5 - Differential Transition Probabilities and Predictions

Summary

Providing the connection between theory and experiment by transforming the invariant amplitude into a prediction for probabilities of final states.

Introduction

In the last chapter we derived the Feynman formalism for computing scattering amplitudes. In this chapter, we take the scattering amplitude and convert it into probability distributions for prediction and comparison with experiment.

Actually, the only example of interacting scalars in nature are the three pions, but their interactions are greatly complicated by non-perturbative effects. Even the magnitude of the effective coupling constant is too large for perturbation theory to make sense beyond the tree level. Hence, we can hardly make any realistic predictions at this point. The formulae of this chapter are introduced now to provide the final stage in the development of scalar field theory from foundations to predictions. Having covered this vertical range of scalar field theory, we will find it a simple matter to make a horizontal extension to include spinor and vector fields.

The central concept for making a prediction from an amplitude is the differential transition probability per unit time. The DTP/T is derived in Section 5.1. Sections 5.2 and 5.3 focus on one factor in the DTP/T, the invariant density of final states, providing convenient formulae for the two-particle and three-particle cases respectively.