Gauge field theory coherent states (GCS): II. Peakedness properties

By T. Thiemann, O. Winkler. 2000

187 citations recorded by the source corpus.

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, Mourao 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 complement these results : First, we explicitly derive the com- plexification 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 semi-classical anal- ysis of canonical quantum gravity and quantum gauge theory coupled to quantum gravity. They also enable error-controlled approximations to difficult analytical cal- culations and therefore set a new starting point for 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.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f1c7b60cc11cc4cd589b525943b38fff3d02f463

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Citations in the corpus

Towards the QFT on curved spacetime limit of QGR: II. A concrete implementation (2002, methodology): “In the present paper, we roughly proceed in three steps: first, we review the coherent state family introduced in [9–11, 19] and fix the parameters in its definition in such a way that the best semiclassical behaviour is obtained for the observables relevant to our considerations.”
Algebraic quantum gravity (AQG): I. Conceptual setup (2006, methodology): “This research project was supported in part by a grant from NSERC of Canada to the Perimeter Institute for Theoretical Physics.”
Gauge field theory coherent states (GCS): I. General properties (2000, methodology): “One can show that these functions are linearly independent.”
Gauge field theory coherent states (GCS): III. Ehrenfest theorems (2000, background): “Then comparing cosh(z) = cosh(z 0 ) cosh(δ) + sinh(z 0 ) sinh(δ) = cosh(z 0 ) + δ sinh(z 0 ) + O(δ 2 ) (3.30) with (3.24) we conclude that δ = −s tr(τ j g ′ ¯ g T ) 2 sinh(z 0 ) (3.31) where the sign of sinh(z 0 ) follows from the formulas given in [41].”
The Phoenix Project: master constraint programme for loop quantum gravity (2003, methodology): “The advantage of having a non-graph changing Master Constraint Operator is that one can quantize it directly as a positive operator on HKin and check its semi-classical properties by testing it with the semi-classical tools developed in [29, 30, 31 , 32, 33, 34], to the best of our knowledge curren”
Algebraic quantum gravity (AQG): IV. Reduced phase space quantization of loop quantum gravity (2007, methodology): “The Gauss invariant [52] kinematical coherent states [18] of unreduced LQG now become physical coherent states.”
Phase space descriptions for simplicial 4D geometries (2008, background): “Of course one can still construct semi–classical states [44] that satisfy these constraints to the zeroth order in Planck’s constant.”
Towards spinfoam cosmology (2010, unknown): “[10] Thomas Thiemann and Oliver Winkler.”