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Bounded invertible real-linear maps preserving the symplectic form B
For a symplectic Hilbert space , let denote the group of bounded invertible real-linear operators on . The symplectic group is
Each preserves the Weyl relations and hence acts on Weyl operators by . Only the restricted symplectic subgroup is unitarily implementable in the associated Fock representation.
A weak symplectic Hilbert space is a real Hilbert space equipped with a continuous, skew-symmetric bilinear form such that
Thus is a continuous symplectic form in the algebraic sense.
A linear operator is a linear map on a vector space .
Let be the standard symplectic matrix
so the bilinear form on is nondegenerate and skew-symmetric. The real symplectic group is
This is a closed subgroup of the general linear group, hence a Lie subgroup by the closed subgroup theorem.
More generally, for any field of characteristic , one defines as the group preserving a fixed nondegenerate alternating form; different choices are conjugate in .
The restricted symplectic group is
where is the restricted general linear group.
In Shale's Theorem 4.1 (Fock–Cook quantization), is unitarily implementable iff , equivalently is Hilbert–Schmidt.