  In this paper we discuss a simplified approach to the symplectic Clifford
algebra, the symplectic Clifford group and the symplectic spinor by first
extending the Heisenberg algebra. We do this by adding a new idempotent element
to the algebra. It turns out that this element is the projection operator onto
the Dirac standard ket. When this algebra is transformed into the Fock algebra,
the corresponding idempotent is the projector onto the vacuum. These additional
elements give a very simple way to write down expressions for symplectic
spinors as elements of the symplectic Clifford algebra. When this algebra
structure is applied to an optical system we find a new way to understand the
Gouy effect. It is seen to arise from the covering group, which in this case is
the symplectic Clifford group, also known as the metaplectic group. We relate
our approach to that of Simon and Mukunda who have shown the relation of the
Gouy phase to the Berry phase.
