A good geometric understanding of the well-known Dirac operator requires the notion of a spinor bundle — a vector bundle associated to a certain principal Spin-bundle by an appropriate representation. The Spin group here is in turn given by the metric structure of the underlying manifold, as a double cover of the Special Orthonormal group.
It turns out that the symplectic structure on some symplectic manifold may allow us to perform an analogous construction, yielding a bundle of Hilbert spaces acted upon by the Metaplectic group — a double cover of the Symplectic group. Due to the similarities, the resulting bundle received the name of symplectic spinor bundle, with "symplectic Dirac operators" acting upon it. However, both procedures differ in several significant points, the most noteworthy being perhaps the replacement of the Clifford algebra used in the classical construction by the Weyl algebra (also known as the symplectic Clifford algebra) and the fact that the Segal-Shale-Weil representation of the Metaplectic group, by which the symplectic spinor bundle is defined, is of infinite rank.