1. Introduction

The word “gauge” is ubiquitous in modern physics. Our best physical theories are described, in various contexts, as “gauge theories.” The “gauge argument” allegedly reveals the underlying “logic of nature” (Martin Reference Martin2002). Our theories regularly exhibit “gauge freedom,” “gauge structure,” and “gauge dependence.” Unfortunately, however, it is far from clear that the term has some univocal meaning across the many contexts in which it appears. It is a bit like “liberal” in American political discourse: it shows up everywhere, and no one knows what it means.

Here I focus on two strands of usage.Footnote ¹ On the first strand, a “gauge theory” is a theory that exhibits excess structure or, in Earman’s words, “descriptive fluff” (Reference Earman2004).Footnote ² On this way of thinking about gauge, there is a mismatch between the mathematical structure used in the theory and the structure we take the world to have, in such a way that (perhaps) one could remove some structure from the theory without affecting its descriptive or representational power. Most famously, Earman and Norton (Reference Earman and Norton1987) argue that the so-called hole argument shows that general relativity is a gauge theory in just this sense; some have taken this as motivation for moving to a different, perhaps undiscovered, formalism for representing space-time (Earman Reference Earman1989).Footnote ³ Similar considerations have motivated some views on Yang-Mills theory (Healey Reference Healey2007; Rosenstock and Weatherall Reference Rosenstock and Weatherall2015).

The second strand of usage concerns a specific class of theories. Here one uses “gauge theory” to refer to various generalizations of classical electromagnetism that share a certain mathematical structure.Footnote ⁴ For instance, Trautman (Reference Trautman and Held1980) defines “gauge theory” as follows: “For me, a gauge theory is any physical theory of a dynamic variable which, at the classical level, may be identified with a connection on a principal bundle” (26). This turns out to be a large class containing most of our fundamental theories, including all Yang-Mills theories, general relativity, and Newton-Cartan theory. It is in this context that physicists seem to speak most often of gauge theories, usually as a synonym for “Yang-Mills theories.”

It is easy to imagine that the two strands are closely related and, in particular, that all gauge theories in the second sense are also gauge theories in the first sense. But as I argue below, this is a mistake. In particular, I will articulate a precise sense in which electromagnetism—the paradigmatic example on both strands—may be understood to have excess structure, and thus to be a gauge theory in the first sense. I then consider whether other theories, such as Yang-Mills theory and general relativity, have excess structure in the same sense. I will argue that they do not. It follows that on at least one precise sense of what it means for a theory to have excess structure, the two strands of usage described above come apart.

2. Two Approaches to Electromagnetism

In what follows, we consider electromagnetism on the fixed background of Minkowski space-time, $(M,{\eta }_{ab})$.Footnote ⁵ In this setting, there are two ways of characterizing models of ordinary electromagnetism.Footnote ⁶

On one characterization, the principal dynamical variable is the electromagnetic field, represented by a two-form $F_{ab}$ on space-time. The electromagnetic field is required to satisfy Maxwell’s equations, which may be expressed as ${\nabla }_{[a}{F}_{bc]}=0$ and ${\nabla }_a{F}^{ab}=J^b$, where $\nabla$ is the Minkowski derivative operator and $J^a$ is a smooth vector field representing the charge-current density on space-time. A model of the theory on this characterization might be written as a triple $(M,{\eta }_{ab},F_{ab})$, where $F_{ab}$ is any closed two form (i.e., any two form satisfying the first of Maxwell’s equations).Footnote ⁷ Call this formulation of the theory EM₁.

On the second characterization, the dynamical field is the four-vector potential, represented by a one-form A_a on space-time. This field is required to satisfy a single differential equation: ${\nabla }_a{\nabla }^a{A}^b-{\nabla }^b{\nabla }_a{A}^a=J^b$, where again J^a is the charge-current density. A model of the theory may again be represented by a triple, $(M,{\eta }_{ab},A_a)$, where A_a is any one form.Footnote ⁸ I call this formulation of the theory EM₂.

These two formulations are systematically related. Given any model $(M,{\eta }_{ab},A_a)$ of EM₂, I can always define an electromagnetic field by $F_{ab}={\nabla }_{[a}{A}_{b]}$. Since any F_ab thus defined is exact, it must also be closed, and thus the resulting triple $(M,{\eta }_{ab},F_{ab})$ is a model of EM₁; moreover, this F_ab is associated with the same charge-current density as A_a. Conversely, given any model $(M,{\eta }_{ab},F_{ab})$ of EM₁, since F_ab is closed, it must also be exact, and thus there exists a one-form A_a such that $F_{ab}={\nabla }_{[a}{A}_{b]}$.Footnote ⁹ The triple $(M,{\eta }_{ab},A_b)$ is then a model of EM₂, again with the same charge-current density.

However, there is an important asymmetry in this relationship. Given any model of EM₂, there exists a unique corresponding model of EM₁, because any smooth one form has a unique exterior derivative. But the converse is not true: given a model $(M,{\eta }_{ab},F_{ab})$ of EM₁, there will generally be many corresponding models of EM₂, since if A_a is such that $F_{ab}={\nabla }_{[a}{A}_{b]}$, then ${A^{\prime }}_a=A_a+{\nabla }_a\chi$, for any smooth scalar field χ, also satisfies $F_{ab}={\nabla }_{[a}{A^{\prime }}_{b]}={\nabla }_{[a}{A}_{b]}+{\nabla }_{[a}{\nabla }_{b]}\chi$, because for any smooth scalar field, ${\nabla }_{[a}{\nabla }_{b]}\chi =0$. Transformations $A_a\mapsto {A^{\prime }}_a$ of this form are sometimes known as gauge transformations.

It is this asymmetry that, I claim, supports the common view that electromagnetism has excess structure. The idea is that EM₁ and EM₂ both have all of the resources necessary to represent classical electromagnetic phenomena. Indeed, in both cases, one may take the empirical content of electromagnetism to be fully exhausted by the electromagnetic field associated with a given model—either directly in the case of EM₁ or as derived above in the case of EM₂. But there are prima facie distinct models of EM₂ associated with the same electromagnetic field. Thus, it would seem that these models of EM₂, although they differ in their mathematical properties, should be taken to have the same representational capacities. Intuitively, then, whatever structure distinguishes models of EM₂ related by gauge transformations must be irrelevant to the representational capacities of the models, at least as long as the empirical content is exhausted by the associated electromagnetic field. It is in this sense that electromagnetism—or really, EM₂—has excess structure.

3. Comparing Structure

In the next section, I make the intuitive argument just given precise. First, however, I take a detour through pure mathematics. Mathematical objects often differ in how much structure they have. For instance, topological spaces have more structure than sets: a topological space (X, τ) consists in a set X, along with something more, namely, a collection τ of open subsets of X satisfying certain properties. Similarly, an inner product space has more structure than a vector space, and a Lie group has more structure than a smooth manifold. In this section, I use some basic category theory to capture these judgments as mathematical relationships between the theories.Footnote ¹⁰

To begin, recall that various mathematical theories may be associated with categories. For instance, there is a category Set, whose objects are sets and whose arrows are functions. There is a category Top, whose objects are topological spaces and whose arrows are continuous functions. There are also functors between such categories. For instance, there is a functor $F:Top\to Set$ that takes every topological space (X, τ) to the set X and takes every continuous map $f:(X,\tau )\to (X^{\prime },{\tau }^{\prime })$ to the function $f:X\to X^{\prime }$. Functors of this sort are often called “forgetful,” because, intuitively speaking, they take objects of one category and forget something about them: in this case, they take topological spaces (X, τ) and forget about τ.

How can one tell whether a given functor is forgetful? There is a simple but insightful theory available, due to Baez, Bartel, and Dolan (Reference Baez, Bartel and Dolan2004; see also Barrett Reference Barrett2013). It requires a few further definitions, concerning properties that a functor $F:C\to D$ may have. First, we will say that F is full if for all objects A, B of $C$, the map $(f:A\to B)\mapsto \left(F\right(f):F(A)\to F(B\left)\right)$ induced by F is surjective. Similarly, F is faithful if for all pairs of objects in $C$, the induced map on arrows is injective. And F is essentially surjective if for every object X of $D$, there is some object A of $C$ and arrows $f:F\left(A\right)\to X$ and $f^{-1}:X\to F\left(A\right)$ such that $f^{-1}\circ f{=1}_{F\left(A\right)}$ and $f\circ f^{-1}{=1}_X$. (Such an arrow f is an isomorphism, so essentially surjective functors are surjective on objects “up to isomorphism.”)

If a functor $F:C\to D$ is full, faithful, and essentially surjective, then the functor is said to realize an equivalence of categories. In such cases, F forgets nothing. Otherwise a functor is forgetful. In particular, a functor forgets (only) structure if it is faithful and essentially surjective but not full. A functor forgets (only) properties if it is full and faithful but not essentially surjective. And a functor forgets (only) stuff if it is full and essentially surjective but not faithful. In general, a given functor may be forgetful in more than one of these ways, but not in any other ways: any functor may be written as the composition of three functors that forget (no more than) structure, properties, and stuff, respectively.

The best way to make this categorization plausible is by considering examples. For instance, the functor we have already considered, $F:Top\to Set$, forgets only structure. This is because every set corresponds to some topological space (or other), which means that F is essentially surjective. Similarly, any distinct continuous functions $f,f^{\prime }:(X,\tau )\to (X^{\prime },{\tau }^{\prime })$ must be distinct as functions, so F is faithful. But F is not full, because not every function $f:X\to X^{\prime }$ is continuous, given topologies on X and X′. So in this case, the classification captures the pretheoretic intuition with which we began.

Similarly, we can define categories Grp and AbGrp, whose objects are groups and Abelian groups, respectively, and whose arrows are group homomorphisms; then, there is a functor $G:AbGrp\to Grp$ that takes Abelian groups and group homomorphisms to themselves. This functor is full and faithful, since it just acts as the identity on group homomorphisms between Abelian groups. But it is not essentially surjective because not every group is Abelian. So this functor forgets only properties—namely, the property of being Abelian. And finally, we can define a functor $H:Set\to 1$, where 1 is the category with one object and one arrow (the identity on the one object). This functor takes every set to the unique object of 1 and every arrow to the unique arrow of 1. It is clearly full and essentially surjective, but not faithful, so it forgets only stuff. To see how, note that we may think of 1 as the category with the empty set as its only object; thus, H forgets all of the elements of the sets.

This classification of functors gives us a criterion for when a mathematical theory T ₁ may be said to have more structure than another theory T ₂, namely, when there exists a functor from the category associated with T ₁ to the category associated with T ₂ that forgets structure. Given two categories, there may be multiple functors between them, and it may be that not all such functors forget structure, even if there exists one that does. This means that comparative judgments of “amount of structure” between theories should be understood as relative to a choice of functor. This flexibility is a virtue: it allows us to explore various ways in which theories may be related.

4. A Diagnostic Tool

I now return to the question of interest. To begin, I use the criterion just developed to make the intuitive argument at the end of section 2 precise. I define two categories, corresponding to the two formulations of electromagnetism already discussed, and then define a functor between them that captures the relationship already discussed between EM₁ and EM₂.

The first category, $E{M}_1$, has models $(M,{\eta }_{ab},F_{ab})$ of EM₁ as objects and as arrows has maps that suitably preserve this structure. For present purposes, we take these to be isometries of Minkowski space-time that preserve the electromagnetic field, so that given two models, $(M,{\eta }_{ab},F_{ab})$ and $(M,{\eta }_{ab},{F^{\prime }}_{ab})$, an arrow $\chi :(M,{\eta }_{ab},F_{ab})\to (M,{\eta }_{ab},{F^{\prime }}_{ab})$ will be an isometry of $(M,{\eta }_{ab})$ such that ${\chi }^{*}\left({F^{\prime }}_{ab}\right)=F_{ab}$. Likewise, we may define a category $E{M}_2$ whose objects are models $(M,{\eta }_{ab},A_a)$ of EM₂ and whose arrows are isometries of Minkowski space-time that preserve the four-vector potential. Given these categories, the map defined above, taking models $(M,{\eta }_{ab},A_a)$ of EM₂ to models $(M,{\eta }_{ab},{\nabla }_{[a}{A}_{b]})$ of EM₁, becomes a functor $F:E{M}_2\to E{M}_1$ that take arrows of $E{M}_2$—which, recall, are isometries of Minkowski space-time with an additional property—to the arrow of $E{M}_1$ corresponding to the same isometry. (This action on arrows is well defined because, given any arrow $\chi :(M,{\eta }_{ab},A_a)\to (M,{\eta }_{ab},{A^{\prime }}_a)$ of $E{M}_2$, ${\nabla }_{[a}{A}_{b]}={\nabla }_{[a}\left({\chi }^{*}\right({A^{\prime }}_{b]}\left)\right)={\chi }^{*}\left({\nabla }_{[a}{A^{\prime }}_{b]}\right)$.)Footnote ¹¹ We then have the following result.

Proposition 1 provides a precise sense in which EM₂ has more structure than EM₁: the functor realizing the natural relationship between the theories forgets (only) structure. Recall that the intuitive argument was that there are distinct models of EM₂ corresponding to a single model of EM₁, and thus there must be features of the models of EM₂ that distinguish them, without making any difference to their empirical content. The present argument, meanwhile, is that there are models of EM₂ that fail to be isomorphic—by the standard of isomorphism used in defining EM₂—even though the corresponding models of EM₁ are isomorphic or even identical. This is captured in the formalism by the fact that there are arrows in $E{M}_1$, which we may interpret as “structure-preserving maps” between models of EM₁, that are not structure-preserving maps between models of EM₂. The structure that these maps do not preserve is the structure that, on the intuitive argument, distinguished models of EM₂.

I take this to be strong evidence that the formal criterion given by forgetful functors captures the sense in which electromagnetism has excess structure. And since electromagnetism is the paradigmatic example of a gauge theory, I take this to be the sense of “excess structure” associated with the first strand of usage.Footnote ¹² Of course, there may be other senses in which a theory might be thought to have excess structure, but I will not consider that question further here. Rather, I stipulate that the criterion developed here is salient and turn to a different question. Do gauge theories in the second sense—that is, the theories Trautman identifies—have excess structure?

No. First, consider electromagnetism, formulated now as a theory whose dynamical variable is a connection on a principal bundle over Minkowski space-time—that is, electromagnetism formulated as a gauge theory in Trautman’s sense. Call this theory EM₃. Models of EM₃ may be written (P, ω_α), where P is the total space of the (unique, trivial) principal bundle $U\left(1\right)\to P\stackrel{\pi }{\to }M$ over Minkowski space-time and ω_α is a principal connection.Footnote ¹³ This theory is closely related to both EM₁ and EM₂ as already discussed: given any (global) section $\sigma :M\to P$, we may define a four-vector potential A_a as the pullback along σ of ω_α: $A_a={\sigma }^{*}\left({\omega }_{\alpha }\right)$. Similarly, we may define an electromagnetic field tensor F_ab as the pullback along σ of the curvature of the connection, defined by ${\Omega }_{\alpha \beta }=d_{\alpha }{\omega }_{\beta }$, where d is the exterior derivative on P: $F_{ab}={\sigma }^{*}\left({\Omega }_{\alpha \beta }\right)$. Thus, A_a and F_ab may be thought of as representatives on M of the connection and curvature on P. In general, A_a will depend on the choice of section σ, whereas F_ab will not depend on that choice because U(1) is an Abelian group.

Given this characterization of the theory, we can define yet another category, $E{M}_3$, as follows: the objects of $E{M}_3$ are models of EM₃, and the arrows are principal bundle isomorphisms $(\Psi ,\psi )$ that preserve both the connection on P and the metric on M: that is, pairs of diffeomorphisms $\Psi :P\to P$ and $\psi :M\to M$ such that ${\psi }^{*}\left({\eta }_{ab}\right)={\eta }_{ab}$, ${\Psi }^{*}\left({\omega }_{\alpha }\right)={\omega }_{\alpha }$, $\pi \circ \Psi =\psi \circ \pi$, and $\Psi \left(xg\right)=\Psi \left(x\right)g$ for any x ∈ P and any g ∈ U(1). Then we may define a functor $\tilde{F}:E{M}_3\to E{M}_1$ as follows: $\tilde{F}$ acts on objects as $(P,{\omega }_{\alpha })\mapsto (M,{\eta }_{ab},{\sigma }^{*}({\Omega }_{\alpha \beta }\left)\right)$, for any global section $\sigma :M\to P$, and $\tilde{F}$ acts on arrows as $(\Psi ,\psi )\mapsto \psi$. (Again, this action on arrows is well defined. Choose any section $\sigma :M\to P$. Then ${\Psi }^{-1}\circ \sigma \circ \psi$ is also a section of P. But since $F_{ab}={\sigma }^{*}\left({\Omega }_{\alpha \beta }\right)$ and ${F^{\prime }}_{ab}={\sigma }^{*}\left({{\Omega }^{\prime }}_{\alpha \beta }\right)$ are independent of the choice of section, $F_{ab}=({\Psi }^{-1}\circ \sigma \circ \psi {)}^{*}({\Omega }_{\alpha \beta })={\psi }^{*}\circ {\sigma }^{*}\circ {\Psi }^{-1*}({\Omega }_{\alpha \beta })={\psi }^{*}\circ {\sigma }^{*}({{\Omega }^{\prime }}_{\alpha \beta })={\psi }^{*}({F^{\prime }}_{ab})$.)

This result shows that EM₃ does not have excess structure in the sense that EM₂ does. To extend this to other gauge theories in the second sense, however, requires more work. The reason is that the criterion we have been using requires us to have two formulations, both of which are taken to be descriptively adequate and empirically equivalent, which we then compare. In other cases of interest, though, such as non-Abelian Yang-Mills theory or general relativity, it is not clear that we have a plausible second theory to consider.

Still, there is something one can say. It concerns the role of “gauge transformations” between models of EM₂, as described at the end of section 2. These are maps that relate models of EM₂ that have the same representational capacities even though they are not isomorphic. The criterion of excess structure described here, meanwhile, requires the existence of a functor between categories of models that fails to be full—or in other words, a standard of comparison between the theories relative to which one formulation has “more” arrows than the other or, alternatively, relative to which one of the formulations is “missing” arrows.Footnote ¹⁴ This suggests a rule of thumb for whether a theory, or a formulation of a theory, has excess structure in the sense described here, namely, the theory has models that are not isomorphic but that nevertheless we interpret as having precisely the same representational content. Indeed, whereas the criterion discussed above tells us when one theory or formulation has more structure than another, this second criterion evaluates whether any alternative formulation could have less structure and still do the same descriptive work—at least without equivocating between physical situations we now think are distinct.

How can we put this rule of thumb to work? Suppose you are given a theory and a collection of maps taking models to models with the same representational capacities—that is, one is presented with a candidate “gauge theory” and a class of “gauge transformations.” One may then ask: are these gauge transformations naturally construed as isomorphisms of the models of the theory, understood as mathematical objects? If the answer is yes, then it would seem that these maps do not signal excess structure, since these maps would not be “missing” from a natural category of models; conversely, if the answer is no, then there likely is excess structure in the formulation.

Applying this diagnostic to some examples of gauge theories in the second sense above, we immediately see that the moral concerning EM₃ generalizes to other Yang-Mills theories. For instance, models of an arbitrary Yang-Mills theory with structure group G may be written $(P,{{\omega }^A}_{\alpha })$, where P is a principal G-bundle over some space-time (M, g_ab) and ${{\omega }^A}_{\alpha }$ is a principal connection on P.Footnote ¹⁵ In this setting, a “gauge transformation” is often defined as a (vertical) principal bundle automorphism $\left(\Psi {,1}_M\right)$ relating models $(P,{{\omega }^A}_{\alpha })$ and $(P,{\Psi }_{*}({{\omega }^A}_{\alpha }\left)\right)$ (see, e.g., Bleecker Reference Bleecker1981, sec. 3.2). But these maps are just a special class of connection- and metric-preserving principal bundle isomorphisms, and so although they do map between models with the same representational resources, they are not “extra” maps, in the sense of the gauge transformations of EM₂. So Yang-Mills theory does not have excess structure in the sense discussed here.

Likewise, for general relativity, we characterize models of the theory as relativistic space-times, (M, g_ab). Here “gauge transformations” are often taken to be diffeomorphisms $\phi :M\to M$ relating models $(M,g_{ab})$ and $(M,{\phi }_{*}(g_{ab}\left)\right)$ (see, e.g., Wald Reference Wald1984; Earman and Norton Reference Earman and Norton1987). But once again, these maps are just isometries (i.e., they are just isomorphisms of Lorentzian manifolds). So here, too, there is no excess structure.

5. Conclusion

I have isolated two strands of usage of the expression “gauge theory” in physics and philosophy of physics. According to one, a gauge theory is a theory that has excess structure; according to the other, a gauge theory is any theory whose dynamical variable is a connection on a principal bundle. I then endeavored to make precise the sense in which the paradigmatic example of a gauge theory (according to both strands)—classical electromagnetism—may be construed as having excess structure. From this discussion, I extracted a general criterion for when a theory has excess structure. From this criterion, I argued that gauge theories in the second sense need not have excess structure—and indeed, Yang-Mills theory and general relativity should not be construed as having excess structure in the sense that one formulation of electromagnetism does.