Let be a metaplectic structure on a symplectic manifold that is, an equivariant lift of the symplectic frame bundle with respect to the double covering
The symplectic spinor bundle is defined [2] to be the Hilbert space bundle
associated to the metaplectic structure via the metaplectic representation also called the Segal–Shale–Weil [3][4][5] representation of Here, the notation denotes the group of unitary operators acting on a Hilbert space
The Segal–Shale–Weil representation [6] is an infinite dimensional unitary representation
of the metaplectic group on the space of all complex
valued square Lebesgue integrable square-integrable functions Because of the infinite dimension,
the Segal–Shale–Weil representation is not so easy to handle.