- #1

- 76

- 0

Let [tex]\Psi_i[/tex], i = 1,2 be two spinor fields, with field equation

[tex]\gamma^{\mu}\partial_{\mu}\Psi_i = - \sum_{j=1}^2 M_{ij} \Psi_j[/tex]

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

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:

[tex]i \gamma^{\mu}\partial_{\mu}\Psi_i - \frac{m_i c}{\hbar}\Psi_i = 0[/tex]

This gives me that the field equations equal to:

[tex]

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

[/tex]

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

[tex]

M \Psi = - \frac{c}{i\hbar} \left(\begin{array}{cc} m_1 e_4 & 0 \\

0 & m_2 e_4\end{array}\right)

[/tex]

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