The Status of our Ordinary Three Dimensions in a Quantum Universe 1

Transcription

1 NOÛS 46:3 (2012) The Status of our Ordinary Three Dimensions in a Quantum Universe 1 ALYSSA NEY University of Rochester 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, threedimensional 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. 1. Introduction For those of us who take our ontological cues from fundamental physics, the dimensionality of the world we inhabit is something about which we have learned to become quite flexible. The world may appear three-dimensional. For example, tables appear to have just the three dimensions of height, width, and depth. People seem to extend out in one dimension from head to toe, C 2010 Wiley Periodicals, Inc. 525

2 526 NOÛS two dimensions in girth, and no more. But if the best physics tell us that the space we inhabit really has four, five, or eleven dimensions, we can, without doing too much damage to our sense of what kind of creatures we are and what kind of world we inhabit, come to understand ourselves as occupying a higher dimensional space. In general, we can be convinced that our world is very high-dimensioned indeed. To be satisfied, we may only demand a story about how the familiar three dimensions of our manifest image 2 are contained within the higher-dimensional world of that theory. We may consider several twentieth century, fundamental theories of physics in order to illustrate our flexibility on this score. Upon learning Einstein s theories of relativity, we may be motivated to view ourselves as inhabiting not a mere three-dimensional space but instead a four-dimensional Minkowski space-time. Many of us are able to take on this revision to our prior conceptual scheme rather easily. We learn that we do not after all have just the three dimensions of height and girth, but in addition, a fourth dimension of temporal extension. Although the proposed revision to our earlier view about our dimensionality may have been surprising, the fact that relativity theory allows us to construe the three dimensions of our manifest image as three of the four dimensions of its postulated structure, makes the resulting theory easy to accept all things considered. Several of the last century s theories attempting to unify the fundamental forces also stipulated facts about our world s dimensionality that challenged ordinary appearances. The first theory along these lines was the Finnish physicist Gunnar Nordström s 1914 attempt to unify electromagnetism with gravity, a predecessor to Kaluza-Klein theory, positing four dimensions of space in addition to the one dimension of time. Surprisingly, when general relativity is modified in this way to incorporate an additional spatial dimension, it is able to make correct predictions regarding electromagnetism (see Smolin 2006, chapter 3). This proposal differs from Einstein s in one way that is relevant to our discussion. According to Kaluza-Klein theory, it is not merely that ordinary objects and people have more dimensions than the ordinary three dimensions we thought they had, but indeed that space itself contains four rather than three dimensions. (In the Minkowskian spacetime that many take to be the metaphysical upshot of special relativity, the total number of dimensions of our universe is four. In Kaluza-Klein theory, there are five total dimensions: four of space, and one of time.) Nevertheless, this theory, like Einstein s, suffers little tension with our prior image of the space we inhabit. After all, the physicist can find ways to account for the fact that although this additional spatial dimension exists, we never noticed it before. For example, one might conjecture (as Oskar Klein in fact did) that the new, posited spatial dimension is wrapped up in such a way that it is too tiny to notice. As Lee Smolin puts it:...we can make the new dimension a circle, so that when we look out, we in effect travel around it and come back to the same place. Then we can make

3 The Status of our Ordinary Three Dimensions in a Quantum Universe 527 the diameter of the circle very small, so that it is hard to see that the extra dimension is there at all. To understand how shrinking something can make it impossible to see, recall that light is made up of waves and each light wave has a wavelength...the wavelength of a light wave limits how small a thing you can see, for you cannot resolve an object smaller than the wavelength of the light you use to see it. (2006, p. 39) On this way of viewing the theory, the new dimension is wrapped up too small to see, and the other three spatial dimensions of the theory are just the ordinary three dimensions of our manifest image. Theories promising to unify the fundamental forces that followed shared these properties of Kaluza-Klein theory. For example, recent versions of string theory describe the world as containing sometimes ten, sometimes eleven spatial dimensions. Although before considering these theories, we might have thought that the world contained just the ordinary three dimensions in which tables have height, width, and depth, we can be quite flexible about revising this belief should we become convinced by the accumulated evidence supporting the theory. This is possible because in string theory, just as in Einstein s theories of relativity, the ordinary three dimensions of our acquaintance are still contained within the theory s structure. Brian Greene, in his defense of string theory, uses the analogy of a garden hose seen at a distance to illustrate how easy it is to reconcile the lessons of string theory with our manifest image of the world:...just like the horizontal extent of the garden hose, our universe has dimensions that are large, extended, and easily visible the three spatial dimensions of common experience. But like the circular girth of a garden hose, the universe may also have additional spatial dimensions that are tightly curled into a tiny space a space so tiny that it has so far eluded detection by even our most refined experimental equipment. (Greene 1999, p. 188) In each case we have considered, the extra dimensions of the theory are either additional, non-spatial dimensions (corresponding to time, for example), or are spatial, but are too small to see. Either way, we have an account of why we may have previously missed them. Contemporary quantum mechanics, however, is a kind of fundamental, physical theory that profoundly tests the limits of our flexibility regarding what we may understand to be the dimensionality of our world. As we will see, on any straightforward ontological reading of quantum mechanics, the theory requires the existence of an object, the wavefunction, that inhabits an extremely high-dimensional space: configuration space. And this, combined with the fact that quantum theory is as well justified as a theory can be, gives us at least prima facie reason to believe that we inhabit this extremely high-dimensional space. The problem this raises, that on which the present paper focuses, is that no three of the many dimensions of configuration space

4 528 NOÛS correspond in any direct way with the three dimensions of our manifest image. It is for this reason challenging to see our world as a quantum world. We are missing the account we desire in order to comfortably view the physical space of our world as higher-than-three-dimensional. We have a well-justified theory in quantum mechanics, but lack an accompanying story about how the familiar three dimensions of our manifest image are contained within the higher-dimensional world of that theory. In the next section, I will say enough about contemporary versions of quantum mechanics so it will be clear why any straightforward quantum ontology suggests that our world includes a physical space distinct from the three-dimensional space of our manifest image. Section 3 expands on the nature of this configuration space of quantum mechanics, along the way clearing up some confusions that often arise in its characterization due to the use of a historically-connected but quite distinct concept of configuration space in classical mechanics. This section also shows why our familiar three dimensions do not correspond to any of the many dimensions of configuration space. In the following sections, I examine two very different strategies for finding ordinary three-dimensional space within the ontology of quantum mechanics albeit in a somewhat less straightforward way than was accomplished for relativity theory and the various unification theories. Section 4 examines a proposal of Tim Maudlin, Valia Allori, and her collaborators to reject what I have been calling the straightforward reading of quantum mechanics. These authors suggest replacing this wavefunction-centered reading of quantum ontology with an ontology closer to that of our manifest image of the world. Section 5 examines a more scientifically conservative strategy of David Albert, Barry Loewer, David Wallace, and others to find our three-dimensional world within the wavefunction ontology of quantum mechanics. This requires a somewhat less straightforward way of locating our ordinary three dimensions than one could find in the cases of special relativity and the unification theories. For, as we will see, no three dimensions of configuration space correspond directly to the three dimensions of our manifest image. Albert and Loewer have found an inventive way of viewing the relationship between quantum ontology and appearances that does give us a way to recover most of our manifest image, but as we will see, it doesn t in the end genuinely allow for the existence of the three-dimensional space they were after. And this raises an important question. In coming to terms with the confusions and ambiguities that beset early versions of quantum mechanics, many philosophers of physics were inspired by the remarkably clear-headed insights of John Bell who in a series of now-classic papers (printed together in Bell 1987), provided several coherent and precise ways of understanding the ontology of quantum mechanics. One thing Bell emphasized again and again was the need for any such ontology to include what he called local beables : that is, entities with well-defined locations in threedimensional space (or four-dimensional space-time). This was a view shared

5 The Status of our Ordinary Three Dimensions in a Quantum Universe 529 by many others, including (arguably) Albert Einstein and Hans Reichenbach. The question is: do we really need to locate the familiar three dimensions of our manifest image within a theory s ontology and structure in order to be able to view ourselves as genuinely inhabiting the world of that theory? 3 Unlike other fundamental physical theories positing spaces of higher dimensions, it appears that quantum mechanics does not give us a natural way of seeing ourselves as genuinely three-dimensional, occupying the threedimensional space we think we do. Nevertheless, the theory is highly justified and can provide us with a clear and precise ontology. And as we ll see, despite the high dimensions of the quantum ontology, it is still possible to give an account of ourselves and the other objects of our manifest image using the resources of quantum mechanics, if not an account that saves three-dimensional space. This paper assumes for the purpose of the discussion that the correct version of quantum mechanics is going to be a realist version of quantum mechanics, in particular, one that takes quantum mechanics to be a comprehensive theory of what the world is like fundamentally: a theory that is able to tell us what sorts of (mind-independent) entities the world contains most fundamentally, and what kind of space these entities inhabit. Several authors have challenged this way of understanding quantum mechanics, arguing instead that quantum mechanics should be viewed as descriptive merely of the kind of information we can have about the world in certain contexts, and about how we should update our beliefs about the world over time (e.g. Fuchs 2003). I will not argue against such anti-realist versions of quantum mechanics here. Indeed, it is possible to read this paper in such a way that it provides someone with one more reason to take an anti-realist stance towards quantum mechanics. For if one takes the belief that our world is three-dimensional to be incorrigible, then the arguments in this paper ought to lead one to believe that quantum mechanics is not a theory that provides an objective description of our world. My own view is that we do not have incorrigible beliefs about the dimensionality of the space we inhabit, but rather that our beliefs about the dimensionality of physical space may be corrected by what fundamental physics tells us. The reaction that most physicists and philosophers have to relativity and Kaluza-Klein theory supports this fact. 2. Realist Versions of Quantum Mechanics By a realist version of quantum mechanics, I mean one that takes the theory to be aimed at providing a true description of a world independent of us as observers. Some approaches to quantum mechanics aren t realist. For example, as we have just noted, many physicists prefer an information-theoretic understanding of quantum mechanics according to which the theory doesn t describe the world independent of us as observers, but rather the evolution

6 530 NOÛS of our states of knowledge as we perform experiments. 4 In general, realist versions of quantum mechanics are intended to be descriptive of an object or objects that exist independently of us or any other observer. I use the term anti-realist to describe versions of quantum mechanics that have it centrally concerning observers or minds. There are several realist versions of quantum mechanics currently on offer. Since the purpose of this paper is not to provide an overview of these approaches but rather to address the physical structure that is common to all of them, my discussion of these theories will be brief. 5 There is now a consensus (at least among most philosophers of physics) that the so-called orthodox Copenhagen account of quantum mechanics is, to put it mildly, not promising. According to this version of quantum mechanics, formulated perhaps best by John von Neumann in 1932, states of quantum systems evolve according to two fundamental laws, the Schrödinger equation and what we will call the collapse postulate. 6 Both laws describe the evolution of quantum systems by describing the evolution of the state of what is called the wavefunction. 7 The laws differ in several ways, the most striking of which being that Schrödinger evolution is completely deterministic, while the collapse postulate is an indeterministic law. In other words, given the state of the wavefunction at one time, t 1, the Schrödinger equation specifies a unique state for the system at any later time, t 2. This is not so if the system is instead obeying the collapse postulate. For given the state of the wavefunction at a time t 1, the collapse postulate gives only chances that the wavefunction of the system will be at any other state at a later time, t 2. Because these laws give different predictions regarding the future states of quantum systems, the question immediately arises: in which circumstances does each law obtain? Von Neumann s version of quantum mechanics stipulates that for the most part, systems obey the Schrödinger equation. However, when a measurement is being performed on the system, it is not the Schrödinger equation, but the collapse postulate that applies. The trouble with this version of quantum mechanics is that it contains no fundamental, physical account of measurement. What kind of physical systems evolve according to Schrödinger dynamics? Which according to collapse dynamics? The theory does not give an answer in physical terms, but rather in imprecise, ambiguous, and seemingly observer-dependent language ( measurement ) that has no place in fundamental physical theory. In Bell s words: The concept of measurement becomes so fuzzy on reflection that it is quite surprising to have it appearing in physical theory at the most fundamental level... And does not any analysis of measurement require concepts more fundamental than measurement? And should not the fundamental theory be about these more fundamental concepts? (1987, pp ) 8 In recent decades, several more promising versions of quantum mechanics have been developed, all of which clearly avoid this problem of the

7 The Status of our Ordinary Three Dimensions in a Quantum Universe 531 Copenhagen view. Three stand out as the subject of serious scrutiny by physicists and philosophers of physics: Everettian quantum mechanics (sometimes called the many worlds view ), Bohmian mechanics, and the spontaneous collapse theory of Ghirardi, Rimini and Weber (hereafter GRW). According to the Everettian view 9, the only fundamental dynamical law governing quantum systems is the Schrödinger equation. As noted above, this is a completely deterministic law describing the evolution of states of the quantum wavefunction over time. There is no collapse law on this version of quantum mechanics, and thus no need to distinguish (using for example, such a problematic term as measurement ) when one law does or does not apply. The theory is thus perfectly unambiguous and precise. According to Bohmian mechanics 10, there are again two fundamental laws. There is the Schrödinger equation which deterministically governs the evolution of the wavefunction over time, and then another deterministic law, one we may call the particle equation 11 that predicts the behavior of something else over time as a function of the state of the wavefunction. The nature of this something else is a matter of debate. Some argue that the particle equation describes the evolution of a system of many particles over time; others that the particle equation only describes the evolution of one particle, sometimes called the world particle, over time. 12 Although this version of quantum mechanics contains two dynamical laws, unlike Copenhagen quantum mechanics, Bohmian mechanics does not suffer that theory s problem of specifying when one or the other law holds. The reason is that each law of Bohmian mechanics describes the behavior of a distinct entity (or system of entities). The Schrödinger equation always describes the behavior of the wavefunction. The particle equation always describes the behavior of the particle or particles. Since the laws describe the evolution of distinct entities, there is no tension in the resulting predictions of the two laws. 13 A third version of quantum mechanics is the GRW spontaneous collapse theory, named after the physicists Giancarlo Ghirardi, Alberto Rimini, and Tullio Weber who developed it in a 1986 paper. 14 Like the Copenhagen and Bohmian approaches, this version of quantum mechanics employs two dynamical laws, and one is again the Schrödinger equation. And, like the Copenhagen view, according to GRW, this law does not universally apply to quantum systems. Instead, for any given time, the second law specifies a precise probability that the system will undergo a collapse, in other words, that the system will momentarily cease obeying the Schrödinger equation, and take on a state with certain precise features. 15 The probability of collapse at any given time is determined by certain features of the system s wavefunction. The resulting picture of the world is indeterministic given a quantum system at a given time, t 1, it cannot be predicted with certainty what the state of the system will be at a later time, t 2. For there is always some chance the system will undergo a collapse. Nevertheless, the theory is precise and like Everettian quantum mechanics and Bohmian mechanics, the GRW

8 532 NOÛS theory does not suffer the problem Bell noted. It never mentions measurement, nor any other similarly problematic concept. According to GRW quantum mechanics, when there is a collapse of the wavefunction, this does not occur because a measurement occurs on the system. Instead, according to GRW, collapses occur randomly, with the precise probabilities of collapse specified by the theory s second law as a function of fundamental physical properties of the system. 16 On the most straightforward, ontological understanding of all of these realist versions of quantum mechanics, we have at least one law, the Schrödinger equation, that describes the behavior of at least one unfamiliar entity: the wavefunction. What do I mean by the most straightforward, ontological understanding of these theories? This may be illustrated by invoking an example from a physics with which we are more familiar. Recall the central dynamical law of Newtonian mechanics: Newton s second law, F = m d2 x. dt 2 On the most straightforward, ontological understanding of this theory, we have a law that describes the behavior of any material object 17 over time. The law asks us to figure out the forces acting on this material object, and from here, once we know the object s mass, we can calculate the change in position this object will undergo, thus knowing where it will be at future times. The Schrödinger equation also has a nice and compact formulation: Ĥ = i t. Just as the left-hand side of Newton s second law concerns the forces acting on a material object, here, the left-hand side of the Schrödinger equation, concerns the total energy of the system in question. This is what the Ĥ operator (the Hamiltonian) signifies. On the basis of information regarding a system s total energy, we can calculate the state of the wavefunction at future times. The signifies just this, the change in the state of the t wavefunction over time. The only other terms on the right-hand-side are the imaginary number i and the constant. Just as on a straightforward, ontological reading, the central, dynamical law of Newtonian physics describes the temporal evolution of a material object s position, the straightforward, ontological reading of the central, dynamical law of realist quantum mechanics describes the temporal evolution of the state of the wavefunction. At least this is the way it has looked to many authors. Peter Lewis puts the point the most succinctly: The wavefunction figures in quantum mechanics in much the same way that particle configurations figure in classical mechanics; its evolution over time successfully explains our observations. So absent some compelling argument to the contrary, the prima facie conclusion is that the wavefunction should be accorded the same status that we used to accord to particle configurations. (2004, p. 714)

9 The Status of our Ordinary Three Dimensions in a Quantum Universe 533 If we look to a central dynamical law of realist versions of quantum mechanics, the straightforward ontological reading of the law is that it is about the wavefunction. It is possible for one to think here that there has been some kind of confusion about the status of the wavefunction in quantum mechanics. One might suggest that what quantum mechanics really describes is the evolution of a system of particles, or bits of matter. The wavefunction is just a name for the overall state of this system of particles at a time. So phrases like the state of the wavefunction are nothing but shorthand for the state of the wavefunction of the system of particles. And when one sees it this way, one views the former phrase as redundant what the Schrödinger equation describes as evolving is just a system of particles over time, not some other mysterious object, the wavefunction. The preceding is a perfectly reasonable thing to think at first. However, it is important to see why this sort of eliminativism about the wavefunction is ultimately untenable. 18 It is not just that the Schrödinger equation superficially looks to just be about this thing, the wavefunction. Quantum mechanics has to invoke the wavefunction because there are certain states, what Schrödinger himself first called entangled states, pervasive in nature, that can only be captured by a physical theory that countenances such an entity as the wavefunction. So, there is also an argument for taking the wavefunction with ontological seriousness, as the (or a) thing theories of quantum mechanics describe. This argument can be summarized in the following way: (1) The laws of quantum mechanics permit the evolution of systems into entangled states. (2) These states cannot be adequately characterized as states of something inhabiting our familiar, three-dimensional space, but rather must be characterized as states of something else spread out in a higher-dimensional (configuration) space. (3) So, (from 1 and 2) there exists something that must be characterized as spread out in a higher-dimensional configuration space: call this the wavefunction. (4) So, the wavefunction exists. In order to grasp the force of this argument, we ll need to better understand the concept of entanglement and be familiar with the nature of the configuration space of quantum mechanics. However, this could take us too far away from the main goals of this paper, so I ve included a more thorough explanation of the argument above in an appendix. 19 The important point to take from this argument for now is just the following. In Newtonian mechanics, it was natural to view the laws as being about the evolution of states of material objects in three-dimensional space because these are the types of things that can be in the kind of position states the laws describe. By contrast, in quantum mechanics, many states the laws

10 534 NOÛS allow (and indeed are pervasive in our universe) cannot be seen as states of material objects in three-dimensional space. 20 In particular, what the laws of quantum mechanics require is the existence of some entity whose states are specifiable in a space of higher dimensions. 21 The wavefunction satisfies this requirement. The straightforward, ontological reading of realist quantum mechanics therefore is that the wavefunction is the object whose behavior the Schrödinger equation describes; it is an object that inhabits this higher-dimensional configuration space. Thus, configuration space should be thought of as at least one (if not the only) fundamental, physical space posited by quantum mechanics; and the wavefunction should be thought of as at least one (if not the only) fundamental object posited by these theories. I say at least one here because, you might recall, although the Schrödinger equation is the only fundamental, dynamical law of Everettian quantum mechanics, and both laws of GRW also concern the behavior of the wavefunction, in Bohmian mechanics there is one other fundamental, dynamical law: the particle equation. I said earlier that I won t take a stand on the disagreement about what this law is about. According to one side of the debate, this law describes the evolution of the positions of a set of many particles in ordinary, three-dimensional space. According to the other side, it describes the evolution of the position of one world particle in configuration space, i.e. the same space the wavefunction inhabits. The following diagram summarizes the different ontologies and spatial structures that are read most straightforwardly off of these several versions of quantum mechanics. Realist versions of quantum mechanics: DYNAMICS FUNDAMENTAL ONTOLOGY Straightforward reading: SPATIAL STRUCTURE Straightforward reading: Everettian quantum mechanics Schrödinger equation The wavefunction Configuration space GRW Schrödinger equation + indeterministic collapse law The wavefunction Configuration space Bohmian mechanics Schrödinger equation + particle equation With a manyparticle reading of the particle equation: The wavefunction + many particles Configuration space + ordinary threedimensional space With a oneparticle reading of the particle equation: The wavefunction + one world particle Configuration space

11 The Status of our Ordinary Three Dimensions in a Quantum Universe 535 One might wonder then, whether we can find the three-dimensional world of our ordinary experience in quantum mechanics if Bohmian mechanics is correct and that theory is given the first, straightforward reading. The idea then would be that the theory posits the existence of two physical spaces: a high-dimensional configuration space occupied by the wavefunction, and a separate, three-dimensional physical space occupied by the many particles that make up our tables, chairs, and the other objects of our manifest image of the world. Call this the two-space reading of Bohmian mechanics. On the two-space reading of Bohmian mechanics, it looks like the central problem of this paper doesn t arise. So perhaps all we need to worry about in the following sections is how to find the ordinary threedimensional space of our manifest image if GRW or Everettian theories are correct, or if the second, one-space reading of Bohmian mechanics is adopted. Perhaps. But it is worth taking a moment to see why many have not thought that this two-space reading of Bohmian mechanics really gives them what they were looking for. 22 While it is undoubtedly clear that there exists a three-dimensional space according to the two-space reading of Bohmian mechanics, it is also unfortunately unclear whether or not this three-dimensional space is the space of our manifest image. For what fundamentally inhabits this three-dimensional space on the two-space reading? Particles. But what are these particles like? In particular, are these the particles that compose you or I, or the tables and chairs of our acquaintance, so that we may be said to also inhabit this three-dimensional space of Bohmian mechanics? The answer is not so clear. For even on this two-space reading of Bohmian mechanics, where the particles are ontologically fundamental, it is still true that it is states of the wavefunction that determine these particles behavior over time. The particles themselves only have positions. And what they do at later times according to this theory, i.e. what positions they move to over time, depends on the wavefunction. The properties (i.e. positions) of these particles at one time, t 1, are not sufficient according to this theory to determine their features at any later time, t 2, nor are facts about these particles even sufficient to determine the chances of the particle s having certain features at any later time, t 2. So, if we tried to view ourselves as constituted ultimately out of these Bohmian particles, we would be forced to conclude that all causal features we have constituting our ability to affect the world, that is, do what it is that we do, are ultimately grounded in features of the wavefunction, not in any features of ourselves or our constituent particles. It then seems like the three-dimensional space of two-space Bohmian mechanics is not the space in which we live. If you believe you are the type of thing that has not just a position, but the ability to affect the world around you, then you must deny that you inhabit the three-dimensional world of Bohmian mechanics. 23 All that inhabits this space is a collection of inert particles. This is one line of reasoning that has convinced many philosophers

12 536 NOÛS and physicists that the two-space reading of Bohmian mechanics does not solve the problem of locating the three-dimensional space of our manifest image within the ontology of quantum mechanics. 24 How to find the threedimensional space of our manifest image within the physical ontology is still a live issue for the two-space reading of Bohmian mechanics, as it is for the one-space reading of Bohmian mechanics, GRW, and Everettian quantum mechanics. Whether the preceding discussion was convincing or not, my suggestion is that we set aside this two-space reading of Bohmian mechanics in what follows. This will allow us to narrow our discussion onto the question of how to locate the three-dimensional space of our manifest image within a science whose most straightforward ontological reading posits only the high-dimensional configuration space as its physical space. 3. How to Think about the Configuration Space of Quantum Mechanics Let s return now to discussing the nature of the configuration space of quantum mechanics. For it has yet to be shown why it is not a simple task to locate the three dimensions of our manifest image within this configuration space. To get a grip on the concept of configuration space, it is useful to ask: how many dimensions are there in configuration space? The correct answer to this question depends on contingent features of the world: the number of independent variables needed to specify the state of the world s wavefunction ( ) as a whole. One less correct, but more ubiquitous answer, an answer that we ll find to be heuristically useful in a moment, is that the dimensionality of configuration space depends on the total number of particles in the system under consideration. The idea is that if the world contains a total of N particles, then the configuration space of the world is 3N-dimensional. The 3N dimensions of the space are understood in the following way. 25 The first, second, and third dimensions of configuration space correspond to the three dimensions of the first particle; the fourth, fifth, and sixth dimensions to the three dimensions of the second particle; the seventh, eighth, and ninth dimensions to the three dimensions of the third particle; and so on. Then say, if the total number of particles in the universe is (near Arthur Eddington s estimate), the wavefunction inhabits a space of dimensions. If this is right, then configuration space is very high-dimensioned indeed. And then, each point in the configuration space can be understood to correspond to a state that specifies all of the particles three-dimensional locations. As Jeffrey Barrett says: One can think of the positions of an N-particle system as being represented by a single point in 3N-dimensional configuration space (since there are N particles and three position coordinates for each particle). One might then picture

13 The Status of our Ordinary Three Dimensions in a Quantum Universe 537 the motions of the particles by considering how that point would be pushed around...as the wavefunction evolves. (1999, p. 61) The preceding is a convenient way to come to grips with the very large dimensionality of configuration space, but it is ultimately not helpful for a couple of reasons. First, as has already been noted, the fundamental ontologies of the realist versions of quantum mechanics under consideration are either the wavefunction and the single world particle, or the wavefunction simpliciter. So, even if there are such things as particles with three-dimensional locations, they are at best derivative entities on these theories constituted in some way or other out of this more fundamental wavefunction (and possibly a single particle), and so they do not seem to be the sorts of things that would constrain the fundamental dimensionality of the configuration space. Even in two-space Bohmian mechanics, where one is able to read the theory as describing a world containing particles, the dimensionality of the configuration space will not be dependent on the existence of the particles. Both the configuration space and the particles in their three-dimensional space exist in their own right as fundamental entities. The fact that this particle characterization of configuration space is not entirely correct is often noted but ignored. Nevertheless, it is important to emphasize, as in the following textbook presentation: We may say, if we like, that [the wavefunction] is spread out over a 3ndimensional coordinate or configuration space in which each point represents a possible configuration of the system as a whole. But the use of such geometrical language is not essential and means merely that, since depends on 3n independent variables, it could be laid out as a point function only in a space having the corresponding number of dimensions. (Kemble 1958, pp. 21 2) As this explanation of configuration space demonstrates, the dimensionality of configuration space depends in a basic way only on the number of independent variables required to completely specify the state of the wavefunction at a time, not on anything having to do with particles in a three-dimensional space. This particle characterization of configuration space is useful as a heuristic, as it will allow us to hone in on approximately the right number of dimensions in our universe s configuration space, but it is not strictly speaking correct. The concept of configuration space is the historical descendent of a related concept of the same name from classical mechanics. The configuration space of classical mechanics is typically not taken to be a genuine, physical space, but rather a purely mathematical representation that is used to conveniently summarize the positions of an entire system at a time. Instead

14 538 NOÛS of representing the locations of N particles in three-dimensional space by N points in a three-dimensional coordinate representation, classical physicists will often use one point in a 3N-dimensional coordinate representation to capture the same information. The concept of configuration space in quantum mechanics is no doubt derived from this other, highly useful concept of classical mechanics, but it is important to be clear that these are distinct concepts aimed at distinct entities. The dimensionality of the fundamental space of quantum mechanics is not determined by the number of particles in the universe. It is not a purely mathematical space invoked merely to summarize the locations of particles in some other, real three-dimensional space. What fundamentally exists in this theory is the wavefunction, and the dimensionality of the space the wavefunction inhabits is not determined by anything more fundamental. 26 Before going any further, it is worth making it explicit that the configuration space of quantum mechanics is not to be confused with Hilbert space the abstract, mathematical structure that is deployed in most textbook presentations of quantum mechanics. Hilbert space is used to represent states of the wavefunction as vectors. It is a convenient representation that allows various measurements we perform on states to be viewed mathematically as cases of vector operations. The Hilbert space does not have the status in quantum mechanics of a genuine physical space, inhabited by a world of vectors, but it is far more typical to see it regarded as a mathematical convenience. 27 We now have the tools to understand why the case of quantum mechanics is so different from the case of relativity and the other theories we discussed above. In those theories, the three-dimensional space of our manifest image was contained within the space of the theories in the sense that three of the dimensions of the theory s space were identical to the three dimensions of our manifest image. Then we could tell a plausible story about how we earlier might not have noticed the other dimensions of the theory s space. However, the configuration space of quantum mechanics isn t a space constituted by the three dimensions of our manifest image plus one, seven, or [( ) 3] more. No three of the dimensions of configuration space correspond to the three dimensions of our manifest image. Let s see why. Imagine we are trying to describe three of the very many woodchips that make up my desk. We may start by coordinatizing the space in which these chips are located using three dimensions: x, y, and z. We may do this in such a way that x, y, and z correspond to the length, width, and height of our manifest image of the desk. And using this coordinate system, it seems we can give a complete description of the locations of these chips. The space in which the chips appear to live is three-dimensional because to specify each of their locations, we only need to specify these three values (their x, y, and z values).

15 The Status of our Ordinary Three Dimensions in a Quantum Universe 539 For example: Chip 1: x = 1 y = 0 z = 2 Chip 2: x = 1 y = 2 z = 2 3 of the desk s wood chips Chip 3: x = 10 y = 2 z = 2. Alternatively, we may represent the locations of the chips using the configuration space of quantum mechanics. To do this, we may exploit the particle characterization of configuration space discussed above. To see how to do this, it is possible to start by representing this system as a wavefunction spread out in a 3N-dimensional space, where N is the number of woodchips. Since the number of chips in this case is three, this wavefunction will be represented as inhabiting a nine-dimensional space. We can then represent the state of the whole system of chips as a point in nine-dimensional space. 28 We will let the first three coordinates correspond to the x, y, and z dimensions of chip 1, the next three correspond to the x, y, and z dimensions of chip 2, and the last three correspond to the x, y, and z dimensions of chip 3. Then we can sketch a nine-dimensional configuration space representation of this system that partially makes up my desk: System: o = 1 p = 0 q = 2 r = 1 s = 2 t = 2 u = 10 v = 2 w= 2 this system s wavefunction Which of these nine coordinates corresponds to the x-coordinate of our threedimensional space? The correct answer is none of them. The o-coordinate corresponds to the x-coordinate-for-chip-1, the r-coordinate corresponds to the x-coordinate-for-chip-2, and so on. But no one of o, r, or u just is the x-dimension. To put this another way, we might ask: which coordinate in the nine-dimensional configuration space corresponds to the height of the desk? Again, it looks like no dimension of the configuration space is the height-dimension of the desk. The second, fifth, and eighth dimensions of the space look like they may in some sense correspond to the height of the

16 540 NOÛS desk, but none just is the height. The p-dimension may correspond to the height of chip 1, the s-dimension to the height of chip 2, and so on, but none corresponds to a common height that we may ascribe to the desk. Another way of recognizing the absence of our familiar three-dimensions from this picture is to note that if there is such a thing as height, then there is an object in the world that is extended in this dimension. This is captured in the three-dimensional graph by the representation of two occupied points at two different locations in the y-direction. It is in virtue of this that it appears that something is extended in this dimension. But as can be seen by the second figure, nothing is represented as being extended in any of the dimensions of the configuration space. So none of the nine dimensions of the configuration space correspond to our ordinary dimension of the height, nor to any of the other two dimensions of our manifest image. Although it not the case that any three of the dimensions of configuration space are the three dimensions of our manifest image, authors have tried to find some other way of locating the ordinary three-dimensional space of our acquaintance within the world we learn about from quantum mechanics. Two clearly divergent strategies have emerged. It is the goal of the next two sections to evaluate them. 4. Reconstruing the Status of the Wavefunction So, let us recap what is at issue. There are several viable, realist versions of quantum mechanics currently on offer. However, on the most straightforward, ontological readings of these theories, they show us that the space of the world we inhabit is not the familiar three-dimensional space we thought. In each case, we have a fundamental physical theory that enjoys an extremely high level of empirical support. Yet on the straightforward readings of these theories, there is no accompanying story (as there was for relativity theory, Kaluza-Klein and string theories) about how our familiar three dimensions are contained within the theory s extremely high-dimensional world. No three dimensions of the configuration space of quantum theory correspond to the three dimensions of our manifest image. One response that has occurred to a number of authors is, on this basis, to simply reject what I have been calling the straightforward ontological readings of these theories. Perhaps one of Bohmian mechanics, Everettian or GRW quantum mechanics is correct. But we should not read an ontology off of these theories in the simple way I charted out above in Section 2. We need to be clear here: no parties to the present debate deny the reality of the wavefunction. 29 Yet these authors insist that the Schrödinger equation is not about the evolution of the wavefunction over time. The theories are about something else. And this something else is something that exists in the three-dimensional space of our manifest image, and so there never really was a tension between the ontology of quantum mechanics and our manifest

17 The Status of our Ordinary Three Dimensions in a Quantum Universe 541 image in this way in the first place. This something else is the collection of fundamental physical objects that constitute us and the ordinary material objects of our acquaintance. So, the straightforward readings of GRW, Everettian, and Bohmian quantum mechanics say the theories are about the evolution of the wavefunction over time. This alternative proposed reading says that these theories are about something else. The first thing one may wonder about this proposal is how this could be when there is a central dynamical law of all of these theories, the Schrödinger equation, which certainly appears to be about the evolution of the wavefunction over time. Moreover, we ve already seen that if one is to give a complete description of the entangled states pervasive at our world, this requires a characterization in a space of a very high number of dimensions; that is, a characterization in configuration space. To make this proposal clearer, Valia Allori and her collaborators (Shelly Goldstein, Roderich Tumulka, and Nino Zanghì), following earlier work by Detlef Dürr, Goldstein, and Zanghì (1992), invoke a distinction between what they call the primitive ontology of a scientific theory (PO): what that theory is about, the basic kinds of entities that are to be the building blocks of everything else [in that theory] ; and the nonprimitive ontology of that theory: phenomena, like laws, to which the theory appeals in order to (but only in order to) explain how the primitive ontology behaves (2008, pp ). Allori et. al. claim that for quantum mechanics, a variety of primitive ontologies are possible 30, but all of these include objects spread out in the ordinary three-dimensional space of our manifest image (or the four-dimensional Minkowski space-time that includes this space). The wavefunction is not part of the primitive ontology of quantum mechanics. They claim: Each of these [realist versions of quantum mechanics] is about matter in spacetime, what might be called a decoration of space-time. Each involves a dual structure (χ, ψ): the PO χ providing the decoration, and the wavefunction ψ governing the PO. The wavefunction in each of these theories, which has the role of generating the dynamics for the PO, has a nomological character utterly absent in the PO. (2008, p. 363) The idea is that although the Schrödinger equation may describe the evolution of an entity, the wavefunction, this entity should not be accorded the status of what quantum mechanics is ultimately about. According to Allori and her collaborators, we can think of the wavefunction as just something that has to be invoked by the theory in order to give a complete account of quantum systems, like a law. It thus follows on this view that the configuration space is not a genuine, physical space. The wavefunction is not what quantum mechanics is about, so the space it is supposed to inhabit is not a genuine space. Indeed as Dürr, Goldstein, and Zanghì put it in 1992:

18 542 NOÛS...insofar as it is a field on configuration space rather than on physical space, the wave function is an abstraction of even higher order than the electromagnetic field. (1992, p. 850) For these authors, only the space of the primitive ontology is a genuine, physical space. Although Allori and her collaborators do not uniformly want to claim that the wavefunction just is a law, it is much like a law in that even if we are realists about laws, we do not take them to inhabit space, in the way we and other material objects inhabit space. 31 This is part of what it means for them to accord to the wavefunction the status of nonprimitive ontology, saying it has a nomological character. 32 Tim Maudlin (2007) has made closely related points regarding the ontology of realist versions of quantum mechanics. Although Maudlin does not make any specific claims about the wavefunction having the sort of nomological status ascribed to it in Allori et al. (2008), like these authors, Maudlin insists that any realist version of the theory must include in its ontology the kinds of things Bell called local beables. By beable, Bell just meant existent or entity. 33 Local beables are those objects which are definitely associated with particular space-time regions (1987, p. 234). As Maudlin puts it, local beables do not merely exist: they exist somewhere (2007, p. 3157). As Bell is commonly interpreted, local beables must exist somewhere in ordinary threedimensional space, or at least four-dimensional space-time. Why is quantum mechanics obviously a theory of local beables, and so obviously a theory whose primitive ontology consists solely of local beables? One answer that can suffice for now is that quantum mechanics, like any candidate fundamental physical theory, is a theory that is intended to explain the behavior of material objects and the particles that make these things up and these are local beables. The wavefunction by contrast is not a local beable. As we saw in the last section, it does not have any location in three-dimensional space; even according to the understanding of it we considered in the previous sections, it exists at best in configuration space. Therefore, the wavefunction is not part of the primitive ontology of quantum mechanics, and so its space is not a physical space. The argument may be summarized in the following way: (1) Quantum mechanics is a theory about local beables, i.e. objects with locations in the ordinary three dimensions of our manifest image. (2) Therefore, the primitive ontology of quantum mechanics must include local beables. (3) If quantum mechanics is correct, then objects can only have locations in the ordinary three dimensions of our manifest image if the space inhabited by the fundamental ontology of quantum mechanics includes these three dimensions.