In this chapter we shall discuss a variety of topics that are not covered by Sewell's scheme. Our main conclusion will be that Sewell's scheme can be extended to cover these topics, and the extension provides adequate answers to the problems of measurement theory in quantum mechanics. We shall then attempt to meet the last three of Wigner's objections listed on page 158, and the problem arising from the failure of localizability in relativistic physics that was stressed by him. The material will be arranged as follows. In Section 11.1 we shall deal with an example of Bell that challenged the notion of quantum mechanics as it has been used so far in this book. In Section 11.2 we shall discuss a few extensions of Sewell's scheme, leaving aside the crucial extension to continuous spectra. That discussion, provided in Section 11.6, will be preceded by short accounts of the results of Araki and Yanase in Section 11.3, the impossibility theorems of Shimony and Busch in Section 11.4, and the Heisenberg cut in Section 11.5. The results of Araki and Yanase were obtained within von Neumann's measurement theory proper,1 whereas Shimony and others based their attempts on a class of modifications of it. Section 11.7 will be devoted to establishing the adequacy of Sewell's scheme, and Section 11.8 to meeting the objections of Wigner and providing our answer to the question that has induced the writing of Part II of this book.
In this chapter we shall discuss a variety of topics that are not covered by Sewell's scheme. Our main conclusion will be that Sewell's scheme can be extended to cover these topics, and the extension provides adequate answers to the problems of measurement theory in quantum mechanics. We shall then attempt to meet the last three of Wigner's objections listed on page 158, and the problem arising from the failure of localizability in relativistic physics that was stressed by him. The material will be arranged as follows. In Section 11.1 we shall deal with an example of Bell that challenged the notion of quantum mechanics as it has been used so far in this book. In Section 11.2 we shall discuss a few extensions of Sewell's scheme, leaving aside the crucial extension to continuous spectra. That discussion, provided in Section 11.6, will be preceded by short accounts of the results of Araki and Yanase in Section 11.3, the impossibility theorems of Shimony and Busch in Section 11.4, and the Heisenberg cut in Section 11.5. The results of Araki and Yanase were obtained within von Neumann's measurement theory proper,1 whereas Shimony and others based their attempts on a class of modifications of it. Section 11.7 will be devoted to establishing the adequacy of Sewell's scheme, and Section 11.8 to meeting the objections of Wigner and providing our answer to the question that has induced the writing of Part II of this book.
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: (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. […]
The newly revised edition of this well-known book categorizes prevalent views on measurement as described by quantum mechanics, discusses basics agreed to by many who on other issues disagree, and describes some of the problems. Although many paragraphs rest unchanged from the first (1991) edition, results found since 1991 are used to reorganize and sharpen the focus of the book, so that even the old paragraphs are now understood differently and more clearly. One advance in the second edition is a new and more general no-go theorem; another is separation of formal derivations from interpretation. The dozen or so definitions essential to following the book even casually are stated clearly. By this means the authors succeed pretty well in achieving their stated desire to make the book comprehensible to those who do not share their philosophic predispositions. While I differ in philosophy from the authors, attending to their discussion uncovered weaknesses in my own thinking and improved it. The mathematical language of quantum mechanics expresses a system under measurement by states (as rays in a Hilbert space) and a hamiltonian operator, distinct from the apparatus that produces outcomes, which is expressed by a positive operator-valued (POV) measure. In this way quantum language separates state preparation from state measurement. Often the Hilbert space is viewed as a tensor product of subspaces, allowing further conceptual separations. The authors take the stance that quantum mechanics calls for splitting the empirical world into parts, the most essential being (1) objective systems S (to be observed) and (2) apparatus A (preparation and registration devices that produces outcomes (or, as the authors write, definite pointer values) of the observed system). In some interpretations, one or both of an observer and an environment appear as additional parts. What is to be included in the system S versus the apparatus A is not specified by the language, but is open to the theoretician; a probe, for example, can be analysed as either part of the apparatus or as part of the system, as was discussed by von Neumann in work cited by the authors. With what should a theory of measurement be concerned? A broad division of approaches is defined by how one answers the question: should one try to do away with the cut between measuring apparatus and system? The authors phrase this as a choice of REFERENT, a term which in my mind puts them on shaky semantic ground; anyway The Quantum Theory of Measurement is concerned with implications of the `yes'. The book unfolds with the power of a tragedy, in which the logical consequences lead, step-by-step, to the statement and proof of theorem 6.2.1, which, roughly speaking, asserts the non-existence of a solution to the problem of getting definite pointer values from measuring apparatus that exhibits the superposition demanded by its description as a quantum system. Possible responses to this theorem are discussed, among them the many-worlds interpretation and a modification of quantum mechanics that elevates decoherence to a principle. Citations of the literature are thorough, and many points of view are crisply summarized. The book is indispensable to both those who work in the broad direction chosen by the authors and to those who wish to set any other direction in context.
Von Neumann's theory of measurement in Quantum Mechanics was spelled out in the last chapter of his book, which was published in 1932. This book was highly mathematical for its time, and in 1939 London and Bauer provided a simplified account of the measurement theory part of it (London and Bauer, 1939). Von Neumann died in 1957. Thirty years after the publication of von Neumann's book, Wigner published a review containing his views on the shortcomings of von Neumann's theory, but omitting any discussion of its mathematical core, namely von Neumann's analysis of composite systems (Wigner, 1963). He also published a set of lecture notes entitled Interpretation of Quantum Mechanics (Wigner, 1983) in which some of his concerns were spelled out in greater detail, and an article addressed ‘to an audience of non-physicists’ (Wigner, 1964). Wigner's own contributions to measurement theory were discussed by Shimony in a talk at the Wigner centennial conference (Shimony, 2002). The English translation of von Neumann's book was published in 1955.1 The account that follows is based on these sources. We shall assume that the reader is acquainted with notions such as wave function collapse and the Heisenberg cut, but we shall not assume familiarity with the technicalities of von Neumann's theory. This chapter is organized accordingly. Section 8.1 explains what we mean by the term von Neumann's measurement theory and gives an overview of the subject. It is followed by Sections 8.2 and 8.3, in which the theory is spelled out in detail. Section 8.4 recounts Wigner's reservations.
It is a pleasure to contribute to this anthology some of my recollections of John Bell. I first met him at SLAC, where we were visitors during 1964–5 when he was on leave from CERN, and I was on leave from the Columbia University Physics Department. Soon I found that we had a common interest in the foundations of quantum mechanics, and we had lively discussions on this subject. We were concerned with the “reduction of the wave packet” after an experiment in quantum mechanics was completed, and wrote a tongue-incheek article on this subject for a festschrift in honor of Viki Weisskopf [1], stating that We emphasize not only that our view is that of a minority but also that current interest in such questions is small. The typical physicist feels that they have long been answered, and that he will fully understand just how, if ever he can spare twenty minutes to think about it. Now, 50 years later, this topic, known as the measurement problem , has become the source of numerous articles representing many different viewpoints. At the time, however, I confess that I did not realize that the main flaw in John von Neumann's presentation of this problem was his assumption that a measuring device can be represented by a pointer with only two quantum states. Actually, a measuring device must be able to record the outcome of an experiment, and for this purpose it has to have an enormous number of quantum states, leading to an irreversible macroscopic transition in the device. A correct discussion of the measurement problemwas given, for example, by Nico vanKampen [2], but recently he told me that he was unable to persuade Bell, who continued to be concerned about this problem for the rest of his life. Indeed, shortly before he died, he wrote a diatribe [3] entitled “Against Measurement,” where he continued to argue that this problem constituted a fundamental flaw in quantum mechanics. Kurt Gottfried, who referred to this problem as Bell's second major theme, called his article “a fervent jeremiad summarizing a lifetime of reflection on what he saw as the fatal flaws of the orthodox theory” [4].
Surely, after 62 years, we should have an exact formulation of some serious part of quantum mechanics? By ‘exact’ I do not of course mean ‘exactly true’. I mean only that the theory should be fully formulated in mathematical terms, with nothing left to the discretion of the theoretical physicist … until workable approximations are needed in applications. By ‘serious’ I mean that some substantial fragment of physics should be covered. Non-relativistic ‘particle’ quantum mechanics, perhaps with the inclusion of the electromagnetic field and a cut-off interaction, is serious enough. For it covers ‘a large part of physics and the whole of chemistry’. I mean too, by ‘serious’, that ‘apparatus’ should not be separated off from the rest of the world into black boxes, as if it were not made of atoms and not ruled by quantum mechanics. The question, ‘… should we not have an exact formulation …?’, is often answered by one or both of two others. I will try to reply to them: Why bother? Why not look it up in a good book ? Why bother? Perhaps the most distinguished of ‘why bother?’ers has been Dirac. He divided the difficulties of quantum mechanics into two classes, those of the first class and those of the second. The second-class difficulties were essentially the infinities of relativistic quantum field theory. Dirac was very disturbed by these, and was not impressed by the ‘renormalisation’ procedures by which they are circumvented. Dirac tried hard to eliminate these second-class difficulties, and urged others to do likewise.