Tag Archives: Classical limit

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

tetraequ8.1

is the metric in these coordinates. The three vectors

tetraequ8.2

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

tetraequ8.3

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

tetraequ8.5

The dihedral angle between two triangles is given by

tetraequ8.6

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

tetraequ8.7

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

tetraequ8.8

The commutator of two angles is:

tetraequ8.9

From this commutation relation, the Heisenberg relation follows:

tetraequ8.12

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

tetraequ8.13

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

tetraequ8.14

since,
tetraequ8.15

we have the Heisenberg relations

tetraequ8.16

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

In the large j limit we have

tetraequ8.22

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,

euler

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

tetraequ8.22c

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.

tetraequ8.24

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

tetraequ8.27

and

tetraequ8.27a

The expansion of these states in terms of Lz eigenstates is

tetraequ8.28

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

tetraequ8.29

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

tetraequ8.29a

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

tetraequ8.32

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

tetraequ8.33

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Towards the graviton from spinfoams: the 3d toy model by Simone Speziale

In this paper Speziale looks at the extraction of the 2-point function of linearised quantum gravity, within the spinfoam formalism. The author that this process relies on the use of a boundary state, which introduces a semi–classical flat geometry on the boundary.

The dynamical tetrahedron

In this paper, Speziale investigates this proposal by considering a toy model in the Riemannian 3d case, where the semi–classical limit is understood. The author shows that in this the semi-classical limit the propagation kernel of the model is that for the for the harmonic oscillator – which leads to expected 1/l behaviour of the 2-point function.

The toy model

The toy model  considered in this paper is a tetrahedron with dynamics described by the Regge action, whose fundamental variables are the edge lengths le. Since there is only   a single tetrahedron, all edges are boundary edges, and the action consists only of the
boundary term, namely it coincides with the Hamilton function of the system:

toy model hamiltonian

Here the θe are the dihedral angles of the tetrahedron, namely the angles between the outward normals to the triangles. They represent a discrete version of the extrinsic curvature,  they satisfy the non–trivial relation

toy model dihedral angle

In this discrete setting, assigning the six edge lengths is equivalent to the assignment of
the boundary gravitational field.

The quantum dynamics  is described by the Ponzano–Regge (PR) model . In the model, the lengths are promoted to operators whose spectrum is labelled by the half–integer j which  labels SU(2) irreducible representations  and the  Casimir operator C^2 = j(j+1). In the model, each tetrahedron has an amplitude given by Wigner’s {6j} symbol for the recoupling theory of SU(2).

propagation kernal toy model