Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
since we have in this case:
ρii(t) = < νi | ˆρ | νi >= ρii(0) =| Uα1 | 2
ρij(t) = < νi | ˆρ | νj >= ρij(0)e −i(Ei−Ej)t = UαiU ∗ αj e−i(Ei−Ej)t
(C.17)
(C.18)
The diagonal terms called ”population” ρii are constant and the non-diagonal
terms ρij (with i = j) called ”coherence” are oscillatingat the Bohr frequence of
the considered transition. Finally, to calculate the survival probability of the νe
neutrino within the density matrix formalism is given by:
P(νe → νe) = < ψνe | ˆρ | ψνe >
= < ψνe | (ρ11 cosθ0 | ν1 > +ρ22 sin θ0 | ν2 > + ρ12 sin θ0 | ν1 > + ρ21 cosθ0 | ν2 >)
= cos 4 θ0 + sin 4 θ0 + cos 2 θ0 sin 2 θ0(e −i(E1−E2)t + e −i(E2−E1)t )
= 1 + 2 cos 2 θ0 sin 2 θ0 (cos((E2 − E1)t) − 1)
= 1 − sin 2 2θ0 sin 2
(E2 − E1)t
2
The statistical mixed states
In certain environments one can be interested to describe the statistical distribution
of the states. This is the case for neutrinos e.g in the Early Universe environment,
where they are at thermal equilibrium and therefore follow the Fermi-Dirac
statistics. To neutrinos νe, νµ, ντ is respectively associated the Fermi-Dirac distribution
fνe, fνµ, fντ respectively. It is thus interesting to use the density matrix
formalism. If we take the mean value of an observable  we obtain:
< Â >=
α=e,µ,τ
fνα < ψνα | Â | ψνα >=
α=e,µ,τ
fναTr(ˆρνα Â) = Tr(ˆρm Â) (C.20)
The statistical mixture of states can be described, with the same rule of calculation
that the mean values for the pure states, by the ”mean” density matrix,
average of the different density matrices of the system studied:
ˆρm =
fνα ˆρνα
(C.21)
α=e,µ,τ
To finish with the example of the Early Universe environment, the density matrix
representing the mixed ensemble of single neutrino states all with momentum p
can be written as the incoherent sum
ρpd 3 p =
dnνα | ψνα >< ψνα |, (C.22)
α=e,µ,τ
where dnνα is the local differential number density of να neutrinos with momentum
p in the d 3 p interval.
171
(C.19)