The volume operator for spin networks with planar or cylindrical symmetry

TL;DR

This paper develops a kinematic basis for symmetric spin networks and analyzes the volume operator's eigenvalues using difference equations and WKBJ methods, focusing on gravitational waves with both polarizations.

Contribution

It introduces a new basis for symmetric spin networks and provides methods to compute the volume operator's eigenvalues for various spins.

Findings

Eigenvalues obtained for low spins using computational methods.

WKBJ approximation used for higher spins.

Applicable to gravitational waves with both polarizations.

Abstract

This paper constructs a kinematic basis for spin networks with planar or cylindrical symmetry, by exploiting the fact that the basis elements are representations of an O(3) subgroup of O(4). The action of the volume operator on this basis gives a difference equation for the eigenvalues and eigenvectors of the volume operator. For basis elements of low spin, the difference equation can be solved readily on a computer. For higher spins, I solve for the eigenvalues using a WKBJ method. This paper considers only the case where the gravitational wave can have both polarizations. The single polarization case is considered in a spearate paper.