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On the Property Structure of Realist Collapse of Quantum Mechanics and the So-Called "Counting Anomaly"

Roman Frigg

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

The aim of this paper is two-fold. Recently, Lewis has presented an argument, now known as the `counting anomaly', that the spontaneous localization approach to quantum mechanics, suggested by Ghirardi, Rimini, and Weber, implies that arithmetic does not apply to ordinary macroscopic objects. I will take this argument as the starting point for a discussion of the property structure of realist collapse interpretations of quantum mechanics in general. At the end of this I present a proof of the fact that the composition principle, which holds true in Standard Quantum Mechanics, fails in all realist collapse interpretations. On the basis of this result I reconsider the counting anomaly and show that what lies at the heart of the anomaly is the failure to appreciate the peculiarities of the property structure of collapse interpretations. Once this flaw is uncovered, the anomaly vanishes.

Related papers

On the property structure of realist collapse interpretations of quantum mechanics and the so-called "counting anomaly"

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The aim of this article is twofold. Recently, Lewis has presented an argument, now known as the "counting anomaly", that the spontaneous localization approach to quantum mechanics, suggested by Ghirardi, Rimini, and Weber, implies that arithmetic does not apply to ordinary macroscopic objects. I will take this argument as the starting point for a discussion of the property structure of realist collapse interpretations of quantum mechanics in general. At the end of this I present a proof of the fact that the composition principle, which holds true in standard quantum mechanics, fails in all realist collapse interpretations. On the basis of this result I reconsider the counting anomaly and show that what lies at the heart of the anomaly is the failure to appreciate the peculiarities of the property structure of such interpretations. Once this flaw is uncovered, the anomaly vanishes.

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Four strategies for dealing with the counting anomaly in spontaneous collapse theories of quantum mechanics

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A few years ago, I argued that according to spontaneous collapse theories of quantum mechanics, arithmetic applies to macroscopic objects only as an approximation. Several authors have written articles defending spontaneous collapse theories against this charge, including Bassi and Ghirardi, Clifton and Monton, and now Frigg. The arguments of these authors are all different and all ingenious, but in the end I think that none of them succeeds, for reasons I elaborate here. I suggest a fourth line of response, based on an analogy with epistemic paradoxes, which I think is the best way to defend spontaneous collapse theories, and which leaves my main thesis intact.

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ABSTRACT This essay aims at reviewing one of the major difficulties affecting the standard formulation of quantum mechanics, that is, the well‐known measurement problem. As it will be explained, the latter originates as a consequence of the quantum theory's postulates. Referring to this, we analyze in particular the conceptual issues implied by the projection (or collapse) rule. After having presented the measurement problem, we then illustrate three possible solutions offered by major interpretations of/alternatives to quantum theory: Bohmian mechanics, Ghirardi–Rimini–Weber theories, and the many‐worlds interpretation. As the reader will see, these theoretical frameworks solve this issue by virtue of their unambiguous ontologies and consistent dynamical laws. Finally, a synopsis of the open debates and future perspectives concerning the measurement problem in relativistic contexts concludes the paper.

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On spontaneous wave function collapse and quantum field theory

Roderich Tumulka

One way of obtaining a version of quantum mechanics without observers, and thus of solving the paradoxes of quantum mechanics, is to modify the Schrödinger evolution by implementing spontaneous collapses of the wave function. An explicit model of this kind was proposed in 1986 by Ghirardi, Rimini & Weber (GRW), involving a nonlinear, stochastic evolution of the wave function. We point out how, by focusing on the essential mathematical structure of the GRW model and a clear ontology, it can be generalized to (regularized) quantum field theories in a simple and natural way.

arXiv:quant-ph/0508230•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Mar 24, 2006•144 citations
The aim of this paper is two-fold. Recently, Lewis has presented an argument, now known as the `counting anomaly', that the spontaneous localization approach to quantum mechanics, suggested by Ghirardi, Rimini, and Weber, implies that arithmetic does not apply to ordinary macroscopic objects. I will take this argument as the starting point for a discussion of the property structure of realist collapse interpretations of quantum mechanics in general. At the end of this I present a proof of the fact that the composition principle, which holds true in Standard Quantum Mechanics, fails in all realist collapse interpretations. On the basis of this result I reconsider the counting anomaly and show that what lies at the heart of the anomaly is the failure to appreciate the peculiarities of the property structure of collapse interpretations. Once this flaw is uncovered, the anomaly vanishes.
The aim of this article is twofold. Recently, Lewis has presented an argument, now known as the "counting anomaly", that the spontaneous localization approach to quantum mechanics, suggested by Ghirardi, Rimini, and Weber, implies that arithmetic does not apply to ordinary macroscopic objects. I will take this argument as the starting point for a discussion of the property structure of realist collapse interpretations of quantum mechanics in general. At the end of this I present a proof of the fact that the composition principle, which holds true in standard quantum mechanics, fails in all realist collapse interpretations. On the basis of this result I reconsider the counting anomaly and show that what lies at the heart of the anomaly is the failure to appreciate the peculiarities of the property structure of such interpretations. Once this flaw is uncovered, the anomaly vanishes.
A few years ago, I argued that according to spontaneous collapse theories of quantum mechanics, arithmetic applies to macroscopic objects only as an approximation. Several authors have written articles defending spontaneous collapse theories against this charge, including Bassi and Ghirardi, Clifton and Monton, and now Frigg. The arguments of these authors are all different and all ingenious, but in the end I think that none of them succeeds, for reasons I elaborate here. I suggest a fourth line of response, based on an analogy with epistemic paradoxes, which I think is the best way to defend spontaneous collapse theories, and which leaves my main thesis intact.
Different attempts to solve the measurement problem of the quantum me-chanics (QM) by denying the collapse principle, and replacing it with changes in the quantum formalism, failed because the changes in the formalism lead to contradictions with QM predictions. To the difference, Ghirardi, Rimini and Weber took the collapse as a real phenomenon, and proposed a calculus by which the wave-function should undergo a sudden localization. Later on, Ghirardi, Pearle and Rimini came with a change of this calculus into the CSL (continuous spontaneous localization) model of collapse. Both these proposals rely on the experimental fact that the reduction of the wave-function occurs when the microscopic system encounters a macroscopic object and involves a big amount of its particles. Both these proposals also change the quantum formalism by introducing in the Schr?dinger equation additional terms with noisy behavior. However, these terms have practically no influence as long as the studied system contains only one or a few components. Only when the amount of components is very big, these terms become significant and lead to the reduction of the wave-function to one of its components. The present work has two purposes: 1) proving that the collapse postulate is unavoidable; 2) ap-plying the CSL model to the process in a detector and showing step by step the modification of the wave-function, until reduction. As a side detail, it is argued here that the noise cannot originate in some classical field, contrary to the thought/hope of some physicists, because no classical field is tailored by the wave-functions of entanglements.
A modified form of quantum mechanics which includes a new mechanism for wavefunction collapse is proposed. The collapse provides a solution to the quantum measurement problem. This modified quantum mechanics is shown to arise naturally from a fully discrete physics, where all physical quantities are discrete rather than continuous. We compare the theory to the spontaneous collapse theories of Ghirardi, Rimini, Weber and Pearle, and argue that the new theory lends itself well to a realist interpretation of the wavefunction.
ABSTRACT This essay aims at reviewing one of the major difficulties affecting the standard formulation of quantum mechanics, that is, the well‐known measurement problem. As it will be explained, the latter originates as a consequence of the quantum theory's postulates. Referring to this, we analyze in particular the conceptual issues implied by the projection (or collapse) rule. After having presented the measurement problem, we then illustrate three possible solutions offered by major interpretations of/alternatives to quantum theory: Bohmian mechanics, Ghirardi–Rimini–Weber theories, and the many‐worlds interpretation. As the reader will see, these theoretical frameworks solve this issue by virtue of their unambiguous ontologies and consistent dynamical laws. Finally, a synopsis of the open debates and future perspectives concerning the measurement problem in relativistic contexts concludes the paper.
One way of obtaining a version of quantum mechanics without observers, and thus of solving the paradoxes of quantum mechanics, is to modify the Schrödinger evolution by implementing spontaneous collapses of the wave function. An explicit model of this kind was proposed in 1986 by Ghirardi, Rimini & Weber (GRW), involving a nonlinear, stochastic evolution of the wave function. We point out how, by focusing on the essential mathematical structure of the GRW model and a clear ontology, it can be generalized to (regularized) quantum field theories in a simple and natural way.