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Decomposition K = M ⊕ M

A choice of real subspace M giving coordinates (x,y) for phase space K

Shale fixes a real subspace M⊂KM\subset K so that K=Λ−1M⊕MK=\Lambda^{-1}M\oplus M, written K=M⊕MK=M\oplus M.

Then every z∈Kz\in K is z=x⊕yz=x\oplus y with x,y∈Mx,y\in M, and Λ(x⊕y)=−y⊕x\Lambda(x\oplus y)=-y\oplus x.

Remarks

Key properties:

  • Positive symplectic operators can be conjugated to block form S−1⊕SS^{-1}\oplus S on M⊕MM\oplus M.
  • Field operators split into "P(x)P(x)" and "Q(x)Q(x)" parts.
Examples
  • For K=R2nK=\mathbb R^{2n}, take M=RnM=\mathbb R^n as the qq-coordinates.