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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