A Short Course on Quantum Mechanics and Methods of Quantization 611.3.3.5 The Moyal product and the quantum to classical transition
The Wigner map allows for the definition of a new algebra structure on the space
of functionsF
(
R^2
)
,theMoyal“∗”-productthat is defined as:f∗g≡Ω−^1(
Ω(̂ f)·Ω(̂ g))
. (1.324)
This product is associativeanddistributivewith respect to the sum, but it is
non-local andnon-commutative(since in generalΩ(̂ f)·Ω(̂ g)=Ω(̂ g)·Ω(̂f)).
Explicitly:
(f∗g)(q,p)=∫
dxdk
2 πexp{−ıω 0 ((x, k),(q,p))/}Tr[
̂Ω(f)·Ω(̂ g)̂W†(x, k)]
,
(1.325)
with
Tr[
Ω(̂f)·Ω(̂ g)̂W†(x, k)]
=
∫
dξdηdξ′dη′
(2π)^2Fs(f)(η,ξ)Fs(g)(η′,ξ′)×Tr[
Ŵ(ξ,η)Ŵ(ξ′,η′)Ŵ†(x, k)]
.
(1.326)
Skipping the details of calculations[ 14 ], it is possible to show that the last expres-
sion can be recast in the form:
(f∗g)(q,p)=4∫
dadbdsdt
(2π)^2f(a, b)g(s, t)×exp{
−
2 ı
[(a−q)(t−p)+(s−q)(p−b)]}
=4
∫
dadbdsdt
(2π)^2f(a, b)g(s, t)×exp{ 2 ıω 0 ((q−a, p−b),(q−s, p−t))/},explicitly exhibiting the non-locality of the Moyal product.
There are several equivalent ways of re-writing such an expression, such as^28 :
(f∗g)(q,p)=∑∞
n,m=0(
i
2)n+m
(−1)n
n!m!{
∂m+nf(a, b)
∂am∂bn∂m+ng(a, b)
∂an∂bm}∣∣
∣∣
a=q,b=p=f(q,p)exp{
i
2[←−
∂
∂q−→∂
∂p−
←−∂
∂p−→∂
∂q]}
g(q,p). (1.327)(^28) All the above expressions for the Moyal product apply of course to functions that are regular
enough for the right-hand side of the defining equations to make sense. In particular, they will
hold whenf,gare Schwartz functions.