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The object of mathematical physics is to describe the physical world in purely mathematical terms. Although it had its origins in the science of ancient Greece, with the work of Archimedes, Euclid and Aristotle, it was not until the discoveries of Galileo and Newton that mathematical physics as we know it today had its true beginnings. Newton’s discovery of the calculus and its application to physics was undoubtedly the defining moment. This was built upon by generations of brilliant mathematicians such as Euler, Lagrange, Hamilton and Gauss, who essentially formulated physical law in terms of differential equations. With the advent of new and unintuitive theories such as relativity and quantum mechanics in the twentieth century, the reliance on mathematics moved to increasingly recondite areas such as abstract algebra, topology, functional analysis and differential geometry. Even classical areas such as the mechanics of Lagrange and Hamilton, as well as classical thermodynamics, can be lifted almost directly into the language of modern differential geometry. Today, the emphasis is often more structural than analytical, and it is commonly believed that finding the right mathematical structure is the most important aspect of any physical theory. Analysis, or the consequences of theories, still has a part to play in mathematical physics – indeed, most research is of this nature – but it is possibly less fundamental in the total overview of the subject.
When we consider the significant achievements of mathematical physics, one cannot help but wonder why the workings of the universe are expressable at all by rigid mathematical laws’. Furthermore. how is it that purely human constructs, in the form of deep and subtle mathematical structures refined over centuries of thought, have any relevance at all? The nineteenth century view of a clockwork universe regulated deterministically by differential equations seems now to have been banished for ever, both through the fundamental appearance of probabilities in quantum mechanics and the indeterminism associated with chaotic systems. These two aspects of physical law, the deterministic and indeterministic, seem to interplay in some astonishing ways, the impact of which has yet to be fully appreciated. It is this interplay, however, that almost certainly gives our world its richness and variety. Some of these questions and challenges may be fundamentally unanswerable, but the fact remains that mathematics seems to be the correct path to understanding the physical world.
The aim of this book is to present the basic mathematical structures used in our subject, and to express some of the most important theories of physics in their appropriate mathematical setting. It is a book designed chiefly for students of physics who have the need for a more rigorous mathematical education. A basic knowledge of calculus and linear algebra, including matrix theory, is assumed throughout, but little else. While different students will of course come to this book with different levels of mathematical sophistication, the reader should be able to determine exactly what they can skip and where they must take pause. Mathematicians, for example, may be interested only in the later chapters, where various theories of physics are expressed in mathematical terms. These theories will not, however, be developed at great length, and their consequences will only be dealt with by way of a few examples.
The most fundamental notion in mathematics is that of a set, or ‘collection of objects’. The subject of this chapter is set theory – the branch of mathematics devoted to the study of sets as abstract objects in their own right. It turns out that every mathematical structure consists of a collection of sets together with some defining relations. Furthermore, as we shall see in Section 1.3, such relations are themselves defined in terms of sets. It is thus a commonly adopted viewpoint that all of mathematics reduces essentially to statements in set theory, and this is the motivation for starting with a chapter on such a basic topic.
The idea of sets as collections of objects has a non-rigorous, or ‘naive’ quality, although it is the form in which most students are introduced to the subject [1–4]. Early in the twentieth century, it was discovered by Bertrand Russell that there are inherent self-contradictions and paradoxes in overly simple versions of set theory. Although of concern to logicians and those mathematicians demanding a totally rigorous basis to their subject, these paradoxes usually involve inordinately large self-referential sets – not the sort of constructs likely to occur in physical contexts. Thus, while special models of set theory have been designed to avoid contradictions, they generally have somewhat artificial attributes and naive set theory should suffice for our purposes. The reader’s attention should be drawn, however, to the remarks at the end of Section 1.5 concerning the possible relevance of fundamental problems of set theory to physics. These problems, while not of overwhelming concern, may at least provide some food for thought
While a basic familiarity with set theory will be assumed throughout this book, it nevertheless seems worthwhile to go over the fundamentals, if only for the sake of completeness and to establish a few conventions. Many physicists do not have a good grounding in set theory, and should find this chapter a useful exercise in developing the kind of rigorous thinking needed for mathematical physics. For mathematicians this is all bread and butter, and if you feel the material of this chapter is well-worn ground, please feel free to pass on quickly.
Contents
- 1.1 Sets and logic
- 1.2 Subsets, unions and intersections
- 1.3 Cartesian products and relations
- 1.4 Mappings
- 1.5 Infinite sets
- 1.6 Structures
- 1.7 Category theory