3.4 Phase Space Fields
49
3.4.3 Distribution Moments
Given the volume form (3.4.45), we can use its Lorentz-invariant momentum factor
to define the moments of f s , i.e. the space-time Lorentz tensors
T
a 1 ...a n
s
(x) :=
d
3 p
(2π) 3 E p,s
f s (x,
p) p
a 1
s . . . p
a n
s ,
n > 0 , p
a
s := (E p,s ,
p) ,
(3.4.46)
of which the n = 1 and n = 2 cases are the particle number current vector and energy
momentum tensor of the s species, respectively, as measured by the observer family
e a . The electric current vector of the s species is then
J
a
s := q s T
a
s .
(3.4.47)
Note that here we focus exclusively on scalar distribution functions, a restriction that
will be justified in Sect. 5. In the absence of interactions, i.e. if L f s = 0 holds, the
moments obey the following conservation equation
∇ a 1 T
a 1 ...a n
s
= nq s F
(a 1
a 1
T
a 2 ...a n )
s
.
(3.4.48)
To see this, first express them as an integral over the 4-momenta p
a
T
a 1 ...a n
s
(x) = 2
p 0 >0
d
4 p
(2π) 4 2πδ
p a p
a
+ m
2
s
f L ,s (x, p) p
a 1 . . . p
a n , (3.4.49)
and use Eqs. (3.4.29) and (3.4.26) to get
∇ a 1 T
a 1 ...an
s
(x) = 2
p 0 >0
d 4 p
(2π) 4 2πδ
p a p
a + m
2
s
p
a 1 ∇
L
a 1
f L ,s (x, p)
p
a 2 . . . p
an (3.4.50)
= −2q s F
b
a 1
(x)
p 0 >0
d 4 p
(2π) 4 2πδ
p a p
a + m
2
s
p
a 1 . . . p
an ∂
∂ p b f L ,s (x, p)
= 2q s F
b
a 1
(x)
p 0 >0
d 4 p
(2π) 4 f L ,s (x, p)
∂
∂ p b
2πδ
p a p
a + m
2
s
p
a 1 . . . p
an
= 2nq s F
(a 1
a 1
(x)
p 0 >0
d 4 p
(2π) 4 f L ,s (x, p) 2πδ
p a p
a + m
2
s
p
a 2 . . . p
an )
≡ nq s F
(a 1
a 1
(x) T
a 2 ...an )
s
(x) .
In the case of the energy momentum tensor, the total one is given by
T
ab
:=
s
T
ab
s + T
ab
EM ,
(3.4.51)
49
3.4.3 Distribution Moments
Given the volume form (3.4.45), we can use its Lorentz-invariant momentum factor
to define the moments of f s , i.e. the space-time Lorentz tensors
T
a 1 ...a n
s
(x) :=
d
3 p
(2π) 3 E p,s
f s (x,
p) p
a 1
s . . . p
a n
s ,
n > 0 , p
a
s := (E p,s ,
p) ,
(3.4.46)
of which the n = 1 and n = 2 cases are the particle number current vector and energy
momentum tensor of the s species, respectively, as measured by the observer family
e a . The electric current vector of the s species is then
J
a
s := q s T
a
s .
(3.4.47)
Note that here we focus exclusively on scalar distribution functions, a restriction that
will be justified in Sect. 5. In the absence of interactions, i.e. if L f s = 0 holds, the
moments obey the following conservation equation
∇ a 1 T
a 1 ...a n
s
= nq s F
(a 1
a 1
T
a 2 ...a n )
s
.
(3.4.48)
To see this, first express them as an integral over the 4-momenta p
a
T
a 1 ...a n
s
(x) = 2
p 0 >0
d
4 p
(2π) 4 2πδ
p a p
a
+ m
2
s
f L ,s (x, p) p
a 1 . . . p
a n , (3.4.49)
and use Eqs. (3.4.29) and (3.4.26) to get
∇ a 1 T
a 1 ...an
s
(x) = 2
p 0 >0
d 4 p
(2π) 4 2πδ
p a p
a + m
2
s
p
a 1 ∇
L
a 1
f L ,s (x, p)
p
a 2 . . . p
an (3.4.50)
= −2q s F
b
a 1
(x)
p 0 >0
d 4 p
(2π) 4 2πδ
p a p
a + m
2
s
p
a 1 . . . p
an ∂
∂ p b f L ,s (x, p)
= 2q s F
b
a 1
(x)
p 0 >0
d 4 p
(2π) 4 f L ,s (x, p)
∂
∂ p b
2πδ
p a p
a + m
2
s
p
a 1 . . . p
an
= 2nq s F
(a 1
a 1
(x)
p 0 >0
d 4 p
(2π) 4 f L ,s (x, p) 2πδ
p a p
a + m
2
s
p
a 2 . . . p
an )
≡ nq s F
(a 1
a 1
(x) T
a 2 ...an )
s
(x) .
In the case of the energy momentum tensor, the total one is given by
T
ab
:=
s
T
ab
s + T
ab
EM ,
(3.4.51)