Exercises

1. Noting that $\alpha_i=-\beta\,\alpha_i\,\beta$ , prove that the $\alpha_i$ and $\beta$ matrices all have zero trace. Hence, deduce that each of these matrices has $n$ eigenvalues $+1$ , and $n$ eigenvalues $-1$ , where $2\,n$ is the dimension of the matrices.
2. Verify that the matrices (1125) and (1126) satisfy Equations (1117)-(1119).
3. Verify that the matrices (1123) and (1124) satisfy the anti-commutation relations (1122).
4. Verify that if
   $$\partial_\mu\,j^{\,\mu} = 0,$$
   where $j^{\,\mu}$ is a 4-vector field, then
   $$\int d^3 x\,j^{\,0}$$
   is Lorentz invariant, where the integral is over all space, and it is assumed that $j^{\,\mu}\rightarrow 0$ as $\vert{\bf x}\vert\rightarrow \infty$ .
5. Verify that (1168) is a solution of (1167).
6. Verify that the $4\times 4$ matrices $\Sigma_i$ , defined in (1189), satisfy the standard anti-commutation relations for Pauli matrices: i.e.,
   $$\{\Sigma_i, \Sigma_j\} = 2\,\delta_{ij}.$$