Single Particle Structure Σ(H)

Given a complex Hilbert space $H$, its single particle structure is $\Sigma(H)=(K,B)$, where $K$ is $H$ viewed as a real Hilbert space with inner product $\Re(\cdot,\cdot)_c$, and $B(z_1,z_2)=\Im(z_1,z_2)_c$.

Key properties:

• $B$ is a symplectic form on $K$ (skew + nondegenerate).
• Symmetries of the free boson field act as $B$-preserving maps on $K$.

Example: $H=L^2(\mathbb R^d)$ gives a real phase space $K$ with $B=\Im(\cdot,\cdot)$.