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Quantum Mechanics in a Time-Asymmetric Universe:
On the Nature of the Initial Quantum State
Eddy Keming Chen*
The British Journal for the Philosophy of Science 74(4), 2021
First published online on October 13, 2018
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 con-
dition.
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, Wentaculus, typicality, unification, foundations of prob-
ability, quantum statistical mechanics, wave function realism, quantum ontology, density
matrix, Weyl Curvature Hypothesis, Humean Supervenience
*Department of Philosophy, University of California, San Diego, 9500 Gilman Dr, La Jolla, CA
92093-0119. Website: www.eddykemingchen.net. Email: eddykemingchen@ucsd.edu
1
Contents
1 Introduction 2
2 Foundations of Quantum Mechanics and Statistical Mechanics 4
2.1 QuantumMechanics ............................... 4
2.2 Quantum Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Density Matrix Realism 9
3.1 W-BohmianMechanics .............................. 10
3.2 W-Everettian and W-GRW Theories . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Field Intepretations of W............................. 13
4 The Initial Projection Hypothesis 14
4.1 ThePastHypothesis................................ 14
4.2 Introducing the Initial Projection Hypothesis . . . . . . . . . . . . . . . . 16
4.3 Connections to the Weyl Curvature Hypothesis . . . . . . . . . . . . . . 18
5 Theoretical Payoffs 19
5.1 Harmony between Statistical Mechanics and Quantum Mechanics . . 19
5.2 Descriptions of the Universe and the Subsystems . . . . . . . . . . . . . 20
6 The Nomological Thesis 21
6.1 TheClassicalCase ................................. 22
6.2 TheQuantumCase ................................ 23
6.3 Humean Supervenience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
7 Conclusion 26
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.1
1See 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 (2017), Callender
(2015), Esfeld and Deckert (2018), Chen (2018, 2017, 2016), Hubert and Romano (2018). For a survey
2
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 in-
strumentalists. As a scientific realist, I do not find their arguments convincing. In
previous papers, I have assumed and defended Wave Function Realism. Neverthe-
less, 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 indis-
pensable 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.2In 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 bound-
of this literature, see Chen (2019b). 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.
2See, for example, Dürr et al. (2005), Maroney (2005), Wallace (2011, 2012), and Wallace (2016).
3
ary condition called the Past Hypothesis.3I 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 t0is 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 hy-
pothesis 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 re-
duction 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 quan-
tum mechanics and quantum statistical mechanics. In §3, I introduce the framework
of Density Matrix Realism and provide some illustrations. In §4, I propose the Ini-
tial 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 ψ. For a spin-less N-particle quantum system in R3, the wave
function is a (square-integrable) function from the configuration space R3Nto the
complex numbers C. Secondly, the wave function evolves in time according to the
the Schrödinger equation:
3For an extended discussion, see Albert (2000).
4
i̵
h∂ψ
∂t=Hψ(1)
Thirdly, the Schrödinger evolution of the wave function is supplemented with col-
lapse 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 Xand 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 Xis given by the Born rule:
P(X)=X∣ψ(x)∣2dx (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.4
Various solutions have been proposed regarding the measurement problem.
Bohmian mechanics (BM) solves it by adding particles to the ontology and an addi-
tional guidance equation for the particles’ motion. Ghirardi-Rimini-Weber (GRW)
theories postulate a spontaneous collapse mechanism. Everettian quantum mechan-
ics (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.5In BM, in addition
to the wave function that evolves unitarily according to the Schrödinger equation,
particles have precise locations, and their configuration Q=(Q1,Q2, ..., QN)follows
the guidance equation:
dQi
dt =̵
h
mi
Im∇iψ(q)
ψ(q)(q=Q)(3)
Moreover, the initial particle distribution is given by the quantum equilibrium
distribution:
ρt0(q)=∣ψ(q,t0)∣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
4See Bell (1990) and Myrvold (2017) for introductions to the quantum measurement problem.
5See Dürr et al. (1992) for a rigorous presentation of BM and its statistical analysis.
5
rule predictions (which are all there is to the observable predictions of quantum
mechanics). For a universe with Nparticles, let us call the wave function of the
universe the universal wave function and denote it by Ψ(q1,q2, ...qN).
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 Nfermions (with N>1020) in a box
Λ=[0,L]3⊂R3and a Hamiltonian ˆ
H. I will first present the “individualistic” view
followed by the “ensemblist” view of quantum statistical mechanics (QSM).6I will
include some brief remarks comparing QSM to classical statistical mechanics (CSM),
the latter of which may be more familiar to some readers.
1. Microstate: at any time t, the microstate of the system is given by a normalized
(and anti-symmetrized) wave function:
ψ(q1, ..., qN)∈Htotal =L2(R3N,Ck),∥ψ∥L2=1,(5)
where Htotal =L2(R3N,Ck)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. Dynamics: the time dependence of ψ(q1, ..., qN;t)is given by the Schrödinger
equation:
i̵
h∂ψ
∂t=Hψ. (6)
(In CSM, the particles move according to the Hamiltonian equations.)
3. Energy shell: the physically relevant part of the total Hilbert space is the
subspace (“the energy shell”):
H⊆Htotal ,H=span{φα∶Eα∈[E,E+δE]},(7)
This is the subspace (of the total Hilbert space) spanned by energy eigenstates
φαwhose eigenvalues Eαbelong to the [E,E+δE]range. Let D=dimH, the
number of energy levels between Eand E+δE.
We only consider wave functions ψin H.
4. Measure: the measure µis given by the normalized surface area measure on
the unit sphere in the energy subspace S(H).
5. Macrostate: with a choice of macro-variables (suitably “rounded” à la Von Neu-
mann (1955)), the energy shell Hcan be orthogonally decomposed into macro-
spaces:
H=⊕νHν,
ν
dimHν=D(8)
6Here I follow the discussions in Goldstein et al. (2010a) and Goldstein and Tumulka (2011).
6
Each Hνcorresponds 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. 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 ψis (in the Hilbert space norm) very close
to a macrostate Hν:⟨ψ∣Pν∣ψ⟩≈1,(9)
where Pνis the projection operator onto Hν. This means that almost all of ∣ψ⟩
lies in Hν. (In CSM, a phase point is always entirely within some macrostate.)
7. Thermal equilibrium: typically, there is a dominant macro-space Heq that has
a dimension that is almost equal to D:
dimHeq
dimH≈1.(10)
A system with wave function ψis in equilibrium if the wave function ψis very
close to Heq in the sense of (9): ⟨ψ∣Peq ∣ψ⟩≈1.
Simple Example. Consider a gas consisting of n=1023 atoms in a box Λ⊆R3.
The system is governed by quantum mechanics. We orthogonally decompose
the Hilbert space Hinto 51 macro-spaces: H0⊕H2⊕H4⊕... ⊕H100,where
Hνis the subspace corresponding to the macrostate such that the number of
atoms in the left half of the box is between (ν−1)% and (ν+1)% of n, with
the endpoints being the exceptions: H0is the interval 0% −1%, and H100 is the
interval 99% −100%. In this example, H50 has the overwhelming majority of
dimensions and is thus the equilibrium macro-space. A system whose wave
function is very close to H50 is in equilibrium (for this choice of macrostates).
8. Boltzmann Entropy: the Boltzmann entropy of a quantum-mechanical system
with wave function ψthat is very close to a macrostate Hνis given by:
SB(ψ)=kBlog(dimHν),(11)
where Hνdenotes the subspace containing almost all of ψin the sense of (9).
The thermal equilibrium state thus has the maximum entropy:
SB(eq)=kBlog(dimHeq)≈kBlog(D),(12)
where Heq 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. 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
7
Hypothesis:
Ψ(t0)∈HPH , dimHPH ≪dimHeq ≈dimH(13)
where HPH is the Past Hypothesis macro-space with dimension much smaller
than that of the equilibrium macro-space.7Hence, 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. A central task of QSM is to establish mathematical results that demonstrate
(or suggest) that for µ−most (maybe even all) wave functions, the small sub-
systems, 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 ˆ
W, a density matrix.8It evolves
according to the von Neumann equation:
i̵
hdˆ
W(t)
dt =[ˆ
H,ˆ
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=ρmc or W=ρcan,(15)
where ρmc is the microcanonical ensemble and ρcan is the canonical ensemble.9
For the QSM individualist, if the microstate ψof a system is close to some macro-
space Hνin the sense of (9), we can say that the macrostate of the system is Hν. The
macrostate is naturally associated with a density matrix:
ˆ
Wν=Iν
dimHν
,(17)
where Iνis the projection operator onto Hν.ˆ
Wνis thus a representation of the
macrostate. It can be decomposed into wave functions, but the decomposition is
7We should assume that HPH is finite-dimensional, in which case we can use the normalized
surface area 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.
8Ensemblists would further insist that it makes no sense to talk about the thermodynamic state
of an individual system.
9The microcanonical ensemble is the projection operator onto the energy shell Hnormalized by
its dimension. The canonical ensemble is:
ρcan =exp(−βˆ
H)
Z,(16)
where Z=tr exp(−βˆ
H), and βis the inverse temperature of the quantum system.
8
not unique. Different measures can give rise to the same density matrix. One such
choice is µ(dψ), the uniform distribution over wave functions:
ˆ
Wν=S(Hν)µ(dψ)∣ψ⟩⟨ψ∣.(18)
In (18), ˆ
Wνis defined with a choice of measure on wave functions in Hν. However,
we should not be misled into thinking that the density matrix is derived from wave
functions. What is intrinsic to a density matrix is its geometrical meaning in the
Hilbert space. In the case of ˆ
Wν, as shown in the canonical description (17), it is just
a normalized projection operator.10
3 Density Matrix Realism
According to Wave Function Realism, the quantum state of the universe is objective
and pure. On this view, Ψis 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 parti-
cle 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 interpre-
tations, the nomological interpretation, and the sui generis interpretation.
On the other hand, we often use W, a density matrix, to represent our ignorance of
ψ, the actual wave function of a quantum system. Wcan also represent a macrostate
in QSM.11
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 Wdoes not
have enough information to be the basis of a realist quantum theory. However, if
we can formulate quantum dynamics directly in terms of Winstead of Ψsuch that
Wguides Bohmian particles, or Wcollapses, or Wrealizes the emergent multiverse,
then we will have good reasons for taking Wto represent something objective in
those theories. At the very least, the reasons for that will be on a par with those for
10Thanks 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.
11In some cases, Wis easier for calculation than Ψ, such as in the case of GRW collapse theories
where there are multiple sources of randomness. Thanks to Roderich Tumulka for discussions here.
9
Wave Function Realism in the Ψ-theories.
However, can we describe the quantum universe with Winstead of Ψ? 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.12 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. 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 tis de-
scribed by (Q(t),Ψ(t)). The universal wave function guides particle motion and
provides the probability distribution of particle configurations. Given the centrality
of Ψin 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):
dQi
dt =̵
h
mi
Im∇qiW(q,q′,t)
W(q,q′,t)(q=q′=Q),(19)
(Here we have set aside spin degrees of freedom. If we include spin, we can add
the partial trace operator trCkbefore each occurrence of “W.”) Finally, we can
impose an initial probability distribution similar to that of the quantum equilibrium
distribution:
P(Q(t0)∈dq)=W(q,q,t0)dq.(20)
The system is also equivariant: if the probability distribution holds at t0, it holds at
all times.13
With the defining equations—the von Neumann equation (14) and the W-guidance
12The 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.
13Equivariance holds because of the following continuity equation:
∂W(q,q,t)
∂t=−div(W(q,q,t)v),
10
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 uni-
verse’s Nparticles. If a universe is accurately described by W-BM, then Wrepresents
the fundamental quantum state in the theory that guides particle motion; it does
not do so via some other entity Ψ. 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 analo-
gous to the quantum equilibrium distribution in BM. With the respective dynamical
equations, both BM and W-BM generate an equivariant Born-rule probability dis-
tribution over all measurement outcomes.14
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.15 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.
where vdenotes the velocity field generated via (19). See Dürr et al. (1992, 2005).
14Here I am assuming that two theories are empirically equivalent if they assign the same proba-
bility 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 fol-
lows from the equivariance property plus the quantum equilibrium distribution. Suppose W-BM is
governed by a universal density matrix Wand 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,t0). 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.
In Chen (2019a), I present the general argument followed by a subsystem analysis of W-BM, in terms
of conditional density matrices.
15Thanks to Roderich Tumulka, Sheldon Goldstein, and Matthias Lienert for discussions here. The
W-GRW formalism was suggested first in Allori et al. (2013).
11
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.16
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)=tr(M(x)W(t)),(21)
where xis a physical space variable, M(x)=∑imiδ(Qi−x)is the mass-density
operator, which is defined via the position operator Qiψ(q1,q2, ...qn)=qiψ(q1,q2, ...qn).
This allows us to determine the mass-density ontology at time tvia 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 Nλ, where λis of
order 10−15 s−1. At a random time when a collapse occur at “particle” kat time T−,
the post-collapse density matrix at time T+is the following:
WT+=Λk(X)1/2WT−Λk(X)1/2
tr(WT−Λk(X)) ,(22)
with Xdistributed by the following probability density:
ρ(x)=tr(WT−Λk(x)),(23)
where WT+is the post-collapse density matrix, WT−is the pre-collapse density matrix,
Xis the center of the actual collapse, and Λk(x)is the collapse rate operator.17
16W-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.
17A collapse rate operator is defined as follows:
Λk(x)=1
(2πσ2)3/2e−(Qk−x)2
2σ2,
where Qkis the position operator of “particle” k, and σis a new constant of nature of order 10−7m
postulated in current GRW theories. Compare W-GRW to Ψ-GRW, where collapses happen at the
same rate, and the post-collapse wave function is the following:
ΨT+=Λk(X)1/2ΨT−
∣∣Λk(X)1/2ΨT−∣∣,(24)
12
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 Ψ, and now they are
played by W. In so far as we have good reasons for Wave Function Realism based on
the roles that Ψplays in the Ψ-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 Ψ. Wave function realists, such as Albert and Loewer,
have offered a concrete physical interpretation: Ψrepresents a physical field on the
high-dimensional configuration space that is taken to be the fundamental physical
space.18
Can we interpret the density matrix in a similar way? Let us start with a math-
ematical representation of the density matrix W(t). It is defined as a positive,
bounded, self-adjoint operator ˆ
W∶H→Hwith tr ˆ
W=1. For W-BM, the configura-
tion space R3N, and a density operator ˆ
W, the relevant Hilbert space is H, which is
a subspace of the total Hilbert space, i.e. H⊆Htotal =L2(R3N,C). Now, the density
matrix ˆ
Wcan also be represented as a function
W∶R3N×R3N→C(25)
(If we include spin, the range will be the endomorphism space End(Ck)of the space
of linear maps from Ckto 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 R6N, and
Wrepresents a field on that space that assigns properties (represented by complex
with the collapse center Xbeing chosen randomly with probability distribution ρ(x)=
∣∣Λk(x)1/2ΨT−∣∣2dx.
18In Chen (2017), 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.
13
numbers) to each point in R6N. In the Bohmian version, Wguides 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 R3Nsubspace.) Second, the fundamental space is R3N,
and Wrepresents a multi-field on that space that assigns properties to every ordered
pair of points (q,q′)in R3N. 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 R3, 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 Npoints in physical space. On this view, the density matrix guides
the motion of Nparticles in physical space.19
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 Was
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 Ψ-quantum theories. However, all of these
theories are time-symmetric, as they obey time-reversal invariance.
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.20 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.”21 Such a low-entropy initial condition explains the arrow of time in
19For discussions about the multi-field interpretation, see Forrest (1988), Belot (2012), Chen (2017),
Chen (ms.) section 3, and Hubert and Romano (2018).
20For an extended discussion, see Boltzmann (1964), Albert (2000), and Callender (2011).
21Feynman et al. (2015), 46-8.
14
thermodynamics.22
Albert (2000) has called this condition the Past Hypothesis (PH). However, his pro-
posal 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 uni-
verse 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.23
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 measure µwe 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.24
22See 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.
23In Chen (2020b), 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.
24For 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 (2020). 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 nomologically fundamental has been debated. See discussions in
Price (1997, 2004), and Callender (2004). However, those challenges are no more threatening to IPH
being a law than PH being a law. We will come back to this point after introducing IPH.
15
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 Ψ(t0).Ψ(t0)has to lie in the
special low-dimensional Hilbert space HPH with dimHPH ≪dimHeq. Moreover,
there are many different choices of initial wave functions in HPH. 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 µon the unit
sphere of HPH. For the finite-dimensional case, it is just the normalized surface area
measure on the unit sphere.
Density Matrix Realism suggests an alternative way to think about the low-
entropy boundary condition. PH pins down the initial macrostate HPH, a special
subspace of the total Hilbert space. Although HPH is compatible with many density
matrices, there is a natural choice—the normalized projection operator onto HPH.
Just as in (17), we can specify it as:
ˆ
WIPH(t0)=IPH
dimHPH
,(26)
where t0represents a temporal boundary of the universe, IPH is the projection opera-
tor onto HPH,dim counts the dimension of the Hilbert space, and dimHPH ≪dimHeq.
Since the quantum state at t0has the lowest entropy, we call t0the 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 WIPH-theories. For example, here are the equations of WIPH-BM:
(A) ˆ
WIPH(t0)=IPH
dimHPH ,
(B) P(Q(t0)∈dq)=WIPH(q,q,t0)dq,
(C) i̵
h∂ˆ
W
∂t=[ˆ
H,ˆ
W],
(D) dQi
dt =̵
h
miIm∇qiWIPH (q,q′,t)
WIPH (q,q′,t)(q=q′=Q).
(A) is IPH and (B)—(D) are the defining equations of W-BM. (Given the initial
quantum state ˆ
WIPH(t0), there is a live possibility that for every particle at t0, 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.25)
Contrast these equations with BM formulated with wave functions and PH (not
including SP for now), which will be called ΨPH-BM:
25Thanks to Sheldon Goldstein and Tim Maudlin for discussions here.
16
(A’) Ψ(t0)∈HPH,
(B’) P(Q(t0)∈dq)=∣Ψ(q,t0)∣2dq,
(C’) i̵
h∂Ψ
∂t=ˆ
HΨ,
(D’) dQi
dt =̵
h
miIm∇qiΨ(q,t)
Ψ(q,t)(Q).
IPH (A) in WIPH-BM plays the same role as PH (A’) in ΨPH-BM. Should IPH be
interpreted as a law of nature in WIPH-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 law26 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 HPH ,
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 WIPH-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 quan-
tum state. The quantum state ˆ
WIPH(t0)is informationally equivalent to the constraint
that PH imposes on the initial microstates. Assuming that PH selects a unique low-
entropy macrostate, ˆ
WIPH(t0)is singled out by the data in PH.27
Secondly, on the universal scale, we do not need to impose an additional probabil-
ity or typicality measure on the Hilbert space. ˆ
WIPH(t0)is mathematically equivalent
to an integral over projection onto each normalized state vectors (wave functions)
compatible with PH with respect to a normalized surface area measure µ. But here we are
not defining ˆ
WIPH(t0)in terms of state vectors. Rather, we are thinking of ˆ
WIPH(t0)
as a geometric object in the Hilbert space: the (normalized) projection operator onto
HPH. That is the intrinsic understanding of the density matrix.28
26See, for example, Feynman (2017), Albert (2000), Loewer (2007) and Loewer (2020).
27The 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
WIPH-theories, as the specific subspace defines WIPH, which determines the dynamics.
28After writing the paper, I discovered that David Wallace has come to a similar idea in a forth-
coming 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). Moreover, it is compatible with there
17
Thirdly, ˆ
WIPH(t0)is simple. Related to the first remark, IPH defines ˆ
WIPH(t0)
explicitly as the normalized projection operator onto HPH. There is a natural cor-
respondence between a subspace and its projection operator. If we specify the
subspace, we know what its projection operator is, and vice versa. Since the projec-
tion operator onto a subspace carries no more information than that subspace itself,
the projection operator is no more complex than HPH. This is different from ΨPH,
which is confined by PH to be a vector inside HPH. A vector carries more information
than the subspace it belongs to, as specifying a subspace is not sufficient to deter-
mine 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 ΨPH (the QSM microstate), it would
have required much more information and become a much more complex posit. PH
as it is determines ΨPH only up to an equivalence class (the QSM macrostate). As
we shall see in §6, the simplicity of ˆ
WIPH(t0)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 Cabcd vanishes at any initial singularity. (Penrose (1979), p.630,
emphasis original)
The Weyl curvature tensor Cabcd is the traceless part of the Riemann curvature tensor
Rabcd. 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
being an underlying pure state. 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), and
there is no underlying pure state. 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 (2020b).
18
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 Cabcd →0. Applying IPH to
such a theory, the initial density matrix will be the normalized projection onto that
subspace (subspaces).29
5 Theoretical Payoffs
WIPH-quantum theories, the result of applying IPH to W-theories, have two theo-
retical 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 inter-
esting differences with the wave-function framework.
5.1 Harmony between Statistical Mechanics and Quantum Me-
chanics
In WIPH-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 WIPH-theories. We will
use WIPH-BM ((A)—(D)) as an example.
WIPH-BM completely specifies the initial quantum state, unlike ΨPH-BM. For ΨPH -
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 WIPH -BM, in contrast, there is no need for something like
SP, as there is only one initial density matrix compatible with IPH—WIPH(t0). It
is guaranteed to evolve to future states that have entropic behaviors. Therefore,
on the universal scale, WIPH -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 WIPH-theories, as it is desirable to unify the
two sources of randomness: quantum-mechanical and statistical-mechanical. Of
29There 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 (1989)). This is because the entire history of the WIPH-Everettian
universe described by WIPH(t), including its initial condition, is fixed by the laws.
19
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 WIPH-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
WIPH-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 con-
tains 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 S1and its environment S2, then for S1, we can define its
conditional wave function:
ψcond(q1)=CΨ(q1,Q2),(27)
where C is a normalization factor and Q2is the actual configuration of S2.ψcond(q1)
always gives the velocity field for the particles in S1according to the guidance
equation. However, for quantum systems with spin, this is not always true. Since
BM is described by (Ψ(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 S1, by plugging in the actual configuration of S2and tracing over the
20
spin components in the wave function associated with S2.30 The conditional density
matrix will guide the particles in S1by 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 Ψ-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 WIPH -Everett theories and WIPH-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 Ψ-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 WIPH-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.
30The conditional density matrix for S1is defined as:
Wconds1
s′
1(q1,q′
1)=1
N∑
s2
Ψs1s2(q1,Q2)Ψ∗
s1s2(q′
1,Q2),(28)
with the normalizing factor:
N=∫Q1
dq1∑
s1s2
Ψs1s2(q1,Q2)Ψ∗
s1s2(q′
1,Q2).(29)
21
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 me-
chanics. 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=(q1(t), ..., qN(t);p1(t), ..., pn(t)),
and the Hamiltonian is H=H(X). Particles move according to the Hamiltonian
equations:
dqi(t)
dt =∂H
∂pi
,dpi(t)
dt =−∂H
∂qi
.(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:
dX
dt =F(X)=FH(X)(31)
Here, FH(X)is H(q,p)with suitable derivative operators. The Hamiltonian equa-
tions have a simple form, because His simple. Hcan be written explicitly as follows:
H=N
i
p2
i
2mi+V,(32)
where Vtakes on this form when we consider electric and gravitational potentials:
V=1
4π0
1≤j≤k≤N
ejek
∣qj−qk∣+
1≤j≤k≤N
Gmjmk
∣qj−qk∣,(33)
That is, the RHS of the Hamiltonian equations, after making the Hamiltonian func-
tion explicit, are still simple. His just a convenient shorthand for (32) and (33).
Moreover, His also fixed by the theory. A classical universe is governed by the
dynamical laws plus the fundamental interactions. If Hwere 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 Has 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 His 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).
22
6.2 The Quantum Case
It is according to the above sense that Dürr et al. (1996), Goldstein and Teufel (2001),
and Goldstein and Zanghì (2013) propose that the universal wave function in BM is
nomological (and governs things in the ontology). With the guidance equation, Ψ
generates the motion of particles. It is of the same form as above:
dX
dt =F(X)=FΨ(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 Ψ
is sufficiently simple, we can afford the same interpretation of Ψas we can for Hin
classical mechanics: both are nomological in the above sense.
WIPH-BM also supports the nomological interpretation of the quantum state but
via a different route. With the W-guidance equation, WIPH generates the motion of
particles. It is of the same form as above:
dX
dt =F(X)=FWIPH (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
Ψ. Instead, we can just appeal to IPH. We have argued in §4.2 that IPH is simple
and that WIPH(t0)is simple. Since the quantum state evolves unitarily by the von
Neumann equation, we can obtain the quantum state at any later time as:
ˆ
WIPH(t)=e−iˆ
Ht/̵
hˆ
WIPH(t0)eiˆ
Ht/̵
h(36)
Since WIPH(t)is a simple function of the time-evolution operator and the initial
density matrix, and since both are simple, WIPH(t)is also simple. So we can think of
WIPH(t)just as a convenient shorthand for (36). (This is not true for ∣Ψ(t)⟩ =ˆ
H∣Ψ(t0)⟩,
as generic ∣Ψ(t0)⟩ is not simple at all.)
The “shorthand” way of thinking about WIPH(t)implies that the equation of
particle motion has a time-dependent form FWIPH (X,t). Does time-dependence un-
dercut the nomological interpretation? It does not in this case, as the FWI 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 the proposal of Dürr
et al. (1996), Goldstein and Teufel (2001), and Goldstein and Zanghì (2013), we do
not need time-independence to argue for the simplicity of the quantum state.
Since WIPH(t0)is fixed by IPH, FWIPH is also fixed by the theory. Let us expand
23
(35) to make it more explicit:
dQi
dt =̵
h
mi
Im∇qiWIPH(q,q′,t)
WIPH(q,q′,t)(Q)=̵
h
mi
Im∇qi⟨q∣e−iˆ
Ht/̵
hˆ
WIPH(t0)eiˆ
Ht/̵
h∣q′⟩
⟨q∣e−iˆ
Ht/̵
hˆ
WIPH(t0)eiˆ
Ht/̵
h∣q′⟩(q=q′=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 WIPH -
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 WIPH-theories with local beables. In WIPH-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 Hin 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 WIPH -GRWm and WIPH-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 WIPH-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
24
properties of individual space-time points.31 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.32
WIPH-theories with local beables offer a way out of the conflict between quan-
tum 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 WIPH-BM as an example:
The WIPH-BM mosaic: particle trajectories Q(t)on physical space-time.
The WIPH-BM best system: four equations—the simplest and strongest axioms sum-
marizing the mosaic:
(A) ˆ
WIPH(t0)=IPH
dimHPH
(B) P(Q(t0)∈dq)=WIPH(q,q,t0)dq,
(C) i̵
h∂ˆ
W
∂t=[ˆ
H,ˆ
W],
(D) dQi
dt =̵
h
miIm∇qiWIPH (q,q′,t)
WIPH (q,q′,t)(q=q′=Q).
Notice that (A)—(D) are simple and informative statements about Q(t). They are
expressed in terms of ˆ
WIPH(t), which via law (C) can be expressed in terms of
ˆ
WIPH(t0). 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 speci-
fying 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 ver-
sion 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 WIPH-theories is different from
the other approaches in the literature: Albert (1996), Loewer (1996), Miller (2014),
Esfeld (2014), Bhogal and Perry (2017), Callender (2015) and Esfeld and Deckert
(2018). In contrast to the high-dimensional proposal of Albert (1996) and Loewer
(1996), our version preserves the fundamentality of physical space.
31This is one reading of David Lewis. Tim Maudlin (2007) calls this thesis “Separability.”
32See Maudlin (2007), Chapter 2.
25
The difference between our version and those of Miller (2014), Esfeld (2014),
Bhogal and Perry (2017), Callender (2015), and Esfeld and Deckert (2018) is more
subtle. They are concerned primarily with Ψ-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 ΨPH-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 ΨPH on
the RHS, but ΨPH can still be part of their best system.33 For us, it is crucial that the
equation remains simple after plugging in WIPH(t0)for it to be in the best system.
Consequently, WIPH(t0)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 ΨPH, 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 do that in Chen (2020a).
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—WIPH-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 WIPH-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
33See Dewar (2017) §5 for some worries about the weaker criterion on the best system.
26
WIPH-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 Phi-
losophy of Science for helpful feedback. I am also grateful for stimulating discus-
sions 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, Craig Callender, Sean Carroll,
Eugene Chua, 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, Kerry McKenzie, Elizabeth Miller, Sebastian Mur-
gueitio, Wayne Myrvold, Jill North, Zee Perry, Davide Romano, Ezra Rubenstein,
Charles Sebens, Jonathan Schaffer, Ted Sider, Joshua Spencer, Noel Swanson, Karim
Thébault, Anncy Thresher, Roderich Tumulka, David Wallace, Isaac Wilhelm, Nino
Zanghì, audiences at Cornell University, University of Western Ontario, University
of Wisconsin-Milwaukee, University of Wisconsin-Madison, University of Califor-
nia, San Diego, and the 2018 Rotman Summer Institute in Philosophy of Cosmology.
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