The chief aim of this chapter is to give brief accounts of topics in nonrelativistic quantum mechanics that are not always treated in elementary texts. We begin with the Hilbert space formulation of quantum mechanics as set down by von Neumann in his book Mathematical Foundations of Quantum Mechanics (von Neumann, 1955), which will henceforth be referred to as von Neumann's book. Much of our concern will be with continuous spectra, which cannot be discussed adequately in the Dirac formalism. The density matrix, which will play a key role in Chapters 8 and 10, will be treated in some detail.

The section on formalism is followed by one on the probability interpretation; the latter is included because Sewell's theory of measurement suggests a subtle reformulation of a part of it. These are followed by sections on superselection rules and the Galilei group, which is the relativity group of quantum mechanics. They are based on the pioneering works of Wigner and Bargmann. The last section is devoted to the fundamental theorems of von Neumann and Stone, and to Reeh's observation on the physical significance of the failure of the Stone–von Neumann uniqueness theorem at the Lie algebra level.

The formalism of quantum mechanics

By quantum mechanics we shall mean the nonrelativistic quantum theory of a system with a finite number, N, of particles. The number N is assumed fixed. It will be convenient, for later recall, to divide the material into subsections.

The aim of this final chapter of Part II is rather different from that of the preceding chapters. It is to provide a glimpse into the generalizations of the formalism of quantum mechanics on Hilbert space that are required to describe the symmetries and dynamics of systems with infinitely many degrees of freedom, owing to the qualitative differences that arise when the number of degrees of freedom tends to infinity.

The burden of the preceding chapters was that the notion of a geometrical point is as meaningful in quantum physics as it is in classical physics. The argument involved a lengthy excursion into quantum-mechanical measurement theory. During this excursion, we found that the notion of design of the experiment played an essential role.

An experiment, no matter how ingenious, does not create laws of nature; it substantiates, or refutes, an assumed law. According to the dichotomy between laws of nature and initial conditions that was posited by Wigner (page 206), the role of human intervention in an experiment is to realize, under controlled and repeatable conditions, a conjunction of initial conditions that may be improbable in nature. Seen in this light:

1. (i) The failure of von Neumann's theory proper to explain wave function collapse is either a failure of the theory to establish control over complicated initial conditions or an indicator of as-yet-undiscovered laws of nature; the success of Sewell's theory suggests that it is the former, and not the latter.

2. […]

In the following we shall assume that the space, time and space-time of physics are made up of ‘points’ in the sense of Euclidean geometry. This assumption has been questioned by Wigner since 1952, and his reservations will be the subject of Part II of this book.

We shall begin by defining the notion of causality (which we shall call Einstein–Weyl causality) on a set of points devoid of any mathematical structure. It will be defined as a partial order on the set. Our point of departure is the observation that the propagation of a light ray determines a total order (the past-future order) on its path. We shall try to build up the causal structure by abstraction from the intersection properties of these paths in Minkowski space, and in real life.

Light rays

The fundamental objects in our scheme will be:

1. (i) A nonempty set of points M.

2. (ii) A distinguished family of subsets of M, called light rays.

The fundamental relation in our scheme will be a total order defined on the light rays.

We shall use the term light ray as a shorthand for the space-time path of a light ray; more precisely, for a mathematical abstraction from the corresponding physical concept. Light rays will generally be denoted by the letter l. A light ray through the point x will be denoted by lx, one through the points x, y by lx, y.

The quantitative data obtained in any physical experiment are recorded as finite, ordered sets of rational numbers. All such sets are discrete. However, when a physicist sits down to make sense of such data, the tools he or she employs are generally based upon the continuum: analytic (or at least smooth) functions, differential equations, Lie groups, and the like. It is the view of many eminent mathematicians that ‘bridging the gap between the domains of discreteness and of continuity … is a central, presumably even the central problem of the foundations of mathematics’, yet Fritz London did not seem to have had the slightest hesitation in writing, in the very first paragraph of his book on superfluidity, ‘that new differential equations were required to describe [the observed behaviour of]… “superfluid” helium…’ The physicist had stepped over the gap which has occupied philosophers for two millenia without even noticing that it existed!

This gap is but a fragment of one that separates theoretical from experimental physics. Some of the most important physicists of the first half of the twentieth century have expressed themselves on the subject, and it is instructive to compare their views. Dirac, for example, had the following to say:

This section provides a thumbnail sketch of those elements of point-set topology (also called general topology or just plain topology) that are used in this book. The subject grew out of attempts to rid the notion of continuity of its traditional dependence on the notion of distance. It turned out that continuity could be defined without using real numbers at all; the subject could be founded, instead, on the calculus of sets. Unfamiliarity with the latter is perhaps the main source of difficulty for the beginner.

Detailed treatments of the material discussed below may be found in standard textbooks such as (Kelley, 1955, Willard, 1970 and Munkres, 1975). Of these, the one by Munkres will perhaps be the easiest for the physicist.

Topological spaces

Point-set topology (usually called topology for short) may be regarded as the study of the notions of convergence of sequences1 and continuity of maps without using the notion of real numbers. In the theory of functions of a real variable, both of these notions are intimately related to that of neighbourhoods of a point. A neighbourhood of a point x on the real line is any subset that contains an open interval (x − a, x + a) around x, where a > 0; usually a is a small number, but it does not have to be.

In a certain sense, object and apparatus were essentially on the same footing in von Neumann's measurement theory. They were systems with k and l degrees of freedom respectively (page 146). There were no constraints on the numbersk and l, which could, for example, be of the same order of magnitude. Quite possibly, it was this lack of differentiation between object and apparatus that led Wigner to assert that ‘the state of the apparatus has no classical description’ (page 137), a state of affairs that produced the infinite von Neumann chain which only ended in the observer's consciousness.

In this and the following chapter we shall break with von Neumann and assume that k is small, i.e., of the order of unity, and that l is large, i.e., within a few orders of magnitude of Avogadro's number. The room for manoeuvre that this provides will allow us to break with Wigner and explore systems with states that do have classical descriptions. It will not surprise the informed reader that the room for manoeuvre created by our assumption will be filled, very substantially, by von Neumann's own work.

This chapter is divided into three sections. Section 9.1 is devoted to a theorem of von Neumann on observables that commute with each other. This prepares the way to our treatment of macroscopic observables, which is based on the commuting approximations to P and Q devised by von Neumann; these results, together with their antecedents, are presented in Section 9.2. In Section 9.3, an attempt is made to resolve some of Wigner's doubts.

Every student of physics is familiar with the Gibbs paradox, and its resolution by ‘correct Boltzmann counting’. The paradox arises because identical particles in classical physics are assumed distinguishable; the question of how they are to be distinguished is not asked. The resolution, correct Boltzmann counting, is equivalent to the assumption that even in classical mechanics, two point-particles with the same mass cannot be distinguished from each other.

The opposite is true in set theory. Take the assembly of geometrical points that constitute an open interval on a straight line, and recall that in Euclidean geometry a point has no structure. Now perform a thought-experiment in which two points are pulled out at random from the interval and shown to an observer in the next room. There is no way in which the observer can tell one from the other. However, admitting two identical objects in a set is a recipe for disaster (see the example on page 242); if one does, then it becomes impossible to define the notion of a function in a sensible manner. A set, in mathematics, has to be a collection of distinct objects, considered as a single entity.

How, then, is one supposed to understand ‘the set of points that constitute the real line’ or the two-dimensional plane? The short answer is: exactly as Gibbs understood an assembly of n classical point-particles – distinguishable. Familiarity with quantum mechanics may have made it counterintuitive to today's physicist, but it was clearly not counterintuitive to Gibbs, and does not appear to be counterintuitive to mathematicians.

The final result of Part I of this book, Theorem 5.10, was stated using the term differentiable manifold. Some complexities of the relation between Liegroup sand Lie algebras were encountered in Sections 7.4.3 and 7.5.2. One needs precise definitions of these mathematical objects to set these questions in their geometrical perspective. In Chapter 12 we made use of Banach and Hilbert bundles. These are special cases of fibre bundles. Fibre bundles also provide deep insights into Lie groups and Lie algebras. The purpose of this appendix is to provide the basic definitions, and brief introductions to these subjects that would suffice for our needs. We begin with the topological concept of a fibre bundle and continue with that of a differentiable manifold. The two are brought together through the notion of tangent spaces to lead us to the desired relation between Lie groups and Lie algebras.

Fibre bundles

The topological spaces called fibre bundles are generalizations of the topological product; they have the product form locally, but not globally. The simplest example is the Möbius strip. The rectangular strip of paper abcd shown in Fig. A8.1 may be formed into a cylinder by glueing the short edges together; a is joined to b, and c to d. However, if one gives the strip a twist and joins a to c and b to d, one obtains a Möbius strip.