In this chapter we are going to discuss some rather esoteric topics. Despite this nature they are of fundamental significance in relativistic quantum theory and have profound consequences. Clearly we could discuss a lot of topics that routinely occur in non-relativistic quantum theory under such a chapter heading. We will not do this, but only consider topics of specific importance in relativistic quantum theory.

We will start this chapter by introducing a new type of operator known as a projection operator. This is an operator that acts on some wave-function and projects out the part of the wavefunction corresponding to particular properties. In particular there are energy projection operators which can project out the positive or negative energy part of a wavefunction and spin projection operators which (surprise surprise!) project out the part of the wavefunction corresponding to a particular spin direction. Such operators form an essential part of the theory of high energy scattering. We will not be using them much in our discussion of scattering because our aim there is towards solid state applications. For further discussion of projection operators, the books by Rose (1961), Bjorken and Drell (1964) and Greiner (1990) are useful.

Secondly, we will look at some symmetries that occur in the Dirac equation.