# The Cosmic Void

Eddy Keming Chen Department of Philosophy, University of California, San
Diego, 9500 Gilman Dr, La Jolla, CA 92093-0119. Website:
www.eddykemingchen.net. Email<EMAIL_ADDRESS>
(Sara Bernstein and Tyron Goldschmidt (eds.), Non-Being: New Essays on the
Metaphysics of Nonexistence, Oxford University Press, 2021)

###### Abstract

What exists at the fundamental level of reality? On the standard picture, the
fundamental reality contains (among other things) fundamental matter, such as
particles, fields, or even the quantum state. Non-fundamental facts are
explained by facts about fundamental matter, at least in part. In this paper,
I introduce a non-standard picture called the “cosmic void” in which the
universe is devoid of any fundamental material ontology. Facts about tables
and chairs are recovered from a special kind of laws that satisfy _strong
determinism_. All non-fundamental facts are completely explained by nomic
facts. I discuss a concrete example of this picture in a strongly
deterministic version of the many-worlds theory of quantum mechanics. I
discuss some philosophical and scientific challenges to this view, as well as
some connections to ontological nihilism.

Keywords: ontological nihilism, objects, emergence, many-worlds, time’s arrow,
Humeanism, non-Humeanism, laws of nature, fundamentality, the Wentaculus

##  1 Introduction

One of the hardest questions in fundamental physics and fundamental
metaphysics is this: what exists at the fundamental level of reality? There is
no consensus in contemporary physics. There is much less consensus in
metaphysics.

Nevertheless, on the physics side, we usually assume that the fundamental
level of reality will be something physical and material. We assume that the
fundamental level will not consist in purely mental ideas. We also assume that
the fundamental level will not be completely devoid of matter, since our world
is manifestly not empty. Different physical theories postulate different kinds
of fundamental matter (and sometimes different versions of the same theory
will differ in the fundamental material ontologies). To see some examples,
here is a preliminary list of fundamental material ontologies in different
physical theories:

  * •

Newtonian mechanics: point particles;

  * •

Maxwellian electrodynamics: charged particles and electromagnetic fields;

  * •

General relativity: matter field on space-time;

  * •

Quantum mechanics: a quantum state and / or local ontologies such as matter
density field, flashes, or particles;

  * •

Quantum field theory: a quantum state and / or local ontologies such as
particles, particle number ontology, or fields;

  * •

Loop quantum gravity: spin-foams;

  * •

String theory: one-dimensional strings.

(This list is certainly incomplete: future theories may have radically
different ontologies; and even theories on this list can be compatible with
other ontologies.)

Describing the fundamental material ontology is an important task in
theoretical physics, since fundamental matter plays crucial roles in a
physical theory. First, the fundamental material ontology in a theory plays an
explanatory and metaphysical role. It is the ultimate explanation for non-
fundamental ontologies and non-fundamental facts. We use fundamental matter as
the ultimate reduction base: we try to reduce the behaviors of macroscopic
systems such as tables and chairs to the behaviors of microscopic constituent
objects such as particles and fields. How tables and chairs move around can be
explained by how the constituent particles and fields behave. How particles
and fields behave can be derived from the laws of physics. If the particles
and fields are part of the theory’s fundamental material ontology, then the
reduction is complete. If they are further reducible to something else in the
fundamental material ontology, such as the quantum state, then more reduction
is in order. In any case, it is hard to see how to carry out the reduction
without the fundamental material ontology.

Second, the fundamental material ontology plays a semantic role by being the
subject matter of the laws of nature. The terms in the laws of nature refer to
properties of the fundamental material ontology. For example, the mass term in
$F=ma$ refers to a property of Newtonian point particles. The acceleration
term describes the change of their velocities. It is hard to see how
fundamental physical laws make sense without the fundamental material
ontology.

Third, the fundamental material ontology plays an informational role in the
physical theory. The physical laws are supposed to be simple. But how can such
simple laws explain such a wide range of complicated phenomena? How does so
much information come from such simple laws? It is made possible by appealing
to the complicated initial conditions and boundary conditions of physical
systems. The complicated data is stored not in the laws but in the matter
distribution, such as the locations of particles and configurations of fields.
By postulating a fundamental material ontology, complicated phenomena can be
derived from simple laws plus certain contingent (and complicated) facts about
fundamental matter. There is a tight connection between fundamental material
ontology and laws of physics that we will try to make more precise later. But
it is worth noting that we usually put the “mess” in the fundamental material
ontology so that the laws can be simple.

Those roles are important and seemingly irreplaceable. Hence, the fundamental
material ontology seems indispensable to any successful physical
theory.111Quantum theory presents an apparent counterexample, as the followers
of the “Copenhagen interpretation” deny the existence of a determinate
microscopic ontology. But we have made sufficient progress about the quantum
measurement problem to understand that there are better ways to understand
quantum theory. See §3.1.  Given those roles fundamental matter seems to play
in successful physical theories, we also have reasons to postulate them in our
fundamental metaphysics.

However, it can be intellectually healthy to reexamine our assumptions. As
Turner, (2010) puts it, “we can come to better understand a proposition by
studying its opposite.” We ask the following question: is a fundamental
material ontology really indispensable to any successful physical theory?
Reflecting on this question does not require us to reject scientific realism.
It may even help us better appreciate the place of fundamental material
ontology in our physical theories and metaphysical frameworks.

In this paper, I discuss a possibility that I call “the cosmic void.” In the
cosmic void scenario, the universe is devoid of any fundamental
matter.222Hence, it is different from the notion of the cosmic void in
astrophysics that the large-scale structure of the universe can be
characterized by mostly empty space between galaxies. I focus on the
informational role of the fundamental matter ontology and suggest that, in
certain theories, it can be played by the laws of physics. To recover the non-
fundamental facts—the manifest image of tables, chairs, and computers—I make
use of _strongly deterministic_ laws of nature that determine a unique history
of the universe. In this scenario, all of the non-fundamental facts are
ultimately explained by nomic facts. I discuss a concrete example of the
cosmic void scenario that arises when we try to put together a many-worlds
theory of quantum mechanics in a time-asymmetric universe. The physical theory
turns out to be strongly deterministic. To be sure, there are both
philosophical and scientific challenges to such a possibility, especially
regarding the metaphysical role and the semantic role of the fundamental
matter ontology. I introduce the cosmic void scenario _not to endorse it but
to draw our attention to an interesting area of logical space that deserves
more scrutiny._

There are some similarities between this project and the anti-object
metaphysics of ontological nihilism (Hawthorne & Cortens, 1995, Turner, 2010),
structural realism (Ladyman et al. , 2007), generalism (Dasgupta, 2009), and
the bare facts framework (Maxwell, 2019). However, the cosmic void possibility
is at the same time more radical and more modest. It is more radical in that
we do not postulate even fundamental properties, general facts, bare facts, or
structural facts in the fundamental ontology. It is more modest in that the
strategy is not supposed to work in general, but only in some special physical
theories where strong determinism holds. Hence, the cosmic void scenario has a
more restricted scope of application.

##  2 The Possibility of the Cosmic Void

What is the cosmic void? It is the scenario in which _nothing material exists
at the fundamental level_. Suppose spacetime is fundamental. Then, in this
scenario, fundamentally speaking the spacetime is completely empty. There are
no fundamental particles, fundamental fields, fundamental quantum states, any
kind of fundamental material distributions, or any non-trivial decorations on
spacetime.

This cosmic void scenario is allowed by both special and general relativity.
However, that is not the intended interpretation here. In the empty solutions
allowed by special and general relativity, non-fundamental facts that there
are tables and chairs are false. In the cosmic void scenario I would like to
describe, those non-fundamental facts remain _true_ , even though there is no
fundamental material ontology.

Figure 1: The standard picture where non-fundamental facts are explained by
fundamental facts about matter and laws of nature.

Figure 2: The non-standard picture where non-fundamental facts are completely
explained by fundamental laws of nature.

On the standard picture (see Figure 1), non-fundamental facts about tables and
chairs are made true by fundamental laws of nature and fundamental facts about
the existence and behavior of fundamental matter, such as point particles.
Fundamental particles constitute tables and chairs, and the existence and
behavior of tables and chairs can be explained by the existence and behavior
of the fundamental material ontology, together with the laws.

On the non-standard picture (see Figure 2), since there is nothing material at
the fundamental level, we have to appeal to something else to explain the non-
fundamental facts. What can that explanation be? A candidate is the
fundamental laws of nature. For the purpose of this paper, we commit to a non-
Humean theory of lawhood. Here we assume that laws are not merely summaries of
the mosaic. There can be non-trivial laws even when the mosaic is completely
empty and un-decorated.

Usually, fundamental laws of nature are _at best_ partial explanations for
most non-fundamental facts. The existence of a table in front of me at this
point is not entailed by the standard laws of physics. At the very least, we
need also the complete state of the universe at some time (or during some time
interval). The complete state of the universe is not encoded in the dynamical
laws, for they are compatible with many different states of the universe. In
standard physical theories, the complete state of the universe refers to the
state of the fundamental material ontology. Hence, the non-standard picture is
not accommodated by the usual laws of physics.

Figure 3: Indeterministic laws of physics. $X0$–$X8$ refer to different
initial conditions of the universe. (Here the illustration is schematic.
Usually there are infinitely many possible initial conditions.) The horizontal
curves correspond to different histories of the universe. Different histories
of the universe can overlap at some time and diverge later. Fixing an initial
condition of the universe does not fix a unique history of the universe.

Figure 4: Deterministic laws of physics. $X0$–$X8$ refer to different initial
conditions of the universe. The horizontal curves correspond to different
histories of the universe. Different histories of the universe cannot overlap
at any point in time. Fixing an initial condition of the universe fixes a
unique history of the universe.

It is useful to distinguish between two types of laws of physics:
indeterministic laws and deterministic laws. If the laws are indeterministic
(see Figure 3), then given a complete state of the universe at some time (or
some duration of time), the laws can allow multiple different pasts and
futures of the universe. In other words, different histories can overlap. If
the laws also assign objective probabilities or chances to the histories, then
given a complete state of the universe at some time, the laws assign a unique
probability distribution over future (or past) histories.

If the laws are deterministic (see Figure 4), then given a complete state of
the universe at some time (or some duration of time), the laws allow only one
past and one future of the universe. In other words, different histories
cannot overlap: no two distinct histories of the universe can overlap (be
exactly the same in terms of the distribution of microphysical properties and
ontologies) at any point in time. Hence, the deterministic theory can be more
informative than an indeterministic theory in the sense that the former but
not the latter uniquely determines the entire history of the universe given
the dynamical laws and a specification of the material ontology at some time
(or some duration of time). In a universe governed by deterministic laws of
physics, fixing the initial condition of the universe is sufficient to fix the
entire history of the universe.

Figure 5: Strongly deterministic laws of physics. $X_{0}$ refers to the unique
initial condition allowed by the laws. The horizontal curve refers to the
unique history of the universe allowed by the laws. The laws completely fix
the history of the universe.

However, information about the exact initial condition itself is not contained
in any standard law of physics. Usually it is the distribution of the
fundamental material ontology, not the laws, that specify the complete initial
condition. After all, usually the exact initial condition is too complicated
to express by any simple law. (As we discuss below, this is true even after
adding the Past Hypothesis as an additional fundamental law of nature, for it
only pins down the macroscopic initial condition of the universe and not the
microscopic initial condition.) Hence, it is in this sense that the standard
deterministic laws of physics are compatible with many different possible
worlds. Usually the laws do not pick out a unique world.

The situation is transformed when the laws are _strongly deterministic_ in the
following sense:

Strong Determinism

The laws of nature pick out a unique history of the universe.

A world is strongly deterministic if its fundamental laws pick out a unique
history of the universe (see Figure 5).333This notion of strong determinism is
introduced in (Penrose, 1989). This is different from the notion of
“superdeterminism” that is sometimes invoked in the context of avoiding Bell
non-locality. See (Chen, 2020a) for an overview.  One way the laws can be
strongly deterministic is by satisfying two conditions:

  1. 1.

The laws are deterministic.

  2. 2.

The laws pick out a unique initial condition.

Here, the unique initial condition refers to an exact specification of the
microphysical details of the universe at $t_{0}$. Typically, it is difficult
to pick out a unique (microscopic) initial condition without making the laws
overly complicated (so that they no longer qualify as good candidate
fundamental laws of physics). Hence, it is highly non-trivial to write down a
_simple_ law (or set of laws) that can do that. In §3, we look at a concrete
example of a law—the Initial Projection Hypothesis—that is not only simple but
also selects a unique (microscopic) initial condition in the Everettian many-
worlds theory of quantum mechanics.

In a strongly deterministic world, at the fundamental level, there is no
contingency. There is only one way the universe could be, in the sense of
nomic possibility. If the actual world is strongly deterministic, then the
actual world is the only one that is nomologically possible. That is, what is
actual is also nomologically necessary. Any sense of contingency would have to
come out at some non-fundamental level.

Let us contrast that with a more familiar theory—classical mechanics. The only
dynamical law $F=ma$ selects a space of possible worlds—the solutions to that
equation. It is possible to add to it a macroscopic initial condition law—the
Past Hypothesis, which says that the universe started in a low-entropy
macrostate.444For discussions of the Past Hypothesis, (Boltzmann, 1964,
Feynman, 2017, Penrose, 1979, Albert, 2000, Goldstein, 2001, Callender, 2004,
Loewer, 2012, 2020). This special macrostate allows the emergence of various
asymmetries in time such as the Second Law of Thermodynamics. However, even
after adding the Past Hypothesis, there is still an infinity of microscopic
initial conditions compatible with the laws. These initial conditions
(compatible with both $F=ma$ and the Past Hypothesis) have smaller measure
than the original set (whose only constraint is $F=ma$). In this scenario,
even though the laws are deterministic, they do not pick out a unique history
of the universe, as there are many possible worlds compatible with the laws.
Hence, the laws are not strongly deterministic.

In a strongly deterministic world, the laws specify the entire history of the
universe without needing any further input. Hence, it is in principle possible
to extract all the information about the world from just the laws alone. It is
this feature of strong determinism that makes possible an empirically adequate
theory of a cosmic void, in the sense of being informationally complete to
recover all the facts. (Is this sense of empirical adequacy strong enough? We
return to this question in §4.) Suppose at the fundamental level we have
strongly deterministic laws. There is only one way the fundamental material
ontology could be like—they have to be arranged exactly according to the only
way allowed by the laws. Such an arrangement of fundamental matter would be
nomologically necessary. Facts about fundamental matter would make true non-
fundamental facts about the locations and behaviors of tables and chairs.

We can go further. The strongly deterministic laws constrain the fundamental
ontology _if it exists_. But now suppose that there is no material ontology at
the fundamental level. All we have are the laws of physics, from which we can
in principle derive all facts about the non-fundamental. Would it make a
difference whether the derivation goes through some postulate about
fundamental matter? It depends on one’s view about ontological dependence:
must non-fundamental facts about tables and chairs bottom out in fundamental
facts about matter? (We return to this question in §4.) Suppose we accept the
possibility that non-fundamental facts about tables and chairs can bottom out
in fundamental facts about laws of nature. That is, all explanation about the
non-fundamental can be purely nomic. We arrive at the cosmic void scenario—in
a strongly deterministic universe, the physical theory can be empirically
adequate without postulating any fundamental material ontology. All there is
at the fundamental level are the laws of nature.

What is this notion of “deriving” non-fundamental facts from laws of nature? I
suggest this does not have to differ from the notion of mathematical-physical
derivation that links the macroscopic facts such as the positions of tables to
the microscopic facts such as the positions of particles. The derivation
combines a variety of techniques, such as mathematical proofs, coarse-
graining, approximations, and idealizations. The kind of examples I have in
mind are those in statistical mechanics concerning the reduction from the
macroscopic phenomena to kinetic theory. But they can also be found in high-
energy physics and quantum theory. However, in the more familiar examples in
physics, since the laws people use are not strongly deterministic, the laws
themselves are insufficient, and the derivation must involve further input
about the contingent initial conditions.

It is worth emphasizing that, since we are assuming a non-Humean theory of
lawhood, strongly deterministic laws are compatible with the existence of
fundamental material ontology as well as their absence. If we do not postulate
fundamental material ontology in a strongly deterministic universe, if the
cosmic void scenario is possible, then there can be non-fundamental facts
about the material ontology that can coincide with the content of the
fundamental material ontology that could be postulated. However, whether or
not the facts about such material ontology are fundamental, they make true the
same set of non-fundamental facts—the locations of tables and chairs. So it is
best to say that the strongly deterministic universe is compatible with two
worlds that coincide in all facts except the fundamental material facts.
Hence, we need to revise our earlier definition of strong determinism:

Strong Determinism*

The laws of nature pick out a unique way to completely specify the fundamental
matter.

However, a strongly deterministic world on this criterion does not need to
specify any fundamental matter—it could realize the cosmic void scenario that
is devoid of fundamental matter. But if we were to completely specify
fundamental matter, then there would be only one way to do so according to the
laws. The qualifier “completely” is important here: if we were to only
partially specify the fundamental matter, then there could be infinitely many
ways to do so. We could, for example, specify only the configuration of matter
in spacetime region $R_{1}$, specify only that in $R_{2}$, and so on.

The cosmic void scenario has many interesting features. In such a world, even
though the fundamental arena (space-time or something analogous to space-time)
is completely empty, we can still recover all the non-fundamental facts from
fundamental laws alone, relying on purely nomic explanations. That would be
surprising if it turns out to be a successful theory. What are some advantages
for maintaining the cosmic void scenario? First, it has a fairly parsimonious
ontological base. All else being equal, we have reasons to prefer a more
parsimonious theory. Of course, not everyone will be convinced that other
things are equal here. Perhaps we lose certain explanatory power. We also
should be careful where and how to apply Ockham’s Razor for reasons mentioned
in (Maudlin, 2007). We will return to this point in §4. Second, the cosmic
void scenario also shares some of the advantages of radical versions of
ontological nihilism, such as the versions developed in (Hawthorne & Cortens,
1995). For example, since there are no material objects at the fundamental
level, the difficult metaphysical disputes about identity of material objects
over time and mereological composition of material objects simply do not
arise, at least for things that exist at the fundamental level. Nevertheless,
neither consideration is decisive. Moreover, the cosmic void scenario may cost
us important intuitions or principles that we cherish (see §4). In the end, it
will require a more comprehensive cost / benefit analysis (one that we do not
have the space to develop in this paper) to reach a reflective equilibrium
about whether the cosmic void scenario should be allowed in logical space. Let
us now turn to a case study of the cosmic void scenario.

##  3 A Case Study

In this section, we provide a concrete example of strong determinism to
illustrate the possibility of the cosmic void. The example relies on a non-
standard way of combining Everettian many-worlds interpretation of quantum
mechanics with the low-entropy initial condition of the universe that is
essential for explaining the arrow of time.

###  3.1 Many-Worlds Interpretation of Quantum Mechanics

Hugh Everett III’s many-worlds interpretation (1957) was an attempt to address
a central problem in the foundation of quantum mechanics—the quantum
measurement problem.555Below is a rough sketch. For a more thorough discussion
about the quantum measurement problem, see (Bell, 1990, Albert, 1992, Myrvold,
2017). Textbook quantum mechanics suggests that we assign a quantum state,
represented by a wave function, to some physical systems of interest, such as
a cat in a box. The wave function obeys two dynamics. First, it obeys a linear
equation of motion—the Schrödinger equation. That equation has the tendency to
spread out the wave function into what is called a superposed state, such as
the cat-being-alive state superposed with the cat-being-dead state. In that
state, it is hard to understand what is going on physically: is the cat alive,
dead, or both? Second, upon “observation” or “measurement,” the system’s wave
function will undergo a random and non-linear collapse to a particular state,
such as the state that the cat is alive.

Despite the predictive success, this recipe for making predictions faces
foundational problems. How can the wave function obey two different dynamics?
If the system is completely described by the wave function, and if both
observers and measurement instruments are physical systems interacting quantum
mechanically, they would always obey the Schrödinger equation and never
collapse into a definite state (such as the cat-being-alive state) from a
superposed state (such as the superposed state of the cat-being-alive and the
cat-being-dead). We can put together the principles that result in a
contradiction:

  * (P1)

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

  * (P2)

The wave function always obeys the Schrödinger equation.

  * (P3)

Every experiment has a unique outcome.

There are three types of solutions to the quantum measurement problem. Each of
them rejects one of the assumptions. The first type of solutions rejects (P1)
and adds additional variables beyond the wave function to represent other
aspects of the system. A prominent theory in this type is Bohmian mechanics,
originally proposed by De Broglie, (1928) and Bohm, (1952) independently. See
(Goldstein, 2017) for an overview. On that theory, there are point particles
with precise locations in addition to the wave function. The cat was made out
of particles guided by the wave function. The particles are always in a
determinate macrostate—either the cat is alive or the cat is dead. The wave
function plays two roles: it guides the motion of particles and it provides a
probability distribution over their precise locations.

The second type of solutions rejects (P2) and replaces the Schrödinger
equation by a spontaneous collapse dynamics. A theory in this type is the GRW
theory, proposed by Ghirardi, Rimini, and Weber (1986). See (Ghirardi, 2018)
for an overview. The GRW theory replaces the measurement-triggered collapse
dynamics with something much less anthropocentric—an objective and spontaneous
collapse mechanism that collapses the wave function with a fixed probability
per unit time per unit particle. Before we open the box to measure the cat,
its wave function has already collapsed into one of the two states: either
alive or dead.

The third type of solutions rejects (P3) and embraces the non-uniqueness that
comes with it. This was proposed by Everett and also known as the many-worlds
interpretation of quantum mechanics. See (Vaidman, 2018) for an overview. On
this theory, the wave function is the complete description of the physical
system. It always obeys the Schrödinger equation. But typically experiments
lead to multiple definite outcomes all at once. When we go and measure the
cat, we observe that it is in a definite state, say, the cat is alive. But
appearances can be misleading. Since nothing has collapsed the wave function
and there are no additional independent variables, the part of the wave
function in which the cat is dead still exists and is equally real as the part
in which the cat is alive. What is going on? On this interpretation, there are
at least two “parallel worlds” after the experiment that correspond to the two
outcomes. Both are real—they are known as two branches of the wave function.
Usually after the experiment, the different branches no longer interact much
with each other. Whether the parallel worlds come from a splitting of a single
world into two or are merely emergent descriptions is a matter of debate. As
with Wallace, (2012), I think it is much more plausible to interpret the
parallel worlds from the emergence perspective—they are not fundamental
properties of the Everettian universe.

_Unlike_ the GRW theory and _like_ Bohmian mechanics, the many-worlds
interpretation is deterministic: given a complete specification of the state
of the universe at one time, all the past and all the future is completely
fixed by the law of motion. In this case, the wave function of the universe at
one time, by the Schrödinger equation, completely fixes the wave function at
all other times. But the many-worlds interpretation differs from Bohmian
mechanics in that the latter has a single world while the former entails a
multiplicity of (emergent) worlds. The reason is that the wave function, which
is ontologically complete in the Everettian picture, introduces a multiplicity
that is resolved in the Bohmian picture by the additional postulate of point
particles with precise locations.

What is the fundamental material ontology on the many-worlds interpretation?
On the most flat-footed way of thinking about it, the fundamental material
ontology consists in the quantum state, represented by the wave function.
However, what the wave function represents in the physical world is a
controversial matter. Realism about the wave function is compatible with many
different views about the underlying nature of the physical reality
represented by the wave function. See (Chen, 2019b) for an overview. There are
two interpretations that will be relevant below. First, we can use the wave
function of the universe to define a matter distribution on physical space-
time. This is known as Sm and proposed first in (Allori et al. , 2010). The
basic idea is that we can use the wave function to define a function $m(x,t)$
whose domain is physical space-time and whose range is the set of real
numbers. This function can then represent a physical field called the matter-
density field in space-time. The higher the “sum” of values of the $m(x,t)$
function in some region $R$, the more stuff there is in $R$. The locations of
tables and chairs at different times can be read off from the history of this
function $m(x,t)$. Second, we can follow Wallace & Timpson, (2010) and
postulate local quantum states for sub-regions of the universe. Given a
partition of the universe into collections of sub-regions, we can use the wave
function of the universe to define sub-region states. (More technically, we
can define reduced density matrices for some local region $R$ by tracing out
the degrees of freedom of the environment $E$ in the universal wave function.)
Because of the holism inherent in quantum entanglement, merely defining the
reduced states for each sub-region is not sufficient to describe the history
of the universe. In general, even after we postulate the reduced state of
$R_{1}$ and the state of $R_{2}$, we will still need to postulate a state of
$R_{1}\bigcup R_{2}$, as its state is in general not a logical sum of the
states of its parts.

We will focus on the many-worlds interpretation of quantum mechanics here
because it has the potential to develop into a strongly deterministic theory
that realizes the possibility of a cosmic void.

###  3.2 The Everettian Wentaculus

The many-worlds interpretation is already deterministic. To make it strongly
deterministic, it suffices to pick a unique microscopic initial condition of
the universe. One idea is to pick a particular wave function and call it the
actual and nomologically necessary initial wave function of the universe. But
that would not work. Typical wave functions of the universe contain too much
information that they will in general be extremely complicated. It is unlikely
that a simple law can pick out any of those complicated wave functions.

In this section, we outline a new version of many-worlds interpretation whose
laws are strongly deterministic _and_ simple. The laws will be: the von Neuman
equation that generalizes the Schrödinger equation, the Initial Project
Hypothesis that replaces the Past Hypothesis, and possibly an additional
matter-density equation.

To appreciate the new version of many worlds interpretation of quantum
mechanics, we need to introduce some background about the arrows of time. We
see the manifestation of arrows of time everywhere. Many processes in nature
typically happen in only one direction: they are time-asymmetric. In room
temperature, the ice cubes in the past are always larger than the ice cubes in
the future; bananas are more ripe in the future than in the past; people are
more wrinkled in the future than in the past. However, the fundamental
dynamical laws of physics are (essentially) symmetric between the past and the
future. Take for example the conjunction of Newton’s equation of motion $F=ma$
and law for universal gravitation $F=Gm_{1}m_{2}/r^{2}$, whatever can happen
forward can as easily happen backward. The same is more or less true for the
Schrödinger equation and special and general relativity. So the explanation
for the arrows of time plausibly come from elsewhere.

One prominent explanation, first proposed by Boltzmann, (1964) (published
first in 1899), invokes what is now called the Past Hypothesis—the universe in
the beginning had very low entropy, where entropy is a measure of disorder.
Given a low-entropy initial macrostate, the universe will typically develop in
such a way to exemplify arrows of time in its typical subsystems. That is,
typically, typical subsystems of the universe are entropic. Modern
developments of Boltzmann’s ideas in the foundations of statistical mechanics
further substantiate this idea. The Past Hypothesis is still compatible with
infinitely many initial microstates. It is reasonable to assume that some of
them will lead to anti-entropic behaviors (e.g. ice cubes will become larger
in room temperature). To solve this problem, it is essential to have the
Statistical Postulate according to which these anti-entropic initial states
have relatively small measure—they are atypical. Albert, (2015) and Loewer,
(2020) call the conjunction of the dynamical law, the Past Hypothesis, and the
Statistical Postulate _the Mentaculus_.

Albert and Loewer’s Mentaculus theory was developed in the classical domain.
In the quantum case, we can postulate something similar to the Mentaculus. In
the quantum case, the Past Hypothesis is now a constraint on initial wave
functions of the universe—all of the nomologically possible ones started in a
low-entropy macrostate $\mathscr{H}_{PH}$, which is a extremely small subset
of all available wave functions. The wave functions inside the macrostate are
macroscopically similar in entropy, temperature, volume, and pressure. Inside
this macrostate, every wave function is equally likely as any other. (To be
more rigorous, we say that there is a uniform probability distribution on wave
functions compatible with $\mathscr{H}_{PH}$ with respect to the normalized
surface area measure on the unit sphere of the Hilbert space subspace picked
out by $\mathscr{H}_{PH}$.) Given the Schrödinger equation, the wave function
of the universe will rotate continuously inside the full energy shell of the
Hilbert space. That will describe (deterministically) the history of the
universe for all times.

In the quantum case, however, another framework is possible to combine the
low-entropy initial macrostate with the quantum dynamics. The alternative
framework, which I call _the Wentaculus_ , has surprising consequences,
including the realization of strong determinism. I introduce this framework in
(Chen, 2018) and develop it in other work such as (Chen, 2019a, 2020b, 2022).
Instead of postulating an initial macrostate compatible with many initial
microstates, we can use the initial macrostate $\mathscr{H}_{PH}$ to select a
unique and natural microstate—the normalized projection onto that subspace.
This is represented by a mixed state density matrix $W$. This statement is
captured in what I call the _Initial Projection Hypothesis_ (IPH), a candidate
new law of nature that replaces the Past Hypothesis in the Mentaculus. (IPH is
as simple as the Past Hypothesis.) Given that there is only one initial
microstate possible under IPH, we no longer need to impose the Statistical
Postulate to neglect abnormal initial states. Moreover, we can replace the
wave function dynamics by the corresponding density matrix dynamics, which is
still deterministic in the case of Everettian theory. In this version of
Everett, then, the dynamics is deterministic and there is only one microscopic
initial condition compatible with the laws. Thus, this version of Everett is
strongly deterministic: given the laws, there is only one microscopic history
of the universe that is possible. That makes the actual history of the
Everettian universe nomologically necessary.666More precisely, we can write
down the postulates of this version of Everett. The Initial Projection
Hypothesis is as follows:
$\hat{W}_{IPH}(t_{0})=\frac{I_{PH}}{dim\mathscr{H}_{PH}}$ (1) where $I_{PH}$
is the projection operator onto the subspace $\mathscr{H}_{PH}$ and “dim”
counts the dimension of that subspace. The Von Neumann Equation is as follows:
$i\hbar\frac{\partial\hat{W}}{\partial t}=[\hat{H},\hat{W}]$ (2) where the
commutator bracket denotes the linear evolution generalized from the
Schrödinger equation.

We arrived at a strongly deterministic version of _the Wentaculus_ (“W” for
the initial density matrix). Not all versions of the Wentaculus are strongly
deterministic. The Bohmian version is not strongly deterministic because there
are additional independent variables representing possible initial locations
of point particles. The GRW version is not strongly deterministic because the
dynamics is stochastic.

The Everettian Wentaculus theory we outlined above admits two interpretations:
one that is compatible with a fundamental material ontology and the other
compatible with the cosmic void scenario. For the first one, we can postulate
that at the fundamental layer of reality there is matter: the quantum state,
the reduced states, or the matter-density ontology represented by
$m(x,t)$.777For the matter-density ontology, we can postulate the following
law—the Matter Density Equation:  $m(x,t)=\text{tr}(M(x)W(t))$ (3) where $x$
is a physical space variable, $M(x)=\sum_{i}m_{i}\delta(Q_{i}-x)$ is the mass-
density operator, which is defined via the position operator
$Q_{i}\psi(q_{1},q_{2},...q_{n})=q_{i}\psi(q_{1},q_{2},...q_{n})$. The non-
fundamental facts and the manifest image are recovered by using facts about
fundamental material ontology as well as the laws.

However, it also supports a second interpretation that is compatible with the
cosmic void scenario. Since the laws are sufficient to determine the actual
microstate of the universe at all times, following the arguments in §2, we no
longer need to postulate facts about fundamental material ontology to explain
the non-fundamental facts. We can bypass the fundamental ontology and to
derive facts about the locations of tables and chairs directly from the
laws—in this case the von Neumann equation that determines how density matrix
evolves and the Initial Projection Hypothesis that fixes the initial density
matrix. However, the language we use here can be misleading. The equation and
the hypothesis seem to assume the existence of some material object
represented by the density matrix. But it does not have to exist at the
fundamental level. All we need is the information stored in the laws of
nature. (What are the laws about if they are not about some fundamental
objects changing in time? We return to this question in §4.)

The Everettian cosmic void scenario, if it can be made empirically adequate,
has the following features:

  1. 1.

The fundamental laws are strongly deterministic. There is no contingency at
the fundamental level.

  2. 2.

There is no material ontology at the fundamental level (no quantum state,
reduced states, or matter density field).

  3. 3.

At the non-fundamental level, there is an emergent multiplicity of worlds that
can be characterized by non-fundamental material ontologies such as the
quantum state, reduced states, or a matter density field.

  4. 4.

The non-fundamental facts can be derived from facts about the fundamental laws
of nature.

  5. 5.

At the non-fundamental level, contingency can reappear as descriptions of the
non-fundamental worlds.

The Everettian Wentaculus provides a concrete example of the possibility of
the cosmic void. In such a world, all there is at the fundamental level are
the strongly-deterministic laws. The non-fundamental facts are derived from
them.888The status of space-time in this theory is an interesting issue. A
defender of the Everettian cosmic void scenario may entertain the possibility
that space-time is not fundamental and is also emergent from the strongly
deterministic laws. She may use ideas from the research program initiated in
(Carroll & Singh, 2018). However, this move will be in tension with the
arguments discussed in (North, 2018).

##  4 Challenges

Can the Everettian cosmic void scenario be made empirically adequate? That
question depends on a further question: can the Everettian picture be made
empirically adequate? Mover, even if the standard Everettian picture can be
empirically adequate, the Everettian cosmic void may suffer from additional
metaphysical and semantic challenges. We discuss some of them in this section.

###  4.1 The Semantic Challenge

We start with a semantic challenge: What are the fundamental laws about if
there is no material ontology at the fundamental level? What do the
theoretical terms refer to?

On the standard picture, the fundamental laws are about the fundamental
material ontology. The terms in the fundamental laws refer to properties of
fundamental matter (e.g. the mass of point particles). Since there is no
material ontology in the cosmic void scenario, this way of thinking about
aboutness and reference need to be revised.

One possible defense of the cosmic void scenario is to invoke certain non-
fundamental objects to make sense of the laws. The Initial Projection
Hypothesis, for example, refers to an initial density matrix. But what is the
meaning of the initial density matrix if there is nothing material at the
fundamental level of which it is about? One possibility is to appeal to the
effects it has on the _non-fundamental_ worlds and objects. To understand the
meaning of the Initial Projection Hypothesis, it may suffice to understand it
in terms of the behaviors of the tables and chairs, which are derived from the
hypothesis. After all, we make sense of fundamental laws by their effects on
observable objects, such as the behaviors of pointers and measurement
instruments. How their behaviors are connected to more fundamental facts
requires theoretical postulates. Our epistemic access to the fundamental
material ontology is rather limited anyways.999Thinking about the metaphysics
of fundamentality, Bernstein, (2020) suggests that we should take seriously
the possibility that the middle level inhabited by medium-sized dry goods such
as tables and chairs is the most fundamental level of reality. Middle level
fundamentalism provides an alternative to the usual debate between
microphysical-fundamentalism and cosmos-fundamentalism. This is not the
metaphysical picture I favor (and neither does Bernstein endorse middle level
fundamentalism). However, if that view is possible, then presumably the
fundamental laws in such a world are primarily about the behaviors of tables
and chairs. Hence, it should also be possible that we can make sense of the
fundamental laws in the cosmic void scenario by looking at their effects on
the non-fundamental objects.

To be sure, some might object that fundamental laws cannot be about anything
other than fundamental things or properties.101010For example, see discussions
in (Lewis, 1983, Sider, 2011). If one holds this view and thinks this view is
justified, one would dismiss the possibility of the cosmic void scenario.
However, one might think it is better if the laws refer only to fundamental
objects and their properties but remain agnostic whether we are justified in
imposing such a condition on all physical theories. One can maintain that the
principle is a desirable theoretical virtue to be balanced with other
considerations. For example, Hicks & Schaffer, (2017) suggest that this
principle is not always true. However, I think that it is a significant cost
if we have to give up this principle to entertain the cosmic void scenario.

###  4.2 The Metaphysical Challenge

Next, we have a metaphysical challenge: How can non-fundamental facts about
tables and chairs depend on facts about non-matter (laws)?

On the standard picture, the non-fundamental facts are ultimately explained
(at least in part) by fundamental facts about fundamental matter. For example,
a typical reductionist would like to reduce facts about tables and chairs into
facts about the arrangements of particles and configurations of fields. In the
cosmic void scenario, the fundamental facts do not include such facts. So the
old way of thinking about ontological dependence needs to be revised if the
non-standard picture is to succeed.

A defender of the cosmic void scenario may point to an ambiguity about
“fundamental facts about fundamental matter.” At the fundamental level, there
may not be any facts about matter. However, at some non-fundamental level,
there can be facts about matter that is as fundamental as it gets in the
cosmic void picture. For example, in the Everettian cosmic void scenario, the
bottom level of reality consists in fundamental laws: the von Neumann
equation, the Initial Projection Hypothesis, and the matter-density equation.
From those equations, we can derive the values of the matter-density field,
which can exist non-fundamentally but at a level that is more fundamental than
any other material objects. Tables and chairs would emerge at a much higher
level from the matter-density field. Therefore, there can be a vestige
remaining about the intuition that non-fundamental facts of tables and chairs
should bottom out in terms of fundamental matter: it is just that the most
fundamental facts about matter will be explained by even more fundamental
facts about laws of nature.

Another metaphysical challenge to the cosmic void scenario is related to the
issue of completeness in the foundations of quantum mechanics. Is the wave
function all there is or is there something else in the material ontology? In
the framework of GRW theory, several answers are available (Allori et al. ,
2008). First, there is the bare GRW theory (GRW0) where the only thing in the
material ontology is the wave function. Second, there is the GRW theory with a
matter density ontology (GRWm) that is similar to the matter-density version
of many-worlds interpretation (Sm) we discussed in §3.1. The definition of the
matter density is the same in both theories. However, they differ in the
dynamics of the wave function. They also have different interpretations of the
nature of probability. Third, there is the GRW theory with a flash ontology
(GRWf), where the matter-density ontology is replaced by discrete events in
space-time.

Both GRWf and GRWm postulate something fundamental and material in physical
space-time. (The flash ontology and the matter-density ontology are called the
_primitive ontology_ in the literature.) The configuration of matter in space-
time are derived from the universal wave function. So in principle we just
need the information about the wave function to derive the information about
everything else. Maudlin, (2007) suggests that even though the GRW wave
function is _informationally complete_ in this sense, it should not be thought
of as _ontologically complete_ in the sense that all there is in the theory is
the wave function. After all, which material ontology is privileged in the GRW
theory? Is it the flash ontology or the matter-density ontology? They are two
different ways to define the material ontology in space-time. A defender of
GRW0 may respond that both ontologies can be taken as derivative physical
structure that we can use to make precise the connection between theory and
evidence. However, whether the response is acceptable may depend on one’s view
about primitive ontology.

We can ask a similar question about the cosmic void scenario. Even though we
can derive everything from the strongly deterministic laws, it only shows that
the laws are informationally complete, and it does not follow that they are
ontologically complete. It requires a further assumption and perhaps
additional justification to postulate that the laws are all there is.

At this point, a defender of the cosmic void scenario may appeal to Ockham’s
razor: it is more parsimonious to postulate just the strongly deterministic
laws than to postulate both the laws plus some material ontology. This is
different from the application of the razor in a merely deterministic
universe: since we can derive all facts about the past and the future from the
laws and the complete state of the universe at some time, perhaps we can get
rid of all the other times (Maudlin, 2007, p. 3153). We can care about
minimizing the kinds of things without caring so much about minimizing the
number of things in the same category. The application of Ockham’s razor in
the strongly deterministic universe gets rid of an entire category of things:
fundamental material ontology. The application of the razor in a merely
deterministic universe only gets rid of the vast number of times saving just
one.

There is a strong intuition that no theory can be entirely satisfactory unless
it postulates some kind of ordinary material ontology. However, if one is
antecedently sympathetic to a _wave function monist_ interpretation of GRW and
accepts GRW0 as a satisfactory physical theory, perhaps she should be more
open to reject the intuition. On GRW0, the quantum state (represented by a
wave function) is the only thing that exists. But it is nothing like the
ordinary material ontology of particles and fields. It is defined on a vastly
high dimensional space, and the recovery of ordinary facts about tables and
chairs is through some abstract mathematical derivations that look nothing
like the ordinary “causal” explanations in space-time. Someone who accepts
GRW0 will be happy to accept that ordinary facts about tables and chairs are
just emergent patterns in the high-dimensional wave function. But the cosmic
void scenario also locates the non-fundamental facts in some unfamiliar
fundamental facts—facts about fundamental laws of nature. If one is happy to
accept the emergence of non-fundamental facts from a radical material
ontology, then one should be more open to accept the emergence of non-
fundamental facts from strongly deterministic laws.

###  4.3 The Empirical Challenge

Finally, we have an empirical challenge: How can the Everettian Wentaculus
theory be empirically adequate when every possible outcome of experiment is
realized? How can the Born rule of probability make sense in such a world?

This is a familiar challenge to many-worlds interpretations of quantum
mechanics. It applies to both the Everettian Mentaculus and the Everettian
Wentaculus (as well as the cosmic void scenario). It is also related to the
emergence of contingency in a strongly deterministic universe. All versions of
the many-worlds interpretation face a general problem of probability: if all
outcomes of the experiments actually occur, how do we make sense of non-
trivial Born rule probabilities? For example, in the case of Schrödinger’s
cat, we can set up the experiment such that quantum mechanics says the
probability that the cat will be alive is 0.3 and the probability that the cat
will be dead is 0.7. But if both outcomes are realized the cat is alive in one
branch and the cat is dead in another, what to make of such probabilities?
Defenders of Everett, such as Deutsch, (1999) and Wallace, (2012), appeal to
Savage-style decision theory. There is also a line of defense developed by
Sebens & Carroll, (2016) that appeal to rational norms governing self-locating
probabilities.111111Everett himself appealed to arguments about long run
frequencies and typicality of branches. See (Barrett, 2017).

But these defenses of Everett make two kinds of assumptions. First, contrary
to the Wentaculus, the quantum state of the universe is assumed to be pure,
represented by a wave function. A technical problem for the Everettian
Wentaculus is how to adapt their arguments in the case when the universe is in
a fundamental mixed state, represented by a density matrix. The technical
challenge, in my opinion, is not difficult to overcome but there is work to be
done. Second, they also postulate some principles of rationality in order to
derive the Born rule. In the case of Wallace, (2012), the Savage-Wallace
axioms of preferences are far from being rationally obligatory. In the case of
Sebens & Carroll, (2016), the epistemic separability principle seems to be
open to counter-examples. So the hard problem is to make a strong case to
justify these assumptions in the proof of the Born rule. I am pessimistic
about the prospects of this project, but perhaps the difficulties can be
overcome. If the technical problem of extension of the argument to mixed
states can be solved, then the Everettian Wentaculus fares no worse than the
standard Everettian theories in the literature, at least with respect to the
problem of probability.

If this project of justifying Born-rule probability in the Everettian
framework can succeed, it can also help answer the question: how can
contingency emerge in a strongly deterministic Everettian universe like the
Everettian Wentaculus? The answer lies in the emergence of the branching
structure with a multiplicity of different macroscopic histories (worlds) and
the interpretation of branch weight as probability. We leave this project to
future work.

##  5 Conclusion

I have introduced a non-standard picture of the physical universe called the
cosmic void. I have discussed the general possibility as well as a concrete
example in the Everettian Wentaculus theory. The possibility of the cosmic
void is tightly connected to the possibility of strong determinism. In the
cosmic void scenario, at the most fundamental level of reality, there are only
fundamental laws of nature and no material ontology. Facts about matter can
emerge at a non-fundamental level: they can be derived from the laws if they
are strongly deterministic. The possibility is certainly unfamiliar and faces
difficult challenges, but it deserves careful and critical scrutiny. I hope
the picture outlined above can interest other people and lead to further
explorations of the topic.

There is an interesting connection between the cosmic void scenario and the
ontologist nihilist’s picture of a fundamental ontology without any objects.
According to ontological nihilism, there are no objects at the fundamental
level. This view is discussed in, for example, (Hawthorne & Cortens, 1995) and
(Turner, 2010). Dasgupta, (2009) develops a related view where facts involving
individual objects are paraphrased away. Maxwell, (2019) also develops a
closely related view according to which objects do not play an indispensable
role in metaphysics. In the nihilist / anti-object framework, ordinary
sentences involving objects are to be paraphrased in terms of non-objectual
language that does not ontologically commit us to the existence of objects.

The cosmic void scenario is compatible with that picture, but the two
approaches are quite different. The heavy lifting in the cosmic void scenario
is done not by revising logic or semantics, but by using a particular kind of
physical theory to derive non-fundamental facts about matter from fundamental
laws alone. Hence, the cosmic void possibility is at the same time more
radical and more modest. It is more radical in that we do not postulate even
fundamental properties, general facts, bare facts, or structural facts
(Ladyman et al. , 2007) in the fundamental ontology. It is more modest in that
the strategy is not supposed to work in general, but only in some special
physical theories where strong determinism holds. Hence, we have two potential
strategies for getting rid of fundamental material ontology: the metaphysical
strategy and the scientific strategy. Both are difficult positions to defend.
On the metaphysical side, we need to show that the relevant kind of semantic
and logical maneuver results in a satisfactory paraphrase of the original
object-language. On the scientific side, we need to show that (1) the actual
laws are strongly deterministic and (2) a strongly deterministic cosmic void
scenario can be empirically adequate.

Can they be made to work? I am not sure. Even if neither strategy works, we
can learn something from these examples: they may give us additional reasons
why material objects are crucial for formulating successful physical theories
and writing the book of the world.

## Acknowledgement

I would like to thank the editors of this volume—Sara Bernstein and Tyron
Goldschmidt—for useful written comments on an earlier draft of this paper. I
am also grateful for helpful discussions with David Chalmers, Sam Elgin, Bixin
Guo, Mario Hubert, Joseph Martinez, Mark Maxwell, Daniel Rubio, Ayoob
Shahmoradi, J. Robert G. Williams, and the audience at the 2020 Central APA
Symposium “Metaphysics Without Ontology.”

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We use the PowerSpectrum task from the python package Turbustat to derive one-dimensional power spectra from the THINGS $\hi$ moment-0 images of rescaled cubes.
We divide the power spectrum of each moment-0 image by the power spectrum of its clean beam.
We select the angular scale range between the largest sensitive scale of 8.82$'$ and 1.5 times the major-axis FWHM of the clean beam, and fit a linear relation between the power and the frequency in the logarithm space.
The slope of this linear relation is the power law index of the power spectrum.

We show the power spectrum and best-fit linear relations in the logarithm space in Figure <ref>.
We list the slopes in Table <ref>. The slopes have a median value of -2.3, and standard deviation of 0.2.
We use these measurements as reference to set the power spectrum of mock images created in Appendix <ref>.

The 1 dimensional power spectra of $\hi$ distribution in THINGS moment-0 images. The solid lines are the directly derived power spectra, and dashed lines of the same colors are the best linear fit in the logarithm space. The different colors represent different galaxies as denoted in the figure.

Power spectral slope

Galaxy      $k$

(1)         (2)

NGC 628     2.17

NGC 925     2.29

NGC 2841    2.30

NGC 2903    2.44

NGC 3198    2.33

NGC 3521    2.80

NGC 5055    2.24

NGC 5194    2.30

NGC 5457    2.59

NGC 7331    2.00

Column (1): Galaxy name.
Column (2): The slope of the 1 dimensional power spectrum of $\hi$ distribution.

§ RELATIONS BETWEEN THE MOMENT IMAGES OF THE DIFFUSE HI, THE DENSE HI AND THE TOTAL HI

We mathematically link the moment images of the total, dense and diffuse $\hi$.
The moment-0 (column density) images of the diffuse $\hi$ is related to those of the total and dense $\hi$ simply as:

\begin{equation}
N_{{\rm HI,diff}}=N_{{\rm HI,tot}}-N_{{\rm HI,dens}}.
\label{eq:mom0}
\end{equation}

The moment-1 (velocity field) images of the diffuse $\hi$ is related to those  of the total and dense $\hi$ as:

\begin{equation}
v_{\rm diff}=\frac{ N_{{\rm HI,tot}} v_{\rm tot}-N_{{\rm HI, dens}} v_{\rm dens}}{N_{{\rm HI, diff}}}.
\label{eq:mom1}
\end{equation}

If we ignore the beam smearing effects, the moment-2 image of the total $\hi$ has four parts in each pixel: the velocity dispersion of the diffuse (dense) $\hi$ around the central velocity of the diffuse (dense) $\hi$, and the offset of central velocity of the diffuse (dense) $\hi$ from the central velocity of the total $\hi$.
So, mathematically, the velocity dispersions of the total, dense and diffuse $\hi$ are related by the following equation.

\begin{align}
N_{{\rm HI, tot}}\sigma_{\rm tot}^2=  {} & N_{{\rm HI,diff}}( \sigma_{\rm diff}^2 + v_{\rm 1,diff}^2) \notag \\
& + N_{{\rm HI,dens}}( \sigma_{\rm dens}^2+ v_{\rm 1,dens}^2)   .
\label{eq:mom2}
\end{align}

where $v_{\rm 1,diff}=v_{\rm diff}-v_{\rm tot}$ and $v_{\rm 1,dens}=v_{\rm dens}-v_{\rm tot}$.
This equation reflects the conservation of kinetic energy in transforming of reference frames and holds regardless of the actual spectral shape, a detailed deduction of which can be found in the appendix <ref>.

These equations hint expectations for cross-checking patterns from images of different $\hi$ components, and propagation of errors dominated by the THINGS data and dependent on $f_{\rm diffuse}$ (or locally on N$_{\rm HI,diff}$/N$_{\rm HI,tot}$).
Due to the relatively high noise levels of the THINGS data, we should avoid extract information more complex than the moment images from the cubes of the diffuse $\hi$.
A future joint deconvolution to coherently take advantages of both datasets, and derive consistent models for both datasets should enable more sophisticated analysis of structural and kinematical components (Lin et al. in prep).

§.§ The Equation Linking the Velocity Dispersions of the Dense HI, Diffuse HI and Total HI

We assume the $\hi$ to be comprised of cloud elements, each having a volume density $n$ and velocity $v'$.
Obviously, for each line of sight, the column density $N$, average velocity $v$, and velocity dispersion $\sigma$ are related to the statistics of the clouds by

\begin{equation}
\begin{split}
N &=\sum n, \\
v &=\frac{\sum n v'}{\sum n}, \\
\sigma^2 & = \frac{\sum n (v'-v)^2}{\sum n}.
\end{split}
\label{eq:dmom2a}
\end{equation}

For each line of sight, we assume the total $\hi$ (denoted with $s$) to be comprised of the dense $\hi$ and the diffuse $\hi$.
Their properties are link as:

\begin{equation}
\begin{split}
N_{\rm tot} \sigma_{\rm tot}^2  = {}& \sum n_s(v_{\rm tot}'-v_{\rm tot})^2  \\
={}& \sum n_{\rm dens}(v_{\rm dens}'-v_{\rm tot})^2 \\
&+ \sum n_{\rm diff}(v_{\rm diff}'-v_{\rm tot})^2.
\end{split}
\label{eq:dmom2b}
\end{equation}

For each gas phase, the average velocity deviate from that of the total $\hi$ by $v_{1,x}=v_x-v_{\rm tot}$, where $x={\rm diff}$ or ${\rm dens}$. So that,

\begin{equation}
\begin{split}
\sum & n_x(v_x'-v_{\rm tot})^2   \\
& =  \sum n_x (v_x'-v_x+v_{1,x})^2   \\
&   =  \sum n_x [ (v_x'-v_x)^2+ v_{1,x}^2 +2(v_x'-v_x)v_{1,x}]  \\
&   =  \sum n_x (v_x'-v_x)^2+ N_x v_{1,x}^2  \\
&   = N_x  ( \sigma_x^2+ v_{1,x}^2).
\end{split}
\label{eq:dmom2c}
\end{equation}

We have used equations <ref> in the above deduction.
Equation <ref> thus becomes Equation <ref>.

§ MOMENT IMAGES OF THE ORIGINAL AND DEGRADED THINGS CUBES

We present moment images of $\hi$ in Figure <ref> to <ref>.
The first row in each figure shows the full-extent moment 0, 1 and 2 images of the FEASTS data.
The second to fourth row shows moment images for the original THINGS data, the degraded dense $\hi$, and the diffuse $\hi$ (see Section <ref>).
The fifth row shows the difference moment-1 image of the diffuse $\hi$, which is the moment-1 image of the resolution-degraded dense $\hi$ minus the moment-1 image of the diffuse $\hi$.

The moment images of the galaxy NGC 628. The sky region displayed is the full region of the FEASTS data.
Row 1: moment 0-2 images of FEASTS data, representing the total $\hi$ at the FAST resolution. The circle in the bottom left of the shows the size of the FAST beam.)
Row 2: moment 0-2 images of THINGS data, representing the dense $\hi$ at the VLA resolution. We do not show the VLA synthesis beam as it is too small compared to the field of view.
Row 3: moment 0-2 images of dense $\hi$ at the FAST resolution.
Row 4: moment 0-2 images of the total $\hi$ minus the dense $\hi$, representing the diffuse $\hi$ at the FAST resolution.
Row 5: the difference in $\hi$ moment-1 images between the dense $\hi$ and diffuse $hi$ at the FAST resolution (the former minus the latter).

Same as Figure <ref>, but for the galaxy NGC 925.

Same as Figure <ref>, but for the galaxy NGC 2841.

Same as Figure <ref>, but for the galaxy NGC 2903.

Same as Figure <ref>, but for the galaxy NGC 3198.

Same as Figure <ref>, but for the galaxy NGC 3521.

Same as Figure <ref>, but for the galaxy NGC 5055.

Same as Figure <ref>, but for the galaxy NGC 5194.

Same as Figure <ref>, but for the galaxy NGC 5457.

Same as Figure <ref>, but for the galaxy NGC 7331.

§ RADIAL PROFILES OF PROPERTIES OF THE MISSED HI

We present profiles of properties of the missed $\hi$, derived in similar ways as for the diffuse $\hi$.
The plots in Figure <ref> are also produced in similar ways as in Figure <ref> for the diffuse $\hi$.
In panel a, most of the missed $\hi$ has column densities below the detection limit of THINGS, with the only exception NGC 5457, which has the largest $\hi$ disk among the sample.
In panel c, the relative velocity differences of NGC 925 and NGC 5055 rise above unity near the center, seemingly implying counter rotation, but the beam smearing and projection effects need be better accounted for in the future before a solid conclusion is drawn.
Otherwise, all panels show that, the distribution and kinematical properties of the missed $\hi$ are qualitatively similar to those of the diffuse $\hi$.

Radial profiles of properties of the missed $\hi$. The figure is similar to Figure <ref>, except that properties of the missed $\hi$ instead of the diffuse $\hi$ are plotted.
In panel a, the dashed lines show the 3-$\sigma$ column density limit of the THINGS data.

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