# Neutrino Oscillation

1. Nov 13, 2008

### Unredeemed

I don't really understand why neutrino oscillation occurs so I looked it up on wikipedia:
http://en.wikipedia.org/wiki/Neutrino_oscillation

I went to the "Theory, graphically" section and it gives quite a good explanation.

However, there are a few details I don't get.

It says: "This flavor state is a combination of mass states" - How can that be?

and "However, each mass state is also made up of flavor states" - How can that be?

In short, it may be due to my lack of understand about flavour (I always assumed it was simply the name given to particles which are the same except in mass), but I don't understand how a mass state can be made up of a flavour state or vice versa.

Can anyone help?

2. Nov 13, 2008

### CompuChip

It's basically a matter of choosing a basis for your Hilbert space of states. The mass states have a well-defined mass (hence the name ). The flavour states are particular linear combinations of these, in the usual QM sense. If we have three mass states and three flavour states, we can write this in matrix form like
$$\begin{pmatrix} \nu_e \\ \nu_\mu \\ \nu_\tau \end{pmatrix} = U \begin{pmatrix} \nu_1 \\ \nu_2 \\ \nu_2 \end{pmatrix}$$
where the numeric indices denote the mass states. From certain symmetry considerations it follows that the 3x3 matrix U must be unitary. This picture is useful when doing oscillation calculations, because our experiments are not sensitive to, say, a neutrino with mass m1, but rather they detect a neutrino which is in the electron flavour state. Of course, we can also invert this relation. If we write $\nu_\mathrm{flavour} = U \nu_\mathrm{mass}$ and we know that $U^\dagger U = 1$ we can also write $\nu_\mathrm{mass} = U^\dagger \nu_\mathrm{flavour}$, expressing the mass eigenstates in terms of the flavour eigenstates. This is useful in theoretical calculations, for example when the actual expression is in terms of the mass eigenstates (which occur naturally in calculations). However, since our experiments detect flavour eigenstates, it is useful to rewrite such an expression in terms of those.

In a sense, the Hilbert space is "three-dimensional" and you can choose any three basis vectors you like, as you are used to in QM. Because in theoretical calculations we would like to have simple mass terms (e.g. terms like $m_1 \nu_1 + m_2 \nu_2 + \cdots$ rather than some complicated matrix expression where the mass terms mix the neutrinos) the mass-basis is useful. I don't think it requires explanation that it is also useful to choose a basis, such that in experiments we measure precisely these basis states directly (i.e. eigenvalues of these basis vectors).

I hope it makes more sense now.

3. Nov 13, 2008

### Unredeemed

Thanks, understood.