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The facts about the theory found in the explorations described in this book have some important consequences on the treatment of some other models, namely the non-linear models of scalar fields usually referred to as the polynomial models and the sigma models. In the next volume of this series we intend to present what is currently known about this. The most important consequences are related to a new insight into and interpretation of the perturbative scheme of approximation for such models, including a critical review of the process known as perturbative renormalization, and the discovery of a connection of the lattice formalism with the metrical geometry of space-time, including the phenomenon of the generation of metric curvature by the quantum fields, under certain circumstances.
Some of the central concepts involved in the theory certainly need further research and development, such as, for example, the process of passing from Euclidean space to Minkowski space and vice-versa. Another central issue is the complete definition of the concept of observables. In the development presented in this book we established some of the necessary conditions for a quantity to be an observable, but the sufficiency of these conditions is still open to question. This topic is certainly related to the concept of the measurement process in the quantum theory, which we have not touched at all, and which is probably one of the most difficult aspects of the theory. The question of the realization of the statistical interpretation of measurements within the structure of the theory is also related to the issue of the process of measurement, and therefore open to further exploration and discussion.
Finally, the extension of these explorations to more realistic types of field, such as vector fields and fermionic fields, would be a very important step towards the completion of the structure. While the realization of vector fields on the lattice is a very well-known subject, the same is not true for fermionic fields, which certainly represents one the major difficulties to be faced in future developments. The work in this area also lacks access to a sufficiently simple non-linear model that would not suffer from the triviality characteristic of the polynomial and sigma models of scalar fields. By the requirement of simplicity we mean a model that could be treated in a precise and complete way with what is currently known about the realization of fields on the lattice, which therefore must exclude fermionic fields. A model like this, containing true interactions between particles, and yet technically manageable on the lattice, would constitute an important tool for the further exploration and development of the theory.
It seems to us that there is ample room for further activity along the lines presented in this book. The use of lattices and of stochastic simulations not only establishes a practical calculational tool for the subject, it also constitutes a language in which one can discuss in a mathematically precise yet simple and clear way the issues and problems of the subject. This language allows for the free and profitable use of the imagination in the exploration of the underlying structure of the theory, followed when necessary by the computational effort needed to obtain precise answers to the questions posed. It is currently true, and may turn out to be the case permanently, that the routine use of computational resources on a large scale is an essential part of the research in this area. Fortunately, we live in a time when the availability of computer resources to the individual is increasing in an exponential way, so that the future prospects are very promising for those who acquire the necessary skills in the world of informatics.