Discreteness of area and volume in quantum gravity

TL;DR

This paper demonstrates that the volume operator in non-perturbative quantum gravity has a discrete spectrum, with eigenstates linked to spin networks, providing insights into quantum geometry at the Planck scale.

Contribution

It introduces a well-defined, background-independent volume operator in loop quantum gravity and explicitly computes its discrete spectrum and eigenstates.

Findings

Volume spectrum is discrete.

Eigenstates correspond to spin networks.

Eigenvalues explicitly computed.

Abstract

We study the operator that corresponds to the measurement of volume, in non-perturbative quantum gravity, and we compute its spectrum. The operator is constructed in the loop representation, via a regularization procedure; it is finite, background independent, and diffeomorphism-invariant, and therefore well defined on the space of diffeomorphism invariant states (knot states). We find that the spectrum of the volume of any physical region is discrete. A family of eigenstates are in one to one correspondence with the spin networks, which were introduced by Penrose in a different context. We compute the corresponding component of the spectrum, and exhibit the eigenvalues explicitly. The other eigenstates are related to a generalization of the spin networks, and their eigenvalues can be computed by diagonalizing finite dimensional matrices. Furthermore, we show that the eigenstates of the volume diagonalize also the area operator. We argue that the spectra of volume and area determined here can be considered as predictions of the loop-representation formulation of quantum gravity on the outcomes of (hypothetical) Planck-scale sensitive measurements of the geometry of space.