In the previous Section, I discussed two of the functions that tools for theory construction can have: a theoretical function and a heuristic function. In this Section, I will discuss those two functions for dualities in string theory, and show how they differ. In Sect. 4.1, I expound the basic distinction, using quotations from the physics literature. In Sect. 4.2, I argue that there is only an apparent tension: not an incompatibility.
The distinction in string theory
In this subsection, I describe how the distinction between the theoretical and the heuristic functions plays out in string theory. To this end, I use quotes from the physics literature (since, as I mentioned, the philosophical literature on dualities has not identified this tension).
After introducing, in Sect. 4.1.1, the string and M theory programmes, I proceed in two steps. In Sect. 4.1.2, I describe the theoretical function of duality. This is the function which the philosophical literature has focused on. I have discussed this theoretical function in detail in De Haro (2018b), to which I refer for details. Then, in Sect. 4.1.3, I argue that there is a second, heuristic, function, which duality plays: that second function is certainly no less important than the theoretical function, and so it deserves philosophical scrutiny.
Motivating duality: string theory and the M theory programe
I first briefly introduce, in this Section, the main ideas behind the string theory and M theory programme: and, in particular, the role of duality within that programme.
String theory is a candidate theory for the unification of general relativity and quantum field theory. Its basic assumption is that matter is made of strings, i.e. extended, one-dimensional objects that can vibrate, move around in spacetime, and interact by joining or by splitting.
For string theory to be mathematically consistent, 10 spacetime dimensions are required for the strings to move in (6 of which are thought to be curled up, so that they are inaccessible to current experiments). In the low-energy limit, string theory is well-approximated by supergravity theories, i.e. supersymmetric extensions of Einstein’s theory of general relativity, which are also 10-dimensional, and compactified down to four dimensions.
Initially, five different string theories were known, differing over the precise details defining the strings. However, significant dualities were found relating them to one another. T duality, for example, relates one type of string theory on a circle of radius R, to another type of string theory on a circle of radius 1 / R. And electric-magnetic duality (so-called S duality) relates some other string theories.
In 1995, Witten conjectured that the five known string theories, plus in addition a sixth known, 11-dimensional, supergravity theory, were all different limits of (approximations to) a single 11-dimensional theory, which he dubbed M theory. Witten assumed the eleventh dimension to be a circle, which could be of one of two kinds. He identified the radius of this circle with the coupling constant ruling the joining and splitting interactions of the strings. For a small circle of the first kind, the string coupling is weak, so that one of the five known 10-dimensional string descriptions describing weakly-coupled strings (the so-called perturbative string theory), is accurate. For a small circle of the second kind, another of the five known versions of perturbative string theory is accurate. The other three string theories are related to these two by T and S dualities.
But, at strong coupling, the eleventh dimension opens up, and the perturbative string descriptions are no longer valid. Eleven-dimensional supergravity provides a semi-classical description in 11 dimensions, valid at strong string coupling but only as long as the length of the fundamental string is small, i.e. in the point-particle limit of the string (or whatever replaces it in eleven dimensions). The challenge is then to find a theory valid away from the point-particle limit: this should be the sought-for M theory.
Since Witten’s conjecture, two main approaches to M theory have been taken. The first is the conjecture by Banks et al. (1997) that M theory is a theory of matrices, with eleven-dimensional supergravity as its low-energy limit.
The second main approach is AdS/CFT, which is a series of conjectured dualities between string theory or M theory in asymptotically anti-de Sitter space (AdS, i.e. a manifold of negative curvature), and a specific quantum field theory at the boundary of this space (where CFT stands for ‘conformal field theory’). Compactifying M theory on e.g. an internal seven-dimensional manifold of positive curvature, the remaining four dimensions have negative curvature (“they are AdS”), and are dual to a three-dimensional CFT, for which exact treatments exist. This approach is more generally called ‘gauge-gravity duality’, because it relates a theory of gravity to a quantum field theory with gauge symmetry.
Details aside, M theory is the main unifying conjecture behind the various versions of string theory, and dualities play a key role in the attempt to formulate M theory. What remains unclear is the precise status that dualities are supposed to have in M theory, once a non-perturbative version for it is found. Should M theory exhibit duality, or should dualities be superseded by the final theory—are they merely “ways towards the formulation of a new theory”?
This question is, of course, not about trying to peek into the future of theories that do not yet exist, but about the heuristic paths of investigation that one may reasonably take dualities to suggest. We will explore the role of dualities within this programme in Sects. 4.1.2 and 4.1.3. Here I anticipate by saying that the answer to this question will come down to a different function of duality.
The conjectural status of most dualities in string theory, and of M theory itself, should not be a reason to dismiss the programme as philosophically irrelevant, or as mere speculation. There are four reasons for this, which I here list:
First, the programme is very influential in physics: and, in the last thirty years or so, it has spawned a large number of new ideas and technical developments which (arguably) no other research programme in high-energy physics has been able to produce. Second, and more importantly, being conjectural does not mean being physically and mathematically unmotivated. For the evidence that is available for some of the string theory dualities is strong and compelling. Third, there are also rigorous results, at various levels of mathematical and physical rigour: especially about the conformal field theories, random matrix models, and quantum field theories involved, fairly rigorous mathematical results exist. Finally, it is of course simply false that philosophy should limit itself to studying theories that are already in final form and that are mathematically completely rigorous: for not only would philosophers then quickly run out of a job, but also because it is their task to clarify and assess whatever fragments of theory are available (cf. Huggett and Wüthrich 2013: p. 284). This is especially true in areas of research such as quantum gravity, where direct observations are so far absent, and so the main guidance is the—apparently very strong—requirement that general relativity and quantum field theory should be reproduced in suitable approximations, and in addition one has the requirement of mathematical consistency and the tools of conceptual analysis at one’s disposal (besides what little available evidence there is from experiments and analogue experiments). Rather than making the quantum gravity attempts uninteresting for philosophers, these four reasons make philosophy relevant, even indispensable, to the programme of quantum gravity.
Duality as exact equivalence: duality’s theoretical function
In this Section, I discuss within string theory the theoretical function of duality, in the sense of Sect. 3.2, where duality is construed as in the Schema from Sect. 2.
The physics literature construes duality as an isomorphism between models. This isomorphism relates the common core that the two models deem physical (i.e. the triple of states, quantities, and dynamics). As such, duality is a formal notion, i.e. a definite relationship between uninterpreted, but physical, models: it is a special case of theoretical equivalence. It relates triples of states, quantities, and dynamics on the two sides, preserving the structure of the models (including the values of the quantities, evaluated on the states). Thus duality is not merely a formal relation, because it deals with physical models, but by itself it makes no reference to interpretation—the latter is the question of what I will call ‘physical equivalence’.
Both physicists and philosophers tend to construe duality this way. Therefore, the theoretical function of dualities, i.e. the function that follows from the nature of duality, as outlined in Sect. 3.2, is to establish theoretical relationships (more specifically: to establish a theoretical equivalence, as a specific kind of isomorphism) between models. These relationships typically entail relating states and quantities in one model, to states and quantities in another model, and also relating the dynamics of one model to the different, but isomorphic, dynamics of the other.
Thus dualities are very strong relationships between two models, since they relate everything that the models deem physical [namely, the model root m that is within the model M in Eq. (2.1)]. Establishing a duality between two models thus presupposes precise knowledge of the elements of the two models (the sets of all the states and quantities, and the complete dynamics), as well as knowledge of the relations in which these elements stand (i.e. there are not only bijections between each of the elements of the triples of the two models, but all physical structure must also be preserved). Thus establishing a duality requires a formulation of a model that captures all of those details, even if perhaps only implicitly. Full transparency of the model, or full understanding of it or perfect computational power, are of course not (and cannot be) required: but duality does require a formulation of the models that is as detailed as just described, within their domains of application. I will say that such a model (i.e. one where all the states and quantities, and the complete dynamics, as well as the complete rules for calculations, are known and are consistent, within the domain of application of the model) is exact.
Notice that this notion of being mathematically well-defined, within a domain of application, is much weaker than the requirement that a model gives a non-vague, good, or succesful description of the domain—the former is a formal requirement, while the latter is interpretative.
Furthermore, when such models are given, and a duality between them exists, we say the duality is exact.Footnote 18
Exactness can be proven for a number of significant dualities in physics. Simple examples are the Fourier transformation in elementary quantum mechanics, harmonic oscillator duality, and electric-magnetic duality in electrodynamics. For more sophisticated dualities in quantum field theory and in quantum gravity, the only case, so far as I know, in which the philosophical literature has proven a duality to be exact is the example of boson-fermion duality in two dimensions (De Haro and Butterfield 2017), though in the physics literature there are other cases. Most dualities in string theory (T duality, gauge-gravity duality, S duality, etc.) are cases of dualities which are conjectural. Nevertheless, it is an important aspect of duality that all dualities are exact—as they must be, according to the above definition.
The physics literature confirms the claim that dualities must be exact: i.e. that the definition of duality entails that they are cases of exact, and not approximate, equivalence, within a domain of application. Also, the physics literature confirms that duality is a case of theoretical equivalence, i.e. of a formal, or mathematical, relationship between two physical models, as in Sect. 2.2. I will now substantiate this consensus some quotations from the physics literature, which also illustrate how physicists think about dualities.
The literature quoted below of course also emphasises the following aspects: seemingly different physics and difference of description, but equivalence (or sameness) of theory; and the exactness of the duality, and of the theories involved, is also denoted as the theory’s being ‘non-perturbative’, i.e. its formulation goes beyond, or does not require, perturbation theory.
(A) In the Glossary of his textbook on string theory, Polchinski (1998, p. 367, my emphasis) defines duality as: ‘the equivalence of seemingly distinct physical systems. Such an equivalence often arises when a single quantum theory has distinct classical limits.’
He describes one specific duality (T duality) as a case of sameness of theory, but difference of description: ‘T-duality is just a different description of the same theory’ (p. 268). ‘[T-]duality is a symmetry not only of string perturbation theory but of the exact theory (p. 248, my emphasis).
(B) In an influential paper putting forward the matrix model conjecture for the definition of M theory (mentioned in Sect. 4.1.1), Banks et al. (1997: Abstract, my emphasis) also regard duality as an exact equivalence. Thus they write: ‘We suggest and motivate a precise equivalence between uncompactified eleven dimensional M-theory and the \(N=\infty \) limit of the supersymmetric matrix quantum mechanics’. ‘If our conjecture is correct, this would be the first nonperturbative formulation of a quantum theory which includes gravity’ (p. 2, my emphasis). And later they say:
‘Our conjecture is thus that M-theory formulated in the infinite momentum frame is exactly equivalent to the \(N\rightarrow \infty \) limit of the supersymmetric quantum mechanics described by the Hamiltonian (4.6). The calculation of any physical quantity in M-theory can be reduced to a calculation in matrix quantum mechanics followed by an extrapolation to large N.’ (p. 11, my emphasis).
(C) In an influential review on gauge-gravity duality (cf. Sect. 4.1.1), Aharony et al. (1999: p. 57, my emphasis) formulate duality in terms of sameness of theoretical description, or theory: ‘Thus, we are led to the conjecture that... Yang–Mills theory in 3 + 1 dimensions is the same as (or dual to)... superstring theory on \(\text{ AdS }_5\times S^5\)’.Footnote 19
They extend this conjecture to a full equivalence between string theory and gauge theory: ‘The strong form of the conjecture, which is the most interesting one and which we will assume here, is that the two theories are exactly the same for all values of \(g_s\) and N [i.e. the string coupling constant and number of colours, respectively].’ (p. 60, my emphasis).
The common thread is clear: these are all cases of conjectured, but exact, equivalences of the theoretical structures (sometimes, in a limit of the physical parameters that is relevant to the theories involved). This is in agreement with the Schema’s definition of duality, given in Sect. 2.2, and it grounds the theoretical function of duality: namely, duality thus construed is a relationship between models that are already there and which were previously thought to be unrelated.
In light of the discussion in Sect. 3.2 on the theoretical function of a tool, we can now understand a conjectured duality as a help in finding more perspicacious formulations of a given model. This is for example the case when physicists use the better-known side of the duality to investigate the lesser-known side. This is akin to solving a problem (even: formulating a model description of a system) in momentum space, and then doing the Fourier transformation back to position space. This use of the Fourier transform, which is a deductive rule that by itself does not add any new degrees of freedom, is a translation of one model description to another, and so it belongs to the theoretical function. Unless the model description A was already known, the Fourier transform would be of no help in getting the model description B via duality. It is only when the model description A is already worked out, that we can find out more about the model description B, in a quasi-mechanical way, using the Fourier transform. The same remarks go through for other dualities, in this kind of use.
But notice the assumption behind string theory dualities: within the theoretical function, the duality relation itself will not change, once the two dual models are formulated to our satisfaction (i.e. as a quadruple, involving the model root and the specific structure: see Eq. 2.1). Rather, the search for a satisfactory formulation of two dual models is a search for two structures that stand in precisely the relation that is described by the duality conjecture. On this view, duality is not to be superseded in the theory one is aiming to construct: rather, establishing duality is the aim of the proof of the duality conjecture. The duality is to be instantiated by the final pair of models: perhaps in a manifest and completely obvious way, on a sufficiently perspicacious formulation of them. I will call the theory, T, thus obtained the common core theory: for this theory contains the core stucture that the models deem physical (usually, a triple of states, quantities, and dynamics, as in Sect. 2.1), and this core structure is isomorphic between dual models, i.e. it is their common core: viz. the model root, Eq. (2.1), of each of the models.
Duality and approximation: duality as a heuristic for theory construction
In this Section, I discuss within string theory the heuristic function of duality, in the sense of Sect. 3.3, and give some quotations from the physics literature supporting the existence, and even the essential role, of this function, in the recent programme of string theory and M theory.
The physics quotations below also emphasises the lack of exactness of the theories involved (viz. they are perturbative) and the use of dualities as heuristics for finding new unifying theories (or new formulations of old theories, describing more physics). The heuristic function, in the context of this literature, is then seen to be strongly linked with the aim of unification. The examples are as follows:
(A) In a review paper about dualities, Dijkgraaf (1997: p. 120, my emphasis) connects the approximate nature of dualities to the suggestion of the existence of new theories: ‘The insight that all perturbative string theories are different expansions of one theory is now known as string duality... It is one of the amazing new insights following from string duality that these theories are all expansions of one and the same theory around different points in the moduli space of vacua.’
‘Expansion... around a point’ should here be taken in the sense of, for example, a Taylor series expansion of a function about a particular point: which is captured by the notion of ‘approximation’, discussed in Sect. 3.3. Dijkgraaf also emphasises the ‘perturbative’ nature of the dual models, i.e. their lack of validity beyond a certain order in such an expansion (a so-called ‘perturbative expansion’). Thus, Dijkgraaf’s picture of dualities is one which regards models as inexact, and dualities as only approximately instantiated, i.e. the dualities are valid only within a limited range of parameters, but are to be superseded by a better theory, namely what he calls ‘one and the same theory’, of which the mutually dual models are expansions, i.e. approximations.
(B) In the paper in which Witten put forward the influential M theory conjecture, he wrote (1995: p. 2, my emphasis): ‘S-duality between weak and strong coupling for the heterotic string in four dimensions... really ought to be a clue for a new formulation of string theory.’
‘Another motivation was to try to relate four-dimensional S-duality to statements or phemonena in more than four dimensions... we are bound to learn something if we succeed’ (p. 2).
‘...in this paper, we will analyze the strong coupling limit of certain string theories in certain dimensions. Many of the phenomena are indeed novel, and many of them are indeed related to dualities’ (p. 2).
‘Combining these statements with the much shakier relations discussed in the present paper, one would have a web of connections between the five string theories and eleven-dimensional supergravity’ (p. 4).
These quotes by Dijkgraaf and Witten underline a related aspect of dualities: they use terms like ‘amazing’, ‘new insights’, ‘clue for a new formulation’, ‘learn something’, ‘novel phenomena’. The emphasis here, unlike the quotes from Sect. 4.1.2, is not on the conjectured equivalence between already existing models: but on the novelty of theory which can arise once a duality between such models is understood.
They also emphasise duality’s pointing to ‘a new formulation of string theory’: where I take it that ‘a new formulation’ is more than just a ‘reformulation’: for a new formulation contains something extra, not only in terms of the mathematical formalism, but also in terms of the physics that is associated with that formalism—as the other quotes confirm, when they talk about novelty of phenomena: ‘we are bound to learn something’ and ‘[m]any of the phenomena are indeed novel’.
Thus, dualities here point to the existence of new theories, but are ultimately bound to be superseded: the new theory, once found, will explain these dualities as being the result of certain approximations, which can be done in different ways, but lead to identical results, as articulated in the duality. But once that new theory is reached, the duality is no longer needed, except for practical purposes: for the resulting theory is a single, complete theory. In other words, establishing duality is here not the goal: rather, it is an intermediate step towards finding a new theory.
In what follows, I will dub that new theory, the one that supersedes the dual models and of which they are particular limits, the successor theory, \(T_{{\mathrm{S}}}\).Footnote 20
These two viewpoints thus lead to different uses of duality in string theory. On the view discussed in Sect. 4.1.2, the goal is to look for a theory, T, that realises the dualities as manifestly as possible. On the view in this Section, the goal is to find the successor theory, \(T_{{\mathrm{S}}}\), that is “behind” the dualities, and which reveals them to be approximations. As I will argue in more detail in the next Section, even if they lead to two different research programmes, the two ideas need not contradict one another, and one could pursue both. But it is important to clearly distinguish the two functions: for otherwise, confusion easily ensues about the nature of duality, and about what one is entitled to expect from a duality conjecture.
Does the distinction imply a tension?
In this Section, I argue that the distinction between the two functions does not necessarily imply a tension.
At first sight, the previous quotes might suggest the distinction as a tension: in the first case (Sect. 4.1.2), string theory and M theory instantiate the dualities exactly, while in the second case (Sect. 4.1.3) dualities are perturbative clues towards finding a new theory, which will not instantiate duality exactly. However, one should interpret these quotations with some care, since they are not very precise (for example, the articles do not even include definitions of what is meant by ‘duality’) and they involve quantum field theories and string theories which are still being developed: therefore, some of the central questions, viz. whether the models as formulated are exactly valid, or whether dualities are exactly instantiated by the models, simply cannot be answered at this stage.
Nevertheless, I argue that the tension does not simply come down to lack of knowledge about the models involved: for the same tension exists for dualities and models which are exact, and well-known.Footnote 21
Here are two important reasons why the two accounts, duality as exact equivalence, and duality as an approximately instantiated equivalence and pointing to new physics, might be thought to be in tension. First, they do not refer to two different levels of explanation or of ontology. Namely, being ‘two dual models of a single theory’ or being ‘approximate dual models of a new underlying theory’ both operate at the level of the formal structure: therefore, this potential resolution (‘the two accounts operate at different levels, and so they do not contradict one another’) is not available. Second, they might be seen to be in tension because the former sense assumes an exact duality, and being an exact instantiation of a theory; while the latter necessitates dualities which are not exactly instantiated, thus pointing to a new (unifying) theory, of which the two models are only approximations.
Nevertheless, I claim that, when made explicit in a language sufficiently precise using the Schema from Sect. 2, the tension turns out to be only apparent, and can be resolved. Namely, one distinguishes two different theories, corresponding to two different ways in which the theory to be constructed can relate to the given duality. Duality is then recognised as having two different functions, which aim at the construction of different kinds of theories, as I will analyse in Sect. 5.Footnote 22