Quantum Field Theory

In this chapter the basic definition of the quantum theory of fields will be presented, following what are essentially traditional lines. The point of view adopted here regarding the nature of the quantum theory is essentially the traditional one, on a conceptual level, although the mathematical tools used are not those commonly employed in the usual presentation of the subject. It is important to point out that this point of view is provisional, and will have to be changed, to some extent, later on.

Later analysis will show that this definition is not complete, due to the fundamental difficulties in dealing with the concepts of the physical observables and of the process of measurement, within the structure of the theory. Due to this some of the statements made here are provisional and will have to be somewhat changed later, such as the statement that the set of $n$-point correlation functions of a model determine all the physics of the model, which is a standard point of view of the traditional approach to the subject.

The formal relation of the theory with the mathematical structure of the statistical mechanics of lattice systems will be pointed out. A detailed mathematical development leading to the tools needed for the solution of the Gaussian model will be presented. This simple model will be solved in detail, including a complete discussion of its correlation functions. The same tools and ideas used here in the solution of this model will also be of much use in the future, for dealing with more complex models by means of approximative schemes such as perturbation theory, the mean-field method and the Gaussian approximation.

The introduction of external sources in the quantum theory will also be discussed in detail. These sources are interpreted as representations of classical objects within the theory, and will lead to the concept of the functional generators of the correlation functions, and ultimately to the important concept of the effective action. Unlike the usual treatment, all these objects will be defined directly on the Euclidean lattice. The physical interpretation of the effective action, as well as its relation to the classical limit of the theory, will be discussed in detail.

The main objective of this chapter is to establish that the quantum theory of fields can be defined and analyzed on the Euclidean lattice on essentially traditional lines, but in a way that is mathematically more solid, constructive and precise than the traditional approach. We will see that using this formalism one can recover all the results of the traditional formalism, in all that concerns the definition and calculation of the set of correlation functions of a given model.

The first two sections of this chapter are of a rather qualitative character, and constitute a survey of the important definitions and known phenomenological facts about the mathematical structure of the theory. The subsequent sections turn to a solid technical approach leading to the solution of the Gaussian model. Unlike the rest of this book, the last two sections of this chapter are developed in quite a general way, with applicability by no means limited to the Gaussian model.