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selfAdjoint
#18
Oct16-03, 11:00 AM
Emeritus
PF Gold
P: 8,147
First, the Coherent State Transform.

In 1994 Brian C. Hall (at UCSD) generalized an old (1962) result of Segal and Bargmann by defining a transform from the Hilbert Space of square integrable fucntions under the Haar measure for a compact group to the space of holomorphic functions on the complexified version of the group, supplied with a measure he defined to make the functions "act nice". This was all pure functional analysis, but it caused breakthroughs in a wide range of fields. So now if you google on Coherent State Transform you will find results on its applications to geodesy, to wavelets, to algebraic varieties, and also to this paper by Ashtekar,Lewandowski, Marolf, Mourao, and Thiemann, gr-qc/9412014 which generalizes the Hall transform and applies it to defining gauge theory on a GR spacetime.

It is this generalized Coherent State Transform which is at the heart of Thiemann's attempt to define quantum theory on the network of LQG.