Self-referential consistency and inconsistency

Universality implies self-referentiality

In the preceding chapters, we have mentioned on several occasions that there are good reasons to consider quantum mechanics as universally valid. Indeed, during the last 70 years quantum mechanics has not been disproved by a single experiment. In spite of numerous attempts to discover the limits of applicability and validity of this theory, there is no indication that the theory should be improved, extended, or reformulated. Moreover, the formal structure of quantum mechanics is based on very few assumptions, and these do not leave much room for alternative formulations. The most radical attempt to justify quantum mechanics, operational quantum logic, begins with the most general preconditions of a scientific language of physical objects, and derives from these preconditions the logico-algebraic structure of quantum mechanical propositions [Mit 78,86], [Sta 80]. There are strong indications that from these structures (orthomodular lattices, Baer*-semigroups, orthomodular posets, etc.) the full quantum mechanics in Hilbert space can be obtained. Although simple application of Piron's representation theorem [Pir 76] does not lead to the desired result, [Kel 80], [Gro 90], there are new and very promising results [Sol 95] which indicate that the intended goal may well be achieved within the next few years. Together with the experimental confirmation and verification of quantum mechanics, these quantum logical results strongly support the hypothesis that quantum mechanics is indeed universally valid.

Historical remarks

The quantum mechanical formalism discovered by Heisenberg [Heis 25] an Schrödinger [Schrö 26] in 1925 was first interpreted in a statistical sense by Born [Born 26]. The formal expressions p(φ,ai) = |(φ,φai)|2, i ∈ N, were interpreted as the probabilities that a quantum system S with preparation φ possesses the value ai that belongs to the state φai. This original Born interpretation, which was formulated for scattering processes, was, however, not tenable in the general case. The probabilities must not be related to the system S in state φ, since in the preparation φ the value ai of an observable A is in general not subjectively unknown but objectively undecided. Instead, one has to interpret the formal expressions p(ai, ai) as the probabilities of finding the value ai after measurement of the observable A of the system S with preparation φ. In this improved version, the statistical or Born interpretation is used in the present-day literature.

On the one hand, the statistical (Born) interpretation of quantum mechanics is usually taken for granted, and the formalism of quantum mechanics is considered as a theory that provides statistical predictions referring to a sufficiently large ensemble of identically prepared systems S(φ) after the measurement of the observable in question. On the other hand, the meaning of the same formal terms p(φ,ai) for an individual system is highly problematic.

In chapter 5 we saw how in quantum mechanics electrons are bound to nuclei so as to form atoms. We now give a rather abbreviated account of how atoms bind together to form molecules. There is more than one type of molecular binding. We shall confine our discussion to the type known as covalent binding. The possibility of this type of binding relies on an effect that is peculiar to quantum mechanics, the tunnel effect, which we have already encountered in chapter 3.

The ionised hydrogen molecule

The simplest molecule is the ionised hydrogen molecule, which consists of two protons and one electron. The Coulomb force between the two protons tends to push them apart; we investigate how the presence of the electron overcomes this repulsion and holds the molecule together.

An exact calculation is difficult, but we can discuss the general features of the bonding by making suitable approximations. As the protons are much heavier than the electron, we may neglect their motion compared with that of the electron, and so regard them as fixed. We show that the expectation value of the energy, considered as a function of the proton separation R, has a minimum for a certain value of R, so that there is a stable equilibrium configuration.

Suppose first that R is so large that in the vicinity of each of the protons the Coulomb field of the other is completely negligible.

This book, on the interpretation of quantum mechanics and the measurement process, has evolved from lectures which I gave at the University of Turku (Finland) in 1991 and later in several improved and extended versions at the University of Cologne. In these lectures as well as in the present book I have aimed to show the intimate relations between quantum mechanics and its interpretation that are induced by the quantum mechanical measurement process. Consequently, the book is concerned both with the philosophical, metatheoretical problems of interpretation and with the more formal problems of quantum object theory.

The book is based on the idea that quantum mechanics is valid not only for microscopic objects but also for the macroscopic apparatus used for quantum mechanical measurements. We illustrate the consequences of this assumption, which turn out to be partly very promising and partly rather disappointing. On the one hand we can give a rigorous justification of some important parts of the interpretation, such as the probability interpretation, by means of object theory (chapter 3). On the other hand, the problem of the objectification of measurement results leads to inconsistencies that cannot be resolved in an obvious way (chapter 4). This open problem has far-reaching consequences for the possibility of recognising an objective reality in physics.

The manuscript of this book was carefully written in TEX by Dipl. Phys. Falko Spiller. In addition, he proposed numerous small corrections and improvements of the first version of the text.

Radiative transitions

When a system changes its energy as the result of the emission or the absorption of a photon, it is said to undergo radiative transition. There are three kinds of radiative transition. In the presence of an electromagnetic field the system can absorb a photon, so that its energy is raised to a higher level. If the system is initially in a state other than the ground state, it can emit a photon and so shed energy. This can happen without the presence of any external electromagnetic field, in which case a spontaneous emission is said to occur. On the other hand, if the excited system is placed in an electromagnetic field that varies in time with the appropriate frequency, the probability that it will emit a photon can be greatly enhanced. In this case the emission process is known as stimulated emission. Stimulated emission is the basis of maser and laser action, which we describe in this chapter.

First, we construct a simple model that allows us to investigate how the internal structure of a quantum system changes as the result of a radiative transition. Although the model is very simple, it will allow us to understand the main features of the proper, more exact treatment. We shall deal mainly with absorption and stimulated emission, the two types of radiative transition where there is an external electromagnetic field.

The concept of measurement

Basic requirements

In this chapter we give a brief account of the quantum theory of measurement. As already mentioned in the preceding chapter, the quantum theory of measurement treats the object system, as well as the measuring apparatus, as proper quantum systems. Here we restrict our considerations to a proper quantum mechanical model of the measuring process that makes use of unitary premeasurements. Furthermore, we will be mainly concerned with ordinary discrete observables of the object system that are measured by an apparatus with a pointer observable which is also assumed to be an ordinary discrete observable. These restrictive assumptions are made here and throughout the entire book in order to simplify the problems as much as possible. The remaining open problems of consistency, completeness, self-referentiality, etc. can then be discussed without unnecessary additional complications.

In order to characterize the concept of measurement in quantum mechanics, we formulate some basic requirements that must be fulfilled by any measuring process. In many situations, one can add further postulates, but these additional requirements are not essential for the concept of measurement. The basic requirements are in accordance with the most general interpretation of quantum mechanics, the minimal interpretation, which has already been mentioned in chapter 1. There is a general interplay between interpretation and the quantum theory of measurement, since the postulates that characterize a given interpretation must be compatible with, and capable of being satisfied by, a corresponding model of the measuring process.