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.
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,, representing three of its sides emanating from a vertex P.
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
is the metric in these coordinates. The three vectors
are normal to the three triangles adjacent to P and their length is the area of these faces. The products
define the matrix hab which is the inverse of the metric h =ea.eb. The volume of the tetrahedron is
The dihedral angle between two triangles is given by
Now we move to the quantum theory. Here, the quantities Ea are quantized as
in terms of the four operators La, which are the hermitian generators of the rotation group:
The commutator of two angles is:
From this commutation relation, the Heisenberg relation follows:
Now we want to look for states whose dispersion is small compared with their expectation value: semiclassical states where
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
we have the Heisenberg relations
In the large j limit we have
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,
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.
For a generic direction n = (nx, ny, nz),therefore:
The expansion of these states in terms of Lz eigenstates is
The most important property of the coherent states is that they provide a resolution of the identity. That is
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
Therefore consider the coherent state;
The resulting state
is the element of Hγ that describes the semiclassical tetrahedron. The projection can be explicitly implemented by integrating over SO(3);
- Quantum tetrahedron and its classical limit by Daniel R. Terno (quantumtetrahedron.wordpress.com)
- Physical boundary state for the quantum tetrahedron by Livine and Speziale (quantumtetrahedron.wordpress.com)
- A new spinfoam vertex for quantum gravity by Livine and Speziale (quantumtetrahedron.wordpress.com)
- Review of the Quantum Tetrahedron – part 1 (quantumtetrahedron.wordpress.com)