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.

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:

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

Area

The area of  the surface A is:

Discreteness of Area and Volume fig 3

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

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