From Classical Mechanics to Quantum Field Theory

A Short Course on Quantum Mechanics and Methods of Quantization 57

Using then the identity:Fs(qnpm)(η,ξ)=2π(−1)mın+mδ(n)(η)δ(m)(ξ), one can
see that Ω sends any monomialqnpm(withn, mintegers) into the operator:

(Ω (qnpm)ψ)(x)=

(

ıd

dξ

)m

[(x+ξ/2)nψ(x+ξ)]|ξ=0

=^1

2 n

∑n

k=0

(

n

k

)

xk

(

ıd

dx

)m[

xn−kψ(x)

]

, (1.300)

i.e.

Ω(qnpm)=Ω(qnpm)=

1

2 n

∑n

k=0

(

n

k

)

[Ω (q)]k·[Ω (p)]m·[Ω (q)]n−k. (1.301)

Let us notice that, forn=m= 1, one has

Ω(qp)=Ω(pq)=

1

2

[Ω (q)·Ω(p)+Ω(p)·Ω(q)] (1.302)

which gives a justification for the symmetrization procedure we have talked about
while discussing the quantization of the harmonic oscillator in Sect. 2, Example 7.
In general, however:

Ω(fg)=

1

2

(Ω (f)·Ω(g)+Ω(g)·Ω(f)), (1.303)

meaning that the so-called “Weyl symmetrization procedure” holds only in very
special cases.

1.3.3.4 The Wigner map

In this subsect. we will see that the Weyl map can be inverted, i.e there exists a
map, called theWigner map:Ω−^1 :Op(H)→F

(

R^2

)

, such that Ω−^1 (Ω (f)) =f.

It is defined as follows: given any operatorÔsuch that Tr

[

ÔŴ(x, k)

]

exists^26 ,
we have:

Ω−^1

(

Ô

)

(q,p)≡

∫

dxdk

2 π

exp{−ıω 0 ((x, k),(q,p))/}Tr

[

Ô̂W†(x, k)

]

. (1.304)

In order to prove Eq. (1.304), we need the expression for the trace:

Tr[̂W(x, k)Ŵ†(ξ,η)] =

∫

dhdh′

〈

h

∣∣

∣̂W(x, k)

∣∣

∣h′

〉〈

h′

∣∣

∣̂W†(ξ,η)

∣∣

∣h

〉

. (1.305)

(^26) AsWis a bounded operator, this is true, e.g., ifAis trace-class.