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 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?
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.
The quantum measurement problem, i.e., the apparent conflict between unitary quantum evolution and non-unitary, stochastic wave-function collapse, remains unresolved a century after the formulation of quantum mechanics. We first review the standard picture, from the Copenhagen prescription with its Heisenberg cut, through von Neumann's movable cut, to environment-induced decoherence, which explains the emergence of stable classical records but still presupposes the Born rule and therefore cannot by itself replace the measurement postulates. We then connect this problem to the Maximum Entropy Principle in two complementary ways. First, decoherence drives the measurement apparatus toward the least biased state compatible with the dynamically protected pointer distribution, so that the emergent collapse may be viewed as a thermodynamic relaxation toward constrained maximum entropy. Second, we propose a Timeless Histories formulation of quantum mechanics in which the primitive objects are ordered sequences of events rather than evolving wave functions. Their conditional probabilities are assigned, relative to specified refinements, by a single Born-Boltzmann rule that combines the quadratic structure of Born's rule with Boltzmann's counting of equally probable microstates. The textbook form of Born's rule, the projection postulate, density operators, and wave functions are then recovered as derived informational constructs, while collapse becomes Bayesian conditioning on recorded events. Time and space enter only subsequently through unitary translation symmetry. Since the probability rule is postulated without reference to any physical collapse or classical observer, the usual circularity of the decoherence program is avoided. Decoherence can then be invoked to explain the stability and approximate additivity of macroscopic records, i.e., the emergence of classicality.
Many textbooks on Quantum Mechanics are not very precise as to the meaning of making a measurement: as a consequence, they frequently make assertions which are not based on a dynamical description of the measurement process. A model proposed by von Neumann allows a dynamical description of measurement in Quantum Mechanics, including the measuring instrument in the formalism.In this article we apply von Neumann’s model to illustrate the measurement of an observable by means of a measuring instrument and show how various results, which are sometimens postulated without a dynamical basis, actually emerge.We also investigate the more complex, intriguing and fundamental problem of two successive measurements in Quantum Mechanics, extending von Neumann’s model to two measuring instruments. We present a description which allows obtaining, in a unified way, various results that have been given in the literature.
We review the famous no-hidden-variables theorem in von Neumann’s 1932 book on the mathematical foundations of quantum mechanics (Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932 ). We describe the notorious gap in von Neumann’s argument, pointed out by Hermann (Abhandlungen der Fries’schen Schule 6:75–152, 1935 ) and, more famously, by Bell (Rev Modern Phys 38:447–452, 1966 ). We disagree with recent papers claiming that Hermann and Bell failed to understand what von Neumann was actually doing.
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.