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.
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
The area of the surface A is:
where Ji is the ji-th representation of SU(2).