A Schema for Duality, Illustrated by Bosonization

Abstract

In this paper we present a schema for describing dualities between physical theories (Sections 2 and 3), and illustrate it in detail with the example of bosonization: a bosonfermion duality in two-dimensional quantum field theory (Sections 4 and 5).The schema develops proposals in De Haro (2016, 2016a): 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 bosonfermion 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. 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'.

Part 2: Theoretical equivalence in physics

“…Physicists often say that such theories are “dual,” or that pairs of such theories exhibit (or are) a “duality.” Dual descriptions of the same physical situations have become particularly important in the context of string theory, which is an ambitious program to unify gravitational physics with the Standard Model of particle physics (Polchinski, ). Still more recently, philosophers of physics have attempted to understand the character and interpretational significance of these dualities (Butterfield, ; De Haro, ; De Haro & Butterfield, ; De Haro, Mayerson, & Butterfield, ; De Haro, Teh, & Butterfield, , ; Le Bihan & Read, ; Matsubara, ; Read, ; Read & Møller‐Nielsen, ; Rickles, , ). We will not give a complete review of the issues related to dualities here; instead, we will focus on how the literature on dualities relates to the issues of equivalence already introduced.…”

Duality and ontology

“…The common core approach purports to identify a possible world “isomorphic” to the mathematical structure common to the dual solutions under consideration. Though this approach is popular in the philosophy of physics literature (see, e.g., De Haro & Butterfield, ; Huggett, ; Matsubara, ; Matsubara & Johansson, ; Rickles, , ), it is not the only live interpretative option purporting to break the underdetermination. Indeed, a distinct position—widely embraced in the physics community in the context of string‐theoretic dualities—is to embed the spaces of solutions of the two dual theories under consideration into that of some deeper, “overarching” theory.…”

“…In this vicinity, cf. also De Haro (), De Haro and Butterfield (), Huggett (), and Matsubara and Johansson (). While it is true that the common core approach is sometimes identified with structural realism, note that the approach is distinct from the two versions of structuralism articulated in §4—for (i) this approach still attempts to explicate the ontology of a given model in terms of extant ontological categories (unlike the version of structuralism considered in the body of §4) and (ii) this approach is not necessarily committed to an “ontology of physical structures” (unlike the version of structuralism considered in footnote 16).…”

“…However, in this paper, we choose to work with the weaker notion of a duality proceeding in terms of empirical equivalence alone, for this will suffice to make all necessary points regarding the interpretation and ontology of duality‐related theories. For more detailed and comprehensive approaches to the definition of dualities, see, for example, De Haro () and De Haro and Butterfield (). Note also that it need not be the case that the duality map is one–one—it might instead be that a class of solutions of the first theory is mapped to a single solution of the second theory under the duality.…”

“…This is a question worthy of considered attention; thankfully, authors in the philosophy of dualities have already done much to clarify these matters. See in particular De Haro and Butterfield (, §2), in which this “common core” is taken to consist of a “bare theory”—in itself understood to be a triple of (respectively) states, quantities, and dynamics, isomorphic representations of which are contained in the structure of each of the dual theories under consideration (along with, potentially, further theory‐specific structure). For our purposes, it suffices to know that the common mathematical core of two dual theories can be constructed in a well‐defined manner.…”