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.
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.
In his book `Mathematical Foundations of Quantum Mechanics', von Neumann asserted the following: the Compton-Simon experiment showed that the state vector must collapse upon measurement of any self-adjoint operator. Comparing von Neumann's account with the Compton-Simon paper, we find that von Neumann had misinterpreted the experiment as consisting of two successive measurements (which gave identical results), whereas the experiment only measured two angles on the same photographic plate. Note, however, that the state vector must collapse upon measurement of an additively-conserved quantity; otherwise the conservation law could be violated. Next, it turns out that the mathematical problem of explaining collapse is not fully defined until one specifies the nature of the apparatus. If the apparatus does not have a `classical description', the problem is insoluble, even if the measurement is only approximate (Fine, Simony, Brown, Simony and Busch); but if it does, the problem is soluble within Schroedinger dynamics (with a time-dependent hamiltonian) for additively-conserved observables. The solution, a modification of Sewell's, shows that the state vector has collapsed, but it does not reveal the eigenvalue of the collapsed state. The collapse is irreversible, and results from the interplay of additive conservation laws with the quantum measurement postulate. Indeed, the quantum measurement problem - as expounded by Wigner - may be better understood as the problem of establishing that the two are compatible with each other; it has little relevance to actual measurements.
In his book Mathematical Foundations of Quantum Mechanics von Neumann asserted that the state vector must collapse upon measurement of any self-adjoint operator. This assertion was based on his reading of the Compton-Simon experiment. However, comparing von Neumann’s account with the Compton-Simon paper itself, we find that von Neumann had completely misinterpreted the experiment. Compton and Simon measured two angles on the same photographic plate; in von Neumann’s reading it became two successive measurements which gave identical results, where the experimenter could choose which measurement to perform first. In short, von Neumann’s case for collapse simply did not exist! In quantum mechanics, superpositions of different eigenvalues of conserved observables are freely admissible, but conservation laws require successive measurements of a conserved quantity upon the same object to return the same value (if the object is isolated and moves under a Schrödinger equation). This can be guaranteed only if the first measurement has caused the state vector to collapse. Thus collapse is a consistency condition which the theory must satisfy. Consistency problems are solved by displaying examples, or models, which fulfil the conditions in question. We therefore have to construct a mathematical model of the measurement process. This requires taking a stand on whether the measuring instrument does (Bohr) or does not (von Neumann-Wigner) have a classical description. If it does not, the ‘quantum measurement problem’ is insoluble. If it does, the problem – for conserved quantities – does have a solution, given by Sewell in 2005. The model displays collapse while evolving under the Schrödinger equation with a time-dependent hamiltonian; it does not have to call upon the observer’s ‘conscious ego’. Its existence shows that additive conservation laws are consistent with the superposition principle and Schrodinger evolution. Measurement of a conserved observable returns an eigenvalue of the observable; measurement of a non-conserved observable returns a diagonal matrix element of the observable in the basis of energy eigenvectors. If ‘classical descriptions’ of the apparatus are essential for these consistency conditions to be satisfied, what does it imply about the relation between classical and quantum mechanics? If the two are on the same footing, is it not possible that ‘dark matter’ is purely classical?
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.
One way of reading this book is as an epilogue-or (more precisely) as a denouement-to a book I wrote more than thirty years ago, which was called Quantum Mechanics and Experience.That earlier book was meant to do two things:1) To introduce its readers to the elementary nuts and bolts of the foundations of quantum mechanics-the idea of superposition, the basics of the standard mathematical formalism of quantum mechanics, the argument of Einstein and Podolsky and Rosen to the effect that that formalism, considered as a description of the fundamental real ity of the world, must be somehow incomplete, Bell's extension of that argument into a general and power ful theorem about locality, the elaboration (by figures like Erwin Schrödinger, and John von Neumann, and Eugene Wigner) of the quantum-mechanical mea surement prob lem, and vari ous proposals (theories of the spontaneous collapse of the wave-function, Bohmian mechanics, the so-called many-worlds interpretations of quantum mechanics, and so on) for solving it.2) To show how the above-mentioned proposals for solving the mea sure ment prob lem, taken together, amount to a
Many works intended to introduce interpretive issues in quantum mechanics present John von Neumann as having a view in which measurement produces a physical collapse in the system being measured. In this paper I argue that such a reading of von Neumann is inconsistent with what von Neumann actually says. I show that much of what he says makes no sense on the physical collapse reading, but falls into place if we assume he does not have such a view. I show that the physical collapse view is based on an understanding of ‘state’ which von Neumann does not share. 1. Introduction2. The standard reading of von Neumann3. The standard reading of von Neumann and Chapter VI4. The Chapter VI argument5. The Chapter V argument6. The Chapters III and IV argument7. Conclusion