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Cambridge University Press eBooks

Von Neumann's theory of measurement

Rathindra Nath Sen

Published Feb 11, 2010• 0 citations
Publisher

Abstract

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.

DOI
10.1017/cbo9780511674761.013

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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.
In 1957, Feyerabend delivered a paper titled ‘On the Quantum-Theory of Measurement’ at the Colston Research Symposium in Bristol to sketch a completion of von Neumann's measurement scheme without collapse, using only unitary quantum dynamics and well-motivated statistical assumptions about macroscopic quantum systems. Feyerabend's paper has been recognised as an early contribution to quantum measurement, anticipating certain aspects of decoherence. Our paper reassesses the physical and philosophical content of Feyerabend's contribution, detailing the technical steps as well as its overall philosophical motivations and consequences. Summarising our results, Feyerabend interpreted collapse as a positivist assumption in quantum mechanics leading to a strict distinction between the uninterpreted formalism of unitary evolution in quantum mechanics and the classically interpreted observational language describing post-measurement outcomes. Thus Feyerabend took the no-collapse completion of the von Neumann measurement scheme to show the dispensability of the positivist assumption, leading the way to a realistic interpretation of quantum theory. We note, however, that there are substantial problems with his account of measurement that bring into question its viability as a legitimate foil to the orthodox view. We further argue that his dissatisfaction with the von Neumann measurement scheme is indicative of early views on theoretical pluralism.
We provide an introduction to the theory of quantum measurements that is centered on the pivotal role played by John von Neumann's model. This introduction is accessible to students and researchers from outside the field of foundations of quantum mechanics and presented within a historical context. We first explain the origins and the meaning of the measurement problem in quantum theory, and why it is not present in classical physics. We perform a chronological review of the quantization of action and explain how this led to successive restrictions on what could be measured in atomic phenomena, until the consolidation of the orthodox interpretation of quantum mechanics. The clear separation between quantum system and classical apparatus that causes these restrictions is subverted in von Neumann's paradigmatic model of quantum measurements, a subject whose concepts we explain, while also providing the mathematical tools necessary to apply it to new problems. We show how this model was important in discussing the interpretations of quantum mechanics and how it is still relevant in modern applications. In particular, we explain in detail how it can be used to describe weak measurements and the surprising results they entail. We also discuss the limitations of von Neumann's model of measurements, and explain how they can be overcome with POVMs and Kraus operators. We provide the mathematical tools necessary to work with these generalized measurements and to derive master equations from them. Finally, we demonstrate how these can be applied in research problems by calculating the Quantum Zeno Effect.
Few things in physics have caused so much hand-wringing as von Neumann's collapse hypothesis. Unable to derive it mathematically, von Neumann attributed it to interaction with the observer's brain! Few physicists agreed, but tweaks of von Neumann's measurement theory did not lead to collapse, and Shimony and Brown proved theorems establishing `the insolubility of the quantum measurement problem'. Many different `interpretations' of quantum mechanics were put forward, none gained a consensus, and some scholars suggested that the foundations of quantum mechanics were flawed to begin with. Yet, in the last ninety years, no-one looked into now von Neumann had arrived at his collapse hypothesis! Von Neumann based his argument on the experiment of Compton and Simon. But, by comparing readings from von Neumann's book and the Compton-Simon paper, we find that the experiment provides no evidence for the collapse hypothesis; von Neumann had misread it completely! We suggest that von Neumann had relied on his phenomenal memory rather than the printed Compton-Simon paper, and his memory had failed him for once. Our finding has considerable implications for physics, which -- briefly sketched here -- will be discussed elsewhere in detail. An Appendix raises some questions for historians of physics.
Review of "The Collected Works of Eugene Paul Wigner", Volume I, III, and VI. Excerpt from the Conclusions: Many of Wigner’s papers on mathematical physics are great classics. Most famous is his work on group representations which is of lasting value for a proper mathematical foundation of quantum theory. The modern development of quantum theory (which is not reflected in Wigner’s work) is in an essential way a representation theory (e.g. representations of kinematical groups, or representations of C*-algebras). This view owes very much to Wigner’s seminal papers on the unitary representations of compact and noncompact groups. Wigner showed much courage in relating the then unresolved questions of the measurement problem to the much deeper problem of consciousness. In view of this very unorthodox proposal it is astonishing that Wigner was very reactionary with respect of the dogmas of orthodox quantum mechanics. In contrast to von Neumann himself, he took the old von-Neumann codification of quantum mechanics as authoritative and not to be questioned. Much of the efforts to interpret the meaning of this codification and to prove no-go theorems, such as the insolubility of the measurement problem or the impossibility of a quantum theory of individual objects, are physically irrelevant since they are based on a codification of quantum mechanics that is valid only for strictly closed systems with finitely many degrees of freedom. However, in nature there are no such systems. Every material system is coupled to the gravitational and to the electromagnetic field – systems which require in a Hamiltonian description infinitely many degrees of freedom. A deeper insight into the conceptual problems of quantum theory is possible only if the modern development of a quantum theory of infinite systems is taken into account.
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.
This short chapter contains basics of the mathematical formalism for thequantum measurement theory. In this book we proceed mainly withthe von Neumann measurement theory in which observables are given byHermitian operators and the state update by projections. However, we alsomention the measurement formalism based on quantum instruments, sinceit gives the general framework for quantum measurements. This formalismis used only in Chapters 10 and 18. The latter chapter is devoted to quantum-likemodeling – the applications of the mathematical formalism and methodologyof quantum mechanics (QM) to cognition, psychology, and decision making.Surprisingly, in such applications even the simplest effects can’t be described bythe von Neumann theory. One should use quantum instruments (compare withquantum physics where the main body of theory can be presented solelywithin the von Neumann measurement theory).