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.

Discreteness of Area and Volume fig 1

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.


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:

Discreteness of Area and Volume fig 2

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


The area of  the surface A is:

Discreteness of Area and Volume fig 3

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


Numerical work with sage 4 – All possible triangulations of a simplicial complex using pachner moves

So what I’ve been able to do using sagemath is triangulate a simplicial complex using Pachner moves. The complete set of all possible triangulations can be recorded, saved and plotted in 3d using jmol. The value of this is that in the path integral formulation it is necessary to be able to calculate the probability of each spin network and use the result to form a partition function.

fixed vertices all possible triangualtions via pachner moves

The next steps following on from this are to add SU(2) spins as edge labels and SU(2) interwiners as node labels. I would also like to investigate moving between simplicial complexes, spin networks and weighted digraphs as representations of discrete spacetimes.

numerical work with sage 4 fig 1

Numerical work with sagemath 3 – tetrahedron graphs

A spin network is a directed graph whose edges are associated with irreducible representations of a compact Lie group and whose vertices are associated with intertwiners of the representations adjacent to it. I have been exploring some of the techniques of graph theory in the context of tetrahedron graphs.

tetrahedron graph 1 tetrahedron graph 2 tetrahedron graph 3


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






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.

quantum polyhedron


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.

Numerical work with sagemath 2

In my  first post about using sagemaths looked at fixed size simplicial complex, this week I have been exploring using varying size simplicial complexes – growing from a single tetrahedron to over over twenty.

sage: r = [(random(),random(),random()) for _ in range(4)]
sage: for i in range(20):
… r = r+[(random(),random(),random()) for _ in range(1)]
… PointConfiguration(r).triangulate().plot(axis=false)

I would ultimately like the growth to be constrained to satisfy  causal constraints at each stage.

simplex growth 1

Animated gif of output from tachyon viewer of sagemath code

I have also been looking at using pachner flips to generate all possible triangulations of a pointset but I haven’t converted the results to a graphical format as yet.

sage: r = [(random(),random(),random()) for _ in range(20)]
sage: points = PointConfiguration(r)
sage: triang = points.triangulate()
sage: triang.plot(axes=False)
sage: p= points.bistellar_flips()
sage: print p

Results are in text form – over 240 pages –  at the moment, so I’m working on code to process file graphically as results are generated:

pachner move output v1

                                          Sample output of pachner move sagemath code –                                                  note that sagemaths calls pachner moves bistellar flips


Angular momentum: An approach to combinatorial space-time by Roger Penrose

This week I have also been reading about Penrose’s “spin networks“. In this paper Penrose attempts to build a purely combinatorial description of spacetime starting from the mathematics of spin-1/2 particles. The appraoch he describes in this paper leads into twistor theory. The spin networks he describe give an interesting theory of space. Penrose’s spin networks are purely combinatorial structures: graphs with edges labelled by numbers j = 0, 1/2, 1, 3/2,… These numbers represent the total angular momentum or “spin”. In the spin networks described in this paper it is required that three edges meet at each vertex, with the corresponding spins j1, j2, j3 adding up to an integer and satisfying the triangle inequalities

|j1 – j2| ≤ j3 ≤ j1 + j2

These rules are motivated by the quantum mechanics of angular momentum in that if we combine a system with spin j1 and a system with spin j2, the spin j3 of the combined system satisfies exactly these constraints. In this paper a spin network represents a quantum state of the geometry of space and this interpretation is justified by computations using a special rule for computing the norm of any spin network.


A spin network