# In Search of New Spacetimes: Topological Redescription and the ISE-
Equivalence Theorem

Daniel Grimmer

###### Abstract

What if it were possible to remove and replace the topological underpinnings
of our spacetime theories just as easily as one can switch between different
coordinate systems? Grimmer (2023) claims this is possible by using his
recently introduced ISE Methodology. Moreover, he claims that 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 some
fundamental substances or relations in the world, we can see 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
is intended to parallel a roughly Humean view of laws and a broadly dynamics-
first view of geometry; 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.

This paper will assess the technical side of this program. Namely, I will
assess the power and scope of the topological redescription techniques offered
to us by the ISE Methodology. I will first identify the range of spacetime
theories which can be output by the ISE Methodology. These include every
spacetime theory which meets a very weak spacetime-kinematic compatibility
condition. Next, I will prove that ISE-relatedness is an equivalence relation
over this range and give an exact characterization of its equivalence classes.
As I will prove, two spacetime theories are ISE-equivalent if and only if 1)
they both have spacetime-kinematic compatibility, and 2) there is a certain
kind of dynamics-preserving isomorphism between their kinematic structures.
Hence, the ISE Methodology allows us to access effectively every possible
spacetime framing of a given theory’s kinematical and dynamical content,
bounded only by a weak spacetime-kinematic compatibility condition. In sum,
the ISE Methodology is a powerful tool for discovering and negotiating between
a vast range of spacetime settings for nearly a given spacetime theory.

##  1 Introduction

In physics, a capacity to freely redescribe our theories in various ways is
often pragmatically useful. Consider our ability to freely switch between
different coordinate systems or to re-axiomatize a theory stated in first-
order logic.111Throughout this paper, it suffices to think of following two
relatively-simple kinds of re-axiomization. Semantically, one set of axioms
$A=\\{A_{1},\dots,A_{n}\\}$ are equivalent to some other set of axioms,
$B=\\{B_{1},\dots,B_{m}\\}$, if and only if they are satisfied by exactly the
same models. Syntactically, $A$ is equivalent to $B$ if and only if for every
$1\leq i\leq n$ and every $1\leq j\leq m$ we have $A\vdash B_{j}$ and $B\vdash
A_{i}$. Past these two simple kinds of logical equivalence, more sophisticated
notions are available. See Barrett and Halvorson (2016) for a discussion of
several alternatives including: definitional equivalence, Morita equivalence,
and categorical equivalence. Philosophically, such capacities for
redescription are useful as they shed light on the age-old question of
scientific realism: Which aspects of our best physical theories should we take
to be reflecting real stuff/structures out there in the world (as opposed to
merely being an artifact of our representational techniques)? Allow me to
briefly review the role that our capacities for axiomatic, coordinate, and
geometric redescription have played in helping us address this question,
before turning to the new topological redescription techniques introduced in
Grimmer (2023), namely the ISE Methodology.

As a first example, consider the following (admittedly non-standard) way of
arriving at a broadly Humean view of the laws of nature.222As Hall (2015)
notes, one ought to distinguish between two aspects of a Humean view of laws.
Firstly, there is Hume’s general ban on necessary connections between distinct
existences. Secondly, there is what Hall (2015) calls the “unofficial guiding
idea” behind Humeanism about laws. Roughly, this is the idea that one ought to
take scientific practice (i.e., arriving at the laws of nature by codifying
the dynamical behavior of matter) and mirror that in our account of the
metaphysics of laws. My focus here is on the codification aspect of Humeanism
about laws rather than its modal aspect. Suppose that one is presented with an
empirically successful physical theory which is stated in first-order logic in
terms of a handful of relatively simple axioms. Several candidates for
ontological commitment may immediately jump out at us: the elements of this
theory’s domain (see Quine (1948)), the properties which are ascribed to those
elements (e.g., understood as Platonic forms or universals), and finally the
axioms themselves (e.g., understood the fundamental laws of nature which
govern the world).

Notice, however, that one can freely re-axiomatize this theory without
changing any of its logical consequences. In light of this, one might grow
suspicious of any ontological status being granted to the theory’s axioms as
fundamental laws of nature.333It should be noted here that my focus is on
axioms-like laws of nature (i.e., fundamental laws) as opposed to theorem-like
laws of nature (i.e., derived laws). Unlike the former, the latter status is
invariant under re-axiomization. Indeed, given our capacity for law-like
redescription one finds some motivation to metaphysically deflate the laws of
nature and to focus instead on the theory’s law-independent content (e.g., the
Humean mosaic, or more generally, the dynamical behavior of matter). From this
law-independent conception of one’s theory, one can then access the complete
range of possible law-like descriptions (most of which will be horrendously
complicated). From this perspective, our theory’s original laws of nature
reflect nothing metaphysically substantial in the world beyond them being one
particularly nice way (among others) of codifying the dynamical behavior of
matter.

As a second example, suppose that one is presented with an empirically
successful spacetime theory which takes a particularly simple form in some
fixed coordinate system, e.g., inertial coordinates. Coordinate systems (more
so than axiom-like laws) are clearly poor candidates for ontological
commitment; They are more obviously a representational device which we project
onto the world. As any good physicist knows, an inelegant choice of
coordinates can lead to coordinate artifacts which simultaneously hide key
bits of physics from view and produce illusory singularities. Fortunately,
these issues can be overcome with a more apt choice of coordinates.

Better yet, one can remove all coordinate artifacts by adopting a stylistic
commitment to general covariance (i.e., a commitment to coordinate-independent
formulations of physics).444I am here following Friedman (1983) in thinking
that adopting the principle of general covariance “merely expresses our
commitment to a certain style of formulating physical theories.” As
Kretschmann pointed out in 1917, any physical theory can be written in a
generally covariant way. See Norton (1993) for a historical review of this
point. Indeed, one can always reformulate one’s spacetime theories in the
coordinate-free language of differential geometry. From this coordinate-free
formulation, one can then access the complete range of possible coordinate
descriptions (most of which will be horrendously complicated). Obviously, our
theory’s original coordinate system reflects nothing metaphysically
substantial in the world beyond it being one particularly nice way (among
others) of codifying the dynamical behavior of matter.

As a bonus, reformulating one’s spacetime theories in a coordinate-free way
has a tendency to reveal their previously hidden geometric structures (e.g.,
$\eta^{\text{ab}}$ in special relativity). These geometric structures are
tempting new candidates for ontological commitment. One might, of course, try
to resist this temptation by developing some geometric redescription
techniques in order to metaphysically deflate these geometric structures (just
as the laws and coordinates were above). See, for instance, the work of
Huggett (2006) and Stevens (2014) on Regularity Relationism. Any such
geometric redescription tools can be employed in support of a dynamics-first
view of geometry:555For a recent overview of the dynamics-first approach to
geometry, see Brown and Read (2018) and references therein, especially O.
Pooley (1999); Brown (2005); Huggett (2006). The fixed geometric structures
which appear in our theories (e.g., $\eta_{\text{ab}}$ in special relativity)
reflect nothing metaphysically substantial in the world beyond them being one
particularly nice way (among others) of codifying the dynamical behavior of
matter. As Norton (2008) notes, however, deflating a theory’s geometric
structures in this way still leaves its topological structure (i.e., the
spacetime manifold itself, $\mathcal{M}$) intact as a serious candidate for
ontological commitment.

In light of Norton’s complaint, it is natural to wonder whether the dynamics-
first view of geometry can be extended to cover the topological structures
which appear ubiquitously in our spacetime theories.666It should be noted
that, throughout this paper, the term “topology” will often be used as a
shorthand for the smooth topology of smooth manifolds. This does not mean,
however, that the smooth manifolds in question need to be continuous or
connected. For instance, consider the group $G=(\mathbb{Z},+)$ equipped with a
discrete topology. This is a 0-dimensional Lie group. Dropping its group
structure yields a 0-dimensional manifold, namely $\mathcal{M}=\mathbb{Z}$.
Note that with this smooth topology, all functions,
$f:\mathcal{M}\to\mathcal{M}$, are continuous as well as smooth. Indeed, this
manifold’s diffeomorphisms, $\text{Diff}(\mathcal{M})$ are exactly its
permutations. As the above discussion has revealed, in a variety of contexts
(e.g., laws, coordinates, geometry) our capacity for redescription impinges
directly upon metaphysical debates regarding what ontological commitments one
ought to draw from our best physical theories. Putting these two points
together, one way to achieve such a dynamics-first view of topology would be
to develop some powerful techniques for topological redescription (ideally
something on par with our capacity for coordinate redescription and re-
axiomization). Grimmer (2023) claims to have done exactly this with his
recently introduced ISE Methodology.

The purpose of this paper is to assess (and ultimately validate) certain
technical aspects of Grimmer (2023)’s claim. Indeed, as Sec. 4 will prove, the
ISE Methodology gives us access to effectively every possible spacetime
framing of a given theory’s kinematical and dynamical content, bounded only by
a weak spacetime-kinematic compatibility condition. Given this over-abundance
of candidate spacetime settings on offer, almost all of them will be
horrendously complicated and ill-suited to the theory in question. But this is
exactly what we want; The same thing is true of our desired analogs:
coordinate redescription and re-axiomization. Indeed, just like our desired
analogs, our new capacity for topological redescription is so complete that if
there exists a particularly good topological framing one’s theory, then it is
accessible to us via the ISE Methodology. Before overviewing how I have
arrived at this assessment, allow me to briefly introduce the ISE Methodology.

###  1.1 Overviewing the ISE Methodology: Internalize, Search, Externalize

The ISE Methodology is a three-step process (Internalize, Search, Externalize,
hence the initials ISE). Firstly, the internalization step allows one to
conceptually divorce a theory’s dynamical and kinematical content from any
assumed topological background structure (i.e., the spacetime manifold,
$\mathcal{M}_{\text{old}}$). This step is similar in spirit to spacetime
algebraicism. In the searching step, one then picks out what Grimmer (2023)
calls a set of pre-spacetime translation operations (PSTOs).777Throughout this
paper, the word “translation operations” ought to be understood quite broadly.
In particular, it should be understood so as to include rotation operations
(e.g., $\text{SO}(3)$ acting on $S^{2}$) as well as arbitrary diffeomorphisms.
These are a certain kind of smooth action on the theory’s states which are, in
a certain sense, structurally indistinguishable from spacetime translations.
An intuitive example would be 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. Grimmer (2023) discusses this
Fourier redescription example at length.

In Grimmer (2023), the exact same topological redescription techniques were
also 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
Fig. 1 in Grimmer (2023)). These smooth continuous transformations are PSTOs
because, when viewed as a Lie group action on the theory’s states, they 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 of the discrete spacetime theory in question (i.e., they
map solutions to solutions). The ISE Methodology allows us to redescribe this
lattice theory in such a way that this hidden dynamical symmetry is re-
expressed as a spacetime symmetry in the new theory. Given this choice of
PSTOs, the old theory’s discrete field states must appear on the new
continuous spacetime setting as bandlimited functions (see Grimmer (2023) for
details). We have as little freedom in mapping the old theory’s states onto
the new theory’s spacetime as we did in the Fourier example; There is simply
no other 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 (see Grimmer (2023) for details).

This paper will consider two more applications of the ISE Methodology in Secs.
3.3 and 3.4. While all of these applications concern theories about scalar
fields, $\varphi:\mathcal{M}\to\mathcal{V}$, it should be noted that the ISE
Methodology is also applicable to theories about particle trajectories,
$x:\mathcal{V}\to\mathcal{M}$, and tensor fields, e.g.,
$\vec{\varphi}:\mathcal{M}\to\text{T}\mathcal{M}$. For brevity, however, this
paper will be focused mainly on the scalar field case. In general, however,
the point remains that two ISE-related spacetime theories differ merely
regarding which smooth transformations of the theory’s states are regarded as
spatiotemporal (i.e., which set of PSTOs are externalized).

###  1.2 Assessing the ISE Methodology: Power and Scope

The aim of this paper is to assess the power and scope of the topological
redescription techniques offered to us by the ISE Methodology. Two key
questions to be answered are as follows: What kinds of spacetime theories can
be taken as input to the ISE Methodology? What kinds of spacetime theories can
be output by the ISE Methodology? Secs. 2 and 3 will answer these questions as
follows. The ISE Methodology can be applied to any spacetime theory which
meets two preliminary assumptions (defined in Sec. 2.1). For brevity, let us
call this scope of spacetime theories the domain of the ISE Methodology. The
spacetime theories which can be output by the ISE Methodology are exactly
those which satisfy these preliminary assumptions and additionally have a
certain level of spacetime-kinematic compatibility (defined in Sec. 2.3). Let
us call this scope of spacetime theories the range of the ISE Methodology.

As I will discuss in Sec. 2.5, spacetime-kinematic compatibility is an
extremely weak assumption. There is a clear sense in which theories lacking
spacetime-kinematic compatibility are unnatural. All such theories exhibit at
least one of the following two flaws: 1) They contain an excessive amount of
spacetime given their kinematics (i.e., they have distinct spacetime points
with exactly matching behavior in every kinematically allowed state), or 2)
they are kinematically inhomogeneous in the sense that the theory’s kinematics
makes a topological distinction between at least two different kinds of
spacetime points. Unsurprisingly, essentially every spacetime theory
considered throughout the history of physics has satisfied these minimal
spacetime-kinematic compatibility criteria. Hence, the domain and the range of
the ISE Methodology are both extremely broad.

Having established both the domain and range of the ISE Methodology, Sec. 4
will define a 2-ary relation over the domain of the ISE Methodology. Namely,
the ISE-relatedness of two spacetime theories is defined by $\text{who}_{1}$
can be redescribed as $\text{who}_{2}$ via the ISE Methodology, at least up to
diffeomorphism. Given that the ISE-Methodology’s domain is larger than its
range (i.e., some inputs cannot be output) ISE-relatedness cannot be an
equivalence relation over its domain. However, as I will prove in Sec. 4, ISE-
relatedness is an equivalence relation over its range. This naturally prompts
the question: Given any spacetime theory within the range of the ISE
Methodology, which other spacetime theories are within its ISE-equivalence
class? Sec. 4 will provide an exact characterization of these ISE-equivalence
classes. Namely, I will there prove the ISE-Equivalence Theorem: Two spacetime
theories are ISE-interrelated if and only if 1) they both theories have
spacetime-kinematic compatibility, and 2) there is a certain kind of dynamics-
preserving isomorphism between their kinematic structures. Hence, the ISE
Methodology is extremely powerful: Using it one can access effectively every
possible spacetime framing of a given theory’s dynamical and kinematical
content. Our capacity for topological redescription is bounded only by a weak
spacetime-kinematic compatibility condition.

The remainder of this paper is organized as follows: Sec. 2 will begin by
discussing the preliminary assumptions which are necessary for the ISE
Methodology to be applicable to a given spacetime theory. This section will
then show how the internalization process applies to any such theory while
introducing some spacetime-kinematic compatibility assumptions along the way.
As I will argue, these additional assumptions are extremely weak. Next, Sec. 3
will discuss the externalization process in general before applying it to two
example theories. Sec. 4 will then define ISE-relatedness and prove the above-
discussed ISE-Equivalence Theorem. Finally, Sec. 5 will conclude.

##  2 The ISE Methodology: Internalization

Before overviewing the ISE Methodology, allow me to first state three example
theories.

> A Quartic Klein Gordon Theory (3+1D QKG) -  Consider a spacetime theory
> about a scalar field,
> $\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$,
> with spacetime, $\mathcal{M}_{\text{old}}\cong\mathbb{R}^{2}\times S^{2}$,
> and value space, $\mathcal{V}_{\text{old}}\cong\mathbb{R}$. For this theory,
> there exists an injective map,
> $C:\mathcal{M}_{\text{old}}\to\mathbb{R}^{5}$, which assigns to each point,
> $p\in\mathcal{M}_{\text{old}}$, some coordinates,
> $C(p)=(t,q,x,y,z)\in\mathbb{R}^{5}$, 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:
>
>
> $\displaystyle(\partial_{t}^{2}-\partial_{q}^{2}-L_{x}^{2}-L_{y}^{2}-L_{z}^{2}+M^{2})\,\varphi_{\text{old}}+\lambda\,\varphi_{\text{old}}^{3}=0,$
> (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.

It will also be helpful in the coming discussion to distinguish between the
following two closely related spacetime theories.

> Periodic Matter on a Non-Periodic Spacetime - Consider a spacetime theory
> about a scalar field,
> $\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$,
> with spacetime, $\mathcal{M}_{\text{old}}\cong\mathbb{R}^{2}$, and value
> space, $\mathcal{V}_{\text{old}}\cong\mathbb{R}$. The theory’s states are
> subject to the following kinematic constraint: In some fixed global
> coordinate system, $(t,x)$, the field, $\varphi_{\text{old}}$, must be
> second-differentiable (and so potentially non-smooth). Moreover, this
> theory’s states are kinematically constrained to be periodic in the
> x-coordinate with some fixed period $L>0$ as,
>
> $\displaystyle\varphi_{\text{old}}(t,x+L)=\varphi_{\text{old}}(t,x).$ (2)
>
> The theory’s states obey the following dynamics:
>
>
> $\displaystyle\partial_{t}\varphi_{\text{old}}(t,x)=\alpha\,\partial_{x}^{2}\varphi_{\text{old}}(t,x),$
> (3)
>
> for some fixed $\alpha>0$.

Note that this theory describes the dynamics of a periodic distribution of
heat, $\varphi_{\text{old}}(t,x)$, evolving on a non-periodic spacetime,
$\mathcal{M}_{\text{old}}\cong\mathbb{R}^{2}$. For brevity, let us refer to
this theory as the periodic matter theory. Contrast this with the following
closely related theory.

> Matter on a Periodic Spacetime - Consider a spacetime theory about a scalar
> field,
> $\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$,
> with spacetime, $\mathcal{M}_{\text{old}}\cong\mathbb{R}\times S^{1}$, and
> value space, $\mathcal{V}_{\text{old}}\cong\mathbb{R}$. For this theory
> there exists a map, $C:\mathbb{R}^{2}\to\mathcal{M}_{\text{old}}$, which
> assigns to each point, $p\in\mathcal{M}_{\text{old}}$, a set of coordinates,
> $C^{-1}(p)=\\{(\tau,q+n\,L)\\}_{n\in\mathbb{Z}}\subset\mathbb{R}^{2}$. The
> theory’s states are subject to the following kinematic constraint: In this
> fixed coordinate system the field, $\varphi_{\text{old}}$, must be second-
> differentiable (and so potentially non-smooth). In this coordinate system,
> the theory’s states obey the following dynamics:
>
>
> $\displaystyle\partial_{\tau}\varphi_{\text{old}}(\tau,q)=\alpha\,\partial_{q}^{2}\varphi_{\text{old}}(\tau,q),$
> (4)
>
> for some fixed $\alpha>0$.

It should be noted that the $q$-coordinate in this theory is periodic with
period $L$, unlike the $x$-coordinate of the previous theory. Hence, this
theory describes the dynamics of a generic distribution of heat,
$\varphi_{\text{old}}(\tau,q)$, evolving on a periodic spacetime,
$\mathcal{M}_{\text{old}}\cong\mathbb{R}\times S^{1}$. For brevity, let us
refer to this theory as the periodic spacetime theory.

As Sec. 4 will discuss, the periodic matter theory is ISE-related to the
periodic spacetime theory, but not vice-versa. Namely, while we can use the
ISE Methodology to redescribe the periodic matter theory as the periodic
spacetime theory, the reverse redescription is impossible. As I will discuss,
this asymmetry is due to the periodic matter theory lacking a certain kind of
spacetime-kinematic compatibility (defined in Sec. 2.3). Roughly, this
theory’s spacetime manifold is excessive given the theory’s kinematics, it has
distinct spacetime points which take on matching values in every kinematically
allowed state.

###  2.1 Two Preliminary Assumptions for Internalization

Past these three example theories, to what breadth of spacetime theories is
the ISE Methodology applicable? In order for the ISE Methodology to be
applicable to a spacetime theory, only two preliminary assumptions must hold.
The first preliminary assumption is that the theory in question must
distinguish between kinematically allowed and dynamically allowed states,
$S_{\text{old}}^{\text{dyn}}\subset S_{\text{old}}^{\text{kin}}\subset
S_{\text{old}}^{\text{all}}$. Let us call this the modal assumption. To make
things concrete, consider a generic spacetime theory about a scalar field,
$\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$. In
this case $S_{\text{old}}^{\text{all}}$ would be the set of all
$\mathcal{V}_{\text{old}}$-valued fields which are definable on
$\mathcal{M}_{\text{old}}$ (even those which are non-smooth and/or
discontinuous). $S_{\text{old}}^{\text{dyn}}$ and
$S_{\text{old}}^{\text{kin}}$ would then be the subsets of
$S_{\text{old}}^{\text{all}}$ which obey the theory’s dynamic and kinematic
constraints respectively.

These two modal sets, $S_{\text{old}}^{\text{dyn}}$ and
$S_{\text{old}}^{\text{kin}}$, are allowed to be equipped with effectively any
additional structures one wishes. (These structured sets will be denoted as
$\mathcal{S}_{\text{old}}^{\text{dyn}}$ and
$\mathcal{S}_{\text{old}}^{\text{kin}}$.) For instance, the two example
theories considered in Grimmer (2023) had $S_{\text{old}}^{\text{kin}}$ being
an algebra or a vector space respectively, namely,
$\mathcal{S}_{\text{old}}^{\text{kin}}=\mathcal{A}_{\text{old}}^{\text{kin}}$
or $\mathcal{S}_{\text{old}}^{\text{kin}}=V_{\text{old}}^{\text{kin}}$. It
should be noted that the common linear structure present in these two examples
is incidental. All that is required for the ISE Methodology to be applicable
is that we can smoothly vary the theory’s kinematically allowed states,
$\mathcal{S}_{\text{old}}^{\text{kin}}$.

This brings us to the second preliminary assumption, the smooth lifting
assumption. This assumption comes in two parts. Firstly, the theory in
question must come along with a map, $*_{\text{lift}}:d\mapsto d^{*}$, which
can lift the theory’s diffeomorphisms,
$d:\mathcal{M}_{\text{old}}\to\mathcal{M}_{\text{old}}$, to act as
$d^{*}:S_{\text{old}}^{\text{all}}\to S_{\text{old}}^{\text{all}}$ on the
theory’s states. This $*_{\text{lift}}$ map must be continuous as well as an
injective group homomorphism, i.e., we must have
$(d_{1}\,d_{2})^{*}=d_{1}^{*}\,d_{2}^{*}$ and that distinct $d_{1}\neq d_{2}$
map to distinct $d_{1}^{*}\neq d_{2}^{*}$. Note that for theories about a
scalar field,
$\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$, we
can simply take $d^{*}\varphi_{\text{old}}\coloneqq\varphi_{\text{old}}\circ
d^{-1}$.

Note that some subset of these lifted diffeomorphisms, $d^{*}$, will preserve
the theory’s kinematic constraints, mapping $S_{\text{old}}^{\text{kin}}$ onto
itself. Let us call this subset the theory’s group of kinematically allowed
diffeomorphisms,

$\displaystyle\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$
$\displaystyle\coloneqq\\{d\in\text{Diff}(\mathcal{M}_{\text{old}})\,|\,\ \
d^{*}(S_{\text{old}}^{\text{kin}})\subset S_{\text{old}}^{\text{kin}}\\}.$ (5)

Note that every kinematically allowed diffeomorphism,
$d\in\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$, can be lifted to
$d^{*}:S_{\text{old}}^{\text{all}}\to S_{\text{old}}^{\text{all}}$ and then
restricted to act on $S_{\text{old}}^{\text{kin}}$ as
$d^{*}|_{\text{kin}}:S_{\text{old}}^{\text{kin}}\to
S_{\text{old}}^{\text{kin}}$.

We are now in a position to state the second part of the smooth lifting
assumption: For every kinematically allowed diffeomorphism,
$d\in\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$, the action of
$d^{*}|_{\text{kin}}$ on $S_{\text{old}}^{\text{kin}}$ must be smooth. Note
that understanding the action of $d^{*}|_{\text{kin}}$ on
$S_{\text{old}}^{\text{kin}}$ as smooth requires that we equip this space with
some smooth topological structure, $\mathcal{T}_{\text{old}}$. We can collect
all of the structures and operations which we have so far equipped
$S_{\text{old}}^{\text{kin}}$ with as follows,

$\displaystyle\mathcal{S}_{\text{old}}^{\text{kin}}=(S_{\text{old}}^{\text{kin}},\mathcal{T}_{\text{old}},\dots,*_{\text{lift}},\mathcal{M}_{\text{old}})$
(6)

The ellipsis here contains the rest of the theory’s dynamical and metaphysical
details (e.g., its algebraic structure if it has any).

In sum, our preliminary assumptions first tell us (via the modal assumption)
that the theory in question distinguishes between kinematically and
dynamically allowed states. The smooth lifting assumption then guarantees two
things: Firstly, that the theory in question comes equipped with some smooth
structure, $\mathcal{T}_{\text{old}}$, for $S_{\text{old}}^{\text{kin}}$; And,
secondly, that it comes equipped with a lifting map, $*_{\text{lift}}:d\mapsto
d^{*}$, such that $d^{*}|_{\text{kin}}$ acts smoothly on
$\mathcal{S}_{\text{old}}^{\text{kin}}$ for all
$d\in\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$. Given these two
preliminary assumptions, we can now discuss how the internalization process
applies in general.

Before this, however, it should be noted that these preliminary assumptions
can also be satisfied by spacetime theories which are not about scalar fields
(i.e., not about
$\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$).
For instance, we could here consider a theory about a particle trajectory,
$x_{\text{old}}:\mathcal{V}_{\text{old}}\to\mathcal{M}_{\text{old}}$, in which
case we would have $d^{*}x_{\text{old}}\coloneqq d\circ x_{\text{old}}$.
Alternatively, we could consider a theory about a tangent vector field,
$\vec{\varphi}_{\text{old}}:\mathcal{M}_{\text{old}}\to\text{T}\mathcal{M}_{\text{old}}$,
in which case we would have $d\in\text{Diff}(\mathcal{M}_{\text{old}})$ lift
to act on $\vec{\varphi}_{\text{old}}$ as $d^{*}\vec{\varphi}_{\text{old}}$
via the push-forward. More generally, the theory in question could be about
global sections, $\sigma_{\text{old}}:\mathcal{M}_{\text{old}}\to
E_{\text{old}}$, of some fiber bundle,
$\pi_{\text{old}}:E_{\text{old}}\to\mathcal{M}_{\text{old}}$, under the
following condition: $E_{\text{old}}$ must be a natural bundle over
$\mathcal{M}_{\text{old}}$. This means that there must be a natural way to
lift any given $d\in\text{Diff}(\mathcal{M}_{\text{old}})$ to a fiber bundle
automorphism of $E_{\text{old}}$, namely
$d^{\ell}\in\text{Auto}(E_{\text{old}})$ with $\pi_{\text{old}}\circ
d^{\ell}=d\circ\pi_{\text{old}}$. Using this automorphism one can define
$d^{*}\sigma_{\text{old}}\coloneqq d^{\ell}\circ\sigma_{\text{old}}\circ
d^{-1}$. For instance, the tangent bundle, $\text{T}\mathcal{M}_{\text{old}}$,
is a natural bundle over $\mathcal{M}_{\text{old}}$ whereas the Möbius strip
is not a natural bundle over the circle.888Relatedly, unlike a tangent bundle
over $\mathcal{M}_{\text{old}}$, a spin bundle over $\mathcal{M}_{\text{old}}$
is not a natural bundle. Hence, some technical upgrades will be needed before
the ISE Methodology can be applied to spinor theories.

For any natural bundle theory or any particle trajectory theory, the
internalization process described below applies unchanged. For the sake of
simplicity, however, this paper will proceed focused mainly on the scalar
field case,
$\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$.
One footnote in Sec. 3.1 will discuss how to generalize the externalization
process outside of the scalar field case.999Specifically, footnote 18 in Sec.
3.1.

###  2.2 Internalization given the Preliminary Assumptions

Recall from Sec. 1 that the purpose of internalization is to break any
connection between the old theory’s states and any assumed topological
background structure (i.e., the spacetime manifold,
$\mathcal{M}_{\text{old}}$). This divorce is to be carried out at the level of
kinematics. Recall that by the modal assumption, our theory comes along with a
set of kinematically allowed states, $S_{\text{old}}^{\text{kin}}\subset
S_{\text{old}}^{\text{all}}$. As it stands, however, these states,
$\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$,
are functions defined on the spacetime manifold, $\mathcal{M}_{\text{old}}$.
Namely, our theory is presently 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,
$\text{eval}(\varphi_{\text{old}},p)=\varphi_{\text{old}}(p)$. This is the
field-to-spacetime connection which the internalization process aims to sever.

This divorce between the theory’s states and its spacetime manifold is to be
achieved in a similar spirit to the algebraic approach to spacetime.101010A
central idea behind spacetime algebraicism is that physical fields exist
without an underlying manifold (e.g., considered as algebraic objects). For
further discussion of spacetime algebraicism see Chen and Fritz (2021); Menon
(2019); Bain (2013); Geroch (1972); Rynasiewicz (1992); Earman (1989);
Rosenstock et al. (2015). Namely, by focusing on the theory’s manifold-
independent structure, one can conceptually divorce the theory’s states from
any underlying manifold (e.g., by dropping the eval operation). As Grimmer
(2023) notes, however, the key difference is that in the ISE Methodology no
special role is given to the theory’s algebraic structure; Indeed, the theory
in question might not even have any algebraic structure. The bare minimum
structures required by the ISE Methodology are those appearing in the smooth
lifting assumption. Namely, we need a smooth structure,
$\mathcal{T}_{\text{old}}$, for the theory’s kinematically allowed states,
$\mathcal{S}_{\text{old}}^{\text{kin}}$, which is connected to
$\mathcal{M}_{\text{old}}$ via a lifting operation, $*_{\text{lift}}:d\mapsto
d^{*}$.

This is not to say, however, that further structural details about the theory
in question are philosophically unimportant. As Grimmer (2023) has discussed,
a crucial first step in the internalization process is to identify which
structural features of one’s theory are dynamically and metaphysically
relevant. In general, we can organize these structures as follows:

$\displaystyle\mathcal{S}_{\text{old}}^{\text{kin}}=(S_{\text{old}}^{\text{kin}},\mathcal{T}_{\text{old}},\text{relevant
structures},\text{irrelevant
structures},\text{eval},*_{\text{lift}},\mathcal{M}_{\text{old}}).$ (7)

An identification of these structures for the Quartic Klein Gordon theory can
be found in Sec. 2.4. To summarize: A theory’s dynamically and metaphysically
relevant structures must include any mathematical structures which are
necessary to state the theory’s dynamics. Hence, for the Quartic Klein Gordon
theory we need addition and scalar multiplication operations
($+_{\text{old}}:S_{\text{old}}^{\text{kin}}\times
S_{\text{old}}^{\text{kin}}\to S_{\text{old}}^{\text{kin}}$ and
$\cdot_{\text{old}}:\mathbb{R}\times S_{\text{old}}^{\text{kin}}\to
S_{\text{old}}^{\text{kin}}$) as well as a product operation,
$\times_{\text{old}}:S_{\text{old}}^{\text{kin}}\times
S_{\text{old}}^{\text{kin}}\to S_{\text{old}}^{\text{kin}}$. Given these we
can define the theory’s dynamically allowed states,
$S_{\text{old}}^{\text{dyn}}\subset S_{\text{old}}^{\text{kin}}$.

Past its dynamically-relevant structure, the Quartic Klein Gordon theory also
comes equipped with a metaphysically-relevant inner product,
$\langle\varphi,\phi\rangle_{\text{old}}$. Namely, we can judge the relative
size and similarity of any two kinematically allowed states, $\varphi,\phi\in
S_{\text{old}}^{\text{kin}}$, using an $L^{2}$ inner product in the
$(t,q,x,y,z)$-coordinate system. Note that while no such inner product appears
in the theory’s dynamical equation, it might nonetheless be crucial in
grounding whatever metaphysical story one wants to tell about this theory.
Consider the importance of the inner product in non-relativistic quantum
mechanics despite not appearing in the Schrödinger equation.

In addition to these dynamically and metaphysically relevant structures, the
theory in question might also come equipped with some additional “irrelevant”
structures (although the term “non-fundamental” is probably more apt). These
would be features of our theory which play no role in the dynamical equations
and, moreover, are not fundamental to the metaphysical story we are trying to
tell about this theory. A central tenet of the ISE Methodology (which it
shares with the algebraic approach to spacetime) is that one can tell a
dynamically and metaphysically satisfying story without reference to a
fundamental spacetime manifold. Hence, $\mathcal{M}_{\text{old}}$ and the
operations which it directly supports ($*_{\text{lift}}$, eval, etc.) are here
deemed “irrelevant”. (They will, however, be recoverable through the
externalization process, see Sec. 3.)

Once these relevant structures have been identified, internalization then
proceeds by applying some forgetful bijection operation, $\mathcal{F}$, to
$\mathcal{S}_{\text{old}}^{\text{kin}}$ as follows:111111Category theorists
would call $\mathcal{F}_{\text{alg}}$ a forgetful functor. I am refraining
from using this terminology, however, because the mathematical overhead of
category theory is simply not relevant here.

$\displaystyle\mathcal{F}:\mathcal{S}_{\text{old}}^{\text{kin}}\to\mathcal{S}_{\text{neutral}}^{\text{kin}}\coloneqq(S_{\text{neutral}}^{\text{kin}},\mathcal{T}_{\text{neutral}},\text{relevant
structures}).$ (8)

Three things have changed here. Firstly, all of the theory’s dynamically and
metaphysically irrelevant structures have been dropped. This includes the
spacetime manifold, $\mathcal{M}_{\text{old}}$, as well as the operations it
directly supports, namely, eval and $*_{\text{lift}}$. Secondly, the old
theory’s kinematically allowed states, $S_{\text{old}}^{\text{kin}}$, have
been replaced via the bijection
$\mathcal{F}:S_{\text{old}}^{\text{kin}}\leftrightarrow
S_{\text{neutral}}^{\text{kin}}$ with an equal-sized set,
$S_{\text{neutral}}^{\text{kin}}$. The key difference here is that whereas
$\varphi_{\text{old}}\in S_{\text{old}}^{\text{kin}}$ is a function valued on
$\mathcal{M}_{\text{old}}$, the corresponding state,
$\bm{\varphi}=\mathcal{F}(\varphi_{\text{old}})\in
S_{\text{neutral}}^{\text{kin}}$, is not. Finally, isomorphic copies of each
of the old theory’s dynamically and metaphysically relevant structures have
been induced in $S_{\text{neutral}}^{\text{kin}}$. Namely, using the
bijection, $\mathcal{F}:S_{\text{old}}^{\text{kin}}\leftrightarrow
S_{\text{neutral}}^{\text{kin}}$, we have defined the relevant structures in
$\mathcal{S}_{\text{neutral}}^{\text{kin}}$ such that we have the appropriate
kind of isomorphism,
$\mathcal{S}_{\text{neutral}}^{\text{kin}}\cong\mathcal{S}_{\text{old}}^{\text{kin}}$.
For an explicit example of how the relevant structures are induced in the
spacetime neutral theory, see Grimmer (2023). Internalizing the theory in this
way completely severs the connection between the old theory’s states,
$\varphi_{\text{old}}$, and the old theory’s spacetime manifold,
$\mathcal{M}_{\text{old}}$.

But what exactly survives internalization? Each of the old theory’s
kinematically allowed states,
$\varphi_{\text{old}}\in\mathcal{S}_{\text{old}}^{\text{kin}}$, has a
corresponding spacetime-neutral state
$\bm{\varphi}\coloneqq\mathcal{F}(\varphi_{\text{old}})$. Hence, we can see
these states as having survived internalization as follows,

$\displaystyle\varphi_{\text{old}}\xrightarrow[\text{int.
as}]{\text{survives}}\bm{\varphi}\coloneqq\mathcal{F}(\varphi_{\text{old}}).$
(9)

Notice that, by design, all of the theory’s dynamically and metaphysically
relevant structures will survive internalization. Hence, we ought to have the
resources within $\mathcal{S}_{\text{neutral}}^{\text{kin}}$ to state the
theory’s dynamics in a spacetime-neutral way. In light of this, the theory’s
dynamically allowed subset survives internalization,

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

For an explicit example of a theory’s dynamics surviving internalization, see
Grimmer (2023).

Just as some (but not all) of theory’s states survive internalization (e.g.,
only $S_{\text{old}}^{\text{kin}}\subset S_{\text{old}}^{\text{all}}$), we
also have some (but not all) of the maps between these states survive
internalization. Let us say that a map
$f_{\text{old}}:S_{\text{old}}^{\text{all}}\to S_{\text{old}}^{\text{all}}$ is
kinematically allowed when it preserves the old theory’s kinematic
constraints, $f_{\text{old}}(S_{\text{old}}^{\text{kin}})\subset
S_{\text{old}}^{\text{kin}}$. For any such map, we can restrict its domain to
$S_{\text{old}}^{\text{kin}}$ as,
$f_{\text{old}}|_{\text{kin}}:S_{\text{old}}^{\text{kin}}\to
S_{\text{old}}^{\text{kin}}$. Note that each of the old theory’s kinematically
allowed maps has a corresponding spacetime-neutral map given by
$f_{\text{neutral}}\coloneqq\mathcal{F}\circ
f_{\text{old}}|_{\text{kin}}\circ\mathcal{F}^{-1}$. Hence, we can see these
maps as surviving internalization as follows,

$\displaystyle f_{\text{old}}|_{\text{kin}}\xrightarrow[\text{int.
as}]{\text{survives}}f_{\text{neutral}}\coloneqq\mathcal{F}\circ
f_{\text{old}}|_{\text{kin}}\circ\mathcal{F}^{-1}.$ (11)

Maps, $f_{\text{old}}:S_{\text{old}}^{\text{all}}\to
S_{\text{old}}^{\text{all}}$ which are not kinematically allowed do not
survive internalization.

So far, the above discussion has demonstrated that theory’s states and
dynamics can be divorced from the old theory’s spacetime manifold (i.e., its
topological background structure). Perhaps surprisingly, however, this does
not mean that all of the old theory’s topological information is simply
deleted by the internalization process. Indeed, as Grimmer (2023) has
demonstrated, a great deal of topological information about the old spacetime
setting actually can survive this topological divorce. The key to seeing how
this happens is the smooth lifting assumption. This guarantees that at least
some of the theory’s diffeomorphisms (namely,
$d\in\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$) can be lifted to act
on the theory’s kinematically allowed states as
$d^{*}|_{\text{kin}}:S_{\text{old}}^{\text{kin}}\to
S_{\text{old}}^{\text{kin}}$. We can then see that these kinematically allowed
diffeomorphisms survive internalization in light of Eq. (11). Namely, we have
that

$\displaystyle
d\in\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})\xrightarrow[\text{int.
as}]{\text{survives}}G(d)\coloneqq\mathcal{F}\circ
d^{*}|_{\text{kin}}\circ\mathcal{F}^{-1}.$ (12)

Thus, we have that every kinematically allowed diffeomorphism survives
internalization. In fact, these diffeomorphisms survive not only individually
but as a Lie group. Namely, it follows from $*_{\text{lift}}:d\mapsto d^{*}$
being a continuous group homomorphism that the $G$ map defined above is a Lie
group homomorphism (for proof, see Appendix A.2).

To review: We can apply the internalization process to any spacetime theory
which satisfies our two preliminary assumptions. For all such theories, the
above discussion has demonstrated that at least some diffeomorphisms (namely,
$\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})\subset\text{Diff}(\mathcal{M}_{\text{old}})$)
will survive the internalization process. Thus, under two very weak
assumptions, at least some topological information regarding the old spacetime
setting survives internalization. Next, I will discuss the conditions under
which complete topological information survives internalization.

###  2.3 Internalization given Spacetime-Kinematic Compatibility

So far we have seen that, for any spacetime theory satisfying our two
preliminary assumptions, any Lie group of kinematically allowed
diffeomorphisms, $H\subset\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$,
will survive internalization as a Lie group, $G(H)$, which acts smoothly on
$\mathcal{S}_{\text{neutral}}^{\text{kin}}$. The following discussion will
demonstrate that given spacetime-kinematic compatibility (definition to
follow) these diffeomorphisms survive internalization faithfully, i.e., with
$G(H)\cong H$. Relatedly, I will demonstrate that under these conditions the
theory’s spacetime manifold, $\mathcal{M}_{\text{old}}$, will also survive
internalization faithfully.

Spacetime-kinematic compatibility is defined as follows:

> Definition: Spacetime-Kinematic Compatibility -  A spacetime theory has
> spacetime-kinematic compatibility just in case:
>
>   1. 1.
>
> it satisfies our two preliminary assumptions (namely, the modal assumption
> and the smooth lifting assumption),
>
>   2. 2.
>
> it is kinematically reduced, and
>
>   3. 3.
>
> it is kinematically homogeneous.
>
>

The definitions of “kinematically reduced” and “kinematically homogeneous”
will be introduced as needed in the coming discussion. As I will discuss in
Sec. 2.5, spacetime-kinematic compatibility is a remarkably weak assumption.

A theory’s kinematically allowed diffeomorphisms,
$\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$, will survive
internalization if and only if the kernel of the $G$ map is trivial. Let us
define $H_{\text{kin}}\coloneqq\text{ker}(G)$ to be the theory’s group of
kinematically trivial diffeomorphisms,

$\displaystyle
H_{\text{kin}}\coloneqq\\{d\in\text{Diff}(\mathcal{M}_{\text{old}})\,|\,\ \
d^{*}|{}_{\text{kin}}\mathcal{}=\mathbbm{1}\\},$ (13)

When lifted, these diffeomorphisms act trivially on every kinematically
allowed state, i.e., they have
$d^{*}\varphi_{\text{old}}=\varphi_{\text{old}}$ for all
$\varphi_{\text{old}}\in S_{\text{old}}^{\text{kin}}$. A spacetime theory will
be called kinematically reduced when this group is trivial,
$H_{\text{kin}}=\\{\mathbbm{1}\\}$.

All but one of the example theories introduced above are kinematically
reduced. Only the Periodic Matter on a Non-Periodic Spacetime Theory is not.
To see this, consider the diffeomorphisms, $d(\Delta t,\Delta x)$, which act
in this theory’s $(t,x)$ coordinate system as $d(\Delta t,\Delta
x):(t,x)\mapsto(t+\Delta t,x+\Delta x)$. Each of these diffeomorphisms is
kinematically allowed, $d(\Delta t,\Delta
x)\in\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$, since each of them
maps kinematically allowed states to kinematically allowed states. Some of
them, however, are kinematically trivial. Namely, when $\Delta t=0$ and
$\Delta x=n\,L$ for $n\in\mathbb{Z}$ these diffeomorphisms act as the identity
on every kinematically allowed state,

$\displaystyle\forall\varphi_{\text{old}}\in\mathcal{S}_{\text{old}}^{\text{kin}}\quad
d(0,n\,L)^{*}\varphi_{\text{old}}=\varphi_{\text{old}}.$ (14)

This is due to the fact that all of the theory’s kinematically allowed states
are periodic in the $x$-coordinate with period $L$. Thus, we have
$d(0,n\,L)^{*}|_{\text{kin}}=\mathbbm{1}$ despite the fact that
$d(0,n\,L)\neq\mathbbm{1}$. Hence this theory is not kinematically reduced.

Let us see how this interferes with the theory’s diffeomorphisms faithfully
surviving internalization. Let the diffeomorphisms, $d(\Delta t,\Delta x)$,
together form a Lie group, $H_{\text{trans}}\cong(\mathbb{R}^{2},+)$.121212The
subscript “trans” here denotes the fact that the Lie group,
$H_{\text{trans}}$, acts transitively over $\mathcal{M}_{\text{old}}$. As I
discuss in Appendix A.2, for every kinematically allowed Lie group,
$H\subset\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$, we have that,

$\displaystyle H\xrightarrow[\text{int. as}]{\text{survives}}G(H)\cong
H/(H_{\text{kin}}\cap H).$ (15)

That is, up to isomorphism,
$H\subset\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$ survives
internalization as a Lie group by being quotiented by $H_{\text{kin}}\cap H$,
its kinematically trivial subgroup.

In the periodic matter theory, the kinematically trivial subgroup of
$H_{\text{trans}}$ is $H_{\text{kin}}\cap H_{\text{trans}}\cong(\mathbb{Z},+)$
containing the $d(0,n\,L)$ maps discussed above. Hence, we have that
$H_{\text{trans}}$ survives internalization albeit unfaithfully as follows:

$\displaystyle H_{\text{trans}}\xrightarrow[\text{int.
as}]{\text{survives}}G_{\text{trans}}\coloneqq
G(H_{\text{trans}})\cong\frac{H_{\text{trans}}}{H_{\text{kin}}\cap
H_{\text{trans}}}\cong\frac{(\mathbb{R}^{2},+)}{(\mathbb{Z},+)}\cong(\mathbb{R},+)\times
U(1).$ (16)

So it goes in general: For kinematically unreduced theories, we will have that
$\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$ and all of its subgroups
survive internalization albeit filtered through the theory’s kinematically
allowed states. For the rest of this section, let us assume that the theory in
question is kinematically reduced. (As I will discuss in Sec. 2.5, this is a
very weak assumption.)

Given this assumption, $\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$
and all of its subgroups will survive internalization faithfully. We are
halfway to our next goal of showing that not only
$\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$ but also
$\mathcal{M}_{\text{old}}$ itself survives internalization faithfully. This
will require us to introduce a bit of mathematics and to make another
assumption relating to the homogeneity of the theory’s spacetime manifold.

A more detailed discussion of smooth homogeneous manifolds can be found in the
appendix of Grimmer (2023). For our purposes, some key properties should be
quickly noted. A smooth manifold, $\mathcal{M}$, is said to be homogeneous if
there exists a Lie group, $G$, and a group action,
$\theta:G\times\mathcal{M}\to\mathcal{M}$, which acts smoothly and
transitively on $\mathcal{M}$.131313Two slightly different definitions of
homogeneity need to be distinguished: One might here either allow or disallow
for infinite-dimensional Lie groups. For an example where this distinction
matters, consider the double-torus,
$\mathcal{M}\cong\mathbb{T}^{2}\\#\mathbb{T}^{2}$. This smooth manifold is
homogeneous with respect to the infinite-dimensional Lie group,
$\text{Diff}(\mathcal{M})$, but not with respect to any finite-dimensional
subgroup thereof. For proof of the latter claim, see Mostow (2005). For what I
expect are merely technical reasons, certain theorems which are central to the
ISE Methodology have only been proven in the finite-dimensional case
(infinite-dimensional Lie groups are very difficult to work with). Indeed, the
theorems proved in an appendix of Grimmer (2023) only deal with the finite-
dimensional notion of homogeneity. Until analogous theorems are proved in the
infinite-dimensional case, the ISE Methodology ought to be understood as
employing the finite-dimensional definition of homogeneity. However, while it
is difficult to prove theorems in the infinite-dimensional case, it is much
easier to simply talk about $\text{Diff}(\mathcal{M})$ rather than quantifying
over all of its finite-dimensional Lie subgroups. Hence, for expositional
convenience, I will sometimes make use of the infinite-dimensional definition
in the main text and clarify in a footnote what changes are needed in the
finite-dimensional case.  On such a manifold all points on $\mathcal{M}$ are
equivalent from the perspective of the Lie group, $G$, in the following sense.
Any point $p\in\mathcal{M}$ can be mapped onto any other point
$q\in\mathcal{M}$ via the smooth action of some $g\in G$ as $\theta(g,p)=q$.
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).

It is well known that the quotient of any two Lie groups is a smooth
homogeneous manifold: For any Lie group, $G_{1}$, and any closed Lie subgroup
thereof, $G_{2}$, their quotient $\mathcal{M}_{\text{G}}\coloneqq G_{1}/G_{2}$
is a smooth homogeneous manifold with $G_{1}$ acting smoothly and transitively
over it. Conversely, it is well known that any homogeneous manifold,
$\mathcal{M}$, is diffeomorphic to the quotient of some two Lie groups.
Concretely, given any such $\mathcal{M}$, one can find Lie groups
$H_{2}\subset H_{1}$ such that
$\mathcal{M}\cong\mathcal{M}_{\text{H}}\coloneqq H_{1}/H_{2}$. In particular,
without loss of generality, one can take
$H_{1}\subset\text{Diff}(\mathcal{M})$ to be any Lie group which acts
transitively on $\mathcal{M}$ and $H_{2}\subset H_{1}$ to be any of $H_{1}$’s
stabilizer subgroups. Given these two results, the class of all smooth
homogeneous manifolds are, up to diffeomorphism, exactly the class of all such
Lie group quotients. A concrete reconstruction of the 2-sphere along these
lines, $S^{2}\cong\text{SO}(3)/\text{SO}(2)$, can be found in the appendix of
Grimmer (2023).

These homogeneous manifold reconstruction results motivate the following
definition. Let us say that a spacetime theory is kinematically homogeneous if
and only if there exists a Lie group of kinematically allowed diffeomorphisms,
$H_{\text{trans}}\subset\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}}),$
which acts transitively on $\mathcal{M}_{\text{old}}$. For the rest of this
section, let us assume that the theory in question is kinematically
homogeneous. (Once again, this is a very weak assumption, see Sec. 2.5.)

Now that we have finished defining spacetime-kinematic compatibility, we are
in a position to see how it facilitates the survival of the old theory’s
spacetime manifold, $\mathcal{M}_{\text{old}}$. It follows from the theory in
question being kinematically homogeneous that its spacetime manifold,
$\mathcal{M}_{\text{old}}$, is homogeneous. Hence we can use the above-
discussed characterization of smooth homogeneous manifolds to reconstruct
$\mathcal{M}_{\text{old}}$ as a quotient of its diffeomorphisms. Namely, there
will be some transitively-acting Lie group,
$H_{\text{trans}}\subset\text{Diff}(\mathcal{M}_{\text{old}})$, such that
$\mathcal{M}_{\text{old}}\cong H_{\text{trans}}/H_{\text{fix}}$ where
$H_{\text{fix}}$ is one of $H_{\text{trans}}$’s stabilizer subgroups.

Indeed, since the theory in question is not only homogeneous but kinematically
homogeneous we can take these two Lie groups to be kinematically allowed,
$H_{\text{fix}}\subset
H_{\text{trans}}\subset\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$.
Recall that it follows from a theory being kinematically reduced that every
Lie group of kinematically allowed diffeomorphisms survives internalization
faithfully. Hence we have that

$\displaystyle H_{\text{trans}}$ $\displaystyle\xrightarrow[\text{int.
as}]{\text{survives}}G_{\text{trans}}\coloneqq G(H_{\text{trans}})\cong
H_{\text{trans}}$ (17) $\displaystyle H_{\text{fix}}$
$\displaystyle\xrightarrow[\text{int.
as}]{\text{survives}}G_{\text{fix}}\coloneqq G(H_{\text{fix}})\cong
H_{\text{fix}}.$

Given that the Lie groups in both the numerator and denominator of
$\mathcal{M}_{\text{old}}\cong H_{\text{trans}}/H_{\text{fix}}$ have
faithfully survived internalization we have that,

$\displaystyle\mathcal{M}_{\text{old}}\cong\mathcal{M}_{\text{H}}\coloneqq\frac{H_{\text{trans}}}{H_{\text{fix}}}\xrightarrow[\text{int.
as}]{\text{survives}}\mathcal{M}_{\text{G}}\coloneqq\frac{G_{\text{trans}}}{G_{\text{fix}}}\cong\frac{H_{\text{trans}}}{H_{\text{fix}}}\cong\mathcal{M}_{\text{old}}.$
(18)

We have thus demonstrated the faithful survival of $\mathcal{M}_{\text{old}}$
for any theory with spacetime-kinematic compatibility.

In addition to surviving internalization faithfully,
$\mathcal{M}_{\text{old}}$ also survives robustly in the following sense. The
surviving quotient manifold, $\mathcal{M}_{\text{G}}$, is diffeomorphic to
$\mathcal{M}_{\text{old}}$ independent of which Lie groups, $H_{\text{trans}}$
and $H_{\text{fix}}$, we used to reconstruct $\mathcal{M}_{\text{old}}$.
Namely, for $H_{\text{trans}}$, we could have picked any kinematically-
allowed, transitively-acting Lie group. For $H_{\text{fix}}$, we could have
picked any stabilizer subgroup of $H_{\text{trans}}$.

In sum, the above discussion has demonstrated how, for any theory with
spacetime-kinematic compatibility, its spacetime manifold,
$\mathcal{M}_{\text{old}}$, survives internalization both faithfully and
robustly. But what goes wrong when our theory lacks spacetime-kinematic
compatibility? Recall that the Periodic Matter on a Non-Periodic Spacetime
theory is not kinematically reduced. Let us see whether or not its spacetime
manifold faithfully survives internalization. Note that the Lie group defined
above, $H_{\text{trans}}\cong(\mathbb{R}^{2},+)$, acts transitively on
$\mathcal{M}_{\text{old}}\cong\mathbb{R}^{2}$. Note that all of
$H_{\text{trans}}$’s stabilizer subgroups are trivial,
$H_{\text{fix}}=\\{\mathbbm{1}\\}$. Using these Lie groups we can reconstruct
$\mathcal{M}_{\text{old}}$ up to diffeomorphism as follows,

$\displaystyle\mathcal{M}_{\text{old}}\cong\mathcal{M}_{\text{H}}\coloneqq\frac{H_{\text{trans}}}{H_{\text{fix}}}\cong\frac{(\mathbb{R}^{2},+)}{\\{\mathbbm{1}\\}}\cong\mathbb{R}^{2}.$
(19)

We have already seen how the Lie group in the numerator survives
internalization unfaithfully. Namely,
$H_{\text{trans}}\cong(\mathbb{R}^{2},+)$ becomes
$G_{\text{trans}}\cong(\mathbb{R},+)\times U(1)$. By contrast, the Lie group
in the denominator, $H_{\text{fix}}$ does survive internalization faithfully
becoming $G_{\text{fix}}\coloneqq G(H_{\text{fix}})=\\{\mathbbm{1}\\}$. Hence,
we have the theory’s spacetime manifold surviving internalization as follows:

$\displaystyle\mathcal{M}_{\text{old}}\cong\mathcal{M}_{\text{H}}\coloneqq\frac{H_{\text{trans}}}{H_{\text{fix}}}\xrightarrow[\text{int.
as}]{\text{survives}}\mathcal{M}_{\text{G}}\coloneqq\frac{G_{\text{trans}}}{G_{\text{fix}}}\cong\frac{(\mathbb{R},+)\times
U(1)}{\\{\mathbbm{1}\\}}\cong\mathbb{R}\times S^{1}.$ (20)

Thus, while $\mathcal{M}_{\text{old}}\cong\mathbb{R}^{2}$ has survived
internalization, it has been done so unfaithfully becoming
$\mathcal{M}_{\text{G}}\cong\mathbb{R}\times S^{1}$.

Given spacetime-kinematic compatibility, however, a theory’s spacetime
manifold will always survive internalization both faithfully and robustly.
This is a strong conclusion from what Sec. 2.5 will argue are very weak
assumptions. Indeed, the spacetime-kinematic compatibility assumptions made in
this section are, in fact, slightly stronger than necessary to achieve the
desired result. As I prove in Appendix A.2, a slightly weakened notion
spacetime-kinematic compatibility is both necessary and sufficient for a
theory’s spacetime manifold to faithfully and robustly survive internalization
in this way. Let us next see a quick example application of the
internalization process.

###  2.4 Example Internalization: A Quartic Klein Gordon Theory

Recall the Quartic Klein Gordon theory (3+1D QKG) introduced in Sec. 1. To
begin internalizing this theory we must first identify its dynamically and
metaphysically relevant structures. One can read some of these structures off
of the theory’s dynamical equation. Namely, in order to state the theory’s
dynamics, we need to define the following three algebraic operations: a
pointwise addition operation, $+_{\text{old}}$, a pointwise scalar
multiplication operation, $\cdot_{\text{old}}$, and a pointwise multiplication
operation, $\times_{\text{old}}$. In addition to these dynamically-relevant
structures, this theory also comes equipped with a metaphysically relevant
inner product, $\langle\varphi,\phi\rangle_{\text{old}}$, namely the $L^{2}$
inner product in its $(t,q,x,y,z)$ coordinate system. Using this inner product
we can judge the relative size and similarity of any pair of kinematically
allowed states, $\varphi,\phi\in S_{\text{old}}^{\text{kin}}$.

Given all of these algebraic structures, the theory’s set of kinematically
allowed states, $S_{\text{old}}^{\text{kin}}$, forms an algebra,

$\displaystyle\mathcal{A}_{\text{old}}^{\text{kin}}\coloneqq(S_{\text{old}}^{\text{kin}},\mathcal{T}_{\text{old}},+_{\text{old}},\,\cdot_{\text{old}}\,,\times_{\text{old}},\,\langle,\rangle_{\text{old}},\text{eval},*_{\text{lift}},\mathcal{M}_{\text{old}}).$
(21)

with a smooth topology, $\mathcal{T}_{\text{old}}$, and some other operations,
$*_{\text{lift}}$ and eval, relating these algebraic structures to
$\mathcal{M}_{\text{old}}$. (See Sec. 2.2, for further discussion of
$\mathcal{T}_{\text{old}}$, eval, and $*_{\text{lift}}$.) As discussed in Sec.
2, we can internalize this theory by applying a forgetful operation,
$\mathcal{F}_{\text{alg}}$, at the level of kinematics which only remembers
only algebraic structures as follows:

$\displaystyle\mathcal{F}_{\text{alg}}:\mathcal{A}_{\text{old}}^{\text{kin}}\to\mathcal{A}_{\text{neutral}}^{\text{kin}}\coloneqq(S_{\text{neutral}}^{\text{kin}},\mathcal{T}_{\text{neutral}},+_{\text{neutral}},\,\cdot_{\text{neutral}}\,,\times_{\text{neutral}},\,\langle,\rangle_{\text{neutral}}).$
(22)

Recall from Sec. 2 that the spacetime neutral states
$\bm{\varphi}\in\mathcal{A}_{\text{neutral}}^{\text{kin}}$ cannot be evaluated
at spacetime points (indeed, there is no long any eval operation which could
relate them to any manifold $\mathcal{M}$). Moreover, it should be noted that
the algebraic structures appearing in
$\mathcal{A}_{\text{neutral}}^{\text{kin}}$ are exactly those induced by the
bijection $\mathcal{F}_{\text{alg}}:S_{\text{old}}^{\text{kin}}\leftrightarrow
S_{\text{neutral}}^{\text{kin}}$. This guarantees that we have the relevant
kind of isomorphism,
$\mathcal{A}_{\text{old}}^{\text{kin}}\cong\mathcal{A}_{\text{neutral}}^{\text{kin}}$,
between these two kinematically allowed spaces. For a more explicit discussion
of how the relevant structures are induced in the spacetime-neutral theory,
see Grimmer (2023).

To finish internalizing this theory, let us see how its spacetime manifold
faithfully survives internalization. First note that this theory has
spacetime-kinematic compatibility; Namely, it satisfies our two preliminary
assumptions, it is kinematically reduced, and it is kinematically homogeneous.
Hence, following Sec. 2.3, we can reconstruct this theory’s spacetime manifold
as a quotient of its diffeomorphisms, $\mathcal{M}_{\text{old}}\cong
H_{\text{trans}}/H_{\text{fix}}$. Here $H_{\text{trans}}$ is any transitively-
acting Lie group which preserves the theory’s kinematic constraint, and
$H_{\text{fix}}$ is any of the stabilizer subgroups of $H_{\text{trans}}$. The
simplest Lie group which satisfies these criteria is the group,
$H_{\text{trans}}\cong(\mathbb{R}^{2},+)\times\text{SO}(3)$, which implements
rigid translations in the old theory’s $(t,q)$ coordinates and rigid rotations
in the old theory’s $(x,y,z)$ coordinates. Note that all of the stabilizer
subgroups for this choice of $H_{\text{trans}}$ are identical,
$H_{\text{fix}}\cong\\{\mathbbm{1}\\}\times\text{SO}(2)$. A concrete
demonstration of the non-trivial part of this quotient,
$\text{SO}(3)/\text{SO}(2)\cong S^{2}$, can be found in the appendix of
Grimmer (2023).

Whichever kinematically allowed 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. Concretely,
$H_{\text{trans}}$ and $H_{\text{fix}}$ become a pair of Lie groups,
$G_{\text{trans}}\coloneqq G(H_{\text{trans}})\cong H_{\text{trans}}$ and
$G_{\text{fix}}\coloneqq G(H_{\text{fix}})\cong H_{\text{fix}}$, which act
smoothly on $\mathcal{A}_{\text{neutral}}^{\text{kin}}$. The fact that the map
$G:H_{\text{trans}}\to G_{\text{trans}}$ is a Lie group isomorphism follows
from this theory being kinematically reduced. Hence, the theory’s old
spacetime manifold, $\mathcal{M}_{\text{old}}\cong
H_{\text{trans}}/H_{\text{fix}}$, also faithfully survives internalization
becoming $\mathcal{M}_{\text{G}}\coloneqq
G_{\text{trans}}/G_{\text{fix}}\cong\mathcal{M}_{\text{old}}$.

###  2.5 The Naturalness of Spacetime Kinematic Compatibility

The remainder of this section will be spent arguing that the above-discussed
spacetime-kinematic compatibility assumptions are extremely weak. There is a
clear sense in which theories lacking spacetime-kinematic compatibility are
unnatural. As I will now discuss, all such theories exhibit at least one of
the following two flaws: 1) They contain an excessive amount of spacetime
given their kinematics (i.e., they have distinct spacetime points which are
nonetheless kinematically identical), or 2) they are kinematically
inhomogeneous in the sense that the theory’s kinematics makes a topological
distinction between at least two kinds of spacetime points. As I will discuss,
failing to have spacetime-kinematic compatibility in either of these ways
would require that the theory in question is subject to some rather severe
kinematic constraints.

Consider a generic spacetime theory about a scalar field,
$\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$,
which is not kinematically reduced (i.e., with
$H_{\text{kin}}\neq\\{\mathbbm{1}\\}$). It follows from the definition of
$H_{\text{kin}}$ that any two spacetime points,
$p_{1},p_{2}\in\mathcal{M}_{\text{old}}$, which are related by some $k\in
H_{\text{kin}}$ take the identical values in all kinematically allowed states:

$\displaystyle\big{(}\exists k\in H_{\text{kin}}\ \
k(p_{1})=p_{2}\big{)}\Longrightarrow\big{(}\forall\varphi_{\text{old}}\in\mathcal{S}_{\text{old}}^{\text{kin}}\
\ \varphi_{\text{old}}(p_{1})=\varphi_{\text{old}}(p_{2})\big{)}.$ (23)

Since $H_{\text{kin}}\neq\\{\mathbbm{1}\\}$ is non-trivial, then it acts non-
trivially somewhere. That is, there exist distinct points $p_{1}\neq p_{2}$
related by some $k\in H_{\text{kin}}$. It then follows that there exist
distinct but kinematically-identical spacetime points.

Hence, every kinematically unreduced theory contains kinematically redundant
spacetime points. Taking the contrapositive, any theory which lacks such
kinematically redundant spacetime points is guaranteed to be kinematically
reduced. Thus any theory which grants a sufficient level of independent
variability to the field values at different spacetime points will
automatically be kinematically reduced. The only way which a spacetime theory
might fail to be kinematically reduced is if it has some rather severe
kinematic constraints which make certain portions of its spacetime
kinematically redundant.

Let us next turn our attention to the assumption of kinematic homogeneity.
Once again, this is also a very weak assumption. It should first be noted that
whether or not a smooth manifold is homogeneous is solely a matter of
topology. Indeed, a spacetime theory can fail to be kinematically homogeneous
for solely topological reasons. Namely, it might be set on a spacetime
manifold, $\mathcal{M}$, which is itself inhomogeneous. This would happen, for
instance, if we had
$\mathcal{M}\cong\mathbb{T}^{2}\,\dot{\cup}\,\mathbb{R}^{2}$ being the
disjoint union of a torus and a two-dimensional plane. This manifold is
inhomogeneous because it contains two distinct kinds of points which no smooth
transformation can relate (namely, torus-points and plane-points).

If one considers all $d\in\text{Diff}(\mathcal{M})$ then this example of
inhomogeneity is, in fact, generic:141414Unfortunately, however, if one
considers only homogeneity with respect to finite-dimensional Lie subgroups,
$H\subset\text{Diff}(\mathcal{M})$, then things get slightly more complicated.
For instance, consider the case of the double-torus,
$\mathcal{M}\cong\mathbb{T}^{2}\\#\mathbb{T}^{2}$ discussed in footnote 13.
This smooth manifold is homogeneous with respect to the infinite-dimensional
Lie group, $\text{Diff}(\mathcal{M})$, but not with respect to any finite-
dimensional subgroup thereof. Hence, the following claim about inhomogeneity
and dissimilar connected components needs some qualification if one is using
the finite-dimensional notion of homogeneity (see footnote 13). Importantly,
however, it is still the case that almost all spacetime manifolds commonly
encountered in physics are homogeneous under this stricter definition. The
only way in which a smooth manifold can be inhomogeneous is by having
dissimilar connected components. Otherwise, $\mathcal{M}$ is homogeneous with
respect to the infinite-dimensional Lie group, $\text{Diff}(\mathcal{M})$.
Thus, all spacetime manifolds commonly encountered in physics are homogeneous.

Simply being set on a homogeneous manifold, however, is not sufficient for a
spacetime theory to qualify as kinematically homogeneous. This would only
guarantee homogeneity with respect to $\text{Diff}(\mathcal{M}_{\text{old}})$
and not $\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$ as
required.151515If one here adopts a finite-dimensional understanding of
homogeneity (see footnote 13) then things get slightly more complicated. Under
this stricter definition, in order for $\mathcal{M}_{\text{old}}$ to qualify
as homogeneous (simpliciter) it must be homogeneous not just with respect to
the infinite-dimensional Lie group, $\text{Diff}(\mathcal{M}_{\text{old}})$,
but to also with respect to some finite-dimensional subgroup thereof,
$H\subset\text{Diff}(\mathcal{M}_{\text{old}})$. Similarly for a finite-
dimensional understanding of kinematic homogeneity. To simplify the
exposition, however, I will continue to talk about
$\text{Diff}(\mathcal{M}_{\text{old}})$ and
$\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$, rather than quantifying
over the finite-dimensional Lie subgroups thereof. Often, however, it will be
the case that every diffeomorphism is kinematically allowed, i.e.,
$\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})=\text{Diff}(\mathcal{M}_{\text{old}})$.
For instance, consider the Quartic Klein Gordon theory where our only
kinematic constraint is that $\varphi_{\text{old}}$ be smooth. Every
diffeomorphism maps smooth states to smooth states. Hence, every
diffeomorphism is kinematically allowed. Similarly, for the periodic spacetime
theory, every diffeomorphism maps second-differentiable states to second-
differentiable. Hence, here too we have
$\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})=\text{Diff}(\mathcal{M}_{\text{old}})$.
It is only for the periodic matter theory that we have
$\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})\neq\text{Diff}(\mathcal{M}_{\text{old}})$.
In this theory, only diffeomorphisms which respect the kinematically-enforced
periodicity of the matter content are kinematically allowed. Even in this
case, however, these kinematically allowed diffeomorphisms are still
sufficient to make the periodic matter theory kinematically homogeneous.

For an example of a kinematically inhomogeneous spacetime theory, consider a
theory about a scalar field,
$\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$,
set on
$\mathcal{M}_{\text{old}}\cong\mathbb{R}^{2}\,\dot{\cup}\,\mathbb{R}^{2}$,
i.e., the disjoint union of two two-dimensional planes. On one of these planes
(but not the other) the theory’s matter content is kinematically constrained
to be periodic in both space and time. While this theory is set on a
homogeneous manifold, it is not kinematically homogeneous. To see this,
consider any point $p_{0}$ on the periodic copy of $\mathbb{R}^{2}$. Note that
by using $d\in\text{Diff}(\mathcal{M}_{\text{old}})$ one can map $p_{0}$ to
anywhere on either plane.161616Indeed, one can map $p_{0}$ to anywhere on
either plane by using only a finite-dimensional subgroup of
$\text{Diff}(\mathcal{M}_{\text{old}})$. Hence, this manifold is also
homogeneous on a finite-dimensional understanding of homogeneity (see footnote
13). Note, however, that none of the kinematically allowed diffeomorphisms,
$d\in\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$, can map $p_{0}$ onto
the non-periodic plane (i.e., all $d$ which do so are kinematically
disallowed). Hence, at the level of kinematics, this theory makes a
topological distinction between two kinds of spacetime points: those reachable
from $p_{0}$ via $\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$ (i.e.,
the periodic plane), and those not reachable from $p_{0}$ via
$\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$ (i.e., the non-periodic
plane). Said differently, when one divides $\mathcal{M}_{\text{old}}$ into
$\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$-orbits one finds multiple
pieces. This aspect of kinematic inhomogeneity is generic:171717On a finite-
dimensional understanding of homogeneity (see footnote 13) this claim should
be understood as follows. For any kinematically inhomogeneous spacetime
theory, and for any finite-dimensional Lie subgroup,
$H\subset\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$, the spacetime
manifold $\mathcal{M}_{\text{old}}$ will be divided into several distinct
$H$-orbits. For any kinematically inhomogeneous spacetime theory, the theory’s
kinematical structure will divide the spacetime manifold,
$\mathcal{M}_{\text{old}}$, into several distinct pieces (namely, into several
distinct $\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$-orbits).

In sum, consider any spacetime theory set on a homogeneous manifold,
$\mathcal{M}_{\text{old}}$. In order for this theory to be kinematically
inhomogeneous it must be subject to rather sever kinematic constraint which
not only disallows some diffeomorphisms,
$\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})\neq\text{Diff}(\mathcal{M}_{\text{old}})$,
but also cuts $\mathcal{M}_{\text{old}}$ into distinct
$\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$-orbits. As the above
examples have shown, common kinematic constraints involving smoothness,
second-differiability, and even periodicity are not sufficient to do this.

##  3 The ISE Methodology: Externalization

The previous section has shown that, for a wide range of spacetime theories,
the theory’s original spacetime manifold, $\mathcal{M}_{\text{old}}$, survives
internalization both faithfully and robustly. Namely, given a very weak
spacetime-kinematic compatibility assumption, it survives as a quotient of two
Lie groups, $G_{\text{trans}}^{\text{(int)}}$ and
$G_{\text{fix}}^{\text{(int)}}$, which act smoothly on
$\mathcal{S}_{\text{neutral}}^{\text{kin}}$. 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.

As Grimmer (2023) has discussed, 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 $\mathcal{S}_{\text{neutral}}^{\text{kin}}$ and satisfy
certain properties (defined below). PSTOs are smooth actions on the theory’s
states which are, in a certain sense, structurally indistinguishable from
spacetime translations. The externalization process will build a new spacetime
setting, $\mathcal{M}_{\text{new}}$, from these hand-picked PSTOs such that
they become honest-to-goodness spacetime translations on
$\mathcal{M}_{\text{new}}$. (N.B.: The term translation is here being used
broadly, such that it includes rotations as well as generic diffeomorphisms.)

Before spelling out the externalization process in detail, however, Sec. 3.1
will first recall the key definitions from Grimmer (2023). Next, Sec. 3.2 will
overview the externalization process in general. Secs. 3.3 and 3.4 will then
apply the externalization process to two example theories. (Two more worked
examples of externalization can be found in Grimmer (2023)). Finally, in Sec.
3.5 I will prove that every spacetime theory produced by the externalization
process is guaranteed to have spacetime-kinematic compatibility. In fact, I
will prove that a spacetime theory is producible via externalization if and
only if it has spacetime-kinematic compatibility.

###  3.1 Defining Pre-Spacetime Translation Operations

The term “pre-spacetime translation operations” was defined in Grimmer (2023)
as follows and will be unpacked here.

> 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.

Faithful construction compatibility was first defined in Grimmer (2023) and
will be restated here momentarily. Before this, however, it should be noted
that in any Lie group quotient, $\mathcal{M}_{\text{G}}=G_{1}/G_{2}$, the
group in the numerator, $G_{1}$, will have a natural group action on the
quotient space, $\mathcal{M}_{\text{G}}$. Namely, the left action of $G_{1}$
on itself can be seen as smoothly permuting its $G_{2}$-cosets (that is,
smoothly permuting the points $[g_{0}]\in\mathcal{M}_{\text{G}}$). Concretely,
we have that $g\in G_{1}$ 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 together form a group
$\bar{g}\in\overline{G}_{1}\subset\text{Diff}(\mathcal{M}_{\text{G}})$.
Faithful construction compatibility is exactly the condition under which
$\overline{G}_{1}\cong G_{1}$ is a faithful representation of $G_{1}$ as
diffeomorphisms on $\mathcal{M}_{\text{G}}$ (see the appendix of Grimmer
(2023) for proof).

> 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 $G_{1}$.)

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
$G_{2}=\\{\mathbbm{1}\\}$.

Before stating the definition of injection compatibility, it should be noted
that so far we have only guaranteed that the structure of $G_{1}$ as a Lie
group is faithfully captured by these diffeomorphisms,
$\bar{g}\in\overline{G}_{1}$, on $\mathcal{M}_{\text{G}}$. Recall that we want
$G_{1}$ to be structurally identical to spacetime translations. Demanding
injection compatibility will additionally guarantee that the structure of
$G_{1}$ as a Lie group action is faithfully captured by the action of the
lifted diffeomorphisms, $\bar{g}^{*}\in\overline{G}_{1}{}^{*}$, on some yet-
to-be-defined states on $\mathcal{M}_{\text{G}}$.

> Definition: Injection Compatibility181818As I mentioned at the end of Sec.
> 2.1, the ISE Methodology can be applied to theories which are not about
> scalar fields,
> $\varphi_{\text{old}}:\mathcal{M}_{\text{old}}\to\mathcal{V}_{\text{old}}$.
> For instance, it can be applied to theories about particle trajectories,
> $x_{\text{old}}:\mathcal{V}_{\text{old}}\to\mathcal{M}_{\text{old}}$, or
> about tensor fields, e.g.,
> $\vec{\varphi}_{\text{old}}:\mathcal{M}_{\text{old}}\to\text{T}\mathcal{M}_{\text{old}}$.
> All that one needs to do to cover these cases is generalize the definition
> of PSTOs or, more specifically, the definition of injection compatibility.
> In this wider context, the definition of injection compatibility appearing
> here must be recognized as injection compatibility for a theory about a
> scalar field. One must then define injection compatibility for a theory
> about a particle trajectory and injection compatibility for a theory about a
> tangent vector field, etc. In general, this will only involve a change in
> the target space of the injective map, $J_{\text{G}}:S_{\text{N}}\to
> S_{\text{G}}^{\text{all}}$, to all definable particle trajectories, or all
> definable vector fields, etc. Eq. (24) and Eq. (25) do not change at all
> besides the fact that the lift map, $*_{\text{lift}}:d\mapsto d^{*}$, will
> vary from context to context. A pair of Lie group actions will be said to
> have injection compatibility simpliciter just in case they have injection
> compatibility for some kind of state definable over
> $\mathcal{M}_{\text{G}}$. Along with generalizing the notion of PSTOs, a
> keen reader might also want to: 1) upgrade the results of Appendix A.3 to
> cover the $x_{\text{old}}$ and $\vec{\varphi}_{\text{old}}$ cases; and 2)
> Argue that spacetime-kinematic compatibility is also a weak assumption in
> these contexts.  \- 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 must exist 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,
>
> $\displaystyle\forall g\ \forall\bm{\sigma}\ \ (J_{\text{G}}\circ
> g)\,\bm{\sigma}=(\bar{g}^{*}\circ J_{\text{G}})\,\bm{\sigma},$ (24)
>
> and, secondly, for all diffeomorphisms,
> $d\in\text{Diff}(\mathcal{M}_{\text{G}})$, we must have,
>
> $\displaystyle\Big{(}\forall\bm{\sigma}\ \
> d^{*}J_{\text{G}}(\bm{\sigma})=J_{\text{G}}(\bm{\sigma})\Big{)}\ \
> \Longleftrightarrow\ \ d=\mathbbm{1}.$ (25)
>
> In both of these expressions $g$ ranges over $G_{1}$ and $\bm{\sigma}$
> ranges over $S_{\text{N}}$.

Eq. (24) says that enacting any $g\in G_{1}$ on any state, $\bm{\sigma}\in
S_{\text{N}}$, and then mapping the result onto $\mathcal{M}_{\text{G}}$ via
$J_{\text{G}}$ is equivalent to first mapping $\bm{\sigma}$ onto
$\mathcal{M}_{\text{G}}$ via $J_{\text{G}}$ and then enacting the
corresponding lifted diffeomorphism, $\bar{g}^{*}$. As was motivated above,
this guarantees that $\overline{G}_{1}{}^{*}$ is a representation of $G_{1}$
not only as a Lie group, but also as a Lie group action on
$S_{\text{G}}^{\text{all}}$. Indeed, this is a faithful representation since
both of the maps $g\mapsto\bar{g}$ and $\bar{g}\mapsto\bar{g}^{*}$ are
injective (assuming we have faithful construction compatibility).

But what is the motivation behind the additional demand (i.e., Eq. (25)) that
injection compatibility makes on $G_{1}$ and $G_{2}$? So far we have
faithfully represented the action of $g\in G_{1}$ on $S_{\text{N}}$ by the
action of $\bar{g}^{*}\in\overline{G}_{1}{}^{*}$ on
$S_{\text{G}}^{\text{all}}$. Demanding Eq. (25) guarantees that this
representation is still faithful if we restrict the action of
$\bar{g}^{*}\in\overline{G}_{1}{}^{*}$ to a smaller set
$J_{\text{G}}(S_{\text{N}})\subset S_{\text{G}}^{\text{all}}$. As it turns
out, the image of the $J_{\text{G}}$ map (namely,
$J_{\text{G}}(S_{\text{N}})\subset S_{\text{G}}^{\text{all}}$) is the analog
in these definitions of the new theory’s kinematically allowed states,
$S_{\text{new}}^{\text{kin}}\subset S_{\text{new}}^{\text{all}}$. Given this,
we would like to demand the following of our PSTOs. The action of $g\in G_{1}$
on $S_{\text{N}}$ ought to be faithfully represented by the action of
$\bar{g}^{*}\in\overline{G}_{1}{}^{*}$ on $J_{\text{G}}(S_{\text{N}})\subset
S_{\text{G}}^{\text{all}}$. To achieve this, we need not only that the maps
$g\mapsto\bar{g}$ and $\bar{g}\mapsto\bar{g}^{*}$ are injective for all $g\in
G_{1}$, but also the map
$\bar{g}^{*}\mapsto\bar{g}^{*}|_{J_{\text{G}}(S_{\text{N}})}$ is too. This
final demand is exactly Eq. (25) except with $d=\bar{g}$ ranging over
$\overline{G}_{1}$.

Of course, Eq. (25) actually makes a stronger demand with $d$ ranging not only
over $\overline{G}_{1}$ but over all of $\text{Diff}(\mathcal{M}_{\text{G}})$.
Indeed, what Eq. (25) actually demands is that the map $d\mapsto
d|_{J_{\text{G}}(S_{\text{N}})}$ is injective for all
$d\in\text{Diff}(\mathcal{M}_{\text{G}})$. As Sec. 3.5 will discuss, this
extra demand is equivalent to demanding that spacetime theory produced by
externalizing $G_{1}$ and $G_{2}$ is kinematically reduced (i.e., that only
the trivial diffeomorphism, $d=\mathbbm{1}$, lifts to act trivially on every
kinematically allowed state). The motivation for making this slightly
stronger-than-required demand of our PSTOs can be found in Sec. 3.5. For now,
it is sufficient to recall from Sec. 2.5 that being kinematically reduced is a
very weak condition to demand to make of our new theory.

###  3.2 Building a New Spacetime Setting from PSTOs

With these definitions in place, let us begin by discussing a rather trivial
example of PSTOs. As the previous section has shown, internalizing any theory
with spacetime-kinematic compatibility yields a pair of Lie groups,
$G_{\text{trans}}^{\text{(int)}}$ and $G_{\text{fix}}^{\text{(int)}}$, which
act smoothly on $\mathcal{S}_{\text{neutral}}^{\text{kin}}$. As I prove in
Appendix A.3 this pair of Lie group actions automatically qualify as PSTOs
(i.e., they have faithful construction compatibility and injection
compatibility). Moreover, by externalizing these PSTOs one can always return
to the theory’s original spacetime setting, at least up to diffeomorphism.
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 $G_{\text{fix}}^{\text{(ext)}}\neq
G_{\text{fix}}^{\text{(int)}}$.

Given some PSTOs ($G_{\text{trans}}^{\text{(ext)}}$ and
$G_{\text{fix}}^{\text{(ext)}}$ acting smoothly on
$\mathcal{S}_{\text{neutral}}^{\text{kin}}$) externalization puts forward
their quotient manifold as the theory’s new spacetime manifold,

$\displaystyle\mathcal{M}_{\text{new}}\cong\mathcal{M}_{\text{G}}\coloneqq
G_{\text{trans}}^{\text{(ext)}}/G_{\text{fix}}^{\text{(ext)}},$ (26)

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 smooth 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,

$\displaystyle\mathcal{M}_{\text{new}}\coloneqq\mathcal{F}_{\text{man}}(G_{\text{trans}}^{\text{(ext)}}/G_{\text{fix}}^{\text{(ext)}}).$
(27)

The quotient $\mathcal{M}_{\text{G}}$ (and hence, $\mathcal{M}_{\text{new}}$)
inherit a unique smooth structure from the Lie group
$G_{\text{trans}}^{\text{(ext)}}$ via the quotient with
$G_{\text{fix}}^{\text{(ext)}}$ (see the appendix of Grimmer (2023) for
details). For notational convenience, the (ext) superscripts will be
suppressed where possible.

In addition to giving $\mathcal{M}_{\text{new}}$ its smooth structure, the Lie
group $G_{\text{trans}}$ is also faithfully represented as diffeomorphisms on
$\mathcal{M}_{\text{new}}$. Recall from Sec. 3.1 that the faithful
construction compatibility of $G_{\text{trans}}$ and $G_{\text{fix}}$
guarantees that each $g\in G_{\text{trans}}$ is faithfully represented as
diffeomorphisms, $\bar{g}\in\overline{G_{\text{trans}}}$, on
$\mathcal{M}_{\text{G}}$. These diffeomorphisms can then be transferred over
into diffeomorphisms on $\mathcal{M}_{\text{new}}$ as,

$\displaystyle
H(g)\coloneqq\mathcal{F}_{\text{man}}\circ\bar{g}\circ\mathcal{F}_{\text{man}}^{-1}.$
(28)

In total therefore we have an injective 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 faithful
representation of $G_{\text{trans}}$ as diffeomorphisms on
$\mathcal{M}_{\text{new}}$. Similarly, $G_{\text{fix}}$ is faithfully
represented by $H_{\text{fix}}^{\text{(ext)}}\coloneqq H(G_{\text{fix}})\cong
G_{\text{fix}}$.

The externalization process cannot stop here, however. After constructing a
new spacetime manifold, externalization must somehow map the theory’s
spacetime-neutral states, $\bm{\varphi}\in S_{\text{neutral}}^{\text{kin}}$,
and their dynamics onto $\mathcal{M}_{\text{new}}$. Let us assume that the new
theory will be about a scalar field,
$\varphi_{\text{new}}:\mathcal{M}_{\text{new}}\to\mathcal{V}_{\text{new}}$. In
order to build this new theory we first need to pick out a new value space,
$\mathcal{V}_{\text{new}}$, for the new field. Having made this selection, we
can then construct $S_{\text{new}}^{\text{all}}$, the set of all
$\mathcal{V}_{\text{new}}$-valued fields definable on
$\mathcal{M}_{\text{new}}$. The new theory will also need a method of lifting
diffeomorphisms, $d\in\text{Diff}(\mathcal{M}_{\text{new}})$, to act on these
scalar field states as $d^{*}:S_{\text{new}}^{\text{all}}\to
S_{\text{new}}^{\text{all}}$. That is, we need a continuous and injective
group homomorphism, $*_{\text{lift}}:d\mapsto d^{*}$. Fortunately, we know how
diffeomorphisms lift to act on scalar field states; We can simply take
$d^{*}\varphi_{\text{old}}\coloneqq\varphi_{\text{old}}\circ d^{-1}$.

Note that $S_{\text{new}}^{\text{all}}$ is presently far too big; It contains
every non-smooth and discontinuous field definable on
$\mathcal{M}_{\text{new}}$. We next need to 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 $S_{\text{new}}^{\text{kin}}\coloneqq
J(S_{\text{neutral}}^{\text{kin}})$.

Importantly, we are not completely free in our choice of injector,
$J:S_{\text{neutral}}^{\text{kin}}\to S_{\text{new}}^{\text{all}}$. Recall
that we would like to make every pre-spacetime translation operation, $g\in
G_{\text{trans}}$, into an honest-to-goodness spacetime translation in the new
theory. We have already seen how externalization generates a diffeomorphism on
$\mathcal{M}_{\text{new}}$ 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, $\varphi_{\text{new}}\coloneqq
J(\bm{\varphi})$. Establishing this will require that we demand some
compatibility between our choice of $J$ and our choice of $G_{\text{trans}}$
and $G_{\text{fix}}$.

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 $\bm{\varphi}\mapsto
g\,\bm{\varphi}$. As discussed above, externalization converts every $g\in
G_{\text{trans}}$ into a diffeomorphism $H(g)\in
H_{\text{trans}}\subset\text{Diff}(\mathcal{M}_{\text{new}})$. Moreover,
externalization converts every spacetime-neutral states, $\bm{\varphi}\in
S_{\text{neutral}}^{\text{kin}}$ into a state on the new spacetime,
$J(\bm{\varphi})\in S_{\text{new}}^{\text{kin}}$. 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:

$\displaystyle\forall g\ \forall\bm{\varphi}\ \ (J\circ g)\,\bm{\varphi}$
$\displaystyle=(H(g)^{*}\circ J)\,\bm{\varphi}.$ (29)

with $g$ and $\bm{\varphi}$ ranging over $G_{\text{trans}}$ and
$S_{\text{neutral}}^{\text{kin}}$ respectively. That is, enacting any $g\in
G_{\text{trans}}$ on any $\bm{\varphi}\in S_{\text{neutral}}^{\text{kin}}$
then externalizing via $J$ ought to be equivalent to first externalizing
$\bm{\varphi}$ with $J$ and then enacting the corresponding lifted
diffeomorphism, $H(g)^{*}$. Demanding Eq. (29) of our injector $J$ ensures
that $\overline{H_{\text{trans}}}{}^{*}$ is a representation of
$G_{\text{trans}}$ not only as a Lie group, but also as a Lie group action on
$S_{\text{new}}^{\text{all}}$. Indeed, this is a faithful representation since
both of the maps $g\mapsto H(g)$ and $H(g)\mapsto H(g)^{*}$ are injective for
all $g\in G_{\text{trans}}$.

Given the close relationship between $S_{\text{neutral}}^{\text{kin}}$ and
$S_{\text{new}}^{\text{kin}}$, however, we would like to make a slightly
stronger demand here. The action of $g\in G_{\text{trans}}$ on
$S_{\text{neutral}}^{\text{kin}}$ ought to be faithfully represented by the
action of $H(g)^{*}\in H_{\text{trans}}{}^{*}$ on the smaller set,
$S_{\text{new}}^{\text{kin}}\subset S_{\text{new}}^{\text{all}}$. At present,
we are guaranteed that distinct maps $g_{1}\neq g_{2}$ are mapped to distinct
diffeomorphisms $H(g_{1})\neq H(g_{2})$ and distinct actions $H(g_{1})^{*}\neq
H(g_{2})^{*}$ on the larger set, $S_{\text{new}}^{\text{all}}$. Conceivably,
however, we could presently have these distinct maps $g_{1}\neq g_{2}$
correspond to identical actions on $S_{\text{new}}^{\text{kin}}$, namely we
could have $H(g_{1})^{*}|_{\text{kin}}=H(g_{2})^{*}|_{\text{kin}}$. If this
were to happen, the $H(g)^{*}$ action on $S_{\text{new}}^{\text{kin}}$ would
not be a faithful representation of the $g$ action on
$S_{\text{neutral}}^{\text{kin}}$. In order to avoid this we must demand that
the map $H(g)\mapsto H(g)^{*}|_{\text{kin}}$ is injective. Namely, we must
demand that for all diffeomorphisms, $h\in
H_{\text{trans}}\subset\text{Diff}(\mathcal{M}_{\text{new}})$, we have,

$\displaystyle\Big{(}\forall\bm{\varphi}\ \
h^{*}J(\bm{\varphi})=J(\bm{\varphi})\Big{)}\ \ \Longleftrightarrow\ \
h=\mathbbm{1},$ (30)

with $\bm{\varphi}$ ranging over $S_{\text{neutral}}^{\text{kin}}$. Under this
condition, the way that $H(g)^{*}$ acts on $S_{\text{new}}^{\text{kin}}$ will
be a faithful representation of the way that $g$ acts on
$S_{\text{neutral}}^{\text{kin}}$.

We have nearly completed the externalization process. By construction, we so
far have a bijection, $J:S_{\text{neutral}}^{\text{kin}}\leftrightarrow
S_{\text{new}}^{\text{kin}}$, between our two characterizations of the
theory’s kinematically allowed states. Recall from Eq. (8) that
$\mathcal{S}_{\text{neutral}}^{\text{kin}}$ comes equipped with all of the
theory’s dynamically and metaphysically relevant structures,

$\displaystyle\mathcal{S}_{\text{neutral}}^{\text{kin}}\coloneqq(S_{\text{neutral}}^{\text{kin}},\mathcal{T}_{\text{neutral}},\text{relevant
structures}).$ (31)

These relevant structures can then be transferred from
$\mathcal{S}_{\text{neutral}}^{\text{kin}}$ to $S_{\text{new}}^{\text{kin}}$
via our choice of $J:S_{\text{neutral}}^{\text{kin}}\to
S_{\text{new}}^{\text{all}}$. Thus, by construction, all of the theory’s
dynamically and metaphysically relevant structures survive faithfully into the
new theory. Explicit demonstrations of how this $J$ map induces these
structures in $\mathcal{S}_{\text{new}}^{\text{kin}}$ can be found in Grimmer
(2023). For now, it suffices to note that these structural details are
mechanically irrelevant throughout the ISE Methodology. This completes the
externalization process.

To review: Externalization builds a new spacetime setting for our theory from
the following materials: A pair of Lie groups, $G_{\text{trans}}$ and
$G_{\text{fix}}$, which act smoothly on
$\mathcal{S}_{\text{neutral}}^{\text{kin}}$ with faithful construction
compatibility; a choice of value space, $\mathcal{V}_{\text{new}}$; and an
injective map, $J:\mathcal{S}_{\text{neutral}}^{\text{kin}}\to
S_{\text{new}}^{\text{all}}$. This construction is successful if Eq. (29) and
Eq. (30) are satisfied. But given some pair of Lie group actions,
$G_{\text{trans}}$ and $G_{\text{fix}}$, how do we know that any such space,
$\mathcal{V}_{\text{new}}$, and injective map, $J$, exists? The existence of
such $\mathcal{V}_{\text{new}}$ and $J$ is straightforwardly guaranteed by the
injection compatibility of $G_{\text{trans}}$ and $G_{\text{fix}}$.

Before discussing some example applications of the ISE Methodology, one final
comment is warranted regarding the externalization process. While injection
compatibility guarantees the existence of some injective map,
$J:S_{\text{neutral}}^{\text{kin}}\to S_{\text{new}}^{\text{all}}$, satisfying
Eq. (29) and Eq. (30), it does not guarantee that this $J$ map is unique.
Indeed, injection compatibility cannot guarantee $J$’s uniqueness since if $J$
works then twice $J$ does too. Grimmer (2023) has investigated the range of
injective maps which are available to us and concluded that our choice of
PSTOs almost entirely fixes our choice of injector, $J$. For example, if we
pick our PSTOs to be rigid shifts in the old theory’s Fourier space then the
new spacetime manifold will effectively be a copy of the old theory’s Fourier
space. When it comes time to map the old theory’s states onto the new
spacetime, we have very little freedom in doing so. Up to some rescaling
freedom, the new theory’s states must be the Fourier transforms of the old
theory’s states. As Grimmer (2023) has argued, what little freedom we have in
picking $J$ is metaphysically innocuous; it only affects how the states are
being represented in the new spacetime setting. Hence, the externalization
process is driven almost entirely by our choice of PSTOs, and not by our
choice of injector $J$.

###  3.3 Example Application 1: Dimensional Reduction

This section will show how the ISE Methodology can be applied to the Quartic
Klein Gordon theory introduced in Sec. 2. Recall that the internalization of
this theory has already been discussed in Sec. 2.4. We can now begin to search
for good PSTOs to externalize, let us call them
$G_{\text{trans}}^{\text{(ext)}}$ and $G_{\text{fix}}^{\text{(ext)}}$. One
pair of PSTOs which immediately come to mind are the Lie groups which were
just produced by the internalization process, namely
$G_{\text{trans}}^{\text{(int)}}$ and $G_{\text{fix}}^{\text{(int)}}$. As I
prove in Appendix A.3, these are, in fact, PSTOs. This follows from the
spacetime-kinematic compatibility of the 3+1D QKG theory. Indeed, as I discuss
in Appendix A.3, one can easily externalize these PSTOs in order to return
back to the theory’s original spacetime setting. Fortunately, however, there
are also more exciting PSTOs available to us.

Let us next search for a non-trivial set of PSTOs. Recall that the theory’s
old spacetime setting, $\mathcal{M}_{\text{new}}\cong\mathbb{R}^{2}\times
S^{2}$, consists of two extended dimensions (spanned by $t$ and $q$) and two
compactified dimensions (spanned by $x$, $y$, and $z$ with
$x^{2}+y^{2}+z^{2}=1$) . After internalizing this theory, one can choose to
leave the old theory’s compactified dimensions internalized and only re-
externalize the old theory’s $t$ and $q$ translations. Intuitively, this
should result in a new spacetime framing of our theory in which is $1+1$
dimensional, i.e., $\mathcal{M}_{\text{new}}\cong\mathbb{R}^{2}$. Let us
embark with this destination in mind.

Recall from Sec. 2.4, the Lie group,
$H_{\text{trans}}^{\text{(int)}}\subset\text{Diff}(\mathcal{M}_{\text{old}})$,
which we used in the internalization process. Recall that this group
$H_{\text{trans}}^{\text{(int)}}\cong(\mathbb{R}^{2},+)\times\text{SO}(3)$
implemented rigid translations in the old theory’s $(t,q)$ coordinates and
rigid rotations in the old theory’s $(x,y,z)$ coordinates. Let us focus on its
Lie subgroup which implements rigid $(t,q)$-translations, namely
$F_{\text{1+1}}\subset H_{\text{trans}}^{\text{(int)}}$. Concretely, let
$f(\Delta t,\Delta q)\in F_{\text{1+1}}\cong(\mathbb{R}^{2},+)$ act in the
$(t,q,x,y,z)$ coordinate system as $f(\Delta t,\Delta
q):(t,q,x,y,z)\mapsto(t+\Delta t,q+\Delta q,x,y,z)$. Since this Lie group is
kinematically allowed, $F_{\text{1+1}}\subset
H_{\text{trans}}^{\text{(int)}}\subset\text{Diff}_{\text{kin}}(\mathcal{M}_{\text{old}})$,
it faithfully survives internalization. Namely, it becomes a Lie group
$G_{\text{1+1}}\coloneqq G(F_{\text{1+1}})\cong
F_{\text{1+1}}\cong(\mathbb{R}^{2},+)$ which acts smoothly on
$\mathcal{A}_{\text{neutral}}^{\text{kin}}$. As I will now discuss, instead of
simply using the PSTOs provided to us by internalization (namely,
$G_{\text{trans}}^{\text{(int)}}$) we can instead choose our PSTOs to be
$G_{\text{trans}}^{\text{(ext)}}\coloneqq G_{\text{1+1}}$.

Intuitively, externalizing with
$G_{\text{trans}}^{\text{(ext)}}=G_{\text{1+1}}$ should yield a new spacetime
framing of our theory which is $1+1$ dimensional, i.e.,
$\mathcal{M}_{\text{new}}\cong\mathbb{R}^{2}$. Before seeing whether or not
this happens, however, we need to complete our choice of PSTOs by picking out
another Lie group, $G_{\text{fix}}^{\text{(ext)}}\subset
G_{\text{trans}}^{\text{(ext)}}$, to externalize. What are our options? Recall
that $G_{\text{fix}}^{\text{(ext)}}$ must have faithful construction
compatibility with $G_{\text{trans}}^{\text{(ext)}}$. In particular,
$G_{\text{fix}}^{\text{(ext)}}$ must be a core-free subgroup of
$G_{\text{trans}}^{\text{(ext)}}=G_{\text{1+1}}$ (i.e., it must contain no
non-trivial normal subgroups of $G_{\text{trans}}^{\text{(ext)}}$). Next, note
that $G_{\text{1+1}}\cong(\mathbb{R},+)$ is abelian and hence all of its
subgroups are normal. This means that the only core-free subgroup of
$G_{\text{1+1}}$ is the trivial group. Hence, given our choice of
$G_{\text{trans}}^{\text{(ext)}}=G_{\text{1+1}}$ we are forced to take
$G_{\text{fix}}^{\text{(ext)}}=\\{\mathbbm{1}\\}$.

Proceeding with this choice of Lie group actions,
$G_{\text{trans}}^{\text{(ext)}}=G_{\text{1+1}}\cong(\mathbb{R}^{2},+)$ and
$G_{\text{fix}}^{\text{(ext)}}=\\{\mathbbm{1}\\}$, externalization will build
a new spacetime manifold from their quotient,

$\displaystyle\mathcal{M}_{\text{new}}\cong
G_{\text{trans}}^{\text{(ext)}}/G_{\text{fix}}^{\text{(ext)}}\cong\mathbb{R}^{2}.$
(32)

As expected, we do end up with a new spacetime manifold consisting of only one
dimension of space and one dimension of time. To see that these two dimensions
deserve the names “space” and “time” respectively, recall that the old
theory’s rigid $(t,q)$-shifts, $f(\Delta t,\Delta q)\in
F_{\text{1+1}}\subset\text{Diff}(\mathcal{M}_{\text{old}})$, have faithfully
survived internalization becoming $g(\Delta t,\Delta q)\coloneqq G(f(\Delta
t,\Delta q))\in G_{\text{1+1}}$ acting on
$\mathcal{A}_{\text{neutral}}^{\text{kin}}$. But this is the Lie group which
we have just externalized. Thus, post-externalization we will have $g(\Delta
t,\Delta q)$ becoming $h(\Delta t,\Delta q)\coloneqq H(g(\Delta t,\Delta
q))\in\text{Diff}(\mathcal{M}_{\text{new}})$, a diffeomorphism on our new
1+1-dimensional spacetime manifold,
$\mathcal{M}_{\text{new}}\cong\mathbb{R}^{2}$. Let $(\tau,\xi)$ be a
coordinate system for $\mathcal{M}_{\text{new}}\cong\mathbb{R}^{2}$ such that
$h(\Delta t,\Delta q)(\tau,\xi)=(\tau+\Delta t,\xi+\Delta q)$. Thus, rigid
$(t,q)$-shifts in the old theory become to rigid $(\tau,\xi)$-shifts in the
new theory. Externalization has turned our choice of PSTOs,
$G_{\text{trans}}^{\text{(ext)}}=G_{\text{1+1}}$, into honest-to-goodness
spacetime translations on $\mathcal{M}_{\text{new}}\cong\mathbb{R}^{2}$.
Namely, $H_{\text{trans}}^{\text{(ext)}}\coloneqq
H(G_{\text{trans}}^{\text{(ext)}})$.

After constructing this new spacetime manifold, $\mathcal{M}_{\text{new}}$,
the next step in the externalization process is to inject the new theory’s
states and dynamics onto this new spacetime setting. Concretely, this is to be
done by picking out an injective map, $J:S_{\text{neutral}}^{\text{kin}}\to
S_{\text{new}}^{\text{all}}$. Here $\varphi_{\text{new}}\in
S_{\text{new}}^{\text{all}}$ is the set of all scalar fields states,
$\varphi_{\text{new}}:\mathcal{M}_{\text{new}}\to\mathcal{V}_{\text{new}}$,
definable on the new spacetime setting for some yet-to-be fixed value space,
$\mathcal{V}_{\text{new}}$. What constrains our choice of $J$? As Sec. 3.2 has
stressed, the new diffeomorphisms, $h(\Delta t,\Delta q)\in
H_{\text{trans}}^{\text{(ext)}}$, must faithfully represent our PSTOs not only
as a Lie group but also as a Lie group action acting on the new theory’s
states. In particular, it follows from Eq. (29) that infinitesimal
$(t,q)$-shifts in the old theory must correspond to infinitesimal
$(\tau,\xi)$-shifts in the new theory. Moreover, this correspondence must hold
even once we lift these diffeomorphisms to act on the old/new theory’s states.
As I will now discuss, it follows from this that a generic planewave in the
old theory’s spacetime, must map onto a corresponding planewave in the new
theory’s spacetime.

Consider a generic planewave in the old theory’s spacetime,

$\displaystyle\varphi_{\text{old}}^{(\omega\,k\,\ell\,m)}(t,q,x,y,z)=Y_{\ell
m}(x,y,z)\ \text{exp}(\mathrm{i}\omega t+\mathrm{i}kq),$ (33)

where $Y_{\ell m}(x,y,z)$ are the spherical harmonics with
$\ell=0,\dots,\infty$ and $-\ell\leq m\leq\ell$. Note that this state is a
simultaneous eigenvector of $\partial_{t}$ and $\partial_{q}$ with eigenvalues
$\mathrm{i}\omega$ and $\mathrm{i}k$ respectively. It follows from Eq. (29)
that the corresponding state in the new theory,
$\varphi_{\text{new}}^{(\omega\,k\,\ell\,m)}(\tau,\xi)\coloneqq
J\circ\mathcal{F}_{\text{alg}}(\varphi_{\text{old}}^{(\omega\,k\,\ell\,m)})$,
must be a simultaneous eigenvector of $\partial_{\tau}$ and $\partial_{\xi}$
with eigenvalues $\mathrm{i}\omega$ and $\mathrm{i}k$ respectively. Namely, we
must have,191919Two analogous applications of the ISE Methodology can be found
in Grimmer (2023). These two examples also include planewave-to-planewave
correspondences analogous to this one. More information about how such
correspondences arise can be found there.

$\displaystyle
J\circ\mathcal{F}_{\text{alg}}:\varphi_{\text{old}}^{(\omega\,k\,\ell\,m)}\mapsto\varphi_{\text{new}}^{(\omega\,k\,\ell\,m)}=a(\omega,k,\ell,m)\
\text{exp}(\mathrm{i}\omega\tau+\mathrm{i}k\xi),$ (34)

for some $a(\omega,k,\ell,m)\neq 0$ in $\mathcal{V}_{\text{new}}$. Thus, up to
some rescaling freedom, the externalization process must map planewaves in the
old theory’s spacetime onto a corresponding planewave in the new theory’s
spacetime.

But what should we make of this sort of planewave-rescaling freedom? Grimmer
(2023) has discussed this issue at length. To summarize: This rescaling
freedom is metaphysically irrelevant. In particular, it only affects how the
theory’s states are being represented in this new spacetime setting. It does
not actually change their relative sizes in any meaningful way. Recall that,
by stipulation, the metaphysically relevant way to judge the relative size and
similarity of the old theory’s states, is with a certain inner product,
$\langle,\rangle_{\text{old}}$. The $\mathcal{F}_{\text{alg}}$ and $J$ maps
combine to induce a corresponding inner product,
$\langle,\rangle_{\text{new}}$, for the new theory’s states. The
metaphysically relevant way to judge the relative size and similarity of the
new theory’s states is with this induced inner product (which may or may not
be the new spacetime’s $L^{2}$ inner product). If one investigates this
induced inner product, one would find that there is absolutely no dependence
on these scale factors, $a(\omega,k,\ell,m)$. Normalized states in the old
theory (with respect to $\langle,\rangle_{\text{old}}$) automatically map onto
normalized states in the new theory (with respect to
$\langle,\rangle_{\text{new}}$).

Given that our choice of $a(\omega,k,\ell,m)$ is merely representational, we
might as well choose a nice one. In particular, we can choose
$a(\omega,k,\ell,m)=\vec{e}_{\ell
m}\in\mathcal{V}_{\text{new}}\cong\mathbb{R}^{\infty}$ where the set
$\\{\vec{e}_{\ell m}\\}$ form a basis for an infinite-dimensional vector
space, $\mathcal{V}_{\text{new}}\cong\mathbb{R}^{\infty}$, where the new field
is valued,
$\varphi_{\text{new}}:\mathcal{M}_{\text{new}}\to\mathcal{V}_{\text{new}}$.
Note that for this choice of $a(\omega,k,\ell,m)$ the new theory’s induced
inner product, $\langle,\rangle_{\text{new}}$, will be the $L^{2}$ inner
product in the $(\tau,\xi)$-coordinate system together with the native inner
product on $\mathcal{V}_{\text{new}}\cong\mathbb{R}^{\infty}$. That is,
whichever basis, $\\{\vec{e}_{\ell m}\\}$, we choose for
$\mathcal{V}_{\text{new}}$ will end up being an orthonormal basis with respect
to this inner product.

Having settled on a choice of scaling, $a(\omega,k,\ell,m)$, we have now fixed
the injective map, $J:S_{\text{neutral}}^{\text{kin}}\to
S_{\text{new}}^{\text{all}}$, which defines the new theory’s states and
dynamics. Indeed, after a straightforward calculation, one finds that the new
theory is defined as follows:202020See Grimmer et al. (2021) for details.

> A Quartic Klein Gordon Theory (1+1D QKG) -  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, $\mathcal{V}_{\text{new}}\cong\mathbb{R}^{\infty}$. For this theory,
> there exists an injective map,
> $C:\mathcal{M}_{\text{new}}\to\mathbb{R}^{2}$, which assigns to each point,
> $p\in\mathcal{M}_{\text{new}}$, some coordinates,
> $C(p)=(\tau,\xi)\in\mathbb{R}^{2}$. Moreover, for this theory’s value space
> there exists a basis $\vec{e}_{\ell m}\in\mathcal{V}_{\text{new}}$ with
> respect to which we can decompose the states as
> $\varphi_{\text{new}}=\sum_{\ell m}\,\varphi_{\text{new}}^{(\ell
> m)}\,\vec{e}_{\ell m}$ with $\ell=0,\dots,\infty$ and $-\ell\leq
> m\leq\ell$.. Adopting a phrase from Grimmer et al. (2021), let us call each
> of these components a “subfield”. This theory’s states are subject to the
> following kinematic constraint: In this fixed coordinate system each field,
> $\varphi_{\text{new}}^{(\ell m)}(\tau,\xi)$, must be smooth. The
> metaphysically relevant way to judge the relative size and similarity of
> this theory’s states is with an $L^{2}$ inner product in the
> $(\tau,\xi)$-coordinate system together with the native inner product on
> $\mathcal{V}_{\text{new}}\cong\mathbb{R}^{\infty}$. In this coordinate
> system, each subfield, $\varphi_{\text{new}}^{(\ell m)}(\tau,\xi)$ obeys the
> following dynamics:
>
> $\displaystyle 0=\ $
> $\displaystyle(\partial_{\tau}^{2}-\partial_{\xi}^{2}+M_{\ell
> m}^{2})\,\varphi_{\text{new}}^{(\ell m)}+\lambda\
> \mathcal{O}\big{(}\varphi_{\text{new}}^{(\ell m)}\
> \varphi_{\text{new}}^{(\ell^{\prime}m^{\prime})}\
> \varphi_{\text{new}}^{(\ell^{\prime\prime}m^{\prime\prime})}\big{)}$ (35)
>
> where $\lambda\in\mathbb{R}$ is some fixed coupling constant and $M_{\ell
> m}^{2}\coloneqq M^{2}+\ell(\ell+1)$ where $M\geq 0$ is some fixed mass
> parameter.

Note that in the non-interacting case (with $\lambda=0$) each subfield evolves
as a massive 1+1 dimensional Klein-Gordon field. From the perspective of the
old theory, the new theory’s subfields acquire their masses, $M_{\ell
m}\coloneqq\sqrt{M^{2}+\ell(\ell+1)}$, partly from the original field’s mass,
$M$ and partly from their geometric confinement. Those familiar with Kaluza
Klein dimensional reduction should recognize what has happened here. We have
traded a theory about a single scalar field, $\varphi_{\text{old}}$, with some
compactified dimensions for a theory about an infinite collection of massive
scalar fields, $\varphi_{\text{new}}^{(\ell m)}$, without any compactified
dimensions.

But what about the interacting case with $\lambda\neq 0$? In the old theory,
the cubic self-interaction term was governed by the old theory’s point-wise
product operation, $\times_{\text{old}}$. One can trace this product operation
through the internalization process to a spacetime-neutral product operation,
$\times_{\text{neutral}}$. One can then follow this product operation through
the externalization process to a new induced product operation,
$\times_{\text{new}}$, in the new theory. Ultimately, $\times_{\text{new}}$
ends up being a point-wise product on $\mathcal{M}_{\text{new}}$ complemented
by a product operation in the new theory’s value space,
$\mathcal{V}_{\text{new}}\cong\mathbb{R}^{\infty}$. This induced product
operation on $\mathcal{V}_{\text{new}}$ is related to the harmonic
decomposition of the products of spherical harmonics. It is ultimately this
induced product operation on $\mathcal{V}_{\text{new}}$ which defines the
cubic coupling between the new theory’s subfields.

In sum, this example has shown how the ISE Methodology can be used to build a
new spacetime setting for the 3+1D Quartic Klein Gordon theory. Beginning from
a relatively large spacetime setting,
$\mathcal{M}_{\text{old}}\cong\mathbb{R}^{2}\times S^{2}$, we could choose to
re-externalize just a part of it yielding a smaller spacetime
$\mathcal{M}_{\text{new}}\cong\mathbb{R}^{2}$. The result is the above-
discussed 1+1D Quartic Klein Gordon theory.

Note that both the input and the output theories have spacetime-kinematic
compatibility. As I will prove in Sec. 4, this means that the ISE Methodology
can be applied equally well in reverse. Namely, if we had started from the
1+1D QKG theory living on $\mathcal{M}_{\text{old}}\cong\mathbb{R}^{2}$ we
could then use the ISE Methodology to redescribe this theory as if it were the
3+1D QKG theory, taking place on a larger spacetime,
$\mathcal{M}_{\text{new}}\cong\mathbb{R}^{2}\times S^{2}$. Sec. 4 will discuss
this possibility further. For now, let’s see one more example application of
the ISE Methodology.

###  3.4 Example Application 2: Periodic Matter to Periodic Spacetime

This section will sketch how the ISE Methodology can be applied to the
Periodic Matter on a Non-Periodic Spacetime theory introduced in Sec. 2. To
begin, allow me to briefly retrace the internalization of this theory
discussed in Sec. 2.3. Recall that the maps, $d(\Delta t,\Delta
x)\in\text{Diff}(\mathcal{M}_{\text{old}})$, implement rigid translations in
the old theory’s $(t,x)$ coordinate system. Together these form a Lie group
$H_{\text{trans}}^{\text{(int)}}\cong(\mathbb{R}^{2},+)$. As we have seen,
while this Lie group does survive internalization, it does so unfaithfully. In
particular, distinct elements of $H_{\text{trans}}^{\text{(int)}}$, namely
$d(\Delta t,\Delta x)\neq d(\Delta t,\Delta x+L)$, end up mapping onto the
same element of $G_{\text{trans}}^{\text{(int)}}\coloneqq
G(H_{\text{trans}}^{\text{(int)}})$, namely $G(d(\Delta t,\Delta
x))=G(d(\Delta t,\Delta x+L))$. In total, we have that
$H_{\text{trans}}^{\text{(int)}}\cong(\mathbb{R}^{2},+)$ is reduced to
$G_{\text{trans}}^{\text{(int)}}\cong(\mathbb{R},+)\times U(1)$.

As it turns out, despite having survived internalization unfaithfully, the Lie
group $G_{\text{trans}}^{\text{(int)}}\cong(\mathbb{R},+)\times U(1)$ is still
structurally indistinguishable from a set of spacetime translation operations
(i.e., they are still PSTOs). Hence, we can take
$G_{\text{trans}}^{\text{(ext)}}=G_{\text{trans}}^{\text{(int)}}$ as our PSTOs
in the externalization process (together with a forced choice of
$G_{\text{fix}}^{\text{(ext)}}=\\{\mathbbm{1}\\}$).212121As in the previous
example, our choice of $G_{\text{fix}}^{\text{(ext)}}=\\{\mathbbm{1}\\}$ is
forced upon us by the fact that $G_{\text{trans}}^{\text{(ext)}}$ is abelian.
Specifically, this follows from the demand that
$G_{\text{trans}}^{\text{(ext)}}$ and $G_{\text{fix}}^{\text{(ext)}}$ have
faithful construction compatibility.. Proceeding with this choice of PSTOs
externalization will build a new spacetime manifold from their quotient,

$\displaystyle\mathcal{M}_{\text{new}}\cong
G_{\text{trans}}^{\text{(ext)}}/G_{\text{fix}}^{\text{(ext)}}\cong\mathbb{R}\times
S^{1}.$ (36)

Perhaps surprisingly, externalizing the PSTOs given to us by the
internalization process (i.e., taking
$G_{\text{trans}}^{\text{(ext)}}=G_{\text{trans}}^{\text{(int)}}$ and
$G_{\text{fix}}^{\text{(ext)}}=G_{\text{fix}}^{\text{(int)}}$) has yielded a
different spacetime manifold $\mathcal{M}_{\text{new}}\cong\mathbb{R}\times
S^{1}$ than we started with $\mathcal{M}_{\text{old}}\cong\mathbb{R}^{2}$. In
Appendix A.3, I prove that one can always return to the theory’s original
spacetime setting under the assumption that the theory in question has
spacetime-kinematic compatibility. This assumption fails here.

This new spacetime manifold, however, is still closely related to the old one.
By construction, the PSTOs which we have chosen (which were diffeomorphisms on
$\mathcal{M}_{\text{old}}\cong\mathbb{R}^{2}$) are naturally represented as
diffeomorphisms on the new spacetime manifold,
$\mathcal{M}_{\text{new}}\cong\mathbb{R}\times S^{1}$. Namely, $g\in
G_{\text{trans}}^{\text{(ext)}}$ becomes $H(g)\in
H_{\text{trans}}^{\text{(ext)}}\coloneqq H(G_{\text{trans}}^{\text{(ext)}})$.
To see this correspondence more clearly, let $(\tau,q)$ be a coordinate system
for $\mathcal{M}_{\text{new}}\cong\mathbb{R}\times S^{1}$ such that
$H(G(d(\Delta t,\Delta x)))(\tau,q)=(\tau+\Delta t,q+\Delta x)$. Note that the
$q$-coordinate is periodic with period $L$ $H$ map is a Lie group isomorphism
and we have $G(d(0,n\,L))=G(d(0,0))$. In sum, rigid $(t,x)$-shifts in the old
theory become to rigid $(\tau,q)$-shifts in the new theory. As before,
externalization has turned our choice of PSTOs,
$G_{\text{trans}}^{\text{(ext)}}=G_{\text{trans}}^{\text{(int)}}$, into
honest-to-goodness spacetime translations on the new spacetime,
$\mathcal{M}_{\text{new}}$.

This new spacetime manifold may strike one as suspiciously similar to the
spacetime from the Matter on a Periodic Spacetime theory introduced in Sec. 2.
In fact, it is not hard to see that the following choice of
$J:S_{\text{neutral}}^{\text{kin}}\to S_{\text{new}}^{\text{all}}$ gives us a
transcription of the periodic matter theory into the periodic spacetime
theory:

$\displaystyle
J\circ\mathcal{F}:\varphi_{\text{old}}(t,x)\mapsto\varphi_{\text{new}}(\tau,q)=\varphi_{\text{old}}(\tau,q).$
(37)

This, we can use the ISE Methodology to redescribe the periodic matter theory
as the periodic spacetime theory. One may wonder whether the reverse
redescription is also possible. Can we use the ISE Methodology to go from the
periodic spacetime theory back to the periodic matter theory? As I will
discuss further in Sec. 4, this reverse redescription is impossible. The issue
is that (as the next section will prove) all spacetime theories which can be
produced by externalization have spacetime-kinematic compatibility. Since the
periodic matter theory lacks this, it cannot be output by the externalization
process.

###  3.5 Externalization Guarantees Spacetime-Kinematic Compatibility

This section will prove that every spacetime theory produced by the
externalization process is guaranteed to have spacetime-kinematic
compatibility. Indeed, I will further prove that the spacetime theories which
are producible via externalization are exactly those with spacetime-kinematic
compatibility.

First, note that the new theory satisfies the modal assumption. By
construction, the new theory has its kinematically allowed states,
$S_{\text{new}}^{\text{kin}}$, picked out as a subset of
$S_{\text{new}}^{\text{all}}$. Next recall from Sec. 2 that there is enough
structure in $\mathcal{S}_{\text{neutral}}^{\text{kin}}$ to identify its
dynamically allowed subspace, $\mathcal{S}_{\text{neutral}}^{\text{dyn}}$. For
instance, we might use a product operation, $\times_{\text{neutral}}$, defined
on $\mathcal{S}_{\text{neutral}}^{\text{kin}}$ to state the dynamics of a non-
linear theory. In this case, externalization will induce an analogous product
operation, $\times_{\text{new}}$, in $\mathcal{S}_{\text{new}}^{\text{kin}}$
via the bijection $J:S_{\text{neutral}}^{\text{kin}}\leftrightarrow
S_{\text{new}}^{\text{kin}}$. Using this we can identify the new theory’s
dynamically allowed subspace $\mathcal{S}_{\text{new}}^{\text{dyn}}$. (See
Grimmer (2023) for a more explicit discussion of how the $J$ map induces this
structure.)

Next, note that the new theory satisfies the smooth lifting assumption. As the
new theory is about a scalar field,
$\varphi_{\text{new}}:\mathcal{M}_{\text{new}}\to\mathcal{V}_{\text{new}}$, we
can lift generic diffeomorphisms $d\in\text{Diff}(\mathcal{M}_{\text{new}})$
to act on them as $d^{*}\varphi_{\text{new}}\coloneqq\varphi_{\text{new}}\circ
d^{-1}$. Note that $*_{\text{lift}}:d\mapsto d^{*}$ is a continuous and
injective group homomorphism.222222Alternatively, as footnote 18 in Sec. 3.1
notes, the new theory could also have been about a particle trajectory,
$x_{\text{new}}:\mathcal{V}_{\text{new}}\to\mathcal{M}_{\text{new}}$, or a
tangent vector field
$\vec{\varphi}_{\text{new}}:\mathcal{M}_{\text{new}}\to\text{T}\mathcal{M}_{\text{new}}$.
In these cases too, we can lift diffeomorphisms using some continuous and
injective group homomorphism $*_{\text{lift}}:d\mapsto d^{*}$. Moreover, the
new theory’s state space $\mathcal{S}_{\text{new}}^{\text{kin}}$ has a smooth
structure, $\mathcal{T}_{\text{new}}$, induced on it via the injector $J$. For
instance, it follows from Eq. (29) and Eq. (30) that the smooth action of our
PSTOs (i.e., the smooth action of $G_{\text{trans}}^{\text{(ext)}}$ on
$\mathcal{S}_{\text{neutral}}^{\text{kin}}$) corresponds exactly to the smooth
actions of $H_{\text{trans}}^{*}$ on $\mathcal{S}_{\text{new}}^{\text{kin}}$.

Let us next confirm that any spacetime theory produced via externalization
will be kinematically homogeneous. To do so, I will prove that
$H_{\text{trans}}^{\text{(ext)}}\coloneqq H(G_{\text{trans}}^{\text{(ext)}})$
is a Lie group which acts transitively and smoothly on
$\mathcal{M}_{\text{new}}$ and is moreover kinematically allowed. All but the
last of these claims follows directly from the fact that the $H$ map defined
in Eq. (28) is a Lie group isomorphism. This itself follows from
$G_{\text{trans}}^{\text{(ext)}}$ and $G_{\text{fix}}^{\text{(ext)}}$ having
faithful construction compatibility.

The final claim (that $H_{\text{trans}}^{\text{(ext)}}$ is kinematically
allowed) follows from Eq. (29). To see this, note that for every $h\in
H_{\text{trans}}^{\text{(ext)}}$ we can find a corresponding $g\in
G_{\text{trans}}^{\text{(ext)}}$ with $h=H(g)$. By Eq. (29), enacting this
diffeomorphism, $h^{*}$, on any kinematically allowed state in the new theory
is equivalent to enacting $g$ pre-externalization and then externalizing with
$J$. But the $J$ map always produces kinematically allowed states by
definition. Hence, every $h^{*}\in H_{\text{trans}}^{\text{(ext)}}{}^{\ *}$
maps the new theory’s kinematically allowed states onto each other. Hence,
every $h\in H_{\text{trans}}^{\text{(ext)}}$ is kinematically allowed.

Finally, let us confirm that the new theory is kinematically reduced, i.e.,
that is has $H_{\text{kin}}=\\{\mathbbm{1}\\}$ or equivalently that the map
$d\mapsto d^{*}|_{\text{kin}}$ is injective. Recalling that
$S_{\text{new}}^{\text{kin}}\coloneqq J(S_{\text{neutral}}^{\text{kin}})$, the
new theory being kinematically reduced follows immediately from the injection
compatibility of $G_{\text{trans}}^{\text{(ext)}}$ and
$G_{\text{fix}}^{\text{(ext)}}$. Namely, it follows directly from Eq. (25).
This completes the proof that spacetime theories produced via externalization
always have spacetime-kinematic compatibility.

A brief comment is here warranted about the last step in this proof. As I
noted at the end of Sec. 3.1, injection compatibility demands that the map
$d\mapsto d^{*}|_{\text{kin}}$ is injective over all
$d\in\text{Diff}(\mathcal{M}_{\text{new}})$ whereas one could conceivably have
gotten away with demanding this for the smaller group, $d\in
H_{\text{trans}}^{\text{(ext)}}$. While this lesser demand would have been
fine from the perspective of externalization, an issue would arise when we
subsequently tried to re-internalize this theory. Namely, if one only makes
the weaker demand then the new spacetime theory might not be kinematically
reduced and hence might not have spacetime-kinematic compatibility. As I
mentioned in Sec. 2.3 (and have proved in Appendix A.2), a theory’s spacetime
manifold will faithfully and robustly survive internalization if and only if
the theory has spacetime-kinematic compatibility.232323Technically, I there
show that a slightly weakened version of spacetime-kinematic compatibility is
necessary and sufficient for a theory’s spacetime manifold will faithfully and
robustly survive internalization. Indeed, we might have this robust kind of
survival for a kinematically unreduced theory, but only under a seemingly
unlikely condition. Namely, $H_{\text{kin}}\neq\\{\mathbbm{1}\\}$ would need
to have a trivial intersection with every finite-dimensional transitively-
acting kinematically-allowed Lie group,