Simplifying the spectral analysis of the volume operator

TL;DR

This paper introduces a method to simplify the spectral analysis of the volume operator in quantum gravity, focusing on states on a cubic lattice and determining spectra for specific states with six-valent intersections.

Contribution

It presents a novel approach using group representation theory to decompose Hilbert space, enabling easier spectral analysis of the volume operator in lattice-based quantum gravity.

Findings

Complete spectrum determined for six-valent intersection states

Method reduces complexity of spectral analysis on cubic lattices

Applicable to states in canonical quantum gravity models

Abstract

The volume operator plays a central role in both the kinematics and dynamics of canonical approaches to quantum gravity which are based on algebras of generalized Wilson loops. We introduce a method for simplifying its spectral analysis, for quantum states that can be realized on a cubic three-dimensional lattice. This involves a decomposition of Hilbert space into sectors transforming according to the irreducible representations of a subgroup of the cubic group. As an application, we determine the complete spectrum for a class of states with six-valent intersections.