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# Introducing the ISE Methodology:A Powerful New Tool for Topological
Redescription

Daniel Grimmer  University of Oxford, Reuben College

###### Abstract

This paper introduces a powerful new tool for topological redescription, the
ISE Methodology. These tools allow us to remove and replace a theory’s
topological underpinnings just as easily as we can switch between different
coordinate systems. Aspirationally, these novel topological redescription
techniques can be used to provide new support for a roughly Kantian view of
space and time; Rather than corresponding to any fundamental substances or
relations, we can see the spacetime manifolds which appear in our theories as
merely being an aspect of how we represent the world. This view of spacetime
topology parallels the dynamic-first view of geometry as well as a Humean view
of laws; The spacetime manifolds which feature in our best theories reflect
nothing metaphysically substantial in the world beyond them it being one
particularly nice way (among others) of codifying the dynamical behavior of
matter.

A parallel publication (namely, Grimmer (2023)) will explicitly characterize
the power and scope of the topological redescription techniques offered to us
by the ISE Methodology. The modest goal of this paper is simply to introduce
the ISE Methodology by applying it to two example theories. Firstly, to
familiarize ourselves with these techniques, I will show how they can be used
to redescribe a spacetime theory via a Fourier transform. Secondly, I will
show how the exact same techniques can be used to redescribe a lattice theory
(i.e., a theory set on a discrete spacetime,
$\mathcal{M}_{\text{old}}\cong\mathbb{R}\times\mathbb{Z}$) as existing on a
continuous spacetime manifold,
$\mathcal{M}_{\text{new}}\cong\mathbb{R}\times\mathbb{R}$.

##  1 Introduction

Since at least Newton, our best physical theories have typically featured some
piece of smooth topological background structure (namely, a spacetime
manifold) either implicitly or explicitly. There are, of course, several ways
in which one can interpret these spacetime manifolds. Three broad classes of
interpretation are as follows: substantivalism, relationalism, and a third
view which I shall here call spacetime codificationism. Views of roughly these
kinds can be traced back to Newton, Leibniz, and Kant respectively.

A substantivalist views spacetime as a collection of topologically
interconnected spacetime points (perhaps understood anti-haecceitistically).
By contrast, relationalists would distance themselves from talk of spacetime
points, instead opting to view the spacetime in terms of spatiotemporal
relation. The third view, spacetime codificationism, holds that rather than
corresponding to some fundamental substances or relations in the world, we
ought to think of the spacetime manifolds which routinely appear in our best
scientific theories as merely being an aspect of how we represent the world.
This view of spacetime topology parallels a broadly dynamic-first view of
geometry as well as a roughly Humean view of laws; The spacetime manifolds
which feature in our best theories reflect nothing metaphysically substantial
in the world beyond them it being one particularly nice way (among others) of
codifying the dynamical behavior of matter.

However one understands the spacetime manifold, the fact remains that our
well-established physical theories near universally feature a designated
spacetime arena within which the world’s events happen (or, at least, are
described to happen). Given the seeming indispensability of the spacetime
manifold in physics, adopting a straightforward realist attitude naturally
pushes one towards some form of spacetime substantivalism. The relationalist
can, of course, push back in a variety of ways. Indeed, much has been written
in the philosophy of spacetime literature about the substantivalism vs
relationalism debate. The goal of this paper, however, is to introduce some
new mathematical techniques which can be used in support of a roughly Kantian
position: spacetime codificationism.

My target shall be the assumed indispensability of the spacetime manifold in
our physical theories. To this end, this paper will introduce some powerful
new tools for topological redescription (namely, the ISE Methodology). As a
piece of mathematical machinery, the goal of the ISE Methodology is to offer
us as many candidate spacetimes as possible, whatever we may think of them. In
particular, I claim that by using the ISE Methodology one can remove and
replace a theory’s topological underpinnings just as easily as one can switch
between different coordinate systems.

Let us suppose for the moment that the ISE Methodology meets its technical
aims (as Grimmer (2023) argues it does). Conceivably, one could adopt a wide
range of different philosophical responses to this sudden over-abundance of
alternative spacetime descriptions. My recommendation would be spacetime
codificationism, but what is this view exactly? Fortunately, a rough analog of
this view is already well known:111As mentioned above, the spacetime
codification view which I am advocating here is also intended to be analogous
to the dynamics-first view of geometry put forward by Brown (2005). Indeed,
this project can be seen as a response to Norton (2008) by extending the
dynamics-first view of geometry to a dynamics-first view of topology. See
Grimmer (2023) for further discussion. namely a broadly Humean view of the
laws of nature. On such a view, the laws of nature which appear in our
theories reflect nothing metaphysically substantial in the world beyond them
being one particularly nice way (among others) of codifying the dynamical
behavior of matter. One might then pick out the theory’s fundamental laws of
nature as being the its best axiomization (in some yet-to-be-specified sense
of best). Note the role that our capacity for logical re-axiomization plays in
supporting this view: Before we can pick out the best axiomization, we first
need access to the complete range of possible axiomizations. The ISE
Methodology aims to play an analogous role in a topological context.

It is beyond the scope of this paper to demonstrate that the ISE Methodology
meets its philosophical aims (i.e., to support a broadly Kantian view of
spacetime which is analogous to a roughly Humean view of laws). Indeed, it is
also beyond the scope of this paper to prove that the ISE Methodology meets
its technical aims (i.e., to give us a capacity for topological redescription
on par with our existing capacity for coordinate redescription and re-
axiomization). A strong argument for this second point can be found in Grimmer
(2023). The modest goal of this paper is merely to get the ISE Methodology on
the table, so to speak. Namely, my goal is simply to introduce these
topological redescription techniques by applying them to two example theories.

Sec. 2 will provide a preview of the ISE Methodology (whose three steps are
Internalize, Search, and Externalize, hence the initials). Secs. 3 and 4 will
then introduce the internalization and externalization processes respectively
by applying them to several example theories. For the sake of pedagogy, the
first example application will be rather tame: It ultimately amounts to taking
a Fourier transform. The real power of these techniques however is in their
scope and generality. In Sec. 5 I will use the exact same topological
redescription techniques to do something more interesting. I will there
redescribe a lattice theory (i.e., a theory set on a discrete spacetime,
$\mathcal{M}_{\text{old}}\cong\mathbb{R}\times\mathbb{Z}$) as existing on a
continuous spacetime manifold,
$\mathcal{M}_{\text{new}}\cong\mathbb{R}\times\mathbb{R}$. Finally, Sec. 6
concludes.

##  2 Previewing the ISE Methodology: Internalize, Search, Externalize

The first step of the ISE Methodology is internalization. Internalization aims
to divorce a theory’s dynamical and kinematical content from any assumed
topological background structure (i.e., the spacetime manifold,
$\mathcal{M}_{\text{old}}$). In spirit, this first step parallels the
algebraic approach to spacetime. A key difference, however, is that the ISE
Methodology can be applied to spacetime theories with no algebraic structure
what-so-ever (see Sec. 3). Indeed, the ISE Methodology does not make use of
any algebraic structure in the construction of new spacetime settings (nor in
the re-construction of old spacetime settings). Instead, new spacetime
settings are to be built solely from our ability to smoothly vary the theory’s
states (see Sec. 4).

This brings us to the second step of the ISE Methodology, searching for the
building blocks of a new spacetime setting. I will call these pre-spacetime
translation operations (PSTOs). PSTOs are a certain kind of smooth
transformation among the theory’s states. Roughly, they are a pair of Lie
groups which act smoothly on the theory’s states and which are, in a sense,
structurally indistinguishable from spacetime translations. (A technical
definition will be given in Sec. 4.) For example, consider translations not in
spacetime, but rather in Fourier space. This and other examples of PSTOs are
sketched below in Fig. 1. Before discussing these, however, allow me to first
quickly overview how these PSTOs can be used to create a new spacetime setting
for the theory in question.

The third step of the ISE Methodology is externalization. Having picked out
some PSTOs, externalization builds a new spacetime setting from them simply by
taking them seriously as spacetime translations. By construction, one’s chosen
pre-spacetime translation operations will end up becoming honest-to-goodness
spacetime translations in the new theory. In the Fourier example, the new
theory’s spacetime is effectively a copy of the old theory’s Fourier space.
Correspondingly, the new theory’s states and dynamics are naturally related to
those of the old theory by a Fourier transform (see Sec. 4).

Let us next see some examples of PSTOs. As a first example, consider smoothly
shifting the theory’s states with respect to the theory’s old spacetime
manifold. To make things concrete, consider a theory about a complex scalar
field, $\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathbb{C}$, with
$\mathcal{M}_{\text{old}}\cong\mathbb{R}^{2}$. In some fixed global coordinate
system, $(t,x)$, we might smoothly shift the theory’s states as
$\varphi_{\text{old}}(t,x)\mapsto\varphi_{\text{old}}(t-\Delta t,x-\Delta x)$
for any $\Delta t,\Delta x\in\mathbb{R}$. The action of this Lie group on
$\varphi_{\text{old}}(t,x)$ is shown in the first column of Fig. 1. In
general, the old theory’s spacetime translations trivially qualify as PSTOs.
As Grimmer (2023) proves, by externalizing these sorts of PSTOs one can easily
return back to the theory’s old spacetime framing. This, however, is not
compulsory. We can choose to externalize different PSTOs.

Figure 1: Three examples of pre-spacetime translation operations (PSTOs) are
shown. The first column shows a wave packet being smoothly translated with
respect to the old theory’s spacetime manifold. The second column shows a wave
packet being smoothly increased in frequency. The third column shows one way
of smoothly implementing a discrete translation operation on a lattice theory.
In all three cases, when viewed as a Lie group action on the theory’s states,
these smooth transformations are structurally indistinguishable from spacetime
translations. Hence, one can use the ISE Methodology to build a new spacetime
setting for the theory in question such that these PSTOs become honest-to-
goodness spacetime translations in the new theory.

As a non-trivial example of PSTOs, consider smoothly increasing the frequency
of every wave state described by the theory in question. Returning to the
above discussed, complex scalar field theory, we might smoothly shift the
theory’s states as $\varphi_{\text{old}}(t,x)\mapsto\exp(-\mathrm{i}\,\Delta
k\,x)\,\varphi_{\text{old}}(t-\Delta t,x)$ for any $\Delta t,\Delta
k\in\mathbb{R}$. The action of this Lie group on $\varphi_{\text{old}}(t,x)$
is shown in the second column of Fig. 1. Why do these pitch-shifting
transformations qualify as PSTOs? While they are not translations in the old
theory’s spacetime, they are nonetheless translation operations somewhere,
namely in the old theory’s Fourier space. Hence, when viewed as a Lie group
action on the theory’s states, these Fourier shift operations are structurally
identical to spacetime translations. Therefore, they qualify as PSTOs.

But what happens when we externalize these Fourier shifts PSTOs?
Externalization builds a new spacetime setting for the theory such that these
PSTOs become honest-to-goodness spacetime translations in the new theory.
Hence, by construction what were translations in Fourier space in the old
theory will become translations in spacetime in the new theory. The new
theory’s spacetime is effectively a copy of the old theory’s Fourier space. In
this example the ISE Methodology effectively implements a Fourier transform
(see Sec. 4).

After Secs. 3 and 4 demonstrate the ISE Methodology in this relatively
familiar context Sec. 5 will use the exact same topological redescription
techniques to do something much more interesting. Namely, I will use the ISE
Methodology to redescribe a lattice theory (i.e., a theory set on a discrete
spacetime, $\mathcal{M}_{\text{old}}\cong\mathbb{R}\times\mathbb{Z}$) as
existing on a continuous spacetime manifold,
$\mathcal{M}_{\text{new}}\cong\mathbb{R}\times\mathbb{R}$. This example
application is inspired by the work of Kempf (2010) entitled “Spacetime could
be simultaneously continuous and discrete, in the same way that information
can be” although I ultimate approach it from a substantially different
direction (see Sec. 5). On my account this discrete-to-continuous spacetime
redescription is facilitated by the fact that the lattice theory in question
admits some continuous pre-spacetime translation operations. These continuous
PSTOs (shown in the third column of Fig. 1) act on the theory’s discrete
states so as to smoothly interpolate between the theory’s discrete spacetime
translation. When viewed as a Lie group action on the theory’s states, these
transformations are structurally identical to spacetime translations. (Namely,
they are structurally identical to spacetime translations in exactly the same
sense as the above-discussed Fourier shifts were). Hence, they too qualify as
PSTOs.

But what prompts us to take these continuous PSTOs seriously as spacetime
translations? As Sec. 5 will discuss the transformations shown in Fig. 1 are
not only PSTOs but also dynamical symmetries (i.e., they map solutions to
solutions). We have thus discovered that these theories have a hidden
dynamical symmetry! The ISE Methodology allows us to redescribe these lattice
theories in such a way that their hidden dynamical symmetries are re-expressed
as spacetime symmetries in the new theory. The process for constructing this
new continuous spacetime is identical to the Fourier example discussed above.

##  3 Surviving Internalization: Dynamics, Diffeomorphisms, and the Spacetime
Manifold

The first step of the ISE Methodology is internalization. Let us see it in
action. Consider the following non-linear theory set on a curved spacetime:

> A Quartic Klein Gordon Theory (2+1D QKG) -  Consider a spacetime theory
> about a scalar field,
> $\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$,
> with spacetime, $\mathcal{M}_{\text{old}}\cong\mathbb{R}\times S^{2}$, and
> value space, $\mathcal{V}_{\text{old}}\cong\mathbb{R}$. For this theory,
> there exists an injective map,
> $C:\mathcal{M}_{\text{old}}\to\mathbb{R}^{4}$, which assigns to each point,
> $p\in\mathcal{M}_{\text{old}}$, coordinates,
> $C(p)=(t,x,y,z)\in\mathbb{R}^{4}$ with $x^{2}+y^{2}+z^{2}=1$. This theory’s
> states are subject to the following kinematic constraint: In this fixed
> coordinate system the field, $\varphi_{\text{old}}$, must be smooth. In this
> coordinate system, the metaphysically relevant way to judge the relative
> size and similarity of this theory’s states is with an $L^{2}$ inner
> product. In this coordinate system, the theory’s states obey the following
> dynamics:
>
>
> $\displaystyle(\partial_{t}^{2}-L_{x}^{2}-L_{y}^{2}-L_{z}^{2}+M^{2})\,\varphi_{\text{old}}+\lambda\,\varphi_{\text{old}}^{3}=0,$
> (1)
>
> where $L_{x}$, $L_{y}$, and $L_{z}$ are the generators of rotations about
> the $x$, $y$, and $z$-axis respectively. $\lambda\in\mathbb{R}$ is some
> fixed self-coupling constant and $M\geq 0$ is some fixed mass parameter.

Let us begin by identifying the modal structure of this theory. At its
broadest level, this theory is about the set, $S_{\text{old}}^{\text{all}}$,
of all $\mathbb{R}$-valued functions definable on
$\mathcal{M}_{\text{old}}\cong\mathbb{R}\times S^{2}$ (even those which are
non-smooth and/or discontinuous). Enforcing the theory’s kinematic
constraints, we can identify within $S_{\text{old}}^{\text{all}}$ a set of
kinematically allowed states, $S_{\text{old}}^{\text{kin}}\subset
S_{\text{old}}^{\text{all}}$. This is the set of all smooth
$\mathbb{R}$-valued functions on $\mathcal{M}_{\text{old}}$. Within this set,
we can then enforce the theory’s dynamics to find its dynamically allowed
states, $S_{\text{old}}^{\text{dyn}}\subset S_{\text{old}}^{\text{kin}}\subset
S_{\text{old}}^{\text{all}}$.

A crucial first step in the internalization process is to identify which
structural features of one’s theory are dynamically and metaphysically
relevant. In particular, the theory’s kinematically allowed states,
$S_{\text{old}}^{\text{kin}}$, must come equipped with all of the resources
needed to state the theory’s dynamics and to make sense of its metaphysics.
For instance, given its dynamics, the Quartic Klein-Gordon theory requires us
to define three point-wise operations on $\mathcal{M}_{\text{old}}$. Namely,
it requires an addition operation,
$+_{\text{old}}:S_{\text{old}}^{\text{kin}}\times
S_{\text{old}}^{\text{kin}}\to S_{\text{old}}^{\text{kin}}$, and a scalar
multiplication operation, $\cdot_{\text{old}}:\mathbb{R}\times
S_{\text{old}}^{\text{kin}}\to S_{\text{old}}^{\text{kin}}$, and finally a
product operation, $\times_{\text{old}}:S_{\text{old}}^{\text{kin}}\times
S_{\text{old}}^{\text{kin}}\to S_{\text{old}}^{\text{kin}}$. Using these we
can state the theory’s dynamical equations and define its dynamically allowed
states, $S_{\text{old}}^{\text{dyn}}\subset S_{\text{old}}^{\text{kin}}$. Past
its dynamically-relevant structure, this theory also comes equipped with a
metaphysically relevant inner product,
$\langle\varphi,\phi\rangle_{\text{old}}$, which we can use to judge the
relative size and similarity of any two kinematically allowed states,
$\varphi,\phi\in S_{\text{old}}^{\text{kin}}$.

Before we can begin internalizing this theory, two more pieces of mathematical
structure need to be pointed out. Recall that the goal of the internalization
process is to divorce the dynamical behavior of matter from the theory’s
assumed topological background structure (i.e., its spacetime manifold). But
where do the dynamical fields,
$\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$,
make contact with the spacetime manifold? Note that each of the modal spaces
discussed above, $S_{\text{old}}^{\text{dyn}}\subset
S_{\text{old}}^{\text{kin}}\subset S_{\text{old}}^{\text{all}}$, is populated
by functions on the spacetime manifold, $\mathcal{M}_{\text{old}}$. Namely,
our theory comes equipped with an operation,
$\text{eval}:S_{\text{old}}^{\text{kin}}\times\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$,
which allows us to evaluate the field states at spacetime points,
$\text{eval}(\varphi_{\text{old}},p)=\varphi_{\text{old}}(p)$. It is this
connection which the internalization process aims to sever.

One final piece of structure must be noted before we internalize this theory.
Note that the Quartic Klein-Gordon theory, admits a map
$*_{\text{lift}}:d\mapsto d^{*}$, which can lift the theory’s diffeomorphisms,
$d\in\text{Diff}(\mathcal{M}_{\text{old}})$, to act on
$S_{\text{old}}^{\text{all}}$ as $d^{*}:S_{\text{old}}^{\text{all}}\to
S_{\text{old}}^{\text{all}}$. Concretely, we have
$d^{*}\varphi_{\text{old}}\coloneqq\varphi_{\text{old}}\circ d^{-1}$. We can
restrict these lifted diffeomorphisms to act only on
$S_{\text{old}}^{\text{kin}}$ as
$d^{*}|_{\text{kin}}:S_{\text{old}}^{\text{kin}}\to
S_{\text{old}}^{\text{kin}}$. Next note that there is an intuitive sense in
which these lifted diffeomorphisms, $d^{*}|_{\text{kin}}$, act smoothly on
$S_{\text{old}}^{\text{kin}}$. But in order to understand the action of
$d^{*}|_{\text{kin}}$ on $\mathcal{S}_{\text{old}}^{\text{kin}}$ as being
smooth, we must be able to equip this theory’s space of kinematically allowed
states, $S_{\text{old}}^{\text{kin}}$, with some smooth topological structure,
$\mathcal{T}_{\text{old}}$. Hence, in general, we can imagine smoothly varying
the theory’s states via some Lie group actions (e.g., any of those shown in
Fig. 1) with $d^{*}|_{\text{kin}}$ being one example.

We can collect together all of the structures associated with the old theory’s
space of kinematically allowed states as follows:

$\displaystyle\mathcal{A}_{\text{old}}^{\text{kin}}=(S_{\text{old}}^{\text{kin}},\mathcal{T}_{\text{old}},+_{\text{old}},\,\cdot_{\text{old}}\,,\times_{\text{old}},\,\langle,\rangle_{\text{old}}\,,\text{eval},*_{\text{lift}},\mathcal{M}_{\text{old}}).$
(2)

This theory is now ready to be internalized. The way in which the ISE
Methodology divorces the theory’s states from the old spacetime setting is
similar in spirit to the algebraic approach to spacetime. Let us first discuss
what these approaches have in common before seeing how the ISE Methodology
goes further. On either approach, we begin by noting that a large portion of
the theory’s structure can be preserved while breaking the field-to-spacetime
connection. In order to break this connection, all we need to do is forget
about $\mathcal{M}_{\text{old}}$ (and relatedly the eval and $*_{\text{lift}}$
operations). We can do this via some forgetful operation,
$\mathcal{F}_{\text{alg}}$, which remembers only algebraic structure as
follows:

$\displaystyle\mathcal{F}_{\text{alg}}:\mathcal{A}_{\text{old}}^{\text{kin}}\to\mathcal{A}_{\text{neutral}}^{\text{kin}}\coloneqq(S_{\text{neutral}}^{\text{kin}},\mathcal{T}_{\text{neutral}},+_{\text{neutral}},\,\cdot_{\text{neutral}}\,,\times_{\text{neutral}},\,\langle,\rangle_{\text{neutral}}\,).$
(3)

Three things have changed here. Firstly, the old theory’s spacetime manifold,
$\mathcal{M}_{\text{old}}$, has been dropped, together with the eval and
$*_{\text{lift}}$ operations it supported. Secondly, the set,
$S_{\text{old}}^{\text{kin}}$, has been replaced with an equal-sized set,
$S_{\text{neutral}}^{\text{kin}}$, whose elements have no connection with
$\mathcal{M}_{\text{old}}$. Namely, whereas $\varphi_{\text{old}}\in
S_{\text{old}}^{\text{kin}}$ is a functions on $\mathcal{M}_{\text{old}}$, the
corresponding element
$\bm{\varphi}\coloneqq\mathcal{F}_{\text{alg}}(\varphi_{\text{old}})\in
S_{\text{neutral}}^{\text{kin}}$ is not. Thirdly, the algebraic operations
appearing in the above expression (e.g., $+$, $\,\cdot\,$, $\times$, and
$\langle,\rangle$) are now defined on $S_{\text{neutral}}^{\text{kin}}$
instead of $S_{\text{old}}^{\text{kin}}$ such that we have an isomorphism,
$\mathcal{A}_{\text{neutral}}^{\text{kin}}\cong\mathcal{A}_{\text{old}}^{\text{kin}}$,
as algebras. Concretely, we have,

$\displaystyle\bm{\varphi}\ +_{\text{neutral}}\ \bm{\phi}$
$\displaystyle\coloneqq\mathcal{F}_{\text{alg}}(\mathcal{F}_{\text{alg}}^{-1}(\bm{\varphi})+_{\text{old}}\mathcal{F}_{\text{alg}}^{-1}(\bm{\phi})),$
(4) $\displaystyle\alpha\,\cdot_{\text{neutral}}\,\bm{\varphi}$
$\displaystyle\coloneqq\mathcal{F}_{\text{alg}}(\alpha\,\cdot_{\text{old}}\,\mathcal{F}_{\text{alg}}^{-1}(\bm{\varphi})),$
$\displaystyle\bm{\varphi}\ \times_{\text{neutral}}\ \bm{\phi}$
$\displaystyle\coloneqq\mathcal{F}_{\text{alg}}(\mathcal{F}_{\text{alg}}^{-1}(\bm{\varphi})\times_{\text{old}}\mathcal{F}_{\text{alg}}^{-1}(\bm{\phi})),$
$\displaystyle\langle\bm{\varphi},\bm{\phi}\rangle_{\text{neutral}}$
$\displaystyle\coloneqq\langle\mathcal{F}_{\text{alg}}^{-1}(\bm{\varphi}),\mathcal{F}_{\text{alg}}^{-1}(\bm{\phi})\rangle_{\text{old}},$

where $\bm{\varphi},\bm{\phi}\in\mathcal{A}_{\text{neutral}}^{\text{kin}}$ are
generic spacetime-neutral states and $\alpha\in\mathbb{R}$. Similarly, the set
of spacetime neutral states $\bm{\varphi}\in S_{\text{neutral}}^{\text{kin}}$
has a smooth structure, $\mathcal{T}_{\text{neutral}}$, which isomorphic to
the old theory’s, $\mathcal{T}_{\text{old}}$. Namely, it has the smooth
structure which is naturally induced by the bijection,
$\mathcal{F}_{\text{alg}}:S_{\text{old}}^{\text{kin}}\leftrightarrow
S_{\text{neutral}}^{\text{kin}}$. In total, $\mathcal{F}_{\text{alg}}$ forgets
everything about the Quartic Klein-Gordon theory except for its algebraic
structure and its smooth topological structure of its kinematically allowed
states.

It is important to note that exactly how forgetful the $\mathcal{F}$ operation
ought to be will vary from theory to theory. Ultimately, the correct level of
forgetfulness for $\mathcal{F}$ will be set by the dynamical and metaphysical
details of the theory in question. Two more examples of internalization will
be discussed in this paper. Grimmer (2023) will discuss the internalization
process in complete generality as well as giving two more concrete examples.
The ISE Methodology allows for a wide variability of the dynamically and
metaphysically relevant structures which are at play. There are, however, some
core structural features which are mechanically necessary for the ISE
Methodology to be applied to some theory.

If one comes from a spacetime algebraicist perspective, one might expect that
some degree of algebraic structure (e.g., $+$, $\,\cdot\,$, $\times$, and
$\langle,\rangle$) will be required later on in constructing a new spacetime
framing of this theory (e.g., if one seeks to identify spacetime points with
certain maximal ideals of some ring). It is at this point, however, that the
ISE Methodology diverges substantially from spacetime algebraicism. Namely,
unlike the algebraic approach to spacetime, the ISE Methodology does not make
use of any algebraic structure in recovering the theory’s old spacetime
settings (or in generating new spacetime settings). Hence, in principle, the
internalization process can be extremely forgetful; Concretely, we might
internalize via a forgetful operation,

$\displaystyle\mathcal{F}_{\text{smooth}}:\mathcal{A}_{\text{old}}^{\text{kin}}\to\mathcal{S}_{\text{neutral}}^{\text{kin}}=(S_{\text{neutral}}^{\text{kin}},\mathcal{T}_{\text{neutral}}),$
(5)

which remembers only the smooth structure of the theory’s kinematically
allowed state space. As noted in Sec. 2, the new spacetime settings offered to
us by the ISE Methodology are to be built solely from our ability to smoothly
vary the theory’s states (recall Fig. 1).

To be clear, there may be dynamical or metaphysical reasons for preserving
other aspects of the theory in question through the internalization process
(e.g., its algebraic structure if it has any). These structures may play a
substantial role in stating the theory’s dynamics, in grounding a satisfying
metaphysical story, and even in choosing the best spacetime setting for the
theory. Nonetheless, any such structures are mechanically irrelevant
throughout the ISE Methodology. They merely come along for the ride so to
speak. It is only the smooth structure, $\mathcal{T}$, of the theory’s state
space which is invoked in the actual construction of new candidate spacetime
settings (see Sec. 4 for details).

Let us return to internalizing the Quartic Klein-Gordon theory, which
incidentally does require us to preserve its algebraic structure. What
survives internalization? The old theory’s kinematically allowed states,
$\varphi_{\text{old}}\in\mathcal{A}_{\text{old}}^{\text{kin}}$, survive
internalization as follows,

$\displaystyle\varphi_{\text{old}}\xrightarrow[\text{int.
as}]{\text{survives}}\bm{\varphi}\coloneqq\mathcal{F}_{\text{alg}}(\varphi_{\text{old}}).$
(6)

Moreover, given that $\mathcal{A}_{\text{old}}^{\text{kin}}$’s algebraic
structure has faithfully survived internalization, we ought to have the
resources within $\mathcal{A}_{\text{neutral}}^{\text{kin}}$ to state the
theory’s dynamics. Let us see how this happens. To begin, recall the theory’s
dynamical equation given by Eq. (1),

$\displaystyle(\partial_{t}^{2}-L_{x}^{2}-L_{y}^{2}-L_{z}^{2}+M^{2})\,\varphi_{\text{old}}+\lambda\,\varphi_{\text{old}}^{3}=0,$
(7)

Each part of this dynamical equation survives internalization as follows:

$\displaystyle\partial_{t}|_{\text{kin}}$
$\displaystyle\xrightarrow[\text{int. as}]{\text{survives}}$ $\displaystyle
D_{0}$
$\displaystyle\coloneqq\mathcal{F}_{\text{alg}}\circ\partial_{t}\circ\mathcal{F}_{\text{alg}}^{-1}$
(8) $\displaystyle L_{x,y,z}|_{\text{kin}}$
$\displaystyle\xrightarrow[\text{int. as}]{\text{survives}}$ $\displaystyle
D_{1,2,3}$ $\displaystyle\coloneqq\mathcal{F}_{\text{alg}}\circ
L_{x,y,z}\circ\mathcal{F}_{\text{alg}}^{-1}$

Hence, we have the dynamical equation itself surviving internalization as
follows:

$\displaystyle(\partial_{t}^{2}-L_{x}^{2}-L_{y}^{2}-L_{z}^{2}+M^{2})\,\varphi_{\text{old}}+\lambda\,\varphi_{\text{old}}^{3}=0$
(9) $\displaystyle\xrightarrow[\text{int.
as}]{\text{survives}}(D_{0}^{2}-D_{1}^{2}-D_{2}^{2}-D_{3}^{2}+M^{2})\,\bm{\varphi}+\lambda\cdot\bm{\varphi}^{3}=0$

where the $\,\cdot\,$, $+$, and $\times$ operations in the spacetime neutral
theory are those induced by the forgetful operation,
$\mathcal{F}_{\text{alg}}$, see Eq. (4). Ultimately, this means that we can
pick out the theory’s dynamically allowed states in a spacetime-neutral way.
Hence, this space too survives internalization,

$\displaystyle\mathcal{S}_{\text{old}}^{\text{dyn}}\subset\mathcal{A}_{\text{old}}^{\text{kin}}$
$\displaystyle\xrightarrow[\text{int.
as}]{\text{survives}}\mathcal{S}_{\text{neutral}}^{\text{dyn}}\subset\mathcal{A}_{\text{neutral}}^{\text{kin}}.$
(10)

We have so far seen how a theory’s states and dynamics can survive being
divorced from the old theory’s topological background structure. Importantly,
however, internalization does not simply delete all of the theory’s
topological information. Surprisingly, a great deal of topological information
about the old spacetime setting can survive internalization. The key to seeing
how any topological information survives internalization the
$*_{\text{lift}}:d\mapsto d^{*}$ map discussed above. Recall that this map
lifts diffeomorphisms,
$d:\mathcal{M}_{\text{old}}\to\mathcal{M}_{\text{old}}$, to act on the
theory’s states as $d^{*}:S_{\text{old}}^{\text{all}}\to
S_{\text{old}}^{\text{all}}$. Once we restrict the action of $d^{*}$ to
kinematically allowed states (e.g.,
$d^{*}|_{\text{kin}}:S_{\text{old}}^{\text{kin}}\to
S_{\text{old}}^{\text{kin}}$), it becomes possible to see it surviving
internalization. That is, we can see them as maps on
$S_{\text{neutral}}^{\text{kin}}$, namely
$G(d)\coloneqq\mathcal{F}_{\text{alg}}\circ
d^{*}|_{\text{kin}}\circ\mathcal{F}_{\text{alg}}^{-1}$.

For the Quartic Klein-Gordon theory, a notable family of diffeomorphism on
$\mathcal{M}_{\text{old}}\cong\mathbb{R}\times S^{2}$ survive internalization
in this way. Let $d_{t}(\epsilon)\in\text{Diff}(\mathcal{M}_{\text{old}})$ be
the diffeomorphism on $\mathcal{M}_{\text{old}}$ which act as
$d_{t}(\epsilon):(t,x,y,z)\mapsto(t+\epsilon,x,y,z)$ in the $C$-coordinate
system. Similarly, let
$d_{x}(\theta),d_{y}(\theta),d_{z}(\theta)\in\text{Diff}(\mathcal{M}_{\text{old}})$,
be the diffeomorphisms which rigidly rotate $\mathcal{M}_{\text{old}}$ around
the $x$, $y$ and $z$-axis in the $C$-coordinate system. Let
$H_{\text{trans}}\cong(\mathbb{R},+)\times\text{SO}(3)$ be the Lie group
formed by these diffeomorphisms. Each of these diffeomorphisms, $d\in
H_{\text{trans}}$ survives internalization as follows:

$\displaystyle
d_{t}^{*}(\epsilon)|_{\text{kin}}=\text{exp}(-\epsilon\,\partial_{t})\xrightarrow[\text{int.
as}]{\text{survives}}G(d_{t}(\epsilon))=\text{exp}(-\epsilon\,D_{0})$ (11)
$\displaystyle d_{x}^{*}(\theta)|_{\text{kin}}=\text{exp}(-\theta
L_{x})\xrightarrow[\text{int.
as}]{\text{survives}}G(d_{x}(\theta))=\text{exp}(-\theta D_{1})$
$\displaystyle d_{y}^{*}(\theta)|_{\text{kin}}=\text{exp}(-\theta
L_{y})\xrightarrow[\text{int.
as}]{\text{survives}}G(d_{y}(\theta))=\text{exp}(-\theta D_{2})$
$\displaystyle d_{z}^{*}(\theta)|_{\text{kin}}=\text{exp}(-\theta
L_{z})\xrightarrow[\text{int.
as}]{\text{survives}}G(d_{z}(\theta))=\text{exp}(-\theta D_{3}).$

Indeed, these diffeomorphisms survive internalization not only individually,
but also as a Lie group (i.e., the $G$ map is a Lie group isomorphism). In
total, therefore, we have that the Lie group
$H_{\text{trans}}\subset\text{Diff}(\mathcal{M}_{\text{old}})$ faithfully
survives internalization as follows:

$\displaystyle H_{\text{trans}}\xrightarrow[\text{int.
as}]{\text{survives}}G_{\text{trans}}\coloneqq G(H_{\text{trans}})$ (12)

with $G_{\text{trans}}\cong
H_{\text{trans}}\cong(\mathbb{R},+)\times\text{SO}(3)$. Thus, at least some
topological information regarding the old spacetime setting,
$\mathcal{M}_{\text{old}}$, has survived internalization.

Perhaps surprisingly, not just partial but complete topological information
about $\mathcal{M}_{\text{old}}$ has faithfully survived internalization.
Seeing this, however, will require that we first understand a bit of
mathematics regarding the quotients of Lie groups, namely, smooth homogeneous
manifolds. A summary of this topic along with proofs of some key results can
be found in Appendix A. In order to see $\mathcal{M}_{\text{old}}$ as
surviving internalization we will first reconstruct it as a quotient of its
diffeomorphisms. To do this, we need a pick out a finite-dimensional Lie
group, $H\subset\text{Diff}(\mathcal{M}_{\text{old}})$, which acts
transitively on $\mathcal{M}_{\text{old}}$. The Lie group
$H_{\text{trans}}\cong(\mathbb{R},+)\times\text{SO}(3)$ discussed above is
just such a Lie group (hence the subscript “trans”). By using the Homogeneous
Manifold Characterization Theorem (proved in Appendix A) we can reconstruct
$\mathcal{M}_{\text{old}}$ up to diffeomorphism as,

$\displaystyle\mathcal{M}_{\text{old}}\cong\mathcal{M}_{\text{H}}\coloneqq\frac{H_{\text{trans}}}{H_{\text{fix}}}\cong\frac{(\mathbb{R},+)\times\text{SO}(3)}{\\{\mathbbm{1}\\}\times\text{SO}(2)},$
(13)

where $H_{\text{fix}}\cong\\{\mathbbm{1}\\}\times\text{SO}(2)$ is one of
$H_{\text{trans}}$’s stabilizer subgroups. A concrete demonstration of the
non-trivial part of this quotient, $\text{SO}(3)/\text{SO}(2)\cong S^{2}$, can
be found in Appendix A.

Having reconstructed the old theory’s spacetime manifold in this way, we can
now see it as faithfully surviving internalization. As noted above, the Lie
group $H_{\text{trans}}$ (and, consequently, $H_{\text{fix}}\subset
H_{\text{trans}}$) faithfully survives internalization:

$\displaystyle H_{\text{trans}}$ $\displaystyle\xrightarrow[\text{int.
as}]{\text{survives}}G_{\text{trans}}\coloneqq G(H_{\text{trans}})\cong
H_{\text{trans}},$ (14) $\displaystyle H_{\text{fix}}$
$\displaystyle\xrightarrow[\text{int.
as}]{\text{survives}}G_{\text{fix}}\coloneqq G(H_{\text{fix}})\cong
H_{\text{fix}},$

becoming some Lie groups, $G_{\text{fix}}\subset G_{\text{trans}}$, acting on
$\mathcal{A}_{\text{neutral}}^{\text{kin}}$. By taking the quotient of these
surviving Lie groups we can faithfully reconstruct $\mathcal{M}_{\text{H}}$
(and, consequently, $\mathcal{M}_{\text{old}}\cong\mathcal{M}_{\text{H}}$)
post-internalization as follows:

$\displaystyle\mathcal{M}_{\text{old}}\cong\mathcal{M}_{\text{H}}\coloneqq\frac{H_{\text{trans}}}{H_{\text{fix}}}\xrightarrow[\text{int.
as}]{\text{survives}}\mathcal{M}_{\text{G}}\coloneqq\frac{G_{\text{trans}}}{G_{\text{fix}}}\cong\frac{H_{\text{trans}}}{H_{\text{fix}}}\cong\mathcal{M}_{\text{old}}.$
(15)

Thus, the Quartic Klein-Gordon theory’s spacetime manifold has faithfully
survived internalization.

This is a surprising conclusion. Recall that the stated purpose of
internalization was to conceptually divorce the theory’s states,
$\varphi_{\text{old}}$, from any assumed topological background structure.
Despite this divorce, it seems that complete topological information about
$\mathcal{M}_{\text{old}}\cong\mathbb{R}\times S^{2}$ has survived
internalization. As Grimmer (2023) proves, for any theory with a sufficient
level of spacetime-kinematic compatibility, there is enough topological
information contained in the surviving Lie groups $G_{\text{trans}}$ and
$G_{\text{fix}}$ to perfectly reconstruct the theory’s original spacetime
setting. Namely, by externalizing these Lie groups, one can reconstruct the
old theory’s spacetime setting, up to diffeomorphism.

Before discussing the externalization process, however, allow me to first
demonstrate how the internalization process applies to one more example
theory. Consider the following theory about a modified quantum harmonic
oscillator (QHO):

> Non-Local QHO - Consider a spacetime theory about a scalar field,
> $\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$,
> with spacetime, $\mathcal{M}_{\text{old}}\cong\mathbb{R}^{2}$, and value
> space, $\mathcal{V}_{\text{old}}\cong\mathbb{C}$. This theory’s states are
> subject to the following kinematic constraint: In some fixed global
> coordinate system, $(t,x)$, each state must be within the usual rigged
> Hilbert space of planewaves and Dirac deltas. In this coordinate system, the
> metaphysically relevant way to judge the relative size and similarity of
> this theory’s states is with an $L^{2}$ inner product. In this coordinate
> system, the theory’s states obey the following dynamics:
>
> $\displaystyle\mathrm{i}\partial_{t}\varphi_{\text{old}}(t,x)$
> $\displaystyle=(-\partial_{x}^{2}+x^{2})\varphi_{\text{old}}(t,x)+\frac{\lambda}{2}\varphi_{\text{old}}(t,x-a)+\frac{\lambda}{2}\varphi_{\text{old}}(t,x+a),$
>
> for some fixed $a>0$ and $\lambda\in\mathbb{R}$.

Note that this is the standard QHO plus a non-local self-coupling.

As with the Quartic Klein-Gordon theory, we ought to begin by identifying this
theory’s dynamically and metaphysically relevant features. Since this theory’s
dynamical equations are linear we need the space of kinematically allowed
states, $V_{\text{old}}^{\text{kin}}$, to have at least a vector space
structure. The space of dynamically allowed states is then a vector subspace
thereof, $V_{\text{old}}^{\text{dyn}}\subset V_{\text{old}}^{\text{kin}}$.
Past its dynamically-relevant linear structure, this theory also has a
metaphysically-relevant inner product,
$\langle\varphi,\phi\rangle_{\text{old}}$, defined for $\varphi,\phi\in
V_{\text{old}}^{\text{kin}}$. As in the previous case, we can internalize this
theory by applying a forgetful operation, $\mathcal{F}_{\text{vec}}$, at the
level of kinematics which only remembers $V_{\text{old}}^{\text{kin}}$’s
vector space structure as follows:

$\displaystyle\mathcal{F}_{\text{vec}}:V_{\text{old}}^{\text{kin}}\to
V_{\text{neutral}}^{\text{kin}}\coloneqq(S_{\text{neutral}}^{\text{kin}},\mathcal{T}_{\text{neutral}},+_{\text{neutral}},\,\cdot_{\text{neutral}}\,,\,\langle,\rangle_{\text{neutral}}).$
(16)

To finish internalizing this theory, let us see how its spacetime manifold
faithfully survives internalization. If $\mathcal{M}_{\text{old}}$ is to
survive internalization we must first reconstruct it as a quotient of its
diffeomorphisms. To do this, we need a pick out a finite-dimensional Lie
group, $H_{\text{trans}}\subset\text{Diff}(\mathcal{M}_{\text{old}})$, which
acts transtively $\mathcal{M}_{\text{old}}$. According to the Homogeneous
Manifold Characterization Theorem it does not matter which Lie group,
$H_{\text{trans}}$, we pick (see Appendix A). The simplest Lie group which
satisfies these criteria is the group,
$H_{\text{trans}}\cong(\mathbb{R}^{2},+)$, which implements rigid translations
in the old theory’s $(t,x)$ coordinate system. Note that the stabilizer
subgroups for this choice of $H_{\text{trans}}$ are all identical,
$H_{\text{fix}}=\\{\mathbbm{1}\\}$.

Whichever Lie groups we use to reconstruct the old theory’s spacetime
manifold, $\mathcal{M}_{\text{old}}\cong H_{\text{trans}}/H_{\text{fix}}$, we
can subsequently see it surviving internalization as follows. The Lie groups
in the numerator and denominator of this quotient each faithfully survive
internalization becoming a pair of Lie groups, $G_{\text{trans}}\cong
H_{\text{trans}}$ and $G_{\text{fix}}\cong H_{\text{fix}}$, which act on
$V_{\text{neutral}}^{\text{kin}}$, the space of kinematically allowed states
in the spacetime-neutral theory. Hence, the theory’s old spacetime manifold,
$\mathcal{M}_{\text{old}}\cong H_{\text{trans}}/H_{\text{fix}}$, also
faithfully survives internalization becoming $\mathcal{M}_{\text{G}}\coloneqq
G_{\text{trans}}/G_{\text{fix}}\cong\mathcal{M}_{\text{old}}$.

##  4 Demonstrating Externalization in a Familiar Setting: Fourier
Redescription

The previous section discussed how a theory’s original spacetime manifold,
$\mathcal{M}_{\text{old}}$, can faithfully survive internalization. Namely, it
can survive as a quotient of two Lie groups, $G_{\text{trans}}^{\text{(int)}}$
and $G_{\text{fix}}^{\text{(int)}}$, acting smoothly on
$\mathcal{S}_{\text{neutral}}^{\text{kin}}$ (e.g., either
$\mathcal{A}_{\text{neutral}}^{\text{kin}}$ or
$V_{\text{neutral}}^{\text{kin}}$). The superscript (int) here indicates that
these Lie groups were produced by the internalization process. Of course,
knowing how a theory’s spacetime manifold survives internalization is very
suggestive of how externalization might be able to generate a new spacetime
manifold for this theory.

This section will introduce the externalization process by using it to build a
new spacetime setting for the non-local QHO theory discussed above. In
particular, the new theory’s spacetime will be effectively a copy of the old
theory’s Fourier space and its states will be related to the old theory’s
states via a Fourier transform. While it may seem like overkill to use the ISE
Methodology to achieve this Fourier redescription, the techniques demonstrated
here are applicable much more generally. See, for instance, Sec. 5 as well as
two additional applications of the ISE Methodology in Grimmer (2023). In fact,
it is there proved that the ISE Methodology is a completely general tool for
topological redescription; Using it one can access every possible spacetime
framing of a given theory, bounded only by a weak spacetime-kinematic
compatibility condition.

Without further ado, let us see the externalization process in action.
Externalization builds new spacetime settings from certain building blocks,
namely, pre-spacetime translation operations (PSTOs). Concretely, these are a
pair of Lie group, $G_{\text{trans}}^{\text{(ext)}}$ and
$G_{\text{fix}}^{\text{(ext)}}$, which act smoothly on
$\mathcal{S}_{\text{neutral}}^{\text{kin}}$. Not just any pair of Lie group
actions will qualify as PSTOs, however, they must be, in a certain sense,
structurally indistinguishable from spacetime translations.

> Definition: Pre-Spacetime Translation Operations (PSTOs) -  Let
> $G_{2}\subset G_{1}$ be a pair of Lie groups which act smoothly on some
> space $\mathcal{S}_{\text{N}}=(S_{\text{N}},\mathcal{T}_{\text{N}},\dots)$.
> This pair of Lie group actions will be called pre-spacetime translation
> operations when they have both Faithful Construction Compatibility and
> Injection Compatibility.

The technical definitions of faithful construction compatibility and injection
compatibility will be given throughout this section. Roughly, faithful
construction compatibility guarantees that our PSTOs,
$G_{\text{trans}}^{\text{(ext)}}$, will end up faithfully represented as
diffeomorphisms on the new spacetime setting, namely, as
$H_{\text{trans}}^{\text{(ext)}}\subset\text{Diff}(\mathcal{M}_{\text{new}})$
with $H_{\text{trans}}^{\text{(ext)}}\cong G_{\text{trans}}^{\text{(ext)}}$.
Injection compatibility will then guarantee that the Lie group action of
$G_{\text{trans}}^{\text{(ext)}}$ on
$\mathcal{S}_{\text{neutral}}^{\text{kin}}$ is faithfully represented by the
lifted actions these diffeomorphisms, $H_{\text{trans}}^{\text{(ext)}}{}^{*}$
on some yet-to-be-defined states on $\mathcal{M}_{\text{new}}$. This justifies
the following gloss of the externalization process: Externalization builds a
new spacetime by taking some hand-picked pre-spacetime translation operations,
$G_{\text{fix}}^{\text{(ext)}}\subset G_{\text{trans}}^{\text{(ext)}}$, and
letting them be honest-to-goodness spacetime translations,
$H_{\text{fix}}^{\text{(ext)}}\subset
H_{\text{trans}}^{\text{(ext)}}\subset\text{Diff}(\mathcal{M}_{\text{new}})$.

Let us begin by briefly revisiting the example PSTOs discussed in Sec. 2 (and
displayed in Fig. 1). In Sec. 2 I claimed that the old theory’s spacetime
translation operations will always qualify as PSTOs (see the first column of
Fig. 1). More technically, this claim is about the corresponding Lie group
actions which have faithfully survived internalization; Namely,
$G_{\text{trans}}^{\text{(int)}}$ and $G_{\text{fix}}^{\text{(int)}}$ acting
on $\mathcal{S}_{\text{neutral}}^{\text{kin}}$. Grimmer (2023) proves that
this pair of Lie group actions will qualify as PSTOs whenever the theory in
question has a sufficient level of spacetime-kinematic compatibility.
Moreover, it is there proved that by externalizing these PSTOs one can easily
return to the theory’s original spacetime setting.

Importantly, however, we are not forced to go back the way we came. We are
free to search for other pre-spacetime translation operations to externalize,
i.e., ones with $G_{\text{trans}}^{\text{(ext)}}\neq
G_{\text{trans}}^{\text{(int)}}$ and/or $G_{\text{fix}}^{\text{(ext)}}\neq
G_{\text{fix}}^{\text{(int)}}$. To have a concrete alternative in mind,
consider the non-local QHO theory introduced in Sec. 3. Let the maps,

$\displaystyle f(\Delta t,\Delta
k)\,\varphi_{\text{old}}(t,x)=e^{-\mathrm{i}\Delta
kx}\,\varphi_{\text{old}}(t-\Delta t,x),$ (17)

form a Lie group $f(\Delta t,\Delta k)\in H_{\text{Fourier}}$ which acts on
$V_{\text{old}}^{\text{kin}}$. Some smooth transformations of this kind are
depicted in the second column of Fig. 1. Note that these $H_{\text{Fourier}}$
transformations faithfully survive internalization becoming a Lie group,
$G_{\text{Fourier}}=\mathcal{F}_{\text{vec}}\circ
H_{\text{Fourier}}\circ\mathcal{F}^{-1}_{\text{vec}}$, which acts on
$V_{\text{neutral}}^{\text{kin}}$.

There is an intuitive sense in which $G_{\text{Fourier}}$’s Lie group action
on $V_{\text{neutral}}^{\text{kin}}$ should qualify as PSTOs; Namely, when
viewed as a Lie group action on the theory’s states, these Fourier shift
operations are structurally identical to spacetime translations. Indeed, while
these aren’t translations in the old theory spacetime, they are nonetheless
translations somewhere, namely in the old theory’s Fourier space. Intuitively,
we should expect that externalizing the PSTOs ought to produce a new theory
which is, roughly speaking, set on a copy of the old theory’s Fourier space.
With this example of PSTOs in mind, let us next see how the externalization
process works more generally.

Recall from Sec. 3 that a theory’s spacetime manifold can faithfully survive
internalization as a quotient of two Lie groups,
$G_{\text{trans}}^{\text{(int)}}$ and $G_{\text{fix}}^{\text{(int)}}$, acting
on $\mathcal{S}_{\text{neutral}}^{\text{kin}}$. Analogously, externalization
begins from a pair of Lie groups $G_{\text{trans}}^{\text{(ext)}}$ and
$G_{\text{fix}}^{\text{(ext)}}$, acting on
$\mathcal{S}_{\text{neutral}}^{\text{kin}}$. Externalization puts forward the
quotient manifold of these Lie groups as the theory’s new spacetime manifold,

$\displaystyle\mathcal{M}_{\text{new}}\cong\mathcal{M}_{\text{G}}\coloneqq
G_{\text{trans}}^{\text{(ext)}}/G_{\text{fix}}^{\text{(ext)}},$ (18)

at least up to diffeomorphism. I say here “up to diffeomorphism” because the
quotient manifold $\mathcal{M}_{\text{G}}$ contains excess structure; Its
elements are sets of maps on $\mathcal{S}_{\text{neutral}}^{\text{kin}}$ as
well as manifold points on $\mathcal{M}_{\text{G}}$. We can remove this excess
by applying a forgetful operation $\mathcal{F}_{\text{man}}$ to
$\mathcal{M}_{\text{G}}$. Namely, this is a diffeomorphism
$\mathcal{F}_{\text{man}}:\mathcal{M}_{\text{G}}\to\mathcal{M}_{\text{new}}$
where $\mathcal{M}_{\text{new}}$ is some smooth manifold which is isomorphic
to $\mathcal{M}_{\text{G}}$, i.e., with
$\mathcal{M}_{\text{new}}\cong\mathcal{M}_{\text{G}}$. In terms of this
forgetful operation, the new spacetime manifold is defined as,

$\displaystyle\mathcal{M}_{\text{new}}\coloneqq\mathcal{F}_{\text{man}}(G_{\text{trans}}^{\text{(ext)}}/G_{\text{fix}}^{\text{(ext)}}).$
(19)

It is worth noting that the new theory’s spacetime,
$\mathcal{M}_{\text{new}}$, ultimately gets its smooth topological structure
from the smoothness of $G_{\text{trans}}^{\text{(ext)}}$’s action on
$\mathcal{S}_{\text{neutral}}^{\text{kin}}$. For notational convenience, let
us now drop the (ext) superscripts.

In addition to giving $\mathcal{M}_{\text{new}}$ its smooth structure, the Lie
group $G_{\text{trans}}$ is also naturally represented as diffeomorphisms on
$\mathcal{M}_{\text{new}}$. To see this, note that the left action of
$G_{\text{trans}}$ on itself can be seen as smoothly permuting its
$G_{\text{fix}}$-cosets (that is, smoothly permuting the points
$[g_{0}]\in\mathcal{M}_{\text{G}}$). Concretely, we have that $g\in
G_{\text{trans}}$ acts smoothly on $\mathcal{M}_{\text{G}}$ as
$\bar{g}([g_{0}])\coloneqq[g\,g_{0}]$. These maps
$\bar{g}:\mathcal{M}_{\text{G}}\to\mathcal{M}_{\text{G}}$ are diffeomorphisms
on $\mathcal{M}_{\text{G}}$ and can be transferred over into diffeomorphisms
on $\mathcal{M}_{\text{new}}$ as,

$\displaystyle
H(g)\coloneqq\mathcal{F}_{\text{man}}\circ\bar{g}\circ\mathcal{F}_{\text{man}}^{-1}.$
(20)

In total therefore we have a Lie group homomorphism,
$H:G_{\text{trans}}\to\text{Diff}(\mathcal{M}_{\text{new}})$, which associates
to every element, $g\in G_{\text{trans}}$, a diffeomorphism,
$H(g)\in\text{Diff}(\mathcal{M}_{\text{new}})$. The image of this map,
$H_{\text{trans}}^{\text{(ext)}}\coloneqq H(G_{\text{trans}})$, is a
representation of $G_{\text{trans}}$ as diffeomorphisms on
$\mathcal{M}_{\text{new}}$.

Recall that our goal is to build a new spacetime setting for the theory in
question such that our PSTOs end up as honest-to-goodness spacetime
translations on $\mathcal{M}_{\text{new}}$. To achieve this, we need some
assurance that the Lie group, $H_{\text{trans}}^{\text{(ext)}}$, will be a
faithful representation of our PSTOs, namely that
$H_{\text{trans}}^{\text{(ext)}}\cong G_{\text{trans}}$. As is proved in
Appendix A, this will happen if and only if $G_{\text{trans}}$ and
$G_{\text{fix}}$ have faithful construction compatibility.

> Definition: Faithful Construction Compatibility -  Let $G_{1}$ and $G_{2}$
> be two Lie groups. These two Lie groups have faithful construction
> compatibility if and only if $G_{2}$ is both a closed subgroup of $G_{1}$ as
> well as a core-free subgroup of $G_{1}$. (N.b., $G_{2}$ is a core-free
> subgroup of $G_{1}$ if and only if $G_{2}$ contains no non-trivial normal
> subgroups of $G_{1}$.)

Let us now return our attention to externalizing the non-local QHO theory with
the following choice of PSTOs, $G_{\text{trans}}=G_{\text{Fourier}}$. In order
to complete our choice of PSTOs, however, we also need to pick out a Lie
subgroup, $G_{\text{fix}}\subset G_{\text{trans}}$, which has faithful
construction compatibility with $G_{\text{trans}}$. What are our options? Note
that faithful construction compatibility requires $G_{\text{fix}}$ to be a
closed core-free subgroup of $G_{\text{trans}}=G_{\text{Fourier}}$. Next, note
that $G_{\text{Fourier}}\cong(\mathbb{R}^{2},+)$ is abelian and hence all of
its subgroups are normal. This means that the only core-free subgroup of
$G_{\text{Fourier}}$ is the trivial group. Hence, given our choice of
$G_{\text{trans}}=G_{\text{Fourier}}$ we are forced to take
$G_{\text{fix}}=\\{\mathbbm{1}\\}$.

Given these PSTOs, the new theory’s spacetime manifold is,

$\displaystyle\mathcal{M}_{\text{new}}\cong
G_{\text{trans}}/G_{\text{fix}}\cong\mathbb{R}^{2}.$ (21)

Note that while this new spacetime is diffeomorphic to the old one,
$\mathcal{M}_{\text{new}}\cong\mathcal{M}_{\text{old}}\cong\mathbb{R}^{2}$ it
is nonetheless distinct,
$\mathcal{M}_{\text{new}}\neq\mathcal{M}_{\text{old}}$. To see the
distinction, recall that Fourier shifts in the old theory, $f(\Delta t,\Delta
k)\in H_{\text{Fourier}}$, have faithfully survived internalization becoming
$g(\Delta t,\Delta k)\in G_{\text{Fourier}}=G_{\text{trans}}$ acting on
$V_{\text{neutral}}^{\text{kin}}$. But this is the Lie group which we have
just externalized. Thus, post-externalization we will have $g(\Delta t,\Delta
k)$ becoming a diffeomorphism, $h(\Delta t,\Delta k)\coloneqq H(g(\Delta
t,\Delta k))\in
H_{\text{trans}}^{\text{(ext)}}\subset\text{Diff}(\mathcal{M}_{\text{new}})$,
on our new two-dimensional manifold. Let $(\tau,q)$ be a coordinate system for
$\mathcal{M}_{\text{new}}\cong\mathbb{R}^{2}$ such that $h(\Delta t,\Delta
k)(\tau,q)=(\tau+\Delta t,q+\Delta k)$. Thus, shifts in Fourier space in the
old theory become shifts in spacetime in the new theory. As promised,
externalization has turned our choice of $G_{\text{trans}}=G_{\text{Fourier}}$
into honest-to-goodness spacetime translations on $\mathcal{M}_{\text{new}}$.

The connection between $\mathcal{M}_{\text{new}}$ and the old theory’s Fourier
space will only become stronger as we proceed to the next step in the
externalization process, namely injecting the old theory’s states and dynamics
into the new spacetime setting. Externalization must somehow map the theory’s
spacetime-neutral states, $\bm{\varphi}\in V_{\text{neutral}}^{\text{kin}}$,
and their dynamics onto $\mathcal{M}_{\text{new}}$. Let us assume that the new
theory will be about a scalar field,
$\varphi_{\text{new}}:\mathcal{M}_{\text{new}}\to\mathcal{V}_{\text{new}}$ for
some value space, $\mathcal{V}_{\text{new}}$. In order to begin identifying
the new theory’s states we must first build $S_{\text{new}}^{\text{all}}$, the
set of all $\mathcal{V}_{\text{new}}$-valued fields definable on
$\mathcal{M}_{\text{new}}$ (even those which are non-smooth and/or
discontinuous). Within this wide-scope we must next pick out the subset of
states which are kinematically allowed in the new theory,
$S_{\text{new}}^{\text{kin}}\subset S_{\text{new}}^{\text{all}}$. Recall that
we already have a spacetime-neutral characterization of the theory’s
kinematically allowed states, namely, $S_{\text{neutral}}^{\text{kin}}$. All
we need to do is to map these states onto the new spacetime setting via some
map $J:\bm{\varphi}\mapsto\varphi_{\text{new}}$. Demanding that distinct
states, $\bm{\varphi}\in S_{\text{neutral}}^{\text{kin}}$, are mapped onto
distinct states, $\varphi_{\text{new}}\in S_{\text{new}}^{\text{all}}$, we are
looking for an injective map, $J:S_{\text{neutral}}^{\text{kin}}\to
S_{\text{new}}^{\text{all}}$. Fixing this injector, $J$, will fix the new
theory’s kinematically allowed states as $S_{\text{new}}^{\text{kin}}\coloneqq
J(S_{\text{neutral}}^{\text{kin}})$.

Given the above-discussed connection between the new theory’s spacetime and
the old theory’s Fourier space, an intuitive choice of $J$ would be the one
which implements a Fourier transform. Namely, we should be able to externalize
via an injective map $J$ which (together with $\mathcal{F}_{\text{vec}}$)
defines $\varphi_{\text{new}}$ as follows:

$\displaystyle
J\circ\mathcal{F}_{\text{vec}}:\varphi_{\text{old}}\mapsto\varphi_{\text{new}}(\tau,q)$
$\displaystyle=\tilde{\varphi}_{\text{old}}(\tau,q).$ (22)

That is, the value of $\varphi_{\text{new}}$ at coordinate $(\tau,q)$ is the
value of $\varphi_{\text{old}}(t,x)$’s Fourier transform (namely,
$\tilde{\varphi}_{\text{old}}(t,k)$) at $(t,k)=(\tau,q)$. At this point one
may rightly wonder, are we somehow forced into this choice of $J$ or are there
other ways in which we could externalize this theory onto
$\mathcal{M}_{\text{new}}$? I will return to this question at the end of this
section. For now, however, let us proceed with this intuitive choice of $J$
and see where it leads us.

With this choice of injector $J:S_{\text{neutral}}^{\text{kin}}\to
S_{\text{new}}^{\text{all}}$, one can straight-forwardly translate the old
theory’s dynamics and kinematics onto its new spacetime setting as follows.

> Cosine QHO - Consider a spacetime theory about a scalar field,
> $\varphi_{\text{new}}:\mathcal{M}_{\text{new}}\to\mathcal{V}_{\text{new}}$,
> with spacetime, $\mathcal{M}_{\text{new}}\cong\mathbb{R}^{2}$, and value
> space, $\mathcal{V}_{\text{new}}\cong\mathbb{C}$. The theory’s states are
> subject to the following kinematic constraint: In some fixed global
> coordinate system, $(\tau,q)$, each state must be within the usual rigged
> Hilbert space of planewaves and Dirac deltas. In this coordinate system, the
> metaphysically relevant way to judge the relative size and similarity of
> this theory’s states is with an $L^{2}$ inner product. In this coordinate
> system, the theory’s states obey the following dynamics:
>
> $\displaystyle\mathrm{i}\partial_{\tau}\varphi_{\text{new}}(\tau,q)$
> $\displaystyle=(-\partial_{q}^{2}+q^{2}+\lambda\,\text{cos}(a\,q))\,\varphi_{\text{new}}(\tau,q),$
> (23)
>
> for some $a>0$ and $\lambda\in\mathbb{R}$.

Note that this is the standard QHO plus a cosine-shaped potential. Note that
whereas the old theory’s dynamics was non-local, this new theory has local
dynamics. In this way, this new spacetime setting is a better fit for this
theory’s dynamics than the old one was.

To finalize the externalization process, let us see how the old theory’s
dynamically and metaphysically structures have survived into the new theory.
Recall from Sec. 3 (specifically, Eq. (16)) that the following features of the
non-local QHO theory survived internalization: an addition operation (which
was point-wise on $\mathcal{M}_{\text{old}}$), an scalar multiplication
operation (which was also point-wise on $\mathcal{M}_{\text{old}}$), an inner
product (which was the $L^{2}$ inner product in the $(t,x)$ coordinate
system), and finally, a smooth topological structure for the theory’s
kinematically allowed states. Note that since the externalization map,
$J:S_{\text{neutral}}^{\text{kin}}\to S_{\text{new}}^{\text{all}}$, is
injective we have a bijection, $J:S_{\text{new}}^{\text{kin}}\leftrightarrow
S_{\text{neutral}}^{\text{kin}}$ between our two characterizations of the
theory’s kinematically allowed states. We can use this bijection to transfer
over all of the old theory’s dynamically and metaphysically relevant structure
onto $S_{\text{new}}^{\text{kin}}$ as follows:

$\displaystyle J:V_{\text{neutral}}^{\text{kin}}\to
V_{\text{new}}^{\text{kin}}\coloneqq(S_{\text{new}}^{\text{kin}},\mathcal{T}_{\text{new}},+_{\text{new}},\,\cdot_{\text{new}}\,,\,\langle,\rangle_{\text{new}}).$
(24)

Note that these new structures on $S_{\text{new}}^{\text{kin}}$ are exactly
those induced by the $J$ map (i.e., they are defined analogously to Eq. (4)).

But what are these induced structures exactly? It follows from familiar
properties of the Fourier transform that $+_{\text{new}}$ and
$\,\cdot_{\text{new}}\,$ are the point-wise addition and scalar
multiplications operations on $\mathcal{M}_{\text{new}}$. Moreover, it follows
that $\,\langle,\rangle_{\text{new}}$ is the $L^{2}$ inner product in the new
theory’s $(\tau,q)$ coordinate system. Thus, given our above choice of $J$ the
new theory’s induced structures all seem to fit very well on its new spacetime
manifold. It should be stressed, however, that this is not guaranteed to
happen in general. For instance, a different choice of $J$ might induce a
different inner product, $\,\langle,\rangle_{\text{new}}$, than the standard
$L^{2}$ inner product. Alternatively, suppose that (like the Quartic Klein-
Gordon theory) the non-local QHO theory had come equipped with a pointwise
product operation, $\times_{\text{old}}$, on $\mathcal{M}_{\text{old}}$. In
this case, the induced product operation in the new theory,
$\times_{\text{new}}$, would have been a convolutional product on
$\mathcal{M}_{\text{new}}$. It is important to note that the new theory’s
induced structures follow solely from our choice of injective map,
$J:S_{\text{neutral}}^{\text{kin}}\to S_{\text{new}}^{\text{all}}$. They are
not arrived at by identifying which operations are natural (e.g., pointwise or
local) on $\mathcal{M}_{\text{new}}$. Although, of course, our choice of PSTOs
might be guided by whether or not a nice $J$ map will be available to us.

###  4.1 But are we forced to have $J$ implement a Fourier transform?

The remainder of this section will discuss in more generality the latter half
of the externalization process (i.e., picking an injector
$J:S_{\text{neutral}}^{\text{kin}}\to S_{\text{new}}^{\text{all}}$ given some
PSTOs). The following questions will be answered in turn: What are the minimal
constraints must our choice of $J$ satisfy? For a generic set of PSTOs, how do
we know that such a $J$ map exists? Given that such a $J$ map exists, to what
extent is it uniquely determined by our PSTOs?

Let us begin by placing some constraints on our choice of $J$. Recall that our
goal is to build a new spacetime setting for the theory in question such that
every pre-spacetime translation operation, $g\in G_{\text{trans}}$, becomes an
honest-to-goodness spacetime translation in the new theory. We have so far
seen how externalization generates a diffeomorphism
$H(g)\in\text{Diff}(\mathcal{M}_{\text{new}})$ from every $g\in
G_{\text{trans}}$. In order for this to truly count as a spacetime
translation, however, it needs to act in the right way on the new theory’s
states, $\varphi_{\text{new}}=J(\bm{\varphi})$. Establishing this will require
that we demand some compatibility between our choice of injector $J$ and our
choice of PSTOs.

For any Lie group action $g\in G_{\text{trans}}$, we already know how it acts
on any spacetime-neutral state, $\bm{\varphi}\in
S_{\text{neutral}}^{\text{kin}}$. Namely, it acts as $\bm{\varphi}\mapsto
g\,\bm{\varphi}$. In addition to converting every $g\in G_{\text{trans}}$ into
a diffeomorphism $H(g)\in
H_{\text{trans}}\subset\text{Diff}(\mathcal{M}_{\text{new}})$, externalization
also converts every spacetime-neutral states, $\bm{\varphi}\in
S_{\text{neutral}}^{\text{kin}}$ into a state on the new spacetime,
$\varphi_{\text{new}}=J(\bm{\varphi})\in S_{\text{new}}^{\text{kin}}$. Our
choice of $J$ must be such that these two externalization maps, $g\mapsto
H(g)$ and $\bm{\varphi}\mapsto J(\bm{\varphi})$, are compatible:

$\displaystyle\forall g\ \forall\bm{\varphi}\ \ (J\circ g)\,\bm{\varphi}$
$\displaystyle=(H(g)^{*}\circ J)\,\bm{\varphi}.$ (25)

with $g$ and $\bm{\varphi}$ ranging over $G_{\text{trans}}$ and
$S_{\text{neutral}}^{\text{kin}}$ respectively. Demanding Eq. (25) of our
injector $J$ ensures that externalization preserves not only the Lie group
structure of $G_{\text{trans}}$, but also its structure as a Lie group action.

We must, however, demand slightly more than Eq. (25) in order to ensure our
desired result, i.e., that the action of $g\in G_{\text{trans}}$ on
$S_{\text{neutral}}^{\text{kin}}$ is faithfully represented by the action of
$H(g)^{*}\in H_{\text{trans}}{}^{*}$ on $S_{\text{neutral}}^{\text{kin}}$.
Given only Eq. (25) we could conceivably presently have distinct $g_{1}\neq
g_{2}$ which correspond to identical actions on
$S_{\text{neutral}}^{\text{kin}}$, namely
$H(g_{1})^{*}|_{\text{kin}}=H(g_{2})^{*}|_{\text{kin}}$. If this were to
happen, then $H(g)^{*}$’s action would not be a faithful representation of the
$g$’s action. In order to avoid this we must demand that for all
diffeomorphisms, $h\in
H_{\text{trans}}\subset\text{Diff}(\mathcal{M}_{\text{new}})$, we have,

$\displaystyle\Big{(}\forall\bm{\varphi}\ \
h^{*}J(\bm{\varphi})=J(\bm{\varphi})\Big{)}\ \ \Longleftrightarrow\ \
h=\mathbbm{1},$ (26)

with $\bm{\varphi}$ ranging over $S_{\text{neutral}}^{\text{kin}}$. One can
easily check that our above-discussed choice of $J$ (i.e., the one which
implements a Fourier transform) satisfies these two conditions.

To review, externalization builds a new spacetime setting for our theory from
the following materials: A pair of Lie groups, $G_{\text{trans}}$ and
$G_{\text{fix}}$, acting on $\mathcal{S}_{\text{neutral}}^{\text{kin}}$ with
faithful construction compatibility; a value space,
$\mathcal{V}_{\text{new}}$, for the new theory; and an injective map,
$J:\mathcal{S}_{\text{neutral}}^{\text{kin}}\to S_{\text{new}}^{\text{all}}$.
This construction is considered successful if Eq. (25) and Eq. (26) are
satisfied. But given some pair of Lie group actions, $G_{\text{trans}}$ and
$G_{\text{fix}}$, how do we know that such value spaces,
$\mathcal{V}_{\text{new}}$, and injective maps, $J$, exist? Their existence is
guaranteed by demanding that $G_{\text{trans}}$ and $G_{\text{fix}}$ have
injection compatibility.

> Definition: Injection Compatibility222A keen reader may note that what is
> demanded in Eq. (28) is slight more than what is needed to satisfy Eq. (26).
> The former quantifies over more diffeomorphisms than the latter does. As
> Grimmer (2023) discusses, making this only very slightly stronger demand
> greatly simplifies our analysis of the ISE Methodology. \- Let $G_{1}$ be a
> Lie group which acts smoothly on some space
> $\mathcal{S}_{\text{N}}=(S_{\text{N}},\mathcal{T}_{\text{N}},\dots)$. Let
> $G_{2}\subset G_{1}$ be a closed subgroup thereof. Let
> $\mathcal{M}_{\text{G}}\coloneqq G_{1}/G_{2}$ be the quotient manifold of
> $G_{2}$ and $G_{1}$. For any value space, $\mathcal{V}_{\text{G}}$, let
> $S_{\text{G}}^{\text{all}}$ be the space of all
> $\mathcal{V}_{\text{G}}$-valued functions definable on
> $\mathcal{M}_{\text{G}}$.
>
> This pair of Lie group actions, $G_{2}\subset G_{1}$, on
> $\mathcal{S}_{\text{N}}$ have injection compatibility if and only if the
> following condition holds. There exists a value space,
> $\mathcal{V}_{\text{G}}$, and an injective map,
> $J_{\text{G}}:S_{\text{N}}\to S_{\text{G}}^{\text{all}}$, with the following
> two properties: Firstly, we must have,
>
> $\displaystyle\forall g\ \forall\bm{\sigma}\ \ \Big{(}(J_{\text{G}}\circ
> g)\,\bm{\sigma}=(\bar{g}^{*}\circ J_{\text{G}})\,\bm{\sigma}\Big{)},$ (27)
>
> and, secondly, for all diffeomorphisms,
> $d\in\text{Diff}(\mathcal{M}_{\text{G}})$, we must have,
>
> $\displaystyle\Big{(}\forall\bm{\sigma}\ \
> d^{*}J_{\text{G}}(\bm{\sigma})=J_{\text{G}}(\bm{\sigma})\Big{)}\ \
> \Longleftrightarrow\ \ d=\mathbbm{1}.$ (28)
>
> In both of these expressions $g$ ranges over $G_{1}$ and $\bm{\sigma}$
> ranges over $\mathcal{S}_{\text{N}}$.

Together with faithful construction compatibility, we are now demanding the
following from our PSTOs. From $G_{\text{trans}}$ and $G_{\text{fix}}$ we can
build a quotient manifold
$\mathcal{M}_{\text{G}}=G_{\text{trans}}/G_{\text{fix}}$ such that: 1) the
acted-upon space, $S_{\text{neutral}}^{\text{kin}}$, can be nicely mapped onto
scalar field states on $\mathcal{M}_{\text{G}}$; and 2) $G_{\text{trans}}$ is
faithfully represented as diffeomorphisms on $\mathcal{M}_{\text{G}}$ which
act on these states in the right way. This justifies the following gloss of
the externalization process: Externalization builds a new spacetime by taking
some hand-picked pre-spacetime translation operations, $G_{\text{fix}}\subset
G_{\text{trans}}$ acting on $S_{\text{neutral}}^{\text{kin}}$, and letting
them be honest-to-goodness spacetime translations, $H_{\text{fix}}\subset
H_{\text{trans}}\subset\text{Diff}(\mathcal{M}_{\text{new}})$, acting on
$S_{\text{new}}^{\text{kin}}$.

Now that we have identified the general conditions which our choice injective
map $J:S_{\text{neutral}}^{\text{kin}}\to S_{\text{new}}^{\text{all}}$ must
satisfy (namely, Eq. (25) and Eq. (26)), we are in a position to ask: Were any
other options for $J$ were available to us in externalizing the non-local QHO
theory? Recall that prior to our choice of $J$ we had already constructed
$\mathcal{M}_{\text{new}}$ from our Fourier-shift PSTOs and noted its close
relationship with the old theory’s Fourier space. Hence, we are really asking
the following: Is there any way to populate the new theory’s spacetime (i.e.,
a copy of the old theory’s Fourier space) other than with the Fourier
transform of the old theory’s states?

The remainder of this section will be spent showing that (up to a
metaphysically irrelevant rescaling freedom) the above-discussed constraints
actually force the intuitive choice of $J$ upon us. To see this, let us take

$\displaystyle\varphi_{\text{old}}^{(\lambda_{1},\lambda_{2})}(t,x)\coloneqq\exp(\mathrm{i}\,\lambda_{1}\,t)\,\delta(x-\lambda_{2})$
(29)

and note that the Fourier transform of this state,

$\displaystyle\tilde{\varphi}_{\text{old}}^{(\lambda_{1},\lambda_{2})}(t,k)=\exp(\mathrm{i}\,\lambda_{1}\,t)\exp(\mathrm{i}\,\lambda_{2}\,k)$
(30)

is a planewave in the old theory’s Fourier space. Namely,
$\tilde{\varphi}_{\text{old}}^{(\lambda_{1},\lambda_{2})}(t,k)$ is a
simultaneous eigenvector of both $\partial_{t}$ and $\partial_{k}$ with
eigenvalues $\mathrm{i}\lambda_{1}$ and $\mathrm{i}\lambda_{2}$ respectively.
These differential operators are, of course, the generators of translations in
the old theory’s Fourier space; We have
$f(\mathrm{d}t,0)=\mathbbm{1}+\partial_{t}$ and
$f(0,\mathrm{d}k)=\mathbbm{1}+\partial_{k}$. Note that
$\varphi_{\text{old}}^{(\lambda_{1},\lambda_{2})}(t,x)$ is a simultaneous
eigenvector of $f(\mathrm{d}t,0)$ and $f(0,\mathrm{d}k)$ with eigenvalues
$1+\mathrm{i}\lambda_{1}$ and $1+\mathrm{i}\lambda_{2}$ respectively.

Per the above discussion, for every Fourier shift in the old theory, $f(\Delta
t,\Delta k)$, there is a corresponding diffeomorphism on
$\mathcal{M}_{\text{new}}$, namely $h(\Delta t,\Delta k)$. In particular,
corresponding to the infinitesimal Fourier shifts, $f(\mathrm{d}t,0)$ and
$f(0,\mathrm{d}k)$, there are corresponding infinitesimal diffeomorphisms on
$\mathcal{M}_{\text{new}}$, namely $h(\mathrm{d}t,0)$ and $h(0,\mathrm{d}k)$.
Lifting these diffeomorphisms in the new spacetime setting we have
$h(\mathrm{d}t,0)^{*}=\mathbbm{1}+\partial_{\tau}$ and
$h(0,\mathrm{d}k)^{*}=\mathbbm{1}+\partial_{q}$. Eq. (25) demands that these
lifted diffeomorphisms act on the new theory’s states in the same way that the
above-discussed Fourier shifts acted on the old theory’s states. In
particular, since $\varphi_{\text{old}}^{(\lambda_{1},\lambda_{2})}$ is a
simultaneous eigenvector of $f(\mathrm{d}t,0)$ and $f(0,\mathrm{d}k)$, it
follows that the new state,

$\displaystyle\varphi_{\text{new}}^{(\lambda_{1},\lambda_{2})}\coloneqq
J\circ\mathcal{F}_{\text{vec}}(\varphi_{\text{old}}^{(\lambda_{1},\lambda_{2})}),$
(31)

must be a simultaneous eigenvector of $h(\mathrm{d}t,0)^{*}$ and
$h(0,\mathrm{d}k)^{*}$. Concretely, we have,

$\displaystyle\partial_{\tau}\,\varphi_{\text{new}}^{(\lambda_{1},\lambda_{2})}=\mathrm{i}\lambda_{1}\,\varphi_{\text{new}}^{(\lambda_{1},\lambda_{2})}\quad\text{and}\quad\partial_{q}\,\varphi_{\text{new}}^{(\lambda_{1},\lambda_{2})}=\mathrm{i}\lambda_{2}\,\varphi_{\text{new}}^{(\lambda_{1},\lambda_{2})}$
(32)

That is, $\varphi_{\text{new}}^{(\lambda_{1},\lambda_{2})}$ must be a
simultaneous eigenvector of $\partial_{\tau}$ and $\partial_{q}$ with
eigenvalues $\mathrm{i}\lambda_{1}$ and $\mathrm{i}\lambda_{2}$. Namely,
$\varphi_{\text{new}}^{(\lambda_{1},\lambda_{2})}$ must be a planewave in the
new theory’s spacetime.

But what does this tell us about which injective maps $J$ are allowed? Recall
that $\varphi_{\text{old}}^{(\lambda_{1},\lambda_{2})}(t,x)$ is a planewave in
the old theory’s Fourier space. We thus have a planewave-to-planewave
correspondence between the old theory’s Fourier space and the new theory’s
spacetime. Concretely, Eq. (25) enforces that we have

$\displaystyle\varphi_{\text{new}}^{(\lambda_{1},\lambda_{2})}(\tau,q)$
$\displaystyle=a(\lambda_{1},\lambda_{2})\,\exp(\mathrm{i}\,\lambda_{1}\,\tau)\,\exp(\mathrm{i}\,\lambda_{2}\,q)$
(33) $\displaystyle=a(\lambda_{1},\lambda_{2})\
\tilde{\varphi}_{\text{old}}^{(\lambda_{1},\lambda_{2})}(\tau,q)$

for some scaling parameter $a(\lambda_{1},\lambda_{2})$. A quick calculation
shows that a generic $\varphi_{\text{old}}(t,x)\in
S_{\text{old}}^{\text{kin}}$ (which has Fourier transform,
$\tilde{\varphi}_{\text{old}}(t,k)$) must be mapped to

$\displaystyle
J\circ\mathcal{F}_{\text{vec}}:\varphi_{\text{old}}\mapsto\varphi_{\text{new}}(\tau,q)$
$\displaystyle=A(\tau,q)*_{\text{conv}}\tilde{\varphi}_{\text{old}}(\tau,q)$
(34)

for some profile $A(\tau,q)$. Recalling that convolution in space corresponds
to point-wise multiplication in Fourier space, this $A(\tau,q)$ freedom
captures our ability to freely rescale each of the new theory’s Fourier modes
via the above-discussed $a(\lambda_{1},\lambda_{2})$. Note that our intuitive
choice of $J$ (namely, Eq. (22)) corresponds to picking a scaling of
$a(\lambda_{1},\lambda_{2})=1$ for each of the new theory’s planewaves and
hence a profile $A(\tau,q)=1$. Up to this rescaling freedom, there is only one
way to map the old theory’s states onto the new theory’s spacetime (namely,
via a Fourier transform).

But what should we make of this rescaling freedom? I will now argue that this
planewave-rescaling freedom is metaphysically irrelevant and that it is merely
a superficial aspect of how we are representing the theory’s states in the new
spacetime setting. To see this, recall that the non-local QHO theory came
equipped with a metaphysically relevant inner product,
$\langle\varphi_{\text{old}},\phi_{\text{old}}\rangle_{\text{old}}$, for
comparing the sizes and similarity of states. In particular, this was the
$L^{2}$ product in the old theory’s $(t,x)$ coordinate system. Note that
according to this inner product, the old theory’s planewave states,
$\varphi_{\text{old}}^{(\lambda_{1},\lambda_{2})}(t,x)$, are normalized (to a
Dirac delta).

Post-internalization a metaphysically relevant inner product,
$\langle\bm{\varphi},\bm{\phi}\rangle_{\text{neutral}}$ is induced on
$V_{\text{neutral}}^{\text{kin}}$. Post-externalization a metaphysically
relevant inner product,
$\langle\varphi_{\text{new}},\phi_{\text{new}}\rangle_{\text{new}}$ is induced
on $V_{\text{new}}^{\text{kin}}$. Perhaps surprisingly, according to this new
inner product, the new planewave states
$\varphi_{\text{new}}^{(\lambda_{1},\lambda_{2})}(\tau,q)$ are normalized no
matter what $a(\lambda_{1},\lambda_{2})$ we choose. Thus, according to the
metaphysically relevant way of judging the relative size and similarity of the
new theory’s states our planewave-rescaling freedom simply does not matter.

As the above discussion has shown, the externalization process is driven
almost entirely by our choice of PSTOs. These determine not only the new
theory’s spacetime manifold (up to diffeomorphism) but also how the old
theory’s states are to be mapped onto this new spacetime setting (up to some
metaphysically irrelevant rescaling of the new theory’s Fourier modes).

##  5 Example Application: Moving from a Lattice Theory to a Continuous
Spacetime Theory

This section will show how one can use the ISE Methodology to redescribe a
lattice theory (i.e., a theory set on a discrete spacetime,
$\mathcal{M}_{\text{old}}\cong\mathbb{R}\times\mathbb{Z}$) as existing on a
continuous spacetime manifold,
$\mathcal{M}_{\text{new}}\cong\mathbb{R}\times\mathbb{R}$. Before this,
however, some mathematical definitions are needed relating to various discrete
approximations of the derivative.

Consider the nearest-neighbor and next-to-nearest-neighbor approximations of
the second derivative,

$\displaystyle\Delta_{(1)}^{2}/\epsilon^{2}:\ \partial_{x}^{2}f(x)$
$\displaystyle\approx\frac{f(x+\epsilon)-2f(x)+f(x-\epsilon)}{\epsilon^{2}}$
(35) $\displaystyle\Delta_{(2)}^{2}/\epsilon^{2}:\ \partial_{x}^{2}f(x)$
$\displaystyle\approx\frac{-f(x+2\epsilon)+16\,f(x+\epsilon)-30\,f(x)+16\,f(x-\epsilon)-f(x+2\epsilon)}{12\,\epsilon^{2}}.$

Closely, associated with these derivative approximations are the following two
infinite matrices,

$\displaystyle\Delta_{(1)}^{2}\coloneqq\text{Toeplitz}($
$\displaystyle\dots,0,0,0,0,1,{\bf-2},1,0,0,0,0,\dots)$ (36)
$\displaystyle\Delta_{(2)}^{2}\coloneqq\text{Toeplitz}($
$\displaystyle\dots,0,0,0,\frac{-1}{12},\frac{16}{12},{\bf\frac{-30}{12}},\frac{16}{12},\frac{-1}{12},0,0,0,\dots).$

Toeplitz matrices are the so-called diagonal-constant matrices with
$[A]_{i,j}=[A]_{i+1,j+1}$. Thus, the values listed in the above expression
give the matrix’s values on either side of the main diagonal. The value in the
middle of the list (in bold) corresponds to the main diagonal.

These two infinite matrices can be easily generalized to, $\Delta_{(n)}^{2}$,
the Toeplitz matrix related to the $\text{(next-to)}^{n}$-nearest-neighbor
approximation of the second derivative. Taking the limit $n\to\infty$ one can
define,

$\displaystyle D^{2}\coloneqq\text{Toeplitz}($
$\displaystyle\dots,\frac{2}{25},\frac{-2}{16},\frac{2}{9},\frac{-2}{4},\frac{2}{1},{\bf\frac{-2\pi^{2}}{6}},\frac{2}{1},\frac{-2}{4},\frac{2}{9},\frac{-2}{16},\frac{2}{25},\dots),$
(37)

as the infinite Toeplitz matrix related to the infinite-range discrete
approximation of the second derivative. One can similarly define an infinite
Toeplitz matrix, $D$, related to the infinite-range discrete approximation of
the first derivative,

$\displaystyle D\coloneqq\text{Toeplitz}($
$\displaystyle\dots,\frac{-1}{5},\frac{1}{4},\frac{-1}{3},\frac{1}{2},-1,{\bf
0},1,\frac{-1}{2},\frac{1}{3},\frac{-1}{4},\frac{1}{5},\dots).$ (38)

As it turns out, $D^{2}$ is the square of $D$.

For later reference, it should be noted how the above four Toeplitz matrices
act on the discrete planewaves,
$\varphi_{\text{old}}^{(\omega,k)}(t,n)\coloneqq\exp(\mathrm{i}\omega
t+\mathrm{i}k\,n)$, with $\omega\in\mathbb{R}$ and $k\in[-\pi,\pi]$,333Note
that these discrete planewaves are period in their wavenumber $k$ with a
period of $2\pi$. Namely, it follows from Euler’s identity that
$\varphi_{\text{old}}^{(\omega,k+2\pi)}$ and
$\varphi_{\text{old}}^{(\omega,k)}$ are literally the same functions of
$(t,n)$.

$\displaystyle\Delta_{(1)}^{2}\,\varphi_{\text{old}}^{(\omega,k)}$
$\displaystyle=(2\cos(k)-2)\ \varphi_{\text{old}}^{(\omega,k)}$ (39)
$\displaystyle\Delta_{(2)}^{2}\,\varphi_{\text{old}}^{(\omega,k)}$
$\displaystyle=\frac{1}{12}(-2\cos(2k)+32\cos(k)-30)\
\varphi_{\text{old}}^{(\omega,k)}$ $\displaystyle
D^{2}\,\varphi_{\text{old}}^{(\omega,k)}$ $\displaystyle=-k^{2}\
\varphi_{\text{old}}^{(\omega,k)}$ $\displaystyle
D\,\varphi_{\text{old}}^{(\omega,k)}$ $\displaystyle=\mathrm{i}k\
\varphi_{\text{old}}^{(\omega,k)}.$

Notably, the discrete planewave, $\varphi_{\text{old}}^{(\omega,k)}$, is an
eigenvector of each of these linear operators with different eigenvalues.
Incredibly, the eigenvalues for $D^{2}$ and $D$ are $-k^{2}$ and $\mathrm{i}k$
exactly matching the eigenvalues of the continuum derivatives,
$\partial_{x}^{2}$ and $\partial_{x}$.

Given these definitions, we can now state the following three lattice
theories:444In these theories, space is discrete whereas time is continuous.
The ISE Methodology can also be applied when both space and time are discrete.
For instance, in Grimmer (2022a) these techniques are applied to a discrete
version of the Klein-Gordon equation.

> Heat Equation on a Discrete Spacetime \- Consider a spacetime theory about a
> scalar field,
> $\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$,
> with spacetime, $\mathcal{M}_{\text{old}}\cong\mathbb{R}\times\mathbb{Z}$,
> and value space, $\mathcal{V}_{\text{old}}\cong\mathbb{R}$. This theory’s
> states are subject to the following kinematic constraint: In some fixed
> global coordinate system, $(t,n)$, each field must be smooth. In this
> coordinate system, the metaphysically relevant way to judge the relative
> size and similarity of this theory’s states is with an $L^{2}$ inner
> product. There are three variants of this theory obeying the following three
> dynamical equations respectively:
>
> $\displaystyle{\bf H1:}$
> $\displaystyle\quad\frac{\mathrm{d}}{\mathrm{d}t}\vec{\varphi}_{\text{old}}(t)=\alpha\,\Delta_{(1)}^{2}\,\vec{\varphi}_{\text{old}}(t)$
> (40) $\displaystyle{\bf H2:}$
> $\displaystyle\quad\frac{\mathrm{d}}{\mathrm{d}t}\vec{\varphi}_{\text{old}}(t)=\alpha\,\Delta_{(2)}^{2}\,\vec{\varphi}_{\text{old}}(t)$
> $\displaystyle{\bf H3:}$
> $\displaystyle\quad\frac{\mathrm{d}}{\mathrm{d}t}\vec{\varphi}_{\text{old}}(t)=\alpha\,D^{2}\,\vec{\varphi}_{\text{old}}(t)$
>
> for some $\alpha>0$.

These are the nearest neighbor (H1), next-to-nearest neighbor (H2), and
infinite-range (H3) discrete heat equations respectively.

For brevity, the internalization of this theory must be skipped over. It
proceeds along very similar grounds to the internalization of the non-local
QHO theory discussed at the end of Sec. 3. After internalization, we next
search for good PSTOs to externalize. Perhaps surprisingly, these lattice
theories also admit continuous pre-spacetime translation operations. In
particular, these are the smooth transformations sketched in the third column
of Fig. 1. As this example will show, by externalizing these PSTOs we can find
new continuous spacetime settings for our formerly discrete theories.

The applicability of (an early version of) the ISE Methodology to these
discrete spacetime theories was first demonstrated in Grimmer (2022b, a). What
follows is a condensed and updated version of this treatment. My analysis of
these discrete spacetime theories is largely inspired by the work of the
mathematical physicist Achim Kempf.555For an overview of Kempf’s work on these
topics see Kempf (2018). See, for instance, Kempf (2010) entitled, “Spacetime
could be simultaneously continuous and discrete, in the same way that
information can be” where such a continuous-to-discrete correspondence is
understood using the Nyquist-Shannon sampling theory.

But how is it that the above-discussed lattice theories can have both
continuous and discrete spacetime representations? Consider first the discrete
shift diffeomorphism, $d_{+m}:(t,n)\mapsto(t,n+m)$ for $m\in\mathbb{Z}$,
acting on the old theory’s spacetime manifold, $\mathcal{M}_{\text{old}}$. Due
to the discreteness of the old spacetime manifold, we cannot promote
$m\in\mathbb{Z}$ to a continuous parameter, $\epsilon\in\mathbb{R}$. Next,
consider lifting these discrete shift maps to act on $\varphi_{\text{old}}\in
V_{\text{old}}^{\text{kin}}$ as $d_{+m}^{*}:V_{\text{old}}^{\text{kin}}\to
V_{\text{old}}^{\text{kin}}$. Unlike the old spacetime,
$\mathcal{M}_{\text{old}}$, the state space, $V_{\text{old}}^{\text{kin}}$, is
continuous; Hence, if we focus on the lifted maps, $d_{+m}^{*}$, we might be
able to promote $m\in\mathbb{Z}$ to a continuous parameter,
$\epsilon\in\mathbb{R}$. This is how we will find our continuous PSTOs.

Recall that $D$ (defined in Eq. (38)) is the best infinite-range discrete
approximation of the derivative. Consider the smooth transformations on
$V_{\text{old}}^{\text{kin}}$ given by $\text{exp}(-\epsilon\,D)$ for
$\epsilon\in\mathbb{R}$. It turns out that for $\epsilon=m\in\mathbb{Z}$ we
have $\text{exp}(-\epsilon\,D)=d_{+m}^{*}$. That is, the continuous family of
smooth transformation, $\text{exp}(-\epsilon D)$, nicely interpolates between
the discrete shift operations. It is the smooth action of these
$\text{exp}(-\epsilon D)$ maps which is shown in the third column of Fig. 1.

Let us now construct a set of continuous PSTOs. Let the maps,

$\displaystyle f(\Delta
t,\epsilon)\,\vec{\varphi}_{\text{old}}(t)\coloneqq\text{exp}(-\epsilon
D)\,\vec{\varphi}_{\text{old}}(t-\Delta t),$ (41)

form a Lie group $f(\Delta t,\epsilon)\in
F_{\text{Cont}}\cong(\mathbb{R}^{2},+)$ which acts on
$V_{\text{old}}^{\text{kin}}$. These transformations faithfully survive
internalization becoming a Lie group $g(\Delta t,\epsilon)\in G_{\text{Cont}}$
acting on $V_{\text{neutral}}^{\text{kin}}$. Let us take
$G_{\text{trans}}^{\text{(ext)}}=G_{\text{Cont}}\cong(\mathbb{R}^{2},+)$ to be
our PSTOs. As in the Fourier example, it follows from
$G_{\text{trans}}^{\text{(ext)}}$ being abelian that we must pick
$G_{\text{fix}}^{\text{(ext)}}=\\{\mathbbm{1}\\}$.

Constructing a new spacetime manifold from these PSTOs we have

$\displaystyle\mathcal{M}_{\text{new}}\cong
G_{\text{trans}}^{\text{(ext)}}/G_{\text{fix}}^{\text{(ext)}}\cong\mathbb{R}^{2}.$
(42)

By construction, the Lie group in the numerator, $g(\Delta t,\epsilon)\in
G_{\text{Cont}}$, will be faithfully represented as a diffeomorphism,
$h(\Delta t,\epsilon)\coloneqq H(g(\Delta
t,\epsilon))\in\text{Diff}(\mathcal{M}_{\text{new}})$, on the new spacetime.
As in the previous examples, it will be helpful to introduce a coordinate
system associated with these diffeomorphisms. Let $(\tau,q)$ be a coordinate
system for $\mathcal{M}_{\text{new}}\cong\mathbb{R}^{2}$ such that such that
$h(\Delta t,\epsilon)(\tau,q)=(\tau+\Delta t,q+a\,\epsilon)$. Note that I have
defined the $(\tau,q)$-coordinates such that a unit shift of $\epsilon$ in the
old spacetime corresponds to a shift of the $q$-coordinate by a distance $a$
in the new theory. Externalization has turned our choice of
$G_{\text{trans}}=G_{\text{Cont}}$ into honest-to-goodness spacetime
translations on $\mathcal{M}_{\text{new}}$.

We next need to map the old theory’s states and dynamics onto this new
spacetime setting by picking out an injective map,
$J:S_{\text{neutral}}^{\text{kin}}\to S_{\text{new}}^{\text{all}}$, which
obeys Eq. (25) and Eq. (26) are satisfied. Namely, the new diffeomorphisms,
$h(\Delta t,\epsilon)\in\text{Diff}(\mathcal{M}_{\text{new}})$, must lift to
act in the right way on the new theory’s states,
$\varphi_{\text{new}}=J(\bm{\varphi})$. As in the Fourier example, an
eigenvector argument will fix our choice of $J$ up to a metaphysically
irrelevant rescaling of the new theory’s Fourier modes.

In particular, it follows from Eq. (25) that infinitesimal action by $f(\Delta
t,\epsilon)$ in the old theory must correspond to infinitesimal action by
$h(\Delta t,\epsilon)$ in the new theory. It follows from this that a generic
discrete planewave in the old theory’s spacetime, must map onto a
corresponding continuous planewave in the new theory’s spacetime. To see this,
consider a generic discrete planewave in the old theory’s spacetime,

$\displaystyle\varphi_{\text{old}}^{(\omega,k)}(t,n)\coloneqq\exp(\mathrm{i}\omega
t+\mathrm{i}k\,n)$ (43)

Note that this state is a simultaneous eigenvector of $\partial_{t}$ and $D$
with eigenvalues $\mathrm{i}\omega$ and $\mathrm{i}k$ respectively. It follows
from Eq. (25) that the corresponding state in the new theory,
$\varphi_{\text{new}}^{(\omega\,k)}(\tau,q)\coloneqq
J\circ\mathcal{F}_{\text{vec}}(\varphi_{\text{old}}^{(\omega\,k)})$, must be a
simultaneous eigenvector of $\partial_{\tau}$ and $\partial_{q}$ with
eigenvalues $\mathrm{i}\omega$ and $\mathrm{i}k/a$ respectively. Namely, we
must have,

$\displaystyle
J\circ\mathcal{F}_{\text{vec}}:\varphi_{\text{old}}^{(\omega\,k)}\mapsto\varphi_{\text{new}}^{(\omega\,k)}(\tau,q)=b(\omega,k)\
\text{exp}(\mathrm{i}\omega\tau+\mathrm{i}kq/a),$ (44)

for some scaling $b(\omega,k)$. Thus, up to some rescaling freedom, the
externalization process must map discrete planewaves in the old theory’s
spacetime onto a corresponding continuous planewave in the new theory’s
spacetime. Note that the old theory’s discrete planewaves (which are exhausted
by the wavenumbers $k\in[-\pi,\pi]$ and $\omega\in\mathbb{R}$) must become
continuous planewaves with wavenumbers $k/a\eqqcolon\kappa\in[-\pi/a,\pi/a]$.
Hence, all of the new theory’s states are guaranteed to be bandlimited with a
bandwidth of $K\coloneqq\pi/a$: Namely, their support in Fourier space is
limited to $\kappa\in[-K,K]$. Note that this conclusion holds independent of
our choice of scalings, $b(\omega,k)$.

A quick calculation shows that a generic state, $\varphi_{\text{old}}(t,n)\in
V_{\text{old}}^{\text{kin}}$, must be mapped onto

$\displaystyle
J\circ\mathcal{F}_{\text{vec}}:\varphi_{\text{old}}\mapsto\varphi_{\text{new}}(\tau,q)$
$\displaystyle=B(\tau,q/a)*_{\text{conv}}\sum_{n\in\mathbb{Z}}\delta(q/a-n)\
\varphi_{\text{old}}(\tau,n)$ (45)

for some profile $B(\tau,q/a)$ given by the Fourier transform of $b(\omega,k)$
extended to vanish for $k\notin[-\pi,\pi]$. This bandlimited profile is
convolved against a direct embedding of the discrete field state,
$\varphi_{\text{old}}(\tau,n)$, as Dirac deltas onto the new continuous
spacetime.

Just as in the Fourier example, there is effectively only one way to map the
old theory’s states onto the new theory’s spacetime. The only freedom we have
in populating the new spacetime is in a free choice of scalings for each of
the new theory’s planewaves, $b(\omega,k)$ for $\omega\in\mathbb{R}$ and
$k\in[-\pi,\pi]$. But what should we make of this planewave-rescaling freedom?
Just as in the Fourier example, this rescaling freedom is metaphysically
irrelevant if we judge the relative size and similarity of states using the
new theory’s induced inner product, $\langle,\rangle_{\text{new}}$. The only
difference this choice makes is in how the theory’s states are being
represented in this new spacetime setting. Hence, we are free to pick
$b(\omega,k)=1$ for all $\omega\in\mathbb{R}$ and all $k\in[-\pi,\pi]$. Noting
that $b(\omega,k)=0$ outside of this region we have,

$\displaystyle
J\circ\mathcal{F}_{\text{vec}}:\varphi_{\text{old}}\mapsto\varphi_{\text{new}}(\tau,q)$
$\displaystyle=\sum_{n\in\mathbb{Z}}\text{sinc}(q/a-n)\
\varphi_{\text{old}}(\tau,n)$ (46)

where $\text{sinc}(z)\coloneqq\sin(\pi\,z)/(\pi\,z)$.

Continuing with this choice of $J$ one can map the old theory’s dynamics into
this new spacetime setting in order to define the following theories:666See
Grimmer (2022b) for details.

> Bandlimited Heat Equation on a Continuous Spacetime - Consider a spacetime
> theory about a scalar field,
> $\varphi_{\text{new}}:\mathcal{M}_{\text{new}}\to\mathcal{V}_{\text{new}}$,
> with spacetime, $\mathcal{M}_{\text{new}}\cong\mathbb{R}^{2}$, and value
> space, $\mathcal{V}_{\text{new}}\cong\mathbb{R}$. This theory’s states are
> subject to the following kinematic constraint: In some fixed global
> coordinate system, $(\tau,q)$, each field must be smooth in the $\tau$
> direction. Moreover, at each $\tau$-coordinate the field must have a well-
> defined Fourier transform, $\tilde{\varphi}_{\text{new}}(t,\kappa)$, with
> support only over $\kappa\in[-\pi/a,\pi/a]$. In this coordinate system, the
> metaphysically relevant way to judge the relative size and similarity of
> this theory’s states is with an $L^{2}$ inner product. There are three
> variants of this theory obeying the following three dynamical equations
> respectively:
>
> $\displaystyle{\bf H1:}$
> $\displaystyle\quad\partial_{\tau}\varphi_{\text{new}}(\tau,q)=\alpha\,[2\text{cosh}(a\partial_{q})-2]\
> \varphi_{\text{new}}(\tau,q)$ (47) $\displaystyle{\bf H2:}$
> $\displaystyle\quad\partial_{\tau}\varphi_{\text{new}}(\tau,q)=\frac{\alpha}{12}[-2\text{cosh}(2a\partial_{q})+32\text{cosh}(a\partial_{q})-30]\varphi_{\text{new}}(\tau,q)$
> $\displaystyle{\bf H3:}$
> $\displaystyle\quad\partial_{\tau}\varphi_{\text{new}}(\tau,q)=\frac{\alpha}{a^{2}}\,\partial_{q}^{2}\,\varphi_{\text{new}}(\tau,q)$
>
> for some $\alpha>0$.

Note that H3 is the standard continuum heat equation (albeit bandlimited). H1
and H2 are similar to the continuum heat equation, but feature some non-
locality at a length scale of order $\sim a$.

One remarkable feature of these bandlimited heat equations should be noted:
They each have a continuous translation symmetry in their $q$-coordinate. In
the old theory, this corresponds to the fact that action by $\exp(-\epsilon
D)$ is a dynamical symmetry. To see that it maps solutions to solutions,
simply note that each of $\Delta_{(1)}^{2}$, $\Delta_{(2)}^{2}$, $D^{2}$, and
$D$ are all diagonal in the same basis (see Eq. (39)). In hindsight, one can
understand this application of the ISE Methodology as follows: In Eq. (41), we
identified a nice set of dynamical symmetries, $f(\Delta t,\epsilon)\in
F_{\text{Cont}}\cong(\mathbb{R}^{2},+)$, in our old lattice-based theory.
Despite not being spacetime translations these transformations on
$V_{\text{old}}^{\text{kin}}$ are structurally indistinguishable from
spacetime translations. Namely, they are PSTOs. The ISE Methodology allows us
to build a new spacetime setting for our theory such that these PSTOs become
honest-to-goodness spacetime translations. Indeed, since these PSTOs are
dynamical symmetries, they turn into spacetime symmetries of the new theory.

##  6 Conclusion and Outlook

This paper has introduced the ISE Methodology and demonstrated how it can be
applied to two example theories. To review, the ISE Methodology is a three-
step process (Internalize, Search, Externalize). The first step,
internalization, allows one to divorce a theory’s dynamical and kinematical
content from any assumed topological background structure (i.e., the spacetime
manifold, $\mathcal{M}_{\text{old}}$). In the searching step, one then picks
out a set of pre-spacetime translation operations (PSTOs). These are a certain
kind of smooth Lie group action on the theory’s states which are, in a certain
sense, structurally indistinguishable from spacetime translations. An
intuitive example explored in Sec. 4 are translations, not in spacetime, but
rather in Fourier space. Finally, the externalization step then builds a new
spacetime setting, $\mathcal{M}_{\text{new}}$, from these hand-picked pre-
spacetime translation operations such that they become honest-to-goodness
spacetime translations on $\mathcal{M}_{\text{new}}$. In the Fourier example,
the new theory’s spacetime is effectively a copy of the old theory’s Fourier
space. Correspondingly, the new theory’s states and dynamics are naturally
related to those of the old theory by a Fourier transform.

In Sec. 5, the exact same topological redescription techniques were used to
redescribe a lattice theory (i.e., a theory set on a discrete spacetime,
$\mathcal{M}_{\text{old}}\cong\mathbb{R}\times\mathbb{Z}$) as existing on a
continuous spacetime manifold,
$\mathcal{M}_{\text{new}}\cong\mathbb{R}\times\mathbb{R}$. This topological
redescription is facilitated by the fact that the discrete spacetime theory in
question admits a set of continuous pre-spacetime translation operations (see
the third column of Fig. 1). What makes these PSTOs is that, when viewed as a
Lie group action on the theory’s states, these continuous transformations are
structurally identical to spacetime translations. Namely, they are
structurally identical to spacetime translations in exactly the same sense as
the above-discussed Fourier shifts were. Hence, they too qualify as PSTOs. In
addition to being PSTOs, these continuous transformations are also dynamical
symmetries (i.e., they map solutions to solutions). The ISE Methodology allows
us to redescribe this lattice theory in such a way that its hidden dynamical
symmetry is re-expressed as a spacetime symmetry in the new theory.

But of what philosophical use are these topological redescription techniques?
I have here claimed that the ISE Methodology is a powerful tool for
discovering and negotiating between a wide range of spacetime settings for a
given spacetime theory. Namely, as Grimmer (2023) proves, by using the ISE
Methodology one can access effectively every possible topological
redescription of a given theory’s dynamical and kinematical content, limited
only by a weak spacetime-kinematic compatibility condition. Arguably, this
puts our newly discovered capacity for topological redescription on par with
our existing capacity for coordinate change (and for re-axiomization of
theories stated in first-order logic).

If an analogy between these various types of redescription can be developed,
it should become possible to defend an analogous interpretation of them. It is
common to view coordinate systems as a representational artifact which we
project onto the world in the process of codifying the dynamical behavior of
matter. Similarly, on a broadly Humean view of the laws of nature (roughly,
fundamental laws as best-axioms) our theory’s laws are thought to reflect
nothing metaphysically substantial in the world beyond them being one
particularly nice way (among others) of codifying the dynamical behavior of
matter. Aspirationally, the ISE Methodology could be used to support an
analogous view of the spacetime manifold: The spacetime manifolds which
appears in our best scientific theories reflect nothing metsphysically
substantial in the world beyond them being one particularly nice way (among
others) of codifying the dynamical behavior of matter.

### Acknowledgements

The author thanks James Read, Tushar Menon, Nick Huggett, Harvey Brown, Oliver
Pooley, Adam Caulton, and Neils Linneman for their helpful feedback.

### Declarations

The author did not receive support from any organization for the submitted
work. The author has no competing interests to declare that are relevant to
the content of this article.

## References

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https://arxiv.org/abs/2205.07701

  * Grimmer (2022b) Grimmer, D. (2022b).  A discrete analog of general covariance: Could the world be fundamentally set on a lattice?
https://arxiv.org/abs/2204.02276

  * Grimmer (2023) Grimmer, D. (2023).  In search of new spacetimes: Topological redescription and the ISE-equivalence theorem.
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https://doi.org/10.1088/1367-2630/12/11/115001

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https://doi.org/10.1007/s10701-018-0163-2

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##  Appendix A An Introduction to Smooth Homogeneous Manifolds

A smooth manifold, $\mathcal{M}$, is said to be homogeneous simpliciter if
there exists a finite-dimensional Lie group, $G$, which it is homogeneous with
respect to. A smooth manifold, $\mathcal{M}$, is said to be homogeneous with
respect to a finite-dimensional Lie group, $G$, if there exists a group
action, $\theta:G\times\mathcal{M}\to\mathcal{M}$, which acts smoothly and
transitively on $\mathcal{M}$. A group action, $\theta$, is said to act
transitively on $\mathcal{M}$ when for every pair of points,
$p,q\in\mathcal{M}$, there is some $g\in G$ with $\theta(g,p)=q$. That is, a
group action is transitive when it can map any point anywhere.

The intuition behind this definition of homogeneity is that all points on such
a manifold, $\mathcal{M}$, are equivalent from the perspective of the Lie
group, $G$: Concretely, any point $q\in\mathcal{M}$ can be replaced by any
other point $p\in\mathcal{M}$ via action by some $g\in G$. Said differently,
there is only one $G$-orbit, namely the whole manifold $\mathcal{M}$. If $G$’s
action on $\mathcal{M}$ wasn’t transitive, then it would divide $\mathcal{M}$
into several distinct pieces (i.e., the several $G$-orbits). Hence, if $G$’s
action on $\mathcal{M}$ fails to be transitive, then from $G$’s perspective
$\mathcal{M}$ would seem inhomogeneous (i.e., with several different kinds of
points, namely $G$-orbits).

For example, $\mathbb{R}^{2}$ is homogeneous under the group of rigid
translations. Similarly, $S^{2}$ is homogeneous under the group of rigid
rotations. It should be noted that while these two examples feature the
geometrically loaded term “rigid”, the sense in which these manifolds are
homogeneous has nothing to do with their geometry. In particular, homogeneous
does not mean constant curvature. Whether or not a smooth manifold is
homogeneous is solely a matter of topology.

A smooth manifold can fail to be homogeneous on purely topological grounds.
For a simple example consider
$\mathcal{M}\cong\mathbb{T}^{2}\,\dot{\cup}\,\mathbb{R}^{2}$, the disjoint
union of a torus and a two-dimensional plane. No smooth transformation on
$\mathcal{M}$ can map a torus-point to a plane-point. For a less trivial
example of a non-homogeneous manifold, consider the double torus,
$\mathcal{M}\cong\mathbb{T}^{2}\\#\mathbb{T}^{2}$. The non-homogeneity of this
manifold hinges crucially upon us restricting our attention to finite-
dimensional Lie groups.777All Lie groups which act smoothly and transitively
over the double torus are infinite-dimensional. This follows from Mostow
(2005)’s proof that all smooth connected manifolds which support smoothly and
transitively-acting finite-dimensional Lie groups actions have Euler
characteristic, $\chi\geq 0$. Note that the double torus has $\chi=-2$.
Namely, if we were to widen the definition of smooth homogeneous manifolds to
allow for infinite-dimensional Lie groups, then the double torus would count
as homogeneous. In fact, under this extended definition, a smooth manifold is
homogeneous if and only if its connected components are pair-wise
diffeomorphic. For any such manifold, the infinite-dimensional Lie group of
diffeomorphisms, $\text{Diff}(\mathcal{M})$, acts smoothly and transitively
over $\mathcal{M}$. If we make this allowance for infinite-dimensional Lie
groups, then the only way in which a smooth manifold can be inhomogeneous is
by having dissimilar connected components. Note that under this extended
definition every smooth connected manifold would be homogeneous. For
mathematical simplicity, however, this dissertation will use the more
restrictive finite-dimensional definition of homogeneity.

The remainder of this appendix is organized as follows. Sec. A.1 will present
several definitions and theorems regarding smooth homogeneous manifolds which
are of central importance to the ISE Methodology.888For a general introduction
to Lie groups and homogeneous spaces, see Arvanitogeōrgos (2003) particularly
Chapter 4. For more technical details relating to the basic theorems of
differential topology invoked below, see Warner (1983); Brickell and Clark
(1970); Kobayashi and Nomizu (1996). To overview, it is well known that the
quotient of any two Lie groups is a smooth homogeneous
manifold.999Technically, not any pair of Lie groups will do. The group in the
denominator needs to be a closed subgroup of the Lie group in the numerator.
Moreover, it is well known that any homogeneous manifold is diffeomorphic to
the quotient of some two Lie groups. Hence, up to diffeomorphism the smooth
homogeneous manifolds are exactly these Lie group quotients. The theorems
needed for the ISE Methodology, however, are slightly stronger. These improved
theorems (stated below) show that, up to diffeomorphism, any smooth
homogeneous manifold, $\mathcal{M}$, can be faithfully reconstructed as a
quotient of a certain kind of Lie groups. Namely, we can take the Lie groups
to be among $\mathcal{M}$’s diffeomorphisms and, moreover, to have a certain
kind of faithful construction compatibility (defined below). These slightly
stronger theorems will be proved in Sec. A.2 and Sec. A.3. Following this,
Sec. A.4 will give a concrete demonstration of these reconstruction techniques
for the 2-sphere, $\mathcal{M}\cong S^{2}$.

###  A.1 Constructing and Characterizing Smooth Homogeneous Manifolds

I should quickly rehearse two old definitions before making a new one.
Firstly, let $H$ be a group with some group action,
$\theta:H\times\mathcal{M}\to\mathcal{M}$, on a smooth manifold,
$\mathcal{M}$. The stabilizer subgroup of $H$ at a point $p_{0}\in\mathcal{M}$
is the subgroup of $H$ whose group action on $\mathcal{M}$ maps $p_{0}$ to
itself as $\theta(h,p_{0})=p_{0}$. That is, the stabilizer subgroup of $H$ at
$p_{0}$ is the subgroup of $H$ which maps $p_{0}$ onto itself.

Secondly, a group $G_{2}$ is said to be a core-free subgroup of $G_{1}$ if and
only if $G_{2}$ contains no normal subgroups of $G_{1}$ (besides the trivial
group, $\\{\mathbbm{1}\\}$). Note that this implies that $G_{2}$ itself is not
a normal subgroup of $G_{1}$ (unless, of course, $G_{2}$ is itself trivial).

The following definition and theorem show how one can construct smooth
homogeneous manifolds from a pair of Lie groups.

> Definition: Faithful Construction Compatibility -  Let $G_{1}$ and $G_{2}$
> be two finite-dimensional Lie groups. These two Lie groups will be said to
> have faithful construction compatibility if and only if $G_{2}$ is both a
> closed subgroup of $G_{1}$ as well as a core-free subgroup of $G_{1}$.

For later reference, a fact regarding abelian groups should be noted. If
$G_{1}$ is abelian, then all of its subgroups, $G_{2}$, are normal, $G_{2}\lhd
G_{1}$. Hence if $G_{1}$ and $G_{2}$ are supposed to have faithful
construction compatibility and $G_{1}$ is abelian then we must have
$G_{2}=\\{\mathbbm{1}\\}$.

> Homogeneous Manifold Construction Theorem \- Let $G_{1}$ be a Lie group and
> $G_{2}\subset G_{1}$ a closed subgroup thereof. The quotient of these
> groups, $\mathcal{M}_{\text{G}}\coloneqq G_{1}/G_{2}$, is a smooth
> homogeneous manifold upon which $G_{1}$ acts smoothly and transitively via
> the following group action. Any $g\in G_{1}$ acts on any
> $[g_{0}]\in\mathcal{M}_{\text{G}}$ as $\bar{g}$ with,
>
> $\displaystyle\bar{g}([g_{0}])=[g\,g_{0}],$ (48)
>
> where $[g^{\prime}]\in\mathcal{M}_{\text{G}}$ is the $G_{2}$-coset in
> $G_{1}$ containing $g^{\prime}\in G_{1}$. Together, these $\bar{g}$ maps
> form a Lie group of diffeomorphisms on $\mathcal{M}_{\text{G}}$, namely,
> $\overline{G}_{1}\subset\text{Diff}(\mathcal{M}_{\text{G}})$. These
> diffeomorphisms, $\overline{G}_{1}$, will be a faithful representation of
> $G_{1}$ (i.e., we will have $\overline{G}_{1}\cong G_{1}$) if and only if
> $G_{1}$ and $G_{2}$ have faithful construction compatibility.

For proof see Sec. A.2. This construction theorem implies the well-known
result that the quotient of a Lie group with one of its closed subgroups
yields a smooth homogeneous manifold. The added content here is the condition
under which the numerator group will be faithfully represented as
diffeomorphisms on the resulting quotient space.

The following theorem gives the converse result.

> Homogeneous Manifold Characterization Theorem \- Let $\mathcal{M}$ be a
> smooth manifold. $\mathcal{M}$ is homogeneous if and only if there exists a
> finite-dimensional Lie group of its own diffeomorphisms,
> $H_{\text{trans}}\subset\text{Diff}(\mathcal{M})$, which acts transitively
> over $\mathcal{M}$. Let $H_{\text{fix}}$ be any stabilizer subgroup of
> $H_{\text{trans}}$. Any such pair of Lie groups, $H_{\text{trans}}$ and
> $H_{\text{fix}}$, will have faithful construction compatibility. Hence, by
> the Homogeneous Manifold Construction Theorem we can build a smooth manifold
> from them. The resulting manifold is diffeomorphic to the original,
>
> $\displaystyle\mathcal{M}\cong\mathcal{M}_{\text{H}}\coloneqq
> H_{\text{trans}}/H_{\text{fix}}.$ (49)
>
> This reconstruction of $\mathcal{M}$ is robust in the sense that, up to
> diffeomorphism, it does not matter which $H_{\text{trans}}$ and
> $H_{\text{fix}}$ we use.

For proof see Sec. A.3. This characterization theorem implies the well-known
result that every smooth homogeneous manifold, $\mathcal{M}$, can be
reconstructed, up to diffeomorphism, as the quotient of two Lie groups. The
added content here is that we can take these Lie groups to have faithful
construction compatibility and, moreover, to be subgroups of $\mathcal{M}$’s
own diffeomorphisms, $H_{\text{fix}}\subset
H_{\text{trans}}\subset\text{Diff}(\mathcal{M})$. For a concrete demonstration
of this reconstruction technique for the 2-sphere,
$S^{2}\cong\text{SO}(3)/\text{SO}(2)$, see Sec. A.4.

###  A.2 Proof of the Homogeneous Manifold Construction Theorem

This section will prove the Homogeneous Manifold Construction Theorem. Given
any finite-dimensional Lie group, $G_{1}$, and closed subgroup thereof,
$G_{2}\subset G_{1}$, one can build a homogeneous manifold as follows. Taking
the quotient of these Lie groups yields the following quotient space,

$\displaystyle\mathcal{M}_{\text{G}}\coloneqq G_{1}/G_{2}.$ (50)

The equivalence classes under this quotient, $[g]=g\,G_{2}$, are
$G_{2}$-cosets. Note that the quotient between these groups is, in general,
not itself a Lie group. This is because the denominator, $G_{2}$, may not be a
normal subgroup of the numerator, $G_{1}$. The quotient is, however, always a
smooth manifold. In particular, this quotient space naturally adopts a smooth
structure from the Lie group in the numerator, $G_{1}$, via its quotient with
$G_{2}$. This follows from the quotient manifold theorem:101010See Theorem 1
of Zikidis (2016)

> Quotient Manifold Theorem -  Suppose $G$ is a Lie group acting smoothly,
> freely, and properly on a smooth manifold $M$. Then the quotient space $M/G$
> has a unique smooth structure with the property that the quotient map
> $\pi:M\to M/G$ is a smooth submersion. (N.b.: If $M$ is a Lie group and $G$
> is a closed subgroup thereof, then $G$ acts smoothly, freely, and properly
> on $M$.)

In order to apply this to our current situation we take the replacements
$G=G_{2}$ and $M=G_{1}$ such that $M/G=\mathcal{M}_{\text{G}}$. The fact that
$G_{2}$ acts smoothly, freely, and properly on $G_{1}$ follows from $G_{2}$
being a closed subgroup of $G_{1}$. Hence we can grant
$\mathcal{M}_{\text{G}}$ the smooth structure which is uniquely picked out by
this theorem.

In terms of diffeomorphisms on $\mathcal{M}_{\text{G}}$ this means the
following: A generic transformation,
$d_{\text{post}}:\mathcal{M}_{\text{G}}\to\mathcal{M}_{\text{G}}$, is smooth
on $\mathcal{M}_{\text{G}}$ if and only if there exists a smooth
transformation, $d_{\text{pre}}:G_{1}\to G_{1}$, which maps the equivalence
classes onto each other in the same way,

$\displaystyle\pi\circ d_{\text{pre}}=d_{\text{post}}\circ\pi.$ (51)

That is, some permutation of the equivalence classes, $[g]\mapsto
d_{\text{post}}([g])$, will be smooth if and only if there is a way to
implement it smoothly on $G_{1}$ prior to taking the quotient, i.e., to
implement it as $[g]\mapsto[d_{\text{pre}}(g)]$.

This quotient space, $\mathcal{M}_{\text{G}}$, is not only a smooth manifold,
but is also homogeneous with $G_{1}$ acting smoothly and transitively over it.
Namely, the following group action,

$\displaystyle\theta:G_{1}\times\mathcal{M}_{\text{G}}\to\mathcal{M}_{\text{G}}$
(52) $\displaystyle\theta(g,[h])=\bar{g}([h])\coloneqq[g\,h].$

acts smoothly and transitively over $\mathcal{M}_{\text{G}}$. This follows
straightforwardly from the left-action of $G_{1}$ on itself being both smooth
and transitive.

Given that this group action is smooth, the above defined $\bar{g}$ maps are
all diffeomorphisms on $\mathcal{M}_{\text{G}}$. Collecting these together we
have a Lie group
$\bar{g}\in\overline{G_{1}}\subset\text{Diff}(\mathcal{M}_{\text{G}})$. In
some sense, the Lie group $\overline{G_{1}}$ is a representation of $G_{1}$ as
diffeomorphisms on $\mathcal{M}_{\text{G}}$. As I will now prove, this will be
a faithful representation (i.e., $\overline{G_{1}}\cong G_{1}$) if and only
$G_{2}$ is a core-free subgroup of $G_{1}$. (Recall that $G_{2}$ is a core-
free subgroup of $G_{1}$ if and only if $G_{2}$ contains no non-trivial normal
subgroups of $G_{1}$.)

The desired congruence result, $\overline{G_{1}}\cong G_{1}$, is equivalent to
the map $g\mapsto\bar{g}$ having a trivial kernel. Namely, it is equivalent to
the claim that $\bar{s}([g])=[g]$ implies $s=\mathbbm{1}$. But
$\bar{s}([g])=[g]$ is equivalent to $[g^{-1}sg]=[e]$ which is itself
equivalent to $g^{-1}sg\in G_{2}$. Hence our desired congruence result,
$\overline{G_{1}}\cong G_{1}$, is equivalent to the following:

$\displaystyle\forall s\left((\forall g\ \ g^{-1}s\,g\in G_{2})\
\Longrightarrow\ s=\mathbbm{1}\right)$ (53)

with $s$ and $g$ both ranging over $G_{1}$. Let $N_{G_{1}}(G_{2})$ denote the
set of all $s\in G_{1}$ satisfying $\forall g\ \ g^{-1}\,s\,g\in G_{2}$. Note
that $N_{G_{1}}(G_{2})\subset G_{2}$ (consider taking $g=\mathbbm{1}$). Note
also that $N_{G_{1}}(G_{2})\lhd G_{1}$ is a normal subgroup of $G_{1}$. In
fact, $N_{G_{1}}(G_{2})$ is the largest normal subgroup of $G_{1}$ which is
contained within $G_{2}$. This is called the normal core of $G_{2}$ in
$G_{1}$. Hence our desired congruence result, $\overline{G_{1}}\cong G_{1}$,
is equivalent to $N_{G_{1}}(G_{2})={\mathbbm{1}}$. This happens exactly when
$G_{2}$ contains no non-trivial normal subgroups of $G_{1}$. That is, when
$G_{2}$ is a core-free subgroup of $G_{1}$.

###  A.3 Proof of the Homogeneous Manifold Characterization Theorem

This section will prove the Homogeneous Manifold Characterization Theorem. By
definition, a smooth manifold $\mathcal{M}$ is homogeneous if and only if
there exists some finite-dimensional Lie group, $G$, which acts smoothly and
transitively over it. Suppose that $G$ acts smoothly on $\mathcal{M}$ with
action $\theta:G\times\mathcal{M}\to\mathcal{M}$. Using this action we can
associate to each $g\in G$ a diffeomorphism $h\in\text{Diff}(\mathcal{M})$
with $h(p)=\theta(g,p)$. Collecting these diffeomorphisms together we have a
finite-dimensional Lie group of diffeomorphisms,
$H_{\text{trans}}\subset\text{Diff}(\mathcal{M})$. Note that
$H_{\text{trans}}$ acts transitively on $\mathcal{M}$ (because $G$ does). Note
also that $H_{\text{trans}}$ automatically acts smoothly on $\mathcal{M}$
since it is a group of diffeomorphisms on $\mathcal{M}$. Hence, $\mathcal{M}$
is homogeneous if and only if there exists some finite-dimensional Lie group
of its diffeomorphisms, $H_{\text{trans}}\subset\text{Diff}(\mathcal{M})$,
which acts transitively over $\mathcal{M}$.

I will now prove that $H_{\text{trans}}$ has faithful construction
compatibility with any of its stabilizer subgroups $H_{\text{fix}}$ (say the
stabilizer subgroup at $p_{0}$). To begin, let us prove that $H_{\text{fix}}$
is a closed subgroup of $H_{\text{trans}}$. Note that the function
$f_{p_{0}}:H_{\text{trans}}\to\mathcal{M}$ given $f(h)=h(p_{0})$ is
continuous. Next note that $H_{\text{fix}}=f^{-1}(p_{0})$. Since the singleton
$\\{p_{0}\\}$ is a closed set and $f$ is continuous, we have that
$H_{\text{fix}}$ is a closed set.

Next, let us prove that $H_{\text{fix}}$ is a core-free subgroup of
$H_{\text{trans}}$. As noted above, this claim is equivalent to the following:

$\displaystyle\forall s\left((\forall h\ \ h^{-1}s\,h\in H_{\text{fix}})\
\Longrightarrow\ s=\mathbbm{1}_{\mathcal{M}}\right)$ (54)

with $s$ and $h$ both ranging over $H_{\text{trans}}$. This equivalent claim
will now be proved by contrapositive. Note that if
$s\neq\mathbbm{1}_{\mathcal{M}}$ then there exists some $q\in\mathcal{M}$ with
$s(q)\neq q$. Since $H_{\text{trans}}$ acts transitively on $\mathcal{M}$
there is some $h\in H_{\text{trans}}$ which maps $p_{0}$ to $q$ as
$q=h(p_{0})$. For this $h$ we have $(s\,h)(p_{0})\neq q$ whereas $h(p_{0})=q$.
Thus we have $(h^{-1}s\,h)(p_{0})\neq p_{0}$ and therefore $h^{-1}s\,h\not\in
H_{\text{fix}}$ as desired. This completes the proof that there is faithful
construction compatibility between any transitively acting Lie group
$H_{\text{trans}}\subset\text{Diff}(\mathcal{M})$ and any of its stabilizer
subgroups, $H_{\text{fix}}$.

Let us next use the above-proved construction theorem to build a homogeneous
manifold from $H_{\text{trans}}$ and $H_{\text{fix}}$, namely
$\mathcal{M}_{\text{H}}\coloneqq H_{\text{trans}}/H_{\text{fix}}$. According
to the characterization theorem, this should be diffeomorphic to the original
manifold. But how do we know? To establish that $\mathcal{M}_{\text{H}}$ and
$\mathcal{M}$ are diffeomorphic we need to find a bijection between them which
is smooth in both directions (i.e., a diffeomorphism). Before picking out such
a bijection, let us rewrite the quotient space
$\mathcal{M}_{\text{H}}\coloneqq H_{\text{trans}}/H_{\text{fix}}$ using the
orbit stabilizer theorem: The $H_{\text{fix}}$-cosets in $H_{\text{trans}}$
are exactly the equivalence classes of the following equivalence relation,

$\displaystyle h_{2}\equiv_{p_{0}}h_{1}\text{ iff }h_{2}(p_{0})=h_{1}(p_{0}).$
(55)

That is, $h_{2}$ and $h_{1}$ are equivalent under $\equiv_{p_{0}}$ if they map
$p_{0}$ to the same place. Given that $hH_{\text{fix}}=[h]_{p_{0}}$, let us
consider the map,

$\displaystyle P:\mathcal{M}_{\text{H}}$ $\displaystyle\to\mathcal{M}$ (56)
$\displaystyle[h]_{p_{0}}$ $\displaystyle\mapsto h(p_{0}).$

This $P$ map sends each equivalence class, $[h]_{p_{0}}$, onto the point,
$h(p_{0})\in\mathcal{M}$, which all of its elements jointly map $p_{0}$ onto.
Note that no two equivalence classes will send $p_{0}$ to the same place (by
definition). Hence, $P$ is injective. Next note that $P$ is surjective because
$H_{\text{trans}}$ acts transitively on $\mathcal{M}$ (i.e., for any $q$ there
is some $h$ with $h(p_{0})=q$). Hence, $P$ a bijection. Thus we have,

$\displaystyle\mathcal{M}\leftrightarrow\mathcal{M}_{\text{H}}\coloneqq
H_{\text{trans}}/H_{\text{fix}}.$ (57)

But is this bijection $P$ between $\mathcal{M}$ and $\mathcal{M}_{\text{H}}$
also a smooth map? It is, in fact, as can be seen from the following
fundamental theorem of differential geometry:111111See Theorem 3 of Zikidis
(2016)

> Unique Smoothness Theorem: Let $G$, $Q$, and $M$ be smooth manifolds and
> $\pi:G\to Q$ a surjective smooth submersion. If $F:G\to M$ is a smooth map
> that is constant across $\pi$ (that is, $\pi(p)=\pi(q)\Longrightarrow
> F(p)=F(q)$) then there exist a unique smooth map $R:Q\to M$ such that
> $R\circ\pi=F$

In order to apply this to our current situation we take the replacements
$G=H_{\text{trans}}$, $Q=\mathcal{M}_{\text{H}}$, $M=\mathcal{M}$, and
$F(h)=h(p_{0})$. Note that each of $G=H_{\text{trans}}$,
$Q=\mathcal{M}_{\text{H}}$, and $M=\mathcal{M}$ are smooth manifolds by either
definition or construction. The map
$\pi:H_{\text{trans}}\to\mathcal{M}_{\text{H}}$ is a smooth submersion by
construction via the quotient manifold theorem. The $F$ map is constant across
$\pi$ by construction.

Under these replacements, this theorem tells us that there exists a unique
smooth map, $R$, such that $R([h]_{p_{0}})=h(p_{0})$. But this is exactly the
$P$ map defined in Eq. (56). Thus we have that $P=R$ is a smooth map as well
as a bijection. To complete our proof that $P$ is a diffeomorphism, we could
next show that $P^{-1}$ is smooth. Alternatively, we could show that $P$ has a
constant rank. This follows from $P$ being equivariant under
$H_{\text{trans}}$’s joint action on $\mathcal{M}$ and
$\mathcal{M}_{\text{H}}$,

$\displaystyle P([g\,h]_{p_{0}})=g\,h\,p_{0}=g\,P([h]_{p_{0}}).$ (58)

Hence, we have that $P$ is a diffeomorphism and therefore that
$\mathcal{M}\cong\mathcal{M}_{\text{H}}=H_{\text{trans}}/H_{\text{fix}}$ as
smooth manifolds. Indeed, we have
$\mathcal{M}=P(H_{\text{trans}}/H_{\text{fix}})$.

###  A.4 Reconstructing the Sphere from its Diffeomorphisms

To explicitly demonstrate how these theorems work, I will now use them to
reconstruct the 2-sphere, $\mathcal{M}\cong S^{2}$. The reader is invited to
think of $\mathcal{M}\cong S^{2}$ as a time-less version of the spacetime
manifold, $\mathcal{M}_{\text{old}}\cong\mathbb{R}\times S^{2}$, from the
Quartic Klein-Gordon theory introduced in Sec. 3. Correspondingly, the
relevant Lie group of diffeomorphisms for the following discussion,
$H_{\text{trans}}\cong\text{SO}(3)$, is a time-less version of the
$H_{\text{trans}}\cong(\mathbb{R},+)\times\text{SO}(3)$ group discussed at the
end of Sec. 3. The removal of the time dimension from this example is merely
for pedagogical reasons; Removing time allows us to visualize the following
discussion in terms of a nice three-dimensional picture, namely Fig. 2.

In order to reconstruct $\mathcal{M}\cong S^{2}$ from $H_{\text{trans}}$ we
must first identify $H_{\text{fix}}$, the stabilizer subgroup of
$H_{\text{trans}}$. Recall that $H_{\text{trans}}\cong\text{SO}(3)$ is the
group of rigid rotations of $\mathcal{M}\cong S^{2}$ in some coordinate
system. If we take $p_{0}=n$ to be the north pole, then we have that
$H_{\text{fix}}\cong\text{SO}(2)$ is the group rotations about the z-axis.
Plugging $H_{\text{trans}}\cong\text{SO}(3)$ and
$H_{\text{fix}}\cong\text{SO}(2)$ into Eq. (49) we have,

$\displaystyle\mathcal{M}_{\text{H}}\coloneqq
H_{\text{trans}}/H_{\text{fix}}\cong SO(3)/SO(2).$ (59)

According to the construction theorem, this quotient should be a smooth
manifold. Moreover, according to the characterization theorem, it should be
diffeomorphic to our original manifold,
$\mathcal{M}_{\text{H}}\cong\mathcal{M}\cong S^{2}$. Let us now check this,
paying careful attention to where the quotient manifold gets its smooth
structure from. See Fig. 2 for an illustrated guide to how this quotient
works.

Figure 2: The Lie group $\text{SO}(3)\cong\text{RP}^{3}$ is depicted here as a
smooth manifold: Namely, as a ball with radius $\pi$ in $\mathbb{R}^{3}$ with
every pair of antipodal points on its surface identified. Concretely, each
point $r\in\mathbb{R}^{3}$ in this figure represents a rotation of
$\theta=|r|$ radians about the $\hat{r}$ axis (where $\hat{r}$ is the unit
vector pointing at $r$). We can recover the sphere $S^{2}$ from $\text{SO}(3)$
by taking its quotient with respect to a certain equivalence relation,
$\equiv_{p_{0}}$. Two rotations are equivalent according to $\equiv_{p_{0}}$
if they map the north-pole, $p_{0}=n$, to the same place. The colored lines
shown are some of the equivalence classes of $\text{SO}(3)$ under
$\equiv_{p_{0}}$. Note that the equator itself is an equivalence class. For
every equivalence class except this one, we can pick out a unique
representative on the $x,y$-plane (see the black points). Taking these points
as our representatives the quotient space becomes a disk with its boundary
compactified to a single point. Namely, this quotient of $\text{SO}(3)$ by
$\equiv_{p_{0}}$ yield the smooth manifold $\text{SO}(3)/\equiv_{p_{0}}\cong
S^{2}$ as desired.

In Eq. (59) we are supposed to regard the Lie group in the numerator,
$H_{\text{trans}}\cong\text{SO}(3)$, as a smooth manifold: But what smooth
manifold is $\text{SO}(3)$? To see this, consider first its Lie algebra,
$\mathfrak{so}(3)$, and note that it is three-dimensional. The group
exponential from $\mathfrak{so}(3)$ to $\text{SO}(3)$ is surjective such that
each element of $\text{SO}(3)$ is represented (possibly multiple times) in the
image of $\mathfrak{so}(3)$ under the group exponential. As I will now
discuss, by identifying these redundancies we can see $\text{SO}(3)$ as a
certain quotient space of $\mathbb{R}^{3}$.

Fig. 2 illustrates exactly how $\mathfrak{so}(3)$ redundantly represents
$\text{SO}(3)$. In Fig. 2 a point, $r\in\mathbb{R}^{3}$, represents a rotation
of $\theta=|r|$ radians about the $\hat{r}$ axis (where $\hat{r}$ is the unit
vector pointing at $r$). In this way, each rotation in $\text{SO}(3)$ can
represented within the region $|r|\leq\pi$ with the following redundancy:
$\pi\,\hat{r}$ is identical to $-\pi\,\hat{r}$ for every $\hat{r}$. That is, a
half-turn about any axis $\hat{r}$ is the same as a half-turn in the reverse
direction.

For example, rotations of the sphere about the $x$-axis correspond to points
on the $x$-axis in Fig. 2 (e.g., the black points). Rotating about the
$x$-axis through an angle just greater than $\pi$ is equivalent to rotating in
the reverse direction by an angle just less than $\pi$. Hence, the antipodal
points on the $x$-axis in Fig. 2 represent the same rotation and so should be
identified. This holds not just the $x$-axis but for every axis. Thus, as a
smooth manifold $\text{SO}(3)$ is diffeomorphic to the unit ball in
$\mathbb{R}^{3}$ with every pair of antipodal points on its surface
identified. That is, as a smooth manifold we have
$\text{SO}(3)\cong\text{RP}^{3}$, the real-projective space in three
dimensions.

Note that $\text{RP}^{3}$ is not diffeomorphic to the manifold,
$\mathcal{M}=S^{2}$, which we are hoping to recover. Indeed, according to Eq.
(59) the desired manifold is a quotient of this manifold by
$H_{\text{fix}}\cong\text{SO}(2)$. The colored lines in Fig. 2 show some of
these $H_{\text{fix}}$-cosets.121212Note that only a subset of these
$H_{\text{fix}}$-cosets (or equivalently, $\equiv_{p_{0}}$-equivalence
classes) are shown in Fig. 2. Namely, only equivalence classes which contain
rotations about the $x$-axis with incrementally larger values of
$\theta\in[-\pi,\pi]$ are shown.

At this point, it will be helpful to use the orbit-stabilizer theorem to
rewrite the quotient manifold, $\mathcal{M}_{\text{H}}\coloneqq
H_{\text{trans}}/H_{\text{fix}}$, in terms of an equivalence relation. This
theorem tells us that $H_{\text{fix}}$-cosets in $H_{\text{trans}}$ are
identical to the equivalence classes of the following equivalence relation
over $H_{\text{trans}}$:

$\displaystyle h_{2}\equiv_{p_{0}}h_{1}\text{ iff }h_{2}(p_{0})=h_{1}(p_{0}),$
(60)

where $p_{0}=n$ is the point at which $H_{\text{fix}}$ is $H_{\text{trans}}$’s
stabilizer subgroup. In other words, two diffeomorphisms, $h_{1},h_{2}\in
H_{\text{trans}}$ are equivalent under $\equiv_{p_{0}}$ if they map $p_{0}=n$
to the same place. The orbit-stabilizer theorem says that we have