Discreteness of area and volume in quantum gravity

TL;DR

This paper shows that in loop quantum gravity the volume and area of regions can be promoted to well-defined, diffeomorphism-invariant operators with discrete spectra. The volume operator acts diagonally on spin-network states, yielding explicit eigenvalues for trivalent nodes and a finite-dimensional matrix structure for higher-valence nodes, while the area operator shares the same eigenbasis and has its own explicit spectrum. The authors interpret these discrete spectra as physical predictions for Planck-scale geometry, arguing that the results are robust kinematical statements independent of the dynamics. By tying the spectra to diffeomorphism-invariant observables defined with matter reference fields, they argue that these predictions survive gauge fixing and have potential implications for fundamental questions in quantum gravity, such as black hole entropy and early-universe physics.

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.

Figures (13)

• Figure 1: A spin network
• Figure 2: The $\hat{T}^{abc}$ operator
• Figure 3: The integers $a, b$ and $c$
• Figure 4: The three possible grasps
• Figure 5: The diagrammatic representation of $T^{abc}$.