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arXiv:quant-ph/9810019

A Beable Interpretation of the Continuous Spontaneous Localization Model

Léa Ferreira dos Santos, Carlos O. Escobar

Published Oct 6, 1998• 1 citation
arXivHTMLPDF

Abstract

We extend the beable interpretation, due to Bell, to the continuous spontaneous localization model (CSL). Results obtained by Vink are generalized to the modified Schrodinger equation of Ghirardi, Pearle and Rimini (GPR), which allows a beable interpretation for position and momentum.
DOI
10.48550/arxiv.quant-ph/9810019

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We extend the beable interpretation, due to Bell, to the Ghirardi, Rimini and Weber [GRW] model of spontaneous collapse of the wave function. In doing so we generalize results obtained by Vink, in the context of a unitary Schrödinger evolution, to the modified Schrödinger equation of Ghirardi, Pearle and Rimini [GPR]. We show that the GPR equation allows a beable interpretation for position and momentum, meaning that stochastic trajectories can be given to these dynamical variables. Keywods: beables, spontaneous collapse models
It has been hypothesized that the time evolution of wavefunctions might include collapses, rather than being governed by the Schrödinger equation. The leading models of such an evolution, Ghirardi–Rimini–Weber (GRW) and continuous spontaneous localization (CSL), have two parameters (or new constants of nature): the collapse width σ and the collapse rate λ. We draw a diagram of the σλ-plane showing the region that is empirically refuted and the region that is philosophically unsatisfactory.
I discuss the interpretation of spontaneous collapse theories, with particular reference to Bell's suggestion that the stochastic jumps in the evolution of the wave function should be considered as local beables of the theory. I develop this analogy in some detail for the case of non-relativistic GRW-type theories, using a generalisation of Bell's notion of beables to POV measures. In the context of CSL-type theories, this strategy appears to fail, and I discuss instead Ghirardi and co-workers' mass-density interpretation and its relation to Schroedinger's original charge-density interpretation. This discussion is extended to relativistic CSL-type theories. A few remarks on Everett's interpretation conclude the paper.
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.
Abstract Before examining experimental consequences of CSL, this chapter discusses the spontaneous localization (SL) theory of Ghirardi, Rimini and Weber (GRW). As for CSL, the chapter presents the structure of the theory, and then specializes it to a non-relativistic theory. SL does not add a term to Schrodinger’s equation. Instead, the state vector is suddenly altered by a “hit” (GRW’s ‘spontaneous localization’), that is, multiplication by an operator, followed by renormalization. The hits occur at random times. The chapter shows the specific choice of operator made by GRW for their non-relativistic theory. Emphasized are two aspects of SL that helped the journey to CSL. One is that the collapse is always acting (one need not “turn it on.” The other is the “smeared” character of the GRW operator. The chapter then shows how the density matrix is calculated. Finally, it discusses the numerical values, chosen by GRW, of the two parameters of SL theory, λ and a, that are provisionally adopted for CSL.
In this volume i.h. of GianCarlo Ghirardi, I discuss my involvement with ideas of dynamical collapse of the state vector. 10 problems are introduced, 9 of which were seen following my initial work. 4 of these problems had a resolution in GianCarlo Ghirardi, Alberto Rimini and Tullio Weber's Spontaneous Localization (SL) model (which added 1 more problem). This stimulated a (somewhat different) resolution of these 5 problems in the Continuous Spontaneous Localization (CSL) model, in which I combined my initial work with SL. In an upcoming volume in h. of Abner Shimony I shall discuss the status of the 5 remaining post-CSL problems.