88
5 General-Relativistic Matrix Kinetic Theory
continuous family of f ss (x,
p) distributions that cannot be “superposed” in some
quantum sense. Thus, the only generalization of the Liouville operator that is LLTinvariant is the trivial generalization of Eq. (3.4.38)
L f ss :=
E p,ss ∂ 0 + p
i ∂ i
f ss
(5.2.4)
+
0i0 E p,ss + q s E i
E p,ss +
0i j − i j0
E p,ss + q s ε i jk B k
p
j − i jk p
j p
k
∂
∂ p i f ss ,
where the energies E p,ss are of the usual form
E p,ss :=
m
2
ss + p 2 ,
(5.2.5)
for some set of masses m ss . Note that the charge array q s is unambiguous here,
because in that case we do have a selection rule (5.2.3). Equation (5.2.4) is therefore
simply the standard Liouville operator for each individual component f ss . Unfortunately, however, the form (5.2.4) treats the off-diagonal terms f s =s as distribution
functions associated with some effective particle of mass m ss , not as a distribution
measuring the quantum superposition of particles s and s
. The classical aspect of
the involved physics in this formalism is therefore at odds with the quantum interpretation of f ss . The fact that the latter is associated with propagation along two
different kinds of paths γ s and γ s because of m s = m s seems to require the notion
of quantum superposition to be somehow present already at the geometric level and
therefore goes beyond the present formalism. Consequently, the Liouville operator
(5.2.4) is bound to miss quantum effects of order O((m s − m s )/E) in this case. In
particular, having lost the quantum nature of f ss in that respect, there is no privileged
way of determining the m ss numbers out of m s .
Nevertheless, one could still hope for some guidance from QFT by noting that
m ss is reminiscent of the mass matrix in flavor space. In that case, the new 1-particle
states, i.e. the “flavor eigenstates”, are related by a unitary matrix
| |
p, s → U
∗
ss | |
p, s
,
(5.2.6)
so the corresponding annihilation operators are related by
a s → U ss a s ,
(5.2.7)
and the free Hamiltonian becomes
H 0 =
d
3 p
(2π) 3 E p,s a
†
p,s a
p,s →
d
3 p
(2π) 3 E p,ss a
†
p,s a
p,s ,
(5.2.8)
i.e. one involving a hermitian matrix of energies
E p,ss := U
∗
rs E p,r U rs .
(5.2.9)
5 General-Relativistic Matrix Kinetic Theory
continuous family of f ss (x,
p) distributions that cannot be “superposed” in some
quantum sense. Thus, the only generalization of the Liouville operator that is LLTinvariant is the trivial generalization of Eq. (3.4.38)
L f ss :=
E p,ss ∂ 0 + p
i ∂ i
f ss
(5.2.4)
+
0i0 E p,ss + q s E i
E p,ss +
0i j − i j0
E p,ss + q s ε i jk B k
p
j − i jk p
j p
k
∂
∂ p i f ss ,
where the energies E p,ss are of the usual form
E p,ss :=
m
2
ss + p 2 ,
(5.2.5)
for some set of masses m ss . Note that the charge array q s is unambiguous here,
because in that case we do have a selection rule (5.2.3). Equation (5.2.4) is therefore
simply the standard Liouville operator for each individual component f ss . Unfortunately, however, the form (5.2.4) treats the off-diagonal terms f s =s as distribution
functions associated with some effective particle of mass m ss , not as a distribution
measuring the quantum superposition of particles s and s
. The classical aspect of
the involved physics in this formalism is therefore at odds with the quantum interpretation of f ss . The fact that the latter is associated with propagation along two
different kinds of paths γ s and γ s because of m s = m s seems to require the notion
of quantum superposition to be somehow present already at the geometric level and
therefore goes beyond the present formalism. Consequently, the Liouville operator
(5.2.4) is bound to miss quantum effects of order O((m s − m s )/E) in this case. In
particular, having lost the quantum nature of f ss in that respect, there is no privileged
way of determining the m ss numbers out of m s .
Nevertheless, one could still hope for some guidance from QFT by noting that
m ss is reminiscent of the mass matrix in flavor space. In that case, the new 1-particle
states, i.e. the “flavor eigenstates”, are related by a unitary matrix
| |
p, s → U
∗
ss | |
p, s
,
(5.2.6)
so the corresponding annihilation operators are related by
a s → U ss a s ,
(5.2.7)
and the free Hamiltonian becomes
H 0 =
d
3 p
(2π) 3 E p,s a
†
p,s a
p,s →
d
3 p
(2π) 3 E p,ss a
†
p,s a
p,s ,
(5.2.8)
i.e. one involving a hermitian matrix of energies
E p,ss := U
∗
rs E p,r U rs .
(5.2.9)