If we quantize a symplectic vector space, then the Weyl algebra retains an action of the symplectic group, however the Hilbert space of states does not have a natural action of the symplectic group. Instead, there is an extension which automatically acts on the space of states. After reviewing this, we will pass from this quantum mechanical system (1-dimensional QFT) to a 3-dimensional (topological) QFT. In this context we will investigate an analogue of the extension from earlier, and the ``space of states'' on which it acts.