Fock–Cook Quantization

Let $H$ be a complex Hilbert space, let $S(H)$ be its symmetric Fock space, and let $C(z)$ and $C^*(z)$ be the creation and annihilation operators associated with $z\in H$ . The Fock–Cook field operator is the closure

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

initially defined on the finite-particle subspace. The corresponding Weyl operators are

V(z)=e^{iR(z)}.

Remarks

The unitaries $V(z)$ satisfy the Weyl canonical commutation relations. In the setting of Shale's theorem, a symplectic transformation $T\in Sp(K)$ is unitarily implementable in this representation exactly when $T\in rSp(K)$ .

Examples

For finite-dimensional $H$ , this is the standard bosonic Fock representation of the canonical commutation relations.