# The Intrinsic Structure of Quantum Mechanics

Eddy Keming Chen Department of Philosophy, 106 Somerset Street, Rutgers
University, New Brunswick, NJ 08901, USA. Website: www.eddykemingchen.net.
Email<EMAIL_ADDRESS>
###### Abstract

The wave function in quantum mechanics presents an interesting challenge to
our understanding of the physical world. In this paper, I show that the wave
function can be understood as four intrinsic relations on physical space. My
account has three desirable features that the standard account lacks: (1) it
does not refer to any abstract mathematical objects, (2) it is free from the
usual arbitrary conventions, and (3) it explains why the wave function has its
gauge degrees of freedom, something that are usually put into the theory by
hand.

Hence, this account has implications for debates in philosophy of mathematics
and philosophy of science. First, by removing references to mathematical
objects, it provides a framework for nominalizing quantum mechanics. Second,
by excising superfluous structure such as overall phase, it reveals the
intrinsic structure postulated by quantum mechanics. Moreover, it also removes
a major obstacle to “wave function realism.”

Keywords: quantum mechanics, wave function, structural realism, phase
structure, mathematical nominalism vs. platonism, foundations of measurement,
intrinsic physical theory, Quine-Putnam indispensability argument, metaphysics
of science.

###### Contents

  1. 1 Introduction
  2. 2 The Two Visions and the Quantum Obstacle
    1. 2.1 The Intrinsicalist Vision
    2. 2.2 The Nominalist Vision
    3. 2.3 Obstacles From Quantum Theory
  3. 3 An Intrinsic and Nominalistic Account of the Quantum State
    1. 3.1 The Mathematics of the Quantum State
    2. 3.2 Quantum State Amplitude
    3. 3.3 Quantum State Phase
    4. 3.4 Comparisons with Balaguer’s Account
  4. 4 “Wave Function Realism”
  5. 5 Conclusion

##  1 Introduction

Quantum mechanics is empirically successful (at least in the non-relativistic
domain). But what it means remains highly controversial. Since its initial
formulation, there have been many debates (in physics and in philosophy) about
the ontology of a quantum-mechanical world. Chief among them is a serious
foundational question about how to understand the quantum-mechanical laws and
the origin of quantum randomness. That is the topic of the quantum measurement
problem. At the time of writing this paper, the following are serious
contenders for being the best solution: Bohmian mechanics (BM), spontaneous
localization theories (GRW0, GRWf, GRWm, CSL), and Everettian quantum
mechanics (EQM and Many-Worlds Interpretation (MWI)).111See [29] for an
updated introduction to the measurement problem and the main solutions.

There are other deep questions about quantum mechanics that have a
philosophical and metaphysical flavor. Opening a standard textbook on quantum
mechanics, we find an abundance of mathematical objects: Hilbert spaces,
operators, matrices, wave functions, and etc. But what do they represent in
the physical world? Are they ontologically serious to the same degree or are
some merely dispensable instruments that facilitate calculations? In recent
debates in metaphysics of quantum mechanics, there is considerable agreement
that the universal wave function, modulo some mathematical degrees of freedom,
represents something objective — the quantum state of the universe.222The
universal quantum state, represented by a universal wave function, can give
rise to wave functions of the subsystems. The clearest examples are the
conditional wave functions in Bohmian mechanics. However, our primary focus
here will be on the wave function of the universe. In contrast, matrices and
operators are merely convenient summaries that do not play the same
fundamental role as the wave function.

However, the meaning of the universal quantum state is far from clear. We know
its mathematical representation very well: the universal wave function, which
is crucially involved in the dynamics of BM, GRW, and EQM. In the position
representation, a scalar-valued wave function is a square-integrable function
from the configuration space $\mathbb{R}^{3N}$ to the complex plane
$\mathbb{C}$. But what does the wave function really mean? There are two ways
of pursuing this question:

  1. 1.

What kind of “thing” does the wave function represent? Does it represent a
physical field on the configuration space, something nomological, or a _sui
generis_ entity in its own ontological category?

  2. 2.

What is the physical basis for the mathematics used for the wave function?
Which mathematical degrees of freedom of the wave function are physically
genuine? What is the metaphysical explanation for the merely mathematical or
gauge degrees of freedom?

Much of the philosophical literature on the metaphysics of the wave function
has pursued the first line of questions.333See, for example, [1, 21, 34, 31,
27, 25, 18, 26, 7]. In this paper, I will primarily pursue the second one, but
I will also show that these two are intimately related.

In particular, I will introduce an intrinsic account of the quantum state. It
answers the second line of questions by picking out four concrete relations on
physical space-time. Thus, it makes explicit the physical basis for the
usefulness of the mathematics of the wave function, and it provides a
metaphysical explanation for why certain degrees of freedom in the wave
function (the scale of the amplitude and the overall phase) are merely gauge.
The intrinsic account also has the feature that the fundamental ontology does
not include abstract mathematical objects such as complex numbers, functions,
vectors, or sets.

The intrinsic account is therefore nominalistic in the sense of Hartry Field
(1980). In his influential monograph _Science Without Numbers: A Defense of
Nominalism_ , Field advances a new approach to philosophy of mathematics by
explicitly constructing nominalistic counterparts of the platonistic physical
theories. In particular, he nominalizes Newtonian gravitation theory.444It is
not quite complete as it leaves out integration.  In the same spirit, Frank
Arntzenius and Cian Dorr (2011) develop a nominalization of differential
manifolds, laying down the foundation of a nominalistic theory of classical
field theories and general relativity. Up until now, however, there has been
no successful nominalization of quantum theory. In fact, it has been an open
problem–both conceptually and mathematically–how it is to be done. The non-
existence of a nominalistic quantum mechanics has encouraged much skepticism
about Field’s program of nominalizing fundamental physics and much optimism
about the Quine-Putnam Indispensability Argument for Mathematical Objects.
Indeed, there is a long-standing conjecture, due to David Malament (1982),
that Field’s nominalism would not succeed in quantum mechanics. Therefore,
being nominalistic, my intrinsic theory of the quantum state would advance
Field’s nominalistic project and provide (the first step of) an answer to
Malament’s skepticism.

Another interesting consequence of the account is that it will make progress
on the first line of questions about the ontology of quantum mechanics. On an
increasingly influential interpretation of the wave function, it represents
something physically significant. One version of this view is the so-called
“wave function realism,” the view that the universal wave function represents
a physical field on a high-dimensional (fundamental) space. That is the
position developed and defended in Albert (1996), Loewer (1996), Ney (2012),
and North (2013). However, Tim Maudlin (2013) has argued that this view leads
to an unpleasant proliferation of possibilities: if the wave function
represents a physical field (like the classical electromagnetic field), then a
change of the wave function by an overall phase transformation will produce a
distinct physical possibility. But the two wave functions will be empirically
equivalent—no experiments can distinguish them, which is the reason why the
overall phase differences are usually regarded as merely _gauge_. Since the
intrinsic account of the wave function I offer here is gauge-free insofar as
overall phase is concerned, it removes a major obstacle to wave function
realism (vis-à-vis Maudlin’s objection).

In this paper, I will first explain (in §2) the two visions for a fundamental
physical theory of the world: the intrinsicalist vision and the nominalistic
vision. I will then discuss why quantum theory may seem to resist the
intrinsic and nominalistic reformulation. Next (in §3), I will write down an
intrinsic and nominalistic theory of the quantum state. Finally (in §4), I
will discuss how this account bears on the nature of phase and the debate
about _wave function realism_.

Along the way, I axiomatize the quantum phase structure as what I shall call a
_periodic difference structure_ and prove a representation theorem and a
uniqueness theorem. These formal results could prove fruitful for further
investigation into the metaphysics of quantum mechanics and theoretical
structure in physical theories.

##  2 The Two Visions and the Quantum Obstacle

There are, broadly speaking, two grand visions for what a fundamental physical
theory of the world should look like. (To be sure, there are many other
visions and aspirations.) The first is what I shall call the intrinsicalist
vision, the requirement that the fundamental theory be written in a form
without any reference to arbitrary conventions such as coordinate systems and
units of scale. The second is the nominalistic vision, the requirement that
the fundamental theory be written without any reference to mathematical
objects. The first one is familiar to mathematical physicists from the
development of synthetic geometry and differential geometry. The second one is
familiar to philosophers of mathematics and philosophers of physics working on
the ontological commitment of physical theories. First, I will describe the
two visions, explain their motivations, and provide some examples. Next, I
will explain why quantum mechanics seems to be an obstacle for both programs.

###  2.1 The Intrinsicalist Vision

Figure 1:  Euclid’s Windmill proof of the Pythagorean Theorem. No coordinate
systems or real numbers were used. Cartesian coordinates were invented much
later to facilitate derivations.

The intrinsicalist vision is best illustrated with some history of Euclidean
geometry. Euclid showed that complex geometrical facts can be demonstrated
using rigorous proof on the basis of simple axioms. However, Euclid’s axioms
do not mention real numbers or coordinate systems, for they were not yet
discovered. They are stated with only qualitative predicates such as the
equality of line segments and the congruence of angles. With these concepts,
Euclid was able to derive a large body of geometrical propositions.

Real numbers and coordinate systems were introduced to facilitate the
derivations. With the full power of real analysis, the metric function defined
on pairs of tuples of coordinate numbers can greatly speed up the
calculations, which usually take up many steps of logical derivation on
Euclid’s approach. But what are the significance of the real numbers and
coordinate systems? When representing a 3-dimensional Euclidean space, a
typical choice is to use $\mathbb{R}^{3}$. It is clear that such a
representation has much surplus (or excess) structure: the origin of the
coordinate system, the orientation of the axis, and the scale are all
arbitrarily chosen (sometimes conveniently chosen for ease of calculation).
There is “more information” or “more structure” in $\mathbb{R}^{3}$ than in
the 3-dimensional Euclidean space. In other words, the $\mathbb{R}^{3}$
representation has gauge degrees of freedom.

The real, intrinsic structure in the 3-dimensional Euclidean space–the
structure that is represented by $\mathbb{R}^{3}$ up to the Euclidean
transformations–can be understood as an axiomatic structure of congruence and
betweenness. In fact, Hilbert 1899 and Tarski 1959 give us ways to make this
statement more precise. After offering a rigorous axiomatization of Euclidean
geometry, they prove a representation theorem: any structure instantiates the
betweenness and congruence axioms of 3-dimensional Euclidean geometry if and
only if there is a 1-1 embedding function from the structure onto
$\mathbb{R}^{3}$ such that if we define a metric function in the usual
Pythagorean way then the metric function is homomorphic: it preserves the
exact structure of betweenness and congruence. Moreover, they prove a
uniqueness theorem: any other embedding function defined on the same domain
satisfies the same conditions of homomorphism if and only if it is a Euclidean
transformation of the original embedding function: a transformation on
$\mathbb{R}^{3}$ that can be obtained by some combination of shift of origin,
reflection, rotation, and positive scaling.

The formal results support the idea that we can think of the genuine,
intrinsic features of 3-dimensional Euclidean space as consisting directly of
betweenness and congruence relations on spatial points, and we can regard the
coordinate system ($\mathbb{R}^{3}$) and the metric function as extrinsic
representational features we bring to facilitate calculations. (Example:
Figure 1. Exercise: prove the Pythagorean Theorem with and without real-
numbered coordinate systems.) The merely representational artifacts are highly
useful but still dispensable.

There are several advantages of having an intrinsic formulation of geometry.
First, it eliminates the need for many arbitrary conventions: where to place
the origin, how to orient the axis, and what scale to use. Second, in the
absence of these arbitrary conventions, we have a theory whose elements
_could_ stand in one-to-one correspondence with elements of reality. In that
case, we can look directly into the real structure of the geometrical objects
without worrying that we are looking at some merely representational artifact
(or gauge degrees of freedom). By eliminating redundant structure in a theory,
an intrinsic formulation gives us a more perspicuous picture of the
geometrical reality.

The lessons we learn from the history of Euclidean geometry can be extended to
other parts of physics. For example, people have long noticed that there are
many gauge degrees of freedom in the representation of both scalar and vector
valued physical quantities: temperature, mass, potential, and field values.
There has been much debate in philosophy of physics about what structure is
physically genuine and and what is merely gauge. It would therefore be helpful
to go beyond the scope of physical geometry and extend the intrinsic approach
to physical theories in general.

Hartry Field (1980), building on previous work by [20], ingeniously extends
the intrinsic approach to Newtonian gravitation theory. The result is an
elimination of arbitrary choices of zero field value and units of mass. His
conjecture is that all physical theories can be “intrinsicalized” in one way
or another.

###  2.2 The Nominalist Vision

As mentioned earlier, Field (1980) provides an intrinsic version of Newtonian
gravitation theory. But the main motivation and the major achievement of his
project is a defense of nominalism, the thesis that there are no abstract
entities, and, in particular, no abstract mathematical entities such as
numbers, functions, and sets.

The background for Field’s nominalistic project is the classic debate between
the mathematical nominalist and the mathematical platonist, the latter of whom
is ontologically committed to the existence of abstract mathematical objects.
Field identifies a main problem of maintaining nominalism is the apparent
indispensability of mathematical objects in formulating our best physical
theories:

> Since I deny that numbers, functions, sets, etc. exist, I deny that it is
> legitimate to use terms that purport to refer to such entities, or variables
> that purport to range over such entities, in our ultimate account of what
> the world is really like.
>
> This appears to raise a problem: for our ultimate account of what the world
> is really like must surely include a physical theory; and in developing
> physical theories one needs to use mathematics; and mathematics is full of
> such references to and quantifications over numbers, functions, sets, and
> the like. It would appear then that nominalism is not a position that can
> reasonably be maintained.555Field (2016), Preliminary Remarks, p.1.

In other words, the main task of defending nominalism would be to respond to
the Quine-Putnam Indispensability Argument:666The argument was originally
proposed by W. V. Quine and later developed by [32]. This version is from
[12].

  1. P1

We ought to be ontologically committed to all (and only) those entities that
are indispensable to our best theories of the world. [Quine’s Criterion of
Ontological Commitment]

  2. P2

Mathematical entities are indispensable to our best theories of the world.
[The Indispensability Thesis]

  3. C

Therefore, we ought to be ontologically committed to mathematical entities.

In particular, Field’s task is to refute the second premise–the
Indispensability Thesis. Field proposes to replace all platonistic physical
theories with attractive nominalistic versions that do not quantify over
mathematical objects

Field’s nominalistic versions of physical theories would have significant
advantages over their platonistic counterparts. First, the nominalistic
versions illuminate what exactly in the physical world provide the
explanations for the usefulness of any particular mathematical representation.
After all, even a platonist might accept that numbers and coordinate systems
do not really exist in the physical world but merely represent some concrete
physical reality. Such an attitude is consistent with the platonist’s
endorsement of the Indispensability Thesis. Second, as Field has argued, the
nominalistic physics seems to provide better explanations than the platonistic
counterparts, for the latter would involve explanation of physical phenomena
by things (such as numbers) external to the physical processes themselves.

Field has partially succeeded by writing down an intrinsic theory of physical
geometry and Newtonian gravitation, as it contains no explicit first-order
quantification over mathematical objects, thus qualifying his theory as
nominalistic. But what about other theories? Despite the initial success of
his project, there has been significant skepticism about whether his project
can extend beyond Newtonian gravitation theory to more advanced theories such
as quantum mechanics.

###  2.3 Obstacles From Quantum Theory

We have looked at the motivations for the two visions for what the fundamental
theory of the world should look like: the intrinsicalist vision and the
nominalistic vision. They should not be thought of as competing against each
other. They often converge on a common project. Indeed, Field’s reformulation
of Newtonian Gravitation Theory is both intrinsic and nominalistic.777However,
the intrinsicalist and nominalistic visions can also come apart. For example,
we can, in the case of mass, adopt an intrinsic yet platonistic theory of mass
ratios. We can also adopt an extrinsic yet nominalistic theory of mass
relations by using some arbitrary object (say, my water bottle) as standing
for unit mass and assigning comparative relations between that arbitrary
object and every other object.

Both have had considerable success in certain segments of classical theories.
But with the rich mathematical structures and abstract formalisms in quantum
mechanics, both seem to run into obstacles. David Malament was one of the
earliest critics of the nominalistic vision. He voiced his skepticism in his
influential review of Field’s book. Malament states his general worry as
follows:

> Suppose Field wants to give some physical theory a nominalistic
> reformulation. Further suppose the theory determines a class of mathematical
> models, each of which consists of a set of “points” together with certain
> mathematical structures defined on them. Field’s nominalization strategy
> cannot be successful unless the objects represented by the points are
> appropriately physical (or non-abstract)…But in lots of cases the
> represented objects _are_ abstract. ([23], pp. 533, emphasis
> original.)888Malament also gives the example of classical Hamiltonian
> mechanics as another specific instance of the general worry. But this is not
> the place to get into classical mechanics. Suffice to say that there are
> several ways to nominalize classical mechanics. Field’s nominalistic
> Newtonian Gravitation Theory is one way. [3] provides another way.

Given his _general_ worry that, often in physical theories, it is abstracta
that are represented in the state spaces, Malament conjectures that, in the
specific case of quantum mechanics, Field’s strategy of nominalization would
not “have a chance”:

> Here [in the context of quantum mechanics] I do not really see how Field can
> get started at all. I suppose one can think of the theory as determining a
> set of models—each a Hilbert space. But what form would the recovery (i.e.,
> representation) theorem take? The only possibility that comes to mind is a
> theorem of the sort sought by Jauch, Piron, _et al_. They start with
> “propositions” (or “eventualities”) and lattice-theoretic relations as
> primitive, and then seek to prove that the lattice of propositions is
> necessarily isomorphic to the lattice of subspaces of some Hilbert space.
> But of course no theorem of this sort would be of any use to Field. What
> could be worse than _propositions_ (or _eventualities_)? ([23], pp. 533-34.)

As I understand it, Malament suggests that there are no good places to start
nominalizing non-relativistic quantum mechanics. This is because the obvious
starting point, according to Malament and other commentators, is the abstract
Hilbert space, $\mathscr{H}$, as it is a state space of the quantum state.

However, there may be other starting points to nominalize quantum mechanics.
For example, the configuration space, $\mathbb{R}^{3N}$, is a good candidate.
In realist quantum theories such as Bohmian mechanics, Everettian quantum
mechanics, and spontaneous localization theories, it is standard to postulate
a (normalized) universal wave function $\Psi(\textbf{x},t)$ defined on the
configuration space(-time) and a dynamical equation governing its temporal
evolution.999Bohmian mechanics postulates additional ontologies—particles with
precise locations in physical space—and an extra law of motion—the guidance
equation. GRW theories postulate an additional stochastic modification of the
Schrödinger equation and, for some versions, additional ontologies such as
flashes and mass densities in physical space. In the deterministic case, the
wave function evolves according to the Schrödinger equation,

$i\hbar\frac{\partial}{\partial
t}\Psi(\textbf{x},t)=[-\sum^{N}_{i=1}\frac{\hbar^{2}}{2m_{i}}\Delta_{i}+V(x)]\Psi(\textbf{x},t):=H\Psi(\textbf{x},t),$

which relates the temporal derivatives of the wave function to its spatial
derivatives. Now, the configuration-space viewpoint can be translated into the
Hilbert space formalism. If we regard the wave function (a square-integrable
function from the configuration space to complex numbers) as a unit vector
$\ket{\Psi(t)}$, then we can form another space—the Hilbert space of the
system.101010This is the Hilbert space $L^{2}(\mathbb{R}^{3N},\mathbb{C})$,
equipped with the inner product $<\psi,\phi>$ of taking the Lebesgue integral
of $\psi^{*}\phi$ over the configuration space, which guarantees Cauchy
Completeness.  Thus, the wave function can be mapped to a state vector, and
vice versa. The state vector then rotates (on the unit sphere in the Hilbert
space) according to a unitary (Hamiltonian) operator,

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

which is another way to express the Schrödinger evolution of the wave
function.

Hence, there is the possibility of carrying out the nominalization project
with the configuration space. In some respects, the configuration-space
viewpoint is more friendly to nominalism, as the configuration space is much
closely related to physical space than the abstract Hilbert space is.111111I
should emphasize that, because of its central role in functional analysis,
Hilbert space is highly important for fascilitating calculations and proving
theorems about quantum mechanics. Nevertheless, we should not regard it as
conclusive evidence for ontological priority. Indeed, as we shall see in §3,
the configuration-space viewpoint provides a natural platform for the
nominalization of the universal wave function. We should also keep in mind
that, at the end of the day, it suffices to show that quantum mechanics can be
successfully nominalized from _some_ viewpoint. Nevertheless, Malament’s
worries still remain, because (_prima facie_) the configuration space is also
quite abstract, and it is unclear how to fit it into the nominalistic
framework. Therefore, at least _prima facie_ , quantum mechanics seems to
frustrate the nominalistic vision.

Moreover, the mathematics of quantum mechanics comes with much conventional
structure that is hard to get rid of. For example, we know that the exact
value of the amplitude of the wave function is not important. For that matter,
we can scale it with any arbitrary positive constant. It is true that we
usually choose the scale such that we get unity when integrating the amplitude
over the entire configuration space. But that is merely conventional. We can,
for example, write down the Born rule with a proportionality constant to get
unity in the probability function:

$P(x\in X)=Z\int_{X}|\Psi(x)|^{2}dx,$

where $Z$ is a normalization constant.

Another example is the overall phase of the wave function. As we learn from
modular arithmetic, the exact value of the phase of the wave function is not
physically significant, as we can add a constant phase factor to every point
in configuration space and the wave function will remain physically the same:
producing exactly the same predictions in terms of probabilities.

All these gauge degrees of freedom are frustrating from the point of view of
the intrinsicalist vision. They are the manifestation of excess structures in
the quantum theory. What exactly is going on in the real world that allows for
these gauge degrees of freedom but not others? What is the most metaphysically
perspicuous picture of the quantum state, represented by the wave function?
Many people would respond that the quantum state is projective, meaning that
the state space for the quantum state is not the Hilbert space, but its
quotient space: the projective Hilbert space. It can be obtained by
quotienting the usual Hilbert space with the equivalence relation $\psi\sim
Re^{i\theta}\psi$. But this is not satisfying; the “quotienting” strategy
raises a similar question: what exactly is going on in the real world that
allows for quotienting with this equivalence relation but not
others?121212These questions, I believe, are in the same spirit as Ted Sider’s
2016 Locke Lecture (ms.), and especially his final lecture on theoretical
equivalence and against what he calls“quotienting by hand.” I should mention
that both Sider and I are really after gauge-free formulations of physical and
metaphysical theories, which are more stringent than merely gauge-independent
formulations. For example, modern differential geometry is gauge-independent
(coordinate-independent) but not gauge-free (coordinate-free): although
manifolds can be defined without privileging any particular coordinate system,
their definition still uses coordinate systems (maps and atlas). No one, as
far as I know, has offered an intrinsic picture of the quantum state, even in
the non-relativistic domain.

In short, at least _prima facie_ , both the intrinsicalist vision and the
nominalist vision are challenged by quantum mechanics.

##  3 An Intrinsic and Nominalistic Account of the Quantum State

In this section, I propose a new account of the quantum state based on some
lessons we learned from the debates about wave function realism.131313Here I’m
taking the “Hard Road” to nominalism. As such, my goal is to (1) reformulate
quantum mechanics (QM) such that _within the theory_ it no longer refers
(under first-order quantifiers) to mathematical objects such as numbers,
functions, or sets and (2) demonstrate that the platonistic version of QM is
conservative over the nominalistic reformulation. To arrive at my theory, and
to state and prove the representation theorems, I refer to some mathematical
objects. But these are parts of the meta-theory to explicate the relation
between my account and the platonistic counterpart and to argue (by
_reductio_) against the indispensability thesis. See Field (2016), Preliminary
Remarks and Ch. 1 for a clear discussion, and [11] for an up-to-date
assessment of the “Easy Road” option. Thanks to Andrea Oldofredi and Ted Sider
for suggesting that I make this clear. As we shall see, it does not take much
to overcome the “quantum obstacle.” For simplicity, I will focus on the case
of a quantum state for a constant number of identical particles without spin.

###  3.1 The Mathematics of the Quantum State

First, let me explain my strategy for nominalizing non-relativistic quantum
mechanics.

  1. 1.

I will start with a Newtonian space-time, whose nominalization is readily
available.141414It is an interesting question what role Galilean relativity
plays in non-relativistic quantum mechanics. I will explore this issue in
future work.

  2. 2.

I will use symmetries as a guide to fundamentality and identify the intrinsic
structure of the universal quantum state on the Newtonian space-time. This
will be the goal for the remaining part of the paper. (Here we focus only on
the quantum state, because it is novel and it seems to resist nominalization.
But the theory leaves room for additional ontologies of particles, fields,
mass densities supplied by specific interpretations of QM; these additional
ontologies are readily nominalizable.)

  3. 3.

In future work, I will develop nominalistic translations of the dynamical
equations and generalize this account to accommodate more complicated quantum
theories.

Before we get into the intrinsic structure of the universal quantum state, we
need to say a bit more about its mathematical structure. For the quantum state
of a spinless system at a time $t$ (see Figure 2), we can represent it with a
scalar-valued wave function:

$\Psi_{t}:\mathbb{R}^{3N}\rightarrow\mathbb{C},$

where $N$ is the number of particles in the system, $\mathbb{R}^{3N}$ is the
configuration space of $N$ particles, and $\mathbb{C}$ is the complex plane.
(For the quantum state of a system with spin, we can use a vector-valued wave
function whose range is the spinor space—$\mathbb{C}^{2^{N}}$.)

My strategy is to start with a Newtonian space-time (which is usually
represented by a Cartesian product of a 3-dimensional Euclidean space and a
1-dimensional time). If we want to nominalize the quantum state, what should
we do with the configuration space $\mathbb{R}^{3N}$? As is now familiar from
the debate about wave function realism, there are two ways of interpreting the
fundamental physical space for a quantum world:

Figure 2: A wave function for a two-particle system in 1-dim space. Source:
www.physics.auckland.ac.nz

  1. 1.

$\mathbb{R}^{3N}$ represents the fundamental physical space; the space
represented by $\mathbb{R}^{3}$ only _appears_ to be real; the quantum state
assigns a complex number to each point in $\mathbb{R}^{3N}$. (See Figure 2.
Analogy: classical field.)

  2. 2.

$\mathbb{R}^{3}$ represents the fundamental physical space; the space
represented by $\mathbb{R}^{3N}$ is a mathematical construction—the
configuration space; the quantum state assigns a complex number to each region
in $\mathbb{R}^{3}$ that contains $N$ points (i.e. the regions will be
irregular and disconnected). (Analogy: multi-field)

Some authors in the debate about wave function realism have argued that given
our current total evidence, option (2) is a better interpretation of non-
relativistic quantum mechanics.151515See, for example, [10] and [19]. I will
not rehearse their arguments here. But one of the key ideas that will help us
here is that we can think of the complex-valued function as really “living on”
the 3-dimensional physical space, in the sense that it assigns a complex
number not to each _point_ but each $N$-element _region_ in physical space. We
call that a “multi-field.”161616This name can be a little confusing. Wave-
function “multi-field” was first used in [6], which was an adaptation of the
name “polyfield” introduced by Forrest (1988). See [3] for a completely
different object called the “multi-field.”

Taking the wave function into a framework friendly for further nominalization,
we can perform the familiar technique of decomposing the complex number
$Re^{i\theta}$ into two real numbers: the amplitude $R$ and the phase
$\theta$. That is, we can think of the compex-valued multi-field in the
physical space as two real-valued multi-fields:

$R(x_{1},x_{2},x_{3},...,x_{N}),\theta(x_{1},x_{2},x_{3},...,x_{N}).$

Here, since we are discussing Newtonian space-time, the $x_{1}.....x_{N}$ are
simultaneous space-time points. We can think of them as:
$(x_{\alpha_{1}},x_{\beta_{1}},x_{\gamma_{1}},x_{t}),$
$(x_{\alpha_{2}},x_{\beta_{2}},x_{\gamma_{2}},x_{t}),......,$
$(x_{\alpha_{N}},x_{\beta_{N}},x_{\gamma_{N}},x_{t})$.

Now the task before us is just to come up with a nominalistic and intrinsic
description of the two multi-fields. In §3.2 and §3.3, we will find two
physical structures (Quantum State Amplitude and Quantum State Phase), which,
via the appropriate representation theorems and uniqueness theorems, justify
the use of complex numbers and explain the gauge degrees of freedom in the
quantum wave function.171717In the case of a vector-valued wave function,
since the wave function value consists in $2^{N}$ complex numbers, where $N$
is the number of particles, we would need to nominalize $2^{N+1}$ real-valued
functions:
$R_{1}(x_{1},x_{2},x_{3},...,x_{N}),\theta_{1}(x_{1},x_{2},x_{3},...,x_{N}),R_{2}(x_{1},x_{2},x_{3},...,x_{N}),\theta_{2}(x_{1},x_{2},x_{3},...,x_{N}),......$

###  3.2 Quantum State Amplitude

The amplitude part of the quantum state is (like mass density) on the ratio
scale, i.e. the physical structure should be invariant under ratio
transformations

$R\rightarrow\alpha R.$

We will start with the Newtonian space-time and help ourselves to the
structure of N-Regions: collection of all regions that contain exactly $N$
simultaneous space-time points (which are irregular and disconnected regions).
We start here because we would like to have a physical realization of the
platonistic configuration space. The solution is to identify configuration
points with certain special regions of the physical space-time.181818Notes on
mereology: As I am taking for granted that quantum mechanics for
indistinguishable particles (sometimes called identical particles) works just
as well as quantum mechanics for distinguishable particles, I do not require
anything more than Atomistic General Extensional Mereology (AGEM). That is,
the mereological system that validate the following principles: Partial
Ordering of Parthood, Strong Supplementation, Unrestricted Fusion, and
Atomicity. See [33] for a detailed discussion. However, I leave open the
possibility for adding structures in N-Regions to distinguish among different
ways of forming regions from the same collection of points, corresponding to
permuted configurations of distinguishable particles. We might need to
introduce additional structure for mereological composition to distinguish
between mereological sums formed from the same atoms but in different orders.
This might also be required when we have entangled quantum states of different
species of particles. To achieve this, we can borrow some ideas from Kit
Fine’s “rigid embodiment” and add primitive ordering relations to enrich the
structure of mereological sums.

In addition to N-Regions, the quantum state amplitude structure will contain
two primitive relations:

  * •

A two-place relation Amplitude–Geq ($\succeq_{A}$).

  * •

A three-place relation Amplitude–Sum ($S$).

Interpretation: $a\succeq_{A}b$ iff the amplitude of N-Region $a$ is greater
than or equal to that of N-Region $b$; $S(a,b,c)$ iff the amplitude of
N-Region $c$ is the sum of those of N-Regions $a$ and $b$.

Define the following short-hand (all quantifiers below range over only
N-Regions):

  1. 1.

$a=_{A}b:=a\succeq_{A}b\text{ and }b\succeq_{A}a.$

  2. 2.

$a\succ_{A}b:=a\succeq_{A}b\text{ and not }b\succeq_{A}a.$

Next, we can write down some axioms for Amplitude–Geq and
Amplitude–Sum.191919Compare with the axioms in Krantz et al. (1971) Defn.3.3:
Let $A$ be a nonempty set, $\succeq$ a binary relation on $A$, $B$ a nonempty
subset of $A\times A$, and $\circ$ a binary function from $B$ into $A$. The
quadruple $<A,\succeq,B,\circ>$ is an extensive structure with no essential
maximum if the following axioms are satisfied for all $a,b,c\in A$:  1.
$<A,\succeq>$ is a weak order. [This is translated as G1 and G2.] 2. If
$(a,b)\in B$ and $(a\circ b,c)\in B$, then $(b,c)\in B,(a,b\circ c)\in B$, and
$(a\circ b)\circ c\succeq_{A}a\circ(b\circ c).$ [This is translated as S1.] 3.
If $(a,c)\in B$ and $a\succeq b$, then $(c,b)\in B$, and $a\circ c\succeq
c\circ b.$ [This is translated as S2.] 4. If $a\succ b$, then $\exists d\in A$
s.t. $(b,d)\in B$ and $a\succeq b\circ d.$ [This is translated as S3.] 5. If
$a\circ b=c$, then $c\succ a$. [This is translated as S4, but allowing
N-Regions to have null amplitudes. The representation function will also be
zero-valued at those regions.] 6. Every strictly bounded standard sequence is
finite, where $a_{1},...,a_{n},...$ is a standard sequence if for
$n=2,..,a_{n}=a_{n-1}\circ a_{1}$, and it is strictly bounded if for some
$b\in A$ and for all $a_{n}$ in the sequence, $b\succ a_{n}$. [This is
translated as S5. The translation uses the fact that Axiom 6 is equivalent to
another formulation of the Archimedean axiom: $\\{n|na\text{ is defined and
}b\succ na\\}$ is finite.] The complications in the nominalistic axioms come
from the fact that there can be more than one N-Regions that are the
Amplitude-Sum of two N-Regions: $\exists a,b,c,d$ s.t. $S(a,b,c)\wedge
S(a,b,d)\wedge c\neq d$. However, in the proof for the representation and
uniqueness theorems, we can easily overcome these complications by taking
equivalence classes of equal amplitude and recover the amplitude addition
function from the Amplitude-Sum relation.  Again, all quantifiers below range
over only N-Regions. $\forall a,b,c:$

  1. G1

(Connectedness) Either $a\succeq_{A}b$ or $b\succeq_{A}a$.

  2. G2

(Transitivity) If $a\succeq_{A}b$ and $b\succeq_{A}c$, then $a\succeq_{A}c$.

  3. S1

(Associativity*) If $\exists x$ $S(a,b,x)$ and $\forall x^{\prime}$ [if
$S(a,b,x^{\prime}))$ then $\exists y$ $S(x^{\prime},c,y)$], then $\exists z$
$S(b,c,z)$ and $\forall z^{\prime}$ [if $S(b,c,z^{\prime}))$ then $\exists w$
$S(a,z^{\prime},w)$] and $\forall f,f^{\prime},g,g^{\prime}$ [if
$S(a,b,f)\wedge S(f,c,f^{\prime})\wedge S(b,c,g)\wedge S(a,g,g^{\prime})$,
then $f^{\prime}\succeq_{A}g^{\prime}$].

  4. S2

(Monotonicity*) If $\exists x$ $S(a,c,x)$ and $a\succeq_{A}b$, then $\exists
y$ $S(c,b,y)$ and $\forall f,f^{\prime}$ [if $S(a,c,f)\wedge
S(c,b,f^{\prime})$ then $f\succeq_{A}f^{\prime}$].

  5. S3

(Density) If $a\succ_{A}b$, then $\exists d,x$ [$S(b,d,x)$ and $\forall f,$ if
$S(b,x,f),$ then $a\succeq_{A}f]$.

  6. S4

(Non-Negativity) If $S(a,b,c)$, then $c\succeq_{A}a$.

  7. S5

(Archimedean Property) $\forall a_{1},b$, if $\neg S(a_{1},a_{1},a_{1})$ and
$\neg S(b,b,b)$, then $\exists a_{1},a_{2},...,a_{n}$ s.t. $b\succ_{A}a_{n}$
and $\forall a_{i}$ [if $b\succ_{A}a_{i}$, then $a_{n}\succeq_{A}a_{i}$],
where $a_{i}$’s, if they exist, have the following properties:
$S(a_{1},a_{1},a_{2})$, $S(a_{1},a_{2},a_{3})$, $S(a_{1},a_{3},a_{4})$, …,
$S(a_{1},a_{n-1},a_{n})$.202020S5 is an infinitary sentence, as the
quantifiers in the consequent should be understood as infinite disjunctions of
quantified sentences. However, S5 can also be formulated with a stronger axiom
called Dedekind Completeness, whose platonistic version says:  Dedekind
Completeness. $\forall M,N\subset A$, if $\forall x\in M,\forall y\in N,y\succ
x$, then $\exists z\in A$ s.t. $\forall x\in M,z\succ x\text{ and }\forall
y\in N,y\succ z.$ The nominalistic translation can be done in two ways. We can
introduce two levels of mereology so as to distinguish between regions of
points and regions of regions of points. Alternatively, as Tom Donaldson,
Jennifer Wang, and Gabriel Uzquiano suggest to me, perhaps one can make do
with plural quantification in the following way. For example ( with $\propto$
for the logical predicate “is one of” ), here is one way to state the Dedekind
Completeness with plural quantification:  Dedekind Completeness Nom Pl.
$\forall mm,nn\in$ N-Regions, if $\forall x\propto mm,\forall y\propto
nn,y\succ x$, then there exists $z\in A$ s.t. $\forall x\propto mm,z\succ
x\text{ and }\forall y\propto nn,y\succ z.$ We only need the Archimedean
property in the proof. Since Dedekind Completeness is stronger, the proof in
Krantz et al. (1971), pp. 84-87 can still go through if we assume Dedekind
Completeness Nom Pl. Such strenghthening of S5 has the virtue of avoiding the
infinitary sentences in S5. Note: this is the point where we have to trade off
certain nice features of first-order logic and standard mereology with the
desiderata of the intrinsic and nominalistic account. (I have no problem with
infinitary sentences in S5. But one is free to choose instead to use plural
quantification to formulate the last axiom as Dedekind Completeness Nom Pl.)
This is related to Field’s worry in _Science Without Numbers_ , Ch. 9, “Logic
and Ontology.”

Since these axioms are the nominalistic translations of a platonistic
structure in Krantz et al. (Defn. 3.3), we can formulate the representation
and uniqueness theorems for the amplitude structure as follows:

######  Theorem 3.1 (Amplitude Representation Theorem)

<N-Regions, Amplitude–Geq,  Amplitude–Sum> satisfies axioms (G1)—(G2) and
(S1)—(S5), only if there is a function $R:\text{
N-Regions}\rightarrow\\{0\\}\cup\mathbb{R}^{+}$ such that $\forall
a,b\in\text{N-Regions}$:

  1. 1.

$a\succeq_{A}b\Leftrightarrow R(a)\geq R(b)$;

  2. 2.

If $\exists x$ s.t. $S(a,b,x)$, then $\forall c$ [if $S(a,b,c)$ then
$R(c)=R(a)+R(b)$].

######  Theorem 3.2 (Amplitude Uniqueness Theorem)

If another function $R^{\prime}$ satisfies the conditions on the RHS of the
Amplitude Representation Theorem, then there exists a real number $\alpha>0$
such that for all nonmaximal element $a\in$ N-Regions, $R^{\prime}(a)=\alpha
R(a).$

Proofs: See [20], Sections 3.4.3, 3.5, pp. 84-87. Note: Krantz et al. use an
addition function $\circ$, while we use a sum relation $S(x,y,z)$, because we
allow there to be distinct N-Regions that have the same amplitude.
Nevertheless, we can use a standard technique to adapt their proof: we can
simply take the equivalence classes N-Regions / $=_{A}$, where $a=_{A}b$ if
$a\succeq_{A}b\wedge b\succeq_{A}a$, on which we can define an addition
function with the Amplitude-Sum relation.

The representation theorem suggests that the intrinsic structure of Amplitude-
Geq and Amplitude-Sum guarantees the existence of a faithful representation
function. But the intrinsic structure makes no essential quantification over
numbers, functions, sets, or matrices. The uniqueness theorem explains why the
gauge degrees of freedom are the positive multiplication transformations and
no further, i.e. why the amplitude function is unique up to a positive
normalization constant.

###  3.3 Quantum State Phase

The phase part of the quantum state is (like angles on a plane) of the
periodic scale, i.e. the intrinsic physical structure should be invariant
under overall phase transformations

$\theta\rightarrow\theta+\phi\text{ mod }2\pi.$

We would like something of the form of a “difference structure.” But we know
that according to standard formalism, just the absolute values of the
differences would not be enough, for time reversal on the quantum state is
implemented by taking the complex conjugation of the wave function, which is
an operation that leaves the absolute values of the differences unchanged. So
we will try to construct a signed difference structure such that standard
operations on the wave function are faithfully preserved.212121Thanks to
Sheldon Goldstein for helpful discussions about this point. David Wallace
points out (p.c.) that it might be a virtue of the nominalistic theory to
display the following choice-point: one can imagine an axiomatization of
quantum state phase that involves only absolute phase differences. This would
require thinking more deeply about the relationship between quantum phases and
temporal structure, as well as a new mathematical axiomatization of the
absolute difference structure for phase.

We will once again start with N-Regions, the collection of all regions that
contain exactly $N$ simultaneous space-time points.

The intrinsic structure of phase consists in two primitive relations:

  * •

A three-place relation Phase–Clockwise–Betweenness ($C_{P}$),

  * •

A four-place relation Phase–Congruence ($\sim_{P}$).

Interpretation: $C_{P}(a,b,c)$ iff the phase of N-Region $b$ is clock-wise
between those of N-Regions $a$ and $c$ (this relation realizes the intuitive
idea that 3 o’clock is clock-wise between 1 o’clock and 6 o’clock, but 3
o’clock is not clock-wise between 6 o’clock and 1 o’clock); $ab\sim_{P}cd$ iff
the signed phase difference between N-Regions $a$ and $b$ is the same as that
between N-Regions $c$ and $d$.

The intrinsic structures of Phase–Clockwise–Betweenness and Phase–Congruence
satisfy the following axioms for what I shall call a “periodic difference
structure”:

All quantifiers below range over only N-Regions. $\forall a,b,c,d,e,f$:

  1. C1

At least one of $C_{P}(a,b,c)$ and $C_{P}(a,c,b)$ holds; if $a,b,c$ are pair-
wise distinct, then exactly one of $C_{P}(a,b,c)$ and $C_{P}(a,c,b)$ holds.

  2. C2

If $C_{P}(a,b,c)$ and $C_{P}(a,c,d)$, then $C_{P}(a,b,d)$; if $C_{P}(a,b,c)$,
then $C_{P}(b,c,a)$.

  3. K1

$ab\sim_{P}ab$.

  4. K2

$ab\sim_{P}cd\Leftrightarrow cd\sim_{P}ab\Leftrightarrow
ba\sim_{P}dc\Leftrightarrow ac\sim_{P}bd$.

  5. K3

If $ab\sim_{P}cd$ and $cd\sim_{P}ef$, then $ab\sim_{P}ef$.

  6. K4

$\exists h,cb\sim_{P}ah$; if $C_{P}(a,b,c)$, then $\exists
d^{\prime},d^{\prime\prime}$ s.t. $ba\sim_{P}d^{\prime}c,$
$ca\sim_{P}d^{\prime\prime}b$; $\exists p,q,C_{P}(a,q,b),C_{P}(a,b,p),$
$ap\sim_{P}pb,bq\sim_{P}qa.$

  7. K5

$ab\sim_{P}cd\Leftrightarrow[\forall e,fd\sim_{P}ae\Leftrightarrow
fc\sim_{P}be].$

  8. K6

$\forall e,f,g,h,$ if $fc\sim_{P}be$ and $gb\sim_{P}ae$, then
$[hf\sim_{P}ae\Leftrightarrow hc\sim_{P}ge].$

  9. K7

If $C_{P}(a,b,c)$, then $\forall e,d,a^{\prime},b^{\prime},c^{\prime}$ [if
$a^{\prime}d\sim_{P}ae,b^{\prime}d\sim_{P}be,c^{\prime}d\sim_{P}ce,$ then
$C(a^{\prime},b^{\prime},c^{\prime})$].

  10. K8

(Archimedean Property) $\forall a,a_{1},b_{1}$, if $C_{P}(a,a_{1},b_{1})$,
then $\exists a_{1},a_{2},...,a_{n},b_{1},b_{2},...,b_{n},c_{1},...c_{m}$ such
that $C_{P}(a,a_{1},a_{n})$ and $C_{P}(a,b_{n},b_{1})$, where
$a_{n}a_{n-1}\sim_{P}a_{n-1}a_{n-2}\sim_{P}...\sim_{P}a_{1}a_{2}$ and
$b_{n}b_{n-1}\sim_{P}b_{n-1}b_{n-2}\sim_{P}...\sim_{P}b_{1}b_{2}$, and that
$a_{1}b_{1}\sim_{P}b_{1}c_{1}\sim_{P}...\sim_{P}c_{n}a_{1}$.222222Here it
might again be desirable to avoid the infinitary sentences / axiom schema by
using plural quantification. See the footnote on Axiom S5.

Axiom (K4) contains several existence assumptions. But such assumptions are
justified for a nominalistic quantum theory. We can see this from the
structure of the platonistic quantum theory. Thanks to the Schrödinger
dynamics, the wave function will spread out continuously over space and time,
which will ensure the richness in the phase structure.

With some work, we can prove the following representation and uniqueness
theorems:

######  Theorem 3.3 (Phase Representation Theorem)

If $<$ N-Regions, Phase–Clockwise–Betweenness, Phase–Congruence$>$ is a
periodic difference structure, i.e. satisfies axioms (C1)—(C2) and (K1)—(K8),
then for any real number $k>0$, there is a function $f:\text{ N-Regions
}\rightarrow[0,k)$ such that $\forall a,b,c,d\in$ N-Regions:

  1. 1.

$C_{P}(c,b,a)\Leftrightarrow f(a)\geq f(b)\geq f(c)\text{ or }f(c)\geq
f(a)\geq f(b)\text{ or }f(b)\geq f(c)\geq f(a)$;

  2. 2.

$ab\sim_{P}cd\Leftrightarrow f(a)-f(b)=f(c)-f(d)$ (mod $k$).

######  Theorem 3.4 (Phase Uniqueness Theorem)

If another function $f^{\prime}$ satisfies the conditions on the RHS of the
Phase Representation Theorem, then there exists a real number $\beta$ such
that for all element $a\in$ N-Regions , $f^{\prime}(a)=f(a)+\beta$ (mod $k$).

Proofs: see Appendix.

Again, the representation theorem suggests that the intrinsic structure of
Phase–Clockwise–Betweenness and Phase–Congruence guarantees the existence of a
faithful representation function of phase. But the intrinsic structure makes
no essential quantification over numbers, functions, sets, or matrices. The
uniqueness theorem explains why the gauge degrees of freedom are the overall
phase transformations and no further, i.e. why the phase function is unique up
to an additive constant.

Therefore, we have written down an intrinsic and nominalistic theory of the
quantum state, consisting in merely four relations on the regions of physical
space-time: Amplitude-Sum, Amplitude-Geq, Phase-Congruence, and Phase-
Clockwise-Betweenness. As mentioned earlier but evident now, the present
account of the quantum state has several desirable features: (1) it does not
refer to any abstract mathematical objects such as complex numbers, (2) it is
free from the usual arbitrary conventions in the wave function representation,
and (3) it explains why the quantum state has its amplitude and phase degrees
of freedom.

###  3.4 Comparisons with Balaguer’s Account

Let us briefly compare my account with Mark Balaguer’s account (1996) of the
nominalization of quantum mechanics.

Balaguer’s account follows Malament’s suggestion of nominalizing quantum
mechanics by taking seriously the Hilbert space structure and the
representation of “quantum events” with closed subspaces of Hilbert spaces.
Following orthodox textbook presentation of quantum mechanics, he suggests
that we take as primitives the _propensities_ of quantum systems as analogous
to probabilities of quantum experimental outcomes.

> I begin by recalling that each quantum state can be thought of as a function
> from events $(A,\Delta)$ to probabilities, i.e., to $[0,1]$. Thus, each
> quantum state specifies a set of ordered pairs $<(A,\Delta),r>$. The next
> thing to notice is that each such ordered pair determines a propensity
> property of quantum systems, namely, an $r-$strengthed propensity to yield a
> value in $\Delta$ for a measurement of $A$. We can denote this propensity
> with “$(A,\Delta,r)$”. (Balaguer, 1996, p.218.)

Balaguer suggests that the propensities are “nominalistically kosher.” By
interpreting the Hilbert space structures as propensities instead of
propositions, Balaguer makes some progress in the direction of making quantum
mechanics “more nominalistic.”

However, Balaguer’s account faces a problem—it is not clear how Balaguer’s
account relates to any mainstream realist interpretation of quantum mechanics.
This is because the realist interpretations—Bohmian Mechanics, GRW spontaneous
collapse theories, and Everettian Quantum Mechanics—crucially involve the
quantum state represented by a wave function, not a function from events to
probabilities.232323See [8] for a discussion about the conflicts between
Balaguer’s account and the modal interpretation of QM. And once we add the
wave function (perhaps in the nominalistic form introduced in this paper), the
probabilities can be calculated (via the Born rule) from the wave function
itself, which makes primitive propensities redundant. If Balaguer’s account is
based on orthodox quantum mechanics, then it would suffer from the dependence
on vague notions such as “measurement,” “observation,” and “observables,”
which should have no place in the fundamental ontology or dynamics of a
physical theory.242424Bell (1989), “Against ‘Measurement,’ ” pp. 215-16.

##  4 “Wave Function Realism”

The intrinsic and nominalistic account of the quantum state provides a natural
response to some of the standard objections to “wave function
realism.”252525“Wave function realists,” such as David Albert, Barry Loewer,
Alyssa Ney, and Jill North, maintain that the fundamental physical space for a
quantum world is 3N-dimensional. In contrast, primitive ontologists, such as
Valia Allori, Detlef Dürr, Sheldon Goldstein, Tim Maudlin, Roderich Tumulka,
and Nino Zanghi, argue that the fundamental physical space is 3-dimensional.
According to David Albert (1996), realism about the wave function naturally
commits one to accept that the wave function is a physical field defined on a
fundamentally 3N-dimensional wave function space. Tim [25] criticizes Albert’s
view partly on the ground that such “naive” realism would commit one to take
as fundamental the gauge degrees of freedom such as the absolute values of the
amplitude and the phase, leaving empirically equivalent formulations as
metaphysically distinct. This “naive” realism is inconsistent with the
physicists’ attitude of understanding the Hilbert space projectively and
thinking of the quantum state as an equivalence class of wave functions
($\psi\sim Re^{i\theta}\psi$). If a defender of wave function realism were to
take the physicists’ attitude, says the opponent, it would be much less
natural to think of the wave function as really a physical _field_ , as
something that assigns physical properties to each point in the 3N-dimensional
space. Defenders of wave function realism have largely responded by biting the
bullet and accepting the costs.

But the situation changes given the present account of the quantum state.
Given the intrinsic theory of the quantum state, one can be realist about the
quantum state by being realist about the four intrinsic relations underlying
the mathematical and gauge-dependent description of the wave function. The
intrinsic relations are invariant under the gauge transformations. Regardless
of whether one believes in a fundamentally high-dimensional space or a
fundamentally low-dimensional space, the intrinsic account will recover the
wave function unique up to the gauge transformations ($\psi\sim
Re^{i\theta}\psi$).

I should emphasize that my intrinsic account of the wave function is
essentially a version of comparativism about quantities. As such, it should be
distinguished from eliminitivism about quantities. Just as a comparativist
about mass does not eliminate mass facts but ground them in comparative mass
relations, my approach does not eliminate wave function facts but ground them
in comparative amplitude and phase relations. My account does not in the least
suggest any anti-realism about the wave function.262626 Thanks to David Glick
for suggesting that I make this clear.

Therefore, my account removes a major obstacle for wave function realism. One
can use the intrinsic account of the quantum state to identify two field-like
entities on the configuration space (by thinking of the N-Regions as points in
the 3N-dimensional space) without committing to the excess structure of
absolute amplitude and overall phase.272727 Unsurprisingly, the present
account also provides some new arsenal for the defenders of the fundamental
3-dimensional space. The intrinsic account of the quantum state fleshes out
some details in the multi-field proposal.

##  5 Conclusion

There are many _prima facie_ reasons for doubting that we can ever find an
intrinsic and nominalistic theory of quantum mechanics. However, in this
paper, we have offered an intrinsic and nominalistic account of the quantum
state, consisting in four relations on regions of physical space:

  1. 1.

Amplitude-Sum ($S$),

  2. 2.

Amplitude-Geq ($\succeq_{A}$),

  3. 3.

Phase-Congruence ($\sim_{P}$),

  4. 4.

Phase-Clockwise-Betweenness ($C_{P}$).

This account, I believe, offers a deeper understanding of the nature of the
quantum state that at the very least complements that of the standard account.
By removing the references to mathematical objects, our account of the quantum
state provides a framework for nominalizing quantum mechanics. By excising
superfluous structure such as overall phase, it reveals the intrinsic
structure postulated by quantum mechanics. Here we have focused on the
universal quantum state. As the origin of quantum non-locality and randomness,
the universal wave function has no classical counterpart and seems to resist
an intrinsic and nominalistic treatment. With the focus on the universal
quantum state, our account still leaves room for including additional
ontologies of particles, fields, mass densities supplied by specific solutions
to the quantum measurement problem such as BM, GRWm, and GRWf; these
additional ontologies are readily nominalizable.

Let us anticipate some directions for future research. First, the intrinsic
structure of the quantum state at different times is constrained by the
quantum dynamics. In the platonistic theory, the dynamics is described by the
Schrödinger equation. To nominalize the dynamics, we can decompose the
Schrödinger equation into two equations, in terms of amplitude and gradient of
phase of the wave function. The key would be to codify the differential
operations (which Field has done for Newtonian Gravitation Theory) in such a
way to be compatible with our phase and amplitude relations. Second, we have
described how to think of the quantum state for a system with constant number
of particles. How should we extend this account to accommodate particle
creation and annihilation in quantum field theories? I think the best way to
answer that question would be to think carefully about the ontology of a
quantum field theory. A possible interpretation is to think of the quantum
state of a variable number of particles as being represented by a complex
valued function whose domain is $\bigcup_{N=0}^{\infty}\mathbb{R}^{3N}$—the
union of all configuration spaces (of different number of particles). In that
case, the extension of our theory would be easy: (1) keep the axioms as they
are and (2) let the quantifiers range over $K$-regions, where the integer $K$
ranges from zero to infinity. Third, we have considered quantum states for
spinless systems. A possible way to extend the present account to accommodate
spinorial degrees of freedom would be to use two comparative relations for
each complex number assigned by the wave function. That strategy is
conceptually similar to the situation in the present account. But it is
certainly not the only strategy, especially considering the gauge degrees of
freedom in the spin space. Fourth, as we have learned from debates about the
relational theories of motion and the comparative theories of quantities,
there is always the possibility of a theory becoming indeterministic when
drawing from only comparative predicates without fixing an absolute
scale.282828See [13, 4, 24], and Field (2016), preface to the second edition,
pp. 41-44.  It would be interesting to investigate whether similar problems of
indeterminism arise in our comparative theory of the quantum state. Finally,
the formal results obtained for the periodic difference structure could be
useful for further investigation into the metaphysics of phase.

The nature of the quantum state is the origin of many deeply puzzling features
of a quantum world. It is especially challenging given realist commitments. I
hope that the account discussed in this paper makes some progress towards a
better understanding of it.

## Acknowledgement

Many thanks to Cian Dorr, Hartry Field, and David Malament for their
encouragements, criticisms, and extensive feedback on this project. For
helpful discussions, I’d like to thank David Albert, Andrew Bacon, Mark
Balaguer, Jeff Barrett, Thomas Barrett, Rhys Borchert, Adam Caulton, Shamik
Dasgupta, Tom Donaldson, Joshua Gert, Sheldon Goldstein, Veronica Gomez,
Jeremy Goodman, Richard Healey, James Ladyman, Barry Loewer, Tim Maudlin,
Richard Pettigrew, Carlo Rovelli, Jeff Russell, Charles Sebens, Ward Struyve,
David Wallace, Jennifer Wang, Nino Zanghì, audiences at the UC Irvine
philosophy of physics reading group, Rutgers metaphysics reading group,
Bristol Center for Science and Philosophy, the 2016 Foundations of Physics
Conference at LSE, the 3rd conference of the Society of Metaphysics of Science
at Fordham, the 4th Black Forest Summer School in Philosophy of Physics in
Saig, Germany, and a symposium at the 2018 Pacific APA meeting in San Diego.
I’d especially like to thank Liam Kofi Bright, Kenny Easwaran, David Glick,
Jill North, Andrea Oldofredi, Bryan Roberts, Ted Sider, and Isaac Wilhelm for
their insightful and detailed written comments.

## Appendix: Proofs of Theorems 3.3 and 3.4.

Step 1. We begin by enriching $<$ N-Regions, Phase–Clockwise–Betweenness,
Phase–Congruence$>$ with some additional structures.

First, to simplify the notations, let us think of N-Regions as a set of
regions, and let us now only consider $\Omega:=$ N-Regions / $=_{P}$, the set
of “equal phase” equivalence classes by quotienting out $=_{P}$. ($a=_{P}b$ if
they form phase intervals the same way: $\forall c\in S,ac\sim_{P}bc$.)

Second, we fix an arbitrary $A_{0}\in\Omega$ to be the “zero phase equivalence
class.”

Third, we define a non-inclusive relation $C$ on $\Omega$ according to $C_{P}$
on N-Regions. ($\forall A,B,C\in\Omega$, $C(A,B,C)$ iff $A,B,C$ are pairwise
distinct and $\forall a\in A,\forall b\in B,\forall c\in C$, $C(a,b,c)$.)

Fourth, we define an addition function $\circ:$
$\Omega\times\Omega\rightarrow\Omega$. $\forall A,B\in\Omega$, $C=A\circ B$ is
the unique element in $\Omega$ such that $CB\sim AA_{0}$, which is guaranteed
to exist by (K4) and provably unique as elements in $\Omega$ form a partition
over N-Regions.

Step 2. We show that the enriched structure $<\Omega,\circ,C>$ with identity
element $A_{0}$ satisfies the axioms for a periodic extensive structure
defined in [22].

Axiom 0. $<\Omega,\circ>$ is an Abelian semigroup.

First, we show that $\circ$ is closed: $\forall A,B\in\Omega$, $A\circ
B\in\Omega$.

This follows from (K4).

Second, we show that $\circ$ is associative: $\forall A,B,C\in\Omega$,
$A\circ(B\circ C)=(A\circ B)\circ C$.

This follows from (K6).

Third, we show that $\circ$ is commutative: $\forall A,B\in\Omega$, $A\circ
B=B\circ A$.

This follows from (K2).

$\forall A,B,C,D\in\Omega$:

Axiom 1. Exactly one of $C(A,B,C)$ or $C(A,C,B)$ holds.

This follows from C1.

Axiom 2. $C(A,B,C)$ implies $C(B,C,A)$.

This follows from C2.

Axiom 3. $C(A,B,C)$ and $C(A,C,D)$ implies $C(A,B,D)$.

This follows from C2.

Axiom 4. $C(A,B,C)$ implies $C(A\circ D,B\circ D,C\circ D)$ and $C(D\circ
A,D\circ B,D\circ C)$.

This follows from (K7).

Axiom 5. If $C(A_{0},A,B)$, then there exists a positive integer $n$ such that
$C(A_{0},A,nA)$ and $C(A_{0},nB,B)$.

This follows from (K8).

Therefore, the enriched structure $<\Omega,\circ,C>$ with identity element
$A_{0}$ satisfies the axioms for a periodic extensive structure defined in
[22].

Step 3. We use the homomorphisms in [22] to find the homomorphisms for <
N-Regions, Phase–Clockwise–Betweenness, Phase–Congruence>.

Since $<\Omega,\circ,C>$ satisfy the axioms for a periodic structure,
Corollary in [22] says that for any real $K>0$, there is a unique function
$\phi$ from $\Omega$ into $[0,K)$ s.t. $\forall A,B,C\in\Omega$:

  1. 1.

$C(C,B,A)$ $\Leftrightarrow$ $\phi(A)>\phi(B)>\phi(C)$ or
$\phi(C)>\phi(A)>\phi(B)$ or $\phi(B)>\phi(C)>\phi(A)$;

  2. 2.

$\phi(A\circ B)=\phi(A)+\phi(B)$ (mod $K$);

  3. 3.

$\phi(A_{0})=0$.

Now, we define $f:$ N-Regions $\rightarrow[0,K)$ as follows: $f(a)=\phi(A)$,
where $a\in A$. So we have $C_{P}(c,b,a)\Leftrightarrow f(a)\geq f(b)\geq
f(c)\text{ or }f(c)\geq f(a)\geq f(b)\text{ or }f(b)\geq f(c)\geq f(a)$.

We can also define $\psi:$ N-Regions $\times$ N-Regions $\rightarrow[0,K)$ as
follows: $\psi(a,b)=\phi(A)-\phi(B)$ (mod $K$), where $a\in A$ and $b\in B$.
Hence, $\forall a,b\in$ N-Regions, $\psi(a,b)=f(a)-f(b)$ (mod $K$).

Moreover, given (K5), $\forall a\in A,b\in B,c\in C,d\in D,$ $ab\sim_{P}cd$

$\Leftrightarrow$ $AB\sim CD$

$\Leftrightarrow$ $A\circ D=B\circ C$

$\Leftrightarrow$ $\phi(A\circ D)=\phi(B\circ C)$

$\Leftrightarrow$ $\phi(A)+\phi(D)=\phi(B)+\phi(C)$ (mod $K$)

$\Leftrightarrow$ $\forall a\in A,b\in B,c\in C,d\in D,$ $f(a)+f(d)=f(b)+f(c)$
(mod $K$)

$\Leftrightarrow$ $\forall a\in A,b\in B,c\in C,d\in D,$ $f(a)-f(b)=f(c)-f(d)$
(mod $K$)

Therefore, we have demonstrated the existence of homomorphisms.

Step 4. We prove the uniqueness theorem.

If another function $f^{\prime}:\text{ N-Regions }\rightarrow[0,K)$ with the
same properties exists, then it satisfies the same homomorphism conditions.
Suppose (for _reductio_) $f^{\prime}$ and $f$ differ by more than a constant
mod K.

Let $D(a,b)$ by the function that measures the differences between $f$ and
$f^{\prime}$ on N-regions:

$D(a,b)=[f(a)-f(b)]-[f^{\prime}(a)-f^{\prime}(b)]\text{ mod }K.$

Without loss of generality, let us suppose that there exist two regions $x,y$
where $D(x,y)\neq 0$. By (K8), there will be a sequence of pairs of regions
that are phase-congruent to $x,y$:
$xy\sim_{P}ya_{1}\sim_{P}a_{1}a_{2}\sim_{P}...\sim_{P}a_{n}x$. Since by
assumption both $f$ and $f^{\prime}$ preserve the structure of phase-
congruence, we have (mod K):

$f(x)-f(y)=f(y)-f(a_{1})=...=f(a_{n})-f(x),$
$f^{\prime}(x)-f^{\prime}(y)=f^{\prime}(y)-f^{\prime}(a_{1})=...=f^{\prime}(a_{n})-f^{\prime}(x).$

Consequently:

$D(x,y)=[f(x)-f(y)]-[f^{\prime}(x)-f^{\prime}(y)]=D(y,a_{1})=...=D(a_{n},x)$

Hence, since the above $D$’s are not zero, they must be either all positive or
all negative. If they are all positive, then the sum of them will be positive:

$D(x,y)+D(y,a_{1})+...+D(a_{n},x)>0$

However, expanding them in terms of $f$ and $f^{\prime}$ we have a telescoping
sum:

$[f(x)-f(y)]-[f^{\prime}(x)-f^{\prime}(y)]+[f(y)-f(a_{1})]-[f^{\prime}(y)-f^{\prime}(a_{1})]+...+[f(a_{n})-f(x)]-[f^{\prime}(a_{n})-f^{\prime}(x)]=0.$

Contradiction. The same argument works for the case when all the $D$’s are
negative.

Therefore, $D(x,y)=0$ for all N-regions. Let $a_{0}$ be where $f$ assigns
zero. Then

$f^{\prime}(a)-f^{\prime}(a_{0})\text{ mod }K=f(a)-f(a_{0})\text{ mod
}K=f(a),$

which entails that

$f^{\prime}(a)=f(a)+\beta\text{ mod }K,$

with the constant $\beta=f^{\prime}(a_{0})$. QED.

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