#  Realism about the Wave Function

Eddy Keming Chen Department of Philosophy MC 0119, University of California,
San Diego, 9500 Gilman Dr, La Jolla, CA 92093-0119. Website:
www.eddykemingchen.net. Email<EMAIL_ADDRESS>
(Forthcoming in _Philosophy Compass_
Penultimate version of June 12, 2019)

###### Abstract

A century after the discovery of quantum mechanics, the meaning of quantum
mechanics still remains elusive. This is largely due to the puzzling nature of
the wave function, the central object in quantum mechanics. If we are realists
about quantum mechanics, how should we understand the wave function? What does
it represent? What is its physical meaning? Answering these questions would
improve our understanding of what it means to be a realist about quantum
mechanics. In this survey article, I review and compare several realist
interpretations of the wave function. They fall into three categories:
ontological interpretations, nomological interpretations, and the _sui
generis_ interpretation. For simplicity, I will focus on non-relativistic
quantum mechanics.

Keywords: quantum mechanics, wave function, quantum state of the universe,
scientific realism, measurement problem, configuration space realism, Hilbert
space realism, multi-field, spacetime state realism, laws of nature

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 The Wave Function
    2. 2.2 The Quantum Measurement Problem
  3. 3 Ontological Interpretations
    1. 3.1 A Field on a High-Dimensional Space
    2. 3.2 A Multi-field on Physical Space
    3. 3.3 Properties of Physical Systems
    4. 3.4 A Vector in Hilbert Space
  4. 4 Nomological Interpretations
    1. 4.1 Strong Nomological Interpretations
    2. 4.2 Weak Nomological Interpretations
  5. 5 The _Sui Generis_ Interpretation
  6. 6 Conclusion

##  1 Introduction

Quantum mechanics is one of the most successful physical theories to date. Not
only has it been confirmed through a wide range of observations and
experiments, but it also has led to technological advances of a breathtaking
scale. From electronics and optics to computing, the applications of quantum
mechanics are ubiquitous in our lives.

As much as it has given rise to technological innovations, the meaning of
quantum mechanics remains elusive. Many curious features of quantum mechanics,
such as entanglement, non-locality, and randomness, are taken to be _prima
facie_ challenges for a clear understanding of quantum mechanics. These
puzzles are related to the _wave function_ , the central object in quantum
mechanics. Understanding the meaning of quantum mechanics seems to require a
good understanding of the meaning of the wave function.

What does the wave function represent? That is the main concern of this survey
article. The answer to that question is complicated by the fact that the wave
function does not look like anything familiar. It is a function defined on a
vastly high-dimensional space, with values in complex numbers, and unique only
up to an “overall phase.” Nevertheless, we have devised many ways of using
wave functions in making predictions and explaining phenomena. We use wave
functions to calculate the probabilities of microscopic and macroscopic
behaviors of physical systems. These led to the successful explanations of the
double-slit experiment, the Stern-Gerlach experiment, and the stability of the
hydrogen atom. The wave function is indispensable for making these
predictions. However, the predictions are probabilistic. (More on this later.)

Roughly speaking, there are three main views about the wave function:

Instrumentalism:

The wave function is merely an instrument for making empirically adequate
predictions.

Epistemicism:

The wave function merely represents the observer’s uncertainty of the physical
situation.111The recently published theorem of [68] shows that a certain class
of epistemic interpretations of the wave function are incompatible with the
empirical facts.

Realism:

The wave function represents something objective and mind-independent.

In this article, I focus on the realist interpretations of the wave function.
They seem to be the most interesting and promising ways of understanding
quantum mechanics.

Let me make four remarks. First, the meaning of the wave function is related
to solutions to the quantum measurement problem. Hence, we will start in §2
with an introduction to that problem, along with some mathematical
preliminaries. Second, I left the definition of _realism_ open-ended. That is
because we will consider proposals for specific versions of realism about the
wave function. The proposals are grouped into three categories: ontological
interpretations (§3), nomological interpretations (§4), and the _sui generis_
interpretation (§5). Third, because of the prevalence of quantum entanglement,
“the wave function” should be understood to refer to the wave function of the
universe, or the universal wave function. The wave functions of the subsystems
are thought to be derivative of the universal one. Fourth, for simplicity, I
will focus on non-relativistic versions of quantum mechanics.222For
complications that arise in the relativistic theories, see [60].

The issues taken up here are continuous with the general question about how to
interpret physical theories. They offer concrete case studies for scientific
realism, and they might be useful for philosophers of science, metaphysicians,
and anyone with an interest in understanding quantum mechanics.

##  2 Background

In this section, we will review some basic facts about the wave function and
its connection to probabilistic predictions. We will then consider the quantum
measurement problem and three realist theories that solve it. The upshot is
that the wave function occupies a central place in their descriptions of
physical reality.

###  2.1 The Wave Function

It will be useful to have a brief review of classical mechanics. To describe a
classical mechanical system of $N$ particles, we can specify the (three-
dimensional) position $\boldsymbol{q_{i}}$ and the (three-dimensional)
momentum $\boldsymbol{p_{i}}$ of each particle in physical space (represented
by $\mathbb{R}^{3}$). We can represent the classical state of an $N$-particle
system in terms of $6N$ numbers, $3N$ for positions and $3N$ for momenta. The
classical state can also be represented as a point in an abstract state space
called the _phase space_ $\mathbb{R}^{6N}$. Once we specify the forces (or
interactions) among the particles, they evolve deterministically, by the
Hamiltonian equations of motion:

$\frac{\partial\boldsymbol{q_{i}}}{\partial t}=\frac{\partial
H}{\partial\boldsymbol{p_{i}}}\text{ ,
}\frac{\partial\boldsymbol{p_{i}}}{\partial t}=-\frac{\partial
H}{\partial\boldsymbol{q_{i}}},$ (1)

where $H$ stands for the Hamiltonian function on the phase space, which is a
short-hand notation that encodes classical interactions such as Newtonian
gravitational potential and Coulomb electric potential. The Hamiltonian
equations are differential equations, and the changes in the particles are
obtained from taking suitable derivatives of $H$. In this sense, $H$ is the
generator of motion. For every point in phase space, $H$ generates a curve
starting from that point. In other words, for every initial condition of the
$N$ particle system, $H$ determines the future trajectories of the particles.

Now let us introduce the quantum mechanical way of describing a system of $N$
“particles.”333In some ways of thinking about quantum mechanics, particles are
not fundamental. Hence the quotation marks.  Instead of describing it in terms
of the positions and the momenta of $N$ particles, we use a wave function for
the system. The wave function represents the _quantum state_ of the system. In
the position representation, the wave function, denoted by $\psi(q)$, is a
particular kind of function from configuration space $\mathbb{R}^{3N}$ to
complex numbers $\mathbb{C}$. Let us elaborate on this definition:

  * •

Domain: the domain of the wave function $\psi$ is $\mathbb{R}^{3N}$, or $N$
copies of physical space $\mathbb{R}^{3}$. $N$ is the total number of
particles in the system. When $N$ is large, $\mathbb{R}^{3N}$ is vastly high-
dimensional. Each point in $\mathbb{R}^{3N}$ is an $N$-tuple
$(\boldsymbol{q_{1}},...,\boldsymbol{q_{N}})$. Each $\boldsymbol{q_{i}}$
corresponds to particle $i$’s position in physical space $\mathbb{R}^{3}$.
Hence, the $N$-tuple lists the positions of $N$ particles. We use a point in
$\mathbb{R}^{3N}$ to represent a particular configuration (arrangement) of $N$
particles in $\mathbb{R}^{3}$. Hence, $\mathbb{R}^{3N}$ is called the
configuration space.444This is the ordered configuration space, in which a
permutation of the particle labels creates a different configuration. If the
particles are indistinguishable, then it is more natural to use the _unordered
configuration space_ , ${}^{N}\mathbb{R}^{3}$. This has implications for the
nature of the wave function. See [21] and the references therein. The wave
function $\psi(\boldsymbol{q_{1}},...,\boldsymbol{q_{N}})$ is a function whose
domain is the configuration space, which is vastly high-dimensional when the
system has many particles.

  * •

Range: the range of the wave function $\psi$, in the simplest case, is the
field of complex numbers $\mathbb{C}$. A complex number has the form $a+bi$,
where $i=\sqrt{-1}$; in polar form, it is $Re^{i\theta}$, where $R$ is the
amplitude and $\theta$ is the phase.555If we include spinorial degrees of
freedom, the range is $\mathbb{C}^{k}$. We set spins aside in this paper.

  * •

Restrictions: the wave function is a particular kind of function from
$\mathbb{R}^{3N}$ to $\mathbb{C}$. It has to be a “nice” function that we can
take certain operations of integration and differentiation.666It has to be
“square-integrable.” That is, if we take the square of the amplitude of the
wave function value at every point, and integrate over the entire
configuration space, we will get a finite value. This is to ensure that we can
normalize the squared value of the wave function to $1$ so that it has
meaningful connections to probabilities. To ensure that we can take suitable
derivatives on the wave function, we often also require the wave functions to
be sufficiently _smooth_.

  * •

Abstract state space: each wave function describes a quantum state of the
system. The space of all possible quantum states is called the _state space_
of quantum mechanics. The state space will include all possible wave functions
for the system, that is, all the “nice” functions from configuration space
$\mathbb{R}^{3N}$ to complex numbers $\mathbb{C}$. Hilbert space is the
abstract mathematical space that we use to describe such a state space.
Hilbert space can be a high-dimensional vector space, in which each wave
function is represented as a vector.

In classical mechanics, the state of a system is represented by the positions
and the momenta of $N$ particles (a point in phase space) that change
deterministically according to (1). If the wave function represents the
quantum state of a system at a time, how does it change over time? It obeys
another differential equation called the _Schrödinger equation_ :

$i\hbar\frac{\partial\psi}{\partial t}=\hat{H}\psi,$ (2)

where $i$ is the complex number $\sqrt{-1}$, $\hbar$ is the Planck constant
divided by $2\pi$, and $\hat{H}$ is the Hamiltonian operator that encodes the
energy and fundamental interactions in nature. It is also deterministic: given
any vector in Hilbert space, the Schrödinger equation (2) produces a
determinate curve in Hilbert space. Another feature of (2) is that it is
linear: if $\psi_{1}$ and $\psi_{2}$ are solutions to the equation, then their
linear combinations are also solutions to the equation. A surprising
consequence of linearity is that, in the Schrödinger’s cat thought experiment,
the system can be in a “superposition” of two macroscopically distinct states:

$\Psi=\frac{1}{\sqrt{2}}\psi_{alive}+\frac{1}{\sqrt{2}}\psi_{dead}$ (3)

A cat in this quantum state is not alive, and it is not dead. The linear
Schrödinger equation (2) ensures that the wave function of the system will not
change into $\psi_{alive}$ (the cat is alive and the atom has not decayed) or
$\psi_{dead}$ (the cat is dead and the atom has decayed). Thus, the
Schrödinger equation does not determine a unique experimental outcome. To
resolve this, textbook quantum mechanics supplements the Schrödinger equation
with additional collapse postulates. Whenever we open the box and “observe”
the cat, the system will suddenly change (collapse) into one of the two
states: $\psi_{alive}$ or $\psi_{dead}$. An important role of the wave
function is determining the probabilities of experimental outcomes, which are
taken to be the results of wave function collapses. For example, the
probability of finding the system in any set of configurations is given by the
Born rule:

$P(q\in A)=\int_{A}|\psi(q)|^{2}dq,$ (4)

where $A$ is a set of points in configuration space, $|\psi(q)|^{2}$ is the
squared amplitude of the wave function, and $dq$ is the Lebesgue measure on
$\mathbb{R}^{3N}$. In the cat example, the probability of finding the cat to
be alive is equal to $\frac{1}{2}$, since
$\int|\frac{1}{\sqrt{2}}\psi_{alive}|^{2}+0=\frac{1}{2}$. The Born rule has
the consequence that wave functions that differ only by an _overall phase_
(multiplied by a complex number $e^{i\theta}$, where $\theta\in[0,2\pi]$) will
give rise to the same observable phenomena
($|\psi|^{2}=|e^{i\theta}\psi|^{2}$). That is called the _overall phase
symmetry_ , which motivates the common view that two wave functions that
differ by an overall phase represent the same quantum state.

###  2.2 The Quantum Measurement Problem

Notwithstanding the empirical success of quantum mechanics, the collapse
postulates introduce a host of difficulties. If the wave function (of the
system and the measurement device) obeys the Schrödinger equation, how can it
also obey the collapse postulates that contradict the linearity of the
Schrödinger equation? But if the wave function does not collapse, how can we
obtain unique experimental outcomes? In short, we have the _quantum
measurement problem_ :

  * (P1)

The wave function is the complete description of the physical system.

  * (P2)

The wave function always obeys the Schrödinger equation.

  * (P3)

Every experiment has a unique outcome.

Each of these three propositions is, on its own, plausible. However, together
they lead to a contradiction. To see the contradiction, let us apply them to
Schrödinger’s cat thought experiment. If P1 is true, the system is completely
described by (3). If P2 is true, the wave function never collapses into one of
the definite states. If P3 is true, the cat is nonetheless in one of the
definite states—either alive or dead.777For a more thorough discussion about
the quantum measurement problem, see [59], [1], and [12].

Since P1—P3 are inconsistent, at least one of them is false. Rejecting P1 or
P2 would require us to develop alternative theories of quantum mechanics,
since we would need to find additional variables omitted by the wave function,
or we would need to modify the Schrödinger equation. Rejecting P3 would lead
to major revisions of our assumptions about the world. There are three main
“interpretations” of quantum mechanics that carry out such strategies. They
all contain significant revisions of quantum mechanics, so we should call them
realist theories of quantum mechanics instead of interpretations.

First, the de Broglie-Bohm theory, or Bohmian mechanics (BM), rejects P1.
According to BM, the wave function is not the complete description of the
physical system. There are actual particles with precise positions in physical
space. The wave function still obeys the Schrödinger equation. But the wave
function also determines the velocity of the particles according to the
_guidance equation_.888 The particles move according to the guidance equation:
$\frac{dQ_{i}}{dt}=\frac{\hbar}{m_{i}}\text{Im}\frac{\nabla_{i}\psi}{\psi},$
(5) where $Q_{i}$ and $m_{i}$ are the position and mass of particle $i$, Im
means taking the imaginary part, and $\nabla_{i}$ means taking the gradient
with respect to the $i$-th particle. The particles are initially distributed
according to the Born rule, and their distribution will always agree with the
Born rule because of the mathematical properties of the Schrödinger equation
and the guidance equation. In the cat example, the cat is made out of
particles in physical space. There is always a determinate configuration of
particles, so the cat is either alive or dead. The probabilities of quantum
mechanics become (more or less) epistemic uncertainties over initial particle
configurations.999 For a survey of BM, see [39]; for the original paper, see
[16]; for a modern version, see [27].

Second, the Ghirardi-Rimini-Weber theories of spontaneous collapse (GRW)
reject P2. According to GRW theories, the wave function $\psi_{t}$ does not
always obey the Schrödinger equation. It undergoes spontaneous collapses with
a fixed rate per particle per unit time. In the cat experiment, given the vast
number of particles in the system, it will quickly collapse into a determinate
state in which the cat is either alive or dead. Collapses are represented by
Gaussian functions with a fixed width in physical space. Due to entanglement,
collapses on a single particle has the effect that the universal wave function
will collapse into a definite state.101010Representing collapses by Gaussian
functions gives rise to the “tails problem.” See [37] §12 and the references
therein.  On the macroscopic scale, the collapse will give rise to
(approximately) Born rule probabilities. Each version of GRW postulates
specific values for the collapse rate and the Gaussian width. Moreover, there
can be additional variables representing ontology in physical space. GRWm adds
a mass-density ontology that specifies the amount of mass in physical space by
a real-valued function $m(x,t)$, where $(x,t)$ is a space-time point.111111The
mass-density function is defined from the wave function:
$m(x,t)=\sum_{i=1}^{N}m_{i}\int_{\mathbb{R}^{3N}}d^{3}x_{1}...d^{3}x_{N}\delta^{3}(x_{i}-x)|\Psi_{t}(x_{1},...,x_{N})|^{2}$
(6) In contrast, GRWf adds a flash ontology that postulates the existence of
space-time events at the center of the Gaussian function. It can be
represented as a function $F(x,t)$ with $x\in\mathbb{R}^{3}$ and $F(x,t)=1$ if
$(x,t)$ is the center of some GRW collapse and $0$ otherwise.121212For a
survey of GRW, see [37]; for the original paper, see [38]; [13], Ch 22,
contains a clear presentation of the theory.

Third, many-worlds interpretations of Everettian quantum mechanics (EQM)
reject P3.131313The many-worlds interpretations are popular among Everettians.
However, [24] has provided textual evidence that it is possible to understand
Everett as endorsing a single-world interpretation of quantum mechanics.
According to these interpretations, there is no need to ensure that there is a
unique outcome in the cat experiment. There simply are two branches of the
wave function, one in which the cat is alive and the atom has not decayed and
the other in which the cat is dead and the atom has decayed. Both branches co-
exist. Because of a property called _decoherence_ , the branches do not
interfere much with each other. The branches of the wave function are emergent
worlds, the wave function is the complete description of the “multi-verse,”
and it always obeys the Schrödinger equation. Similarly to GRWm, we can devise
a version of EQM with a mass-density ontology. This is called Sm and was first
proposed by [7]. A challenge for any version of EQM is how to make sense of
probability in a world in which every possible outcome of every quantum
experiment happens with certainty.141414For a survey of EQM, see [72]; for the
original paper, see [32]; for an updated book-length development of the
theory, see [74]. There has been significant progress in addressing the
probability challenge with the tools of typicality, decision theory, and self-
locating probabilities. For some recent examples, see [11], [74], [69], and
the references therein.

The upshot is that the wave function figures prominently in all three realist
quantum theories. In BM, although the wave function is not the complete
description of the system, it is still part of the description. Moreover, the
wave function guides particle motion. In GRW, the wave function collapses and
gives rise to unique outcomes of experiments. In (many-worlds interpretations
of) EQM, the wave function never collapses but gives rise to emergent parallel
worlds. For quantum theories with additional ontology, such as BM, GRWm, GRWf,
and Sm, the wave function is also tied to the dynamics of the additional
ontology. But their relationship is different in these theories. Bohmian
particles have independent dynamics: even if the wave function were not to
change, Bohmian particles would still move in a non-trivial fashion. That is
not the case in GRWm, GRWf, and Sm. Had there been no change to the wave
function, the additional ontology would not change either. It is in this sense
that the dynamics of mass-densities and flashes are not independent of the
dynamics of the wave function.

##  3 Ontological Interpretations

In this section, I review four _ontological_ interpretations of the wave
function. However, the label “ontological” could be misleading. These four
interpretations share the feature that the wave function is interpreted as
part of the fundamental _material ontology_ , on a par with particles, fields,
space-time events or properties, which are the kind of microscopic things that
make up macroscopic objects such as tables and chairs. In §4 and §5, we will
review nomological interpretations and the _sui generis_ interpretation of the
wave function, which are compatible with the position that the wave function
is part of the ontology but just not in the same ontological category as
particles or fields.

###  3.1 A Field on a High-Dimensional Space

According to the first ontological interpretation, the fundamental space is a
high-dimensional space, and the wave function is a field in that space. This
was introduced by [2]. Albert calls this view _wave function realism_.151515It
has been developed and defended by [47], [61, 62], and [67], although North is
primarily concerned with the first part of the thesis, i.e. the fundamental
space is a high-dimensional space. However, as we shall see in the later
sections, that label is no longer appropriate given the abundance of other
approaches that are also realist about the wave function.

It is counterintuitive how the fundamental space can be high-dimensional. It
might help to compare this idea with something familiar—classical physics. In
classical field theories such as Maxwellian electrodynamics, electromagnetic
fields are fields on the four-dimensional physical space-time. A field on
physical space-time can be interpreted as an assignment of monadic properties
(field strength and direction) to each point in space-time. Such an assignment
is contrained by Maxwell’s equations and certain boundary conditions.

In a similar way, the wave function can be interpreted as a physical field.
However, the wave function cannot be interpreted as a field on physical space,
as its domain is the high-dimensional configuration space, represented by
$\mathbb{R}^{3N}$. If we take configuration space to be the fundamental space,
then the wave function can be interpreted as a field that assigns properties
to each point in configuration space. The properties assigned by the field,
represented as complex numbers, change according to the Schrödinger equation.
On this view, the high-dimensional configuration space is ontologically prior
to physical space(time), and the latter somehow comes out of the fundamental
structure. This is despite the fact that we call the high-dimensional space
“configuration space,” which seems to imply the reverse order of ontological
dependence.161616There are three versions of this view:  • Bohmian version:
the fundamental space is represented by $\mathbb{R}^{3N}$. The fundamental
ontology consists in a point particle located in that space and a field that
assigns properties to points of that space. The field always evolves by the
Schrödinger equation. The point particle moves along in the field according to
the guidance equation, much like corks move along in a river. Here we see a
dis-analogy with the classical field. In classical physics, the field and the
particles satisfy the action-reaction principle; the fields and the particles
can influence each other. In Bohmian mechanics, the wave function interpreted
as a field can influence the particles but not the other way around. • GRW
version: the fundamental space is represented by $\mathbb{R}^{3N}$. The
fundamental ontology consists in a field that assigns properties to points of
that space. The field evolves by the Schrödinger equation most of the time but
sometimes collapses by the GRW collapse mechanism. • Everettian version: the
fundamental space is represented by $\mathbb{R}^{3N}$. The fundamental
ontology consists in a field that assigns properties to points of that space.
The field always evolves by the Schrödinger equation. The high-dimensional
field interpretation of the wave function is incompatible with GRWm, GRWf, or
Sm.

The high-dimensional field interpretation prioritizes the structure of the
wave function and its dynamics. The fundamental physical events are those that
happen on the high-dimensional space. A key challenge to this view is to
explain our apparent experiences in a three-dimensional space. That is not
just a question about recovering the manifest image, but it is also about
whether such an interpretation of quantum mechanics can be “empirically
coherent,” in the sense that if our evidence for quantum mechanics comes from
instrument readings in the three-dimensional space, the theory should not
undermine such evidence. It should explain how the appearances of three-
dimensional objects come out of the high-dimensional fundamental
space.171717See [10] and [9].

Albert (1996) suggests that the explanation lies in the dynamics—in the
structure of the Hamiltonian operator. Although all the $3N$ dimensions are
metaphysically on a par:

$\\{q_{1},q_{2},q_{3},q_{4},q_{5},q_{6},...,q_{3N-2},q_{3N-1},q_{3N}\\}$ (7)

the Hamiltonian operator has a term that encodes fundamental interactions and
it takes on a particular form:

$\mathop{\sum\sum}_{1\leq i<j\leq
N}V_{ij}[(q_{3i-2}-q_{3j-2})^{2}+(q_{3i-1}-q_{3j-1})^{2}+(q_{3i}-q_{3j})^{2}]$
(8)

The Hamiltonian operator groups the coordinates in the $3N$-dimensional
configuration space into triplets, such that there might be emergent objects
that have the same functional profile as what we take to be ordinary objects
in the 3-dimensional space. This provides reasons to believe that there might
be an emergent $3$-dimensional physical space. For a discussion of Albert’s
proposed explanation of emergence, see [56], [45], [61], [21, 29], and [63].

[52] criticizes the high-dimensional field interpretation on the ground that
it reifies redundant structure. The common view (§2.1) holds that two wave
functions that differ only by a multiplicative constant represent the same
physical state. But if we interpret the wave function as a field that assigns
numbers (that represent physical magnitudes) to points in configuration space,
then we would distinguish two wave functions related by a constant, for the
numbers assigned to the points are different.181818This problem can be avoided
if we adopt an _intrinsic_ (or gauge-free) characterization of the wave
function, in terms of comparative relations that are invariant under any
change by a multiplicative constant. On the comparativist view, the
fundamental structure consists in not monadic properties but comparative
relations holding between points of the high-dimensional space. See [22] for a
proposal.

###  3.2 A Multi-field on Physical Space

The high-dimensional field interpretation of the wave function faces
difficulties, primarily because it privileges configuration space over
physical space. There are many good reasons to take physical space to be
ontologically more basic. First, it underlies many important symmetries in
physics. Second, it is much easier for a theory to be empirically coherent if
it does not undermine the relative fundamentality of physical space(time).

These difficulties are avoided in the second ontological interpretation,
according to which the fundamental space is the ordinary physical space(time).
On this view, the wave function is not a field in the traditional sense, but a
_multi-field_ on physical space. (See [34], [14], [22, 21], [44].) A multi-
field is similar to a field. However, unlike fields, multi-fields assign
properties not to individual points but to _regions_ of points in space. Such
regions can be connected or disconnected. Mathematically, the wave function is
a function from $N$ copies of $\mathbb{R}^{3}$ to complex numbers. Instead of
thinking of it as a field that assigns properties to every point in
$\mathbb{R}^{3N}$, we can think of it as a “multi-field” that assigns
properties to every region of $\mathbb{R}^{3}$ that is composed of $N$ points.
The multi-field interpretation is a more faithful representation for
“indistinguishable particles,” for which particle labels do not matter. This
is because spatial regions understood as $N$-element subsets of
$\mathbb{R}^{3}$ (or mereological fusions of $N$ points in $\mathbb{R}^{3}$)
are _unordered_. Thus, the multi-field interpretation has the additional
advantage of automatically enforcing what is called “permutation invariance”:
mere permutations of a configuration of $N$ particles do not change the
physical state.

###  3.3 Properties of Physical Systems

The third ontological interpretation, proposed by [75], affirms the (relative)
fundamentality of the physical space(time). On this view, we assume that the
universe has a compositional structure, i.e. it is divided into subsystems
that occupy some spatial-temporal regions.191919This assumption is important
and can turn out to fail in some theories. For the Everettian interpretation
according to which the quantum state or the wave function represents
everything there is, the compositional structure may need to be emergent and
is certainly not fundamental. See §3.4 for a discussion of that possibility.
Larger systems can be made out of unions of smaller systems. And the universe
is the union of all systems. Although not every system has a wave function
(because of entanglement), we can still associate to each system a determinate
property represented by what is called a _density matrix_. A density matrix
encodes much dynamical information of the system.202020For example, if the
universe consists in two systems $A$ and $B$, and if their joint quantum state
is this:
$\ket{\Psi_{AB}}=\frac{1}{\sqrt{2}}(\ket{\alpha}_{A}\ket{\beta}_{B}-\ket{\beta}_{A}\ket{\alpha}_{B})$
(9) then the density matrix associated with system $A$ (called the _reduced
density matrix_) will be:
$\rho_{A}=\frac{1}{2}(\ket{\alpha}_{A}\bra{\alpha}_{A}+\ket{\beta}_{A}\bra{\beta}_{A})$
(10) On this view, $\ket{\Psi_{AB}}$ and $\rho_{A}$ are understood as
representing intrinsic properties of the systems or the space-time locations
of the systems.  This view was introduced as an alternative to the high-
dimensional field interpretation. It is also an alternative to the low-
dimensional multi-field interpretation. However, it is still a version of
realism about the wave function, since the universal wave function (or the
universal density matrix) is to be found in the ontology—the property of the
entire universe.

Wallace and Timpson call this approach _spacetime state realism_. Compared to
the field interpretations in §3.1 and §3.2, spacetime state realism seems more
minimal, in the sense that it is neutral with respect to the specific
structures of quantum theories. As long as the quantum theory comes with a
compositional structure (a decomposition of larger systems into smaller
systems), we have a well-defined procedure of ascribing properties to physical
systems. They argue that it also has significant advantages in handling
relativistic invariance. See [71] for some discussions about the relativistic
extensions.

A question about spacetime state realism is whether the fundamental ontology
contains redundant structure. If we help ourselves to a decomposition of the
universe into subsystems, and if we have the quantum state of the universe,
then we can obtain density matrices of the subsystems by a purely mathematical
procedure of tracing out the environmental degrees of freedom. Since they can
be derived from the quantum state of the universe, the properties of the
subsystem need not be placed in the fundamental ontology. Hence, spacetime
state realism is in tension with the principle of parsimony that we should
minimize postulating more fundamental structure than we need. If we get rid of
the subsystem properties and only keep the universal property (the universal
density matrix), then this approach would be in the same spirit as the low-
dimensional multi-field approach in §3.2. However, if one is interested in
finding the most _perspicuous_ interpretation (as suggested by Wallace and
Timpson), and not the adequate ontology that is the most parsimonious and
simple, perhaps space-time state realism is the right choice. On the other
hand, one might object that an ontology of properties represented by density
matrices, which is not metaphysically impossible, may be much less perspicuous
than an ontology of properties represented by numbers and vectors.212121See
[57] and [58] for another view that interprets the wave function as properties
of physical systems.

###  3.4 A Vector in Hilbert Space

The final ontological interpretation of the wave function takes the abstract
Hilbert space more seriously. Recall that the wave function is represented as
a vector in Hilbert space, and the Schrödinger equation can be represented as
an equation for vector rotation in that space. [20] suggest that Hilbert space
is the fundamental space, and the wave function is just a vector in that
space. Physical goings-on in the world will correspond to some particular
directions the vector is pointing at.

Since the Everettian interpretation of QM is the most natural place for this
view, [20] call this approach _Mad-Dog Everettianism_. In their words, the
label is “to emphasize that it is as far as we can imagine taking the program
of stripping down quantum mechanics to its most pure, minimal elements.”

It is already difficult to recover ordinary objects from configuration space.
It is even more difficult to recover them from Hilbert space. For one thing,
there is no space-time structure in Hilbert space. The state of the world
corresponds to a vector, which is just like every other vector. How can
anything familiar, such as space, time, and ordinary objects, come out of a
vector in a high-dimensional Hilbert space? Like Albert (1996), Carroll and
Singh propose that the answer lies in the structure of the Hamiltonian
operator. The Hamiltonian provides a privileged way to decompose the total
Hilbert space into smaller spaces, which may explain the emergent
structure.222222Their analysis is restricted to _locally finite-dimensional
Hilbert spaces_. See [25] and [8] for more details. Carroll and Singh’s
proposal is more speculative than the high-dimensional field interpretation,
since Hilbert space is more abstract than configuration space. However, it is
in part motivated by the non-fundamentality of space-time in several theories
of quantum gravity. As such, it could be a fruitful project to explore.

##  4 Nomological Interpretations

According to the previous ontological interpretations, the wave function is
part of the fundamental material ontology of the world. The wave function in
quantum mechanics is given the same status as particles and fields in
classical mechanics. In contrast, nomological interpretations hold that the
wave function is nomological, i.e. on a par with laws of nature. In this
section, I survey two kinds of nomological interpretations of the wave
function: the strong nomological interpretations and the weak nomological
interpretations.

These interpretations are most compelling from a Bohmian point of view.
However, they might be adaptable for some versions of GRW theories and
Everettian theories with additional ontologies.

###  4.1 Strong Nomological Interpretations

The guiding idea of nomological interpretations is that the wave function is
on a par with laws of nature ([28, 40, 41]). To appreciate the strong
nomological interpretations, it would be helpful to review the status of the
Hamiltonian function in classical mechanics. As mentioned in §2.1, the
Hamiltonian equations (1) govern the motion of classical particles in physical
space, represented by a curve in phase space. The Hamiltonian function is the
generator of such motion. It encodes the total energy of the system. In one of
the more familiar cases, it is a short hand for the kinetic energy term and
the pair-wise interactions of the particles (including gravitational,
electric, and magnetic interactions). We can write out $H$ explicitly as a
function (of position and momentum) on the right hand sides of the equations.
For the Hamiltonian equations to be simple _laws_ of nature, $H$ has to be a
simple function. In this sense, we give $H$ a nomological interpretation.
Although it is a function on phase space, we do not treat it as part of the
material ontology.

Let us now consider Bohmian mechanics. In this theory, the guidance equation
governs the motion of particles in physical space, represented by a curve in
configuration space. The wave function is the generator of such motion.
Moreover, there is no back reaction from the particles to the wave function.
If the wave function turns out to be a simple function that is somehow fixed
by the theory, then we can write out $\Psi$ explicitly as a function (of
configuration variables) on the right hand side of the equation. In that case,
we can give it an analogous nomological interpretation. Although it is a
function on configuration space, we can treat it as not part of the ontology
but only part of the law system. I call this the _strong nomological
interpretation_ , for it affords the same status to the wave function as it
does to the classical Hamiltonian function. Just as both Humeans and non-
Humeans about laws of nature can embrace the nomological interpretation of the
Hamiltonian function in classical mechanics, both Humeans and non-Humeans can
embrace the strong nomological interpretation of the wave function.

However, the analogy between the Hamiltonian function in classical mechanics
and the wave function in Bohmian mechanics is not perfect. While the
Hamiltonian function is time-independent (it does not change over time), the
wave function is typically time-dependent (it changes over time). For
something to be nomological or to be like a fundamental law, one might expect
it to be time-independent. However, there are reasons to be optimistic. [28,
40], and [41] have offered one. When we eventually extend quantum mechanics to
quantum gravity, the wave function of the universe may turn out to be time-
independent. This is seen in the Wheeler-DeWitt equation of canonical quantum
gravity:

$\hat{H}\Psi=0$ (11)

If we understand (11) as telling us about the time evolution of the wave
function, then it tells us that the wave function does not change over time,
i.e. it is time-independent.232323 Since the Schrödinger equation governs how
the wave function changes over time, it is to be treated not as a fundamental
equation but only as an effective equation—describing the behavior of
subsystems. It is worth emphasizing that a time-independent universal wave
function does not entail that there is no change in the universe (cf: the
problem of time in quantum gravity). In the Bohmian theory, given a time-
independent wave function, the particles can still move in a non-trivial way,
since a time-independent wave function can generate a non-trivial velocity
field in configuration space.

The strong nomological interpretation faces some challenges. First, it is an
open question whether the Wheeler-DeWitt equation governs the universal wave
function. For example, there are research programs in quantum gravity that do
not presuppose it. Second, since the wave function does not change over time,
it requires some revisions about how we think about the arrow of
time.242424The problem is that in standard Boltzmannian quantum statistical
mechanics, the arrow of time is associated with the increase of entropy of the
quantum system, which is a property of the wave function. If the universal
wave function is time-independent, then there is no increase of Boltzmann
entropy. Perhaps the Bohmian approach can help by providing an alternative
definition of entropy (or “effective entropy”) in terms of particle
configurations.

Why should fundamental laws be time-independent? One might appeal to one’s
intuition that laws are eternal and unchanging. But I think a better reason is
that time-independent laws are more likely to be simple, and simplicity is the
more fundamental consideration.252525Here I am supposing that the demand for
simplicity of the fundamental laws is neutral between Humeanism and non-
Humeanism about laws of nature. For Humeans, theoretical virtues such as
simplicity and informativeness (and the balance of the two) are constitutive
of some facts being Humean laws of nature (that are the best summaries of the
Humean mosaic). For non-Humeans, simplicity and informativeness are our
epistemic guides to discover which facts are non-Humean laws. Both parties, it
seems to me, should respect the pre-theoretical intuitions that fundamental
laws should be as simple as possible. And the complexity of some principle is
a serious strike against it being a fundamental law.  A time-dependent
function may be more complicated than a time-independent one. And if a wave
function $\Psi$ is time-independent, as suggested in (11), then $\Psi$
presumably has many symmetries, so that the different contributions under
time-evolution will cancel out. Such symmetries _may_ ensure that the wave
function is simple. For example, a translationally invariant function on
physical space can only be a constant function, which is relatively
simple.262626Relatedly, [5] has proposed a new argument for nomological
interpretations based on symmetry principles. However, if simplicity is the
more fundamental desideratum, we may consider whether time-independence is
only a defeasible guide and whether there may be simple yet time-dependent
laws.

Can we find a simple (but maybe time-dependent) quantum state that plays the
same role as the classical Hamiltonian? An example can be found in [23]. Chen
proposes a choice of the initial quantum state that is as simple as the choice
of the low-entropy macrostate specified in the Past Hypothesis (Albert 2000).
The Past Hypothesis has been taken to be a candidate fundamental law of
nature.272727See discussions in [33, 3, 18, 49, 73, 74] and [50]. If the Past
Hypothesis is simple enough to be a law, then Chen’s quantum state is simple
enough to be nomological. However, in that approach, the fundamental quantum
state is a mixed state rather than a pure state (it has to be represented by a
density matrix rather than a wave function). Moreover, the generator of motion
becomes time-dependent.

The strong nomological interpretation is most compelling in the Bohmian
framework. However, In Everettian and GRW theories with additional ontologies,
it may also be possible to make a case for the strong nomological
interpretation.

###  4.2 Weak Nomological Interpretations

The literature on the nomological interpretation of the wave function is
expanding. However, much of that is directed at a weaker thesis, which I call
the _weak nomological interpretation_. On that view, the wave function $\Psi$
does not need to be like the classical Hamiltonian to fit into the law system.
In particular, $\Psi$ does not need to be time-independent or simple. This
interpretation recommends a weaker criterion for being nomological. The idea
is most plausible in some extended Humean framework. In the original Humean
framework, laws of nature are the axioms of the best system that summarizes
the mosaic. In [48], the Humean framework has been extended to allow for
deterministic “chances.” In [42], it has been further extended to allow
intrinsic properties such as mass and charge to be non-fundamental and to be
merely part of the best system.

According to the weak nomological interpretation (Humean version), what is
fundamental is just matter (particles or fields) in the four-dimensional
spacetime, and the wave function is just a dynamical variable that assists in
a simple and informative summary of the mosaic. See [55, 30, 15, 19], and
[31]. Although the wave function is part of the best system, it does not need
to be time-independent. Moreover, it does not need to be simple _simpliciter_.
It just needs to be the simplest one among all competitors. Even though the
exact specification of the wave function is complicated, the best system
involving the wave function might still be the simplest overall. Albert
(p.c.), Maudlin (p.c.), and [26] have raised the worry that the complete
specification of particle trajectories, which will form another system, seems
to postulate much less information than the wave function. This is because the
particle trajectories form a single curve in configuration space, while the
wave function assigns values to every point in configuration space. Moreover,
they have raised the worry that the wave function does not supervene on the
particle trajectories, since _prima facie_ the particle trajectories do not
determine the exact values of the wave function. However, it is true that
physicists who have access only to facts about the locations of pointers and
experimental instruments nonetheless postulate wave functions to make accurate
explanations and predictions, and they often agree on the exact wave function
of the system. So the best system comparisons and the issue about
supervenience may be more complicated than the debate has assumed.

At any rate, the weak nomological interpretation demands less of the wave
function of the universe. It does not need to be a time-independent function,
a simple function, or a function determined in a simple way. It can be time-
dependent and highly complex, as long as it is the simplest among all the
choices. One could argue that the weak nomological interpretation is less
realist than the previous approaches,282828One argument is that the weak
nomological interpretation is “too cheap,” i.e. too easy to satisfy. We can
presumably extend it to any piece of problematic ontology and put it in the
law system. There is no difficulty, for example, of extending the
interpretation to classical physics and move the classical electromagnetic
fields from fundamental ontology to the law system. So the fundamental
ontology will only consist in particles, not particles plus fields. That would
be an unwelcome result, as we can test whether an interpretive framework is
realist enough by testing whether it affirms, in classical field theory, the
ontological reality of fields.  but it could still be realist if the extended
Humean framework can be understood as a realist view about laws and
properties. So far the weak nomological interpretation has only been defended
in the extended Humean framework. It is an open question whether it can be
made plausible in some non-Humean framework.

##  5 The _Sui Generis_ Interpretation

It is possible to be not persuaded by any of the above strategies. The high-
dimensional field interpretation and Hilbert space interpretation require
sophisticated stories about the emergence of the apparent three-dimensional
objects and ordinary space-time. The low-dimensional multi-field
interpretation and the subsystem property interpretation may seem to be trying
too hard to squeeze the wave function into familiar ontological categories.

Perhaps the lesson of quantum mechanics is that the wave function does not fit
into any familiar categories of things; it is a new kind of entity. Perhaps it
is neither ontological nor nomological. In that case, the wave function has
its own category of existence that is distinct from anything we have
considered. In other words, the wave function is ontologically _sui generis_.
[52] suggests that we should be open to that possibility.

Interpretation | BM | GRW0 | GRWm | GRWf | S0 | Sm | HD | LD
---|---|---|---|---|---|---|---|---
High-Dimensional Field | ✓ | ✓ | ✗ | ✗ | ✓ | ✗ | ✓ | ✗
Low-Dimensional Multi-field | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✗ | ✓
Properties of Systems | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✗ | ✓
Vector in Hilbert Space | ? | ✓ | ? | ? | ✓ | ? | ✓ | ✗
Strongly Nomological | ✓ | ? | ? | ? | ? | ✓ | ✓ | ✓
Weakly Nomological | ✓ | ? | ✓ | ✓ | ? | ✓ | ✓ | ✓
_Sui Generis_ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓
Table 1: The first column lists all the realist interpretations reviewed in
this article. In the first row, we have the main solutions to the quantum
measurement problem: BM (Bohmian mechanics with a particle ontology), GRW0
(GRW theory without additional ontologies), GRWm (GRW theory with a mass-
density ontology), GRWf (GRW theory with a flash ontology), S0 (Everettian
theory without additional ontologies), and Sm (Everettian theory with a mass-
density ontology). “HD” stands for the view that the fundamental physical
space is high-dimensional ($10^{80}$ dimensions in configuration space
fundamentalism or possibly infinity in Hilbert space fundamentalism), and “LD”
stands for the view that the fundamental space is low-dimensional ($3$
dimensions of ordinary physical space). We mark their compatibility with a
check (compatible), a cross (incompatible), or a question mark (unknown
compatibility).

##  6 Conclusion

In this article, we have surveyed three kinds of realist interpretations of
the wave function: ontological interpretations, nomological interpretations,
and the _sui generis_ interpretation. (See Table 1 for a summary.) A century
after the discovery of quantum mechanics, although there is no consensus on
what it means, we have made significant progress in constructing several
realist interpretations. Almost every interpretation requires further
developments, and it is too early to say which one is the best or the most
fruitful. It is also too early to think that those are exhaustive of all the
options available to the realist. In all likelihood, there will be other ways
to think about the wave function from the realist perspective that we have
never considered.292929For book-length treatments of the subject, see [65],
[46], [35], and [64]. For clear introductions to the foundations of quantum
mechanics, see [13, 1, 36, 17], [66], and [53]. For an alternative perspective
on realism about the wave function, see [43].

## Acknowledgement

I am grateful for useful comments from the editors and two anonymous referees
of _Philosophy Compass_ and helpful discussions with David Albert, Valia
Allori, Sean Carroll, Christina Conroy, Alexander Ehmann, Sheldon Goldstein,
Veronica Gomez, Mario Hubert, Barry Loewer, Ricardo Martinez, Bradley Monton,
Bradley Rettler, Thad Roberts, Davide Romano, and Ashmeet Singh.

## References

  * Albert, [1992] Albert, D. Z. (1992).  Quantum Mechanics and Experience.  Harvard University Press.
  * Albert, [1996] Albert, D. Z. (1996).  Elementary quantum metaphysics.  In Cushing, J. T., Fine, A., and Goldstein, S., editors, Bohmian Mechanics and Quantum Theory: An Appraisal, pages 277–84. Kluwer Academic Publishers, Dordrecht.
  * Albert, [2000] Albert, D. Z. (2000).  Time and chance.  Harvard University Press.
  * Allori, [2013] Allori, V. (2013).  Primitive ontology and the structure of fundamental physical theories.  The Wave Function: Essays on the Metaphysics of Quantum Mechanics, pages 58–75.
  * Allori, [2017] Allori, V. (2017).  A new argument for the nomological interpretation of the wave function: The galilean group and the classical limit of nonrelativistic quantum mechanics.  International Studies in the Philosophy of Science, 31(2):177–188.
  * Allori et al., [2008] Allori, V., Goldstein, S., Tumulka, R., and Zanghì, N. (2008).  On the common structure of bohmian mechanics and the Ghirardi–Rimini–Weber theory: Dedicated to Giancarlo Ghirardi on the occasion of his 70th birthday.  The British Journal for the Philosophy of Science, 59(3):353–389.
  * Allori et al., [2010] Allori, V., Goldstein, S., Tumulka, R., and Zanghì, N. (2010).  Many worlds and schrödinger’s first quantum theory.  British Journal for the Philosophy of Science, 62(1):1–27.
  * Bao et al., [2017] Bao, N., Carroll, S. M., and Singh, A. (2017).  The hilbert space of quantum gravity is locally finite-dimensional.  International Journal of Modern Physics D, 26(12):1743013.
  * Barrett, [1996] Barrett, J. A. (1996).  Empirical adequacy and the availability of reliable records in quantum mechanics.  Philosophy of Science, 63(1):49–64.
  * Barrett, [1999] Barrett, J. A. (1999).  The quantum mechanics of minds and worlds.  OUP Oxford.
  * Barrett, [2017] Barrett, J. A. (2017).  Typical worlds.  Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 58:31–40.
  * Bell, [1990] Bell, J. (1990).  Against ‘measurement’.  Physics world, 3(8):33.
  * Bell, [2004] Bell, J. S. (2004).  Speakable and Unspeakable in Quantum Physics: Collected Papers on Quantum Philosophy.  Cambridge University Press.
  * Belot, [2012] Belot, G. (2012).  Quantum states for primitive ontologists.  European Journal for Philosophy of Science, 2(1):67–83.
  * Bhogal and Perry, [2015] Bhogal, H. and Perry, Z. (2015).  What the humean should say about entanglement.  Noûs.
  * Bohm, [1952] Bohm, D. (1952).  A suggested interpretation in terms of “hidden variables”: Part i and part ii.  Phys. Rev, 85(166179):180193.
  * Bricmont, [2016] Bricmont, J. (2016).  Making sense of quantum mechanics, volume 37.  Springer.
  * Callender, [2004] Callender, C. (2004).  Measures, explanations and the past: Should ‘special’ initial conditions be explained?  The British journal for the philosophy of science, 55(2):195–217.
  * Callender, [2015] Callender, C. (2015).  One world, one beable.  Synthese, 192(10):3153–3177.
  * Carroll and Singh, [2018] Carroll, S. M. and Singh, A. (2018).  Mad-dog Everettianism: Quantum mechanics at its most minimal.  arXiv preprint arXiv:1801.08132.
  * Chen, [2017] Chen, E. K. (2017).  Our fundamental physical space: An essay on the metaphysics of the wave function.  Journal of Philosophy, 114: 7.
  * [22] Chen, E. K. (2018a).  The intrinsic structure of quantum mechanics.  Philsci-archive preprint:15140.
  * [23] Chen, E. K. (2018b).  Quantum mechanics in a time-asymmetric universe: On the nature of the initial quantum state.  The British Journal for the Philosophy of Science, forthcoming.
  * Conroy, [2012] Conroy, C. (2012).  The relative facts interpretation and Everett’s note added in proof.  Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 43(2):112–120.
  * Cotler et al., [2017] Cotler, J. S., Penington, G. R., and Ranard, D. H. (2017).  Locality from the spectrum.  arXiv preprint arXiv:1702.06142.
  * Dewar, [2017] Dewar, N. (2017).  La Bohume.  Philsci-archive preprint:13235.
  * Dürr et al., [1992] Dürr, D., Goldstein, S., and Zanghì, N. (1992).  Quantum equilibrium and the origin of absolute uncertainty.  Journal of Statistical Physics, 67(5-6):843–907.
  * Dürr et al., [1996] Dürr, D., Goldstein, S., and Zanghì, N. (1996).  Bohmian mechanics and the meaning of the wave function.  In in Experimental Metaphysics: Quantum Mechanical Studies in honor of Abner Shimony.
  * Emery, [2017] Emery, N. (2017).  Against radical quantum ontologies.  Philosophy and Phenomenological Research, 95(3):564–591.
  * Esfeld, [2014] Esfeld, M. (2014).  Quantum humeanism, or: physicalism without properties.  The Philosophical Quarterly, 64(256):453–470.
  * Esfeld and Deckert, [2017] Esfeld, M. and Deckert, D.-A. (2017).  A minimalist ontology of the natural world.  Routledge.
  * Everett III, [1957] Everett III, H. (1957).  “Relative state” formulation of quantum mechanics.  Reviews of modern physics, 29(3):454.
  * Feynman, [2017] Feynman, R. (2017).  The Character of Physical Law.  MIT press.
  * Forrest, [1988] Forrest, P. (1988).  Quantum metaphysics.  Blackwell Publisher.
  * Gao, [2017] Gao, S. (2017).  The Meaning of the Wave Function: In Search of the Ontology of Quantum Mechanics.  Cambridge University Press.
  * Ghirardi, [2005] Ghirardi, G. (2005).  Sneaking a look at God’s cards: unraveling the mysteries of quantum mechanics.  Princeton University Press.
  * Ghirardi, [2018] Ghirardi, G. (2018).  Collapse theories.  In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, fall 2018 edition.
  * Ghirardi et al., [1986] Ghirardi, G. C., Rimini, A., and Weber, T. (1986).  Unified dynamics for microscopic and macroscopic systems.  Physical Review D, 34(2):470.
  * Goldstein, [2017] Goldstein, S. (2017).  Bohmian mechanics.  In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, summer 2017 edition.
  * Goldstein and Teufel, [2001] Goldstein, S. and Teufel, S. (2001).  Quantum spacetime without observers: ontological clarity and the conceptual foundations of quantum gravity.  Physics meets Philosophy at the Planck scale, pages 275–289.
  * Goldstein and Zanghì, [2013] Goldstein, S. and Zanghì, N. (2013).  Reality and the role of the wave function in quantum theory.  The wave function: Essays on the metaphysics of quantum mechanics, pages 91–109.
  * Hall, [2015] Hall, N. (2015).  Humean reductionism about laws of nature.  A companion to David Lewis, 57:262.
  * Halvorson, [2018] Halvorson, H. (2018).  To be a realist about quantum theory. _Philsci-archive preprint:14310_.
  * Hubert and Romano, [2018] Hubert, M. and Romano, D. (2018).  The wave-function as a multi-field.  European Journal for Philosophy of Science, 8(3):521–537.
  * Lewis, [2004] Lewis, P. J. (2004).  Life in configuration space.  The British journal for the philosophy of science, 55(4):713–729.
  * Lewis, [2016] Lewis, P. J. (2016).  Quantum Ontology: A Guide to the Metaphysics of Quantum Mechanics.  Oxford University Press.
  * Loewer, [1996] Loewer, B. (1996).  Humean supervenience.  Philosophical Topics, 24:101–27.
  * Loewer, [2001] Loewer, B. (2001).  Determinism and chance.  Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 32(4):609–620.
  * Loewer, [2007] Loewer, B. (2007).  Counterfactuals and the second law.  In Price, H. and Corry, R., editors, Causation, Physics, and the Constitution of Reality: Russell’s Republic Revisited. Oxford University Press.
  * Loewer, [2016] Loewer, B. (2016).  The Mentaculus Vision. In Barry Loewer, Brad Weslake, and Eric Winsberg (eds.) _Time’s Arrow and World’s Probability Structure_. Harvard University Press, forthcoming.
  * Matarese, [2018] Matarese, V. (2018).  A challenge for super-humeanism: the problem of immanent comparisons.  Synthese, pages 1–20.
  * Maudlin, [2013] Maudlin, T. (2013).  The nature of the quantum state.  The wave function: Essays on the metaphysics of quantum mechanics, pages 126–53.
  * Maudlin, [2019] Maudlin, T. (2019).  Philosophy of physics: Quantum Theory, volume 9.  Princeton University Press.
  * Maudlin, [2007] Maudlin, T. W. (2007).  Completeness, supervenience and ontology.  Journal of Physics A: Mathematical and Theoretical, 40(12):3151.
  * Miller, [2014] Miller, E. (2014).  Quantum entanglement, bohmian mechanics, and humean supervenience.  Australasian Journal of Philosophy, 92(3):567–583.
  * Monton, [2002] Monton, B. (2002).  Wave function ontology.  Synthese, 130(2):265–277.
  * Monton, [2006] Monton, B. (2006).  Quantum mechanics and 3n-dimensional space.  Philosophy of science, 73(5):778–789.
  * Monton, [2013] Monton, B. (2013).  Against 3n-dimensional space.  The wave function: essays on the metaphysics of quantum mechanics, 154.
  * Myrvold, [2017] Myrvold, W. (2017).  Philosophical issues in quantum theory.  In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, spring 2017 edition.
  * Myrvold, [2015] Myrvold, W. C. (2015).  What is a wavefunction?  Synthese, 192(10):3247–3274.
  * Ney, [2012] Ney, A. (2012).  The status of our ordinary three dimensions in a quantum universe.  Noûs, 46(3):525–560.
  * Ney, [2013] Ney, A. (2013).  Ontological reduction and the wave function ontology.  The wave function: Essays on the metaphysics of quantum mechanics, 168.
  * Ney, [2017] Ney, A. (2017).  Finding the world in the wave function: Some strategies for solving the macro-object problem.  Synthese, pages 1–23.
  * Ney, [2018] Ney, A. (2018).  Finding the World in the Wave Function.  Unpublished Manuscript.
  * Ney and Albert, [2013] Ney, A. and Albert, D. Z. (2013).  The wave function: Essays on the metaphysics of quantum mechanics.  Oxford University Press.
  * Norsen, [2017] Norsen, T. (2017).  Foundations of Quantum Mechanics: An Exploration of the Physical Meaning of Quantum Theory.  Springer.
  * North, [2013] North, J. (2013).  The structure of a quantum world.  In Albert, D. Z. and Ney, A., editors, The Wavefunction: Essays in the Metaphysics of Quantum Mechanics. Oxford University Press, Oxford.  Forthcoming.
  * Pusey et al., [2012] Pusey, M. F., Barrett, J., and Rudolph, T. (2012).  On the reality of the quantum state.  Nature Physics, 8(6):475–478.
  * Sebens and Carroll, [2016] Sebens, C. T. and Carroll, S. M. (2016).  Self-locating uncertainty and the origin of probability in everettian quantum mechanics.  The British Journal for the Philosophy of Science, 69(1):25–74.
  * [70] Sider, T. (forthcoming).  The tools of metaphysics and the metaphysics of science.  Oxford University Press, Oxford.
  * Swanson, [2018] Swanson, N. (2018).  How to be a relativistic spacetime state realist. _Philsci-archive preprint:14412_.
  * Vaidman, [2018] Vaidman, L. (2018).  Many-worlds interpretation of quantum mechanics.  In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, fall 2018 edition.
  * Wallace, [2011] Wallace, D. (2011).  The logic of the past hypothesis.
  * Wallace, [2012] Wallace, D. (2012).  The emergent multiverse: Quantum theory according to the Everett interpretation.  Oxford University Press.
  * Wallace and Timpson, [2010] Wallace, D. and Timpson, C. G. (2010).  Quantum mechanics on spacetime I: Spacetime state realism.  The British Journal for the Philosophy of Science, 61(4):697–727.