7 - Fields with spin

Summary

Non-relativistic quantum states of any given angular momentum S can be built out of the direct product of spin ½ states. Suitable linear combinations of these product states can be formed to produce the irreducible representation of the rotation group which describes angular momentum S. In a similar fashion, relativistic quantum fields describing particles with spin S can be built out of spin ½ fields. We shall, therefore, dwell on the nature of spin ½ fields at some length. Moreover, it is the basic spin ½ field, the Weyl spinor, which is used in the modern electro-weak theory of weak interactions, and so these spinors will be discussed extensively with the more commonly known Dirac field related to them. Since the behavior of integer-spin fields, such as the spin-one photon field, is described by tensor fields that should be familiar, not much attention will be paid to them. Throughout this chapter, we shall work in our observable world of four dimensions. Higher dimensional spaces, however, are needed in the regularization procedure produced by dimensional continuation which we employ throughout this book. Moreover, there has been considerable theoretical speculation which involves higher-dimension space times. Much of the machinery for the extension to higher dimensions appears in the problems at the end of this chapter. The problems also deal with some of the peculiarities of a world with two space-time dimensions.