Our purpose in this chapter is to present the key concepts of quantum mechanics in the language of Hilbert spaces. The reader who has not previously met the physical ideas motivating quantum mechanics, and some of the more elementary applications of Schrödinger's equation, is encouraged to read any of a number of excellent texts on the subject such as [1–4]. Otherwise, the statements given here must to a large extent be taken on trust – not an altogether easy thing to do, since the basic assertions of quantum theory are frequently counterintuitive to anyone steeped in the classical view of physics. Quantum mechanics is frequently presented in the form of several postulates, as though it were an axiomatic system such as Euclidean geometry. As often presented, these postulates may not meet the standards of mathematical rigour required for a strictly logical set of axioms, so that little is gained by such an approach. We will do things a little more informally here. For those only interested in the mathematical aspects of quantum mechanics and the role of Hilbert space see [5–8].

Many of the standard applications, such as the hydrogen atom, will be omitted here as they can be found in all standard textbooks, and we leave aside the enormous topic of measurement theory and interpretations of quantum mechanics. This is not to say that we need be totally comfortable with quantum theory as it stands. Undoubtedly, there are some philosophically disquieting features in the theory, often expressed in the form of socalled paradoxes.

Some algebraic structures have more than one law of composition. These must be connected by some kind of distributive laws, else the separate laws of composition are simply independent structures on the same set. The most elementary algebraic structures of this kind are known as rings and fields, and by combining fields and abelian groups we create vector spaces [1–7].

For the rest of this book, vector spaces will never be far away. For example, Hilbert spaces are structured vector spaces that form the basis of quantum mechanics. Even in non-linear theories such as classical mechanics and general relativity there exist local vector spaces known as the tangent space at each point, which are needed to formulate the dynamical equations. It is hard to think of a branch of physics that does not use vector spaces in some aspect of its formulation.

Rings and fields

A ringR is a set with two laws of composition called addition and multiplication, denoted a + b and ab respectively. It is required that R is an abelian group with respect to +, with identity element 0 and inverses denoted −a. With respect to multiplication R is to be a commutative semigroup, so that the identity and inverses are not necessarily present. In detail, the requirements of a ring are:

(R1) Addition is associative, (a + b) + c = a + (b + c).

(R2) Addition is commutative, a + b = b + a.

(R3) There is an element 0 such that a + 0 = a for all a ∈ R.

The cornerstone of modern algebra is the concept of a group. Groups are one of the simplest algebraic structures to possess a rich and interesting theory, and they are found embedded in almost all algebraic structures that occur in mathematics [1–3]. Furthermore, they are important for our understanding of some fundamental notions in mathematical physics, particularly those relating to symmetries [4].

The concept of a group has its origins in the work of Evariste Galois (1811–1832) and Niels Henrik Abel (1802–1829) on the solution of algebraic equations by radicals. The latter mathematician is honoured with the name of a special class of groups, known as abelian, which satisfy the commutative law. In more recent times, Emmy Noether (1888–1935) discovered that every group of symmetries of a set of equations arising from an action principle gives rise to conserved quantities. For example, energy, momentum and angular momentum arise from the symmetries of time translations, spatial translations and rotations, respectively. In elementary particle physics there are further conservation laws related to exotic groups such as SU(3), and their understanding has led to the discovery of new particles. This chapter presents the fundamental ideas of group theory and some examples of how they arise in physical contexts.

Elements of group theory

A group is a set G together with a law of composition that assigns to any pair of elements g, h ∈ G an element gh ∈ G, called their product, satisfying the following three conditions:

(Gp1) The associative law holds: g(hk) = (gh)k, for all g, h, k ∈ G.

After some twenty years of teaching different topics in the Department of Mathematical Physics at the University of Adelaide I conceived the rather foolhardy project of putting all my undergraduate notes together in one single volume under the title Mathematical Physics. This undertaking turned out to be considerably more ambitious than I had originally expected, and it was not until my recent retirement that I found the time to complete it.

Over the years I have sometimes found myself in the midst of a vigorous and at times quite acrimonious debate on the difference between theoretical and mathematical physics. This book is symptomatic of the difference. I believe that mathematical physicists put the mathematics first, while for theoretical physicists it is the physics which is uppermost. The latter seek out those areas of mathematics for the use they may be put to, while the former have a more unified view of the two disciplines. I don't want to say one is better than the other – it is simply a different outlook. In the big scheme of things both have their place but, as this book no doubt demonstrates, my personal preference is to view mathematical physics as a branch of mathematics.

The classical texts on mathematical physics which I was originally brought up on, such as Morse and Feshbach [7], Courant and Hilbert [1], and Jeffreys and Jeffreys [6] are essentially books on differential equations and linear algebra. The flavour of the present book is quite different.

Topology does not depend on the notion of ‘size’. We do not need to know the length, area or volume of subsets of a given set to understand the topological structure. Measure theory is that area of mathematics concerned with the attribution of precisely these sorts of properties. The structure that tells us which subsets are measurable is called a measure space. It is somewhat analogous with a topological structure, telling us which sets are open, and indeed there is a certain amount of interaction between measure theory and topology. A measure space requires firstly an algebraic structure known as a σ-algebra imposed on the power set of the underlying space. A measure is a positive-valued real function on the σ-algebra that is countably additive, whereby the measure of a union of disjoint measurable sets is the sum of their measures. The measure of a set may well be zero or infinite. Full introductions to this subject are given in [1–5], while the flavour of the subject can be found in [6–8].

It is important that measure be not just finitely additive, else it is not far-reaching enough, yet to allow it to be additive on arbitrary unions of disjoint sets would lead to certain contradictions – either all sets would have to be assigned zero measure, or the measure of a set would not be well-defined. By general reckoning the broadest useful measure on the real line or its cartesian products is that due to Lebesgue (1875–1941), and Lebesgue's theory of integration based on this theory is in most ways the best definition of integration available.

For much of physics and mathematics the concept of a continuous map, provided by topology, is not sufficient. What is often required is a notion of differentiable or smooth maps between spaces. For this, our spaces will need a structure something like that of a surface in Euclidean space ℝn. The key ingredient is the concept of a differentiable manifold, which can be thought of as topological space that is ‘locally Euclidean’ at every point. Differential geometry is the area of mathematics dealing with these structures. Of the many excellent books on the subject, the reader is referred in particular to [1–14].

Think of the surface of the Earth. Since it is a sphere, it is neither metrically nor topologically identical with the Euclidean plane ℝ2. A typical atlas of the world consists of separate pages called charts, each representing different regions of the Earth. This representation is not metrically correct since the curved surface of the Earth must be flattened out to conform with a sheet of paper, but it is at least smoothly continuous. Each chart has regions where it connects with other charts – a part of France may find itself on a map of Germany, for example – and the correspondence between the charts in the overlapping regions should be continuous and smooth. Some charts may even find themselves entirely inside others; for example, a map of Italy will reappear on a separate page devoted entirely to Europe.

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’.