Single Particle Structure Σ(H) - Imported Knowlpedia Content

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 $B(z_1,z_2)=\Im(z_1,z_2)_c$ .

Remarks

Key properties:

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

Examples

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