94
5 General-Relativistic Matrix Kinetic Theory
V := (2π)
3
δ
(3)
(
p = 0) ≡
(2π)
3
d 3 p
,
(5.3.15)
is the “volume” of the mesoscopic space-time, a singular constant that drops out
of the physical quantities. Making explicit the x
μ dependence of f
in,out
s
through the
one of ρ in,out , we get that f
in,out
s
(x,
p) has the interpretation of the average number
density of s-particles in phase space at T → ±∞, i.e. it is the Boltzmann distribution
f s (x,
p) before and after scattering. Now note that (5.3.14) appears as the diagonal
of the “correlation” function
f
in,out
ss
(
p,
p
) := V
−1
a
†
p ,s a
p,s ρ in,out .
(5.3.16)
Remember, however, that only homogeneous states should be considered in the mesoscopic space-time, meaning that
[
P, ρ] = 0 ,
(5.3.17)
where
P is the momentum operator, a condition that is consistently preserved under
evolution in T , because of the Jacobi identity of the commutator and the conservation
of
P
∂
∂T
[
P, ρ] = −i[
P, [H int. , ρ]] ≡ −[H int. , [ρ,
P]] − [ρ, [
P, H int. ]] = −[H, [ρ,
P]] − [ρ, [
P, H ]] = 0 .
(5.3.18)
Equation (5.3.17) then implies that the |ψ states appearing in Eq. (5.3.1) are eigenstates of
P, so
f
in,out
ss
(
p,
p
) ∼ δ
(3)
(
p − −
p
) .
(5.3.19)
In contrast, nothing keeps the s index from mixing, so we must consider the hermitian
matrix distribution
f
in,out
ss
(
p) := V
−1
N
p,s s ρ in,out ,
(5.3.20)
where we have defined
N
p,ss := a
†
p,s a
p,s ,
(5.3.21)
and we now understand the singular normalization V as canceling the one coming
from the Dirac delta in Eq. (5.3.19) evaluated at
p = =
p
. Note that, contrary to N
p,s ,
these operators do not necessarily commute among themselves
N
p,ss , N
q,rr
= (2π)
3
δ
(3)
(
p − −
q)
N
p,sr δ rs − N
p,rs δ sr
.
(5.3.22)
Moreover, their time-evolution in the interaction picture is
N
p,ss (T
) = e
i(E p,s −E p,s )(T
−T ) N
p,ss (T ) .
(5.3.23)
5 General-Relativistic Matrix Kinetic Theory
V := (2π)
3
δ
(3)
(
p = 0) ≡
(2π)
3
d 3 p
,
(5.3.15)
is the “volume” of the mesoscopic space-time, a singular constant that drops out
of the physical quantities. Making explicit the x
μ dependence of f
in,out
s
through the
one of ρ in,out , we get that f
in,out
s
(x,
p) has the interpretation of the average number
density of s-particles in phase space at T → ±∞, i.e. it is the Boltzmann distribution
f s (x,
p) before and after scattering. Now note that (5.3.14) appears as the diagonal
of the “correlation” function
f
in,out
ss
(
p,
p
) := V
−1
a
†
p ,s a
p,s ρ in,out .
(5.3.16)
Remember, however, that only homogeneous states should be considered in the mesoscopic space-time, meaning that
[
P, ρ] = 0 ,
(5.3.17)
where
P is the momentum operator, a condition that is consistently preserved under
evolution in T , because of the Jacobi identity of the commutator and the conservation
of
P
∂
∂T
[
P, ρ] = −i[
P, [H int. , ρ]] ≡ −[H int. , [ρ,
P]] − [ρ, [
P, H int. ]] = −[H, [ρ,
P]] − [ρ, [
P, H ]] = 0 .
(5.3.18)
Equation (5.3.17) then implies that the |ψ states appearing in Eq. (5.3.1) are eigenstates of
P, so
f
in,out
ss
(
p,
p
) ∼ δ
(3)
(
p − −
p
) .
(5.3.19)
In contrast, nothing keeps the s index from mixing, so we must consider the hermitian
matrix distribution
f
in,out
ss
(
p) := V
−1
N
p,s s ρ in,out ,
(5.3.20)
where we have defined
N
p,ss := a
†
p,s a
p,s ,
(5.3.21)
and we now understand the singular normalization V as canceling the one coming
from the Dirac delta in Eq. (5.3.19) evaluated at
p = =
p
. Note that, contrary to N
p,s ,
these operators do not necessarily commute among themselves
N
p,ss , N
q,rr
= (2π)
3
δ
(3)
(
p − −
q)
N
p,sr δ rs − N
p,rs δ sr
.
(5.3.22)
Moreover, their time-evolution in the interaction picture is
N
p,ss (T
) = e
i(E p,s −E p,s )(T
−T ) N
p,ss (T ) .
(5.3.23)