The book starts with the definition of the quartic polynomial model of a single real scalar field, using the mathematical structure described in the previous book The Gaussian Model. Using the experience acquired in that earlier work, a heuristic solution of the model is proposed and explained. The critical behavior of the model is described, as well as the phenomenon of spontaneous symmetry breaking. A qualitative version of the critical diagram of the model is drawn.
The perturbation theory of the model is developed, and shown to be more in the character of a Gaussian approximation than of an approximative series. The heuristic solution is confirmed in this approximation, as well as the spontaneous symmetry breaking. The mathematical origin of the infinities which appear in the perturbative expansion is revealed in a very clear way. The physical implications of the phenomenon are also discussed. Some heuristic indications of triviality are pointed out.
The infinite-coupling limit of the model is worked out, leading to the non-linear sigma models, which is this simple case reduce to the Ising model. The mean-field approximation is introduced, and its use for the determination of the critical behavior of the models is discussed. Some extensions of the mean-field approximation are presented, leading to a discussion of the different types of boundary conditions that may be used in the definition of the models.
The perturbative calculation of the coupling constant of the polynomial model is examined, leading to a critical discussion of the concept of perturbative renormalization. The important physical distinction between a laboratory model such as this one and a truly physical model is pointed out. The relation of perturbative renormalization with the triviality of the model is discussed.
The above is the part that exists so far. What follows are the current plans for the completion of the book.
A study of the problem of triviality, using an approach based on external sources and the effective action. This provides motivation which leads to the introduction of vector fields.
A study of the generation of non-flat metrical geometry in the lattice, by the quantization of non-linear fields. This leads to the discussion of the relation of the theory with gravitation.