5.2 Quantum Superposition and Matrix Distributions
91
information is needed in order to evolve the quantities of interest and the cosmological observables are only sensitive to the trace of f since we cannot directly measure
the cosmic neutrino background polarization. In the absence of neutrino interactions,
other than gravitational, the only quantities that are required for closing the evolution
equations are the neutrino moments (5.2.18), which obey themselves a closed set of
hierarchical evolution equations. The f s =s information thus effectively drops out.
We can therefore accurately describe neutrinos in this formalism if we evolve all of
f up to the middle era and then simply retain Tr f for the rest of the integration.
5.3 Matrix Distribution from a QFT Density Matrix
We now provide the concrete relation between the matrix distribution f ss and the
mesoscopic QFT. We consider a given macroscopic space-time point x
μ and focus on
the QFT that lives in the corresponding mesoscopic space-time with coordinates X
a .
We will therefore omit the x
μ dependencies in what follows, although one should keep
in mind that all of the objects that we are about to define and use do depend on that
variable. This is simply because the LLTs are by definition Lorentz transformations
that can be different at every x
μ , so their action through unitary transformations on
quantum states or operators also depends on x
μ . Thus, these objects will depend on
x
μ for a generic observer family e a (x).
In the statistical context, the state of the system is described by a density matrix
ρ ≡ ρ(x), which can be formally written as
ρ =
ψ
p ψ |ψψ| ,
(5.3.1)
for some orthonormal basis |ψ. The p ψ correspond to the statistical probability of the
system being in the quantum state |ψ, so it is constant in mesoscopic time T := X
0 .
Consequently, ρ is T -dependent in the Schrödinger picture and T -independent is the
Heisenberg one, i.e. contrary to usual operators. Since the p ψ are probabilities we
have
Tr ρ ≡
ψ
p ψ ≡ 1 ,
ρ
†
≡ ρ ,
ψ|ρ|ψ ≡ p ψ ≥ 0 , ∀ψ , (5.3.2)
and the quantum statistical expectation value of some observable O is thus given by
O ρ := Tr [ρ O] ≡
ψ
p ψ ψ|O|ψ .
(5.3.3)
Here we choose to work in the interaction picture, so ρ evolves as
ρ(T
) = U (T
, T ) ρ(T ) U
†
(T
, T ) ,
(5.3.4)
91
information is needed in order to evolve the quantities of interest and the cosmological observables are only sensitive to the trace of f since we cannot directly measure
the cosmic neutrino background polarization. In the absence of neutrino interactions,
other than gravitational, the only quantities that are required for closing the evolution
equations are the neutrino moments (5.2.18), which obey themselves a closed set of
hierarchical evolution equations. The f s =s information thus effectively drops out.
We can therefore accurately describe neutrinos in this formalism if we evolve all of
f up to the middle era and then simply retain Tr f for the rest of the integration.
5.3 Matrix Distribution from a QFT Density Matrix
We now provide the concrete relation between the matrix distribution f ss and the
mesoscopic QFT. We consider a given macroscopic space-time point x
μ and focus on
the QFT that lives in the corresponding mesoscopic space-time with coordinates X
a .
We will therefore omit the x
μ dependencies in what follows, although one should keep
in mind that all of the objects that we are about to define and use do depend on that
variable. This is simply because the LLTs are by definition Lorentz transformations
that can be different at every x
μ , so their action through unitary transformations on
quantum states or operators also depends on x
μ . Thus, these objects will depend on
x
μ for a generic observer family e a (x).
In the statistical context, the state of the system is described by a density matrix
ρ ≡ ρ(x), which can be formally written as
ρ =
ψ
p ψ |ψψ| ,
(5.3.1)
for some orthonormal basis |ψ. The p ψ correspond to the statistical probability of the
system being in the quantum state |ψ, so it is constant in mesoscopic time T := X
0 .
Consequently, ρ is T -dependent in the Schrödinger picture and T -independent is the
Heisenberg one, i.e. contrary to usual operators. Since the p ψ are probabilities we
have
Tr ρ ≡
ψ
p ψ ≡ 1 ,
ρ
†
≡ ρ ,
ψ|ρ|ψ ≡ p ψ ≥ 0 , ∀ψ , (5.3.2)
and the quantum statistical expectation value of some observable O is thus given by
O ρ := Tr [ρ O] ≡
ψ
p ψ ψ|O|ψ .
(5.3.3)
Here we choose to work in the interaction picture, so ρ evolves as
ρ(T
) = U (T
, T ) ρ(T ) U
†
(T
, T ) ,
(5.3.4)