5.2 Quantum Superposition and Matrix Distributions
89
The flavor eigenstates therefore mix under free evolution, leading in particular to a
forward scattering term [7, 14] in the evolution of f (in matrix notation)
∼ i
E p , f (x,
p)
.
(5.2.10)
This could therefore seem as a natural candidate for the energy matrix entering
(5.2.4). There are, however, two reasons why this cannot be the case. First, in the
limit case of a diagonal E p matrix we have that the f s =s components propagate at
the speed of light, independently of the mass scales on the diagonal. This situation is
clearly unphysical, as one would expect the off-diagonal terms to have a qualitatively
intermediate behavior, not a completely different one.
Second, this kind of energy matrix is not related to a mass matrix through some
on-shell condition holding individually for each component (5.2.5). Instead, it is
related to the corresponding mass matrix
(m
2
) ss := U
∗
rs m
2
r U rs ,
(5.2.11)
through a matrix relation
E p =
m 2 + +
p 2 ,
(5.2.12)
i.e. which mixes the matrix components, as opposed to Eq. (5.2.5). In particular,
this equation is not Lorentz-invariant, because such a transformation would lead
to a matrix of momenta as well.
4 In fact, choosing another basis than the mass
eigenstates is not a “Lorentz-friendly” operation to begin with, because under a
Lorentz transformation the flavor eigenstates become a superposition of kets with
different momenta
U
∗
ss | |
p, s
→ U
∗
ss |
i
j p
j
+
i
0 E p,s , s
.
(5.2.13)
In retrospect, we can now understand the problem as the impossibility of building
a Liouville operator that is both LLT-invariant and covariant under unitary transformations of the s index (change of Hilbert space basis)
f → U f U
†
,
(5.2.14)
if the components that are mixed belong to different mass shells. LLT-invariance
requires the form (5.2.4) and the relation (5.2.5), whereas the Hilbert basisindependence requires matrix-type multiplications of f ss and E p,ss and the matrix
relation (5.2.12). For this reason, one can only obtain a consistent generalized BUU
equation in the case where there are selection rules forbidding the superposition of
mass eigenstates with different mass, i.e.
4 The Hamiltonian operator (5.2.8) of course still transforms as the time-component of a Lorentz
vector, thanks to a non-trivial transformation of the ladder operators, but the E p,ss components do
not.
89
The flavor eigenstates therefore mix under free evolution, leading in particular to a
forward scattering term [7, 14] in the evolution of f (in matrix notation)
∼ i
E p , f (x,
p)
.
(5.2.10)
This could therefore seem as a natural candidate for the energy matrix entering
(5.2.4). There are, however, two reasons why this cannot be the case. First, in the
limit case of a diagonal E p matrix we have that the f s =s components propagate at
the speed of light, independently of the mass scales on the diagonal. This situation is
clearly unphysical, as one would expect the off-diagonal terms to have a qualitatively
intermediate behavior, not a completely different one.
Second, this kind of energy matrix is not related to a mass matrix through some
on-shell condition holding individually for each component (5.2.5). Instead, it is
related to the corresponding mass matrix
(m
2
) ss := U
∗
rs m
2
r U rs ,
(5.2.11)
through a matrix relation
E p =
m 2 + +
p 2 ,
(5.2.12)
i.e. which mixes the matrix components, as opposed to Eq. (5.2.5). In particular,
this equation is not Lorentz-invariant, because such a transformation would lead
to a matrix of momenta as well.
4 In fact, choosing another basis than the mass
eigenstates is not a “Lorentz-friendly” operation to begin with, because under a
Lorentz transformation the flavor eigenstates become a superposition of kets with
different momenta
U
∗
ss | |
p, s
→ U
∗
ss |
i
j p
j
+
i
0 E p,s , s
.
(5.2.13)
In retrospect, we can now understand the problem as the impossibility of building
a Liouville operator that is both LLT-invariant and covariant under unitary transformations of the s index (change of Hilbert space basis)
f → U f U
†
,
(5.2.14)
if the components that are mixed belong to different mass shells. LLT-invariance
requires the form (5.2.4) and the relation (5.2.5), whereas the Hilbert basisindependence requires matrix-type multiplications of f ss and E p,ss and the matrix
relation (5.2.12). For this reason, one can only obtain a consistent generalized BUU
equation in the case where there are selection rules forbidding the superposition of
mass eigenstates with different mass, i.e.
4 The Hamiltonian operator (5.2.8) of course still transforms as the time-component of a Lorentz
vector, thanks to a non-trivial transformation of the ladder operators, but the E p,ss components do
not.