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.