#  Same-diff?
Conceptual similarities between gauge transformations and diffeomorphisms
Part III: Representational conventions and relationism

Henrique de A<EMAIL_ADDRESS>
University of Oxford
Oriel College Oxford  OX2 1TQ  UK

###### Abstract

The following questions are germane to our understanding of gauge-(in)variant
quantities and physical possibility: in which ways are gauge transformations
and spacetime diffeomorphisms similar, and in which are they different?
Sophistication is the most popular attitude towards some of these questions:
roughly, it takes models related by these symmetries to represent the same
physical possibility. In the previous paper in this series, I discussed
obstacles to sophistication and then showed how these obstacles are overcome
by theories that fulfill three Desiderata (i-iii). But this resolution still
leaves open two main worries about sophistication: (a) it allows the
individuation of structure-tokens to remain intractably prolix and thus of
limited use, which is why practising physicists frequently invoke ‘relational,
symmetry-invariant observables’; and (b) it leaves us with no formal framework
for expressing counterfactual statements about the world. Here I will show
that a third Desideratum, (iii), answers these worries. The new Desideratum
requires a ‘relational’ understanding _of coordinates_ (or frames, etc).

###### Contents

  1. 1 Introduction
    1. 1.1 Recap
    2. 1.2 The remaining worries
  2. 2 The challenge of individuating structural content
    1. 2.1 The identification of structure-tokens
    2. 2.2 Can the sceptic ever be satisfied?
  3. 3 Representational conventions
    1. 3.1 Representational conventions conceptually summarised
      1. A brief history of representational conventions
    2. 3.2 Representational conventions: general definitions
    3. 3.3 Representational conventions in field theories: relation to gauge-fixing
  4. 4 How to choose a representational convention?
    1. 4.1 Healey’s argument from functional roles
    2. 4.2 Refuting Healey’s alleged distinction using representational conventions
    3. 4.3 Examples
      1. 4.3.a Center of mass for non-relativistic point-particles
      2. 4.3.b Electromagnetism: Lorenz gauge
      3. 4.3.c General relativity: DeDonder gauge
      4. 4.3.d General Relativity: Komar-Kretschmann coordinates
    4. 4.4 Whence these choices?
  5. 5 Representational conventions resolving Worry (2-ii)
    1. 5.1 Basic formalism
    2. 5.2 A snag
    3. 5.3 Changing counterparts
  6. 6 Summing up
    1. 6.1 Further objections.
    2. 6.2 Summary.

##  1 Introduction

###  1.1 Recap

In the previous paper in this series, Gomes (20213), we mounted a defense of
sophistication—an attitude towaards symmetry-related models in modern
classical field theories—in general terms. When applied to spacetime
diffeomorphisms and to gauge symmetries, this defence resulted in anti-
haecceitism about spacetime and anti-quidditism about gauge fields. Thus we
recovered the familiar or standard view of physical equivalence between
isomorphic models—labeled _Leibniz equivalence_ in the philosophical
literature (cf. Earman Norton (1987)).

That defense required us to first clarify the obstacles to the doctrine of
sophistication, even in the case of spacetime. Thus I demanded two desiderata
in order for sophistication to apply in the envisaged way.

The first Desideratum, which encompasses different criteria for sophistication
that are encountered in the literature (see e.g. (Earman, 1989, p. 45-47),
(Belot, 2013, p. 9), (Jacobs, 20212, Ch. 4, 8)), was a version of Earman’s SP
principles: not only should symmetries be construed as isomorphisms of some
structure, but these isomorphisms should arise from the induced action of the
automorphisms of an underlying space. Thus, for instance, the isomorphisms of
Lorentzian metrics over some fixed smooth manifold arise from the induced
actions of the automorphisms of this smooth manifold: namely, the isometries
are given by the pull-back of the diffeomorphisms of the manifold on the
fields. The same holds for Yang-Mills theory, whose model isomorphisms are
induced by the action of the vertical automorphisms of the principal bundle on
the Ehresmann connection forms. Moreover, for these two theories, though
perhaps not in general, the satisfaction of Desideratum (i) implied that we
could construe the symmetries of the theories as notational redundancies, in
the sense that there is a 1-1 relation between the isomorphisms and changes of
coordinates (as elaborated in (Gomes, 20214, Sec. 5)).

The second Desideratum required us to first understand the invariant structure
‘internally’. That Desideratum was satisfied if the theory admitted an
axiomatic formulation of the mathematical structure endowed with a perspicuous
(meta)physical interpretation; I called such a formulation _synthetic_. This
synthetic formulation was secured by requiring the axioms to be understood as
plausible and basic physical posits of the given theory. In this way, we
clarified the isomorphism-invariant mathematical structure while connecting
that structure to the basic ontology of the theory. However, I conceded that
this second Desideratum is a very high-bar: most physicists and
philosophers—including me—accept an intuitive understanding of the invariant
structure independently of the fulfilment of this Desideratum.

Nonetheless, certain questions—or better: worries—about this resolution
remained, and here I will answer to some of them.222I label them worries and
not questions or shortcomings. A worry connotes a temporary challenge: it is a
short-lived shortcoming; and it does not need to be formulated in terms of a
question.

###  1.2 The remaining worries

There are three main ‘worries’ about sophistication as an attitude towards
symmetry-related models that the previous paper Gomes (20213) left unanswered.
The present paper aims to resolve two of these, that are ramifications of what
I earlier called Worry (2):

  1. (i)

Though sophistication may give us a handle on what the underlying ontology of
the theory is as a whole, it is unhelpful when we seek to describe individual
physical possibilities, and individual facts of the manifold, in an invariant
manner. In other words, sophistication gives us a criterion of physical
equivalence between entire models; it provides sufficient conditions of
identification (or contrapositively: necessary conditions of non-
identification). But these conditions fall short in several ways. For given
two models, the sufficient condition is not, _prima facie_ , tractable; one
cannot try out every diffeomorphism—or, more generally, automorphism of the
models’ common base set—to see if one will bring the two models into
coincidence. In the mathematical jargon to be developed in Section 3.2 (see
also (Gomes, 20214, Sec. 2)), sophistication illuminates the equivalence
relation, $\sim$, but does not help to characterise the individual equivalence
classes, $[\bullet]$, in a concise way. Of course, the issue would be solved
by a formalism that describes individual physical possibilities without
redundancy, but such a formalism is not required by, and indeed would obviate
the need for, sophistication. This shortcoming motivates theoretical
physicists—especially those working in quantum gravity—to seek more
perspicuous, relational, characterizations of the symmetry-invariant ontology;
even while endorsing sophistication at a more abstract level (as described in
Belot Earman (1999, 2001)).

  2. (ii)

Sophistication’s commitment to a strictly qualitative ontology deflates the
physical significance of the set-theoretical elements involved in the usual or
standard representation of the models—elements such as the points representing
spacetime events. For instance, once we accept sophistication’s sister
doctrine of anti-haecceitism and its denial of primitive identity between
points that belong to non-isomorphic spacetimes, we have no notion of
correspondence between spacetime points. But such correspondence relations are
crucial to express counterfactuals and modal intuitions, so how are
counterfactuals expressed within sophistication’s commitment to a strictly
qualitative ontology?

Worry (3), which will not be addressed here, was argued forcefully by Belot
(2018). It is that in certain sectors of general relativity, and contrary to
sophistication, there is a general consensus that isomorphisms relate
_different_ physical possibilities. Namely, in the context of spatially
asymptotically flat spacetimes, some diffeomorphisms—those that preserve the
asymptotic conditions but don’t asymptote to the identity map—are interpreted
as relating different physical possibilities. The same occurs in Yang-Mills
theory (Giulini, 1995). But the reasons given for sophistication (see Gomes
(20213)) were general; they applied at the level of the whole theory. So
sophistication, as a doctrine for the whole theory, i.e. all its sectors,
seems refuted.

I will single out and elaborate Worry (2-i) in Section 2. In order to respond
to this Worry, I will introduce the idea of _representational conventions_ ,
in Section 3. Such conventions are a reconstrual of choices of gauge-fixings
as choices of physical relations with which we can completely describe each
physical possibility without redundancy.

In Section 4 I discuss how choices of representational conventions are
possible, and how they are made. Unlike the standard reasons given for
particular coordinate systems—such as being adapted to a given
configuration—these choices must be applicable to large sets of
configurations.

The resolution of Worry (2-i) via representational conventions offers not only
tractable conditions of identity for the entire universe: they also provide
local correspondence relations between non-isomorphic models. This
correspondence is associated to countepart relations, as described in Gomes
Butterfield (2023).333 Counterpart theory is the philosophical doctrine, due
to David Lewis (D. Lewis (1968), (D. Lewis, 1973, p. 38-43) and (DK. Lewis,
1986, Ch. 4)) that any two objects—in particular, spacetime points—in two
different possibilities (in philosophical jargon: possible worlds) are never
strictly identical. They are distinct, though of course similar to each other
in various, perhaps many, respects. And what makes true a proposition that the
object $a$ could have had property $F$ (though in fact it lacks $F$) is—not
that in another possible world, $a$ itself is $F$—but that in another possible
world, an object appropriately similar to $a$ is $F$. That object is called
the counterpart, at this other possible world, of $a$. Representational
conventions allow us to ‘get-inside’ the models, and provide explict
counterpart relations between _points or regions_ belonging to non-isomorphic
models.

In Section 5 I will develop the theory of counterparthood based on
representational conventions and show that resolving Worry (2-i) resolves
Worry (2-ii). And in Section 6 I conclude.

An answer to the third Worry requires the application of representational
conventions to subsystems, and so deserves a separate treatment (which I give
in (Gomes, 2022); see also Gomes (20211, 20212)).

##  2 The challenge of individuating structural content

In Section 2.1 I will further discuss Worry 2, which is the focus of this
paper, and whose resolution lies beyond what sophistication can illuminate:
namely, the identification of structure-tokens. In Section 2.2 I will diagnose
the roots of this obscurity.

###  2.1 The identification of structure-tokens

Even once we have characterized to our full satisfaction _the type_ of
invariant structure, we may not have a perspicuous way to identify the
_structure tokens_. That is, even admitting sophistication about isomorphisms,
given two models of the theory, how do we in practice adjudicate whether they
represent the same physical possibility? In other words, two models may
instatiate the same type of structure—say semi-Riemannian geometry—but
different structure tokens, represented by two non-isomorphic models. What are
tractable conditions of structural identity for tokens?

The sophisticated interpretation—augmented as it was in Gomes (20213) by
further formal and interpretive desiderata—provides a condition of physical
identity between models, but it is not a tractable condition. To show that two
given models are non-isomorphic, sophistication would have us scan the entire
infinite-dimensional space of gauge transformations or diffeomorphisms: a
clearly intractable procedure. To be more explicit about this problem: in
coordinates, given two descriptions of metrics
$g^{1}_{\mu\nu},g^{2}_{\mu\nu}$, we would either have to stumble upon a
coordinate change that relates them, or else find some coordinate invariant
function that can be defined on all spacetimes and that distinguishes the two
metrics. And more importantly, failing on both accounts, we could not
decisively conclude that they correspond to different geometries. There is no
algorithm we can follow here to tell whether the two models in fact represent
the same structure token.

An analogy to Leibniz’s Principle of the Identity of Indiscernibles (PII) may
be helpful here. In order to apply the PII, we cannot check whether Fred and
George have _all_ the same properties—even all the same qualitative
properties! No, instead we apply a tractable condition of identity: whether
George and Fred share the same bodies, for instance, or the same exact
fingerprints.444 In our case, the analogy doesn’t go very far, since the
properties used for our (tractable) conditions of identity below also provide
a basis of all the qualitative properties.

It is true that sophistication suffices _to recognise_ properties555As in
Gomes (20213), I mean ‘properties’ in the more general sense: not solely as
unary, or 1-place relations, but general n-place relations. This was
traditionally dubbed an ‘attribute’, but I think this term is ever so slightly
more abstract than ‘property’, and has fallen out of use; and so I will stick
with ‘property’.  that are invariant, or structural. But it does not suffice
to individuate tokens of the invariant structure. For illustration, take a
familiar list of isomorphism-invariant properties. For instance, you could
single out models that have “a two-dimensional surface, of area $X$, bounded
by two geodesics, of length, $L$”; or “the manifold is geodesically convex”;
or “for all points $x$, there exists a unique point $y$ whose geodesic
distance from $x$ is greater than all other points”. All of these properties
are invariant under isometries. The last property is instantiated say, in the
two-sphere, where it describes anti-podal points, and it is not instantiated
on a plane. It is only this last property that, given the number of dimensions
and the signature of the metric, fully characterises the geometry, and that is
because the round sphere is a highly unusual, highly symmetric geometry. In
the first two cases we are far from uniquely characterising a physical
possibility, or even points or regions of the world in which these properties
hold. To do so in general we would need an infinite, intractably holistic and
prolix list of properties as the ones above.666My use of natural language here
is not the source of the obstruction: I could have equally well phrased these
questions in a more mathematical language: the point, as we will see, is that
there is no God-given basis of natural invariant relations with which to
describe isomorphism-equivalence classes.

An attempt to single out a structure token in this piecewise way brings up the
related, but distinct, Worry (2-ii), also left obscure by sophistication: how
are we to individuate points, regions, or properties within a given physical
possibility? That is, how can we give points or regions tractable conditions
of identity? Again, we can take for granted the standard anti-haecceitist
argument that one can only individuate points as places in a structure. But
that assumption takes us nowhere closer to an exact description of how this
individuation is to proceed; all we have is still a vague allusion to an
intractably prolix conjunction of properties.777 And such an intractably
prolix characterization poses a conceptual obstacle also for the
metaphysician. For example, Dasgupta (2011, p. 151)’s second challenge (see
(Gomes, 20213, Sec. 2.2)) requires a description of all _individualistic
facts_ through purely qualitative terms. In my interpretation here, though we
may have clearly articulated “what the underlying qualitative facts _are like_
” (my emphasis), thus satisfying Dasgupta’s first challenge, we have not yet
shown “that they are sufficient to explain individualistic facts about the
manifold.”

Even accepting that the ontology is firmly about e.g. distances between
spacetime points, and so being clear about the type of structure, we have no
concise way of describing how these distances are distributed, since we have
no concise way of designating the points that stand in these relations.
Relatedly, the literature on sophisitication, and on symmetry and equivalence
more generally, has frequent but vague allusions to how e.g. spacetime points
can only be individuated as places in a structure, by their relations to other
points. But what are these relations, and how do they successfully individuate
points? This is Worry (2-ii), and this paper can be seen as a successful
attempt to make precise what such relations have to be, and when they succeed.
And in making it precise, it becomes apparent that there are many different
sets of relations that can fully ‘individuate points as places in a
structure’—a plurality that is essentially absent from the vague allusions to
individuation in the literature on symmetry and equivalence.

Note also that a resolution of Worry (2-ii) is necessary in order to discuss
local spacetime counterfactuals: modal intuitions—such as that Earth _could
have been_ closer to the Sun—invoke _some_ relation between spacetime regions
across physical possibilities. Since sophistication strips spacetime points of
primitive identity across physical possibilities, we need an alternative way
to establish local inter-world comparisons.

Though these arguments and counterarguments have been given within the context
of general relativity, there are closely analogous ones for Yang-Mills theory:
as shown in (Gomes, 20213, Sec. 3.2b), the type of structure can be
axiomatized by taking parallel transport around closed spacetime curves as
primitive Barrett (1991). But, since the space of closed curves on spacetime
is non-separable—has no enumerable basis—the individual structure tokens have
likewise remained opaque, since inconcise.

In sum, I have here fine-grained Worry (2) to focus on two shortcomings of a
sophisticated attitude towards symmetry-related models, briefly described in
Section 1.2. Worry (2-i) is that we have not specified how individual physical
possibilities are to be tractably individuated, or uniquely characterised.
Worry (2-ii) is that we have not provided invariant descriptions of objects
within each physical possibility, and correspondence—or rather, ‘counterpart
relations’; cf. Gomes Butterfield (2023)—between objects across
possibilities.888And since the individuation of physical possibilities should
be tractably concise, we cannot straightforwardly apply mereological arguments
to say that a resolution of Worry (2-ii) provides a resolution of (2-i).
Nonetheless, my resolution of Worry (2-i) will go via a resolution of Worry
(2-ii).

###  2.2 Can the sceptic ever be satisfied?

The answer to this subsection’s title is: it depends on the sceptic. My
envisaged sceptic is reasonable and defeasible, and here I’ll be precise about
what it would take to satisfy him. First I will adduce evidence that the
sceptic’s worry is not completely ill-founded. This evidence will help us
diagnose what is at the root of our previous failure to resolve his worry.

The sceptic I am envisaging was originally disappointed by the somewhat
implicit characterization of the symmetry-invariant structure that
sophisticated substantivalism furnishes—as ‘that which remains invariant under
isomorphism’; he was disappointed by what has been labeled ‘external
sophistication’ (see e.g. Dewar (2017); Martens Read (2020); Jacobs (20211)).
In (Gomes, 20213, Sec. 4), I reported axiomatic characterisation of the
invariant structures of general relativity and Yang-Mills theory that complied
with the strictures of _internal_ sophistication. I take that to have resolved
the original sceptical worry (Worry 1).

But now the skeptic has resurfaced more ambitious. He has accepted an
explication of the _type_ of invariant structure at hand—which amounts to a
characterisation of the physical equivalence relation between models,
‘$\sim$’—but has moved the goalposts by asking for a concise criteria of
individuation for each structure _token_ , $[\bullet]$: this is Worry (2-i),
as described in Section 2.1.

Far from being a debate of merely metaphysical interest, the skeptic contends,
the lack of such a characterisation has implications for the practitioner of
quantum gravity, as most thoroughly described in Belot Earman (1999, 2001).
They write:

> Far from dismissing the hole argument as a simple-minded mistake which is
> irrelevant to understanding general relativity, many physicists see it as
> providing crucial insight into the physical content of general relativity.
> (Belot Earman, 1999, p. 169)

The hole argument arises from the existence of isomorphic models, and the
allusion here is to the physicist’s yearning for a description of the
invariant content of general relativity, usually labeled ‘relational’, since
it involves relations between the values of different fields or objects and at
different points.

And here is Isham, on the difficulty of explicitly characterizing invariant
structure (“what constitutes an observable”):

> The diffeomorphism group moves points around. Invariance under such an
> active group of transformations robs the individual points of $M$ of any
> fundamental ontological significance . . . [the argument] is closely related
> to the question of what constitutes an observable in general relativity—a
> surprisingly contentious issue that has generated much debate over the
> years. (Isham, 1992, p. 170)

To buttress his point that even internal sophistication leaves some aspects of
the invariant structure obscure, the skeptic adduces the difficulties in
finding an _explicit_ basis of ‘relational’, isomorphism-invariant quantities
in general relativity and quantum gravity. Since these difficulties are major
hurdles to all approaches to quantum gravity, the sceptic reasons, the
obscurity has consequences even in the practice of theoretical physics.999See
e.g. Thiemann (2003); Rovelli (2007); Donnelly Giddings (2016) for general
arguments surrounding this difficult issue of ‘gravitational observables’, and
Harlow Wu (2021) for an explicit, and rather complicated basis in the
simplified context of a two-dimensional gravitational theory called Jackiw-
Teitelboim gravity.

But what was the problem with my previous attempt at answering the sceptic,
through a list of invariant properties of the model? I said it was
‘intractably prolix’. And here is why: given a list of properties, e.g. ‘there
are such and such surfaces of area A, and with such and such spacetime
signature, that are bounded by geodesics that intersect at such and such
points…’ I will generally have only incompletely described any given region of
spacetime which these objects inhabit. In other words, in general a
description such as the one just given does not fully determine all the
geometric content of a given region. Only in highly homogeneous cases will
such a finite descrition be complete. And it is extremely unclear what kind of
mereology we can apply to these descriptions: can such objects inhabit the
same spacetime region? And do such relations suffice to, as we say
‘individuate spacetime points by their pattern of relations to other points’,
as required by Worry (2-ii)?

In contrast, a metric tensor $g_{ab}$ is a tractably concise description of
the invariant structure—i.e. the geometry—of a region where it is defined. The
metric is a local tensor: given metric tensors of neighboring regions we
straightforwardly obtain a criterion for their composition: smoothness.
Presumably, a sufficiently inhomogeneous metric tensor also provides
sufficient relations to identify each spacetime point by its pattern of metric
relations to other points. This is unlike the list of invariant quantities
given above, whose conditions of compossibility are not clear, and which may
not suffice to individuate points. The problem, as described in Section 2.1,
is that the metric tensor has no simple conditions of structural identity:
given two metric tensors, it is not obvious whether they represent the same
structure.

So, to summarise, here is a set of abstract conditions which would satisfy the
sceptic, which I will call a

######  Definition 1 (Representational convention)

A representational convention for a given structure is a tractably concise,
complete description of the invariant structure that has simple conditions of
identity and compossibility.

The paradigmatically successful example of a characterization that would meet
the sceptic’s Worry (2-i) is that of vector spaces, where specifying the field
of scalars and a single number—the dimension of the space—fully characterizes
each structure token. But it is not the finiteness of the characterization
that makes it successful: a scalar field in a fixed background, whose state is
located in an infinite-dimensional space of possibilities, such as a
temperature or density, could be considered by the skeptic I am here
envisioning to be an equally transparent characterization of the ontology. In
these cases, the ontology still admits a direct description that is concise,
complete, has simple conditions of identity—numerical identity—and
compossibility: two distributions are compossible if they occupy distant
regions of space. The descriptions are completely _local_ , in the sense that
two temperature distributions are physically identical iff they numerically
match at each point of spacetime. Likewise, for spacetime: if we find
tractable criteria of identity for regions or points (belonging to isomorphic
or non-isomorphic models), and if these criteria employ only local fields—that
are mereological simple in the same way as temperature—we will have answered
the sceptic.

And so, I will judge the sceptic to be satisfied once we have, similarly to
the case of temperature, provided a representational convention for our models
that satisfies Definition 1.

Now, it is not quite right to pin our discontent with the original attempt to
describe invariant structure through a list of invariants on a kind of
‘holism’ or non-locality. For, as we will see, generically, representational
conventions will include some level of non-locality in their representation of
structure.

The idea that physical content can be holistic has a noble tradition in
quantum gravity: Torre (1993) has shown that, for the spatially closed
universe, there are no local symmetry-invariant quantities (i.e. quantities
which can be written as integrals of the canonical variables and a finite
number of their derivatives over the Cauchy surface). Diffeomorphism-invariant
functions are usually non-local: this is a notorious problem for quantum
gravity (see Thiemann (2003); Rovelli (2007); Donnelly Giddings (2016)). Thus
Isham (1992, p. 170) continues the quotation I gave above:

> In the present context, the natural objects [that constitute ‘observables’
> in general relativity] are ${\mathrm{Diff}}(M)$-invariant spacetime
> integrals […] Thus the ‘observables’ of quantum gravity are intrinsically
> non-local.

Of course, the spacetime covariant form of the equations of motion for either
general relativity of Yang-Mills theories respect relativistic causality. In
mathematical jargon, we have _quasi-hyperbolic_ equations of motion, for
either the Lorentzian metric or the gauge potential. Given initial data for
this kind of equation, its solutions are not uniquely determined: for any
given solution of the equations for that initial data, any isomorphism that
leaves the initial data intact but is otherwise arbitrary on the rest of the
solution, gives another solution. This lack of uniqueness is just a
reformulation of gauge freedom in terms of differential equations. A local
symmetry implies, via Noether’s second theorem (cf. Brading Brown (2000) for a
conceptual exposition), that the equations of motion of the theory are not all
independent, and thus only uniquely determine the evolution of a subset of the
original degrees of freedom. There is thus a certain freedom in choosing which
‘components of the fields’ will be uniquely propagated—each choice corresponds
to _a choice of gauge_.101010And it is clear that relativistic causality holds
for the isomorphism-invariant facts, since (quasi-) hyperbolicity of the
equations of motion ensures causality is respected for one choice of metric or
gauge potential representing the isomorphism-equivalence class, and the
isomorphism-invariant facts supervene only on the metric or gauge potential.

In general, the function that takes the original local degrees of freedom to
this uniquely propagated subset of local degrees of freedom is non-local: the
value of the subset at point $x$ depends on the values of the original degrees
of freedom at other points. But once we have agreed on what kind of subset of
degrees of freedom this should be, we have unique, tractably concise
characterisations of the invariant structure, that will admit simple
conditions of compossibility and identity. This will be the topic of the next
Section.

Representational conventions generally require this particular (rather benign)
type of holism, or non-separability, that applies to theories with elliptic
initial value constraints, such as general relativity and Yang-Mills theories;
but does not apply to theories like Klein-Gordon in a fixed Minkowski
background, nor to any ‘gauge theory’ whose gauge transformations involve no
derivatives. (See also (Gomes Butterfield, 2022, Sec. 1.1, point (3)) for more
discussion about this sort of (non-signalling) classical non-locality.)111111
As I will argue at further length in the accompanying paper Gomes (20215), in
practice, ellipticity means that boundary value problems require only the
boundary configuration of the field, i.e. they do not also require the field’s
rate of change at the boundary. The solution of these equations exists _on
each simultaneity surface_ ; and so the solution does not correspond to the
propagation of a field, as would a solution to a hyperbolic equation. The
source of the non-locality is that the allowed values of the fields at a point
depend on the values of the field away from the point. This (classical and
non-signalling) non-locality can be quantified to a certain extent, cf. Gomes
(20211); Gomes Butterfield (2022).

In sum: though holism may be at the root of our discontent with describing
invariant structure via a list of invariants, some level of non-locality, or
non-separability is necessary in order to satisfy Definition 1. In the next
Section, we will see how representational conventions correspond to choices of
subsets of our original variables—e.g. composites of the original fields—which
uniquely describe structure in a manner that simultaneously resolves Worries
(2-i) and (2-ii). A representational convention will be provided by imposing
further conditions on the fields, e.g. an auxiliary set of differential
equations besides their equations of motion. The local construction will admit
of an alternative, non-local, invariant interpretation, that takes non-local
field composites—dubbed _dressed fields_ —as relational bases for the
parametrization of physical states, which satisfies Definition 1.

##  3 Representational conventions

The quotes of Section 2.2 illustrate a common theme within approaches to
quantizing the gravitational field. There we see much discussion about finding
‘observables’: quantities that are invariant under the symmetries of the
theory, that we could use to describe physical content without redundancy.
These quantities are usually called ‘relational’ and ‘non-local’, because they
often involve comparisons of values of fields at different spacetime points.

And unlike the philosophical discussion we have focussed on so far, physicists
mostly seek symmetry-invariant, ‘relational’ quantities within the context of
quantization. The technical reasons for this can be very easily illustrated in
the path integral, or sum-over-histories approach. There, the tools of
perturbative field theory fail if we do not treat the symmetry-related
histories as being, in some sense, physically identical.121212In slightly more
detail: since the Hessian of the action vanishes, the propagator around any
classical solution diverges, and therefore one obtains divergences at every
order of the perturbation series. Finally, these divergences cannot be neatly
separated into energy scales: they are immune to the machinery of effective
field theories. One solution to this problem is to reconstrue the path
integral in terms of symmetry-invariant, relational variables: this is the
‘purist’s’ approach; common among theorists who more often come to the field
from the side of general relativity.

In contrast, particle physicists working on gauge theory will not spend much
time discussing relational invariant observables. There the usual approach to
redundancy is to just “fix the gauge”. To fix the gauge is to implement
auxiliary conditions on the representation of the physical state that are
satisfied by a single element in each class of symmetry-related models. Gauge
fixing is very useful in practice and widely employed in the actual
calculations of gauge theory and (classical and perturbative) gravity. But it
is also often (erronously) assumed to do violence to the original symmetry of
the theory.

The contrast is clear: in the less applied, more conceptually-driven domain of
(non-perturbative) quantum gravity, gauge-fixing is often overlooked in lieu
of a choice of ‘relational observables’ and, more recently, in lieu of more
specific ‘dressed fields’, which is a particular type of relational
observable.131313Dressed fields are a type of relational observable _field_
that are symmetry-invariant composites of the original degrees of freedom.
This is an auspicious label, in view of the sophisticationist’s recommendation
that we cannot individuate spacetime points independently of their ‘dressing’.
That is, sophisticationists heed Kaplan (1979, p. 96)’s warning about how it
may be psychologically appealing to adopt a metaphysics in which points have
primitive identity irrespective of their properties and relations, i.e.
irrespective of their “clothing”, or “dressing”:  “[T]he use of models as
representatives of possible worlds has become so natural for logicians that
they sometimes take seriously what are really only artifacts of the model. In
particular, they are led almost unconsciously to adopt a bare particular
metaphysics. Why? Because the model so nicely separates the bare particular
from its clothing. The elements of the universe of discourse of a model have
an existence which is quite independent of whatever properties the model
happens to tack onto them.

But I maintain that this assumption, this contrast sketched above, between
gauge-fixing and relational observables, is misleading. For gauge fixings can
be understood in terms of dressed fields, and therefore do no violence to the
original symmetry of the theory. Here I will give a conceptually unified
account of dressed quantities, material reference frames, gauge-fixings,
relational observables, and a more flexible tool called _relational
connection-forms_ (which was technically developed in Gomes Riello (2017,
2018); Gomes . (2019); Gomes Riello (2021); and see Gomes (2019, 20211) and
(Gomes Butterfield, 2023, Appendix A) for philosophical introductions). And I
will connect this constellation of ideas to Worries (2-i, 2-ii); to
counterpart theory; and to Einstein’s conception of ‘physical reference
frames’.

Here is a quick summary of the relationship between these mathematical
objects, as I will develop them in the next Sections: the space of states of a
given theory is seen as a principal bundle (where each fibre comprises an
isomorphism class of models), and a choice of a _representational convention_
is like a choice of section of this bundle: a surface that intersects each
symmetry orbit exactly once; it _fixes the gauge_. The relational connection-
forms are a distribution of infinitesimal gauge-fixing surfaces, with
appropriate covariance properties;141414That is, those connections define an
isomorphism-covariant decomposition of all the linear perturbations of each
model (the tangent space at the model), as a direct sum between the
perturbations along the fiber and along a transversal complement to the fiber.
and finally, dressings are a projection operator from the entire space of
models to the gauge-fixing surface. And to illustrate—and presage!—the
relationship between dressing and gauge-fixing, that I will describe in
Sections 3.2 and 3.3 (see also the quote by Harlow and Wu below): if we
‘dress’ electrons with their ‘Coulombic tails’, we obtain a gauge-invariant
composite that solves the Gauss constraint; if we dress the gauge potential
with the same kind of dressing, we obtain a (Lie-algebra-valued) one-form that
is (gauge-covariantly) divergenceless. Thus this choice of dressing is, in a
very precise sense, related to the Coulomb gauge-fixing; see Gomes Butterfield
(2022) for an in-depth conceptual analysis of this case. Thus representational
conventions provide a technical and philosophical reflection on gauge-fixing.

This resolution of Worry (2) is in line with the modern construal of symmetry-
invariant treatments of field theories (cf. Berghofer . (2022)). As described
recently in a paper dealing with the symmetry invariant content of modern
gravitational theories (Harlow Wu, 2021, p. 3):

> It has long been understood that diffeomorphism symmetry must be a gauge
> symmetry, and that physical observables must therefore be invariant under
> almost all diffeomorphisms. […] There is an analogous situation in
> electromagnetism: fields which carry electric charge are unphysical unless
> they are “dressed” with Wilson lines attaching them either to other fields
> with the opposite charge or to the boundary of spacetime. One way to think
> about dressed observables in electromagnetism is that they create both a
> charged particle and its associated Coulomb field, which ensures that the
> resulting configuration obeys the Gauss constraint. Similarly in gravity any
> local observable by itself will not be diffeomorphism-invariant, so we must
> dress it […creating] for it a gravitational field that obeys the constraint
> equations of gravity. In practice such observables are usually constructed
> by a “relational” approach: rather than saying we study an observable at
> some fixed coordinate location, we instead define its location relative to
> some other features of the state.

But I am getting ahead of myself.

Here is how we will proceed. In Section 3.1 I will provide a conceptual
summary of representational conventions, that encapsulates their uses and
advantages. In Section 3.2, we introduce the mathematics of representational
conventions. A representational convention is a choice of physical relations
with which we choose to describe the world. Following up on this idea, in
Section 3.3 I will make the relationship between representational conventions,
gauge-fixings, and dressings explicit.

###  3.1 Representational conventions conceptually summarised

To recap: in both the gravitational and the gauge cases, there are many ways
in which we can represent qualitatively identical patterns of e.g. parallel
transport; each of these ways correspond to one of many _isomorphic models_ of
the theory. But we can pick a particular representation by resorting to a
_representational convention_.

Representational conventions introduce a notion of ‘relationism’ into the
formalism. This is not the notion that is usually attributed to Mach,
Poincaré, Barbour, etc. All I mean by it is that it requires relative values
of fields. But in practice, representational conventions _will_ invoke notions
that are relational in the traditional sense: for instance, they could invoke
inter-particle distances for Newtonian systems. One main difference to my
meaning is that I will invoke multiple choices of sets of relations: one set
can only be _pragmatically_ preferred to others. So representational
conventions can be seen as a choice of physical features with respect to which
we parametrize the physical states of a target system completely; each choice
of representational convention can be identified as a choice of a basis of
relations that can concisely describe physical content without redundancy, and
which thus satisfies simple conditions of compossibility and identity—i.e.
satisfying Definition 1.151515By ‘parametrize’, I mean ‘to convey in exact
words the particular physical state by stating what are the values of
meaningful, but relational, quantities for the given physical state’.

For instance, in non-relativistic particle mechanics, the relational basis
could be the set of all constituent particles’ distance to a center of mass
and some preferred frame therein (e.g. as a function of an anisotropic
distribution of mass). In field theory, the relational basis is given by some
possibly non-local function of all the fields. These relations are usually
‘non-local’, because they often involve comparisons of values of fields at
different spacetime points.161616For example: we can use generalisations of
Green’s functions to compare fields at different points. This is the basis of
our examples (cf. also (Gomes Butterfield, 2023, Appendix)). All these
examples will be fleshed out in Section 4.3.

Each _representational convention_ will completely represent each physical
state of the world by a unique model; thus the conceptual similarity to ideas
of ‘fixing the gauge’, or fixing the representation.

To foreshadow the technical aspects of a representational convention: it
requires us to see the space of models of a theory—which I will call
${{\Phi}}$—as a (principal) bundle, where fibers are the orbits of the group
of isomorphisms, $\mathcal{G}$. Then finding a section $\mathcal{F}$ for this
action—a surface that intersects each orbit exactly once—we have a way of
fixing the representation: this is essentially a fixing of the gauge. But we
can also construct from such a section the relational observables that are
invariant under the action of the group of isomorphisms: this is the dressed
quantity corresponding to the section, i.e. an invariant quantity that
satisfies the same conditions than the section.

As was to be expected from gauge-fixings, using a fixed choice of
representational convention we have a tractable criterion for when two generic
models instantiate the same qualitative, or gauge-invariant, pattern of
parallel transport. Namely, when the values of the dressed fields
(equivalently, of all the composites in a basis, or equivalently, the gauge-
fixed values) numerically match. We reduce matters of structural identity to
numerical coincidence. As we will see, since qualitative equality can then be
applied piecewise to spacetime, representational conventions allow us “to get
inside the models” and characterize individual elements of each world
qualitatively.

Once we are thus able to characterise regions of space via their invariant
attributes, we can give a new mathematical gloss on counterpart theory. Each
representational convention gives us a particular inter-world counterpart
relation between qualitatively different patterns, i.e. between non-isomorphic
models. Roughly, the idea is that the qualitative relational basis singled out
by one convention provides similarity relations across physical possibilities.
So questions such as: ‘what is the counterpart of event $p$ of our world in a
qualitatively different pattern?’ makes sense under a representational
convention, since there will be a unique point $q$ at a different world that
shares the subset of properties that are “selected as salient” by the
convention (see Gomes Butterfield (2023) for a complete treatment of
counterparthood and its relations to Lewis’s doctrines, which we will touch on
in Section 5).

To pick a convention we employ intra-theoretic resources, by which the
convention provides perspicuous characterizations of structure. Isomorphisms
of a theory can then be understood as changes in the relations and properties
we select to represent a given physical structure; or, more prosaically: as
changes of gauge. That is, changes between representational conventions can be
interpreted both actively and passively. In one way of looking at them, they
represent different choices of composite, or relational fields—an often
pragmatic choice, suited to the problem at hand. This is the ‘passive’ view.
On the other hand, mapping from one representation to the other has an active
intepretation, since points with the same coordinates under different
representations will have different qualitative (i.e. physical) properties.
Indeed, the best way of understanding changes between representational
conventions is almost identical to how Einstein at one point conceived of
coordinate transformations (see Norton (1989)).

In sum, representational conventions can be construed as choices of relational
bases that:

  1. (I)

provide concise, albeit non-unique, complete characterizations of structure-
tokens, i.e. of the structure that is: common to each class of isomorphic
models, and different for different classes of isomorphic models. They give a
tractable criterion of individuation of structure-tokens.

  2. (II)

allow us to describe qualitative counterpart relations for points, regions, or
properties, across isomorphic and non-isomorphic models, while entirely
denying any primitive—i.e. convention-independent—identification of points or
properties across models. These counterpart relations make precise the
frequent but vague allusions in the literature on sophistication to ‘points
being only individuated by their pattern of relations to other points’.

And so, to complement my first two Desiderata for sophistication of Gomes
(20213), I here introduce Desideratum (iii): that the theory’s symmetries
admit representational conventions. Fulfillment of this Desideratum will
resolve the first two Worries of Section 1.2, left over from Gomes (20213).
But the fact that Desideratum (iii) is a non-trivial requirement will only be
explained in Section 6.1.

##### A brief history of representational conventions

Representational conventions as we understand them here were first introduced
for subsystems in gauge theory and general relativity, in Gomes Riello (2017)
(where they were described as ‘abstract reference frames’, or ‘relational
connection-forms’). This initial general treatment was illustrated with
explicit examples in Gomes Riello (2018); Gomes . (2019); Gomes Riello (2021),
where these frames were also compared to dressings in gauge theory and
gravity; in Gomes (2019, 20211) I gave a conceptual treatment of
representational conventions, or ‘abstract reference frames’.

The reason these representational conventions were first introduced in the
context of subsystems was that there are subtleties in construing the
isomorphisms of subsystem states as dynamical symmetries, subtleties which can
be treated by appealing to properties of representational conventions. In
particular, there are subtleties about the symmetry-invariance of a bounded
subsystem’s own dynamical structures, such as its intrinsic Hamiltonian,
symplectic structure, and variational principles in general. The conceptual
reason behind this obstacle is that gauge theories manifest a type of non-
locality (see Section 2.2 and (Gomes, 20215, Sec. 4)). Thus the global,
physical phase space (or the corresponding global physical Hilbert space, in
the quantum theory) is not factorizable into the physical phase spaces of the
composing regions.171717For a reference that explores non-factorizability in
the context of the holonomy formalism, see Buividovich Polikarpov (2008). For
the relation between different symmetries and locality in gauge theory, see
Berghofer . (2022), and for the non-locality of gravitational invariants see
e.g. Torre (1993); Donnelly Giddings (2016); for more recent use of this non-
factorizability in the black hole information paradox, see Jacobson Nguyen
(2019).  The modern guise of these issues was first highlighted in Donnelly
Freidel (2016) (which also introduced ‘edge-modes’), which caused a flurry of
papers on subsystems in gauge theory (cf. e.g. Ramirez Teh (2019); Geiller
(2017); Donnelly Freidel (2016); Gomes Riello (2021); Riello (2021); Gomes
(2019); Geiller Jai-akson (2020); Carrozza Hoehn (2021) and references
therein).

###  3.2 Representational conventions: general definitions

At the abstract level explored in this and the next Sections, there are no
differences between diffeomorphisms and the symmetries of gauge theories, and
so we treat them uniformly as the group ${\mathcal{G}}$.

To recapitulate from Gomes (20213), symmetries will be here represented as
groups (which could be infinite-dimensional), denoted ${\mathcal{G}}$, such
that, given the space of models of a theory, ${{{\Phi}}}$, there is an action
of ${\mathcal{G}}$ on ${{{\Phi}}}$, a map
${{{\Phi}}}:{\mathcal{G}}\times{{{\Phi}}}\rightarrow{{{\Phi}}}$, that
preserves the action functional, and any further structure on ${{\Phi}}$ (e.g.
symplectic, smooth, etc) and so is an $S$-symmetry. More formally: there is a
structure-preserving map, $\mu$, on ${{\Phi}}$ that can be characterized
element-wise, for $g\in{\mathcal{G}}$ and $\varphi\in{{\Phi}}$, as follows:

$\displaystyle\mu:{\mathcal{G}}\times{{\Phi}}$ $\displaystyle\rightarrow$
$\displaystyle{{\Phi}}$ $\displaystyle(g,\varphi)$ $\displaystyle\mapsto$
$\displaystyle\mu(g,\varphi)=:\varphi^{g}.$ (3.1)

The symmetry group partitions the state space into equivalence classes in
accordance with an equivalence relation, $\sim$, where
$\varphi\sim\varphi^{\prime}$ iff for some $g$,
$\varphi^{\prime}=\varphi^{g}$. We denote the equivalence classes under this
relation by square brackets $[\varphi]$ and the orbit of $\varphi$ under
$\mathcal{G}$ by $\mathcal{O}_{\varphi}:=\\{\varphi^{g},g\in\mathcal{G}\\}$.
Though there is a one-to-one correspondence between $[\varphi]$ and
$\mathcal{O}_{\varphi}$, the latter is rather seen as an embedded manifold of
${{\Phi}}$, whereas the former exists abstractly, outside of ${{\Phi}}$ (see
Figure 1).

Figure 1:  The space of states, ‘foliated’ by the action of some group
$\mathcal{G}$ that preserves the value of some relevant quantity, $S$, and the
space of equivalence classes. In field theory, when $S$ is the value of the
action functional, the spaces $\mathcal{G}$, ${{\Phi}}$, and $[{{\Phi}}]$ are
infinite-dimensional (Frechét) manifolds.

More mathematically, were we to write the canonical projection operator onto
the equivalence classes,
$\mathsf{pr}:{{{\Phi}}}\rightarrow{{{\Phi}}}/{\mathcal{G}}$, taking
$\varphi\mapsto[\varphi]$, then the orbit ${\cal O}_{\varphi}$ is the pre-
image of this projection, i.e.
$\mathcal{O}_{\varphi}:=\mathsf{pr}^{-1}([\varphi])$.

I can now be more precise about what I called _structure-type_ and _structure-
token_ : the structure-type is the equivalence relation $\sim$ and the
structure-tokens are the individual $[\varphi]$. It is clear that knowing the
structure-type does not give us _tractable_ conditions of identity for the
structure-tokens; but now we will provide these conditions.

With these tools in place, we can improve on Definition 1 with the more
precise:

######  Definition 2 (Representational convention)

A representational convention is an injective map

$\displaystyle\sigma:[{{\Phi}}]$ $\displaystyle\rightarrow{{\Phi}}$ (3.2)
$\displaystyle[\varphi]$ $\displaystyle\mapsto\sigma([\varphi])$

that respects the required mathematical structures of ${{{\Phi}}}$, e.g.
smoothness or differentiability and is such that
$\mathsf{pr}(\sigma([\varphi]))=[\varphi]$, where $\mathsf{pr}$ is the
canonical projection map onto the equivalence classes.

Armed with such a choice of representative for each orbit, a generic state
$\varphi$ could be written uniquely as some doublet
$\varphi=([\varphi],g)_{\sigma}:=\sigma([\varphi])^{g}$, which of course
satisfies:
$\varphi^{g^{\prime}}=(\sigma([\varphi]))^{gg^{\prime}}=([\varphi],gg^{\prime})_{\sigma}$.181818Our
notation is slightly different than Wallace (2019, p. 9)’s, who denotes these
doublets as $(O,g)$ (in our notation $([\varphi],g)$), and labels the choice
of representative (or gauge-fixing) as $\varphi_{O}$ (our $\varphi_{\sigma}$).
We prefer the latter notation, since it makes it clear that there is a choice
to be made. As with coordinate systems, the interesting quantities will be
invariant under these choices; nonetheless, we need to keep them fixed. This
requirement becomes nuanced when we are comparing different subsystems, with
each other and with the joint system (cf. Gomes (20211)). Thus we identify
${{{\Phi}}}\simeq[{{{\Phi}}}]\times{\mathcal{G}}$ via the diffeomorphism:

$\displaystyle\overline{\sigma}:[{{{\Phi}}}]\times{\mathcal{G}}$
$\displaystyle\rightarrow{{\Phi}}$ (3.3) $\displaystyle([\varphi],g)$
$\displaystyle\mapsto\sigma([\varphi])^{g}$

Now, as I mentioned, the space $[{{\Phi}}]$ is abstract, or only defined
implicitly. Since we cannot usually represent elements $[\varphi]$ of
$[{{\Phi}}]$ intrinsically, we in practice replace $\sigma$ by an equivalent
projection operator that takes any element of a given orbit to the image of
$\sigma$, as in:

######  Definition 3 (Projection operator)

A map

$\displaystyle h_{\sigma}:{{\Phi}}$ $\displaystyle\rightarrow{{\Phi}}$
$\displaystyle\varphi$ $\displaystyle\mapsto
h_{\sigma}(\varphi)=\sigma([\varphi]),$ (3.4)

is called the projection operator for the representational convention,
$\sigma$. The end result, $h_{\sigma}(\varphi)$, is called a _dressed_
$\varphi$.

Since $[\varphi^{g}]=[\varphi]$, we must have

$h_{\sigma}(\varphi^{g})=h_{\sigma}(\varphi).$ (3.5)

Moreover, $h_{\sigma}(\varphi)$ can be seen explicitly, as a map on
${{\Phi}}$, i.e. as a function of $\varphi$. As such, an $h_{\sigma}$ uniquely
and concretely represents structural content; and so projection operators
provide a valuable mathematical tool with which we can discuss physical
possibility in terms of structure. In other words, two given representatives,
$\varphi,\varphi^{\prime}$, that are in principle unrelated, are physically
the same, i.e. give the same value for all symmetry-invariant quantities iff
they numerically match once dressed:
$h_{\sigma}(\varphi)=h_{\sigma}(\varphi^{\prime})$. For fields, this will mean
that $\varphi$ anf $\varphi^{\prime}$ are physically identical iff the dressed
field values match at _every_ spacetime point. Thus a a dressed field over a
certain region gives a complete physical description of that region, without
redundancy. A projection reduces problems of identity of physical states to
numerical identity over spacetime; and the dressed fields for two disconnected
regions are compossible.

I have called $h_{\sigma}(\varphi)$ the dressed field. What is it dressed
with? With the gauge transformation $g_{\sigma}(\varphi)$ required to
transform $\varphi$ to a configuration $\varphi^{g_{\sigma}(\varphi)}$ which
belongs to the gauge-fixing section $\sigma$. This is called _the dressing
function_.191919This is not the most general type of dressing function.
Firstly, there is the infinitesimal version, given by the relational
connection-form $\varpi$ (cf. Gomes Riello (2017, 2018, 2021) and Gomes .
(2019); and Gomes (20211, 2019) and (Gomes Butterfield, 2023, Appendix A) for
philosophical introductions). Second, there is a more complete theory of
dressings and dressed fields, which only requires the right covariance
properties and can restrict to subgroups of the gauge group: see François
(2021); François . (2021) for a review.  In more detail, we have

$g_{\sigma}:{{\Phi}}\rightarrow{\mathcal{G}}$ (3.6)

such that:

$\varphi^{g_{\sigma}(\varphi)}={\sigma}([\varphi]).$ (3.7)

Thus, under the group action, $\varphi\mapsto\varphi^{g}$, and since
${\sigma}([\varphi])={\sigma}([\varphi^{g}])$ (since
$[\varphi^{g}]=[\varphi]$), we get:

$\varphi^{g_{\sigma}(\varphi)}=(\varphi{}^{g})^{g_{\sigma}(\varphi^{g})},$
(3.8)

and we obtain the following equivariance property for $g_{\sigma}$:202020These
equations are schematically identical to the ones we find for Yang-Mills
theory in the finite-dimensional case. In the Yang-Mills case, a section
$\sigma$ in the space of models is essentially a physical-state-dependent
section $\sigma$ of the finite-dimensional principal bundle, $P$.

$g_{\sigma}(\varphi^{g})=g^{-1}g_{\sigma}(\varphi).$ (3.9)

To render things more explicit: in the case of general relativity,
$g_{\sigma}$ would take each metric to an element of the diffeomorphism
group—understood as the element that drags any field distribution on spacetime
to a particular representative distribution. In the Yang-Mills case,
$g_{\sigma}$ would take a field configuration to a gauge transformation.

In possession of a dressing function $g_{\sigma}$, it is convenient to rewrite
the dressed field $h_{\sigma}$ of (3.4) as
$h_{\sigma}(\bullet):=\bullet^{g_{\sigma}(\bullet)}$, i.e:

$\displaystyle h_{\sigma}:{{\Phi}}$ $\displaystyle\rightarrow{{\Phi}}$
$\displaystyle\varphi$ $\displaystyle\mapsto
h_{\sigma}(\varphi):=\varphi^{g_{\sigma}(\varphi)}$ (3.10)

And, as expected, from (3.8), $h_{\sigma}(\varphi)$ is a gauge-invariant
functional, in the sense that
$h_{\sigma}(\varphi^{g})=h_{\sigma}(\varphi)$.212121But note that reducible
states, i.e. states that have stabilizers as per footnote 22 are not “wrinkly
enough”, do not have features that vary enough, to completely fix the
representation. Stabilizers inevitably produce degeneracies in the
representational convention: they foil uniqueness, but for physical reasons.

###  3.3 Representational conventions in field theories: relation to gauge-
fixing

As I said: to illustrate the relationship between a representational
convention and a gauge-fixing, we require some of the mathematical tools of
fiber bundles, as described in detail in (Gomes, 20214, Sec. 4), since a
representational convention, like a gauge-fixing, requires a choice of section
of the configuration space, seen as a (possibly infinite-dimensional)
principal bundle.

In practice, the gauge-fixing procedure relies on the given representational
convention $\sigma([\varphi])$ satisfying some auxiliary condition. But in
(3.3), I oversimplified: there is no such global product form, and even a
local product form is not guaranteed to exist, as it is in the finite-
dimensional case (see (Gomes Butterfield, 2023, Sec.2)).

Nonetheless, it is in fact true that the space of models ${{\Phi}}$—for both
Riemannian metrics and gauge connection-forms—is mathematically very similar
to a principal fiber bundle, with ${\mathcal{G}}$ as its structure group
(resp. diffeomorphisms and vertical automorphisms). But there are important
differences between the infinite-dimensional and the finite-dimensional case.
As described in (Gomes, 20214, Sec. 4), in the finite-dimensional case, it
suffices that the action of a group $G$ on the given manifold $P$ be free (and
proper) for that manifold to have a principal ${G}$-bundle structure, usually
written as ${G}\hookrightarrow P\rightarrow P/{G}$. In the infinite-
dimensional case, these properties of the group action are not enough to
guarantee the necessary fibered, or local product structure: one has to
construct that structure by first defining a _section_.222222 In fact, in the
case of field theories, such as general relativity and Yang-Mills, not even
such a local product structure exists everywhere: ${{\Phi}}$ is not an
infinite-dimensional principal $\mathcal{G}$-bundle, i.e.
$\mathcal{G}\hookrightarrow{{\Phi}}\rightarrow{{\Phi}}/\mathcal{G}$. The
obstacle is that there are special states—called _reducible_ —that have
stabilizers, i.e. elements $\tilde{g}\in{\mathcal{G}}$ such that
$\tilde{\varphi}^{\tilde{g}}=\tilde{\varphi}$. Moreover, the statement holds
for the entire group orbit, as it is easy to show that for some
$\tilde{\varphi}^{\prime}:=\tilde{\varphi}^{g}\in\mathcal{O}_{\tilde{\varphi}}$,
$\tilde{\varphi}^{\prime
g^{-1}\tilde{g}g}=\tilde{\varphi}^{\tilde{g}g}=\tilde{\varphi}^{g}=\tilde{\varphi}^{\prime}$
and so all the elements of the orbit are also reducible (with stabilizers
related by the co-adjoint action of the group). And so orbits are of
‘different sizes’, and not isomorphic to the structure group. Nonetheless,
there is a generalization of a section, called a slice, that provides a close
cousin of the required product structure (see (Gomes Butterfield, 2023, Sec.
2.1, and footnotes 4,5)). As has been shown using different techniques and at
different levels of mathematical rigour, Ebin (1970); Palais (1961); Mitter
Viallet (1981); Isenberg Marsden (1982); Kondracki Rogulski (1983); Wilkins
(1989); Diez Rudolph (2019) both the Yang-Mills configuration space and the
configuration space of Riemannian metrics (called $\mathrm{Riem}(M)$), admit
slices. These slices endow the quotient space with a _stratified_ structure.
That is, the state space can be organised into orbits of states that possess
different numbers of stabilizers; with the orbits with more stabilizers being
at the boundary of the orbits of states with fewer stabilizers. For both
general relativity and non-Abelian gauge theories, reducible configurations
form a meagre set. Meagre sets are those that arise as countable unions of
nowhere dense sets. In particular, a small perturbation will get you out of
the set (and this is true of the reducible states in the state spaces of those
theories, according to the standard field-space metric topology (the Inverse-
Limit-Hilbert topology cf. e.g. Kondracki Rogulski (1983); Fischer Marsden
(1979)). In this respect, Abelian theories, such as electromagnetism, are an
exception: all their configurations are reducible, possessing the constant
gauge transformation as a stabilizer. And apart from this obstruction to the
product structure—i.e. even if we were to restrict attention to the generic
configurations in the case of non-Abelian field theories—one can have at most
a _local_ product structure: no representational convention, or section,
giving something like (3.3), is global (this is known as the _Gribov
obstruction_ ; see Gribov (1978); Singer (1978)). Unfortunately, the space of
Lorentzian metrics is not known to have such a structure: it has only been
shown for the space of Einstein metrics that admit a constant-mean-curvature
(CMC) foliation.

A choice of section is essentially a choice of embedded submanifold on the
state space ${{\Phi}}$ that intersects each orbit exactly once. In figure 1
above, this would be represented by a ‘nowhere-vertical’ curve: a submanifold
that is transversal to all the ${\mathcal{G}}$-orbits and intersects each
orbit once.

That is, we impose further functional equations that the state in the aimed-
for representation must satisfy; this is like defining a submanifold
indirectly, through the regular value theorem: E.g. defining a co-dimension
$k$ surface $\Sigma\subset N$ for some $n>k$-dimensional manifold $N$, as
$\mathcal{F}^{-1}(c)$, for $c\in\mathds{R}^{k}$, and $\mathcal{F}$ a smooth
and regular function, i.e. $\mathcal{F}:N\rightarrow\mathds{R}^{k}$ such that
$\mathsf{ker}(T\mathcal{F})=0$.232323In the infinite-dimensional case, both
the dimension and the co-dimension of a regular value surface can be infinite,
and it becomes trickier to construct a section: roughly, one starts by
endowing ${{\Phi}}$ with some ${\mathcal{G}}$-invariant (super)metric, and
then finds the orthogonal complement to the orbits, $\cal{O}_{\varphi}$, with
respect to this supermetric. But here the intersection of the orbit with its
orthogonal complement cannot be assumed to vanish, as it does in the finite-
dimensional case. Nonetheless, in the cases at hand, that intersection is
given by the kernel of an elliptic operator, and one therefore can invoke the
‘Fredholm Alternative’ (see (Gilbarg Trudinger, 2001, Sec. 5.3 and 5.9)) to
show that that intersection is at most finite-dimensional, but generically is
zero, and thus the generic orbit has the ‘splitting’ property: the total
tangent space decomposes into a direct sum of the tangent space to the orbit
and its orthogonal complement. Now we must extend the directions transverse to
the orbit, so as to construct a small patch that intersects the neighboring
orbits only once. But the space of metrics is not a vector or even an affine
space, and so we cannot just linearly extend the directions normal to the
orbit and hope for the best. And so we extend the normal directions by using
the Riemann normal exponential map with respect to the supermetric (cf. Gil-
Medrano Michor (1991)), and thus conclude that, for a sufficiently small
radius, the resulting submanifold is transverse to the neighboring orbits and
has no caustics. Finally, to show that this ‘section’ is not only transverse
to the orbits, but indeed that it intersects neighboring orbits only once, the
orbits must be embedded manifolds, and not just local immersions: this is
guaranteed if the group action is proper, cf. Ebin (1970).

Once the surface is defined, $\sigma$ can be seen as the embedding map with
range $\sigma([{{{\Phi}}}])=\mathcal{F}^{-1}(0)\subset{{{\Phi}}}$. The next
step is to find a gauge-invariant projection map, $h_{\sigma}$, that dresses,
or projects, any configuration to this surface.

Here we encounter the dual interpretations of gauge-fixings and dressed
fields: the relational, non-local interpretation, is as the image of this
projection: this is the dressed field. The local interpretation is as a field
that satisfies an auxiliary set of differential equations, by belonging to
$\mathcal{F}$: this is the gauge-fixed field. We will briefly come back to
this point in the gravitational case, in Section 4.3.d (see, in particular,
the second long quotation from Komar).

Ideally, a suitable real-valued function $\mathcal{F}$ that defines the
section (surface) $\sigma([{{\Phi}}])$ as its level surfaces
$\mathcal{F}^{-1}(0)$ must satisfy two conditions:
$\bullet$ Universality (or existence): For all $\varphi\in{{\Phi}}$, the
equation $\mathcal{F}(\varphi^{g})=0$ must be solvable by a dressing
$g_{\sigma}:{{\Phi}}\rightarrow{\mathcal{G}}$.

That is:

$g_{\sigma}:{{\Phi}}\rightarrow{\mathcal{G}},\,\,\text{is such
that}\,\,\mathcal{F}(\varphi^{g_{\sigma}(\varphi)})=0,\,\,\text{for
all}\,\,\varphi\in{{\Phi}}.$ (3.11)

This condition ensures that $\mathcal{F}$ doesn’t forbid certain states, i.e.
that each orbit possesses at least one intersection with the gauge-fixing
section.

$\bullet$ Uniqueness: If $g_{\sigma}$ as above satisfies
$\mathcal{F}(\varphi^{g_{\sigma}(\varphi)})=0$, then
$\varphi^{g_{\sigma}(\varphi)}=\varphi^{\prime}{}^{g_{\sigma}(\varphi^{\prime})}$
if and only if $\varphi\sim\varphi^{\prime}$, meaning that only isomorphic
models have the same projection onto the section.

As I alluded to above, both conditions on $g_{\sigma}$ are idealised, and
cannot hold absolutely. _Uniqueness_ fails due to the presence of stabilizers,
and _Universality_ fails due to the presence of Gribov obstructions, as
described in footnote 22. Nonetheless, as described in footnotes 22 and 23, a
representational convention $\mathcal{F}$ is still well-defined locally on
${{\Phi}}$.

Thus, using representational conventions (and the notation $\cal F$ for gauge-
fixing) we could, for example, give a definition of structure-tokens, such as:

######  Definition 4 (Structure-tokens according to $\cal F$)

Each structure token $[\varphi]$ is represented uniquely, according to $\cal
F$, by the corresponding value of the map $h:{{\Phi}}\rightarrow{{\Phi}}$,
where $h(\varphi)$ satisfies some extra condition, $\mathcal{F}(h)=0$.

By (3.11) and (3.10), this is an invariant characterization. And as mentioned
above, the invariant function will usually have a dual interpretation: it is
non-local on the domain of the projection, but it is local on its image, since
it is just a field respecting an extra local (usually differential) condition.

For instance, in the case of Lorentzian geometries $\langle
M,\mathbf{g}\rangle$ (omitting the indices of $g_{ab}$), given a suitable
$\cal F$, the dressed field $h_{ab}(\mathbf{g})(x)$ is a local field
satisfying some extra condition ${\cal F}(h)=0$ but, given an arbitrary
$\mathbf{g}$, in order to find the preferred representation, we require a non-
local function of $\mathbf{g}$ (the value of $h_{ab}(\mathbf{g})(x)$ depends
on the values of $\mathbf{g}$ at points far away from $x$). That is, the
projection function takes its arguments non-locally, but the value of the
projection is just another (by definition local) model of the theory.

If $\cal F$ is suitable, Definition 4 gives a complete, local, tractable
condition of structural identity: any two models $\mathbf{g}$ and
$\mathbf{g}^{\prime}$ are isomorphic iff
$h_{ab}(\mathbf{g}^{\prime})(x)=h_{ab}(\mathbf{g})(x)$ for all $x\in M$. And
similarly, whether or not two such projections match in some region is a
gauge-invariant question: if they do match, they match for all isomorphic
models, and if they don’t, they don’t for all isomorphic models.

Of course, we can still change the representational convention itself, i.e.
act with the group on the image of $h_{\sigma}$: Because
$h:{{\Phi}}\rightarrow{{\Phi}}$ is a projection, as opposed to a reduction
($\mathsf{pr}:{{\Phi}}\rightarrow[{{\Phi}}]$), and thus
$h_{\sigma}(\varphi)\in{{\Phi}}$, we could consider
$(h_{\sigma}(\varphi))^{g}$. But the new, transformed model, would no longer
obey the conditions defining the previous representational convention: if
these conditions are taken to define, for extra-theoretical reasons,
‘acceptable’ representations, the new model would not be an ‘acceptable’
representation of that orbit.

Given two representational conventions $\sigma,\sigma^{\prime}$, we obtain a
relation:

$h_{\sigma}(\varphi)=(h_{\sigma^{\prime}}(\varphi))^{\mathfrak{t}_{\sigma\sigma^{\prime}}(\varphi)},$
(3.12)

where
$\mathfrak{t}_{\sigma\sigma^{\prime}}(\varphi):=g_{\sigma^{\prime}}(h_{\sigma}(\varphi))$
is a state-dependent isomorphism (the analog of the transition map between
sections of a principal bundle). Since
$h_{\sigma}(\varphi^{g})=h_{\sigma}(\varphi)$, the transition map depends only
on the orbit $\mathcal{O}_{\varphi}$ of $\varphi$.

To circle back to the quotes from physicists about invariant observables in
Section 2.2, many, if not all of the examples of ‘relational gravitational
observables’ used by physicists employ just such projections and
representational conventions (though the physicist’s nomenclature does not
involve the term ‘representational conventions’). These are the type of
‘relational dressings’ mentioned in the Harlow Wu (2021, p.3) quote of Section
2.2. We will look at some of these examples in Section 4.3.

##  4 How to choose a representational convention?

Before answering the question of this Section, I will motivate it with an
argument from Healey (2007). More specifically, I take issue with one argument
in Healey’s (otherwise outstanding!) book on the philosophical interpretation
of gauge theories. It occurs in his Chapter on classical gauge theories
(Chapter 4); and it is directed at an interpretation of the theories that
Healey calls ‘the localized gauge potential properties view’.

In the previous Sections, my treatment of representational conventions has
remained rather abstract. Here, by answering Healey, I will provide more
explicit examples and (superficially) connect the ideas to a popular topic in
analytic philosophy: functionalism.

I will start in Section 4.1 by recapitulating Healey’s argument. Then in
Section 4.2 I will show that, _contra_ Healey, intra-theoretic resources
enable us to pick out gauge representatives. I will complement this answer in
Section 4.3, by providing more examples, in other theories, of explicit
representational conventions, alongside brief physical interpretations.
Finally, I conclude this Section in Section 4.4, by describing the entirely
pragmatic warrant for the different choices of representational convention.

###  4.1 Healey’s argument from functional roles

To set the scene, Healey defines the ‘localized gauge potentials properties
view’ as saying that the gauge potential $A$ at a spacetime point $x$
represents a physically real property of, or at, $x$. This is, in essence,
viewpoint labeled _quidditist_ , according to which properties and relations
have a nature that outstrips their patterns of instantiation in objects and
co-instantiation with each other. The viewpoint has been defended, for
instance in (Arntzenius, 2012, Ch. 6), but (Healey, 2007, Section 4.2) is
devoted to rebutting this view. And although I also reject this quidditist
view, I disagree with Healey’s arguments, which are distinctively
philosophical.

For in making his argument, Healey touches on what he takes to be a difference
between the tenability of gauge-fixing in general relativity and in Yang-Mills
theory. That is, a difference between specifying, amongst the infinitely many
physically equivalent representatives, a particular spacetime distribution of
the gauge potentials or of the metric. As I understand Healey, he posits that
this specification is easy for the metric, but impossible for the gauge
potential. And with this alleged contrast I will disagree. My disagreement
will shed light on how representational conventions are chosen.

To spell out Healey’s argument in more detail, I will now indulge in a bit of
‘Healey exegesis’. Healey admits that within a theory there may be many terms
standing for unobservable items; but unobservability by itself is not bad
news, since we can employ D. Lewis (1970, 1972)’s construction, that applies
not only to the observable-unobservable distinction, but, more broadly, to a
distinction between purely theoretical, or less understood terms, and the
better understood, or ‘old’ terms. The idea is that we can simultaneously
specify what several theoretical items—labeleled T-terms—refer to, by their
each uniquely satisfying some description (usually called “functional role”)
that can be formulated in terms of each other and of the better understood
terms, labeled O-terms.

But, even in principle, how could we use the theory at hand to functionally
single out any item if the theory treats it and others, at least in certain
respects, as being on a par, i.e. related by some symmetry? As described by D.
Lewis (2009), the response is to appeal to patterns of facts of “geography” to
break the underdetermination:

> Should we worry about symmetries, for instance the symmetry between positive
> and negative charge? No: even if positive and negative charge were exactly
> alike in their nomological roles, it would still be true that negative
> charge is found in the outlying parts of atoms hereabouts, and positive
> charge is found in the central parts. O-language has the resources to say
> so, and we may assume that the postulate mentions whatever it takes to break
> such symmetries. Thus the theoretical roles of positive and negative charge
> are not purely nomological roles; they are locational roles as well. [my
> italic]

But Healey argues that, in gauge theories, even this Lewisian strategy is
bound to be plagued by under-determination. I interpret Healey as saying that
one can functionally specify a spacetime metric using only O-terms, but cannot
specify a gauge potential. More precisely, here is Healey’s argument that the
functionalist methodology applies so as to single out spacetime metrics, but
not to single out gauge potentials:

> The idea seems to be to secure unique realization of the terms [e.g. the
> gauge potential $A$…] in face of the assumed symmetry of the fundamental
> theory in which they figure by adding one or more sentences [namely, $S$]
> stating what might be thought of as “initial conditions” to the laws of that
> theory […] to break the symmetry of how these terms figure in $T$. They
> would do this by applying further constraints […namely, $S$…] Those
> constraints would then fix the actual denotation of the […symmetry-related
> terms…] in $T$ so that, subject to these further constraints, $T$ is
> uniquely realized. […But] The gauge symmetry of the theory would prevent us
> from being able to say or otherwise specify which among an infinity of
> distinct distributions so represented or described is realized in that
> situation. This is of course, not the case for general relativity. (Healey,
> 2007, p. 93) [my italics]

But I would ask: Why is this “of course not the case for general relativity”?
And why does Healey see a contrast between general relativity and gauge
theory?

I do not see in this entire passage an attempt to draw a distinction regarding
Leibniz-equivalence and sophistication about symmetries between the two cases.
I think the interesting question being alluded to here is whether we can use
features of the world around us to single out a unique model of the theory by
its unique features. This other question is more interesting because, in
practice, we _do_ select some model over others when we represent a given
physical situation, and therefore we must somehow ‘break the symmetry’ between
all of the models. In other words, if we accept that the basic ontology is
symmetry-invariant, we are faced with a mystery: how do we select particular
representatives of the ontology—i.e. models—over others? Under what conditions
could we be justified in choosing for the gauge potential one spacetime
distribution over another? Selecting such a representative involves a tension
between: (a) a structural construal of physical properties—as ones that are
invariant under the symmetries in question—and (b) in practice selecting
unique representative distribution, among the infinitely many isomorphic
representatives of the same situation. At first sight, these two requirements,
(a) and (b), are inimical, if not contradictory, for (a) implies we can have
no physical guidance for accomplishing (b)!

Getting representational conventions off the ground of course depends on a
positive resolution of this paradox. For though no particular choice is
mandatory, each must be based on something: physical features, indexicals,
ostension, etc.

Thus I take the more interesting interpretation of Healey’s passage here to be
that this ‘singling out’ of particular models is possible for gravity, but not
for gauge theory, where the choice is unthethered from any feature of the
physical world: it is entirely arbitrary, according to him. If this were so—if
there was nothing contingent on which we could base our representational
conventions for gauge theory—we would indeed be more motivated to seek out
reduction or elimination for gauge theory than for gravity.242424And indeed,
later in the book Healey uses this distinction as a motivation for seeking a
different, symmetry-invariant ontology of gauge theory, based on holonomies.
But, as argued in Section 2.1, these are not perspicuous formulations of the
structure-tokens of the theory. Answering this question is thus crucial in
order to clarify the scope of representational conventions.

###  4.2 Refuting Healey’s alleged distinction using representational
conventions

_Contra_ Healey, I will argue that having some physical “hook” with which to
choose representatives does _not_ imply that we are breaking the symmetry at a
fundamental level. Different choices of representational conventions would be
equally capable of representing a given state of affairs; some may just be
more cumbersome than others. Or perhaps some representations obscure matters
for the purposes at hand, even while they may shine light on complementary
aspects of that state of affairs. And, in this sense, by choosing different
conventions we shift our focus to different features of the world, according
to our interest. For instance, the center of mass of the solar system does not
break the translation invariance of Newton’s equations of motion; but it
provides a hook with which we can uniquely represent a given solution of the
equations. More germanely for our case studies here, the Cosmic Microwave
Background does not break diffeomorphism symmetry, but, for applications to
cosmology, it does provide a particularly perspicuous hook for representing
spacetimes uniquely. We find particular representational conventions that are
suited to it, by e.g. taking it to define a foliation of the spacetime metric,
in which the surfaces of simultaneity are taken as those with constant energy
density.

Now I will show that we can construct a particular representative of the gauge
potential as fulfilling a given role, and explicitly check that such a notion
is invariant under isomorphisms. I will complement this answer in Section 4.3,
by providing more examples, in other theories, of explicit representational
conventions, alongside brief physical interpretations.

I will resolve the tension between (a) and (b) described above, with explicit
examples; by, in Healey’s words: ‘breaking the symmetries’, by providing
‘further constraints’.

Note to begin with that, according to Healey’s standards, we are justified in
including in our O-vocabulary all the ‘locational roles’, which describe
contingent, happenstantial facts about ‘where and when’ specified events
happen; and which I will loosely interpret as ‘referring to spacetime’. Thus I
free myself to include in the O-vocabulary, and thereby use in the
specification of the roles, the differential geometry of spacetime. In short,
I will assume reference to spacetime is ‘old’ or already understood.

In the simple example of electromagnetism, we require the model to satisfy
certain relations among the parts of the field. For example, in our hierarchy
of extra-empirical theoretical virtues, we could place Lorentz covariance very
highly (cf. Mulder (2021) and Mattingly (2006) for advocacy of this criterion
and choice of gauge) and therefore prefer an explicitly Lorentz-covariant
choice of convention:

$\mathcal{F}(A):=\nabla^{a}A_{a}\equiv\nabla^{a}(\sigma^{*}\omega)_{a}=0.$
(4.1)

With this choice, the equations of motion for $A_{a}$, the Maxwell equations
of motion become the hyperbolic equation (which gives a well-posed initial
value problem: see Gomes (20215) for more details about the relation between
the IVP and gauge freedom):

$\square A_{a}=0,$ (4.2)

where $\square:=\nabla^{a}\nabla_{a}$ is the d’Alembertian.252525Equation
(4.1) still has the gauge-freedom corresponding to $\phi$ such that
$\square\phi=0$. To fix $\phi$ uniquely, one requires e.g. elliptic equations,
which carry the non-local behavior we have alluded to (cf. Gomes (20215)).
Thus this example is merely illustrative, as in the Lorentzian setting this
gauge-fixing is not complete and we would require additional imput about the
initial state. This failure will be remedied by the examples in Section 4.3;
for now, if we replace the metric in (4.1) by one of Euclidean signature,
everything in the main text is strictly true.

Here the value of the connection $\omega$ is fixed: all we are trying to do is
to determine a particular, state-dependent section $\sigma(A)$ (seen as a
state-dependent submanifold of the bundle $P$). The only extra constraints
that we have imposed in this equation, namely, that the spacetime divergence
of the particular representative of $\omega$ vanishes, use only the
O-vocabulary that Healey would grant us.262626There is a subtle point here: we
are in effect assuming the active-passive correspondence of (Gomes, 20214,
Sec. 5), to say that fixing a choice of section of the bundle is equivalent to
picking out a unique model within an isomorphism class.

We can also explicitly display in $O$-vocabulary the projection of an
arbitrary representative of $\omega$ into a representative satisfying (4.1).
Namely, given any representative of any equivalence class, $\mathbf{A}$, we
define, as in Equation (3.4):

$h(\mathbf{A})_{a}:=A_{a}-\nabla_{a}(\square^{-2}\nabla^{b}A_{b}),$ (4.3)

where $A_{a}$ is any 1-form and $\square^{-1}$ is a propagator (which would
require more input to be fully determined: cf. footnote 25). We will derive
this form of the projected potential in Section 4.3, when we discuss examples
in more detail. We can thus state that $h$ is a composite, whose parts are
different components of the gauge potential.

Much as in other representations of gauge-invariant quantities—such as in the
holonomy interpretation—fixing the gauge is non-local in the following sense:
just as $\oint_{\gamma}A$ requires the value of of $A$ at several points
simultaneously as an input—namely all along the closed loop $\gamma$—the
projected state $h_{\sigma}(A)$ generically requires the value of $A$
throughout the region in which it is defined as input. This is just a
reflection of the non-local aspects of gauge-invariant functions as described
in Section 2.2 (cf. also (Earman, 1987, p. 460), (Healey, 2007, Ch. 4.5),
(Strocchi, 2015; Gomes, 2019), and the appendix of Gomes Butterfield
(2022)).272727In the particular case of (4.1), while the gauge potential is
‘propagated’ from an initial value hypersurface, it still requires us to
establish conditions at infinity that are not Lorentz invariant. See footnote
25.

Of course we still have the freedom to change the section $\sigma$, but a
different section would not satisfy the original condition that uniquely
specified $\sigma$.

And, consistently with my arguments, as we will see in the following Section,
it follows from this construction that the determination of the representative
is structural, in the sense that, though $h(\mathbf{A})_{a}$ is, like
$\mathbf{A}$, a Lie-algebra valued one-form, $h(\mathbf{A})_{a}$ is a gauge-
invariant functional of $\mathbf{A}$. That is, for a gauge-related
$A^{\prime}_{a}=A_{a}+\nabla_{a}\chi$, we obtain, from a simple computation,
that $h(\mathbf{A}^{\prime})_{a}=h(\mathbf{A})_{a}$. And therefore
$h(\mathbf{A})_{a}$ can be construed as a structural property of the field.
Moreover, it provides a complete and local description: two models are
isomorphic if and only if the value of their projections are numerically
identical for every point.

In sum, the representational convention is defined by a gauge-invariant
specification. Nonetheless, it successfully pins down a representation for the
electromagnetic potential. This construction thus explicitly contradicts the
letter of Healey’s under-determination argument in the quotation above. This
concludes my response to what I called Healey’s ‘interesting challenge’:
showing that intra-theoretic resources enable us to choose representational
conventions and pick out representatives. In the next Section, I will give
more examples of dressings and dressed variables in all kinds of theories with
symmetries

###  4.3 Examples

Representational conventions, as introduced in Section 3, are a very general
construction, applicable to any theory whose state space admits a version of
symmetry as defined in (Gomes, 20213, Sec. 2). So it behoves us to provide
examples not only for electromagnetism, as follows.

####  4.3.a Center of mass for non-relativistic point-particles

Take Newtonian mechanics for a system with $N$ particles, written in
configuration space as trajectories
$\varphi(t)=q_{\alpha}(t)\in\mathds{R}^{3N}$ (with $\alpha=1,\ldots,N$
labeling the particles), and with shift symmetry:

$\varphi^{g}=(q^{\alpha}(t)+g),\quad\text{with }g\in\mathbb{R}^{3}.$ (4.4)

Now we take the section $\mathcal{F}$ to be defined by the center of mass:

$\mathcal{F}(q)=\sum_{\alpha}q^{\alpha}m^{\alpha}=0.$ (4.5)

The most straightforward way to find the projection $h(q)$, is to find the
dressing as a functional of the configuration. That is, the horizontal
projection here acts on each particle, as

$h^{\alpha}(q)=q^{\alpha}+g_{\sigma}(q),$ (4.6)

where we solve

$\mathcal{F}(h(q))=0\quad\text{ for
}g_{\sigma}:\Phi\to\mathbb{R}^{3}\,\,\text{and arbitrary }\,\,q.$ (4.7)

We obtain:

$\displaystyle\sum_{\alpha}(q^{\alpha}+g_{\sigma}(q))m^{\alpha}=0\Rightarrow
g_{\sigma}(q)=-\frac{\sum_{\alpha}q^{\alpha}m^{\alpha}}{\sum_{\alpha}m^{\alpha}}.$

Clearly,

$g_{\sigma}(\varphi^{g})=g_{\sigma}(q+g)=g_{\sigma}(q)-g.$ (4.8)

Thus if follows that the new set of configuration variables, written as
functionals of arbitrary configuration variables, is gauge-invariant (now
including indices for both particles and components in $\mathds{R}^{3}$, to be
completely explicit):

$h_{i}^{\alpha}(q^{\alpha}+g)=(q_{i}^{\alpha}(t))+g+g_{\sigma}(q)-g=h_{i}^{\alpha}(q^{\alpha}).$
(4.9)

And it is also trivial to see that, if
$h^{\alpha}(q)=h^{\alpha}(q^{\prime}{})$, then
$q^{\alpha}-q^{\prime}{}^{\alpha}$ is given by a shift, and so the two
original configurations are symmetry-related. That is,
$h^{\alpha}(q)=h^{\alpha}(q^{\prime}{})$ iff $q$ and $q^{\prime}{}$ are
isomorphic.

Of course, examples abound: beyond the case of translations and boosts, we
could fix a frame in $\mathds{R}^{3}$ by diagonalizing the moment of inertia
tensor around the center of mass; in this case $g_{\sigma}$ would be a state-
dependent rotation (see (Gomes, 20212, Sec. 4) for
details).282828Configurations that are collinear, or are spherically
symmetric, etc. would be unable to fix the representation: as per footnotes 22
and 21, these are _reducible_ configurations; they have stabilizers that
cannot be fixed by any feature of the state.

####  4.3.b Electromagnetism: Lorenz gauge

The choice of (4.1) is explicitly Lorentz covariant (and has its virtues
thoroughly extolled by Mattingly (2006), who argues that it should be
considered as Maudlin (1998)’s “ONE TRUE GAUGE”). Maudlin (2018) himself
endorses a different choice, Coulomb gauge.

This particular example of $\mathcal{F}$ (and $\sigma$, $h_{\sigma}$, and
$g_{\sigma}$) in field theory is fully worked out by Gomes Butterfield (2022),
alongside its physical interpretation. This choice arises from a split of the
electric field into a component that is purely Coulombic, or determined by the
synchronic distribution of charges, and another component that is purely
radiative. This particular choice of convention is generally applicable and
has a clear physical intepretation. Let us see how it gives rise to an
isomorphism-invariant description of the degrees of freedom. First, we write:

$h(\mathbf{A})_{a}:=A_{a}+\nabla_{a}g_{\sigma}(\mathbf{A}),$ (4.10)

where $g_{\sigma}:{{\Phi}}\rightarrow C^{\infty}(M)$, and $M$ is the spacetime
manifold, with Euclidean signature, and ${{\Phi}}$ is the space of gauge
potentials over $M$ satisfying appropriate boundary conditions (which I will
not here discuss: for more details, about boundary conditions and the point
that this gauge is not complete in the Lorentzian signature, see footnote 25
and Gomes Riello (2021)). The condition to be satisfied by $g_{\sigma}$ will
be obtained from our definition of $\mathcal{F}$:

$\mathcal{F}(\mathbf{A})=\nabla^{a}A_{a}=0.$ (4.11)

That is, we obtain:

$\mathcal{F}(h(\mathbf{A}))=\nabla^{a}(A_{a}+\nabla_{a}g_{\sigma}(\mathbf{A}))=0,$
(4.12)

which can be readily worked out to be:

$g_{\sigma}(\mathbf{A})=-\square^{-2}\nabla^{b}A_{b},$ (4.13)

and which clearly satisfies the covariance equation (3.9), namely:

$g_{\sigma}(\mathbf{A}^{g})=-\square^{-2}\nabla^{b}(A_{b}+\nabla_{b}g)=g_{\sigma}(\mathbf{A})-g.$
(4.14)

Now it is trivial to find the dressed functional given in (4.3):

$h(\mathbf{A})_{a}:=A_{a}-\nabla_{a}(\square^{-2}\nabla^{b}A_{b})$ (4.15)

where $\square^{-2}$ is the Green’s function, or propagator, for the
D’Alembertian. We can easily verify that this dressed functional is gauge-
invariant:

$\displaystyle h(A+\nabla\chi)_{a}$ $\displaystyle=$ $\displaystyle
A_{a}+\nabla_{a}\chi-\nabla_{a}(\square^{-2}\nabla^{b}(A_{b}+\nabla_{b}\chi))$
(4.16) $\displaystyle=$ $\displaystyle
A_{a}-\nabla_{a}(\square^{-2}\nabla^{b}A_{b})+\nabla_{a}\chi-\nabla_{a}(\square^{-2}\square\chi)$
$\displaystyle=$ $\displaystyle h(A)_{a}$

And of course, from (4.10), it is clear that if two dressed potentials match,
they are dressings of gauge-related potentials.

As required by _universality_ (cf. Section 3), the choice is not physically
restrictive. Given any $\mathbf{A}$, we can translate it along the fibers,
looking for a representation of the field that satisfies equation (4.1); and
as required by _existence_ , for any $\mathbf{A}$, $h(\mathbf{A})_{a}$ is a
potential that is related to $\mathbf{A}$.

There are many other, physically motivated, choices we could have made. For
instance, we may also want to highlight the helicity degrees of freedom of the
theory, in which case we would use temporal gauge. Of course, in the more
realistic Lorentzian signature, both temporal and Coulomb gauge depend on a
foliation of spacetime by spacelike surfaces, and so, strictly speaking,
cannot be given a univocal physical interpretation unless coupled to a
representational convention about spacetime foliations. We turn to this next.

####  4.3.c General relativity: DeDonder gauge

In the case of Lagrangian general relativity, it is common to have $\sigma$
correspond to De Donder gauge (see Section 3.3), which fixes coordinates so
that the densitized metric is divergence-free. Using that Section’s notation,
$\mathcal{F}$, for the functional whose value fixes a gauge (or section), we
have in this example:

$\mathcal{F}(g)={\partial}_{\mu}(g^{\mu\nu}\sqrt{g})=0.$ (4.17)

Geometrically, De Donder gauge anchors our coordinate systems on spacetime
waves (solutions of the relativistic wave equation, written as $\square
x^{\mu}=0$). I will not bore the reader with yet another demonstration that
this gauge gives rise to dressed gravitational fields with the expected
properties.292929As in footnote 26, we are here also in effect assuming the
active-passive correspondence of (Gomes, 20214, Sec. 5), to say that fixing a
choice of coordinates is equivalent to picking out a unique model within an
isomorphism class. Of course, more generally, in order to fix a given choice
coordinates, any imposed condition such as DeDonder gauge cannot be fully
coordinate-invariant: it must be satisfied only by the specific coordinate
system that we are seeking to define. Moreover, the use of coordinates implies
that in general such conditions are only local. Giving a global, or geometric
version of deDonder gauge can accomplished using an auxiliary metric, a
measure of distance between metrics, and a diffeomorphism that is used to
compare them (see (Landsman, 2021, Sec. 7.6)). This strategy—of using
auxiliary metrics—is often employed in converting results originally expressed
in particular coordinate systems into explicitly covariant results.

Another common choice is to physically anchor time coordinates using ‘CMC
gauge’: this choice adjusts clocks and simultaneity surfaces so that
simultaneous observers measure the same local expansion of the universe. In
this case, $\mathcal{F}(g_{ij},\pi^{ij})=g^{ij}\pi_{ij}=$const, where $g_{ij}$
is the spatial metric and $\pi^{ij}$ is its conjugate momentum. To take some
other recent examples, the relational observables of MB. Fröb Lima (2022); M.
Fröb (2018) can be seen as projections onto the gauge-fixing surface of a non-
linear generalization of de Donder gauge, if one looks just at the invariant
metric perturbation. The idea is to construct relational observables using
‘field-dependent coordinates’ (see e.g. Eq. (2.22)). Similarly, light-cone
coordinates and ‘geometric clocks’ (see e.g. Giesel . (2018)) anchor the
representation, or the coordinates, to the state. The ‘dressed’
diffeomorphism-invariant observables of Donnelly Giddings (2016) are
relativized to a state since the quantities are anchored on non-null spacetime
curves.303030 For a general relationship between different types of gauge
symmetry and non-locality, see Berghofer . (2022). For the constraints on
finding local observables in the Hamiltonian context, see also Torre (1993).

####  4.3.d General Relativity: Komar-Kretschmann coordinates

This Section gives an example that aptly illustrates the constructions of this
paper. For this example, we must first restrict our attention to spacetimes
that are not homogeneous, i.e. generic spacetimes (i.e. excluding Pirani’s
type II and III spaces of pure radiation, in addition to excluding symmetric
type I spacetimes). Once this is done, take the four scalar functions
$R^{(\mu)}(g_{ab}(x)),\,\,\mu=1,\cdots 4$, formed by certain real scalar
functions of the Riemman tensor. The use of an index in parenthesis emphasizes
that this is just a list, and not a vector field, but we will dropt it in what
follows, since we will use these scalars _to define_ a coordinate system.

Komar (1958) finds these real scalars through an eigenvalue problem:

$(R_{{a}{b}{c}{d}}-\lambda(g_{{a}{c}}g_{{b}{d}}+g_{{a}{d}}g_{{b}{c}}))V^{{c}{d}}=0,$

where $V^{{c}{d}}$ is an anti-symmetric tensor. The requirement ensures
solutions $\lambda$, whose existence we assume, are independent real scalar
functions. Komar (1958, p.1183) takes these scalars to be preferred, “since
they are the only nontrivial scalars which are of least possible order in
derivatives of the metric, thus making them the simplest and most natural
choice.”

In the corresponding spacetimes, we _define_ coordinates (omitting the
dependence on the metric on the left-hand-side and the index’s parenthesis):

$\overline{x}^{\mu}:=R^{\mu}(g_{ab}(x)).$ (4.18)

In a less coordinate-centric language, the idea here is to fix the identity of
spacetime points through a certain choice of their qualitative properties,
quantifying existentially over the points of $M$. That is, we define point
$\overline{x}(g_{ab})$ through an inverse relation, in which $g_{ab}$ is the
variable argument (as in the above: ‘$\overline{x}(g)$ is the point in which a
given list of curvature scalars $R^{(\mu)}(g)$, $\mu=1,\cdots,4$ takes a
specific list of values, $(a_{1},\cdots,a_{4})$’). This inverse relation
specifying points in terms of their qualitative properties is in a sense
holistic (since it quantifies over points in $M$), and yet it furnishes a
specific qualitative identity _for spacetime points_ across isomorphic models;
(and indeed, it can also furnish a qualitative identity relation across non-
isomorphic models, cf. (Gomes Butterfield, 2021)). As we will see in Section
5, all of our examples furnish a similar notion of counterparhood between
models.313131 Curiel (2018, p. 468) construes qualitative identity of points
similarly:  Once one has the identification of spacetime points with
equivalence classes of values of scalar fields, one can as easily say that the
points are the objects with primitive ontological significance, and the
physical systems are defined by the values of fields at those points, those
values being attributes of their associated points only per accidens. But he
does not construe diffeomorphisms as intra-theoretic changes of convention
about the choices of scalar fields, as we do.

Now, given some metric tensor $g^{\kappa\gamma}$ in coordinates $x^{\kappa}$,
we can compute the metric in the new, $\overline{x}^{\mu}$ coordinate system
as:

$\overline{g}^{\mu\nu}=\frac{{\partial}R^{\mu}}{{\partial}x^{\kappa}}\frac{{\partial}R^{\nu}}{{\partial}x^{\gamma}}g^{\kappa\gamma}.$
(4.19)

This is just a family of 10 scalar functions indexed by $\mu$ and $\nu$. The
interpretation of $R^{\mu}$ in (4.19) is precisely that of the $g_{\sigma}$,
given in Equation (3.11) and used in (5.1), which is here valued in the group
of transformations of the theory. It is a projection onto a section
$\mathcal{F}$, with all the properties we have endowed it with; in the words
of (Komar, 1958, p. 1183):

> [Equation (4.19)] is component by component a well defined scalar
> constructed from the metric tensor and its derivatives. If we consider two
> metric tensor fields and ask whether they represent the same physical
> situation, differing perhaps by being viewed in different coordinate
> systems, we now have a ready criterion for determining the answer. Clearly,
> at corresponding points in any identification of the two spaces, the values
> of all scalars must agree if the spaces are to be equivalent. […] Thus we
> find that the functional form of the 10 scalars [i.e.
> $\overline{g}^{\mu\nu}$] as functions of the four scalars [i.e. $R^{\mu}$]:
> (a) is uniquely determined by the metric space independently of any choice
> of coordinate system, and furthermore (b) uniquely characterizes the space.

Much as we did in this paper, Komar sees an alternative rendition of (4.18) as
a gauge-fixing, or coordinate condition. Namely, instead of defining new
coordinates through (4.18), we write out $g_{ab}$ in coordinates and, unlike
(4.18), have those same coordinates appear on the left and right hand side of
the equation, namely:

$x^{\mu}=R^{(\mu)}(g_{\kappa\gamma}(x)),\quad\text{or
equivalently}\quad{\partial}_{\nu}R^{(\mu)}(g_{\kappa\gamma}(x))=\delta_{\nu}^{\mu},$
(4.20)

an equation that is to be solved _for the coordinates_ $x^{\mu}$, seen as
functions of the metric. And also in the same spirit as us, (Komar, 1958,
p.1186) highlights the difference between (what we would call) a projection
operator and the idea that we are somehow ‘breaking [gauge] covariance’:

> The usual argument, that employing coordinate conditions may destroy the
> general covariance […] does not apply in this case. For considering (4.20)
> as a coordinate condition [as opposed to the definition of 10 scalar
> functions of the metric] is only a heuristic device to make it easier to
> visualize how to manipulate the quantities with which we are dealing. In
> point of fact, we know how to interpret these quantities as true
> observables. (An apt analog in electromagnetic theory may clarify this point
> of view. The transverse components of the vector potential may be considered
> as the gauge-invariant true observables; or they may be considered as the
> components of the vector potential in a particular gauge, namely the
> radiation gauge. )

Here, Komar conveniently alludes to the difference between: restricting the
models to belong to a given gauge-fixing section $\mathcal{F}$—e.g. ‘the
radiation gauge’—and projection to that section—‘the gauge-invariant
observables’; a distinction we have also emphasized (see Section 3.3). In one
interpretation we have auxiliary local differential equations, and in the
other we have a non-local projection operator.

Here is a good place to illustrate some of the assertions of Section 3.3.
Suppose we have, as above, a sequence of four scalar quantities,
$\mathcal{R}_{g}^{(i)}$, with which we define a map:

$\mathcal{R}_{g}=(\mathcal{R}_{g}^{(1)},\cdots\mathcal{R}_{g}^{(4)}):M\rightarrow\mathds{R}^{4}.$
(4.21)

In each model along an orbit, each value of
$(\mathcal{R}_{g}^{(1)},\cdots\mathcal{R}_{g}^{(4)})$ will occur in a
different $x\in M$. So, to pick out points $x\in M$ by the value of the
quadruple, i.e. we invert the map (4.21).

Assuming that the map (4.21) is a diffeomorphism—in general it is not—there is
a unique value, for all of the models, of
$g_{ab}(\mathcal{R}_{g}^{-1})$.323232More commonly, for each $g_{ab}$, there
will be only a subset $U\subset M$ that is mapped diffeomorphically to
$\mathbb{R}^{4}$. That is, applying the chain rule for the transformations of
$\mathcal{R}_{g}$,

$\forall f\in Diff(M),\quad
g_{ab}\circ\mathcal{R}_{g}^{-1}=f^{*}g_{ab}\circ\mathcal{R}_{f^{*}g}^{-1}.$
(4.22)

So, using this quadruple, we have a unique representation of the metric on
$\mathds{R}^{4}$. This is the general idea of a representational convention.

We can also isolate the non-locality from the expression of the models. In
general, for a given representational convention $\sigma$, the dressing
$f^{\sigma}:Lor\rightarrow Diff(M);f^{\sigma}\mapsto f^{\sigma}_{g}$, is
clearly model-dependent. And indeed, it will in general depend non-locally on
$g_{ab}$, in the sense that $f^{\sigma}_{g}(x^{\prime})$ does not depend
solely on $g_{ab}(x^{\prime})$, but also on $g_{ab}(x)\,\,\forall x\in M$. And
while for any given $g_{ab}$, $f^{\sigma}_{g}$, will depend non-locally on
$g_{ab}$, we can ‘forget’ this dependence on the argument of $f^{\sigma}$: the
image under the projection is just another model of the theory.

###  4.4 Whence these choices?

One could still ask what conditions could possibly suggest a choice such as
(4.1), or any of the other choices displayed above. The answer, made explicit
in the description of the examples, is: different pragmatic, explanatory, and
theoretical requirements. Of course, the representational convention cannot be
empirically mandated, since, by assumption, the symmetries leave all empirical
matters invariant, in both the gravitational and the gauge cases. No, each
choice is at most suggested by being _suitable_ for certain types of questions
one might want to ask about the system. For instance, a uniquely specifying
functional role for $A$, such as Equation (4.1), should be one of mere
preference for particular representations of the field; e.g. so that we can
track its propagating features. The point of the multitude of choices is that
any one of them is ‘minimally good’. Or, at least, any one satisfying the
conditions of Section 3.3, and thus yielding a fully gauge-invariant and
complete projection operator. Different choices merely represent different
lenses through which we capture the invariant structure of the states.

We should not be disappointed by this lack of uniqueness. It should be seen
instead as a _flexibility_ that is also explanatory. Just as it is easy to
explain the Larmor effect by a Lorentz boost between different frames, it is
easy to explain that a given process in quantum electrodynamics involves just
two physical polarisation states and that it is Lorentz invariant. In both the
special relativistic and the gauge scenarios, two different ‘frames’—for
quantum field theory, the temporal and Lorenz gauge—are necessary to easily
explain two different aspects of a given phenomenon.

##  5 Representational conventions resolving Worry (2-ii)

I take Worry (2-i)—left unresolved in Gomes (20213)—to have been now
completely resolved in Section 3, and clarified in Section 4. Namely, the
issue described in Section 2: that sophistication as elaborated in Gomes
(20213)—which required a physically-interpreted axiomatic understanding of the
invariant type of structure of each theory—left open the matter of concisely
or perspicuoulsy describing _structure-tokens_. The issue, fully described in
Section 2.1, affects general relativity and Yang-Mills theory, since it has
its roots in holism or non-locality (as described in 2.2). Representational
conventions resolve this by supplying (non-unique) relational bases—or bases
of certain field-composites—with which to fully and concisely describe
individual structure tokens.

But we still have not explicitly addressed how representational conventions
also resolve Worry (2-ii). This will be the aim of this Section. We will
address the Worry through a construal of Lewisian countepart theory (to be
fully elaborated in Gomes Butterfield (2023)). This resolution also makes
precise the kind of relations required to make precise a frequent but vague
claim in the literature on sophistication, that ‘spacetime points can be only
idenfied by their pattern of relations to other points’.

To recap, the second unresolved issue is that (cf. Section 1.2):

  1. 2.

The strictly qualitative description of theories strips set-theoretical
elements of any primitive identity across worlds. But these elements and their
identity are useful in order to express counterfactuals and modal intuitions.
So how are counterfactuals expressed qualitatively?

The answer is simple, and already discussed in the literature: counterpart
theory. As described in footnote 3, counterpart theory is the philosophical
doctrine, due to David Lewis (D. Lewis (1968), (D. Lewis, 1973, p. 38-43) and
(DK. Lewis, 1986, Ch. 4)) that any two objects—in particular, spacetime
points—in two different possibilities (in philosophical jargon: possible
worlds) are never strictly identical. They are distinct, though of course
similar to each other in various, perhaps many, respects. And what makes true
a proposition that the object $a$ could have had property $F$ (though in fact
it lacks $F$) is—not that in another possible world, $a$ itself is $F$—but
that in another possible world, an object appropriately similar to $a$ is $F$.
That object is called the counterpart, at this other possible world, of $a$.
Denying trans-world identity fits sophistcation’s denial of any sort of
primitive identity, but counterparthood furnishes a relation that
sophistication lacks, allowing us to discuss counterfactuals.

Here, by providing a basis of physical relations with which to express the
qualitative ontology, we are also able to express qualitative counterpart
relations. Each representational convention will supply its own counterpart
relation. This idea will be developed in fullness in Gomes Butterfield (2023);
here I will give only a sketch. In Section 5.1 I provide the basic formalism
for conceiving of counterpart relations as given by representational
conventions; in Section 5.2 I describe obstructions to construing counterpart
relations in this manner—obstructions due to homogeneity of the states; and in
Section 5.3 I discuss how changing representational conventions affects the
counterpart relations.

###  5.1 Basic formalism

Mathematically, a counterpart relation can be seen as a group element
$g\in{\mathcal{G}}$ that relates two different states $\varphi$ and
$\varphi_{2}$, not necessarily isomorphic. For instance, given two models for
vacuum general relativity, $\langle M,g^{1}_{ab}\rangle$ and $\langle
M,g^{2}_{ab}\rangle$, a diffeomorphism $f\in{\mathrm{Diff}}(M)$ will give us a
counterpart relation between the spacetime points of $M$ that ‘inhabit’ the
two models. Similarly, a vertical automorphism between two models of Yang-
Mills theory: $\langle P,\omega_{1}\rangle$ and $\langle P,\omega_{2}\rangle$,
will give us a counterpart relation for $P$.

Given a section $\mathcal{F}$, as described in Section 3.3, we have just such
an element: $g_{\sigma}:{{\Phi}}\rightarrow{\mathcal{G}}$, given in Equation
(3.11). Using this convenient mathematical operator, the counterpart relation
between $\varphi$ and $\varphi_{2}$ is given by:

$\text{Counter}_{\sigma}(\varphi_{1},\varphi_{2}):=g_{\sigma}(\varphi)g_{\sigma}(\varphi_{2})^{-1}.$
(5.1)

Generally, i.e. for models that are not along the section, the relation is
given by the group element that takes the first model down to the gauge-fixing
section and then back up towards the second model. In particular, we see that
the counterpart relation between any two states that already lie at the gauge-
fixing section is just the identity.

In the sense that the projection to the gauge-fixing section is gauge-
invariant, each section gives a qualitative, e.g. diffeomorphism-invariant,
counterpart relation between the dressed spacetime points of distinct physical
possibilities.

Counterpart relations obey an important consistency check. Namely, due to
covariance, i.e. Equation (3.9),

${\rm
Counter}_{\sigma}(\varphi^{g}_{1},\varphi^{g}_{2})=g^{-1}g_{\sigma}(\varphi_{1})(g^{-1}g_{\sigma}(\varphi_{2})^{-1}=g^{-1}\,{\rm
Counter}_{\sigma}(\varphi_{1},\varphi_{2})g\;.$ (5.2)

Moreover, two models that lie in the same orbit will always be related by the
unique isomorphism that connects them. That is: if
$\varphi_{2}=\varphi^{g}_{1}$, then ${\rm
Counter}_{\sigma}(\varphi_{1},\varphi_{2})=g$, even if neither $\varphi_{1}$
nor $\varphi_{2}$ lie in the section $\sigma$. For with the definitions above,

${\rm
Counter}_{\sigma}(\varphi_{1},\varphi^{g}_{1})=g_{\sigma}(\varphi_{1})(g^{-1}g_{\sigma}(\varphi_{1}))^{-1}=g$
(5.3)

Thus, for example, in the spacetime case, if the unique counterpart of $p$ in
model $\langle M,g_{ab}\rangle$ is $q$ in model $\langle
M,g^{\prime}_{ab}\rangle$, then the counterpart of $p$ in model $\langle
M,f^{*}g^{\prime}_{ab}\rangle$ will be $f(q)$. And this particular property is
independent of which convention $\sigma$ we choose.

In other words, though there are several distinct choices of counterparts,
each choice must identify the same spacetime points across isomorphic models.
In the spacetime context, this is essentially Field (1984, p. 77)’s
observation that

> “individuation of objects across possible worlds” is sufficiently tied to
> their qualitative characteristics so that if there is a unique 1-1
> correspondence between the space-time of world A and the space-time of world
> B that preserves all geometric properties and relations (including geometric
> relations among the regions, and occupancy properties like being occupied by
> a round red object), then it makes no sense to suppose that identification
> of space-time regions across these worlds goes via anything other than this
> isomorphism.

Indeed, this criterion, that we can here call ‘the drag-along’ interpretation
of isomorphic models (cf. Butterfield Gomes (2022)), was the centerpiece of
Weatherall (2018): the paper that sparked a recent surge of interest in the
hole argument in general relativity. Weatherall (2018); Bradley Weatherall
(2022) associate the ‘the drag-along’ idea less with Field and more with Mundy
(1992), who distinguishes a theory’s synthetic language—that is only able to
expresses qualitative, non-singular facts—from a theory’s metalanguage, in
which we are able to talk about points singularly, i.e. without a definite
description (see (Gomes, 20213, Sec. 3.2a)). But the gist is the same:
isomorphisms— expressed in the metalanguage in Mundy’s case—will map objects
singled out by the same description into each other. Here is Mundy (1992, p.
520):

> Thus an element $p$ of $M$ satisfies a description $D$ of $L_{R}$ [i.e. a
> qualitative description] in $S$ iff the corresponding element $f(p)$
> satisfies $D$ in $Sf$: $p$ and $fp$ occupy the same structural roles in
> these two isomorphic models, so every statement true of $p$ in $S$ is true
> of $f(p)$ in $Sf$. Therefore, since a theory identifies elements of its
> domain only by descriptions expressed in its language, two such models
> describe the same theoretical world. Only in our semantic metalanguage can
> we state a difference between them, but this difference does not reflect any
> fact expressible in the theoretical language.

And here we see that this criterion is satisfied by our notion of qualitative
counterpart relations using representational conventions. But we do not always
have universal, unique, counterpart relations, and so Field’s criterion does
not always apply, as we will see in Section 5.3.

###  5.2 A snag

So far in our discussion, we made a ‘simplifying’ assumption: that the web of
properties and relations in each of our models is sufficiently
complex.333333In the Komar example of Section 4.3.d, we stipulated explicitly
that the representational convention would succeed only in generic spacetimes.
But unless a similar assumption is made for the other examples, the
consequences discussed in this present Section would also follow. The reason
is that, on homogeneous states, $\tilde{\varphi}$, a representational
convention—seen as a map $\sigma:[\Phi]\rightarrow\Phi$—will fail to uniquely
specify the $g_{\sigma}$ of (3.11); $g_{\sigma}$ fails to satisfy the property
of _Uniqueness_ of Section 3.3, as explained in footnote 22. There are
particular $\tilde{g}\in{\mathcal{G}}$ such that
$\tilde{\varphi}^{\tilde{g}}=\tilde{\varphi}$, and so, in particular, we
cannot fulfill the covariance Equation (3.9).

To visualize this consequence of homogeneity it is convenient to introduce a
simple example using non-relativistic particles. As briefly described in
Section 4.3, we could not only fix the boosts by using a center of mass
convention, but indeed we could fix an entire frame in $\mathds{R}^{3}$ by
diagonalizing the moment of inertia tensor around the center of mass; in this
case $g_{\sigma}$ would be a state-dependent rotation and translation. And if
two distinct configurations are already described in the given convention,
then their two frames are counterparts of each other with respect to this
convention. They are already ‘best-matched’ according to this convention. To
be blunt: East in one state is identified with East in the other. Or a top
quark at $x$ in one world is identified with a top quark at $x$ at another.
And, as long as the configurations are sufficiently complex, all
representational conventions will agree about which objects are identified
across isomorphic models (in accord with Field’s criterion, above).

But it is not a given that objects (e.g. spacetime points) _can be_ singled
out, i.e. uniquely specified or individuated, by a web of properties and
relations: if this web is not sufficiently complex, the specification will
fail. For instance, as I mentioned in footnote 28, configurations that have
some extra degree of symmetry, e.g. are spherically symmetric, etc, will not
be able to fix the representation: as per footnotes 22 and 21, these are
_reducible_ configurations; they have stabilizers that cannot be fixed by any
feature of the state. To be blunt: we cannot stipulate a convention for East
if there aren’t enough surrounding features to single it out.

In other words, in the case of homogeneous patterns, i.e. patterns that are
not complex enough for individuation of objects, a counterpart relation
between models will not be unique, even within a single representational
convention. Using Field’s example, if $\tilde{g}_{ab}$ of world A is such that
there exists an $\tilde{f}\in{\mathrm{Diff}}(M)$ such that
$\tilde{f}^{*}\tilde{g}_{ab}=\tilde{g}_{ab}$, and world B’s geometry is given
by $g_{ab}^{\prime}=f^{*}\tilde{g}_{ab}$, for some $f\in{\mathrm{Diff}}(M)$,
then there are more than one isomorphisms between the spacetime of world A and
the spacetime of world B (namely, $f$ composed with any power of $\tilde{f}$).

This type of obstruction to individuation due to homogeneity is well known,
and related to Black (1952)’s criticism of Leibniz’s Principle of the Identity
of Indiscernibles—the principle that motivates Leibniz equivalence (see
(Pooley, 2022, Sec. 3.3) for a thorough exposition). The problem now is that,
if, with Weatherall (2018), one is tempted to identify elements of one model
with those of the other by the isomorphism relation, the transitivity of
identity would lead to the disastrous conclusion that homogeneous
spacetimes—whose isometry group can take any spacetime point to any
other—‘truly’ contain only one point (cf. Wüthrich (2009))!

Even though representational conventions are still well-defined and valid when
used for very homogeneous states, $g_{\sigma}$ is not, and so the counterpart
relation (5.1) is no longer valid and will not recover Field’s criterion
above. The underlying reason is that ‘”individuation of objects across
possible worlds” is _not_ sufficiently tied to their qualitative
characteristics’, and so the correspondence between the spacetimes is not 1-1.

In this way the interpretation of isomorphic models through sophistication and
Leibniz equivalence, and the use of representational conventions, encompass
but are not limited to ‘the drag-along’ response to the hole argument, and to
the problem of individuating physical possibility more generally. Homogeneous
models fail to satisfy Field’s criterion, by failing to satisfy its
antecedent, but Leibniz equivalence is holistic and survives unscathed. For it
merely states that physical possibility is entirely determined by qualitative
properties and relations _of each entire world or model_ , and even a
homogeneous world can be qualitatively described perfectly well (cf. also the
related Muller (2011)’s counterargument to Wüthrich (2009)).343434For
instance, maximally homogeneous spacetimes are singled out by having the
maximal number of Killing vector fields: a completely qualitative description.
In sum, one can consistently endorse qualitative ontology without also
endorsing the ‘drag-along’ response.353535Bradley Weatherall (2022) tie the
drag-along response to Mundy (1992)’s constructive axiomatization for
Lorentzian geometry; but this tie can be broken: Mundy (1992) implies only
that worlds are qualitatively individuated.

###  5.3 Changing counterparts

We can reconceive of passive transformations as changes of representational
convention; these transformations can also be construed as ‘actively’ shifting
points around in an underlying geometry (see (Gomes, 20214, Sec. 5)).

To illustrate this with an example, let us go back to the Komar-Kretschmann
scalars, discussed in Section 4.3.d. We could easily choose different set of
scalars $R^{(\mu)}$, provided we preserved their functional independence. For
example, we could have multiplied $R^{(1)}$ by two and taken the third power
of $R^{(2)}$. Or we could take different linear combinations of $R^{(\mu)}$;
and so on. Ultimately, we could take any diffeomorphisms of $\mathds{R}^{4}$
to yield a different convention.363636And to fix a given choice, we would use
a coordinate-based function such as De Donder gauge.

There are here two important things to note. First, that going from one choice
to another will determine an (active) diffeomorphism of the base set that is
not understood in the ‘drag-along’ manner of Section 5 (see also Butterfield
Gomes (2022)). For instance, the point (1,2,1,0) in one Komar basis has
different physical properties than the point labeled by those same coordinates
in another basis.

The second point to note is that each spacetime point may have _many_
distinct, uniquely individuating properties. Just like the tallest man in
Sweden can also be individuated by his passport number, as well as by his
height and location. Likewise, ‘relationism’ is a broad church, admitting many
physical bases for describing the world.

Thus, when Einstein famously resolved his questions about the ‘hole argument’
by appealing to ‘point coincidences’ (cf. Giovanelli (2021)), he assumed there
was no further question as to what these points were coincidences of. For, as
we have seen, different choices of matter fields or composites can yield very
different physical descriptions of given state of affairs: very different
coincidences, which equally well coordinitize a given region of spacetime.
Thus the availability of individuating features for spacetime points allows us
to represent active diffeomorphisms on spacetime that truly shift the points
around, and do not just drag them along.

##  6 Summing up

In this Section, I conclude, by first, in Section 6.1, listing a couple of
idealizations inherent to representational conventions, which I have so far
skatted over; and in Section 6.2, I offer a very brief summary.

###  6.1 Further objections.

The above comparison to Einstein’s thought brings to the fore a limitation of
representational conventions as actual charts, which we have not yet touched
on. Take Komar variables. Even if their domain is a generic spacetime, that
spacetime will (generically) not be globally isomorphic to $\mathds{R}^{4}$.
As we know, there are topological limitations to such bijections, and there
might be other obstructions as well. For instance, a spacetime may simply not
have any point at which all of the Komar invariants vanish.

If we remember Section 2, representational conventions were introduced ro
respond to a challenge of individuating structure-tokens, a very common goal
for theorists working in quantum gravity, and in the (non-perturbative)
quantization of Yang-Mills theory in general. And indeed, this limitation of
representational conventions is a major obstacle for the quantization of these
theories. For instance, in quantum gravity, the infamous “problem of time” is
often portrayed as a problem of finding suitably general ‘internal clocks’
(cf. e.g. (Isham, 1992, Sec. 3.4)). It is hard to find a physical definition
of surfaces of simultaneity that would apply to every spacetime. In our
language: the given representational convention does not cover the entirety of
${{\Phi}}$.373737Similarly, in general more than one representational
convention is required to cover the entire spacetime manifold. We reiterate
that the introduction of transition functions is generally necessary because
global sections do not exist, also in the gauge case, unless the bundle is
trivial, i.e. unless $P=M\times G$ globally not just locally. In the trivial
case, and only in the trivial case, all transition functions can be
trivialized to be the identity, i.e.
$\mathfrak{t}_{\beta\alpha}=g_{\beta}g_{\alpha}^{-1}$ for some choices of
$g_{\alpha}$’s. This fact has important consequences for the treatment of
subsystems for theories with local symmetries, since it obliges us to consider
field-dependent transition functions. This is the reason representational
conventions were originally introduced for subsystems in gauge theory and
general relativity (cf. last paragraph of Section 3.1, for a brief history).

This failure is not only due to Gribov ambiguities; sometimes, our physical
anchors find nothing to get hooked to. This is why Desideratum (iii)—that the
theory’s symmetries admit a representational convention—is non-trivial. In
fact, it is not perfectly satisfied even for the well-known examples of non-
Abelian Yang-Mills and general relativity! But it is satisfied in the Abelian
case, and its general features should be present even with the caveats above.

For instance, in the original treatment Gomes Riello (2017, 2018); Gomes .
(2019); Gomes Riello (2021), one avoids the problem of Gribov ambiguities, for
instance, by introducing a more flexible tool than representational
conventions. This more general treatment introduces a distribution on
${{\Phi}}$ that is a transversal complement to the infinitesimal gauge
transformations at each state, but which is not necessarily integrable into
‘slices’ (see footnote 23). It is therefore a much weaker notion than a
representational convention, but which nonetheless serves many of the same
purposes (and which therefore might provide a fruitful mathematical tool in
approaches to quantization of theories with local non-Abelian symmetries).

###  6.2 Summary.

To summarize, I here introduced representational conventions as an answer to
an objection to sophistication, about the transparency of structure-tokens. I
also responded to new concerns—namely Healey’s objection: what warrants a
choice of representational convention, if all physical features are deemed, by