Coherent state functional integrals in quantum cosmology

Coherent state functional integrals for the minisuperspace models of quantum cosmology are studied. By the well-established canonical theories, the transition amplitudes in the path-integral representations of Wheeler-DeWitt quantum cosmology and loop quantum cosmology can be formulated through group averaging. The effective action and Hamiltonian with higher-order quantum corrections are thus obtained in both models within the scheme of Gaussian coherent states. It turns out that for a nonsymmetric Hamiltonian constraint operator, the Moyal *-product emerges naturally in the effective Hamiltonian. This reveals the intrinsic relation among coherent state functional integral, effective theory, and Moyal *-product. Moreover, both the resulted effective theories imply a possible quantum cosmological effect in large scale limit under certain condition.