Quantum Gravity Vacuum and Invariants of Embedded Spin Networks

TL;DR

This paper connects 3D SU(2) BF theory with quantum gravity wavefunctions, defining invariants of embedded spin networks and proposing a flat metric spacetime vacuum state within a spin foam framework.

Contribution

It introduces a new interpretation of the path integral as a loop transform of quantum gravity wavefunctions and constructs invariants for embedded spin networks in a three-manifold.

Findings

Invariants of embedded spin networks are derived from the quantum SU(2) group.

A flat connection vacuum state is constructed in the q-deformed spin network basis.

A modification is proposed to obtain a vacuum state corresponding to flat metric spacetime.

Abstract

We show that the path integral for the three-dimensional SU(2) BF theory with a Wilson loop or a spin network function inserted can be understood as the Rovelli-Smolin loop transform of a wavefunction in the Ashtekar connection representation, where the wavefunction satisfies the constraints of quantum general relativity with zero cosmological constant. This wavefunction is given as a product of the delta functions of the SU(2) field strength and therefore it can be naturally associated to a flat connection spacetime. The loop transform can be defined rigorously via the quantum SU(2) group, as a spin foam state sum model, so that one obtains invariants of spin networks embedded in a three-manifold. These invariants define a flat connection vacuum state in the q-deformed spin network basis. We then propose a modification of this construction in order to obtain a vacuum state corresponding to the flat metric spacetime.