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.

2004-03-18