Abstract
In this paper we present a schema for describing dualities between physical theories (Sects. 2 and 3), and illustrate it in detail with the example of bosonization: a boson-fermion duality in two-dimensional quantum field theory (Sects. 4 and 5). The schema develops proposals in De Haro (Space and Time after Quantum Gravity, 2016 [15]; Duality and Physical Equivalence, 2016a [16]): these proposals include construals of notions related to duality, like representation, model, symmetry and interpretation. The aim of the schema is to give a more precise criterion for duality than has so far been considered. The bosonization example, or boson-fermion duality, has the feature of being simple yet rich enough to illustrate the most relevant aspects of our schema, which also apply to more sophisticated dualities. The richness of the example consists, mainly, in its concern with two non-trivial quantum field theories: including massive Thirring-sine-Gordon duality, and non-abelian bosonization. This prompts two comparisons with the recent philosophical literature on dualities. (a) Unlike the standard cases of duality in quantum field theory and string theory, where only specific simplifying limits of the theories are explicitly known, the boson-fermion duality is known to hold exactly. This exactness can be exhibited explicitly. (b) The bosonization example illustrates both the cases of isomorphic and non-isomorphic models: which we believe the literature on dualities has not so far discussed.
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Notes
- 1.
Again, we can only touch on the vast literature. For Hilbert’s Problems of 1900, cf. e.g. [32, 33]. For the sixth Problem, cf. [11,12,13,14, 51]. For some context for Hilbert’s famous ‘beer-mug’ remark, cf. [38]. Finally, we note that Gray [34] makes an interesting case that this broad development represented a rise of ‘modernism’, in a sense analogous to that in art and literature: cf. also [35].
- 2.
The Hilbertian idea has, of course, other important facets: for example, in fostering the idea that an axiom system—or more generally, a doctrine expressed in language—‘implicitly defines’ its terms. This has been very influential in the foundations of logic and mathematics, beginning with Hilbert’s debate with Frege. It has also of course been contested: in the face of non-categoricity, the claim to ‘define’ terms by a body of doctrine containing them is questionable.
- 3.
For some ‘post-Hilbert’ history of axiomatisation as ‘deepening the foundations’, cf. [52, 53]. But we should add that we do not endorse the logical empiricist project of distinguishing, once and for all, the factual and conventional parts of a theory: our misgivings are essentially those of Putnam [47].
- 4.
We agreed that for our notion, the word ‘model’ has disadvantages. But note that other words also have disadvantages. For example: ‘formulation’ connotes that any two formulations of a theory are ‘notational variants’, i.e. fully equivalent: they say exactly the same thing about the world. But that is far from true for our notion (and this matches the connotations of ‘model’): for us, two models of a bare theory are in general not isomorphic, and not in any sense equivalent; and so typically, it is surprising to find two isomorphic models, i.e. to find a duality. Other examples: ‘realization’, ‘instance’ and ‘instantiation’ connote being part of the physical world, as in ‘the mechanism/hardware which realizes some specific function/software’, or ‘the object is an instance/instantiation of the predicate’—which is the misleading connotation (iii) above.
Notice that in theoretical physics, the use of model is, roughly, between: (a) our use, and (b) (ii) and (iii) above: e.g. the ‘massive Thirring model’ or the ‘sine-Gordon model’.
- 5.
- 6.
Despite the name of ‘symmetry’, mirror symmetry falls under what we here call a duality. That mirror symmetry is a case of duality—of two different models, rather than a single model, being related—is uncontroversial, and reflected in the literature.
- 7.
- 8.
In T-duality and mirror symmetry, it is not the size of the couplings that is inverted by the duality map but, roughly speaking, the sizes of the spaces.
- 9.
- 10.
Agreed, pure mathematicians sometimes work with uninterpreted theories; and duality is a grand theme in mathematics, just as it is in physics. But although comparing duality in mathematics and in physics would be a very worthwhile project, we set it aside. Cf. [10].
- 11.
Besides, in physics examples the actual mathematical structure of a model is often very rich, e.g. in gauge-gravity dualities the metric of a \((d+1)\)-dimensional spacetime will belong to the specific structure \(\bar{M}\), but this metric will be a fibration over the metric of a d-dimensional spacetime, which belongs to \(T_M\): see [15, Sect. 2.1], [16, Sect. 2.2].
- 12.
Also agreed: it is common, and mathematically natural, to consider the set \(\mathcal S\) of all states, with quantities as extra structure on \(\mathcal S\): e.g. in classical mechanics, as real-valued functions on the phase space \(\mathcal S\), and in quantum mechanics, as linear operators on the Hilbert space \(\mathcal S\). But this does not make it compulsory to treat states as the basic objects in a semantics of a physical theory. For it is equally legitimate, though less common in textbooks, to start with the set \(\mathcal Q\) of quantities, and take states as extra structure on \(\mathcal Q\). And the legitimacy of both these approaches shows that au fond, a state is an assignment of numerical values to all quantities; and mutatis mutandis a quantity is an assignment of numerical values to all states. For now, the point is just that if one is asked to classify states and quantities in either of the philosophical categories of ‘object’ and ‘ property’, undoubtedly one should classify both states and quantities as properties.
- 13.
We say ‘appropriate’ so as to signal that of course, for any T or M, not every possible world has a context rich enough to determine reference for all T’s or M’s elements. Indeed: for many worlds, all their contexts will determine reference for none of T’s or M’s elements. For example, take T or M to be supersymmetric theories, and a world with no supersymmetric physics. So whatever our precise definition of the domain (i.e. set of arguments) of the map \(I_{ \text{ Ext }}\), the map will surely be partial, i.e. undefined on some, maybe the majority, of its arguments. But that is no problem. Formal semantics and philosophy of language in the Frege-Carnap-Lewis framework have long had various proposals for how to treat words and phrases that lack extensions (called ‘bearerless terms’); and these proposals can be adapted to our T or M.
- 14.
This line of thought is not only widespread, but has a long tradition: for thousands of years in philosophical accounts of abstraction; and for a hundred and fifty years in mathematics, with e.g. Frege’s proposal to define notions as equivalence classes (e.g. a direction as an equivalence class of parallel lines).
- 15.
The first two maps will be isomorphisms, the last an equivariance condition: we will say more about this in Sect. 3.2.1.
- 16.
Similarly, if we apply this proposal to models: all models of a theory thus defined are isomorphic. Of course: we expect that since a model M has specific structure \(\bar{M}\) going beyond its model root m, isomorphism for models will in general be stronger—i.e. lead to smaller equivalence classes—than does isomorphism of model roots. But in this Section we will not need to linger on this model versus model root contrast. For our main concern is defining a theory using isomorphism of model roots. As we will argue below (at the end of this Subsection), there is a natural constraint that model roots must be sufficiently varied. For mistaking the presence of accidental similarities between the model roots one happens to have at hand for necessary similarities between all the model roots of the theory one is trying to define, leads to unnecessarily, or perhaps undesirably, restrictive theories. This is a fortiori true of the models \(M_i\) of the theory: since the specific structure \(\bar{M}_i\) is specific to \(M_i\), and so in general not shared with the another model.
- 17.
Agreed: one does not always get to ‘choose’ one’s model roots (or models), and so this constraint cannot always be implemented. Thus there is judgment involved in this process of abstraction, viz. of (i) how many, and how varied, the model roots should be, to provide representations of one’s theory, and (ii) how to make the distinction, for a given model, between model root and specific structure (since part of the specific structure of a model could be mistaken for e.g. additional information about the theory). We therefore maintain that this reverse approach, from model roots to theory, is not deductive but inductive—which brings us back to our Hilbertian theme from Sect. 1. It only stops when one is happy with the theory—based on whatever independent criteria one uses to judge one’s theory and models.
- 18.
For the moment, we just note that it is also common to think that a symmetry as a map on states is ‘active’, i.e. the image-state must be a different physical state of affairs (so the question’s answer is ‘No’), while a symmetry as a map on quantities is ‘passive’, i.e. the image-quantity and the argument-quantity (each with their common value) describe the single given physical state (so that now the question’s answer is ‘Yes’).
We will deny this. There is no universal association of symmetry as a map on states as ‘active’, and symmetry as a map on quantities as ‘passive’. The reason lies, essentially, in the distinction between a mathematical state and a physical state: (in the jargon of ‘gauge’, the latter is a gauge-equivalence class of the former). That is: we of course concede that a symmetry as a map on states is ‘active’, in the sense that it changes the states. That is a tautology: (except for the degenerate case where the symmetry is given as being the identity map!). But this concession does not imply that a symmetry as a map on states must change the physical state of affairs represented: for the states in question could yet be ‘merely’ mathematical. That is: one still needs a further argument why a difference of these states must imply a difference of physical state (and thus why the question’s answer is ‘No’). This distinction, between a mathematical and a physical symmetry, was labelled, in De Haro et al. [20, Sect. 2], with the label (Redundant); and in De Haro [16, Sect. 1.1.2.b], as (Physical)-(Redundant). It also roughly corresponds to the distinction, in Caulton [8], between an ‘analytic’ and a ‘synthetic’ symmetry.
- 19.
If the dynamics is deterministic, we can write \(s(t) = D_{t, t_0}(s(t_0))\) where \(D_{t, t_0}\) represents the deterministic dynamics; and then ‘preserving the dynamics’ is equivalent to the commutation i.e. equivariance condition, \(a(s(t)) \equiv a(D_{t, t_0}(s(t_0))) = D_{t, t_0}(a(s(t_0)))\).
- 20.
Together with a set of maps to the real numbers, to express evaluation of the quantities. But for simplicity we ignore these maps for the moment.
- 21.
Agreed, to impose this commutation condition for every symmetry and every interpretation is contentious. It seems best justified when we envisage that the bare theory or model describes the whole universe; so that for the example of three point particles, there are no other material bodies in the universe. But in this paper, we do not need to assess exactly when the commutation condition is justified. See De Haro [16, Sect. 1.3–1.4], especially the condition called ‘unextendability’.
- 22.
As we mentioned in Sect. 2.2.1, ‘appropriate structure’ here refers to: (i) the structure of the sets of spaces and quantities, (ii) the rules for evaluating quantities, (iii) the structure which the dynamics satisfies, (iv) the set of symmetries of the theory. We can now be more specific about these, for the examples of quantum theories, which will illustrate our schema: (ia) the set of states will be a separable Hilbert space; (ib) the quantities will be elements (normally the self-adjoint, renormalisable elements) of an algebra; (ii) the rules for evaluating quantities are maps to the appropriate field: for most quantum theories, the inner product on the Hilbert space, and the usual rules for evaluating matrix elements; (iii) dynamical evolution will usually be a (unitary) map, satisfying appropriate commuting diagrams with the other maps in the theory; (iv) the group of symmetries will comprise the automorphisms of the algebra: and possibly additional symmetries, on the states and on the quantities. For classical theories, these comments get modified in familiar ways: e.g. (ia) would say that the set of states is a manifold, with structure appropriate to e.g. Lagrangian or Hamiltonian mechanics.
- 23.
Our proposal does not depend on the formulation of models as triples. A model root can be presented in many different forms, and the isomorphism should then preserve the corresponding structure. Even for triples, one can envisage isomorphisms which do not respect the triple structure, though they map the model roots isomorphically. Compare Sect. 3.2.2. But it will suffice for our purposes to restrict to model roots defined as triples, whose structure is preserved by the duality.
- 24.
We should put this last point more precisely, since our notion of bare theory is logically weak, with even a group or an algebra, together with a set of maps to the real numbers, counting as a legitimate bare theory: (cf. (1) in Sect. 2.1). And it is in general not surprising, nor likely to be empirically fruitful, to learn that two very disparate models are both groups, or both algebras: (unless the maps to the real numbers are so disparate that the existence of an isomorphism is not easy to guess). Thus the point here, more precisely, is that, for a given degree of detail or logical strength in the bare theory (and the more, the better!): the more disparate its models (considered in their entirety), the more surprising, and one hopes fruitful, is their both realizing the bare theory. That is: the more surprising is the duality.
- 25.
This simpler idea of a triple was used in our earlier—cruder!—discussion of duality: cf. [20, Sect. 3.2]. There, the simplicity engendered no errors, since our general description of duality was but a preamble to a specialist topic: an assessment of gauge symmetries in gauge-gravity duality.
- 26.
Therefore duality will imply unitary equivalence.
- 27.
See footnote 23 and Sect. 3.2.2 for a brief discussion of more general cases.
- 28.
At least, this is what we would in general expect. Agreed, one might interpret a model without interpreting all of the specific structure \(\bar{M}\): recall footnote 13 on the need to allow the interpretation maps to be partial, i.e. to deliver no value for certain arguments. For more details, see Sect. 1.1.2.a of De Haro [16].
- 29.
In Sect. 3.2.2, we emphasised the fact that only the model triples, and not the specific structure, are physically significant. When we now consider external interpretations that do give a physical meaning to the specific structure, we have to say that these interpretations change the physical content of the model (its physical degrees of freedom). This is correct, because external interpretations do not need to preserve the structure of the model as a quadruple.
- 30.
- 31.
The results in Minkowski signature are readily obtained by a Wick rotation, as we discuss in Sect. 4.1.
- 32.
Classically, we may indeed require the solutions to be holomorphic and anti-holomorphic functions. Quantum mechanically, there are singularities which are both inevitable and the source of interesting physics, as we will see. Thus we will allow \(\phi (z)\) and \(\bar{\phi }(\bar{z})\) to have isolated singularities, hence we will allow them to be meromorphic and anti-meromorphic functions (operators, in the quantum version of the model), respectively.
- 33.
They are called ‘primary’ because all other fields, which are called ‘descendants’, can be obtained from them, through successive application of derivatives. See De Haro et al. [15, Sect. 3].
- 34.
As remarked in footnote 32, in the quantum case we allow for (anti-) meromorphic, rather than (anti-) holomorphic, operators.
- 35.
In more detail: \(h+\bar{h}\) is the eigenvalue of the dilatation operator, and \(h-\bar{h}\) is the eigenvalue of the (Euclidean) rotation operator. Hence, the conformal weights contain information about the mass and the Euclidean spin of a field.
- 36.
Notice that (25) satisfies the property that the level k can be changed by a rescaling of J. Thus, in the simple case we are dealing with here, in which the affine Lie algebra is based on the commutative Lie group U(1), the level has no real meaning. This is not important for us, since we will not use it: rather, our analysis in Sect. 5 will be based on the fact that we are here dealing with a special case of the general enveloping algebra of the affine Lie algebra.
- 37.
- 38.
Here, \(\nu ={1\over 2}\,\) for periodic boundary conditions, and \(\nu =0\) for anti-periodic.
- 39.
- 40.
Our use of the word ‘Euclidean spin’ here follows the jargon in the physics literature, for the eigenvalue under Euclidean rotations, as we mentioned in Sect. 4.1. It is questionable whether such jargon is justified by the physical interpretation in 1+1 dimensions. But we will not need to dwell on this point, since our main aim in this Section is formal.
- 41.
As an example, consider the bosonic and fermionic models, but now weakened by the stipulation that the Virasoro algebra belong to the model triple, while the affine Kac-Moody algebra [(and the third line in Eq. (25)] belongs to the specific structure. As we argued in Sect. 3.2.2, this stipulation changes the physical content of the models, and so it is not innocuous. The models thus obtained contain different numbers of (uninterpreted) physical degrees of freedom, and so cannot describe the bosons or the fermions of Sect. 4. This is because the boson and the fermion CFTs (even before they are physically interpreted) treat the Kac-Moody degrees of freedom not as ‘accidental commonalities’, in the sense of Sect. 2.4: but as physical, and related to the Virasoro generators by the Sugawara construction. (For example, if we drop the chiral symmetry on the fermionic model, we lose the reason to restrict to chiral quantities only: cf. Sect. 5.2.2; and likewise for the boson’s affine current symmetry algebra.) Thus the boson and fermion models are not dual, if based on just the Virasoro algebra. We thank Josh Hunt for bringing up this example.
- 42.
Notice that the ambiguity here is in the best definition of the theory, not of the duality:
- 43.
- 44.
The scale \(\mu \) is already present in the massless theory. But it does not play any important role, since it is just an overall renormalisation constant.
- 45.
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Acknowledgements
We thank Joseph Kouneiher, not least for his patience! We also thank an anonymous referee for comments, the participants at the Symmetries and Asymmetries in Physics conference in Hannover, and especially Josh Hunt for comments. SDH thanks several audiences: the British Society for the Philosophy of Science 2016 annual conference, the Oxford philosophy of physics group, LSE’s Sigma Club, the Munich Center for Mathematical Philosophy, and DICE2016. SDH’s work was supported by the Tarner scholarship in Philosophy of Science and History of Ideas, held at Trinity College, Cambridge.
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Appendix: Some Elements of Conformal Field Theory
Appendix: Some Elements of Conformal Field Theory
In Sect. 4.1, we used the notion of a primary field. A primary field of conformal weight \((h,\bar{h})\) is defined to transform, under a conformal transformation (16), as follows:
This is in analogy with the transformation law for covariant tensors in ordinary QFTs: it takes the transformation property of the field under conformal transformations as defining for the class of primary fields. The physical significance of primary fields is discussed around Eq. (20).
In our analysis in Sects. 4 abd 5, an essential role was played by the enveloping Virasoro algebra (25), with \(c=1\) and \(k=1\). This algebra is a special case of the following general enveloping algebra of an affine Lie algebra:
Here, c is the central charge and k is the level, and \(f^{ab}{}_c\) are the structure constants of the underlying Lie algebra of the affine Lie algebra. Notice that the above algebra contains, in the first line, the ordinary Virasoro algebra. And the last line is the affine Lie algebra. The middle line gives the commutation relation between generators of the two algebras.Footnote 45
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De Haro, S., Butterfield, J. (2018). A Schema for Duality, Illustrated by Bosonization. In: Kouneiher, J. (eds) Foundations of Mathematics and Physics One Century After Hilbert. Springer, Cham. https://doi.org/10.1007/978-3-319-64813-2_12
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