52 From Classical Mechanics to Quantum Field Theory. A Tutorial
whichissuchthat:
F†Q̂F=−P,̂ F†P̂F=Q.̂ (1.263)It is simple to verify that this is indeed the action induced by the liner map onR^2 :
(q,p)→(−p, q), (1.264)for which
Û(q)=Ŵ((q,0))→̂W((0,−p)) =V̂(−p)V̂(p)=̂W((0,p))→̂W((q,0)) =Û(q). (1.265)
Suppose now to have a one-parameter group{Tλ}λ∈Rof linear symplectic trans-
formations, which is generated by the linear vector field Γ. Invariance of the sym-
plectic form is encoded in the infinitesimal relation: LΓω=0,whereLdenotes
the Lie derivative. This implies that there exists a globally defined functiongsuch
that:
iΓω=dg , (1.266)which, in addition, will be a quadratic function of the coordinates.
According to (1.260), to the family{Tλ}we can associate a strongly continuous
one-parameter group{Uλ}λ∈Rof unitary operators such that:
Ŵ(z(λ)) =Ûλ†Ŵ(z)Ûλ, (1.267)wherez(λ)=Tλ(z). Now, through Stone’s theorem, we can obtain the self-adjoint
generatorĜ:
Ûλ=exp{
−iλG/̂}
, (1.268)
representing the quantum counterpart of the quadratic functiong.
Notice that, in this way, we have achieved a way to quantize all quadratic
functions.
Let us consider now a general, not necessarily symplectic, linear transformation
T :S→S. Denoting withω 0 the standard symplectic form onS whichina
Darboux chart is written asω 0 =dqj∧dpj, we define a new symplectic structure
ωTvia:
ωT(z,z′)≡ω 0 (Tz,Tz′). (1.269)We leave to the reader to prove that we obtain in this way a new Weyl system for
(S,ωT),whichisdefinedby:
̂WT(z)≡Ŵ(Tz) (1.270)