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:

lengthequ1

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,

lengthequ2

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,

lengthequ22

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

lengthequ23

where

lengthequ24

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:

lengthequ33

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}:

lengthequ43

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:

lengthequ46

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:

lengthequ47

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

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

The operator lengthequ47ahas the following representation:

lengthequ54

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

lengthequ55

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

lengthequ58

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

lengthequ59

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

lengthequ60

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:

lengthequ61

and the matrix elements are given by:

lengthequ65

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:

lengthequ66

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:

lengthequ66a

lengthfig5

For jo = 1 we have:

lengthequ66b

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)

and

(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:

lengthequ72

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.

lengthequ73

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.

lengthfig6

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:

Conclusions

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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