5.8 Liouville-Transported Wave-Functions and Tensor Distributions
115
to the intrinsic spin direction when the fermion fluid is polarized (or in this case
“magnetized”). Indeed, Tr
γ
a
γ
5 f
±
is the statistical analogue of the spin pseudocurrent of the Dirac field given in Eq. (3.2.77). In terms of these variables, the BUU
equation (5.8.32) reads
LI
±
= Tr C
±
,
LS
±
a = Tr
γ a γ
5 C
±
,
(5.8.37)
where in the latter L is the Liouville operator acting on Lorentz vectors (3.4.42). As
in the case of the photon distribution, here too both sides are consistently normal to
p
a , because the latter commutes with L.
5.9 Intrinsic Moment Sources
Until now we have considered only one way in which the matter distribution f ss (x,
p)
affects the space-time fields e
a
μ (x) and A μ (x) − through the total number of particles
in phase space. Indeed, in both the energy momentum tensor T
ab and the electric
current J
a it is the trace of each block of f ss that is involved (the I components in the
language of the previous section) and this is nothing but the statistical expectation
value of the QFT number operators N
p,s := a
†
p,s a
p,s . For example, a Dirac particle of
mass m and charge q corresponds to the following sources for the space-time fields
T
ab
(x) =
d
3 p
(2π) 3 E p
p
a p
b I (x,
p) ,
J
a
(x) = q
d
3 p
(2π) 3 E p
p
a I (x,
p) .
(5.9.1)
The question therefore arises of whether, and if so how, the polarization components
affect the space-time fields as well. To get some intuition about this issue, note that
in the case of electromagnetism, the above current will generate both electric and
magnetic fields. However, the latter will only be due to the motion of the total charge,
not to the intrinsic magnetic moment that comes from spin, and which lies in S
a ,
not I .
In the examples we considered, we saw that the polarization components arise as
Lorentz tensors V , S
a , P
ab , which means that they should be treated as “intrinsic”
(unresolved) multipole moments from the viewpoint of the space-time fields. We
must therefore look for effective couplings of the space-time fields to such moments
at the action level in order to derive the corresponding contributions at the level of
the equations of motion. As in any multipole expansion, these moments couple to
derivatives of e
a
μ and A μ such as the spin connection
ab
μ , the field strengths F ab and
R abcd and derivatives thereof. Consequently, the corresponding terms in the sources
T
ab and J
a come with more space-time derivatives than the ∼ I contributions of Eq.
(5.9.1). Since we are working with long wave-length modes, these new terms would
therefore appear as “small” corrections to (5.9.1). However, this is really the case only
if there are no “cancellations” in I . In the case of T ab , the energy is a positive-definite
115
to the intrinsic spin direction when the fermion fluid is polarized (or in this case
“magnetized”). Indeed, Tr
γ
a
γ
5 f
±
is the statistical analogue of the spin pseudocurrent of the Dirac field given in Eq. (3.2.77). In terms of these variables, the BUU
equation (5.8.32) reads
LI
±
= Tr C
±
,
LS
±
a = Tr
γ a γ
5 C
±
,
(5.8.37)
where in the latter L is the Liouville operator acting on Lorentz vectors (3.4.42). As
in the case of the photon distribution, here too both sides are consistently normal to
p
a , because the latter commutes with L.
5.9 Intrinsic Moment Sources
Until now we have considered only one way in which the matter distribution f ss (x,
p)
affects the space-time fields e
a
μ (x) and A μ (x) − through the total number of particles
in phase space. Indeed, in both the energy momentum tensor T
ab and the electric
current J
a it is the trace of each block of f ss that is involved (the I components in the
language of the previous section) and this is nothing but the statistical expectation
value of the QFT number operators N
p,s := a
†
p,s a
p,s . For example, a Dirac particle of
mass m and charge q corresponds to the following sources for the space-time fields
T
ab
(x) =
d
3 p
(2π) 3 E p
p
a p
b I (x,
p) ,
J
a
(x) = q
d
3 p
(2π) 3 E p
p
a I (x,
p) .
(5.9.1)
The question therefore arises of whether, and if so how, the polarization components
affect the space-time fields as well. To get some intuition about this issue, note that
in the case of electromagnetism, the above current will generate both electric and
magnetic fields. However, the latter will only be due to the motion of the total charge,
not to the intrinsic magnetic moment that comes from spin, and which lies in S
a ,
not I .
In the examples we considered, we saw that the polarization components arise as
Lorentz tensors V , S
a , P
ab , which means that they should be treated as “intrinsic”
(unresolved) multipole moments from the viewpoint of the space-time fields. We
must therefore look for effective couplings of the space-time fields to such moments
at the action level in order to derive the corresponding contributions at the level of
the equations of motion. As in any multipole expansion, these moments couple to
derivatives of e
a
μ and A μ such as the spin connection
ab
μ , the field strengths F ab and
R abcd and derivatives thereof. Consequently, the corresponding terms in the sources
T
ab and J
a come with more space-time derivatives than the ∼ I contributions of Eq.
(5.9.1). Since we are working with long wave-length modes, these new terms would
therefore appear as “small” corrections to (5.9.1). However, this is really the case only
if there are no “cancellations” in I . In the case of T ab , the energy is a positive-definite