Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
C.4 The polarization vector formalism
The last formalism we would like to discuss for the evolution of neutrinos is the
Bloch vector formalism, also called the polarization vector formalism. Note that
we present here the case of only two flavors, therefore the Bloch vector is expanded
on a SU(2) basis, i.e the Pauli matrices. In 3 flavors, one can expand the Bloch
vector on the SU(3) basis, namely the Gell-Mann matrices. The Bloch vector
formalism is deeply connected to the density matrix one. Indeed, its definition is
via the density matrix:
ρ(t, p) = 1
2 [P0(t, p) + σ · P(t, p)] = 1
2
P0 + Pz Px − iPy
Px + iPy P0 − Pz
(C.23)
where σ is a vector made of the Pauli matrices.
⎛ ⎞
0 1
⎜ 1 0 ⎟
⎛ ⎞ ⎜ ⎟
σx ⎜ ⎟
⎜
σ = ⎝ σy ⎠ = ⎜ 0 −i ⎟
⎜
σz
⎜ i 0 ⎟ . (C.24)
⎟
⎜ ⎟
⎜ ⎟
⎝ 1 0 ⎠
0 −1
and the polarization vector P is :
⎛
P = ⎝
Px
Py
Pz
⎞
⎠. (C.25)
With Eq.(C.23), one can immediately see that P0 represent the trace of the density
matrix, Tr(ρ(t, p)) = P0, therefore P0 is the total neutrino density number
for a given momentum p. In an environment where the neutrinos are in equilibrium,
P0 is constant and can be normalized to 1 and Eq.(C.23) becomes then:
ρ(t, p) = 1
[1 + σ · P(t,p)] with 1 the unit matrix. Note that one can also define
2
a normalized polarization vector using:
ρ(t, p) = P0(t, p)
[1 + σ · P(t,p)]. (C.26)
2
Let us write the evolution equation for the polarization vector. If we expand
the two-flavor mixing matrix U on the Pauli matrices basis, one obtains:
U = cosθ0I + i sin θσy
(C.27)
Therefore in the polarization formalism the vacuum hamiltonian is :
H = ∆m2
− cos 2θ0 sin 2θ0
≡
4E sin 2θ0 cos 2θ0
∆m2
⎛ ⎞
− cos 2θ0
⎝ 0 ⎠ . (C.28)
4E
sin 2θ0
172