     Next: Lorentz Boosts Up: Proof of Covariance Previous: Proof of Covariance

## Rotations

Consider a rotation through an angle about the -axis: (5.140)

The matrix has all zero elements accept for , or . Thus (5.141)

Therefore (5.142)

For a rotation about an arbitrary axis , we write (5.143)

The appearance of the half-angle is an expression of the double-valuedness of the spinor law of rotation; it takes a rotation of to return to its original value. This is a characteristic of 1/2 integer spin. Therefore physical observables in the Dirac theory must be bilinear, or an even power in . By using the rotation operator upon the solution for the Dirac particle at rest and polarized in the -direction, it is possible to form states of any arbitrary direction. Since is hermitian, for spatial rotations. We shall henceforth call the wave function of the Dirac theory a spinor.     Next: Lorentz Boosts Up: Proof of Covariance Previous: Proof of Covariance
Douglas M. Gingrich (gingrich@ ualberta.ca)
2004-03-18