Imposing det E > 0 in discrete quantum gravity

TL;DR

This paper explores the role of the inequality det E > 0 in discrete quantum gravity, distinguishing it from gauge theories and analyzing the eigenstructure of the associated quantum operator.

Contribution

It introduces a criterion det E > 0 to differentiate kinematical phases and classifies eigenvectors using octagonal group representations in lattice quantum gravity.

Findings

Eigenvectors classified by octagonal group representations.

Eigenvalues include positive, negative, and zero.

Clarification of the phase space structure in discrete quantum gravity.

Abstract

We point out that the inequality det E > 0 distinguishes the kinematical phase space of canonical connection gravity from that of a gauge field theory, and characterize the eigenvectors with positive, negative and zero-eigenvalue of the corresponding quantum operator in a lattice-discretized version of the theory. The diagonalization of the operator det E is simplified by classifying its eigenvectors according to the irreducible representations of the octagonal group.