58 From Classical Mechanics to Quantum Field Theory. A Tutorial
Using Eq. (1.254), we have: Tr
[
Ŵ(x, k)̂W†(ξ,η)]
=2πδ(x−ξ)δ(k−η),which, inserted into (1.304), gives:
Ω−^1 (Ω (f)) (q,p)=∫
dξdη
2 πFs(η,ξ)exp{−ıω((ξ,η),(q,p))}=f(q,p).It may be useful to have an expression for the Wigner map directly in terms
of the matrix elements of the operators, which for plane waves reads[ 14 ]:
Ω−^1(
Ô
)
(q,p)=∫
dkeıqk〈
−p/+k/ 2 |Ô|−p/−k/ 2〉
. (1.306)
Also, from the very definition, it is not difficult to prove that
Ω−^1(
Ŵ(q′,,p′))
(q,p)=exp{ıω 0 ((q,p),(q′,p′))/}. (1.307)Introducing now two resolutions of the identity relative to the coordinates, we can
write:
Ω−^1(
Ô
)
(q,p)=∫
dkdxdx′eıqk〈
−p/+k/ 2 |x〉〈x|Ô|x′〉〈x′|−p/−k/ 2〉
,
(1.308)
where the integration overkcan be explicitly performed, yielding a delta-function.
Thus one obtains the celebrated Wigner formula:
Ω−^1(
Ô
)
(q,p)=∫
dξeıpξ/〈
q+ξ/ 2 |Ô|q−ξ/ 2〉
. (1.309)
Notice also that the Wigner transform inverts to:
〈
x|Ô|x′〉
=
∫ dp2 πexp{−ıp(x−x′)/}Ω−^1(
Ô
)(x+x′
2,p)
. (1.310)
Example 1.3.9.
(1) IfÔ=−P̂,sinceP̂|m〉=m|m〉,wehave:
〈
−p/+k/ 2 |(−P̂)|−p/−k/ 2〉
=(p+k/2)〈
−p/+k/ 2 |−p/−k/ 2〉
=pδ(k)and we find, as expected:Ω−^1(
(−P̂)
)
(q,p)=p. (1.311)