Weyl CCR Quantization

Let $(K,B)$ be a real symplectic Hilbert space. A Weyl representation of $(K,B)$ on a complex Hilbert space $\mathcal H$ is a map

V:K\longrightarrow U(\mathcal H)

satisfying the Weyl relations

V(z_1)V(z_2)=e^{-iB(z_1,z_2)/2}\,V(z_1+z_2).

It is regular if, for every $z\in K$ , the one-parameter unitary group $t\mapsto V(tz)$ is strongly continuous. A regular Weyl representation is the exponentiated form of a quantization of the canonical commutation relations.

Infinitesimal generators

Regularity is what permits the field operators to be defined. By Stone's theorem, for each $z\in K$ there is a unique self-adjoint operator $R(z)$ such that

V(tz)=e^{itR(z)}.

In particular $V(z)=e^{iR(z)}$ . The unbounded operators $R(z)$ need not share a domain, so the Weyl relations are the domain-free formulation of the canonical commutation relations.

Continuity convention

For unitary-valued maps, strong and weak continuity of each one-parameter subgroup are equivalent. Regularity may equivalently be stated as weak continuity of $V$ on every finite-dimensional subspace of $K$ ; this implies, and in this setting is equivalent to, the one-parameter condition above. Compare weak continuity of a representation.

Examples

For the Schrödinger representation on $L^2(\mathbb R)$ , let $Q$ be multiplication by $x$ and let $P=-i\,d/dx$ . With

B\bigl((q,p),(q',p')\bigr)=q p'-p q',

the unitaries

V(q,p)=e^{i(qQ+pP)}

give a regular Weyl representation with the phase convention used above.