Reduced Phase Space Quantization of Black Holes: Path Integrals, and Effective Dynamics

We consider the loop quantum theory of the spherically symmetric model of gravity coupled to Gaussian dust fields, where the Gaussian dust fields provide a material reference frame of the space and time to deparameterize gravity. This theory, used to study the quantum features of the spherically symmetric black hole, is constructed based on a 1-dimensional lattice $\gamma\subset\mathbb R$. Taking advantage of the path integral formulation, we investigate the quantum dynamics and obtain an effective action. With this action, we get an effective continuous description of this quantum lattice system which is not the same as the one described by the effective Hamiltonian used in arXiv:2012.05729, i.e. the classical Hamiltonian with the holonomy correction. It turns out that the Hamiltonian derived in this paper returns that used in arXiv:2012.05729 only for macro black holes since the lattice $\gamma$ is required to be sufficiently fine. Indeed, it is necessary to propose this fine-grained lattice structure in order to well describe the underlying lattice theory by the continuous description.