ABSTRACT
The relativistic wave equations and field theories encountered in chapter 7 described only the properties of free, non-interacting particles. The wave equation for a free particle is always of the form (differential operator)φ = 0, and therefore the corresponding Lagrangians are quadratic in the fields. We have already seen that gauge theories give rise, in a natural way, to Lagrangians that contain terms of higher than quadratic order in the fields, and these terms describe interactions. In (8.41), for example, ψ¯ /Aψ describes an interaction between a fermion and a gauge field, while the higher-order terms in FaµνFaµν describe interactions of the gauge fields amongst themselves. It is, of course, only in the presence of interactions that physically interesting events can occur. At the same time, the physical interpretation of interacting quantum field theories is rather difficult. The interpretation of free field theories is based on expansions such as (7.80) in terms of solutions of the appropriate wave equation, the coefficients being interpreted as creation and annihilation operators. When a fermion interacts with a gauge field, the Dirac equation is modified as in (8.13). If the gauge field is itself an operator, this equation cannot be solved for ψ alone, and the plane-wave solutions of the free theory have no definite significance. It is, of course, possible to write the field as a Fourier transform, but the momentum kµ no longer satisfies the constraint kµkµ = m2. Although field operators still have the canonical commutation relations, such as (7.87) for Dirac spinors, the coefficients in the Fourier expansion no longer have the simple commutation relations required for creation and annihilation operators.
