92
5 General-Relativistic Matrix Kinetic Theory
where
U (T
, T ) ≡ T exp
−i
T
T
d ˜
T H int. ( ˜
T )
,
(5.3.5)
T is the time-ordering operator and H int. is the interaction Hamiltonian in the interaction picture, thus made of freely evolving fields.
Remember that we assume each mesoscopic space-time to consist of a homogeneous thermodynamic system, thus harboring a large number of particles, but that
is dilute enough so that collisions are rare. Put differently, the typical time spent in
free motion L free is much larger than the typical time scale of the collision event
L coll . Here this translates in the fact that, between two consecutive collisions, the
states have the time to reach asymptotic states, i.e. states that behave as free states.
Each collision can therefore be treated as a standard scattering process, i.e. from
asymptotic “in” state to asymptotic “out” state
ρ in := lim
T →−∞
ρ(T )
ρ out := lim
T →∞
ρ(T ) .
(5.3.6)
The states |ψ involved in ρ in,out are superpositions of Fock states | |
p 1 , s 1 , . . . ,
p n , s n
defined through creation operators a
†
p,s acting on a vacuum state |0
| |
p 1 , s 1 , . . . ,
p n , s n :=
2E p n ,s n a
†
p n ,s n
. . .
2E p 1 ,s 1 a
†
p 1 ,s 1
|0 ,
a
p,s |0 ≡ 0 .
(5.3.7)
Here s ∈ {1, . . . , D} is the aforementioned discrete index collectively parametrizing
spin states, flavor, species, particle/anti-particle pairs, etc. The ladder operators obey
canonical (anti-)commutation relations
[a
p,s , a
†
p ,s ] |s||s | = (2π)
3
δ
(3)
(
p − −
p
) δ ss ,
[a
p,s , a
p ,s ] |s||s | = 0 , (5.3.8)
where
[A, B] n := AB − (−1)
n B A ,
(5.3.9)
which then imply the following symmetries
| |
p 1 , s 1 , . . . ,
p k , s k , . . . ,
p l , s l , . . . ,
p n , s n = (−1)
|s k ||s l | | |
p 1 , s 1 , . . . ,
p l , s l , . . . ,
p k , s k , . . . ,
p n , s n .
(5.3.10)
The asymptotic states are eigenstates of the “asymptotic” Hamiltonian
H asy. :=
s
d
3 p
(2π) 3 E p,s N
p,s ,
E p,s :=
m 2
s + +
p 2 ,
(5.3.11)
where
N
p,s := a
†
p,s a
p,s ,
(5.3.12)
5 General-Relativistic Matrix Kinetic Theory
where
U (T
, T ) ≡ T exp
−i
T
T
d ˜
T H int. ( ˜
T )
,
(5.3.5)
T is the time-ordering operator and H int. is the interaction Hamiltonian in the interaction picture, thus made of freely evolving fields.
Remember that we assume each mesoscopic space-time to consist of a homogeneous thermodynamic system, thus harboring a large number of particles, but that
is dilute enough so that collisions are rare. Put differently, the typical time spent in
free motion L free is much larger than the typical time scale of the collision event
L coll . Here this translates in the fact that, between two consecutive collisions, the
states have the time to reach asymptotic states, i.e. states that behave as free states.
Each collision can therefore be treated as a standard scattering process, i.e. from
asymptotic “in” state to asymptotic “out” state
ρ in := lim
T →−∞
ρ(T )
ρ out := lim
T →∞
ρ(T ) .
(5.3.6)
The states |ψ involved in ρ in,out are superpositions of Fock states | |
p 1 , s 1 , . . . ,
p n , s n
defined through creation operators a
†
p,s acting on a vacuum state |0
| |
p 1 , s 1 , . . . ,
p n , s n :=
2E p n ,s n a
†
p n ,s n
. . .
2E p 1 ,s 1 a
†
p 1 ,s 1
|0 ,
a
p,s |0 ≡ 0 .
(5.3.7)
Here s ∈ {1, . . . , D} is the aforementioned discrete index collectively parametrizing
spin states, flavor, species, particle/anti-particle pairs, etc. The ladder operators obey
canonical (anti-)commutation relations
[a
p,s , a
†
p ,s ] |s||s | = (2π)
3
δ
(3)
(
p − −
p
) δ ss ,
[a
p,s , a
p ,s ] |s||s | = 0 , (5.3.8)
where
[A, B] n := AB − (−1)
n B A ,
(5.3.9)
which then imply the following symmetries
| |
p 1 , s 1 , . . . ,
p k , s k , . . . ,
p l , s l , . . . ,
p n , s n = (−1)
|s k ||s l | | |
p 1 , s 1 , . . . ,
p l , s l , . . . ,
p k , s k , . . . ,
p n , s n .
(5.3.10)
The asymptotic states are eigenstates of the “asymptotic” Hamiltonian
H asy. :=
s
d
3 p
(2π) 3 E p,s N
p,s ,
E p,s :=
m 2
s + +
p 2 ,
(5.3.11)
where
N
p,s := a
†
p,s a
p,s ,
(5.3.12)