj12 such that:


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:


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


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.




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The length operator in Loop Quantum Gravity by Bianchi

This week I have been reading about the curvature operator in Loop Quantum Gravity. In this post I want to look at the length operator in LQG by Bianchi – which is important in understanding the curvature operator. This related to the previous posts:

In this paper the author, discusses the dual picture of quantum geometry provided by a spin network state is discussed. From this perspective, the author introduces a new operator in Loop Quantum Gravity – the length operator.

This paper describes its quantum geometrical meaning and derives some of its properties. In particular it shows that the operator has a discrete spectrum and is diagonalized by appropriate superpositions of spin network states. A series of eigenstates and eigenvalues is presented and  its semiclassical properties are reviewed.

A remarkable feature of the loop approach to the problem of quantum gravity is the prediction of a quantum discreteness of space at the Planck scale. Such discreteness manifests itself in the analysis of the spectrum of geometric operators describing the volume of a region of space or the area of a surface separating two such regions. In this paper the author introduces a new operator – the length operator – study its properties and show that it has a discrete spectrum and an appropriate semiclassical behaviour.

In this paper the author describes the picture of quantum geometry coming from Loop Quantum Gravity and the role played by the length in this picture. It reviews the standard procedure used in Loop Quantum Gravity when introducing an operator corresponding to a given classical geometrical quantity.

The dual picture of quantum geometry

In Loop Quantum Gravity,  the state of the 3-geometry can be given in terms of a linear superposition of spin network states. Such spin network states consist of a graph embedded in a 3-manifold and a
coloring of its edges and its nodes in terms of SU(2) irreducible representations and of SU(2) intertwiners. Thanks to the existence of a volume operator and an area operator, the following dual picture of the quantum geometry of a spin network state is available  a node of the spin network corresponds to a chunk of space with definite volume while a link connecting two nodes corresponds to an interface of definite area which separates two chunks.

lengthfig1Moreover, a node connected to two other nodes identifies two surfaces which intersect at a curve. The operator introduced in this paper corresponds to the length of this curve.

Quantization of 3-geometric observables

At the classical level ,in general relativity ,volume and area are functions of the metric which is the dynamical variable. Generally speaking, in quantum geometry approaches, the metric is promoted to an operator on a Hilbert space. Therefore,  one can introduce for instance a volume operator as a function of the metric operator
and study its eigenstates and its spectrum. Such mathematical control is available in Loop Quantum Gravity thanks to the existence of the Ashtekar-Lewandowski measure on the space of connections.

The starting point for quantization is canonical general relativity written in terms of a real SU(2) connection A(x) and its conjugate momentum E(x), i.e. in terms of the so-called Ashtekar-Barbero
variables with real Immirzi parameter. All the information about the geometry of space is encoded in the field E(x). Being the momentum conjugate to a connection, it is called the ‘electric field’. It is a density of weight one which corresponds to the inverse densitized triad. For instance, the volume of a region R is a functional of the electric field lengthequ1a and is given by:


The essential assumption in Loop Quantum Gravity is that the mathematically well-defined operators acting on the Hilbert space are the holonomy of the connection along a curve e, namely the flux of the electric field through the surface S,


Every operator is to be considered as a function of such fundamental quantities.

Construction of the length operator

The starting point is a classical expression, the expression for the length of a curve. Given a curve embedded in the 3-manifold Σ,


the length L(γ ) of the curve is a functional of the electric field Ea given by the  one-dimensional integral




External regularization of the length of a curve

The external regularization of the length of a curve is used  as the starting point for quantization. The regularization goes through the following steps:

  1. The one-dimensional integral  is replaced by the limit of a Riemann sum. lengthequ25
  2. The first step of the fluxization procedure, is to write Gi(s) in terms of surface integrals: lengthequ26
  3. The second step of the fluxization procedure, is to write the surface integrals in as Riemann sums of fluxes: lengthequ29 where Y is given by: lengthequ30Then the length of the segment γ can be defined in terms of Gi and is given by:lengthequ31As a result we the length of the curve can be written as the limit s→ 0 of a sum of terms depending on s only implicitly:lengthequ32

Having constructed a sequence of regularized expressions having the appropriate classical limit, can now attempt to promote L(γ) to a quantum operator by invoking the known action of the holonomy
and of the flux on cylindrical functions, namely:


The final step of the construction is to build an operator corresponding to the quantity L defined by:

lengthequ31out of the two-hand operator:lengthequ34andlengthequ36

the local inverse-volume operator.  Both of them, the two-hand operator and the local inverse-volume operator, admit a dual description in terms of nodes and links of the graph of a spin network state. Such a  dual description matches with the  one described for the length operator. The operator built in such manner is the ‘elementary’ length operator.

lengthequ39Properties of the ‘elementary’ length operator

The ‘elementary’ length operator measures the length of a curve defined in the following intrinsic way. Starting with a spin network state with graph Γ and focus on a node and a couple of links originating at that node. Then considering two surfaces dual to the two links. These two surfaces intersect at a curve. The ‘elementary’ length operator measures the length of this curve.

More formally, consider the Hilbert space Ko(Γ) and – for each wedge ω = {n, e, e′} of the graph Γ – we define an operator L( γ) which measures the length of a curve γ associated to the wedge ω.

The operator L( γ)  acts on the L links e1, . . , eL originating at the node n distinguishing the links e, e′ from the remaining L − 2 links.  We
have that for the wedge ω12 = {n, e1, e2}:


that is the action of the operator is completely encoded into the matrix Lω.

In this section of the paper the author computes the matrix elements of this operator for nodes which are four-valent and discusses some of its properties. In the particular  a number of its eigenvalues and eigenstates are computed, non- commutation relations are discussed  and  its semiclassical behaviour analysed.

Let’s review some basics about the intertwiner vector space. The intertwiner basis can be written in the following way:lengthequ44

In the trivalent case L = 3, for admissible spins j1, j2, j3, the
intertwiner vector space is one-dimensional and the unique intertwining tensor is given by the Wigner 3j symbol:


The intertwining tensor vanishes if the three spins are not admissible. The spins j1, j2, j3 are said to be admissible if j1 + j2 + j3 is integer and they satisfy the triangular inequality:

|j1 − j2| ≤ j3 ≤ j1 + j2.

These two conditions are equivalent to the requirement that there are three integers a, b, c such that:

j1 = (a + b)/2, j2 = (a + c)/2, j3 = (b + c)/2.

In the four-valent case, the intertwiner vector space is less trivial. An orthonormal basis can be given in terms of the coupling of two three-valent intertwiners:


Given the operator L( γ ), in order to derive its matrix elements we need two ingredients

  • the matrix elements of the operator lengthequ47a
  • the matrix elements of the inverse-volume operatorlengthequ47b.

The operator lengthequ47ahas the following representation:


with the coefficients c(k, j1, j2) given by:


A second ingredient we need is the operator Q {e1,e2,e3}. On the basis |k>12  the operator has the following form:


The coefficients a(k, j1, j2, j3, j4) are real and have the following explicit expression:


The Q {e1,e2,e3} operator can be diagonalized:


and the eigenvalueslengthequ60a are real and non-degenerate. The eigenstates |qi> with i = 1, . . ,K provide an orthonormal basis of the intertwiner vector space. The non-zero eigenvalues always appear in pairs with opposite sign. A zero eigenvalue is present only when the dimension of the intertwiner vector space is odd – see the post, Numerical work with sage 7: eigenvalues of the volume operator in loop quantum gravity.

As a result the volume operator on the intertwiner vector space
is simply given by:


and the matrix elements are given by:


Spectrum and eigenstates: the four-valent monochromatic case

The ‘elementary’ length operator L( γ ) is diagonalized by spin network states which have the node n labelled by eigenstates of the operator L:


The eigenvalues of the ‘elementary’ length operator in the case of a four- valent node with spins which are all equal, j1 = j2 = j3 = j4 ≡ jo. The dimension of the four-valent monochromatic intertwiner vector space Vo is K = 2jo + 1.

For jo=½ we have:



For jo = 1 we have:


Semiclassical behaviour

Identifying semiclassical states in Loop Quantum Gravity is a difficult issue. To ease this we focus on the Hilbert space Ko(Γ) spanned by spin network states with graph Γ and look for states Ψ, which have the following two properties:

(a) they are required to be peaked on a given expectation value of the geometric operators available on Ko(Γ)


(b) have vanishing relative uncertainty

A state ψ ∈ Ko(Γ) satisfying the semiclassical requirements (a) and (b) can be found taking large spins on the links of Γ and labeling four-valent nodes with the semiclassical intertwiner:


The mean value of k is given as a function of jo by ko = 2 jo/√3 and phase  φo =π/2.

The expectation value of the elementary volume operator as a function of jo scales as jo to the power of 3/2.

Recalling that the eigenvalues of the area are given by lengthequ72awe have that the volume scales as the power 3/2 of the area as expected for a regular classical tetrahedron.

The state  ψ ∈ Ko(Γ)  provides a  setting for testing the semiclassical
behaviour of the elementary length operator introduced in this paper. Numerical investigations indicate that the expectation value of the elementary length operator for a wedge of ω on the stateΨ scales as the square root of jo.


Therefore it scales exactly as the classical length of an edge of the
tetrahedron, i.e. as the square root of the area of one of its faces.


This set of results strongly strengthens the relation between the quantum geometry of a spin network state and the classical simplicial geometry of piecewise-flat 3-metrics, see the posts:


The operator is constructed starting from the classical expression for the length of a curve. The quantization procedure goes through an external regularization of the classical quantity, a canonical
quantization of the regularized expression and an analysis of the existence of the limit in the Hilbert space topology. The operator constructed in this way has a number of properties which are :

  • The length operator fits into the dual picture of quantum geometry proper of Loop Quantum Gravity. For given spin network graph, the operator measures the length of a curve in the dual graph.
  • An ‘elementary’ length operator can be introduced. It measures the length of a curve defined in the following intrinsic way. A node of the spin network graph is dual to a region of space and a couple of links at such node are dual to two surfaces which intersect at a curve. The ‘elementary’ length operator measures the length of this curve.
  • The ‘elementary’ length operator has a discrete spectrum.
  • The ‘elementary’ length operator has non-trivial commutators with other geometric operators
  • A semiclassical analysis shows that the ‘elementary’ length operator has the appropriate semiclassical behaviour: on a state peaked on the geometry of a classical tetrahedron it measures the length of one of its edges.
  • For given spin network graph, the length operator for a curve in the dual graph can be written in terms of a sum of ‘elementary’ length operators.


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Pentahedral volume, chaos, and quantum gravity by Haggard

This week I have been reviewing a couple of papers by Hal Haggard:

  • Pentahedral volume, chaos, and quantum gravity
  • Dynamical Chaos and the Volume Gap

These are related to the posts:

In these papers the author shows that chaotic classical dynamics associated to the volume of discrete grains of space leads to quantal spectra that are gapped between zero and nonzero volume. This strengthens the connection between spectral discreteness in the quantum geometry of gravity and tame ultraviolet behavior.  A detailed analysis of the geometry of a pentahedron, provides new insights into the volume operator and evidence of classical chaos in the dynamics it generates.

An outgrowth of quantum gravity has been the discovery that convex polyhedra can be endowed with a dynamical phase space structure. This structure was utilized to perform a Bohr-Sommerfeld quantization of the volume of a tetrahedron, yielding a novel
route to spatial discreteness and new insights into the spectral properties of discrete grains of space. Many approaches to quantum gravity rely on discretization of space or spacetime. This allows one to control, and limit, the number of degrees of freedom of the gravitational field being studied. Attention is often restricted to simplices. These papers study of grains of space more complex than simplices.

The Bohr-Sommerfeld quantization relied on the
integrability of the underlying classical volume dynamics, that is, the dynamics generated by taking as Hamiltonian the volume, H = Vtet. In general, integrability is a special property of a dynamical system exhibiting a high degree of symmetry. Instead, Hamiltonians with
two or more degrees of freedom are generically chaotic. Polyhedra with more than four faces are associated to systems with two or more degrees of freedom and so ‘Are their volume dynamics chaotic?’

The answer to this question has important physical consequences for quantum gravity. Prominent among these is that chaotic volume dynamics implies that there is generically a gap in the volume spectrum separating the zero volume eigenvalue from its
nearest neighbors. In loop gravity, it is convenient to
work with a polyhedral discretization of space because it
allows concrete study of a few degrees of freedom of the
gravitational fi eld;


however, what is key is the spectral discreteness of the geometrical operators of the theory. This is because the partition functions and transition amplitudes that defi ne the theory are expressed as sums over these area and volume eigenvalues. The generic presence
of gaps in the spectra of these operators ensures that these sums will not diverge as smaller and smaller quanta are considered; such a theory should be well behaved in the ultraviolet regime.

The author looks at the classical volume associated to a single pentahedral grain of space. He provides  evidences that the pentahedral volume dynamics is chaotic.  A new formula for the volume of a pentahedron in terms of its face areas and normals is found and it is shown that the volume dynamics is adjacency changing, General results from random matrix theory are used to argue that a chaotic volume dynamics implies the generic presence of a volume gap.

Consider a single pentahedral grain of space. An examination
of the classical volume dynamics of pentahedra relies on turning the space of convex polyhedra living in Euclidean three-space into a phase space. This is accomplished with the aid of two results:

  • Minkowski’s theorem states that the shape of a polyhedron is completely characterized by the face areas A and face normals n. More precisely, a convex polyhedron is uniquely determined, up to rotations, by its area vectors. The space of shapes of polyhedra with N faces of given areas A is:



The space PN naturally carries the structure of a phase space, with Poisson brackets,


This is the usual Lie-Poisson bracket if the A are interpreted physically as angular momenta, i.e. as generators of rotations.

To study the pentahedral volume dynamics on P5, with
H = Vpent, it is  necessary to find this volume as a function of the area vectors. This can be done as shown in the diagram below:


The main findings of the paper are summarized in a pentahedral phase diagram:



The presence of a volume gap in the integrable case of a tetrahedron has already been established. Furthermore, a chaotic pentahedral volume dynamics strongly suggest that there will be chaos for polyhedra with more faces, which have an even richer structure in their phase space. Consequently, the argument presented in this paper provides a very general mechanism that would ensure a volume gap for all discrete grains of space.


In this paper the authors have completely solved the geometry of
a pentahedron speci ed by its area vectors and defined its
volume as a function of these variables. By performing a numerical integration of the corresponding volume dynamics they have given early indicators that it generates a chaotic flow in phase space. These results uncover a new mechanism for the presence of a volume gap in the spectrum of quantum gravity: the level repulsion of quantum
systems corresponding to classically chaotic dynamics. The generic presence of a volume gap further strengthens the expected ultraviolet finiteness of quantum gravity theories built on spectral discreteness.


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Complete Loop Quantum Gravity Graviton Propagator by Emanuele Alesci

This week I have been reading a couple of PhD thesis

They are both quite nice works and contain content relevant to my work on the quantum tetrahedron. In this post I be looking at some of Alesci’s work whilst I will write about Perini’s work in a later post.

Quantum tetrahedron in 3d
Here the author describes the quantum geometry of a tetrahedron in 3 dimensions as given by a spin network  independently from the LQG approach. This description is in the context of spinfoam models and is used in the calculation of the graviton propagator.
Consider a compact, oriented, triangulated 3-manifold , and the complex Δ∗ dual to it, so having one node for each tetrahedron in  and one link for each face  – a triangle.

Considering a single tetrahedron in a 3d reference system R³; its geometry is uniquely determined by the assignment of its 4 vertexes. The same geometry can be determined by a set of 4 bivectors Ei – elements of ∧²R³ obtained taking the wedge product of the
displacement vectors of the vertexes) normal to each of the 4 triangles satisfying the closure constraint

where the last constraint simply says that the triangles close to form a tetrahedron. The quantum picture proceed as follows. Each bivector corresponds uniquely to an angular momentum operator in 3 dimensions , so an element of SU(2) (using the isomorphism between
∧²R³ and so(3)), and we can consider the Hilbert space of states of a quantum bivector as given by H = ⊕j, where j indicates the spin-j representation space of SU(2). Considering the tensor product of 4 copies of this Hilbert space, we have the following operators acting
on it:


with I = 1, 2, 3, and the closure constraint is given by


But now the closure constraint indicates nothing but the invariance under SU(2) of the state ψ so that the Hilbert space of a quantum tetrahedron is given by:


or in other words the sum, for all the possible irreducible representations assigned to the triangles in the tetrahedron, of all the possible invariant tensors of them, i.e. all the possible intertwiners.

It isalo possible to define 4 area operators;


and a volume operator;


and find that all are diagonal on;


the area having eigenvalue;


We can think of a spin network living in the dual complex Δ∗ of the triangulation, and so with one 4-valent node, labelled with an intertwiner, inside each tetrahedron, and one link, labelled with an irreducible representation of SU(2), intersecting exactly one
triangle of the tetrahedron. We then immediately recognize that this spin network completely characterizes a state of the quantum tetrahedron, so a state in T , and gives volume to it and areas to its faces – matching the results from LQG.

Now we look at an approach leading to the Barrett-Crane model; based on the geometrical interpretation of the discretized B fields on a fixed triangulation.
The first step is to turn the bivectors associated to the triangles into operators. To do this use is made of the isomorphism between the space of bivectors ∧²R4 and so(4) identifying the bivectors associated with the triangles with the generators of the algebra:


Then we proceed turning these variables into operators by associating to the different triangles t an irreducible representation ρt of the group and the corresponding representation space, so that the the generators of the algebra act on it  as derivative operators.  Unitary representations are chosen.

In the Riemannian case we use the group Spin(4). We can then define the Hilbert space of a quantum SO(4) bivector to be;


in the Riemannian case. However, this is not yet the Hilbert space of a quantum geometric triangle, simply because a set of bivectors does not describe a simplicial geometry unless it satisfies constraints equal to the Plebanski one for the discretized B fields. The task is then to translate these constraints in the language of group representation theory into the quantum domain.

Of the constraints given for the bivectors,only

alesciequ1.52refer to a triangle alone, and these are then enough to characterize a quantum triangle.

The most important constraint is however the simplicity constraint, that forces the bivectors to be simple, i..e formed as wedge product of two edge vectors. Classically this was expressed by: Bt ·∗Bt = 0. Using the quantization map and substituting the bivectors with
the Lie algebra elements in the representation ρt for the triangle t, we obtain the condition:

The condition is then of having vanishing pseudoscalar Casimir i.e the condition is to have simple representations.

The simplicity constraint forces the selfdual and anti-selfdual parts to be equal, and we are restricted to considering only representations of the type (j, j).

The Hilbert space of a quantum triangle is given by:


In this way the geometrical meaning of the parameters labelling the representations becomes clear: Classically the area of a triangle is given in terms of the associated bivector by:


that in the quantum case translates into


i.e. the area operator is the scalar Casimir and it is diagonal on each representation space associated to the triangle. Its eigenvalues are then in the Riemannian case, for simple representations


The representation label characterizes the quantum area of the triangle to which that representation is associated.

States of the quantum theory are assigned to 3-dimensional hypersurfaces embedded in the 4-dimensional spacetime, and are thus formed by tetrahedra glued along common triangles. We then have to define a state associated to each tetrahedron in the triangulation, and then define a tensor product of these states to obtain any given state of the theory. In turn, the Hilbert space of quantum states for the tetrahedron has to be obtained from the Hilbert space of its triangles, since these are the basic building blocks at our disposal corresponding to the basic variables of the theory.

Each tetrahedron is formed by gluing 4 triangles along common edges, and this gluing is naturally represented by the tensor product of the corresponding representation spaces; Spin(4) decomposes into irreducible representations which do not necessarily satisfy the simplicity constraint.

The third of the constraints composition on bivectors translates at the quantum level as the requirement that only simple representations appear in this decomposition. Therefore we are considering for each tetrahedron with given representations assigned to its triangles a tensor in the tensor product:


of the four representation spaces for its faces, with the condition that the tensored spaces decompose pairwise into vector spaces for simple representations only. We have still to impose the closure constraint; this is an expression of the invariance under the gauge group of the tensor we assign to the tetrahedron, thus in order to fulfill the closure constraint we have to associate to the tetrahedron an invariant tensor, i.e. an intertwiner between the four simple representations associated to its faces:


Therefore the Hilbert space of a quantum tetrahedron is given by:


with Hi being the Hilbert space for the i-th triangle defined above. Each state in this Hilbert space turns out to be the Barratt-Crane intertwiner.

This construction was rigorously perfomed  in the Riemannian case, using geometric quantization methods. In any case, this invariant tensor represents is the state of a tetrahedron whose faces are labelled by the given representations; it can be represented graphically as:


A vertex corresponding to a quantum tetrahedron in 4d, with links labelled by the representations and state labels associated to its four boundary triangles.

The result is that a quantum tetrahedron is characterized uniquely by 4 parameters, i.e. the 4 irreducible simple representations of so(4) assigned to the 4 triangles in it, which in turn are interpretable as the oriented areas of the triangles. This result is geometrically rather puzzling, since the geometry of a tetrahedron is classically determined by its 6 edge lengths, so imposing only the values of
the 4 triangle areas should leave 2 degrees of freedom, i.e. a 2-dimensional moduli space of tetrahedra with given triangle areas. For example, this is what would happen in 3 dimensions, where we have to specify 6 parameters also at the quantum level. So why does a tetrahedron have fewer degrees of freedom in 4 dimensions than in 3 dimensions, at the quantum level, so that its quantum geometry is characterized by only 4 parameters? The answer is that  the essential difference between the 3-dimensional and 4-dimensional cases is represented by the simplicity constraints that have to be imposed on the bivectors in 4 dimensions. At the quantum level these additional constraints reduce significantly the number of degrees of freedom for the tetrahedron, as can be shown using geometric quantization see the post The Quantum Tetrahedron in 3 and 4 Dimensions , leaving us at the end with a 1-dimensional state space for each assignment of simple irreps to the faces of the tetrahedron, i.e. with a unique quantum state up to normalization.

Then the question is: what is the classical geometry corresponding to this state? In addition to the four triangle areas operators, there are two other operators that can be characterized just in terms of the representations assigned to the triangles: one can consider the parallelograms with vertices at the midpoints of the edges of a tetrahedron and their areas, and these are given by the representations entering in the decomposition of the tensor product of representations labelling the neighbouring triangles. Analyzing the commutation relations of the quantum operators corresponding to the triangle and parallelogram area operators, it turns out that while the 4 triangle area operators commute with each other and with the parallelogram areas operators among which only two are independent, the last ones have non-vanishing commutators among themselves. This implies that we are free to specify 4 labels for the 4 faces of the tetrahedron, giving 4 triangle areas, and then only one additional parameter, corresponding to one of the parallelogram areas, so that only 5 parameters determine the state of the tetrahedron itself, the other one being completely randomized, because of the uncertainty principle.

Consequently, we can say that a quantum tetrahedron does not have a unique metric geometry, since there are geometrical quantities whose value cannot be determined even if the system is in a well-defined quantum state. In the context of the Barrett-Crane spin
foam model, this means that a complete characterization of two glued 4-simplices at the quantum level does not imply that we can have all the informations about the geometry of the tetrahedron they share. This is an example of the kind of quantum uncertainty relations
that we can expect to find in a quantum gravity theory.

A generic quantum gravity state is to be associated to a 3-dimensional hypersurface in spacetime, and this will be triangulated by several tetrahedra glued along common triangles; therefore a generic state will live in the tensor product of the Hilbert spaces of the tetrahedra of the hypersurface, and in terms of the Barrett-Crane intertwiners it will be given by a product of one intertwiner for each tetrahedron with a sum over the parameters -angular
momentum projections – labelling the particular triangle state for the common triangles -the triangles along which the tetrahedra are glued. The resulting object will be a function of the simple representations labelling the triangles, intertwined by the Barrett-Crane intertwiner to ensure gauge invariance, and it will be given by a graph which has representations of Spin(4) labelling its links and the Barrett-crane intertwiner at its nodes; in other words, it will be
given by a simple spin network.


We are left with a last ingredient of a quantum geometry still to be determined: the quantum amplitude for a 4-simplex χ, interpreted as an elementary change in the quantum geometry and thus encoding the dynamics of the theory, and representing the fundamental building block for the partition function and the transition amplitudes of the theory.

This amplitude can be constructed out of the tensors associated to the tetrahedra in the 4-simplex, so that it immediately fulfills the conditions necessary to describe the geometry of the simplicial manifold, at both the classical and quantum level, and has to be  invariant under the gauge group of  spacetime Spin(4). The natural choice is to obtain a C-number for each 4-simplex, a function of the 10 representations labelling its triangles, by fully contracting the tensors associated to its five tetrahedra pairwise summing over the parameters associated to the common triangles and respecting the symmetries of the 4-simplex, so:


The amplitude for a quantum 4-simplex is thus, in the Riemannian case:


So we obtain the amplitude as the 10j-symbol.

This amplitude is for fixed representations associated to the triangles, i.e. for fixed triangle areas; the full amplitude involves a sum over these representations with the above amplitude as a weight for each configuration. In general, the partition function of the quantum theory describing the quantum geometry of a simplicial complex made of a certain number of 4- simplices, will be given by a product of these 4-simplex amplitudes, one for each 4-simplex in the triangulation, and possibly additional weights for the other elements of it, triangles and tetrahedra, with a sum over all the representations assigned to triangles. We can then envisage a model of the form:


or, if one sees all the data as assigned to the 2–complex Δ∗ dual to the triangulation  with faces f dual to triangles, edges e dual to tetrahedra, and vertices v dual to 4-simplices:


clearly with the general structure of a spin foam model.
The last aspect that needs to be implemented is sum over triangulations; the restriction to a fixed triangulation  of spacetime has to be lifted, since in this non-topological case it represents a restriction of the dynamical degrees of freedom of the quantum spacetime, and a suitably defined sum over triangulations has to be implemented.

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Review of the Quantum Tetrahedron – part II

Intrinsic coherent states

A class of wave-packet states is given by the coherent states, which are  states labelled by classical variables (position and momenta) that minimize the spread of both. Coherent states are the basic tool for studying the classical limit in quantum gravity. They connect  quantum theory with classical general relativity. Coherent states in the Hilbert space of the theory can be used in proving the large distance behavior of the vertex amplitude and connecting it to the Einstein’s equations.

Given a classical tetrahedron, we can find a quantum state in Hγ such that all the dihedral angles are minimally spread around the classical values, these are the intrinsic coherent states

Tetrahedron geometry

Consider the geometry of a classical tetrahedron reviewed in Review of the Quantum Tetrahedron – part I . A tetrahedron in flat space can be determined by giving three vectors,tetraequ8.1a, representing three of its sides emanating from a vertex P.

Fig 1

Forming a non-orthogonal coordinate system where the axes are along these vectors and the vectors determine the unit of coordinate length, then ei  is the triad and


is the metric in these coordinates. The three vectors


are normal to the three triangles adjacent to P and their length is the area of these faces. The products


define the matrix hab which is the inverse of the metric h =ea.eb. The volume of the tetrahedron is

tetraequ8.4Extending the range of the index a to 1, 2, 3, 4, and denote all the four normals, normalised to the area, as Ea. These satisfy the closure condition


The dihedral angle between two triangles is given by


Now we move to the quantum theory. Here, the quantities Ea are quantized as


in terms of the four operators La, which are the hermitian generators of the rotation group:


The commutator of two angles is:


From this commutation relation, the Heisenberg relation follows:


Now we want to look for states whose dispersion is small compared with their expectation value: semiclassical states where


SU(2) coherent states
Consider a single rotating particle. How do we write a state for which the dispersion of its angular momentum is minimized? If j is the quantum number of its total angular momentum, a basis of states is



we have the Heisenberg relations


Every state satisfies this inequality. A state |j,j> that saturatestetraequ8.15is one  for whichtetraequ8.16a.

In the large j limit we have


Therefore this state becomes sharp for large j.

The geometrical picture corresponding to this calculation is that the state |j, j> represents a spherical harmonic maximally concentrated on the North pole of the sphere, and the ratio between the spread and the radius decreases with the spin.

Other coherent states  are  obtained rotating the state |j, j> into an arbitrary direction n. Introducing Euler angles θ,Φ  to label rotations,


Then let tetraequ8.22aand define the matrix R in SO(3) of the form  , tetraequ8.22bWith this, define:


The states |j,n> form a family of states, labelled by the continuous parameter n, which saturate the uncertainty relations for the angles. Some of their properties are the following.


For a generic direction n = (nx, ny, nz),therefore:




The expansion of these states in terms of Lz eigenstates is


The most important property of the coherent states is that they provide a resolution of the identity. That is


The left hand side is the identity in Hj. The integral is over all normalized vectors, therefore over a two sphere, with the standard R3 measure restricted to the unit sphere.

Observe that by taking tensor products of coherent states, we obtain coherent states. This follows from the properties of mean values and variance under tensor product.

Livine-Speziale coherent intertwiners

Now  introduce “coherent tetrahedra” states. A classical tetrahedron is defined by the four areas Aa and the four normalized normals na, up to rotations. These satisfy


Therefore consider the coherent state;

tetraequ8.30                               in         reviewpart1equ1.15

and project it down to its invariant part in the projectiontetraequ8.31

The resulting state


is the element of Hγ that describes the semiclassical tetrahedron. The projection can be explicitly implemented by integrating over SO(3);


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