Symmetry, empirical equivalence, and identity

Simon Friederich

email@simonfriederich.eu

Universität Göttingen, Philosophisches Seminar, Humboldtallee 19, D-37073
Göttingen, Germany

Abstract:
The paper proposes a novel approach to the much discussed question of which
symmetries have direct empirical significance and which do not. The approach is
based on a development of a recently proposed framework by Hilary Greaves and
David Wallace, who claim that, contrary to the standard folklore among philoso-
phers of physics, local symmetries may have direct empirical significance no less
than global ones. Partly vindicating the standard folklore, a result is derived here
from a number of quite plausible assumptions, which states that local symmetries
can indeed have no direct empirical significance. Ways to interpret the result are
considered and possible morals are outlined.

Contents

1 Introduction 2

2 Greaves and Wallace on interior vs. non-interior symmetries 5

3 Elaborating on the Greaves/Wallace framework 9

4 The result 13

5 Problems with ’t Hooft’s beam splitter? 15

6 Summary and conclusion 21

1



1 Introduction

The importance of symmetries in physics has been a recurring topic in the philos-
ophy of science in recent years.1 According to a (first and imprecise) character-
isation, symmetries are (or induce) mappings of a theory’s state space onto itself
which connect states that are in some sense “physically equivalent”. Philosophical
debates about symmetries often start from recognising that “physical equivalence”
can have (at least) two different meanings in this context. Distinguishing between
these two meanings by deciding which one applies in which case is perhaps the
main challenge for the philosophical analysis of symmetries.

According to the first meaning of “physical equivalence”, symmetries are de-
scriptive redundancies in that any two states related by a symmetry transformation
represent one and the same physical states of affairs in mathematically distinct
ways. According to the second, symmetries operate between physically distinct
states of affairs, but in such a way that there is no empirically detectable differ-
ence between states connected by symmetries for observers who can only make
observations inside the region where the symmetry transformations operate.2 How
to classify symmetries in actual physical theories in terms of this distinction is a
nontrivial task with respect to which there are controversial views. The debate is
often formulated in terms of the question of which symmetries have “direct empir-
ical significance” and which do not. Roughly speaking, those symmetries which
connect physically identical states of affairs are (or correspond to) those which do
not have any direct empirical significance, whereas those which operate between
physically distinct states of affairs are (or correspond to) those which have some.

The distinction between symmetries which have direct empirical significance
and those which do not is often linked to that between global and local symme-
tries. The received view—inasmuch as there is one in this debate—is that only
global, but not local, symmetries can have direct empirical significance. Global
symmetries, roughly speaking, are those which act in a “globally” uniform way on
the whole of space-time (typically parametrised by a single real or discrete param-
eter), whereas local symmetries are defined in terms of real-valued functions χ(x)
from the infinitely many (“local”) points x of space-time to the reals. Examples of
global symmetries include the Galileo transformations in Newtonian mechanics,

1See (Brading and Castellani [2003]) for a useful anthology.
2For the purposes of the present paper I will simply assume that the question of which state space

automorphisms relate states between which there is no empirically detectable difference has been
settled in advance. In practice, the task of determining the symmetries of a theory in the sense of
singling out these automorphisms can be highly nontrivial. See (Belot [2011]) for a recent study of
“symmetry” and “physical equivalence”, which argues that we do not have any formal (i.e. purely
mathematical) criterion for symmetries that goes well with the idea that symmetries are those state
space automorphisms which operate between physically equivalent states.

2



the Poincaré transformations in special relativity (with the Lorentz transformations
as a subgroup), and the constant shifts of the electrostatic potential in Coulomb
electrostatics. Examples of symmetries which are counted as local symmetries in-
clude the gauge symmetries of the four-potential Aµ(x) in electrodynamics and its
relatives in non-abelian gauge theories (such as those underlying our currently most
successful theories of elementary particle physics) as well as the automorphisms
of the metric- and mass/energy-configurations induced by coordinate diffeomor-
phisms in general relativity (referred to simply as “diffeomorphisms” in what fol-
lows).

Many philosophers of physics defend what may be called the standard view,
namely, that certain global symmetries (though not necessarily all of them) have
direct empirical significance, whereas local symmetries generally don’t.3 Hilary
Greaves and David Wallace, however, have recently mounted a forceful challenge
against this standard view in terms of a formal framework that is meant to allow to
distinguish systematically between those symmetries which have direct empirical
significance and those which don’t, see (Greaves and Wallace [forthcoming]).

Greaves and Wallace mention three closely related aspects of symmetries and
their role in theories which they regard as indicating problems for the standard
view: first, any theory which has local symmetries must have global ones (i.e.
globally constant ones) as a subgroup. If the local symmetries are parametrised in
terms of space-time-dependent functions χ(x), the global symmetries correspond
to the spatio-temporally constant functions χ(x) = α with α constant. Greaves and
Wallace note that ‘it would be highly mysterious if global symmetries managed to
have empirical significance while no other symmetries were around, but somehow
lost this capacity once the full local group of transformations appeared as symme-
tries.’ (Greaves and Wallace [forthcoming], p. 2) This observation seems to point
to an inadequacy of the view that only global, but not local, symmetries can have
direct empirical significance.

The second observation which, according to them, suggests a related problem
for the received view is that theories which have a global symmetry are, histori-
cally, often replaced by theories where that symmetry is ‘localised’ (Greaves and
Wallace [forthcoming], p. 2) in that the parameter labelling global symmetry trans-
formations in the earlier theory is replaced by a space-time-dependent function in
the later theory. For example, the group of constant potential shifts in Coulomb
electrostatics is replaced by the group of local gauge transformations of the four
potential Aµ in the full theory of electromagnetism. Greaves and Wallace argue

3See, for instance, (Kosso [2000]; Redhead [2002]; Brading and Brown [2004]; Lyre [2004];
Healey [2007], Healey [2009]), for such views, and (Maudlin [1998]) for a criticism, to which
(Healey [1998]) replies.

3



that this would seem particularly troubling from the perspective of scientific real-
ism: ‘if the explanatory successes of old theories aren’t reproduced in their succes-
sors, much of the justification for realism is undermined.’ (Greaves and Wallace
[forthcoming], p. 3)

The third observation made by Greaves and Wallace concerns the explanatory
role of symmetries with respect to empirically indistinguishable situations. For
example, the phenomenon that—as famously pointed out by Galilei– it is empiri-
cally impossible to distinguish from the inside of a ship’s cabin whether the ship
is at rest or at uniform motion with respect to the sea is explained in Newtonian
physics in terms of global space-time symmetries (so-called “boosts”). These op-
erate, in an “active” interpretation, between systems that are at uniform relative
motion with respect to each other. The main claim brought forward by Greaves
and Wallace with respect to scenarios where a symmetry helps explain an empir-
ical equivalence of such sorts is that one can ‘sketch several examples that seem,
prima facie at least, to be perfect analogs of the Galileo-ship scenario for cases of
local symmetry.’ (Greaves and Wallace [forthcoming], p. 3) Other such scenarios
include Faraday’s cage and a version of the famous two-slit experiment they refer
to as “’t Hooft’s beam splitter”. This latter setup is discussed in detail in Section
5 of this paper. According to Greaves and Wallace, these analogies are accommo-
dated most naturally if one take the local symmetries used in these explanations to
have direct empirical significance no less than the global ones.

Inspired by this challenge to the standard philosophers’ view as regards sym-
metries, Greaves and Wallace develop a framework which is meant to permit a
more systematic distinction between symmetries which have direct empirical sig-
nificance and symmetries which don’t. According to them, the main lesson from
applying this framework to a wide range of examples is that ‘nothing like the
global/local distinction tells us which symmetries can in general have empirical
significance—and in particular, it is false that local symmetries are in general un-
observable.’ (Greaves and Wallace [forthcoming], p. 3)

The aim of the present paper is to show that, on a natural development of
the Greaves/Wallace framework, a version of the standard view can be vindicated,
which says that only global symmetries can have direct empirical significance. In
conformity with the strategy proposed by Greaves and Wallace, the present devel-
opment of their framework focuses on the symmetries of subsystems of the universe
rather than on symmetries of the universe itself. Unlike their analysis, however, the
present investigation applies the relation of representing the same physical state
of affairs also to the subsystem states themselves, not only to the universe states.
Using a small number of arguably rather plausible assumptions a result is derived
which says that local symmetries—unlike global ones—are indeed without any di-
rect empirical significance, more or less in agreement with the standard folklore

4



and contrary to the claims made by Greaves and Wallace.
The structure of the remaining sections of this paper is as follows: Section 2

introduces the framework proposed by Greaves and Wallace and reviews their core
claims. Section 3 elaborates on this framework by specifying the suggested as-
sumptions concerning subsystem symmetries and the relation of representing the
same physical state of affairs among subsystem states. Section 4 formulates the
result that, given the assumptions specified in Section 3, local symmetries cannot
have any direct empirical significance. Section 5 discusses an apparent contradic-
tion between this result and a claim made by Greaves and Wallace in connection
with the setup referred to as “’t Hooft’s beam splitter”. The paper closes in Section
6 with a brief summary and conclusion.

2 Greaves and Wallace on interior vs. non-interior sym-
metries

According to the leading interpretations of our most successful theories of space-
time—the more sophisticated versions of relationalism and substantivalism—any
two states of the universe as a whole which are connected by a space-time sym-
metry represent one and the same physical situation.4 For example, two universe
states u1 and u2 in special relativity which are linked by a Lorentz transformation
(for example, a rotation or a boost) are regarded as representing one and the same
global physical state of affairs, though described in different mathematical terms. If
one accepts this perspective yet wants to maintain a nontrivial distinction between
symmetries with and without direct empirical significance, one will therefore ex-
pect this distinction to hold between symmetries operating between the states of
subsystems of the universe itself.5

The idea to focus on subsystem symmetries to elucidate matters of direct em-
pirical significance is suggested already in (Brown and Sypel [1995]), taken up in
(Kosso [2000]; Brading and Brown [2004]; Greaves and Wallace [forthcoming]),
and adopted in the present paper as well. It goes well with the fact that in ex-
amples such as Galileo’s ship the symmetries in question are in fact subsystem
symmetries: in the case of Galileo’s ship it is only the cabin of the ship, not the

4See (Baker [2011]) for detailed considerations in support of this claim.
5See (Guilini [1995]) for a defence of the view that symmetry transformations which act differ-

ently from the identity transformation at space-time-like infinity in a gauge theory are “physical”
rather than “gauge”, which might be taken to contradict the view mentioned here with respect to
space-time symmetries. According to Healey, however, inasmuch as Giulini is right, his arguments
do not establish that these transformations ‘are empirical’ (Healey [2007] p. 182) and do therefore
not demonstrate that these symmetries have any direct empirical significance.

5



surrounding ocean, which is represented by a symmetry transformed state in the
situation where the ship is moving as compared to where it is at rest.6

The framework suggested by Greaves and Wallace is meant to have a wide
range of applications, both in quantum and classical contexts and for various dif-
ferent types of subsystems.7 However, for the purposes of the present investigation
it suffices to consider only classical theories and to assume that the subsystems un-
der consideration are finite, compact space-time regions. The examples discussed
by Greaves and Wallace can all be treated in that setting, so the question of whether
or not the present considerations remain valid in the absence of this restriction need
not bother us here.

When one analyses questions concerning the direct empirical significance of
symmetries, one would like to be able to ask whether two mathematically distinct
states represent one and the same physical state of affairs. It is therefore crucial to
employ a mathematical (rather than physical) notion of state. The framework sug-
gested by Greaves and Wallace is based on distinguishing between subsystem and
environment (mathematical) states, where state spaces S and E are postulated for
the subsystem and the environment, respectively. Elements u ∈ U of the universe
state space U are assumed to be uniquely decomposable in terms of subsystem
states s ∈ S and environment states e ∈ E. The operation of combining a subsys-
tem with an environment state is denoted by “∗” (that is, u = s ∗ e is the universe
state which arises from combining s and e).

Arbitrary pairs of subsystem and environment states s and e need not in general
give rise to a well-defined universe state u = s ∗ e. For example, if s and e denote
field configurations of a finite space-time region and its environment in a classical
field theory, they do not in general coincide on the subsystem boundary, and in case
they do, their derivatives may not coincide. In these cases, their composition need
not be well-defined (depending on whether higher derivatives are required to exist
etc.). However, in those cases where the composition u = s ∗ e of states s ∈ S and
e ∈E is well-defined, it is assumed to be unique.

Greaves and Wallace introduce subsystem and environment symmetries σS and
σE as restrictions of universe symmetries σ to the states spaces S and E. They
require that ‘for all s ∈ S, e ∈ E, σ(s ∗ e) = σ|S (s) ∗ σ|E (e) for some maps σ|S
and σ|E [such that t]he symmetries ΣS of S and ΣE of E are just the sets of all
such σ|S and σ|E respectively.’ (Greaves and Wallace [forthcoming], p. 9) Using
this formal machinery, the essential idea of their analysis is to consider arbitrary
universe states u = s ∗ e, apply some subsystem symmetry σS to the subsystem

6See (Healey [2007], Chapter 6) and (Healey [2009]) for a framework which proposes a related
classification, but without the focus on subsystem states.

7Greaves and Wallace [forthcoming], pp. 8 f.

6



component s in u, and see whether the resulting universe state u′ = σ(s) ∗ e—if
it exists! (see below for the case where it doesn’t)—represents the same physical
state of affairs as the original u. Their (unsurprising) suggestion is that σS does
not have any direct empirical significance if, for all s ∈ S and e ∈ E for which
u = s∗e and u′ = σ(s)∗e are defined, u and u′ represent the same physical state of
affairs (in which case I shall write u ∼ u′); and that σS does have direct empirical
significance if for some u = s ∗ e and u′ = σ(s) ∗ e, these two universe states are
physically distinct, i.e. u / u′.

This simple idea is already sufficient to distinguish between the two following
types of subsystem symmetries σS ∈ ΣS in terms of whether they have any direct
empirical significance:

First, there are those symmetries σS where for all s ∈ S and e ∈ E such that
both s∗e and σS (s)∗e are defined the universe states s∗e and σS (s)∗e ‘represent
the same possible world as one another.’ (Greaves and Wallace [forthcoming], p.
11) In other words, these are the symmetries for which s∗e ∼ σS (s)∗e for all s ∈S
and e ∈ E where both s ∗ e and σS (s) ∗ e are defined. In line with the considera-
tions just presented, Greaves and Wallace argue that these symmetries do not have
any direct empirical significance. Since these symmetries differ from the identity
transformation only in the interior of the subsystem S (and reduce to the identity
transformation on its boundary), they refer to them as “interior symmetries”:

‘Since performing an interior symmetry transformation on the subsys-
tem state and leaving the environment state alone results in a redescrip-
tion of the same possible world, such a subsystem transformation does
not lead to a distinct situation, hence no (nontrivial) empirical sym-
metry is associated with such transformations.’ (Greaves and Wallace
[forthcoming], pp. 11 f.)

Greaves and Wallace argue that local gauge transformations in gauge theories and
space-time diffeomorphisms in general relativity which differ from the identity
transformation only in the interior of the subsystem belong to these class of sym-
metries and hence have no direct empirical significance.

The second type of symmetries σS in the classification proposed by Greaves
and Wallace are those where, for some s ∈ S and e ∈ E where s ∗ e and σS (s) ∗ e
are defined, s∗e / σS (s)∗e. For these symmetries, in other words, universe states
s ∗ e and σS (s) ∗ e represent physically distinct states of affairs and the symmetry
σS is therefore taken to have direct empirical significance. This is in accordance
with the “transformation condition” for symmetries having direct empirical signif-
icance suggested by Brading and Brown, which says that ‘the transformation of a
subsystem of the universe with respect to a reference system must yield an empir-
ically distinguishable scenario’ (Brading and Brown [2004] p. 646) for the whole

7



universe.
Greaves and Wallace argue that the symmetry which has direct empirical sig-

nificance ‘in this case will be purely relational, in the sense that the intrinsic prop-
erties of both subsystem and environment separately are entirely unaffected, and it
is only the relations between the two [...] that are changed by the transformation.’
(Greaves and Wallace [forthcoming], p. 13, the emphasis is due to Greaves and
Wallace.) Since there are subsystem and environment states in Newtonian mechan-
ics and special relativity where the subsystem is effectively isolated from its envi-
ronment, the space-time transformations such as translations, rotations, and boosts
as restricted to proper subsystems of the universe belong to this type. Greaves and
Wallace do not stop here, however, but make the further claim that even certain
local symmetry transformations in gauge theories and diffeomorphisms in gen-
eral relativity as restricted to subsystems belong to this class and thus have direct
empirical significance. This claim will be critically examined in Section 5 when
discussing the setup referred to as “’t Hooft’s beam splitter”.

Symmetries of a third type pose at least prima facie problems for the approach
suggested by Greaves and Wallace: namely, those subsystem symmetries where,
if σS differs from the identity transformation, for all pairs of s ∈ S and e ∈ E,
either s ∗ e or σS (s) ∗ e is undefined. This is indeed the typical case in theories
with local symmetries, but it holds also in Coulomb electrostatics.8 Unfortunately,
most symmetries with respect to which it is controversial whether they have direct
empirical significance belong to this class. Even though their framework gives no
clear-cut verdict in these cases, Greaves and Wallace argue that σS has direct em-
pirical significance here nevertheless, since there typically is some suitably chosen
environment state e′ , e for which s ∗ e / σS (s) ∗ e′. Their main motivation for
this move seem to be their worries concerning the standard view that only global
symmetries can have direct empirical significance, which were outlined in Section
1. So, even though their stipulation that σS has direct empirical significance in
these cases is well-motivated, it is by no means mandatory. In particular, if one
acknowledges that a physical difference exists between s ∗ e and σS (s) ∗ e′, one
may always regard it as arising from a physical contrast between e and e′, in which
case there would be no compelling reason for regarding σS as having any direct
empirical significance. This move could only be blocked if Greaves and Wallace
came up with a well-motivated recipe for obtaining e′ from e such that the physical
difference between s ∗ e and σS (s) ∗ e′ could not longer be blamed on the choice
of e′.

The problem of deciding whether σS has any direct empirical significance in
these cases arises of course directly from the fact that, for any combination of s ∈S

8See (Greaves and Wallace [forthcoming], pp. 18-20 and p. 14).

8



and e ∈E, either s ∗ e or σS (s) ∗ e is undefined. In the following section, I suggest
an approach which allows us to overcome this problem. Its core idea is to apply the
relation of representing the same physical state of affairs not only to universe states
but also to subsystem states. The next section formulates a few simple and arguably
very plausible assumptions as regards which subsystem states represent the same
physical state of affairs. Based on these assumptions, a result can be derived—
and need not be stipulated—which allows us to answer the question whether the
symmetries for which s ∗ e and σS (s) ∗ e are not both defined have any direct
empirical significance for many cases of interest. As it turns out, the answer is
(typically) negative.

3 Elaborating on the Greaves/Wallace framework

Greaves and Wallace presuppose that a sharp distinction exists between pairs of
universe states which represent physically distinct states of affairs and pairs of
universe states which represent identical physical states of affairs. They do not
apply this distinction to pairs of subsystem states and would perhaps be unwilling
to do so. However, at least prima facie there seems to be no reason for withholding
this distinction from subsystem states and, therefore, to accept that well-defined
subsystem physical states of affairs exist no less than well-defined universe states
of affairs.

It may seem, however, as if well-defined identities for subsystem physical
states of affairs could not help us much in our investigation of the direct empir-
ical significance of symmetries, simply because, just as for universe states, any
two subsystem states which are related by a subsystem symmetry would have to
be regarded as representing exactly the same subsystem physical state of affairs.
This would make it trivial that all subsystem symmetries do not have any direct
empirical significance—surely an unwanted result.

All the intrinsic physical properties of a subsystem are, by definition, the same
in two symmetry-related subsystem states. However, unlike for universe states, for
subsystem states one may consider the relations to systems external to the subsys-
tem. And indeed, from the perspective of these external systems there may well be
a fact of the matter as to which of two symmetry-related states correctly describes
the subsystem. (Consider, for example, an observer at the shore, who is looking at
Galileo’s ship. For her, whether or not the ship is at rest or in motion surely makes
a physical difference.) Furthermore, this physical difference, if it exists, may well
be regarded as grounded in a physical difference in (what one may call) the ex-
trinsic physical properties of the subsystem at issue.9 If one takes these extrinsic

9The notion of an extrinsic property is of course an intricate one. Here its only purpose is to ges-

9



properties into account as well, the relation of representing the same physical state
of affairs may well become non-trivial with respect to subsystem states. In fact,
as I shall argue, it can then be used in a natural way to define what it means for a
subsystem symmetry to have direct empirical significance.

In what follows, I shall use the symbol “∼” to denote the relation of repre-
senting the same physical state of affairs for universe as well as for subsystem
states. We can now define what it means for a subsystem symmetry to have direct
empirical significance, simply by saying that the symmetry has direct empirical
significance if and only if it connects at least one pair of subsystem states between
which the relation ∼ does not hold. To sum up (where “DES” stands for “direct
empirical significance”):

Assumption (DES)
A subsystem symmetry σS has direct empirical significance iff σS (s) / s for some
s ∈S.

In other words, according to (DES), σS does not have any direct empirical signifi-
cance iff σS (s) ∼ s for all s ∈S.

How do facts as regards which mathematically distinct states represent identi-
cal physical states relate to each other for pairs of universe states on the one hand
and pairs of subsystem states on the other? A natural answer to this question,
which essentially reproduces the claims made by Greaves and Wallace as regards
the first two types of symmetries discussed in the previous section, is as follows:
As explained, Greaves and Wallace claim that σS does not have any direct empiri-
cal significance if s∗e ∼ σS (s)∗e for all s and e where both s∗e and σS (s)∗e are
defined, and that σS does have direct empirical significance if s ∗ e / σS (s) ∗ e for
some s and e where both s∗e and σS (s)∗e are defined. Using (DES), we can trans-
late this into the statement that σS does not have any direct empirical significance
just in case s∗e ∼ σS (s)∗e for all s ∈S and e ∈E where both s∗e and σS (s)∗e are
defined. Since this should hold for arbitrary subsystem symmetries σS , it seems
natural to generalise it as follows (where “SUL” stands for “subsystem-universe
link”):

Assumption (SUL)
For all s, s′ ∈S:
s ∼ s′, iff s ∗ e ∼ s′ ∗ e for all e ∈E for which s ∗ e and s′ ∗ e are defined.

Assumption (SUL) allows us to connect facts about the ∼-relation for universe
states with facts about it for subsystem states. It is natural to ask whether we can say

ture at the possibility of defining a more ambitious relation of physical equivalence among subsys-
tem states in addition to the trivial one according of which symmetry-related states are by definition
equivalent.

10



anything as to how facts about the ∼-relation are related for different subsystems. I
shall argue that there is a further plausible assumption we can make, which allows
us to answer this question:

Consider a subsystem S and decompose it into two non-overlapping sub-sub-
systems S 1 and S 2. Now consider the state spaces S1 and S2 of these sub-sub-
systems. The question I propose to ask is whether for all pairs of sub-subsystem
states s1, s′1 ∈ S1 and s2, s

′
2 ∈ S2 such that s1 / s

′
1 (and either s2 ∼ s

′
2 or s2 / s

′
2)

the inequivalence s1 ∗ s2 / s′1 ∗ s
′
2 must obtain. In other words, the question

is whether the complete physical state of affairs of the subsystem S determines
uniquely the physical states of affairs of the sub-subsystems S 1 and S 2.

Arguably, on a reading of “physical states of affairs” that conforms to our intu-
itive (“pre-theoretic”) understanding of what constitutes “physical states of affairs”
the question should be answered affirmatively: if s′1 is obtained from s1 by means
of a change in physical situation (where “change” is meant metaphorically, not as a
process in time), we cannot in general “compensate” for this change by altering s2
to a physically distinct state s′2 with the result that, in the end, the original universe
physical state of affairs s1 ∗ s2 is reproduced in the form s′1 ∗ s

′
2. The main rea-

son for why it is arguably plausible that s1 ∗ s2 is physically different from s′1 ∗ s
′
2

unless s1 ∼ s′1 and s2 ∼ s
′
2 lies in the fact that, in order for them to be physically

identical, replacing s1 ∗ s2 by s′1 ∗ s
′
2 must have no net effect on the overall physical

universe state for arbitrary environment states e with which s1 ∗ s2 and s′1 ∗ s
′
2 may

be combined.
To understand how important the arbitrariness of the environment state e is in

this context (and how much it bears on the plausibility of the assumption I am going
to make), consider what may seem to be the scenario where the claim just made
seems most likely to fail, namely, a pair of equally large spherical sub-subsystems
S 1 and S 2 one of which can be seen as the “mirror-image” of the other. It may seem
that, with respect to these systems, a “state swapping”, i.e. the choice s1 = s′2 and
s2 = s′1, would lead to identical physical states of affairs s1 ∗ s2 and s

′
1 ∗ s

′
2 = s2 ∗ s1

for the combined state, contrary to the claim just made. However, even in this
highly special case there exist countless environment states e which disturb the
envisaged mirror-symmetry between S 1 and S 2 (one of them might represent, for
example, a human agent who points with a finger at S 1 and with no finger at S 2).
In fact, for all environment states e which do not exhibit the same symmetry as
the two spherical systems together, one has s1 ∗ s2 ∗ e / s2 ∗ s1 ∗ e and therefore
s1 ∗ s2 / s2 ∗ s1.

To sum up these considerations, I suggest the following assumption (where
“MAH” stands for “modest anti-holism”):

Assumption (MAH)

11



For all s1, s′1 ∈S1 and s2, s
′
2 ∈S2,

if s1 ∗ s2 ∗e ∼ s′1 ∗ s
′
2 ∗e for all e ∈E for which s1 ∗ s2 ∗e and s

′
1 ∗ s

′
2 ∗e are defined,

then s1 ∼ s′1 and s2 ∼ s
′
2.

This assumption is an “anti-holism” inasmuch as it rejects the holistic idea that one
and the same physical state of affairs of a subsystem S = S 1 ∪S 2 can be “reduced”
(or “decomposed”) in more than one way to physical states of affairs of its (sub-)
subsystems. I have already argued for why it merits the predicate “modest” as well.

To derive the announced result about the (lack of) direct empirical significance
of local symmetries, we now need an interesting general feature of theories which
have interior symmetries. According to Greaves and Wallace, these include gauge
theories such as Maxwell electrodynamics and theories with diffeomorphism in-
variance such as general relativity. In all these cases, the interior symmetries form
a subclass of a larger class of symmetries (consisting of local gauge symmetries
or generic diffeomorphisms), which, for the sake of brevity, will be uniformly re-
ferred to as the “local” symmetries of the theory in what follows. The feature of
such theories which I have in mind is the following (with the label “Ext” standing
for “extendability”):

Assumption (Ext)
Any local symmetry σS defined on the subsystem state space S can be extended to
an interior symmetry defined on the state space V of a larger subsystem V ⊃ S .

In other words, according to (Ext), for any σS on S there exists a symmetry σV on
V which does not have any direct empirical significance and which reduces to σS
on S.

To see why (Ext) holds for theories with local symmetries (under the assump-
tion, made by Greaves and Wallace, that the local symmetries include interior ones
which differ from the identity), consider as an example Maxwell electrodynamics
as formulated in terms of the four-vector potential Aµ. It is invariant under local
gauge transformations of the form

Aµ(x) 7→ Aµ(x) + ∂µχ(x) , (1)

where χ(x) is a (differentiable) real-valued function of the space-time variable x.
Now, if S is a finite and compact space-time volume and χ(x) is a function that is
defined within S and induces a symmetry transformation σS , it is always possible
to extend χ(x) in such a way outside S that its value becomes constant (say zero)
towards the boundary of a larger subsystem V which contains S as a proper sub-
system and remains at that constant for the “outward” rest of space-time. Thus,
the resulting extended symmetry transformation σV (which acts on the state space

12



associated with V ) acts differently from the identity only in the interior of V , but
not on its boundary and outside of it. By the assumptions made by Greaves and
Wallace, the symmetry transformation σV is interior. Since no special assumptions
about χ(x) inside S were made, this establishes (Ext) for arbitrary local subsystem
symmetries σS of the form (1) (assuming that subsystem symmetries which re-
duce to the identity at the subsystem boundary are indeed interior symmetries with
respect to that subsystem).

Evidently, the assumption (Ext) can be avoided for a theory by denying that it
has any interior symmetries (other than perhaps the identity), which do not have
any direct empirical significance in the first place. For example, on a non-standard
view of general relativity according to which diffeomorphisms which differ from
the identity only in the interior of a finite space-time volume (the “hole” of the hole
argument) may still relate physically distinct states of affairs, the assumption (Ext)
does not hold. Non-standard views aside, however, the assumption (Ext) seems
quite plausible with respect to the theories formulated in terms of local symme-
tries. In the following section I shall highlight what follows from assuming (Ext)
in conjunction with (DES), (SUL), and (MAH).

4 The result

After the conceptual groundwork of the previous sections I can now formulate
the following result concerning the (lack of) direct empirical significance of local
symmetries:

Proposition Given (DES), (SUL), (MAH), and (Ext), local symmetries do not
have any direct empirical significance.

The proposition is easily demonstrated as follows:

Proof. Let σS be a local symmetry defined on a subsystem state space S. By (Ext),
we can extend it to an interior symmetry σV on V, the state space associated with
a larger subsystem V ⊃ S . Assume V = S ∪ S ′ with disjoint space-time regions
S and S ′ and let S′ be the state space associated with S ′. Now, the fact that σV is
interior means that for arbitrary states s ∈S and s′ ∈S′ for which s∗ s′ is defined:

σV (s ∗ s
′) ∼ s ∗ s′ . (2)

Using (SUL), one obtains that for all e ∈ E for which σV (s ∗ s′) ∗ e and s ∗ s′ ∗ e
are defined

σV (s ∗ s
′) ∗ e ∼ s ∗ s′ ∗ e . (3)

13



Decomposing σV in terms of its restrictions σS (the symmetry we were originally
interested in) and σS ′ on the state spaces S and S′, we find that for all e ∈ E for
which σV (s ∗ s′) ∗ e and s ∗ s′ ∗ e are defined:

σS (s) ∗σS ′(s
′) ∗ e ∼ s ∗ s′ ∗ e . (4)

Using (MAH), it follows that

σS (s) ∼ s . (5)

Since s was chosen arbitrary, Eq. (5) holds for all s ∈S. By (DES), this means that
σS doesn’t have any direct empirical significance. �

The result that local symmetries cannot have any direct empirical significance
is in direct opposition to the claim made by Greaves and Wallace that subsystem
local symmetries which do not reduce to the identity on the subsystem bound-
ary do have direct empirical significance. Let us briefly discuss the prospects for
evading the result by dropping one of the assumptions (DES), (SUL), (MAH), or
(Ext)—either by rejecting it or by saying that it does not apply to the theories under
consideration.

(DES) may be rejected in either of two ways: either by saying that the relation
of “representing the same physical states of affairs” (i.e. the relation “∼”) cannot
be meaningfully applied to subsystem states; or by accepting that it applies to sub-
system states while disputing that it has anything to do with the question of which
subsystem symmetries have direct empirical significance. The first option—which
rejects the relation of representing the same physical state of affairs for subsystem
states—is perhaps the one Greaves and Wallace themselves would choose. I see no
reason to doubt its coherence, but it comes at the price of denying that subsystem
physical states of affairs have well-defined identities at all.

The second option of rejecting (DES) is to accept the ∼-relation between sub-
system states and to dispute that it has anything to do with the question of which
subsystem symmetries have direct empirical significance. This seems to be the op-
tion of choice for those who believe in well-defined subsystem physical states of
affairs, while holding that any two symmetry-related subsystem states represent the
same physical state of affairs. The drawback of this position, as explained above, is
that it ignores the possibility that symmetry-related subsystem states may differ in
their extrinsic physical properties. Therefore, I would not recommend this option
as very attractive.

Rejecting (SUL) while preserving (DES) is a logically viable option, but I see
no strong reasons for it. If one accepts the ∼-relation as a non-trivial relation be-
tween subsystem states, then (SUL), in conjunction with (DES), immediately re-
produces the plausible verdicts by Greaves and Wallace on the first two types of

14



symmetries they consider, while being tacit about those of the more controversial
third type. Therefore, there seem to be very good reasons for accepting it (unless,
of course, one rejects (DES) as well).

Rejecting (MAH) while preserving (DES) is a logically viable option as well,
but it commits one to a strong holism. It implies that the physical state of a universe
subsystem S does not uniquely determine the physical state of its sub-subsystems
S 1 ⊂ S and S 2 ⊂ S , so that two pairs s1, s′1 ∈ S1 and s2, s

′
2 ∈ S2 of pairwise

physically distinct states (i.e. s1 / s′1 and s2 / s
′
2) can combine to yield one and

the same “whole”, i.e. one and the same physical state of affairs represented by
s1 ∗ s2 ∼ s′1 ∗ s

′
2. This is implausible, as argued above, due to the abundance of

environment states which allow to distinguish between the two systems S 1 and
S 2, which means that one cannot simply reproduce the state s1 ∗ s2 in the form
s′1 ∗ s

′
2 = s2 ∗ s1.

As already remarked, whether one accepts the assumption (Ext) for some the-
ory depends on whether one holds that the “local symmetries” of the theory include
any interior ones, which do not have any direct empirical significance. Greaves and
Wallace claim explicitly with respect to theories such as Maxwell electrodynamics
and general relativity that they do have such interior symmetries. This seems to
make it difficult for them to avoid the assumption (Ext). In contrast, those who are
willing to accept the extreme view that all symmetries in these theories have direct
empirical significance will see no reason to accept (Ext) in the first place. Such a
view seems quite daring, however, as it implies that any distinct configuration of
the gauge potentials in a gauge theory and any distinct coordinate configuration in
general relativity represents a distinct physical states of affairs. Proponents of this
view (and probably also of slightly weakened versions of it) clearly have to address
such challenges as the hole argument.

Independent reasons for denying (Ext) in its full generality for some theory
may perhaps arise if the underlying space-time has nontrivial topological structure.
However, I am unable to address this suggestion at present and recommend it as
an interesting topic for future work. Fortunately, none of the scenarios discussed
by Greaves and Wallace is set in a space-time with nontrivial topological structure.
Arguably, with respect to the scenarios they discuss the four assumptions (DES),
(SUL), (MAH), and (Ext) are all quite attractive.

5 Problems with ’t Hooft’s beam splitter?

In this section I discuss a more specific point of contradiction between the result
stated in the previous section and a more specific claim made by Greaves and Wal-
lace. Readers who are willing to accept the result of the previous section without

15



worrying about its correctness may directly turn to the concluding section.
As already mentioned, for most of the local subsystem symmetries σS which,

according to Greaves and Wallace, have direct empirical significance there are no
pairs of states s ∈ S and e ∈ E for which both s ∗ e and σS (s) ∗ e are defined. Ac-
cording to them, in these cases “in order to realise [the direct empirical significance
of σS ] the environment state must be altered” (Greaves and Wallace [forthcoming],
p. 14), that is, one must change the environment state e to another state e′ with the
result that an inequivalence s ∗ e / σS (s) ∗ e′ holds. As remarked, with respect
to these cases Greaves and Wallace essentially stipulate that this inequivalence
s ∗ e / σS (s) ∗ e′ indicates the direct empirical significance of σS . They do not
derive the result that σS has direct empirical significance from any of the assump-
tions of their framework. Therefore, there is no consistency problem between what
they directly establish and the result stated in the previous section, which says that,
given (DES), (SUL), (MAH), and (Ext), σS does not have any direct empirical
significance.

However, with respect to a more narrow class of local symmetries Greaves and
Wallace claim that there are subsystem and environment states s ∈ S and e ∈ E
such that for a subsystem symmetry σS both s ∗ e and σS (s) ∗ e are defined and
represent physically distinct states of affairs. According to them, in other words,
the inequivalence s∗e / σS (s)∗e holds for these local symmetries. Let us assume
that this is true. Then, for appropriate states s ∈ S and e ∈ E, it follows from
(SUL) that s / σS (s) and further, from (DES), that σS has direct empirical signif-
icance, contradicting the result presented in the previous section and establishing
an inconsistency between the assumptions involved.

An example of a theory which, according to Greaves and Wallace, has local
symmetries σS with the property that s ∗ e / σS (s) ∗ e is what they call “Klein-
Gordon-Maxwell electrodynamics”. In this theory, the four-vector potential Aµ as
familiar from Maxwell electrodynamics is coupled to a complex scalar field ψ. The
Lagrangian of the theory is given by

L = (∂µψ− iqAµψ)
∗(∂µψ− iqAµψ) − m2ψ∗ψ + LEM . (6)

Here LEM is the Lagrangian of Maxwell electrodynamics without matter. The
Lagrangian L itself is invariant under local gauge transformations of the form

ψ(x) 7→ exp (iqχ(x)) ψ(x) , (7)

Aµ(x) 7→ Aµ(x) + ∂µχ(x) . (8)

The argument brought forward by Greaves and Wallace for the claim that some of
these local symmetry transformations (conceived as subsystem symmetries) have

16



direct empirical significance is based on a setup that realises a version of the cele-
brated two-slit interference experiment. In this version of the two-slit setup, often
referred to as “’t Hooft’s beam splitter”10, a phase-shifter is inserted behind the
upper, but not the behind the lower, of the two slits. The essential empirical ob-
servation is that the interference pattern that emerges on a screen behind the slits
‘depends nontrivially on whether or not the phase-shifter is present.’ (Greaves and
Wallace [forthcoming], p. 6) Greaves and Wallace interpret this fact as demonstrat-
ing that subsystem local gauge transformations have direct empirical significance
in this case.

The crucial move in their argument is to treat the upper half-beam as a subsys-
tem in the sense of their framework and to argue that ‘the phase-shifter precisely
implements a transformation ψ 7→ eiθψ on the upper half-beam while leaving the
electromagnetic potential unchanged (as gauge transformations with constant χ
do).’ (Greaves and Wallace [forthcoming], p. 7) Their suggestion is that, if the sit-
uation where the phase-shifter is absent corresponds to some state s∗e, the situation
where the phase-shifter is present corresponds to some state σS (s) ∗ e, where σS
is some transformation ψ 7→ eiθψ with constant θ. As they point out, it is possible
to let both s ∗ e and σS (s) ∗ e be well-defined in this case by choosing ψ such that
it vanishes in a neighbourhood of the boundary which separates the subsystem and
the environment. Since the physical state of affairs depends on whether or not the
phase-shifter is present, they conclude that σS has direct empirical significance in
this case.

However, their assumption that inserting the phase-shifter “implements a trans-
formation ψ 7→ eiθψ on the upper half-beam” without having an impact on the
physical state of the environment is untenable. If the upper half-beam plays the
role of the subsystem S and the rest of the setup plays the role of the environment
E, then, if two situations with different interference pattern are compared, they
must evidently be represented by physically distinct environment states e / e′.
The physical situation of the screen is different, and this must be accounted for by
a physical difference between these states e and e′. Insisting that the state of the
lower half-beam, excluding the screen, is unchanged by inserting a phase shifter in
the upper half-beam does not help, since the physical difference between the two
situations of the screen must still be accounted for by the environment state. So,
the situation where the phase-shifter is present and the one where it is absent are
not represented by states s ∗ e / σS (s) ∗ e with a fixed environment state e.11

10The same setup is also discussed in (Brading and Brown [2004] and Lyre [2004]), who main-
tain that it does not provide an example of a situation where a local symmetry has direct empirical
significance.

11Essentially the same point is made by Brading and Brown, who emphasise that ‘an interference
pattern occurs only where ψI and ψI I overlap, and clearly these conditions [i.e. a non-boundary

17



In response to this argument, Greaves and Wallace may claim that, in order
to realise the physical difference between ψ and eiθψ, the screen behind the setup
need not be actually present, but only hypothetically.12 The idea in this case would
be that the difference between the two interference patterns on the screen is a mere
reflection of a preceding physical difference between ψ and eiθψ which had been
there all along in the absence of the screen. The purpose of invoking the hypothet-
ical screen, this argument goes, is to make the physical difference between ψ and
eiθψ vivid and transparent, not to include the screen as an actual ingredient of the
environment system.

In more formal terms, this argument takes the following form: Consider, first,
a situation where the two beams have vanishing overlap and can be described by
the field configurations (ψI, Aµ,I ) and (ψI I, Aµ,I I ), respectively. Consider, second, a
situation where they are described by (eiθψI, Aµ,I ) and (ψI I, Aµ,I I ) with θ , 0, 2π.
The question we would like to answer is whether the two situations are physically
distinct or identical. However, as long as there is no overlap between the two
beams (i.e. no region where both ψI and ψI I are non-zero), there is no interference
pattern and therefore no manifest physical difference between the two situations.
The suggested idea is to consider a modification of both situations by introducing a
hypothetical region of overlap (“the screen”), to arrive at two manifestly physically
different situations, and to conclude from that difference that a physical difference
between (ψI, Aµ,I ) and (ψI I, Aµ,I I ) had been there all along.

Now let us try to write down the field configurations for two physically different
situations in the (hypothetical) case where a region of overlap is present such that
the physical difference between them appears as a consequence of the physical
difference between ψI and eiθψI .

The first situation, where the phase-shifter is absent, —call it “situation A”—
can be taken to be one where the upper half-beam is described by (ψI, Aµ,I ), the
lower by (ψI I, Aµ,I I ), and the overlap region by a superposition of the two:(

ψI,A(x) + ψI I,A(x), Aµ,A
)

(9)

with ψI,A and ψI I,A in the overlap region chosen as smooth continuations of ψI and
ψI I in the upper and lower half-beams.

If a phase-shifter is inserted in the upper half-beam, however, a physically dif-
ferent situation arises—call it “situation B”—, where the upper half-beam can be
described by (eiθψI, Aµ,I ), the lower (unchanged) by (ψI I, Aµ,I I ), and the overlap

preserving local gauge transformation which acts only on the state of the upper half-beam] on [sic]
cannot be met in such a region.’ (Brading and Brown [2004] p. 653. The emphasis is due to Brading
and Brown.)

12I would like to thank two anonymous referees for suggesting this possible move.

18



region by the superposition(
eiθψI,A(x) + ψI I,A(x), Aµ,B

)
(10)

with the same ψI,A and ψI I,A as before.
For non-trivial values of the constant θ the two field configurations (9) and (10)

are indeed physically different—at least if the plausible assumption is made that
|ψ|2 = ψ∗ψ corresponds to a physical quantity such as the matter or charge density.
The suggested argument now appeals to the physical difference between these two
situations A and B, which appears in the presence of an overlap region, and sees it
as reflecting a physical difference between the two situations described by (ψI, Aµ,I )
and (eiθψI, Aµ,I ) in the upper half-beam—both with and without the region of over-
lap. The idea here is that the physical difference between the situations A and B in
the presence of an overlap region arises precisely as a dynamical consequence of
the physical difference between (ψI, Aµ,I ) and (eiθψI, Aµ,I ) in the upper half-beam
and thereby testifies to the fact that this difference has been there all along.

Contrary to this suggestion, however, the physical difference between the sit-
uations A and B should not be taken to reflect a physical difference between the
upper half-beam configurations (ψI, Aµ,I ) and (eiθψI, Aµ,I ). To see this, it suffices
to note that the field configuration (eiθψI, Aµ,I ) can also describe the situation A,
and the field configuration (ψI, Aµ,I ) can also describe the situation B in the upper
half-beam, while the field configuration of the lower half-beam remains (ψI I, Aµ,I I )
in both situations.

To be specific, the situation A can alternatively be described by the field con-
figuration (eiθψI, Aµ,I ) in the upper half-beam, (ψI I, Aµ,I I ) in the lower half-beam,
and by the gauge-transformed(

eiθλ(x)
(
ψI,A(x) + ψI I,A(x)

)
, Aµ,A +

θ

q
∂µλ(x)

)
(11)

in the region of overlap, where λ(x) is a smooth real-valued function with λ(x) = 1
at the boundary between the upper half-beam and the region of overlap and λ(x) =
0 at the boundary between the lower half-beam and the region of overlap.

Similarly, situation B can alternatively be described by the field configuration
(ψI, Aµ,I ) in the upper half-beam, (ψI I, Aµ,I I ) in the lower half-beam, and by the
gauge-transformed(

e−iθλ(x)
(
eiθψI,A(x) + ψI I,A(x)

)
, Aµ,B −

θ

q
∂µλ(x)

)
(12)

in the region of overlap. The point of writing down these alternative field con-
figurations is of course that they describe exactly the same situations A and B as
before, but now with reversed ascriptions of ψI and eiθψI to the upper half-beam.

19



The upshot of these considerations is that one should not take the difference
between the two interference patterns in the situations A and B to reflect a physical
difference between ψI and eiθψI . In fact, both situations A and B can equally well
be described in terms of either ψI or eiθψI for the upper half-beam. From the point
of view based on (DES), (SUL), (MAH), and (Ext), developed here, this makes
perfect sense: Both ψI and eiθψI describe the same physical state of affairs, and the
acknowledged physical difference between the situations A and B in the overlap
region concerns merely the environment state, not the state of the upper half-beam.

In other words, while the interference pattern dynamically reflects whether or
not the phase shifter is present, it does not dynamically reflect whether the state
of the upper-half beam is ψI or eiθψI . The phase shifter and the screen are both
features of the environment (excluding both beams), and the dynamical correlation
between them, according to the present analysis, is a feature which concerns only
the environment in that the physical states of the beams are the same in both situa-
tions. Whether or not this means that the physical situation on the screen depends
non-locally on the presence of the phase shifter is a complicated question that I
wish to leave aside in the context of the present investigation, for its (by no means
obvious) answer requires a detailed investigation of the specific dynamical proper-
ties of the theory under considerations.13 To conclude, the hypothetical presence
of a region where the half-beams overlap does not give us any reason to regard ψI
and eiθψI as representing different subsystem physical states of affairs.

Finally, it should be noted that the present argument for the physical identity of
ψI and eiθψI applies only in virtue of the fact that the symmetry (7) is a local rather
than a global one. To see this, consider, as an example, a version of Galileo’s ship
where a screen, located at the shore, registers different interference pattern of the
water waves, depending on whether the ship is at rest or in constant relative mo-
tion with respect to the shore. Do the different interference patterns on the screen
correspond one-to-one to physically different, symmetry related states s and (its
boosted analogon) σS (s) of the ship in this setup? The answer depends crucially of
whether we assess the problem in the framework of a theory which contains only
global symmetries (such as special relativity) or also local ones (such as general
relativity). If the theory has only global symmetries, once some coordinate system
has been chosen (with the shore at rest, say), the question of whether the ship is in
s or in σS (s) has a determinate answer; if it has also local symmetries, both s and
σS (s) can be chosen to describe the ship both at rest and in uniform motion, and
the physical difference between the two situations, as it manifests itself in the two

13See (Vaidman [2012]) for an illuminating discussion of the conceptually related Aharonov-
Bohm effect, which lends some plausibility to the idea that a non-local influence need not be as-
sumed.

20



different interference patterns on the screen, is accounted for entirely by a physi-
cal difference in the states used to describe the (complete) environment of the ship.
This goes well with the conclusion of the above discussion of ’t Hooft’s beam split-
ter, where the symmetry (7) is also local. So, claiming that the local symmetries in
the case of ’t Hooft’s beam splitter lack any direct empirical significance does not
commit one to the view that global symmetries do not have any direct empirical
significance either.

6 Summary and conclusion

The present paper has proposed a development of a recently suggested framework
by Hilary Greaves and David Wallace to distinguish between symmetries which
have direct empirical significance and symmetries which do not. A result has been
presented, based on four rather plausible assumptions, according to which subsys-
tem local symmetries do not have any direct empirical significance, contrary to the
claims made by Greaves and Wallace.

The assumptions, to recapitulate, are the following: first, that the question of
direct empirical significance for subsystem symmetries is the same as that of which
symmetries connect physically identical subsystem states of affairs (assumption
(DES)); second, that two subsystem states s and s′ correspond to the same physical
state of affairs just in case their combinations with arbitrary environment states e
(if defined) yield identical universe states of affairs (assumption (SUL)); third, that
the combined physical state of affairs of two subsystems determines uniquely the
physical states of the individual subsystems (assumption (MAH)); and, fourth, that
local symmetries defined on subsystem state spaces can always be extended to
interior symmetries (i.e. symmetries without any direct empirical significance) on
the state spaces of larger subsystems (assumption (Ext)). None of the assumptions
is indisputable—in particular, the assumption (Ext) may not be adequate for some
theories with local symmetries—, but all four can be motivated very well for the
scenarios considered by Greaves and Wallace.

An implication of the Greaves/Wallace framework together with (DES), (SUL),
(MAH), and (Ext) is that subsystem symmetries in theories with only global sym-
metries may have direct empirical significance, whereas their (“globally constant”)
counterparts in theories which have local symmetries do not. For example, from
the perspective of an interpretation which accepts the assumption (Ext) for dif-
feomorphisms in general relativity, subsystem-wise globally constant diffeomor-
phisms do not have any direct empirical significance, whereas boosts as employed
in the Newtonian explanation of Galileo’s ship may. According to Greaves and
Wallace, such a perspective is odd, since ‘it would be highly mysterious if global

21



symmetries managed to have empirical significance while no other symmetries
were around, but somehow lost this capacity once the full local group of trans-
formations appeared as symmetries.’ (Greaves and Wallace [forthcoming], p. 2)
In response to these qualms, those who defend the assumptions (DES), (SUL),
(MAH), and (Ext) may respond as follows:

It is true that, from the perspective of an account which accepts the Greaves/-
Wallace framework together with (DES), (SUL), (MAH), and (Ext), it appears that
subsystem global symmetries “lose” their direct empirical significance once the
global symmetry group of the earlier theory is replaced by the local symmetry
group of the later one. However, there is nothing odd or “mysterious” about this
formal “loss”, for all that it means is that what appear to be two physically distinct
yet empirically indistinguishable subsystem situations in the earlier theory turns
out to be one single physical subsystem situation in the later theory (which, how-
ever, can be related in physically distinct ways to its environment). As a matter
of fact, it is difficult to imagine a more elegant explanation of the empirical equiv-
alence!14 So, far from being worrisome for the scientific realist (as suggested by
Greaves and Wallace, see Greaves and Wallace [forthcoming], p. 3.), the loss of
direct empirical significance due to the “localisation” of global symmetries in the
switch from one theory to another can in fact be seen as a gain in our explanatory
resources.

Acknowledgements

I would like to thank Holger Lyre for fueling my interest in symmetries and for
alerting me of the work of Graves and Wallace. I am particularly grateful to
Nazim Bouatta for many fruitful meetings dedicated to the understanding of sym-
metries and the Greaves/Wallace framework. Furthermore, I would like to thank
two anonymous referees of the British Journal for the Philosophy of Science for

14In (Healey [2009], Section 5), Richard Healey presents an account of how empirical equiva-
lences between physical situations (“empirical symmetries”) are sometimes accounted for as “strong”
in an earlier theory and as “perfect” in a later one. For example, as Healey points out and as famously
noted by Einstein, pre-relativistic Maxwell electrodynamics gives conceptually different accounts of
electrodynamic processes, depending on which systems are regarded as rest and which are regarded
as moving, while the predictions it makes are empirically equivalent. Special relativity accounts for
this empirical equivalence by revealing that, as Healey puts it, ‘distinct but symmetrically related sit-
uations are in fact duplicates of one another.’ (Healey [2009] p. 709) The present account, based on
the Greaves/Wallace framework together with (DES), (SUL), (MAH), and (Ext), permits to see the
replacement of the global symmetries of the earlier theory by the local symmetries of the later one
in an analogous light: what appeared to be physically distinct, yet empirically equivalent, subsystem
situations from the point of view of the earlier theory is revealed to be one single subsystem situation
from the point of view of the later one.

22



their very useful comments.

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