The general principles of relativistic quantum mechanics described in the previous chapter have so far been applied here only to states of a single stable particle. Such one-particle states by themselves are not very exciting — it is only when two or more particles interact with each other that anything interesting can happen. But experiments do not generally follow the detailed course of events in particle interactions. Rather, the paradigmatic experiment (at least in nuclear or elementary particle physics) is one in which several particles approach each other from a macroscopically large distance, and interact in a microscopically small region, after which the products of the interaction travel out again to a macroscopically large distance. The physical states before and after the collision consist of particles that are so far apart that they are effectively non-interacting, so they can be described as direct products of the one-particle states discussed in the previous chapter. In such an experiment, all that is measured is the probability distribution, or ‘cross-sections’, for transitions between the initial and final states of distant and effectively non-interacting particles. This chapter will outline the formalism used for calculating these probabilities and cross-sections.

‘In’ and ‘Out’ States

A state consisting of several non-interacting particles may be regarded as one that transforms under the inhomogeneous Lorentz group as a direct product of one-particle states. To label the one-particle states we use their four-momenta pμ, spin z-component (or, for massless particles, helicity) σ, and, since we now may be dealing with more than one species of particle, an additional discrete label n for the particle type, which includes a specification of its mass, spin, charge, etc.