2.1 General Background

This Element is about physical symmetry: that is to say, the notion of symmetry that is implicated in the empirical practice of physics. Within this practice, we pursue an understanding of certain aspects of the world – an understanding that is correlative to prediction and confirmation – by representing these aspects in particular ways, and by employing and elaborating on (one might even say: by performing) these representations. Thus, while the theme of “representation” is not my main topic, it is an essential piece of background for appreciating the notion of physical symmetry that I am after. Without some grasp of what “physical representation” – namely, the kind of representation practiced by physicists – amounts to, it would be impossible to latch onto the distinctive subject of physical symmetry (as opposed to mathematical symmetry, or symmetry in some other representational practice such as painting or sculpture), and it would be impossible to formulate the distinctive philosophical questions that pertain to this subject.

I would thus like to begin by briefly sketching my views on physical representation, in anticipation of Section 3’s discussion of how symmetry comes to play a role in physical representation.Footnote ¹ Since my view is highly analogous to the view that Reference PodroPodro (1998) advanced in the case of depiction (especially for drawings, paintings, and relief sculptures), it will help to introduce the view by first considering representation rather more generally, and getting a feel for how “idealization” – a theme of great interest in contemporary philosophy of science – plays out in an especially accessible case such as a painting.

With that in mind, let us begin with the idea of “representation” or “imitation” or “imaging” more generally. The locus classicus for a philosophical discussion of representation is of course Book X of Plato’s Republic: why, Socrates archly inquires of the slow-witted Thrasymachus, would anyone want to view a representation of a subject if they were in a position to view the actual subject? The implicit assumption here, which Thrasymachus does not think to question, is that representations are mere (albeit unachievable) attempts to provide a literal or “isomorphic” copy of the subject, and are thus always inferior to directly experiencing the subject.

The reason Socrates’ question is liable to strike us as absurd is that it flies in the face of so much of what we know about how representations are designed to work, and how they are appropriately received. Take the case of a representation such as the following Holbein sketch of St Thomas More at Windsor Castle (Figure 1).

Figure 1 Portrait Study of Sir Thomas More. Black and colored chalks on unprimed paper, $38\times 25.8$  cm, Royal Collection, Windsor

Here the stroke is loose, open, and relaxed; the blending is soft, and the tint is light; furthermore, these various elements come together so that – as Reference MartzMartz (1990) notes – we see in the sketch “the face of a man unguarded, open, vulnerable, seeking, devotional in its mood” (it is instructive to compare this with Holbein’s more famous portrait of More at the Frick, which conveys perfect composure and an iron-clad determination).

Pace Socrates, what I would like to draw your attention to is the implausibility of construing Holbein’s sketch as aiming at literal verisimilitude – as approximating the visual experience of gazing at the actual More. To understand this point, consider first the sketch’s medium (chalk and brown wash on paper) and its procedures (the loose stroke that deliberately omits detail), and then consider how these are put to representational use. The sketch does not try to disguise its medium and its procedures as it would if it were aimed at simulating the experience of seeing More; on the contrary, when we contemplate the sketch, it is an intended part of our experience that we are aware of the medium – and the quality of its strokes – as a medium; we are conscious that an aspect of what confronts us is a surface marked with very particular qualities of stroke.

An important moral here is that while the medium and its procedures (as such) can be said to have their own “meanings” (the impulse of this stroke, the torsion of that spiral, the impasto of the oil paint), the proper or “literal” domain of these meanings is the marked surface. On the other hand, in a representation these meanings are nonliterally or metaphorically transferred to a new domain, namely the subject (St Thomas More, in this case). Thus, the sketch can be said to “idealize” or “distort,” because the qualities of the medium are in the literal sense inapplicable to the representation’s subject, and must thus suffer loss in relation to more literally veridical modes of confronting its subject (actually seeing More, for instance).

In the typical case of a sketch, we are meant to understand that the meanings of the medium do not literally apply to the subject; furthermore, as Reference VasariVasari (1900) first observed in his discussion of artistic “design” or di segno, the representational power of the sketch is rooted in this understanding, as well as our grasp of the literal/proper meanings of the medium: for instance, it is through our following the formulation of Holbein’s stroke as loose, open, and relaxed that we come to experience the sketch’s distinctive power to disclose something about its subject, namely More’s openness and vulnerability. The point is that through this particular metaphorical transference of meaning, we come to perceive or think the unity that is the representation – we do not experience “the medium’s procedures” and “the face of St Thomas More” separately (nor indeed this in addition to some long chain of conditions relating the two in order to simulate a unity), but rather a fusion of the two, whose unity governs our responses. As Scruton writes (of portraits in general), “...I respond to the flowing lines and flesh-tints with emotions and expectations that derive from my experience of faces, and to the face with emotions and expectations that arise from my interest in colour, harmony, and expressive line.” (p. 87 of Reference ScrutonScruton (1997)).

It is this metaphorical transfer, I would suggest, that gives rise to much of the distinctively powerful content of representations – a content that is not propositional, and cannot be had in this particular way apart from the embodiment of thought in the representation, and a content that is (among other things) received in the kind of thought that we call contemplation. In Reference Podro and HarrisonPodro’s (1987) poetic rendering of this point, a representation “ $\dots$ directs itself to the mind of the perceiver, who sees the subject remade within it, sees a new world which exists only in the [representation] and can be seen only by the spectator who attends to the procedures of [the medium]”.Footnote ² And this, ultimately, is why we find it absurd of Socrates to suggest that a representation such as Holbein’s sketch aims to copy the literal appearances: Socrates is denying the very feature – the intended idealization and metaphorical transfer effected by the distinctive procedures of the medium and our awareness of them – that gives rise to the power of these representations.

Of course, the case can be considerably more complicated than that of Holbein’s portrait of More, as Reference PodroPodro (1998) discusses at length in his monograph Depiction. A particular choice of medium and its procedures (e.g. the technique of linear perspective) conditions the representational possibilities that are open to the artist, foreclosing some of these to a greater or lesser extent and at the same time opening up or suggesting new possibilities; indeed a certain drawing or painting procedure (think of linear perspective or pointillism) may take on a life of its own, lending a seemingly inescapable momentum to the act of representation and helping to extend our thought about the subject. And it may be that this conditioning itself complicates the act of representation and the search for a compelling representation of the subject (consider for instance the struggle to maintain realism of animation in the transition from the Florentine trecento to the more optically accurate paintings of the quattrocento). Nonetheless, some of the most remarkable representational successes in painting have been borne out of cases where these complications have been bent toward achieving a compelling representation of the subject – in Sections 4–6, we will see how a central thread in the development of physical symmetry can be understood in this way.

There is of course much more that could be said about how drawings and paintings represent. But the example of Holbein’s sketch will suffice to illustrate the general features of representation that will inform my approach to physical representation.

1. (i) A representation involves a medium and a subject.

2. (ii) A representation is an embodied, performative kind of thought – we use the medium and its procedures to think the subject in the representation, and in so doing, we reenact the pattern of attention of the representation’s maker(s), itself embodied in that medium and its procedures.

3. (iii) The di segno thesis: in a representation, we come to understand the subject by following the procedures of the medium.Footnote ³ It is through this fruitful interplay between the represented subject and our awareness of the medium qua medium that a representation distinctively extends and mobilizes our thought about its subject: it absorbs the particular momentum of the medium and its procedures and uses these to reconstruct the subject as can only be done in the world of that representation.

Let us see how this conception plays out in the case of physical representation. First, let us fill in (i) by specifying the “subject” and “medium” of physical representation. I take it that the focal subject of a physical representation is the dynamics of an empirical scenario, namely the description of how certain empirical degrees of freedom evolve over time. These dynamical degrees of freedom could be particle positions and velocities, or field configurations, or even quantities in theories whose formal description does not involve dynamical equations of motion (such as thermodynamics).Footnote ⁴

The particular forms that such an empirical scenario could take are various, but very roughly, all of them should have some conception of the target degrees of freedom whose evolution we would like to keep track of – call these the subsystem degrees of freedom – and some reference standard that makes the description and identification of the subsystem degrees of freedom an empirical matter – call the system that defines this reference standard the environment (note that on this conception, the specification of an environment is correlative to the subsystem that one wants to model). The environment system might consist of spatially localized and separated degrees of freedom from the target subsystem (e.g. some material body far away from the subsystem) as is often assumed in textbook mechanics, and – for historical reasons – this is indeed the case that I will focus on in the rest of this Element. At the same time, it is important to note that my use of the term is in principle much more general: it can encompass nonspatially separated (from the subsystem) fields of various kinds and local inertial frames – anything that provides a reasonable reference standard in the context of some measurement setup will do, and the particular details of what an environment system looks like will be highly theory dependent.

Since these are empirical scenarios, the subsystem-environment configuration needs to form what Reference CartwrightCartwright (1999) calls a “nomological machine”: a “ $\dots$ fixed (enough) arrangement of components, or factors, with stable (enough) capacities that in the right sort of stable (enough) environment will, with repeated operation, give rise to the kind of behavior that we represent $\dots$ ” Furthermore, the subsystem degrees of freedom should be thought of as implicitly indexed by a certain scale (e.g. length or energy scale) in relation to the environment and our measurement scale – typically, the ratio of scales set by this relationship that marks out the physical regime of interest and makes some particular set of degrees of freedom an empirically sensible description (it also gives us a qualitative grip on what “enough” means in the quote from Cartwright). This is not to say, of course, that in latching on to the subject one needs to have highly theoretical thoughts about how to quantify different regimes and describe precisely how they are related (as we do, for instance, when using the set of techniques known as effective field theory), but anyone seriously involved in emperically effective representation nonetheless has a loose – perhaps quite informal – sense that there is a regime involved, and of some of the bounds of this regime. In order for these representations to be taken up in use, we assume that there is an observer with access to the kinds of scales that define the measurement relationship between the subsystem and the environment; in the context of our subsequent exploration of physical symmetry, we will see that it also makes sense to introduce a subsystem observer, whose measurements are confined to the characteristic scale of the subsystem.Footnote ⁵

The subject of a physical representation tends to be something that is informally conveyed in physics classes – especially those that emphasize empirical know-how and the interface between theory and experiment – and the laboratory, whereas textbook presentations tend to emphasize the medium of a physical representation: the mathematical apparatus (differential equations, geometry, coordinate frames, functions, sections of bundles, etc.) that is used to model the subject. Nonetheless, it is important to recall the point that I made earlier about a representation being a unity: we experience a physical representation as a unity, that is not as the mathematics of the model and the empirical subject merely conjoined, but rather as inextricably fused – in the world of the model, we understand and manipulate various pure mathematical objects (geometries, symmetries, equations) in a way that is inflected by our understanding of experimental protocols and know-how, and we respond to and conceptualize empirical scenarios in a way that is driven by our interest in mathematical objects.Footnote ⁶ Of course, in the process of describing, analyzing, or teaching some part of our representational experience, it is good and natural to home in on particular aspects of it (the mathematical framework, or some aspect of the experimental configuration etc.) – similar to what one does in a close reading of literature – but this does not change the fact that one’s experience is of the whole.

With this in mind, let us proceed to the version of point (ii) that applies to physical representation: a physical representation is a kind of performative, embodied thought, in which we use the mathematical medium and its procedures (geometry, partial differential equations, the symmetries of these objects, etc.) to embody a certain pattern of attention toward an empirical scenario – the kind of attention that is concerned with understanding how to model and predict and measure, and to extend that understanding to novel scenarios. This is especially obvious when we consider how we are to receive a “dynamical equation” in a physics textbook: the mathematics that we confront is really a prompt to perform a characteristic action of a physicist, namely to use that equation (and boundary conditions, analysis of the stability and regularity of solutions etc.) and know-how about the link between theory, and experiment to model the empirical subsystem degrees of freedom – this is what it is to think in the representation.Footnote ⁷

The point is well-illustrated by considering even the simplest subject, such as the motion of some medium-sized dry good (the subsystem object) with respect to some reference object in an environment, where (let us assume) speeds are small relative to the speed of light, and the scale of the investigation is one for which the subsystem object is going to count as “free” or “isolated,” and its extension matters little. This is a paradigmatic scenario for physical representation, and one in which context we introduce the formal mathematical medium of a function $x^i:R\to R^3$ and an ordinary differential equation ${\ddot{x}}^i=0$ whose solutions are straight lines in $R^3$ .Footnote ⁸ What it means, however, for us to actually be able to receive this physical representation is to take it up in use as physics: to understand that $x^i\left(t\right)$ represents the position of the subsystem object at time $t$ and that a solution to ${\ddot{x}}^i=0$ represents an empirically possible trajectory given a certain timescale; to understand that this position should not be taken literally as a position of a point (it is not part of the content of the representation that “there is an object that has no extension...”) but rather as modelling the insight that the object’s extension does not matter in the regime of interest to us; to understand that the environment has been left implicit in the formal medium, and that modelling space as $R^3$ is not to represent it as infinite (as a confused literalistic reading might suppose) but is instead a way of dramatizing the “isolated” property of the subsystem; and to understand much else in a similar vein, including the relationship between the mathematical medium and how we are to actually prepare and measure the position and velocity of the subsystem object, that we are not actually positing that quantities have real-number values, and so on.

Finally, point (iii) – the di segno thesis – takes us deeper into the relationship between mathematical medium and subject in a physical representation, and how this relationship works to extend our physical understanding of the subject, and even to reimagine and transform the subject quite radically within the world of physical representation. The idea here is that mathematics has its own rhythms, textures, and procedures (e.g. its own impetus to generalize its objects, such as moving from flat space to curved space) and that we come in part to perceive the subject – the empirical scenario – within the representation through appreciating the medium and procedures of a physical representation as mathematical, and through the metaphorical transfer of these (mathematical) meanings to the subject, which yields the distinctive content of the representation.Footnote ⁹

To give an example of which Reference BrownH. Brown (2005) makes much, and which will be a persistent theme in much of what follows, it is easy to consider the equation ${\ddot{x}}^i=0$ and notice – at a mathematical level – that its form does not change under rotations, uniform boosts and translations in three-dimensional Euclidean space; and that these transformations take solutions to solutions. This is an example of mathematical symmetry (invariance of a mathematical object under mathematical transformations), but when it is taken up into the kind of understanding involved in physical representation, we come to perceive something physical in following the mathematical procedure of symmetry: the way in which various empirically possible subsystem trajectories are related to each other, and further – as we will discuss in Section 3 – the inability of a subsystem observer to discriminate between these trajectories.Footnote ¹⁰ In a manner analogous to Holbein sketch – where we come to perceive the openness of More’s disposition by following the openness of Holbein’s stroke – here we come to understand an empirical invariance (of what one can observe at a certain scale) by following the formulation of a mathematical invariance.

In the next section, I will say more along these lines about the historically and conceptually central case of symmetry in physics, namely the RP and how it exemplifies the notion of physical representation that we have just been discussing. On the other hand, recall the more subtle and interesting possibility that I discussed earlier in the case of painting, namely that introducing a novel procedure or modification of a representation’s medium might create a fruitful complication for the work of resolving the representation’s subject; of seeking and finding the subject again in the act of representing with novel technique. One might thus wonder: Are there examples of this kind of scenario playing out in the case of physical representation? In Sections 4–6, I will make the case that a pivotal episode in the development of physical symmetry (and its resolution) is best understood as a scenario of just this kind. The episode that I am thinking of is Einstein’s attempt to introduce a novel procedure of the mathematical medium (local spacetime symmetry) and to seek and rework the subject of the RP within this new representation, as well as Emmy Noether’s subsequent triumph: a supremely fluent rearticulation of the mathematical procedure that Einstein initiated.

To conclude this sketch of physical representation, let me say a few words about how the notions of “idealization” and “physical theories” fit into this picture. First, on idealization: the term “idealization” is used in many ways in the philosophy of science literature, but here is what I take to be a fairly neutral and uncontroversial description from Reference Frigg, Hartmann and ZaltaFrigg and Hartmann (2020): “idealized models are models that involve a deliberate simplification or distortion of something complicated with the objective of making it more tractable or understandable.” Notice that this broad description is not yet committed to any particular theory of what the content of an idealized representation is, and whether we should understand that content as making a false claim about the target of the representation.

From what I have said thus far, it should be clear that idealization in this broad sense is inherent to what I call a physical representation, because any choice of medium will introduce a divergence between the subject-as-represented and the subject itself. Nonetheless, on my account, this sense of idealization is not aptly fleshed out as a “representation as if,” that is “representing the subject as if it were something it in fact is not” in virtue of idealizing. To use an example that I already raised earlier, it is not part of the representational content of a formal point particle model of a ball that the ball is in fact a point; on the other hand, it is part of the (perhaps formally implicit) representational content that the empirical regime we want to model – the subject of the representation – is one such that it makes sense to invoke the formal device of a point.

I should note that some writers, for example Reference PotochnikPotochnik (2020), do offer an account of idealization as “representation as if ” and thus take the important phenomenon of idealization to count against the verisimilitude of the theory. While I agree with Potochnik about the importance and pervasiveness of idealization, on my view of what physical representation is, the problem is not verisimilitude but rather a literalistic conception of verisimilitude: one that fixates on the mathematical syntax of (the medium of) a physical representation and then takes that to naively disclose the content or ontological claims of a representation (“the represented object is a point...”). By contrast, in my view the focal content of a physical representation is nonliteral and one should adopt a conception of verisimilitude that aligns with that content.Footnote ¹¹

Second, on physical theories: I have said quite a bit about physical representation thus far, and not much about “physical theories” at all. Broadly speaking, I agree with Reference WallaceWallace (2021a) that as used in physics, the notion of a “theory” is “ $\dots$ ambiguous between various elements of a complex hierarchy”; furthermore, the notion tends to be used very formally (and without much concern for representation) in some quarters of the literature. For instance, on one level, one might define a theory as the collection of solutions to some equation of motion having fixed the boundary conditions (a formalistic approach to this definition is quite popular in some parts of the philosophical literature, where the “solutions” are referred to as ‘models’). And at the next level up, one might take a theory to allow for different choices of boundary conditions (thus modelling different subsystem-environment relationships). These and more general uses all have their role to play in the practice of physics, provided that one does not proceed purely formally and always considers the empirical-representational role that the relevant mathematical devices are playing. For our purposes, it will suffice to think of “a theory” as a collection of physical representations in the above sense, joined by a particular subject matter (e.g. gravity) and accompanied by a keen “know-how” about what we can do with such representations and how they are related to each other.

In the subsequent sections, we will be plunging deep into what I earlier called “the margins of analysis” of a representation, where – in an analogous way to any serious analysis of a painting – we will inevitably find ourselves immersed in commentary on subtle and technical features of the mathematical medium, while trying not to lose our sense of the representation as a whole. When we are in the throes of such necessary – but potentially disruptive – analysis, I ask the reader to bear in mind that what ultimately interests us is physical representation in the sense that I have just laid out.

2.2 The Dynamical View

In the next section, we will be entering more deeply into a discussion of how the concept of symmetry enters into physical representation, and in particular, one that draws heavily on Harvey Brown’s work on the (representational) interpretation of physical symmetry (Reference Brown and SypelH. R. Brown and Sypel, 1995). Although it is well-known that this is only one of the two key strands of Brown’s work – the other being “the dynamical approach to spacetime” – the relationship between the two is rarely remarked on, even by Brown himself. In this brief interlude, I would like to point out that the understanding of physical representation that I have sketched above can be understood as capturing some of the core insights in Brown’s “dynamical approach to spacetime,” as advanced in Reference BrownH. Brown (2005) – indeed I take it to be a way of elaborating on Brown’s view that is not only deeply faithful to the motivations behind the dynamical approach, but explains its continuity with Brown’s views on physical symmetry and will thus provide important background for our discussion of the RP in Section 3. Nonetheless, I should acknowledge that my gloss on Brown runs contrary to the mainstream interpretation of the dynamical view by subsequent commentators such as Reference Read, to C. Beisbart, Sauer and WuthrichRead (2018) (on the other hand, it is somewhat closely related to the interpretation of Brown given by Reference StevensStevens (2020), albeit without the Humeanism).

The dynamical approach is most simply stated for the theory of Special Relativity (SR), so I will confine myself to this setting here. Reference Read, to C. Beisbart, Sauer and WuthrichRead (2018), a prominent expositor of the dynamical approach, introduces it by first defining the view it is opposed to, namely what he calls the “geometrical approach”: according to Read, the geometrical approach holds that “...the Minkowski metric field ${\eta }_{ab}$ of SR is ontologically autonomous and primitive, and [...] constrains the possible form of dynamical equations for matter, such that metric symmetries coincide with dynamical symmetries.” (Here, what Read means by “metric symmetries” is those mathematical symmetries that leave ${\eta }_{ab}$ invariant, collectively known as the Poincare group.) By contrast, Read writes of the dynamical approach that on “ $\dots$ this view, the metric field ${\eta }_{ab}$ is not ontologically autonomous and primitive; rather, it is a codification of the symmetry properties of the dynamical equations governing matter fields.” (Here, the relevant frame-based special relativistic equations are left form-invariant under the action of the Poincare group, which is of course of a piece with the fact that ${\eta }_{ab}$ is used to write them in a frame-invariant form.) According to the dynamical approach, “dynamics” (in a sense that we will soon explore) is prior to, and is codified by, our interpretation of the mathematical entity ${\eta }_{ab}$ .

The point that I wish to highlight is that there are two ways to further elaborate on the dynamical view beyond the negative claim that the mathematical entity ${\eta }_{ab}$ (the Minkowski metric) should not be taken to (quite literalistically, I assume) represent some ontologically autonomous entity. The first way is already suggested by Read when he writes that “One may, therefore, understand the dynamical approach to SR—and to theories with fixed metric structure more generally—as an ontological thesis; as a form of relationalism.” Read is suggesting that the dynamical approach is itself making a positive ontological claim, where this might perhaps be further cashed out in terms of some ontological entities that are represented (literalistically, let us assume) by the dynamical equations and their symmetry properties, but not by the metric.

Nonetheless, there is a second way of understanding Brown’s claim that in SR, the Minkowski metric ${\eta }_{ab}$ is merely a codification of the dynamics, and that is to understand “dynamics” here in a representational mode: as the evolution of the subsystem degrees of freedom of an empirical scenario (which presumes the background context of an environment, a particular regime of interest involving certain length scales, time scales, and measurement accuracy, etc.). From this point of view, the metric ${\eta }_{ab}$ , the equations of motion, the relevant boundary conditions, and the Poincare symmetries – various aspects of the mathematical medium of the representation – are all codifications of the dynamics of an empirical scenario. Furthermore, while this representational reading of Brown’s dynamical approach is compatible with various ontologies, one thing that is clear on this reading is that the mathematical medium of a physical representation should not be literalistically interpreted as suggesting some particular ontology. Understood in this way, the dynamical approach is not a statement about ontology at all.Footnote ¹²

Our treatment of symmetry will be very much of a piece with this understanding of the dynamical view: the mathematics of symmetry plays a key role in the (nonliteralistic) representation of an empirical scenario, and in this capacity, it should not be taken to suggest any particular ontology (although it might in fact be compatible with various ontologies).

2.3 Further Reading

The view of physical representation that I have sketched here has a close affinity with those of several other authors writing about “modeling and representation” in the general philosophy of science. For instance, it is very much the spirit of the chapter titled “Fables and models” in Reference CartwrightCartwright (1999), and also related to certain themes in Reference ChangChang (2022), Reference Van FraassenVan Fraassen (2010), and Reference AnscombeAnscombe (1971). As a reference on the philosophy of scientific representation, I recommend Reference Frigg and NguyenFrigg and Nguyen (2020), and for a detailed study of how representation works in painting (by which my sketch has been greatly influenced), I refer the reader to Reference PodroPodro (1998). Finally, for a discussion of the concept of “spacetime” that is very sensitive to its use in physical representation, I recommend Reference BrownH. Brown (2005).

3.1 Yuyi’s Boat

I should like to initiate our discussion of physical symmetry proper by attending to – what is to the best of my knowledge – the earliest clear articulation of how the intuitive notion of physical symmetry figures in our grasp of the subject of a physical representation. In 1118, the Song dynasty poet Chen Yuyi penned the following stanza about sailing on the Grand Canal to Xiangyi, a suburb of Bianjing (here I use the translation offered by Reference McCrawMcCraw (1986)):

Putting aside the intricacies of poesy, the remarkable thing to notice is that Yuyi has here articulated a core aspect of the “ship thought experiment” discussed by Reference GalileiGalilei (1967), and that subsequent generations of commentators would discuss under the rubric of the RP.Footnote ¹³ Put in a more familiar idiom, Yuyi has noticed that in this scenario, there is an observational scale (the “subsystem scale”) according to which a traveler on the boat cannot tell the difference between being in a particular constant velocity state and an alternate scenario in which we “boost” or transform the boat into a distinct constant velocity state; and on the other hand, the poet also implicitly recognizes that there is an environment (the bank of the Grand Canal) equipped with an observational scale that is sensitive to the velocity of the boat relative to the bank, and according to which an external observer can discriminate between the boat being stationary, and the boat moving East with some nonzero velocity. In other words, from the perspective of the subsystem (and its observational scale), there is a subsystem symmetry – namely an invariance of what the “internal” or subsystem observer can detect under constant velocity boosts of his boat – but from the perspective of the total subsystem-environment configuration, there is a sense in which a symmetry fails to obtain; in a locution that I will only return to in the Epilogue, one might even say that there is a sense in which (from the total system perspective) a symmetry is “broken.”

Evidently, the symmetry (and the lack thereof) that Yuyi has in mind here is a feature of the dynamics of the empirical scenario: of the composite subsystem–environment degrees of freedom. And this is true regardless of whether – as in Yuyi’s case – one has a relatively rough and untheorized sense of this empirical dynamics, or whether, as in the cases that we are about to consider, one is viewing the dynamics under the more theorized aspect of “the equations of motion.” It is no exaggeration to say that much of the history of physics has been an attempt to articulate and rework this understanding of symmetry, which is now referred to as the RP.

How is the RP incarnated in what I earlier called a physical representation (see the discussion of Section 2)? In particular, how can some particular mathematical procedure be recruited so that in following its formulation as mathematics – and in understanding that it does not literally apply to the empirical subject – we come to reconstruct RP within the world of the representation, as a physical thought about a subsystem-environment configuration? I submit that we can find an answer to these questions in the work of Reference Brown and SypelH. R. Brown and Sypel (1995) on the philosophy of symmetry (a line of inquiry that has been carried out somewhat independently of Brown’s work on the dynamical approach to spacetime, but is in fact closely related to it on my representational understanding of the dynamical approach, cf. Section 2.2).

In their rich and learned investigation, Reference Brown and SypelH. R. Brown and Sypel (1995) trace the history of attempts to articulate the RP from Galileo onwards, through Newton, Huygens, and Einstein. For instance, they note that in Corollary V of the Principia, Newton asserts a version of the RP: “The motions of bodies included in a given space [i.e. a subsystem, in our interpretation] are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion.” The context here is that Newton was trying to recover Yuyi’s observational subsystem symmetry from within the ambit of his chosen mathematical medium for representing the dynamics of certain empirical subsystem degrees of freedom, namely the equation of motion ${\ddot{x}}^i=0$ .Footnote ¹⁴ There is a remarkable continuity between Newton’s statement and what we find Einstein writing almost two centuries later in the context of SR, viz. “The [equations of motion] by which the states of physical systems undergo changes are independent of whether these changes of states are referred to one or the other of two coordinate systems moving relatively to each other in uniform translational motion.” (Reference EinsteinEinstein, 1905).

Of course, there are differences of detail here. For Einstein, the mathematical symmetry that expresses the subsystem observer’s observational invariance is the Poincare group (the group of symmetries that leaves the Minkowski metric $\eta$ invariant), whereas in Newtonian gravity the mathematical symmetry that expresses the subsystem observer’s observational invariance is the Galilean group (the group of symmetries that leaves invariant the degenerate Galilean metric and a compatible flat connection).Footnote ¹⁵ But what Newton and Einstein’s representational strategies have in common is much more important than their differences – they both take a certain procedure of mathematical symmetry (their medium) and use it to dramatize a particular dynamical feature of an empirical scenario, namely the observational invariance of the subsystem observer and the corresponding observational variance of the external observer.

To give an explicit and very elementary example of the mathematics of symmetry that provides a medium for understanding RP (in the Einsteinian case, let us say), consider a coordinate-free (abstract index) version of the equation of motion for a free scalar field on a Minkowski background:

${\eta }^{ab}{\nabla }_a{\nabla }_b\varphi =0.$(1)

When we choose to write this equation in inertial coordinates (in which $\nabla$ becomes an ordinary partial derivative), we see that although a general Poincare transformation will take us to a different inertial coordinate frame, the form of the equation is left invariant – a mathematical feature that is referred to as the “invariance” of the equation under the symmetry.

At this point, it is instructive to note that one could give a literalistic reading of the form-invariance of this equation under Poincare transformations: what we perceive in this representation is a lonely scalar field in the universe (represented by spatially infinite Minkowski space), and the different inertial coordinate frames are just further choices of representational convention on which no physical significance hangs; thus, no physical significance can be attributed to the form-invariance of an equation under transformations from one inertial frame to another. In consequence, either this form-invariance cannot be the expression of a physically contentful idea like RP, or – if it is – then RP cannot have any physical content. But this would be obtuse – as obtuse as looking at Holbein’s sketch of More in the previous section and interpreting it as aspiring to photorealism (and judging it as such).

By contrast, Reference Brown and SypelH. R. Brown and Sypel (1995) offer a more perceptive account of how we come to recognize the RP by following the mathematical form-invariance of the scalar field equation (in an inertial frame). They emphasize two key points:

1. (1) By their lights, when Einstein writes that the special relativistic equations of motion (as expressed in an inertial frame) are form-invariant under the Poincare transformations (taking from one inertial frame to another), this mathematical statement is really meant to dramatize the observational-invariance of the outcomes of subsystem experiments set up with the same initial and boundary conditions.Footnote ¹⁶

   In the language of the previous section, we can say that Einstein has created a physical representation in which we are intended to grasp the observational invariance of Yuyi’s empirical scenario (adapted to SR) through following the mathematical formulation of “form-invariance of the SR equations of motion under Poincare symmetries,” or alternatively, “the Poincare-invariance of the Minkowski metric used to formulate the SR equations of motion.” Implicit in this representational understanding is the thought that “inertial frames” here are not mere coordinate systems, but also a stand-in for the subsystem observer – these inertial frames represent the subsystem observer’s state of motion relative to the environment frame.

2. (2) They also stress that the formulation of Einstein’s RP – and those of his seventeenth-century progenitors – assumes that the physical symmetry transformations (i.e. changes in the inertial state of motion) are performed on an isolated subsystem and are thus testable.

   In other words, Brown and Sypel are highlighting the fact that it is not just any subsystem-environment that one is considering in a Yuyi’s boat scenario, but rather one in which – due to the isolation of the subsystem – it makes sense to perform a subsystem symmetry that does not disturb the environment, and thus to compare the relational difference in the total (composite subsystem and environment) state before and after the subsystem symmetry is effected.

I will call the interpretation of RP given in (1) and (2) Representational RP because of its emphasis on how the mathematical symmetries (of equations of motion, in this case) need to be understood not literalistically, but as the medium of what I have called a physical representation: when we receive this representation appropriately, we use its mathematical procedure to reenact a certain pattern of attention – the one spelled out in (1) and (2) – toward an empirical subject. Such an understanding also makes it manifest why the RP is a physical principle and not a metaphysical or an a priori one.Footnote ¹⁷

I now turn to a more recent discussion of Representational RP by Reference Greaves and WallaceGreaves and Wallace (2014), in which Reference Brown and SypelH. R. Brown and Sypel’s analysis is re-articulated in greater generality, so as to highlight the generic conditions under which the mathematical symmetries of a physical theory can be used to represent “empirical significance” in the manner of Yuyi’s boat. For instance, in the case of RP that we have just been discussing above, we focused on rigid spacetime symmetries, that is symmetries of a fixed background (non-dynamical) metric which are rigid in the sense that they are not functions of space and time. Reference Greaves and WallaceGreaves and Wallace’s analysis clarifies the conditions under which any rigid symmetry (which need not be a spacetime symmetry) of a theory’s Lagrangian or equations of motion can be used to exhibit empirical significance in the manner of RP. For the moment, we will follow Greaves and Wallace in considering symmetries of a theory’s equations of motion, but in the next three sections, we will switch over completely to the more typical contemporary approach (within physics) of considering symmetries of Lagrangians – either way, the important point to take note of for now is that such symmetries send solutions (of the equations of motion) to solutions.

Let us now see how Greaves and Wallace’s schema for Yuyi’s boat-type empirical significance goes, to a first approximation. First, they introduce $S$ , the space of subsystem solutions to the theory’s subsystem equations of motion (equipped with “isolated” boundary conditions), and $E$ , the space of environment solutions to the theory’s equations of motion, as well as the space of solutions $T$ of the total (composite subsystem and environment) system; the rigid symmetries of $S$ , $E$ , and $T$ take solutions to solutions in each of these spaces, respectively.Footnote ¹⁸ Greaves and Wallace then tell us that a subsystem symmetry (that is, a symmetry of $S$ ) of an isolated subsystem has (Yuyi’s boat-type) empirical significance just in case (i) it preserves the “isolated” boundary conditions (since these are a defining condition of that subsystem); and (ii) the solutions of $T$ prior to, and after, performing the subsystem transformation are not related by a symmetry of $T$ .

Although this sounds like a mouthful, it is simply capturing Yuyi’s observation – and Representational RP – in more formal and schematic terms, in which equations of motion and their solutions are used to represent the dynamics of an empirical scenario from the get-go.Footnote ¹⁹ To link Greaves and Wallace’s schema back to point (1) of what I call the Representational RP in Section 3.1, we should note that implicit in their notion of a subsystem symmetry that preserve the boundary conditions – their condition (i) – is a “subsystem observer scale” with respect to which the subsystem observer cannot distinguish between subsystem experiments (set up with the same initial and boundary conditions) run in the pre- and post-transformed state. This can also be dramatized by saying that with respect to the measurement scale of the subsystem observer, the subsystem symmetry seems like a universe symmetry, which transforms everything in the same way and thus lead to any physical differences. And to explicitly link Greaves and Wallace’s schema to point (2) of the Representational RP, their condition (ii) is simply a formal way of encoding the relational difference in the total system state before and after subsystem symmetry is performed.

It may help to give a concrete example at this point: consider a Newtonian particle (represented by a uniform velocity solution) deep inside a subsystem and far from the spatial boundary that divides the subsystem from the environment, in which there is, let us assume, some other reference particle. We can imagine boosting the subsystem particle to a different uniform velocity solution (relative to the reference particle) without changing anything significant (relative to the scales of interest) about the Newtonian gravitational potential at the spatial boundary; since we would have changed the relative velocity (between the subsystem and reference particles) by performing the subsystem boost, there is no total system symmetry (i.e. boost of the composite subsystem-environment system) that connects the relative velocity pre-subsystem-boost to the relative velocity post-subsystem-boost.Footnote ²⁰ Thus, this lack of a total system symmetry is a manifestation of the relational differences that can be detected by an external observer.

Now, it is fairly uncontroversial that the rigid symmetry cases of Yuyi’s boat have “empirical significance” along the lines of what Reference Greaves and WallaceGreaves and Wallace (2014) say. However, they are further concerned with two kinds of (seemingly unrelated) novel cases in which one intuitively wants to say that symmetries have some kind of empirical significance, but where it is not clear that the standard understanding of Representational RP, or Yuyi’s boat-type empirical significance, straightforwardly applies.

The first case – and the important one for this Element – is one in which the symmetries (of the equations of motion or Lagrangian) are no longer rigid, but are now allowed to be “local,” in the sense that they are nontrivial functions of space and time. To say this case is an important one in contemporary physics would be an understatement: it is this case that Einstein had to confront head-on in developing the theory of GR, and it is this case that obtains in electromagnetism, the Standard Model, and more generally, in various gauge theories that are used in both high energy and condensed matter physics. Reference Greaves and WallaceGreaves and Wallace (2014) claim that their Yuyi’s boat-type schema can indeed be extended to apply to this case, provided that one appropriately specifies asymptotic boundary conditions that are preserved by the local subsystem symmetries (see Reference TehTeh (2016); Reference WallaceWallace (2021b) for a further discussion of this extension, and the relevant boundary conditions).

The second case has a much older pedigree; it is already mentioned by Newton as Corollary VI of his Principia:

The idea is that here again there is observational invariance for the internal/subsystem observer, but not just with respect to uniform velocity boosts; in this novel empirical scenario, the subsystem observer is unable to discriminate between different (spatially) uniform accelerations of the subsystem with respect to an environment. Furthermore, such a transformation is not obviously a symmetry of Newtonian gravitation, at least when the theory is naively written.Footnote ²¹ Greaves and Wallace say that this case indeed has empirical significance, but of a type that is distinct from Yuyi’s boat-type scenarios: they take its empirical significance to consist in the fact that the subsystem symmetry transformation (the uniform acceleration) does not preserve the subsystem boundary condition and is accompanied by a corresponding environment transformation such that the composite (subsystem-environment) state post-transformation belongs in $T$ and is physically distinct from the original composite state.

I have discussed an interpretation of this claim in Reference Ramírez and TehRamírez and Teh (2020) and a further refinement has also been given by Reference WallaceWallace (2021b), but space constraints force me to put the Corollary VI-type case aside in the rest of this Element, at least until the Epilogue. In what follows, we will enter more deeply into the central question underlying the first case, namely the question of whether local symmetries can be used to extend our conception of Representational RP – sufficient unto the day is this representational complication introduced by the (mathematical) medium of local symmetry!

3.2 Our Task

I agree with Greaves and Wallace that understanding the empirical significance of local symmetry (e.g. internal gauge symmetry or diffeomorphism symmetry) is central to any philosophical attempt to understand physical symmetry as it is used in the representational practices of contemporary physics. However, instead of focusing on their abstract schema, I wish to adopt a somewhat different tack.Footnote ²²

There are three related reasons for this. First, focusing on this sort of abstract schema tends to obscure several important elements of how the mathematics of symmetry is bent to representational ends in the practice of contemporary physics: it abstracts away from the important notion of Lagrangian symmetries and its associated Noetherian machinery of currents and charges, which is common coin amongst most contemporary physicists.

Second, it elides an important way in which contemporary practice conceptualizes an isolated subsystem, namely as a subsystem that has certain kinds of conserved charges (by virtue of being isolated from external disturbances that would result in a “leakage” of such quantities into the environment). I should note that the term “charge” is being used here in a highly general way, that is not just to include electric charge, but also quantities such as linear and angular momentum, certain notions of mass, and so on. As a matter of fact, contemporary practice not only puts much theoretical weight on this concept of charge, but also put a huge operational emphasis – these subsystem charges play the role of observables that we can probe in various experiments.

Third, I want to focus on the practicing physicist’s phenomenology of representation, and so instead of adopting an overly schematized – and thus potentially deracinated – way of articulating how local symmetries might produce complications for Representational RP, I would like to center our inquiry on an actual historical episode in which these complications became salient. Until the very early twentieth century, the tradition of physics had a relatively straightforward understanding of how the mathematics of rigid symmetry (symmetries with constant parameters such as uniform boosts) serves to embody our understanding of Yuyi’s boat-type scenarios, or the Representational RP. But shortly thereafter, this Arcadian state of affairs was disrupted by the very internal momentum driving the use of this subtle medium – symmetry – in physical representation, and which led to the introduction of the local symmetries that we have just been discussing. In the episode that I wish to direct our attention to, no less a figure than Einstein struggled with the task of understanding how – not just despite, but because of – the complications of local symmetries, he could re-construct within this novel medium the subject of Yuyi’s ship in a manner at once familiar and strange.

It is perhaps a good time to flag for the reader that in most of the representational tradition within which Einstein is working (and indeed in large swathes of present-day physics) there is no explicit modelling of the empirical scenario’s environment in the mathematics of the representation, although the environment is of course still represented, albeit by means of the boundary conditions of a subsystem (and the operational understanding of how those boundary conditions incorporate a standard of reference that comes from the environment). In fact, we already saw an example of this in Reference Brown and SypelH. R. Brown and Sypel’s discussion of how we should interpret Representational RP as embodied in the mathematical form-invariance of the scalar field equation on a Minkowski background – even though they have in mind an environment for the subsystem, this environment is only implicitly represented in the mathematical formalism; by contrast, Reference Greaves and WallaceGreaves and Wallace’s discussion of Yuyi’s boat makes this environment explicit, albeit in a highly schematized way.

In the next three sections, we will follow in the footsteps of Einstein, and thus we too will adopt the practice of incorporating our assumptions about the environment (and the subsystem-environment relationship) into the boundary conditions of the subsystem. In this sense, our representations will be less explicit than Greaves and Wallace’s schema. Nonetheless, in a different sense, they will include much physically relevant detail that is absent from Greaves and Wallace’s schema, namely detail about the structure of the space of solutions and the conserved quantities of a subsystem. These details are not only routinely appealed to by physicists (both formally and informally) but – as we will soon see – they are also essential for entering more deeply into the hermeneutics of local symmetry.

3.3 Some Orthogonal Themes

In the final part of this section, I should like to briefly explain why a certain subset of work within the philosophy of symmetry literature (which is typically presented as having “metaphysical” purport) is largely orthogonal to my topic of interest – and my line of inquiry – in this Element. On the one hand, it would be helpful to clarify for the student the reasons for this divergence; on the other hand, there is a sense in which – when suitably interpreted – some of these themes have analogs within our inquiry, and I will point to these connections.

In this Element, a physical “theory” tends to be associated (or even identified) with a class of mathematical models. These models are initially conceived of as “kinematically possible models,” which are “tuples of specified geometrical objects” (cf. Reference Martens and ReadMartens and Read (2020)) that are unconstrained by equations of motion or a Lagrangian (meaning that we do not yet constrain them to be solutions to equations of motion, or critical points of the action defined by that Lagrangian). Given such a class of kinematically possible models, one can then further restrict to a set of mathematical objects called “dynamical models,” which is the subspace of the kinematically possible models carved out by constraining the models to be solutions of the equations of motion, or to lie in the critical locus of an action.

I now turn to Reference Martens and ReadMartens and Read (2020) for a lucid presentation of a particular view about “symmetry” in terms of the above definitions, as well as several distinctions concerning symmetry that this literature finds to be important. First of all, Read and Martens are clear in their exposition that they initially treat the above definitions of models purely formally, namely as mathematics without any representational purport. And it is worth pointing out that this is already a different kind of treatment from what one is intuitively doing when one introduces a “kinematical space of fields” in a full-blooded physics setting: here, even though there is not yet any formal dynamics in the sense of equations of motion or an action, one is motivated to introduce such kinematical structures in order to represent the dynamics of an empirical scenario (even if, for now, that empirical scenario is a rather hypothetical or schematic one). In other words, the choice of kinematic structures is already one that is adapted to the project of representing a certain kind of dynamics.

Next, in order to avoid controversy surrounding what a symmetry is, Read and Martens introduce the minimal notion of a symmetry as transformations “...which (whether by definition or otherwise) are regarded as relating empirically equivalent models” (p. 7 of Reference Martens and ReadMartens and Read (2020)). At this point, representation is clearly on the scene, but if it is to be physical representation in the sense that I have established, then one needs to hear a lot more about the subject of the representation (in particular its subsystem-environment structure and the relevant scales in play) before one can arrive at a sensible judgment about whether two solutions are in fact “empirically equivalent.” Since much of the literature in this vein seems content to ignore these details in its investigation of symmetry, its ends seem largely orthogonal to my own, a point that will further emerge as we now turn to the consideration of two pairs of approaches to symmetry that are discussed in this Element.

Again following the summary in Reference Martens and ReadMartens and Read (2020), I now go over the two pairs of approaches, followed by a brief discussion of their relationship to the topic of this Element, namely the philosophy of symmetry as it relates to physical representation.

First, the “interpretational” versus the “motivational” approach to symmetry. According to the interpretational approach, “two symmetry-related models of a theory typically may be regarded ab initio as representing the same possible world, even in the absence of a coherent explication of their common ontology” (p. 7 of Reference Martens and ReadMartens and Read (2020)). By contrast, according to the motivational approach, “the existence of symmetry-related models at most motivates us to provide an explication of the shared ontology of these symmetry-related models, but only once such an explication is provided is it legitimate to regard those models as representing the same possible world.” (p. 8 of Reference Martens and ReadMartens and Read (2020)). If we take this description at face value – in which case empirical equivalence is a given prior to introducing the distinction and the notions of “possible world” and “sameness” are metaphysically loaded, then this dispute is one that is orthogonal to my representational ends.

Second, the so-called “sophistication” versus “reduction” approach to symmetry. This distinction is supposed to come into play when a physical theory has some mathematical structure – and typically symmetry structure – that one regards as “surplus”; an example that is often given (but which I disagree with, cf. Reference Nguyen, Teh and WellsNguyen, Teh, and Wells (2020)) is the local $U\left(1\right)$ symmetry of electromagnetism. “Sophistication” is the position that, when confronted with this scenario, one should reformulate the theory’s mathematics so that the models related by a symmetry are isomorphic; by contrast, “reduction” is the position that one should modify the theory’s mathematics forming a naive quotient of the space of models by that symmetry.

Since this second pair of positions does bear some relationship to the project of physical representation as I understand it (and also some relationship to a long tradition of mathematical and physical theorizing prior to the introduction of these position), there is somewhat more to comment on concerning this distinction. First, there is the issue of how one should interpret the question of whether a certain structure is “surplus” or not. If the question is one of whether, for instance, the local $U\left(1\right)$ symmetry of electromagnetism plays a role in physical representation (as opposed to a role ascribed to it by a priori metaphysical or semantic considerations), then not only is the question relevant to the present inquiry, but it has also been a central topic of discussion and interpretation within physics in the last century. On this conception, whether a structure is to be classified as “surplus” is something that needs to be adjudicated with respect to the structure’s representational use in the context of various empirical subsystem-environment configurations. And as Reference WallaceWallace (2019) has recently argued, from this practice-oriented perspective, reduction is only justified when one is trying to represent the internal degrees of freedom of a subsystem independently of any consideration of representing the coupling between this subsystem and other subsystems – in other words, it is justified (and useful) in a fairly limited range of contexts relative to our representational practice as a whole.Footnote ²³ Furthermore, since Grothendieck’s “Pursuing Stacks” and the introduction of the Batalin–Vilkovisky formalism roughly 40 years ago, mathematicians and mathematical physicists have on the whole shied away from taking the naive quotient of a space with symmetries acting on it, preferring instead to work with what is known as a “quotient stack” (or at the infinitesimal level, with an $L^{\infty }$ -algebra), which retains information about the different ways in which isomorphic objects are related (see e.g. Reference Nguyen, Teh and WellsNguyen et al. (2020) for an elementary introduction to the philosophy of this topic).Footnote ²⁴

3.4 Further Reading

For a further discussion of symmetry in this vein, I refer the reader to Reference Brown and SypelH. R. Brown and Sypel (1995), Reference Greaves and WallaceGreaves and Wallace (2014), Reference TehTeh (2016), Reference WallaceWallace (2019), Reference WallaceWallace (2021a), Reference WallaceWallace (2021b), and Reference Ramírez and TehRamírez and Teh (2020). The “subsystem-environment” view of symmetry has been challenged in Reference BelotBelot (2018) on the basis of a particular interpretation of the moduli space of t’Hooft-Polyakov monopoles; this interpretation has in turn been criticized in Reference WallaceWallace (2021b), but more work clearly remains to be done on both sides.

4.1 A Novel Mathematical Procedure

Let us put ourselves in Einstein’s shoes post SR as he contemplates the mathematical medium of symmetry en route to his formulation of GR. It is already clear to him how Representational RP (the representational interpretation of the Principle of Relativity that we discussed in Section 3.1) works in the context of SR and he has a rich physics of inertial frames in terms of which he understands RP; in other words, he knows how important the subsystem-environment conception is for understanding symmetry in physics.Footnote ²⁵ Also clear to him is the associated mathematics of symmetry – the rigid Poincare symmetries that preserve the Minkowski metric and the form of the SR equations of motion, and which relate (mathematical) inertial coordinate-frames.

Einstein’s goal is to develop a novel mathematical procedure for the physical representation of gravity, and we join him at a stage of his journey when he is fully prepared to deform the rigid Poincare symmetries of SR to what we now call diffeomorphism symmetry, that is smooth transformations between the (mathematical) object called a spacetime manifold and itself, and which are thus now local functions of the spacetime parameters. (I thus blithely skip over the prior episode of the Entwurf theory, when he was briefly prepared to give up on diffeomorphism symmetry!) Einstein’s term for the resulting formal property was “general covariance” and he initially intended it to be an adequacy condition that would single out the theory of GR.

Einstein’s physical motivations for pursuing general covariance were various and somewhat in flux. On the one hand, a key motivation came from the physics of inertial frames: certain states of motion – but not others – are privileged by the inertial frame structure of SR (which, recall, are those frames related by the Poincare group) and Einstein wanted to construct a theory in which all states of motion were on a par – thus his attempt to “enlarge” the Poincare group to the diffeomorphism group. On the other hand, the physics of inertial frames (as Einstein clearly recognized in his treatment of the RP in SR) requires an understanding of subsystem-environment relationships along the lines of Yuyi’s boat and Representational RP, and this is a point that Einstein seems to have neglected in pursuing a different conception of general covariance, wherein a generally covariant theory is one that can be formulated in such a way that the theory is expressible in any coordinate system (where the different coordinate systems are related by diffeomorphisms).Footnote ²⁶

The problem with this latter notion of general covariance, as Reference KretschmannKretschmann (1918) pointedly remarked, is that it does not achieve Einstein’s goal of singling out a particular theory – just about any theory is generally covariant according to this definition, including Newtonian mechanics, as Cartan showed in his invariant, curved geometric formulation of Newtonian gravitation (often called Newton-Cartan theory). In light of Kretschmann’s response to Einstein, it is now commonly accepted (see Reference Norton, Brading and CastellaniNorton (2003) and references therein for discussion) that this latter notion of general covariance was a misstep. Instead, one should try to provide a “physically contentful” or substantive notion of general covariance that really does pick out some proper subset of physical theories, and perhaps GR in particular. This has led to a small cottage industry of trying to articulate a notion of substantive general covariance that distinguishes it from the mere freedom to express a theory’s fields, equations of motion, and solutions in arbitrary frames (see Reference Pooley, Suárez, Dorato and RédeiPooley (2010) for a review and Reference Freidel, Teh, Read and TehFreidel and Teh (2022) for my own views on the matter).

I will return to the issue of how to distinguish substantive general covariance (or more generally: gauge symmetry) from mere expressive freedom in Section 4.3. At present, however, I would simply like to get a minimal and uncontroversial conception of general covariance on the table in order to fix ideas. Before stating this version, let me provide one more stipulation about when we are joining Einstein in his intellectual journey, namely a point at which he is using the mathematical materials of the Lagrangian formalism in order to formulate the theory of GR. Many readers will be familiar with this formalism from a course on classical mechanics: the idea there is that instead of formulating a particle theory using the equations of motion, one introduces the integral of a functional called a Lagrangian that one tries to optimize over the space of possible particle trajectories while holding the initial and final data fixed; the optimal trajectory describes the motion of said particle (and is thus a solution to the corresponding equations of motion). In the next section, we will employ this apparatus in the case of field theory but without fixing the initial and final data, so that it describes the space of solutions and not just a particular solution of the theory.

With that out of the way, let us turn to stating the “minimal” version of general covariance for a Lagrangian theory:

BGC is a notion of general covariance that is adapted to variational formulations of physical theories, and it is one that Einstein would have found uncontroversially necessary by the time he had started working on variational formulations of GR (NB: by which I do not mean that he would have found it contentful enough to count as substantive). It is also what physicists invariably mean when they speak of a “diffeomorphism invariant” Lagrangian theory.

By extension, I will also use BGC to refer to the analogous case in which diffeomorphisms are replaced by internal gauge symmetries; which case I mean will be obvious from the context. Although there are of course important differences between diffeomorphism symmetry and internal gauge symmetry, the important contrast for our purposes is between theories whose actions have only rigid symmetries (and thus do not satisfy BGC) and theories with BGC, that is whose actions have symmetries that are functions of spacetime (such as diffeomorphism symmetry and internal gauge symmetry).

As we are about to see, the novel mathematical procedure of Einstein’s general covariance, that is the local variational symmetries of BGC, led to both a complication of Einstein’s search for physical representation and – ultimately – to a profound reconceptualization of his desired subject: a novel kind of empirical scenario instantiating Representational RP. In the next subsection, we turn to the complications caused by this alteration in the mathematical medium.

4.2 Complications of the Medium

There are two closely related features that Einstein took to be crucial in imparting a sense of the subject in the physical representations of SR (where recall, the “subject” is an empirical scenario), and which he correspondingly sought through the novel mathematical procedure of BGC. First, a conception of an isolated subsystem along the lines of Yuyi’s boat or the Representational RP (which is an empirical description for the reasons that I mention in Section 1.2.1). And second, a conception of an isolated subsystem as having conserved charges, whose mathematical aspect lies in the fact that the rigid Poincare symmetries can be used to construct such mathematical quantities.

Thus, we can conceptualize part of Einstein’s quest to achieve physical representation in GR as the struggle to understand how, through the materials of BGC, one might in some sense recover these features. The reason I speak of a “struggle” is that – as we are about to see – the materials of BGC themselves produce complications for trying to represent the empirical scenario associated with RP and the subsystem charges. At this point, I should again emphasize that in much of this tradition of representation, the environment is not represented explicitly; thus, although both the physical understanding of the RP and the notion of charge require an environment reference standard, we will not be explicitly modelling the environmental degrees of freedom in what follows. I turn now to the complications.

4.2.2 The Klein-Einstein Dispute

The second complication arises when, like Einstein, we ask the question of whether the mathematical materials of general covariance can be used to represent the kind of empirical scenario – paradigmatically, an isolated subsystem! – that has physically meaningful conserved charges, such as energy. For instance, GR is a theory whose Lagrangian has the diffeomorphism symmetries of BGC: does it by virtue of this have physically meaningful charges? (We will see in Section 6 that this question is in fact of a piece with our previous question in Section 4.2.1 concerning the RP.)

Historically speaking, this question found its genesis in Hilbert’s variational treatment of GR (see Reference RoweRowe (2019, Reference Rowe2021)for a masterful treatment of the history, which I follow here). In addition to deriving the field equations in his treatment, Hilbert’s main contribution (which he called “the most important goal of his theory”) was the formulation of an invariant “energy vector” and a proof of its conservation. In response, Einstein provided his own slightly different variational approach to GR; in this note he derived an energy current that can be written in modern notation as $J_X=C_X+d{U}_X$ , where $X$ is a local symmetry, $C_X$ is a quantityFootnote ³³ that vanishes on-shell (meaning that it vanishes when the equations of motion are satisfied, that is when we are considering solutions to these equations), and $U_X$ is called the “charge aspect” or “superpotential” (these formulae are generic for any theory with local symmetry – including GR – but we will provide explicit details for electromagnetism in the next section).

It was this series of developments that set the stage for Emmy Noether’s contribution to the mathematical foundations of GR, in terms of which the complications of BGC for Einstein’s representational project are best understood. While working as Felix Klein’s research assistant, Noether worked out the exact relationship between Hilbert’s energy vector and Einstein’s energy current. This analysis, in conjunction with Klein’s own work on the topic, led to an open letter from Klein to Hilbert in which Klein explained that the energy current $J_X$ could always be decomposed as the sum of two parts, the first of which vanishes on-shell (meaning that it vanishes when the equations of motion are satisfied) and the second of whose divergence is identically zero, regardless of whether the equations of motion are satisfied (physicists usually refer to such a property as an “off-shell” property).

On the basis of this analysis, Klein asserted that the statement that $d{J}_X=0$ is merely a mathematical identity and thus does not have any physical content – hence the complication for Einstein’s attempt to represent energy conservation for isolated systems using the subtle medium of BGC. Klein’s point here, I take it, is that (as we will see more clearly in the next section), in the normal case of theories with a rigid symmetry, conservation laws hold on-shell only, that is $dJ\approx 0$ (where “ $\approx$ ” denotes equality when the equations of motion are satisfied, also called “on-shell equality”) and are in this sense consequences of the particular dynamics of the relevant physical theory; by contrast, in the generally covariant (or the gauge theory) case, the conservation law seems to hold independently of the dynamics one is trying to model and thus appears physically vacuous.

Upon reading this letter, Einstein wrote Klein to express admiration for his insights, but also to protest that “I regard what you remark about my formulation of the conservation laws as incorrect.” According to Einstein, the conservation of the current was not itself a mathematical identity, but was instead the consequence of a mathematical identity and the equations of motion. Einstein further argued that the physical interpretation of such a conservation statement was analogous to the integral form of Gauss’s law. To see what Einstein meant, it will help to have before us the cartoon of a spacetime subsystem displayed in Figure 2. Recalling that the codimension of a $d$ -dimensional submanifold in an ambient four-dimensional spacetime is given by $(4-d)$ , we see here that the initial Cauchy surface $\Sigma$ is of codimension 1, $\Gamma$ is a time-like boundary of codimension 1, and $\partial \Sigma$ is a codimension 2 submanifold that is typically called a “corner”.

Figure 2 A subsystem with corner $\partial \Sigma$

Here is how Einstein reasoned: since the current $J_X$ can be shown to be on-shell exact, that is $J_X\approx d{U}_X$ (where I remind the reader that “ $\approx$ ” denotes equality when the equations of motion are satisfied), one can use Stokes’ theorem to write the on-shell charge (the current integrated over $\Sigma$ ) as a corner quantity, namely a quantity that is only integrated over $\partial \Sigma$ . In symbols, we have ${\int }_{\Sigma }d{U}_X={\int }_{\partial \Sigma }{U}_X$ , where $\Sigma$ is a Cauchy surface and its boundary $\partial \Sigma$ is the so-called “corner” of the manifold. Einstein then proceeded to sketch a particular subsystem for whose boundary conditions (essentially a primitive version of asymptotic flatness) such a computation of charge made sense. It may help the reader here to point out the analogy between such charges and more mundane systems in which the charge of a subsystem is computed by means of quantities on its spatial boundary (where we represent the subsystem as isolated from any relevant interference), as in the use of Gauss’ Law in electrostatics.

As regards Einstein and Klein, their further exchange only led to a stalemate, with Einstein insisting that his conception of energy conservation in GR was physically contentful, and Klein denying this claim on the grounds that “mathematical identities” cannot be physically contentful statements. Who was right? While Klein’s general line of argumentation is tendentious at best (we frequently use mathematical identities to model various physical happenings), he has nonetheless identified a genuine disanalogy between how dynamics enters our understanding of conserved quantities in theories with rigid symmetries, as opposed to the role it plays in theories with gauge symmetries – a point that complicates matters for Einstein, who very much wished to maintain that analogy. We will revisit their exchange in Section 6 after we have developed the tools in Section 5 to make more sense of it (in particular, we will derive all the formulae mentioned in this subsection).

For now, what matters is that against the background of this impasse concerning physical interpretation, further progress was made with regard to the mathematical medium of GR: in 1918, Noether published her seminal paper “Invariante Variationsprobleme” Reference NoetherNoether (1918). Herein, she proved two theorems – referred to in the physics community as Noether’s first theorem and Noether’s second theorem respectively – that would eventually shape how just about every physicist thinks about symmetry, and she derived a corollary (the so-called “Hilbertian assertion”) that bore directly on the Einstein-Klein debate. More specifically, the second theorem powerfully reframes the notion of BGC in terms of what we would today call “Noether identities,” and the corollary states that (in our terminology) Lagrangian field theories with BGC only have (in virtue of that BGC) trivial currents, where we remind the reader that a “trivial current” is one that can be written as the sum of an exact form and a term that vanishes on-shell (i.e. our $C_X$ and $d{U}_X$ , respectively). Thus, the on-shell current is identically conserved and is said to have an “improper conservation law.”

It would be a mistake to understand Noether here as weighing in on the question of physical representation or interpretation; rather, one should understand her as offering a mathematician’s conceptual analysis through her corollary: given that you think the concepts of BGC and trivial currents are physically relevant, you should be aware of the following mathematical relationship between them. Read this way, Noether’s corollary does not answer our question about charges, but it does prompt the following specific version of it: “Does a theory with BGC have – on that basis – physically meaningful conserved charges?” where we should now understand the latter as “physically meaningful charges that stem from trivial currents,” since (as Noether showed) nontrivial currents cannot be derived from the diffeomorphism (or gauge) symmetry of BGC.

4.3 The Einstein Test for “Substantive”

Let me sum up the foregoing. Introducing the mathematical medium of general covariance led to two profound representational complications for Einstein – two obstacles to what Aristotle would have called theorein. As a result, Einstein was embroiled in a struggle to refashion within this new image – this theoria – the kind of subject that he took to be a focal case of physical representation: an empirical scenario exemplifying the RP (or Yuyi’s boat) and whose isolated subsystems have measurable conserved charges such as energy.

I would now like to suggest that it is fruitful to use Einstein’s representational desiderata to furnish us with a test for whether a particular conception of general covariance counts as substantive. The loose notion of “substantive-ness” that I would like to begin with is just some notion of general covariance that is physically significant in a sense that goes beyond the mere freedom to formulate a theory in any coordinate system. Of course, there may be many ways of specifying such a notion, and I certainly do not make any pretense at an exhaustive classification. However, I take Einstein’s desiderata to gesture at two ways in which a notion of general covariance can reasonably be said to go beyond the mere freedom to use arbitrary coordinates. First, if the general covariance (or gauge/diffeomorphism symmetry) in question plays an essential role in defining measurable charges for a subsystem; and second, if – given a specification of isolated boundary conditions for subsystem – the general covariance extends the RP relative to those boundary conditions.

Although I have not yet given you any reason to think that these two ways are related, they are in fact deeply interwoven, as we shall see in the next section. In anticipation of their relationship, let me introduces what I will call the “Einstein test” for a substantive notion of general covariance:Footnote ³⁴

1. 1. The “corner charge” part of the Einstein test: does the candidate for substantive general covariance yield nontrivial corner charges?Footnote ³⁵ Here, I include the qualifier “corner” in order to anticipate a result of the next section, namely that the charges associated to a gauge theory are necessarily corner charges.

2. 2. The “Extending RP” part of the Einstein test: does the candidate for substantive general covariance allow us to extend the principle of relativity (in the sense of Representational RP that we discussed)?

If the answer to both questions is “yes,” then the candidate notion counts as substantively generally covariant for the purposes of our investigation.

Given the status of BGC as the “received” version of general covariance (when formulated variationally), and Einstein, Klein, and Noether’s own recognition of the relevance of BGC to the question of conserved charges, it is natural for us to try to run the Einstein test on BGC. To that end, the next section will explain how the raw mathematical materials of Noether’s theorems can be taken up – in the art of the physicist – to develop a particularly trenchant formulation of this test. To presage our results: we will find in Section 6 that BGC does not pass the test, but also that the framework of Noether’s second theorem suggests a version of substantive general covariance that does.

5.1 The Covariant Phase Space

First, let me sketch the (mathematical) spacetime geometry of the subsystems that we will consider. It will suffice to consider the elementary case of a contractible four-dimensional spacetime $M$ equipped with Minkowski metric $\eta$ , and whose boundary has two kinds of pieces – a timelike boundary $\Gamma$ , on the one hand, and (spacelike) initial and final data surfaces $\Sigma$ and ${\Sigma }^{\prime }$ , respectively, on the other (see Figure 2 for an illustration of this geometry).Footnote ³⁷ Introducing such a timelike boundary $\Gamma$ is crucial for modelling the boundary conditions of a subsystem; furthermore, one can think of such a finite boundary model as a regularized version of a subsystem with asymptotic boundary conditions.Footnote ³⁸ I note that $\Sigma$ meets $\Gamma$ at a two-dimensional manifold $\partial \Sigma$ , which is typically called a “corner”; mathematical objects supported on $\partial \Sigma$ are correspondingly called “corner quantities.”

In this section, we will take a Lagrangian to be a spacetime four-form $L$ ; its integral on $M$ is called the action $S={\int }_{M}L$ . Consider then the equivalence class $\left[L\right]$ of Lagrangians on $M$ , where the equivalence relation is given by $L\sim L+d\ell$ , that is two representatives of the class differ by a boundary Lagrangian $\ell$ (on $\Gamma$ ). Given one of these representatives (say $L$ for concreteness) and fixing boundary conditions for the subsystem on $\Gamma$ , we characterize the solutions (or dynamical states) of the subsystem as those field configurations such that an arbitrary variation of the action is stationary up to terms that are supported at $\Sigma$ and ${\Sigma }^{\prime }$ , that is

$\delta S=\delta {\int }_{M}L={\int }_{\Sigma }\Psi -{\int }_{{\Sigma }^{\prime }}\Psi ,$(2)

where we assume that the variation satisfies the boundary conditions on $\Gamma$ and $\Psi$ is a local function of dynamical and background fields on $\Sigma$ and ${\Sigma }^{\prime }$ . The reason that stationarity has this form – as opposed to the more familiar $\delta S=0$  – is that we would like to use it to construct an entire space of possible solutions (corresponding to different initial data) and thus we do not place any boundary conditions on $\Sigma$ and ${\Sigma }^{\prime }$ .

To see why stationary field configurations are solutions to equations of motion in the usual sense, let us first take note of the fundamental variational formula (a version of which appeared in Reference NoetherNoether’s Reference Noether1918 paper)

$\delta L=-E+d\theta ,$(3)

where $\delta$ should here be thought of as an exterior derivative on the space of fields, $d$ is the usual spacetime exterior derivative, $E:=E_{\varphi }\delta \varphi$ is a field space $1$ -form (and spacetime $4$ -form) whose coefficient $E_{\varphi }$ is the Euler-Lagrange quantities, and $\theta$ is a field space $1$ -form (and spacetime $3$ -form) called the pre-symplectic potential current.Footnote ³⁹ Upon integrating (3) over $M$ , we see that in order for (2) to be satisfied for arbitrary variations that satisfy the boundary conditions on $\Gamma$ , the configuration about which we are varying needs to satisfy the equation of motion $E_{\varphi }=0$ (furthermore, changing $L$ by a $\Gamma$ boundary term to obtain a different representative of $\left[L\right]$ yields the same bulk equation of motion).

Matters are slightly more subtle for the $d\theta$ term in (3), which descends to $\Gamma$ , $\Sigma$ , and ${\Sigma }^{\prime }$ when integrated, by means of Stokes’ theorem. In order to choose boundary conditions for $\theta {|}_{\Gamma }$ such that (2) is satisfied, it will help to note that $\theta$ has the generic form $A\delta B$ , and thus there are two kinds of boundary conditions that we can impose in order to guarantee that $\theta {|}_{\Gamma }$ vanishes) – for instance, we can set $B{|}_{\Gamma }$ to any of some set of fixed values which kills the variation (thus yielding a family of boundary conditions) or we can simply set $A$ to zero; in what follows we will think of boundary conditions in the former “family” sense.Footnote ⁴⁰ For instance, if $\theta =A\delta \varphi$ , then introducing a particular Dirichlet boundary condition $\varphi ={\varphi }_0$ on $\Gamma$ would amount to singling out a “leaf ” in the family of Dirichlet boundary conditions on $\Gamma$ . In this sense, $\theta$ is associated with a family of boundary conditions (often called a “polarization”), and since in our scheme each Lagrangian $L\in \left[L\right]$ picks out a unique $\theta$ , $L$ is also associated with that family of boundary conditions.

The covariant (pre-)phase space $P$ is the space of subsystem solutions that we have just constructed by means of a variational principle (i.e. field configurations that satisfy $E_{\varphi }=0$ ). However, it is important to notice that we will not be thinking of $P$ as a mere set – that does not carry further structure – of subsystem solutions satisfying certain boundary conditions on $\Gamma$ (as I discussed in Section 2, this is how Reference Greaves and WallaceGreaves and Wallace (2014) think of the data of a subsystem). This is because the mathematical procedure that we have just used to construct $P$ also endows it with the rich structure of a pre-symplectic manifold, in the sense that $\delta \theta$ can be integrated over an initial data surface $\Sigma$ to construct the presymplectic field space 2-form $\Omega :={\int }_{\Sigma }\delta \theta$ on the space of solutions.

At an intuitive physical level, $\Omega$ is a mathematical structure that keeps track of the distinct physical degrees of freedom of the subsystem. For instance, when a vector field (on field space) is in the kernel of $\Omega$ it can be quotiented out to eliminate the degeneracy and to arrive at a correct count of the physical degrees of freedom of the system.Footnote ⁴¹ In the absence of boundary conditions on $\Gamma$ , therefore, there is no reason to expect that $\Omega$ will be independent of which data surface $\Sigma$ it is defined on; it could well change from surface to surface, depending on the subsystem’s interactions with the environment – such changes are usually referred to in the literature as the “leakage of symplectic flux through the boundary $\Gamma$ .” Nonetheless, in our setting, we have imposed “closed” boundary conditions on $\Gamma$ that ensure that there is no leakage of symplectic flux, and thus that $\Omega$ is independent of which $\Sigma$ we are integrating over. In other words, in our setting, the pre-symplectic form $\Omega$ is a property of the entire closed subsystem.

I now proceed to introduce Noether’s theorems. In order to do so, let us consider the case where a vector field on field space $\xi$ is a variational symmetry of a Lagrangian $L$ , meaning that $L_{\xi }L=d{\ell }_{\xi }$ , where $L_{\xi }$ denotes the Lie derivative (with respect to $\xi$ ) on the space of fields.Footnote ⁴² The Noether current is defined as (the spacetime 3-form) $J_{\xi }:=I_{\xi }\theta -{\ell }_{\xi }$ , where $I_{\xi }$ is the field space interior product which is here being used to contract the field space $1$ -form $\theta$ with the field space vector field $\xi$ so as to yield the field space $0$ -form $J_{\xi }$ . The Noether charge can then be constructed by integrating $J_{\xi }$ over a Cauchy surface $\Sigma$ , that is as $Q_{\xi }:={\int }_{\Sigma }{J}_{\xi }$ .

Suppose that our Lagrangian $L$ has a variational symmetry $\xi$ . We can then use Cartan’s magic formula ( $L_{\xi }=\delta I_{\xi }+I_{\xi }\delta$ ) and (3) to derive the formula $d{J}_{\xi }=I_{\xi }E$ . The part of Noether’s first theorem that I would like to consider here now follows immediately from noticing that if $\xi$ is a rigid symmetry, then $J_{\xi }$ is a nontrivial Noether current in the sense discussed at the beginning of this section, that is it cannot be written as the sum of an exact form and a term that vanishes on-shell. Thus, we have a part of Noether’s first theorem, namely that given a rigid symmetry and its associated current $J$ , when we go on-shell (i.e. set $E_{\varphi }=0$ ), we obtain the corresponding conservation law $dJ=0$ . This is what Klein meant when he said that (in our paraphrase), in the case of rigid symmetries, the conservation laws are a consequence of the dynamics of the physical theory.

On the other hand, when the symmetry $\xi$ is a local symmetry, that is a nonconstant function of spacetime, and thus has an infinite number of parameters, we find that we can write $I_{\xi }E=d{C}_{\xi }+N_{\xi }$ where $C_{\xi }$ is a constraint term that vanishes on-shell, and $N_{\xi }$ is the sum of terms linear in $\xi$ , each of which contains a differential operator acting on the Euler-Lagrange quantities (in the next section, I will carry this out explicitly for the case of electromagnetism).Footnote ⁴³ We can then use the locality of $\xi$ to argue that $N$ must vanish as an identity – this yields the nontrivial Noether identity corresponding to the local symmetry $\xi$ .Footnote ⁴⁴ By combining this observation with $L_{\xi }L=d{l}_{\xi }$ and (3), we can conclude that $d(J_{\xi }-C_{\xi })=0$ and so the Noether current takes the form $J_{\xi }=C_{\xi }+d{U}_{\xi }$ , where the “superpotential” or “charge aspect” $U_{\xi }$ depends on the gauge parameter $\xi$ in a linear but nontrivial manner. In other words, we have just demonstrated a part of Noether’s second theorem, viz. that a gauge symmetry gives rise to a “trivial” Noether current, that is a current that can be written as the sum of an exact form and a term that vanishes on-shell.

There are two extremely important points to emphasize about the superpotential $U_{\xi }$ at this stage:

• The superpotential $U_{\xi }$ is uniquely determined by our choice of Lagrangian in the equivalence class $\left[L\right]$ and the symmetry generator $\xi$ .Footnote ⁴⁵

• For gauge theories, that theories with BGC, the Noether charge $Q_{\xi }^{\Sigma }:={\int }_{\Sigma }{J}_{\xi }={\int }_{\partial \Sigma }{U}_{\xi }$ is always a corner quantity (in the sense that it is being integrated over the corner $\partial \Sigma$ ) because the Noether current is on-shell exact, and thus by Stokes’ theorem, the charge is determined entirely by the superpotential $U_{\xi }$ and the geometry of the corner.

To round off our discussion of Noether’s theorems, let me note that at a highly schematic level, the chief moral of Noether’s two theorems for Lagrangian field theory is that the space of such theories can be partitioned according to whether or not the theories possess two features, which in turn correspond to the symmetry-structure of the theories’ Lagrangians. These two features are:

• Noether currents, denoted by $J_{\xi }$ . A nontrivial Noether current is one that cannot be written as the sum of an exact form and a term that vanishes on-shell. A trivial Noether current is one that can be so written.

• Noether identities – denoted $N_{\xi }$  – which are relations between the Euler-Lagrange quantities that take the form $N^i\left[\frac{\delta L}{\delta {\varphi }^i}\right]=0$ , where the coefficient $N^i$ is a differential operator (note that since these are mathematical identities, they hold off-shell). A nontrivial Noether identity is one whose coefficients $N^i$ do not vanish on-shell. A trivial Noether identity is one whose coefficients vanish on-shell.

Furthermore, the correspondence between Noether currents/identities and a theory’s symmetry-structure is given by Noether’s two theorems (and their modern interpretations), as follows:Footnote ⁴⁶

• (Noether’s first theorem) Rigid symmetries (symmetries with a finite set of parameters) are in 1-1 correspondence with nontrivial Noether currents. Furthermore, these currents are conserved when we assume the on-shell property in computing the divergence of the current.

• (Noether’s second theorem) Local symmetries (symmetries with an infinite set of parameters, that is symmetries in the sense of BGC) are in 1-1 correspondence with nontrivial Noether identities. Furthermore, they yield trivial currents whose on-shell expression is conserved as a mathematical identity, that is without using the on-shell property in computing the divergence.

Thus, instead of speaking of the local symmetries – or BGC – of a physical theory, we could just as well speak of its nontrivial Noether identities; instead of speaking of the rigid symmetries of a physical theory, we could just as well speak of its nontrivial Noether currents (in fact, this substitution has become standard in physics subfields such as Quantum Field Theory).

With these observations in hand, we can see that it is the “charge” – indeed “corner charge,” since we are discussing gauge theories! – part of the Einstein test which is more fundamental than the “Extending RP” part of the test: if the corner charges for a gauge theory are trivial (i.e. vanishing), then there is no sense in which the RP can be extended (from a rigid precursor theory) to that gauge theory. If, on the other hand, the corner charges are nontrivial, then subject to understanding what kinds of boundary conditions model an “isolated system,” it may be possible to obtain a generalized version of RP in that gauge theory.

5.2 Charges and their Relationship to the GW Schema

With the insights from Noether’s theorems (as absorbed into the covariant phase space formalism) in hand, we are in a position to see why I used the term “corner charge” in formulating the first part of the Einstein test for substantive general covariance in Section 4.3: this is because the Noether charges stemming from local symmetry are always corner quantities, that is quantities defined on the co-dimension 2 manifold $\partial \Sigma$ . This is one of the distinctive features of the mathematical medium of local symmetry that makes it so different from rigid symmetry.

We are also now in a position to understand why the “corner charge” part of the Einstein test (for substantive general covariance) and the “Extending RP” part of the test are deeply interwoven, as I claimed in Section 4.3. In order to appreciate this point in summary, let us recall a basic idea from Hamiltonian classical mechanics, where a symplectic manifold (a $2n$ -dimensional smooth manifold $P$ endowed with a symplectic form $\omega$ ) and a Hamiltonian $H:P\to R$ are used to describe the time evolution of a particle. In this case, we say that $H$ is a “Hamiltonian charge” because it satisfies the following equation:

$-I_{X_H}\omega =\delta H,$(4)

where $X_H$ is the vector field associated to $H$ by $\omega$ , $\delta$ is the exterior derivative on $P$ , and $I$ is a contraction on $P$ . Furthermore, we say that the Hamiltonian charge $H$ generates time evolution because the integral curves of $X_H$ describe the time evolution of the particle. Analogously, in the covariant phase space description of classical field theory, we can define the Hamiltonian generator of a subsystem symmetry as a charge that satisfies the analog of (4), and in virtue of which it is called a “Hamiltonian charge.” The link, then, between the first and second parts of the Einstein test is that a nontrivial corner charge (available via general covariance if it passes the first part of the test) is a candidate for playing the role of a Hamiltonian charge that generates a subsystem symmetry that “extends RP,” thus passing the second part of the Einstein test, and recovering Representational RP in theories with local symmetry.

Let us briefly see how the connection between charges and Greaves and Wallace’s schema for Yuyi’s boat (their version of Representational RP) plays out in a case in which the relevant charge is not a corner quantity, namely the case of rigid symmetry.

First, recall how the GW schema for Yuyi’s boat goes: the empirically significant subsystem symmetry has to preserve the isolated boundary conditions of the subsystem, and it has to be such that the composite (subsystem and environment) state prior to and after the subsystem symmetry are not related by a symmetry of the composite system. And recall too an important point that I have already commented on, but wish to emphasize yet again: there is of course a mathematical gap between the GW schema and the covariant phase space framework as we have just been developing it, because the latter is not in the business of explicitly modelling the environment. Nonetheless, one is still representing an environment (albeit implicitly) in our use of this framework.Footnote ⁴⁷

With that in mind, let us now explain how Noether’s charges for rigid symmetries play into this schema. For concreteness, it will help to consider an example such as the free complex scalar field Lagrangian $L=-1/2d{\varphi }^{*}\wedge \star d\varphi$ and its rigid $U\left(1\right)$ internal symmetry , where $\alpha$ is a constant parameter. Let $\xi =-i\alpha \delta /\delta \varphi$ be such an infinitesimal rigid symmetry represented as a tangent vector on field space. From Noether’s first theorem, the fact that $\xi$ is a variational symmetry of $L$ implies that we have an on-shell conserved Noether current $J_{\xi }$ which can be used to define a charge $Q_{\xi }={\int }_{\Sigma }{J}_{\xi }$ (albeit one that does not descend to a corner like gauge charges).

Having put in place the above as well as our understanding of an implicitly represented environment (which carries a reference standard for the phase of $\varphi$ ), it is now easy to spell out the connection to Greaves and Wallace’s schema: the Noether charge $Q_{\xi }$ is the Hamiltonian generator of the (phase space) subsystem symmetries of Greaves and Wallace’s schema, in the sense that it satisfies the following analog of (4) on-shell:

$\delta Q_{\xi }=-I_{\xi }\Omega ,$(5)

and the flow of its associated vector field implements the symmetry as a canonical transformation on phase space. In other words the Hamiltonian charge $Q_{\xi }$ generates the relative phase difference between subsystem and environment that is discussed in Reference Greaves and WallaceGreaves and Wallace (2014). A similar point could be made for other Lagrangian theories with rigid symmetries, and to which Greaves and Wallace’s schema for Yuyi’s boat applies.

What then of the (variational) local or gauge symmetries of Lagrangian theories? Do these also invariably yield nontrivial corner charges that act as the Hamiltonian generator of the relevant subsystem symmetries? Supposing that local symmetry here yields a nontrivial corner charge (a point that we shall investigate on further in the next section) this matter is somewhat subtle: as Reference Chandrasekaran, Flanagan, Shehzad and SperanzaChandrasekaran et al. (2021) and Reference Freidel, Oliveri, Pranzetti and SpezialeFreidel et al. (2021) argue, in the absence of any boundary conditions, one does not obtain (5) but instead the charge-flux relation:

$\delta Q_{\xi }-F_{\xi }=-I_{\xi }\Omega ,$(6)

where the symplectic flux term $F_{\xi }$ measures the degree to which the corner charge $Q_{\xi }$ fails to be Hamiltonian. In our setting, we have imposed boundary conditions that kill off this symplectic flux term, thus making the charge Hamiltonian relative to that boundary condition. Thus, assuming that we have chosen the right “isolated” boundary conditions for Representational RP, our discussion provides a general strategy for reconstructing RP in the case of generally covariant theories.

It is worth pausing to reflect on our earlier characterization of physical symmetry as a coarse-grained description of the empirical dynamics of a subsystem vis-a-vis a possibly implicit environment. Greaves and Wallace’s schema – which represents dynamics under the aspect of the equations of motion – is one expression of this characterization, and we have just seen that the notion of variational symmetries – in which context dynamics is represented under the aspect of a Lagrangian – offers yet another expression; the two are linked by a sufficiently rich representational understanding of what it means for a subsystem charge to generate a subsystem symmetry.

5.3 Further Reading

For an introduction to the covariant phase space from a physicist’s perspective, I recommend Reference Donnelly and FreidelDonnelly and Freidel (2016), Reference Freidel, Geiller and PranzettiFreidel et al. (2020, Reference Freidel, Oliveri, Pranzetti and Speziale2021), and Reference Harlow and WuHarlow and Wu (2020). A somewhat more careful treatment of analytical issues is given in Reference KhavkineKhavkine (2014).

In Section 5.2, we have been assuming the imposition of non-gauge-invariant boundary conditions; for a discussion of gauge-invariant boundary conditions, see Reference Carrozza and HoehnCarrozza and Hoehn (2021); Reference Mathieu, Murray, Schenkel and TehMathieu, Murray, Schenkel, and Teh (2019); Reference Mathieu and TehMathieu and Teh (2021); Reference Gomes and RielloGomes and Riello (2017) were the first to give a general discussion of abstract gauge-invariant boundary conditions within the physics context. The most general treatment of “virtual” or Segal boundaries for gluing has been given in Reference Cattaneo, Mnev and ReshetikhinCattaneo et al. (2014), and a particular case (in homological degree zero) is discussed in Reference Gomes, Hopfmüller and RielloGomes, Hopfmüller, and Riello (2019).

6.1 Test Cases

Let us now run the “corner charge” part of the Einstein test on BGC, that is pose the question: does a theory with BGC thereby have nontrivial corner charges? To that end, it will be helpful to use as our test cases some of the toy models that Reference Pooley, Suárez, Dorato and RédeiPooley (2010, Reference Pooley, Lehmkuhl, Schiemann and Scholz2017) and others use to test the validity of different definitions of general covariance, although I warn the reader that for reasons of simplicity, I have opted for a slightly different labeling convention from that of Reference Pooley, Lehmkuhl, Schiemann and ScholzPooley (2017).

Consider first the following theory of a free scalar field on Minkowski space that we will callFootnote ⁴⁸ SR1Footnote ⁴⁹:

$L(\varphi ,\eta )=-\frac{1}{2}{\eta }^{ab}{\partial }_a\varphi {\partial }_b\varphi ,$(7)

where the scalar field $\varphi$ is the dynamical variable and the Minkowski metric $\eta$ is a nondynamical (or background) structure. Our variational formula (3) for this theory is $\delta L=E_{\varphi }\delta \varphi +{\nabla }_a{\theta }^a$ , where the Euler-Lagrange expression is $E_{\varphi }=\square \varphi$ and the pre-symplectic potential current is ${\theta }^a=-{\nabla }^a\varphi \delta \varphi$ . Notice that the conservation of the stress-energy tensor $T^{ab}$ follows from applying Noether’s first theorem to the rigid spacetime symmetries of this theory.

Clearly, this theory does not satisfy BGC, because diffeomorphisms are not a variational symmetry of $\varphi$ . In our next example, I will modify SR1 slightly to show that one can produce an example of a theory that satisfies BGC, but which fails the “corner charge” part of the Einstein test, because it only has a trivial superpotential, and thus the corner charge vanishes.

6.1.1 BGC with Trivial Superpotential

Consider now a theory that we will call SR2 which is given by,

$L^{\text{SR2}}(\varphi ,Y;\eta )=L(\varphi ,Y^{*}\eta ),$(8)

where $Y:R^d\to M$ is a parametrization field that captures our freedom to choose coordinates on the spacetime $M$ .Footnote ⁵⁰ In particular, we can use $Y$ to express this special relativistic theory in non-inertial frames if we wish. Due to this interpretation of $Y$ , it is appropriate to call SR2 a “Kretschmannized” version of SR1. Our dynamical variables in this theory will be $\varphi$ and $Y$ , whereas the Minkowski metric $\eta$ still retains its status as a non-dynamical background structure.

We can again compute the variational formula (3) for this theory, thus obtaining $\delta L^{\text{SR2}}=E_{\varphi }\delta \varphi +E_a{\chi }^a+{\nabla }_a{\theta }^a$ , where $\chi :=\delta Y\circ Y^{-1}$ , $E_a=-2{\nabla }_b{T}^{ba}$ , and ${\theta }^a=2T^{ab}{\chi }_b-{\partial }^a\varphi \delta \varphi$ . Notice that in addition to the previous equation of motion $E_{\varphi }=0$ , we now have a new equation of motion $E_a=0$ , that is the conservation equation for the stress-energy tensor. Furthermore, by inspecting $\theta$ , we can see that in addition to the original symplectic pair $\varphi$ and its momentum ${\nabla }^a\varphi$ , we now have a new symplectic pair $Y^a$ and its momentum $T^{ab}$ .

We now draw the reader’s attention to an interesting point: clearly, $L^{\text{SR2}}$ is generally covariant in the sense of BGC (this is true even though $\eta$ remains a non-dynamical field and is not varied) and thus this theory must have a nontrivial Noether identity. By turning the crank of the Noether machine, we find that this identity is $E_{\varphi }{\partial }_a\varphi -2{\nabla }_b{T}^{ba}=0$ (that this is an identity can be straightforwardly checked from the definition of $T_{ab}$ ). In other words, the content of the Noether identity SR2 is actually the content of Noether’s first theorem for SR1 – one might call this the transmutation of Noether’s first theorem into Noether’s second theorem via Kretschmannization!

It is also interesting to compute the (trivial) Noether current, i.e. $J_{\xi }^a=I_{\xi }\theta -{\iota }_{\xi }{L}^{\text{SR2}}=0$ . In other words, the current vanishes identically off-shell, and we can deduce that the superpotential $U_{\xi }$  – and thus the corner charge algebra – is trivial. Hence, the theory SR2 demonstrates why BGC fails the “corner charge” part of the Einstein test for substantive general covariance (and in consequence fails the “extending RP” part of the test): even though this theory clearly has BGC, it nonetheless fails to deliver nontrivial corner charges.

The example is exemplary, and it is easy to construct analogous examples of Kretschmannized theories with a similar structure. For instance, consider the case of a scalar field theory coupled to a flat background $U\left(1\right)$ gauge field, but now equipped with a parametrization field that captures our freedom to change our typical representation of this background gauge field from $A=0$ by switching to a “non-inertial” choice of section (which one can think of as an “internal frame” if one wishes). The Lagrangian corresponding to this case is $L(\varphi ,\theta )=-1/2d_{d\theta }\varphi \wedge \star d_{d\theta }\varphi$ , where $\theta$ is a “parameterization field” that acts on a gauge field $A$ as ${\theta }^{*}A=A-d\theta$ (thus shifting our representation from the usual choice $A=0$ to $A=-d\theta$ , for instance) and which transforms under the $U\left(1\right)$ gauge transformation $A\mapsto A+d\chi$ as $\theta \mapsto \theta +\chi$ ; and where $d_{d\theta }$ denotes the gauge covariant derivative. (Note that in this case, the “inertial frames” are precisely the ones where $\theta$ is constant – thus stabilizing $A=0$  – and one simply has the ordinary exterior derivative, whereas the non-inertial frames are the ones where $\theta$ is nonconstant and so one is forced to introduce the covariant derivative.) Upon performing the covariant phase space analysis of this example, one finds that it replicates the structure of the previous example, in the sense that it clearly satisfies BGC (for $U\left(1\right)$ gauge symmetry) but only yields a trivial superpotential and thus also a trivial corner charge.

To sum up, this section has established P2, namely that there are two examples of theories which satisfy BGC (because they are Kretschmannized versions of theories that do not satisfy BGC), but which nonetheless only have trivial corner charges.

6.1.2 BGC with Nontrivial Superpotential

Next, we turn to a familiar example in which BGC does yield a nontrivial corner charge (and thus passes the “corner charge” part of the Einstein test). Consider the Maxwell gauge theory Lagrangian

$L=-\frac{1}{4}F\wedge \star F,$(9)

where we have chosen a representative $L\in \left[L\right]$ with vanishing boundary Lagrangian, that is $\ell =0$ . The variational formula (3) is

$\delta L=-E\delta A+d\theta ,$(10)

where $\delta$ is the exterior derivative on the space of fields, $E=d\star F$ is the Euler-Lagrange expression, and $\theta =\star F\delta A$ is the presymplectic potential current.

In this case, the gauge symmetry vector field on field space has the form $\hat{X}=dX(x,t)\delta /\delta A$ , and $L_{\hat{X}}$ is the field space Lie derivative with respect to this vector field, although we will now abuse notation by using $X$ to denote both the field space vector field and the local gauge parameter. Since $L_{X}L=0$ , we can repeat the general analysis summarized in Section 3.1 to obtain

$0=L_{X}L=d\left(I_X\theta \right)-I_X\left(E\delta A\right)$(11)

and then by using integration by parts and the definition of the current $J_X=I_X\theta$ , we have

$d(J_X-EX)=-XdE.$(12)

Since $X$ is an arbitrary gauge symmetry (i.e. a local function of spacetime), we can integrate both sides over a domain with boundary, apply Stokes’ theorem to the LHS, and – by assuming that $X$ and its derivatives vanish on the boundary – deduce that $dE=0$ as an identity. This is a simple example of a nontrivial Noether identity (here the Noether operator $N$ is just the exterior derivative $d$ , which does not vanish on-shell). We note that $dE=dd*F=0$ is clearly a mathematical identity since it follows from $d^2=0$ .

By applying the Noether identity $dE=0$ and the Poincare lemma, we see from (12) that

$J_X=EX+d{U}_X\approx d{U}_X,$(13)

where “ $\approx$ ” denotes “on-shell equality.”Footnote ⁵¹ In other words, we see that the current $J_X$ that is associated with the gauge symmetry is trivial, although – as we are about to see – this of course does not imply that the Noether charge is trivial!

Given that we have a current $J$ , we can define a charge by integrating $J$ over a spacelike Cauchy surface $\Sigma$ to obtain:

$Q^{\Sigma }={\int }_{\Sigma }J.$(14)

We now remind the reader that a unique possibility arises for trivial currents (but not nontrivial currents), which is a direct consequence of the nontrivial Noether identity: by Stokes’ theorem, a trivial on-shell current $J_X\approx dU$ can be converted into a purely boundary Noether charge

$Q_X^{\Sigma }\approx {\int }_{\Sigma }d{U}_X={\int }_{\partial \Sigma }{U}_X={\int }_{\partial \Sigma }X(\star F),$(15)

where in the last equality we used the explicit form of the superpotential $U_X$ (fixed by $X$ and our choice of $L$ ), which – unlike the case of SR2 – is nontrivial and leads to a nontrivial corner charge algebra.

To go further and say more about the conservation of this corner charge and its role as Hamiltonian generator, one needs to explicitly spell out the subsystem (canonical) boundary condition that we have been assuming in our analysis, namely that the pullback of $\theta$ to $\Gamma$ vanishes. Recall that such a boundary condition corresponds to our particular choice of $L$ , and also rules out the leakage of “symplectic flux” from the subsystem (see Reference Freidel, Oliveri, Pranzetti and SpezialeFreidel et al. (2021); Reference Harlow and WuHarlow and Wu (2020) for a more comprehensive discussion of flux leakage).Footnote ⁵² In the context of the present example, the pre-symplectic potential current is $\theta =\star F\delta A$ , and so the corresponding boundary condition is $\delta A{|}_{\Gamma }=0$ .Footnote ⁵³ This is a Dirichlet boundary condition for the gauge field $A$ , whose standard interpretation is precisely as representing an isolated subsystem, for which the electric charge should be conserved.Footnote ⁵⁴

The symmetries of the subsystem need to preserve this boundary condition, and so we should set $dX{|}_{\partial M}=0$ . Evidently, when $X{|}_{\partial M}=0$ , the symmetry is in the kernel of $\Omega$ and so does not represent a physical symmetry generated by a charge, leaving non-zero constant transformations at the boundary $X$ as the symmetries generated by the corner charge $Q_X^{\Sigma }$ (notice that these are precisely the nontrivial stabilizers of the boundary condition).Footnote ⁵⁵ Thus, we find that in the present example, the Noether corner charge is also the Hamiltonian charge associated with our boundary condition of interest – the Dirichlet boundary condition – and it generates “rigid on the boundary” $U\left(1\right)$ symmetry transformations in phase space. We have thus produced an example in which we are able to connect the nontrivial corner charges of a gauge subsystem to the subsystem symmetry transformations of Greaves and Wallace’s schema in the manner discussed in Section 5.2 – indeed, such boundary rigid subsystem symmetries are precisely what they appeal to when they argue that gauge theories can instantiate Yuyi’s boat-type scenarios. However, as we will soon see, there can be more complex scenarios in which a Hamiltonian charge associated with a particular boundary condition is the Noether charge of a shifted Lagrangian $L+d\ell$ , but not of the original Lagrangian $L$ that we considered. In Section 6.3, I will briefly discuss how one can resolve this tension between Noether and Hamiltonian charges – which is relevant to the “extending RP” part of the Einstein test for substantive general covariance – but let us first collect results from the present discussion.

6.2 The RP and Charges

We have just seen that, due to examples such as SR2, theories with BGC might still fail the “corner charge” part of the Einstein test for substantive general covariance. On the other hand, we have also seen that central examples of theories with BGC (electromagnetism, and by a simple extension: Yang–Mills theory and GR) do yield nontrivial corner charges. The general moral is that while BGC is insufficient for the existence of nontrivial corner charges, it is nonetheless necessary, because one needs a trivial Noether current in order to obtain a corner charge. What then needs to be added to BGC to obtain a nontrivial corner charge?

As it turns out, this is one of the cases where the analysis of the difficulty (by means of the Noether machine) also makes the remedy obvious: what needs to be added is precisely the requirement that the theory’s Lagrangian $L$ and the variational symmetry $\xi$ be such that the resulting superpotential is nontrivial (I again remind the reader that a choice of $L$ and its variational symmetry $\xi$ also fixes the superpotential $U_{\xi }$ ). In Reference Freidel, Teh, Read and TehFreidel and Teh (2022), Laurent Freidel and I thus proposed the following definition of substantive general covariance in response to the “corner charge” part of the Einstein test:

It is worth pausing to reflect on how Noether’s second theorem has brought us to this insight. On the one hand, the potency of Noether’s second theorem here is not that of a purely mathematical result. However, just as the character of a line or a stroke can begin to suggest the animation or dynamism of a figure in painting, so too can mathematics be taken up into the mobilization of physical thought. What Noether’s second theorem does here is to offer physicists a heightened awareness of what the mathematical medium of field theory (understood through the lens of the variational calculus and Noether’s theorems) is capable of when it is pressed into the service of physical representation: our discussion thus far is precisely an illustration of how the second theorem can be so recruited! Through that discussion, we come to see that the substantiveness of general covariance is not just about the local symmetry of a Lagrangian, but also about how – through that symmetry – some kinds of Lagrangians determine a nontrivial corner charge algebra.

I now turn to the “Extending RP” part of the Einstein test, for which (Corner-SGC) is a necessary but insufficient condition, because of the aforementioned potential tension between the Noether charge and the Hamiltonian charge relative to some boundary condition. Note that for a gauge theory (where I mean to include both GR and Yang–Mills theory), whatever the symmetries of generalized Representational RP are, they will need to be generated by the nontrivial corner charges from (Corner-SGC). However, there is an added complication present in generalizing RP: we are now considering the symmetries of dynamically isolated subsystems, which are generated by the Hamiltonian charges for the boundary conditions that model dynamical isolation – thus, we will only succeed in extending RP if the corner charges coming from the trivial Noether current of some Lagrangian are also Hamiltonian charges relative to these isolated boundary conditions. In Section 6.1.2, we saw an example (Maxwell gauge theory with Dirichlet boundary conditions) where this coincidence clearly obtains: we can think of it as a regularized (finite) version of a case in which the asymptotically rigid $U\left(1\right)$ symmetries of some subsystem provide a generalized version of RP relative to an environment subsystem (where the environment has been idealized away into the boundary conditions). However, as we are about to see in the next section, the corner charges from some particular Lagrangian can also fail to be Hamiltonian charges relative to our desired choice of dynamically isolated boundary conditions. I thus propose the following more demanding notion of substantive general covariance in order to satisfy the “extending RP” part of the Einstein test:

In the next subsection, we will consider the case of GR, which illustrates the subtleties of RP-SGC.

6.3 Revisiting the Case of General Relativity

In Reference Freidel, Oliveri, Pranzetti and SpezialeFreidel et al. (2021), the general way in which Noether charges can fail to be Hamiltonian charges is considered, and this framework is then applied to the case of GR (Section 3.4 of Reference Freidel, Oliveri, Pranzetti and SpezialeFreidel et al. (2021)). Here I only summarize the relevant features for us to revisit the Einstein-Klein dispute, namely the second of the “complications of the medium” that I mentioned in Section 3.

First, consider the Einstein-Hilbert Lagrangian $L_{EH}$ . We can turn the crank of the covariant phase space recipe sketched in Section 5 and compute the (nontrivial) superpotential corresponding to $L_{EH}$ , which yields the Noether (corner) charge that is known as the Komar charge (see (2.38) of Reference Freidel, Geiller and PranzettiFreidel et al. (2020) for a definition). This shows that we have satisfied (Corner-SGC).

Next, let us consider whether this Lagrangian satisfies (RP-SGC). To do so, suppose that we wish to use Dirichlet boundary conditions to model an isolated system (or a finite boundary regularization thereof): that is to say, we wish to set the induced metric ${\bar{g}}_{ab}$ on the timelike boundary $\Gamma$ to a fixed value, thereby also setting $\delta {\bar{g}}_{ab}\stackrel{\Gamma }{=}0$ . More specifically, and as per standard practice, we will take ${\bar{g}}_{ab}$ to be the Minkowski metric, thus leading to an “asymptotically flat” solution in the limit where we take the boundary to infinity.Footnote ⁵⁷ It is well-known that given this Dirichlet boundary condition, one can construct a Hamiltonian charge; however, this is not the Komar charge, but a different quasi-local charge known as the Brown-York charge (see (2.31) of Reference Freidel, Geiller and PranzettiFreidel et al. (2020) for a definition).

Thus, we have just encountered the complication that we discussed earlier: if one chooses $L_{EH}$ as one’s Lagrangian and takes the Dirichlet boundary condition as the “isolated boundary condition” in the “Extending RP” part of the Einstein test, then one ends up with a mismatch between the Noether charge (the Komar charge) corresponding to $L_{EH}$ and the Hamiltonian charge (the Brown-York charge) corresponding to the Dirichlet boundary condition. The resolution of this complication, which is discussed in Reference Harlow and WuHarlow and Wu (2020); Reference Freidel, Oliveri, Pranzetti and SpezialeFreidel et al. (2021), is to choose a different Lagrangian – which is associated with a different boundary condition – in order to satisfy (RP-SGC). In particular, if we choose the Lagrangian $L=L_{EH}+d{\ell }_{GHY}$ , where ${\ell }_{GHY}$ is the celebrated Gibbons-Hawking-York boundary term, then the corresponding Noether charge is precisely the Brown-York charge, so we have now brought our Lagrangian into alignment with the boundary condition – and the Hamiltonian charge – that we are after. Furthermore, as Reference Harlow and WuHarlow and Wu (2020) shows, if we assume that the metric tends toward the Minkowski metric in the asymptotic limit, then we recover precisely the ADM charges for an asymptotically flat spacetime, which is precisely what one expects of an isolated subsystem in GR. In particular, these charges generate the “asymptotic Poincare symmetries” at spatial infinity, which are the generalized analogs of the rigid Poincare symmetries (from the standard RP) in this context.Footnote ⁵⁸ We thus see that $L=L_{EH}+d{\ell }_{GHY}$ satisfies (RP-SGC), but $L_{EH}$ does not. In other words, despite the subtlety associated with making a careful choice of Lagrangian here – and which (RP-SGC) is designed to take into account – we can conclude that the subject of Yuyi’s boat can indeed be reconstructed within the setting of GR, and that it once again connects up with the Greaves and Wallace schema in the manner discussed in Section 5.2 (namely, the Brown-York charge is the Hamiltonian generator of subsystem symmetries of the isolated subsystem).

With these clarifications in hand, let me close this subsection by returning to an assessment of the aforementioned dispute between Einstein and Klein (I refer the reader to Reference De HaroDe Haro (2021) for a complementary analysis of this dispute, with an emphasis on holographic renormalization). To what extent was Einstein confused about the status of conserved charges in GR, and to what extent did he have the representational insights into whose service we have just pressed Noether’s second theorem? To answer this question, let us consider several key points from Einstein’s March 1918 response to Klein (letter 480 in Reference EinsteinEinstein (1998)):

• Einstein says the importance of the quantity $J_{\xi }=T+t$ (the on-shell exact Noether current) is its physical interpretation as the energy of a point mass when we go very far away from the subsystem; in other words, when we treat the subsystem as dynamically isolated from an environment.

• He also says that the possibility of this physical interpretation is underwritten by the fact that $J_{\xi }$ (on-shell) is an exact term, presumably because he knows that via Stokes’ theorem, a bulk integral of an exact term can be transformed into a boundary integral (which is why we call such charges “quasi-local”).

• Einstein emphasizes that the equations of motion are necessary in order to show that $J_{\xi }$ is exact, and thus $J_{\xi }$ should be regarded as physically meaningful.

And then – a little over a week later – in letter 492 to Klein, Reference EinsteinEinstein (1998) goes on to place a heuristic “isolated” boundary condition on the subsystem that allows for the construction of a conserved charge!

We can now ask ourselves how Einstein’s more physically oriented (and more mathematically imprecise) strategy looks with the clarity of hindsight. It would seem that Einstein already understood much of the intuitive physical content that can be expressed by means of Noether’s second theorem: certainly, he understood that the distinctive qualitative feature of a gauge theory (implied by Noether’s second theorem) is the on-shell exactness of the Noether current, and that this feature leads to a boundary charge; and he was undoubtedly sensitive to the need to choose boundary conditions in order to construct such a charge. In all this, I believe that Einstein was vindicated in resisting Klein’s criticisms.

On the other hand, the place where Einstein’s understanding was most lacking was in his grasp of (i) the relationship between a choice of a Lagrangian $L$ and a choice of boundary conditions, (ii) the potential mismatch between the Noether charge corresponding to some $L$ and the Hamiltonian charge corresponding to some choice of boundary conditions, and (iii) the potential appearance of infinite-dimensional symmetries (such as the BMS groupFootnote ⁵⁹ at null infinity) in the asymptotic limit. Indeed it is perhaps this last feature and – more generally – the physics of null infinity for an asymptotically flat subsystem that inclines me to say that the search for Yuyi’s boat in GR does not so much result in a replication of the subject, but a novel and radical reworking of it. Yes, we discern very clearly the contours of Yuyi’s boat at the spatial boundary of the subsystem (as we have just seen above), but at the same time, asymptotically flat subsystems bring with them the physics of gravitational radiation, and as Reference WielandWieland (2021) has recently shown, the corresponding interpretation of null infinity as an open Hamiltonian system.Footnote ⁶⁰ And yet, such novelty is precisely what one should expect from the impulse that the procedures of mathematics can give to the task of physical representation: for the reworking of a subject like Yuyi’s boat is not something essentially static – a replication as it were – but a dynamic mobilization of the physical imagination in search of new understanding, and of new phenomena to understand.

6.4 A Response to Kretschmann

In this last subsection, I should like to return to Kretschmann’s critique of a conception of “general covariance” as the ability to express a physical theory in an arbitrary frame, whether we are talking about a spacetime coordinate frame (as in the case of a special relativistic scalar field theory) or an internal gauge frame (as in the case of a scalar field coupled to a flat gauge potential). Recall that Kretschmann’s objection is that, on this conception, there is nothing about general covariance that would distinguish the ones we would like to call gauge theories (or diffeomorphism-invariant theories), because all physical theories (in which a notion of frame applies) can be expressed in an arbitrary frame.

For instance, in the case of a scalar field theory formulated on a background Minkowski space, we can certainly choose to formulate its equations of motion in a non-inertial frame obtained via pulling back the metric and fields by a general diffeomorphism – the resulting equations would not have the simple form that we are used to from working in inertial frames (for they would pick up connection coefficients), but nothing would have changed concerning the physics that such a formalism is trying to represent. And as I mentioned at the end of Section 6.1.1, we can make a similar point in the case of a scalar field theory formulated relative to a background flat $U\left(1\right)$ gauge field $A$ : there is an analog of “inertial” internal frames, viz. those gauge frames in which the exact (here I assume a contractible spacetime) $U\left(1\right)$ gauge field vanishes, and these are of course all connected by global $U\left(1\right)$ transformations which are the stabilizers of the flat background gauge field, just as the Poincare transformations are the stabilizers of the Minkowski metric.Footnote ⁶¹ But here again, and similarly to the spacetime case, one has the freedom to re-formulate the equation of motion $d\star d\varphi =0$ in a non-inertial frame obtained via transforming $A=0$ by an arbitrary $U\left(1\right)$ gauge transformation. The new equations would not take the simple form that we are used to (they would now in general involve a gauge covariant derivative and a non-vanishing exact background gauge field) but the physics would not have changed for all that.

Kretschmann is right so far as this goes. But equipped with the materials of the previous subsections, we can now make a deeper response.

The response begins by emphasizing that a substantive form of general covariance is not after all the ability to express a physical theory in an arbitrary frame – this is not how Einstein used the diffeomorphism symmetry of GR in practice, and it is not how future generations of physicists would (again in practice!) use the gauge symmetry of Yang–Mills theory and other gauge theories. One route toward this point stems from the results of the previous subsections: ${xx}^{\prime }$ consider a minimal description of general covariance/gauge invariance/diffeomorphism invariance such as BGC, and note that one can satisfy this description by simply rewriting a theory with rigid symmetries using parametrization fields (as we saw in Section 6.1.1) and then allowing oneself to vary the parametrization fields and to take into account their local gauge symmetries. But as we saw earlier, this kind of repackaging does not have the capacity to yield the nontrivial corner charges that would allow us to (given appropriate boundary conditions implementing subystem isolation) generate the empirically significant symmetries of a Yuyi’s boat type scenario – plausibly, any such Kretschmannizing will yield trivial corner charges. Thus, we can conclude that for the practicing physicist, substantive general covariance requires that, in addition to BGC, the theory have at least the representational capacity for the construction of nontrivial corner charges.

Further to this point, is it possible to say anything more incisive about what – formally speaking – really distinguishes theories with “substantive” gauge symmetry from the mere freedom to rewrite the theory in an arbitrary frame? For instance, what kind of formal standard of comparison might one use to articulate this point? While the full details of an account lie beyond the scope of this Element, it is possible to sketch a strategy for answering these questions, based on the work of Reference Cattaneo, Mnev and ReshetikhinCattaneo et al. (2014) and Reference Mathieu, Murray, Schenkel and TehMathieu et al. (2019).

Recall that in the previous Subsections, Noether’s second theorem (as it is conceptualized within the covariant phase space formalism) gave us control over the more subtle distinctions that we needed in order to distinguish Kretschmannized theories with local symmetry from theories with physically substantive forms of local symmetry – the latter marked by their non-vanishing corner charges. Noether’s theorems spring from the manipulation (and geometrical interpretation) of the fundamental variational formula $\delta L=-E+d\theta$ ; in order to go further in pursuing our strategy, we will need to embark on a radical reimagining of the concept of a variation itself and what it means for a variation to vanish.

We are already familiar with the idea of finding the space of solutions in the standard variational calculus (without boundaries): one sets $\delta S=\delta {\int }_{M}L=0$ and thereby goes “on-shell.” The radical reimagining in question is motivated by the insight that this naive way of making $\delta S$ “coincide” with zero is unduly sensitive to mathematically different – but physically equivalent – reformulations of the space of fields, in the sense that these reformulations will in general produce mathematically distinct spaces of solutions (at least according to naively chosen mathematical standards of distinctness), unaccompanied by the (mathematical) expressive resources to describe their physical equivalence. For instance, in the case of the Kretschmannized theories that we have discussed, setting $\delta S=0$ clearly produces a distinct space of solutions from their un-Kretschmannized versions. The resolution of this formal difficulty is that one can try to construct the “space” of solutions by means of derived intersection theory, in which case this novel “space” – called the derived critical locus – will have the mathematical form of a “stack” within which one really does have the mathematical resources to express the kind of physical equivalence that we have been discussing.Footnote ⁶² In particular, this framework has the formal resources to express the observables of a theory – and thus the Noether charges – in a “homological” way that is invariant under superficial mathematical reformulations. The strategy, then, for applying this framework to our scenario would be to introduce a precise enough definition of Krestchmannization to prove a result along the following lines: any Krestchmannized theory gives rise to the same “solution stack” as its pre-Krestchmannized version, and in particular, these theories share the same observables.

6.5 Further Reading

The examination of our test cases originated in a context in which substantive general covariance was being discussed in relation to the notion of “background independence.” For the early discussion of these cases, see Reference Pooley, Suárez, Dorato and RédeiPooley (2010), Reference Pooley, Lehmkuhl, Schiemann and Scholz(2017)and references therein, and see Reference Freidel, Teh, Read and TehFreidel and Teh (2022) for a reassessment of the relationship between general covariance and background independence in light of the results of this section.

7.1 Donnelly-Freidel Edge Modes

First, the reader might well wonder how the covariant phase space framework might need to be modified if – hypothetically – one did not place any (gauge-non-invariant) boundary conditions on the fields. This is precisely the investigation that Reference Donnelly and FreidelDonnelly and Freidel (2016) undertook in the case of Yang–Mills theory and GR. What they found is that if one allows arbitrary field-independent gauge transformations at the boundary $\Gamma$ of a spacetime $M$ , then one ends up with a “symplectic anomaly” – the pre-symplectic form $\Omega$ is no longer invariant under gauge transformations. To restore gauge-invariance, they extended the phase space by introduced new “edge mode” degrees of freedom (and their momenta), as well as a corresponding corner symplectic form. They also argued that a new kind of physical “boundary symmetry” should manifest itself at the corner (call this the DF-type boundary symmetry).

Donnelly and Freidel’s extension of the phase space was based on requirement of restoring gauge-invariance, but they did not specify the boundary Lagrangian that led to these edge mode degrees of freedom. Subsequently, in Reference Mathieu, Murray, Schenkel and TehMathieu et al. (2019) (see also Reference Geiller and Jai-AksonGeiller and Jai-Akson (2020)), we showed that kinematically, these edge modes come from imposing a topological boundary condition (i.e. a fixed trival boundary bundle) on the subsystem, and that the extended symplectic structure is produced by specifying a particular boundary (edge mode) Lagrangian on $\Gamma$ . Furthermore, Reference Mathieu and TehMathieu and Teh (2021) argued that this boundary Lagrangian could be understood as a kind of spontaneous symmetry breaking on the boundary, and that the DF-type symmetries could be interpreted as the result of acting on the boundary condition by means of an “external automorphism,” in the sense that the transformation is carried out by an observer in the environment of the subsystem (or whose observational scale includes the environment).

7.2 Classical Spontaneous Symmetry Breaking

The next part of the story involves the notion of what Reference StrocchiStrocchi (2011) calls classical spontaneous symmetry breaking (SSB). To start with, consider the conceptual elements of what we would generally regard as an SSB scenario in condensed matter physics: there is a degenerate set of ground states – often taken to define “superselection sectors” because they cannot be changed by any subsystem operation – all connected by the “broken” symmetry transformations; there is an “internal” (or subsystem) observational scale which is unable to discriminate between which of these ground states the subsystem observer is in (although the subsystem observer can detect small fluctuations around the ground state); and there is an “external” (or environmental) scale with respect to which these ground states are in fact distinguishable. A key objective in applying the SSB formalism to this kind of scenario is to produce an effective field theory of the small fluctuations about a ground state; indeed this can be done very generally using the “coset construction,” which allows us to construct an effective Lagrangian using only the symmetry-breaking pattern (and without detailed knowledge of the underlying dynamics, such as one has to hand in the classic example of Ginzburg-Landau theory).

This framework can be carried over quite straightforwardly to the setting of classical field theory, with the modification that here one should think of a particular subsystem boundary condition of a certain type (e.g. Dirichlet) as defining a superselection sector, in the sense that the subsystem observer is not (let us arrange) able to perform any operations that can change the boundary condition. Then, if the relevant boundary conditions are related by a subgroup of the symmetry group of the theory, we can apply the standard SSB apparatus in this context. Thus, for instance, we can the SSB coset construction to construct a classical free point-particle Lagrangian as the effective dynamics of a scenario where boosts and translations are spontaneously broken. This background will be very useful to understand the next part of the story.

7.3 Edge Modes from Reference Frames

More recently, Reference Carrozza and HoehnCarrozza and Hoehn (2021) have posed the question of how one might understand the operational meaning of the boundary edge modes (and extended phase space) introduced in Reference Donnelly and FreidelDonnelly and Freidel (2016). The following scenario – which can be generalized quite greatly – is exemplary of the kind of answer that they give.

Let $M$ be a subsystem spacetime with timelike boundary $\Gamma$ , which divides it from an environment spacetime $\bar{M}$ (strictly speaking, the environment spacetime is the complement of $M$ in an ambient spacetime). One of the examples that Reference Carrozza and HoehnCarrozza and Hoehn consider is a $U\left(1\right)$ gauge theory set on the total spacetime $M\cup \bar{M}$ , where $\bar{M}$ is assumed to have an asymptotic boundary where gauge transformations fall off sufficiently rapidly – this boundary provides a non-dynamical anchor for Wilson lines and can be used to represent for example the heavy degrees of freedom of a measurement apparatus. They start with a dynamical environment gauge field $\bar{A}$ on $\bar{M}$ and observe that by taking a Wilson line of $\bar{A}$ from a reference point on the asymptotic boundary to a point $x\in \Gamma$ , one thereby constructs a $U\left(1\right)$ -valued reference frame on $\Gamma$ , which is precisely what Donnelly and Freidel call an edge mode. This edge mode on $\Gamma$ , then, is a way of “internalizing” to $\Gamma$ the dynamical reference frame degrees of freedom provided by environment, so that even if one proceeds to omit $\bar{M}$ from the formalism of the representation, one can still has a formal way of expressing how the field degrees of freedom on $M$ relate to the field degrees of freedom on $\bar{M}$  – in particular, the edge mode helps us to encode the relational information (between subsystem and environment) that allows us to reconstruct all the gauge-invariant observables with support on $M\cup \bar{M}$ (including ones that we would not have been able to reconstruct from just the individual regional data).

In the $U\left(1\right)$ case, it is more convenient to work on the Lie algebra level, in which case the edge mode can be expressed as $\phi \in {\Omega }^0\left(\Gamma \right)$ (the group-valued reference frame being $U=exp\left(i\phi \right)$ ) and transforms as $\phi \mapsto \phi +\chi$ under a boundary gauge transformation $A\mapsto A+d\chi$ . As discussed in Reference Donnelly and FreidelDonnelly and Freidel (2016); Reference Mathieu, Murray, Schenkel and TehMathieu et al. (2019) one can use this edge mode field to dress the boundary gauge field, thus obtaining a (boundary) gauge-invariant field $a:=A{|}_{\Gamma }-d\phi$ (which was already introduced in Reference Mathieu, Murray, Schenkel and TehMathieu et al. (2019) in order to write the boundary symmetry-breaking action). As Reference Carrozza and HoehnCarrozza and Hoehn (2021) observe, this dressed field $a$ is a relational observable, describing the value of $A_{|}\Gamma$ (a subsystem degree of freedom) when the $U\left(1\right)$ dynamical reference frame (defined using environment degrees of freedom) is in some orientation.

Reference Carrozza and HoehnCarrozza and Hoehn go on to describe the process of imposing gauge-invariant boundary conditions, that is boundary conditions for the dressed field $a$ , on the subsystem. They do this for Dirichlet, Neumann, and mixed boundary conditions, but here I will simply illustrate the Dirichlet case: take the space of total solutions $S$ over $M\cup \bar{M}$ and foliate it into a set of leaves, each of which is defined by the gauge-invariant Dirichlet boundary conditions $a=X_0$ (where $X_0$ is some fixed background $1$ -form field), so that $S=\underset{X_0}{⨆}{S}_{X_0}$ , where a leaf $S_{X_0}$ is the set of field configurations in $S$ such that $a=X_0$ . They then note that we can proceed to focus just on the subsystem dynamics over $M$ , defined by a particular boundary condition $a=X_0$ : “Since the [boundary condition defines] the dynamical theory for $M$ [...] we can think of each leaf in the foliation of $S$ as a particular subregion theory. As such, the global solution space $S$ assumes the role of a space of subregion theories, in this sense constituting a meta-theory for the local subregions.”

Indeed, once we follow Reference Carrozza and HoehnCarrozza and Hoehn (2021) in the last step, we see that there are two kinds of transformations that are supported on the subsystem boundary $\Gamma$ . First, we have the gauge-transformations $A\mapsto A+d\chi$ , $\phi \mapsto \phi +\chi$ , $a\mapsto a$ , which lie in the kernel of the extended pre-symplectic form constructed in Reference Donnelly and FreidelDonnelly and Freidel (2016) and Reference Mathieu, Murray, Schenkel and TehMathieu et al. (2019) – these do not encode any relational difference between the subsystem and the environment because one is transforming the gauge field and the edge mode simultaneously. Second, we have the reference frame re-orientations $A\mapsto A$ , $\phi \mapsto \phi -\rho$ , $a\mapsto a+d\rho$ , which can be further divided into two types. When $\rho$ stabilizes $a$ (i.e. is a constant, in this case), it preserve the gauge-invariant Dirichlet boundary condition, and we have what I earlier called a DF-type boundary symmetry – this symmetry does explicitly capture a relational difference between the subsystem and environment degrees of freedom (where the latter are encoded in the edge mode). On the other hand, a generic spacetime-dependent $\rho$ will not stabilize $a$ : it will instead take us from the leaf whose boundary condition is $a=X_0$ into a different leaf in whose boundary condition is $a=X_0+d\rho$ . Reference Carrozza and HoehnCarrozza and Hoehn call these “meta-symmetries” because they do not transform a solution to a solution within a subregion theory, but instead transform one subregion theory into a different subregion theory (in Reference Carrozza and HoehnCarrozza and Hoehn (2021)’s use of “theory”).

But now this picture should look somewhat familiar based on our previous sketch of Classical SSB: as Teh and Mathieu (forthcoming) have recently observed, in this case, Reference Carrozza and HoehnCarrozza and Hoehn have essentially defined a Classical SSB scenario for $U\left(1\right)$ gauge theory, where the subsystem leaves define a degenerate set of superselection sectors of the scenario that are related by nonconstant local $U\left(1\right)$ gauge transformations, and each superselection sector is stabilized by a global $U\left(1\right)$ subgroup. In other words, the symmetry-breaking pattern is that of local $U\left(1\right)$ to global $U\left(1\right)$ . As a sanity check, we can go in the reverse direction: we can start with just this symmetry-breaking pattern and easily compute the effective dynamics within a superselection sector by means of the coset construction. Reassuringly, at lowest order, the effective Lagrangian is precisely that of a free $U\left(1\right)$ gauge theory – precisely the dynamics from which (with the gauge-invariant Dirichlet boundary condition) we defined this symmetry-breaking pattern to begin with!

7.4 Yuyi’s Boat and Corollary VI

What does any of this have to do with Yuyi’s boat-type scenarios and Newton’s Corollary VI? To see the connection with Yuyi’s boat, I refer the reader to toy mechanical model for edge modes introduced in Reference Carrozza and HoehnCarrozza and Hoehn (2021), where they use a particle degree of freedom in the environment to define an edge mode reference frame, and use another particle as a stand-in for the “boundary.” For gauge-invariant Dirichlet boundary conditions on the subsystem particle degree of freedom $x^i\left(t\right)$ , they show that any uniform (Galilean) boost of the particle relative to the edge mode will take one boundary condition to another, and so in the classical SSB terminology that we have been developing, this is a scenario where the Galilean boost symmetry is completely broken (since there are no nontrivial stabilizers). Nonetheless, an internal observer (Yuyi running short-range experiments on this boat) is still unable to distinguish between different superselection sectors.

To see the connection with Newton’s Corollary VI, consider that the (pure) potential part of the Newtonian gravitational Lagrangian

$S={\int }_{R\times M}\left(\frac{1}{2}m{\dot{\text{u}}}^2-m\varphi \left(\text{r},t\right)\right){\delta }^{\left(3\right)}\left(\text{u}\left(t\right)-\text{r}\right)-\frac{1}{8\pi G}{\left(\nabla \varphi \right)}^2{d}^3\text{r}dt$(16)

is very reminiscent of the $U\left(1\right)$ gauge theory Lagrangian, except that in the former the exterior derivative and Hodge star are purely spatial, and the resulting equations of motion are elliptic (and fixed completely by the boundary data). By running an analysis that is analogous to the $U\left(1\right)$ gauge theory analysis that we just sketched, one can define an edge mode (using the frame of a body in the environment) for a Newtonian gravitational subsystem and show that gauge-invariant Dirichlet boundary conditions for the potential lead to a Classical SSB scenario from the local in time boosts of Newton’s Corollary VI to the constant (or global) Galilean boosts of Yuyi’s boat. And, in fact, when we think hard about it, what else is Newton’s Corollary VI but a statement of SSB in this sense? For consider again what Newton writes: “If bodies are moved in any way among themselves, and are urged by equal accelerative forces along parallel lines, they will all continue to move among themselves in the same way as if they were not acted on by those forces.” Newton of course knew that in speaking of this physical indistinguishability, he was describing it from the perspective of a subsystem observer equipped with a certain measurement scale (think of local measurements made in the vicinity of Jupiter and its moons) and that there could also be an “external” observational scale according to which different uniform accelerations could be discriminated (say, observations of the Jupiter subsystem relative to the sun). The indistinguishability that Newton is discussing is thus the inability of the subsystem observer to discriminate between a degenerate set of SSB superselection sectors, which are related to each other by (nonconstant) time-dependent boosts, and stabilized by Galilean boosts. And again very reassuringly, the coset construction tells us that the symmetry-breaking pattern from time-local to time-global boosts results precisely in the dynamics of the Poisson equation at lowest order. Thus, we see that our present line of thought has managed to unify both Yuyi’s boat and Newton’s Corollary VI under the rubric of edge modes and Classical SSB.