  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.
