Geometry Eigenvalues and Scalar Product from Recoupling Theory in Loop Quantum Gravity

TL;DR

This paper reviews the loop representation of quantum gravity, utilizing recoupling theory to derive spectra of geometric operators, compute volume eigenvalues, and establish a scalar product that makes these operators symmetric and states orthogonal.

Contribution

It introduces a recoupling theory approach to derive spectra of quantum geometric operators and defines a scalar product ensuring their symmetry and orthogonality of states.

Findings

Derived general expressions for area and volume spectra.

Computed explicit volume eigenvalues.

Established a scalar product for symmetric operators and orthogonal states.

Abstract

We summarize the basics of the loop representation of quantum gravity and describe the main aspects of the formalism, including its latest developments, in a reorganized and consistent form. Recoupling theory, in its graphical Temperley-Lieb-Kauffman formulation, provides a powerful calculation tool in this context. We describe its application to the loop representation in detail. Using recoupling theory, we derive general expressions for the spectrum of the quantum area and the quantum volume operators. We compute several volume eigenvalues explicitly. We introduce a scalar product with respect to which area and volume are symmetric operators, and (the trivalent expansions of) the spin network states are orthogonal.