Angle and Volume Studies in Quantized Space

TL;DR

This paper investigates the properties of angle and volume operators in loop quantum gravity, showing how they relate to classical space and providing numerical and analytical insights into their behavior.

Contribution

It presents new analytical and numerical results on the eigenvalues and scaling of angle and volume operators in spin networks within loop quantum gravity.

Findings

Minimum observable angle decreases inversely with square root of total spin

Angle operator can reproduce classical angle distribution

Volume scales as the 3/2 power of boundary surface

Abstract

The search for a quantum theory of gravity is one of the major challenges facing theoretical physics today. While no complete theory exists, a promising avenue of research is the loop quantum gravity approach. In this approach, quantum states are represented by spin networks, essentially graphs with weighted edges. Since general relativity predicts the structure of space, any quantum theory of gravity must do so as well; thus, "spatial observables" such as area, volume, and angle are given by the eigenvalues of Hermitian operators on the spin network states. We present results obtained in our investigations of the angle and volume operators, two operators which act on the vertices of spin networks. We find that the minimum observable angle is inversely proportional to the square root of the total spin of the vertex, a fairly slow decrease to zero. We also present numerical results indicating that the angle operator can reproduce the classical angle distribution. The volume operator is significantly harder to investigate analytically; however, we present analytical and numerical results indicating that the volume of a region scales as the 3/2 power of its bounding surface, which corresponds to the classical model of space.