The geometry of quantum spin networks

TL;DR

This paper explores the extension of quantum geometry in loop quantum gravity through q-deformation, analyzing how it affects operators like volume and the structure of spin networks.

Contribution

It introduces a q-deformation framework for quantum geometry, diagonalizes key operators, and demonstrates how degeneracy is broken in spin networks.

Findings

Eigenstates expressed via q-deformed spin networks

Volume operator degeneracy is reduced by q-deformation

Trivalent spin-networks acquire non-zero volume

Abstract

The discrete picture of geometry arising from the loop representation of quantum gravity can be extended by a quantum deformation. The operators for area and volume defined in the q-deformation of the theory are partly diagonalized. The eigenstates are expressed in terms of q-deformed spin networks. The q-deformation breaks some of the degeneracy of the volume operator so that trivalent spin-networks have non-zero volume. These computations are facilitated by use of a technique based on the recoupling theory of SU(2)_q, which simplifies the construction of these and other operators through diffeomorphism invariant regularization procedures.