Symplectic Form

A symplectic form on a real vector space $K$ is a bilinear form $B:K\times K\to\mathbb R$ with $B(x,y)=-B(y,x)$ and nondegeneracy: $B(x,\cdot)=0\Rightarrow x=0$.

Key properties (used here):

• Defines the Weyl phase $e^{-iB(z_1,z_2)/2}$ in the Weyl CCR .
• Symplectic operators satisfy $B(Tx,Ty)=B(x,y)$.

Example: On $\mathbb R^{2n}$, $B((p,q),(p',q'))=p\cdot q'-q\cdot p'$.