In quantum mechanics, the wavefunction predicts probabilities of possible measurement outcomes, but not which individual outcome is realised in each run of an experiment. This suggests that it describes an ensemble of states with different values of a hidden variable. Here, we analyse this idea with reference to currently known theorems and experiments. We argue that the ψ-ontic/epistemic distinction fails to properly identify ensemble interpretations and propose a more useful definition. We then show that all local ψ-ensemble interpretations which reproduce (...) quantum mechanics violate Statistical Independence. Theories with this property are commonly referred to as superdeterministic or retrocausal. Finally, we explain how this interpretation helps make sense of some otherwise puzzling phenomena in quantum mechanics, such as the delayed choice experiment, the Elitzur-Vaidman bomb detector, and the Extended Wigner's Friends Scenario. (shrink)
It is generally argued that if the wave-function in the de Broglie–Bohm theory is a physical field, it must be a field in configuration space. Nevertheless, it is possible to interpret the wave-function as a multi-field in three-dimensional space. This approach hasn’t received the attention yet it really deserves. The aim of this paper is threefold: first, we show that the wave-function is naturally and straightforwardly construed as a multi-field; second, we show why this (...) interpretation is superior to other interpretations discussed in the literature; third, we clarify common misconceptions. (shrink)
This is a new volume of original essays on the metaphysics of quantum mechanics. The essays address questions such as: What fundamental metaphysics is best motivated by quantum mechanics? What is the ontological status of the wavefunction? What is the nature of the fundamental space (or space-time manifold) of quantum mechanics?
This is a new volume of original essays on the metaphysics of quantum mechanics. The essays address questions such as: What fundamental metaphysics is best motivated by quantum mechanics? What is the ontological status of the wavefunction? Does quantum mechanics support the existence of any other fundamental entities, e.g. particles? What is the nature of the fundamental space of quantum mechanics? What is the relationship between the fundamental ontology of quantum mechanics and ordinary, macroscopic objects like tables, (...) chairs, and persons? This collection includes a comprehensive introduction with a history of quantum mechanics and the debate over its metaphysical interpretation focusing especially on the main realist alternatives. (shrink)
I argue that the wavefunction ontology for quantum mechanics is an undesirable ontology. This ontology holds that the fundamental space in which entities evolve is not three-dimensional, but instead 3N-dimensional, where N is the number of particles standardly thought to exist in three-dimensional space. I show that the state of three-dimensional objects does not supervene on the state of objects in 3N-dimensional space. I also show that the only way to guarantee the existence of the appropriate mental (...) states in the wavefunction ontology has undesirable metaphysical baggage: either mind/body dualism is true, or circumstances which we take to be logically possible turn out to be logically impossible. (shrink)
An overview of the collapse theories of quantum mechanics. Written by distinguished physicists and philosophers of physics, it discusses the origin and implications of wave-function collapse, the controversies around collapse models and their ontologies, and new arguments for the reality of wavefunction collapse.
A century after the discovery of quantum mechanics, the meaning of quantum mechanics still remains elusive. This is largely due to the puzzling nature of the wavefunction, the central object in quantum mechanics. If we are realists about quantum mechanics, how should we understand the wavefunction? What does it represent? What is its physical meaning? Answering these questions would improve our understanding of what it means to be a realist about quantum mechanics. In this (...) survey article, I review and compare several realist interpretations of the wavefunction. They fall into three categories: ontological interpretations, nomological interpretations, and the sui generis interpretation. For simplicity, I will focus on non-relativistic quantum mechanics. (shrink)
In this paper, I critically assess different interpretations of Bohmian mechanics that are not committed to an ontology based on the wavefunction being an actual physical object that inhabits configuration space. More specifically, my aim is to explore the connection between the denial of configuration space realism and another interpretive debate that is specific to Bohmian mechanics: the quantum potential versus guidance approaches. Whereas defenders of the quantum potential approach to the theory claim that Bohmian mechanics is (...) better formulated as quasi-Newtonian, via the postulation of forces proportional to acceleration; advocates of the guidance approach defend the notion that the theory is essentially first-order and incorporates some concepts akin to those of Aristotelian physics. Here I analyze whether the desideratum of an interpretation of Bohmian mechanics that is both explanatorily adequate and not committed to configuration space realism favors one of these two approaches to the theory over the other. Contrary to some recent claims in the literature, I argue that the quasi-Newtonian approach based on the idea of a quantum potential does not come out the winner. (shrink)
There is not much of a consensus on almost anything about quantum mechanics. I take it, however, that the minimum consensus is that "although quantum mechanics is empirically successful, quantum mechanics is hard to understand." Quantum mechanics, in the way it is presented in most textbooks, does indeed not provide a clear picture of reality that would make it a theory to be understood. In her new book, "The World in the WaveFunction: A Metaphysics for Quantum Physics," (...) Alyssa Ney tries to make this blurry picture of reality more precise, even if this picture will turn out to be heterodox and unfamiliar. (shrink)
Scientific realism is the view that our best scientific theories can be regarded as (approximately) true. This is connected with the view that science, physics in particular, and metaphysics could (and should) inform one another: on the one hand, science tells us what the world is like, and on the other hand, metaphysical principles allow us to select between the various possible theories which are underdetermined by the data. Nonetheless, quantum mechanics has always been regarded as, at best, puzzling, if (...) not contradictory. As such, it has been considered for a long time at odds with scientific realism, and thus a naturalized quantum metaphysics was deemed impossible. Luckily, now we have many quantum theories compatible with a realist interpretation. However, scientific realists assumed that the wave-function, regarded as the principal ingredient of quantum theories, had to represent a physical entity, and because of this they struggled with quantum superpositions. In this paper I discuss a particular approach which makes quantum mechanics compatible with scientific realism without doing that. In this approach, the wave-function does not represent matter which is instead represented by some spatio-temporal entity dubbed the primitive ontology: point-particles, continuous matter fields, space-time events. I argue how within this framework one develops a distinctive theory-construction schema, which allows to perform a more informed theory evaluation by analyzing the various ingredients of the approach and their inter-relations. (shrink)
The meaning of the wavefunction has been a hot topic of debate since the early days of quantum mechanics. Recent years have witnessed a growing interest in this long-standing question. Is the wavefunction ontic, directly representing a state of reality, or epistemic, merely representing a state of knowledge, or something else? If the wavefunction is not ontic, then what, if any, is the underlying state of reality? If the wave (...) class='Hi'>function is indeed ontic, then exactly what physical state does it represent? In this book, I aim to make sense of the wavefunction in quantum mechanics and find the ontological content of the theory. The book can be divided into three parts. The first part addresses the question of the nature of the wavefunction. After giving a comprehensive and critical review of the competing views of the wavefunction, I present a new argument for the ontic view in terms of protective measurements. In addition, I also analyze the origin of the wavefunction by deriving the free Schroedinger equation. The second part analyzes the ontological meaning of the wavefunction. I propose a new ontological interpretation of the wavefunction in terms of random discontinuous motion of particles, and give two main arguments supporting this interpretation. The third part investigates whether the suggested quantum ontology is complete in accounting for our definite experience and whether it needs to be revised in the relativistic domain. (shrink)
The meaning of the wavefunction and its evolution are investigated. First, we argue that the wavefunction in quantum mechanics is a description of random discontinuous motion of particles, and the modulus square of the wavefunction gives the probability density of the particles being in certain locations in space. Next, we show that the linear non-relativistic evolution of the wavefunction of an isolated system obeys the free Schrödinger equation due (...) to the requirements of spacetime translation invariance and relativistic invariance. Thirdly, we argue that the random discontinuous motion of particles may lead to a stochastic, nonlinear collapse evolution of the wavefunction. A discrete model of energy-conserved wavefunction collapse is proposed and shown consistent with existing experiments and our macroscopic experience. Besides, we also give a critical analysis of the de Broglie-Bohm theory, the many-worlds interpretation and other dynamical collapse theories, and briefly discuss the issues of unifying quantum mechanics and relativity. (shrink)
In quantum mechanics, the wavefunction of a N-body system is a mathematical function defined in a 3N-dimensional configuration space. We argue that wavefunction realism implies particle ontology when assuming: (1) the wavefunction of a N-body system describes N physical entities; (2) each triple of the 3N coordinates of a point in configuration space that relates to one physical entity represents a point in ordinary three-dimensional space. Moreover, the motion of particles (...) is random and discontinuous. (shrink)
Realists wanting to capture the facts of quantum entanglement in a metaphysical interpretation find themselves faced with several options: to grant some species of fundamental nonseparability, adopt holism, or to view localized spacetime systems as ultimately reducible to a higher-dimensional entity, the quantum state or wavefunction. Those adopting the latter approach and hoping to view the macroscopic world as grounded in the quantum wavefunction face the macro-object problem. The challenge is to articulate the metaphysical (...) relation obtaining between three-dimensional macro-objects and the wavefunction so that the latter may be seen in some sense as constituting the former. This paper distinguishes several strategies for doing so and defends one based on a notion of partial instantiation. (shrink)
We investigate the meaning of the wavefunction by analyzing the mass and charge density distributions of a quantum system. According to protective measurement, a charged quantum system has effective mass and charge density distributing in space, proportional to the square of the absolute value of its wavefunction. In a realistic interpretation, the wavefunction of a quantum system can be taken as a description of either a physical field or the ergodic motion (...) of a particle. The essential difference between a field and the ergodic motion of a particle lies in the property of simultaneity; a field exists throughout space simultaneously, whereas the ergodic motion of a particle exists throughout space in a time-divided way. If the wavefunction is a physical field, then the mass and charge density will be distributed in space simultaneously for a charged quantum system, and thus there will exist gravitational and electrostatic self-interactions of its wavefunction. This not only violates the superposition principle of quantum mechanics but also contradicts experimental observations. Thus the wavefunction cannot be a description of a physical field but a description of the ergodic motion of a particle. For the later there is only a localized particle with mass and charge at every instant, and thus there will not exist any self-interaction for the wavefunction. Which kind of ergodic motion of particles then? It is argued that the classical ergodic models, which assume continuous motion of particles, cannot be consistent with quantum mechanics. Based on the negative result, we suggest that the wavefunction is a description of the quantum motion of particles, which is random and discontinuous in nature. On this interpretation, the square of the absolute value of the wavefunction not only gives the probability of the particle being found in certain locations, but also gives the probability of the particle being there. We show that this new interpretation of the wavefunction provides a natural realistic alternative to the orthodox interpretation, and its implications for other realistic interpretations of quantum mechanics are also briefly discussed. (shrink)
The ontology of Bohmian mechanics includes both the universal wavefunction and particles. Proposals for understanding the physical significance of the wavefunction in this theory have included the idea of regarding it as a physically-real field in its 3N-dimensional space, as well as the idea of regarding it as a law of nature. Here we introduce and explore a third possibility in which the configuration space wavefunction is simply eliminated—replaced by a set (...) of single-particle pilot-wave fields living in ordinary physical space. Such a re-formulation of the Bohmian pilot-wave theory can exactly reproduce the statistical predictions of ordinary quantum theory. But this comes at the rather high ontological price of introducing an infinite network of interacting potential fields which influence the particles’ motion through the pilot-wave fields. We thus introduce an alternative approach which aims at achieving empirical adequacy with a more modest ontological complexity, and provide some preliminary evidence for optimism regarding the program of trying to replace the configuration space wavefunction with a set of fields in ordinary physical space. (shrink)
The mathematical structure of realist quantum theories has given rise to a debate about how our ordinary 3-dimensional space is related to the 3N-dimensional configuration space on which the wavefunction is defined. Which of the two spaces is our (more) fundamental physical space? I review the debate between 3N-Fundamentalists and 3D-Fundamentalists and evaluate it based on three criteria. I argue that when we consider which view leads to a deeper understanding of the physical world, especially given the (...) deeper topological explanation from the unordered configurations to the Symmetrization Postulate, we have strong reasons in favor of 3D-Fundamentalism. I conclude that our evidence favors the view that our fundamental physical space in a quantum world is 3-dimensional rather than 3N-dimensional. I outline lines of future research where the evidential balance can be restored or reversed. Finally, I draw lessons from this case study to the debate about theoretical equivalence. (shrink)
It is well known that Niels Bohr insisted on the necessity of classical concepts in the account of quantum phenomena. But there is little consensus concerning his reasons, and what he exactly meant by this. In this paper, I re-examine Bohr’s interpretation of quantum mechanics, and argue that the necessity of the classical can be seen as part of his response to the measurement problem. More generally, I attempt to clarify Bohr’s view on the classical/quantum divide, arguing that the relation (...) between the two theories is that of mutual dependence. An important element in this clarification consists in distinguishing Bohr’s idea of the wavefunction as symbolic from both a purely epistemic and an ontological interpretation. Together with new evidence concerning Bohr’s conception of the wavefunction collapse, this sets his interpretation apart from both standard versions of the Copenhagen interpretation, and from some of the reconstructions of his view found in the literature. I conclude with a few remarks on how Bohr’s ideas make much sense also when modern developments in quantum gravity and early universe cosmology are taken into account. (shrink)
This paper defends wavefunction realism against the charge that the view is empirically incoherent because our evidence for quantum theory involves facts about objects in three-dimensional space or space-time . It also criticizes previous attempts to defend wavefunction realism against this charge by claiming that the wavefunction is capable of grounding local beables as elements of a derivative ontology.
Does consciousness collapse the quantum wavefunction? This idea was taken seriously by John von Neumann and Eugene Wigner but is now widely dismissed. We develop the idea by combining a mathematical theory of consciousness (integrated information theory) with an account of quantum collapse dynamics (continuous spontaneous localization). Simple versions of the theory are falsified by the quantum Zeno effect, but more complex versions remain compatible with empirical evidence. In principle, versions of the theory can be tested by (...) experiments with quantum computers. The upshot is not that consciousness-collapse interpretations are clearly correct, but that there is a research program here worth exploring. (shrink)
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 wavefunction collapse, the best known of which is the Ghirardi–Rimini–Weber theory. We show mathematically that in GRW theory 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 versions of quantum mechanics, such as Bohmian mechanics. Most of our results also apply to observer-induced collapses as in orthodox quantum mechanics. 1 Introduction1.1 Known examples of limitations to knowledge1.2 Remarks2 Brief Review of GRW Theories2.1 The GRW process2.2 GRWm2.3 GRWf3 First Examples of Limitations to Knowledge in GRW Theories4 Measurements of Flashes in GRWf, or of Collapses in GRWm4.1 An example in which ψ is known4.2 Other choices of ψ4.3 Experiments beginning before t24.4 If ψ is random4.5 Optimal way of distinguishing two density matrices4.6 If ψ is unknown5 Measurements of m in GRWmAppendix. (shrink)
It is argued that the many-worlds interpretation is by far the best interpretation of quantum mechanics. The key points of this view are viewing the wave functions of worlds in three dimensions and understanding probability through self-locating uncertainty.
Some wave functions separate into two or more distinct regions in phase space. Each region is characterized by a trajectory and a spread about that trajectory. The trajectory is the quantum mechanical current. We show that these regions correspond to parts of the wavefunction and that these parts are generally nonorthogonal.
It is pointed out that ordinary quantum mechanics as a classical field theory cannot account for the wavefunction collapse if it is not seen within the framework of field quantization. That is needed to understand the particle structure of matter during wavefunction evolution and to explain the collapse as symmetry breakdown by detection. The decay of a two-particle bound s state and the Stern-Gerlach experiment serve as examples. The absence of the nonlocality problem in (...) Bohm’s version of the EPR arrangement favours the approach described. (shrink)
The possibility of consistency between the basic quantum principles of quantum mechanics and wavefunction collapse is reexamined. A specific interpretation of environment is proposed for this aim and is applied to decoherence. When the organization of a measuring apparatus is taken into account, this approach leads also to an interpretation of wavefunction collapse, which would result in principle from the same interactions with environment as decoherence. This proposal is shown consistent with the non-separable character (...) of quantum mechanics. (shrink)
I survey the options for understanding the nature of the wave-function in the setting of the relativistic collapse models recently developed by Tumulka. Some of the options involve surprising features, such as backwards causation or locality.
In this paper I investigate, within the framework of realistic interpretations of the wavefunction in nonrelativistic quantum mechanics, the mathematical and physical nature of the wavefunction. I argue against the view that mathematically the wavefunction is a two-component scalar field on configuration space. First, I review how this view makes quantum mechanics non- Galilei invariant and yields the wrong classical limit. Moreover, I argue that interpreting the wavefunction as (...) a ray, in agreement many physicists, Galilei invariance is preserved. In addition, I discuss how the wavefunction behaves more similarly to a gauge potential than to a field. Finally I show how this favors a nomological rather than an ontological view of the wavefunction. (shrink)
The wavefunction is the central mathematical entity of quantum mechanics, but it still lacks a universally accepted interpretation. Much effort is spent on attempts to probe its fundamental nature. Here I investigate the consequences of a matter ontology applied to spherical masses of constant bulk density. The governing equation for the center-of-mass wavefunction is derived and solved numerically. The ground state wavefunctions and resulting matter densities are investigated. A lowering of the density from its bulk value is found for low (...) masses due to increased spatial spreading. A discussion of the possibility to experimentally observe these effects is given and the possible consequences for choosing an ontological interpretation for quantum mechanics are commented upon. (shrink)
In this paper I review three different positions on the wavefunction, namely: nomological realism, dispositionalism, and configuration space realism by regarding as essential their capacity to account for the world of our experience. I conclude that the first two positions are committed to regard the wavefunction as an abstract entity. The third position will be shown to be a merely speculative attempt to derive a primitive ontology from a reified mathematical space. Without entering any (...) discussion about nominalism, I conclude that an elimination of abstract entities from one’s ontology commits one to instrumentalism about the wavefunction, a position that therefore is not as unmotivated as it has seemed to be to many philosophers. (shrink)
The main claim of the paper is that one can be ‘realist’ (in some sense) about quantum mechanics without requiring any form of realism about the wavefunction. We begin by discussing various forms of realism about the wavefunction, namely Albert’s configuration-space realism, Dürr Zanghi and Goldstein’s nomological realism about Ψ, Esfeld’s dispositional reading of Ψ Pusey Barrett and Rudolph’s realism about the quantum state. By discussing the articulation of these four positions, and their interrelation, (...) we conclude that instrumentalism about Ψ is by itself not sufficient to choose one over the other interpretations of quantum mechanics, thereby confirming in a different way the indetermination of the metaphysical interpretations of quantum mechanics. -/- Key words: . (shrink)
What is the meaning of the wave-function? After almost 100 years since the inception of quantum mechanics, is it still possible to say something new on what the wave-function is supposed to be? Yes, it is. And Shan Gao managed to do so with his newest book. Here we learn what contemporary physicists and philosophers think about the wave-function; we learn about the de Broglie-Bohm theory, the GRW collapse theory, the gravity-induced collapse theory by (...) Roger Penrose, and the famous PBR theorem; we learn about Schrödinger's original idea that the wave-function represents charge densities; we learn about the notorious measurement problem and its consequences; we learn about the challenges to find a consistent relativistic quantum theory; and we learn, of course, Gao's own suggestion for the status of the wave-function. Above all, Gao shows us the significance of protective measurements for our search of the ontology of quantum mechanics. Still not widely recognized among physicists and philosophers, protective measurements let us look deeper into quantum mechanics. For Gao this is the main tool to settle the issue on the ontological status of the wave-function: the wave-function is real because one can measure it. (shrink)
The gravity-related model of spontaneous wavefunction collapse, a longtime hypothesis, damps the massive Schrödinger Cat states in quantum theory. We extend the hypothesis and assume that spontaneous wavefunction collapses are responsible for the emergence of Newton interaction. Superfluid helium would then show significant and testable gravitational anomalies.
What is the quantum state of the universe? Although there have been several interesting suggestions, the question remains open. In this paper, I consider a natural choice for the universal quantum state arising from the Past Hypothesis, a boundary condition that accounts for the time-asymmetry of the universe. The natural choice is given not by a wavefunction but by a density matrix. I begin by classifying quantum theories into two types: theories with a fundamental wave (...) class='Hi'>function and theories with a fundamental density matrix. The Past Hypothesis is compatible with infinitely many initial wave functions, none of which seems to be particularly natural. However, once we turn to density matrices, the Past Hypothesis provides a natural choice---the normalized projection onto the Past Hypothesis subspace in the Hilbert space. Nevertheless, the two types of theories can be empirically equivalent. To provide a concrete understanding of the empirical equivalence, I provide a novel subsystem analysis in the context of Bohmian theories. Given the empirical equivalence, it seems empirically underdetermined whether the universe is in a pure state or a mixed state. Finally, I discuss some theoretical payoffs of the density-matrix theories and present some open problems for future research. (Bibliographic note: the thesis was submitted for the Master of Science in mathematics at Rutgers University.). (shrink)
Multi-time wave functions are wave functions for multi-particle quantum systems that involve several time variables. In this paper we contrast them with solutions of wave equations on a space–time with multiple timelike dimensions, i.e., on a pseudo-Riemannian manifold whose metric has signature such as \ or \, instead of \. Despite the superficial similarity, the two behave very differently: whereas wave equations in multiple timelike dimensions are typically mathematically ill-posed and presumably unphysical, relevant Schrödinger equations for (...) multi-time wave functions possess for every initial datum a unique solution on the spacelike configurations and form a natural covariant representation of quantum states. (shrink)
I address the question whether the wavefunction in quantum theory exists as a real quantity or not. For this purpose, I discuss the essentials of the quantum formalism and emphasize the central role of the superposition principle. I then explain the measurement problem and discuss the process of decoherence. Finally, I address the special features that the quantization of gravity brings into the game. From all of this I conclude that the wavefunction really exists, (...) that is, it is a real feature of Nature. (shrink)
Scientific endeavour has often tried to localize superior cerebral functions either in areas like the ones described by Broca as being those connected with language in the left hemisphere, or in the huge array of the hundred billion of interconnected neurons. But in this last case the coined description of the grandmother neuron, tends to show humorously that hopes have fallen short of their target.Along the same lines, the specific timing of electric neural activity is known to take place around (...) a few milliseconds, which seems to be insufficient to account for the high potential speed necessary to sustain the very massive and complex process which is involved in mental activity. (shrink)
We show that the physical meaning of the wavefunction can be derived based on the established parts of quantum mechanics. It turns out that the wavefunction represents the state of random discontinuous motion of particles, and its modulus square determines the probability density of the particles appearing in certain positions in space.
The most puzzling issue in the foundations of quantum mechanics is perhaps that of the status of the wavefunction of a system in a quantum universe. Is the wavefunction objective or subjective? Does it represent the physical state of the system or merely our information about the system? And if the former, does it provide a complete description of the system or only a partial description? We shall address these questions here mainly from a (...) Bohmian perspective, and shall argue that part of the difficulty in ascertaining the status of the wavefunction in quantum mechanics arises from the fact that there are two different sorts of wave functions involved. The most fundamental wavefunction is that of the universe. From it, together with the configuration of the universe, one can define the wavefunction of a subsystem. We argue that the fundamental wavefunction, the wavefunction of the universe, has a law-like character. (shrink)
Testable predictions of quantum mechanics are invariant under time reversal. But the evolution of the quantum state in time is not so, neither in the collapse nor in the no-collapse interpretations of the theory. This is a fact that challenges any realistic interpretation of the quantum state. On the other hand, this fact raises no difficulty if we interpret the quantum state as a mere calculation device, bookkeeping past real quantum events.
This paper focuses on a constructive treatment of the mathematical formalism of quantum theory and a possible role of constructivist philosophy in resolving the foundational problems of quantum mechanics, particularly, the controversy over the meaning of the wavefunction of the universe. As it is demonstrated in the paper, unless the number of the universe’s degrees of freedom is fundamentally upper bounded or hypercomputation is physically realizable, the universal wavefunction is a non-constructive entity in the (...) sense of constructive recursive mathematics. This means that even if such a function might exist, basic mathematical operations on it would be undefinable and subsequently the only content one would be able to deduce from this function would be pure symbolical. (shrink)
We investigate and develop further two models, the GRW model and the K model, in which the Schrödinger evolution of the wavefunction is spontaneously and repeatedly interrupted by random, approximate localizations, also called “self-reductions” below. In these models the center of mass of a macroscopic solid body is well localized even if one disregards the interactions with the environment. The motion of the body shows a small departure from the classical motion. We discuss the prospects and the (...) difficulties of observing this anomaly. As far a the influence of the surroundings on submacroscopic objects (like dust particles) is concerned, we show that the estimates obtained recently in the theory of continuous measurements and in the K model are compatible. Also, we elaborate upon the relationship between the models. Firstly, borrowing a line of thought from the K model, we find the transition region between macroscopic and microscopic behaviors in the GRW model. Secondly, we refine the propagation rule of the wavefunction in the K model with the help of the time-evolution equation proposed in the GRW model. (shrink)