  In this paper we provide the representation of the symplectic group $Sp(2n,
\mathbb{R})$ in polymer quantum mechanics. We derive the propagator of the
polymer free particle and the polymer harmonic oscillator without considering a
polymer scale. The polymer scale is then introduced to reconcile our results
with those expressions for the polymer free particle. The propagator for the
polymer harmonic oscillator implies non-unitary evolution.
