# Numerical work with sage 5 – Wigner {3j}, {6j}, {9j} symbol evaluation

I’ve been working on weighed digraphs, Dynkin diagrams and various other graph theory concepts over the last week. The reason for this is that as we saw in “Towards the graviton from spinfoams: the 3d toy model by Simone Speziale”  in  the Ponzano–Regge (PR) model  each tetrahedron has an amplitude given by Wigner’s {6j} symbol for the recoupling theory of SU(2).

This use of Wigner symbols is a common feature of Quantum Gravity models. Fortunately, Sagemaths provides a simple and straightforward way to evaluate Wigner symbols {3j}, {j6}, or {9j}. I am still working on evaluating {10j}, {15j} and {20j} Wigner symbols which are used in 3d and 4d Loop Quantum Gravity .

The table below taken from Quantum Gravity By Rovelli shows how the Wigner symbols are used in various quantum gravity models.

# Discreteness of Area and Volume in Quantum Gravity by Carlo Rovelli and Lee Smolin

Absolute classical paper, by two great physicists – Carlo Rovelli and Lee Smolin. Its well worth watching their lectures at the Perimeter Institute:

In this paper Rovelli and Smolin study the operator that measures  volume, in non-perturbative quantum gravity and  compute its spectrum, which they find is discrete. They construct an operator in the loop representation  finding  that it is finite, background independent, and diffeomorphism-invariant –  well defined on the space of knot states. They find that the eigenstates are in one to one correspondence with the spin networks.

They argue that the spectra of volume and area  can be considered as predictions of quantum gravity about Planck-scale  measurements of the geometry of space.

Volume

Let pi, qi, ri be the colors of the links adjacent to the i-th node of the spin network, and
let ai, bi, ci be defined by pi = ai +bi, qi = bi+ci, ri = ci +ai, where ai, bi, ci are integers then the volume V of a region R containing the nodes is given by:

where lp is the Planck length and the sum runs over all the nodes i contained in the region R

Area

The area of  the surface A is:

where Ji is the ji-th representation of SU(2).

# 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.

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:

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

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).

# Polyhedra in loop quantum gravity by Bianchia , Dona and Speziale

This week I have been reading about a recent generalisation of the quantum tetrahedron – the quantum polyhedron.  The authors state that interwiners are the building blocks of spin-network states and that the  space of intertwiners is the quantization of a classical symplectic manifold. They show that a theorem by Minkowski allows them to interpret conﬁgurations in this space as bounded convex polyhedra in R3. Given that a polyhedron is uniquely described by the areas and normals to its faces, theys are able to  give formulas for the edge lengths, the volume and the adjacency of faces of polyhedra. At the quantum level, this correspondence allows them  to identify an intertwiner with the state of a quantum polyhedron –  generalizing the notion of quantum tetrahedron.

In the loop quantum gravity, they find that coherent spin-network states with nodes of arbitrary valence represent a collection of semiclassical polyhedra. They also introduce an

operator that measures the volume of a quantum polyhedron and examine its relation
with the standard volume operator of loop quantum gravity.

# The Area of the Medial Parallelogram of a Tetrahedron by David N. Yetter

This  is a nice paper which  finds a simple formula for the area of the medial parallelogram of a tetrahedron in terms of the lengths of the six edges. This is interesting to me because in  simplicial models for quantum gravity, the formula is needed to deal the problem of length operators.

The paper finds that given a pair of non-incident edges in a tetrahedron, the medial parallelogram determined by the pair is the parallelogram whose vertices are the mid-points of the remaining four edges.

The area of the medial parallelogram determined by the edges of lengths d and e in the tetrahedron is