Topspin Networks in Loop Quantum Gravity

TL;DR

This paper extends loop quantum gravity by introducing topspin networks, which incorporate topological information into spin networks, potentially enriching the theory's background independence and opening new research directions.

Contribution

It proposes a minimal extension to the existing framework of loop quantum gravity to include topology via topspin networks, affecting phase space, operators, and background independence.

Findings

Topspin networks encode topological data in spin networks.

Minimal modifications are needed to incorporate topology.

Operators like area and Hamiltonian depend on topology.

Abstract

We discuss the extension of loop quantum gravity to topspin networks, a proposal which allows topological information to be encoded in spin networks. We will show that this requires minimal changes to the phase space, C*-algebra and Hilbert space of cylindrical functions. We will also discuss the area and Hamiltonian operators, and show how they depend on the topology. This extends the idea of "background independence" in loop quantum gravity to include topology as well as geometry. It is hoped this work will confirm the usefulness of the topspin network formalism and open up several new avenues for research into quantum gravity.