The volume operator for singly polarized gravity waves with planar or cylindrical symmetry

TL;DR

This paper develops a quantum operator for singly polarized gravitational waves with planar or cylindrical symmetry, analyzing its spectrum and eigenvectors, and discussing implications for additional constraints in quantum gravity models.

Contribution

It constructs the volume operator for singly polarized gravitational waves with specific symmetries and analyzes its spectral properties, advancing understanding of quantum gravitational states.

Findings

Spectrum contains a manageable number of zero eigenvalues

Zero eigenvectors can be explicitly constructed

Further non-local constraints are anticipated but not analyzed

Abstract

A previous paper constructed a kinematic basis for spin networks with planar or cylindrical symmetry and arbitrary polarization. This paper imposes a constraint which limits the gravitational wave to a single polarization. The spectrum of the constraint contains a physically reasonable number of zero eigenvalues, and the zero eigenvectors can be constructed explicitly. Commutation of the constraint with the Hamiltonian is expected to lead to a further constraint. This new constraint is not investigated in this paper, but I argue it will be non-local, relating states at two or more neighboring vertices.