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Noûs

The Status of our Ordinary Three Dimensions in a Quantum Universe1

Alyssa Ney

Published Dec 15, 2010• 77 citations
Publisher

Abstract

Abstract There are now several, realist versions of quantum mechanics on offer. On their most straightforward, ontological interpretation, these theories require the existence of an object, the wavefunction, which inhabits an extremely high‐dimensional space known as configuration space. This raises the question of how the ordinary three‐dimensional space of our acquaintance fits into the ontology of quantum mechanics. Recently, two strategies to address this question have emerged. First, Tim Maudlin, Valia Allori, and her collaborators argue that what I have just called the ‘most straightforward’ interpretation of quantum mechanics is not the correct one. Rather, the correct interpretation of realist quantum mechanics has it describing the world as containing objects that inhabit the ordinary three‐dimensional space of our manifest image. By contrast, David Albert and Barry Loewer maintain the straightforward, wavefunction ontology of quantum mechanics, but attempt to show how ordinary, three‐dimensional space may in a sense be contained within the high‐dimensional configuration space the wavefunction inhabits. This paper critically examines these attempts to locate the ordinary, three‐dimensional space of our manifest image “within” the ontology of quantum mechanics. I argue that we can recover most of our manifest image, even if we cannot recover our familiar three‐dimensional space.

DOI
10.1111/j.1468-0068.2010.00797.x

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According to the high-dimensional approach to quantum mechanics (a.k.a. wavefunction realism), the fundamental space of our world has an unfathomably large number of dimensions. This account is empirically adequate only if the three-dimensional manifest image can somehow be recovered from high-dimensional reality. A proper understanding of inter-level metaphysics (a.k.a. metaphysical explanation, grounding, etc.) shows that the manifest image can indeed be recovered, and answers the most concerning objections to high-dimensionalism. But it also shows that high-dimensionalism has disturbing consequences about the objectivity of the manifest image.

The Journal of Philosophy•Jan 1, 2024•1 citation

The World in the Wave Function

David J. Wallace

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.

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Alyssa Ney

The purpose of the present chapter is to respond to a thread of recent criticism against one candidate framework for interpreting quantum theories, a framework introduced and defended by David Albert and Barry Loewer: wave function realism, a framework for interpreting the ontology of quantum theories according to which what appears to be a nonseparable metaphysics ofentangled objects acting instantaneously across spatial distances is a manifestation of a more fundamental separable and local metaphysics in higher dimensions. Thechapterconsiders strategies for extending the wave function realist interpretation of quantum mechanics to the case of relativistic quantum theories, responding to arguments that this cannot be done.

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The Structure of a Quantum World*

Jill North

I argue that the fundamental space of a quantum mechanical world is the wavefunction's space. I argue for this using some very general principles that guide our inferences to the fundamental nature of a world, for any fundamental physical theory. I suggest that ordinary three-dimensional space exists in such a world, but is non-fundamental; it emerges from the fundamental space of the wavefunction.

Oxford University Press eBooks•Apr 8, 2013•121 citations
Abstract There are now several, realist versions of quantum mechanics on offer. On their most straightforward, ontological interpretation, these theories require the existence of an object, the wavefunction, which inhabits an extremely high‐dimensional space known as configuration space. This raises the question of how the ordinary three‐dimensional space of our acquaintance fits into the ontology of quantum mechanics. Recently, two strategies to address this question have emerged. First, Tim Maudlin, Valia Allori, and her collaborators argue that what I have just called the ‘most straightforward’ interpretation of quantum mechanics is not the correct one. Rather, the correct interpretation of realist quantum mechanics has it describing the world as containing objects that inhabit the ordinary three‐dimensional space of our manifest image. By contrast, David Albert and Barry Loewer maintain the straightforward, wavefunction ontology of quantum mechanics, but attempt to show how ordinary, three‐dimensional space may in a sense be contained within the high‐dimensional configuration space the wavefunction inhabits. This paper critically examines these attempts to locate the ordinary, three‐dimensional space of our manifest image “within” the ontology of quantum mechanics. I argue that we can recover most of our manifest image, even if we cannot recover our familiar three‐dimensional space.
According to the high-dimensional approach to quantum mechanics (a.k.a. wavefunction realism), the fundamental space of our world has an unfathomably large number of dimensions. This account is empirically adequate only if the three-dimensional manifest image can somehow be recovered from high-dimensional reality. A proper understanding of inter-level metaphysics (a.k.a. metaphysical explanation, grounding, etc.) shows that the manifest image can indeed be recovered, and answers the most concerning objections to high-dimensionalism. But it also shows that high-dimensionalism has disturbing consequences about the objectivity of the manifest image.
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.
Abstract The three major interpretations of quantum mechanics—Bohm, GRW, and many worlds—all treat the wave function as a physical entity. But this entity apparently resides in a high-dimensional configuration space, with three coordinates for each particle in the universe. Some conclude on this basis that the three-dimensionality of the world of experience is an illusion. Others respond by adding three-dimensional ontology to the theory or by reinterpreting the wave function either as a law or as an indirect representation of properties in three-dimensional space. It is argued that the latter possibility makes it unnecessary to postulate either additional ontology or a fundamentally high-dimensional world.
The purpose of the present chapter is to respond to a thread of recent criticism against one candidate framework for interpreting quantum theories, a framework introduced and defended by David Albert and Barry Loewer: wave function realism, a framework for interpreting the ontology of quantum theories according to which what appears to be a nonseparable metaphysics ofentangled objects acting instantaneously across spatial distances is a manifestation of a more fundamental separable and local metaphysics in higher dimensions. Thechapterconsiders strategies for extending the wave function realist interpretation of quantum mechanics to the case of relativistic quantum theories, responding to arguments that this cannot be done.
Abstract What are the ontological implications of quantum theories, that is, what do they tell us about the fundamental objects that make up our world? How should quantum theories make us reevaluate our classical conceptions of the basic constitution of material objects and ourselves? Is there fundamental quantum nonlocality? This book articulates several rival approaches to answering these questions, ultimately defending the wave function realist approach. Wave function realism is a way of interpreting quantum theories so that the central object they describe is the quantum wave function, interpreted as a field in an extremely high-dimensional space. According to this approach, the nonseparability and nonlocality we seem to find in quantum mechanics are ultimately manifestations of a more intuitive, separable, and local picture in higher dimensions.