Automorphisms in Loop Quantum Gravity

TL;DR

This paper explores an extended automorphism group in Loop Quantum Gravity, revealing how it enlarges the symmetry group beyond traditional diffeomorphisms and impacts the structure of the quantum state space.

Contribution

It introduces a distributional extension of the spatial diffeomorphism group via automorphisms of the path groupoid, expanding the symmetry framework in LQG.

Findings

Automorphisms can map graphs with different knotting classes.

The automorphism-invariant Hilbert space is characterized.

Potential for a combinatorial formulation of LQG is discussed.

Abstract

We investigate a certain distributional extension of the group of spatial diffeomorphisms in Loop Quantum Gravity. This extension, which is given by the automorphisms Aut(P) of the path groupoid P, was proposed by Velhinho and is inspired by category theory. This group is much larger than the group of piecewise analytic diffeomorphisms. In particular, we will show that graphs with the same combinatorics but different generalized knotting classes can be mapped into each other. We describe the automorphism-invariant Hilbert space and comment on how a combinatorial formulation of LQG might arise.