Gauge-invariant coherent states for Loop Quantum Gravity II: Non-abelian gauge groups

TL;DR

This paper extends the study of gauge-invariant coherent states in Loop Quantum Gravity to non-abelian SU(2) gauge groups, analyzing their properties on special graphs and revealing how gauge space geometry affects their semiclassical behavior.

Contribution

It provides a detailed analysis of gauge-invariant coherent states for SU(2) in Loop Quantum Gravity, including their overlap properties and effects of gauge space singularities.

Findings

Overlaps are Gauss-peaked in gauge-invariant quantities.

Degenerate gauge orbits cause plateau structures in overlaps.

Semiclassical properties are influenced by gauge-invariant phase space geometry.

Abstract

This is the second paper concerning gauge-invariant coherent states for Loop Quantum Gravity. Here, we deal with the gauge group SU(2), this being a significant complication compared to the abelian U(1) case encountered in the previous article. We study gauge-invariant coherent states on certain special graphs by analytical and numerical methods. We find that their overlap is Gauss-peaked in gauge-invariant quantities, as long as states are not labeled by degenerate gauge orbits, i.e. points where the gauge-invariant configuration space has singularities. In these cases the overlaps are still concentrated around these points, but the peak profile exhibits a plateau structure. This shows how the semiclassical properties of the states are influenced by the geometry of the gauge-invariant phase space.