I.3 Moyal Bracket
523
However, from Eqs. (I.1) and (I.4), the commutator in Eq. (I.20) is given by
[e
i
¯
h (τ 1 ˆ
p+φ 1 ˆ
q) , e
i
¯
h (τ 2 ˆ
p+φ 2 ˆ
q) ] = 2i e
i
¯
h ((τ 1 +τ 2 ) ˆ
p+(φ 1 +φ 2 ) ˆ
q) sin
1
2 ¯
h
(τ 1 φ 2 − τ 2 φ 1 )
.
(I.21)
If we substitute Eq. (I.21) into Eq. (I.20) and take the expectation value of Eq. (I.20)
with respect to the state |ψ, we obtain
i ¯
h
1
2π ¯
h
∞
−∞
∞
−∞
dτ 0 dφ 0 c(τ 0 , φ 0 ) M(τ 0 , φ 0 )
= 2i
1
2π ¯
h
2 ∞
−∞
. . .
∞
−∞
dτ 1 dτ 2 dφ 1 dφ 2 a(τ 1 , φ 1 ) b(τ 2 , φ 2 )
× sin
1
2 ¯
h
(τ 1 φ 2 − τ 2 φ 1 )
M(τ 1 + τ 2 , φ 1 + φ 2 ).
(I.22)
We next use Eqs. (I.6) and (I.8) to find
i ¯
h
1
2π ¯
h
∞
−∞
∞
−∞
dp 0 dq 0 C(p 0 , q 0 ) f (p 0 , q 0 )
= 2i
1
2π ¯
h
3 ∞
−∞
. . .
∞
−∞
dτ 1 dτ 2 dφ 1 dφ 2 dpdq f (p, q)
× sin
1
2 ¯
h
(τ 1 φ 2 − τ 2 φ 1 )
× a(τ 1 , φ 1 ) e
i
¯
h (τ 1 p+φ 1 q) b(τ 2 , φ 2 ) e
i
¯
h (τ 2 p+φ 2 q)
=
2i
2π ¯
h
∞
−∞
∞
−∞
dpdq f (p, q) sin
¯
h
2
∂
∂p B
∂
∂q A
−
∂
∂p A
∂
∂q B
× A(p, q) B(p, q),
(I.23)
where
∂
∂p A
and
∂
∂q A
operate on the function A(p, q) and
∂
∂p B
and
∂
∂p B
operate on the
function B(p, q). From Eq. (I.23), we find that the phase function equivalent to the
commutator, Eq. (I.19), is the phase function relation
i ¯
hC(p, q) = (2i) sin
¯
h
2
∂
∂q A
∂
∂p B
−
∂
∂p A
∂
∂q B
A(p, q)B(p, q).
(I.24)
Equation (I.24) was first derived by Moyal and called the Moyal bracket. In the limit
¯
h → 0, Eq. (I.24) reduces to