Asymptotic Analysis of Spin Networks with Applications to Quantum Gravity by Hal Haggard

This week I have been reading a couple of PhD thesis. The first is ‘The Chiral Structure of Loop Quantum Gravity‘ by Wolfgang  Wieland, the other is Hal Haggard’s PhD thesis, ‘Asymptotic Analysis of Spin Networks with Applications to Quantum Gravity‘ .

I’m going to focus in this post on a small section of  Haggard’s work about  the quantization of space.

In loop gravity quantum states are built upon a spin coloured graph. The nodes of this graph represent three dimensional grains
of space and the links of the graph encode the adjacency of these regions. In the semiclassical description of the grains of space – see the post:

these grains are  interpreted as giving rise to convex polyhedra in this limit. This geometrical picture suggests a natural proposal for the quantum volume operator of a grain of space: it should be an operator that corresponds to the volume of the associated classical polyhedron. This section provides a detailed analysis of such an operator, for a single 4-valent node of the graph. The valency of the node determines the number of faces of the polyhedron and so we will be focusing on classical and quantum tetrahedra.

The space of convex polyhedra with fixed face areas has a natural phase space and symplectic structure.  Using this kinematics we can study the classical volume operator and perform a Bohr-Sommerfeld quantization of its spectrum. This quantization
is in good agreement with previous studies of the volume operator spectrum and provides a simplified derivation.

This section of the thesis looks at:

  • The node Hilbert Spaces Kn for general valency.
  • The classical phase space and  its Poisson structure.
  • The volume operator  in loop gravity.
  •  The quantum tetrahedron
  • Phase space for the classical tetrahedron and its Bohr-Sommerfeld quantization.
  • Wavefunctions for the volume operator

 Setup

Focus on a single node n and its Hilbert space
Kn. The space Kn is defined as the subspace of the tensor
product Hj1⊗ · · ·  ⊗HjF that is invariant under global SU(2) transformations

halequ1

We call Kn the space of intertwiners. The diagonal action is generated by the operatorJ,

halequ2

States of Kn are called intertwiners and can be expanded as:

halequ3

The components i transform as a tensor under SU(2) transformations in such a way that the condition

halequ4

is satisfied. These are precisely the invariant tensors captured graphically by spin networks.The finite dimensional intertwiner space Kn can be understood as the quantization of a classical phase space.

Begin with Minkowski’s theorem, which states the following: given F vectors Ar ∈ R³ (r = 1, . . . , F) whose sum is zero

halequ5

 

then up to rotations and translations there exists a unique convex polyhedron with F faces associated to these vectors. These convex polyhedra are our semiclassical interpretation of the loop gravity grains of space.

The second step  is to associate a classical phase space to these
polyhedra

We call this the space of shapes and denote it P(A1, . . . ,AF ) or more briefly PF.  Interpret the partial sums halequ6

as generators of rotations about the μk = A1 + · · · + Ak+1 axis.This interpretation follows naturally from considering each of the Ar vectors to be a classical angular momentum, so that really Ar ∈ Λr ≅R³.

There is is a natural Poisson structure on each r and this extends to the bracket

halequ7

With this Poisson bracket the μk do in fact generate rotations about the axis A1 + · · ·Ak+1. This geometrical interpretation suggests a natural conjugate coordinate, namely the angle φk of the rotation. Let k be the angle between the vectors

halequ8

then

halequ9

The pairs (μk,φk) are canonical coordinates for a classical phase space of dimension 2(F −3). Thisis called  the space of shapes and denote it P(A1, . . . ,AF ) or more briefly PF.

In the final step we  show that that quantization of PF is the Hilbert space Kn of an F-valent node n. The constraint

halequ11

beomes

halequ12

which is precisely the gauge invariance condition. This is where the classical vector model of angular momentum originates.

 Volume operators in loop gravity

Most of the loop gravity research on the volume operator has been done in the context of the original canonical quantization of general relativity.  In Ashtekar’s formulation of classical general relativity the gravitational field is described in terms of the triad variables E. This triad corresponds to a 3-metric h and is called the electric field. The elementary quantum operator that measures the geometry of space
corresponds to the flux of the electric field through a surface S. When such a surface is punctured by a link of the spin network graph Γ the flux can be parallel transported, back along the link, to the node using the second of Ashtekar’s variables, the
Ashtekar-Barbero connection. This results in an SU(2) operator that acts on the intertwiner space Kn at the node n.The parallel transported flux operator at the node is proportional to the generator of SU(2) transformations

halequ13

where γ is a free parameter of the theory called the Barbero-Immirzi parameter and Pl is the Planck length.

The volume of a region of space R is obtained by regularizing and quantizing the classical expression

halequ14

using the operators Er. The total volume is obtained by summing the contributions from each node of the spin network graph Γcontained in the region R.

There are different proposals for the volume operator at a node. The operator originally proposed by Rovelli and Smolin is

halequ15

A second operator introduced by Ashtekar and Lewandowski is

halequ16

Both the Rovelli-Smolin and the Ashtekar-Lewandowski proposals have classical versions. This results in two distinct functions on phase space:

halequ17

A third proposal for the volume operator at a node has emerged,
Dona, Bianchi and Speziale suggest the promotion of the classical volume of the polyhedron associated to {Ar} to an operator

halequ18

In the case of a 4-valent node all three of these proposals agree.

The volume of a quantum tetrahedron

In the case of a node with four links, F = 4, all the proposals for the volume operator discussed above coincide and match the operator introduced by Barbieri for the volume of a quantum tetrahedron.

– see post Quantum tetrahedra and simplicial spin networks  by A.Barbieri

The Hilbert space K4 of a quantum tetrahedron is the intertwiner space of four representations of SU(2),

halequ19Introduce  basis into this Hilbert space using the recoupling channel Hj1 ⊗Hj2 and call these basis states |k>. The basis vectors are defined as

halequ20

where the tensor ik is defined in terms of the Wigner 3j-symbols as

halequ21The index k ranges from kmin to kmax in integer steps with

halequ22

The dimension d of the Hilbert space K4 is finite and given by

halequ23
The states |k> form an orthonormal basis of eigenstates of the operator Er · Es. This operator measures the dihedral angle between the faces r and s of the quantum tetrahedron.
The operator The operator √Er · Er measures the area of the rth face of the quantum tetrahedron and states in K4 are area eigenstates with eigenvalues halequ24a,

halequ24

The volume operator introduced by Barbieri is

halequ25

and because of the closure relation

halequ26

this operator coincides with the Rovelli-Smolin operator for                  α = 2√2/3. The volume operator introduce by Barbieri can be understood as a special case of the volume of a quantum polyhedron.

In order to compute the spectrum of the volume operator, it is useful to introduce the operator Q defined as

halequ27

It represents the square of the oriented volume. The matrix elements of this operator are  computed in the post:

The eigenstates |q> of the operator Q,

halequ28

are also eigenstates of the volume. The eigenvalues of the volume are simply given by the square-root of the modulus of q,

halequ30

The matrix elements of the operator Q in the basis |k> are given by

halequ31

The function Δ(a, b, c) returns the area of a triangle with sides of length (a, b, c) and is conveniently expressed in terms of Heron’s formula

halequ32

This can be done numerically and we can  compare the eigenevalues calculated in this manner to the results of the Bohr-Sommerfeld quantization.

There are a number of properties of the spectrum of Q and therefore of V that can be determined analytically.

  • The spectrum of Q is non-degenerate: it contains d distinct real eigenvalues. This is a consequence of the fact that the matrix elements of Q on the basis |k> determine a d × d Hermitian matrix of the formhalequ33

with real coefficients ai.

  • The non-vanishing eigenvalues of Q come in pairs ±q. A vanishing eigenvalue is present only when the dimension d of the intertwiner space is odd.
  • For given spins j1, . . . , j4, the maximum volume eigenvalue can be estimated  using Gershgorin’s circle theorem and wefind that it scales as halequ35a   where jmax is the largest of the four spins jr.
  • The minimum non-vanishing eigenvalue (volume gap)
    scales as halequ35b

Tetrahedral volume on shape space

The starting point for the  Bohr-Sommerfeld analysis is the volume of a tetrahedron as a function on the shape phase space P(A1, . . . ,A4)≡  P4.

The Minkowski theorem guarantees the existence and uniqueness
of a tetrahedron associated to any four vectors Ar, (r = 1, . . . , 4) that satisfy A1 + · · · + A4 = 0. The magnitudes Ar ≡ |Ar|, (i = 1, . . . , 4) are interpreted as the face areas. A condition for the existence of a tetrahedron is that A1 +A2 +A3  ≥ A4, equality giving a flat -zero volume tetrahedron. The space of tetrahedra with four fixed face areas P(A1,A2,A3,A4) ≡ P4 is a sphere.

halfig1

Consider the classical volume,

halequ37it will be more straightforward to work with the squared classical volume,

halequ38

Writing the triple product of Q as the determinant of a matrix M = (A1,A2,A3) whose columns are the vectors A1,A2 and A3 and squaring yields,

halequ42

Bohr-Sommerfeld quantization of tetrahedra

The Bohr-Sommerfeld quantization condition is expressed in
terms of the action I associated to the each of these orbits:

halequ47

halfig2

Wavefunctions

Just as the Bohr-Sommerfeld approximation can be used for finding the eigenvalues of the volume operator. Semiclassical techniques can
also be used to find the volume wavefunctions.

halequ97

where,

halequ93

and

halequ94

Conclusions
At the Planck scale, a quantum behaviour of the geometry of space is expected. Loop gravity provides a specific realization of this expectation: it predicts a granularity of space with each grain having a quantum behaviour.

Based on semiclassical arguments applied to the simplest model for a grain of space, a Euclidean tetrahedron, and is closely related to Regge’s discretization of gravity and to more recent ideas about
general relativity and quantum geometry. The spectrum has been computed by applying Bohr-Sommerfeld quantization to the volume of a tetrahedron seen as an observable on the phase space of shapes.
There is quantitative agreement of the spectrum calculated here and
the spectrum of the volume in loop gravity. This result lends credibility to the intricate derivation of the volume spectrum in loop gravity, showing that it matches an elementary semiclassical approach.

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