Gauge Field Theory Coherent States (GCS) : III. Ehrenfest Theorems
  In the preceding paper of this series of articles we established peakedness properties of a family of coherent states that were introduced by Hall for any compact gauge group and were later generalized to gauge field theory by Ashtekar, Lewandowski, Marolf, Mour\~ao and Thiemann.   In this paper we establish the ``Ehrenfest Property'' of these states which are labelled by a point (A,E), a connection and an electric field, in the classical phase space. By this we mean that i) The expectation value of {\it all} elementary quantum operators $\hat{O}$ with respect to the coherent state with label (A,E) is given to zeroth order in $\hbar$ by the value of the corresponding classical function O evaluated at the phase space point (A,E) and ii) The expectation value of the commutator between two elementary quantum operators $[\hat{O}_1,\hat{O}_2]/(i\hbar)$ divided by $i\hbar$ with respect to the coherent state with label (A,E) is given to zeroth order in $\hbar$ by the value of the Poisson bracket between the corresponding classical functions $\{O_1,O_2\}$ evaluated at the phase space point (A,E).   These results can be extended to all polynomials of elementary operators and to a certain non-polynomial function of the elementary operators associated with the volume operator of quantum general relativity. It follows that the infinitesimal quantum dynamics of quantum general relativity is to zeroth order in $\hbar$ indeed given by classical general relativity. 
T. Thiemann, O. Winkler