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Fock–Cook Quantization

The standard bosonic CCR representation built from creation/annihilation operators
Fock–Cook Quantization

In the Fock–Cook representation on S(H)S(H), the field operator is

R(z)=12 (C(z)+C∗(z))∼, R(z)=\frac1{\sqrt2}\,(C(z)+C^*(z))^{\sim},

where (⋅)∼(\cdot)^{\sim} denotes closure as an unbounded operator.

Key properties (paper use):

  • Produces Weyl unitaries V(z)=eiR(z)V(z)=e^{iR(z)} satisfying .
  • Shale’s main result: T∈Sp(K)T\in Sp(K) is implementable iff T∈rSp(K)T\in rSp(K).

Example: For finite-dimensional HH, this is the usual Schrödinger/Fock CCR representation.