116
5 General-Relativistic Matrix Kinetic Theory
quantity and therefore all contributions add up constructively. In J
a , however, the
fact that q can have either sign may indeed lead to a total J
a that is many orders of
magnitude smaller that the individual contributions. This is particularly the case in
cosmology, since the universe is electrically neutral on large scales. For this reason,
one could a priori have that the intrinsic magnetization contribution ∼ S
a is of the
same order of magnitude as the ∼ I contribution. This is especially relevant, given
that there exist large scale magnetic fields whose origin still remains a mystery (see
for instance [20] and references therein). Here we will therefore work out the leading
order ∼ S
a part of J
a , but we will ignore the polarization corrections to T
ab .
The coupling of interest in the action is eF ab M
ab
/2, where M
ab
≡ −M
ba is a
magnetic moment, because this is relativistic generalization of the usual magnetic
dipole coupling
B ·
M, with M
i
:= ε
i jk M
jk
/2. Varying with respect to A μ , such a
term modifies the current of a Dirac particle as follows
J
a
→ q
d
3 p
(2π) 3 E p
p
a I + ∇ b M
ba
,
(5.9.2)
which remains conserved thanks to the antisymmetry of M
ab . We must next build
M
ab out of S
a . To that end, we note that the spin charge is ∼ ε abcd p
c S
d , since S
a
is the spin pseudo-vector and p
a selects the “time-component” in phase space. The
“spin moment” is therefore given by
S ab (x) := ε abcd
d
3 p
(2π) 3 E p
p
c S
d
(x,
p) .
(5.9.3)
To obtain a more familiar relation, we can use p a S
a
(x,
p) ≡ 0 to find
1
2
ε
i jk S
jk
(x) =
d
3 p
(2π) 3
δ
i j
−
p
i p
j
E 2
p
S
j
(x,
p) ≈
d
3 p
(2π) 3 S
i
(x,
p) , (5.9.4)
where the approximation holds if S
i
(x,
p) is supported on non-relativistic momenta
p E p . The magnetic moment is then given by the standard relation to the angular
momentum
M
ab
:=
gq
2m
S
ab
,
(5.9.5)
where g is the “gyromagnetic ratio” (g ≈ 2). Thus, the total source current for the
electromagnetic field receives a contribution (5.9.2) for each Dirac particle species.
5.10 Example: The Photon-Electron-Proton Fluid
As our particle content here we consider photons (γ), electrons (e) and protons
( p), neglecting the internal structure of the latter, thus effectively treating it as a
5 General-Relativistic Matrix Kinetic Theory
quantity and therefore all contributions add up constructively. In J
a , however, the
fact that q can have either sign may indeed lead to a total J
a that is many orders of
magnitude smaller that the individual contributions. This is particularly the case in
cosmology, since the universe is electrically neutral on large scales. For this reason,
one could a priori have that the intrinsic magnetization contribution ∼ S
a is of the
same order of magnitude as the ∼ I contribution. This is especially relevant, given
that there exist large scale magnetic fields whose origin still remains a mystery (see
for instance [20] and references therein). Here we will therefore work out the leading
order ∼ S
a part of J
a , but we will ignore the polarization corrections to T
ab .
The coupling of interest in the action is eF ab M
ab
/2, where M
ab
≡ −M
ba is a
magnetic moment, because this is relativistic generalization of the usual magnetic
dipole coupling
B ·
M, with M
i
:= ε
i jk M
jk
/2. Varying with respect to A μ , such a
term modifies the current of a Dirac particle as follows
J
a
→ q
d
3 p
(2π) 3 E p
p
a I + ∇ b M
ba
,
(5.9.2)
which remains conserved thanks to the antisymmetry of M
ab . We must next build
M
ab out of S
a . To that end, we note that the spin charge is ∼ ε abcd p
c S
d , since S
a
is the spin pseudo-vector and p
a selects the “time-component” in phase space. The
“spin moment” is therefore given by
S ab (x) := ε abcd
d
3 p
(2π) 3 E p
p
c S
d
(x,
p) .
(5.9.3)
To obtain a more familiar relation, we can use p a S
a
(x,
p) ≡ 0 to find
1
2
ε
i jk S
jk
(x) =
d
3 p
(2π) 3
δ
i j
−
p
i p
j
E 2
p
S
j
(x,
p) ≈
d
3 p
(2π) 3 S
i
(x,
p) , (5.9.4)
where the approximation holds if S
i
(x,
p) is supported on non-relativistic momenta
p E p . The magnetic moment is then given by the standard relation to the angular
momentum
M
ab
:=
gq
2m
S
ab
,
(5.9.5)
where g is the “gyromagnetic ratio” (g ≈ 2). Thus, the total source current for the
electromagnetic field receives a contribution (5.9.2) for each Dirac particle species.
5.10 Example: The Photon-Electron-Proton Fluid
As our particle content here we consider photons (γ), electrons (e) and protons
( p), neglecting the internal structure of the latter, thus effectively treating it as a