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- Noting that , prove that the and matrices all have zero trace. Hence, deduce that each of these matrices has eigenvalues , and eigenvalues , where is the dimension of the matrices.
- Verify that the matrices (1125) and (1126) satisfy Equations (1117)-(1119).
- Verify that the matrices (1123) and (1124) satisfy the anti-commutation relations (1122).
- Verify that if

where is a 4-vector field, then

is Lorentz invariant, where the integral is over all space, and it is assumed that as . - Verify that (1168) is a solution of (1167).
- Verify that the matrices , defined in (1189), satisfy the standard anti-commutation relations for Pauli matrices: i.e.,