# Our Fundamental Physical Space:
An Essay on the Metaphysics of the Wave Function

Eddy Keming Chen Department of Philosophy, University of California, San
Diego, 9500 Gilman Dr, La Jolla, CA 92093-0119. Website:
www.eddykemingchen.net. Email<EMAIL_ADDRESS>
( _The Journal of Philosophy_ , 114(7), 2017: 333-65)

###### Abstract

The mathematical structure of realist quantum theories has given rise to a
debate about how our ordinary 3-dimensional space is related to the
3N-dimensional configuration space on which the wave function is defined.
Which of the two spaces is our (more) fundamental physical space? I review the
debate between 3N-Fundamentalists and 3D-Fundamentalists and evaluate it based
on three criteria. I argue that when we consider which view leads to a deeper
understanding of the physical world, especially given the deeper topological
explanation from the unordered configurations to the Symmetrization Postulate,
we have strong reasons in favor of 3D-Fundamentalism. I conclude that our
evidence favors the view that our fundamental physical space in a quantum
world is 3-dimensional rather than 3N-dimensional. I outline lines of future
research where the evidential balance can be restored or reversed. Finally, I
draw lessons from this case study to the debate about theoretical equivalence.

> Already in my original paper I stressed the circumstance that I was unable
> to give a logical reason for the exclusion principle or to deduce it from
> more general assumptions. I had always the feeling, and I still have it
> today, that this is a deficiency.
>
> Wolfgang Pauli (1946 Nobel Lecture)

Keywords: wave function, foundations of quantum mechanics, Bohmian mechanics,
GRW, Everett, fundamentality, spacetime, identical particles, Symmetrization
Postulate, Manifest Image

###### Contents

  1. 1 Evidence #1: The Dynamical Structure of Quantum Mechanics
    1. 1.1 The Argument for 3N-Fundametalism
    2. 1.2 The Assumptions in Premise 2
  2. 2 Evidence #2: Our Ordinary Perceptual Experiences
    1. 2.1 The Argument for 3D-Fundamentalism
    2. 2.2 The Assumptions in Premise 4
  3. 3 Evidence #3: Mathematical Symmetries in the Wave Function
    1. 3.1 Another Argument for 3D-Fundamentalism
    2. 3.2 Justifying Premise 6
    3. 3.3 Deep versus Shallow Explanations
    4. 3.4 What about ${}^{N}\mathbb{R}^{3}$-Fundamentalism?
  4. 4 Conclusion
  5. 5 Appendix: A Topological Explanation of the Symmetrization Postulate

## Introduction

This is an essay about the metaphysics of quantum mechanics. In particular, it
is about the metaphysics of the quantum wave function and what it says about
our fundamental physical space.

To be sure, the discussions about the metaphysics within quantum mechanics
have come a long way. In the heyday of the Copenhagen Interpretation, Niels
Bohr and his followers trumpeted the instrumentalist reading of quantum
mechanics and the principles of complementarity, indeterminacy, measurement
recipes, and various other revisionary metaphysics. During the past three
decades, largely due to the influential work of J. S. Bell, the foundations of
quantum mechanics have been much clarified and freed from the Copenhagen
hegemony.111For good philosophical and historical analyses about this issue,
see Bell (2004) for and Cushing (1994). Currently, there are physicists,
mathematicians, and philosophers of physics working on solving the measurement
problem by proposing and analyzing various realist quantum theories, such as
Bohmian mechanics (BM), Ghirardi-Rimini-Weber theory (GRW), and Everettian /
Many-World Interpretation as well as trying to extend them to the relativistic
domains with particle creation and annihilation. However, with all the
conceptual and the mathematical developments in quantum mechanics (QM), its
central object—the wave function—remains mysterious.

Recently, philosophers of physics and metaphysicians have focused on this
difficult issue. Having understood several clear solutions to the measurement
problem, they can clearly formulate their questions about the wave function
and their disagreements about its nature. Roughly speaking, there are those
who take the wave function to represent some mind-dependent thing, such as our
ignorance; there are also people who take it to represent some mind-
independent reality, such as a physical object, a physical field, or a law of
nature.

In this paper, I shall assume realism about the wave function—that is, what
the wave function represents is something real, objective, and physical.
(Thus, it is not purely epistemic or subjective).222I take it that there are
good reasons to be a realist about the wave function in this sense. One reason
is its important role in the quantum dynamics. See Albert (1996) and Ney
(2012). The recently proven PBR theorem lends further support for this
position. See the original PBR paper Pusey et al. (2012) and Matthew Leifer’s
excellent review article Leifer (2014). I shall conduct the discussion in non-
relativistic quantum mechanics, noting that the lessons we learn here may well
apply to the relativistic domain.333See Myrvold (2014) for an insightful
discussion on the complications when we transfer the discussion to the
relativistic domain with particle creation and annihiliation. I take it that a
clear ontology of QFT will provide a natural arena for conducting a similar
debate. Since a consistent QFT is still an incomplete work-in-progress
(although an effective quantum field theory is highly useful and predictively
accurate, and there have been proposals with clearer ontology such as Bell-
type QFT), I believe that there are many values in conducting the discussion
as below—in the non-relativistic QM—although we know that it is only an
approximation to the final theory. The clarity of the non-relativistic QM, if
for no other purposes, makes many issues more transparent. To disentangle the
debate from some unfortunately misleading terminologies, I shall adopt the
following convention.444I am indebted to Maudlin (2013) for the following
distinctions. I use _the quantum state_ to refer to the physical object and
reserve the definite description _the wave function_ for its mathematical
representation, $\Psi$. In the position representation (which I use throughout
this paper), the domain of the wave function is all the ways that fundamental
particles can be arranged in space and the codomain is the complex field
$\mathbb{C}$.555It is reasonable to wonder whether our definition applies to
“flashy” GRW (GRWf) or mass-density versions of GRW (GRWm). Although particles
are not fundamental in these theories, the definitions of GRWf and GRWm and
the mathematical structures of these theories can be captured by using wave
functions defined on the particle-location configuration space. _When
interpreting the physical theory_ (either according to the low-dimensional
view or the high-dimensional view described below), if we think of the wave
function as representing something _in addition to_ the material ontology, we
can regard the wave function space as having a structure that is independent
from the material ontology (and the configurations of material stuffs). We can
call the domain of the wave function “the configuration space,” but we do not
need to understand it literally as the space of configurations of the material
ontology. How the material ontology is connected to the wave function on “the
configuration space” is explained by the definitions (such as the mass-density
function $m(x,t)$ in GRWm) and the dynamical laws. Interpreted this way, many
arguments that rely on the configuration space of particle locations,
including those in Sections 2 and 3, apply directly to GRWf and GRWm. The
exceptions are those arguments that rely on a _justification_ or an
_explanation_ of the configuration space. Therefore, our arguments in Section
4.2 apply most smoothly to theories with a particle ontology such as Bohmian
Mechanics and much less smoothly to GRW theories. (For convenience and
simplicity, I leave out spin when discussing the value of the wave function,
noting that it can be added as additional degrees of freedom (spinor values).
Hence, $\Psi\in L^{2}(\mathbb{R}^{3N},\mathbb{C}).$) The domain of the wave
function is often called _the configuration space_ , and it is usually
represented by $\mathbb{R}^{3N}$, whose dimension is 3 times N, where N is the
total number of the fundamental particles in the universe. (I shall represent
the configuration space by $\mathbb{R}^{3N}$ instead of $\mathbb{E}^{3N}$,
noting that the latter would be more physical than the former.) For the wave
function of a two-particle system in $\mathbb{R}^{3}$, the configuration space
is $\mathbb{R}^{6}$. For the wave function of our universe, whose total number
of fundamental particles most likely exceeds $10^{80}$, the configuration
space has at least as many dimensions as $10^{80}$.

Our key question is: given realism about the quantum state, how is the
configuration space related to our familiar 3-dimensional space? There are,
roughly speaking, two views corresponding to two ways to answer that
question.666I leave out the middle position between the two views, which says
that both the 3N space and the 3D space are fundamental. See Dorr (2009) for
an exploration of that view.

3N-Fundamentalism

$\mathbb{R}^{3N}$ represents the fundamental physical space in quantum
mechanics. Our ordinary space, represented by $\mathbb{R}^{3}$, is less
fundamental than and stands in some grounding relations (such as emergence) to
$\mathbb{R}^{3N}$.777This view is often called Wave Function Realism, although
we should remember that this is unfortunately misleading terminology, for its
opponents are also realists about the wave function. Under my classification,
3N-Fundamentalists include David Albert, Barry Loewer, Alyssa Ney, Jill North,
and _arguably_ , the time-slice of J. S. Bell that wrote “Quantum Mechanics
for Cosmologists:” “ _No one can understand this theory until he is willing to
think of $\psi$ as a real objective field rather than just a ‘probability
amplitude’. Even though it propagates not in 3-space but in 3N-space._” (Bell
(1987) p.128, his emphasis.)

3D-Fundamentalism

$\mathbb{R}^{3}$ represents the fundamental physical space in quantum
mechanics. The configuration space, represented by $\mathbb{R}^{3N}$, is less
fundamental than and stands in some grounding relations (such as mathematical
construction) to $\mathbb{R}^{3}$.888This view is often taken to be a
commitment to Primitive Ontology or Primary Ontology. Under my classification,
3D-Fundamentalists include Bradley Monton, Detlef Dürr, Sheldon Goldstein,
Nino Zanghì, Roderich Tumulka, James Taylor, Ward Struvye, Valia Allori, Tim
Maudlin, Michael Esfeld, CianCarlo Ghirardi, and Angelo Bassi. Depending on
how one thinks about the nature of the wave function, there are three ways to
flesh out 3D-Fundamentalism: (1) the wave function is a physical object—a
multi-field (more later); (2) the wave function is nomological or quasi-
nomological; (3) the wave function is a _sui generis_ entity in its own
category. No matter how one fleshes out the view, the following discussion
should be relevant to all, although (1) is the most natural viewpoint to
conduct the discussion. In a companion paper, I discuss the viability of (2)
in connection to David Lewis’s thesis of Humean supervenience. In a future
paper, I plan to discuss the viability of (3). In any case, defending (2) and
(3) takes much more work and might in the end decrease the plausibility of
3D-Fundamentalism. If the following discussion is right, since (1) is the most
plausible competitor with 3N-Fundamentalism with a high-dimensional field, it
has been premature for 3D-Fundamentalists to give up the defense of (1).
Indeed, a careful examination of the pros and the cons reveals its superiority
over 3N-Fundamentalism (and its internal competitors (2) and (3)). I shall not
defend this claim here, and for this paper the reader does not need to take
sides in this internal debate among 3D-Fundamentalists.

3N-Fundamentalism and 3D-Fundamentalism are the targets of the recent debates
under the name “The Metaphysics of the Wave Function,” “Wave Function Realism
versus Primitive Ontology,” and “Configuration Space Realism versus Local
Beables.” (To be sure, these debates often branch into discussions about other
important issues, such as the nature of the quantum state: whether $\Psi$ is a
field, a law, or a _sui generis_ physical object.) In those debates, the most
common considerations include: (1) the dynamical structure of the quantum
theory, and (2) the role of our ordinary experiences. Usually, the opposing
thinkers are taken to assign different weights to these considerations.

For example, Jill North (2013) summarizes the disagreement as follows:

> This brings us to a basic disagreement between wave function space and
> ordinary space views: how much to emphasize the dynamics in figuring out the
> fundamental nature of the world. Three-space views prioritize our evidence
> from ordinary experience, claiming that the world appears three-dimensional
> because it is fundamentally three-dimensional. Wave function space views
> prioritize our inferences from the dynamics, claiming that the world is
> fundamentally high-dimensional because the dynamical laws indicate that it
> is.999North (2013), p. 196.

However, as I shall argue, the common arguments based on (1) and (2) are
either unsound or incomplete. Completing the arguments, it seems to me, render
the overall considerations based on (1) and (2) _roughly_ in favor of
3D-Fundamentalism. However, there is another relevant consideration: (3) the
deep explanations of striking phenomena. Here, we find a more decisive
argument in favor of 3D-Fundamentalism as it leads to a deeper understanding
of the mathematical symmetries in the wave function, known as the
Symmetrization Postulate. Since 3D-Fundamentalism explains the Symmetrization
Postulate much better than 3N-Fundamentalism does, we have strong reasons to
accept 3D-Fundamentalism.

I shall argue that the considerations (1), (2), and (3), taken together,
favors 3D-Fundamentalism over 3N-Fundamentalism. Here is the roadmap. In §1, I
shall argue, against the 3N-Fundamentalist, that the common argument from the
dynamical structure of quantum mechanics is unsound, and that after proper
refinement it no longer provides strong reason in favor of 3N-Fundamentalism.
(As a bonus, I will provide a direct answer to David Albert’s question to the
primitive ontologists: what is their criterion for something to be the
fundamental physical space?) In §2, I shall argue, against the typical
3D-Fundamentalist, that the common argument from the Manifest Image can be
resisted with spatial functionalism, but such a response can be weakened by
examining the details of the functionalist strategies. This suggests that the
argument from the Manifest Image, though no longer decisive, still has some
force. In §3, I shall explore which position leads to a better understanding
of the physical world and formulate a new argument based on the recent
mathematical results about the relationship between identical particles and
the symmetries in the wave function. The deeper topological explanation
offered by 3D-Fundamentalism strongly suggests that our fundamental physical
space in a quantum world is 3-dimensional rather than 3N-dimensional.

Finally, I shall offer future directions of research, point out how a
3N-Fundamentalist might be able to respond to the last two considerations, and
offer additional explanations to restore or reverse the evidential balance
between 3D-Fundamentalism and 3N-Fundamentalism.

As for the scope, this essay is more or less orthogonal to the debate between
3D and 4D views about time and persistence. However, the analysis in this
essay should be of interest to participants in those debates as well as to
philosophers working on the nature of the quantum state, the fundamentality /
emergence of space-time, and mathematical explanation in physics. As we shall
see, our case study also motivates a particular way of understanding the
equivalence of theories in terms of explanatory equivalence.

##  1 Evidence #1: The Dynamical Structure of Quantum Mechanics

At first glance, 3N-Fundamentalism looks like a highly surprising idea. What
can possibly motivate such a revisionary metaphysical thesis? The initial
motivation was the reality of the non-local and non-separable quantum state
and the phenomena of quantum entanglement—causal influences that are
unweakened by arbitrary spatial separation. Perhaps what we see in front of us
(the interference patterns on the screen during the double-slit experiment),
the idea goes, are merely projections from something like a “Platonic
Heaven”—some higher-dimensional reality on which the dynamics is perfectly
local.

The initial motivation leads to our first piece of evidence in the
investigation of our fundamental physical space. Such evidence is based on the
dynamical laws and the dynamical objects in quantum mechanics. This seems to
be the strongest evidence in favor of 3N-Fundamentalism. In this section, I
suggest that the dynamical structure of the quantum theory, under closer
examination, does not support the view that the fundamental physical space in
quantum mechanics is 3N-dimensional.

###  1.1 The Argument for 3N-Fundametalism

Here is one way to write down the argument for 3N-Fundamentalism based on
considerations of the dynamics.101010Strictly speaking, the following are
based on considerations of both the kinematics and the dynamics of the quantum
theory. Nevertheless, the kinematical structure of the quantum wave function
plays an important role in determining the dynamics of the material ontology
(Bohmian particles, mass densities, etc) that move in the physical space.

P1

We have (defeasible) reasons to infer a fundamental structure of the world
that matches the dynamical structure of our fundamental physical theory. [The
Dynamical Matching Principle]

P2

The dynamical structure of quantum mechanics (a candidate fundamental physical
theory) includes the quantum state which is defined over a 3N-dimensional
configuration space. [The 3N-Structure of the Quantum State]

C1

We have (defeasible) reasons to infer that our fundamental physical space is
3N-dimensional.

P1—the Dynamical Matching Principle—follows from a more general principle of
inference from physical theories to metaphysical theories:

The Matching Principle

We have (defeasible) reasons to infer a fundamental structure of the world
that matches the structure of our fundamental physical theory. We should infer
no more and no less fundamental structure of the world than what is needed to
support the structure of our fundamental physical theory.

The Matching Principle can be a highly useful guide for empirically-minded
metaphysicians.111111This principle comes from North (2013) and North
(forthcoming). See also Albert (1996) and Albert (2015) for ideas in a similar
spirit. For example, we can use it to infer that the classical space-time is
Galilean rather than Newtonian. This inference agrees with our intuitions.
Moreover, the inference is not based on controversial empiricist (perhaps
verificationalist) assumptions about eliminating unobservable or unverifiable
structure.

###  1.2 The Assumptions in Premise 2

Suppose that something like the Matching Principle is true and the particular
application with the Dynamical Matching Principle is justified. Then the
argument would rest on P2. However, does the quantum state, represented by the
wave function $\Psi$, really have a 3N-dimensional structure? In other words,
does the wave function have to be defined over a high-dimensional
configuration space?

Many people would answer positively. How else can we define the wave function,
if not on a high-dimensional configuration space? For surely the wave function
has to take into account quantum systems with entangled subsystems that have
non-local influences on each other, which have to be represented by wave
functions that are non-separable. To take these into account, the usual
reasoning goes, the wave function of an N-particle system (say, our universe
with $10^{80}$ particles), in general, has to be defined as a field over on a
3N-dimensional space. Alyssa Ney (2012), for example, endorses this line of
reasoning:

> …[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.121212Ney
> (2012), p. 556.

North’s reasoning is similar:

> In quantum mechanics, however, we must formulate the dynamics on a high-
> dimensional space. This is because quantum mechanical systems can be in
> entangled states, for which the wave function is nonseparable. Such a wave
> function cannot be broken down into individual three-dimensional wave
> functions, corresponding to what we think of as particles in three-
> dimensional space. That would leave out information about correlations among
> different parts of the system, correlations that have experimentally
> observed effects. Only the entire wave function, defined over the entire
> high-dimensional space, contains all the information that factors into the
> future evolution of quantum mechanical systems.131313North (2013), p. 190

There are two assumptions that are worth making explicit in the above
reasonings (to be sure, North and Ney discuss them in their respective
papers):

$\Psi$-as-a-Field

The quantum state, represented by the wave function $\Psi$, is a field.

Field-Value-on-Points

The space that a field “lives on” is the smallest space where it assigns a
value for each point in that space.

The two assumptions141414The qualification “smallest” is added to ensure that
the principles render a unique space that a field lives on. are explicit in
David Albert’s argument in his important paper “Elementary Quantum
Metaphysics”:

> The sorts of physical objects that wave functions are, on this way of
> thinking, _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 specifies (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.151515 Albert (1996), p. 278

However, as I shall argue, there are good reasons to reject both assumptions.

To deny $\Psi$-as-a-Field, one might instead propose that $\Psi$ is not
strictly speaking a field (in any classical sense). For example, an
electromagnetic field assigns a field value to every point in the
3-dimensional space, and the values are meaningful partly because multiplying
the field by a constant will result in a different field and different
dynamics. (The gauge freedom in the electric potential therefore indicates its
relation to something more physical and more fundamental, such as the electric
field.161616Thanks to Michael Townsen Hicks for alerting me to the
connection.) However, the quantum wave function, multiplied by a global phase
factor ($\Psi\Rightarrow e^{i\theta}\Psi$), remains physically the same.
Insisting on $\Psi$-as-a-Field will lead one to recognize physically
indistinguishable features (that play no additional explanatory role) as
physically real and meaningful. It is desirable, therefore, that we do not
recognize $\Psi$ as a fundamental field.171717However, it is possible to
provide a gauge-free account of the wave function by giving a nominalistic and
intrinsic account of quantum mechanics. In a future paper, I will provide the
beginning of such an account. But the next point holds regardless of the
success of that program.

However, even if one were to embrace the indistinguishable features brought
about by $\Psi$-as-a-Field, one can still deny Field-Value-on-Points. We
usually think, as Albert describes, that the fundamental physical space is the
fundamental space that the dynamical object “lives on.” In particular, the
fundamental space that a field lives on is one in which we specify the state
of the field at a time by specifying the values at _each point_. This is
indeed the case for the electromagnetic field. But why think that this is an
essential feature for a physical field?

Suppose that a fundamental field is mathematically represented by a function
from one space to another space. Such a function can be as general as one
likes. A special case, of course, is the electromagnetic field that assigns
values to each point in $\mathbb{R}^{3}$. But another one can be a quantum
multi-field that assigns values to every N-tuple of points in
$\mathbb{R}^{3}$. That is, the function takes N arguments from
$\mathbb{R}^{3}$. If we think of the electromagnetic field as assigning
properties to each point in $\mathbb{R}^{3}$, we can think of the quantum
multi-field as assigning properties to each plurality of N points in
$\mathbb{R}^{3}$ (N-plurality). The multi-field defined over N-pluralities of
points in $\mathbb{R}^{3}$ is, unsurprisingly, equivalent to the quantum wave
function defined over each point in $\mathbb{R}^{3N}$. 181818This approach of
taking the wave function as a “multi-field” is explained in Forrest (1988) Ch.
6 and Belot (2012). I have heard that for many working mathematical physicists
in the Bohmian tradition, this has always been how they think about the
wavefunction.

However, this approach has a disadvantage. On this conception of the multi-
field, we lose Albert’s very clear criterion for something to be the
fundamental physical space (based on the space that the fundamental field
“lives on”). Indeed, Albert asks,191919Albert raised this question during his
talk “On Primitive Ontology” at the 2014 Black Forest Summer School in
Philosophy of Physics. having given up his criterion, what would be an
alternative conception of the fundamental physical space that the
3D-Fundamentalists can give?

No one, as far as I know, has provided an alternative criterion for something
to be the fundamental physical space. But would it be terrible if there were
none? Some people would not worry, for the following reason. When we work on
the metaphysics of physical theories, we have in mind complete physical
theories—theories that specify their fundamental physical spaces; that is, for
each theory, we do not need to infer its fundamental space from the other
parts of the theory. For each complete theory, the fundamental space is given.
Our task is to decide which one (among the complete theories) is the best
theory, by considering their theoretical virtues and vices (especially their
super-empirical virtues such as simplicity, parsimony, explanatory depth, and
so on). Albert’s question is not an urgent one, because it is about a
different feature of theories: how we arrive at them in the first place. We
could wonder whether we have inferred in the right way when we say that the
fundamental space of the theory is 3-dimensional. But we could also just ask
which theory is more plausible: the quantum theory defined on $\mathbb{R}^{3}$
or the quantum theory defined on $\mathbb{R}^{3N}$.

However, we might construe Albert’s question in a different way, as asking
about a distinctive theoretical virtue, namely, internal coherence. In
contrast to the previous construal, this perspective would make Albert’s worry
highly relevant to the interpretive projects in the metaphysics of quantum
mechanics. As I think of it, internal coherence is distinct from logical
consistency. A metaphysical / physical theory can be logically consistent
without being internally coherent, in the sense that it might contain
theoretical parts that are in serious tension (though no logical
incompatibility) with each other. What the tension is can be precisified
according to the types of the theory. For example, a theory with a field-like
entity defined not on individual points but on non-trivial regions in the
fundamental space has theoretical parts that are in tension. Hence, one way
for a theory to lack internal coherence is for it to fail Albert’s criterion
of Field-Value-on-Points.

Construed in that way, Field-Value-on-Points is a precisification of the
theoretical virtue of internal coherence. However, I shall argue that Albert’s
argument against 3D-Fundamentalism will not work even if internal coherence is
a theoretical virtue, for there is another equally valid way to precisify
internal coherence.

Let us consider the following proposal for a first pass:

Field-Value-on-Pluralities

The space that a field “lives on” is the smallest space where it assigns
values for pluralities of points in that space.

The modified criterion no longer requires the field to take on values at each
point in the space. But this seems to lead to another disastrous consequence
for the 3D-Fundamentalist who holds the multi-field view, for the smallest
space where it assigns values for a plurality of points is not
$\mathbb{R}^{3}$ but $\mathbb{R}^{1}$ (a linear subspace of $\mathbb{R}^{3}$)
where 3N fundamental particles move around (in a 1-dimensional
space).202020Thanks to David Albert for pointing out this worry. Should the
3D-Fundamentalist, based on the above criterion, accept $\mathbb{R}^{1}$ as
the fundamental physical space?

That initial worry dissolves itself if we take the particles to be
indistinguishable and physically identical, and we take the configuration
space to be an unordered configuration space (we will return to this proposal
in §3). If the fundamental physical space is $\mathbb{R}^{1}$, then two
particles cannot pass through each other (see diagram below). For two
particles to pass through each other (which we would like to make possible
given the degrees of freedom of the particles in the 3D space and the
complexity of our world), there must be a moment in time where they occupy the
same point, which means that the wave function will be undefined at that
moment. (Recall that we represent an unordered configuration as a set instead
of an ordered tuple, and that the wave function is defined as functions from
sets of N points in $\mathbb{R}^{3}$ to complex numbers. At the instant where
the two particles occupy the same point (as the following diagram in
$\mathbb{R}^{1}$ shows), the configuration will lose at least one point,
dropping down to a configuration of 3N-1 points. The wave function would
therefore be undefined.) However, things would be different as soon as we go
to a bigger space, say, $\mathbb{R}^{2}$. A particle can “pass through”
another by missing it by a small distance (such as particle 2* in the second
diagram below) or going around it, perhaps in a non-straight line, with
arbitrarily small positive distance $\epsilon$ (such as particle
2).212121Thanks to Sheldon Goldstein and Nino Zanghì for pointing this out to
me. Thus, the bigger space $\mathbb{R}^{2}$ provides more “expressive power”
than $\mathbb{R}^{1}$.

particle 1particle 2$\mathbb{R}^{1}$

particle 1particle 2particle 2*$\mathbb{R}^{2}$

Why not, then, take $\mathbb{R}^{2}$ instead of $\mathbb{R}^{3}$ as our
fundamental physical space, since it is also a linear subspace of
$\mathbb{R}^{3}$? There are two reasons. First, we do not know whether the
total number of apparent particles in 3D space is even, which is required if
the fundamental dynamics is given by 3N/2 many particles in $\mathbb{R}^{2}$.
However, we do know (as the reader can easily verify) that the number of
fundamental particles times the dimensions of the fundamental space is a
multiple of 3. It seems plausible that, at least in the quantum theory, the
fundamental physical space should remain neutral about whether there are $2k$
or $2k-1$ particles. Second, even more than the dynamics of 3N particles in
$\mathbb{R}^{1}$, the dynamics of 3N/2 many particles, especially the
Hamiltonian function, (even if we assume N is even) is unlikely to be
natural.222222There may be many other reasons why the 3-dimensional space is
special. In §3.2, we consider the case of identical particles, for which
spaces with fewer than 3 dimensions would allow fractional statistics that
correspond to anyons, in addition to fermions and bosons. It is appropriate to
add a neutrality constraint and a naturality constraint to the previous
criterion of the fundamental physical space:

Field-Value-on-Pluralities*

The space that a field “lives on” is the smallest and most natural space that
is neutral to the parity of the total number of fundamental particles and that
the field assigns values for pluralities of points in that space.

If we regard Albert’s question as tracking the theoretical virtue of what I
call “internal coherence,” then I take Field-Value-on-Pluralities* as an
equally attractive way of precisifying internal coherence. Given this
precisification, 3D-Fundamentalism wins, as $\mathbb{R}^{3}$ satisfies this
criterion and is the space where the quantum multi-field lives on. Without
offering strong reasons to favor his precisification, Albert’s argument falls
short of delivering the conclusion.

In this section, I have examined and improved the Argument against
3D-Fundamentalism and still found its key premise unjustified. Therefore, our
first piece of evidence from the dynamical structure of quantum mechanics
underdetermines our choice between 3N-Fundamentalism and 3D-Fundamentalism.

##  2 Evidence #2: Our Ordinary Perceptual Experiences

In this debate, many people have pointed out the obvious—our Manifest
Image—the tables, chairs, and experimental devices in front of us. How can
these apparently 3-dimensional objects exist if reality is vastly high-
dimensional? If the high-dimensional quantum mechanics could not explain the
apparent reality of pointer readings in 3 dimensions, from which we come to be
justified in believing in quantum mechanics, the theory would surely be self-
undermining.

Hence it is common for 3D-Fundamentalists to use the second piece of
evidence—our ordinary perceptual experiences—to argue in favor of their view.
In this section, I suggest that although the evidence seems to support their
view over the alternative—3N-Fundamentalism, the evidential force depends
crucially on the fate of functionalism.

###  2.1 The Argument for 3D-Fundamentalism

Here is one way to write down the argument for 3D-Fundamentalism based on
considerations of the Manifest Image of 3-dimensional observers and pointers.
(I shall use relatively weak assumptions to generate a relatively strong
argument.)

P3

If we cannot locate the Manifest Image of human observers and pointer readings
in an empirical (fundamental) scientific theory, then we have (defeasible)
reasons against that theory.

P4

We cannot locate the Manifest Image of human observers and pointer readings in
the quantum theory with 3N-Fundamentalism, which is an empirical (fundamental)
scientific theory.

C2

We have (defeasible) reasons against the quantum theory on 3N-Fundamentalism.

The first premise—P3—does not depend on an insistence of respecting the common
sense or the reality of the Manifest Image. Instead, it follows from a more
general principle:

Self-Undermining

If we arrive at an empirical (fundamental) theory T based solely on evidence
E, and either we see that T entails that E is false without providing a
plausible error theory, or we see that T entails that our acquiring E is
produced by an unreliable process, or we see an ineliminable explanatory gap
from T to the truth of E (or some nearby propositions), then we have
(defeasible) reasons against T.

To see how P3 follows from Self-Undermining, we need to rewrite P3 in more
precise terms.

P3*

If an empirical (fundamental) scientific theory contains an ineliminable
explanatory gap for the true claims (or some nearby propositions) about human
observers’ pointer readings after experiments, then we have (defeasible)
reasons against that theory.

P3* seems highly plausible.232323This is similar to the demand for “empirical
coherence” in Barrett (1999) and Healey (2002). The principle, I take it, is
weaker than the often cited principle in Maudlin (2007b) that the physical
theory makes contact with evidence via “local beables.” Huggett and Wüthrich
(2013) discuss, in light of the recent developments of quantum gravity and
string theory, the complications of that requirement. The key idea is that our
belief in a fundamental scientific theory is not justified in a void, or
properly basic, or merely coherent with respect to the rest of our
metaphysical beliefs. Rather, we are justified in believing in it because we
(or the scientific community as a whole) have acquired sufficient empirical
evidence through the pointer readings connected to particular experimental
set-ups. These pointer readings can be in principle reduced to the positions
of entities in a 3-dimensional space. And we acquire such crucial pieces of
evidence through perception and testimony. Each person in the society may not
acquire the evidence in a direct way. However, there has to be some direct
perception of the 3-dimensional recordings of the positions that justifies our
beliefs in the scientific theory. That is, we come to believe in the
scientific theory on the basis of our observations about the arrangement of
macroscopic objects in the 3-dimensional space. Now, if the scientific theory
cannot explain the truth of such observations of pointer readings, or if it
turns out that we would be systematically wrong about the arrangements of
these macroscopic objects in the 3-dimensional space, then we acquire an
undermining defeater: the theory suggests that our evidence for the theory is
false. This is especially objectionable for scientific theories that attempt
to give a comprehensive explanation for the empirical world that includes the
human observers, pointers, and laboratory set-ups.

###  2.2 The Assumptions in Premise 4

If we accept P3*, then the success of the argument depends on the second
premise—P4. P4 is the claim that we cannot find the Manifest Image of human
observers and pointer readings in the quantum theory on 3N-Fundamentalism. In
other words, there is an ineliminable explanatory gap for the true claims (or
some nearby propositions) about human observers’ position readings of the
experimental set-ups.

Many people sympathetic to 3D-Fundamentalism take this premise to be straight-
forward.242424See, for example, Allori (2013) and Maudlin (2013). However,
3N-Fundamentalists have offered a proposal about how to close the explanatory
gap. Their general strategy goes by the name of “functionalism.”252525It is
different from functionalism in philosophy of mind, as it is concerned not
with minds but macroscopic physical objects, but the two theories have
interesting similarities that are worth exploring further.

In our context here, “functionalism” refers to a formulation of the sufficient
condition for what counts as emergent and real objects:

Functionalism

If we have a fundamental theory of the world in which we can define a mapping
from the fundamental degrees of freedom to $Q_{1},Q_{2},...Q_{n}$, and
$Q_{1},Q_{2},...Q_{n}$ have the same counterfactual profile of what we take to
be the constituents of the Manifest Image, then $Q_{1},Q_{2},...Q_{n}$ are
real, emergent entities from that fundamental theory and our discourse about
the Manifest Image is made true by the fundamental theory via the
functionalist mapping.262626Thanks to an anonymous referee, I realize that
this definition of functionalism leaves open the question how to understand
the notion of “counterfactual profiles” in certain theories. Take GRWm for
example: when we evaluate a counterfactual situation by changing the mass
densities within a bounded spatial region $R$, it is not at all clear how we
should change the mass densities outside of $R$ or how we should change the
wave function to reflect the changes in the mass densities as the mappings are
many-to-one. There may be some ways to handle this worry (and we invite the
defenders of 3N-Fundamentalism to address this worry), but the referee is
right that it raises another sort of worry on the viability of the
functionalist program.

More specifically, we take the 3N-Fundamentalist to be adopting the following
sufficient condition:

Functionalism about pointers

If we have a fundamental theory of the world in which we can define a mapping
from the fundamental degrees of freedom to $Q_{1},Q_{2},...Q_{n}$, and
$Q_{1},Q_{2},...Q_{n}$ have the same counterfactual profile of what we take to
be the 3-dimensional pointers in the Manifest Image, then
$Q_{1},Q_{2},...Q_{n}$ are real, emergent entities from that fundamental
theory and our discourse about pointer readings is made true by the
fundamental theory via the functionalist mapping.

Three questions arise. First, can we find such a mapping? Answer: yes, it is
available to the 3N-Fundamentalist. The most obvious solution would be to
choose just the three degrees of freedom associated with each apparent
particle from which we construct the configuration space. The 3-dimensional
particles are not fundamental, but we recognize that there is a mapping from
the 3N-dimensional space to N particles in the 3-dimensional space. Therefore,
our discourse about the pointers and human observers that are made out of
particles in the 3-dimensional space is grounded via the functionalist mapping
from the fundamental picture (in the Bohmian 3N picture: one Marvelous Point
moving in a high-dimensional space).

Second, is the mapping unique? Answer: it depends. We can, for example, use
the dynamical structure to privilege one set of mappings. Albert (1996) takes
this strategy and suggests that there is a uniquely preferred mapping because
of the contingent structure of the Hamiltonian of the universe that governs
the dynamics of the fundamental degrees of freedom:

$H=\sum_{i}^{\Omega}\frac{p_{i}^{2}}{2m_{i}}+\sum_{k,j=1;k\neq
j}^{\Omega/3}V_{j,k}([(x_{3j-2}-x_{3k-2})^{2}+(x_{3j-1}-x_{3k-1})^{2}+(x_{3j}-x_{3k})^{2}])$

The idea is that the fundamental degrees of freedom are contingently grouped
into triples in the Hamiltonian potential term, instantiating a relation just
like the familiar Pythagorean distance relation, which gives rise to the
3-dimensional Euclidean metric. Each triple is what we take to be a particle
in the 3-dimensional space. Even in a classical world, the Pythagorean
distance relation gives rise to a structure that is invariant under rotation,
translation, and reflection symmetries, and grounds our claims about the
3-dimensional space. In a relativisitic world, the Minkowski metric, according
to many people, gives rise to a genuinely 4-dimensional spacetime:

$d(a,b)^{2}=-(c\delta t)^{2}+(\delta x)^{2}+(\delta y)^{2}+(\delta z)^{2}$

Therefore, the reasoning goes, the mathematical structure in the Hamiltonian
potential term gives rise to an emergent, distinguished, and invariant
structure of 3-dimensionality—$\mathbb{R}^{3}$—that grounds our discourse
about pointer readings and human observers. Moreover, the Hamiltonian, in a
unique way, puts the fundamental degrees of freedom into $\Omega/3$ triples.

Third, isn’t the sufficient condition too permissive? If what suffices to be
emergent and real is just to stand in a mathematical mapping relation to the
fundamental dynamical objects, then (as Tim Maudlin272727Personal
communication February 2015. and John Hawthorne282828See Hawthorne (2010),
pp.147-152. observe independently) there will be many more emergent objects
than we realize, a consequence that is highly implausible. Under
Functionalism, simply by defining a mathematical mapping from the location of
every object in the world to three feet north of that object, there will be
emergent objects interacting with other emergent objects unbeknown to all of
us! Since the 3-feet-north mapping is completely arbitrary, there will be
infinitely many mappings, each of which will realize a different set of
emergent entities. As a consequence, just by defining trivial mappings like
these, we can create metaphysical monstrosities of an unimaginable scale.

The 3-feet-north objection is a special case of a general objection to
structuralism and functionalism: the mere existence of certain structure,
counterfactual dependence, or causal relations is sometimes insufficient
ground for establishing the reality of the emergent entities. In our context,
the 3-feet-north objection suggests that the functionalist criterion is
insufficient, and hence the proposed mapping from the fundamental degrees of
freedom in 3N-Fundamentalism does not _explain_ the pointers or observers in
the 3-dimensional space.

I see two potential responses. First, the 3N-Fundamentalist can bite the
bullet, embrace the seemingly absurd consequence of the functionalist
criterion, and include in her ontology infinitely many sets of emergent
entities. To show that the consequences are tolerable, she can give an
argument that shows that all sets of emergent entities are in fact equivalent
in some way. One obvious argument makes use of relationalism about space. If
relationalism is true, then there is no container (the substantial space) in
addition to spatial relations among objects. Take any set of emergent entities
(for example, the emergent entities that are 3 feet north of everything in the
world), the spatial relations instantiated among them are the same spatial
relations instantiated in any other set (for example, the emergent entities
that are 5 feet west of everything in the world). The sets of emergent
entities related by such mappings are just different ways of describing the
same relational world. Call this The Relationalist Approach.

The Relationalist Approach, I think, is the most promising response on behalf
of the 3N-Fundamentalist. However, it faces a problem. Functionalism, if true,
seems to be a necessary truth. Relationalism, on the other hand, does not seem
to be a necessary truth. The mere possibility of the failure of relationalism
implies that there are possible worlds in which the functionalist criterion
generates too many emergent entities. Since those possible worlds are not too
remote, our modal intuitions seem robust enough to account for them. The
metaphysical monstrosity seems highly implausible. So it seems that
Functionalism is still false. At this point, the 3N-Fundamentalist can reply
that Functionalism is in fact a contingent truth that holds only among
relational worlds. It would take future work to show why this is true or how
it follows from other commitments of Functionalism.

Instead of suggesting that the thesis is contingent, the 3N-Fundamentalist can
reply that there are ways of restricting the functionalist criterion such that
it does not lead to the absurd consequences. One way to restrict the criterion
is to say that the functionalist mapping has to be an identity
mapping.292929Barry Loewer suggests this in personal communication, but he
does not necessarily endorse it. The reason that projection mappings from
$\mathbb{R}^{3N}$ to $\mathbb{R}^{3}$ are insufficient is that a projection
map composed with a spatial translation is another a projection map; since the
spatial translamtion can be completely arbitrary, there will be an infinite
number of projection mappings that are equally good. However, if we let the
functionalist relation to be the identity mapping, we can eliminate this
problem. Applied to our case, the relation takes the triplets of degrees of
freedom in the configuration space as _identical_ with the 3-dimensional
particles. For example, take the Bohmian “marvelous point” in the
3N-dimensional configuration space: its coordinates $x_{1},y_{1},z_{1}$ just
are a 3-dimensional particle. Identity is a strict relation that is not
preserved by arbitrary mappings. Neither does the strategy rely on
relationalism about space. Call this The Identity Approach.

However, not only does it have difficulty extending to GRW theories, the
Identity Approach gets rid of the metaphysical monstrosities at the expense of
eliminating the Manifest Image altogether. To see this, let us borrow some
idioms from the grounding literature. The grounding relation, as commonly
conceived,303030For example, see Rosen (2010). is an umbrella term for several
kinds of metaphysical dependence relations. One widely-accepted feature of
grounding is that if a set of facts or entities, $\Sigma$, grounds another set
of facts or entities $\Gamma$, then it is not the case that $\Gamma$ grounds
$\Sigma$. Such an asymmetry is a defining feature of the grounding relation.
If the particles and their electromagnetic interactions ground the existence
of a table, it is not the case that the table’s existence grounds the
particles and their electromagnetic interactions. Let us examine the
suggestion that the functionalist mapping has to be an identity mapping. As I
understand it, the functionalist mapping is a metaphysical dependence relation
that falls under the umbrella of the grounding relation. Hence, the mapping
has to be asymmetric. However, the identity relation is symmetric. We have
arrived at a contradiction. Therefore, one of the assumptions has to go. Since
the functionalist mapping is supposed to metaphysically explain the emergence
of 3-dimensional objects such as pointers and observers, it is best understood
as a metaphysical explanation relation that is asymmetric. So it seems to me
that the 3N-Fundamentalist should reject the suggestion that the functionalist
criterion only allows identity mappings.

There is another way to restrict the functionalist criterion. Instead of
counting mathematical mappings of any sort, a 3N-Fundamentalist can restrict
to mappings between different spaces. For this smaller class of mathematical
mappings, the 3-feet-north counterexamples do not arise, for the dynamics and
causal behaviors on different levels are likely to be quite different. Call
this The Different Space Approach. In addition to being obviously _ad hoc_ and
unprincipled, the Different Space Approach does not block composite mappings.
The composite mappings first from $\mathbb{R}^{3N}$ to $\mathbb{R}^{3}$ then
from $\mathbb{R}^{3}$ to itself are not ruled out by the restriction and still
generate the same counterexamples.

The above discussions of different approaches of functionalism—the
Relationalist Approach, the Identity Approach, and the Different Space
Approach—suggest that the current functionalist criterion is too permissive
and leads to disastrous results for the 3N-Fundamentalists. Thus, we are right
to doubt whether there can be any principled way to close the apparent
explanatory gap in 3N-Fundamentalism. However, Functionalism, just like
Structuralism, is still being developed, and its application here is novel and
potentially promising. Given the recent work on the emergence of space-time
and structural realism, there may well be future work that suggests better
proposals than the ones we have considered. Anecdotal evidence suggests that
many philosophers are not deterred by the above counterexamples and are still
searching for some version of Functionalism that closes the explanatory gap in
a principled way.

So far our discussions have focused on the two main considerations in the
literature: the dynamical structure of the quantum theory and the successful
explanation of the Manifest Image. If I am right, then the common arguments
based on these considerations are either unsound or in need of refinement.
After locating the crucial premises, I show how they can be resisted on either
side. While the 3D-Fundamentalist can respond line by line by giving an
alternative criterion of fundamental physical space, the 3N-Fundamentalist
cannot provide a satisfactory functionalist criterion. So far, the
considerations based on the dynamical structure and the Manifest Image are
roughly in favor of 3D-Fundamentalism. However, anecdotal evidence suggests
that this has not persuaded many 3N-Fundamentalists to switch sides, as they
are often inclined to bite the bullet and swallow the costs of counter-
intuitiveness.

I take this to suggest that the most common philosophical arguments do not
fully settle the debate between the two views. It looks like we have reached a
stalemate. However, I suggest that we have not. In the next section, by
looking into which view leads to a deeper understanding of the quantum world,
I argue that we can find a new class of powerful arguments that favor
3D-Fundamentalism over 3N-Fundamentalism.

##  3 Evidence #3: Mathematical Symmetries in the Wave Function

Having seen that the common arguments from the dynamical structure and
ordinary experiences do not fully settle the debate between 3D-Fundamentalism
and 3N-Fundamentalism, I suggest we look at the debate from a different angle.
As we are examining scientifically motivated metaphysical positions, we would
like to see which one leads to a better or deeper understanding of the
scientific phenomena. In this section, I offer an argument for
3D-Fundamentalism on the basis that it provides a deeper mathematical
explanation of certain symmetries in the quantum theory. Since I do not take
this to be the final word on the issue, I offer this as a first step toward an
open research program to explore various mathematical explanations313131Here I
do not take sides whether such mathematical explanations are non-causal
explanations. in the quantum theories.

###  3.1 Another Argument for 3D-Fundamentalism

P5

If a fundamental theory T explains S while T’ does not, and S should be
explained rather than postulated (other things being equal), then we have
(defeasible) reasons to infer that T is more likely than its alternative to be
the fundamental theory.

P6

3D-Fundamentalism explains the Symmetrization Postulate but 3N-Fundamentalism
does not.

P7

The Symmetrization Postulate should be explained rather than postulated.

C3

We have (defeasible) reasons to infer that 3D-Fundamentalism is more likely to
be the fundamental theory than 3N-Fundamentalism.

The first premise—P5—seems to me highly plausible. When we compare two
fundamental theories, we measure them not just by empirical adequacy and
internal coherence but also by explanatory depth. Say that a theory T provides
a deeper explanation for X than T’ does if T explains X from its axioms and T’
postulates X as an axiom.

For example, if a fundamental physical theory T says that the Symmetrization
Postulate is true (there are two groups of fundamental particles and one group
has symmetric wave functions and the other group has anti-symmetric wave
functions), if (other things being equal) we would like to understand why that
is the case, and if we find out that its alternative S provides an explanation
for that fact, then we have defeasible reasons to favor S over T.

###  3.2 Justifying Premise 6

If we accept P5, then the success of the argument depends on P6 and P7. It is
important to note that P6 is not the unique premise that delivers the
conclusion. There probably are many mathematical explanations that favor
3D-Fundamentalism over 3N-Fundamentalism, including Lorentz symmetry.323232For
example, see Allori (2013), pp. 72-73. (Indeed, I take it to be an open
research question whether there are _good_ mathematical explanations that
support 3N-Fundamentalism. To be sure, what counts as a good explanation and
what counts as something that should be explained can be controversial.)

In any case, P6 focuses on a particular mathematical explanation that arises
from the mathematical study of identical particles and the nature of their
configuration space.333333In writing this section, I am grateful for many
extensive discussions with Sheldon Goldstein and Roderich Tumulka about their
papers on the Symmetrization Postulate. 343434I do not include this
explanation in the “inferences from the dynamics” category, for it concerns
the mathematical construction of the theory.

Roughly speaking, identical particles share the same intrinsic physical
properties, that is, the same qualitative properties recognized by the
physical theory. For example, all electrons can be regarded as identical
(which in this context means _physically indistinguishable_) since they have
the same charge, mass, and spin. (To be sure, their different positions in
space cannot correspond to their intrinsic properties, for if nothing else
they would not be able to move.) Suppose we take the ordinary configuration
space $\mathbb{R}^{3N}$ for N electrons. Then the configuration space is
ordered:

$(x_{1},y_{1},z_{1},x_{2},y_{2},z_{2},...,x_{n},y_{n},z_{n})$

Each point in the configuration space is an ordered 3N-tuple and can be read
as the $x,y,z$ coordinates of electron 1, those of electron 2, those of
electron 3, and all the way up to the $x,y,z$ coordinates of electron N. The
numeral labels provide names (and thus distinguishability) to the electrons.
Therefore, the configuration in which electron 1 and electron 2 are exchanged
will be a different configuration and hence will be represented by a different
point in the configuration space.

However, the distinct points resulting from permutations of the electrons do
not give rise to any real physical differences recognized by the physical
laws. If we were to have the simplest ontology supporting the laws of quantum
mechanics (or indeed any atomistic theory, including the Democritean atomism
and classical mechanics), we should have a physical space representing all and
only the real physical differences, excluding differences resulting from
permutations of identical particles. Therefore, we can introduce the notion of
an unordered configuration space for N particles in 3-dimensional space as:

${}^{N}\mathbb{R}^{3}:=\\{S\subset R^{3}|\text{ cardinality}(S)=N\\}$

So a point in this configuration space would be:

$\\{(x_{1},y_{1},z_{1}),(x_{2},y_{2},z_{2}),...,(x_{n},y_{n},z_{n})\\}$

Notice that the points are not 3N-tuples but sets of N elements, which are
unordered (given the extensionality axiom of ZF set theory). In fact, it is a
set of tuples, but the ordering corresponds to the coordinatization of the
3-dimensional space, which can be ultimately eliminated by using
$\mathbb{E}^{3}$ instead of $\mathbb{R}^{3}$.

Even in classical mechanics, we can use the unordered configuration space
${}^{N}\mathbb{R}^{3}$ (or the relevant unordered phase space) instead of the
ordered configuration space $\mathbb{R}^{3N}$. Such a choice would not lead to
any physical differences, as the two spaces share the same _local_ properties.
They differ, however, in their _global_ topological properties, which become
essential in quantum mechanics. In particular, ${}^{N}\mathbb{R}^{3}$ (like
the circle $S^{1}$) is topologically non-trivial in that it is not simply-
connected, as not every closed loop is contractible to a point, while
$\mathbb{R}^{3N}$ (like the real line $\mathbb{R}^{1}$) is simply connected—a
much more trivial topology. As we will explain below, their global topological
differences allow us to derive the deep and useful principle known as the
Symmetrization Postulate.

Conventionally, a quantum theory with wave functions defined over an ordered
configuration space needs to postulate an additional requirement—the
Symmetrization Postulate—to rule out certain types of particles. It says that
there are only two kinds of particles: symmetric wave functions for bosons and
anti-symmetric wave functions for fermions. The “symmetry” and “anti-symmetry”
refer to the behavior of the wave functions under position exchange. Stated
for particles in 3-dimensions:

Symmetrization Postulate:

There are only two kinds of wave functions:

$\text{(Bosons)
}\qquad\psi_{B}(\textbf{x}_{\sigma(1)},...,\textbf{x}_{\sigma(N)})=\psi_{B}(\textbf{x}_{1},...,\textbf{x}_{N}),$
$\text{(Fermions)
}\quad\psi_{F}(\textbf{x}_{\sigma(1)},...,\textbf{x}_{\sigma(N)})=(-1)^{\sigma}\psi_{F}(\textbf{x}_{1},...,\textbf{x}_{N}),$

where $\sigma$ is a permutation of $\\{1,2,...,N\\}$ in the permutation group
$S_{N}$, $(-1)^{\sigma}$ denotes the sign of $\sigma$,
$\textbf{x}_{i}\in\mathbb{R}^{3}$ for $i=1,2,...,N.$

This is an extremely deep and highly useful principle. To have a more
intuitive grasp of it, we can recall the more familiar Pauli Exclusion
Principle, which says that no two electrons can occupy the same quantum state
(usually characterized by quantum numbers). For one direction of implication,
consider two electrons (fermions) with a wave function that is totally anti-
symmetric under position exchange. It follows that they cannot occupy the same
position, that is:

$\forall
x_{1},x_{2}\in\mathbb{R}^{3},\psi(x_{1},x_{2})=-\psi(x_{2},x_{1})\Longrightarrow\forall
x\in\mathbb{R}^{3},\psi(x,x)=0$

Surprisingly, the Symmetrization Postulate, being deep and useful, emerges as
a result of a beautiful mathematical analysis about the topological properties
of the unordered configuration space.353535For the classic papers in the
mathematical physics literature, see Dowker (1972) and Leinaas and Myrheim
(1977). But Dürr et al. (2006) and Dürr et al. (2007) carry out the
explanation much more thoroughly and successfully in Bohmian mechanics. The
outline in the next paragraph is a summary of their more technical derivation
in the case of scalar wave functions. Here we sketch only an outline of the
derivation; we refer interested readers to the Appendix for more technical
details.

Let us take Bohmian Mechanics (BM) as the background theory, in which there
really are particles with precise trajectories, guided by a universal wave
function. (Let us use scalar-valued wave functions to avoid further
technicalities. We will later return to the question whether such an
explanation is available on the Copenhagen intepretation and GRW theories.) In
BM, the universal wave function evolves according to the Schrödinger equation,
and particles move according to the universal wave function and the guidance
equation. The configuration space for N identical particles in
$\mathbb{R}^{3}$ is ${}^{N}\mathbb{R}^{3}$, not $\mathbb{R}^{3N}$. However,
they are intimately related. In fact, $\mathbb{R}^{3N}$ is what is called the
“universal covering space” of ${}^{N}\mathbb{R}^{3}$. There is a natural
projection mapping from $\mathbb{R}^{3N}$ to ${}^{N}\mathbb{R}^{3}$ that
forgets the ordering of particles. Since the physical configuration lies in
${}^{N}\mathbb{R}^{3}$, the velocity field needs to be defined there, whereas
the wave function can still very well be defined on $\mathbb{R}^{3N}$, as we
can project its velocity field from $\mathbb{R}^{3N}$ to the
${}^{N}\mathbb{R}^{3}$. However, not every velocity field can be so projected.
To ensure that it can , we impose a natural “periodicity condition” on the
wave function on $\mathbb{R}^{3N}$:

$\forall\hat{q}\in\mathbb{R}^{3N},\sigma\in
S_{N},\psi(\sigma\hat{q})=\gamma_{\sigma}\psi(\hat{q}).$

Since the fundamental group $S_{N}$ is a finite group, the topological factor
$\gamma_{\sigma}$ has to be a group character (see Unitarity Theorem in
Appendix). But $S_{N}$ has only two characters: (1) the trivial character
$\gamma_{\sigma}=1$ and (2) the alternating character
$\gamma_{\sigma}=sign(\sigma)=1\text{ or }-1$ depending on whether $\sigma\in
S_{N}$ is an even or an odd permutation. The former corresponds to the
symmetric wave functions of bosons and the latter to the anti-symmetric wave
functions of fermions. Since any other topological factors are banned, we have
ruled out other types of particles such as anyons. This result is equivalent
to the Symmetrization Postulate.

We have seen that given some natural assumptions about identical particles in
$\mathbb{R}^{3}$, the unordered configuration space ${}^{N}\mathbb{R}^{3}$,
and a globally well-defined velocity field, we have very easily arrived at an
explanation for the Symmetrization Postulate. With Bohmian Mechanics in the
background, the derivation is well-motivated at each step. The rigorous
mathematical work is done by Bohmian researchers in two beautiful papers Dürr
et al. (2006) and Dürr et al. (2007), of which the above discussion is a
simplification (of the simple case of scalar-valued wave functions). (For
technical details, please see the Appendix.)

The above explanation might still work in the context of standard textbook
quantum mechanics and the Copenhagen interpretation, according to which we
should not take it seriously that particles are things that have precise
locations and velocities simultaneously. Usually, in this context, we are told
to act as if there are particles and to construct various mathematical objects
from the degrees of freedom of the particles (such as the use of the
configuration space). From a foundational point of view, it is not clear why
the configuration space should be useful in such a theory. But if we play the
usual game of acting _as if_ there are particles, perhaps we can also act _as
if_ they are really identical, _as if_ their states are given by a point in
the unordered configuration space, and _as if_ the above assumptions in the
Bohmian theory are also justified in this theory. Indeed, this seems to be the
implicit attitude taken in two pioneer studies—Dowker (1972) and Leinaas and
Myrheim (1977)—about quantum mechanics for identical particles and unordered
configuration space.

What about other solutions to the measurement problem, such as GRWm (a
spontaneous collapse theory with a mass-density ontology)? Since its
fundamental material ontology consists in a mass-density field (not particles)
in the 3-dimensional space, it is unclear why we should use the configuration
space of particle locations. However, usually the wave function and the mass-
density field are defined with the help of the configuration space. 363636For
example, when defining the mass-density function at a time $t$, we start with
a $|\Psi_{t}|^{2}$-distribution in the particle-location configuration space
$\mathbb{R}^{3N}$, and obtain the value by using the marginal distribution
weighted by the particle masses and summing over all particle degrees of
freedom:  $\forall
x\in\mathbb{R}^{3},m(x,t)=\sum_{i=1}^{N}m_{i}\int_{\mathbb{R}^{3N}}d^{3}x_{1}...d^{3}x_{N}\delta^{3}(x_{i}-x)|\Psi_{t}(x_{1},...,x_{N})|^{2}$
Nevertheless, it seems inconsistent with the spirit of GRWm to act as if there
are fundamental particles or as if the above-mentioned Bohmian assumptions
hold in the theory. In particular, there do not seem to be any compelling
reasons to consider ${}^{N}\mathbb{R}^{3}$, a crucial piece in the above
derivation, unlike in the case of BM. Hence, we do not think that the previous
argument applies smoothly to the case of GRW theories or the Many-Worlds
interpretation with a mass-density ontology.373737We thank an anonymous
referee for helping us make this clear. Unlike the arguments in the previous
sections, the argument we offer here does depend on the specific
interpretation of quantum mechanics, namely Bohmian Mechanics. Whether we can
make similar topological arguments in the context of GRW and MWI would require
thinking more deeply about the structures of their state spaces. (This stands
in contrast with the situation in §1 and §2, where the arguments do not
require a justification of the structure of the configuration space.)

Returning to the construction of the unordered configuration space
${}^{N}\mathbb{R}^{3}$, we observe that it stands in a special relation to
$\mathbb{R}^{3}$, namely, mathematical construction. In this sense, a quantum
system of N-identical-particles in $\mathbb{R}^{3}$ naturally give rise to
${}^{N}\mathbb{R}^{3}$, which in turn naturally give rise to the mathematical
explanation of the Symmetrization Postulate.

However, this explanation is not available for the quantum theory on
3N-Fundamentalism. Although in the above explanation we use $\mathbb{R}^{3N}$
in defining the wave function (as it is the universal covering space of
${}^{N}\mathbb{R}^{3}$), simply starting with $\mathbb{R}^{3N}$ gets us
nowhere near to the characters of the permutation group. Starting with a wave
function and a “marvelous point” (the only fundamental particle) on
$\mathbb{R}^{3N}$, we do not have any motivation to consider
${}^{N}\mathbb{R}^{3}$, the natural configuration space for $N$ particles in
$\mathbb{R}^{3}$. Consequently, we cannot make use of its explanatory power in
the derivation of the Symmetrization Postulate. (That is because, given a
particle moving in $\mathbb{R}^{3N}$, we do not have any motivation to use the
covering-space construction to derive the topological factors of the wave
function, and we do not have an explanation for the Symmetrization Postulate.)

Therefore, we have justified P6—the second premise in the argument.

###  3.3 Deep versus Shallow Explanations

What about P7, the idea that the Symmetrization Postulate should be explained
rather than postulated?383838P7 shows the difference between my argument here
and Maudlin’s argument in Maudlin (2013) pp.140-42. The difference in our
arguments lies in our use of different explananda. Maudlin suggests that for,
say, an 8-particle universe, the natural configuration space is
${}^{8}\mathbb{R}^{3}$ instead of $\mathbb{R}^{24}$, but that cries out for an
explanation: why is this particular space ${}^{8}\mathbb{R}^{3}$ the
fundamental space instead of some other space with a different factorization
of 24 (such as ${}^{4}\mathbb{R}^{6}$)? This is an insightful argument, but I
do not think it would _convince_ many defenders of 3N-Fundamentalism, as on
their view the fundamental space does not need further explanation. In
contrast, our argument is quite different: we are not asking for an
explanation of the fundamental space, but for an explanation of the
Symmetrization Postulate, something that everyone agrees to be a deep and
puzzling symmetry in quantum mechanics. Thus, I include Pauli’s quote as it
beautifully conveys the deep puzzlement.  Here are some compelling reasons to
endorse it:

  * •

Its form is very different from the other laws of QM such as the Schrödinger
equation. It is a direct restriction on the possible quantum states of systems
with indistinguishable particles.

  * •

It has high explanatory power. The Symmetrization Postulate explains Pauli’s
Exclusion Principle and the Periodic Table. It also explains why there are
exactly two kinds of particles in QM.

  * •

In the words of Pauli, at the occasion of his 1946 Nobel Lecture, as quoted in
the epigraph: “Already in my original paper I stressed the circumstance that I
was unable to give a logical reason for the exclusion principle or to deduce
it from more general assumptions. I had always the feeling, and I still have
it today, that this is a deficiency.”393939Pauli et al. (1994), p.171.

Granted, one might still deny P7 at this point. But it seems to me (and to
many working physicists) that P7 is much more likely to be true than its
denial. The above reasons suggest that an explanation of P7 would point us to
something deep in the physical nature of the world. It is true that in an
ultimately axiomatic theory of physics, the explanatory relations can be
reversed (just as logical theories can be axiomatized in different ways
corresponding to different explanatory relations). However, I think it is
reasonable to believe that the final quantum theory ought not to postulate the
Symmetrization Postulate as fundamental but rather as something that follows
from simpler axioms.

###  3.4 What about ${}^{N}\mathbb{R}^{3}$-Fundamentalism?

In this essay, I have assumed that the 3N-Fundamentalist is really a
$\mathbb{R}^{3N}$-Fundamentalist. She can of course modify her thesis and
believe instead in a fundamentally multiply-connected physical space,
${}^{N}\mathbb{R}^{3}$. The 3D-Fundamentalist can respond by charging her
opponent’s move as _ad hoc_ or unnatural, since ${}^{N}\mathbb{R}^{3}$ is
“clearly” about $N$ identical particles in $\mathbb{R}^{3}$. But similar
criticisms have not prevented anyone from taking $\mathbb{R}^{3N}$ as
fundamental, even though in some sense it is “clearly” about $N$ particles in
$\mathbb{R}^{3}$.

3D-Fundamentalism | 3N-Fundamentalism
---|---
$\mathbb{R}^{3}$+distinguishable particles |
| $\mathbb{R}^{3N}$
|
$\mathbb{R}^{3}$+indistinguishable particles | ${}^{N}\mathbb{R}^{3}$

There is, however, another argument against the 3N-Fundamentalist’s maneuver
that is even less controversial. My main goal in this paper has been to
evaluate the evidential support for 3D-Fundamentalism and 3N-Fundamentalism.
Given 3N-Fundamentalism, the fundamental physical space has two main
possibilities: (1) a simply-connected space $\mathbb{R}^{3N}$ and (2) a
multiply-connected space that is a quotient space of $\mathbb{R}^{3N}$ by the
permutation group of $k$ objects where $k$ divides $3N$. I believe that (1) is
intrinsically more likely than (2). But suppose we grant them equal
probability. Still, (2) comes in many further possibilities, and
${}^{N}\mathbb{R}^{3}$ is intrinsically as likely as any other quotient space.
So we should assign equal probability to each possibility in (2). Given a
large $N$, ${}^{N}\mathbb{R}^{3}$ will receive a very small probability
compared to that of $\mathbb{R}^{3N}$.404040Thanks to Gordon Belot for helpful
discussions here. In the table above, the ratio of the two areas for
$\mathbb{R}^{3N}$ and ${}^{N}\mathbb{R}^{3}$ roughly corresponds to the
relative confidence that we have in them.

Given 3D-Fundamentalism, there is only one main candidate for the fundamental
physical space–$\mathbb{R}^{3}$ and two main candidates for the configuration
space–$\mathbb{R}^{3N}$ and ${}^{N}\mathbb{R}^{3}$. Although an ontology of
identical particles is simpler than that of non-identical particles, for the
sake of the argument, we can give these two possibilities equal weight. Hence,
in the table above, the two candidates divide the area roughly in half.

If we take as a datum that the Symmetrization Postulate is true, assume that
${}^{N}\mathbb{R}^{3}$ is the correct route to explain it, and we allow
explanatory strength to come in degrees, then overall (considering all
versions) 3D-Fundamentalism explains it better than 3N-Fundamentalism does. So
although both 3D-Fundamentalism and 3N-Fundamentalism contain versions of them
that explain the Symmetrization Postulate, 3D-Fundamentalism is better
supported by the successful explanation. Therefore, the evidence “disconfirms”
3N-Fundamentalism over 3D-Fundamentalism. (That is, if we assume a standard
way of thinking about update and Bayesian confirmation theory.414141In a
different context (the problem of evil) at the Rutgers Religious Epistemology
Workshop in May 2014, Lara Buchak discussed a different approach to rational
updating—updating by conditionals. It is up to the defender of
3N-Fundamentalism to develop and apply that approach here. Even if that were
to succeed, however, I think it would be a significant concession that
3N-Fundametnalism is disconfirmed in the “Bayesian” sense.) In fact, we can
reformulate the original argument by replacing P6 with a weaker premise:

P6*

3D-Fundamentalism explains the Symmetrization Postulate significantly better
than 3N-Fundamentalism.

Therefore, even if ${}^{N}\mathbb{R}^{3}$-Fundamentalism is an option for the
3N-Fundamentalism, her view is still not well supported by the explanation.
The above argument fleshes out our intuition that the 3N-Fundamentalist
maneuver is _ad hoc_ or unnatural.

However, this should reflect only our current state of knowledge and should
not be taken as the final word on the issue. Indeed, there are several ways
that future research can go:

  1. 1.

We discover more mathematical explanations only available to
3D-Fundamentalism, which gives us more reason to favor 3D-Fundamentalism over
3N-Fundamentalism.

  2. 2.

We discover some mathematical explanations only available to
3N-Fundamentalism, which restores or reverses the evidential balance between
3D-Fundamentalism and 3N-Fundamentalism.

  3. 3.

We discover that there exists a mathematical explanation of the Symmetrization
Postulate from other resources in 3N-Fundamentalism that is independent from
the topological considerations as discussed above, which restores the
evidential balance between 3D-Fundamentalism and 3N-Fundamentalism.

Considerations | 3D-Fundamentalism | 3N-Fundamentalism
---|---|---
Dynamics | 0/+ | +
Manifest Image | + | -/0
Explanatory Depth | + | -/0
Table 1: Summary of pros and cons; where “+” means “strongly in favor,” “-”
means “strongly against,” and “0” means “roughly neutral.”

##  4 Conclusion

Based on the above evaluation of the three kinds of evidence (see Table 1), I
conclude that, given our current knowledge, it is more likely that the
fundamental physical space in quantum mechanics is 3-dimensional rather than
3N-dimensional. However, as noted above, there are future directions of
research where the 3N-Fundamentalists can restore or even reverse the
evidential balance.

Our debate here is related to the discussions about theoretical equivalence;
we observe the following:

  1. 1.

3N-Fundamentalism and 3D-Fundamentalism are highly equivalent in theoretical
structure–the mathematics used is exactly the same;

  2. 2.

The debate between 3N-Fundamentalists and 3D-Fundamentalists almost reaches a
stalemate and hence might be considered as non-substantive;

  3. 3.

An important new piece of evidence lies not in the inferences from the
dynamics or ordinary experiences but in the mathematical symmetries in the
wave function. What breaks the tie is the fact that 3N-Fundamentalism and
3D-Fundamentalism are explanatorily inequivalent, and the latter explains the
Symmetrization Postulate better than the former.

Therefore, the discussion about the wave function provides another useful case
for the ongoing debate about theoretical equivalence and structure. The
connection should prove fruitful for future research.424242Thanks to David
Schroeren for discussing this last point with me.

##  5 Appendix: A Topological Explanation of the Symmetrization Postulate

Here, we provide the technical details of our argument in §3.2 by following
the mathematical derivations in Dürr et al. (2006) and Dürr et al. (2007)
regarding the case of scalar-valued wave functions. We omit the case of
vector-valued wave functions as it is mathematically more complicated but
conceptually similar to the scalar case.

Our goal is to arrive at all possible Bohmian dynamics for $N$ identical
particles moving in $\mathbb{R}^{3}$, guided by a scalar wave function. We
want to show that, given some natural assumptions, there are only two possible
dynamics, corresponding to the symmetric wave functions for bosons and the
anti-symmetric wave functions for fermions. We will use the covering-space
construction and examine the permissible topological factors.434343This
argument, of course, is not intended as a mathematical demonstration, as we
appeal to considerations of naturalness and simplicity. But these assumptions
are relatively weak, and they are guided by strong intuitions of the
practicing mathematical physicists. Even better, in our case of
${}^{N}\mathbb{R}^{3}$, we can prove the Unitarity Theorem—that the
topological factor in the periodicity condition has to be a character of the
fundamental group.

The natural configuration space for $N$ identical particles in
$\mathbb{R}^{3}$ is:

${}^{N}\mathbb{R}^{3}:=\\{S\subset R^{3}|\text{ cardinality}(S)=N\\}$

Since it is a multiply-connected topological space, we follow the usual
covering-space construction to define the dynamics on its universal covering
space and project it down to ${}^{N}\mathbb{R}^{3}$. Its universal covering
space is, unsurprisingly, the simply-connected $\mathbb{R}^{3N}$. Let
$Cov(\mathbb{R}^{3N},^{N}\mathbb{R}^{3})$ denote the covering group of the
base space ${}^{N}\mathbb{R}^{3}$ and its universal covering space
$\mathbb{R}^{3N}$. Given a map
$\gamma:Cov(\mathbb{R}^{3N},^{N}\mathbb{R}^{3})\rightarrow\mathbb{C}$, we
define a wave function on this space:
$\psi:\mathbb{R}^{3N}\rightarrow\mathbb{C}$ with the following periodicity
condition associated with $\gamma$:

$\forall\hat{q}\in\mathbb{R}^{3N},\forall\sigma\in
Cov(\mathbb{R}^{3N},^{N}\mathbb{R}^{3}),\psi(\sigma\hat{q})=\gamma_{\sigma}\psi(\hat{q}).$

First, we show that if the wave function does not identically vanish (which is
a natural assumption), the topological factor $\gamma$ is a representation.
Given any $\sigma_{1},\sigma_{1}\in Cov(\mathbb{R}^{3N},^{N}\mathbb{R}^{3})$:

$\gamma_{\sigma_{1}\sigma_{2}}\psi(\hat{q})=\psi(\sigma_{1}\sigma_{2}\hat{q})=\gamma_{\sigma_{1}}\psi(\sigma_{2}\hat{q})=\gamma_{\sigma_{1}}\gamma_{\sigma_{2}}\psi(\hat{q}).$

Hence, $\gamma_{\sigma_{1}\sigma_{2}}=\gamma_{\sigma_{1}}\gamma_{\sigma_{2}}.$
Therefore, $\gamma$ is a structure-preserving map and a representation of the
covering group.

It is a well-known fact that the covering group is canonically isomorphic to
the fundamental group $\pi_{1}(Q,q)$, where $Q$ is the base space. In our
case, the fundamental group $\pi_{1}(^{N}\mathbb{R}^{3},q)$ is $S_{N}$, the
group of permutations of $N$ objects. It has only two characters: (1) the
trivial character $\gamma_{\sigma}=1$ and (2) the alternating character
$\gamma_{\sigma}=sign(\sigma)=1\text{ or }-1$ depending on whether $\sigma\in
S_{N}$ is an even or an odd permutation. The former corresponds to the
symmetric wave functions of bosons and the latter to the anti-symmetric wave
functions of fermions. If we can justify the use of the periodicity condition
and a ban on any other topological factors in the periodicity condition, we
would be able to rule out other types of particles such as anyons. This result
would be equivalent to the Symmetrization Postulate.

The periodicity condition is the most natural condition to require if we want
a projectable velocity field given by the wave function on the covering space.
The periodicity condition implies that
$\nabla\psi(\sigma\hat{q})=\gamma_{\sigma}\sigma^{*}\nabla\psi(\hat{q}),$
where $\sigma^{*}$ is the push-forward action of $\sigma$ on tangent vectors.
Hence, if we define the velocity field on $\mathbb{R}^{3N}$ in the usual way:

$\hat{v}^{\psi}(\hat{q}):=\hbar\text{Im}\frac{\nabla\psi}{\psi}(\hat{q}),$

then it is related to other levels of the covering space:

$\hat{v}^{\psi}(\sigma\hat{q})=\sigma^{*}\hat{v}^{\psi}(\hat{q}).$

This makes $\hat{v}$ projectable to a well-defined velocity field on the base
space ${}^{N}\mathbb{R}^{3}$:

$v^{\psi}(q)=\pi^{*}\hat{v}^{\psi}(\hat{q}),$

where $\hat{q}$ is an arbitrary point in $\mathbb{R}^{3N}$ that projects down
to $q$ via the projection map $\pi$.

Moreover, the periodicity condition is natural because it is preserved by the
Schrödinger evolution. Therefore, given an initial point $q_{0}$ in the
unordered configuration space ${}^{N}\mathbb{R}^{3}$, we can choose an
arbitrary point $\hat{q_{0}}$ in the ordered configuration space
$\mathbb{R}^{3N}$ that projects to $q_{0}$ via $\pi$, evolve $\hat{q_{0}}$ by
the usual guidance equation in $\mathbb{R}^{3N}$ until time $t$, and get the
final point $q_{t}=\pi(\hat{q_{t}})$. The final point
$q_{t}\in^{N}\mathbb{R}^{3}$ is independent of the choice of the initial
$\hat{q_{0}}\in\mathbb{R}^{3N}$.

We explain why it is crucial for the topological factor to be not only a
representation of the fundamental group but also unitary, as it is required to
ensure that the probability distribution is equivariant. Given the periodicity
condition, we have

$|\psi(\sigma\hat{q})|^{2}=|\gamma_{\sigma}|^{2}|\psi(\hat{q})|^{2}.$

To make $|\psi(\hat{q})|^{2}$ projectable to a function on
${}^{N}\mathbb{R}^{3}$, we require that $\forall\sigma\in
S_{N},|\gamma_{\sigma}|^{2}=1$. This is equivalent to $\gamma$ being a
character (a unitary representation) for the fundamental group. Given the
Schrödinger equation on $\mathbb{R}^{3N}$ and the projection of
$|\psi(\hat{q})|^{2}$ to $|\psi|^{2}(q),$ we have:

$\frac{\partial|\psi_{t}|^{2}(q)}{\partial
t}=-\nabla\cdot(|\psi_{t}|^{2}(q)v^{\psi_{t}}(q)).$

Compare this with the transport equation for a probability density $\rho$ on
${}^{N}\mathbb{R}^{3}$:

$\frac{\partial\rho_{t}(q)}{\partial
t}=-\nabla\cdot(\rho_{t}(q)v^{\psi_{t}}(q)).$

Therefore, if $\rho_{t_{0}}(q)=|\psi_{t_{0}}|^{2}(q)$ at the initial time
$t_{0}$, then $\rho_{t}(q)=|\psi_{t}|^{2}(q)$ at all later times. We have
arrived at equivariance.

The above argument works for the general case of multiply-connected spaces as
well as the particular case of ${}^{N}\mathbb{R}^{3}$. In our case of
${}^{N}\mathbb{R}^{3}$, we can prove the following simple theorem that the
topological factor $\gamma$ must be a unitary representation, i.e. a group
character.

###### Unitarity Theorem

Let $\sigma\in Cov(\mathbb{R}^{3N},^{N}\mathbb{R}^{3})$,
$\gamma:Cov(\mathbb{R}^{3N},^{N}\mathbb{R}^{3})\rightarrow\mathbb{C}$. If
$\gamma_{\sigma}\neq 0$ and
$\gamma_{\sigma_{1}\sigma_{2}}=\gamma_{\sigma_{1}}\gamma_{\sigma_{2}},$
$\forall\sigma_{1},\sigma_{2}\in Cov(\mathbb{R}^{3N},^{N}\mathbb{R}^{3})$,
then $|\gamma_{\sigma}|=1$.

  * Proof

The fundamental group of ${}^{N}\mathbb{R}^{3}$ is the permutation group
$S_{N}$, which has $N!$ elements. We obtain that $\gamma_{Id}=1$, because
$\gamma_{\sigma}=\gamma_{(\sigma\ast Id)}=\gamma_{\sigma}\gamma_{Id}$ and
$\gamma_{\sigma}\neq 0$. It is a general fact that in a finite group with $k$
elements, every element $\sigma$ satisfies $\sigma^{k}=Id.$ Therefore,
rewriting $\gamma_{\sigma}=Re^{i\theta}$, we have
$1e^{0i}=1=\gamma_{Id}=\gamma_{\sigma^{k}}=(\gamma_{\sigma})^{k}=R^{k}e^{ik\theta}$.
So we have: $|\gamma_{\sigma}|=1$, which makes $\gamma_{\sigma}$ a unitary
representation of the covering group, which is a character of the fundamental
group. $\square$

Therefore, the periodicity condition associated with the topological factor:

$\forall\hat{q}\in\mathbb{R}^{3N},\forall\sigma\in
S_{N},\psi(\sigma\hat{q})=\gamma_{\sigma}\psi(\hat{q})$

is a highly natural and simple condition that guarantees well-defined dynamics
on ${}^{N}\mathbb{R}^{3}$, and the topological factors are the characters of
the fundamental group.444444In our particular case of ${}^{N}\mathbb{R}^{3}$,
we have the Unitarity Theorem to explain why the topological factors have to
be characters of the fundamental group. In the general case, even without such
a theorem there are still many good reasons why we should restrict the
topological factors to the group characters. See Dürr et al. (2007) §9 “The
Character Quantization Principle.” In the case of ${}^{N}\mathbb{R}^{3}$, the
fundamental group is the permutation group of $N$ objects: $S_{N}$. Recall
that it has only two characters: (1) the trivial character $\gamma_{\sigma}=1$
and (2) the alternating character $\gamma_{\sigma}=sign(\sigma)=1\text{ or
}-1$ depending on whether $\sigma\in S_{N}$ is an even or an odd permutation.
This leads to two possible dynamics corresponding to the symmetric and the
anti-symmetric wave functions (and no more):

$\text{(Bosons)
}\qquad\psi_{B}(\textbf{x}_{\sigma(1)},...,\textbf{x}_{\sigma(N)})=\psi_{B}(\textbf{x}_{1},...,\textbf{x}_{N}),$
$\text{(Fermions)
}\quad\psi_{F}(\textbf{x}_{\sigma(1)},...,\textbf{x}_{\sigma(N)})=(-1)^{\sigma}\psi_{F}(\textbf{x}_{1},...,\textbf{x}_{N}),$

where $\sigma$ is a permutation of $\\{1,2,...,N\\}$ in the permutation group
$S_{N}$, $(-1)^{\sigma}$ denotes the sign of $\sigma$,
$\textbf{x}_{i}\in\mathbb{R}^{3}$ for $i=1,2,...,N.$ Therefore, we have
arrived at the statement of the Symmetrization Postulate for the 3-dimensional
physical space. Interestingly, the same argument would predict that there are
more possibilities than fermions and bosons if we go to a smaller space. For
$N$ identical particles in a 2-dimensional space, there is the additional
possibility of fractional statistics, corresponding to anyons.454545Perhaps
this is yet another reason why the 3-dimensionality of the physical space is a
distinguished feature of reality.

## Acknowledgement

I am grateful for insightful comments from the anonymous reviewers at _The
Journal of Philosophy_ that led to not only significant improvements
throughout the paper but also the use of Pauli’s quote in the epigraph, and
for helpful discussions with David Albert, Barry Loewer, Roderich Tumulka,
Sheldon Goldstein, Nino Zanghì, Tim Maudlin, Gordon Belot, Jonathan Schaffer,
Michael Townsen Hicks, Jill North, Isaac Wilhelm, Thad Roberts, Bradley
Monton, audiences at the Rutgers Metaphysics Reading Group, the 2015 Black
Forest Summer School in Philosophy of Physics (Saig), a joint meeting with the
Beyond Spacetime Project at University of Illinois–Chicago and University of
Geneva, the 2015 Annual Meeting of the Society for the Metaphysics of Science
(Newark), where Vishnya Maudlin provided insightful comments that led to much
rethinking about §1, and a symposium discussion at the 2016 Central APA
Meeting (Chicago), where Nick Huggett provided eight pages of helpful comments
that led to many improvements in §2 and §3.

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