Although there is a complete consensus among working physicists with respect to the practical and operational meanings of quantum states, and also a rather loosely formulated general philosophic view called the Copenhagen interpretation, a great deal of confusion and divergence of opinions exist as to the details of the measurement process and its effects upon quantum states. This paper reviews the current expositions of the measurement problem, limiting itself for lack of space primarily to the writings of physicists; it calls attention to inconsistencies and proposes resolutions. Except for a summary of the properties of statistical matrices which are needed in Part II, the first part is non-mathematical and deals largely with two kinds of probability, reducible and irreducible probabilities, which need to be distinguished for a proper understanding of the measurement act.
Although there is a complete consensus among working physicists with respect to the practical and operational meanings of quantum states, and also a rather loosely formulated general philosophic view called the Copenhagen interpretation, a great deal of confusion and divergence of opinions exist as to the details of the measurement process and its effects upon quantum states. This paper reviews the current expositions of the measurement problem, limiting itself for lack of space primarily to the writings of physicists; it calls attention to inconsistencies and proposes resolutions. Except for a summary of the properties of statistical matrices which are needed in Part II, the first part is non-mathematical and deals largely with two kinds of probability, reducible and irreducible probabilities, which need to be distinguished for a proper understanding of the measurement act.
This is the second, mathematically more detailed part of a paper consisting of two articles, the first having appeared in the immediately preceding issue of this Journal∗. It shows that a measurement converts a pure case into a mixture with reducible probabilities. The measurement as such permits no inference whatever as to the state of the physical system subjected to measurement after the measurement has been performed. But because the probabilities after the act are classical and therefore reducible, it is often possible to adjust them so that Von Neumann's projection postulate is true. Among the more specific features dealt with in Part II is the occurrence of negative joint probabilities for the measurement of non-commuting operators in certain (not all!) quantum states. The general conclusions reached are stated at the end of the article.
Introduction In this chapter I attempt to review a few somewhat loosely connected themes, which figure in recent work by physicists on decoherence and irreversibility in quantum mechanics. The looseness is forced upon us by the subject – any proper discussion inevitably drags in questions about thermodynamic reversibility and the various arrows of time, about the foundations of quantum mechanics and quantum measurement, about the nature of quantum coherence, and about the relationship between quantum and classical mechanics. There are inevitably many gaps in our understanding of these questions, both for the physicist and for the philosopher. Some philosophers may feel that the subject is too immature to be interesting to them – that it is not yet ripe for philosophical analysis. Indeed I have recently heard the opinion expressed that quantum mechanics in its entirety was still not amenable to such analysis – that if the physicists could not yet understand it, there was no point in philosophers trying! I hope this feeling is not too widespread amongst philosophers of science – I suspect that it may arise in part because the philosophical community (along with a good part of the scientific community) is unaware of some important advances that have been made in the last decade in understanding the physics of quantum processes at the macroscopic level, as well as the relation between classical and quantum mechanics. Much of the chapter is therefore unashamedly tutorial in nature; however, given the wide range of topics covered, I imagine that even physicists will find at least some of the material to be novel.
In most textbooks, measurement does not receive the full attention it deserves and sometimes is even not treated at all, apart a brief and cryptic mention of the “reduction of the wave packet.” However, in the last decades, the situation has profoundly changed and it is time to consider measurement a fundamental part of quantum mechanics, even, to a certain extent, an important generalization of the traditional theory (see also Chs. 14–15). This chapter consists of three major parts. In the first block (Secs. 9.1–9.4) we develop the main physical features of the measurement process: the heart of the argument is here represented by the aspects related to the interpretation. In the second part (Secs. 9.5–9.8) we discuss several special (and partly interdependent) topics of measurement: the heart here is represented by experimental aspects. In the third part (Secs. 9.9–9.12) we deal with the measurement process on a more formal plane, making use, in particular, of the generalization represented by the concepts of effect and positive operator valued measure (POVM). As we have said, the measurement problem is one of the most fundamental issues in the conceptual structure of quantum mechanics (as described in Sec. 9.1) and has a long history that will be examined in Sec. 9.2. In this context, the existence of apparently paradoxical quantum states comes about: the so-called Schrödinger cat states (see Sec. 9.3).
This paper presents the measurement problem from the point of view of the thermal interpretation of quantum physics introduced in Part II. The measurement of a Hermitian quantity A is regarded as giving an uncertain value approximating the q-expectation ⟨A⟩ rather than (as tradition wanted to have it) as an exact revelation of an eigenvalue of A. Single observations of microscopic systems are (except under special circumstances) very uncertain measurements only. The thermal interpretation * treats detection events as a statistical measurement of particle beam intensity. * claims that the particle concept is only asymptotically valid, under conditions where particles are essentially free. * claims that the unmodeled environment influences the results enough to cause all randomness in quantum physics. * allows one to derive Born's rule for scattering and in the limit of ideal measurements; but in general, only part of Born's rule holds exactly: Whenever a quantity A with zero uncertainty is measured exactly, its value is an eigenvalue of A. * has no explicit collapse -- the latter emerges approximately in non-isolated subsystems. * gives a valid interpretation of systems modeled by a quantum-classical dynamics. * explains the peculiar features of the Copenhagen interpretation (lacking realism between measurements) and the minimal statistical interpretation (lacking realism for the single case) where these interpretations apply -- in the microscopic domain. The thermal interpretation is an interpretation of quantum physics that is in principle refutable by theoretical arguments leading to a negative answer to a number of open issues collected at the end of the paper, since there is plenty of experimental evidence for each of the points mentioned there.
THE conceptual problems raised by the discovery of quantum effects have not been fully resolved after half a century. Part of the reason for this is undoubtedly to be found in the mutual ignorance which prevails between physicists and philosophers. In his book Heisenberg brings together a philosophically inclined temperament with an unrivalled knowledge of physics. The result is a book of very great interest, however much one might disagree with his conclusions. The collection of essays of which the other book is made up is intended to bring crucial conceptual problems in physics especially to the notice of philosophers. The design and intention of the collection is admirable for this is the proper way to overcome the departmentalization of interest that has left the deep conceptual problems still unresolved. Men like Heisenberg who combine philosophical temperament with scientic knowledge of great distinction are rare, and yet the difficulties can only be cleared up by the application of philosophical techniques of analysis to problems which are an intimate and difficult part of physics. A great deal of the interest of Heisenberg's book lies in the very comprehensive account he gives of the origin and growth of the Copenhagen school, and the way the new phenomena forced a largely unwelcome picture of nature upon Bohr and his friends. The most important chapter from a philosophical point of view is that in which Heisenberg defends this interpretation against the attacks to which it has been subjected. The essence of his argument for the interpretation can be summed up in two propositions. (i) The Copenhagen interpretation of physical theory, which depends upon the acceptance of the so-called Principle of Complementarity, is an inevitable and final interpretation. The reason for this is that there are only two possible general models for understanding given physical phenomena; a particle model or a wave model. These are exhaustive and exclusive because the former depends upon discontinuous and the latter upon continuous (using substance here in the traditional philosophical sense). If we find, as we do, that the two fundamental constituents of the universe, matter and radiation, each exhibit two classes of phenomena, one explainable only on a particle model and one on a wave model, then we are compelled to adopt a complementarity hypothesis and give up the idea of finding a single and uniform model for the understanding of phenomena. There are only two kinds of models; they are mutually exclusive; both are required; hence we either despair before an insoluble problem or adopt the Copenhagen interpretation. (2) Any serious criticism of this interpretation must offer an alternative
Ever since the beginning of quantum mechanics, the measurement problem has been a subject of sometimes discontinued but nevertheless recurrent concern. As students, most physicists have raised questions about it, but the spectacular and renewed successes of quantum theory were a strong incentive for them to work out applications without further worrying about foundations. In each generation, however, some dissatisfaction reemerges, which leads to slow progress, and we may expect that the measurement problem will some day be cleared up rather than forgotten. The situation recalls statistical mechanics, the foundations of which still give rise to scientific discussions, in spite of more than one century’s achievements. Actually, the paradoxes of irreversibility and of Maxwell’s demon share some common features with quantum measurements, to wit, they involve probabilities in a crucial fashion and their explanation requires elucidating the role of the observer.