Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
Appendix C
Several possible formalisms for
the neutrino evolution equations
In the study of the neutrino propagation, people are indistinctly using at least four
kinds of formalism, all perfectly equivalent, to describe the evolution of neutrinos:
the Schrödinger equation for wave functions, the Schrödinger equation for the
evolution operators, the Liouville-Von Neumann equation for the density matrix,
and finally polarization vectors. Each of these formalisms are useful depending
on the specific problem to tackle. Wave functions may appear more natural to
use, but density matrices are more compact. Eventually, the polarization vector
formalism is currently the best way to interpret the complex behavior of neutrinos
in supernovae when they experience neutrino-neutrino interactions. We discuss
here only the vacuum case for simplicity . However the discussion can be easily
extended to the case where matter and neutrino-neutrino interactions are present.
C.1 The Schrödinger equation for wave functions
The evolution equation of neutrinos is actually a Schrödinger-like equation because,
unlike the Schrödinger equation, we are concerned with flavor evolution at
fixed energy (or fixed momentum) of relativistic leptons. The evolution equation
in vacuum in the flavour basis is then:
i ∂
⎛
⎝
∂t
Ψe
Ψµ
Ψτ
⎞
⎠ =
⎡
⎣T23T13T12
⎛
⎝
E1 0 0
0 E2 0
0 0 E3
⎞
⎠ T †
12T †
13T †
23
⎤⎛
⎦⎝
Ψe
Ψµ
Ψτ
⎞
⎠ , (C.1)
where T23, T13 and T12 where given in the first chapter. As said in the previous
appendix, for antineutrinos a global minus sign appear, the equation (C.1)
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