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arXiv:1307.0827 · The British Journal for the Philosophy of Science

Epistemology of Wave Function Collapse in Quantum Physics

Charles Wesley Cowan, Roderich Tumulka

Published Dec 23, 2014• 24 citations

Abstract

Among several possibilities for what reality could be like in view of the empirical facts of quantum mechanics, one is provided by theories of spontaneous wave function collapse, the best known of which is the Ghirardi–Rimini–Weber (GRW) theory. We show mathematically that in GRW theory (and similar theories) there are limitations to knowledge, that is, inhabitants of a GRW universe cannot find out all the facts true of their universe. As a specific example, they cannot accurately measure the number of collapses that a given physical system undergoes during a given time interval; in fact, they cannot reliably measure whether one or zero collapses occur. Put differently, in a GRW universe certain meaningful, factual questions are empirically undecidable. We discuss several types of limitations to knowledge and compare them with those in other (no-collapse) versions of quantum mechanics, such as Bohmian mechanics. Most of our results also apply to observer-induced collapses as in orthodox quantum mechanics (as opposed to the spontaneous collapses of GRW theory). 1 Introduction 1.1 Known examples of limitations to knowledge 1.2 Remarks2 Brief Review of GRW Theories 2.1 The GRW process 2.2 GRWm 2.3 GRWf3 First Examples of Limitations to Knowledge in GRW Theories4 Measurements of Flashes in GRWf, or of Collapses in GRWm 4.1 An example in which ψ is known 4.2 Other choices of ψ 4.3 Experiments beginning before t 2 4.4 If ψ is random 4.5 Optimal way of distinguishing two density matrices 4.6 If ψ is unknown5 Measurements of m(x) in GRWmAppendix

DOI: 10.1093/bjps/axu038

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Among several possibilities for what reality could be like in view of the empirical facts of quantum mechanics, one is provided by theories of spontaneous wave function collapse, the best known of which is the Ghirardi–Rimini–Weber (GRW) theory. We show mathematically that in GRW theory (and similar theories) there are limitations to knowledge, that is, inhabitants of a GRW universe cannot find out all the facts true of their universe. As a specific example, they cannot accurately measure the number of collapses that a given physical system undergoes during a given time interval; in fact, they cannot reliably measure whether one or zero collapses occur. Put differently, in a GRW universe certain meaningful, factual questions are empirically undecidable. We discuss several types of limitations to knowledge and compare them with those in other (no-collapse) versions of quantum mechanics, such as Bohmian mechanics. Most of our results also apply to observer-induced collapses as in orthodox quantum mechanics (as opposed to the spontaneous collapses of GRW theory). 1 Introduction 1.1 Known examples of limitations to knowledge 1.2 Remarks2 Brief Review of GRW Theories 2.1 The GRW process 2.2 GRWm 2.3 GRWf3 First Examples of Limitations to Knowledge in GRW Theories4 Measurements of Flashes in GRWf, or of Collapses in GRWm 4.1 An example in which ψ is known 4.2 Other choices of ψ 4.3 Experiments beginning before t 2 4.4 If ψ is random 4.5 Optimal way of distinguishing two density matrices 4.6 If ψ is unknown5 Measurements of m(x) in GRWmAppendix

The Ghirardi–Rimini–Weber (GRW) theory is a physical theory that, when combined with a suitable ontology, provides an explanation of quantum mechanics. The so-called collapse of the wave function is problematic in conventional quantum theory but not in the GRW theory, in which it is governed by a stochastic law. A possible ontology is the flash ontology, according to which matter consists of random points in space-time, called flashes. The joint distribution of these points, a point process in space-time, is the topic of this work. The mathematical results concern mainly the existence and uniqueness of this distribution for several variants of the theory. Particular attention is paid to the relativistic version of the GRW theory that was developed in 2004.

Bohmian mechanics and the Ghirardi–Rimini–Weber theory provide opposite resolutions of the quantum measurement problem: the former postulates additional variables (the particle positions) besides the wave function, whereas the latter implements spontaneous collapses of the wave function by a nonlinear and stochastic modification of Schrödinger's equation. Still, both theories, when understood appropriately, share the following structure: They are ultimately not about wave functions but about ‘matter’ moving in space, represented by either particle trajectories, fields on space-time, or a discrete set of space-time points. The role of the wave function then is to govern the motion of the matter. 1. Introduction2. Bohmian Mechanics3. Ghirardi, Rimini, and Weber3.1. GRWm3.2. GRWf3.3. Empirical equivalence between GRWm and GRWf4. Primitive Ontology4.1. Primitive ontology and physical equivalence4.2. Primitive ontology and symmetry4.3. Without primitive ontology4.4. Primitive ontology and quantum state5. Differences between BM and GRW5.1. Primitive ontology and quadratic functionals5.2. Primitive ontology and equivariance6. A Plethora of Theories6.1. Particles, fields, and flashes6.2. Schrödinger wave functions and many-worlds7. The Flexible Wave Function7.1. GRWf without collapse7.2. Bohmian mechanics with collapse7.3. Empirical equivalence and equivariance8. What is a Quantum Theory without Observers?

Ever since we have been in the possession of quantum theories without observers, such as Bohmian mechanics or the Ghirardi‐Rimini‐Weber (GRW) theory of spontaneous wave function collapse, a major challenge in the foundations of quantum mechanics is to devise a relativistic quantum theory without observers. One of the difficulties is to reconcile nonlocality with relativity. I report about recent progress in this direction based on the GRW model: A relativistic version of the model has been devised for the case of N noninteracting (but entangled) particles. A key ingredient was to focus not on the evolution of the wave function but rather on the evolution of the matter in space‐time as determined by the wave function.

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

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.

Spontaneous collapse models are proposed modifications to quantum mechanics which aim to solve the measurement problem. In this article, we will consider models which attempt to extend a specific spontaneous collapse model, the Ghirardi-Rimini-Weber model (GRW), to be consistent with special relativity. We will present a condition that a relativistic GRW model must meet for three cases: for a single particle, for

N

distinguishable particles, and for indistinguishable particles. We will then show that this relativistic condition implies that one can have a relativistic GRW model for a single particles or for distinguishable noninteracting, nonentangled particles but not otherwise.