Two roads diverged in a wood, and I – I took the one less traveled by , And that has made all the difference . Robert Frost (1916) The explanatory role of empty waves in quantum theory In this chapter we are concerned with two classes of interpretations of quantum mechanics: the epistemological (the historically dominant view) and the ontological. The first views the wave function as just a repository of (statistical) information on a physical system. The other treats the wave function primarily as an element of physical reality, whilst generally retaining the statistical interpretation as a secondary property. There is as yet only theoretical justification for the program of modelling quantum matter in terms of an objective spacetime process; that some way of imagining how the quantum world works between measurements is surely better than none. Indeed, a benefit of such an approach can be that “measurements” lose their talismanic aspect and become just typical processes described by the theory. In the quest to model quantum systems one notes that, whilst the formalism makes reference to “particle” properties such as mass, the linearly evolving wave function ψ( x ) does not generally exhibit any feature that could be put into correspondence with a localized particle structure. To turn quantum mechanics into a theory of matter and motion, with real atoms and molecules consisting of particles structured by potentials and forces, it is necessary to bring in independent physical elements not represented in the basic formalism. The notion of an “empty wave” is peculiar to those representatives of this class of extended theories which postulate that the additional physical element is a corpuscle-like entity or point particle. For clarity, we shall develop the discussion in terms of a definite model of this kind whose properties are well understood and which it is established reproduces the empirical content of quantum mechanics: the de Broglie–Bohm theory, a prominent representative of the class of ontological interpretations (Holland, 1993).
Two roads diverged in a wood, and I – I took the one less traveled by , And that has made all the difference . Robert Frost (1916) The explanatory role of empty waves in quantum theory In this chapter we are concerned with two classes of interpretations of quantum mechanics: the epistemological (the historically dominant view) and the ontological. The first views the wave function as just a repository of (statistical) information on a physical system. The other treats the wave function primarily as an element of physical reality, whilst generally retaining the statistical interpretation as a secondary property. There is as yet only theoretical justification for the program of modelling quantum matter in terms of an objective spacetime process; that some way of imagining how the quantum world works between measurements is surely better than none. Indeed, a benefit of such an approach can be that “measurements” lose their talismanic aspect and become just typical processes described by the theory. In the quest to model quantum systems one notes that, whilst the formalism makes reference to “particle” properties such as mass, the linearly evolving wave function ψ( x ) does not generally exhibit any feature that could be put into correspondence with a localized particle structure. To turn quantum mechanics into a theory of matter and motion, with real atoms and molecules consisting of particles structured by potentials and forces, it is necessary to bring in independent physical elements not represented in the basic formalism. The notion of an “empty wave” is peculiar to those representatives of this class of extended theories which postulate that the additional physical element is a corpuscle-like entity or point particle. For clarity, we shall develop the discussion in terms of a definite model of this kind whose properties are well understood and which it is established reproduces the empirical content of quantum mechanics: the de Broglie–Bohm theory, a prominent representative of the class of ontological interpretations (Holland, 1993).
Within the class of ontological interpretations of quantum theory where a physical system comprises a particle and a field (wavefunction) guiding it, an empty wave is a segment of the wavefunction not containing the particle. We examine the impact of this concept on the debate between the epistemological and ontological viewpoints. The theoretical merits of the empty wave in avoiding the wavefunction collapse hypothesis, and in supplying conceptual precision in the application of quantum mechanics, are emphasized. Particular reference is made to path detection in interferometry and it is shown how a consistent application of the particle and empty wave concepts dispels the suggestion that this is a theory. The problem of how the reality of an empty wave might be demonstrated by its effect on other systems is examined and general arguments advanced against this possibility. However, these arguments are not conclusive and we describe how the technique of protective measurement suggests a method to probe the empirical implications of empty waves and potentially distinguish between the epistemological and ontological perspectives.
In this book, I have argued that the wave function in quantum mechanics is real, and it represents the state of random discontinuous motion (RDM) of particles in three-dimensional space. Moreover, this picture of quantum ontology is complete in accounting for our definite experience, but it requires that the quantum dynamics be revised to include a stochastic, nonlinear evolution term resulting from the RDM of particles. Obviously I took a road less traveled by researchers in the foundations of quantum mechanics. In order to convince more readers that this road is deserved to be taken, I will review the main results of this book and discuss them in a broader context. The starting point of my road is protective measurements (Chapter 1). In 1993, this new method of measurement in quantum mechanics was discovered (Aharonov and Vaidman, 1993; Aharonov, Anandan, and Vaidman, 1993). Distinct from conventional projective measurements, protective measurement is a method for measuring the expectation value of an observable on a single quantum system. By a series of protective measurements, one can even measure the wave function of a single quantum system. Therefore, besides the Born rule for projective measurements, protective measurements provide another, more direct connection between the wave function and results of measurements, which is not probabilistic but definite. It can be expected that a definite connection between the wave function and results of measurements is more important for understanding the meaning of the wave function and searching for the ontology of quantum mechanics. However, it seems that this connection provided by protective measurements is still less well known today, and its significance has not been fully realized by most researchers, either. The first stop on my road is the reality of the wave function (Chapter 4). It is correct to say that protective measurements alone do not imply the reality of the wave function. An additional connection between the state of reality and results of measurements is needed.
Quantum mechanics is our most successful physical theory. It predicts the behavior of atoms, molecules, and light to ten decimal places. No experiment has ever contradicted it. It also contains a crack at its foundation. The theory’s central equation—the Schrodinger equation—describes a smooth, continuous, perfectly predictable evolution of the quantum state. But whenever anyone performs a measurement, they get a single, definite result. Which result they get appears to be random. The transition from smooth evolution to random outcome is called the “collapse of the wavefunction,” and after nearly a century of analysis, there is no agreed-upon explanation for what causes it, when it occurs, or whether it occurs at all. This is the measurement problem. The measurement problem is not a minor puzzle at the margins of the theory. It is a failure to explain the most basic operation in science: looking at the result of an experiment. Around it cluster other puzzles. Light behaves like a wave in some experiments and like a particle in others—two contradictory descriptions. Two particles that have interacted remain correlated across arbitrary distances, as though communicating faster than light. The probabilities that govern measurement outcomes appear as a postulate—the Born rule—inserted by hand with no deeper justification. And the theory offers no unified picture of reality: quantum mechanics, general relativity, and the standard model of particle physics speak incompatible mathematical languages. A century of effort has produced many interpretations of quantum mechanics. Copenhagen. Many-worlds. De Broglie-Bohm. Objective collapse. Quantum Bayesianism. None has achieved consensus. The measurement problem remains. This document proposes that the source of these puzzles is not quantum mechanics itself. It is a single, invisible assumption made at the very beginning of the theory in 1900—an assumption so deeply embedded that nobody noticed they were making it. Change that assumption, and the puzzles do not get solved. They dissolve. They stop being puzzles. They become predictable consequences of geometry.
David Lewis famously quipped that he was willing to take metaphysical lessons from quantum mechanics only when it cleaned up its own act by providing interpretations or modifications of the formalism shorn of appeal to consciousness, irreducibly macroscopic notions, or outright instrumentalism. By the first decade of the twenty-first century, multiple approaches to quantum mechanics had been developed to a degree which met Lewis’s challenge: dynamical-collapse theories, Bohm-style hidden-variable theories, and Everett-style many-worlds theories, for all their respective problems and challenges, all offer ways of understanding quantum mechanics that metaphysicians can reasonably engage with. And so the last twenty years have seen a resurgence in the metaphysics of quantum mechanics, the question of what ontology and ideology is appropriate to a quantum universe. The metaphysics of quantum mechanics is not independent of the quantum measurement problem but does not coincide with it, either: there is a widespread view that adopting, say, the Everett interpretation still leaves significant ontological questions undecided, and conversely, a given answer to those questions might answer them for a dynamical-collapse theory too.Wave function realism has become a leading contender for the metaphysics of quantum mechanics, particularly in the case of dynamical-collapse and Everettian approaches. The starting point is nonrelativistic quantum mechanics expressed in configuration space: mathematically, the subject matter of the theory is a complex function of 3N variables, evolving under the Schrödinger equation (perhaps interrupted by occasional stochastic collapses). Traditionally those 3N variables are interpreted as the three coordinates of position of N particles—but in wave function realism, they are interpreted as points in a new, high-dimensional space. Quantum mechanics, according to the wave function realist, fundamentally concerns the undulations of the wave function in this space; three-dimensional goings-on are emergent, high-level, derivative.Alyssa Ney has been a leading figure in the development of wave function realism for most of its existence. Her articles advocating and developing the position have been highly influential, and the present book can be seen as the culmination of this line of her research and is the definitive statement of wave function realism as of 2021. Despite its comparative brevity, the book is too rich to be adequately described—far less critiqued—in this brief review; my goal is to give an overview of the main themes and arguments, and to make a few critical remarks along the way.The World in the Wavefunction is not formally divided into parts, but it can profitably be thought of that way. Chapters 1 through 3 develop the core case for wave function realism. Chapter 4 defends the position from the frequent criticism that it is restricted to nonrelativistic physics, by providing a version of wave function realism appropriate to quantum field theory. And chapters 5 through 7 are an extended exploration of how the observed three-dimensional world can be understood as emerging from a high-dimensional fundamental reality.The case Ney makes in chapters 1 through 3 for wavefunction realism is modest and cautious. In chapter 1 she provides a fairly standard—though elegantly expressed—case for the prima facie plausibility of wave function realism simply based on the fact that standard formulations of quantum mechanics are written in 3N dimensions, and on the (related) fact that entanglement means we cannot interpret quantum mechanics simply in terms of spatially localized particles and nothing else. But in chapter 2 she explicitly argues against any attempt to strengthen these plausibility arguments into reasons why wave function realism must be adopted; indeed, she provides a detailed and sympathetic taxonomy of the alternatives to wave function realism that have been developed, all of which she regards as entirely viable. (Chapter 2 of the book could profitably be used as introductory reading for a general graduate course on quantum metaphysics.) For Ney, the question of which metaphysics to adopt is a standard metaphysician’s question, to be answered by weighing strengths and weaknesses of the various alternatives against one another.In chapter 3, Ney identifies two such strengths for wave function realism. First, it is separable: we can specify the complete state of the universe point-by-point on the fundamental space, without any need to appeal to irreducible holism. Second, it is dynamically local: expressed in 3N-dimensional space, the Schrödinger dynamics involve no action at a distance.Ney makes the case for separability at length. Her case for dynamical locality is brief: she regards it as a straightforward consequence of the fact that the Schrödinger equation is a differential equation, so that the rate of change of the wave function at a point x depends only on the wave function arbitrarily close to x. But I think there is more worth investigating here: after all, the nonrelativistic Schrödinger equation may be a differential equation, but it is a parabolic differential equation, with no maximum speed of propagation: the rate of change of the wave function at x may depend only on a neighborhood of x, but the value of the wavefunction at x an arbitrarily short time later depends on the wave function everywhere in space. (And this is not—or not straightforwardly—just an artifact of ignoring relativity.)But in any case, why think that a separable, local theory is preferable to the alternative? Ney claims that separability and locality are more intuitive and lead to a more comprehensible ontology—and she makes an unabashed defense of intuitiveness and explicability as theoretical virtues. She is quite explicit that her reasons are pragmatic: intuitive theories benefit students and teachers; they are better suited to help us develop new theories; they speak better to the reasons why many students (Ney included) want to study physics in the first place. The implied metaphilosophy here is fascinating and it would be good to see it developed further in subsequent work: Ney seems to reject the mainstream modern approach to ontology (in which these pragmatic virtues are not obviously relevant to the truth of wave function realism) in favor of something neo-Carnapian where (so far as I can see) the goal of a quantum ontology is to help us get a deeper understanding of quantum mechanics, and that’s all there is to it. This also fits naturally with the pluralism she urges, in which approaches other than hers can also be valuable.Chapter 4—on relativistic generalizations of wavefunction realism—attempts both a conceptual and a technical task. The conceptual task is to argue that wavefunction realism per se is not wedded to any particular theory or choice of Hilbert-space basis within the theory: rather, wave function realism is any strategy that reifies a high-dimensional space in order to produce a separable local quantum ontology. If so, there is no block in principle to a relativistic generalization of the approach: we just need to find some appropriate high-dimensional space to describe the quantum state in relativistic quantum theory.The general philosophical point is well taken and convincingly defended (though it’s important to note that while arbitrary choices of Hilbert-space basis will give a separable theory, there is no reason to assume they will give a dynamically local theory: even something as simple as shifting from the position to the momentum basis in nonrelativistic quantum mechanics transforms the Schrödinger equation into something explicitly nonlocal). Unfortunately, I don’t think Ney succeeds in giving a clear statement of the high-dimensional space she has in mind for quantum field theory. This is one place where her laudable goal of making the book intelligible to nonspecialists becomes problematic: she makes many suggestive and interesting comments on quantum field theory, but there is no explicit, precise, mathematical statement of her view that can be contrasted with the explicit, precise, mathematical representation of the state in nonrelativistic quantum mechanics as a complex function on 3N-dimensional space.In the last part of the book—concerned with the emergence of ordinary three-dimensional ontology—Ney begins (chapter 5) by rebutting influential arguments by Tim Maudlin, Valia Allori, Sheldon Goldstein, and others that it is empirically incoherent for a physical theory not to make precise, nonemergent, nonderivative statements about matter in three-dimensional space (for a theory not to have a primitive ontology, to use the standard term for it). Ney argues—persuasively, to me—that it is perfectly compatible with the way science usually works for a theory to make statements about the observable only in an indirect, derivative way, so that while it might be a desideratum for a theory to have a primitive ontology, it is not compulsory for it to do so in order to be empirically adequate. She goes on, though, to note that a satisfactory quantum metaphysics had better recover the existence of ordinary three-dimensional objects, on Moorean grounds: we are really confident that those objects exist!Chapter 6 is largely critical: an extended exegesis of the functionalist strategy for recovering higher-order ontology developed by David Albert, myself, and others. She raises two main criticisms of the functionalist strategy: it fails to distinguish between the wave function simulating three-dimensional objects and constituting those objects; it is restricted to recovering classical three-dimensional ontology, whereas we want to recover quantum-mechanical phenomena in three dimensions too. (The former objection is a development of criticisms by Bradley Monton, Tim Maudlin, and others; the latter is novel so far as I know.) For reasons of space, I won’t attempt to reply on behalf of the functionalist, but Ney certainly puts the challenge with admirable clarity.The final chapter of the book presents Ney’s positive account of emergent ontology, based not on functionalism but on considerations of symmetry (though she concedes that the separation is not completely sharp). Simplifying Ney’s subtle position: the permutation symmetries of quantum mechanics give a clear dynamical reason to regard certain decompositions of 3N-dimensional space (intuitively: those that decompose it into the N-fold product of Euclidean space) as preferred; that then licenses the metaphysical hypothesis that a localized wave function in 3N-dimensional space has, as its parts, particles in three-dimensional space. If the wave function is not localized (as will be the case generically, at least for the Everett interpretation), then we can think of those particles as being partially localized in various different regions of space.I have to say that I am not convinced this works, at least without a dynamical-collapse mechanism: in the Everett interpretation, it will yield an ontology where all the particles are completely delocalized. We only see a somewhat localized description of matter in three dimensions when we restrict to an Everett branch, and that restriction is normally motivated on functionalist grounds. But this may simply reflect subtleties of Ney’s intriguing proposal that I am not fully grasping at present.In summary, I congratulate Ney for a rich, subtle, and provocative account of wave function realism. Her edited collection of papers (coedited with Albert), The Wave Function, set the agenda for much of the metaphysics of quantum mechanics in the 2010s; The World in the Wave Function has a good chance of doing the same for the 2020s.
The meaning of the wave function 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 wave function ontic, directly representing a state of reality, or epistemic, merely representing a state of (incomplete) knowledge, or something else? If the wave function is not ontic, then what, if any, is the underlying state of reality? If the wave function is indeed ontic, then exactly what physical state does it represent? In this book, I aim to make sense of the wave function 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 wave function (Chapters 1-5). After giving a comprehensive and critical review of the competing views of the wave function, I present a new argument for the ontic view in terms of protective measurements. In addition, I also analyze the origin of the wave function by deriving the free Schroedinger equation. The second part analyzes the ontological meaning of the wave function (Chapters 6, 7). I propose a new ontological interpretation of the wave function 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 (Chapters 8, 9).
The wave function gives not the density of stuff, but gives rather (on squaring its modulus) the density of probability. Probability of what, exactly? Not of the electron being there, but of the electron being found there, if its position is ‘measured’. Why this aversion to ‘being’ and insistence on ‘finding’? The founding fathers were unable to form a clear picture of things on the remote atomic scale . John S. Bell (1990) Introduction The physical meaning of the wave function is an important interpretative problem of quantum mechanics. Notwithstanding nearly ninety years of development of the theory, it is still an unsolved issue. During recent years, more and more research has been done on the ontological status and meaning of the wave function (see, e.g. Monton, 2002; Lewis, 2004; Gao, 2011a, 2011b; Pusey, Barrett and Rudolph, 2012; Ney and Albert, 2013). In particular, Pusey, Barrett and Rudolph (2012) demonstrated that under certain non-trivial assumptions such as the preparation independence assumption, the wave function of a quantum system is a representation of the physical state of the system. This poses a further question, namely whether the reality of the wave function can be argued without resorting to non-trivial assumptions. Moreover, a harder problem is to determine the ontological meaning of the wave function, which is still a hot topic of debate in the realistic alternatives to quantum mechanics such as the de Broglie–Bohm theory or Bohmian mechanics (Belot, 2012). In this chapter, we will first give a clearer argument for the reality of the wave function in terms of protective measurements, which does not depend on non-trivial assumptions and can also overcome existing objections. Next, based on an analysis of the mass and charge properties of a quantum system, we will propose a new ontological interpretation of the wave function. According to this interpretation, the wave function of an N -body system represents the state of ergodic motion of N particles.