We define what it means for a function on ω1 to be a collapsing function for λ and show that if there exists a collapsing function for +, then there is no precipitous ideal on ω1. We show that a collapsing function for ω2 can be added by forcing. We define what it means to be a weakly ω1-Erdös cardinal and show that in L[E], there is a collapsing function for λ (...) iff λ is less than the least weakly ω1-Erdös cardinal. As a corollary to our results and a theorem of Neeman, the existence of a Woodin limit of Woodin cardinals does not imply the existence of precipitous ideals on ω1. We also show that the following statements hold in L[E]. The least cardinal λ with the Chang property ↠ is equal to the least ω1-Erdös cardinal. In particular, if j is a generic elementary embedding that arises from non-stationary tower forcing up to a Woodin cardinal, then the minimum possible value of j is the least ω1-Erdös cardinal. (shrink)

It is shown how the strong ordinal notation systems that figure in proof theory and have been previously defined by employing large cardinals, can be developed directly on the basis of their recursively large counterparts. Thereby we provide a completely new approach to well-ordering proofs as will be exemplified by determining the proof-theoretic ordinal of the systemKPM of [R91].

In the previous paper of this volume, Kojman and Shelah solved our long standing problem of collapsing cardinal κ0 to ω1 by the forcing for singular κ with countable cofinality. The aim of the present paper is to give an explicit construction of the Boolean matrix for this collapse.

Does consciousness collapse the quantum wave function? 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)

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 wave function collapse.

The following introduction offers a broad survey of the history of quantum physics. It then outlines the position of each contributor in this Special Focus Section concerning the collapse of the quantum wave function and defines three important terms (Hilbert space, Schrödinger’s cat, and decoherence) used in discussing this topic.

Experiments are described, using electroencephalography (EEG) and simple tests of performance, which support the hypothesis that collapse of a quantum field is of importance to the functioning of the brain. The theoretical basis of our experiments is derived from Penrose (1989) who suggested that conscious decision-making is a manifestation of the outcome of quantum computation in the brain involving collapse of some relevant wave function. He also proposed that collapse of any wave function depends on a gravitational criterion. (...) As different brain areas are known to subserve different functions, we argue that `Penrose collapse' must occur in a particular brain area when performing a task that uses it. Further, taking an EEG from the area should amplify the gravitational prerequisite for collapse, so affecting task performance. There are no non-quantum theories which could lead one to expect that taking an EEG could directly affect task performance by subjects. The results of both pilot and main experiments indicated that task performance was indeed influenced by taking an EEG from relevant brain areas. Control experiments suggested that the influence was quantum mechanical in origin, and was not due to any experimental artefact. The results are statistically significant and merit attempts at replication in an independent laboratory, preferably with more sophisticated equipment than was available to us. (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 wave function 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)

Probability distributions are seen to be observer dependent. The probability function ψ†ψ can be put into an observer-dependent form. This eliminates the acausal behavior of the collapse of the wave function.

The following introduction offers a broad survey of the history of quantum physics. It then outlines the position of each contributor in this Special Focus Section concerning the collapse of the quantum wave function and defines three important terms (Hilbert space, Schrödinger’s cat, and decoherence) used in discussing this topic.

The possibility of consistency between the basic quantum principles of quantum mechanics and wave function 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 wave function 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)

The gravity-related model of spontaneous wave function collapse, a longtime hypothesis, damps the massive Schrödinger Cat states in quantum theory. We extend the hypothesis and assume that spontaneous wave function collapses are responsible for the emergence of Newton interaction. Superfluid helium would then show significant and testable gravitational anomalies.

It is pointed out that ordinary quantum mechanics as a classical field theory cannot account for the wave function collapse if it is not seen within the framework of field quantization. That is needed to understand the particle structure of matter during wave function 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)

Early Chinese argumentative texts are full of historical anecdotes. These short accounts of events in Chinese history enhance the appeal of the text, but they also have an important rhetorical function in helping the reader understand, accept, and remember the arguments propounded in the text. In this paper I examine the rhetorical function of historical anecdotes in two argumentative texts of the Western Han dynasty (202 BCE-9 CE): Han’s Illustrations of the Odes for Outsiders and The Master of (...) Huainan. These two texts found creative use for anecdotes, namely as illustrations of quotations from canonical sources. Through case studies of several combinations of anecdotes and quotations, I argue that the combinations serve to present the creators of these texts as beacons of knowledge with profound understanding of historical events and canonical literature, and with the necessary skills to fruitfully combine the two. (shrink)

The black hole information paradox arises from an apparent conflict between the Hawking black hole radiation and the fact that time evolution in quantum mechanics is unitary. The trouble is that while the former suggests that information of a system falling into a black hole disappears, the latter implies that information must be conserved. In this work we discuss the current divergence in views regarding the paradox, we evaluate the role that objective collapse theories could play in its resolution and (...) we propose a link between spontaneous collapse events and microscopic virtual black holes. (shrink)

The many-Hilbert-spaces approach to the measurement problem in quantum mechanics is reviewed, and the notion of wave function collapse by measurement is formulated as a dephasing process between the two branch waves of an interfering particle. Following the approach originally proposed in Ref. 1, we introduce a “decoherence parameter,” which yields aquantitative description of the degree of coherence between the two branch waves of an interfering particle. By discussing the difference between the wave function collapse and the orthogonality (...) of the apparatus' wave functions, we analyze critically two proposals, recently appeared in the literature, (2, 3) and argue that neither one describes a dephasing process. We conclude that the concept of “wave function collapse,” according to the conventional Copenhagen interpretation, is to be replaced by that of a statisticallydefined dephasing process. (shrink)

Collapse models predict the spontaneous collapse of the wave function, in order to avoid the emergence of macroscopic superpositions. In their mass-dependent formulation, they claim that the collapse of any system’s wave function depends on its mass. Neutral K, D, B mesons are oscillating systems that are given by Nature as superposition of two distinct mass eigenstates. Thus they are unique laboratory for testing collapse models that are sensitive to the mass. In this paper we derive—for the single (...) mesons and bipartite entangled mesons—the effect of the mass-proportional CSL (Continuous Spontaneous Localization) collapse model on the dynamics on neutral mesons. We compare the theoretical prediction with experimental data from different accelerator facilities. (shrink)

We give a new argument supporting a gravitational role in quantum collapse. It is demonstrated that the discreteness of space-time, which results from the proper combination of quantum theory and general relativity, may inevitably result in the dynamical collapse of thewave function. Moreover, the minimum size of discrete space-time yields a plausible collapse criterion consistent with experiments. By assuming that the source to collapse the wave function is the inherent random motion of particles described by the wave (...) class='Hi'>function, we further propose a concrete model of wavefunction collapse in the discrete space-time. It is shown that the model is consistent with the existing experiments and macroscopic experiences. (shrink)

The aim of this paper is to refute the hypothesis that the observer’s consciousness is necessary in the quantum mechanics measurement process. In order to achieve our target, we propose and investigate a variation of the Schrödinger’s cat thought experiment called “DAP”, short for “Dead-Alive Physicist”, in which a human being replaces the cat. This strategy enables us to logically disprove the consistency of the above hypothesis, and to oblige its supporters either to be trapped in solipsism or to rely (...) on an alternative interpretation of quantum mechanics in which the conscious observer plays the sole role of acknowledging the experimental results. Our analysis hence provides support to clarify the relationship between the observer and the objects of her/his experimental observation; this and a few other implications are discussed in the fourth section and in the conclusions. (shrink)

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 Schrödinger's original charge-density interpretation. This discussion is extended to relativistic CSL-type theories. A few remarks on Everett's interpretation conclude the paper. (shrink)

The primary quantum mechanical equation of motion entails that measurements typically do not have determinate outcomes, but result in superpositions of all possible outcomes. Dynamical collapse theories (e.g. GRW) supplement this equation with a stochastic Gaussian collapse function, intended to collapse the superposition of outcomes into one outcome. But the Gaussian collapses are imperfect in a way that leaves the superpositions intact. This is the tails problem. There are several ways of making this problem more precise. But many authors (...) dismiss the problem without considering the more severe formulations. Here I distinguish four distinct tails problems. The first (bare tails problem) and second (structured tails problem) exist in the literature. I argue that while the first is a pseudo-problem, the second has not been adequately addressed. The third (multiverse tails problem) reformulates the second to account for recently discovered dynamical consequences of collapse. Finally the fourth (tails problem dilemma) shows that solving the third by replacing the Gaussian with a non-Gaussian collapse function introduces new conflict with relativity theory. (shrink)

Despite growing interest in risks of societal collapse due to anthropogenic climate change, there exists no consensus about how collapse should be understood. In this article, we critically examine existing definitions and argue that none adequately address the challenges for conceptualizing collapse that climate change presents. We therefore propose an alternative conception, which regards collapse as a reduction of collective capacity resulting in a pervasive and difficult-to-reverse loss of basic functionality. Our conception is dynamic in that it focuses on the (...) interrelations of constituent subsystems. It also distinguishes collapse from transformations needed to address climate change and provides insight into the relationship between collapse and sustainability. [Open access]. (shrink)

Collapse is a term that has attracted much attention in social science literature in recent years, but there remain substantial areas of disagreement about how it should be understood in historical contexts. More specifically, the use of the term collapse often merely serves to dramatize long-past events, to push human actors into the background, and to mystify the past intellectually. At the same time, since human societies are complex systems, the alternative involves grasping the challenges that a holistic analysis presents, (...) taking account of the many different levels and paces at which societies function, and developing appropriate methods that help to integrate science and history. Often neglected elements in considerations of collapse are the perceptions and beliefs of a historical society and how a given society deals with change; an important facet of this, almost entirely ignored in the discussion, is the understanding of time held by the individuals and social groups affected by change; and from this perspective ‘collapse’ depends very much on perception, including the perceptions of the modern commentator. With this in mind, this article challenges simplistic notions of ‘collapse’ in an effort to encourage a more nuanced understanding of the impact and process of both social and environmental change on past human societies. (shrink)

In ordinal analysis of impredicative theories so-called collapsing functions are of central importance. Unfortunately, the definition procedure of these functions makes essential use of uncountable cardinals whereas the notation system that they call into being corresponds to a recursive ordinal. It has long been claimed that, instead, one should manage to develop such functions directly on the basis of admissible ordinals. This paper is meant to show how this can be done. Interpreting the collapsing functions as operating directly (...) on admissible sets also renders a new and perspicuous approach to well-ordering proofs possible. MSC: 03F15, 03F35. (shrink)

We compute the effect of collapse models on neutrino oscillations. The effect of the collapse is to modify the evolution of the spatial part of the wave function and we will show that this indirectly amounts to a change on the flavor components. For the analysis we use the mass proportional CSL model, and perform the calculation to second order perturbation theory. As we will show, the CSL effect is very small—mainly due to the very small mass of neutrinos—and (...) practically undetectable. (shrink)

GRW theory offers precise laws for the collapse of the wave function. These collapses are characterized by two new constants, \ and \ . Recent work has put experimental upper bounds on the collapse rate, \ . Lower bounds on \ have been more controversial since GRW begins to take on a many-worlds character for small values of \ . Here I examine GRW in this odd region of parameter space where collapse events act as natural disasters that destroy (...) branches of the wave function along with their occupants. Our continued survival provides evidence that we don’t live in a universe like that. I offer a quantitative analysis of how such evidence can be used to assess versions of GRW with small collapse rates in an effort to move towards more principled and experimentally-informed lower bounds for \. (shrink)

A correction to this paper has been published: https://doi.org/10.1007/s10701-021-00451-y.

Many non-physicalists, including Chalmers, hold that the zombie argument succeeds in rejecting the physicalist view of consciousness. Some non-physicalists, including, again, Chalmers, hold that quantum collapse interactionism, i.e., the idea that non-physical consciousness causes collapse of the wave function in phenomena such as quantum measurement, is a viable interactionist solution for the problem of the relationship between the physical world and the non-physical consciousness. In this paper, I argue that if QCI is true, the zombie argument fails. In particular, (...) I show that if QCI is true, a zombie world physically identical to our world is impossible because there is at least one law of nature, a fundamental law of physics in particular, that exist only in the zombie world but not in our world. This shows that philosophers like Chalmers are committing an error in endorsing the zombie argument and QCI at the same time. (shrink)

The existence of spacetime singularities is one of the biggest problems of nowadays physics. According to Penrose, each physical singularity should be covered by a “cosmic censor” which prevents any external observer from perceiving their existence. However, classical models describing the gravitational collapse usually results in strong curvature singularities, which can also remain “naked” for a finite amount of advanced time. This proceedings studies the modifications induced by asymptotically safe gravity on the gravitational collapse of generic Vaidya spacetimes. It will (...) be shown that, for any possible choice of the mass function, quantum gravity makes the internal singularity gravitationally weak, thus allowing a continuous extension of the spacetime beyond the singularity. (shrink)

In this article we attempt to bring out some significant general aspects of what we call collapse-induced quantum nonlocal effects resulting from the use of the hypothesis of wave function collapse.

We consider a well-known partial order of Prikry for producing a collapsing function of minimal degree. Assuming MA + ≠ CH, every new real constructs the collapsing map.

One of the most serious challenges (if not the most serious challenge) for interactive psycho-physical dualism (henceforth interactive dualism or ID) is the so-called ‘interaction problem’. It has two facets, one of which this article focuses on, namely the apparent tension between interactions of non-physical minds in the physical world and physical laws of nature. One family of approaches to alleviate or even dissolve this tension is based on a collapse solution (‘consciousness collapse/CC) of the measurement problem in quantum mechanics (...) (QM). The idea is that the mind brings about the collapse of a superposed wave function onto one of its eigenstates. Thus, it is claimed, can the mind change the course of things without violating any law figuring in physical theory. I will first show that this hope is premature because energy and momentum are probably not conserved in collapse processes, and that even if this can be dealt with, the violations are either severe or produce further ontological problems. Second, I point out several conceptual difficulties for interactionist CC. I will also present solutions for those problems, but it will become clear that those solutions come at a high cost. Third, I shall briefly list some empirical problems which make life even harder for interactionist CC. I conclude with remarks about why no- collapse interpretations of QM don’t help either and what the present study has shown is the real issue for ID: namely to find a plausible integrative view of dualistic mental causation and laws of nature. (shrink)

We consider a well-known partial order of Prikry for producing a collapsing function of minimal degree. Assuming $MA + \neq CH$, every new real constructs the collapsing map.

A familiar interpretation of quantum mechanics (one of a number of views sometimes labeled the "Copenhagen interpretation'"), takes its empirical apparatus at face value, holding that the quantum wave function evolves by the Schrödinger equation except on certain occasions of measurement, when it collapses into a new state according to the Born rule. This interpretation is widely rejected, primarily because it faces the measurement problem: "measurement" is too imprecise for use in a fundamental physical theory. We argue that this (...) is a weak objection, as there may be many ways of making "measurement" precise. However, measurement-collapse interpretations face a more serious objection: a dilemma tied to the quantum Zeno effect. Is measurement itself an observable that can enter superpositions? If yes, then the standard measurement-collapse dynamics is ill-defined. If no, then (at least if measurement is an observable), measurements can never start or finish. The best way out is to deny that measurement is an observable, but this leads to strong and revisionary consequences. This reinforces the view that there is no nonrevisionary interpretation of quantum mechanics. (shrink)

In this chapter, I will discuss what it takes for a dynamical collapse theory to provide a reasonable description of the actual world. I will start with discussions of what is required, in general, of the ontology of a physical theory, and then apply it to the quantum case. One issue of interest is whether a collapse theory can be a quantum state monist theory, adding nothing to the quantum state and changing only its dynamics. Although this was one of (...) the motivations for advancing such theories, its viability has been questioned, and it has been argued that, in order to provide an account of the world, a collapse theory must supplement the quantum state with additional ontology, making such theories more like hidden-variables theories than would first appear. I will make a case for quantum state monism as an adequate ontology, and, indeed, the only sensible ontology for collapse theories. This will involve taking dynamical variables to possess, not sharp values, as in classical physics, but distributions of values. (shrink)

Dynamical collapse models embody the idea of a physical collapse of the wave function in a mathematically well-defined way. They involve modifications to the standard rules of quantum theory in order to describe collapse as a physical process. This appears to introduce a time reversal asymmetry into the dynamics since the state at any given time depends on collapses in the past but not in the future. Here we challenge this conclusion by demonstrating that, subject to specified model constraints, (...) collapse models can be given a structurally time symmetric formulation in which the collapse events are the primitive objects of the theory. Three different examples of time asymmetries associated with collapse models are then examined and in each case it is shown that the same dynamical rule determining the collapse events works in both the forward and backward in time directions. Any physically observed time asymmetries that arise in such models are due to the asymmetric imposition of initial or final time boundary conditions, rather than from an inherent asymmetry in the dynamical law. This is the standard explanation of time asymmetric behaviour resulting from time symmetric laws. (shrink)

No one really believes that literally all functions of the entire brain must be lost for an individual to be dead. A better definition of death involves a higher brain orientation.

A simple quantum model describing the onset of time is presented. This is combined with a simple quantum model of the onset of space. A major purpose is to explore the interpretational issues which arise. The state vector is a superposition of states representing different “instants.” The sample space and probability measure are discussed. Critical to the dynamics is state vector collapse: it is argued that a tenable interpretation is not possible without it. Collapse provides a mechanism whereby the universe (...) size, like a clock, is narrowly correlated with the quantized time eigenvalues. (shrink)

Position measurement-induced collapse states are shown to provide a unified quantum description of diffraction of particles passing through a single slit. These states, which we here call ‘quantum location states’, are represented by the conventional rectangular wave function at the initial time of position measurement. We expand this state in terms of the position eigenstates, which in turn can be represented as a linear combination of energy eigenfunctions of the problem, using the closure property. The time-evolution of the location (...) states in the case of free particles is shown to have position probability density patterns closely resembling diffraction patterns in the Fresnel region for small times and the same in Fraunhofer region for large times. Using the quantum trajectory representations in the de Broglie–Bohm, modified de Broglie–Bohm and Floyd–Faraggi–Matone formalisms, we show that Fresnel and Fraunhofer diffractions can be described using a single expression. We also discuss how to obtain the probability density of location states for the case of particles moving in a general potential, detected at some arbitrary point. In the case of the harmonic oscillator potential, we find that they have oscillatory properties similar to that of coherent states. (shrink)

In this paper, I argue that explanations just ARE those sorts of things that, under the right circumstances and in the right sort of way, bring about understanding. This raises the question of why such a seemingly simple account of explanation, if correct, would not have been identified and agreed upon decades ago. The answer is that only recently has it been made possible to analyze explanation in terms of understanding without the risk of collapsing both to merely phenomenological (...) states. For the most part, theories of explanation were for 50 years held hostage to the historical accident that they far outstripped in sophistication corresponding accounts of understanding. (shrink)

The meaning of the wave function and its evolution are investigated. First, we argue that the wave function in quantum mechanics is a description of random discontinuous motion of particles, and the modulus square of the wave function gives the probability density of the particles being in certain locations in space. Next, we show that the linear non-relativistic evolution of the wave function 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 wave function. 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)

I examine a major objection to the mechanistic view of concrete computation, stemming from an apparent tension between the abstract nature of computational explanation and the tenets of the mechanistic framework: while computational explanation is medium-independent, the mechanistic framework insists on the importance of providing some degree of structural detail about the systems target of the explanation. I show that a common reply to the objection, i.e. that mechanistic explanation of computational systems involves only weak structural constraints, is not enough (...) to save the standard mechanistic view of computation—it trivialises the appeal to mechanism, and thus makes the account collapse into a purely functional view. I claim, however, that the objection can be put to rest once the account is appropriately amended: computational individuation is indeed functional, while mechanistic explanation plays a role in accounting for computational implementation. Since individuation and implementation are crucial elements in a satisfying account of computation in physical systems, mechanism keeps its central importance in the theory of concrete computation. Finally, I argue that my version of the mechanistic view helps to provide a convincing reply to a powerful objection against non-semantic theories of concrete computation: the argument from the multiplicity of computations. (shrink)

Quantum phenomena are notoriously difficult to grasp. The present paper first reviews the most important quantum concepts in a non-technical manner: superposition, uncertainty, collapse of the wave function, entanglement and non-locality. It then tries to clarify these concepts by examining their analogues in complex, self-organizing systems. These include bifurcations, attractors, emergent constraints, order parameters and non-local correlations. They are illustrated with concrete examples that include Rayleigh–Bénard convection, social self-organization and Gestalt perception of ambiguous figures. In both cases, quantum and (...) self-organizing, the core process appears to be a symmetry breaking that irreversibly and unpredictably “collapses” an ambiguous state into one of a number of initially equivalent “eigenstates” or “attractors”. Some speculations are proposed about the non-linear amplification of quantum fluctuations of the vacuum being ultimately responsible for such symmetry breaking. (shrink)