#  Quantum Mechanics in a Time-Asymmetric Universe:
On the Nature of the Initial Quantum State

Eddy Keming Chen Department of Philosophy, 106 Somerset Street, Rutgers
University, New Brunswick, NJ 08901, USA. Website: www.eddykemingchen.net.
Email<EMAIL_ADDRESS>
(Forthcoming in _The British Journal for the Philosophy of Science_
Draft of )

###### Abstract

In a quantum universe with a strong arrow of time, we postulate a low-entropy
boundary condition (the Past Hypothesis) to account for the temporal
asymmetry. In this paper, I show that the Past Hypothesis also contains enough
information to simplify the quantum ontology and define a natural initial
condition.

First, I introduce _Density Matrix Realism_ , the thesis that the quantum
state of the universe is objective and impure. This stands in sharp contrast
to _Wave Function Realism_ , the thesis that the quantum state of the universe
is objective and pure.

Second, I suggest that the Past Hypothesis is sufficient to determine a
natural density matrix, which is simple and unique. This is achieved by what I
call the _Initial Projection Hypothesis_ : the initial density matrix of the
universe is the (normalized) projection onto the Past Hypothesis subspace (in
the Hilbert space).

Third, because the initial quantum state is unique and simple, we have a
strong case for the _Nomological Thesis_ : the initial quantum state of the
universe is on a par with laws of nature.

This new package of ideas has several interesting implications, including on
the harmony between statistical mechanics and quantum mechanics, theoretical
unity of the universe and the subsystems, and the alleged conflict between
Humean supervenience and quantum entanglement.

Keywords: quantum state of the universe, time’s arrow, Past Hypothesis,
Statistical Postulate, the Mentaculus Vision, typicality, unification,
foundations of probability, quantum statistical mechanics, wave function
realism, quantum ontology, density matrix, Weyl Curvature Hypothesis, Humean
Supervenience

###### Contents

  1. 1 Introduction
  2. 2 Foundations of Quantum Mechanics and Statistical Mechanics
    1. 2.1 Quantum Mechanics
    2. 2.2 Quantum Statistical Mechanics
  3. 3 Density Matrix Realism
    1. 3.1 W-Bohmian Mechanics
    2. 3.2 W-Everettian and W-GRW Theories
    3. 3.3 Field Intepretations of $W$
  4. 4 The Initial Projection Hypothesis
    1. 4.1 The Past Hypothesis
    2. 4.2 Introducing the Initial Projection Hypothesis
    3. 4.3 Connections to the Weyl Curvature Hypothesis
  5. 5 Theoretical Payoffs
    1. 5.1 Harmony between Statistical Mechanics and Quantum Mechanics
    2. 5.2 Descriptions of the Universe and the Subsystems
  6. 6 The Nomological Thesis
    1. 6.1 The Classical Case
    2. 6.2 The Quantum Case
    3. 6.3 Humean Supervenience
  7. 7 Conclusion

##  1 Introduction

In the foundations of quantum mechanics, it has been argued that the wave
function (pure state) of the universe represents something objective and not
something merely epistemic. Let us call this view _Wave Function Realism_.
There are many realist proposals for how to understand the wave function. Some
argue that it represents things in the ontology, either a physical field
propagating on a fundamental high-dimensional space, or a multi-field
propagating on the three-dimensional physical space. Others argue that it is
in the “nomology”—having the same status as laws of nature. Still others argue
that it might belong to a new ontological category.111See Albert, (1996),
Loewer, (1996), Wallace and Timpson, (2010), Ney, (2012), North, (2013),
Maudlin, (2013), Goldstein and Zanghì, (2013), Miller, (2014), Esfeld, (2014),
Bhogal and Perry, (2015), Callender, (2015), Esfeld and Deckert, (2017), Chen,
2017a , Chen, 2017b , Chen, (2016), Hubert and Romano, (2018). For a survey on
this literature, see Chen, 2018a . Notice that this is not how Albert, Loewer,
or Ney characterizes wave function realism. For them, to be a wave function
realist is to be a realist about the wave function and a fundamental high-
dimensional space—the “configuration space.” For the purpose of this paper,
let us use _Wave Function Realism_ to designate just the commitment that the
wave function represents something objective.

Wave Function Realism has generated much debate. In fact, it has been rejected
by many people, notably by quantum Bayesians, and various anti-realists and
instrumentalists. As a scientific realist, I do not find their arguments
convincing. In previous papers, I have assumed and defended Wave Function
Realism. Nevertheless, in this paper I want to argue for a different
perspective, for reasons related to the origin of time-asymmetry in a quantum
universe.

To be sure, realism about the universal wave function is highly natural in the
context of standard quantum mechanics and various realist quantum theories
such as Bohmian mechanics (BM), GRW spontaneous collapse theories, and
Everettian quantum mechanics (EQM). In those theories, the universal wave
function is indispensable to the kinematics and the dynamics of the quantum
system. However, as I would like to emphasize in this paper, our world is not
just quantum-mechanical. We also live in a world with a strong arrow of time
(large entropy gradient). There are thermodynamic phenomena that we hope to
explain with quantum mechanics and quantum statistical mechanics. A central
theme of this paper is to suggest that quantum statistical mechanics is highly
relevant for assessing the fundamentality and reality of the universal wave
function.

We will take a close look at the connections between the foundations of
quantum statistical mechanics and various solutions to the quantum measurement
problem. When we do, we realize that we do not _need_ to postulate a universal
wave function. We need only certain “coarse-grained” information about the
quantum macrostate, which can be represented by a density matrix. A natural
question is: can we understand the universal quantum state as a density matrix
rather than a wave function? That is, can we take an “ontic” rather than an
“epistemic” attitude towards the density matrix?

The first step of this paper is to argue that we can. I call this view
_Density Matrix Realism_ , the thesis that the actual quantum state of the
universe is objective (as opposed to subjective or epistemic) and impure
(mixed). This idea may be unfamiliar to some people, as we are used to take
the mixed states to represent our _epistemic uncertainties of the actual pure
state_ (a wave function). The proposal here is that the density matrix
directly represents the actual quantum state of the universe; there is no
further fact about which is the actual wave function. In this sense, the
density matrix is “fundamental.” In fact, this idea has come up in the
foundations of physics.222See, for example, Dürr et al., (2005), Maroney,
(2005), Wallace, (2011, 2012), and Wallace, (2016). In the first step, we
provide a systematic discussion of Density Matrix Realism by reformulating
Bohmian mechanics, GRW theories, and Everettian quantum mechanics in terms of
a _fundamental_ density matrix.

The second step is to point out that Density Matrix Realism allows us to
combine quantum ontology with time-asymmetry in a new way. In classical and
quantum statistical mechanics, thermodynamic time-asymmetry arises from a
special boundary condition called the Past Hypothesis.333For an extended
discussion, see Albert, (2000). I suggest that the information in the _Past
Hypothesis_ is sufficient to determine a natural density matrix. I postulate
the _Initial Projection Hypothesis_ : the quantum state of the universe at
$t_{0}$ is given by the (normalized) projection onto the Past Hypothesis
subspace, which is a particular low-dimensional subspace in the total Hilbert
space. The conjunction of this hypothesis with Density Matrix Realism pins
down a _unique_ initial quantum state. Since the Initial Projection Hypothesis
is as simple as the Past Hypothesis, we can use arguments for the simplicity
of the latter (which is necessary for it to be a law of nature) to argue for
the simplicity of the former. We can thus infer that the initial quantum state
is very _simple_.

The third step is to show that, because of the simplicity and the uniqueness
of the initial quantum state (now given by a fundamental density matrix), we
have a strong case for the _Nomological Thesis_ : the initial quantum state of
the world is on a par with laws of nature. It is a modal thesis. It implies
that the initial quantum state of our world is nomologically necessary; it
could not have been otherwise.

As we shall see, this package of views has interesting implications for the
reduction of statistical mechanical probabilities to quantum mechanics, the
dynamic and kinematic unity of the universe and the subsystems, the nature of
the initial quantum state, and Humean supervenience in a quantum world.

Here is the roadmap of the paper. First, in §2, I review the foundations of
quantum mechanics and quantum statistical mechanics. In §3, I introduce the
framework of Density Matrix Realism and provide some illustrations. In §4, I
propose the Initial Projection Hypothesis in the framework of Density Matrix
Realism. In §5, I discuss their implications for statistical mechanics,
dynamic unity, and kinematic unity. In §6, I suggest that they provide a
strong case for the Nomological Thesis and a new solution to the conflict
between quantum entanglement and Humean supervenience.

##  2 Foundations of Quantum Mechanics and Statistical Mechanics

In this section, we first review the foundations of quantum mechanics and
statistical mechanics. As we shall see in the next section, they suggest an
alternative to Wave Function Realism.

###  2.1 Quantum Mechanics

Standard quantum mechanics is often presented with a set of axioms and rules
about measurement. Firstly, there is a quantum state of the system,
represented by a wave function $\psi$. For a spin-less $N$-particle quantum
system in $\mathbb{R}^{3}$, the wave function is a (square-integrable)
function from the configuration space $\mathbb{R}^{3N}$ to the complex numbers
$\mathbb{C}$. Secondly, the wave function evolves in time according to the the
Schrödinger equation:

$i\hbar\frac{\partial\psi}{\partial t}=H\psi$ (1)

Thirdly, the Schrödinger evolution of the wave function is supplemented with
collapse rules. The wave function typically evolves into superpositions of
macrostates, such as the cat being alive and the cat being dead. This can be
represented by wave functions on the configuration space with disjoint
macroscopic supports $X$ and $Y$. During measurements, which are not precisely
defined processes in the standard formalism, the wave function undergoes
collapses. Moreover, the probability that it collapses into any particular
macrostate $X$ is given by the Born rule:

$P(X)=\int_{X}|\psi(x)|^{2}dx$ (2)

As such, quantum mechanics is not a candidate for a fundamental physical
theory. It has two dynamical laws: the deterministic Schrödinger equation and
the stochastic collapse rule. What are the conditions for applying the former,
and what are the conditions for applying the latter? Measurements and
observations are extremely vague concepts. Take a concrete experimental
apparatus for example. When should we treat it as part of the quantum system
that evolves linearly and when should we treat it as an “observer,” i.e.
something that stands outside the quantum system and collapses the wave
function? That is, in short, the quantum measurement problem.444See Bell,
(1990) and Myrvold, (2017) for introductions to the quantum measurement
problem.

Various solutions have been proposed regarding the measurement problem.
Bohmian mechanics (BM) solves it by adding particles to the ontology and an
additional guidance equation for the particles’ motion. Ghirardi-Rimini-Weber
(GRW) theories postulate a spontaneous collapse mechanism. Everettian quantum
mechanics (EQM) simply removes the collapse rules from standard quantum
mechanics and suggest that there are many (emergent) worlds, corresponding to
the branches of the wave function, which are all real. My aim here is not to
adjudicate among these theories. Suffice it to say that they are all quantum
theories that remove the centrality of observations and observers.

To simplify the discussions, I will use BM as a key example.555See Dürr et
al., (1992) for a rigorous presentation of BM and its statistical analysis. In
BM, in addition to the wave function that evolves unitarily according to the
Schrödinger equation, particles have precise locations, and their
configuration $Q=(Q_{1},Q_{2},...,Q_{N})$ follows the guidance equation:

$\frac{dQ_{i}}{dt}=\frac{\hbar}{m_{i}}\text{Im}\frac{\nabla_{i}\psi(q)}{\psi(q)}(q=Q)$
(3)

Moreover, the initial particle distribution is given by the quantum
equilibrium distribution:

$\rho_{t_{0}}(q)=|\psi(q,t_{0})|^{2}$ (4)

By equivariance, if this condition holds at the initial time, then it holds at
all times. Consequently, BM agrees with standard quantum mechanics with
respect to the Born rule predictions (which are all there is to the observable
predictions of quantum mechanics). For a universe with $N$ particles, let us
call the wave function of the universe the _universal wave function_ and
denote it by $\Psi(\boldsymbol{q_{1},q_{2},...q_{N}})$.

###  2.2 Quantum Statistical Mechanics

Statistical mechanics concerns macroscopic systems such as gas in a box. It is
an important subject for understanding the arrow of time. For concreteness,
let us consider a quantum-mechanical system with $N$ fermions (with
$N>10^{20}$) in a box $\Lambda=[0,L]^{3}\subset\mathbb{R}^{3}$ and a
Hamiltonian $\hat{H}$. I will first present the “individualistic” view
followed by the “ensemblist” view of quantum statistical mechanics
(QSM).666Here I follow the discussions in Goldstein et al., 2010a  and
Goldstein and Tumulka, (2011). I will include some brief remarks comparing QSM
to classical statistical mechanics (CSM), the latter of which may be more
familiar to some readers.

  1. 1.

Microstate: at any time $t$, the microstate of the system is given by a
normalized (and anti-symmetrized) wave function:

$\psi(\boldsymbol{q_{1}},...,\boldsymbol{q_{N}})\in\mathscr{H}_{total}=L^{2}(\mathbb{R}^{3N},\mathbb{C}^{k})\text{
, }\parallel\psi\parallel_{L^{2}}=1,$ (5)

where $\mathscr{H}_{total}=L^{2}(\mathbb{R}^{3N},\mathbb{C}^{k})$ is the total
Hilbert space of the system. (In CSM, the microstate is given by the positions
and the momenta of all the particles, represented by a point in phase space.)

  2. 2.

Dynamics: the time dependence of
$\psi(\boldsymbol{q_{1}},...,\boldsymbol{q_{N}};t)$ is given by the
Schrödinger equation:

$i\hbar\frac{\partial\psi}{\partial t}=H\psi.$ (6)

(In CSM, the particles move according to the Hamiltonian equations.)

  3. 3.

Energy shell: the physically relevant part of the total Hilbert space is the
subspace (“the energy shell”):

$\mathscr{H}\subseteq\mathscr{H}_{total}\text{ ,
}\mathscr{H}=\text{span}\\{\phi_{\alpha}:E_{\alpha}\in[E,E+\delta E]\\},$ (7)

This is the subspace (of the total Hilbert space) spanned by energy
eigenstates $\phi_{\alpha}$ whose eigenvalues $E_{\alpha}$ belong to the
$[E,E+\delta E]$ range. Let $D=\text{dim}\mathscr{H}$, the number of energy
levels between $E$ and $E+\delta E$.

We only consider wave functions $\psi$ in $\mathscr{H}$.

  4. 4.

Measure: the measure $\mu$ is given by the standard Lebesgue measure on the
unit sphere in the energy subspace $\mathscr{S}(\mathscr{H})$.

  5. 5.

Macrostate: with a choice of macro-variables (suitably “rounded” _à la_ Von
Neumann, (1955)), the energy shell $\mathscr{H}$ can be orthogonally
decomposed into macro-spaces:

$\mathscr{H}=\oplus_{\nu}\mathscr{H}_{\nu}\text{ ,
}\sum_{\nu}\text{dim}\mathscr{H}_{\nu}=D$ (8)

Each $\mathscr{H}_{\nu}$ corresponds more or less to small ranges of values of
macro-variables that we have chosen in advance. (In CSM, the phase space can
be partitioned into sets of phase points. They will be the macrostates.)

  6. 6.

Non-unique correspondence: typically, a wave function is in a superposition of
macrostates and is not entirely in any one of the macrospaces. However, we can
make sense of situations where $\psi$ is (in the Hilbert space norm) very
close to a macrostate $\mathscr{H}_{\nu}$:

$\bra{\psi}P_{\nu}\ket{\psi}\approx 1,$ (9)

where $P_{\nu}$ is the projection operator onto $\mathscr{H}_{\nu}$. This
means that almost all of $\ket{\psi}$ lies in $\mathscr{H}_{\nu}$. (In CSM, a
phase point is always entirely within some macrostate.)

  7. 7.

Thermal equilibrium: typically, there is a dominant macro-space
$\mathscr{H}_{eq}$ that has a dimension that is almost equal to D:

$\frac{\text{dim}\mathscr{H}_{eq}}{\text{dim}\mathscr{H}}\approx 1.$ (10)

A system with wave function $\psi$ is in equilibrium if the wave function
$\psi$ is very close to $\mathscr{H}_{eq}$ in the sense of (9):
$\bra{\psi}P_{eq}\ket{\psi}\approx 1.$

_Simple Example._ Consider a gas consisting of $n=10^{23}$ atoms in a box
$\Lambda\subseteq\mathbb{R}^{3}.$ The system is governed by quantum mechanics.
We orthogonally decompose the Hilbert space $\mathscr{H}$ into 51 macro-
spaces:
$\mathscr{H}_{0}\oplus\mathscr{H}_{2}\oplus\mathscr{H}_{4}\oplus...\oplus\mathscr{H}_{100},$
where $\mathscr{H}_{\nu}$ is the subspace corresponding to the macrostate such
that the number of atoms in the left half of the box is between $(\nu-1)\%$
and $(\nu+1)\%$ of $n$. In this example, $\mathscr{H}_{50}$ has the
overwhelming majority of dimensions and is thus the equilibrium macro-space. A
system whose wave function is very close to $\mathscr{H}_{50}$ is in
equilibrium (for this choice of macrostates).

  8. 8.

Boltzmann Entropy: the Boltzmann entropy of a quantum-mechanical system with
wave function $\psi$ that is very close to a macrostate $\mathscr{H}_{\nu}$ is
given by:

$S_{B}(\psi)=k_{B}\text{log}(\text{dim}\mathscr{H}_{\nu}),$ (11)

where $\mathscr{H}_{\nu}$ denotes the subspace containing almost all of $\psi$
in the sense of (9). The thermal equilibrium state thus has the maximum
entropy:

$S_{B}(eq)=k_{B}\text{log}(\text{dim}\mathscr{H}_{eq})\approx
k_{B}\text{log}(D),$ (12)

where $\mathscr{H}_{eq}$ denotes the equilibrium macrostate. (In CSM,
Boltzmann entropy of a phase point is proportional to the logarithm of the
volume measure of the macrostate it belongs to.)

  9. 9.

Low-Entropy Initial Condition: when we consider the universe as a quantum-
mechanical system, we postulate a special low-entropy boundary condition on
the universal wave function—the quantum-mechanical version of the _Past
Hypothesis_ :

$\Psi(t_{0})\in\mathscr{H}_{PH}\text{ ,
}\text{dim}\mathscr{H}_{PH}\ll\text{dim}\mathscr{H}_{eq}\approx\text{dim}\mathscr{H}$
(13)

where $\mathscr{H}_{PH}$ is the Past Hypothesis macro-space with dimension
much smaller than that of the equilibrium macro-space.777We should assume that
$\mathscr{H}_{PH}$ is finite-dimensional, in which case we can use the
Lebesgue measure on the unit sphere as the typicality measure for # 10. It
remains an open question in QSM about how to formulate the low-entropy initial
condition when the initial macro-space is infinite-dimensional. Hence, the
initial state has very low entropy in the sense of (11). (In CSM, the Past
Hypothesis says that the initial microstate is in a low-entropy macrostate
with very small volume.)

  10. 10.

A central task of QSM is to establish mathematical results that demonstrate
(or suggest) that $\mu-$most (maybe even all) wave functions of small
subsystems, such as gas in a box, will approach thermal equilibrium.

Above is the individualistic view of QSM in a nutshell. In contrast, the
ensemblist view of QSM differs in several ways. First, on the ensemblist view,
instead of focusing on the wave function of an individual system, the focus is
on an ensemble of systems that have the same statistical state $\hat{W}$, a
density matrix.888Ensemblists would further insist that it makes no sense to
talk about the thermodynamic state of an individual system. It evolves
according to the von Neumann equation:

$i\hbar\frac{d\hat{W}(t)}{dt}=[\hat{H},\hat{W}].$ (14)

The crucial difference between the individualistic and the ensemblist views of
QSM lies in the definition of thermal equilibrium. On the ensemblist view, a
system is in thermal equilibrium if:

$W=\rho_{mc}\text{ or }W=\rho_{can},$ (15)

where $\rho_{mc}$ is the microcanonical ensemble and $\rho_{can}$ is the
canonical ensemble.999The microcanonical ensemble is the projection operator
onto the energy shell $\mathscr{H}$ normalized by its dimension. The canonical
ensemble is:  $\rho_{can}=\frac{\text{exp}(-\beta\hat{H})}{Z},$ (16) where
$Z=\text{tr }\text{exp}(-\beta\hat{H})$, and $\beta$ is the inverse
temperature of the quantum system.

For the QSM individualist, if the microstate $\psi$ of a system is close to
some macro-space $\mathscr{H}_{\nu}$ in the sense of (9), we can say that the
macrostate of the system is $\mathscr{H}_{\nu}$. The macrostate is naturally
associated with a density matrix:

$\hat{W}_{\nu}=\frac{I_{\nu}}{dim\mathscr{H}_{\nu}},$ (17)

where $I_{\nu}$ is the projection operator onto $\mathscr{H}_{\nu}$.
$\hat{W}_{\nu}$ is thus a representation of the macrostate. It can be
decomposed into wave functions, but the decomposition is not unique. Different
measures can give rise to the same density matrix. One such choice is
$\mu(d\psi)$, the uniform distribution over wave functions:

$\hat{W}_{\nu}=\int_{\mathscr{S}(\mathscr{H_{\nu}})}\mu(d\psi)\ket{\psi}\bra{\psi}.$
(18)

In (18), $\hat{W}_{\nu}$ is defined with a choice of measure on wave functions
in $\mathscr{H_{\nu}}$. However, we should not be misled into thinking that
the density matrix is derivative of wave functions. What is intrinsic to a
density matrix is its geometrical meaning in the Hilbert space. In the case of
$\hat{W}_{\nu}$, as shown in the canonical description (17), it is just a
normalized projection operator.101010Thanks to Sheldon Goldstein for helping
me appreciate the intrinsic meaning of density matrices. That was instrumental
in the final formulation of the Initial Projection Hypothesis in §4.2.

##  3 Density Matrix Realism

According to Wave Function Realism, the quantum state of the universe is
objective and pure. On this view, $\Psi$ is both the microstate of QSM and a
dynamical object of QM.

Let us recall the arguments for Wave Function Realism. Why do we attribute
objective status to the quantum state represented by a wave function? It is
because the wave function plays crucial roles in the realist quantum theories.
In BM, the wave function appears in the fundamental dynamical equations and
guides particle motion. In GRW, the wave function spontaneously collapses and
gives rise to macroscopic configurations of tables and chairs. In EQM, the
wave function is the whole world. If the universe is accurately described by
BM, GRW, or EQM, then the wave function is an active “agent” that makes a
difference in the world. The wave function cannot represent just our
ignorance. It has to be objective, so the arguments go. But what is the nature
of the quantum state that it represents? As mentioned in the beginning of this
paper, there are several interpretations: the two field interpretations, the
nomological interpretation, and the _sui generis_ interpretation.

On the other hand, we often use $W$, a density matrix, to represent our
ignorance of $\psi$, the actual wave function of a quantum system. $W$ can
also represent a macrostate in QSM.111111In some cases, $W$ is easier for
calculation than $\Psi$, such as in the case of GRW collapse theories where
there are multiple sources of randomness. Thanks to Roderich Tumulka for
discussions here.

Is it possible to be a realist about the density matrix of the universe and
attribute objective status to the quantum state it represents? That depends on
whether we can write down realist quantum theories directly in terms of $W$.
Perhaps $W$ does not have enough information to be the basis of a realist
quantum theory. However, if we can formulate quantum dynamics directly in
terms of $W$ instead of $\Psi$ such that $W$ guides Bohmian particles, or $W$
collapses, or $W$ realizes the emergent multiverse, then we will have good
reasons for taking $W$ to represent something objective in those theories. At
the very least, the reasons for that will be on a par with those for Wave
Function Realism in the $\Psi$-theories.

However, can we describe the quantum universe with $W$ instead of $\Psi$? The
answer is yes. Dürr et al., (2005) has worked out the Bohmian version. In this
section, I describe how. Let us call this new framework _Density Matrix
Realism_.121212The possibility that the universe can be described by a
fundamental density matrix (mixed state) has been suggested by multiple
authors and explored to various extents (see Footnote #2). What is new in this
paper is the combination of Density Matrix Realism with the Initial Projection
Hypothesis (§4) and the argument for the Nomological Thesis (§6) based on
that. However, Density Matrix Realism is unfamiliar enough to warrant some
clarifications and developments, and the GRW versions are new. I will use
W-Bohmian Mechanics as the main example and explain how a fundamental density
matrix can be empirically adequate for describing a quantum world. We can also
construct W-Everett theories and W-GRW theories (which have not appeared in
print so far). Similar to Wave Function Realism, Density Matrix Realism is
open to several interpretations. At the end of this section, I will provide
three field interpretations of $W$. In §6, I discuss and motivate a
nomological interpretation.

###  3.1 W-Bohmian Mechanics

First, we illustrate the differences between Wave Function Realism and Density
Matrix Realism by thinking about two different Bohmian theories.

In standard Bohmian mechanics (BM), an $N$-particle universe at a time $t$ is
described by ($Q(t)$, $\Psi(t)$). The universal wave function guides particle
motion and provides the probability distribution of particle configurations.
Given the centrality of $\Psi$ in BM, we take the wave function to represent
something objective (and it is open to several realist interpretations).

It is somewhat surprising that we can formulate a Bohmian theory with only $W$
and $Q$. This was introduced as W-Bohmian Mechanics (W-BM) in Dürr et al.,
(2005). The fundamental density matrix $W(t)$ is governed by the von Neumann
equation (14). Next, the particle configuration $Q(t)$ evolves according to an
analogue of the guidance equation (W-guidance equation):

$\frac{dQ_{i}}{dt}=\frac{\hbar}{m_{i}}\text{Im}\frac{\nabla_{q_{i}}W(q,q^{\prime},t)}{W(q,q^{\prime},t)}(q=q^{\prime}=Q),$
(19)

(Here we have set aside spin degrees of freedom. If we include spin, we can
add the partial trace operator $\text{tr}_{\mathbb{C}^{k}}$ before each
occurrence of “$W$.”) Finally, we can impose an initial probability
distribution similar to that of the quantum equilibrium distribution:

$P(Q(t_{0})\in dq)=W(q,q,t_{0})dq.$ (20)

The system is also equivariant: if the probability distribution holds at
$t_{0}$, it holds at all times.131313Equivariance holds because of the
following continuity equation:  $\frac{\partial W(q,q,t)}{\partial
t}=-\text{div}(W(q,q,t)v),$ where $v$ denotes the velocity field generated via
(19). See Dürr et al., (1992, 2005).

With the defining equations—the von Neumann equation (14) and the W-guidance
equation (19)—and the initial probability distribution (20), we have a theory
that directly uses a density matrix $W(t)$ to characterize the trajectories
$Q(t)$ of the universe’s $N$ particles. If a universe is accurately described
by W-BM, then $W$ represents the fundamental quantum state in the theory that
guides particle motion; it does not do so via some other entity $\Psi$. If we
have good reasons to be a wave function realist in BM, then we have equally
good reasons to be a density matrix realist in W-BM.

W-BM is empirically equivalent to BM with respect to the observable quantum
phenomena, that is, pointer readings in quantum-mechanical experiments. By the
usual typicality analysis (Dürr et al., (1992)), this follows from (20), which
is analogous to the quantum equilibrium distribution in BM. With the
respective dynamical equations, both BM and W-BM generate an equivariant Born-
rule probability distribution over all measurement outcomes.141414Here I am
assuming that two theories are empirically equivalent if they assign the same
probability distribution to all possible outcomes of experiments. This is the
criterion used in the standard Bohmian statistical analysis (Dürr et al.,
(1992)). Empirical equivalence between BM and W-BM follows from the
equivariance property plus the quantum equilibrium distribution. Suppose W-BM
is governed by a universal density matrix $W$ and suppose BM is governed by a
universal wave function chosen at random whose statistical density matrix is
$W$. Then the initial particle distributions on both theories are the same:
$W(q,q,t_{0})$. By equivariance, the particle distributions will always be the
same. Hence, they always agree on what is typical. See Dürr et al., (2005).
This is a general argument. A more detailed subsystem statistical analysis, in
terms of conditional density matrices, would be illuminating. I leave it for
future work.

###  3.2 W-Everettian and W-GRW Theories

W-BM is a quantum theory in which the density matrix is objective. In this
theory, realism about the universal density matrix is based on the central
role it plays in the laws of a W-Bohmian universe: it appears in the
fundamental dynamical equations and it guides particle motion. (In §3.3, we
will provide three concrete physical interpretations of $W$.) What about other
quantum theories, such as Everettian and GRW theories? Is it possible to
“replace” their universal wave functions with universal density matrices? We
show that such suggestions are also possible.151515Thanks to Roderich Tumulka
for suggesting the three equations below and to Sheldon Goldstein and Matthias
Lienert for discussions. The W-GRW formalism, I believe, has not appeared
elsewhere in print. First, let us define _local beables_ (à la Bell, (2004)).
Local beables are the part of the ontology that is localized (to some bounded
region) in physical space. Neither the total energy function nor the wave
function is a local beable. Candidate local beables include particles, space-
time events (flashes), and matter density ($m(x,t)$).

For the Everettian theory with no local beables (S0), we can postulate that
the fundamental quantum state is represented by a density matrix $W(t)$ that
evolves unitarily by the von Neumann equation (14). Let us call this theory
W-Everett theory (W-S0). Since there are no additional variables in the
theory, the density matrix represents the entire quantum universe. The density
matrix will give rise to many branches that (for all practical purposes) do
not interfere with each other. The difference is that there will be (in some
sense) more branches in the W-Everett quantum state than in the Everett
quantum state. In the W-Everett universe, the world history will be described
by the undulation of the density matrix.161616W-S0 is a novel version of
Everettian theory, one that will require more mathematical analysis to fully
justify the emergence of macroscopic branching structure. It faces the
familiar preferred-basis problem as standard Everett does. In addition, on
W-S0 there will be some non-uniqueness in the decompositions of the Hilbert
space into macrospaces. I leave the analysis for future work.

It is difficult to find tables and chairs in a universe described only by a
quantum state. One proposal is to add “local beables” to the theory in the
form of a mass-density ontology $m(x,t)$. The wave-function version was
introduced as Sm by Allori et al., (2010). The idea is that the wave function
evolves by the Schrödinger equation and determines the shape of the mass
density. This idea can be used to construct a density-matrix version (W-Sm).
In this theory, $W(t)$ will evolve unitarily by the von Neumann equation.
Next, we can define the mass-density function directly in terms of $W(t)$:

$m(x,t)=\text{tr}(M(x)W(t)),$ (21)

where $x$ is a physical space variable, $M(x)=\sum_{i}m_{i}\delta(Q_{i}-x)$ is
the mass-density operator, which is defined via the position operator
$Q_{i}\psi(q_{1},q_{2},...q_{n})=q_{i}\psi(q_{1},q_{2},...q_{n})$. This allows
us to determine the mass-density ontology at time $t$ via $W(t)$.

For the density-matrix version of GRW theory with just a quantum state
(W-GRW0), we need to introduce the collapse of a density matrix. Similar to
the wave function in GRW0, between collapses, the density matrix in W-GRW0
will evolve unitarily according to the von Neumann equation. It collapses
randomly, where the random time for an $N$-particle system is distributed with
rate $N\lambda$, where $\lambda$ is of order $10^{-15}$ s-1. At a random time
when a collapse occur at “particle” $k$, the post-collapse density matrix is
the following:

$W_{T^{+}}=\frac{\Lambda_{I_{k}}(X)^{1/2}W_{T^{-}}\Lambda_{I_{k}}(X)^{1/2}}{\text{tr}(W_{T^{-}}\Lambda_{I_{k}}(X))},$
(22)

with $X$ distributed by the following probability density:

$\rho(x)=\text{tr}(W_{T^{-}}\Lambda_{I_{k}}(x)),$ (23)

where $W_{T^{+}}$ is the post-collapse density matrix, $W_{T^{-}}$ is the pre-
collapse density matrix, $X$ is the center of the actual collapse, and
$\Lambda_{I_{k}}(x)$ is the collapse rate operator.171717A collapse rate
operator is defined as follows:
$\Lambda_{I_{k}}(x)=\frac{1}{(2\pi\sigma^{2})^{3/2}}e^{-\frac{(Q_{k}-x)^{2}}{2\sigma^{2}}},$
where $Q_{k}$ is the position operator of “particle” $k$, and $\sigma$ is a
new constant of nature of order $10^{-7}$ m postulated in current GRW
theories. Compare W-GRW to $\Psi$-GRW, where collapses happen at the same
rate, and the post-collapse wave function is the following:
$\Psi_{T^{+}}=\frac{\Lambda_{I_{k}}(X)^{1/2}\Psi_{T^{-}}}{||\Lambda_{I_{k}}(X)^{1/2}\Psi_{T^{-}}||},$
(24) with the collapse center $X$ being chosen randomly with probability
distribution $\rho(x)=||\Lambda_{i_{k}}(x)^{1/2}\Psi_{T^{-}}||^{2}dx$.

For the GRW theory (W-GRWm) with both a quantum state $W(t)$ and a mass-
density ontology $m(x,t)$, we can combine the above steps: $W(t)$ evolves by
the von Neumann equation that is randomly interrupted by collapses (22) and
$m(x,t)$ is defined by (21). We can define GRW with a flash-ontology (W-GRWf)
in a similar way, by using $W(t)$ to characterize the distribution of flashes
in physical space-time. The flashes are the space-time events at the centers
($X$) of the W-GRW collapses.

To sum up: in W-S0, the entire world history is described by $W(t)$; in W-Sm,
the local beables (mass-density) is determined by $W(t)$; in W-GRW theories,
$W(t)$ spontaneously collapses. These roles were originally played by $\Psi$,
and now they are played by W. In so far as we have good reasons for Wave
Function Realism based on the roles that $\Psi$ plays in the $\Psi$-theories,
we have equally good reasons for Density Matrix Realism if the universe is
accurately described by W-theories.

###  3.3 Field Intepretations of $W$

Realism about the density matrix only implies that it is objective and not
epistemic. Realism is compatible with a wide range of concrete interpretations
of what the density matrix represents. In this section, I provide three field
interpretations of the density matrix. But they do not exhaust all available
options. In §6, I motivate a nomological interpretation of the density matrix
that is also realist.

In debates about the metaphysics of the wave function, realists have offered
several interpretations of $\Psi$. Wave function realists, such as Albert and
Loewer, have offered a concrete physical interpretation: $\Psi$ represents a
physical field on the high-dimensional configuration space that is taken to be
the fundamental physical space.181818In Chen, 2017b , I argue against this
view and suggest that there are many good reasons—internal and external to
quantum mechanics—for taking the low-dimensional physical space-time to be
fundamental.

Can we interpret the density matrix in a similar way? Let us start with a
mathematical representation of the density matrix $W(t)$. It is defined as a
positive, bounded, self-adjoint operator
$\hat{W}:\mathscr{H}\rightarrow\mathscr{H}$ with $\text{tr}\hat{W}=1$. For
W-BM, the configuration space $\mathbb{R}^{3N}$, and a density operator
$\hat{W}$, the relevant Hilbert space is $\mathscr{H}$, which is a subspace of
the total Hilbert space, i.e.
$\mathscr{H}\subseteq\mathscr{H}_{total}=L^{2}(\mathbb{R}^{3N},\mathbb{C})$.
Now, the density matrix $\hat{W}$ can also be represented as a function

$W:\mathbb{R}^{3N}\times\mathbb{R}^{3N}\rightarrow\mathbb{C}$ (25)

(If we include spin, the range will be the endomorphism space
$\text{End}(\mathbb{C}^{k})$ of the space of linear maps from $\mathbb{C}^{k}$
to itself. Notice that we have already used the position representation in
(19) and (20).)

This representation enables three field interpretations of the density matrix.
Let us use W-BM as an example. First, the fundamental space is represented by
$\mathbb{R}^{6N}$, and $W$ represents a field on that space that assigns
properties (represented by complex numbers) to each point in
$\mathbb{R}^{6N}$. In the Bohmian version, $W$ guides the motion of a “world
particle” like a river guides the motion of a ping pong ball. (However, the
world particle only moves in a $\mathbb{R}^{3N}$ subspace.) Second, the
fundamental space is $\mathbb{R}^{3N}$, and $W$ represents a multi-field on
that space that assigns properties to every ordered pair of points
$(q,q^{\prime})$ in $\mathbb{R}^{3N}$. The world particle moves according to
the gradient taken with respect to the first variable of the multi-field.
Third, the fundamental space is the physical space represented by
$\mathbb{R}^{3}$, and the density matrix represents a multi-field that assigns
properties to every ordered pair of $N$-regions, where each $N$-region is
composed of $N$ points in physical space. On this view, the density matrix
guides the motion of $N$ particles in physical space.191919For discussions
about the multi-field interpretation, see Forrest, (1988), Belot, (2012), Chen
(2017), Chen (ms.) section 3, and Hubert and Romano, (2018).

These three field interpretations are available to the density matrix
realists. In so far as we have good grounds for accepting the field
interpretations of wave function realism, we have equally good grounds for
accepting these interpretations for the W-theories. These physical
interpretations, I hope, can provide further reasons for wave function
realists to take seriously the idea that density matrices _can_ represent
something physically significant. In §6, we introduce a new interpretation of
$W$ as something nomological, and we will motivate that with the new Initial
Projection Hypothesis. That, I believe, is the most interesting realist
interpretation of the universal density matrix all things considered.

##  4 The Initial Projection Hypothesis

W-quantum theories are alternatives to $\Psi$-quantum theories. However, all
of these theories are time-symmetric, as they obey time-reversal invariance
$t\rightarrow-t$.

In statistical mechanics, a fundamental postulate is added to the time-
symmetric dynamics: the Past Hypothesis, which is a low-entropy boundary
condition of the universe. In this section, we will first discuss the wave-
function version of the Past Hypothesis. Then we will use it to pick out a
special density matrix. I call this the _Initial Projection Hypothesis_.
Finally, we point out some connections between the Initial Projection
Hypothesis and Penrose’s Weyl Curvature Hypothesis.

###  4.1 The Past Hypothesis

The history of the Past Hypothesis goes back to Ludwig Boltzmann.202020For an
extended discussion, see Boltzmann, (2012), Albert, (2000), and Callender,
(2011). To explain time asymmetry in a universe governed by time-symmetric
equations, Botlzmann’s solution is to add a boundary condition: the universe
started in a special state of very low-entropy. Richard Feynman agrees, “For
some reason, the universe at one time had a very low entropy for its energy
content, and since then the entropy has increased.”212121Feynman et al.,
(2015), 46-8.  Such a low-entropy initial condition explains the arrow of time
in thermodynamics.222222See Lebowitz, (2008), Ehrenfest and Ehrenfest, (2002)
and Penrose, (1979) for more discussions about a low-entropy initial
condition. See Earman, (2006) for worries about the Past Hypothesis as an
initial condition for the universe. See Goldstein et al., (2016) for a
discussion about the possibility, and some recent examples, of explaining the
arrow of time without the Past Hypothesis.

David Albert (2000) has called this condition the _Past Hypothesis_ (PH).
However, his proposal is stronger than the usual one concerning a low-entropy
initial condition. The usual one just postulates that the universe started in
some low-entropy macrostate. It can be any of the many macrostates, so long as
it has sufficiently low entropy. Albert’s PH postulates that there is a
_particular_ low-entropy macrostate that the universe starts in—the one that
underlies the reliability of our inferences to the past. It is the task of
cosmology to discover that initial macrostate. In what follows, I refer to the
strong version of PH unless indicated otherwise.232323In Chen, 2018b , a
companion paper, I discuss different versions of the Past Hypothesis—the
strong, the weak, and the fuzzy—as well as their implications for the
uniqueness of the initial quantum state that we will come to soon. The upshot
is that in all cases it will be sufficiently unique for eliminating
statistical mechanical probabilities.

In QSM, PH takes the form of §2.2 #9. That is, the microstate (a wave
function) starts in a particular low-dimensional subspace in Hilbert space
(the PH-subspace). However, it does not pin down a unique microstate. There is
still a continuous infinity of possible microstates compatible with the PH-
subspace.

It is plausible to think that, for PH to work as a successful explanation for
the Second Law, it has to be on a par with other fundamental laws of nature.
That is, we should take PH to be a law of nature and not just a contingent
initial condition, for otherwise it might be highly unlikely that our past was
in lower entropy and that our inferences to the past are reliable. Already in
the context of a weaker version of PH, Feynman, (2017) suggests that the low-
entropy initial condition should be understood as a law of nature. However, PH
by itself is not enough. Since there are anti-thermodynamic exceptions even
for trajectories starting from the PH-subspace, it is crucial to impose
another law about a uniform probability distribution on the subspace. This is
the quantum analog of what Albert (2000) calls the Statistical Postulate (SP).
It corresponds to the Lebesgue measure we specified in §2.2 #4. We used it to
state the typicality statement in #10. Barry Loewer calls the joint system—the
package of laws that includes PH and SP in addition to the dynamical laws of
physics—the Mentaculus Vision.242424For developments and defenses of the
nomological account of the Past Hypothesis and the Statistical Postulate, see
Albert, (2000), Loewer, (2007), Wallace, (2011, 2012) and Loewer, (2016).
Albert and Loewer are writing mainly in the context of CSM. The Mentaculus
Vision is supposed to provide a “probability map of the world.” As such, it
requires one to take the probability distribution very seriously. To be sure,
the view that PH is nomological has been challenged. See, for example, Sklar,
(1995) and Callender, (2004). However, these arguments are no more threatening
to IPH being a law than PH being a law. We will come back to this point after
introducing IPH.

###  4.2 Introducing the Initial Projection Hypothesis

The Past Hypothesis uses a low-entropy macrostate (PH-subspace) to constrain
the microstate of the system (a state vector in QSM). This is natural from the
perspective of Wave Function Realism, according to which the state vector (the
wave function) represents the physical degrees of freedom of the system. The
initial state of the system is described by a normalized wave function
$\Psi(t_{0})$. $\Psi(t_{0})$ has to lie in the special low-dimensional Hilbert
space $\mathscr{H}_{PH}$ with $dim\mathscr{H}_{PH}\ll dim\mathscr{H}_{eq}$.
Moreover, there are many different choices of initial wave functions in
$\mathscr{H}_{PH}$. That is, PH is compatible with many different low-entropy
wave functions. Furthermore, for stating the typicality statements, we also
need to specify a measure $\mu$ on the unit sphere of $\mathscr{H}_{PH}$. For
the finite-dimensional case, it is just the uniform Lebesgue measure.

Density Matrix Realism suggests an alternative way to think about the low-
entropy boundary condition. PH pins down the initial macrostate
$\mathscr{H}_{PH}$, a special subspace of the total Hilbert space. Although
$\mathscr{H}_{PH}$ is compatible with many density matrices, there is a
natural choice—the normalized projection operator onto $\mathscr{H}_{PH}$.
Just as in (17), we can specify it as:

$\hat{W}_{PH}(t_{0})=\frac{I_{PH}}{dim\mathscr{H}_{PH}},$ (26)

where $t_{0}$ represents a temporal boundary of the universe, $I_{PH}$ is the
projection operator onto $\mathscr{H}_{PH}$, $dim$ counts the dimension of the
Hilbert space, and $dim\mathscr{H}_{PH}\ll dim\mathscr{H}_{eq}$. Since the
quantum state at $t_{0}$ has the lowest entropy, we call $t_{0}$ the initial
time. We shall call (26) the _Initial Projection Hypothesis_ (IPH). In words:
the initial density matrix of the universe is the normalized projection onto
the PH-subspace.

I propose that we add IPH to any W-quantum theory. The resultant theories will
be called $W_{PH}$-theories. For example, here are the equations of
$W_{PH}$-BM:

  * (A)

$\hat{W}_{PH}(t_{0})=\frac{I_{PH}}{dim\mathscr{H}_{PH}},$

  * (B)

$P(Q(t_{0})\in dq)=W_{PH}(q,q,t_{0})dq,$

  * (C)

$i\hbar\frac{\partial\hat{W}}{\partial t}=[\hat{H},\hat{W}],$

  * (D)

$\frac{dQ_{i}}{dt}=\frac{\hbar}{m_{i}}\text{Im}\frac{\nabla_{q_{i}}W_{PH}(q,q^{\prime},t)}{W_{PH}(q,q^{\prime},t)}(q=q^{\prime}=Q).$

(A) is IPH and (B)—(D) are the defining equations of W-BM. (Given the initial
quantum state $\hat{W}_{PH}(t_{0})$, there is a live possibility that for
every particle at $t_{0}$, its velocity is zero. However, even in this
possibility, as long as the initial quantum state “spreads out” later, as we
assume it would, the particle configuration will typically start moving at a
later time. This is true because of equivariance.252525Thanks to Sheldon
Goldstein and Tim Maudlin for discussions here.)

Contrast these equations with BM formulated with wave functions and PH (not
including SP for now), which will be called $\Psi_{PH}$-BM:

  * (A’)

$\Psi(t_{0})\in\mathscr{H}_{PH},$

  * (B’)

$P(Q(t_{0})\in dq)=|\Psi(q,t_{0})|^{2}dq,$

  * (C’)

$i\hbar\frac{\partial\Psi}{\partial t}=\hat{H}\Psi,$

  * (D’)

$\frac{dQ_{i}}{dt}=\frac{\hbar}{m_{i}}\text{Im}\frac{\nabla_{q_{i}}\Psi(q,t)}{\Psi(q,t)}(Q).$

IPH (A) in $W_{PH}$-BM plays the same role as PH (A’) in $\Psi_{PH}$-BM.
Should IPH be interpreted as a law of nature in $W_{PH}$-theories? I think it
should be, for the same reason that PH should be interpreted as a law of
nature in the corresponding theories. The reason that PH should be interpreted
as a law262626See, for example, Feynman, (2017), Albert, (2000), Loewer,
(2007) and Loewer, (2016). is because it is a particularly simple and
informative statement that accounts for the widespread thermodynamic asymmetry
in time. PH is simple because it characterizes a simple macrostate
$\mathscr{H}_{PH}$, of which the initial wave function is a vector. PH is
informative because with PH the dynamical equations predict time asymmetry and
without PH the dynamical equations cannot. Similarly, IPH is simple because it
provides crucial resources for explaining the arrow of time. IPH is
informative because it is essential for explaining the time asymmetry in a
quantum universe described by a density matrix. (This is in addition to the
fact that IPH helps determine the $W_{PH}$-version of the guidance equation
(D).) To be sure, PH and IPH as laws face the same worries: both are
statements about boundary conditions but we usually think of laws as dynamical
equations. However, these worries are no more threatening to IPH being a law
than PH being a law.

Let us make three remarks about IPH. Firstly, IPH defines a unique initial
quantum state. The quantum state $\hat{W}_{PH}(t_{0})$ is informationally
equivalent to the constraint that PH imposes on the initial microstates.
Assuming that PH selects a unique low-entropy macrostate,
$\hat{W}_{PH}(t_{0})$ is singled out by the data in PH.272727The weaker
versions of PH are vague about the exact initial low-entropy macrostate. It is
vague because, even with a choice of macro-variables, there may be many
subspaces that can play the role of a low-entropy initial condition. It would
be arbitrary, from the viewpoint of wave-function theories, to pick a specific
subspace. In contrast, it would not be arbitrary from the viewpoint of
$W_{PH}$-theories, as the specific subspace defines $W_{PH}$, which determines
the dynamics.

Secondly, on the universal scale, we do not need to impose an additional
probability or typicality measure on the Hilbert space. $\hat{W}_{PH}(t_{0})$
is mathematically equivalent to an integral over projection onto each
normalized state vectors (wave functions) compatible with PH _with respect to
a Lebesgue measure_. But here we are not defining $\hat{W}_{PH}(t_{0})$ in
terms of state vectors. Rather, we are thinking of $\hat{W}_{PH}(t_{0})$ as a
geometric object in the Hilbert space: the (normalized) projection operator
onto $\mathscr{H}_{PH}$. That is the _intrinsic_ understanding of the density
matrix.282828After writing the paper, I discovered that David Wallace has come
to a similar idea in a forthcoming paper. There are some subtle differences.
He proposes that we can reinterpret probability distributions in QSM as actual
mixed states. Consequently, the problem of statistical mechanical probability
is “radically transformed” (if not eliminated) in QSM. Wallace’s proposal is
compatible with different probability distributions and hence different mixed
states of the system. It does not require one to choose a particular quantum
state such as (26). In contrast, I propose a particular, natural initial
quantum state of the universe based on the PH subspace—the normalized
projection onto the PH subspace (26). As we discuss in §5.1, this also leads
to the elimination of statistical mechanical probability, since the initial
state is fixed in the theory. Moreover, as we discuss below, the natural state
inherits the simplicity of the PH subspace, which has implications for the
nature of the quantum state. For a more detailed comparison, see Chen, 2018b .

Thirdly, $\hat{W}_{PH}(t_{0})$ is simple. Related to the first remark, IPH
defines $\hat{W}_{PH}(t_{0})$ explicitly as the normalized projection operator
onto $\mathscr{H}_{PH}$. There is a natural correspondence between a subspace
and its projection operator. If we specify the subspace, we know what its
projection operator is, and vice versa. Since the projection operator onto a
subspace carries no more information than that subspace itself, the projection
operator is no more complex than $\mathscr{H}_{PH}$. This is different from
$\Psi_{PH}$, which is confined by PH to be a vector inside $\mathscr{H}_{PH}$.
A vector carries more information than the subspace it belongs to, as
specifying a subspace is not sufficient to determine a vector. For example, to
determine a vector in an 18-dimensional subspace of a 36-dimensional vector
space, we need 18 coordinates in addition to specifying the subspace. The
higher the dimension of the subspace, the more information is needed to
specify the vector. If PH had fixed $\Psi_{PH}$ (the QSM microstate), it would
have required much more information and become a much more complex posit. PH
as it is determines $\Psi_{PH}$ only up to an equivalence class (the QSM
macrostate). As we shall see in §6, the simplicity of $\hat{W}_{PH}(t_{0})$
will be a crucial ingredient for a new version of the nomological
interpretation of the quantum state.

###  4.3 Connections to the Weyl Curvature Hypothesis

Let us point out some connections between our Initial Projection Hypothesis
(IPH) and the Weyl Curvature Hypothesis (WCH) proposed by Penrose, (1979).
Thinking about the origin of the Second Law of Thermodynamics in the early
universe with high homogeneity and isotropy, and the relationship between
space-time geometry and entropy, Penrose proposes a low-entropy hypothesis:

> I propose, then, that there should be complete lack of chaos in the initial
> _geometry_. We need, in any case, some kind of low-entropy constraint on the
> initial state. But thermal equilibrium apparently held (at least very
> closely so) for the _matter_ (including radiation) in the early stages. So
> the ‘lowness’ of the initial entropy was not a result of some special matter
> distribution, but, instead, of some very special initial spacetime geometry.
> The indications of [previous sections], in particular, are that this
> restriction on the early geometry should be something like: _the Weyl
> curvature $C_{abcd}$ vanishes at any initial singularity_. (Penrose, (1979),
> p.630, emphasis original)

The Weyl curvature tensor $C_{abcd}$ is the traceless part of the Riemann
curvature tensor $R_{abcd}$. It is not fixed completely by the stress-energy
tensor and thus has independent degrees of freedom in Einstein’s general
theory of relativity. Since the entropy of the matter distribution is quite
high, the origin of thermodynamic asymmetry should be due to the low entropy
in geometry, which corresponds very roughly to the vanishing of the Weyl
curvature tensor.

WCH is an elegant and simple way of encoding the initial low-entropy boundary
condition in the classical spacetime geometry. If WCH could be extended to a
quantum theory of gravity, presumably it would pick out a simple subspace (or
subspaces) of the total Hilbert space that corresponds to $C_{abcd}\rightarrow
0$. Applying IPH to such a theory, the initial density matrix will be the
normalized projection onto that subspace (subspaces).292929There is another
connection between the current project and Penrose’s work. The W-Everettian
theory that we considered in §3.2 combined with the Initial Projection
Hypothesis is a theory that satisfies _strong determinism_ (Penrose, (1999)).
This is because the entire history of the WPH-Everettian universe described by
$W_{PH}(t)$, including its initial condition, is fixed by the laws.

##  5 Theoretical Payoffs

WPH-quantum theories, the result of applying IPH to W-theories, have two
theoretical payoffs, which we explore in this section. These are by no means
decisive arguments in favor of the density-matrix framework, but they display
some interesting differences with the wave-function framework.

###  5.1 Harmony between Statistical Mechanics and Quantum Mechanics

In WPH-quantum theories, statistical mechanics is made more harmonious with
quantum mechanics. As we pointed out earlier, standard QM and QSM contain the
wave function in addition to the density matrix, and they require the addition
of both the Past Hypothesis (PH) and the Statistical Postulate (SP) to the
dynamical laws. In particular, we have two kinds of probabilities: the
quantum-mechanical ones (Born rule probabilities) and the statistical
mechanical ones (SP). The situation is quite different in our framework. This
is true for all the WPH-theories. We will use WPH-BM ((A)—(D)) as an example.

WPH-BM completely specifies the initial quantum state, unlike $\Psi_{PH}$-BM.
For $\Psi_{PH}$-BM, because of time-reversal invariance, some initial wave
functions compatible with PH will evolve to lower entropy. These are called
_anti-entropic_ exceptions. However, the uniform probability distribution (SP)
assigns low probability to these exceptions. Hence, we expect that with
overwhelming probability the actual wave function is entropic. For WPH-BM, in
contrast, there is no need for something like SP, as there is only one initial
density matrix compatible with IPH—$W_{PH}(t_{0})$. It is guaranteed to evolve
to future states that have entropic behaviors. Therefore, on the universal
scale, WPH-BM eliminates the need for SP and thus the need for a
probability/typicality measure that is in addition to the quantum-mechanical
measure (B). This is a nice feature of WPH-theories, as it is desirable to
unify the two sources of randomness: quantum-mechanical and statistical-
mechanical. Of course, wave functions and statistical-mechanical probabilities
are still useful to analyze subsystems such as gas in a box, but they no
longer play fundamental roles in WPH-theories. Another strategy to eliminate
SP has been explored in the context of GRW jumps by Albert, (2000). Wallace,
(2011, 2012) has proposed a replacement of SP with a non-probabilistic
constraint on the microstate, giving rise to the _Simple Dynamical
Conjecture_. These are quite different proposals, all of which deserve further
developments.

###  5.2 Descriptions of the Universe and the Subsystems

WPH-quantum theories also bring more unity to the kinematics and the dynamics
of the universe and the subsystems.

Let us start with a quantum-mechanical universe $U$. Suppose it contains many
subsystems. Some of them will be interacting heavily with the environment,
while others will be effectively isolated from the environment. For a universe
that contain some quasi-isolated subsystems (interactions with the environment
effectively vanish), the following is a desirable property:

Dynamic Unity The dynamical laws of the universe are the same as the effective
laws of most quasi-isolated subsystems.

Dynamic Unity is a property that can come in degrees, rather than an “on-or-
off” property. Theory A has more dynamic unity than Theory B, if the
fundamental equations in A are valid in more subsystems than those in B. This
property is desirable, but not indispensable. It is desirable because law
systems that apply both at the universal level and at the subsystem level are
unifying and explanatory.

W-BM has more dynamic unity than BM formulated with a universal wave function.
For quantum systems without spin, we can always follow Dürr et al., (1992) to
define _conditional wave functions_ in BM. For example, if the universe is
partitioned into a system $S_{1}$ and its environment $S_{2}$, then for
$S_{1}$, we can define its conditional wave function:

$\psi_{cond}(q_{1})=C\Psi(q_{1},Q_{2}),$ (27)

where C is a normalization factor and $Q_{2}$ is the actual configuration of
$S_{2}$. $\psi_{cond}(q_{1})$ always gives the velocity field for the
particles in $S_{1}$ according to the guidance equation. However, for quantum
systems with spin, this is not always true. Since BM is described by
($\Psi(t),Q(t)$), it does not contain actual values of spin. Since there are
no actual spins to plug into the spin indices of the wave function, we cannot
always define conditional wave functions in an analogous way. Nevertheless, in
those circumstances, we can follow Dürr et al., (2005) to define a
_conditional density matrix_ for $S_{1}$, by plugging in the actual
configuration of $S_{2}$ and tracing over the spin components in the wave
function associated with $S_{2}$.303030 The conditional density matrix for
$S_{1}$ is defined as:
${W_{cond}}_{s_{1}^{\prime}}^{s_{1}}(q_{1},q_{1}^{\prime})=\frac{1}{N}\sum_{s_{2}}\Psi^{s_{1}s_{2}}(q_{1},Q_{2})\Psi_{s_{1}s_{2}}^{*}(q_{1}^{\prime},Q_{2}),$
(28) with the normalizing factor:
$N=\int_{\mathcal{Q}_{1}}dq_{1}\sum_{s_{1}s_{2}}\Psi^{s_{1}s_{2}}(q_{1},Q_{2})\Psi_{s_{1}s_{2}}^{*}(q_{1}^{\prime},Q_{2}).$
(29) The conditional density matrix will guide the particles in $S_{1}$ by the
W-guidance equation (the spin version with the partial trace operator).

In W-BM, the W-guidance equation is always valid for the universe and the
subsystems. In BM, sometimes subsystems do not have conditional wave
functions, and thus the wave-function version of the guidance equation is not
always valid. In this sense, the W-BM equations are valid in more
circumstances than the BM equations. However, this point does not rely on IPH.

What about Everettian and GRW theories? Since GRW and Everettian theories do
not have fundamental particles, we cannot obtain conditional wave functions
for subsystems as in BM. However, even in the $\Psi$-versions of GRW and
Everett, many subsystems will not have pure-state descriptions by wave
functions due to the prevalence of entanglement. Most subsystems can be
described only by a mixed-state density matrix, even when the universe as a
whole is described by a wave function. In contrast, in WPH-Everett theories
and WPH-GRW theories, there is more uniformity across the subsystem level and
the universal level: the universe as a whole as well as most subsystems are
described by the same kind of object—a (mixed-state) density matrix. Since
state descriptions concern the kinematics of a theory, we say that W-Everett
and W-GRW theories have more _kinematic unity_ than their $\Psi$-counterparts:

Kinematic Unity The state description of the universe is of the same kind as
the state descriptions of most quasi-isolated subsystems.

So far, my main goal has been to show that Density Matrix Realism + IPH is a
viable position. They have theoretical payoffs that are interestingly
different from those in the original package (Wave Function Realism + PH). In
the next section, we look at their relevance to the nature of the quantum
state.

##  6 The Nomological Thesis

Combining Density Matrix Realism with IPH gives us WPH-quantum theories that
have interesting theoretical payoffs. We have also argued that the initial
quantum state in such theories would be simple and unique. In this section, we
show that the latter fact lends support to the nomological interpretation of
the quantum state:

The Nomological Thesis:

The initial quantum state of the world is nomological.

However, “nomological” has several senses and has been used in several ways in
the literature. We will start with some clarifications.

###  6.1 The Classical Case

We can clarify the sense of the“nomological” by taking another look at
classical mechanics. In classical $N$-particle Hamiltonian mechanics, it is
widely accepted that the Hamiltonian function is nomological, and that the
ontology consists in particles with positions and momenta. Their state is
given by
$X=(\boldsymbol{q_{1}}(t),...,\boldsymbol{q_{N}}(t);\boldsymbol{p_{1}}(t),...,\boldsymbol{p_{n}}(t))$,
and the Hamiltonian is $H=H(X)$. Particles move according to the Hamiltonian
equations:

$\frac{d\boldsymbol{q_{i}}(t)}{dt}=\frac{\partial
H}{\partial\boldsymbol{p_{i}}}\text{ ,
}\frac{d\boldsymbol{p_{i}}(t)}{dt}=-\frac{\partial
H}{\partial\boldsymbol{q_{i}}}.$ (30)

Their motion corresponds to a trajectory in phase space. The velocity field on
phase space is obtained by taking suitable derivatives of the Hamiltonian
function $H$. The equations have the form:

$\frac{dX}{dt}=F(X)=F^{H}(X)$ (31)

Here, $F^{H}(X)$ is $H(q,p)$ with suitable derivative operators. The
Hamiltonian equations have a simple form, because $H$ is simple. $H$ can be
written explicitly as follows:

$H=\sum^{N}_{i}\frac{p_{i}^{2}}{2m_{i}}+V,$ (32)

where $V$ takes on this form when we consider electric and gravitational
potentials:

$V=\frac{1}{4\pi\epsilon_{0}}\sum_{1\leq j\leq k\leq
N}\frac{e_{j}e_{k}}{|q_{j}-q_{k}|}+\sum_{1\leq j\leq k\leq
N}\frac{Gm_{j}m_{k}}{|q_{j}-q_{k}|},$ (33)

That is, the RHS of the Hamiltonian equations, after making the Hamiltonian
function explicit, are still simple. $H$ is just a convenient shorthand for
(32) and (33). Moreover, $H$ is also fixed by the theory. A classical universe
is governed by the dynamical laws plus the fundamental interactions. If $H$
were different in (31), then we would have a different physical theory (though
it would still belong to the class of theories called classical mechanics).
For example, we can add another term in (33) to encode another fundamental
interaction, which will result in a different theory.

Consequently, it is standard to interpret $H$ as a function in (30) that does
not represent things or properties of the ontology. Expressed in terms of $H$,
the equations of motion take a particularly simple form. The sense that $H$ is
nomological is that (i) it generates motion, (ii) it is simple, (iii) it is
fixed by the theory (nomologically necessary), and (iv) it does not represent
things in the ontology. In contrast, the position and momentum variables in
(30) are “ontological” in that they represent things and properties of the
ontology, take on complicated values, change according to $H$, and are not
completely fixed by the theory (contingent).

###  6.2 The Quantum Case

It is according to the above sense that Goldstein and Zanghì, (2013) proposes
that the universal wave function in BM is nomological (and governs things in
the ontology). With the guidance equation, $\Psi$ generates the motion of
particles. It is of the same form as above:

$\frac{dX}{dt}=F(X)=F^{\Psi}(X).$ (34)

Why is it simple? Generic wave functions are not simple. However, they observe
that, in some formulations of quantum gravity, the universal wave function
satisfies the Wheeler-DeWitt equation and is therefore stationary. To be
stationary, the wave function does not have time-dependence and probably has
many symmetries, in which case it _could_ be quite simple. The Bohmian theory
then will explicitly stipulate what the universal wave function is. Therefore,
in these theories, provided that $\Psi$ is sufficiently simple, we can afford
the same interpretation of $\Psi$ as we can for $H$ in classical mechanics:
both are nomological in the above sense.

$W_{PH}$-BM also supports the nomological interpretation of the quantum state
but via a different route. With the W-guidance equation, $W_{PH}$ generates
the motion of particles. It is of the same form as above:

$\frac{dX}{dt}=F(X)=F^{W_{PH}}(X).$ (35)

Why is it simple? Here we do not need to appeal to specific versions of
quantum gravity, which are still to be worked out and may not guarantee the
simplicity of $\Psi$. Instead, we can just appeal to IPH. We have argued in
§4.2 that IPH is simple and that $W_{PH}(t_{0})$ is simple. Since the quantum
state evolves unitarily by the von Neumann equation, we can obtain the quantum
state at any later time as:

$\hat{W}_{PH}(t)=e^{-i\hat{H}t/\hbar}\hat{W}_{PH}(t_{0})e^{i\hat{H}t/\hbar}$
(36)

Since $W_{PH}(t)$ is a simple function of the time-evolution operator and the
initial density matrix, and since both are simple, $W_{PH}(t)$ is also simple.
So we can think of $W_{PH}(t)$ just as a convenient shorthand for (36). (This
is not true for $\ket{\Psi(t)}=\hat{H}\ket{\Psi(t_{0})}$, as generic
$\ket{\Psi(t_{0})}$ is not simple at all.)

The “shorthand” way of thinking about $W_{PH}(t)$ implies that the equation of
particle motion has a time-dependent form $F^{W_{PH}}(X,t)$. Does time-
dependence undercut the nomological interpretation? It does not in this case,
as the $F^{W_{PH}}(X,t)$ is still simple even with time-dependence. It is true
that time-independence is often a hallmark of a nomological object, but it is
not always the case. In this case, we have simplicity without time-
independence. Moreover, unlike Goldstein and Zanghì, (2013), we do not need
time-independence to argue for the simplicity of the quantum state.

Since $W_{PH}(t_{0})$ is fixed by IPH, $F^{W_{PH}}$ is also fixed by the
theory. Let us expand (35) to make it more explicit:

$\frac{dQ_{i}}{dt}=\frac{\hbar}{m_{i}}\text{Im}\frac{\nabla_{q_{i}}W_{PH}(q,q^{\prime},t)}{W_{PH}(q,q^{\prime},t)}(Q)=\frac{\hbar}{m_{i}}\text{Im}\frac{\nabla_{q_{i}}\bra{q}e^{-i\hat{H}t/\hbar}\hat{W}_{PH}(t_{0})e^{i\hat{H}t/\hbar}\ket{q^{\prime}}}{\bra{q}e^{-i\hat{H}t/\hbar}\hat{W}_{PH}(t_{0})e^{i\hat{H}t/\hbar}\ket{q^{\prime}}}(q=q^{\prime}=Q)$
(37)

The initial quantum state (multiplied by the time-evolution operators)
generates motion, has a simple form, and is fixed by the boundary condition
(IPH) in $W_{PH}$-BM. Therefore, it is nomological. This is of course a modal
thesis. The initial quantum state, which is completely specified by IPH, could
not have been different.

Let us consider other $W_{PH}$-theories with local beables. In $W_{PH}$-Sm,
the initial quantum state has the same simple form and is fixed by IPH. It
does not generate a velocity field, since there are no fundamental particles
in the theory. Instead, it determines the configuration of the mass-density
field on physical space. This is arguably different from the sense of
nomological that $H$ in classical mechanics displays. Nevertheless, the mass-
density field and the Bohmian particles play a similar role—they are “local
beables” that make up tables and chairs, and they are governed by the quantum
state. In $W_{PH}$-GRWm and $W_{PH}$-GRWf, the initial quantum state has the
same simple form and is fixed by IPH. It does not generate a velocity field,
and it evolves stochastically. This will determine a probability distribution
over configurations of local beables—mass densities or flashes—on physical
space. The initial quantum state in these theories can be given an _extended
nomological interpretation_ , in the sense that condition (i) is extended such
that it covers other kinds of ontologies and dynamics: (i’) the quantum state
determines (deterministically or stochastically) the configuration of local
beables.

The $W_{PH}$-theories with local beables support the nomological
interpretation of the initial quantum state. It can be interpreted in non-
Humean ways and Humean ways. On the non-Humean proposal, we can think of the
initial quantum state as an additional nomological entity that _explains_ the
distribution of particles, fields, or flashes. On the Humean proposal, in
contrast, we can think of the initial quantum state as something that
_summarizes_ a separable mosaic. This leads to reconciliation between Humean
supervenience and quantum entanglement.

###  6.3 Humean Supervenience

Recall that according to Humean supervenience (HS), the ”vast mosaic of local
matters of particular fact” is a _supervenience base_ for everything else in
the world, the _metaphysical ground floor_ on which everything else depends.
On this view, laws of physics are nothing over and above the “mosaic.” They
are just the axioms in the simplest and most informative summaries of the
local matters of particular fact. A consequence of HS is that the complete
physical state of the universe is determined by the properties and
spatiotemporal arrangement of the local matters (suitably extended to account
for vector-valued magnititudes) of particular facts. It follows that there
should not be any state of the universe that fails to be determined by the
properties of individual space-time points.313131This is one reading of David
Lewis. Tim Maudlin (2007) calls this thesis “Separability.” Quantum
entanglement, if it were in the fundamental ontology, would present an
obstacle to HS, because entanglement is not determined by the properties of
space-time points. The consideration above suggests a strong _prima facie_
conflict between HS and quantum physics. On the basis of quantum non-
separability, Tim Maudlin has proposed an influential argument against
HS.323232See Maudlin, (2007), Chapter 2.

$W_{PH}$-theories with local beables offer a way out of the conflict between
quantum entanglement and Humean supervenience. A Humean can interpret the laws
(including the IPH) as the axioms in the best system that summarize a
separable mosaic. Take WPH-BM as an example:

The WPH-BM mosaic:

particle trajectories $Q(t)$ on physical space-time.

The WPH-BM best system:

four equations—the simplest and strongest axioms summarizing the mosaic:

  * (A)

$\hat{W}_{PH}(t_{0})=\frac{I_{PH}}{dim\mathscr{H}_{PH}}$

  * (B)

$P(Q(t_{0})\in dq)=W_{PH}(q,q,t_{0})dq,$

  * (C)

$i\hbar\frac{\partial\hat{W}}{\partial t}=[\hat{H},\hat{W}],$

  * (D)

$\frac{dQ_{i}}{dt}=\frac{\hbar}{m_{i}}\text{Im}\frac{\nabla_{q_{i}}W_{PH}(q,q^{\prime},t)}{W_{PH}(q,q^{\prime},t)}(q=q^{\prime}=Q).$

Notice that (A)—(D) are simple and informative statements about $Q(t)$. They
are expressed in terms of $\hat{W}_{PH}(t)$, which via law (C) can be
expressed in terms of $\hat{W}_{PH}(t_{0})$. We have argued previously that
the initial quantum state can be given a nomological interpretation. The
Humean maneuver is that the law statements are to be understood as axioms of
the best summaries of the mosaic. The mosaic described above is completely
separable, while the best system, completely specifying the quantum state and
the dynamical laws, contains all the information about quantum entanglement
and superpositions. The entanglement facts are no longer fundamental. As on
the original version of Humean supervenience, the best system consisting of
(A)—(D) supervenes on the mosaic. Hence, this proposal reconciles Humean
supervenience with quantum entanglement. As it turns out, the above version of
Quantum Humeanism also achieves more theoretical harmony, dynamical unity, and
kinematic unity (§5), which are desirable from the Humean best-system
viewpoint. We can perform similar “Humeanization” maneuvers on the density
matrix in other quantum theories with local beables—W-GRWm, W-GRWf, and W-Sm
(although such procedures might not be as compelling).

This version of Quantum Humeanism based on $W_{PH}$-theories is different from
the other approaches in the literature: Albert, (1996), Loewer, (1996),
Miller, (2014), Esfeld, (2014), Bhogal and Perry, (2015), Callender, (2015)
and Esfeld and Deckert, (2017). In contrast to the high-dimensional proposal
of Albert, (1996) and Loewer, (1996), our version preserves the fundamentality
of physical space.

The difference between our version and those of Miller, (2014), Esfeld,
(2014), Bhogal and Perry, (2015), Callender, (2015), and Esfeld and Deckert,
(2017) is more subtle. They are concerned primarily with $\Psi$-BM. We can
summarize their views as follows (although they do not agree on all the
details). There are several parts to their proposals. First, the wave function
is merely part of the best system. It is more like parameters in the laws such
as mass and charge. Second, just like the rest of the best system, the wave
function supervenes on the mosaic of particle trajectories. Third, the wave
function does not have to be very simple. The Humean theorizer, on this view,
just needs to find the simplest and strongest summary of the particle
histories, but the resultant system can be complex _simpliciter_. One
interpretation of this view is that the best system for $\Psi_{PH}$-BM is just
(A’)—(D’) in §4.2 (although they do not explicitly consider (A’)), such that
neither the mosaic nor the best system specifies the exact values of the
universal wave function. In contrast, our best system completely specifies the
universal quantum state. The key difference between our approaches is that
their interpretation of the wave function places much weaker constraints than
our nomological interpretation does. It is much easier for something to count
as being part of the best system on their approach than on ours. While they do
not require the quantum state to be simple, we do. For them, the Bohmian
guidance equation is likely very complex after plugging in the actual wave
function $\Psi_{PH}$ on the RHS, but $\Psi_{PH}$ can still be part of their
best system.333333See Dewar, (2017) §5 for some worries about the weaker
criterion on the best system. For us, it is crucial that the equation remains
simple after plugging in $W_{PH}(t_{0})$ for it to be in the best system.
Consequently, $W_{PH}(t_{0})$ is nomological in the sense spelled out in §6.1,
and we can give it a Humean interpretation similar to that of the Hamiltonian
function in CM. Generic $\Psi_{PH}$, on the other hand, cannot be nomological
in our sense. But that is ok for them, as their best-system interpretation
does not require the strong nomological condition that we use. Here we do not
attempt to provide a detailed comparison; we will do that in future work.

##  7 Conclusion

I have introduced a new package of views: Density Matrix Realism, the Initial
Projection Hypothesis, and the Nomological Thesis. In the first two steps, we
introduced a new class of quantum theories—WPH-theories. In the final step, we
argue that it is a theory in which the initial quantum state _can_ be given a
nomological interpretation. Each is interesting in its own right, and they do
not need to be taken together. However, they fit together quite well. They
provide alternatives to standard versions of realism about quantum mechanics,
a new way to get rid of statistical-mechanical probabilities, and a new
solution to the conflict between quantum entanglement and Humean
Supervenience. To be sure, there are many other features of WPH-theories in
general and the nomological interpretation in particular that are worth
exploring further.

The most interesting feature of the new package, I think, is that it brings
together the foundations of quantum mechanics and quantum statistical
mechanics. In WPH-theories, the arrow of time becomes intimately connected to
the quantum-mechanical phenomena in nature. It is satisfying to see that
nature is so unified.

## Acknowledgement

I would like to thank the editors and referees of _The British Journal for the
Philosophy of Science_ for helpful feedback. I am also grateful for
stimulating discussions with David Albert, Sheldon Goldstein, and Barry
Loewer. I have received helpful feedback from Abhay Ashtekar, Karen Bennett,
Harjit Bhogal, Max Bialek, Miren Boehm, Robert Brandenberger, Tad Brennan,
Sean Carroll, Juliusz Doboszewski, Detlef Dürr, Denise Dykstra, Michael
Esfeld, Veronica Gomez, Hans Halvorson, Harold Hodes, Mario Hubert, Michael
Kiessling, Dustin Lazarovici, Stephen Leeds, Matthias Lienert, Niels
Linnemann, Chuang Liu, Vera Matarese, Tim Maudlin, Elizabeth Miller, Sebastian
Murgueitio, Wayne Myrvold, Jill North, Zee Perry, Davide Romano, Ezra
Rubenstein, Charles Sebens, Ted Sider, Joshua Spencer, Noel Swanson, Karim
Thébault, Roderich Tumulka, David Wallace, Isaac Wilhelm, Nino Zanghì,
audiences at Cornell University, the University of Western Ontario, the
University of Wisconsin-Milwaukee, and the 2018 Rotman Summer Institute in
Philosophy of Cosmology.

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