Gauge Field Theory Coherent States (GCS) : II. Peakedness Properties

TL;DR

This paper rigorously analyzes the peakedness properties of gauge field theory coherent states, establishing their suitability for semi-classical analysis in quantum gravity and gauge theories, and clarifying their complexification structure.

Contribution

It proves that these states satisfy key semi-classical properties, addressing previous gaps regarding their peakedness, overcompleteness, and complexification in the phase space.

Findings

States satisfy peakedness in configuration, momentum, and phase space

States saturate the Heisenberg uncertainty bound

States form an overcomplete basis

Abstract

In this article we apply the methods outlined in the previous paper of this series to the particular set of states obtained by choosing the complexifier to be a Laplace operator for each edge of a graph. The corresponding coherent state transform was introduced by Hall for one edge and generalized by Ashtekar, Lewandowski, Marolf, Mour\~ao and Thiemann to arbitrary, finite, piecewise analytic graphs. However, both of these works were incomplete with respect to the following two issues : (a) The focus was on the unitarity of the transform and left the properties of the corresponding coherent states themselves untouched. (b) While these states depend in some sense on complexified connections, it remained unclear what the complexification was in terms of the coordinates of the underlying real phase space. In this paper we resolve these issues, in particular, we prove that this family of states satisfies all the usual properties : i) Peakedness in the configuration, momentum and phase space (or Bargmann-Segal) representation, ii) Saturation of the unquenched Heisenberg uncertainty bound. iii) (Over)completeness. These states therefore comprise a candidate family for the semi-classical analysis of canonical quantum gravity and quantum gauge theory coupled to quantum gravity, enable error-controlled approximations and set a new starting point for {\it numerical canonical quantum general relativity and gauge theory}. The text is supplemented by an appendix which contains extensive graphics in order to give a feeling for the so far unknown peakedness properties of the states constructed.