#  Time’s Arrow in a Quantum Universe:
On the Status of Statistical Mechanical Probabilities

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 Valia Allori (ed.), _Statistical Mechanics and Scientific
Explanation: Determinism, Indeterminism and Laws of Nature_ , World Scientific
)

###### Abstract

In a quantum universe with a strong arrow of time, it is standard to postulate
that the initial wave function started in a particular macrostate—the special
low-entropy macrostate selected by the Past Hypothesis. Moreover, there is an
additional postulate about statistical mechanical probabilities according to
which the initial wave function is a “typical” choice in the macrostate (the
Statistical Postulate). Together, they support a probabilistic version of the
Second Law of Thermodynamics: typical initial wave functions will increase in
entropy. Hence, there are two sources of randomness in such a universe: the
quantum-mechanical probabilities of the Born rule and the statistical
mechanical probabilities of the Statistical Postulate.

I propose a new way to understand time’s arrow in a quantum universe. It is
based on what I call the Thermodynamic Theories of Quantum Mechanics.
According to this perspective, there is a _natural_ choice for the initial
quantum state of the universe, which is given by not a wave function but by a
density matrix. The density matrix plays a microscopic role: it appears in the
fundamental dynamical equations of those theories. The density matrix also
plays a macroscopic / thermodynamic role: it is exactly the (normalized)
projection operator onto the Past Hypothesis subspace (of the Hilbert space of
the universe). Thus, given an initial subspace, we obtain a unique choice of
the initial density matrix. I call this property _the conditional uniqueness_
of the initial quantum state. The conditional uniqueness provides a new and
general strategy to eliminate statistical mechanical probabilities in the
fundamental physical theories, by which we can reduce the two sources of
randomness to only the quantum mechanical one. I also explore the idea of an
_absolutely unique_ initial quantum state, in a way that might realize
Penrose’s idea (1989) of a strongly deterministic universe.

Keywords: 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, initial condition of the universe, strong determinism

###### Contents

  1. 1 Introduction
  2. 2 Statistical Mechanics and Time’s Arrow
    1. 2.1 Classical Statistical Mechanics
    2. 2.2 Quantum Statistical Mechanics
    3. 2.3 $\Psi_{PH}$-Quantum Theories
  3. 3 The Thermodynamic Theories of Quantum Mechanics
    1. 3.1 The Initial Projection Hypothesis
    2. 3.2 Conditional Uniqueness
      1. 3.2.1 Eliminating Statistical Postulate with Conditional Uniqueness
      2. 3.2.2 Three Versions of PH and IPH
      3. 3.2.3 The Emergence of Statistical Mechanical Probabilities
    3. 3.3 Absolute Uniqueness
    4. 3.4 Other Applications
    5. 3.5 Generalizations
    6. 3.6 Other Proposals
  4. 4 Conclusion

##  1 Introduction

In a quantum universe with a strong arrow of time (large entropy gradient), it
is standard to attribute the temporal asymmetry to a special boundary
condition. This boundary condition is a macrostate of extremely low entropy.
David Albert (2000) calls it the Past Hypothesis. The quantum version of the
Past Hypothesis dictates that the initial wave function of the universe lies
within the low-entropy macrostate. Mathematically, the macrostate is
represented by a low-dimensional subspace in the total Hilbert space, which
corresponds to low Boltzmann entropy.

The Past Hypothesis is accompanied by another postulate about statistical
mechanical probabilities according to which the initial wave function is a
“typical” choice in the macrostate (the Statistical Postulate). Together, the
Past Hypothesis and the Statistical Postulate support a probabilistic version
of the Second Law of Thermodynamics: typical initial wave functions will
increase in entropy.

The standard theory, which has its origin in Boltzmann’s statistical mechanics
and has been substantially developed in the last century, is a simple and
elegant way of understanding time’s arrow in a quantum universe. It has two
features:

  1. 1.

Although the theory restricts the choices of initial quantum states of the
universe, it does not select a unique one.

  2. 2.

The theory postulates statistical mechanical probabilities (or a typicality
measure) on the fundamental level. They are _prima facie_ on a par with and in
addition to the quantum mechanical probabilities.

Hence, there are two sources of randomness in such a universe: the quantum-
mechanical probabilities in the Born rule and the statistical mechanical
probabilities in the Statistical Postulate. It is an interesting conceptual
question how we should understand the two kinds of probabilities in the
standard theory.

In this paper, I propose a new way to understand time’s arrow in a quantum
universe. It is based on what I call the Thermodynamic Theories of Quantum
Mechanics (TQM). According to this perspective, there is a _natural_ choice
for the initial quantum state of the universe, which is given not by a wave
function but by a density matrix. Moreover, the density matrix enters into the
fundamental equations of quantum mechanics. Hence, it plays a microscopic
role. Furthermore, the density matrix is exactly the (normalized) projection
operator onto the Past Hypothesis macrostate, represented by a subspace in the
Hilbert space of the universe.

In Chen, 2018b , I introduced the Initial Projection Hypothesis (IPH) as a new
postulate about a natural initial density matrix of the universe. In that
paper, I focused on its relevance to the debate about the nature of the
quantum state. In Chen, (2019), I focused on its empirical equivalence with
the Past Hypothesis. In this paper, I focus on its relevance to the status of
statistical mechanical probabilities. According to the Initial Projection
Hypothesis, given a choice of an initial subspace, we have a unique choice of
the initial density matrix. This property is called _conditional uniqueness_ ,
which will be sufficient to eliminate statistical mechanical probabilities at
the level of fundamental physics. Thus, in contrast to the standard theory,
TQM has the following two features:

  1. 1.

The theory selects a unique initial quantum state of the universe, given a
choice of the Past Hypothesis subspace. [Conditional Uniqueness]

  2. 2.

Because of conditional uniqueness, the theory does not postulate statistical
mechanical probabilities (or a typicality measure) at the level of fundamental
physics.

That is, TQM provides a new and general strategy for eliminating statistical
mechanical probabilities at the fundamental level. To be sure, statistical
mechanical probabilities can still emerge as useful tools of analysis. TQM
satisfies conditional uniqueness, which is sufficient for the purpose at hand.
But we might also wonder whether TQM may lead to absolute uniqueness, in the
sense that there is a unique initial quantum state of the universe that does
not depend on a prior selection of a unique initial subspace. Such absolute
uniqueness will go some way towards realizing a strongly deterministic
universe in the sense of Penrose, (1989).

Here is the roadmap of the paper. In §2, I will review the standard
(Boltzmannian) way of understanding temporal asymmetry in classical and
quantum mechanics. In §3, I introduce the Thermodynamic Theories of Quantum
Mechanics. First, I formulate the Initial Projection Hypothesis, which is the
key component of TQM and an alternative to the Past Hypothesis. Next, I focus
on the property of conditional uniqueness, the elimination of statistical
mechanical probabilities at the fundamental level, the emergence of such
probabilities as useful tools of analysis, and the possibility of absolute
uniqueness and strong determinism. I also discuss the (lack of) analogues in
classical statistical mechanics, implications for the status of the quantum
state, Lorentz invariance, theoretical unity, generalizations to other
cosmological initial conditions, and relation to other proposals in the
literature.

##  2 Statistical Mechanics and Time’s Arrow

Statistical mechanics concerns macroscopic systems such as a gas in a box. It
is an important subject for understanding the arrow of time, since
thermodynamic systems are irreversible. A gas in a box can be described as a
system of $N$ particles, with $N>10^{20}$. If the system is governed by
classical mechanics, although it is difficult to solve the equations exactly,
we can still use classical statistical mechanics (CSM) to describe its
statistical behaviors, such as approach to thermal equilibrium suggested by
the Second Law of Thermodynamics. Similarly, if the system is governed by
quantum mechanics, we can use quantum statistical mechanics (QSM) to describe
its statistical behaviors. Generally speaking, there are two different views
on CSM: the individualistic view and the ensemblist view. For concreteness, we
will adopt the individualistic view in this paper, and we will start with a
review of CSM, which should be more familiar to researchers in the foundations
of physics.

###  2.1 Classical Statistical Mechanics

Let us review the basic elements of CSM on the individualistic view.111Here I
follow the discussion in Goldstein and Tumulka, (2011). §2.1 and §2.2 do not
intend to be rigorous axiomatizations of CSM and QSM. They are only summaries
of the key concepts that are important for appreciating the main ideas in the
later sections. For concreteness, let us consider a classical-mechanical
system with $N$ particles in a box $\Lambda=[0,L]^{3}\subset\mathbb{R}^{3}$
and a Hamiltonian
$H=H(X)=H(\boldsymbol{q_{1}},...,\boldsymbol{q_{N}};\boldsymbol{p_{1}},...,\boldsymbol{p_{n}})$.

  1. 1.

Microstate: at any time $t$, the microstate of the system is given by a point
in a $6N$-dimensional phase space,

$X=(\boldsymbol{q_{1}},...,\boldsymbol{q_{N}};\boldsymbol{p_{1}},...,\boldsymbol{p_{n}})\in\Gamma_{total}\subseteq\mathbb{R}^{6N},$
(1)

where $\Gamma_{total}$ is the total phase space of the system.

  2. 2.

Dynamics: the time dependence of
$X_{t}=(\boldsymbol{q_{1}}(t),...,\boldsymbol{q_{N}}(t);\boldsymbol{p_{1}}(t),...,\boldsymbol{p_{n}}(t))$
is given by the Hamiltonian equations of motion:

$\frac{d\boldsymbol{q_{i}}}{dt}=\frac{\partial
H}{\partial\boldsymbol{p_{i}}}\text{ ,
}\frac{d\boldsymbol{p_{i}}}{dt}=-\frac{\partial
H}{\partial\boldsymbol{q_{i}}}.$ (2)

  3. 3.

Energy shell: the physically relevant part of the total phase space is the
energy shell $\Gamma\subseteq\Gamma_{total}$ defined as:

$\Gamma=\\{X\in\Gamma_{total}:E\leq H(x)\leq E+\delta E\\}.$ (3)

We only consider microstates in $\Gamma.$

  4. 4.

Measure: the measure $\mu_{V}$ is the standard Lebesgue measure on phase
space, which is the volume measure on $\mathbb{R}^{6N}$.

  5. 5.

Macrostate: with a choice of macro-variables, the energy shell $\Gamma$ can be
partitioned into macrostates $\Gamma_{\nu}$:

$\Gamma=\bigcup_{\nu}\Gamma_{\nu}.$ (4)

A macrostate is composed of microstates that share similar macroscopic
features (similar values of the macro-variables), such as volume, density, and
pressure.

  6. 6.

Unique correspondence: every phase point $X$ belongs to one and only one
$\Gamma_{\nu}$. (This point is implied by (5). But we make it explicit to
better contrast it with the situation in QSM.)

  7. 7.

Thermal equilibrium: typically, there is a dominant macrostate $\Gamma_{eq}$
that has almost the entire volume with respect to $\mu_{V}$:

$\frac{\mu_{V}(\Gamma_{eq})}{\mu_{V}(\Gamma)}\approx 1.$ (5)

A system is in thermal equilibrium if its phase point $X\in\Gamma_{eq}.$

  8. 8.

Boltzmann Entropy: the Boltzmann entropy of a classical-mechanical system in
microstate $X$ is given by:

$S_{B}(X)=k_{B}\text{log}(\mu_{V}(\Gamma(X))),$ (6)

where $\Gamma(X)$ denotes the macrostate containing $X$. The thermal
equilibrium state thus has the maximum entropy.

  9. 9.

Low-Entropy Initial Condition: when we consider the universe as a classical-
mechanical system, we postulate a special low-entropy boundary condition,
which David Albert calls _the Past Hypothesis_ :

$X_{t_{0}}\in\Gamma_{PH}\text{ ,
}\mu_{V}(\Gamma_{PH})\ll\mu_{V}(\Gamma_{eq})\approx\mu_{V}(\Gamma),$ (7)

where $\Gamma_{PH}$ is the Past Hypothesis macrostate with volume much smaller
than that of the equilibrium macrostate. Hence, $S_{B}(X_{t_{0}})$, the
Boltzmann entropy of the microstate at the boundary, is very small compared to
that of thermal equilibrium.

  10. 10.

A central task of CSM is to establish mathematical results that demonstrate
(or suggest) that $\mu_{V}-$most microstates will approach thermal
equilibrium.

###  2.2 Quantum Statistical Mechanics

Let us now turn to quantum statistical mechanics (QSM). 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 $\hat{H}$.
For concreteness, I will focus on the individualistic view of QSM. The main
difference with CSM is that they employ different state spaces. The classical
state space is the phase space while the quantum state space is the Hilbert
space. However, CSM and QSM are conceptually similar, as we can see from
below.222Here I follow the discussions in Goldstein et al., 2010a  and
Goldstein and Tumulka, (2011).

Figure 1: QM and CM employ different state spaces. These pictures are not
drawn to scale, and they are drawn with much fewer dimensions. In (a), we only
draw a three-dimensional equilibrium subspace, which is embedded in a higher-
dimensional energy shell. In (b), we only draw a two-dimensional energy
surface, which includes a large equilibrium set.

  1. 1.

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

$\psi(\boldsymbol{q_{1}},...,\boldsymbol{q_{N}})\in\mathscr{H}_{total}=L^{2}(\Lambda^{N},\mathbb{C}^{k})\text{
, }\parallel\psi\parallel_{L^{2}}=1,$ (8)

where $\mathscr{H}_{total}=L^{2}(\Lambda^{N},\mathbb{C}^{k})$ is the total
Hilbert space of the system.

  2. 2.

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

$i\hbar\frac{\partial\psi}{\partial t}=\hat{H}\psi.$ (9)

  3. 3.

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

$\mathscr{H}\subseteq\mathscr{H}_{total}\text{ ,
}\mathscr{H}=\text{span}\\{\phi_{\alpha}:E_{\alpha}\in[E,E+\delta E]\\},$ (10)

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: given a subspace $\mathscr{H}$, the measure $\mu_{S}$ is the surface
area measure on the unit sphere in that subspace
$\mathscr{S}(\mathscr{H})$.333For simplicity, here we assume that the
subspaces we deal with are finite-dimensional. In cases where the Hilbert
space is infinite-dimensional, it is an open and challenging technical
question. For example, we could use Gaussian measures in infinite-dimensional
spaces, but we no longer have uniform probability distributions.

  5. 5.

Macrostate: with a choice of macro-variables,444For technical reasons, Von
Neumann, (1955) suggests that we round up these macro-variables (represented
by quantum observables) so as to make the observables commute. See Goldstein
et al., 2010c  §2.2 for a discussion of von Neumann’s ideas.  the energy shell
$\mathscr{H}$ can be orthogonally decomposed into macro-spaces:

$\mathscr{H}=\oplus_{\nu}\mathscr{H}_{\nu}\text{ ,
}\sum_{\nu}\text{dim}\mathscr{H}_{\nu}=D$ (11)

Each $\mathscr{H}_{\nu}$ corresponds more or less to small ranges of values of
macro-variables that we have chosen in advance.

  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 $\mathscr{H}_{\nu}$:

$\bra{\psi}P_{\nu}\ket{\psi}\approx 1,$ (12)

where $P_{\nu}$ is the projection operator into $\mathscr{H}_{\nu}$. This
means that $\ket{\psi}$ lies almost entirely in $\mathscr{H}_{\nu}$. (This is
different from CSM, where every phase point lies 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:

$\frac{\text{dim}\mathscr{H}_{eq}}{\text{dim}\mathscr{H}}\approx 1.$ (13)

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 (12):
$\bra{\psi}P_{eq}\ket{\psi}\approx 1.$

_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 $n$, with the endpoints being the exceptions:
$\mathscr{H}_{0}$ is the interval of $0\%-1\%$ and $\mathscr{H}_{100}$ is the
interval of $99\%-100\%$. 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 $\nu$ is given by:

$S_{B}(\psi)=k_{B}\text{log}(\text{dim}\mathscr{H}_{\nu}),$ (14)

where $\mathscr{H}_{\nu}$ denotes the subspace containing almost all of $\psi$
in the sense of (12). The thermal equilibrium state thus has the maximum
entropy:

$S_{B}(eq)=k_{B}\text{log}(\text{dim}\mathscr{H}_{eq})\approx
k_{B}\text{log}(D),$ (15)

where _eq_ denotes the equilibrium macrostate.

  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_ :

$\Psi(t_{0})\in\mathscr{H}_{PH}\text{ ,
}\text{dim}\mathscr{H}_{PH}\ll\text{dim}\mathscr{H}_{eq}\approx\text{dim}\mathscr{H}$
(16)

where $\mathscr{H}_{PH}$ is the Past Hypothesis macro-space with dimension
much smaller than that of the equilibrium macro-space.555Again, we assume that
$\mathscr{H}_{PH}$ is finite-dimensional, in which case we can use the 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. Hence, the
initial state has very low entropy in the sense of (23).

  10. 10.

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

If the microstate $\psi$ of a system is close to some macro-space
$\mathscr{H}_{\nu}$ in the sense of (12), we can say that the macrostate of
the system is $\mathscr{H}_{\nu}$. The macrostate $\mathscr{H}_{\nu}$ 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 $\mathscr{H}_{\nu}$. The
density matrix also obeys a dynamical equation; it evolves according to the
von Neumann equation:

$i\hbar\frac{d\hat{W}(t)}{dt}=[\hat{H},\hat{W}].$ (18)

Given the canonical correspondence between a subspace and its normalized
projection operator, $\hat{W}_{\nu}$ is also a representation of the subspace
$\mathscr{H}_{\nu}$. 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 $\mu_{S}(d\psi)$, the uniform distribution
over wave functions:

$\hat{W}_{\nu}=\int_{\mathscr{S}(\mathscr{H_{\nu}})}\mu_{S}(d\psi)\ket{\psi}\bra{\psi}.$
(19)

In (19), $\hat{W}_{\nu}$ 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.

###  2.3 $\Psi_{PH}$-Quantum Theories

If we treat the universe as a quantum system, then we can distill a picture of
the fundamental physical reality from the standard Boltzmannian QSM
(individualistic perspective). The universe is described by a quantum state
represented by a universal wave function. It starts in a Past Hypothesis
macrostate (§2.2 #9) and is selected randomly according to the probability
specified by the Statistical Postulate (§2.2 #4). It evolves by the quantum
dynamics—the Schrödinger equation (§2.2 #2). That is, it has three fundamental
postulates (fundamental lawlike statements666Barry 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. For 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.):

  1. 1.

The Schrödinger equation.

  2. 2.

The Past Hypothesis.

  3. 3.

The Statistical Postulate.

However, this theory by itself faces the measurement problem. To solve the
measurement problem, we can combine it with three well-known strategies:

  * •

Bohmian mechanics (BM): the fundamental ontology in addition to the quantum
state also includes point particles with precise locations, and the
fundamental dynamics also includes a guidance equation that relates the wave
function to the velocity of the particles.

  * •

Everettian mechanics (S0): the fundamental ontology consists in just the
quantum state evolving unitarily according to the Schrödinger equation.

  * •

GRW spontaneous collapse theory (GRW0): the fundamental ontology consists in
just the quantum state evolving by the Schrödinger equation, but the unitary
evolution is interrupted by a spontaneous collapse mechanism.

The universal wave function $\Psi$ is central to standard formulations of the
above theories. The Past Hypothesis and the Statistical Postulate need to be
added to them to account for the arrow of time. Let us label them
$\Psi_{PH}$-BM, $\Psi_{PH}$-S0, and $\Psi_{PH}$-GRW0. They are all instances
of what I call $\Psi_{PH}$-quantum theories.

There are also Everettian and GRW theories with additional fundamental
ontologies in physical space-time, such as Everettian theory with a mass-
density ontology (S0), GRW theory with a mass-density ontology (GRWm), and GRW
theory with a flash ontology (GRWf). See Allori et al., (2008, 2010) for
discussions. Thus, we also have $\Psi_{PH}$-Sm, $\Psi_{PH}$-GRWm, and
$\Psi_{PH}$-GRWf.

##  3 The Thermodynamic Theories of Quantum Mechanics

The $\Psi_{PH}$-quantum theories attempt to solve the measurement problem and
account for the arrow of time. However, each $\Psi_{PH}$-quantum theory admits
many choices for the initial quantum state. This is because the Past
Hypothesis subspace $\mathscr{H}_{PH}$, although being small compared to the
full energy shell, is still compatible with many wave functions. One might
wonder whether there is some hypothesis that could determine a unique initial
quantum state of the universe. There is an interesting consequence if that can
be done: it may bring us closer to what Penrose, (1989) calls _strong
determinism_ , the idea that the entire history of the universe is fixed by
the theory, with laws that specify both a deterministic dynamics and a
(nomologically) unique initial state of the universe (see §3.5). As we shall
see, there are other benefits of having a unique initial quantum state of the
universe, such as the elimination of statistical mechanical probabilities
(§3.2.1).

Is it possible to narrow down the initial choices to exactly one? I believe
that we can, but we need to make some conceptual changes to the standard
theory. Our proposal consists in three steps:

  1. 1.

Allowing the universe to be in a fundamental mixed state represented by a
density matrix.

  2. 2.

Letting the density matrix enter into the fundamental dynamical equations.

  3. 3.

Choosing a natural density matrix associated with the Past Hypothesis subspace
$\mathscr{H}_{PH}$.

The natural density matrix together with its fundamental dynamical equations
define a new class of quantum theories, which we will call the _Thermodynamic
Theories of Quantum Mechanics_. As we explain below, all of them will have a
conditionally unique initial quantum state.

###  3.1 The Initial Projection Hypothesis

The standard approach to the foundations of quantum mechanics assumes that the
universe is described by a _pure_ quantum state $\Psi(t)$, which is
represented as a vector in the (energy shell of the) Hilbert space of the
system. However, it is also possible to describe the universe by a _mixed_
quantum state $W(t)$, which has its own geometric meaning in the Hilbert
space.

On the latter perspective, the quantum state of the universe is in a mixed
state $W$, which becomes the central object of quantum theory. $W$ obeys its
own unitary dynamics—the von Neumann equation (18), which is a generalization
of the Schrödinger equation (9). Moreover, we can write down density-matrix
versions of Bohmian mechanics, Everettian mechanics, and GRW spontaneous
collapse theory. (See Appendix for the mathematical details.) Given these
equations, the density matrix of the universe plays a microscopic role: it
enters into the fundamental dynamical equations of those theories. As a
result, it guides Bohmian particles, gives rise to GRW collapses, and realizes
the emergent multiverse of Everett. I call this view _Density Matrix Realism_
, in contrast with Wave Function Realism, the view that the central object of
quantum theory is the universal wave function.777Density Matrix Realism 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.
See, for example, Dürr et al., (2005), Maroney, (2005), Wallace, (2011, 2012),
and Wallace, (2016). I discuss this idea in more detail in Chen, 2018b  §3.

According Density Matrix Realism, the initial quantum state is also
represented by a density matrix. Now, we know that to account for the arrow of
time we need to postulate a low-entropy initial condition—the Past Hypothesis.
Moreover, under Wave Function Realism, it is also necessary to postulate a
Statistical Postulate.

Now, we make a crucial observation. If the initial quantum state is a density
matrix, then there is a natural choice for it given the Past Hypothesis
subspace. The natural choice is the normalized projection operator onto
$\mathscr{H}_{PH}$:

$\hat{W}_{IPH}(t_{0})=\frac{I_{PH}}{dim\mathscr{H}_{PH}},$ (20)

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. In Chen, 2018b , I call (20) the _Initial Projection Hypothesis_ (IPH).
In words: the initial density matrix of the universe is the normalized
projection onto the PH-subspace.

The projection operator onto $\mathscr{H}_{PH}$ contains no more and no less
information than what is contained in the subspace itself. 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).

I propose, then, that we replace the Past Hypothesis with the Initial
Projection Hypothesis, the Schrödinger equation with the von Neumann equation,
and the universal wave function with a universal density matrix. In contrast
to $\Psi_{PH}$ theories, there are only two corresponding fundamental
postulates (fundamental laws of physics) in the new theory:

  1. 1.

The von Neumann equation.

  2. 2.

The Initial Projection Hypothesis.

Notice that IPH defines a unique initial quantum state, given a choice of the
initial PH subspace. There is only one state that is possible, not a
collection of states, in the PH subspace. We call this _conditional
uniqueness_. The quantum state $\hat{W}_{IPH}(t_{0})$ is informationally
equivalent to the constraint that PH imposes on the initial microstates. Given
the PH subspace, $\hat{W}_{IPH}(t_{0})$ is singled out by the data in PH (we
will come back to this point in §3.2). Consequently, on the universal scale,
we do not need to impose an additional probability or typicality measure on
the Hilbert space. $\hat{W}_{IPH}(t_{0})$ is mathematically equivalent to an
integral over projection onto each normalized state vectors (wave functions)
compatible with PH _with respect to $\mu_{S}(d\psi)$_. Of course, we are not
defining $\hat{W}_{IPH}(t_{0})$ in terms of state vectors. Rather, we are
thinking of $\hat{W}_{IPH}(t_{0})$ 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. (In §3.2 and §3.3 we will
discuss different versions of PH and their relevance to conditional
uniqueness. In §3.5 we will discuss why this strategy does not work so well in
CSM.)

As before, the above theory by itself faces the measurement problem. To solve
the measurement problem, we can combine it with the well-known strategies of
BM, EQM, and GRW. We will label them as WIPH-BM, WIPH-EQM, and WIPH-GRW. (See
the Appendix for mathematical details.) Together, we call them _Density-Matrix
Quantum Theories with the Initial Projection Hypothesis_ (WIPH-QM). Moreover,
since they are motivated by considerations of the thermodynamic macrostate of
the early universe, we also call them the _Thermodynamic Theories of Quantum
Mechanics_ (TQM).

In WIPH-QM, the density matrix takes on a central role as the quantum
microstate. Besides the low-entropy initial condition, it is also necessary to
reformulate some definitions in quantum statistical mechanics:

  * 6’

Non-unique correspondence: typically, a density matrix is in a superposition
of macrostates and is not entirely in any one of the macrospaces. However, we
can make sense of situations where $W$ is very close to a macrostate
$\mathscr{H}_{\nu}$:

$\text{tr}(WP_{\nu})\approx 1,$ (21)

where $P_{\nu}$ is the projection operator onto $\mathscr{H}_{\nu}$. This
means that almost all of $W$ is in $\mathscr{H}_{\nu}$.

  * 7’

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

$\frac{\text{dim}\mathscr{H}_{eq}}{\text{dim}\mathscr{H}}\approx 1.$ (22)

A system with density matrix $W$ is in equilibrium if $W$ is very close to
$\mathscr{H}_{eq}$ in the sense of (21): $\text{tr}(WP_{eq})\approx 1$.

  * 8’

Boltzmann Entropy: the Boltzmann entropy of a quantum-mechanical system with
density matrix $W$ that is very close to a macrostate $\nu$ is given by:

$S_{B}(W)=k_{B}\text{log}(\text{dim}\mathscr{H}_{\nu}),$ (23)

where $\mathscr{H}_{\nu}$ denotes the subspace containing almost all of $W$ in
the sense of (21).

For the Bohmian version, we have the additional resource of the particle
configuration, which can enable us to further define the effective macrostate
and the effective Boltzmann entropy of a quantum system in a way that is
analogous to the effective wave function of the system.888Here is a flat-
footed way to do it. First, we start from the wave-function picture of BM. If
$\Psi_{i}$ is the effective wave function of the universe, and if $\Psi_{i}$
is almost entirely in $\mathscr{H}_{\nu}$, then the universe is in an
effective macrostate of $\mathscr{H}_{\nu}$ and it has effective Boltzmann
entropy of $k_{B}\text{log}(\text{dim}\mathscr{H}_{\nu})$. Second, we note
that we can define an analogous notion of the effective density matrix of the
universe. Third, we can transpose the first point to the density-matrix
perspective. If $W_{i}$ is the effective density matrix of the universe, and
if $W_{i}$ is almost entirely in $\mathscr{H}_{\nu}$, then the universe is in
an effective macrostate of $\mathscr{H}_{\nu}$ and it has effective Boltzmann
entropy of $k_{B}\text{log}(\text{dim}\mathscr{H}_{\nu})$. To be sure, further
analysis is required.

###  3.2 Conditional Uniqueness

####  3.2.1 Eliminating Statistical Postulate with Conditional Uniqueness

The Past Hypothesis, given a choice of an initial subspace, is compatible with
a continuous infinity of initial wave functions. However, the Initial
Projection Hypothesis determines a unique initial density matrix given an
initial subspace; the initial quantum state is the normalized projection onto
the subspace we select. I call this feature the _conditional uniqueness_ of
the initial quantum state in $W_{IPH}$ theories.

In the standard theory, the Statistical Postulate determines a uniform
probability distribution ($\mu_{S}$ with a normalization constant) over wave
functions once we fix an initial subspace. This is because there are some wave
functions compatible with PH that are anti-entropic, i.e. they will evolve
into lower-entropy states. Having a uniform probability distribution will
provide a way to say that they are unlikely, since there are very few of them
compared to the entropic wave functions. In the new theory ($W_{IPH}$), if we
fix an initial subspace, there is only one choice for the initial quantum
state because of conditional uniqueness. If typical (in the sense of
$\mu_{S}$) wave functions starting in $\mathscr{H}_{PH}$ will increase in
entropy, then we know that almost all of the normalized projection onto
$\mathscr{H}_{PH}$ will increase in entropy.

Therefore, in $W_{IPH}$ theories, we no longer need the Statistical Postulate;
we no longer need a probability / typicality distribution over initial quantum
states, since only one state is possible (given an initial subspace). The only
kind of probability in those theories will be the quantum mechanical
probabilities. We expect the following to be true. In $W_{IPH}$-BM, by the
W-version of the quantum equilibrium postulate (see Appendix), most initial
particle configurations will increase in entropy. In $W_{IPH}$-GRW, by the
W-version of the spontaneous collapse postulate, most likely the future
collapses will result in higher-entropy states. In $W_{IPH}$-EQM, most
emergent branches / worlds will evolve into higher-entropy ones.999Much more
work need to be done to show these results rigorously. A step in this
direction is to show the empirical equivalence between $\Psi_{PH}$ theories
and $W_{IPH}$ theories. For an overview of some existing results and some new
analysis, see Chen, (2019).

The property “conditional uniqueness" is a conditional property: the
uniqueness of the initial quantum state depends on a choice of an initial
subspace, representing the PH low-entropy macrostate. There are reasons to
believe that the choice of the initial subspace will not be unique in the
standard wave-function theory $\Psi_{PH}$. After all, the PH selects a low-
entropy macrostate given macroscopic variables such as volume, density, and
volume (§2.2 #5). The values of the macroscopic variables do not have precise
correspondence with subspaces in the Hilbert space. Two slightly different
subspaces $\mathscr{H}_{1}$ and $\mathscr{H}_{2}$ that share most wave
functions but differ regarding some wave functions will not make a difference
to the role that PH plays. The role of PH is to explain the macroscopic
regularity of entropic arrow of time and perhaps other arrows of time, which
by themselves do not require a maximal level of precision. (Macroscopic facts
are notoriously vague and imprecise. See the sorites problem surveyed in Hyde
and Raffman, (2018).) The precision in the subspace will not correspond to
anything in the physical world. If the actual wave function $\Psi$ is
contained in two slightly different subspaces $\mathscr{H}_{1}$ and
$\mathscr{H}_{2}$, the actual velocity of Bohmian particles, the actual
collapse probability distribution, and the structure of emergent Everettian
branches are all determined by $\Psi$. The slight differences between the two
subspaces (i.e. slight differences in the boundary of the macrostate) have no
influence on the microscopic facts in the world, since any microscopic
influence is screened off by the actual wave function. Moreover, the non-
unique correspondence in QSM (§2.2 $6) makes the boundaries between
macrostates even more fuzzy. It would seem highly arbitrary to pick a precise
boundary of the macrostate. If there are many choices of the initial subspace,
all differing from each other by a small amount, then perhaps there will be
many choices of the initial projection operators that are hard to distinguish
on a macroscopic level.

However, if there are merely multiple choices of the initial subspace, and it
is _determinate_ which ones are the admissible choices, perhaps we can just
collect all the admissible choices of the initial subspace and make PH a
disjunctive statement: it is one of the admissible choices. But the problem is
that what is admissible and what is not admissible will also be
_indeterminate._ This is analogous to the higher-order problem in vagueness
(see Sorensen, (2018)). So the most realistic version of PH will be allowing
the fuzzy boundary of the initial macrostate. We discuss these different
versions in more detail below.

####  3.2.2 Three Versions of PH and IPH

In what follows, I formulate three versions of PH, with different levels of
precision: the Strong PH, the Weak PH, and the Fuzzy PH. For each version, I
formulate a version of IPH that matches the level of precision in PH. Although
the Fuzzy PH might be the best choice for the standard theory, the Strong IPH
might be viable for the new theories of $W_{IPH}$.

The Strong Past Hypothesis relies on a particular decomposition of the energy
shell into macrostates (§2.2 #5). It selects a unique initial
subspace—$\mathscr{H}_{PH}$. The initial wave function has to start from
$\mathscr{H}_{PH}$.

Strong Past Hypothesis

$\Psi(t_{0})\in\mathscr{H}_{PH}$ (24)

Moreover, it is a well-defined probability space on which we can impose the
surface area measure $\mu_{S}$ (on the unit sphere of vectors). If the Strong
PH makes into the $\Psi_{PH}$ theories, then we can impose a similarly strong
IPH in the $W_{IPH}$ theories:

Strong Initial Projection Hypothesis

$\hat{W}_{IPH}(t_{0})=\frac{I_{PH}}{dim\mathscr{H}_{PH}}$ (25)

The Strong IPH selects a unique initial quantum state, which does not require
a further specification of statistical mechanical probabilities.

As mentioned before, there may be several low-dimensional subspaces that can
do the job of low-entropy initial condition equally well. They do not have to
be orthogonal to each other. The Past Hypothesis is formulated in macroscopic
language which is not maximally precise. The inherent imprecision can result
in indeterminacy of the initial subspace. There are two kinds of indeterminacy
here: (1) there is a determinate set of admissible subspaces to choose from,
and (2) the boundary between what is admissible and what is not is also fuzzy;
some subspaces are admissible but some others are borderline cases.

The first kind of indeterminacy corresponds to the Weak Past Hypothesis:

Weak Past Hypothesis

$\Psi(t_{0})\in\mathscr{H}_{1}\text{ or }\Psi(t_{0})\in\mathscr{H}_{2}\text{
or ... or }\Psi(t_{0})\in\mathscr{H}_{k}$ (26)

To be sure, the set of admissible subspaces may be infinite. For simplicity we
shall assume it is finite for now. For the Weak PH, there is the corresponding
Weak IPH:

Weak Initial Projection Hypothesis

$\hat{W}_{WPH_{i}}(t_{0})=\frac{I_{\mathscr{H}_{i}}}{dim\mathscr{H}_{i}}$ (27)

where $\mathscr{H}_{WPH_{i}}$ is the normalized projection onto the subspace
$\mathscr{H}_{i}$. For each admissible initial subspace, there is exactly one
choice of the initial quantum state. Even though there is only one actual
initial density matrix, there are many possible ones that slightly differ from
each other ($\hat{W}_{WPH_{1}}(t_{0})$, $\hat{W}_{WPH_{2}}(t_{0})$, …,
$\hat{W}_{WPH_{k}}(t_{0})$). Weak IPH does not need to be supplemented with an
additional probability distribution over possible initial quantum states in
that subspace.

The second kind of indeterminacy corresponds to the Fuzzy Past Hypothesis:

Fuzzy Past Hypothesis

The universal wave function started in some low-entropy state. The boundary of
the macrostate is vague. (The Fuzzy PH does not determinately pick out any set
of subspaces.)

Given the Fuzzy PH, we can formulate a Fuzzy IPH:

Fuzzy Initial Projection Hypothesis

$\hat{W}_{FPH}(t_{0})=\frac{I_{\mathscr{H}_{i}}}{dim\mathscr{H}_{i}}$ (28)

where $\mathscr{H}_{i}$ is an admissible precisification of the Fuzzy PH. Just
like the Weak IPH, the Fuzzy IPH does not select a unique initial quantum
state, since there are many admissible ones. (There is an actual initial
density matrix, and it is compatible with the Fuzzy IPH.) However, unlike Weak
IPH, there is no sharp cut-off between what is admissible and what is not
admissible. Hence, for theories that include the Fuzzy IPH, it may be
indeterminate whether some normalized projection operator is possible or not
possible.

In any case, neither the Weak IPH nor the Fuzzy IPH requires the further
specification of statistical mechanical probabilities. To see why, consider
what is required for the Statistical Postulate. It requires a probability
space to define the measure. The measure can only be defined relative to an
admissible precisification. That is, the measure is over possible state
vectors in a precise state space. In other words, the statistical mechanical
probabilities are only meaningful relative to a precisification of the initial
subspace. To be sure, the exact probability may be empirically inaccessible to
us — what we have access to may only be certain interval-valued
probabilities.101010See Bradley, (2016) for a review of imprecise
probabilities. However, without the precisification of the boundary, we do not
even have an interval-valued probabilities, as where to draw the boundary is
unsettled. Given this fact, we can see that the original statistical
mechanical probabilities require admissible precisifications—the statement
will have to be:

  * •

For a choice of an initial subspace, and given a uniform measure $\mu_{S}$ on
it, $\mu_{S}-$most (maybe even all) wave functions in that subspace will
approach thermal equilibrium.

So the corresponding statement for the Weak IPH and the Fuzzy IPH will be:

  * •

Every admissible density matrix will approach thermal equilibrium, where an
admissible density matrix is a normalized projection operator onto an
admissible initial subspace.

Since every admissible density matrix will approach thermal equilibrium, there
is no need for an additional probability measure over initial quantum states
in order to neglect the anti-entropic states. What is different on Fuzzy IPH
is that what is admissible is a vague matter.

To summarize: all three versions of the IPH get rid of the fundamental
statistical mechanical probabilities. The Strong IPH selects a unique initial
quantum state. Although the Weak IPH and the Fuzzy IPH do not, the initial
state is unique given an admissible precisification of the PH subspace. Hence,
all three versions—Strong IPH, Weak IPH, and Fuzzy IPH—satisfy conditional
uniqueness of the initial quantum state. That is, given a choice of an initial
subspace, there is a unique initial quantum state, i.e. its normalized
projection. Since conditional uniqueness is sufficient to eliminate
statistical mechanical probabilities at the fundamental level, all three
versions will be able to reduce the two kinds of probabilities to just the
quantum mechanical probabilities. However, for Strong IPH, the choice of the
initial subspace is unique while it is not unique for Weak IPH or Fuzzy IPH.
Therefore, Strong IPH trivially satisfies conditional uniqueness.

####  3.2.3 The Emergence of Statistical Mechanical Probabilities

Although the $W_{IPH}$ theories (with any of the three versions of IPH)
eliminate statistical mechanical probabilities at the fundamental level, they
can still emerge as useful tools for analysis. As we saw in (19), the
normalized projection onto a subspace can be decomposed as an integral of pure
state density matrices with respect to the uniform probability distribution on
the unit sphere. So mathematical results proven for a statistical ensemble of
wave functions can be directly applied to normalized projections. The
statistical perspective could prove useful for analysis of typicality and
long-time behaviors of a statistical ensemble of wave functions, which could
be translated to the long-time behavior of the mixed-state density matrix.

However, we recall that the decomposition into an integral of pure states is
not unique. For the same density matrix in (19), it can be decomposed as a sum
of pure states that are the orthonormal basis vectors, which gives us a
different statistical ensemble of wave functions (a discrete probability
distribution). Different statistical ensembles and the different emergent
probabilities can be useful for different purposes. The non-uniqueness in
decomposition further illustrates the idea that, in $W_{IPH}$ theories, given
an objective (real) density matrix $W$, the statistical mechanical
probabilities over initial wave functions are not fundamental but are only
emergent at the level of analysis.

###  3.3 Absolute Uniqueness

All three versions of IPH satisfy the property called _conditional uniqueness_
: given a choice of the initial subspace, there is a unique choice of the
initial quantum state—the normalized projection onto that subspace. The Strong
IPH satisfies that trivially. Indeed, the Strong IPH satisfies a stronger
property that I call _absolute uniqueness_ : there is only a unique choice of
the initial quantum state. This is because Strong IPH builds in a selection of
a unique initial subspace.

As we mentioned in §3.2.1, the Strong PH in the $\Psi_{PH}$ theories, which is
the counterpart to the Strong IPH in $W_{IPH}$ theories, is highly
implausible. The precision of the Strong PH corresponds to a precise low-
entropy initial macrostate, which has no correspondence to any precision in
the microscopic histories of particles, collapses, or field values. The PH
macrostate plays merely a macroscopic role, and the actual microstate, i.e. a
wave function, screens off the microscopic influences of the PH macrostate.

However, the situation is quite different in $W_{IPH}$ theories with a Strong
IPH. Here, the initial density matrix $W_{IPH}(t_{0})$ plays two roles. First,
it plays the usual macroscopic role corresponding to the low-entropy initial
condition. Second, and more importantly, it plays the microscopic role since
it enters into the fundamental dynamical equations of $W_{IPH}$ quantum
mechanics (see Appendix for details). $W_{IPH}(t_{0})$ directly guides Bohmian
particles, undergoes GRW collapses, and realizes the emergent branches of
Everett. Different choices of $W_{IPH}(t_{0})$ will leave different
microscopic traces in the world, in the form of different velocity fields for
Bohmian particles, different probabilities for GRW collapses, and emergent
branches with microscopic differences. Hence, the extra precision that sets
Strong IPH apart from Weak or Fuzzy IPH has a worldly correspondence: it
matches the microscopic history of the universe. Even if we were to think that
different choices of the initial subspaces might not make an observable
differences (at least to our current observational technologies), there is
still a fact of the matter in the world about the precise microscopic history.
Hence, it seems to me it is quite justifiable to postulate the Strong IPH in
$W_{IPH}$ theories, even though it is not justifiable to postulate the Strong
PH in $\Psi_{PH}$ theories.

The Strong IPH has another interesting consequence. It will make the
Everettian theory strongly deterministic in the sense of Penrose, (1989):
Strong determinism “is not just a matter of the future being determined by the
past; _the entire history of the universe is fixed_ , according to some
precise mathematical scheme, _for all time._ ” This is because the Everettian
theory is deterministic for all beables (local or non-local). With the help of
a unique initial state, there is no ambiguity at all about what the history
could be—there is only one way for it to go: starting from
$\hat{W}_{IPH}(t_{0})$ (on Strong IPH) and evolving by the von Neumann
equation (18).

###  3.4 Other Applications

Getting rid of statistical mechanical probabilities at the fundamental level
is not the only advantage of $W_{IPH}$-quantum theories over
$\Psi_{PH}$-quantum theories. I discuss them in more detail in Chen, 2018b ,
Chen, 2018c . Here I briefly summarize the other features of $W_{IPH}$-quantum
theories:

  1. 1.

The meaning of the quantum state.

It has been a long-standing puzzle regarding how to understand the meaning of
the quantum state, especially because it is defined as a function on a high-
dimensional space and it is a non-separable object in physical space. A
particularly attractive proposal is to understand it as nomological—being on a
par with laws of nature. However, it faces a significant problem since typical
wave functions are highly complex and not simple enough to be nomological. In
contrast, if the quantum state is given by IPH, the initial quantum state will
inherit the simplicity of the PH subspace. If PH is simple enough to be a law,
then the initial quantum state is simple enough to be nomological. This
feature of simplicity supports the nomological interpretation without relying
on specific proposals about quantum gravity (cf: Goldstein and Zanghì,
(2013)). However, I should emphasize that the nomological interpretation is
not the only way to understand the density matrix theory, as other proposals
are also valid, such as the high-dimensional field and the low-dimensional
multi-field interpretations (Chen, 2018b  §3.3).

  2. 2.

Lorentz invariance.

David Albert, (2015) observes that there is a conflict among three properties:
quantum entanglement, Lorentz invariance, and what he calls narratability. A
world is narratable just in case its entire history can be narrated in a
linear sequence and every other sequence is merely a geometrical
transformation from that. Since narratability is highly plausible, denying it
carries significant cost. Hence, the real conflict is between quantum
entanglement (a purely kinematic notion) and Lorentz invariance. This applies
to all quantum theories that take quantum entanglement to be fundamental.
However, given IPH, it is possible to take the nomological interpretation of
the quantum state and remove entanglement facts among the facts about the
distribution of physical matter. This is especially natural for $W_{IPH}$-Sm,
somewhat less naturally to $W_{IPH}$-GRWm and $W_{IPH}$-GRWf, and potentially
applicable to a fully Lorentz-invariant version of $W_{IPH}$-BM.

  3. 3.

Kinematic and dynamic unity.

In $\Psi_{PH}$-quantum theories, 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-
quantum 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 WIPH-
quantum 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.

Moreover, in a universe described by $\Psi_{PH}$-BM, subsystems sometimes do
not have conditional wave functions due to the presence of spin. In contrast,
in a universe described by $W_{IPH}$-BM, the universe and the subsystems
always have quantum states given by density matrices. This is because we can
always define conditional density matrix for a Bohmian subsystem (Dürr et al.,
(2005)). That is, in $W_{IPH}$-BM, the W-guidance equation is always valid for
the universe and the subsystems. In $\Psi_{PH}$-BM, the wave-function version
of the guidance equation is not always valid. Thus, the W-BM equations are
valid in more circumstances than the BM equations. We say that $W_{IPH}$-BM
has more dynamic unity than $\Psi_{PH}$-BM:

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

Both kinematic unity and dynamic unity come in degrees, and they are only
defeasible reasons to favor one theory over another. But it is nonetheless
interesting that merely adopting the density-matrix framework will make the
theory more “unified” in the above senses.

###  3.5 Generalizations

IPH is not the only principle that leads to the selection of a unique or
effectively unique initial quantum state of the universe. It is just one
example of a simple principle. It is easy to generalize the strategy discussed
here to other simple principles about the cosmological initial condition:

  * •

Start from the full Hilbert space (energy shell) $\mathscr{H}$.

  * •

Use simple principles (if there are any) to determine an initial subspace
$\mathscr{H}_{0}\subset\mathscr{H}$.

  * •

Choose the natural quantum state in that subspace—the normalized projection
$\hat{W}_{0}(t_{0})=\frac{I_{0}}{dim\mathscr{H}_{0}}$.

  * •

The natural choice will be simple and unique.

Another related but different example is the quantum version of the Weyl
Curvature Hypothesis proposed by Ashtekar and Gupt, (2016) based on the
proposal of the classical version in Penrose, (1989). However, Ashtekar and
Gupt’s hypothesis results in an infinite-dimensional unit ball of initial wave
functions, which may not be normalizable. It is not clear to me whether there
will be significant dimension reduction if we intersect it with the energy
shell. In any case, the problem of non-normalizability is a general problem in
cosmology which may require an independent solution.

###  3.6 Other Proposals

I would like to contrast my proposal with three other proposals in the
literature.

Albert (2000) proposes that it is plausible that $\Psi_{PH}$-GRW theories do
not need the Statistical Postulate. This is because the GRW jumps may be large
enough (in Hilbert space) to render every initial wave function entropic in a
short time. An anti-entropic wave function of a macroscopic system that
evolves forward will be quickly hit by a GRW jump. As long as the GRW jump has
a certain width that is large compared to the width of the anti-entropic set,
the wave function will collapse into an entropic one. This relies on a
conjecture about GRW theory: the final and empirically adequate GRW models
will have a collapse width that is large enough. If that conjecture can be
established, then, for every initial wave function, it is with a GRW
probability that it will be entropic. This may correspond to the right form of
the probabilistic version of the Second Law of Thermodynamics. I believe this
is a very plausible conjecture. But it is an additional conjecture
nonetheless, and it only works for collapse theories. It is an empirically
open question whether GRW will survive experimental tests in the next 30
years. In contrast, my proposal is fully general—it works for GRW theories,
Bohmian theories, and Everettian theories, and it does not rely on additional
conjectures beyond those already postulated in QSM.

Wallace, (2011) proposes that we can replace the Past Hypothesis and the
Statistical Postulate with a Simple Dynamical Conjecture. In essence, it says
that every Simple wave function will evolve to higher entropy. Here, “Simple”
is a technical notion here meaning that the wave function is simple to specify
and not specified by using time-reverse of an anti-entropic wave function. The
idea is to replace the statistical postulate, which gives us reasons to
neglect certain initial wave functions, with another postulate about
simplicity, which also gives us reasons to set aside certain initial wave
functions. This is a very interesting conjecture, which I think one should
seriously investigate. But it is an additional conjecture nonetheless,
although it has applicability to all quantum theories.

Wallace, (2016) proposes that we can allow quantum states to be either pure or
mixed. Moreover, he suggests that we can reinterpret probability distributions
over wave functions as part of the state description and not an additional
postulate about probability. There is much in common between Wallace’s (2016)
proposal and my proposal. However, the way that I get rid of statistical
mechanical probabilities is not by way of a reinterpretation strategy but by
using the uniqueness (or conditional uniqueness) of the initial quantum state.
To be sure, there are many ways to achieve the goal, and the two approaches
are quite related.

I would like to briefly mention the possibility of implementing my proposal in
CSM. The upshot is that it is much less natural to do so in CSM than in QSM,
for several reasons. On the classical mechanical phase space, the object that
plays a similar role to the density matrix is the probability function
$\rho(x)$. However, it is not clear how to understand its meaning as something
ontic. If it is to be understood as a high-dimensional field and the only
object in the ontology, then we have a Many-Worlds theory, unless we revise
the Hamiltonian dynamics and make it stochastic. If it is to be understood as
a low-dimensional multi-field (cf: Chen, (2017) and Hubert and Romano, (2018)
and the references therein), it is not clear what corresponds to the momentum
degrees of freedom. Second, the dynamics is not as natural in CSM. If it is to
be understood as a high-dimensional field guiding a point particle (the phase
point), it is not clear why we increase the complexity of the ontology. If it
is to be understood as a nomological object, it is not clear what role it
plays in the dynamics—it certainly does not give rise to a velocity field,
because that is the job of the Hamiltonian function.111111See McCoy, (2018)
for a recent attempt in this direction. Despite what I say here, I believe it
is worth exploring, as McCoy does, alternative interpretations of the
statistical state in CSM and QSM.

##  4 Conclusion

The Thermodynamic Theories of Quantum Mechanics (TQM or $W_{IPH}$-QM) provide
a new strategy to eliminate statistical mechanical probabilities from the
fundamental postulates of the physical theory. They do so in a simple way not
relying on any further conjectures about quantum mechanics or statistical
mechanics. Moreover, they lead to several other applications and
generalizations, which we only touched on briefly in this paper. Most
importantly, we have found another deep connection between the foundations of
quantum mechanics and the foundations of statistical mechanics.

## Appendix

### (1) $\Psi_{PH}$-Quantum Theories

####  $\Psi_{PH}$-Bohmian mechanics: ($Q,\Psi_{PH}$)

The Past Hypothesis:

$\Psi_{PH}(t_{0})\in\mathscr{H}_{PH}$ (29)

The Initial Particle Distribution:

$\rho_{t_{0}}(q)=|\psi(q,t_{0})|^{2}$ (30)

The Schrödinger Equation:

$\ i\hbar\frac{\partial\psi}{\partial t}=H\psi$ (31)

The Guidance Equation (Dürr et al. 1992):

$\frac{dQ_{i}}{dt}=\frac{\hbar}{m_{i}}\text{Im}\frac{\nabla_{i}\psi(q)}{\psi(q)}(q=Q)$
(32)

####  $\Psi_{PH}$-Everettian mechanics

The Past Hypothesis:

$\Psi_{PH}(t_{0})\in\mathscr{H}_{PH}$ (33)

The Schrödinger Equation:

$i\hbar\frac{\partial\psi}{\partial t}=H\psi$ (34)

The Mass Density Equation:

$m(x,t)=\bra{\psi(t)}M(x)\ket{\psi(t)},$ (35)

$W_{PH}$-S0: only $\Psi_{PH}$.

$W_{PH}$-Sm: $m(x,t)$ and $\Psi_{PH}$.

####  $\Psi_{PH}$-GRW theory

The Past Hypothesis:

$\Psi_{PH}(t_{0})\in\mathscr{H}_{PH}$ (36)

The linear evolution of the wave function is interrupted randomly (with rate
$N\lambda$, where $\lambda$ is of order $10^{-15}$ s-1) by collapses:

$\Psi_{T^{+}}=\frac{\Lambda_{I_{k}}(X)^{1/2}\Psi_{T^{-}}}{||\Lambda_{I_{k}}(X)^{1/2}\Psi_{T^{-}}||},$
(37)

with the collapse center $X$ being chosen randomly with probability
distribution $\rho(x)=||\Lambda_{i_{k}}(x)^{1/2}\Psi_{T^{-}}||^{2}dx$, where
the collapse rate operator is defined as:

$\Lambda_{I_{k}}(x)=\frac{1}{(2\pi\sigma^{2})^{3/2}}e^{-\frac{(Q_{k}-x)^{2}}{2\sigma^{2}}}$
(38)

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.

$\Psi_{PH}$-GRWm and $\Psi_{PH}$-GRWf: defined with local beables $m(x,t)$ and
$F$.

### (2) $W_{IPH}$-Quantum Theories

####  $W_{IPH}$-Bohmian mechanics: ($Q,W_{IPH}$)

The Initial Projection Hypothesis:

$\hat{W}_{IPH}(t_{0})=\frac{I_{PH}}{dim\mathscr{H}_{PH}}$ (39)

The Initial Particle Distribution:

$P(Q(t_{0})\in dq)=W_{IPH}(q,q,t_{0})dq$ (40)

The Von Neumann Equation:

$i\hbar\frac{\partial\hat{W}}{\partial t}=[\hat{H},\hat{W}]$ (41)

The $W_{PH}$-Guidance Equation (Dürr et al. 2005):

$\frac{dQ_{i}}{dt}=\frac{\hbar}{m_{i}}\text{Im}\frac{\nabla_{q_{i}}W_{IPH}(q,q^{\prime},t)}{W_{IPH}(q,q^{\prime},t)}(q=q^{\prime}=Q)$
(42)

####  $W_{IPH}$-Everettian mechanics

The Initial Projection Hypothesis:

$\hat{W}_{IPH}(t_{0})=\frac{I_{PH}}{dim\mathscr{H}_{PH}}$ (43)

The Von Neumann Equation:

$i\hbar\frac{\partial\hat{W}}{\partial t}=[\hat{H},\hat{W}]$ (44)

The Mass Density Equation:

$m(x,t)=\text{tr}(M(x)W(t)),$ (45)

$W_{IPH}$-S0: only $W_{IPH}$.

$W_{IPH}$-Sm: $m(x,t)$ and $W_{IPH}$.

####  $W_{IPH}$-GRW theory

The Initial Projection Hypothesis:

$\hat{W}_{IPH}(t_{0})=\frac{I_{PH}}{dim\mathscr{H}_{PH}}$ (46)

The linear evolution of the density matrix is interrupted randomly (with rate
$N\lambda$, where $\lambda$ is of order $10^{-15}$ s-1) by collapses:

$W_{T^{+}}=\frac{\Lambda_{I_{k}}(X)^{1/2}W_{T^{-}}\Lambda_{I_{k}}(X)^{1/2}}{\text{tr}(W_{T^{-}}\Lambda_{I_{k}}(X))}$
(47)

with $X$ distributed by the following probability density:

$\rho(x)=\text{tr}(W_{T^{-}}\Lambda_{I_{k}}(x))$ (48)

where the collapse rate operator is defined as:

$\Lambda_{I_{k}}(x)=\frac{1}{(2\pi\sigma^{2})^{3/2}}e^{-\frac{(Q_{k}-x)^{2}}{2\sigma^{2}}}$
(49)

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. The dynamical equations of W-GRW were introduced in Allori et al.,
(2013) §4.5.

$W_{IPH}$-GRWm and $W_{IPH}$-GRWf: defined with local beables $m(x,t)$ and
$F$.

## Acknowledgement

I would like to thank two anonymous reviewers of this volume for their helpful
feedback on an earlier draft of this paper. I am also grateful for stimulating
discussions with Valia Allori, Sean Carroll, Detlef Dürr, Michael Esfeld,
Veronica Gomez, Dustin Lazarovici, Matthias Lienert, Tim Maudlin, Sebastian
Murgueitio, Wayne Myrvold, Jill North, Daniel Rubio, Ted Sider, Roderich
Tumulka, David Wallace, Isaac Wilhelm, Nino Zanghì, and especially David
Albert, Sheldon Goldstein, and Barry Loewer.

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