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arXiv:1407.4746

Life and death in the tails of the GRW wave function

Wallace, David

Published Jul 17, 2014• 20 citations
arXivHTMLPDF

Abstract

It is often assumed that the only effect of the Ghirardi-Rimini-Weber (`GRW') dynamical collapse mechanism on the `tails' of the wavefunction (that is, the components of superpositions on which the collapse is \emph{not} centred) is to reduce their weight. In consequence the tails are often thought to behave exactly as do the various branches in the Everett interpretation except for their much lower weight. These assumptions are demonstrably inaccurate: the collapse mechanism has substantial and detectable effects within the tails. The relevance of this misconception for the dynamical-collapse theories is debatable, though.

DOI
10.48550/arxiv.1407.4746

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It is often assumed that the only effect of the Ghirardi-Rimini-Weber (GRW') dynamical collapse mechanism on the tails' of the wavefunction (that is, the components of superpositions on which the collapse is \emph{not} centred) is to reduce their weight. In consequence the tails are often thought to behave exactly as do the various branches in the Everett interpretation except for their much lower weight. These assumptions are demonstrably inaccurate: the collapse mechanism has substantial and detectable effects within the tails. The relevance of this misconception for the dynamical-collapse theories is debatable, though.
It is often assumed that the only effect of the Ghirardi-Rimini-Weber (`GRW') dynamical collapse mechanism on the `tails' of the wavefunction (that is, the components of superpositions on which the collapse is \emph{not} centred) is to reduce their weight. In consequence it seems to be generally accepted that the tails behave exactly as do the various branches in the Everett interpretation except for their much lower weight. These assumptions are demonstrably inaccurate: the collapse mechanism has substantial and detectable effects within the tails. The relevance of this misconception for the dynamical-collapse theories is debatable, though.
Carrying out a research program outlined by John S. Bell in 1987, we arrive at a relativistic version of the Ghirardi–Rimini–Weber (GRW) model of spon-taneous wavefunction collapse. The GRW model was proposed as a solution of the measurement problem of quantum mechanics and involves a stochastic and nonlinear modification of the Schrödinger equation. It deviates very little from the Schrödinger equation for microscopic systems but efficiently suppresses, for macroscopic systems, superpositions of macroscopically different states. As sug-gested by Bell, we take the primitive ontology, or local beables, of our model to be a discrete set of space-time points, at which the collapses are centered. This set is random with distribution determined by the initial wavefunction. Our model is nonlocal and violates Bell’s inequality though it does not make use of a preferred slicing of space-time or any other sort of synchronization of spacelike separated points. Like the GRW model, it reproduces the quantum probabilities in all cases presently testable, though it entails deviations from the quantum formalism that are in principle testable. Our model works in Minkowski space-time as well as in (well-behaved) curved background space-times. PACS numbers: 03.65.Ta; 03.65.Ud; 03.30.+p. Key words: spontaneous wave-function collapse; relativity; quantum theory without observers. 1
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
We study the effect that the state-vector reduction of Ghirardi, Rimini, and Weber (GRW) [Phys. Rev. D 34, 470 (1986)] has on the quantum fluctuation-dissipation process. A conventional fluctuation-dissipation process stems from the interaction between a system of interest and a macroscopic number of microscopic variables, thereby suggesting the possible occurrence of an enhancement of the GRW wave-function collapse. We carry out an exact treatment of the joint action of conventional quantum dissipation and GRW wave-function collapse in the special case where the system of interest is a harmonic oscillator. We prove that the GRW dynamics does not lead to any significant departure from the predictions of ordinary quantum mechanics, if the region of astronomically large times is disregarded, where the energy of the system diverges as an effect of a slow but steady increase caused by the GRW collapses.
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.
Peter Lewis ([1997]) has recently argued that the wavefunction collapse theory of GRW (Chirardi, Rimini and Weber [1986]) can only solve the problem of wavefunction tails at the expense of predicting that arithmetic does not apply to ordinary macroscopic objects. More specifically, Lewis argues that the GRW theory must violate the enumeration principle: that 'if marble 1 is in the box and marble 2 is in the box and so on through marble n, then all n marbles are in the box' ([1997], p. 321). Ghirardi and Bassi ([1999]) have replied that it is meaningless to say that the enumeration principle is violated because the wavefunction Lewis uses to exhibit the violation cannot persist, according to the GRW theory, for more than a split second ([1999], p. 709). On the contrary, we argue that Lewis's argument survives Ghirardi and Bassi's criticism unscathed. We then go on to show that, while the enumeration principle can fail in the GRW theory, the theory itself guarantees that the principle can never be empirically falsified, leaving the applicability of arithmetical reasoning to both micro- and macroscopic objects intact.