56 From Classical Mechanics to Quantum Field Theory. A Tutorial
gives a map Ω from the space of functionsF
(
R^2
)
to operatorsOp(H), i.e.:
Ω(f)≡
∫ dξdη
2 π
[
1
Fs(f)
(
η
,
ξ
)]
̂W(ξ,η). (1.293)
It is simple to show that:
(i) Ω (f) is at least a symmetric operator.
This follows from the identity: [Fs(f)(η,ξ)]∗=Fs(f)(−η,−ξ)which
holds whenfis real.
(ii) The action on a wave functionψ(x) is explicitly given by:
(Ω (f)ψ)(x)=
∫
dξdη
2 π
Fs(f)(η,ξ)exp[ıη(x+ξ/2)]ψ(x+ξ),
(1.294)
as it can be proven by taking into account that:
(
Ŵ(ξ,η)ψ
)
(x)=exp{ıη[x+ξ/2]/}ψ(x+ξ). (1.295)
(iii) The matrix elements of the Weyl operator Ω (f) are given by the expres-
sion:
〈φ|Ω(f)|ψ〉=
∫
dxdξdη
2 π
Fs(f)(η,ξ)eıη(x+ξ/2)φ∗(x)ψ(x+ξ),
(1.296)
as it is found directly from Eq. (1.294). In particular, in a plane-wave
basis:
〈k′|Ω(f)|k〉=
∫
dξ
2 π
Fs(f)(k′−k,ξ)exp{ıξ(k+k′)/ 2 }. (1.297)
As expected, forf=qandf=p, Eq. (1.294) yields^25 :
(Ω (q)ψ)(x)=xψ(x)
(Ω (p)ψ)(x)=ı
dψ
dx
, (1.298)
i.e.
Ω(q)=Q̂
Ω(p)=−P̂
. (1.299)
(^25) Allowing for distribution-valued transforms, the result follows form the identities:
Fs(q)(η, ξ)=2πıδ′(η)δ(ξ)andFs(p)(η, ξ)=− 2 πı δ(η)δ′(ξ).