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Motivating dualities

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Abstract

There exists a common view that for theories related by a ‘duality’, dual models typically may be taken ab initio to represent the same physical state of affairs, i.e. to correspond to the same possible world. We question this view, by drawing a parallel with the distinction between ‘interpretational’ and ‘motivational’ approaches to symmetries.

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Notes

  1. For the definition of ‘dynamically possible model’, see Sect. 2. In this paper, the term ‘model’ is understood in the sense of the semantic conception of scientific theories—see e.g. Van Fraassen (1980, ch. 2), and Sect. 2.

  2. In this regard, we follow the methodology of Belot (2017).

  3. In this regard, this paper may be viewed as continuous with Møller-Nielsen (2017), offering further reasons to endorse the motivational approach, as well as providing an extended application of the interpretation/motivation distinction to the case of dualities.

  4. One should distinguish the claim that a given theory has an associated class of models from the (more controversial) claim that a theory should be identified with such a class of models. In this paper, we embrace the former, but remain agnostic on the latter.

  5. Van Fraassen identifies a model of a theory as “Any structure which satisfies the axioms of [that] theory” (Van Fraassen 1980, p. 53). In the language of this paper, it is natural to identify models in Van Fraassen’s sense with dynamically possible models (see below). Following e.g. Pooley (2013, 2017), we understand the notion of a model in a broader sense.

  6. Throughout this paper, abstract (i.e. coordinate-independent) indices are written in Latin script, and we set \(G_N=c=1\).

  7. Two points here are in order. First, one should avoid, at this stage, asserting M to be the spacetime manifold, for to do so is to conflate the mathematical model under consideration with the possible world to which that model is ultimately interpreted as corresponding. Second, and relatedly, in light of the debate over the hole argument (Earman and Norton 1987), it is not necessarily correct to interpret M as representing substantival spacetime at all—though this issue will here be set aside.

  8. These are the Einstein field equations with vanishing cosmological constant \(\Lambda \). For \(\Lambda \ne 0\), the field equations read \(G_{ab} + \Lambda g_{ab} = 8 \pi T_{ab}\).

  9. Strictly, independence of these dynamical equations from the Einstein field equations depends on the case in question—see Brown (2005, §9.3) and Misner et al. (1973, §20.6).

  10. It should be stressed that the term ‘gauge redundancy’ is deployed in this paper in a broader sense than that typically found in the physics literature, where the term is often reserved for certain ‘internal’ symmetries associated with Yang-Mills type theories. For philosophical discussion, see e.g. Wallace (2015), Weatherall (2016a) and Healey (2009).

  11. For a concise expression of these points in the language of category theory, see Weatherall (2016a, c), Weatherall (2017).

  12. One assumes that two models cannot be gauge-equivalent if they satisfy different dynamics. While one might worry that this understanding of gauge redundancies effaces the possibility that two models with different dynamics may correspond to the same possible world (and thereby pose problems for the interpretation of dualities—see Sect. 4 below), this is not correct, for nothing in the above precludes the possibility that there exist other relations which may allow for inter-theoretic model identification.

  13. Clearly, such a claim has substance only once an appropriate definition of a ‘symmetry transformation’ is provided; this matter is addressed in detail below.

  14. What is meant by such an explication will be made explicit over the following subsections. This explication must cohere both internally, and with the structure of the models under consideration (it is, therefore, insufficient to simply assert that the two models under consideration be interpreted as corresponding to some arbitrary possible world).

  15. Here, we say potentially, for there does not necessarily exist any pressure to construct such a \(\tilde{\mathcal {K}}\). To illustrate, consider the case of models related by a hole diffeomorphism in GR: even one who interprets such models as corresponding to the same possible world is not obliged to construct such a reduced theory. Cf. Sect. 3.2.6 below.

  16. For a discussion of such issues, see Read (2016b, §2); in this paper (modulo some brief considerations in Sect. 5.3), we set these complications aside by considering only symmetry transformations which act ‘globally’ upon the \(O_i\) of KPMs of \(\mathcal {T}\), rather than upon proper subsystems in those models.

  17. Our thanks to Neil Dewar and an anonymous referee for helpful discussion on this point.

  18. This definition of a symmetry transformation has the merit of being broadly analogous with our construal of dualities, presented in Sect. 4.1.

  19. Each of Caulton (2015), Dasgupta (2016) and Ismael and Fraassen (2003) offer more nuanced ways of cashing out the ‘empirical equivalence’ criterion in the above definition of a symmetry transformation—for example, Dasgupta appeals both to a notion of ‘how things look’ (Dasgupta 2016, §6.3), and to Quinean ‘observation sentences’ (Dasgupta 2016, §6.3) [for details of such observation sentences, see Quine (1975, 1970)].

  20. For rigorous presentations of this theory, see e.g. Earman (1989), Friedman (1983), Pooley (2015) and Malament (2012).

  21. Strictly, neither \(t_{ab}\) nor \(h^{ab}\) is a metric field—see e.g. Malament (2012, p. 250). Insofar as they are not metric fields, \(t_{ab}\) and \(h^{ab}\) are still tensor fields of rank \(\left( 0,2\right) \) and \(\left( 2,0\right) \), respectively.

  22. I.e. is timelike, in the sense of footnote 23.

  23. Given any vector \(\theta ^a\) at a point \(p \in M\), we can take its ‘temporal length’ to be \(\left( t_{ab} \theta ^a \theta ^b\right) ^{1/2}\). We further classify \(\theta ^a\) as either ‘timelike’ or ‘spacelike’, depending on whether its temporal length is positive or zero, respectively. We understand a smooth curve to be ‘timelike’ (respectively ‘spacelike’) if its tangent vectors are of this character at every point along the curve. Note that (3.4) ensures that \(\sigma ^a\) is a timelike vector field.

  24. One may question whether these laws faithfully represent Newton’s thinking on these matters, since they make no reference to the persisting point of absolute space, as picked out by \(\sigma ^a\). For an arguably less anachronistic presentation of the laws of NGT set in Newtonian spacetime, see Pooley (2015, §4.4). The presentation of the dynamical laws of this subsection will suffice for the purposes of this paper.

  25. The terms ‘static shift’ and ‘kinematic shift’ are relatively standard in the literature, and are originally due to Maudlin (1993, §3). The term ‘dynamic shift’ is slightly less standard, and is due to Huggett (1999, §8.3).

  26. One worry regarding speaking of the ‘centre of mass of the universe’ is the following: this notion may only be well-defined under a certain restricted set of circumstances (for example, when the mass density \(\rho \) is asymptotically zero at infinity). Given this, it may be preferable to resort to the following fix: use instead the centre of mass of some arbitrary body of matter. Our thanks to Neil Dewar for raising this point.

  27. Following Earman (1989, §2.3), the Maxwell group of transformations is defined as \(\mathbf {x} \rightarrow \mathbf {x}' = \mathbf{{R}}\mathbf {x} + \mathbf {a}\left( t\right) \); \(t\rightarrow t^{\prime } = t + d\). The ‘internal’ transformation on \(\varphi \) is defined as \(\varphi \rightarrow \varphi ' = \varphi - \mathbf {x} \cdot \ddot{\mathbf {a}} + f \left( t\right) \). For further details, see Knox (2014).

  28. By ‘ratios of’ distances and velocities, we have in mind such notions relative to a pre-defined standard of measurement—e.g. the Parisian ‘metre rod’.

  29. One may here understand ‘mathematical reformulation’ to mean: an alteration of the space of models of the theory (whether KPMs or DPMs). This will become clear through the examples presented in this subsection.

  30. Here, ‘object’ refers to any substructure of the model in question—rather than (necessarily) to the geometric objects \(O_i\) introduced in the KPMs of a generic theory \(\mathcal {T}\) in Sect. 2.

  31. Note that isomorphism of two spaces of models (e.g. \(\tilde{\mathcal {D}}_1\) and \(\tilde{\mathcal {D}}_2\), associated respectively to two theories \(\mathcal {T}_1\) and \(\mathcal {T}_2\))—as introduced in Sect. 1, and discussed further in Sect. 4.1 below—should not be confused with isomorphism of a given pair of models themselves. It is the latter that is under consideration here.

  32. See, e.g., Pooley (2013, p. 575). Other names for this view include ‘moderate structural realism’ about spacetime (Esfeld and Lam 2011, pp. 31–32) and ‘non-reductive relationalism’ (Saunders 2003, §5).

  33. Here we ignore the related (but distinct) ‘indeterminism’ objection to substantivalism in the context of GR raised at Earman and Norton (1987, §5). The reasons for this are twofold. First, this objection is not directly related to the static shift argument in NGT. Second, sophisticated substantivalism also seems sufficient as a response [for more on this latter point, see Pooley (2002, §4.1.4)].

  34. Cf. footnote 15. Of course, sophisticated substantivalism constitutes just one of many positions available in the vicinity of discussions of the hole argument. For a recent review of the literature, see Pooley (2017, §7).

  35. For the parallel point in the case of dualities, see Read (2016b, §5.3). Cf. Sect. 5.1.

  36. Note that, since the mathematically reformulated theory will have a different space of models to the original theory (cf. footnote 29), it may best be regarded as a new theory, distinct from the original (on the setup of Sect. 2).

  37. For further discussion, see e.g. Earman (1989, §2.4) and Maudlin (2012, pp. 54–66). Although such a reformulation of NGT may appear trivial from a modern four-dimensional, differentio-geometric perspective, it certainly would not have appeared so to Newton or his contemporaries. This appearance of triviality is arguably reinforced by the fact that, in setting up NGT, we have [following the canonical literature on this subject, in particular Friedman (1983, pp. 71–94)] formulated the laws directly in terms of \(\nabla _a\), rather than \(\sigma ^a\). For more on this point, see Pooley (2015, p. 134); for a discussion of NGT which puts particular emphasis on the non-triviality of the move to Galilean spacetime, see Maudlin (2012, pp. 54–66).

  38. \(\hat{\nabla }_a\) is related to \(\nabla _a\) by , with ; bracket notation for derivative operators means that \(\hat{\nabla }_a\), \(\nabla _a\), and are related by ; and \(t_a\) is a covector field which may (locally) be defined from \(t_{ab}\) via \(t_{ab} = t_a t_b\) in a ‘temporally orientable’ spacetime—for details, see Malament (2012, pp. 250–251).

  39. We say ‘partly’ rather than ‘fully’ dynamical in light of the compatibility conditions (3.2) and (3.3), which hold also for \(\hat{\nabla }_a\).

  40. For original sources, see Trautman (1965); for contemporary discussion and proofs, see Malament (2012), p. 267ff.

  41. For further details, see Knox (2014). Note also that Saunders (2013) has argued on the basis of similar considerations that the appropriate spacetime setting for Newtonian theory is in fact ‘Newton-Huygens spacetime’, a close relative of what Earman (1989, §2.3) has dubbed ‘Maxwellian spacetime’. For illuminating discussion of Saunders’ paper, see Dewar (2018), Weatherall (2016b), Wallace (2017) and Teh (2018).

  42. A similar moral applies in the case of moving to Galilean spacetime as a response to NGT’s boost invariance. Many thanks to David Wallace for pushing us on this point.

  43. We draw the term ‘moderate structuralism’ from Esfeld and Lam (2011); compare also the ‘modest structuralism’ of Pooley (2006, p. 102). According to this view, objects (e.g. points of spacetime) are construed as being nothing more (or less) than ‘nodes’ in the relational structures in which they are embedded; and the possibility of purely haecceitistic distinctions between worlds is denied. Construed in this way, moderate structuralism encompasses sophisticated substantivalism—but is a stronger thesis due to the latter clause.

  44. For example, the staunchest advocate of the interpretational approach would likely not be motivated to consider whether NGT can be reformulated in terms of Galilean spacetime, or NCT: the bare assertion that models of NGT related by kinematic or dynamic shifts are physically equivalent effectively eliminates motivation for the advocate of the interpretational approach to pursue this research programme.

  45. Cf. footnote 14. Here, the advocate of the interpretational approach may be guided by overarching, a priori principles, connecting certain features of the symmetry-related models under consideration with their physical equivalence. However, unless a necessary connection between such features and the physical equivalence of the models can be forged, the point in the body of this paragraph stands.

  46. We owe the nomenclature of ‘confident’ versus ‘cautious’ versions of the motivational approach to Jeremy Butterfield.

  47. Recall from Sect. 2 that, for a given theory \(\mathcal {T}\), \(\tilde{\mathcal {D}}\) denotes the gauge-reduced space of DPMs of \(\mathcal {T}\). Note also that one may introduce a graded notion of theoretical equivalence, by imposing restrictions on the structure of the models preserved by this isomorphism. Though important to note, this latter point will be set aside in this paper.

  48. Each \(\tilde{\mathcal {M}}_1 \in \tilde{\mathcal {D}}_1\) and \(\tilde{\mathcal {M}}_2 \in \tilde{\mathcal {D}}_2\) which correspond to the same empirical predictions in this way may be said to be ‘empirically equivalent’—cf. Sect. 3.1. Note that two models may be empirically equivalent without the theories to which they belong being empirically equivalent, in the sense given below.

  49. The map between \(\tilde{\mathcal {M}}_1 \in \tilde{\mathcal {D}}_1\) and \(\tilde{\mathcal {M}}_2 \in \tilde{\mathcal {D}}_2\) may not be one-one in the absence of formal equivalence.

  50. For philosophically-oriented introductions to dualities—including all the theories and their respective dualities mentioned in this paragraph—see e.g. Polchinski (2017) and Rickles (2011). For more recent and nuanced philosophical approaches to dualities, see De Haro (2016); De Haro and Butterfield (2018) and Butterfield (2018).

  51. For a philosophical introduction to mirror symmetry, see Rickles (2013b).

  52. I.e. to the effect that each \(\tilde{\mathcal {M}}_1 \in \tilde{\mathcal {D}}_1\) and its associated \(\tilde{\mathcal {M}}_2 \in \tilde{\mathcal {D}}_2\) must correspond to the same world.

  53. For a critical discussion of the Quinean approach to theoretical equivalence, see Barrett and Halvorson (2016a). With the conclusion of that paper—“If one takes Quine equivalence as the standard for theoretical equivalence, one underestimates the threat of underdetermination” (Barrett and Halvorson 2016a, p. 483)—we are in agreement. Cf. also Barrett and Halvorson (2016b).

  54. To allay any possible misunderstanding: Quine’s view on this matter is not that theories with isomorphic spaces of solutions are always theoretically, or even empirically, equivalent. Rather, Quine holds a strictly stronger view on what it is for two theories to be theoretically equivalent: if theories are related by a suitable reconstrual of predicates, then such theories are theoretically equivalent. This, plausibly, entails that their respective spaces of solutions are isomorphic. But, for Quine, the fact that theories have isomorphic spaces of solutions does not by itself entail that they are theoretically or even empirically equivalent.

  55. The concern, therefore, is over the adequacy of formal equivalence—which is, indeed, a formal notion—to capture an informal or semantic notion: that of two models representing the same possible world. Our thanks to Jeremy Butterfield for suggesting that we put the matter in this way.

  56. This said, Rickles has recently suggested that cases such as the AdS/CFT correspondence may give rise to structural underdetermination (Rickles 2011, 2013a, 2017). (We concur with this view; cf. footnote 57 below.) Note that if this is so, then even the structuralist may not be able to argue that such pairs of formally equivalent models correspond to the same possible world.

  57. In our view, adopting structural realism as a means of identifying symmetry-related models succeeds only if the models in question are ‘naïvely’ understood as representing at most haecceitistically distinct possible worlds. In that case, it is clear how adopting structural realism allows us to identify such (putatively) distinct physically possibilities as (actually) not distinct after all. However, if the models in question are ‘naïvely’ understood as representing more than haecceitistically distinct possible worlds, then adopting structural realism (by itself) is insufficient to provide grounds for understanding the models in question as corresponding to the same possible world.

  58. If this is correct, then we concur with De Haro et al. that “we will need to allow that formal isomorphisms do not in general imply sameness of content” (De Haro et al. 2017, §3.1). Cf. footnote 55.

  59. For further clear expression of this position, see e.g. Matsubara and Johansson (2016) and Matsubara and Smeenk (2016).

  60. One might argue that the final sentence here is in line with the motivational approach. Even if this is true, however, in our view it is not correct to read Rickles as endorsing the motivational approach, in light of the preceding sentences in the quote—see below.

  61. Here, the same points from Sect. 3.1 regarding the ‘typicality’ clause arise again.

  62. This view appears more popular in the literature; cf. again e.g. Rickles (2017, p. 62, and footnote 60).

  63. This can be considered the analogy of Dewar’s approach to symmetries in the case of dualities (Dewar 2015, p. 322).

  64. Or quantum field theories, in the case of e.g. the AdS/CFT correspondence.

  65. Perhaps it is true that many recent philosophical authors’ views on dualities are more subtle than a straightforward endorsement of the interpretational approach, à la Rickles (Rickles 2017, p. 62). This notwithstanding, however, a reader unfamiliar with the literature on dualities may obtain the impression that the interpretational approach is widely embraced. Here is some prima facie evidence to support this claim: “[D]ual [theories] should be understood as giving physically equivalent descriptions” (Huggett 2017, pp. 87–88); “In all dualities, it is the theories that are equivalent. ... [C]ertain transformations ‘don’t matter’. The only difference between these [dual] transformations and standard gauge symmetries is that they seem to relate things that look like they really should matter!” (Rickles 2017, p. 64); “[Our] conception of duality meshes with two dual theories being ‘gauge-related’, in the general philosophical sense of being physically equivalent. For a string duality, such as T-duality and gauge/gravity duality, this means taking such features as the radius of a compact dimension, and the dimensionality of spacetime, to be ‘gauge”’ (De Haro et al. 2017, p. 68); “The stance adopted [in this paper] is ... to avoid a literal reading of the elementary/composite interchange and, on this basis, to avoid mixing the question of its meaning with the question of physical fundamentality. The attitude is analogous to the one shared in this volume [a recent special issue of Studies in the History and Philosophy of Modern Physics devoted to dualities] about how to understand apparently puzzling features such as the interchange of tiny and huge dimensions connected with T-duality in string theory, or the duality of dimension under the AdS/CFT (gauge/gravity) correspondence. The underlying idea is that, what the dual descriptions do not agree upon, should not be attributed a real physical significance. In fact, this means nothing else than saying that the physics (including its ontology) remains the same under the duality. What changes, is just the way of looking at it” (Castellani 2017, p. 101).

  66. See below for further discussion of this theory.

  67. An anonymous reviewer has questioned whether our nomenclature for these two positions is entirely apt. This is for two main reasons: (i) It appears to obscure the fact that both the interpretational and motivational approaches are, broadly speaking, possible ways of ‘interpreting’ theories; (ii) It might suggest that the interpretationalist never agrees with the motivationalist on the issue of whether one is motivated to reformulate a given theory in the presence of certain symmetries. With regard to (i): We agree that both approaches are, in some sense, possible ways of ‘interpreting’ physical theories, although we disagree that our nomenclature truly obscures this fact. (Our claim is simply that someone who endorses the interpretational approach to symmetries is not ‘interpreting’ symmetries the right way; the right way to ‘interpret’ them is motivationally!) With regard to (ii): As noted in the main text, although the interpretationalist might agree with the motivationalist on the issue of motivation in certain cases, she need not always do so. Motivation is not an essential component of the interpretationalist’s position, as it is for the motivationalist.

  68. That is, the formal structure preserved across duality-related models—cf. De Haro (2016) and Dewar (2017). This is the ‘common core’ of the two dual models, in the sense given above.

  69. Such an approach to dualities is also implicit in Matsubara and Johansson (2016) and Matsubara and Smeenk (2016). In these passages, by a “quantum system”, Polchinski means a quantum theory; by two “classical limits”, he means two theories for which perturbation theory is applicable, which may be defined from the original quantum theory—see Polchinski (2017, p. 6ff).

  70. The facto here being the construction of the two original, dual theories.

  71. Specifically, vis-à-vis both their theoretical equivalence and their empirical equivalence.

  72. If one endorses these views, then one will argue that more needs to be done to demonstrate the physical equivalence of duality-related models of certain string theories—in terms of exploring the mathematics and interpretation of M-theory—before such physical equivalence can be declared by appeal to this (conjectured) theory.

  73. For further details, see De Haro (2016, §1). Cf. also Dewar (2017), in which a very similar distinction is drawn.

  74. For further discussion of these matters in the symmetries literature, see Greaves and Wallace (2014) and Teh (2016); cf. also footnote 16.

  75. We thank Jeremy Butterfield and Sebastian De Haro for discussion on these matters; in private communication, both have indicated that they favour the cautious motivational approach.

  76. There is some analogy here with van Fraassen’s constructive empiricism (Van Fraassen 1980), often justified on the grounds that “belief in the empirical adequacy of accepted theories [is] the weakest attitude one can attribute to scientists at the same time that one is still able to make sense of their scientific activity” (Monton and Mohler 2017, §2.2).

  77. For philosophical discussions drawing upon Vafa’s definition of a duality, see Matsubara and Smeenk (2016, §3) and Read (2016b, §2).

  78. Here, by a ‘physical system’ Q, we can understand Vafa to mean a physical theory \(\mathcal {T}\); one may then identify \(\mathcal {M}\) with our reduced space of DPMs, \(\tilde{\mathcal {D}}\) (note that the tilde here refers to the quotienting of \(\mathcal {D}\) by gauge-equivalent models, rather than to the space of DPMs of the dual theory). By \(\mathcal {O}\), we understand Vafa to mean the set of observables \(\mathcal {O}_\alpha \) for the theory in question.

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Acknowledgements

We are indebted to Jeremy Butterfield (thrice over!), Sebastian De Haro, Neil Dewar, Niels Martens, Keizo Matsubara, Tushar Menon, Teruji Thomas, and several anonymous referees, for valuable comments on drafts of this paper, and are also grateful to Oliver Pooley and David Wallace for helpful discussions. J.R. is supported by an AHRC studentship at the University of Oxford, and is also grateful to Hertford College, Oxford for a graduate senior scholarship. T.M.-N. is supported by a stipendiary lectureship at Oriel College, Oxford.

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    James Read & Thomas Møller-Nielsen

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Read, J., Møller-Nielsen, T. Motivating dualities. Synthese 197, 263–291 (2020). https://doi.org/10.1007/s11229-018-1817-5

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