1. Introduction

The past decade has seen a surge of interest in the (in)equivalence of theories among philosophers of science.Footnote ¹ During the same period, philosophers of physics have begun to study a theoretical relationship, widely discussed by physicists, known as “duality.”Footnote ² Physicists working in this subject often characterize duality as a kind of equivalence between distinct descriptions of the world—a characterization that philosophers of physics have tended to endorse as well. And yet, the (philosophical) literature on duality and the (philosophical) literature on equivalence have proceeded largely in parallel, with very little contact.

My goal here is to bring these two literatures together and to make a simple (although perhaps controversial) claim: dualities are not (generally) theoretical equivalences.Footnote ³ The reason is that dualities do not preserve empirical content—at least, not in the sense that has usually been required in the theoretical equivalence literature—and philosophers have generally taken empirical equivalence to be a necessary, although not sufficient, condition for theoretical equivalence of any sort.

This claim might seem surprising, at least if one is familiar with the literature in philosophy of physics on dualities, where the claim that dualities do preserve empirical content (and, often, a whole lot more) is regularly made. For instance, in an early philosophical foray into string theory, Dean Rickles writes, “Roughly, two theories are dual whenever they determine the same physics: same correlation functions, same physical spectra, etc. … Dualities map one theory onto another in a way that preserves all ‘physical’ predictions” (Reference Rickles2011, 55). Similarly, and more recently, string theorist–turned-philosopher Richard Dawid writes, “String dualities establish empirical equivalence between theories that often look entirely different with respect to their basic ontology and physical structure (Reference Dawid2017, 21).Footnote ⁴

I argue in what follows that all of these distinguished philosophers of physics—among many others—are wrong about whether dualities realize empirical equivalences. But there is an important proviso: my argument will be that dualities do not preserve empirical content in a particular sense that has been required in the theoretical equivalence literature. Hence, my arguments, if sound, can be used to support two different positions: either that dual theories are not really equivalent or that philosophers of science working on theoretical equivalence have gotten empirical equivalence wrong. I will give reasons why I think the first of these is more attractive, but I take it that both positions would be of interest.

The remainder of the article is structured as follows. I begin by describing empirical equivalence as it has been understood in the theoretical equivalence literature. I then introduce a particular example of a duality—electromagnetic duality, which is a special case of what is known as S-duality or Montonen-Olive duality—to which a widely discussed criterion of theoretical equivalence can be readily applied. As I will show, electromagnetic duality does not preserve empirical content. I conclude by discussing what one would need to do to recover empirical equivalence in this case and exploring why one might not be satisfied by such a move.

2. Theoretical Equivalence

Over the past half decade, philosophers of science have considered several potential criteria for theoretical equivalence. For instance, Glymour (Reference Glymour1970, Reference Glymour1980) proposed that definitional equivalence, a notion of equivalence borrowed from first-order logic, provides a natural criterion of equivalence of theories. Two theories are definitionally equivalent if, roughly speaking, the nonlogical vocabulary of each theory can be defined using the nonlogical vocabulary of the other, in such a way that any true assertion of one theory can be translated into a true assertion of the other, and vice versa, in an invertible manner. To avoid certain technical limitations of definitional equivalence, related to multisorted logic, Barrett and Halvorson (Reference Barrett and Halvorson2016) have proposed a generalization of definitional equivalence that they call Morita equivalence. Halvorson (Reference Halvorson2012) and Weatherall (Reference Weatherall2016), meanwhile, have proposed categorical equivalence (described below), a notion of equivalence borrowed from category theory, as a more attractive, more “structural” notion of equivalence. And so on.

These criteria are generally named for mathematical relationships between (formal representations of) theories. But these mathematical relationships are taken to be necessary, but not sufficient, conditions for equivalence.Footnote ⁵ For two physical theories to be theoretically equivalent, on any of the proposals just mentioned, one also requires that they be empirically equivalent, in a way that is compatible with the “formal” equivalence realized by the mathematical relationships. More precisely, each of the mathematical criteria of equivalence under consideration involves mappings between formal structures that satisfy some stipulated conditions. We also suppose that given a pair of physical theories that may or may not be equivalent to one another, there is some (fixed, prior) way of associating structures in the theories with the world.Footnote ⁶ Empirical equivalence, in the current context, is the requirement that these mappings commute.

To illustrate what I mean, consider the case of categorical equivalence.Footnote ⁷ To apply that criterion of equivalence to a pair of theories, we begin by defining categories 𝒯₁ and 𝒯₂. The objects of each of these categories are models of each theory, respectively, and the arrows are appropriate “structure preserving maps.” These categories represent these theories.Footnote ⁸ In addition, one identifies a collection W of “stylized phenomena” or “predictions” that one can associate with the theory, where each model of the theory will be associated with particular predictions corresponding to what one might observe (or what the world might be like) if the world were as represented by that model. We may take W to be a category, trivially, by endowing each object with an identity arrow.Footnote ⁹ Finally, we fix functors $E_1:{𝒯}_1\to W$ and $E_2:{𝒯}_2\to W$, relating these models to the “predictions” of the theories. Note that on this way of setting things up, it is already possible to associate the models of both theories with predictions in a single category W. This is not as strong a condition as it appears, insofar as one could take W to be the union of the “stylized phenomena” associated with each theory separately, in which case the functors E ₁ and E ₂ could still be defined, even though the theories made incomparable predictions.

To fix these ideas with an example, we might represent (vacuum) electromagnetism in Minkowski space-time as follows. We define a category ℰℳ

1. 1. whose objects are Faraday tensors F _ab on Minkowski space-time satisfying the vacuum Maxwell equations and

2. 2. whose arrows are isometries χ of Minkowski space-time that preserve F _ab.

We may then take W to consist of a specification of the empirical consequences of a particular Faraday tensor F _ab: its associated energy-momentum tensor, a collection of Lorentz force curves for test matter of different charge-to-mass ratios, and so on. Finally, we may define a “functor” $E:\mathrm{ℰℳ}\to W$ that takes Faraday tensors to the observable quantities associated with that tensor.

With this machinery in place, we may define categorical equivalence. We will say that two theories are categorically equivalent if there exist functors $F_1:{𝒯}_1\to {𝒯}_2$ and $F_2:{𝒯}_2\to {𝒯}_1$ such that

1. 1. F ₁ and F ₂ are essentially inverses (i.e., they realize an equivalence of categories), and

2. 2. $E_2\circ F_1=E_1$ and $E_1\circ F_2=E_2$.

In other words, the categories representing the two theories are equivalent (as categories), and the functors realizing the equivalence commute with the maps taking models to their predictions. Thus, if I begin with a model of the first theory and I ask what phenomena are associated with that model, and then I take the corresponding model of the second theory under the equivalence, it will be associated with the same predictions as the model with which I began.

3. Electromagnetic Duality

With this background from the theoretical equivalence literature in place, we now turn to dualities. In fact, physicists discuss many different examples of dualities. For instance, one widely discussed example that has played an important role in string theory since it was introduced more than 2 decades ago by Maldacena (Reference Maldacena1999) is the AdS/CFT correspondence, also known as the gauge-gravity duality. This duality relates a theory of gravity in an (asymptotically) anti–de Sitter space to a conformal field theory on the boundary of that space. Another, even older, example is T-duality, which relates string theories defined on a manifold with radius R to string theories defined on a manifold with radius $1/R$ (Giveon, Porrati, and Rabinovici Reference Giveon, Porrati and Rabinovici1994; Alvarez, Alvarez-Gaumé, and Lozano Reference Alvarez, Alvarez-Gaumé and Lozano1995). Other, recently introduced, dualities are the particle-vortex dualities discovered by Karch and Tong (Reference Karch and Tong2016) and Seiberg et al. (Reference Seiberg, Senthil, Wang and Witten2016). And so on.

My focus here is on yet another “classic” string duality, known as S-duality or Montonen-Olive duality (Montonen and Olive Reference Montonen and Olive1977; Seiberg and Witten Reference Seiberg and Witten1994; Alvarez-Gaumé and Hassan Reference Alvarez-Gaumé and Hassan1997; Figueroa-O’Farrill Reference Figueroa-O’Farrill1998). S-duality relates certain (supersymmetric) Yang-Mills theories whose coupling constants are large to other ones whose coupling constants are small. This feature makes the duality extremely useful, as strongly coupled theories are generally more difficult to work with than weakly coupled ones. But it is also of interest for other, more foundational reasons. For instance, it relates theories with gauge group G to theories whose gauge group is the Langlands dual to G, which suggests a physical connection to the geometric Langlands program (Frenkel Reference Frenkel and Cartier2007).

I will not consider S-duality in full generality. Instead, I focus on the simplest case of S-duality, known as electromagnetic duality. The rest of what I say will apply to electromagnetic duality.Footnote ¹⁰

At this point two concerns might arise. First, one might worry that by focusing on just one example of a duality what I say will lack generality. In fact, I am sympathetic to this worry, but I would go further: the word “duality” is used in many different ways in physics, and it is not at all clear that all “dualities” one encounters in the physics literature should (or can) be seen as instances of a single species. Certainly one does not find a clear or precise definition of “duality” in the literature or attempts to argue that such and such example satisfies such a definition.Footnote ¹¹ Still, I think the classic string dualities mentioned above (S-duality, T-duality, gauge-gravity duality, etc.) are meant to refer to importantly similar relationships, and I think, although I will not argue for this, that what I say about S-duality applies to these, at least, as well.Footnote ¹² But strictly speaking, whether this is true does not affect my argument, as my goal is to establish an existence claim: that is, there exist dualities that are not theoretical equivalences in the sense philosophers of science have considered.

The second concern is similar but stronger: one might think that, by focusing just on electromagnetic duality, I have moved away from the characteristic features of dualities. I am not considering quantum field theories, much less string theory; I am not making use of any of the machinery usually used to show that a duality exists; and so on. To this I respond: true, but in these regards electromagnetic duality offers many advantages, especially regarding questions about empirical significance, because it is much clearer how one should understand classical electromagnetism to represent the world than string theory. More, electromagnetic duality is widely cited as a (motivating) instance of S-duality (cf. Alvarez-Gaumé and Hassan Reference Alvarez-Gaumé and Hassan1997; Figueroa-O’Farrill Reference Figueroa-O’Farrill1998), and so one should expect the relevant features to be similar. And, once again, even if this turns out to be false, my point is to make an existence claim. To rebut it, one needs to either refute my argument concerning electromagnetic duality or else deny that electromagnetic duality is, indeed, a duality. And this latter move, I suspect, would be difficult to make without also denying that electromagnetic duality is an instance of S-duality.

We now move on to define electromagnetic duality. For simplicity, we work on Minkowski space-time (M, η_ab). Mathematically, electromagnetic duality is a mapping from 2-forms to 2-forms given by:

$\begin{array}{}& F_{\mathrm{ab}}\mapsto \star F_{\mathrm{ab}},\end{array}$

where ⋆ is the Hodge star operation. Now observe that this mapping preserves Maxwell’s equations, in the sense that if ${\nabla }_a{F}^{\mathrm{ab}}=0$ and ${\nabla }_{[a}{F}_{\mathrm{bc}]}=0$, then ${\nabla }_a(\star F^{\mathrm{ab}})=0$ and ${\nabla }_{[a}\star F_{\mathrm{bc}]}=0$.Footnote ¹³ It follows that if F _ab is a source-free solution to Maxwell’s equations, then so is ⋆F _ab. One way to see how this duality acts is to fix an observer and decompose F _ab at a point, relative to that observer’s state of motion, into electric and magnetic fields E ^a and B ^a. In that case, the duality acts as $(E^a,B^a)\mapsto (B^a,-E^a)$.

We can express this relationship in the framework of categorical equivalence. Consider again the category ℰℳ described above. Then electromagnetic duality defines a functor $D:\mathrm{ℰℳ}\to \mathrm{ℰℳ}$, as follows: D acts on objects as $F_{\mathrm{ab}}\mapsto \star F_{\mathrm{ab}}$ and on arrows as $\chi \mapsto \chi$.Footnote ¹⁴ Defined thus, it follows immediately that D is an autoequivalence of ℰℳ.Footnote ¹⁵

I can now state my central claim more precisely. The functor D does not preserve empirical content, in the follow sense. Recall that in the previous section, I defined an “empirical significance” functor, $E:\mathrm{ℰℳ}\to W$, that takes Faraday tensors to stylized phenomena. The claim is that D does not commute with $E:E\circ D\ne E$. In other words, the electromagnetic duality maps Faraday tensors to Faraday tensors that give rise to different predictions. The theories are, in this sense, empirically distinguishable and thus cannot be theoretically equivalent.

4. Manna Lelyalde Duality?

Thus far we have argued that dualities do not (necessarily) preserve empirical content, in the following sense: fix, in advance, how one’s mathematical formalism(s) will correspond to observable phenomena; then the duality transformation will not necessarily preserve those phenomena. We will now consider some reactions one might have to this argument.

One reaction is to say that we should not apply the same “empirical significance” functor both before and after applying the duality. Instead, to capture the sense in which the duality realizes an empirical equivalence, one needs to consider both our original empirical significance functor and another “dual” empirical equivalence functor that properly assigns predictions to Faraday tensors under the duality.

One consideration strongly favoring this perspective is that the stylized predications of electromagnetism, as described above, involve theoretical concepts that should be transformed under the duality. In particular, the predictions concern the trajectories of (electrically) charged “test particles.” And charge-current density should not be invariant under the duality: although we have been considering the vacuum case so far, were we to include sources, we would expect the duality to act by taking electric charge-current density to magnetic charge-current density and vice verse. Although test matter is not, by definition, source matter, we might nonetheless insist that it is an inconsistent application of the duality transformation to keep our test matter fixed while transforming fields (and, if appropriate, sources). Thus, the argument goes, what we really need is an empirical significance functor Ẽ that associates each Faraday tensor with the trajectories of accelerated test (magnetic) monopoles instead of test charges.

This response has merit, but I see at least two counterarguments. The first is that employing this dual empirical significance functor seems in tension with a natural rough-and-ready criterion of empirical inequivalence, where two theories are empirically inequivalent if I can build a device that would permit me to empirically distinguish them. In the current case, one could build such a device: in effect, one could build a box that, on its outside, had a digital display giving a value of the electric and magnetic fields inside the box.Footnote ¹⁶ And any such device, if used to measure the electric and magnetic fields associated with a Faraday tensor F _ab and its dual ⋆F _ab would, in general, yield different results. In this sense the theory and its dual are empirically distinguishable by direct experimental test.

Second, while it is true that one can always find a different “empirical significance” functor, Ẽ, with the property that $E=\tilde{E}\circ D^{\mathrm{-1}}$, its properties should give us pause. In fact, that D is an automorphism of the category ℰℳ ensures that this functor Ẽ is uniquely given by $\tilde{E}=E\circ D^{\mathrm{-1}}$. If we use this “dual” empirical significance functor, we regain a sense in which the duality preserves empirical significance. But we should be cautious about this conclusion, because, as is evident from the definition of Ẽ, we get to claim equivalence only by inverting the duality and then applying our old empirical equivalence functor, E. In other words, we get equivalence only by systematically redefining all of the terms so as to effectively “undo” the duality. We can indiscriminately swap electric and magnetic fields as long as we simultaneously redefine “electric field” to refer to what we used to mean by “magnetic field,” redefine “magnetic field” to mean what we used to mean by (negative) “electric field,” redefine “magnetic monopole” to what we used to mean by “point charge,” and so on.Footnote ¹⁷

Given these objections, the sense of empirical equivalence given by appealing to Ẽ might seem trivial. And in a sense it is: if we permit arbitrary reinterpretations of our theories to force empirical equivalence whenever we have some mathematical equivalence, then empirical equivalence adds nothing to a (sufficiently strong) notion of mathematical equivalence. But this is not to say that the duality is trivial. In fact, that the duality map exists reflects a highly nontrivial mathematical fact about the formalism of electromagnetism—and, likewise, for the more sophisticated dualities described above. Even the existence of Ẽ is nontrivial, insofar as it captures the sense in which, given the duality D, it is possible to systematically and completely perform the reinterpretation necessary to give a different, but in a sense equally good, recipe for extracting empirical significance from the formalism of electromagnetism.

There is another line of objection that is related to the foregoing discussion but that I think is stronger.Footnote ¹⁸ One might think that there is a basic problem with the starting point for the analysis given here. Following the empirical equivalence literature, I have assumed that it is possible, given a theory, to fix, once and for all, how that theory makes predictions. In fact, how we interpret the mathematics we use in physics itself depends on contingent facts about physical systems, relationships between different sets of laws, and in many cases, the history of how we developed the mathematics for particular applications. It is not something we “fix,” by fiat, given some collection of mathematical structures, nor is it something we can arbitrarily revise. Now, after the long and messy work of building theories, applying them, and gradually learning how to interpret them is done, one might be able to summarize the results using something like the functor E. But E is not given to us, prior to and independent of that work. From this perspective, then, it is wrongheaded to try to use E to extract empirical information from the dual theory, D(ℰℳ), because to do so is to stipulate meanings for a putatively “new theory.” Really, we would need to imagine trying to apply the dual theory to real world phenomena and ask how we would come to understand how to interpret the formalism from that starting point. Doing this would very likely lead one to Ẽ as a natural way of association the formalism with physical situations.Footnote ¹⁹ So taking E to reflect the “default” interpretation of the formalism is misguided.

This response further supports the suggestion above, that moving to a different interpretation may be well motivated in some cases. But it does not undermine the claim that E and Ẽ are different—or that we could not empirically distinguish electromagnetism from its dual.

Where does this leave us? The upshot is that there is a sense in which dual theories are empirically equivalent and a (stronger) sense in which they are not, at least in general. They are not (in general) empirically equivalent in the sense usually required for theoretical equivalence: they do not necessarily preserve “prior” empirical content. They are empirically equivalent in a weaker sense, where we permit reinterpretations that force the equivalence. In other words, a theory is, in general, equivalent to its dual theory only with a compensating “antidual” empirical interpretation. How well motivated this reinterpretation is will depend on context.