The success of loop quantum cosmology to resolve classical singularities of homogeneous models has led to its application to classical Schwarszchild black hole interior which takes the form of a homogeneous, Kantowski-Sachs, model. First steps were done in pure quantum mechanical terms hinting at the traversable character of the would be classical singularity and then others were performed using effective heuristic models capturing quantum effects that allowed a geometrical description closer to the classical one but avoiding its singularity.However, the problem to establish the link between the quantum and effective descriptions was left open. In this work we propose to fill in this gap by considering the path integral approach to the loop quantization of the Kantowski-Sachs model corresponding to the Schwarzschild black hole interior. We show the transition amplitude can be expressed as a path integration over the imaginary exponential of an effective action which just coincides, under some simplifying assumptions, with the heuristic one. Additionally we further explore the consequences of the effective dynamics. We prove first such dynamics imply some rather simple bounds for phase space variables and in turn, remarkably, in an analytical way, they imply various phase space functions that were singular in the classical model are now well behaved. In particular, the expansion rate, its time derivative, and shear become bounded and hence the Raychauduri equation is finite term by term thus resolving the singularities of classical geodesic congruences. Moreover, all effective scalar polynomial invariants turn out to be bounded.

A recent proposal to connect the loop quantization with the spin foam model for cosmology via the path integral is hereby adapted to the case of mechanical systems within the framework of the so called polymer quantum mechanics. The mechanical models we consider are deparametrized and thus the group averaging technique is used to deal with the corresponding constraints. The transition amplitudes are written in a vertex expansion form used in the spin foam models, where here a vertex is actually a jump in position. Polymer propagators previously obtained by spectral methods for a nonrelativistic polymer particle, both free and in a box, are regained with this method and as a new result we obtain the polymer propagator of the relativistic particle. All of them reduce to their standard form in the continuum limit for which the length scale parameter of the polymer quantization is taken to be small. Our results are robust thanks to their analytic and exact character which in turn come from the fact that presented models are solvable. They lend support to the vertex expansion scheme of the polymer path integral explored before in a formal way for cosmological models. Some possible future developments are commented upon in the discussion.

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