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# Chapter 1: Introduction: The emergence of spacetime

Nick Huggett and Christian Wüthrich This is a chapter of the planned monograph
_Out of Nowhere: The Emergence of Spacetime in Quantum Theories of Gravity_ ,
co-authored by Nick Huggett and Christian Wüthrich and under contract with
Oxford University Press. More information at www.beyondspacetime.net. This
work was supported financially by the ACLS and the John Templeton Foundation
(the views expressed are those of the authors not necessarily those of the
sponsors.

(18 January 2021)

###### Contents

  1. 1 Quantum gravity and philosophy
  2. 2 Worlds without spacetime?
  3. 3 Challenges to spacetime emergence
  4. 4 Physical salience
  5. 5 Non-commutative geometry
  6. 6 Spacetime functionalism
  7. 7 The role of philosophy in physics
  8. 8 The plan for the book

“Big Bang Machine Could Destroy Earth”

…ran an attention grabbing headline in The Sunday Times (Leake 1999),
regarding the new Relativistic Heavy Ion Collider (RHIC) at Brookhaven
National Laboratory. To be fair, the main apocalyptic concern of the paper was
that the RHIC would create a form of matter in which strange quarks eat the up
and down quarks found in ordinary matter. But it also discussed the
possibility that experiments involving high-energy collision of gold ions
could create microscopic blackholes, which would pull all the matter in the
world into them. Such scenarios were taken seriously enough that they were
evaluated by a panel of elders, who concluded that the chances of any such
events were utterly minuscule (Busza et al. 1999)—happily, up to the time of
writing, they have not been contradicted by events at Brookhaven!

Let’s look into the business of black hole formation more carefully to explain
why physics needs an account of quantum gravity. First, the RHIC was built to
probe how matter behaves under intense temperatures and pressures—in effect
recreating in a tiny region the state of the universe within the first second
of its existence, when quarks and gluons flowed in a plasma rather than
binding to form particles. The predictions tested here are largely those of
quantum chromodynamics, the quantum theory of the strong force binding
nucleons and their constituents. That is, the collisions between heavy nuclei
such as gold in the RHIC are governed by the laws of quantum mechanics (QM).

The concern over black holes, however, arises when one asks how general
relativity (GR)—the classical, non-quantum, theory of gravity, gets into the
picture. According to GR, the spacetime metric outside a sphere of mass $M$
takes the form:

$\mathrm{d}s^{2}=c^{2}\mathrm{d}t^{2}(1-\frac{2GM}{rc^{2}})-\mathrm{d}r^{2}/(1-\frac{2GM}{rc^{2}}),$
(1)

where $G$ is Newton’s gravitational constant, $c$ is the speed of light, and
$r$ the distance from the center. ($\mathrm{d}s^{2}$ is the infinitesimal
spacetime ‘distance’ squared—the ‘interval’—between two radial points
separated in time by $\mathrm{d}t$ and space by $\mathrm{d}r$, in suitable co-
ordinates; though an exact understanding is not crucial here.) What is
important is that this quantity blows up when $2GM=rc^{2}$, or $r=2GM/c^{2}$.
Understanding this occurrence was an important issue in the early development
of GR, but what it actually signifies is the presence of an ‘event horizon’
around the mass, from which neither matter nor light can escape—the boundary
of a black hole. Of course this story only makes sense if the mass is all
located within a radius of $2GM/c^{2}$, since the metric formula only holds
outside the mass. So one can equally well say that if one has a mass $M$ it
will only form a black hole if it is all located within a radius less than
$2GM/c^{2}$.

So finally, the question posed by the panel at Brookhaven was how small a
region would the amount of energy to be created in collisions be located (see
page 7 of the report)? Acting cautiously, they assumed the best conditions for
black hole formation, supposing that all the energy produced by the collision
contributes to the mass: about 50 times that of a gold atom. For a black hole
of this mass the event horizon has a radius of $10^{-39}$m. On the other hand,
a gold atom has a radius of around $10^{-12}$m, so even supposing that all the
energy is concentrated in a region the size of a suitably Lorentz-contracted
nucleus, general relativity predicts that collisions will be many, many orders
of magnitude from creating a black hole.

Phew.111Don’t be confused if you have read of black holes being created at
RHIC. In fact what has (perhaps) been observed is the Unruh effect, which is
formally equivalent to Hawking radiation, but does not involve black holes,
but acceleration, which is in a sense indistinguishable from a gravitational
force according to GR. See Nastase (2005).

##  1 Quantum gravity and philosophy

What we have then is an argument that the physics of Brookhaven lies within
the domain of relativistic QM, but that the gravitational effects of the
collisions are utterly negligible. Perhaps the world is just ‘dappled’ in this
way: in some domains, such as the motions of the planets (GR explains the
perihelion of Mercury, for instance) GR holds and QM is irrelevant; in others,
as in RHIC, it is QM that holds sway, with GR entering only to provide a
background geometry determined by ambient bodies, not by the system under
consideration. But the very argument here shows that it is possible to bring
considerations from both theories to bear on a single system, making clear
that one can sensibly ask whether there are domains in which both theories
apply. On our current understanding of the universe indeed there are. In the
first place, there is the big bang, which is entailed by GR given the current
state of the universe, in which matter becomes so hot and compact that QM
effects will necessarily occur. (And conversely, if the inflationary
hypothesis is correct, between the first $10^{-33}$ to $10^{-32}$s quantum
fields provide the energy which drives the expansion of the universe,
according to the laws of GR.) In the second, Hawking radiation is predicted to
occur around black holes as a QM effect resulting from the geometry of
spacetime given by GR (Hawking 1974). Indeed, assuming that the local
equivalence of acceleration and gravity—the ‘equivalence principle’—holds in
the quantum domain, then RHIC does provide tests for this physics, since the
radiation produced by decelerating ions can be measured (see footnote 1):
under the assumption, it provides indirect tests of the overlap of gravity and
QM.

Thus, we say, the ultimate need for a theory that in some way unifies QM and
GR—a theory of _quantum gravity_ (QG)— arises from the existence of phenomena
in our universe in which the domains of the two overlap. There are other, more
theoretical, arguments for such a theory and concerning the form it should
take. For a critical evaluation of these, arguing that overlap is best reason
to seek quantum gravity, and that empirical considerations best dictate its
form see Callender and Huggett (2001); for further discussion Wüthrich
(2005).222This situation may change if the technology improves enough over the
next few years to carry out the experiment proposed by Bose et al. (2017).

There are then good reasons for physicists to investigate QG. This book is
predicated on the view that it also has an important call on the efforts of
philosophers. Indeed, we would like this work to encourage, by example, our
colleagues to be more adventurous in their choice of topics of enquiry.
Philosophy of physics, we suggest, has a tendency to look too much to the
past, and to the metaphysics of well-established physics (and of course to
internecine disputes): classical statistical mechanics, classical spacetime
theory and non-relativistic quantum mechanics are ‘so twentieth (or even
nineteenth) century’, and yet have a virtual lock on the discipline. While
quantum field theory (QFT) is becoming a significant topic, that is still at
least half a century behind the physics!

We’re overstating things somewhat for effect here: of course, even old
theories do face important foundational problems, and their consequences for
our broader understanding of the world take considerable elucidation. And of
course it is unfair to suggest that no philosophers of physics show an
interest in contemporary physics. Indeed, since we started writing this book,
there has been an explosion of interest in QG, especially amongst a younger
generation of scholars, which we find very exciting. Still, we do say that
collectively the discipline pays insufficient attention to cutting edge
physics, and hope this book in some way serves as an impetus to greater
engagement.

Our point is not that novelty is good for its own sake, nor that philosophy is
a ‘hand-maiden’, who should dutifully follow the fashions of physics. Rather,
we are inspired by recent work in the history and philosophy of science to
believe that it is central to the business of philosophy to engage with
developing physical theories—both because the search for philosophical
knowledge must be responsive to empirical discoveries, and because philosophy
has important contributions to make to the development of physics (and other
sciences). We will discuss this point at greater length below (§7), to explain
our aims and motivations.

First, in part to make that discussion more concrete, in §2 we will very
briefly introduce some theories of QG (or in its vicinity). We especially want
to focus on a rather generic feature of them—that in various ways they do not
contain familiar spacetime at a fundamental level, but rather it ‘emerges’ (in
a sense to be discussed) in a higher, non-fundamental domain. That will lead
us to a discussion (in §3) of the challenges to the very idea that something a
seemingly fundamental as spacetime could be derivative, from even more
fundamental, yet non-spatiotemporal, physics. We then address these
challenges, analyzing how they can be overcome, and illustrating this process
in a historical case (§4); and in a short contemporary example (§5). This
discussion encapsulates much of the work of the book: the chapters introduce,
for philosophers, several proposals for a theory of QG, discuss the ways in
which they eliminate spatiotemporal structures, and investigates the ways in
which they are recovered as effective, apparent structures. What we propose is
a form of ‘functionalism’, so in §6 we explain that position.

In a final section (§8) we will give an overview of the different strategies
one might take towards quantizing gravity, to relate the different proposals
that we will consider in the book.

##  2 Worlds without spacetime?

All theories of QG are, to a large extent, speculative; some of the examples
that follow are more speculative than others. However, as we shall explain
below, there are good reasons to think that they may still teach important
lessons in the search for QG. The first three examples are the focus of the
following chapters; the remaining two are also illuminating, but we have
discussed them elsewhere.

  * •

Causal Set Theory (CST): As we will see in chapters 2 and 3, CST makes liberal
use of GR as a vantage point for its research programme. In fact, it takes its
most important motivation from theorems stating that given the causal
structure of a spacetime, its metric is determined up to a conformal factor.
In other words, the causal structure determines the geometry of a
spacetime—but not its ‘size’. Taking this cue, CST posits that the fundamental
structure is a set of elementary events which are locally finite, partially
ordered by a basic causal relation. In other words, the fundamental structure
is a causal set. The assumption of local finitarity is nothing but the formal
demand that the fundamental structure—whatever else it is—is discrete.
Together with the demand of Lorenz invariance at the derived level, the
discreteness of causal sets forces a rather odd locality structure onto the
elementary events of the causal set (§3.5). Furthermore, although the
fundamental relation of causal precedence can double up as something akin to
temporal precedence, space is altogether lost in a causal set (§2.3). Jointly,
these facts entail that the structure we are facing in causal set theory is
also rather different from the spacetime encountered in GR. In fact, the
quantum nature of the causal sets yet to be incorporated into causal set
theory is bound to further complicate the picture and to remove the resulting
structure from that of relativistic spacetimes.

  * •

Loop Quantum Gravity (LQG): LQG starts out from a Hamiltonian formulation of
GR and attempts to use a recipe for cooking up a quantum from a classical
theory that has been utilized with great success in other areas of physics.
This recipe is the so-called canonical quantization. The goal of applying the
canonical quantization procedure is to find the physical Hilbert space, i.e.
the space of admissible physical states, and the operators defined on it that
correspond to genuinely physical quantities. As will be seen in chapters 4-5,
following this recipe leads rather straightforwardly into a morass of deep
conceptual, interpretative, and technical issues concerning the dynamics of
the theory as well as on time quite generally. We find that the states in the
‘kinematic’ Hilbert space afford a natural geometric interpretation: its
elements are states that give rise to physical space, yet are discrete
structures with a disordered locality structure. At least in one basis of this
Hilbert space, the states appear to be states of a granular structure, welding
together tiny ‘atoms’ of space(time). It is crucial to this picture that these
atoms of space(time) are atoms in the original meaning of the word: they are
the truly indivisible smallest pieces of space(time). The smooth space(time)
of the classical can thus be seen to be supplanted by a discrete quantum
structure. Moreover, since generically a state of this structure will be a
superposition of basis states with a determinate geometry, generic states will
not possess determinate geometric properties. If continuity, locality, or
determinate geometry was an essential property of spacetime, then whatever the
fundamental structure is, it is not spacetime. In this sense, spacetime is
eliminated from the fundamental theory.

  * •

String Theory: According to string theory (chapter 6) tiny one-dimensional
objects move around space, wiggling as they go—the different kinds of
vibration correspond to different masses (charges and spins) and hence to
different subatomic particles. So it sounds as if spacetime is built into the
theory in a pretty straight-forward way; however, we will see that things are
not so simple. First, in chapter 7 we will see that various versions which are
intuitively very different in fact correspond to the same physics—are ‘ _dual_
’. For instance, suppose that at least one of the dimensions of space is
‘compactified’, or circular. Then it turns out that a theory in which the
circumference of the dimension is $C$ has the same collection of values for
physical quantities as a theory in which the circumference is $1/C$: i.e.,
that theories in which compactified dimensions are small are physically
indistinguishable from—or ‘dual’ to— those in which they are large. Other
dualities relate spaces of different topologies. These facts raise important
questions about whether the space in which the string lives is the one we
observe, since that is definitely large though the string space could be
small, or even conventional. Second, in chapters 8-9, we will see how the
geometrical structure of spacetime—and indeed GR—arises from the behavior of
large collections of strings in a ‘graviton’ state, the quantum particle that
mediates gravitational forces.

  * •

Group Field Theory: Consider rotations, by and angle $\theta$, in the plane: a
different one for each value of $0\leq\theta<2\pi$. These form a group under
composition: a rotation by $\alpha$ followed by an angle $\beta$ is just a
rotation by $\alpha+\beta$ (and for example, a rotation by $2\pi-\alpha$
undoes a rotation by $\alpha$). Similarly for rotations in three dimensions,
and indeed similarly for Lorentz transformations (which are in fact nothing
rotations in Minkowski spacetime), and so on. For each, composition yields a
different function from any pair to third. We can thus characterize the
abstract group structure simply by the action of this function of an entirely
arbitrary set of elements—forget that we started with rotations, and let the
elements be anything, with a composition rule isomorphic to that of the
rotations. One could take then a set of such blank elements, which compose
like rotations in the plane, but which should not be thought of as literal
rotations: no plane at all is postulated, the only manifold is that formed by
the group elements themselves—the circle of points with labels $0-2\pi$, not a
2-dimensional plane.333More accurately, the space that is assumed is not the
space of relativistic physics, or that of planar rotations, but four copies of
the group of Minkowski rotations. And finally one can introduce a field on
this group manifold, a real number for each group element, with a dynamical
law for its evolution; and indeed quantize this field. As we have emphasized,
while physical space was used to guide the construction of this theory, it is
not an explicit component of it, and yet models of general relativity can be
derived from it. This example is discussed in greater detail in Oriti (2014)
and Huggett (2018).

  * •

Non-Commutative Geometry: Picture a Euclidean rectangle whose sides lie along
two coordinate axes, so that the lengths of its sides are $x$ and $y$—its area
is $x\cdot y=y\cdot x$. But what if such products fail to commute, $xy\neq
yx$, so that area is no longer a sensible quantity? How can we understand such
a thing—a _non-commutative geometry_? By abandoning ordinary images of
geometry in terms of a literal space (such as the plane) and presenting it in
an alternative, algebraic way. In fact, our example already starts to do so:
even thinking about areas as products of co-ordinates uses Descartes’
algebraic approach to Euclidean geometry. Once we have entered the realm of
algebra, all kinds of possible modifications arise. Especially, an abstract
algebra $\mathcal{A}$ requires an operation of ‘multiplication’, $\star$, but
this can be a quite general map from pairs of elements,
$\star:\mathcal{A}\times\mathcal{A}\to\mathcal{A}$, _which need not be
commutative_! For instance, one could define ‘multiplication’ to satisfy
$x\star y-y\star x=\theta$ (a small number), and use it to generate
polynomials in $x$ and $y$; these carry geometric information. Such a thing is
perfectly comprehensible from the abstract point of view of algebra, but it
cannot be given a familiar Euclidean interpretation via Cartesian geometry.
So, such a theory seems to describe a world that is fundamentally algebraic,
not spatial (in the ordinary sense)—there is $x\star y$ and $y\star x$ but no
literal rectangle. If there is thus fundamentally nothing ‘in’ space, is the
ultimate ontology ‘structural’, based on algebraic relations only? And how
could an appearance of familiar (commutative!) space arise; especially, what
significance could point-valued quantities have? We will return to this
example in §3.

All of these examples are speculative to some extent or other, and none can
claim to be a complete quantum theory of gravity (and none has convincing,
currently testable, novel predictions!), yet all have some claim to model
relevant physical features of QG, worth exploring. In particular, we have
emphasized in each case how spacetime features are missing in the theories (to
be spelled out in detail in later chapters). This situation thus appears to be
a common condition of many approaches to QG, in which case we say that
classical, relativistic spacetime is ‘emergent’. We emphasize (as we have
elsewhere) that we do not use this term in its strongest philosophical sense
to indicate the _inexplicability_ of X from Y: as some have claimed life or
mind emerges from matter. On the contrary, we argue that classical spacetime
structures _can_ be explained in more fundamental terms: indeed, it was
largely to explicate how physicists do so that we wrote the book. Some might
then say that spacetime ‘reduces’ to non-spatiotemporal QG, but we prefer to
stick with the notions of ‘explanation’ or ‘derivation’, because there are
many notions of ‘reduction’, some of which are too strict. But we are also
happy to speak of (weak) ‘emergence’ even when spacetime is derived, because
the gulf between a theory that does not assume spacetime, and one that does is
so great. Having spacetime or not makes a huge formal and conceptual
difference, in particular because in almost all theories prior to QG classical
spacetime has apparently been one of the most basic posits. Indeed, this very
gulf makes one wonder what it could mean to derive spacetime, and whether it
is possible at all.

Before we proceed, we need to introduce some terminology to keep the
discussion straight. The issue is that the theories of QG often contain some
object referred to as ‘space’, even when they do not assume ‘space’ in the
ordinary sense. For instance, there may be a ‘Hilbert space’, or a ‘dual
space’, or ‘Weyl space’, or ‘group space’. So we will refer to spacetime in
the ordinary sense as ‘classical’, or ‘relativistic’, or sometimes just
‘space’ or ‘spacetime’ when the context makes matters clear. (We eschew the
phrase ‘physical space’, since the other ‘spaces’ may well be part of the
fundamental physical furniture. We have previously used ‘phenomenal space’, to
indicate that classical space is that of observable phenomena, according to
the physicist’s use of ‘phenomenological’. However, this leads to confusion
with the philosophical doctrine of ‘phenomenalism’, so we have dropped it.)

By classical or relativistic spacetime, we mean that theorized in QM
(especially QFT) and relativity, approximated in non-relativistic mechanics,
and ultimately implicated in our observations of the physical world. As
stated, that is not an entirely homogeneous concept, so we will say more later
(§6) about exactly what features of classical spacetime are emergent from our
theories of QG. First, we turn to some challenges to the project of deriving
spacetime.

##  3 Challenges to spacetime emergence

Space and time are so basic to both our manifest and scientific images of the
world that at first the mind boggles at the thought that they might be mere
‘appearances’ or ‘phenomena’, of some deeper, more fundamental, non-
spatiotemporal reality. Is a physics without spacetime even intelligible? And
if it is, is spacetime the kind of thing whose existence could be explained?
At its core, this book seeks to address these questions: on the one hand
explicating the worlds described by theories of QG, while on the other showing
how spacetime can be derived from them. But to understand the nature and
methodology of that project, it is important here to unpack the mind boggling,
vertiginous panic about the very idea. Larry Sklar (1983) gave expression to
this all too common sentiment among philosophers (and physicists) when he
wrote444A note on terminology: we take ‘Platonists’ to be committed to the
existence of abstract entities, such as propositions, sets, love, and justice,
but also to the existence of the concrete, physical, and spatiotemporal world.
Those who maintain that our world is fundamentally mathematical in nature and
thus entirely consists in ultimately abstract entities or structures are often
labelled as ‘Pythagoreans’. Since we are interested not in whether there exist
abstracta, but in the possibility that all physical existence is grounded in
non-spatiotemporal structures, we will refer to those who maintain that a
fundamentally non-spatiotemporal physical world is not devoid of “real
being”—no doubt historically inaccurately—as _Pythagoreans_. We take this
Pythagoreanism to be Sklar’s target—and the one of this monograph.

> What could possibly constitute a more essential, a more ineliminable,
> component of our conceptual framework than that ordering of phenomena which
> places them in space and time? The spatiality and temporality of things is,
> we feel, the very condition of their existing at all and having other, less
> primordial, features. A world devoid of color, smell or taste we could,
> perhaps, imagine. Similarly a world stripped of what we take to be essential
> theoretical properties also seems conceivable to us. We could imagine a
> world without electrical charge, without the atomic constitution of matter,
> perhaps without matter at all. But a world not in time? A world not spatial?
> Except to some Platonists, I suppose, such a world seems devoid of real
> being altogether. (45)

According to Sklar, a non-spatiotemporal world is inconceivable, and thus
presumably not even metaphysically possible, let alone physically. This
monograph is concerned with establishing the possibility of a fundamentally
non-spatiotemporal world, articulating the consequences of such a possibility,
and defending the idea that spacetime may be merely emergent in a perfectly
acceptable scientific explanation of the manifest world. So in this section we
will discuss various more precise ways that one might doubt the possibility of
deriving spacetime.

First, in Huggett and Wüthrich (2013) we discussed the idea that a theory
without spacetime might be ‘empirically incoherent’. That is, any theory which
entails that the observations apparently supporting it are impossible, cannot
receive empirical support (Barrett 1996)—it undermines the very grounds for
believing it. Since all observations ultimately involve events localized in
spacetime, it might seem that theories without spacetime in their basic
formulation are threatened with empirical incoherence; the confirmation of
such a theory might be ruled out a priori. However, it is clear that a
conflation is involved. (More) fundamental theories of QG are non-
spatiotemporal in the sense that spatiotemporal structure is missing in their
more furniture; but it is perfectly consistent to think that spacetime is
present as an effective object, arising from the more fundamental ones. (And
that observation events can be identified within effective spacetime.) That
is, QG will not be empirically incoherent if the appearance of spacetime can
be adequately explained, and of course that is exactly what our case studies
aim to do.

Second, while this book concerns the idea of ‘emergent’ spacetime as it arises
in QG, one of our key concerns has already arisen in discussions of a
different kind of spacetime emergence. The quantum mechanical wavefunction of
$N$ particles is not a function in ordinary space, but of the positions of all
the particles:
$\Psi(x_{1},y_{1},z_{1};x_{2},y_{2},z_{2};\dots;x_{N},y_{N},z_{N})$. Thus
$\Psi$ lives in ‘configuration’ space, in which there are three dimensions for
each particle. Albert (1996) has argued that we should take the wavefunction
‘seriously’ as the ontology of the theory, and conclude that configuration
space is more fundamental than regular space—that the three dimensions of
experience are mere appearances of the $3N$ dimensions of reality. Whatever
the merits of that view, the general idea has been attacked by Tim Maudlin
(2007). In particular he argues as follows: one might

> derive a physical structure with the form of local beables from a basic
> ontology that does not postulate them. This would allow the theory to make
> contact with evidence still at the level of local beables, but would also
> insist that, at a fundamental level, the local structure is not itself
> primitive. … This approach turns critically on what such a derivation of
> something isomorphic to local structure would look like, _where the derived
> structure deserves to be regarded as physically salient_ (rather than merely
> mathematically definable). Until we know how to identify physically serious
> derivative structure, it is not clear how to implement this strategy. (3161)

We have italicized the key phrase here. Suppose that one managed to show
formally that certain derivative quantities in a non-spatiotemporal theory
took on values corresponding to the values of classical spatiotemporal
quantities; one would then be in a position to make predictions about derived
space. However, according to the passage quoted, such a derivation (even if
the predictions were correct) would not show that spacetime had been
explained. In addition, we have to be assured that the formally derived
structure is ‘ _physically salient_ ’. We agree with Maudlin that physical
salience is required of proper—one can say ‘explanatory’—derivations:
otherwise one simply has a formal, instrumental book-keeping of the phenomena.
Indeed, we agree with him that the issue is particularly pressing in theories
of emergent spacetime. But we think that it can be addressed in QG: one of the
goals of this book is to investigate the (novel) principles of physical
salience for theories of QG, the principles whose satisfaction makes the
derivations of spacetime physically salient. In the following chapters we will
look in detail at the derivations, to make clear the assumptions and forms of
reasoning that lie behind them. In the concluding chapter 10 we will analyze
what have learned, to start to explicate what makes a derivation of spacetime
in QG physically salient.

But to explain that project—and its relevance to philosophy—we need to unpack
the very notion of physical salience, as we understand it.555We are grateful
to Maudlin for conversations on this topic. We believe that we capture the
essence of his idea, even if we might differ in details; and especially
regarding the depth of the problem in the case of spacetime emergence. We do,
however, want to point out an important difference between the cases of
emergence from QG and from configuration space: in the latter, but not the
former, there is a way to formulate the theory in 3-space (as single particle
wavefunctions with a tensor product). Thus in QM (but not QG) Maudlin can
argue that the derivation isn’t physically salient, because the formulation
from which it is derived is unnecessary in the first place. That the crucial
difference between QG on the one hand and QM (and GR) on the other lies in
there being no alternatives translates into a different status for spacetime
functionalism in the two cases has been argued by Lam and Wüthrich
(forthcoming).

##  4 Physical salience

There is a subtlety about the way that Maudlin makes the point, however (which
we did not clearly address in Huggett and Wüthrich (2013)). For the target
derived structure in itself is prima facie physically salient: it is the
physical datum to which the more fundamental theory is answerable. (Perhaps a
more fundamental theory will show that some less fundamental theory is
profoundly confused; but more generally one expects that existing, well-
confirmed theories have latched onto some genuine physical structures, and
that new, better theories will simply explain how, by subsuming the old in
some broad sense.) So in that sense there is really no question of the
physical salience of the ‘derived structure’—in the sense of the structure _to
be_ derived.

Rather, Maudlin is talking about a formal derivation within a proposed new
theory, and the question of whether what is at present simply a mathematical
structure, in numerical agreement with the target structure, in fact explains
it, and isn’t merely an instrument for generating predictions. We would break
this question down into two interconnected parts (which will also help
illuminate what is involved in explanation here). First, the question of
whether and how the basal objects or structures of the more fundamental theory
accurately represent physically salient objects and structures. As we shall
see shortly, that question becomes far more pressing when none of the putative
objects or structures are supposed to be in spacetime. Second, does the formal
derivation of the phenomenal from the more fundamental make physical sense?
That the derivation exists shows that it makes sense at the level of the
formalism, and especially that the derivation is compatible with the
mathematical laws. But, as Maudlin suggests, there is more to the question of
physical salience than that. And the question is especially pointed when one
wonders how the spatiotemporal could ever be ‘made’ of the non-spatiotemporal.
We will illustrate these ideas with a homely (and idealized in many ways)
example.666The following has also been discussed in Huggett (2018).

The ideal gas law tells that for a gas (in a box of fixed volume) pressure
$\propto$ temperature. Ideal gas theory says nothing about the microscopic
composition of gases, so these are (among) the primitive quantities of the
theory, operationalized via pressure gauges (relying on forces measured via
Hooke’s law for springs), and thermometers (so relying on the linear expansion
with temperature of some substance). This is the phenomenon to be explained by
the more fundamental theory, the kinetic gas model, according to which the gas
is composed of atoms with mass $m$, whose degrees of freedom are their
positions and velocities. The latter can be expressed by a vector $\vec{V}$,
with 3$n$ components: for each of the $n$ atoms that make up the gas, three
components, to describe the speed with respect to each of the three dimensions
of space. Each atom has a kinetic energy associated with its velocity
($1/2m\vec{v}^{2}$); the average kinetic energy is simply their sum, divided
by $n$: denote this quantity

$T(\vec{V})\equiv\overline{\frac{1}{2}m\vec{V}^{2}}.$ (2)

Now one computes the atoms’ momentum change (per second per unit area)
resulting from their collisions with the sides of the box: assuming that the
collisions are elastic, and that the atoms are distributed evenly throughout
the box and with respect to their velocities, one formally derives that

$P(\vec{V})\equiv\frac{2n}{3v}\overline{\frac{1}{2}m\vec{V}^{2}}.$ (3)

Clearly the two quantities are proportional:

$P(\vec{V})\propto T(\vec{V}),$ (4)

which has the form of the ideal gas law. However (and despite the suggestive
names, $P$ and $T$) we have so far said nothing to justify identifying the
quantities with the pressure and temperature of the ideal gas law; we have
only noted a formal proportionality.

From this example we can abstract the following schema:

> If (a) fundamental quantities $X$ can be ‘aggregated’ into $\alpha(X)$ and
> $\beta(X)$, such that (b) $f(\alpha(X))=g(\beta(X))$ follows from
> fundamental laws, then the law $f(A)=g(B)$ relating less fundamental
> quantities $A$ and $B$ is _formally derived_.

The term ‘aggregated’ is supposed to be vague, in order to accommodate the
many ways a derivation might proceed. But the underlying idea is that the more
fundamental theory has (many) more degrees of freedom than the less
fundamental, and somehow the more fundamental must be ‘summarized’ by the
less, for example by averaging, or by coarse-graining.

Maudlin’s claim is that formal derivability does not suffice to properly
derive phenomena: in particular, 3-dimensional space can be formally derived
from the full $3N$-dimensional configuration space, but for Maudlin that does
not make it a plausible, more fundamental alternative to ordinary space. And
more generally, one should worry that a merely formal condition does not
distinguish instrumental calculi from serious physical accounts. And indeed,
further analysis of the derivation of the ideal gas law shows that
considerations of physical salience are at play.

In particular, $P(\vec{V})$ is derived by assuming that the atoms are striking
the sides of the box, and exerting a force there: so acting exactly at the
place and in the way that would produce a reading on a pressure gauge. And
$T(\vec{V})$ is (according to the randomness assumption) the amount of energy
in any macroscopic region of the box, say the location of the bulb of a
thermometer: and collisions with the bulb will transfer kinetic energy to the
molecules of the thermometer, causing thermal expansion. Imagine if instead
that $P(\vec{V})$ only referred to the center of the box; or if $T(\vec{V})$
referred to a single atom in the box. Then the formal derivation would not be
convincing. Or suppose that instead of the atomic gas model we imagined that a
gas was a continuous object, whose degrees of freedom were somehow described
by $\vec{V}$, but not as the velocities of anything (certainly not atoms).
Then the formal consequences of kinetic gas theory could still be taken to
hold, but they would no longer have the interpretation that they do in the
kinetic gas model; the whole derivation would go through, but its physical
meaning would be obscure. In short, the reason, in addition to their
proportionality, that we find $P(\vec{V})$ and $T(\vec{V})$ convincing as
pressure and temperature, and not just quantities following a similar law, is
that they are spatiotemporally coincident with those quantities, and involve
processes capable of producing the phenomena associated with those quantities.
The derivation is not merely formal, but also physically salient.

Continuing our schema:

> A (non-instrumental) derivation of phenomena requires, in addition to a
> formal derivation, that (c) the derivation have _physical salience_.

Of course, this schema does not tell us what it is to be physically salient,
but the example of the ideal gas above illustrates two very important aspects,
spatiotemporal coincidence, and the action of a physically accepted mechanism.
And this observation immediately reveals the problem for the emergence of
spacetime, because such criteria simply cannot be satisfied by derivations
from non-spatiotemporal theories, because they are explicitly spatiotemporal
criteria. For instance, it makes no fundamental sense in such a theory to even
ask where a structure is. So if such criteria are a priori constraints on
science, then the QG program, to the extent that it involves non-
spatiotemporal theories, is in some serious trouble. However, a second example
indicates the contextuality of physical salience, and thereby the way in which
QG can hope to achieve physical salience in its derivations.

Figure 1: Descartes’ and Newton’s competing images of gravity. On the left is
pictured Descartes’ vortex model: each cell represents a ball of rotating
matter, with lines to indicate the direction of rotation (e.g., those
surrounding $f,L,Y$ rotate about axes in the plane shown, while those
surrounding $D,F,S$ rotate about axes perpendicular to the plane). The bodies
at the center of a cell represent suns: $S$ is ours. On the right is the
diagram from Newton’s Proposition I.1 proof of Kepler’s equal areas law for a
central force (essentially, conservation of angular momentum). All that
matters is the direction of the force (towards the point $S$), not any
‘hypothesis’ about its nature. Ultimately Newton will apply the proposition to
the case in which $S$ is our Sun. (Public domain, via Wikimedia Commons and
Google Books.)

Consider the competing Cartesian and Newtonian accounts of gravity,
exemplified by illustrations from the _Principles of Philosophy_ (Descartes
1644) and the _Mathematical Principles of Natural Philosophy_ (Newton 1726),
respectively: see figure 1. On the one hand we have the vortices of Descartes,
which aimed to provide a mechanical account of gravity, in terms of the
motions and collisions of particles. On the other there is Newtonian action at
a distance, which allowed him (as in Proposition I.1) to formulate and use his
mathematical principles. We pass over Newton’s own ambiguous attitude towards
the causes of gravitation (his refusal to ‘feign hypotheses’ on the one hand,
but his speculations in the _Optiks_ (Newton 1730) on the other). The point to
which we draw attention is the controversy between the Newtonians and
Cartesians regarding the need for mechanical explanation.777Note especially
that we strictly distort the logic of Newton’s _Principia_ here: as far as
Proposition I.1 is concerned, the forces could be impulses directed towards
the point $S$. However, though Newton’s reader may not at that stage know the
nature of the force, for Newton the figure represents the action of universal
gravitation. For the latter, Newton might have captured the effects of gravity
in a formally accurate way, but offered no scientific explanation for the
phenomena. For example, consider Leibniz’s clear statement to Clarke:

> If God would cause a body to move [round a] fixed centre, without any
> [created thing] acting upon it …it cannot be explained by the nature of
> bodies. For, a free body does naturally recede from a curve in the tangent.
> And therefore …the attraction of bodies …is a miraculous thing, since it
> cannot be explained by the nature of bodies. (_Leibniz-Clarke
> Correspondence_ in Clarke et al. (1956))

We take Leibniz’s complaint to be exactly that Newton’s derivation of the
phenomena lacks physical salience, because only mechanical causes are
physically salient explanations of unnatural motions.

Of course, the Newtonians were ultimately victorious, and this Cartesian
condition of physical salience was replaced by one that allows action at a
distance, because of the success of universal gravity, and the failure of
mechanical alternatives, such as Leibniz’s. But that was not the end of the
story: through the development and empirical success of electromagnetic
theory, culminating in the development of special relativity, action at a
distance was again rejected, with contact action replaced by the demand for
local field interactions—and hence the replacement of Newtonian gravity with
general relativity. Again, we understand this demand as a criterion of
physical salience, required for more than merely formal accounts. But even
that is not the end of the story, for quantum mechanics experimentally
conflicts with that concept of locality, and so quantum non-locality must be
accommodated in some way. (Hesse 1961 is a classic telling of this tale.)

By now, three points are indicated by this story: first, questions of physical
salience, here in the form of the principles of locality, are genuine,
controversial components of scientific enquiry. Second, such principles are
historically contingent, changing in step with major advances in physics.
Third, such changes are ultimately settled by, and epistemically justified by,
empirical success: one of the things that we learn in a scientific revolution
is a set of criteria of physical salience for explanation appropriate to the
new domain of enquiry. Put this way, we see principles of physical salience as
part of what Kuhn called the ‘disciplinary matrix’ in the _Postscript_ to the
second edition of (1962), or what Friedman (2001) refers to as the ‘relative,
constitutive a priori’. Though changes in the principles change wholesale what
theories are even candidate explanations, we don’t infer any catastrophic
incommensurability here: as we said, innovations in physical salience are
grounded in empirical success, like all other scientific knowledge.

So we have a general answer to the problem raised earlier. _How can a
derivation of spacetime from a non-spatiotemporal theory ever be physically
salient?_ Well, it cannot satisfy the standards of physical salience that
apply to theories with classical spacetime, but we should expect a non-
spatiotemporal theory to require new standards. And so the real question is
what are those new principles? Like Friedman, we see that question, and the
development of such new principles as a foundational, interpretational,
conceptual—hence _philosophical_ —endeavor. We shall elaborate on how such a
project is to be conducted in §7. We will see throughout the book how this
endeavor is ineliminably philosophical in the different approaches to QG. For
now we want to illustrate the problem with an example.

##  5 Non-commutative geometry

Of necessity, this section is somewhat more technical than the others, and
could be skipped by those not requiring a concrete illustration of how
interpretational considerations come into play in elevating a formal
derivation into one (potentially) having physical salience. It elaborates an
example of a non-spatiotemporal theory already given, to show how one might
come to view it as a theory from which spacetime emerges.

We start with familiar, commutative geometry, for which $xy=yx$, in a smooth
manifold of points; let it be 2-dimensional for simplicity.888This section is
based on Huggett et al. (forthcoming). See also Lizzi (2009) for a more
mathematical survey. Consider polynomials $\mathcal{P}(x,y)$ of $x$ and $y$.
These are ‘fields’, meaning that they return a numerical value at each point
$(x,y)$. They form an algebra with respect to multiplication: this just means
that when you multiply two polynomials together, the result is another
polynomial. Moreover, because $xy=yx$ we have that
$\mathcal{P}(x,y)\mathcal{Q}(x,y)=\mathcal{Q}(x,y)\mathcal{P}(x,y)$, so that
the algebra is commutative. (Check with $\mathcal{P}(x,y)=xy$ and
$\mathcal{Q}(x,y)=x^{2}+y^{2}$ if you like.)

It may seem like a rather uninteresting structure, but in fact such algebraic
relations alone contain geometric information about the space: in this case,
that it is smooth, that it is 2-dimensional, and whether it is open or closed.
This fact is shown by the important Gelfand-Naimark theorem (1943), which is
the foundation of ‘algebraic geometry’. Indeed, the whole structure of
differential geometry can be recast in algebraic terms. (An interesting
application is Geroch’s (1972) formulation of general relativity as an
‘Einstein algebra’; discussed by Earman (1989, §9.9) as a possible response to
the hole argument.)

For a mathematician, the question of what happens when the algebra is
‘deformed’ so that it is no longer commutative is irresistible: so one sets
$xy-yx=i\theta$ and sees what happens. (And similarly in spaces of any
dimensions.) Surprisingly, one finds that the structure necessary to cast
geometry in algebraic terms remains (at bottom, one can still define a
derivative on the algebra, in terms of which the other structure is defined).
Moreover, the Euler-Langrange equation and Noether’s theorem do not require
commutativity, and so the structure of modern physics is preserved, even in
such a ‘non-commutative spacetime’—in a purely algebraic formulation.

But suppose such a physics were correct: how could it explain spacetime as it
appears to us? Specifically, how are we to understand events localized in
space in terms of an abstract algebra? When the algebra is commutative, the
Gelfand-Naimark theorem lets us interpret the elements as fields,
$\mathcal{P}(x,y)$ related to regions of space; but what about the non-
commutative case? The question is just that which has concerned us in this
chapter (and indeed the whole book): how can we derive the appearance of
classical spacetime from a non-spatiotemporal theory, in a physically salient
way?

The obvious thing to try is to (a) interpret $x$ and $y$ not as elements of an
abstract algebra, but as fields in an ordinary plane: taking the value of the
$x$ and $y$ coordinates at any point $(x,y)$. Then (b) define a new binary
operation, $\star$, such that $x\star y-y\star x=i\theta$. Then (c) construct
the algebra of polynomial fields, but with $\star$-multiplication instead of
regular (point-wise) multiplication. Indeed, this is exactly how one typically
proceeds in non-commutative geometry: in one formulation, the operation is
‘Moyal-$\star$’ multiplication999$\phi\star\psi\equiv\phi\cdot\psi\
+\sum^{\infty}_{n=1}(\frac{i}{2})^{n}\frac{1}{n!}\theta^{i_{1}j_{1}}\dots\theta^{i_{n}j_{n}}\partial_{i_{1}}\dots\partial_{i_{n}}\phi\cdot\partial_{j_{1}}\dots\partial_{j_{n}}\psi$.,
and the fields form the ‘Weyl representation’ of the algebra.

The algebra of the fields with respect to $\star$ will be that of the abstract
non-commutative algebra, and now we have referred that algebra to objects in
an ordinary manifold. In particular, one could talk about the local region in
which such-and-such a field has values less than 1, say. Indeed, one might now
wonder whether we should throw away the abstract algebra, and just treat
physics in ‘non-commutative geometry’ as really physics in commutative
geometry, but with an unfamiliar multiplication operation. In other words,
wonder whether classical spacetime needs to be recovered at all?

Huggett, Lizzi, and Menon (forthcoming) argue that indeed it must be, for the
Weyl representation has formal representational structure that exceeds its
meaningful, physical content. In particular, the concept of a region with an
area smaller than $\theta$—a forteriori that of a point—is undefinable in the
theory. This can be seen in a couple of ways, but for instance the attempt to
measure positions more accurately leads to unphysical results. The conclusion
is that, although the Weyl representation contains points and arbitrarily
small regions, they are purely formal, and do not represent anything real:
non-commutative geometry—even the Weyl representation—is physically
‘pointless’.

As a result, we cannot understand a point value of the Weyl fields as having
any physical meaning. Rather we need to understand the fields as complete
configurations: the unit of physical meaning for a field in non-commutative
space is the _function_ from each point to a value,
$\mathcal{P}:(x,y)\to\mathbb{R}$; not its _value_ ,
$\mathcal{P}(\mathrm{x,y})$ at any particular point $(\mathrm{x,y})$. But the
full configuration is equivalent to the place of the field in the abstract
algebra, and so we are back to the question of deriving locality.

Here is one way to proceed, using an ansatz proposed by Chaichian, Demichev,
and Presnajder 2000 (discussed further in Huggett et al. forthcoming). They
propose that an ordinary, commuting field—the kind observed in classical
spacetime—be related to a Weyl field $W(x,y)$ by an operation of ‘smearing’.
One multiplies $W(x,y)$ by a $\theta$-sized ‘bell function’ about $(X,Y)$, and
integrates over the Weyl space coordinates $x$ and
$y$.101010$\Omega(X,Y)\propto\int\big{(}e^{-((X-x)^{2}+(Y-y)^{2})/\theta}\cdot
W(x,y)\big{)}\ \mathrm{d}x\mathrm{d}y$. The result is a new field
$\Omega(X,Y)$. Extrapolating from this ‘CDP ansatz’, the result of smearing is
to introduce classical space into the theory. $W$ lives in Weyl space, whose
status, we argue, is only that of a formal representation of the fundamental
algebra, while $\Omega$ should be interpreted as living in the physical space
that we observe. We thus interpret smearing as relating a _function on one
space_ to a _value on another space_ : it relates the non-commuting field $W$,
represented as a function over Weyl space points $(x,y)$, to the _value_ of an
observed, commuting field $\Omega$ at physical space point $(X,Y)$. That it
takes a function to a value is just mathematics; that it relates Weyl and
physical spaces is a substantive physical postulate.

However, it still makes no sense to consider $\Omega$ in regions smaller than
$\theta$: we have in fact erased the unphysical information at such scales by
smearing $W$. So strictly a single coordinate pair $(X,Y)$ does not label a
physical point. Rather, the proposal is that these smeared fields are
approximated by observable fields over regions greater than $\theta$; thereby
formally deriving the latter, spatially localized objects from the former,
purely algebraic objects. (In this case, we have smearing as ‘aggregating’, in
a very loose sense.) Of course, in our existing theories, fields live in a
full commuting spacetime, but that is an extrapolation from our actual
observations of fields, which are always over finite regions, to date larger
than $\theta$. On the proposed interpretation, then, any information contained
in $\Omega$ about regions less than $\theta$ is not only unobserved, but
unphysical, surplus representational ‘fluff’.

Now, nothing in the theory forces this picture as a physical story—it is
merely an interpretational postulate. (Though we claim that it is conceptually
coherent.) However, it has empirical consequences: the dynamics magnifies the
$\theta$-scale non-commutativity to observable scales (e.g., Carroll et al.
2001). If those predictions are successful, then we have evidence that the
underlying non-commutative field theory _and_ the interpretational postulate
are correct. Imagining that situation then, we claim that the situation is
exactly analogous to that of the Newtonians regarding action at a distance.
That is, we would be justified in accepting the CBP ansatz and our
interpretational postulate as novel principles of physical salience: they
regulate what constitutes a physically salient derivation in the theory. In
both cases, the final ground is the empirical success of the theory.

So it should be clear how the example illustrates our points about physical
salience and our scheme proposed above. In the first place we have argued that
non-commutative geometry is non-spatial, in the sense that it is ‘pointless’,
and so must be understood as a purely algebraic theory. Then we have
explicated a possible formal derivation of localizable fields from this more
fundamental theory. And finally, we have sketched a scenario in which such a
derivation leads to successful predictions, and hence to the conclusion that
the formal derivation is physically salient, in fact _explaining_ the
appearance of localized fields, and ultimately classical spacetime. Of course,
we highlight that this discovery constitutes a _change_ in what derivations
‘deserve to be regarded as physically salient (rather than merely
mathematically definable)’, to paraphrase Maudlin.

This is the pattern that we will see in more detail in the examples of the
following chapters, and to which we will return in the conclusion. In the
following two sections we will investigate further what is achieved by such a
derivation (§6), and use our account of physical salience to indicate what
philosophy should aim to do when engaging emerging physics such as QG (§7).

##  6 Spacetime functionalism

The schema that we presented for a physically salient derivation relates
closely (but not exactly) to Lewis’ (1972) account of functional
identification. Since ‘spacetime functionalism’, of various varieties, has
been recently discussed it is worth drawing the comparison, to better
understand the nature of the proposed emergence. Suppose, in idealization,
that a theory $T$ is formulated as a postulate ‘$T[t]$’. $t$ represents what
was traditionally called the ‘theoretical’ terms, though we prefer
‘troublesome’ (Walsh and Button 2018, §3.1): the idea is that these are the
new terms introduced by the theory, and the ‘trouble’ is the question of how
they garner meaning. ‘$T[\cdot]$’ also involves (traditionally)
‘observational’, or (according to Lewis) ‘old’, or (with Walsh and Button)
‘okay’ terms, and Lewis proposes that the $t$ are defined in terms of them by

$t=\rotatebox[origin={c}]{180.0}{$\iota$}x\ T[x].$ (5)

That is, ‘the $t$ are (if anything) the extant, unique things that satisfy the
theory postulate’. The $t$ are thus defined in terms of their nomic relations
to one another and to the okay terms—i.e., in terms of their ‘functional’
relations—and so (5) is a functional definition.111111Lewis proposes in
passing that the actual definition be modified to allow for _approximate_
satisfaction of $T[\cdot]$. In our opinion that is always going to be the case
in actual theories, so this modification is not optional but required, and his
discussion is a significant idealization. The harder question of how exactly
the modification is to be implemented is not carefully addressed by Lewis. As
a result, the terms $t$ are rendered semantically okay (though they may remain
metaphysically problematic).

Suppose that we also hold a postulate $R[r]$, where $r$ and $t$ do not
overlap, so that our acceptance $R$ does not depend on our views on the
troublesome terms of $T$.121212Acceptance of a theory requires that it be
meaningful, hence acceptance of $R[r]$ requires that the $r$ be referential,
and thus that any troublesome $r$ can be functionally defined in terms of the
okay $r$ as in (5), mutatis mutandis. We accept this assumption for Lewis’
cases, but we will see that things are more complex in the case of QG. Further
suppose that we come to believe $T[r]$. Now,

$T[r],\ t=\rotatebox[origin={c}]{180.0}{$\iota$}x\ T[x]\ \vDash\ r=t,$ (6)

so, by definition of the $t$, $T[r]$ deductively entails the identity of the
objects of $R$ and $T$. Lewis’ point is the epistemic one that such functional
identifications are thus not inductive: given a functional definition, once
one accepts that the objects of $R$ play the same roles as those of $T$,
_logic and meaning alone commit one to accepting their identity_.

Lewis offered electromagnetic waves and light as an example of the scheme, but
of course his point was that ‘when’ neuroscience showed that neural states
played the functional roles of mental states, then they would—as a matter of
logic and definition—be identified. The subject of this book also broadly fits
Lewis’ scheme: theory $T$ is our spacetime theory assumed by QFT and
relativity theory, while $R$ is a theory of QG. Denote them $ST$ and $QG$,
respectively, and use $ST$ to functionally define any troublesome spacetime
terms. Then, according to the schema of §4, a physically salient derivation of
$ST$ from $QG$ shows that ‘aggregates’ of $QG$, described using its terms $q$,
satisfy $ST[\cdot]$—that they indeed play the functional roles of the objects
of $ST$. So the identity follows. However, there are differences to Lewis’
functionalism, which we will explain presently.

Now, how to turn Lewis’ scheme into a concrete plan for the functional
reduction of spacetime? The spacetime functionalism recently introduced by Lam
and Wüthrich (2018, forthcoming) is based on the general scheme in the spirit
of Kim (2005, 101f) according to which a functional reduction of higher-level
properties or entities to lower-level properties or entities consists in two
necessary and jointly sufficient steps:131313Kim’s model involves three steps,
where the second is to identify the entities in the reduction base that
perform the role at stake, and the third is to construct a theory explaining
how these fundamental entities perform that role. We subsume these two steps
in our second stage.

  1. (FR1)

The higher-level entities/properties/states to be reduced are
‘functionalized’; i.e., one specifies the causal roles that identify them,
effectively making (5) explicit.

  2. (FR2)

An explanation is given of how the lower-level entities/properties/states fill
this functional role, so that we come to accept $T[r]$.

If these two steps are fulfilled, then it follows that the higher-level
entities/properties/states are realized by the lower level ones.

Applying the template of functional reduction to the case of the emergence of
spacetime in QG, the two steps above become:

  1. (SF1)

Spacetime entities/properties/states, $s$, are functionalized by specifying
their identifying roles, such as spacetime localization, dimensionality,
interval, etc. Effectively, one makes explicit
$s=\rotatebox[origin={c}]{180.0}{$\iota$}x\ ST[x]$.

  2. (SF2)

An explanation is given of how the fundamental entities/properties/states,
$q$, postulated by the theory of quantum gravity fill these roles, so that we
come to accept $ST[q]$.

Again, if these two steps are fulfilled, it follows that the (perhaps
aggregated) QG entities/properties/states _are_ the spacetime
entities/properties/states. In the following chapters, after explaining each
theory of QG and its conceptual foundations, we will follow this scheme in our
discussions: on the one hand describing the functional roles of spacetime
entities/properties/states, and on the other showing how the theory of QG
proposes that those roles are played. Of course, given the evidential state of
QG, we do not claim that that these proposals are correct: we only describe
how the theories _may_ functionally reduce spacetime, not—as far as we
currently know–how they _do_.

Several remarks are in order regarding the functionalist approach to
spacetime. First, it should be made clear that we take emergence and reduction
to be compatible with one another, and hence functional _reduction_ may serve
as a template to explain the _emergence_ of a higher-level feature, i.e., the
fact that higher-level entities exhibit novel and robust behaviour not
encountered or anticipated at the more fundamental level.141414As restated
many times in our earlier publications, and in agreement with what we take to
be the consensus in philosophy of physics as stated, e.g., in Butterfield
(2011a, b) and Crowther (2016, §2).

Second, there is a sense in which a functionalism about spacetime must start
from a broader conception of functional reduction than is usual in the
familiar functionalisms in the philosophy of mind or the philosophy of the
special sciences. There, a mental or biological or other higher-level property
is understood to be determined by—indeed, usually _identified with_ —its
_causal role_ within the relevant network such as the network of mental or
biological activities. If in spacetime functionalism the roles are still
supposed to be _causal_ , then a much broader notion of ‘causal’ must be at
work, one that does not in any way depend on the prior existence of spacetime.
As it is not clear what that would be, it is preferable to formulate a notion
of functionalism devoid of any insistence that the functional roles be causal.

Third, the central claim of spacetime functionalism is that it is sufficient
to establish only the functionally relevant aspects of spacetime. In
particular, it is therefore not necessary to somehow derive relativistic
spacetime in its full glory and in its every aspect in order to discharge the
task. Naturally, this raises the question of what these functionally relevant
aspects of spacetime are—the task of (SF1). As we will see in the following
chapters, different approaches to QG take different stances on what functions
are to be recovered, though broadly speaking, all aim to recover functions
sufficient for the empirical significance of basic metrical and topological
properties. Our stance will be that the list of functions cannot be determined
_a priori_ from conceptual analysis of classical spacetime theories, but by
the twin demands of the empirical, and of the resources of the proposed
reducing theory. In short, part of the work of each chapter will be to
identify the spacetime functions recovered in the different approaches, and
indicate how they relate to observation.

Fourth, the scheme permits a form of ‘multiple realizability’, as is typical
of functionalism also in the philosophy of mind or the philosophy of the
special sciences: (SF2) allows that different (kinds of) fundamental entities
might play one and the same functional role, i.e., that the ‘realizer’ of
spacetime might have been by something other than what it in fact is. This
liberal stance spurs a concern that functionalism is too weak a condition to
secure the emergence of spacetime, that the true nature of spacetime is not
exhausted by its functional roles, so that none of the mere functional
realizers could ever truly be spacetime. In particular, the worry continues, a
rash reliance on functionalism misses the qualitative nature of spacetime—some
kind of spacetime ‘qualia’, as it were—and it is precisely such qualitative
features that make spacetime what it is, and which cannot be recovered by mere
functional realization. However, the case of spacetime is disanalogous to that
of mind: we agree with Knox (2014) who states that where “the fan of qualia
[in the philosophy of mind] has introspection, the fan of the [spacetime]
container has only metaphor” (16), and with Lam and Wüthrich (2018) who agree
that the “nature and status of the evidence in favour of [mental] qualia may
be equivocal, but the alleged ineliminable intrinsically spatiotemporal but
ineffable quality of spacetime substance remains positively elusive” (43f). We
conclude with them that the qualia worry in this form gets little if any
traction in the spacetime case.151515We also concur with Lam and Wüthrich
(2018) in their rejection of the version of this concern articulated in Ney
(2015), who worries that if the fundamental entities are not already
appropriately (spatio)temporal in their nature, they cannot ‘build up’ or
constitute spacetime as they are not the right kind of stuff (see also Hagar
and Hemmo 2013). As diagnosed by Lam and Wüthrich, advocates of this worry
seem to rely on an unreasonably narrow concept of constitution. We might also
object that if we surrendered to this worry, there would be no principled
reason to think that it would not also annihilate all other cases of presumed
emergence and amount to an unyielding dualism.

Borrowing a distinction from Le Bihan (2018) between a “hard” and an “easy
problem” of spacetime emergence, spacetime functionalism amounts to the denial
that there is a hard problem of an unbridgeable explanatory gap between the
fundamental, non-spatiotemporal and the emergent, spatiotemporal realm. For
the functionalist, what is to be shown—by a physically salient derivation—is
how the fundamental degrees of freedom can collectively behave in ways such
that they play the required spacetime roles. And nothing more. No special
character, or essence, or metaphysical nature need be accounted for.
Functional identification requires no ‘luminosity’ of light beyond the
behavior of electromagnetic waves, or ‘consciousness’ beyond the functioning
of neurons. Or in our case, no special ‘spatiotemporality’ that the non-
spatiotemporal could never obtain. Once one has shown that the non-
spatiotemporal plays the roles of the spatiotemporal—and so _is_ the
spatiotemporal—no more need be said: one has a full scientific account of the
emergence of spacetime, and no ‘explanatory gap’ remains.

Fifth, functionalism shows how the goal of reduction can be the scientific
explanation of the functional roles of higher level entities/structures/states
by lower level entities/structures/states (and to nothing more). But, it is
debatable to what exactly spacetime functionalism is ontologically committed:
substances, relations, entities, structures, states, or something else. We
will not further pursue this debate as we believe it to be orthogonal to the
concerns of this book. Thus we hope that the reader will forgive our switching
between speaking of the spatiotemporal as if it were an entity, or a
structure, or a state, or something different yet again. We simply aim to
avoid torturing English more than necessary, and no deep philosophical
commitment should, for instance, be read into our using ‘spacetime’ as a noun.

Sixth, we are far from the first to suggest functionally defining space or
spacetime. DiSalle (2006, chapter 2) reads Newton’s Scholium to the definition
in much this way (though Huggett 2012 disagrees). Functionalist strategies
have also become very visible in the philosophy of non-relativistic quantum
mechanics, where Wallace (2012) deploys it in his defence of an Everettian
interpretation and Albert (2015, Ch. 6) in support of wave function monism.
Those latter applications differ from ours because they are concerned with
recovering three-dimensional physical space. In contrast, spacetime
functionalism in QG is commissioned with functionally recovering 4-dimensional
spacetime, and so relates to work by Knox (2013, 2014, 2019) in the context of
classical spacetime physics. For her, something ‘plays spacetime’s role’ and
thus _is_ spacetime “just in case it describes the structure of inertial
frames, and the coordinate systems associated with these” (2014, 15). In GR,
the metric field performs spacetime’s role in this sense and thus is
identified with spacetime by her. As the metric may itself not be fundamental
but instead emerge from the collective behavior of more fundamental degrees of
freedom, she explicitly leaves open the possibility that the realizers of
spacetime’s functions may themselves not be fundamental (Knox 2013, 18). As
the relationship between the fundamental degrees of freedom and the emergent
spacetime realizer is left untouched by Knox’s inertial frame functionalism,
the latter does also not shed any light on it.161616Cf. Lam and Wüthrich
(2018, 40) and Lam and Wüthrich (forthcoming, §3) for a more detailed
discussion of inertial frame functionalism and how it relates to our project.

Seventh and finally, there is an important but subtle difference in the
application of Lewis’s scheme to QG from that in the cases he has in mind.
Suppose we accept a spacetime theory $ST[s]$, where whatever it is that
performs the spacetime functions is denoted by the troublesome terms, $s$.
These, following Lewis, we take to be defined by

$s=\rotatebox[origin={c}]{180.0}{$\iota$}x\ ST[x].$ (7)

The okay terms appearing in $ST[\cdot]$ would refer to matter of various
kinds, its relative motions and point-coincidences: so, for instance, the
metric in GR might be defined locally in terms of its role in determining
motions under gravity or scattering amplitudes. We think that this part of
Lewis’ picture—which corresponds to (SF1)—fits our cases well. But what about
the second part of his scheme, involving $R[r]$? Although the result is still
a functional identification, its significance has shifted somewhat, as we
shall now explain.

Butterfield and Gomes (2020a; 2020b) analyze recent proposals for spacetime
functionalism in explicitly Lewisian terms. They emphasize, as we have, that
in Lewis’ scheme theoretical identification follows by definition alone (once
the $r$s are known to play the role of the $t$s), and that functional
identification is a species of reduction. But they also show how various
spacetime and temporal functionalisms follow the ‘Canberra plan’, according to
which the troublesome $t$s are not only defined by $T$, but are also
‘vindicated’ by their functional identification as $r$s. For instance, as
mental states, perhaps, turn out to be neural states so, in their examples, a
temporal metric might be identified with purely spatial structure; then, if
neural states or spatial structures are on a firm (or firmer) ontological
footing than mental states or time, the identifications show that the latter
are equally well grounded. They are, that is, vindicated against any
metaphysical suspicions raised against them. That vindication is not by itself
achieved by the functional definition (5) of the $t$s; that merely makes the
terms referential, so that they can be meaningfully employed. Put another way,
(FR1) alone does not vindicate the mental, for instance; (FR2) is also needed,
to show how the mental is part of the physical.171717Or put yet another way,
the $t$ are often troublesome both semantically and ontologically: the
functional definition takes care of the first problem, while the functional
identification takes care of the second. When we use ‘troublesome’ we always
mean semantically. Regarding these cases, we are in agreement with Butterfield
and Gomes emphasis of this important distinction, and its applicability to the
cases that they discuss.

However, in our cases, for which $R$ is some $QG$, the second step, while
still involving a functional identification, does not follow the Canberra
plan, because the troublesome terms, $q$, of $QG$ are non-spatiotemporal, and
so on a _weaker_ , not firmer, footing—the ontological and semantic correlate
of empirical incoherence.181818Butterfield and Gomes do not claim otherwise,
and indeed acknowledge that QG will look different (2020b, 3). Ontologically,
as we have discussed, our physical and metaphysical categories assume
spatiotemporality, and so the natures of the $q$ are mysterious. Semantically,
we can expect an attempt to functionally define the $q$s as
$q=\rotatebox[origin={c}]{180.0}{$\iota$}x\ QG[x]$ to fail. Lewis’ scheme for
functional definition requires that a theory have sufficient okay terms to
_uniquely_ define the troublesome ones: if many collections of terms satisfy
the putative definition, then it fails to establish reference. But that is
what one expects in a theory that breaks from established categories as
radically as a non-spatiotemporal one; the terms that we take to be okay are
systematically spatiotemporal in some way, and so are expected not to appear
in $QG$. And indeed, we contend that the theoretical concepts of the theories
we consider in this book cannot be defined without appeal to spatiotemporal
concepts external to the basic formulation of the theory.

Given this situation, the significance of functional reduction is different
from the way in which Lewis (and Kim) proposed. Rather than following the
Canberra plan of vindicating spacetime objects by reduction, in our approach
to QG things are _reversed_ : the non-spatiotemporal objects of $QG$ are
vindicated via their identifications with spatiotemporal objects. Clearly this
approach only works to the extent that the spatiotemporal is itself on a firm
ontological footing, which of course is a topic of endless debate. To skirt
such debates in this book we will remain as neutral as possible, and not take
any stand on the metaphysical nature of spacetime features such as topology or
metricity, so that our conclusions remain valid for anyone who accepts them
under whatever interpretation.

Within Lewis’ framework, the vindication of the $q$ works as follows. Suppose
that non-spatiotemporal $QG[q]$ has been proposed. As explained, the $q$ are
semantically troublesome and ontologically suspect. Moreover, until we accept
that a derivation of (at least a fragment of) $ST$ is physically salient, we
have no empirical grounds for accepting $QG$. Such a derivation will provide,
not only grounds for $QG$, but also define and vindicate the $q$. Introducing
the ‘aggregate operator’ $\alpha(\cdot)$, according to our schema, when we
have a physically salient derivation of spacetime properties, then we accept

$\alpha(q)=\rotatebox[origin={c}]{180.0}{$\iota$}x\ ST[x].$ (8)

In conjunction with $ST[s]$ this entails that

$\alpha(q)=s$ (9)

more-or-less as for Lewis. However, the reversal of the Canberra plan makes
several things different.

First, semantics. As noted, the $q$ were not antecedently defined, but now can
be through their—or rather the $\alpha(q)$’s—role as spacetime
entities/structures/states. In other words, (8) is in part definitional of the
$q$: the troublesome non-spatiotemporal terms of $QG$ can only be defined with
reference to spatiotemporal terms not native in $QG$. Moreover, (8) only
succeeds in defining the $q$ if in physical fact they play the ascribed roles,
and do merely mimic them formally; something that the physical salience of the
derivation will secure.191919(8) is not purely definitional, since it also
also involves an existential commitment that the $q$s exist. And it need not
fully define $q$; we also still have that
$q=\rotatebox[origin={c}]{180.0}{$\iota$}x\ QG[q]$ by definition. Second,
ontology. The $q$s are placed on a firm ontological footing—are
vindicated—when we accept that the $\alpha(q)$ are in physical fact those
entities/structures/states that play the spacetime role.202020In Huggett and
Wüthrich (2013, 284), we described this approach to vindication as physical
salience flowing down to the $q$ ‘from above’. Once again, acceptance of the
physical salience of the derivation secures just that.

Finally, epistemology. In Lewis’ scheme, we have independently accepted
theories of, say, neuronal and mental states, and _later_ discover that they
play the same functional roles, entailing that they are identical. In our
case, the acceptance that $QG$’s objects (or rather their aggregates) play the
same roles as $ST$’s objects, and hence are identical with them, is
_simultaneous_ with our acceptance of $QG$. In general terms, the evidence for
$R[r]$ is no longer antecedent (or independent) of the evidence for $T[r]$,
but rather the very same evidence. As such the epistemic calculus is
different. In one case, observations of neuronal states can be made
independently of mental states, and we only have to show that they perform the
relevant functions: producing suitable behaviors, for instance. In the other,
observations are not independent of spacetime states, and have to support both
the truth of a theory of QG, and that its objects perform the right functions.
To give evidence, that is, that the formal derivation of those functions is
indeed physically salient. As a result, although the deductive logic is the
same, the empirical inference to the premises of the identification is
different, and indeed weaker. However, as we say, it is of the normal
empirical kind, and we fully expect it to be made for a successful theory of
QG. There is no special ground for skepticism.

So much for the functionalism that lies behind the investigations of this
book. But why is finding a functional reduction in any way a philosophical
task, rather than one for physics?

##  7 The role of philosophy in physics

As we noted, the theories that we plan to investigate are all speculative at
present, faced with considerable formal and empirical uncertainties. So what
can we hope to learn from a philosophical enquiry into something that is at
worst likely false, or at best a work in progress? We see the situation as
characteristic of emerging fundamental physics (and perhaps other sciences).
The process of discovery takes place along various fronts: obviously, new
empirical work constrains theory and requires explanation; also obviously, new
mathematical formalisms are tried out and explored; less obviously, but just
as importantly, conceptual analysis of the emerging theory is undertaken. In
particular, we want to stress that this last kind of work is carried out
concurrently with the empirical and theoretical. One should not view
interpretation as something that merely happens after an uninterpreted formal
structure is presented, but as an inextricable aspect of the process of
discovery. As such, it is something that has to be carried out on inchoate
theories, in order to help their development into a finished product.

We claim that this view is supported by the historical record: we have in fact
already seen this for Newtonian gravity. But one can equally well point to the
absolute-relative debate in the development of the concept of motion, or 19th
century efforts to come to grips with the physical significance of non-
Euclidean geometry. These debates did not wait until after a theory was
developed to clarify its concepts; rather they had to be carried out
simultaneously, as an integral part of the development of the theory (see
DiSalle (2006)). Of course we are hardly the first to realize that such
philosophical issues have to be addressed together with the empirical and
theoretical ones. Many of Kuhn’s (1962) arguments illustrate this point, and
more recently it is a major theme of Friedman (2001). But while we agree with
their focus on philosophical, conceptual analysis as an essential part of
theory construction, we don’t intend to get involved in issues involving the a
priori or incommensurability, instead we want to emphasize the practical role
for analysis in the development of QG.

In the search for a new fundamental theory, the goal is—as it was for Einstein
and for Newton—a new formalism plus an interpretation that connects parts of
the formalism to antecedently understood aspects of the physical world,
especially to the empirical realm. That is, an interpretation of how the more
fundamental plays the functional roles of the less fundamental. And of course
that means undertaking the project that we have been talking about in this
section, of deriving spatiotemporal predictions from theories of QG. But one
never simply co-opts or invents formalism without some eye on the question of
how it represents existing physics of interest; and as the formalism is
developed it becomes possible to see more clearly how and what the new
formalism represents. Addressing this question is of on-going importance for
finding the right formalism for the area under study. Moreover, constructing
such a formalism does not typically proceed in a monolithic fashion; instead
different fragments of theory are proposed, investigated, developed or
abandoned. For example, think of the development of the standard model of QFT
from the early days of quantum mechanics. So the analysis of concepts of the
new theory in terms of existing physics is often faced with a range of half-
baked theories and models. All the same, lessons about how a more developed,
less fragmented theory can be found depend on asking how the fragments
represent known physics—the answers are potential clues to how the finished
product could do so.

We believe that contemporary QG should be thought of in just this
way—certainly the fragmentation is real! Our primary goal is to look at a
range of the half-baked fragments and ask how they connect to spatiotemporal
phenomena. Since they do not do so in a familiar way, in terms of a continuous
manifold of points, the question becomes ‘how does spacetime emerge from the
underlying physics?’. We hope, therefore, that by concentrating on the
question of emergence, aside from all the other issues involved in the search
for a theory of QG, we will be performing a service to physicists working in
QG, by focussing their attention on what is already known—and reminding them
that success depends on making it part of the search. Naturally, we do not
expect to find solutions of the order of Newton or Einstein! Indeed, a lot of
what we shall do is draw out answers already given by physicists; we believe
that careful philosophical analysis of these answers can help clarify them to
reveal strengths and weaknesses, and hence aid progress. (Moreover, because we
are focussed on this quite narrow issue, we can survey a wider range of
approaches than most physicists actively study, and so provide a helpful
overview of the topic.) And hence we believe that in the examples we will
consider there are important clues for the development of QG which
philosophical analysis can reveal.

##  8 The plan for the book

Thus, our primary aim is to see how spacetime disappears and re-emerges in
several approaches to QG, and to show how this is not just a technical issue
for physicists to solve, but instead elicits numerous foundational and
philosophical problems. As we work through three such approaches—causal set
theory (CST), loop quantum gravity (LQG), and string theory, which were all
briefly introduced in §2—, we bring these philosophical issues to the fore and
will concentrate our discussion on them.

It is common to divide approaches to QG into those which start out from GR and
attempt to convert it into a quantum theory of gravity in different ways and
into those departing from the standard model of particle physics and aim to
add gravity to the other three forces of the standard model. In the former
approaches such as CST and LQG, we would not expect the resulting theories to
fold in the physics of the standard model, whereas the latter, such as string
theory, will presumably deliver more encompassing, unifying theories. It is
clear that either way, a theory of QG needs to address how the geometrical
degrees of freedom of spacetime interact with the matter degrees of freedom
present in the world. But it is also clear that both kinds of degrees of
freedom may well look very differently from what we are used to from other
theories.

The first two chapters after this one focus on CST. Chapter 2 introduced the
basic kinematic axiom of the theory and shows how in it at least space
disappears rather radically from the fundamental ontology, but also that
temporal aspects do not all survive. This raises the immediate question of the
relationship between the fundamental ontology of causal sets with that of
relativistic spacetimes, a question we start to address in chapter 2. Although
some functions of space can tentatively be recovered, what is needed is a more
systematic understanding of how causal sets generically give rise to worlds
which appear to be spatiotemporal in ways described, to good approximation, by
GR. The way in which this ‘derivation’ of spacetime is attempted in CST is
sketched and discussed in chapter 3. In this chapter, we will discuss the role
played by introducing a dynamics for the theory. We will argue that the
emergence of spacetime in CST is closely tied to deeply philosophical
questions regarding the metaphysics of space and time.

Chapter 4 and 5 turn to LQG, retracing the disappearance and emergence of
spacetime in this approach. Just as CST, LQG builds a research program around
what it takes to be GR’s central lesson. In the case of LQG, this is the
insight that GR postulates a truly dynamical spacetime, interacting with other
fields. The demand is encoded in the theory’s general covariance. LQG seeks to
articulate a theory of QG by delicately applying known quantization procedures
to a Hamiltonian formulation of GR. Chapter 4 chronicles and discusses whether
and, if so, how this approach leads to the disappearance of spacetime. Unlike
CST, it is time whose existence is much threatened in LQG than space. Chapter
5 seeks to understand how relativistic spacetime then emerges from the
fundamental theory, finding, again, close ties to philosophical questions.

Other approaches apply the strategies of perturbative QFT—so successful in
understanding the other forces—to quantize gravity. The technique calls for
starting a system in which the fields do not interact to build up a space of
states: a lowest, vacuum state, and states of discrete, particle-like
‘quanta’. Generally such a system is solved exactly, and the vacuum describes
an obvious classical state. Then one introduces a small interaction, and uses
approximation techniques to study the behavior of fields: especially the
scattering of quanta. This approach was applied to gravity early on: Minkowski
spacetime is a natural vacuum, and the gravitational field has quanta known as
‘gravitons’, very analogous to photons, the quanta of the electromagnetic
field. Indeed, quite a lot is known about the quantized gravitational field
through such methods, and this knowledge is taken as a constraint on a
successful theory of QG. However, divergences prevent the theory from being
generally applied; moreover, these divergence cannot be adequately resolved by
‘renormalization’ as they can for other QFTs.212121See Kiefer (2004, chapter
2) for a very nice survey of QFT of the gravitational field.

String theory works within this approach, but with one important tweak:
instead of quantized point like particles, it deals in quantized
1-dimensional, string-like objects. This, it appears, makes all the difference
to the finiteness of the theory. Chapters 6-9, address the emergence of
spacetime in string theory. Chapter 6 is a fairly technical introduction of
the theory, aimed at philosophers of physics: it aims to be more intuitive,
and more explicit about the conceptual and physical framework than physics
textbooks usually are. For those who have some familiarity with classical and
quantum field theory, it will tell you what you need to know about strings.
Chapter 7 deals with string ‘dualities’: some fascinating and powerful
symmetries that arise when space has an interesting topology (a cylinder,
say). We argue that they are the kind of symmetries are not merely
observational, but ‘go all the way down’, showing that string theory does not
possess, in its basic objects, familiar spacetime properties, such as definite
size or topology; it is for largely that reason that spacetime ‘emerges’.
Chapter 8 is again fairly technical, explaining and analyzing in some detail
the derivation of the Einstein field equation for gravity, from string theory.
This is a central part of emergence, for it derives the spacetime metric,
giving empirical content to spacetime geometry, and gives rise to GR. Finally,
chapter 9 draws on the material of the previous chapters to argue that indeed
spacetime emerges in string theory, how this happens, and what ‘principles of
physical salience’ are required.

The final, concluding, chapter draws on the results of the previous ones to
return to the question of this introduction. How can we see that the
derivations of spacetime that we have investigated are themselves physically
salient, and what principles can we extract from them that might be helpful in
the search for QG?

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