Knowlpedia

Symplectic Hilbert Space (K,B)

A real Hilbert space K equipped with a continuous nondegenerate skew form B

A symplectic Hilbert space is a real KK with a continuous BB.

In Shale's paper, KK comes from a complex HH by realification, with B=ℑ(⋅,⋅)cB=\Im(\cdot,\cdot)_c.

Remarks

Key properties:

  • Linear symmetries are elements of .
  • Implementability in Fock space depends on "restricted" size conditions (Hilbert–Schmidt).
Examples
  • K=HRK=H_{\mathbb R} for complex HH, B(z1,z2)=ℑ(z1,z2)cB(z_1,z_2)=\Im(z_1,z_2)_c.