From Classical Mechanics to Quantum Field Theory

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).