Choice of examples:

our choice has been guided by our aim of giving an even treatment along the following three axes:

1. (i) familiar and less familiar examples,

2. (ii) elementary and more technically demanding examples, and

3. (iii) examples that are representative of a variety of branches of physics: quantum mechanics, statistical mechanics, electrodynamics, condensed matter physics, quantum field theory, and quantum gravity.

Thus, for example, we include a discussion of Kramers–Wannier duality (Section 3.2) for a two-dimensional Ising lattice. Although this duality is elementary, it has received very little attention in the philosophical literature on dualities, and so it will be less familiar to philosophers.Footnote ⁷ However, Kramers–Wannier duality is in some sense a paradigmatic example of the kind of physical phenomena that are associated with dualities, for it allows the precise study of the phase transition between a ferromagnetic and a paramagnetic phase, symmetry breaking, and the emergence of an ordered state of matter.

Other dualities that build on this elementary example, and that show similar behaviour, are particle–soliton dualities in three dimensions (two dimensions of space and one of time: see Section 4.1.1), which are associated with the Berezinskii–Kosterlitz–Thouless phase transition between a high-temperature phase of dissociated vortices and a low-temperature phase of condensated pairs of vortices and anti-vortices.

Point (iii) leads to some unexpected connections. For example, there are (under the umbrella of AdS-CFT) surprising dualities that map solid state systems to (quantum) gravity systems (Mauri et al. Reference Mauri, Smit, Golden and Stoof2024). This is surprising, because, apart from the fact that the solid state systems in question are actually realized in the lab (Smit et al. Reference Smit, Mauri and Bawden2021), the duality relation goes well beyond the analogies that have been discussed in the philosophical literatureFootnote ⁸: in particular, it provides a much closer connection between condensed matter physics and high-energy physics than previously considered examples.Footnote ⁹

In the philosophical literature, inter-theoretic relations such as theoretical equivalence, reduction and emergence have traditionally been discussed between similar, or at least closely related, branches of physics. Standard examples of theoretical equivalence are the relations between different versions of the Maxwell theory, between different versions of Newtonian mechanics, and between different versions of general relativity; and standard examples of reduction and emergence are the relations between statistical mechanics and thermodynamics and between high-energy and low-energy quantum field theories (arguably, the emergence of classical mechanics from quantum mechanics is one example where the distances are greater). Indeed, the discussion of inter-theoretic relations between more distant branches of physics, where the interpretations of the two theories differ more, has often been mostly in discussions of formal analogies, for example, between condensed matter physics and quantum field theory.Footnote ¹⁰ What strikes us as novel about the examples of dualities, such as the example of condensed matter and AdS-CFT as previously discussed, is that these relations are not formal analogies but isomorphisms (or close to isomorphisms): and furthermore that, even if the physics on the two sides of a duality sometimes looks very different, there is nevertheless a deep match of the physics across the duality, that is, not only of formulas and formalisms, but also of detailed physical mechanisms, which can be used to answer questions about, and even to (re)construct, the mechanisms in the dual model. This ‘matching of physical mechanisms’, and the related ideas of modelling, simulation and use of analogies across dualities, surely deserve further philosophical analysis.

Preliminary note and prospectus.

This Element draws from De Haro and Butterfield (Reference De Haro and Butterfield2025) in: (a) its general approach to dualities (namely, the introduction of a Schema for dualities), (b) its choice of examples, and (c) its philosophical discussions. For this reason, we will not always refer back to De Haro and Butterfield (Reference De Haro and Butterfield2025), except when we use specific results. But the reader can rest assured that, on all the topics (except for the role of emergence in bulk reconstruction, see Sections 4.3 and 5.3), more detailed results and explanations can be found there.

Having said that, this Element also contains novel aspects, and aspects that are presented slightly differently from how they are discussed in De Haro and Butterfield (Reference De Haro and Butterfield2025). For example, the discussion, at the end of Section 3.3, of the common core theory for the simple electric–magnetic duality of the Maxwell theory in vacuum, differs from the one in De Haro and Butterfield (Reference De Haro and Butterfield2025) in that it is formulated for the Maxwell theory written in components, rather than for the Lorentz-invariant theory. Also, the discussion, in Section 4.3, of bulk reconstruction in AdS-CFT, and its relation to emergence discussed in Section 5.3, is wholly new. Finally, Section 5.4 contains a number of FAQs about dualities that may be convenient for readers looking for quick answers to specific conceptual questions about dualities.

The plan of the rest of the Element is as follows. Section 2 introduces the notion of duality and its main features. Sections 3 and 4 are a brief introduction to a selection of examples of dualities: Section 3 introduces classic, more elementary, examples, and Section 4 gives physically more advanced examples. Then Section 5 discusses a selection of philosophical questions associated with dualities, and also includes answers to a number of FAQs. Section 6 concludes.

Duality and symmetry.

It is natural to contrast duality and symmetry, understood as a map between solutions (more generally, an automorphism of state-spaces and quantities) that, at least in certain cases, is expected to leave the physical content of the theory invariant. This feature of symmetries does indeed make them apparently very similar to dualities, because both are formal equivalence maps that may allow a verdict of physical equivalence, depending on the specific physical system or theory under consideration, and on our interpretative choices regarding these equivalences. Indeed, we can think of dualities and symmetries as being related, and of a duality as a giant symmetry (De Haro and Butterfield, Reference De Haro and Butterfield2025). The main formal difference between the two is that dualities map whole theories, while symmetries are automorphisms of state-spaces and-or of sets of quantities (Caulton Reference Caulton2015).

Theories and models.

The previous observation motivates the first important point to make in introducing the Schema: namely, in the presence of dualities, it is sensible to take the standard philosophical terminology of theory and model (which in philosophy of physics usually takes the place of the previously-mentioned ‘solutions’) and bring it one level up. By ‘one level up’, we mean that model will mean one of the ‘theories’ (in the old sense) that are related by the duality map, while theory will mean a hypothetical theory that is behind the two duals. This usage is one level up because ‘model’ usually means solutions of theories, while ‘theory’ means the theories themselves; on the other hand, in the case of dualities, we call ‘models’ the theories themselves, and we call ‘theory’ a further theory, where the duality is a manifest equivalence and which stands to the duals in a relation analogous to that between models and theories in the standard picture. More generally, and regardless of dualities, models are representations (in the mathematical not philosophical sense) or instantiations of theories.

An example will be useful to get acquainted with this jargon. During the development of quantum mechanics, two rival theories became popular to account for quantum phenomena: Heisenberg’s matrix mechanics, which described quantum phenomena using matrices and linear algebra, and Schrödinger’s wave mechanics, which described quantum phenomena in terms of waves. These two theories were considered rival attempts at formulating a theory of quantum mechanics, with various arguments given in favour of one or the other,Footnote ¹⁴ until 1932, when von Neumann, in his seminal work on the mathematical foundations of quantum mechanics (John von Neumann Reference von Neumann1932), showed that matrix and wave mechanics are two equivalent ways of presenting the same physical content, now expressed in terms of Hilbert spaces. In this example, the models are the two ‘theories’ (old sense!) of wave and matrix mechanics, which are models of the underlying theory of quantum mechanics expressed in terms of operators on Hilbert spaces of states, and where the duality between wave and matrix mechanics is the isomorphism between the corresponding Hilbert spaces, with their respective sets of operators.

Theories as triples of states, quantities, and dynamics.

With our usage of ‘theory’ and ‘model’ clarified, we now introduce a more formal, but otherwise standard, definition of what a scientific theory or model minimally amounts to, from a mathematical point of view (and for the moment, regardless of physical interpretation)Footnote ¹⁵: a theory or model is a triple $⟨S,Q,D⟩$ of set of states $S$ or state-space, set of quantities $Q$ , and dynamics $D$ . ‘Set of states’ here means, broadly, the space of variables that, once interpreted, will describe the possible configurations of a physical system or sets of systems that we are interested in, formally represented in the triple by things like, for eample, points in phase space or rays in Hilbert space. ‘Quantities’ are the variables that, once interpreted, will describe the observable physical properties of the system: in the triple, these properties are formally represented by, for example, linear operators (in a quantum theory) or, in classical mechanics, by functions on phase space. Finally, ‘dynamics’ is some set of equations that, once interpreted, describe the evolution of the system we are studying, such as the Schrödinger equation in quantum mechanics or the Einstein field equations in General Relativity.Footnote ¹⁶ Furthermore, the state-space and the set of quantities are both usually equipped with additional structures (e.g. symmetries) and rules for evaluating the values of quantities on physical states. These triples $⟨S,Q,D⟩$ encode the basic structure of physical models, and will serve us in articulating the Schema.Footnote ¹⁷

The conception of duality.

Given a theory and a pair of models thus formulated as triples, we define a duality as a bijective, structure-preserving, map (i.e. an isomorphism),Footnote ¹⁸ $d$ , between the models’ sets of states $S$ , and between the models’ sets of quantities $Q$ .Footnote ¹⁹ We define the structure to be preserved as follows: there is an isomorphism of state-spaces and an isomorphism of sets of quantities,Footnote ²⁰ which are (i) equivariant with respect to the dynamics $D$ ; and (ii) preserve the values of the quantities. We can visualize these relations through the commutative diagram in Figure 1.Footnote ²¹

Figure 1

Equivariance of duality and dynamics, for states and quantities.

The isomorphism condition for duality secures that states and quantities are in a precise correspondence across the duality map, while the equivariance condition for the dynamics secures that the dynamics of the two models are compatible. Value-preservation for quantities is also required. (This is usually interpreted as the two theories having, for all possible measurement outcomes related by duality, the same values.)

The common core theory.

We have defined a duality as an isomorphism between models. We now return to our earlier discussion of theories and models, where duals are, in our jargon, models. This raises the natural question of what theory these duals are models of. We shall call this theory a common core theory for the dual models, that is, one that contains the structure that is common to the duals. Thus we require that the duals are instantiations, usually mathematical representations, of this common core theory. This formal condition secures that the structure that is common to the models is capable of physical interpretation, and so is physically significant. That it is a ‘theory’ (albeit uninterpreted at this stage) means that it is again a triple $⟨S,Q,D⟩$ .Footnote ²²

Since at this formal stage this is an uninterpreted theory, we also call it the bare theory. The requirement that the common core is a theory means that being a duality is a strong requirement, because the common structure should amount to a triple of set of states, quantities, and dynamics. This implies that not any old partial isomorphism between models is a duality. Furthermore, the role of the common core theory is also that, especially in cases where the two duals are very different, it makes the isomoprhism explicit. This theory deals with structure that is invariant under the duality map $d$ , similarly to how we would define a theory that is invariant under the action of a certain symmetry relating two models of that theory.Footnote ²³ We then recover the original duals by representing the common core theory using specific structure, that is further, non-duality invariant bits of formalism that are used to individuate and define the models. Using an analogy from De Haro and Butterfield (Reference De Haro and Butterfield2025), we can think of the relation between dual models and their common core, in terms of the relation that obtains between, for example, a group and its representations. Here, the common core is like the abstract group, and each dual model is like a representation of the group: thus there is a duality if two such representations are isomorphic.

Interpreting duals.

We have so far discussed models and theories as formal, that is,  uninterpreted triples. However, as we mentioned in our discussion of empirical and theoretical equivalence, to understand the roles of dualities in physical theories we cannot ignore questions about the relation between the duals and the systems they model, that is, about their interpretation. For our purposes, it is useful, and also a widespread philosophical practice, to think of an interpretation as a mapping of a theory or model (here, a triple $⟨S,Q,D⟩$ ) into a domain of application $D$ in the world. The domain of application is the relevant collection of physical systems that our theory or model describes, identified through observational and experimental procedures, specific limits, and so on, as they are familiar from the practice of science. In other words, the domain of application of a theory or model is a part or domain of the world, of physical reality, that is suitable to be described by a given triple $⟨S,Q,D⟩$ .

To sum up: an interpretation is a map $i$ from a triple $⟨S,Q,D⟩$ to a domain of application $D$ . Since we are interested in studying how interpretations interact with dualities, we should ask what sorts of maps are allowed for dual models of the type described by the Schema. Based on the structure that these maps map, there are two possible kinds of interpretationFootnote ²⁴:

• Internal interpretations: These are interpretations that only interpret the structure that is common to the duals, that is, the common core theory. In other words, the interpretation map $i$ maps duals into a single domain of application $D$ , so that $D$ ’s features depend on the structure of the common core only, and not on further structure typical of either dual, that is, their specific structures. Since internal interpretations only deal with structure that is invariant under the duality map, they give interpretations of the common core theory.

• External interpretations: These are interpretations that, besides mapping the structure that is common to the duals (i.e. the common core theory), also map the specific structure into the domain of application $D$ . This entails that the specific structure is part of the physical content of our theories, that is, it is part of the structure that represents the theory’s domain of application. As a consequence, since dual models do not have this specific structure in common, any two models that are interpreted according to their external interpretations are mapped into different domains of application.

The external vs. internal contrast exhausts the options for interpreting duals. For either we interpret each dual with its specific structure, independent of the other dual’s non-matching specific structure (as in an external interpretation), or we do not interpret the specific structure, and we interpret only the common core (as in an internal interpretation). Thus going back to the Schema described previously, there is no further structure that we could interpret: and so, for duals and interpretations thus defined according to the Schema, we have exhausted the different ways to interpret dualities.

We have so far given a general introduction to dualities, without delving into specific examples. In the following sections, we will give several examples that illustrate these ideas, and which show the variety of interesting philosophical issues that dualities raise. In the next section, we first introduce a number of interesting philosophical and conceptual themes that emerge from the study of dualities, and which illustrate dualities’ many roles in physics: these themes will guide our analysis of various examples in the rest of this Element.

Hard-easy.

The contrast ‘hard-easy’ means that a problem that is difficult to solve in one dual, is easy to solve in the other dual. For example, a difficult calculation becomes tractable after a duality transformation. This is similar to how changes of variables can be valuable tools in solving integrals or differential equations.

To see this more precisely, consider how dualities map the coupling constants of physical theories.Footnote ²⁵ Recall that coupling constants characterise the strength of interactions: the larger the value of the theory’s coupling constant, the stronger the interactions. Thus we will call a (regime of a) theory where interactions are weak, weakly coupled. Likewise, a (regime of a) theory where interactions are strong is called strongly coupled. Furthermore, if we take the coupling constant of a theory to be an arbitrary positive real number then, according to our usage of ‘theory’ and ‘model’, each of these regimes gives a model, that is, a specific realization of the theory. Each such model is identified by its specific structure: here, a choice of coupling.Footnote ²⁶ While the weakly-coupled regime of a theory usually allows perturbative calculations, the strongly-coupled regime rarely allows this, and often remains poorly understood.

Perturbation theory is a method to approximate the value of physical quantities for an (unknown) solution, by taking their values in another (known) solution and adding small perturbations. ‘Small’ is here defined by a parameter such as a coupling constant. This amounts to expanding the value of the quantity, for example a scattering amplitude $A(g)$ with coupling constant $g$ , as a power series in $g$ around the value at zero coupling, that is, $A(g=0)$ :

$\begin{array}{r}A(g)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{g}^n.\end{array}$(1)

Since the theory can be solved exactly at zero coupling, this method gives us a way to approximate unknown solutions of quantum field theory to the ones at zero coupling. The main limitation of perturbation theory is that the series (1) is often asymptotic, that is, it diverges at large values of $n$ . This requires us to truncate the expansion (1) at a finite value of $n$ , thus rendering the result of this procedure an approximation rather than a convergent series.

The expansion Eq. (1) usually only works in a small neighbourhood of the known solution, $A(0)$ . For large values of the coupling constant, that is in the strongly-coupled regime, the expansion Eq. (1) breaks down, and we cannot rely on perturbation theory. For this reason, the strongly-coupled regime is also called the non-perturbative regime of the theory. Since perturbation theory is the main method to construct solutions of quantum field theories, this implies that the strongly-coupled regime is in general beyond the scope of current calculational techniques, which almost always rely on perturbation theory.

With this in hand, it is straightforward to illustrate the hard-easy theme of dualities. The point of this contrast, and the reason why dualities are important calculational and descriptive tools, is that they allow us to relate a perturbative, weakly-coupled, description in one dual, to a strongly-coupled, non-perturbative, description in the other dual. Indeed, as we will discuss in Sections 3 and 4, dualities often map the coupling constant $g$ of one dual to the inverse coupling constant, $1/g$ , of the other dual. And by our previous discussion, this means that a weakly-coupled calculation is dual to a strongly-coupled one. Thus using dualities we can extend the regime of applicability of our calculational techniques to non-perturbative phenomena, thus opening a new avenue for the study of the strongly-coupled regime of quantum field theories.Footnote ²⁷

Elementary-composite.

This is another interesting theme associated with dualities. Although we will give a more detailed treatment of this theme in Section 4.1, we briefly sketch it here. Dualities sometimes map objects that are simple, that is, that cannot be decomposed into more elementary components in a given model, into objects that are composites, that is, that can be decomposed in the dual model. So by this theme, dualities map objects that lack mereological structure,Footnote ²⁸ into objects that have it. And this casts doubt on the fundamentality of such mereological notions, and on their ability to ‘carve nature at the joints’ (see Castellani (Reference Castellani2017) for more on this point).

An especially important example of this theme is particle–soliton duality, which maps a fundamental degree of freedom in one model to a soliton in the other model, that is, a topologically non-trivial configuration of the fundamental degrees of freedom of the dual. This relation is clearly an example of our theme of elementary-composite, since it relates a fundamental, elementary object, a particle, to a complex, composite one, that is, a soliton. A simple illustration of this phenomenon is the Sine Gordon-Massive Thirring model duality.

Without going into the details,Footnote ²⁹ the point of such a duality is that we relate the fundamental fermionic degrees of freedom of the Massive Thirring model, to a topologically complex configuration of the Sine Gordon modelFootnote ³⁰: and vice versa, we relate the fundamental bosonic degrees of freedom of the Sine Gordon model to a composite of two fermion fields in the Massive Thirring model. Due to this mapping between bosons and fermions, this duality is an example of bosonization. The relation between the Sine Gordon and Massive Thirring models illustrates the theme of elementary-composite, since it relates simple fundamental entities to composite ones, and vice versa, thus raising important questions about the meaningfulness of such a distinction.

Exact-effective.

In some cases, what looks like a duality may in fact not be a precise isomorphism between models. For example, the map may only map some quantities from one model to the other, but not all of them, as is required for a duality. Also, while a duality is by definition exact, that is, valid without approximations, there is an effective duality when the map between models approximates a duality in a regime of parameters. More generally, a quasi-duality is any relation between models that is similar to or close to a duality, but is not a duality, that is,  a precise isomorphism of the type defined in Section 2.1. Most examples of this theme are somewhat technical, and so we will not here give an intuitive introduction like we did for the other themes (important examples of this are in Section 4.1: particle–soliton duality, dual superconductivity, and electric–magnetic duality for $N=2$ supersymmetric Yang–Mills theory). However, to get an idea of the type of physics involved, note that arguably the most important and famous instance of effective duality is the relation between perturbative string theory and non-perturbative M-theory, that is, the fundamental theory underlying the five different superstring theories, which we will discuss in Section 4.2.

An immediate consequence of the fact that quasi-dualities are not isomorphisms of models, so that there is no precise common structure of models that they should preserve, is that there is no straightforward common core theory underlying the two quasi-dual models. (We say ‘no straightforward common core theory’, since a precise common core theory for duals might exist in some limit of parameters in which we do recover a duality.)

The main reason for the interest in quasi-dualities, which we will explore in Section 5.2, is that they suggest a novel way to represent the structure of scientific theories. Indeed, quasi-dualities suggest that scientific theories are not best represented as, for example, collections or sets of models (as in the semantic conception of theories), but rather as differential manifolds or more general geometric objects (such as algebraic varieties). We will call this view of physical theories the geometric view of theories.

We next discuss some of the roles that dualities play in physical theorizing. Here the main distinction, introduced in De Haro (Reference De Haro2019), is between the theoretical and heuristic roles of dualities. In simple terms, the relevant distinction is between the role of dualities as well-established theoretical relations that can be used to study the properties of scientific theories (their theoretical role), and dualities as clues for the construction of more fundamental theories (their heuristic role).

It is worth mentioning two central philosophical topics to which dualities, in their theoretical role, are closely related: (i) theoretical equivalence, which will be the topic of Section 5.1; and (ii) scientific realism, in particular in connection with issues of under-determination. While we will not discuss topic (ii) in detail in this Element, we refer the reader to Matsubara (Reference Matsubara2013), Read (Reference Read2016), Le Bihan and Read (Reference Le Bihan and Read2018), De Haro (Reference De Haro2023), and De Haro and Butterfield (Reference De Haro and Butterfield2025) for thorough treatments.

About the heuristic role, it is worth mentioning that the natural heuristic use of dualities is as guesses for the minimal structure of a more fundamental theory, of which duals are expected to be some kind of approximation or limit: we call this deeper theory a successor theory. If the duals are approximations or limits, then it is natural, from the point of view of the more fundamental theory that we are constructing, to think of the duality as being a limit of an effective duality or a quasi-duality. Thus in their heuristic role, dualities can often be taken to point towards a more fundamental theory.

Finally, we discuss the special types of dualities that one finds in the literature, and which we will discuss in this Element. There are two special types: quantum dualities and self-dualities.

A quantum duality is a duality between quantum mechanical models whose classical limits are not duals. These dual quantum models are different representations of a single quantum theory, and so the two classical models can often be seen as different classical limits of a single quantum theory.

A self-duality maps a given model onto another model of the same kind, or onto the same model, while it also changes a parameter (possibly with other additional changes that compensate for this variation). An example of this is the self-duality of $N=4$ supersymmetric Yang-Mills theory that we will discuss in Section 4.1, where the relevant parameter is the coupling $g$ , and the additional changes are the exchange of the theory’s gauge group $G$ with its Langlands dual group $G^{\vee }$ (but this last change will not play any role in our discussion). The reason for the name ‘self-duality’ is the physics convention of treating theories that differ only by a change in a parameter as being the same theory.Footnote ³¹ Thus, as with symmetries, such transformations can be seen as automorphisms of the state-space and set of quatities (as we defined them previously, following Caulton (Reference Caulton2015)); hence the self in ‘self-duality’, and in this case we just talk about ‘the theory’.

This said about the themes, roles, and types of dualities, the next two sections will illustrate these general ideas in several examples. Then we will return, in Section 5, to a general discussion of the philosophical questions.

Fourier duality and the Schema.

We now discuss how Fourier duality counts as a duality according to the Schema. First, take the Hilbert spaces in the position and momentum variables, discussed previously, to be our dual models (in the sense of ‘model’ in Section 2). Second, take the duality map between these two models to be the Fourier transformation, that is,  $d_S=F$ . Then: (i) In virtue of $F$ ’s being a unitary equivalence of Hilbert spaces, it is an isomorphism of the state-spaces $S$ . (ii) $F$ also induces an isomorphism between the sets of operators, that is, the sets of quantities $Q$ . For example, the Hamiltonian in the momentum representation and the Hamiltonian in the position representation are related as follows: $H_{\text{momentum}}=F{H}_{\text{position}}{F}^{-1}$ . (iii) $F$ is also equivariant with respect to the dynamics of the system. This can be seen from the relation between the Hamiltonian in the position and momentum representation given under (ii), Footnote ³⁵ which also maps the Schrödinger equation of one model to the Schrödinger equation of the other model. Therefore, this establishes the dynamical equivalence of the models. (iv) Finally, note that the preservation of inner products, secured by the unitarity of $F$ , guarantees that the values assigned by the states $S$ to each quantity $Q$ are the same in both representations.

Having discussed how Fourier duality illustrates the Schema, it is interesting to discuss in more detail the common core theory underlying the two Fourier dual models. The answer to this question has been known since von Neumann’s seminal book (von Neumann Reference von Neumann1932): namely, quantum mechanics as described in terms of Hilbert spaces. For once we start representing quantum mechanics in terms of Hilbert spaces, it is natural to associate a specific quantum system with a Hilbert space of states for that system, which we will call $H$ . The position and momentum representations are then realized as two different bases of the Hilbert space, and the Fourier duality between them is a change of basis. The Hilbert space is itself independent of the choice of a basis, so that it can be taken to be the state-space $S$ of the common core theory, of which the two basis-dependent state-spaces (whose states are written in Eqs. (2) and (3)) are representations. Likewise, the algebra of operatorsFootnote ³⁶ associated with that Hilbert space can be taken to be the set of quantities, $Q$ , of the common core theory, of which the position and momentum operators, written above in terms of a choice of basis, are representations. Finally, the dynamics $D$ is given by the Hamiltonian (i.e. the time-evolution) operator on the Hilbert space. Thus, the common core theory for Fourier duality is Hilbert space quantum mechanics.

Fourier duality also illustrates our theme of hard-easy from Section 2.2: namely, that some problems that are hard to solve in one representation, are easy (or at least easier) to solve in the other. Depending on the form of the Hamiltonian, this is the case when the Schrödinger equation is easier to solve in one representation than in the other. For example, for constant potentials (such as the one-dimensional infinite square well, or the step potential) the Schrödinger equation is easier to solve in the position representation. For the time-independent Schrödinger equation is then the free wave equation, and the boundary conditions have a clearer interpretation in position space. The momentum representation has advantages in other cases: for example, in three-dimensional problems with symmetries, like the rigid rotor, it is easier to think about the wave-functions as eigenstates of the angular momentum operators. And for scattering problems on lattices, it is also often easier to work with momentum vectors on the reciprocal lattice and then Fourier transform to the original lattice.

The common core theory.

For the simple Maxwell theory, perhaps the simplest way to formulate a common core theory is by defining the following complex vector field:

$E:=\mathbf{\text{E}}/c+i\mathbf{\text{B}}.$

In terms of this complex vector, the Maxwell equations in vacuum, Eqs. (7), reduce to the condition that $E$ is divergence-free, and satisfies the following linear wave equation:

$\mathrm{\nabla }\times E=\frac{i}{c}\frac{\mathrm{\partial }E}{\mathrm{\partial }t},$

which in turn implies that $E$ satisfies the Klein–Gordon equation.

The duality map rotates this vector field over $\pi /2$ in the complex plane: $d_S:E\mapsto -iE$ .Footnote ⁴² This means that, formally, we can think of the duality transformation as a rotation of the vector in the complex plane: or, alternatively, we can adopt a passive interpretation where the duality transformation corresponds to a choice of coordinates (specifically, a choice of complex structure on the complex plane). In what follows, we will adopt this latter interpretation.

The states of this theory are states of a complex, transverse-polarised, vector field whose four-momenta are null vectors (so that each plane wave propagates at the speed of light). Thus we can interpret these properties as properties of an electromagnetic theory, but without committing to identifying ‘purely electric’ or ‘purely magnetic’ properties, which could only be defined through a choice of complex structure–and we do not need to make such a choice in the common core theory. A change of complex structure changes the putative ‘purely electric’ or ‘purely magnetic’ properties.

Thus the specific structure that a model adds to the common core theory is a choice of complex structure on the plane. This choice determines what properties count, in a given model, as ‘purely electric’ or ‘purely magnetic’.

The common core theory itself only knows about electromagnetic properties and excitations, that is, excitations of a complex vector field that are transverse, propagate at the speed of light, and satisfy a dynamical equation that (for any choice of complex structure) is equivalent to the Maxwell equations.

This clarifies the sense in which a common core ontology ‘says less’ than each of the models. Nevertheless, we see that the specific structure that is added to the common core theory to get a specific model or, alternatively, that is ‘erased’ from a set of models to get the common core theory, is not a matter of ‘deleting degrees of freedom’ or ‘describing the same physics with fewer variables’. Rather, what the specific structure comes down to is the specification of some choice out of a number of possibilities allowed by the common core theory: this choice then bears on how the theory is interpreted.

That we here speak of ‘electromagnetic properties’ without a specification, in the common core theory, of ‘purely electric’ or ‘purely magnetic’ properties, may sound familiar. Indeed, regardless of dualities, the Lorentz transformations map electric and magnetic fields onto each other under a change of coordinates (and this symmetry of electric and magnetic properties was of course the starting point for Einstein’s formulation of special relativity).

As Read (Reference Read2016: p. 224) has noted, this is analogous to sophisticated substantivalism in the discussion of the hole argument. Thus considered, sophisticated substantivalism is a balancing act that (i) keeps the points of a manifold in the ontology of a spacetime theory, that is, spacetime points are genuine entities, while (ii) it denies in some sense their individuality, by denying that spacetimes that are related by a diffeomorphism can differ solely by the properties that are assigned to the same spacetime points.

By analogy, an internal interpretation of the common core of electric-magnetic duals is a balancing act that (i) keeps electric and magnetic properties of fields in the ontology, that is, electric and magnetic properties are genuine properties, while (ii) it denies in some sense their individuality, by denying that electromagnetic fields that are related by a duality transformation can differ solely by the properties that are called electric and the ones that are called magnetic.

This broadly aligns with Dewar’s (Reference Dewar2019) account of sophistication. In order to interpret putatively isomorphic models as physically equivalent, instead of moving to a reduced formalism, we formulate a new theory (here, the common core theory) where the models are distinct but are rendered isomorphic. Then we interpret that theory anti-quidditistically, that is, we deny that two different possibilities can differ just by a permutation of fundamental (here, electric and magnetic) properties.Footnote ⁴³ (For more on this discussion, see the FAQs in Section 5.4.)

4.1.1 Particle–Soliton Duality in Three Dimensions

Before we discuss electric–magnetic duality in quantum field theories, it will be useful to discuss a simpler case: namely, the Maxwell theory in three dimensions (two dimensions of space and one of time), coupled to a complex scalar. This theory is often called the three-dimensional Abelian Higgs model, where the complex scalar field plays the role of the Higgs field in an Abelian theory, that is, the gauge group is U(1). This example is of great importance for condensed matter systems, and its behaviour gives a blueprint for the more advanced dualities in quantum field theory (e.g. quark confinement) and string theory. In particular, it illustrates the theme, from Section 2.1, of elementary-composite.

The leading idea is going to be that, in models where the gauge field is coupled to a scalar, there are soliton solutions that are magnetically charged and that are related by duality to non-solitonic, that is, particle-like, solutions that are electrically charged. (In line with our theme, we will interpret the solitonic solutions as composite, and the particle-like solutions as elementary.)

A soliton is, in short, a solution of the (non-linear) equations of motion that has finite energy and is spatially extended, that is, it is localized within a finite region, and it is topologically stable. For example, solitary water waves of exceptional stability, such as waves of translation, are solitons.

Our leading idea thus combines electric–magnetic duality with the interchange of particles and solitons. These two exchanges are required because, while electric–magnetic duality does not obtain for models with scalars, when combined with the exchange of a set of solutions (the particle solutions) with another set of solutions (the soliton solutions), one can get an electric–magnetic duality or a quasi-duality that exchanges particles and solitons.

We will first discuss the idea of particle–vortex duality and the role of vortices in generating a phase transition. Then we will discuss the duality between the topological and Noether currents, associated with solitons. (In this section, we discuss quasi-dualities that are valid only by approximation. There will be no confusion in our calling them ‘dualities’.)

Noether and topological currents.

We have discussed that vortices and other solitons usually have topological fluxes and charges. These are like ‘winding numbers’, that is, they are integers that characterise topologically non-trivial properties of the magnetic field around a circle or closed surface. They can usually be obtained by integrating a topological current over a circle or closed surface surrounding the soliton.

This topological current has two important properties: (i) it is not a Noether current for some continous symmetry; (ii) it is magnetic, because it does not couple to the gauge field (as an electric, Noether, current would do, through a coupling $A{J}_{\text{Noether}}$ ), but to the Hodge dual field. (Recall that the electric flux is given by the electric field across a surface, which is contained in the Faraday tensor: while the magnetic flux is given by the magnetic field across a surface, which is contained in the Hodge dual of the Faraday tensor.) This explains why these kinds of dualities usually require the exchange of both elementary and solitonic solutions, and of electric and magnetic charges.

Electric–magnetic duality can be used to simplify the description of these magnetically charged solitons, thus aligning the elementary-composite theme with the easy-hard theme: the topological current associated with them can be mapped to a Noether current that couples electrically to a (dual) gauge field. The Noether current is the current corresponding to the U(1) symmetry of this dual gauge field.

For example, three-dimensional vortices have the following current associated with them:

$J_{\text{vortex}}^{\lambda }=\frac{1}{2\pi }{ϵ}^{\mu \nu \lambda }{\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\nu }\chi ,$(15)

where $\chi$ is the phase of the complex scalar field around the vortex.Footnote ⁴⁶

The above current is related by a Hodge-type (quasi-) duality to the Noether current of the scalar field:

$J_{\mu }^{\text{Noether}}=e\text{Im}({\varphi }^{*}{\mathrm{\partial }}_{\mu }\varphi ),$(16)

which couples in the usual way to the gauge field, while the vortex current couples to the Hodge dual of the field.

This ability to dualize soliton currents, and the gauge fields they couple to, into dual Noether currents coupled to dual gauge fields, opens the possibility of learning about the solitonic phase of a model by studying its dual, non-solitonic, phase. This is reminiscent of the Ising model, where Kramers–Wannier duality allowed the study of the high-temperature phase by dualizing it into the low-temperature phase.

4.1.2 Monopoles and Confinement of Colour Charge

In this section, we briefly illustrate the heuristic power of dualities, harnessed in 1975 by ’t Hooft and Mandelstam to propose a mechanism for quark confinement. The mechanism involves the condensation of monopoles, and is suggested by considering the electric-magnetic quasi-dual of the confinement of the magnetic field in a superconductor to the interior of vortices. Just as, in a superconductor, the magnetic field outside the vortices is screened by the Meissner effect, due to the condensation electrons into Cooper pairs: so they proposed that, in a non-abelian theory, the electric colour charge is screened by the condensation of magnetic monopoles. Furthermore, the discussion of the properties of magnetic monopoles as soliton solutions of classical field theories will pave the way to discussing, in Section 4.1.3, electric–magnetic duality in quantum field theory.

Thus this section takes three steps: it first discusses the confinement of magnetic charge in a superconductor, then it discusses the confinement of electric (colour) charge in a dual superconductor, and then it discusses magnetic monopoles.

Magnetic monopoles.

The previous study of electric-magnetic duality requires the discussion of magnetic monopole solutions, which are the carriers of magnetic charge. Monopoles in non-abelian gauge theories share key formal properties with the vortices in the previous subsection. Like vortices, they have an associated topological current (see Eq. (15)), whose integral is an integer that is topologically invariant (namely, the degree of a certain map between spheres), and their charge is quantised according to Dirac’s quantisation condition.

Although, from a distance, monopoles in non-abelian gauge theories look like Dirac monopoles, they are in fact very different. For, unlike Dirac monopoles, they have a smooth core. Also, unlike Dirac monopoles, the mass of a monopole in a non-abelian gauge theory is predicted by the theory, and satisfies the Bogomolʼnyi bound that relates the mass, $M$ , and its magnetic charge, $g$ , as follows:

$\begin{array}{r}M\ge |vg|,\end{array}$

where $v$ is the vaccum expectation value of the Higgs field. Thus magnetic monopoles are very natural solitonic solutions of non-abelian quantum field theories.

Note that, in general, there is no duality between the electrically charged particle excitations, or states, of a quantum field theory and its magnetic monopole states. Thus a general quantum field theory does not satisfy electric–magnetic duality.

However, in the next section we will discuss important examples of quantum field theories, especially those involving supersymmetry, that do enjoy a version of electric–magnetic duality. In such cases, duality is a powerful tool that allows us to study the theory’s spectrum.

4.1.3 Electric–Magnetic Duality in Supersymmetric Yang–Mills Theory

In this section, we will discuss two examples of electric–magnetic duality for quantum field theories. Both concern a supersymmetric version of Yang–Mills theory with gauge group SU(2), although we will not require details about supersymmetry. The first example is a theory that is mapped onto itself under electric–magnetic duality, that is, it is a self-duality (see Section 2.2). The second example is a case of a (quasi-) duality that relates different (low-energy) models of a given theory.

Supersymmetry is a spacetime symmetry that extends the Poincaré symmetry, such that each boson has a supersymmetric fermionic partner, and vice versa. It does so by adding fermionic generators to the Poincaré algebra, thus getting a super-Poincaré algebra.Footnote ⁴⁷

One first point to clarify is that, in general, supersymmetry is not itself a duality (nor is it a case of self-duality). This is because, although supersymmetry maps bosonic states and quantities into corresponding fermionic states and quantities of the same theory, it does so without preserving the structure of the state-space and the values of the quantities. Thus, it is in general not an isomorphism of supersymmetric theories.

Because supersymmetry relates the bosonic and the fermionic states, and is encoded in the super-Poincaré algebra, the supersymmetry algebra contains a great deal of information about the states. In particular, it contains information about the masses, charges, and spins of the theory’s normal, that is, perturbative, particle states, which are in general electrically charged: and also about the theory’s solitonic states, such as magnetically charged (monopole) states.Footnote ⁴⁸

Thus if one is looking for a theory that, as in our first example, is mapped onto itself under electric–magnetic duality, one should first check whether there are electric and magnetic states in the supersymmetry algebra that have the right quantum numbers for them to enter in an electric–magnetic duality relation. In particular, for each electric state there should be a magnetic state in the spectrum with the same mass and spin, and vice versa.

In what follows, $N$ will indicate the ‘amount of supersymmetry’: namely, the number of ways in which one can exchange a boson and a fermion under supersymmetry. Models with higher $N$ have more supersymmetry, because there are more ways to exchange bosons and fermions.

String theory and dualities.

String theory is an attempt to construct a theory of quantum gravity, that is, to provide a quantum description of general relativity. Furthermore, string theory is also sometimes referred to as a theory of everything. For, in attempting to give a quantum description of gravity, it includes and unifies general relativity with the standard model of particle physics, thus describing all four fundamental forces in a single theory. The basic idea behind string theory is that the fundamental building blocks of reality are not, as traditionally thought, particles, that is,  $0+1$ -dimensional objects, but rather extended strings, that is,  $1+1$ -dimensional objects. This idea is formalized in perturbative string theory, that is, the study of string theory for small values of the string coupling constant $g_s$ . This theory is by now reasonably well-understood, and displays a variety of interesting dualities, of which T-duality is one. Finally, let us mention that, while we have so far only spoken of ‘string theory’, there are actually five distinct perturbative string theories. Furthermore, while these are distinct theories at the perturbative level, there is a web of dualities (see Figure 2) that relates them to one another, and that is usually understood as hinting to the existence of a single non-perturbative formulation of string theory.

Figure 2

Relations between the five superstring theories and eleven-dimensional supergravity. At the centre, M-Theory is highlighted as providing a natural unification of the various string theories and with eleven-dimensional supergravity. The five perturbative string theories, together with eleven-dimensional supergravity, are expected to be specific limits of the more fundamental M-Theory.

Nevertheless, the non-perturbative definition of string theory is much less understood, that is, the formulation that is relevant to strongly-coupled gravitational phenomena like the endpoint of black hole evaporation and the recovery of the information. The theory that is supposed to encode the non-perturbative data of string theory is called M-theory.Footnote ⁵⁸ Here, the current understanding of string theory is significantly less developed: concrete formulations of (corners of) this non-perturbative M-theory are the BFSS matrix model of (Banks et al. Reference Banks, Fischler, Shenker, Susskind and Duff1999) and the AdS-CFT correspondence (Witten Reference Witten1998; Maldacena Reference Maldacena1999), which will be discussed in the next section. However, the main hint towards the existence and nature of M-theory is given by the web of dualities that are believed to obtain between the five perturbative superstring theories (more precisely, between the five superstring theories and eleven-dimensional supergravity, which would be a low-energy effective theory for M-theory), and which establishes their equivalence. This equivalence is then understood, as per the discussion in Section 2, in terms of a successor theory, encompassing and extending the structure of the five perturbative superstring theories.Footnote ⁵⁹ Insofar as this theory exists, it is natural to think of it as the sought-after M-theory. Interesting as these questions are, both for the physics and the philosophy of dualities in string theory, in this section we will confine ourselves to perturbative string theory, and explore the basics of T-duality there.

Perturbative string theory.

We first briefly introduce the basic features of perturbative string theory.Footnote ⁶⁰ At the perturbative level, string theory is naturally understood in terms of so-called $\sigma$ -models. A $\sigma$ -model, in the present context, is a two-dimensional field theory, whose fields are maps from the two-dimensional manifold that is interpreted as the worldsheet swept out by the string as it propagates, to a higher dimensional manifold, called target manifold or spacetime, which is the ambient space where the string moves. In string theory, the target manifold is a ten-dimensional spacetime that is a solution of supergravity, that is, the supersymmetric extension of Einstein’s general relativity. This spacetime gives the background against which one sets up the perturbative expansion constituting perturbative string theory, while the fields of the $\sigma$ -model describe how the two-dimensional worldsheet is embedded into the ten-dimensional target space. Hence, string theory describes, via a $\sigma$ -model, the dynamics of strings propagating in a ten-dimensional spacetime. In particular, the action of the two-dimensional field theory defining the $\sigma$ -model is the Polyakov action, which reads as follows:

$S_{\text{Poly}}\left[h,X\right]=\frac{1}{{\ell }_s^2}\int {\text{d}}^2\sigma \sqrt{-h}{h}^{\alpha \beta }{\mathrm{\partial }}_{\alpha }{X}^{\mu }{\mathrm{\partial }}_{\beta }{X}^{\nu }{g}_{\mu \nu }(X^{\rho }),$(18)

where ${\ell }_s$ is the string length, that is, the length scale for the extension of the string, ${\sigma }^0=\tau$ and ${\sigma }^1=\sigma$ are the world-sheet coordinates, $h^{\alpha \beta }$ and $h$ are respectively the inverse metric and the determinant of the world-sheet metric $h_{\alpha \beta }$ , which describes the geometry of the world-sheet. $X^{\mu }\left(\sigma \right)$ is a map between the string world-sheet and the target space, that is, the spacetime in which the string propagates, while $g_{\mu \nu }(X)$ is both the coupling function of the string interactions and the metric on the target space.

Here, the embedding of the worldsheet into the target spacetime is given by the $X^{\mu }$ fields, each of which gives the position of the string in one of the dimensions of the target space. Hence, we have ten $X^{\mu }$ fields in string theory, one for each dimension of the spacetime where the string propagates. This action has conformal symmetry under conformal transformations of the metric $h_{\alpha \beta }$ , and it describes a two-dimensional conformal field theory. This observation is important, since the conformal symmetry of the Polyakov action is not preserved upon quantization, giving rise to what in quantum field theory is called an anomaly, that is, a classical symmetry broken by quantization. It is required that this type of anomaly cancels, that is, one adds terms to the action such that they cancel the contribution from the term creating the anomaly. In string theory, cancelling the conformal anomaly is equivalent to requiring that the target space is governed by the Einstein equations (plus quantum corrections) in their appropriate extension to cover ten-dimensional supergravity, and hence it is crucial to string theory’s claim to be a theory of gravity, since it implies that it actually describes gravity in the classical limit. However, cancelling the conformal anomaly requires (once supersymmetry is included) ten $X^{\mu }$ fields, that is,  $\mu =0,\dots ,9$ , so that the target spacetime is ten-dimensional.Footnote ⁶¹ This is the origin of the requirement, in string theory, that there are extra dimensions beyond the four dimensions of relativistic spacetime.

Compactification.

At first sight, the requirement that there are extra dimensions poses a phenomenological problem for string theory, for we seem to be living in a four-dimensional, not a ten-dimensional, spacetime. One way out of this predicament is to postulate that these extra dimensions are compact (in the physicists’, not the mathematicians’, sense). To see what physicists mean by ‘compactness’, think of a five-dimensional universe in which one of the dimensions has the topology of a circle $S^1$ . We can describe phenomena whose characteristic length scale is much larger than the circle’s radius by using an effective field theory in a four-dimensional spacetime with some additional degrees of freedom (encoding the residual physics coming from the fifth dimension), thus in effect removing the fifth dimension or, at least, making it invisible at large distance scales. We say that the fifth dimension has been compactified, and the mechanism for this compactification is the Kaluza–Klein mechanism, after the two physicists who initially introduced it (Theodor, Reference Theodor1921, Klein, Reference Klein1926). In string theory, the compactified dimensions are six rather than one, and their topology and geometry are much more complicated than a circle: a well-studied example is that of compactifications on six-dimensional Calabi–Yau manifolds, which give rise to supersymmetric extensions of the standard model of particle physics. Nonetheless, the basic idea is the same as the five-dimensional example we have discussed.

In the context of T-duality, it will be enough for us to think of a ten-dimensional spacetime that is a solution of perturbative string theory, with a single dimension compactified on a circle $S^1$ , without having to look at the complexities of Calabi–Yau compactifications. We highlight two properties that are especially relevant for the description of a string propagating in the compact dimension: the momentum $P$ of the string and its winding number $n$ . The momentum is the usual quantum mechanical observable that we encountered earlier in this Element, and given that we are studying propagation along a compact dimension, upon quantization it will have a discrete spectrum. By contrast, the winding number is a purely string-theoretic property, since it counts the number of times that the string winds around the compact dimension: a property that no particle could have, since it lacks extension in space and hence it cannot be wound around anything. Also the winding number of the string has, upon quantization, a discrete spectrum. Finally, we also need to keep track of the radius of the circle, that is,  $R$ .

Invariance of the spectrum under T-duality.

Recall, from Section 2.1, that a duality maps states between models, such that the values of the quantities match. Here, we will show that under the exchange of quantum numbers $n$ and $m$ , the masses of the states indeed remain invariant. Thus consider the mass formula for a string propagating in the compact dimension. It takes the following form:

$M^2=\frac{n^2}{R^2}+m^2{R}^2+\text{oscillations},$(19)

where $n$ is the quantum number characterizing momentum modes, $m$ is the quantum number characterizing winding modes, $R$ is the radius of the compact dimension, that is, the circle, and ‘oscillations’ refers to the oscillation modes of the string, which we suppress because they are not relevant for our argument. We also follow the widespread convention of setting the string length equal to one, that is,  $l_{\text{s}}=1$ .

Notice in particular the first two terms, given by the momentum and winding numbers. We can see that we can get a mass equivalent to that in Eq. (19) if we carry out the following change of variables:

$m\leftrightarrow n,R\leftrightarrow \frac{1}{R}.$(20)

In other words, the mass formula for a string propagating along a compact circular dimension is invariant under a transformation that exchanges the quantum numbers of winding $m$ and momentum $n$ , and changes the radius of the compact dimension from $R$ to $\frac{1}{R}$ . This equivalence illustrates a more general fact in string theory: namely, pairs of solutions with compact dimensions, whose radii have reciprocal values, are equivalent. This equivalence extends to all other quantities, and it is what we call T-duality.

Thus we have an equivalence between a string theory defined in a spacetime with a compact dimension with radius $R$ and one defined in a spacetime with a compact dimension with radius $\frac{1}{R}$ . This implies that, when we have a model in a spacetime that has a small radius and hence a singular compact dimension, we can, by an appropriate change of variables, that is, by exchanging winding and momentum quantum numbers, get a model where the compact dimension has a large radius and is non-singular. Note that this equivalence is only the simplest case of T-duality: in general, a spacetime can have any number of compact dimensions, on each of which we can apply a T-duality map. Furthermore, related to T-duality is a more general duality called mirror symmetry (Hori and Vafa Reference Hori and Vafa2000), which not only concerns the size of the compact dimensions, but also their topology, and which has proven fruitful, beyond its string-theoretic applications, in pure mathematics (Kontsevich Reference Kontsevich and Chatterji1995; Strominger, Yau, and Zaslow Reference Strominger, Yau and Zaslow1996).Footnote ⁶²

Roles of T-duality.

An interesting feature of T-duality is its role within the web of string dualities that are taken to point to the existence of M-theory, which we discussed at the beginning of this Section. Going back to Figure 2, we see that T-duality relates Type IIA and Type IIB string theories. This equivalence is an instance of what we described in Section 2 as the heuristic role of dualities, that is, dualities’ role as clues towards a more fundamental and encompassing theory which is expected to supersede the two dual models, as their successor theory. T-duality is in fact part of the motivation for developing M-theory, which is a fundamental, non-perturbative formulation of string theory: and note that, as one expects from a successor theory, M-theory indeed goes beyond the common core of T-duality, since it encompasses all five superstring theories and eleven-dimensional supergravity (thus it goes well beyond the T-dual models). Furthermore, insofar as M-theory is well-understood, its basic structure is expected to go beyond that of superstring theory, which might only be recovered in a specific limit of M-theory. Thus T-duality is used heuristically to find clues about M-theory. For example, by studying how T-duality maps the Type IIA model as its coupling increases, one hopes to get clues about some of the corners of M-theory. By allowing the coupling of Type IIA to be large, we get the corner of M-theory describing the strong coupling behaviour of Type IIA. If we compactify this corner of M-theory on a torus, and apply T-duality to one of the two circles which make up the torus and then decompactify the compact dimensions, we get the corner of M-theory describing the strong-coupling behaviour of Type IIB string theory, that is, F-theory (Vafa Reference Vafa1996).Footnote ⁶³ While a detailed introduction to F-theory goes beyond the scope of this book, its discovery and relation to M-theory through T-duality is a powerful reminder of the heuristic power of dualities.

The ontology of T-duals.

Before moving on, let us comment on an influential argument about the ontology of T-dual models due to Huggett (Reference Huggett2017). Huggett (Reference Huggett2017) argues that T-duality gives rise to a distinction between the physical space of string theory and its target space. Roughly, the idea is that, since the size of the compact dimensions is not invariant under the T-duality transformation, the extra compact dimension should not count as part of the physical space where a string propagates. Support for this idea is drawn from a thought experiment, due to Brandenberger and Vafa (Reference Brandenberger and Vafa1989), that envisages a procedure for determining the size of the extra dimension. This thought experiment suggests that any measurement of the extra dimension that we could make will return the result that it is large, regardless of whether the extra dimension of the target space is large or small (with a correspondingly inverse size for the space of winding modes of the string). In the language of the Schema, the argument says that the compact extra dimension is not part of the common core of T-duality: more precisely, the information about the size of the extra dimension is not in the common core, but the fact that an extra dimension exists is true in both models and is part of the common core, possibly together with, for example, the topology of the extra dimension.

T-duality is an interesting case of a duality transformation in perturbative string theory. However, it does not have any explicit information about the non-perturbative regime: at best, we might want to extrapolate T-duality to that regime, but it does not describe it directly. In the following section, we will look at an attempt to define the non-perturbative behaviour of quantum gravity, and string theory in particular, in terms of a duality relation: the AdS-CFT correspondence.

The AdS side of the duality, or ‘bulk’.

AdS spacetime is the maximally symmetric solution of the Einstein Field Equations with a negative cosmological constant. By ‘asymptotically AdS’, we mean a spacetime that approaches AdS locally near asymptotic infinity, while in the interior the spacetime metric may deviate from AdS, for example by there being a black hole (for simplicity, below we will often use ‘AdS’ for asymptotically locally AdS).

An important feature of AdS spacetime, especially in the context of AdS-CFT, is that in AdS there is a timelike boundary, so that a beam of light travelling out to infinity along one of the spatial directions will reach a boundary; indeed, for an $n+1$ -dimensional Lorentzian AdS spacetime, this boundary is an $n$ -dimensional Lorentzian manifold. It is a conformally flat manifold, that is, it is locally Minkowski spacetime up to a conformal transformation. In other words, the AdS boundary has both spatial and temporal extension, which is the reason why we call the boundary ‘timelike’. We can use the boundary of AdS to gain some geometric intuition about AdS-CFT: we can think of the dual CFT as being defined on the boundary manifold of the AdS spacetime. For this reason, we will sometimes speak of a boundary CFT. Also, we will sometimes simply call the AdS spacetime the bulk spacetime, or bulk. These observations both explain the fact that the CFT is in one less dimension than the AdS spacetime, and also the holographic terminology for this duality: since we are mapping an $n+1$ -dimensional theory into a theory defined on its $n$ -dimensional boundary, much like a hologram encodes a three-dimensional theory into its two-dimensional hologram.

The CFT side of the duality.

A CFT is a quantum field theory that is invariant under conformal transformations, that is, under (local) changes of scale. In other words, in a conformal field theory all the physical content of the theory is encoded in the angles between trajectories in spacetime, while lengths do not play any role, since they can always be changed under a local rescaling, and the (classical) theory is defined to be invariant under such transformations. An important point, for present purposes, is that a CFT is just a normal quantum field theory, and in particular it does not describe the gravitational force: thus it is not a theory of quantum gravity, and it is more similar to the quantum field theory of the Standard Model of particle physics (though the latter does not have conformal symmetry, while CFTs do). This explains the sense in which we can say that AdS-CFT is an instance of gauge–gravity duality: since we are mapping a theory of quantum gravity, that is, quantum gravity in AdS spacetime, into a non-gravitational field theory, that is, the boundary CFT, we are showing that gravitational and field theoretic (gauge) degrees of freedom are equivalent.

The duality map.

We can schematically represent the duality map in AdS-CFT by the following expression:

$Z_{\text{CFT}}(M)=\int DgD\varphi e^{-iS[g,\varphi ]},$(21)

which relates: (a) on the left-hand side, the partition function of the boundary CFT, defined on the boundary manifold $M$ , encodes the field theory’s physical data; and (b) on the right-hand side, the path integral of quantum gravity in AdS spacetime, is an informal expression for the physical content of quantum gravity in these spacetimes. Furthermore, the validity of the perturbative expansions associated with the gravity path integral, and with the CFT is controlled by the ’t Hooft coupling $\lambda =g_{YM}^{2}N$ , where $g_{YM}$ is the coupling constant of the CFT and $N$ is the rank of its gauge group, which counts its degrees of freedom.Footnote ⁶⁵ In particular, small $\lambda$ corresponds to a regime where the CFT perturbative expansion is reliable, while strong coupling in $\lambda$ corresponds to the regime of validity of gravitational perturbation theory. This will be relevant when we discuss how AdS-CFT illustrates the themes and types of dualities introduced in Section 2.2.

This equivalence between the CFT partition function and the gravitational path integral in AdS, if true, secures that AdS-CFT provides an isomorphism between quantities $Q$ and states $S$ on either side of the duality, since both notions can be defined starting from the CFT partition function/gravity path integral, Eq. (21). Furthermore, since both the partition function of the CFT and the path integral of the gravity theory encode their respective dynamical information, Eq. (21) also secures that the duality map is compatible with the models’ dynamics, hence realizing the structure of the Schema for dualities, as introduced in Section 2.

Thus AdS-CFT has been conjectured to be a duality between quantum gravity and a conformal field theory, and in particular it shows that, at least in the context of spacetimes that are asymptotically AdS, the problem of defining a theory of quantum gravity is formally the same as the problem of identifying an appropriate CFT. Given that our current understanding of quantum field theory is much better than our understanding of quantum gravity, this duality allows us to make significant progress in our quest towards a theory of quantum gravity. In particular, AdS-CFT gives, at least in principle, a non-perturbative formulation of quantum gravity for AdS spacetimes, since the CFT is, at least in principle, non-perturbatively well-defined. We say ‘in principle’, because the problem of giving a rigorous and general non-perturbative definition of quantum field theory is still open.Footnote ⁶⁶ However, we do have some control over the relevant non-perturbative physics in quantum field theory, especially in simple examples. Furthermore, there is no known reason why there should be obstructions to defining quantum field theory non-perturbatively. In other words, it seems reasonable to say that we do know the basic principles governing quantum field theory, even if we do not yet know how to formulate the theory non-perturbatively. Hence, our in principle qualification. Note that the analogous statements for quantum gravity are indeed unknown, and in particular that in quantum gravity the limitation towards a full formulation, at present, is not mathematical like in QFT, but rather that we do not really know the basic principles governing such a theory. Hence the progress implicit in a duality like AdS-CFT.

A crucial step in identifying an appropriate CFT dual to quantum gravity in AdS is the identification of CFT quantities that are dual to local quantities in the semiclassical approximation to quantum gravity in AdS spacetime. This is necessary to ensure that the boundary CFT is actually dual to quantum gravity in AdS: without such a mapping, we would fail to have an isomorphism between the two models, in the sense of the Schema. The programme of identifying CFT expressions for semiclassical, local bulk quantities is the programme of bulk reconstruction (Harlow Reference Harlow2018).

The crucial concept for the bulk reconstruction programme is entanglement wedge reconstruction. Let us briefly explain what this is. In order to define the entanglement wedge of a given boundary region $R$ , we first need to define the quantum extremal surface (QES) associated with that region. As follows:

1. (QES) A quantum extremal surface $\chi$ is defined as a surface satisfying two conditions:

   1. (i) Homology Constraint: given a boundary region $B$ , a surface $\chi$ satisfies the homology constraint if, for $C$ a space-like hypersurface, $\chi \cup B=\mathrm{\partial }C$ , that is, the union of $\chi$ with a boundary region $B$ is the boundary of some space-like region $C$ . $C$ is called homology hypersurface.

   2. (ii) Extremize Generalised Entropy: the surface $\chi$ should be a surface that extremises the generalised entropy:

      $S_{\text{gen}}\left(\chi \right)=\text{ext}\left[\frac{A(\chi )}{4G_N}+S_{\text{bulk}}(\chi )\right],$(22)
      where $S_{\text{bulk}}(\chi )$ is the von Neumann entropyFootnote ⁶⁷ of the bulk fields contained in $\chi \cup B$ and $A(\chi )$ is the area of the hypersurface $\chi$ .

We then define the quantum Ryu–Takayanagi or HRT surface as the QES associated with the boundary region $R$ , which minimises the generalised entropy. Given the notion of HRT surface, we define an Entanglement Wedge (EW) as follows:

1. (EW) Let $B$ be a boundary spatial subregion of an asymptotically-AdS spacetime. The entanglement wedge of $B$ , which we denote by $W[B]$ , is the bulk domain of dependence $D[C]$ Footnote ⁶⁸ of the homology hypersurface $C$ delimited by the HRT surface $\chi$ .

We can now introduce the crucial tool for bulk reconstruction, which is the entanglement wedge reconstruction conjecture, which says that:

1. (EWR) Entanglement wedge reconstruction: all physical quantities in $W[B]$ , that is, the entanglement wedge of a spatial subregion $B$ , are represented in the CFT by operators in $B$ .

With the notion of entanglement wedge reconstruction, we have a way to represent local, semiclassical, bulk quantities in the language of the boundary CFT. In particular, the relation between $W[R]$ and $R$ is given by a quantum error-correcting code,Footnote ⁶⁹ which is required to encode the locality properties of the bulk degrees of freedom in their CFT description. While the details of this go beyond our present purposes, what is essential to the present discussion is that the Hilbert space of the semiclassical bulk gravitational theory in $W[R]$ is mapped not to the entire Hilbert space of the boundary region $R$ , but only to a specific subspace, which we call the ‘code subspace’. This code subspace is the space of bulk semiclassical quantities and represents semiclassical gravitational degrees of freedom in the CFT. The quantum error-correcting code gives us a map between bulk semiclassical quantities and their representation in the CFT.Footnote ⁷⁰

Philosophical questions.

With these ideas about AdS-CFT in hand, we can look at how holography connects to more general philosophical questions. Indeed, AdS-CFT is a powerful test case for ideas about dualities. We will see its relevance for emergence in the next section: here, we focus on the themes instantiated by AdS-CFT. We will focus on two aspects of AdS-CFT that are relevant to the discussion in Section 2.2: the fact that AdS-CFT is a quantum duality, and the fact that it instantiates the hard-easy theme of dualities.

Quantum duality.

That AdS-CFT is a quantum duality is clear from the fact that the semiclassical limits of both quantum gravity in AdS and a CFT are theories that are not equivalent in any obvious sense. In AdS-CFT, the regimes of the coupling parameter, namely the ‘t Hooft coupling $\lambda$ , where semiclassical gravity in AdS and the semiclassical CFT are respectively valid, are incompatible. For semiclassical gravity in AdS is valid at strong ‘t Hooft coupling, while the semiclassical CFT is valid at weak ‘t Hooft coupling. Thus these two semi-classical limits cannot be equivalent, because each is well-defined for a different range of values of the coupling constant. This behaviour is the hallmark of a quantum duality, that is, a duality that is realized only at the quantum level, which displays two distinct and non-equivalent semiclassical limits.

Hard-easy.

It is clear from the previous discussion how AdS-CFT illustrates the theme hard-easy. Since the perturbative expansions of the CFT and of the gravity theory are well-defined at reciprocal values of $\lambda$ , it follows that easy, perturbative calculations in the CFT correspond to difficult, non-perturbative calculations in the gravity theory, and vice versa. This property allows AdS-CFT to provide concrete insights into the non-perturbative, highly quantum dynamics of gravity, since these are mapped to perturbative calculations in the CFT, which are computationally much more tractable than their gravitational counterpart. Indeed, an important example of AdS-CFT’s ability to probe non-perturbative properties of gravity is the microscopic counting of black hole entropy in string theory of Strominger and Vafa, which can be realized using holographic methods, thereby making essential use of the hard-easy properties of holography (Strominger and Vafa Reference Strominger and Vafa1996).Footnote ⁷¹ Another important application of AdS-CFT to black holes, where the hard-easy theme stands out, are recent holographic calculations of the entropy curve, that is, the Page curve, for evaporating black holes (Almheiri et al. Reference Almheiri, Engelhardt, Marolf and Maxfield2019; Penington Reference Penington2020). These are crucial to the resolution of the black hole information paradox (Hawking Reference Hawking1976) and the firewall paradox (Almheiri et al. Reference Almheiri, Marolf, Polchinski and Sully2013).Footnote ⁷²

Real-world physics?

Before concluding this section, let us remark on one interesting recent development in AdS-CFT. It is often remarked that AdS-CFT is not relevant to real-world physics. The reason for this claim is that AdS spacetime has a negative cosmological constant, while the local region of the universe that we inhabit is thought to have a positive cosmological constant. Thus it is unclear how AdS-CFT could describe physics in our universe. However, despite this fact, recent results indicate the possible relevance of AdS-CFT to real-world physics, not though the gravitational physics: rather, it is the CFT side of the duality which has proven useful to real-world experiments. In more detail, we can use CFTs with holographic duals to model certain properties of strange metals, a kind of strongly-coupled condensed matter system. It has proven necessary, in order to study various properties of the strange metals, to study the system not through its CFT description, which is largely intractable, but rather using an AdS dual system, which takes the form of an electrically charged black hole.Footnote ⁷³ Even more interesting, various properties of strange metals described in this way have been detected, are currently in the process of being detected, in laboratory experiments.

Does this open a window into testing AdS-CFT in the laboratory? Could these experiments on condensed matter systems tell us anything useful about quantum gravity? If so, this would be remarkable. However, more generally, it is not completely clear what the role of the gravitational dual in these experiments is: does it explain features of the strange metals? Or is it only a calculational tool? Should we be realists about it or not? Does it make sense to speak about emergence of the gravitational system from the strange metal, or vice versa, or neither? These are philosophical questions, concerned with topics such as scientific realism, explanation, and emergence: and it is indeed to these kinds of philosophical questions emerging from the study of dualities that we will turn in the next section.

5.1.1 Theoretical Equivalence in Philosophy of Science and Some Interpretative Options

‘Theoretical equivalence’ is a broad term, which different authors have different ways of making precise. In general, ‘theoretical equivalence’ is understood as ‘full equivalence’ of (physical) theories. Theories that are theoretically equivalent ‘say the same thing, in different words’: their disagreements are merely verbal, they are equivalent formulations, or ‘mere reformulations’, of one and the same theory.

In philosophy of science, two influential criteria that have been proposed as conditions for theoretical equivalence are due to Glymour (Reference Glymour1970: p. 279) and Quine (Reference Quine1975: p. 320).Footnote ⁷⁹ Glymour’s inter-translatability criterion says that two theories are definitionally equivalent iff they have a common definitional extension, that is, if the signatures of each of these theories can be extended to include the symbols of the other theory. (‘Signature’ here means the non-logical vocabulary, for example, predicates, as against logical connectives like ‘and’ and ‘or’.) If the thus extended theories are logically equivalent, then the original theories are definitionally equivalent, that is, they are inter-translatable.

Quine, rather than extending the signatures of the two theories, maps one theory into the other by a reconstrual of the predicates of one theory in terms of the predicates of the other theory, such that the reconstrual of one theory is logically equivalent to the other theory.

It is worth noting, and this will feed into our analysis of dualities below, that neither Glymour’s nor Quine’s criteria are (what is sometimes called) ‘purely formal’ criteria, for two reasons. First, both criteria include a semantic component: the requirement that the extended theories (in Glymour’s case) or the reconstrued theories (in Quine’s case) are logically equivalent often involves considering their models. For, while first-order logic’s completeness allows us to reformulate logical equivalence in terms of provability of sentences, the traditional philosophical interpretation includes the semantic aspect.Footnote ⁸⁰

Second, Glymour is explicit that his criterion is only a necessary, but not a sufficient, condition for theoretical equivalence: and this is indeed how any such criteria have usually been interpreted in the philosophy of science literature.

As Section 2 emphasised, we can think of a duality as a formal relation between models. Therefore, as our examples of dualities in the previous section have illustrated, the interpretations of dual models can take a rich variety of forms. This richness is expressed by our labels ‘hard-easy’, ‘elementary-composite’, and so on, for the contrasting behaviour of duals. What does this imply for the question of the theoretical equivalence of duals?

The answer is that the possibility of theoretical equivalence depends on various factors, which we will discuss in Section 5.1.2: and so, that the verdicts differ by case. There are three broad types of possible cases, of which the first two are extremes, and the third one is an intermediate case that for us will be the most interesting one: (i) some cases where duals can clearly be, and are commonly, taken to be theoretically equivalent, (ii) some cases where duals are clearly not theoretically equivalent (for none of the systems they can possibly describe, that is, there is no system that is jointly described by both duals), and (iii) intermediate cases, where the possibility of theoretical equivalence requires further analysis. Let us note that, although cases (i) and (iii) have clear instantiations, it is not clear that there are any examples of type (ii). Next, we will suggest an example that may be of this type.

(i) In the first case, there is an internal interpretation (see the end of Section 2.1) on which dual models are standardly taken to be theoretically equivalent. For example, although models of quantum mechanics in the position and in the momentum representation look very different, once they are formulated in a basis-independent formalism, it is clear that they share a common core with an internal interpretation, according to which they use different variables to describe the same physical situations (see Section 3.1).

(ii) As an example of the second case, we can take the Kramers–Wannier high vs. low temperature duality of the two-dimensional Ising model (see Section 3.2), as it is usually interpreted. On the standard way of interpreting the Ising model in statistical mechanics, an internal interpretation in terms of a single physical situation or theory does not seem to exist: since a lattice at high temperature is clearly not physically equivalent to a lattice at low temperature. In such cases, duals are not empirically equivalent, and a fortiori they are not theoretically equivalent.

(iii) Most other cases we have discussed in our examples are of the third type, that is, they are in-between these two extremes, so that further analysis is required to decide whether, and under what conditions, there is theoretical equivalence. This includes examples of dualities where an internal interpretation is the most plausible interpretation, and examples where a fully worked out internal interpretation for a given type of system does not exist.

For example, while it seems prima facie reasonable to say that electric and magnetic duals describe different physical situations, one can imagine a possible world where ‘purely electric’ and ‘purely magnetic’ states do not exist, and so where the duals are in fact theoretically equivalent (see the discussion at the end of Section 3.3).

But it is also clear that, in this kind of case, it takes some work to develop the internal interpretation, because it is not a priori clear that the common core theory has an ontology that corresponds to the kind of physical system that one aims at describing.Footnote ⁸¹

5.1.2 A Formal and an Interpretative Criterion of Equivalence

The upshot of the previous discussion is that, although formal equivalence (in particular, duality) is a necessary condition for theoretical equivalence, it is not sufficient. Thus we require, in addition, an interpretative criterion. Summing up, theoretical equivalence requires two conditions:

(A) A formal criterion of equivalence: namely, duality.

(B) An interpretative criterion of equivalence: namely, duals have the same domain of application. More strongly, the requirement is that there is an internal interpretation, compatible with the duality.

Let us say a bit more about the motivation for adopting these two criteria, and about how the criteria are to be understood.

About (A): by ‘formal criterion’, we mean a non-interpretative criterion of equivalence. Depending on how scientific theories are formulated, this criterion can be either syntactic or semantic, or both: as we discussed earlier, Glymour and Quine’s criteria both have a syntactic and a semantic component, yet they are formal that is, non-interpretative in that, by themselves, they do not introduce additional requirements that the physical interpretations of the theories must satisfy (which is done by condition (B)).Footnote ⁸²

For physical theories, which are normally presented by mathematical physicists using set theory, the natural criterion of formal equivalence is the isomorphism criterion. For, as De Haro and Butterfield and De Haro and Butterfield (Reference De Haro and Butterfield2025) argue, some of the main critiques of the isomorphism criterion that have been given for physical theories stumble, because (i) either they rely on incorrect construals of the spaces that are required to be isomorphic; or (ii) they give incorrect treatments of the interpretative criteria, that is, of point (B).Footnote ⁸³

For physical theories as standardly formulated by mathematical physicists in set theory, physicists take duality, as an appropriate isomorphism between physical theories, to be the correct criterion of formal equivalence. We argue that the wealth of worked-out examples of significant dualities in physics justifies our endorsing this verdict.

Reformulating physical theories using other mathematical tools (e.g. of formal logic, category theory, etc.) may allow for more mathematically precise criteria of formal equivalence: but it seems safe to say that any such improvements will be precisely that–reformulations of physical theories in a mathematical language that is more sophisticated. What we argue is that, given a physical theory formulated using set theory, as is standard in mathematical physics, duality is the correct formal criterion of equivalence.Footnote ⁸⁴

About (B): the requirement is that equivalent models have the same physical semantics, that is, the same domain of application (i.e. one requires numerical identity), and also that elements (i.e. states and quantities) that are formally equivalent (i.e. duals) are mapped into the same elements in the domain of application. In other words, the duality and the interpretation, both construed as particular kinds of maps, commute.

We will discuss interpretation further below. Let us here make a comment about the relation between the requirements (A) and (B): the literature on theoretical equivalence has mostly focussed on (A), where various formal criteria have been investigated.Footnote ⁸⁵ When the literature has focussed on (B) in detail, this has sometimes been at the expense of (A). For example, Coffey (Reference Coffey2014) has argued that judgements of theoretical equivalence boil down to judgements of interpretative equivalence: ‘claims of theoretical equivalence are normative claims about how theoretical formalisms ought to be interpreted’ (p. 823). Formal considerations are relevant only in so far as they shape interpretative judgements.

By contrast, the recent philosophical literature on dualities has argued that both possible extremes, that is, defining theoretical equivalence solely in terms of (A) or solely in terms of (B), are too weak, and that in general the only correct way to define theoretical equivalence is by using a mixed approach that requires both (A) and (B).Footnote ⁸⁶

One way to argue for the need for a mixed approach is from what is usually required for a conception of a scientific theory and, more specifically, a physical theory, especially in theoretical physics. For, even though there are different (even conflicting) views of what a scientific theory is and how it is best formulated, major accounts such as the syntactic and the semantic conceptions of theories all require that scientific (more specifically, physical) theories have a formal or mathematical component and an interpretation (and, in particular, scientific statements have both a formal and an interpretative component). Thus for two theories to be equivalent, it is not sufficient to require that their interpretative components match: their formal components must also match. Therefore, a conception of the equivalence of theories must include both (A) and (B). (We will return to this topic in the FAQs in Section 5.4.)

FAQ1. Is the common core ontology of two duals obtained by ‘deleting those variables or features that the duals do not share’?

Answer: No, it is not quite as simple as that. Building an ontology for the common core, that is, constructing an internal interpretation, may require a more sophisticated treatment of the features that are not shared by the duals than a simple ‘deletion’. For example, electric and magnetic duals do not share ‘pure electric’ and ‘pure magnetic’ states: but this does not mean that the common core theory has no electric and magnetic variables and features at all, since it does have an electromagnetic field. We can use the analogy with the Lorentz transformations, which mix ‘purely electric’ and ‘purely magnetic’ fields (and forces) into an electromagnetic field that has both electric and magnetic properties combined.Footnote ⁹⁸

In the philosophical literature, this often goes under the name of ‘sophistication’, and the standard example is gauge symmetries. If one wishes to interpret variables, such as gauge potentials related by a gauge transformation, as representing the same physical possibility, it is not necessary to use a reduced formalism that only trades in equivalence classes of gauge potentials. Instead, one can use a principal bundle, which keeps gauge-related gauge potentials, with gauge transformations as vertical automorphisms of the bundle. For a discussion of sophistication, see Dewar (Reference Dewar2019) and Martens and Read (Reference Martens and Read2021). (For more details, see the discussion at the end of Section 3.3.)

FAQ2. Why is condition (A) for theoretical equivalence (i.e. having a formal criterion of equivalence, as in Section 5.1.2) needed? Is condition (B) (i.e. having an interpretative criterion) not enough?

Answer: No, condition (B) is not enough. This is because a theory in the natural sciences only says something if it uses an appropriate formalism: without a formalism, a physical theory of the usual type does not say (almost) anything, for example, it does not make any precise predictions. For example, the description and prediction of phenomena requires a formalism, and cannot be given in ‘purely interpretative’ or ‘purely physical’ terms. Reducing a theory to only its interpretative consequences, free of formalism, is hopeless, since it deprives one of the resources required to correctly express the concepts that are relevant to describe the envisaged domain of application. (The point here is not about using mathematics, as against some other language, in the theory’s formalism. Rather, the point is that a formalism with sufficient expressive power, and a precise notion of formal equivalence, are required for theoretical equivalence: whether this formalism is written in a mathematical, or in a different, language.) Namely, the representing side of the interpretation relation determines the level of detail at which a theory can describe a domain of application.

As a consequence, it is a requirement that the mathematical structure of theoretically equivalent formulations should match, because this is required for an appropriate ‘inter-translation’, so that the two theory formulations have the same level of detail and can ‘say the same thing’.

A maverick philosopher might claim that he or she can stipulate that the letter $E$ represents the theory of everything–on the grounds that, by a trivial semantic convention, any symbol can stand for anything: so that, in particular, $E$ stands for the (content of the) theory of everything.

But this misunderstands what is required of a scientific theory: the symbol $E$ , alone and by itself, even if endowed with an ‘interpretation’ of some type, but free of mathematical formalism, is not a physical theory unless we are given the resources to make predictions and give explanations: namely, by adding a calculus, that is, a piece of mathematics. For the symbol $E$ , without further syntactic rules, does not have the expressive power that enables the prediction of the cross-section of the collision between an electron and a positron, or of the half-life of carbon-14. In addition to interpretation, symbols require a calculus and rules of evaluation: which is what the formal aspects of scientific theories (as used in (A)) provide. (We also emphasised this aspect in Section 2, where we required that a physical theory is formulated as a triple of states, quantities and dynamics: some such formulation is required for a physical theory.)

Thus, despite the idea of trivial semantic conventionality, formal inter-translation is required for theoretical equivalence, to secure that the two theories have the same predictive and descriptive power.

FAQ3. Why is condition (B) for theoretical equivalence (i.e. having an interpretative criterion of equivalence, as in Section 5.1.2) needed? Is condition (A) (i.e. having a formal criterion of equivalence) not enough?

Answer: No, condition (A) is not enough. This is because, except on a structural realist position, the semantics of a scientific theory is not only formal structure: it is not always, or not just, a model-theoretic semantics, with interpretations as models that are mathematical structures: even for the arch-anti-realist van Fraassen, the mathematical structure does not exhaust the semantics.

This distinction, between model-theoretic semantics and the rest of the physical semantics, is of course shiftable and not absolute. Also, formalization helps to articulate and develop the physical interpretation.

But the distinction remains relevant within any given physical theory: see, for example, the hole argument, where given a certain type of spatiotemporal structure, there are significant disagreements about how this structure is to be interpreted, with what seem like various legitimate interpretative options.

FAQ4. Surely two duals are automatically theoretically equivalent?

Some readers who are physicists might be surprised by our denying that duals are automatically theoretically equivalent (namely, whether they are theoretically equivalent is conditional on an explication of the common ontology). After all, one might argue, duality entails that any possible prediction, that one dual makes, can be ‘translated’ into the corresponding prediction for the other dual, and vice versa. So, what else could duals possibly disagree on? It surely cannot be anything physical?

Another way of asking this question would be: setting aside those theories that, like the Ising model, only give partial descriptions of a physical system, how could duals that describe all the physical aspects of the system be inequivalent?

Answer: This is a version of FAQ3, and so one answer is ‘go to FAQ3’: but since that was not meant as an exhaustive answer, we will here add some additional elements for a reply, which can be made at various levels.

About the ‘other way of asking this question’: some authors have indeed required that one is justified in making a verdict of theoretical equivalence for duals that somehow ‘describe the whole world’.Footnote ⁹⁹ This is a reasonable requirement, because theories that only describe partial systems might look equivalent on the partial systems, but might be inequivalent on the full system.

For example, the electric–magnetic duality of the Maxwell equations in vacuum is an exact formal equivalence of the Maxwell theory and its dual, but expresses a symmetry only of regions of spacetime where there are no particles, but only the electromagnetic field. The duality is not valid in regions of space where there are electrical charges (the duality extends to such regions only if magnetic monopoles exist).

FAQ5. Do dualities favour structural realism or anti-realism?

Answer: No, they do not, because they are compatible with realism about the common core theory (see FAQ1).

However, it is worth noting that one could construe the common core strategy as being structuralist in nature. Structural realism, as introduced in philosophy of science by Worrall (Reference Worrall1989), is the belief that scientific theories tell us only about the form or structure of the unobservable world and not about its nature. It comes in two varieties (Ladyman Reference Ladyman1998): an ontic one, where structure is all there is in the world, and an epistemic one, where structure is all we can come to know about the unobservable world. In both cases, it emphasizes the belief in the shared mathematical structure of theories and models, rather than in the specific objects to which they are ontologically committed. In the context of dualities, the common core theory characterises the shared mathematical structure between dual models. Therefore, the explicit construction of a shared structure can be seen as congenial to structural realism, which focusses on the preservation of mathematical relationships and structures across different theoretical frameworks. However, in order for this strategy to amount to a structural realist position, an additional step is required: namely, committing ourselves to the common core of duals while rejecting either that the nature of unobservable objects is correctly described by this common core (epistemic structural realism) or that there are unobservable objects at all (ontic structural realism).

Thus, while dualities do not intrinsically favour structural realism or anti-realism, the common core strategy, by emphasizing the importance of the shared mathematical structure, aligns with the principles of structural realism, especially in its ‘retention of shared structure’.

Dualities do bear on the question of scientific realism along a different direction. Namely, as we discussed in Section 5.1.2, the duality-based account of theoretical equivalence bears on the question of the structure and individuation of scientific theories. Thus this account highlights how dualities can shape our understanding of what constitutes a scientific theory and how theories are individuated based on their mathematical content and their interpretation.