- Demonstrate that the four operators
transform under Lorentz transformation
as the covariant components of 4-vector, whereas the four operators
transform
as the contravariant components of the same vector.
- Demonstrate that Equation (11.29) is equivalent to Equations (11.24)-(11.26).
- 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 (11.30) and (11.31) satisfy the anti-commutation relations (11.29).
- Verify that the matrices (11.32) and (11.33) satisfy Equations (11.24)-(11.26).
- Verify that if
- Verify that Equation (11.77) is a solution of Equations (11.76).
- A Lorentz transformation between frames
and
takes the form
- Show that the transformation rule for spinor wavefunctions associated with a Lorentz transformation from frame
to some frame
moving with velocity
with respect to
is
, where
- Consider the spinors

Verify that the are solutions of the Dirac equation in free space corresponding to electrons of energy , momentum , and spin angular momentum parallel to the -axis , where for , and for . - Show that the four solutions of the Dirac equation corresponding to an electron
of energy
and momentum
moving in free space take the form

Here, . Demonstrate that these spinors become identical to the of the previous exercise in the limit that . - Verify that the
matrices
, defined in Equation (11.98), satisfy the standard anti-commutation
relations for Pauli matrices: that is,