Most of the work in this book will be strictly perturbative. However it is important not to consider perturbation theory as the be-all and end-all of field theory. Rather, it must be looked on only as a systematic method of approximating a complete quantum field theory, with the errors under control. So in this chapter we will review the foundations of quantum field theory starting from the functional integral.

The purpose of this review is partly to set out the results on which the rest of the book is based. It will also introduce our notation. We will also list a number of standard field theories which will be used throughout the book. Some examples are physical theories of the real world; others are simpler theories whose only purpose will be to illustrate methods in the absence of complications.

The use of functional integration is not absolutely essential. Its use is to provide a systematic basis for the rest of our work: the functional integral gives an explicit solution of any given field theory. Our task will be to investigate a certain class of properties of the solution.

For more details the reader should consult a standard textbook on field theory.