Knowlpedia

Field Automorphism θ(T) from a Symplectic Map

The *-automorphism induced by sending Weyl operators V(z) to V(Tz)

For T∈Sp(K)T\in Sp(K), θ(T)\theta(T) is the unique *-automorphism of the CCR C∗C^*-algebra A\mathfrak A such that

θ(T) eiR(z)=eiR(Tz).\theta(T)\,e^{iR(z)} = e^{iR(Tz)}.
Remarks

It induces an action on states by pullback: θ∗(T)E(X)=E(θ(T)−1X)\theta^*(T)E(X)=E(\theta(T)^{-1}X).

Key property (paper use):

  • E∼θ∗(T)EE\sim \theta^*(T)E holds exactly when TT is unitarily implementable in the chosen quantization.
Examples
  • In finite dimensions, metaplectic operators implement θ(T)\theta(T) projectively.