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

NOÛS 46:3 (2012) 525–560
The Status of our Ordinary Three Dimensions
in a Quantum Universe1
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,
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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 image2 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
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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
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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
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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
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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. 117–118)8
In recent decades, several more promising versions of quantum mechanics have been developed, all of which clearly avoid this problem of the
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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 view9 , 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 mechanics10 , 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
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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
2
dynamical law of Newtonian mechanics: Newton’s second law, F = m ddtx2 .
On the most straightforward, ontological understanding of this theory, we
have a law that describes the behavior of any material object17 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 ‘ ∂t∂ ’ signifies just this, the change in the state of the
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)
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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
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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
Everettian
quantum
mechanics
Schr ödinger
equation
GRW
Bohmian mechanics
Schr ödinger
equation +
indeterministic
collapse law
Schr ödinger equation + particle
equation
With a oneWith a manyparticle reading particle reading
of the particle
of the particle
equation:
equation:
FUNDAMENTAL
ONTOLOGY
Straightforward
reading:
The
wavefunction
The
wavefunction
The wavefunction + many
particles
The wavefunction + one
“world”
particle
SPATIAL
STRUCTURE
Straightforward
reading:
Configuration
space
Configuration
space
Configuration
space +
ordinary threedimensional
space
Configuration
space
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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
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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 1080 (near Arthur Eddington’s
estimate), the wavefunction inhabits a space of 3 × 1080 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
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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
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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 × 1080 ) −
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).
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
Chip 3:
x = 10
y=2
z = 2.
3 of the desk’s wood chips
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
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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
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 Zanghı̀), following earlier work by
Detlef Dürr, Goldstein, and Zanghı̀ (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. 363–5). Allori et. al. claim that for quantum mechanics, a variety of
primitive ontologies are possible30 , 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 Zanghı̀ put it in 1992:
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. . . 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.
The Status of our Ordinary Three Dimensions in a Quantum Universe
543
Therefore,
(4) The space inhabited by the fundamental ontology of quantum mechanics
includes these three dimensions.
The suggestion is then that the space of the world described by quantum
mechanics does and must after all include the three dimensions of our ordinary experience. Since the wavefunction does not inhabit any such space, it
is taken to be an object with a nonprimitive status – invoked by the theory
to help (in some way) describe the local beables. Is this then the correct way
to think about the ontological implications of realist versions of quantum
mechanics?
The main question that must be addressed if one is to adequately evaluate
this proposal is: how plausible is it to take the wavefunction to have this nonprimitive status, as something that exists but nevertheless is not something
that inhabits a physical space? John Bell seems to have thought that this
is not plausible. He once said, speaking of a Bohmian version of quantum
mechanics:
No one can understand this theory until he is willing to think of [the wavefunction] as a real objective field . . . Even though it propagates not in 3-space
but in 3N-space. (1987, p. 128)
And, as David Albert puts it:
The sorts of physical objects that wave functions are . . . are (plainly) fields –
which is to say that they are the sorts of objects whose states one specifies
by specifying the values of some set of numbers at every point in the space
where they live, the sorts of objects whose states one specified (in this case) by
specifying the values of two numbers (one of which is usually referred to as an
amplitude, and the other as a phase) at every point in the universe’s so-called
configuration space. (1996, p. 278)
In quantum mechanics, states of the wavefunction appear very much like
states of real things like electromagnetic fields. The wavefunction is characterized as spread out in configuration space with phase and amplitudes
at each point in that space. Maudlin, Allori, and her collaborators want to
argue that the wavefunction is not like a field, and thus does not have amplitudes at locations in a genuine physical space. We need an account however
of why we are justified in thinking electromagnetic fields are the sorts of
things that inhabit a genuine space, but the wavefunction is not.
Prima facie, the wavefunction would not appear to have the abstract or
nomological character Allori and her collaborators wish to ascribe it. It
is natural to think of laws as entities that describe or govern the causal
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efficacy of other things. They are not themselves the things that have the
causal efficacy.34 But certainly at least on the Bohmian version of quantum
mechanics the wavefunction appears to have at least some efficacy, guiding
the particle or particles into specific states.
And the wavefunction has more analogies with electromagnetic fields as
well. Just as there is a set of physical laws, Maxwell’s equations, that describe the evolution of electromagnetic fields over time, there exists a law, the
Schrödinger equation, that describes the evolution of the wavefunction over
time. One might think that this doesn’t establish that either electromagnetic
fields or the wavefunction exist. But note that the question isn’t whether or
not the wavefunction exists. All sides to the present debate agree it exists.
The wavefunction must exist in order to capture the facts of entanglement.
The question is whether it is part of quantum mechanics’ primitive ontology
or nonprimitive ontology. Since, in general, we do not think that elements of
a theory’s nonprimitive ontology, i.e. its laws, change, it would seem that the
wavefunction belongs to quantum mechanics’ primitive ontology.35 After all,
it is the type of entity whose state evolves over time, in conformity with the
Schrödinger equation.
One might object that physicists have recently raised the possibility that
the fundamental laws of physics might themselves be evolving over time.
Perhaps there is a meta-law that would govern this evolution in the other
laws. Then we could view the Schrödinger equation as having a status like
such a meta-law. It doesn’t describe how the primitive ontology of the theory
evolves, but rather how the nonprimitive ontology that governs the primitive
ontology evolves.
When physicists talk about the laws of physics evolving over the history
of our universe, however, what they usually have in mind is the possibility
that what we thought were constants in nature (e.g. the speed of light or the
charge of the electron) are really variables whose values change over time.36
What does this mean? It doesn’t seem to really mean that the fundamental
laws of the universe are changing, but rather that the fundamental laws
of the universe are different than we thought they were. For example, one of
Einstein’s postulates in his special theory of relativity was that the speed of
light in a vacuum (c) is constant. Now, if c has actually changed its value
since the beginning of time, then this postulate is incorrect. My point is: it
is not really so clear that there exist precedents for thinking that elements of
a theory’s nonprimitive ontology are changing. Thus it seems that perhaps
the wavefunction is more like a field (part of a primitive ontology) than
a law (part of a nonprimitive ontology). It is not good reason to deny
the wavefunction primitive status simply because it fails to be matter in
space-time. Such a principle would have us similarly rejecting the primitive
status of electromagnetic fields.37
In any event, these authors certainly have a reason to want to reject these
claims of analogies with electromagnetism and deny that the wavefunction is
The Status of our Ordinary Three Dimensions in a Quantum Universe
545
part of the primitive ontology of quantum mechanics. For they think that in
order to account for the appearance of a three-dimensional world, the fundamental physical ontology must be three-dimensional. What we therefore
need to explore is whether it is possible to capture the appearances – our
perception of tables, chairs, measurement apparatus, and so on – in a world
whose fundamental space is configuration space. We’ll be discussing such
work in the next section. The upshot of this for the present section will be
to remove the motivation for the proposal of Maudlin, Allori, and her collaborators. We may be able to account for the appearances in a fashion that
is more scientifically conservative. Instead of rejecting the straightforward
ontology read off of realist versions of quantum mechanics and replacing it
with something closer to our manifest image of the world, we may be able to
accept the straightforward ontology of quantum theory – the wavefunction
in configuration space – and capture the appearances using those resources
alone.38
5. Three-dimensional Space as a Non-Fundamental, Enacted Space
An alternative, less revisionary strategy for recovering parts of our manifest
image from the ontology of quantum mechanics has been proposed and
developed in work by David Albert and Barry Loewer in discussion of GRW
quantum mechanics and David Wallace on Everettian quantum mechanics.
The idea is this. Accept the straightforward, ontological reading of these
realist versions of quantum mechanics.39 In other words, accept that all there
is fundamentally is a wavefunction in configuration space. Then the claim
is that in the actual world, the behavior of the wavefunction over time is
such that it is able to play the functional role we ordinarily associate with
material objects in a three-dimensional space. In Albert’s terminology, the
wavefunction is thereby able to “enact” the existence of material objects in a
three-dimensional space. These authors thus accept the first premise of the
preceding section’s argument:
(1) Quantum mechanics is a theory about local beables, i.e. objects with locations in the ordinary three dimensions of our manifest image.
However, they deny this argument’s third premise:
(3) 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.
How does this work? We begin by being functionalists about the material
objects of our manifest image – all that is required for there to be a chair
is for there to be something that can play the functional role of a chair.
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For there to be a person, there just must be something that can play the
functional role of a person. Albert suggests that any physics that is going to
have a chance at describing our world as we experience it is going to have to
describe a wavefunction that evolves in such a way that it is able to play the
functional role of a universe with tables and chairs and people in it (1996,
pp. 279–280). If this is right, then we may avoid the need to move to the less
straightforward reading of quantum mechanics offered by Maudlin, Allori,
and her collaborators in order to account for the appearances. We can allow
that the fundamental space of quantum mechanics is the high-dimensional
configuration space, but also claim that there is a derivative, functionallyenacted three-dimensional space occupied by tables, chairs, and people. This
derivative space arises by the behavior over time of the wavefunction in the
configuration space.40
This suggestion raises two questions. First, are the ordinary material objects of our manifest image the sorts of things that can be functionally
enacted by the wavefunction? And second, is the three-dimensional space of
our manifest image itself the sort of thing that can be functionally enacted
by the wavefunction?
For the purposes of this paper let us concede the point that the movement
of areas of high amplitude of the wavefunction through the configuration
space can enact something like a person, or a table, or a measuring device. That is, let us concede that these are the sorts of things that are not
fundamental entities, but rather are constituted out of more fundamental entities’ playing the right kind of causal roles.41 This concession has extremely
interesting consequences for our theories of reduction, realization and constitution; our theories, that is, of what relation the medium-to-large-sized
objects of our manifest image bear to what fundamentally exists according
to our best physical theories. In particular, an influential picture of reduction
advocated by Paul Oppenheim and Hilary Putnam (1958, p. 8) according to
which material objects relate to the fundamental physical ontology by mereological relations of part and whole (the fundamental physical ontology being
the parts of everything else), would seem to have things backwards. The
objects of fundamental physics are not parts of tables and chairs, but rather
tables and chairs would appear instead to be parts of the wavefunction.42 It is
very hard to wrap our heads around the idea that we ourselves are ultimately
constituted by the dynamical behavior of a wavefunction spread out in a very
high-dimensional configuration space. The diagrams in Section 3 may help to
make this point vivid. If we appear to see a system of many particles spread
out to make up our world in three-dimensional space, then (assuming the
wavefunction is in nearly an eigenstate of position43 ) this is realized by one
nearly point-sized region of high wavefunction amplitude in configuration
space. The entire world ultimately amounts to a near-point-sized speck in
configuration space. More work needs to be done to determine whether our
concepts of reduction, realization, and constitution can accommodate the
The Status of our Ordinary Three Dimensions in a Quantum Universe
547
live epistemic possibility that material objects are realized by, constituted by,
or reduce to mere specks in configuration space.
Still, granting this functionalism about ordinary objects is coherent (you,
me, tables, chairs, planets, and galaxies could all be enacted by the wavefunction), we now need to ask: could space, or space-time be like that? Could a
genuine, physical three-dimensional space be a functionally-enacted object?
Now, someone could think space was this sort of thing. This is similar to
how Leibniz thought of space. His view was that there isn’t this fundamental,
physical thing, space, that exists in its own right. Rather, space exists when
there are other substances standing in certain relations to each other. It
wasn’t Leibniz’s view that space existed insofar as there were specifically
causal relations obtaining between objects, but the views are still similar in
this way: they ground the existence of space in the obtaining of relations
between something or things more fundamental. I do not want to deny that
one could have this view about three-dimensional space. What I want to deny
is that one ought to think this is a view granting genuine, physical status to
three-dimensional space when one also believes that there is another space
that exists in its own right as a fundamental, physical substance.
The predicament for those who try to find three-dimensional space in
the ontology of quantum mechanics by viewing it as functionally enacted
out of the behavior of the wavefunction is that they want to hold that
this three-dimensional enacted space is real (though not fundamental) to
get local beables and at the same time accept that there is a separate nonenacted, fundamental configuration space. Here is how Albert understands
the configuration space. It is:
an arena within which the dynamics does its work, a stage on which whatever
theory we happen to be entertaining at the moment depicts the world as unfolding: a space (that is) in which a specification of the local conditions at every
address at some particular time . . . amounts to a complete specification of the
physical situation of the world. (1996, pp. 282–3)
Configuration space is the genuine physical space here. It is not something
that arises from the playing out of dynamical roles; it is the fundamental
structure required for the dynamical roles to be played in the first place. By
contrast, the three dimensions of our manifest image on this view are akin
to the images that may seem to be appear when we are viewing a movie with
3D glasses.
I am claiming that while functionally-enacted chairs are chairs, and
functionally-enacted people are people, for a substantivalist, functionallyenacted space is nothing more than a simulation. There are several reasons
to think this. Following John Earman, we may consider “two time-honored
tests for substance” (Earman 1989, p. 111). These are not plausible accounts
for existence simpliciter (for many objects of our manifest image, like tables
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and chairs may fail to satisfy them), but certainly seem to be features commonly associated with the nature of substantival space. First, substantivalist
space is supposed to exist independently of other objects.44 But if threedimensional space is functionally enacted, it depends for its existence on
other objects implementing causal roles, and thus would seem not to be a
genuine, substantival space. In addition, according to Earman, substances
are supposed to be themselves causally active. Whether or not functionallyenacted objects can themselves be coherently taken to be causally active
has been a matter of heated debate in the philosophy of mind. A common
verdict one finds, e.g. in Kim (1998) is that if what it is for something x
to exist is for there to be something else y playing a certain causal role,
then it seems that unless the x and the y are identified, there is no causal
work left for the x to do. So, it is at least controversial that a functionallyenacted three-dimensional space could be causally active in the sense Earman appears to have in mind. We may also add to Earman’s list a third
test to see if a space is legitimately viewed as the substantivalist would have
it: substantival space is usually taken to be a background against which
other events in our universe play out. For example, for Newton, space is
the background required for objects to have absolute accelerations.45 But
if three-dimensional space is functionally enacted, then it is not the background for fundamental interactions, but is instead constructed out of these
interactions.
If one wants to claim the three-dimensional space that is functionally enacted out of the behavior of the wavefunction is a genuine physical space,
this would appear to threaten the substantivalism about configuration space
Albert and others seem to endorse. If one takes a substantivalist attitude
toward configuration space, then the approach of Albert and Loewer fails to
provide a plausible way of finding three-dimensional space as (genuinely) existing within the ontology of quantum mechanics. The kind of functionalism
that is plausible for ordinary, material objects fails according to substantivalists about space or space-time structures. The three-dimensional space of
Albert and Loewer is nothing more than a simulation of space, a mirage.46
Albert himself gestures at the idea that perhaps this three-dimensional space
is just space in a different sense. He distinguishes the physical space that
is “the arena within which the dynamics does its work,” the configuration
space of quantum mechanics, from the physical space that is “the space of
possible interactive distances,” the three-dimensional space of our manifest
image (1996, p. 282). He calls the former “the more fundamental one”. What
I wish to argue here is that it is the only genuine concept of space, at least
for a substantivalist.
I conclude that neither account we have considered gives us a plausible
way of understanding three-dimensional space as (genuinely) existing within
the ontology of quantum mechanics. In this last section, I’d like to return to
the argument of Section 4 and investigate whether there is an overwhelming
The Status of our Ordinary Three Dimensions in a Quantum Universe
549
reason to recover the three-dimensional space of our manifest image from
the ontology of quantum mechanics.
6. Must We Recover Three-Dimensional Space?
Those authors we discussed in Section 4 were motivated to rethink the ontological status of the wavefunction (and its space) by their conviction that
quantum mechanics is a theory about local beables, and the fact that the
wavefunction is not a local beable.
Recall the argument:
(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.
Therefore,
(4) The space inhabited by the fundamental ontology of quantum mechanics
includes these three dimensions.
Local beables are believed to be the subject of the theory because they provide
the simplest explanation of the following facts. First, we think quantum
mechanics is about a world containing things like tables and chairs, objects
that are something like the way we ordinarily take them to be. Another
reason we have not up to this point considered is that the theory is justified
by a wealth of data involving the status of material objects like pointers
of measuring devices.47 The simplest explanation of both facts is that there
really are such material objects inhabiting the kind of space we think they
do. Now perhaps these points can be explained construing material objects
functionally in the way Wallace, Albert, and Loewer suggest. For there to
be a measuring device, there just needs to be something that can play the
functional role of a measuring device. For there to be a table, there just needs
to something that can play the functional role of a table.
What is not required, I would like to suggest however, is the existence
of a real, three-dimensional space. For there to be something that plays
the causal roles of tables, chairs, people, and pointer readings, to ground the
appearances and the confirmation of the theory, it does seem that there needs
to be something located in some physical space or other. We need an arena
in which the dynamics of our theory can unfold. And indeed we can concede
that this be the kind of thing that can simulate heights, widths, and depths.
However, there seems to be no reason why the physical ontology should
entail that the appearance of three-dimensional space (or four-dimensional
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space-time) be anything more than a simulation. My suggestion, in other
words, is that we should reject the first premise of the preceding argument:
(1) Quantum mechanics is a theory about local beables, i.e. objects with locations in the ordinary three dimensions of our manifest image.
Why would we need anything more than a simulation of a threedimensional space? According to Maudlin:
The contact between theory and evidence is made exactly at the point of some
local beables: beables that are predictable according to the theory and intuitively
observable as well . . . Collections of atoms or regions of strong field . . . because
they are local beables, can unproblematically be rock-shaped and move in reasonably precise trajectories. If the theory says that this is what rocks really are,
then we know how to translate the observable phenomena into the language of
theory, and so make contact with the theoretical predictions. (2007, p. 3159)
I am not arguing that on a straightforward, ontological reading of realist
versions of quantum mechanics that there are not rocks. I agree with Albert,
Loewer, and Wallace that there are. However, I claim that this does not imply
that there is anything that is genuinely rock-shaped (if this implies occupying
locations in a physical, three-dimensional space) or that moves in precise
trajectories through such a space. Since we can allow that the wavefunction
simulates the behavior of something in a three-dimensional space, we have
a way of making sense of our observations in terms of the language of the
theory. Even if this isn’t the simplest account of our observations, it is the
most scientifically conservative.
Most of the authors writing on this topic have been inspired by Bell
to assume that there must be local beables in any fundamental, physical
ontology. Bell defined ‘local beable’ in the following way:
We will be particularly concerned with local beables, those which . . . can be
assigned to some bounded space-time region . . . It is in terms of local beables
that we can hope to formulate some notion of local causality. (1987, p. 53)
According to Bell, it is desirable to have objects with precise locations and
trajectories in the physical space of a theory to be able to make sense of local
causality. For there to be causal interactions at precise locations in a theory
seemed to him (plausibly) to require the existence of entities with definite
locations at which such causal interactions take place.
The straightforward reading of realist versions of quantum mechanics
gives us such objects: parts of the wavefunction with high amplitude that
move and accelerate through the configuration space over time with precise
locations and trajectories. So in the sense that really matters, we get local
The Status of our Ordinary Three Dimensions in a Quantum Universe
551
beables: real, physical entities with precise locations in space. It is likely that
Bell wanted more, that he meant by ‘local beables’: real, physical entities with
precise locations in the space of our manifest image.48 But this doesn’t seem
required by the physics, nor the grounding of our manifest image. Moreover,
this seems unmotivated by Bell’s own emphasis on the issue of local causality.
If we want a physical account that includes local physical interactions, there
ought to be locations in some space or other where these interactions are
actually taking place. But it isn’t clear why Bell should require these to
be locations in a three-dimensional space. Without question, we need to be
able to make sense of the fact that it seems to us as if these interactions are
taking place in a three-dimensional space. But as has already been stated,
this can be accomplished without there actually being such a space.
Hans Reichenbach also argued that as a matter of empirical fact, to
ground local causality, the space of our physical ontology must be threedimensional:
The three-dimensionality of space has often been looked upon as a function
of the human perceptual apparatus, which can visualize spatial relations only
in this fashion. Poincaré tried to find a physiological foundation for this number . . . Even if this physiological explanation were tenable, it completely overlooks the fact that the number 3 of dimensions represents primarily a fact
concerning the objective world and that the function of the visual apparatus is
due to a developmental adaptation to the physical environment . . . It is the characteristic of three-dimensionality that it and only it leads to continuous causal
laws for physical reality. (Reichenbach 1928, p. 274)
Reichenbach’s argument for this claim proceeded as follows. First, he claimed
that any tenable physical theory must be a causally local theory in the
sense that it obeys the following principle. “Causal effects cannot reach
distant points of space without having previously passed through intermediate points” (Reichenbach 1928, p. 275). The current physics of his day was,
according to him, a theory that is causally local in that sense. Moreover,
Reichenbach argued that any transformation of one physical theory positing
a space of n dimensions into another with a space of n = n dimensions
will disrupt the original theory’s preservation of local causality. This was an
instance of a more general thesis of Reichenbach’s that whenever one transformed one geometry into another with distinct topological features, at least
some causal relationships that turned out to satisfy a locality principle when
viewed in the initial geometry would no longer be local in the geometry that
resulted from the transformation. The dimensionality of a space is one of its
topological features. We have a physical theory that satisfies local causality
and posits a three-dimensional physical space. Therefore, it is not possible
for there to be any other theory that preserves this theory’s physical content
while positing a distinct number of spatial dimensions and is similarly tenable
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(in the sense of preserving local causality). Therefore, Reichenbach claimed,
no tenable physical theory can posit anything other than a three-dimensional
physical space.
There is a lot to say about this argument. The most relevant point for
our purposes however is that although it was reasonable when Reichenbach
wrote this to argue that the current physics preserved local causality within
a three-dimensional spatial geometry, when it comes to quantum mechanics and what we have learned about it up to now, we know that at least
when formulated as a theory about objects in a three-dimensional physical
space, this theory does not obey Reichenbach’s principle of local causality.
This is a result that was shown conclusively by Bell (see his 1987, and for
further discussion Maudlin 1994). Interestingly enough, local causality does
appear to obtain according to one version of quantum mechanics we have
discussed: the Everettian account. And if this is correct, then given that the
most sophisticated and well-worked-out understanding of Everettian quantum mechanics, that of Wallace and his colleagues, posits a physical space
of extremely high dimensionality, we look to get, using Reichenbach’s own
reasoning, the conclusion that any tenable physical theory cannot posit a
physical geometry with only three dimensions.
So, there appear to be two reasons for insisting that we must somehow include in our physical ontology the existence of a genuine, three-dimensional
space. First: our perceptual interaction with the world and justification of
our scientific theories proceeds through interaction with material objects
that appear three-dimensional. In response to this, I have pointed out that
Albert, Loewer, and Wallace have already argued convincingly that a highdimensional quantum ontology can ground these appearances. There really
are material objects, even if their three-dimensionality is a mirage, and they
are ultimately grounded in the behavior of the wavefunction in configuration space. The second defense of local beables derives from the claim that
local causality requires grounding in a genuine three-dimensional space. I
have argued that Reichenbach’s reason for this looks to fail for reasons he
could not have anticipated, and anyway, local causality seems only to require
that one’s physical ontology include objects with precise locations in some
space or other. This space need not be the three-dimensional one of our
acquaintance.
In the beginning of this paper, I described a satisfying process by which
we are sometimes able to adjust our picture of the world in light of scientific
evidence to see it as more highly dimensioned than we may have earlier
thought. This required seeing our familiar three dimensions as three of the
many dimensions of this new, physical orthodoxy. I have argued that this
cannot be accomplished for realist versions of quantum mechanics, and
this is part of what makes quantum mechanics such a puzzling theory for
those of us who want to understand the fundamental ontology it presents.
This forces us to reject our earlier notions of reduction and realization, and
The Status of our Ordinary Three Dimensions in a Quantum Universe
553
challenges our intuitive sense of being extended in three dimensions. But
just because we cannot find our ordinary three dimensions in the world of
quantum mechanics, this does not mean that we may not be able to locate
ourselves in a quantum world. If we are functionalists about material objects
and ourselves, perhaps this is enough to see ourselves as inhabiting such a
high-dimensional world as quantum mechanics demands.
Appendix:
From Entangled States to the Existence of a Wavefunction in Configuration
Space
The quantum state of a system is the state of the system’s wavefunction
(). We will use the standard Dirac notation to express quantum states. For
example, the quantum state of a system of just one particle that is at location
(4, 0, 0) will be represented by:
1 = |(4, 0, 0) >
The most interesting fact about quantum mechanics is that it allows for
systems to evolve into states that are superpositions of a feature; for example,
the state in which there is a single particle in a superposition of being at
location (4, 0, 0) and being at location (7, 0, 0). (When a state is not in a
superposition of position, it is said to be in an eigenstate of position.) This
superposition will be represented in the following way:
1
1
| (4, 0, 0) > +
| (7, 0, 0) >
2 =
2
2
To say that a system is in this quantum state is not to say that the particle is
at location (4, 0, 0), nor is it to say that it is at location (7, 0, 0). It is also
not to say that the particle is at both location (4, 0, 0) and at location (7, 0,
0); nor is it to say that the particle is at neither location (4, 0, 0) nor location
(7, 0, 0). One thing that is true of a particle that is in this quantum state is
that if the particle’s location is measured, then there is a 0.5 chance that it
will be found at location (4, 0, 0). And if the particle’s position is measured,
there is a 0.5 chance that it will be found at location (7, 0, 0). The probability
is 0 that the particle will be found at any other location at that time. These
probabilities are given by the square of the coefficients in the representation
of the quantum state.
Entangled states may be described as superpositions involving multiparticle systems. For example, we might consider a system of two particles that
are entangled with respect to their position:
1
1
| (4, 0, 0) >1 | (7, 0, 0) >2 +
| (7, 0, 0) >1 | (4, 0, 0) >2
3 =
2
2
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What is true of particles in such a state? On the assumption that the quantum state is ontologically complete, it is not true that either particle has a
determinate location, either of being at (4, 0, 0) or of being at (7, 0, 0).
However, it is true that if one were to successfully measure the positions of
the particles, they would be found in one of the following two states. There
is a 0.5 chance that particle 1 would be found at (4, 0, 0) and particle 2
would be found at (7, 0, 0). And there is a 0.5 chance that particle 1 would
be found at (7, 0, 0) and particle 2 would be found at (4, 0, 0). And these are
the only possibilities. It is natural to conclude from this fact that although
neither particle in 3 has a determinate position, the relation between the
positions of the two particles is determinate. For, there is a probability 1
that the particles will be found at a distance of 3 from each other in the
x-dimension.
How should one describe this state as being instantiated in a threedimensional space? One way is by representing at each point in the space the
chances that a particle is at that location by a peak with a certain amplitude.
Then to represent 3 , we will have two peaks of amplitude 0.5 at each of
the two locations in three-dimensional space. In the figure below, we have
marked the chances that each particle will be found at each location. Peaks
for particle 1 are gray. Particle 2’s are white.
y
x
z
To see why this does not give us an adequate characterization of 3 , consider
the distinct quantum state, 4 :
4 =
√1
√
/2 | (4, 0, 0) >1 | (4, 0, 0) >2 + 1/2 | (7, 0, 0) >1 | (7, 0, 0) >2
4 is clearly a distinct quantum state than 3 . For if a system is in state 3 ,
then the particles may be correctly described as being at different locations a
distance of 3 apart in the x-dimension (even though it is not necessarily true
that either particle is at either location). However, if a system is in 4 , this
is not the case. Here, the particles would not be truly said to be at different
locations. For there is a probability of 1 that both particles will be found
upon measurement to be in the same location. Nevertheless, despite these
quantum states being clearly distinct and indeed empirically distinguishable,
the three-dimensional representation of 4 will be the same as what was
given above for 3 . The reader may check this against the above diagram. If
The Status of our Ordinary Three Dimensions in a Quantum Universe
555
we want an adequate characterization of either state, one that distinguishes
3 from 4 , we will need to move to a higher dimensional configuration
space.
The configuration space that is used to represent quantum states in general has 3N dimensions, where N is the number of particles in the system
to be characterized. Since the present examples each involve two entangled
particles, the configuration space we will need to use in order to represent
these states will be 6-dimensional. The first three coordinates in the configuration space correspond to x, y, and z coordinates for particle 1, and the
second three coordinates in the configuration space correspond to x, y, and
z coordinates for particle 2. 3 can be represented by peaks at locations
(4, 0, 0, 7, 0, 0) and (7, 0, 0, 4, 0, 0) in the configuration space and the
following diagram:
x2
y2
y1
z1
x1
z2
Here, the peaks are used to represent locations where there is some nonzero
chance that the entire system will be found. The two peaks correspond to
the two terms in the superposition. In configuration space, 4 constitutes a
completely different state
x2
y2
x1
y1
z1
z2
556
NOÛS
For a system in 4 , the peaks in the configuration space representation are
at locations (4, 0, 0, 4, 0, 0) and (7, 0, 0, 7, 0, 0).
As has been shown, entangled states can only be distinguished, and hence
completely characterized in a higher-than-3-dimensional configuration space.
They are states of something that can only be adequately characterized as
inhabiting this higher-dimensional space. This is the quantum wavefunction.
As I discuss in the third section above, none of the dimensions of configuration space are dimensions of the three-dimensional space of our manifest
image.
Notes
1
I am deeply indebted to David Albert, Valia Allori, John Bennett, Barry Loewer, and Jill
North for extensive comments on an earlier draft, as well as to audiences at the University of
Rochester, Vassar College, and Rutgers University for valuable discussion.
2 Following Wilfrid Sellars, I use ‘manifest image’ to describe our way of understanding
our place in the world that relies upon what we learn from our perceptual interaction with it
and introspection. This is intended as a contrast to the ‘scientific image’ (Sellars 1962).
3 As will become clearer I hope later on, this is the question of whether we need a physical
ontology that includes local beables.
4 The formalism that results if one assumes such an anti-realism is different than the kind of
formalism one finds in realist versions, so I call these different ‘versions’ of quantum mechanics,
not different ‘interpretations’.
5 There are several places to look for a more comprehensive introduction to these theories.
The best place to begin is probably Bell’s (1987) collected papers. He discusses Bohmian mechanics in ch. 17, the spontaneous collapse theory of Ghirardi, Rimini, and Weber in ch. 22,
and the Everettian view in ch. 11. More references may be found in the footnotes below.
6 Von Neumann himself was not Danish, but Hungarian. Still, it is the version of quantum
mechanics formulated by von Neumann, after being first suggested by Niels Bohr and his
students, that is now commonly referred to as ‘Copenhagen’ quantum mechanics.
7 What is this thing, the system’s wavefunction? For some, a sufficient answer is this: it is
the object whose evolution over time the fundamental laws of quantum mechanics describe.
More will be said about this below.
8 See Bell 1987, ch. 23 for a more extensive discussion of the problem with a fundamental
physical theory’s invoking concepts like measurement.
9 This is often called the ‘Everettian view’ because of its association with Hugh Everett
III, who proposed confining the dynamics of quantum mechanics solely to the Schrödinger
equation in his 1957 paper. It has since been developed into a clear and promising position in
recent work by David Wallace (2002) and Simon Saunders (1995) among others.
10 An early variant on this version of quantum mechanics was proposed by David Bohm
(1952), hence its current moniker. It has since been elaborated and clarified by Shelly Goldstein
and his collaborators. See, for example, his (2006).
11 This is usually called ‘the guidance equation’, the thought being that this law describes
how the wavefunction “guides” the motion of the particle or particles the theory posits.
12 The details of this debate are beyond the scope of this paper. For more, see Albert (1996),
Monton (2000), and Maudlin (2007).
13 It is also the case that (arguably) the particle equation can itself be derived from the
Schrödinger equation plus some natural symmetry considerations.
14 Philip Pearle developed a similar approach earlier (see Pearle (1976)).
15 The precise nature of the collapse isn’t important for the purposes of this paper. However,
it is worth emphasizing that the theory doesn’t say that the system will undergo random behavior
The Status of our Ordinary Three Dimensions in a Quantum Universe
557
at the time of the collapse. Rather the theory specifies a very specific kind of change in the
system’s wavefunction at times of collapse.
16 Sometimes this version of quantum mechanics is presented not as one involving two laws,
Schrödinger’s equation and another higher-level law saying when the Schrödinger equation fails
to obtain, but rather as involving just one law that is a probabilistic revision of Schrödinger
dynamics. I am here following Bell’s presentation of the theory: “The idea is that while a
wavefunction . . . normally evolves according to the Schrödinger equation, from time to time it
makes a jump” (1987, p. 202).
17 We know this, because anything with mass (m) is a material object according to Newton.
In his Principia Mathematica, mass is defined as quantity of matter.
18 I say this view is untenable; this does not mean it is not still advocated in the physics
community. This sort of eliminativism was defended most recently in the pages of Physics
Today by David Mermin (2009), and raised a quite lively debate. Mermin argued that reifying
the wavefunction was the manifestation of a more general bad habit that “makes life harder than
it needs to be” of taking abstractions useful as calculational devices to be “real”. Reifying the
wavefunction, according to Mermin, makes life harder by inducing people to write books and
organize conferences on the topic of the measurement problem, or worry about faster-than-light
signals produced as a result of quantum non-locality.
19 See also my (2010) for a more comprehensive discussion.
20 This much is true according to all of the realist versions of quantum mechanics we have
considered (aside from certain Bohr-inspired variants of the problematic Copenhagen account)
and anti-realist versions of quantum mechanics. On these accounts, it is not material objects
that enter entangled states. It is the wavefunction that does so. This is something on which all
parties to the present debate can agree, including my opponents in the following sections of this
paper.
21 As I’ll discuss a bit more below, in the classical mechanics that followed Newton’s
early presentation, use is also made of a higher-dimensional space. However, that space is
not indispensable in order to completely describe the states of classical systems. This higherdimensional classical space instead has the status of an alternative representational framework
that is used for convenience. This is why, unlike the configuration space of quantum mechanics,
this classical high-dimensional space does not have the place in the theory of a genuine, physical
space – the space anything actually inhabits.
22 This doubt is shared by all of the physicists and philosophers I discuss in the following
sections of this paper, with the exception perhaps of Bell. (It is not clear to me from his work
whether Bell understood Bohmian mechanics in this way or not).
23 It might help some philosophers to think of occasionalism here. The two-space picture
is much like this view of causal interaction advocated by Malebranche except this time the role
of God is played by the wavefunction.
24 There is also a more basic point that it is not clear even how to write down a law
that expresses how the wavefunction may direct the behavior of the particles if one adopts
this understanding of Bohmian mechanics. In general, physical theories describe interactions
by inter alia specifying the geometrical relations that must obtain between the two entities for
an interaction to take place. In this case, where the wavefunction and particles are supposed
to exist in altogether distinct spaces, one cannot do this, for there are no geometrical relations
obtaining between such objects in two separate spaces. I am grateful to David Albert for this
point.
25 This way of specifying the dimensionality of configuration space is given by Bell (1987),
Albert (1996), Lewis (2004), and many others.
26 Ideally, those endorsing this sort of realism about the wavefunction would introduce a
new, distinct name for the space it is supposed to inhabit, but the name ‘configuration space’ is
entrenched.
27 The question of the status of such vector spaces gets raised most explicitly when one
starts to investigate the matter of describing quantum systems with various properties such as
558
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spin, charge, and the like. Since it is peripheral to the present topic and the issue of configuration
space, I will set it aside here.
28 The reason why the chips may be represented by a single point, as opposed to a wavefunction smeared out over more of the configuration space, is that I am assuming our system is
not in an entangled state. See the appendix for more discussion of these concepts and my (2010)
for why this is not a generally accurate assumption for quantum systems.
29 So this proposal is not anything like what was considered briefly in the middle of
Section 2, p. 9.
30 For Bohmian mechanics, the primitive ontology might consist of particles. For GRW
and Everettian quantum mechanics, primitive ontologies of mass densities have been explored
among other things.
31 As Zanghı̀ and Roderich Tumulka have emphasized in conversation, the precise status of
the wavefunction really depends on which version of quantum mechanics is under consideration.
For example, in Goldstein and Stefan Teufel’s (2001) it is claimed that in Bohmian mechanics,
the wavefunction ought to be understood as a “field on the abstract space of all possible
[particle] configurations.” In GRW quantum mechanics (which itself may be formulated in
several versions), its status may be different, as the theory does not fundamentally involve
particle configurations. What is important for our discussion is not what according to this view
the precise status of the wavefunction is, but rather what its status is not – namely an entity like
a field spread out over a genuine, physical space.
32 I have heard this position often confused with the two-space reading of Bohmian mechanics described at the end of Section 2 above. However, note that this is a distinct interpretation of
quantum mechanics. The suggestion Allori and her collaborators wish to make is not that there
are two physical spaces – ordinary three-dimensional space and configuration space. Their view
is that there is only one space: the space inhabited by a material primitive ontology that does
not include the wavefunction.
33 In introducing this concept, Bell was primarily interested in moving away from talk of
observables in discussions of quantum mechanics, i.e. what can be observed according to the
theory, as Heisenberg commonly insisted, and toward a discussion of beables, i.e. what exists
according to the theory.
34 I am grateful to John Bennett for helpful comments on this point.
35 This worry might be removed if certain theories of quantum gravity turn out to be correct.
As Goldstein and Teufel (2001) have argued, on a theory of quantum gravity incorporating the
Wheeler-Dewitt equation, the wavefunction of the universe would actually be static, making it
more akin to what we normally think of as a law.
36 This seems to be a common view in the physics community, for example, in Uzan
(2003), p. 403: “Indeed, it is difficult to imagine a change of the form of physical laws (e.g. a
Newtonian gravitation force behaving as the inverse of the square of the distance on Earth and
as another power somewhere else) but a smooth change in the physical constants is much easier
to conceive.”
37 Though perhaps this is what some of these authors intend. See especially Dürr et. al.
1992.
38 As Allori has pointed out to me in comments to this paper, what is scientifically conservative may be open to interpretation. For although the approach she and her collaborators
favor revises quantum ontology as it may be straightforwardly read off the formalism, it is
scientifically conservative in another sense, namely in remaining closer to the ontology that one
finds naturally in classical mechanics.
39 Let’s set aside Bohmian mechanics for the moment.
40 Albert has a very nice story about how the wavefunction of our world is able to play this
role. I won’t reproduce it in detail here, but see his (1996, pp. 280–281). The crux of the story is
that, in order to support our quasi-Newtonian manifest image, it must be the case that when the
amplitude of the wavefunction gets high at certain locations in configuration space, the system
begins to accelerate more quickly than it does at other kinds of locations. These locations are
The Status of our Ordinary Three Dimensions in a Quantum Universe
559
ones that should be occupied by regions of the wavefunction with high amplitude in order to
ground the existence of material objects coming into contact. The diagrams in Section 3 can
help to illustrate the idea. According to Albert, when the wavefunction moves into a state with
high amplitude around the location (1, 2, 3, 1, 2, 3, 1, 2, 3) for example, we should expect the
movement of regions of high amplitude to accelerate quite rapidly through configuration space.
This makes sense since this corresponds roughly in the three-dimensional representation to a
system of three particles with locations (1, 2, 3), (1, 2, 3), and (1, 2, 3), i.e. particles that are
moving close enough to impact each other. We would expect this to create an acceleration in the
movement of the particles in three-dimensional space. This corresponds to an expectation that
the regions of high-amplitude wavefunction should accelerate in configuration space around
locations like (1, 2, 3, 1, 2, 3, 1, 2, 3).
41 This is something very compellingly defended by David Wallace in his (2003), inspired by
the work of Daniel Dennett, especially Dennett’s (1991). A similar view is endorsed by Albert
and Loewer (1996), though they disagree with Wallace that the wavefunction could play such
a role in Everettian quantum mechanics, as opposed to GRW quantum mechanics. There is
disagreement between these authors also on the issue of whether the wavefunction can play
the functional role to enact the material objects of our manifest image in Bohmian mechanics.
Wallace argues (in collaboration with Harvey Brown in their 2005) that it can, and so the
particle(s) of Bohmian mechanics are superfluous.
42 Jonathan Schaffer (2008) defends a version of ontological monism partly based on this
point.
43 See the appendix for more clarification on the meaning of ‘eigenstate’.
44 Although Newton was not a substantivalist, he also viewed physical space in this way.
Thus he claimed in the Scholium to his Principia that “absolute space, in its own nature, without
relation to anything external, always remains similar and immovable”.
45 This is also suggested by the above passage from the Scholium to Newton’s Principia.
46 I am trying to find an analogy that will more clearly illustrate the problem. The best I
can do is the following. Many philosophers believe that qualia – the qualitative features that
make for phenomenal consciousness are basic, intrinsic features (e.g. Chalmers 1996). Now try
to imagine someone holding this position about qualia, and then also allowing at the same
time that there are also these other features that are realized by certain entities playing some
functional role, and that these features are also qualia. This position seems puzzling in precisely
the same way as the position of Albert and Loewer on three-dimensional space.
47 See Maudlin (2007), pp. 3158–3161.
48 I am grateful to David Albert for pressing me on this point.
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