  In this paper, we show the relation between $sp(4,\mathbb{R})$, the Lie
algebra of the symplectic group, and the elements of $Sp(4,\mathbb{R})$. We use
this result to obtain some special cases of symplectic matrices relevant to the
study of squeezed states. In this regard, we provide some applications in
quantum mechanics and analyze the squeezed polymer states obtained from the
polymer representation of the symplectic group. Remarkably, the polymer's
dispersions are the same as those obtained for the squeezed states in the usual
representation.
