Solve Neutrino Oscillations Homework: Eigenspinors & Particle Masses

[Pietjuh]

I have to make a homework problem about neutrino oscillations, but I already don't know how to answer the first question.
Let $$\Psi_i$$, i = 1,2 be two spinor fields, with field equation
$$\gamma^{\mu}\partial_{\mu}\Psi_i = - \sum_{j=1}^2 M_{ij} \Psi_j$$

where $$M_{ij}$$ is a hermitian matrix. Suppose this matrix has eigenvalues m1 and m2. Show that the eigenspinors of this matrix represent particles with mass $$m_1 = \frac{\hbar\mu_1}{c}$$ and $$m_2 = \frac{\hbar\mu_2}{c}$$

I know that each of the two spinorfields have to satisfy the dirac equation. But the field equations of these spinorfields are coupled equations, so I can't just make the correspondence. From the dirac equation I know that:

$$i \gamma^{\mu}\partial_{\mu}\Psi_i - \frac{m_i c}{\hbar}\Psi_i = 0$$

This gives me that the field equations equal to:

$$\sum_{j=1}^2 M_{ij}\Psi_j = - \frac{m_i c}{i\hbar}\Psi_i$$

But if we now compose a new vector, consisting of the two spinorfields joined in one, we get the following:

$$M \Psi = - \frac{c}{i\hbar} \left(\begin{array}{cc} m_1 e_4 & 0 \\<br /> 0 & m_2 e_4\end{array}\right)$$

But I don't really know how we can determine from this the masses of the particles. Can someone give me a hint?

 

[Physics Monkey]

The spinor fields $$\Psi_i$$ do not satisfy the usual Dirac equation. You have their equations of motion given to you. However, in order to interpret things as particles, you would like to put the equations of motion into the Dirac form. The key observation is that the derivative term in the equation of motion is just the identity in the 1,2 space. Hint: how does the derivative term act on the eigenspinors of the mass matrix?

Let me know if you need further help.