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Exercises

1. 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.

2. Demonstrate that Equation (11.29) is equivalent to Equations (11.24)-(11.26).

3. 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.

4. Verify that the matrices (11.30) and (11.31) satisfy the anti-commutation relations (11.29).

5. Verify that the matrices (11.32) and (11.33) satisfy Equations (11.24)-(11.26).

6. 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 .

7. Verify that Equation (11.77) is a solution of Equations (11.76).

8. A Lorentz transformation between frames and takes the form

If

where , and otherwise, then the transformation corresponds to an infinitesimal velocity boost, , parallel to the -axis. Show that if a finite boost is built up from a great many such boosts then the transformation matrix becomes

where is the velocity of frame relative to frame . Show that the corresponding transformation rule for spinor wavefunctions is , where

9. 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

and .

10. Consider the spinors

for . Here, for , and for . Moreover,

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 .

11. Show that the four solutions of the Dirac equation corresponding to an electron of energy and momentum moving in free space take the form

where

Here, . Demonstrate that these spinors become identical to the of the previous exercise in the limit that .

12. Verify that the matrices , defined in Equation (11.98), satisfy the standard anti-commutation relations for Pauli matrices: that is,

Next: Physical Constants Up: Relativistic Electron Theory Previous: Positron Theory
Richard Fitzpatrick 2016-01-22