In general, perturbation theory is a mathematical method that is used to find approximative solutions to problems that cannot be solved exactly. Therefore, the starting point of perturbation theory is the exact solution to a related problem. Perturbation theory can be applied if the original problem can be reformulated by adding a ‘small’ term to the exactly solvable problem. Thus, perturbation theory gives rise to an expression for the original solution in terms of a power series in some ‘small’ parameter, which measures the deviation from the exactly solvable problem. The leading term in this power series is the solution to the exactly solvable problem, whereas the higher-order terms may be found by some iterative procedure. In order for the perturbation theory to work properly, the higher-order terms in the ‘small’ parameter need to become successively smaller, i.e. the power series should converge. However, normally in quantum field theory, the terms will not anymore become successively smaller at some specific order, since the number of possible Feynman diagrams will grow so fast that the terms will instead become successively larger. Thus, perturbation theory will only be successful up to this order, and then, it will break down.

This chapter is devoted to the study of perturbation theory, which is, nevertheless, a very powerful tool in quantum field theory. Note that the discussion in Sections 11.1–11.6 will mainly be performed for a real scalar field, i.e. a neutral Klein–Gordon field, but it can, of course, be naturally extended to other types of fields, such as a Dirac field, if the appropriate changes are made.