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Abstract:
In this paper 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, Mourăo and Thiemann to arbitrary, finite, piecewise-analytic graphs.
However, both of these works were incomplete with respect to the following two issues.
The focus was on the unitarity of the transform and left the properties of the corresponding coherent states themselves untouched.
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 complement these results: first, we explicitly derive the complexification of the configuration space underlying these heat kernel coherent states and, secondly, 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 semiclassical analysis of canonical quantum gravity and quantum gauge theory coupled to quantum gravity. They also enable error-controlled approximations to difficult analytical calculations and therefore set a new starting point for numerical, semiclassical 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.
