2.1. Symmetries of Physical Theories

In the contemporary debate, the symmetries of a physical theory are standardly taken to be transformations that act on the set of (mathematical) states of the theory and that preserve the theory’s laws of motion. This idea can be spelled out in various ways. Since our main examples will come from classical (Yang-Mills) gauge field theory, we mostly adopt the standard picture of a symmetry acting on the space of fields and leaving the theory’s action invariant—this invariance ensures that such symmetries will also preserve the equations of motion (EOM), that is, map field configurations that are solutions of the EOM to other such solutions.Footnote ⁴ One can also introduce an analysis of symmetries in terms of constraints, which classifies symmetries in various ways and in particular characterizes the sense in which gauge symmetries relate redundant descriptions: it will be useful for us to apply this analysis in section 4.1.

In order to proceed, we need to recall the standard taxonomy of symmetries in modern physics. This is usually broken into two logically independent distinctions. So a symmetry will generally be described by two predicates: one from each pair.

First, there is the external (also called space-time) versus internal symmetry distinction. External symmetries are those transformations that effect a change in the space-time parameters (e.g., x, y, z, t) of a theory. But, transformations that do not affect the space-time parameters are said to change only ‘internal parameters’ and are thus called the internal symmetries of a theory.Footnote ⁵

Next, there is the global versus local symmetry distinction.Footnote ⁶ Very roughly speaking, global symmetries are transformations that do not depend on space-time parameters, whereas local symmetries are functions from space-time parameters to a ‘gauge group’ G of transformations, for example, g : ℝ^3,1 → G, where ℝ^3,1 is four-dimensional Minkowski space-time.

This rough (albeit pervasive, in both the physics and philosophy literature) characterization in fact obscures a subtle and important point about the global versus local distinction, which will turn out to be a source of philosophical confusion in the debate. I thus give a more careful characterization of this distinction at the end of section 2.2, in the context of gauge theories.

Following standard practice, we say that a global internal symmetry transformation g ∈ G (where G is some Lie group) is localized when we turn g into a function g(x) from space-time to G (i.e., it is a gauge transformation in the general sense). Physicists often speak of ‘localizing’ a symmetry as a method of moving from a theory with a global symmetry to a theory with a corresponding local symmetry. But, we refer to the reverse procedure—replacing functions into G with constant elements of G—as the ‘globalizing’ of a gauge symmetry.

Three caveats must be issued about the use of ‘local’ and ‘gauge’ in this article. First, some authors use the terms ‘local symmetry’ and ‘gauge symmetry’ synonymously.Footnote ⁷ However, in keeping with the most popular use of the term in physics, we will take gauge symmetry to refer only to the internal, local symmetries of Yang-Mills type theories, which we review in section 2.2.Footnote ⁸

The second caveat is of greater significance. Even after restricting ‘gauge symmetry’ to the internal local symmetries of Yang-Mills theory, the use of this notion in much of the philosophical/physics literature is ambiguous between two different ideas: first, there is the general and formal notion that we have just been using:

And second, we can further demand that the functional role of ‘representing redundancy’ is satisfied by Formal gauge transformations:

It is easy to slide between Formal and Redundant gauge transformations, or simply identify them, in casual conversation. But there is no reason to expect these concepts to align in general. On the one hand, satisfying the functional role of redundancy is clearly not a sufficient condition for being a gauge transformation since—as we note below—global symmetries of the universe also satisfy the functional role of representing redundancy. On the other hand, we will see in section 4 that not all Formal gauge transformations are Redundant.

Third, note that the term ‘local’ is also used in a different sense in discussions of geometry and topology. To disambiguate, we use local to refer to properties that can be predicated of an arbitrarily small, and in particular topologically trivial, patch of a smooth manifold. By contrast, global properties are those that hold of an entire (possibly topologically nontrivial) manifold.

Many (although not all) agree that, regardless of the type of a symmetry, it exhibits Identical when it acts on the entire universe.Footnote ¹⁰ After all, what physical difference can it make to simultaneously transform everything in the universe in the same way? There is also a consensus that, when an external global symmetry acts on a proper subsystem of the universe, it can in principle exhibit Empirical, just as in the case of Galileo’s ship.Footnote ¹¹ This is because such a transformation can effect a (genuine, physical) relational difference between the subsystem and its environment—and thus a genuine difference between the pre- and posttransformation states of the universe.

However, apart from this, controversies about Empirical abound.Footnote ¹² This article’s focus is a specific controversy about whether gauge symmetries can have direct empirical significance: Greaves and Wallace claim that this is possible, but their claim is denied by Kosso, Brading and Brown, and Healey (and many others), who thus represent the orthodoxy in this debate.

Let us now recall how the terms direct and indirect empirical significance have been used in this literature:

DES is formulated in such a way as to capture the sense of Empirical that is characteristic of Galileo’s ship scenarios. IES, however, covers a far wider range of possible implications of a physical symmetry under its rubric of ‘indirect’. For us, the most important implication of IES is the idea that a charge is associated with each global symmetry via Noether’s theorem. A charge in this highly general sense is some attribute of a dynamical object (e.g., a field) that determines how this object will respond to some particular force. Indeed, each charge is typically ‘conjugate’ to some force and serves to characterize the force, as well as to provide a constant conserved over time by the dynamics. This aspect of IES is closely related to DES: in a Galileo’s ship scenario, the relational difference between the transformed subsystem and its environment is precisely a relative difference in a generalized charge of this kind. Thus, in order to effect such a scenario, a subsystem symmetry must give rise to a charge that is well defined and conserved.

While some parts of the debate have been conducted at a rather high (i.e., non-theory-specific) level of generality (see, e.g., Kosso Reference Kosso2000; Greaves and Wallace Reference Greaves and Wallace2014), much can be learned by exploring these issues in the case of one large and particularly important class of theories, that is, Yang-Mills (often referred to simply as gauge) theories. The reasons are several. First, field theories—and in particular gauge theories—are radically different from particle theories, not least because they allow for nontrivial long-range interactions: as we will see in the case of gauge fields. Second, gauge theories have highly complex symmetry transformations (i.e., maps into non-Abelian Lie groups), which allows us to appreciate the richness and novelty of local symmetry, while still being more tractable than the local symmetry of general relativity. Third, gauge theories are hardly speculative or inessential physics: they account for three of the four fundamental forces with which we have discovered. At any rate, the examination of whether Yang-Mills gauge symmetries can display DES dovetails neatly with one of Greaves and Wallace’s four main examples (i.e., a scalar field coupled to a U(1) gauge field).

2.2. Gauge Theory

Yang-Mills gauge theory, like any other field theory, has fields (i.e., maps from space-time to some abstract value space) as its dynamical variables. In the sketch that follows, we are concerned with the local aspects of gauge theory (i.e., its properties on a small topologically trivial patch U of a space-time M), except at the end of section 2.2, where I explain one important global fact about gauge theory.

Physicists often speak of the ‘gauge group’ of a theory—let us stress that this concept is not to be confused with ‘the group of gauge transformations’; it instead refers to the group (usually a Lie group such as SU(n) or SO(n)) in which the gauge transformations take values. Locally, a ‘pure’ gauge theory with gauge group G describes the dynamics of a gauge field A : U → Ω¹(U) ⊗ . In other words, A is a map from a patch of space-time U to an abstract value space of -valued 1 forms.Footnote ¹⁵

The gauge symmetry transformations of a gauge theory are usually defined locally, in a way that exemplifies Formal of section 2.1. They are maps g from space-time into the gauge group G, which act on a gauge field in the following manner:Footnote ¹⁶

where μ is a space-time index. This symmetry is local (since g is space-time dependent) and internal (since g does not act on space-time parameters). Furthermore, it is clear that at this local level, the global transformations are a special case (i.e., just the constant functions) of the gauge transformations.

By extending the notion of a derivative to a gauge covariant derivative, we can define the field strength or curvature field F (in coordinates): F _μν ≔ ∂ _μA _ν − ∂_νA _μ + [A _μ, A _ν], where F is often taken to be the ‘physical field’ in elementary treatments of U(1) gauge theory. However, various experimental and theoretical arguments (e.g., the Aharonov-Bohm effect) have shown that the physical degrees of freedom of a gauge theory are rather more subtle and should instead be identified with various topological classes or objects (e.g., Wilson loops) that can be associated with the gauge field A. Thus, despite the representational redundancies of the A field, physicists tend to treat it as the fundamental dynamical field variable of gauge theory.

We now proceed to dynamics. The curvature F enters directly into the description of the dynamics of the theory, which is specified by the action S = Tr(∫ F ∧ *F), where Tr is a nondegenerate inner product on the Lie algebra of the gauge group G. The theory’s EOM can be derived from this action. Furthermore, the action is left invariant by gauge transformations, which ensures that such transformations will map a solution of the theory to another solution.Footnote ¹⁷

The main examples in this article come not from pure gauge theory but from coupling a gauge field to some other field. For instance, Greaves and Wallace’s example of a Galileo’s ship scenario for gauge theory comes from a model of this sort (which they call Klein-Gordon-Maxwell [KGM] electrodynamics; cf. sec. 3.2). In section 4 we consider a closely related model called Yang-Mills-Higgs (YMH) theory: this extension is obtained by coupling a scalar field ϕ to the gauge field A and writing down an appropriate action to specify the dynamics of the theory.

The YMH action is . As such, this theory has two interacting dynamical variables (i.e., the fields A _μ and ϕ) on which gauge transformations act. We will not need the detailed form of the potential V(ϕ), apart from noting that it is invariant under gauge transformations and that its minimum energy submanifold (called the vacuum manifold) in the space of fields ϕ has the topology of a sphere S ².

Our primary reason for introducing YMH theory is that in addition to linearized solutions about the vacuum manifold, it also contains a special class of topological soliton solutions called ‘monopoles’. Like other topological solitons, monopoles are defined in terms of topologically nontrivial properties of a field configuration, which also accounts for their particle-like properties. We will see that such solutions have interesting implications for the question whether gauge symmetry exhibits DES.

Finally, let us touch in a preliminary way on the relationship between global transformations and gauge transformations. First, recall that global symmetries play an important role in specifying the empirical content of a field theory, because they (via Noether’s first theorem) give rise to a conserved current, which can in turn be integrated over a spatial slice to give rise to a conserved charge. In section 2.1’s taxonomy of ‘empirical significance’, this is classified as IES; however, it is not any less important for that. Indeed, if one hopes to construct an instance of a Galileo’s ship scenario (i.e., exhibit DES) for such theories, then one needs the notion of a relative difference in charge or phase between a subsystem and its environment. Furthermore, one needs such charges to be conserved, on pain of introducing unpredictable irregularities in the dynamics of charged objects.

How does this connection between symmetries and charges look when we move to a general gauge theory? As discussed in Brading and Brown (Reference Brading and Brown2000), one can still define conserved currents for a gauge theory (albeit with some novel aspects to the derivation). However, when one tries to integrate such currents over a spatial slice in order to obtain a charge, one sees that either (i) the spatial slice has no boundary and the charges vanish identically or (ii) the spatial slice does have a boundary and the charges are ill defined, because there is an arbitrariness in the object that one is integrating over the boundary.Footnote ¹⁸

The moral of this observation is that in order for gauge theories to have a nontrivial well-defined charge, one needs to specify appropriate asymptotic boundary conditions. This was first recognized in an informal way in Abbott and Deser (Reference Abbott and Deser1982), where the existence of charges for Yang-Mills theory is related to boundary conditions with certain symmetries, but the study of how boundary conditions give rise to asymptotic charges has become a deep and technical subject (see, e.g., Barnich and Brandt [Reference Barnich and Brandt2002] for a review). Thus, one sees that unlike global symmetries, this aspect of the empirical content of a gauge theory depends on boundary conditions in complex and subtle ways.

The foregoing results should already give one pause about drawing too tight a connection between the empirical content of a global theory, on the one hand, and a gauge theory, on the other hand. For they tell us that, even if DES-exhibiting symmetries can be constructed from gauge symmetry, they will in general require the presence of special boundary conditions (i.e., those that allow one to construct a well-defined charge). In section 4, we see that precisely because of such conditions, the DES-exhibiting symmetries that emerge need not be the globalization of the gauge symmetry.

A yet more dramatic rift between global and gauge symmetry appears if one is prepared to consider scenarios in which a space-time M has a nontrivial global topology: we see that the globalization of a gauge symmetry is not in general well defined. Let us pause to explain this point.

In this general scenario, one needs to consider gauge transformations γ _i : U _i → G, where the U_i are different (possibly intersecting) topologically trivial patches that form a cover of the space-time M. Suppose i = 1, 2. Then the relationship between γ ₁ and γ ₂ on a nonempty intersection U ₁ ∩ U ₂ is given by

where g ₁₂ is a transition function from U ₁ ∩ U ₂ to G. Such transition functions will be nontrivial (i.e., different from the identity) when M is topologically nontrivial, and when combined with the fact that G can be non-Abelian, this implies that the globalization of a gauge symmetry (i.e., setting γ ₁ = γ ₂ = constant) will not exist in general. More precisely, for an arbitrary gauge group G, (2) can only be satisfied, and the globalization exist, when at least one of the following conditions is met: (a) space-time is topologically trivial (i.e., only one contractible patch U_i is required to describe it) and (b) when the gauge group is Abelian (e.g., U(1)).

The case of non-Abelian gauge theories on topologically nontrivial space-times is important because it helps us appreciate just how fundamentally different gauge symmetry is from global symmetry. But there are other ways in which topology can be physically significant; for example, one can consider a topologically nontrivial boundary of a subsystem (which in turn has implications for the system’s boundary conditions) that is accessible to an observer outside the subsystem. The morals of such subsystems for our topic is explored in the appendix, available online only.

3.1. Greaves and Wallace’s Objections to the Orthodoxy

Greaves and Wallace offer three objections to the claim that gauge symmetries cannot have DES, which in turn provide a motivation for their positive proposal to construct analogs of Galileo’s ship within gauge theory. I now criticize two of these objections (which I do not take to be crucial to their positive proposal), in order to refine and develop their position.

The first objection is based on the mantra that ‘global symmetry is contained in (or is a special case of) local symmetry’—call this GlobalInLocal (cf. Greaves and Wallace Reference Greaves and Wallace2014, 61). Thus, if global symmetries can have DES, so too can gauge symmetries.

But as we have seen at the end of section 2.2, the mantra is in general false. GlobalInLocal fails when one carefully considers the concept of a local—specifically gauge—symmetry transformation, because such transformations are in general required to satisfy a global consistency condition (i.e., eq. [2]). Is it then in any sense true that global symmetries are contained in local symmetries? As discussed in section 2.2, this is true at the level of local symmetry transformations. Thus, in order for Greaves and Wallace’s objection to work, they would have to restrict it to the limited class of theories for which only local symmetry is relevant.Footnote ²⁰ Furthermore, we should bear in mind that such local transformations (and the associated conserved currents) will not in general be sufficient for constructing Galileo’s ship scenarios: what is needed for the latter is a relative difference in charge, and the definition of such a charge in gauge theory depends crucially on the boundary conditions of the system, including the possibly nontrivial topology of a subsystem boundary.

Greaves and Wallace’s second objection is very closely related to the first. They suggest LocalizeEmpirical: the localization of a global theory should have the same DES-exhibiting symmetry group as the global theory, because this explains the ‘preservation of empirical content’ in the localization procedure. As we will see, this is true of the U(1) case but generically false for gauge theory.Footnote ²¹

3.2. Greaves and Wallace’s Framework

We now review Greaves and Wallace’s proposed framework for analyzing symmetries, and in particular for thinking about the abstract conditions under which Galileo’s ship scenarios can arise. I adapt some of their notation and introduce the framework informally (for an axiomatization, see the final section of Greaves and Wallace [Reference Greaves and Wallace2014]). Throughout this section, ‘gauge transformation’ is used in a Formal sense, in keeping with Greaves and Wallace’s usage (cf. the beginning of sec. 3)—we will consider its relation to Redundant in section 4.

Let be the set of dynamically (including kinematically) possible states of the universe, also known as the total system, with a set of universe symmetries acting on it.Footnote ²² So, for instance, in the case of YMH fields over the entire universe, is the configuration space of fields ≔ {(A, ϕ)} on a space-time M (which represents the universe), and is the group of gauge symmetries acting on .

The cases of interest to us are ones in which can be decomposed into two sets of states: (i) a subsystem , which is dynamically isolated from (ii) its environment ℰ. In general, an arbitrary element of the Cartesian product × ℰ will not be ‘physical’ because the pair 〈s, e〉 (where s ∈ and e ∈ ℰ) might not be compatible; that is, they might not satisfy the right boundary conditions to form a dynamically possible state of the universe. Greaves and Wallace assume that s and e are dynamically compatible, just in case there is a universe state u ∈ ⊂ × ℰ that projects onto these two factor states respectively. We can thus introduce the notation u ≡ s ⊔ e to indicate that u is the composite, or conjunction, of states from and ℰ respectively.

In Greaves and Wallace’s framework, one starts with a primitive set of universe symmetries

and uses restrictions of these to

and ℰ to define their respective sets of subsystem symmetries

and Σ_ℰ. In particular, we require that for all s ∈

and e ∈ ℰ, a universe symmetry σ ∈ Σ_U acting on the universe state s ⊔ e = u ∈

restricts to subsystems in the following manner,

and thus defines the subsystem symmetries

and

.

Following Greaves and Wallace, we also define the composition of two subsystem symmetries

and l _ℰ ∈ Σ_ℰ; that is, for all s ∈

and e ∈ ℰ,

where we assume that the right-hand side is a well-defined state of the universe. Notice a very important feature of this definition: the composition of two subsystem symmetries need not be a universe symmetry. In other words,

:

might be an automorphism of

that is not an element of

. The importance of this feature emerges in Greaves and Wallace’s gloss on the definition of DES: recall that a universe symmetry σ cannot effect a real physical difference; thus, a subsystem symmetry

can only exhibit DES if it is the restriction of a universe map that is not a universe symmetry.

Let us now illustrate how the subsystems and ℰ are defined in gauge theory. As Greaves and Wallace note, in the case of field theories, a dynamically isolated subsystem can sometimes be defined by the restriction of fields (satisfying certain conditions) to some compact region of space or space-time.Footnote ²³ So, for instance, might be explicitly defined as field configurations on a compact region S within a space-time M; the environment system ℰ is then defined as field configurations on M − S. Subsystem symmetries are then defined as the (spatiotemporal) restriction of the group of gauge symmetries on the entire space-time M, to S and to E, respectively.

However, a crucial point in what follows is that not all field-theoretic subsystems can be described in this way. Indeed, some field subsystems have asymptotic properties that are highly nontrivial, a perspicuous description of which requires us to represent the boundary ∂S of a spatial region S as being located out at ‘spatial infinity’. Of course, the use of ‘infinity’ here is merely an idealization: it does not mean that ∂S is literally infinitely far away from the interior of S but rather that, relative to some scale set by the physics, ∂S is very far away from the ‘core’ of the field configuration. Nonetheless, the use of such infinities allows one to deploy a powerful mathematical description of the asymptotics of such field configurations. The point is that the formal description of some field subsystems will not fit the simple model invoked by Greaves and Wallace, namely, the restriction of a field configuration to some compact region of space-time.

Let us now discuss the topic of boundary conditions of a state. We need to clarify a terminological point: Greaves and Wallace use ‘the boundary condition B_s of the state s ∈ ’ to refer to the set of all e ∈ ℰ, such that s ⊔ e is dynamically possible, in the sense that the factor states are dynamically compatible; similarly, given e ∈ ℰ, C_e denotes the set of all s ∈ such that s ⊔ e is dynamically possible. However, in field theory it is much more common to have the boundary conditions be (roughly) the—perhaps asymptotic—field values on the geometric boundary ∂S meshing with S and not the equivalence class of all e ∈ ℰ that mesh with s. For us, the most important case is when a field ϕ on three-dimensional space takes certain asymptotic values on the ‘sphere at infinity’, denoted , of this space. We use the notation ϕ _∞ to represent the asymptotic values of such fields (i.e., the values they take on ). Such asymptotic values will of course determine boundary conditions in Greaves and Wallace’s sense; however, they are not defined in terms of the ℰ states that are compatible with a given state.

We now turn to Greaves and Wallace’s basic taxonomy of symmetries of the subsystem . They introduce two (logically independent) types:

• • An interior symmetry is one such that, for all and e ∈ ℰ such that s ⊔ e is defined,

  
  is also a universe symmetry, where 1_ℰ is the identity on ℰ. This condition implies that is dynamically possible.

• • A boundary-preserving symmetry of a state s is a subsystem symmetry such that for all e ∈ ℰ compatible with s, s and are elements of the boundary condition C_e.

According to Greaves and Wallace, interior symmetries are precisely those symmetries that cannot exhibit DES. They reason as follows: by definition, the interior symmetry acting on the subsystem is equivalent to letting act on the universe, and a universe symmetry does not effect a genuine physical difference. Thus, does not have DES. In particular, one cannot use it to effect a relational difference of the sort that is characteristic of Galileo’s ship scenarios.

But, Greaves and Wallace argue that the essence of Galileo’s ship scenarios is captured by subsystem symmetries that are noninterior but are boundary preserving. First, by noninteriority, such a symmetry does not, for all , combine with 1_ℰ to form a universe symmetry (so it is not subject to the last paragraph’s argument against exhibiting DES). Second, by boundary-preserving-ness, one can combine with 1_ℰ, because boundary-preserving subsystem symmetries will not change the dynamical compatibility relations between states in and ℰ. Thus, it is possible to construct a composite transformation of the universe that is nonetheless not a symmetry of the universe, meaning that can effect a physical difference and display DES. Furthermore, since each subsystem symmetry does not change the intrinsic properties of its respective subsystems, this physical difference is purely relational (i.e., a Galileo’s ship scenario).

It is straightforward to construct a Galileo’s ship scenario for a scalar field ϕ with global U(1) symmetry. By localizing this theory, we can obtain the model that Greaves and Wallace call KGM electrodynamics.Footnote ²⁴ This is their example of a noninterior, boundary-preserving subsystem symmetry in gauge theory. The Lagrangian density of this theory is

The space of states on the entire space-time manifold M is

≡ {(A, ϕ)} (i.e., the space of field configurations satisfying the EOM), and gauge transformations act on the fields as follows:

where χ(t, x) is a smooth ℝ-valued function that parameterizes the gauge transformations.

Greaves and Wallace take to be the space of fields on a compact space-time region S with (connected) boundary ∂S, and they take ℰ to be the space of fields on the space-time region E = M − S. They also interpret the condition that is a dynamically isolated subsystem to mean that the fields (A, ϕ) vanish on ∂S.

Evidently, if we restrict our interest to gauge symmetries, then the subsystem transformation of interest on is represented by the function χ_S : S → ℝ. Greaves and Wallace claim that χ_S will represent a merely interior symmetry if it is zero on a small neighborhood of ∂S, meaning that the gauge symmetry g = exp(−iqχ) tends smoothly to the identity near ∂S.Footnote ²⁵

This claim may seem innocuous, but it in fact contains the assumption Trivial → Interior: a gauge symmetry of a system S is asymptotically trivial (in the sense that it tends to Id on the ∂S) only if it is an interior symmetry.Footnote ²⁶ In other words, the composability of g with 1_ℰ (as a smooth function on the entire system) is taken to imply that g ⊔ 1_ℰ is a universe symmetry. Trivial → Interior happens to be true in the above example, but in section 4.2 we see that it is false in general.

Greaves and Wallace further claim that gauge transformations can be noninterior and boundary preserving. For instance, consider an ‘asymptotic’ gauge transformation parameterized by χ_∞, which tends toward a nonzero value on a neighborhood of ∂S. First, χ_∞ is noninterior since it is easy to find field configurations s in for which s ⊔ e is dynamically possible, but not χ_∞(s) ⊔ e.Footnote ²⁷ Second, in this ‘dynamically isolated’ scenario, χ_∞ is also boundary preserving on any state s that satisfies ϕ∣ _∂S = 0. This is because the environment states compatible with s, and their boundary class, are determined by the smooth gluing of ϕ∣ _E with ϕ∣ _S along ∂S, and given the boundary condition ϕ∣ _∂S = 0, none of these data are changed by an arbitrary gauge transformation on S.

Thus, we see that the example adduced by Greaves and Wallace indeed meets all their desiderata for a Galileo’s ship scenario with respect to gauge symmetry (and more generally, local symmetry). Subject to the appropriate boundary conditions on ∂S, one can find an asymptotic subsystem transformation that changes the phase or the ‘charge orientation’ of the ϕ field within S and thus creates a relational difference between the field configuration on S and the field configuration on E.Footnote ²⁸

3.3. Collecting Results, and the Real Issue

Let us take stock and consider how the results of section 3.2 relate to those of section 3.1. According to GlobalInLocal and LocalizeEmpirical, there is a global symmetry contained in each gauge symmetry, and since the global symmetry can exhibit DES, so too can the gauge symmetry—by means of the global symmetry sitting inside it. Greaves and Wallace’s framework is supposed to give us an abstract way of picking out this DES-exhibiting global part of a gauge symmetry: we identify it as those subsystem symmetries that are noninterior and boundary preserving. It then emerges that the global symmetries have the property of being asymptotically defined.

But as we saw in section 3.1, it is in general false that a global symmetry is contained in a gauge symmetry, or that the definability of a charge from a global symmetry entails the definability of that same charge from the corresponding local symmetry. So for the general case, either (i) Greaves and Wallace’s characterization of DES-exhibiting symmetries is wrong or (ii) their characterization does not in fact pick out a global symmetry that is embedded inside a local symmetry. In section 4 we will see that ii is indeed the case, because there are examples in which the DES symmetry differs from the globalization of the gauge symmetry. In the atypical U(1) example, the two happen to coincide, but this fact should not be taken as a guide to the general connection between gauge theories and Galileo’s ship.

4.1. Gauge and Redundancy

What exactly is the formal characterization of a gauge symmetry that corresponds to Redundant? Many field theories are initially formulated in terms of a Lagrangian, and indeed considering the matter from this perspective may be particularly apt if one is interested in the space-time geometry of field theory. However, the most straightforward approach is to consider the implications of constrained Hamiltonian analysis for a proper subsystem of the universe.Footnote ³⁰

First, recall that our naive space of states is the space of field configurations that satisfy the field equations and have finite energy. The evolution of a dynamical system is then given by a time-dependent curve γ through . For instance, in the case of YMH theory, we have . But there are redundancies in the YMH state space . For instance, one of the consequences of the EOM is Gauss’s Law, which implies that the time-component A ₀ of A _μ is determined by A_i, ϕ, and their derivatives. Furthermore, it is easy to see that a set of initial data does not determine a unique trajectory γ(t) through . This is because gauge transformations that are trivial at some initial time can become nontrivial at future times while remaining consistent with the EOM, and one can act on γ(t) with these to produce a whole class of solutions to the EOM that have the same initial data.

We now sketch how the constrained Hamiltonian analysis of gauge systems eliminates such redundancies.Footnote ³¹ First, recall that a Lagrangian is defined on a space whose coordinates are the fields and their derivatives, that is, , where q represents a field configuration. From the Legendre transformation , we obtain the Hamiltonian phase space, whose coordinates q and p represent fields and field momenta, respectively. The crucial point here is that, in gauge theories, this map is not surjective; instead, the image of the map is a submanifold (called the constraint surface) of P that is defined by the vanishing of a set of quantities C_j(q, p) = 0, j = 1, 2, 3, …, which are known as constraints.

We recall that the phase space P has an algebraic structure, the Poisson bracket {·, ·}, which is preserved by canonical transformations such as the dynamical evolution and the symmetries of the system. A function f is said to generate a canonical transformation on a variable h if its Poisson bracket with h is equal to an infinitesimal transformation of h; exponentiating this infinitesimal transformation then results in a finite canonical transformation.

Let us now return to consider the constraints C_j = 0. Since these do not vanish everywhere on P, there exist functions f whose Poisson bracket with C_j does not vanish on the constraint surface; in other words, they generate canonical transformations that move points out of the constraint surface. Because the equation C_j = 0 does not remain valid if we take Poisson brackets {f, ·} on both sides of it, we say that it vanishes weakly, and we denote this by C_j ≈ 0. This is a very fruitful approach to analyzing the dynamics of a physical system. For instance, one can show that in order for the dynamics to be consistent the Poisson brackets of the constraints C_j and the Hamiltonian H are required to vanish weakly. This requirement may in turn lead to other constraints, and by iterating the process of taking Poisson brackets in this way, one obtains the complete set of constraints that is required for the consistency of the dynamics—let us continue to denote this set by {C_k ≈ 0}.

It is conventional wisdom in constrained Hamiltonian dynamics that the Redundant gauge transformations are generated by the first-class constraints, that is, those constraints K_l ≈ 0 for which {K_l, C_k} ≈ 0.Footnote ³² In the YMH case there are two first-class constraints: the momentum constraint π ⁰ ≈ 0 and the Gauss’s Law constraint D_i π ⁱ ≈ 0, where π ^μ = F ^μ0.

The formal elimination of the redundancies can now be carried out in the Hamiltonian approach. Just as in section 2.2, one introduces an equivalence relation between points connected by transformations generated by the first-class constraints—the equivalence classes are the orbits of the action of these transformations. Physical (often called gauge-invariant) quantities are then taken to be the functions that are constant on these orbits. The space of orbits is thus free of redundancies and has a deterministic initial value problem.

Among other things, the constrained Hamiltonian analysis offers a framework within which one can give a formal characterization of Redundant gauge transformations. In particular, Balachandran (Reference Balachandran1994) has used it to argue that these transformations must take certain asymptotic values on the boundary of a subsystem, on pain of the Poisson brackets between first-class constraints and physical quantities of interest being ill defined. For instance, in the case of YMH theory, the argument runs as follows: smear the Gauss constraint with a test function in order to obtain a generator of gauge transformations (which we can identify by the action of its Poisson brackets on A_i and E_i). Next, evaluate the Poisson bracket of the smeared Gauss constraint with a physical quantity like J_i, the generator of rotations—one sees that there are two different ways of evaluating the Poisson bracket (based on two possible orders of integration), and the answers do not agree unless asymptotic conditions are imposed on the gauge transformations. Balachandran (see 21–22) showed that the simplest such condition is

Furthermore, the transformations generated by this constraint can be smoothly deformed to the identity Id ∈ G—transformations that have this property are called ‘topologically trivial’ or ‘small’.Footnote ³³ By contrast, the ‘large’ gauge transformations are defined as the ones that cannot be smoothly deformed to the identity. We use

to denote the group of Redundant gauge transformations, that is, those small gauge transformations that satisfy (9).

Let us pause to consider the morals of the constrained Hamiltonian analysis for our topic. As we noted in section 2.1 above, it is widely acknowledged—on philosophical grounds—that every symmetry transformation (and not just gauge transformations) of the universe connects redundant descriptions. But, when it comes to proper subsystems of the universe, many share the intuition that there is a fundamental difference between global transformations and gauge transformations: the former describe operations that can (directly or indirectly) be performed on the subsystem (and constitute a physical change in relation to its environment), whereas the latter are just mathematical artifacts of redundant descriptions of the subsystem.

The constrained Hamiltonian analysis of ‘gauge’ teaches us how to square these two ideas. First, it uses physical reasoning (in the form of the consistency of Poisson brackets) to identify as the subgroup of Formal gauge transformations that is responsible for mathematical redundancy. Second, it is natural to introduce the assumption that since elements of are deformable into the identity, they are interior symmetries in Greaves and Wallace’s sense. In other words, they are subsystem symmetries that extend to symmetries of the universe: one thereby squares the two prima facie distinct notions of ‘redundancy through being a symmetry of the universe’ and ‘redundancy through being a gauge symmetry’. Indeed, one might try to use analyze the latter type of redundancy in terms of the former; that is, Redundant gauge symmetries are redundant in virtue of the fact that they are a restriction of universe symmetries.Footnote ³⁴

To recapitulate: the constrained Hamiltonian formalism shows that given some proper subsystem, the specific type of redundancy that comes from the notion of ‘gauge transformation’ is not captured by the Formal group of gauge transformations but rather the restriction to its subgroup . In the next section, we see that is the smallest of three physically relevant subgroups of , each of which plays an important conceptual role in interpreting gauge symmetry.

4.2. Different Groups of Gauge Transformations

In what follows, we will only be interested in time-independent solutions to YMH theory (since the standard strategy for constructing solitons is to find such solutions and then boost them in order to introduce time dependence). As such, gauge transformations will be maps from space (which we take to be ℝ³) to a gauge group G.

The Formal group of gauge transformations that we first naively introduced in section 2.2 is simply

without any asymptotic conditions imposed. Next, and in keeping with the insights of the previous section, let us define the asymptotically trivial group of gauge transformations

As previously stated, constrained Hamiltonian analysis shows us that the Redundant group of gauge transformations is

, that is, the subgroup of

that is smoothly connected to the identity. We are now in a position to see why Greaves and Wallace’s claim that Trivial → Interior is false: in some cases, it can turn out that

, but in general (e.g., for the SU(2) monopoles discussed below) asymptotically trivial transformations can turn out to be topologically nontrivial; thus, the two groups cannot be identified.

In fact, these three groups do not exhaust the set of physically relevant distinctions that one might draw with respect to the ‘group of gauge transformations’, because there is a fourth possibility that was already anticipated in our discussion of KGM electrodynamics at the end of section 3.2. In the KGM example, the boundary condition of the subsystem was specified by the requirement that the field ϕ vanish in a neighborhood of the (space-time) boundary ∂S of S. In general, such boundary conditions (in conjunction with the initial conditions and EOM) need to be laid down in order to specify a well-defined dynamical system and should be treated as part of the framework of the physical problem that one wants to solve. Thus, the dynamical evolution and the symmetry transformations of the system should leave such boundary conditions invariant. In the KGM case, it was trivial to satisfy the requirement that a gauge transformation leave the boundary condition ϕ∣ _{∂ S} = 0 invariant. But it is entirely conceivable that this requirement will not be so trivially satisfied when one is dealing with more complex theories, in which case we are led to schematically define our fourth group of gauge transformations: the invariance group , which leaves invariant the boundary conditions of the fields, whatever they might be.

Notice that these four groups form a hierarchy of subgroups:

We can now see the punch line. The group of (subsystem) gauge symmetries that is a candidate for exhibiting DES must satisfy two desiderata: (1) it must leave the boundary conditions of the fields fixed (otherwise the transformation would change the type of field configuration whose properties were being studied), and (2) it must exclude the interior subsystem symmetries (i.e., the Redundant part of the gauge transformations), which cannot exhibit DES. We need the structure of

to satisfy 1, but we also need to quotient this by

to satisfy 2. Thus, the DES-exhibiting group of symmetries must be the quotient group

An informal version of this result was already present in Greaves and Wallace (Reference Greaves and Wallace2014, 87), but a more careful statement will allow us to see why one should refrain from identifying the DES-exhibiting part of a gauge symmetry with its globalization.

In order to forestall a possible misunderstanding, it is instructive to contrast this result with the case of global symmetries, which can exhibit DES (as in the original Galileo’s ship scenario). In particular, one might wonder whether (i) the idea that global symmetries that both relate redundant descriptions (when they are symmetries of the universe) and also have the potential to exhibit DES conflicts with (ii) the idea that Redundant gauge transformations need to be eliminated in order to obtain a DES-exhibiting group of symmetries. The answer is that there is no such conflict, because (nontrivial) global symmetries can only exhibit DES when they are symmetries of proper subsystems, and in this context, they do not extend to symmetries of the universe—thus there are no such redundancies to be eliminated. By contrast, the notion of a gauge symmetry allows for a rich structure of nontrivial subsystem transformations that are also symmetries of the universe—this is precisely the Redundant part of that does not exhibit DES.

Let us now revisit the KGM example with the tools that we have developed in this section. In that case, G_I is the group of gauge transformations that asymptotically tend toward a constant value on ∂S. And is easily seen to be U(1) by performing the quotient group computation in (13), which might tempt one to generally identify the DES-exhibiting symmetry of a gauge theory with the globalization of the theory’s Formal gauge symmetry on the boundary ∂S. But this would be a mistake: we can see from (13) that such an identification would be false if had a different structure. Such is the case with the ’t Hooft-Polyakov monopole, whose analysis we consider in the appendix.