Symplectic Hilbert Space (K,B)

A symplectic Hilbert space is a real Hilbert space $K$ with a continuous symplectic form $B$.

In Shale’s paper, $K$ comes from a complex $H$ by realification, with $B=\Im(\cdot,\cdot)_c$.

Key properties:

• Linear symmetries are elements of Sp(K) .
• Implementability in Fock space depends on “restricted” size conditions (Hilbert–Schmidt).

Example: $K=H_{\mathbb R}$ for complex $H$, $B(z_1,z_2)=\Im(z_1,z_2)_c$.