Loop Quantum Gravity on Non-Compact Spaces

TL;DR

This paper introduces a method to construct Hilbert spaces for loop quantum gravity on non-compact spaces using background states and the Gel'fand-Naimark-Segal construction, enabling analysis of quantum fluctuations around classical geometries.

Contribution

It provides a novel procedure to build Hilbert spaces for non-compact manifolds in loop quantum gravity using background states and the GNS construction.

Findings

Constructed Hilbert spaces for non-compact geometries.

Demonstrated a background state approximating flat space.

Enabled analysis of quantum fluctuations around classical backgrounds.

Abstract

We present a general procedure for constructing new Hilbert spaces for loop quantum gravity on non-compact spatial manifolds. Given any fixed background state representing a non-compact spatial geometry, we use the Gel'fand-Naimark-Segal construction to obtain a representation of the algebra of observables. The resulting Hilbert space can be interpreted as describing fluctuation of compact support around this background state. We also give an example of a state which approximates classical flat space and can be used as a background state for our construction.