Appendix 3 - Spin properties of fields and wave equations

Summary

This is not a book on field theory, so we do not wish to get involved in a comprehensive discussion of field equations. But the transformation laws for particle states examined in Section 2.4 shed an interesting light upon the problem of constructing fields for arbitrary spin-particles and upon the wave equations they satisfy.

In particular, concerning the Dirac equation, many readers will have followed the beautiful derivation by Dirac of his famous equation for spin-1/2 particles (See Dirac, 1947). Here we shall look at the Dirac equation from a different point of view which provides an alternative insight into the origin and meaning of the equation.

Relativistic quantum fields

The essence of the physical states that were discussed in Chapter 1 is that for a particle at rest they transform irreducibly under rotations. It would be possible to deal with quantum field operators that also had this property (Weinberg, 1964a), i.e. spin-s fields, which have only 2s + 1 components. This, as we shall see, is not very convenient for constructing Lagrangians and building-in symmetry properties so that, for example, we normally use a four-component field for spin-1/2 Dirac particles and a 4-vector A_μ to describe spin-1 mesons or photons etc.