114
5 General-Relativistic Matrix Kinetic Theory
¯
f
±
(x,
p) ≡ f
±
(x,
p) ,
( p ∓ m) f
±
(x,
p) ≡ f
±
(x,
p) ( p ∓ m) ≡ 0 .
(5.8.29)
Their sum is the Fourier transform of the correlation function of free quantum fields
ψ
a
(X ) ¯
ψ
b
(Y ) ρ , so they are invariant under U(1)GTs in particular
˜
f
±
(x,
p) = e
−iqθ(x) f
±
(x,
p) e
iqθ(x)
≡ f
±
(x,
p) .
(5.8.30)
As in the photon case, we demand that the wave-functions u
±
s be Liouvilletransported
Lu
±
s := p
a
∂ a −
iba p
b
+ q F ia
∂
∂ p i +
1
4
bca γ
b
γ
c
u
±
s = 0 ,
(5.8.31)
for the corresponding BUU equations to become
L f
±
= ±
1
2m
C
±
ss u
±
s ¯
u
±
s ,
(5.8.32)
where now L is a straightforward generalization of (3.4.42) to two Dirac indices (the
second one in the conjugate representation)
L f
±
:= p
a
∂ a −
iba p
b
+ q F ia
∂
∂ p i
f
±
+
1
4
p
a
bca
γ
b
γ
c
, f
±
. (5.8.33)
However, as already noted in Sect. 3.4.2, any pair of Dirac indices can be turned
into a Lorentz index. More precisely, a bar-hermitian Dirac matrix such as f
± can
be decomposed in the Clifford algebra basis {1, γ
a
, iγ
[a
γ
b]
, γ
a
γ
5
, iγ
5
}. The general
solution to the algebraic conditions (5.8.29) takes the form [14]
f
±
(x,
p) =
1
4
I
±
(x,
p) + γ
a
γ
5 S
±
a (x,
p)
1 ±
p
m
,
(5.8.34)
where
p
a S
±
a (x,
p) ≡ 0 .
(5.8.35)
Indeed, this condition makes f
± bar-hermitian and it also implies that the (1± p/m)
factor in (5.8.34) can be put on either side of the square bracket, thus satisfying both
of the last two equations in (5.8.29). That this is the general solution is then due to
the fact that I
± and S
±
a have four independent components, for each sign, just like
the original matrices f
±
ss . The inverse relation is then simply
I
±
≡ Tr f
±
,
S
±
a ≡ Tr
γ a γ
5 f
±
.
(5.8.36)
Since Tr f
±
≡ f
±
ss , the I
± are the total number density in phase space of particles
and anti-particles, respectively, while the space-like pseudo-vectors S
±
a correspond