On knottings in the physical Hilbert space of LQG as given by the EPRL model

TL;DR

This paper investigates the invariance properties of the EPRL spin foam amplitude in Loop Quantum Gravity and demonstrates that the physical Hilbert space contains no knotting classes of graphs due to certain invariance under deformations.

Contribution

It introduces a definition of face- and edge amplitudes that ensure invariance under trivial subdivisions and deformations, leading to the removal of knotting classes in the physical Hilbert space.

Findings

Spin foam amplitudes are invariant under certain deformations.

Knotting classes are absent in the physical Hilbert space.

Invariance is achieved through specific amplitude definitions.

Abstract

We consider the EPRL spin foam amplitude for arbitrary embedded two-complexes. Choosing a definition of the face- and edge amplitudes which lead to spin foam amplitudes invariant under trivial subdivisions, we investigate invariance properties of the amplitude under consistent deformations, which are deformations of the embedded two-complex where faces are allowed to pass through each other in a controlled way. Using this surprising invariance, we are able to show that in the physical Hilbert space as defined by the sum over all spin foams contains no knotting classes of graphs anymore.