5.2 Quantum Superposition and Matrix Distributions
87
|s| = |s
|
⇒
f ss | s =s ≡ 0 ,
(5.2.2)
where |s| ∈ {0, 1} denotes the Grassmann parity
3 of the particle s, or of states with
different charge
q s = q s
⇒
f ss | s =s ≡ 0 .
(5.2.3)
Because of these rules, the f ss matrix will generically be in block diagonal form, with
each block corresponding to a subspace of superposable 1-particle states. Another
natural question is whether one should also double the
p entries in f ss (x,
p), since
p labels quantum oscillators just as s. As we will shortly see, the reason only one
p
dependence remains is nothing but the translational invariance in
X that we require
at the mesoscopic level.
To our knowledge, the corresponding matrix kinetic theory for f ss has been
initially developed on flat space-time in the context of neutrino flavor mixing [5–9]
and has also been applied to curved space-time for the case of CMB polarization [10–
13] and the fermionic case [14, 15], using either the background space-time approach
or the intermediate tetrad field approach (see also [16] for a summary of matrix
kinetic theory techniques and further references.). In the following subsections, we
will derive a generalization of the BUU equation for matrix distributions that includes
all the possible microscopic QFT processes. However, before we proceed, we must
first discuss some limitations of the formalism that are due to the classical nature of
the Liouville operator.
Remember that the Liouville operator (3.4.38) depends on a mass m, through E p ,
and even the distribution f (x,
p) := f L (x, E p ,
p) implicitly depends on a mass.
Thus, in the presence of two s indices in the distribution f ss , there is an ambiguity in
the choice of mass parameter, since the available ones form an array m s , not a matrix
m ss . If, in any set of superposable 1-particle states, all particles have the same mass,
then there is no ambiguity, but what if this is not the case? One could a priori think
that the QFT would then have a selection rule forbidding the superposition of such
particles, thus leading to a consistent f ss | s =s = 0 throughout evolution, if the initial
conditions satisfy that condition. However, it turns out that nature provides us with
at least one counter-example, which is relevant at cosmological scales: neutrinos.
Indeed, the mass eigenstates of neutrinos | |
p, s, i.e. those that do not mix under free
evolution, are mixed by the weak interactions. Thus, starting with f ss | s =s = 0 at
some time, we will have f ss | s =s = 0 at latter times. The question therefore remains:
what mass should one associate to the matrix element f ss given the array m s ?
A first guiding remark is that LLT covariance forces us to consider a definite mass
m ss for every f ss component. To see this, note that active LLTs (3.4.34) bring in
a mass dependence through E p and, for this to be a representation of the Lorentz
symmetry, that energy must be of the form E p =
m 2 + p 2 for some mass m, i.e.
to derive from the Lorentz-invariant condition p a p
a
+ m
2
= 0. Even if we chose to
work with the off-shell distribution f L ,ss (x, p), Liouville’s theorem guarantees that
evolution will not mix different mass shells, so we would just be working with a
3 That is, |s| = 0 for bosons and |s| = 1 for fermions.
87
|s| = |s
|
⇒
f ss | s =s ≡ 0 ,
(5.2.2)
where |s| ∈ {0, 1} denotes the Grassmann parity
3 of the particle s, or of states with
different charge
q s = q s
⇒
f ss | s =s ≡ 0 .
(5.2.3)
Because of these rules, the f ss matrix will generically be in block diagonal form, with
each block corresponding to a subspace of superposable 1-particle states. Another
natural question is whether one should also double the
p entries in f ss (x,
p), since
p labels quantum oscillators just as s. As we will shortly see, the reason only one
p
dependence remains is nothing but the translational invariance in
X that we require
at the mesoscopic level.
To our knowledge, the corresponding matrix kinetic theory for f ss has been
initially developed on flat space-time in the context of neutrino flavor mixing [5–9]
and has also been applied to curved space-time for the case of CMB polarization [10–
13] and the fermionic case [14, 15], using either the background space-time approach
or the intermediate tetrad field approach (see also [16] for a summary of matrix
kinetic theory techniques and further references.). In the following subsections, we
will derive a generalization of the BUU equation for matrix distributions that includes
all the possible microscopic QFT processes. However, before we proceed, we must
first discuss some limitations of the formalism that are due to the classical nature of
the Liouville operator.
Remember that the Liouville operator (3.4.38) depends on a mass m, through E p ,
and even the distribution f (x,
p) := f L (x, E p ,
p) implicitly depends on a mass.
Thus, in the presence of two s indices in the distribution f ss , there is an ambiguity in
the choice of mass parameter, since the available ones form an array m s , not a matrix
m ss . If, in any set of superposable 1-particle states, all particles have the same mass,
then there is no ambiguity, but what if this is not the case? One could a priori think
that the QFT would then have a selection rule forbidding the superposition of such
particles, thus leading to a consistent f ss | s =s = 0 throughout evolution, if the initial
conditions satisfy that condition. However, it turns out that nature provides us with
at least one counter-example, which is relevant at cosmological scales: neutrinos.
Indeed, the mass eigenstates of neutrinos | |
p, s, i.e. those that do not mix under free
evolution, are mixed by the weak interactions. Thus, starting with f ss | s =s = 0 at
some time, we will have f ss | s =s = 0 at latter times. The question therefore remains:
what mass should one associate to the matrix element f ss given the array m s ?
A first guiding remark is that LLT covariance forces us to consider a definite mass
m ss for every f ss component. To see this, note that active LLTs (3.4.34) bring in
a mass dependence through E p and, for this to be a representation of the Lorentz
symmetry, that energy must be of the form E p =
m 2 + p 2 for some mass m, i.e.
to derive from the Lorentz-invariant condition p a p
a
+ m
2
= 0. Even if we chose to
work with the off-shell distribution f L ,ss (x, p), Liouville’s theorem guarantees that
evolution will not mix different mass shells, so we would just be working with a
3 That is, |s| = 0 for bosons and |s| = 1 for fermions.