5.5 The Collision Term
103
We can now compute C ss (x,
p) in Eq. (5.5.4). To find the commutator
N
p,s s , ,
,
where is given by Eq. (5.5.9), we use the Leibniz rule [N , AB] ≡ [N , A] B +
A [N , B], thanks to which we only need the elementary commutators
[N
p,s s , a
†
q,r ] ≡ (2π)
3 δ
(3) (
p − −
q) δ rs a
†
p,s ,
[N
p,s s , a
q,r ] ≡ −(2π)
3 δ
(3) (
p − −
q) δ rs a
p,s .
(5.5.13)
Given the symmetries of A c , all the commutators involving a creation (resp. annihilation) operator contribute the same, thus simply giving rise to a factorial weight.
The result can be written as
C ss (
p) ≡ C
+
ss (
p) − C
−
ss (
p) ,
(5.5.14)
where
C
+
ss (
p) :=
E p,s
√
2V T
∞
n,n ,m,m =0
1
n!m!n !m !
(5.5.15)
×
⎛
⎝
n
k=1
n
k =1
m
l=1
m
l =1
d 3 p k d 3 p
k d 3 q l d 3 q
l
(2π) 3 (2π) 3 (2π) 3 (2π) 3
2E p k ,s k 2E p
k ,s
k
2E q l ,r l 2E q
l ,r
l
⎞
⎠
×(2π) 4 δ (4)
⎛
⎝
n
k=1
p k −
m
l=1
q l
⎞
⎠ (2π) 4 δ (4)
⎛
⎝ p +
n
k =1
p
k −
m
l =1
q
l
⎞
⎠
×A ∗
c ( q 1 , r 1 , . . . ,
q m , r m → →
p 1 , s 1 , . . . ,
p n , s n )
×A c ( q
1 , r
1 , . . . ,
q
m , r
m → →
p
1 , s
1 , . . . ,
p
n , s
n ,
p, s)
××a
†
q m ,r m
. . . a
†
q 1 ,r 1
a
p 1 ,s 1
. . . a
p n ,s n a
†
p,s a
†
p
n ,s
n
. . . a
†
p
1 ,s
1
a
q
1 ,r
1
. . . a
q
m ,r
m
ρ ,
and
C
−
ss (
p) :=
E p,s
√
2V T
∞
n,n ,m,m =0
1
n!m!n !m !
(5.5.16)
×
⎛
⎝
n
k=1
n
k =1
m
l=1
m
l =1
d 3 p k d 3 p
k d 3 q l d 3 q
l
(2π) 3 (2π) 3 (2π) 3 (2π) 3
2E p k ,s k 2E p
k ,s
k
2E q l ,r l 2E q
l ,r
l
⎞
⎠
×(2π) 4 δ (4)
⎛
⎝
n
k=1
p k −
m
l=1
q l
⎞
⎠ (2π) 4 δ (4)
⎛
⎝
n
k =1
p
k −
m
l =1
q
l − p
⎞
⎠
×A ∗
c ( q 1 , r 1 , . . . ,
q m , r m → →
p 1 , s 1 , . . . ,
p n , s n )
×A c (
p, s ,
q
1 , r
1 , . . . ,
q
m , r
m → →
p
1 , s
1 , . . . ,
p
n , s
n )
××a
†
q m ,r m
. . . a
†
q 1 ,r 1
a
p 1 ,s 1
. . . a
p n ,s n a
†
p
n ,s
n
. . . a
†
p
1 ,s
1
a
p,s a
q
1 ,r
1
. . . a
q
m ,r
m
ρ .
By looking at the position of the (
p, s) entry in the scattering amplitude A c , we
recognize in C
± the “creation” and “annihilation” terms of the collision, respectively.
Here the 4-momenta appearing in the Dirac deltas are on-shell
103
We can now compute C ss (x,
p) in Eq. (5.5.4). To find the commutator
N
p,s s , ,
,
where is given by Eq. (5.5.9), we use the Leibniz rule [N , AB] ≡ [N , A] B +
A [N , B], thanks to which we only need the elementary commutators
[N
p,s s , a
†
q,r ] ≡ (2π)
3 δ
(3) (
p − −
q) δ rs a
†
p,s ,
[N
p,s s , a
q,r ] ≡ −(2π)
3 δ
(3) (
p − −
q) δ rs a
p,s .
(5.5.13)
Given the symmetries of A c , all the commutators involving a creation (resp. annihilation) operator contribute the same, thus simply giving rise to a factorial weight.
The result can be written as
C ss (
p) ≡ C
+
ss (
p) − C
−
ss (
p) ,
(5.5.14)
where
C
+
ss (
p) :=
E p,s
√
2V T
∞
n,n ,m,m =0
1
n!m!n !m !
(5.5.15)
×
⎛
⎝
n
k=1
n
k =1
m
l=1
m
l =1
d 3 p k d 3 p
k d 3 q l d 3 q
l
(2π) 3 (2π) 3 (2π) 3 (2π) 3
2E p k ,s k 2E p
k ,s
k
2E q l ,r l 2E q
l ,r
l
⎞
⎠
×(2π) 4 δ (4)
⎛
⎝
n
k=1
p k −
m
l=1
q l
⎞
⎠ (2π) 4 δ (4)
⎛
⎝ p +
n
k =1
p
k −
m
l =1
q
l
⎞
⎠
×A ∗
c ( q 1 , r 1 , . . . ,
q m , r m → →
p 1 , s 1 , . . . ,
p n , s n )
×A c ( q
1 , r
1 , . . . ,
q
m , r
m → →
p
1 , s
1 , . . . ,
p
n , s
n ,
p, s)
××a
†
q m ,r m
. . . a
†
q 1 ,r 1
a
p 1 ,s 1
. . . a
p n ,s n a
†
p,s a
†
p
n ,s
n
. . . a
†
p
1 ,s
1
a
q
1 ,r
1
. . . a
q
m ,r
m
ρ ,
and
C
−
ss (
p) :=
E p,s
√
2V T
∞
n,n ,m,m =0
1
n!m!n !m !
(5.5.16)
×
⎛
⎝
n
k=1
n
k =1
m
l=1
m
l =1
d 3 p k d 3 p
k d 3 q l d 3 q
l
(2π) 3 (2π) 3 (2π) 3 (2π) 3
2E p k ,s k 2E p
k ,s
k
2E q l ,r l 2E q
l ,r
l
⎞
⎠
×(2π) 4 δ (4)
⎛
⎝
n
k=1
p k −
m
l=1
q l
⎞
⎠ (2π) 4 δ (4)
⎛
⎝
n
k =1
p
k −
m
l =1
q
l − p
⎞
⎠
×A ∗
c ( q 1 , r 1 , . . . ,
q m , r m → →
p 1 , s 1 , . . . ,
p n , s n )
×A c (
p, s ,
q
1 , r
1 , . . . ,
q
m , r
m → →
p
1 , s
1 , . . . ,
p
n , s
n )
××a
†
q m ,r m
. . . a
†
q 1 ,r 1
a
p 1 ,s 1
. . . a
p n ,s n a
†
p
n ,s
n
. . . a
†
p
1 ,s
1
a
p,s a
q
1 ,r
1
. . . a
q
m ,r
m
ρ .
By looking at the position of the (
p, s) entry in the scattering amplitude A c , we
recognize in C
± the “creation” and “annihilation” terms of the collision, respectively.
Here the 4-momenta appearing in the Dirac deltas are on-shell