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, Mourao 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 the expectation values of the elementary operators (and of their commutators divided by i (h) over bar, respectively) in a coherent state labelled by the (A, E) are, to zeroth order in (h) over bar, given by the values of the corresponding elementary functions (and of their Poisson brackets, respectively) at the 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. These findings are another step towards establishing that the infinitesimal quantum dynamics of quantum general relativity might, to lowest order in (h) over bar, indeed be given by classical general relativity.