Decomposition K = M ⊕ M

Shale fixes a real subspace $M\subset K$ so that $K=\Lambda^{-1}M\oplus M$, written $K=M\oplus M$.

Then every $z\in K$ is $z=x\oplus y$ with $x,y\in M$, and $\Lambda(x\oplus y)=-y\oplus x$.

Key properties:

• Positive symplectic operators can be conjugated to block form $S^{-1}\oplus S$ on $M\oplus M$.
• Field operators split into “$P(x)$” and “$Q(x)$” parts.

Example: For $K=\mathbb R^{2n}$, take $M=\mathbb R^n$ as the $q$-coordinates.