  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.
