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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