Cosmological Coherent State Expectation Values in LQG I. Isotropic Kinematics

Abstract: This is the first paper of a series dedicated to LQG coherent states and cosmology. The concept is based on the effective dynamics program of Loop Quantum Cosmology, where the classical dynamics generated by the expectation value of the Hamiltonian on semiclassical states is found to be in agreement with the quantum evolution of such states. We ask the question of whether this expectation value agrees with the one obtained in the full theory. The answer is in the negative. This series of papers is dedicated to detailing the computations that lead to that surprising result. In the current paper, we construct the family of coherent states in LQG which represent flat ($k=0$) Robertson-Walker spacetimes, and present the tools needed to compute expectation values of polynomial operators in holonomy and flux on such states. These tools will be applied to the LQG Hamiltonian operator (in Thiemann regularization) in the second paper of the series. The third paper will present an extension to $k\neq0$ cosmologies and a comparison with alternative regularizations of the Hamiltonian.