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This is the first of a series of a few short books about the foundations of quantum field theory. It is not meant as a traditional textbook on that subject. Its intention is not to exhaust the subject, but to probe deeply into just a few essential topics of a conceptual nature. These topics are treated in a way that is mathematically more solid than what is usually the case in the subject without, however, any intention to aim at extreme generality or extreme mathematical rigor. The use of the mathematics is kept as elementary as possible throughout the text.
This first volume has as its subject the Gaussian model, which is the Euclidean version of the theory of the free scalar field. This model is treated mostly by analytical means, which are within reach of anyone with a solid background in physics. Only a very limited use of computation is involved, mostly in an auxiliary role. Since the Gaussian model is the only model that can be solved completely by analytical means, in all the space-time dimensions we will be interested in, it is essential that we take maximum advantage of it, in our exploration of the conceptual foundations of the theory.
The second volume of the series will address some non-linear models of scalar fields which can be understood as generalizations of the Gaussian model, namely the polynomial models and the sigma models. In this second volume the ideas and techniques developed in the present volume will be applied to these more complex models, in the same spirit in which they are used here, to probe into the foundations of the theory. Since in these non-linear models no exact analytical solutions can be found, the use of numerical computation will have a much more extensive role to play in the second volume.
The strong dependence on the use of rather large computational resources for the very large scale stochastic simulations needed to deal with this subject is a serious stumbling block for its teaching and dissemination. In fact, one of the criteria for the contents of the current volume is that they do not depend on such large-scale stochastic simulations. A third volume of the series will eventually cover the technical aspects of programming and of the use of computers and of free software, which are essential for the acquisition of a real measure of technical control over the subject.
This subject is still an open and incomplete chapter of physics, and is full of misunderstandings and misconceptions. The line of thought developed here is meant to point a way out of the state of confusion in which one currently finds the subject. However, this line of thought is very far from finished, and it is likely that most of the important work on it still lies ahead. These volumes are being written and published freely through the Internet in the hope that they may be useful for physicists involved with this difficult subject. Students and researchers troubled by the deficiencies of the usual approach to the subject may find here some food for thought.
All the source code of the programs used to produce the data and results presented in this book will be made available through the network. At a time when progress in science is so dependent on computer work, sometimes on massive amounts of it, it is a question of scientific integrity and of intellectual honesty that the source code used in science be made openly available to all those that may be interested in it. It is important that the source code be made available, not only in order to allow for its free and open criticism, but also to allow it to be used and improved by others, thus promoting cooperation in this most crucially important human activity.
Considering that the historical development of science, and consequently of the technology that springs from it, has given us the means to promote very easily and cheaply the publication of truly massive amounts of information, as well as the tools needed to retrieve very fast whatever part of this information we find relevant, it is only appropriate that we use this new technology for the dissemination and development of scientific ideas, and hence for the furthering of science.
We would like to acknowledge here the contribution to parts of the material contained within this book by former students and collaborators who were active in this subject in the past. The work on the discontinuity of the fields was done in collaboration with Dr. Timothy Edward Gallivan and Dr. See Kit Foong. Some of the work on states of particles was done in cooperation with Dr. Silvana Perez. Some contributions to specific things were made by Mr. Arnaldo Gomes de Oliveira Filho. When appropriate, due credit will be given in footnotes along the text. We would also like to thank those that helped by reading and correcting the manuscript at various stages of its production.