Spectrum of the Volume Operator in Quantum Gravity

TL;DR

This paper develops a general algorithm to compute the spectrum of the volume operator in quantum gravity, explicitly evaluates some eigenvalues, and discusses the influence of Mandelstam constraints on four-valent spin network states.

Contribution

It introduces a novel algorithm for spectrum calculation of the volume operator in quantum gravity and analyzes the effects of Mandelstam constraints.

Findings

Explicit eigenvalue formulas for the volume operator

Algorithm for spectrum computation on four-valent states

Insights into Mandelstam constraints' role

Abstract

The volume operator is an important kinematical quantity in the non-perturbative approach to four-dimensional quantum gravity in the connection formulation. We give a general algorithm for computing its spectrum when acting on four-valent spin network states, evaluate some of the eigenvalue formulae explicitly, and discuss the role played by the Mandelstam constraints.