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The British Journal for the Philosophy of Science

The Coalescence Approach to Inequivalent Representation: Pre-QM∞ Parallels

Caspar Jacobs

Published Apr 27, 2021• 1 citation
Publisher

Abstract

In Interpreting Quantum Theories, Ruetsche argues that the occurrence of unitarily inequivalent representations in quantum theories with infinitely many degrees of freedom poses a novel interpretational problem. According to Ruetsche, such theories compel us to reject the so-called ideal of pristine interpretation; she puts forward the ‘coalescence approach’ as an alternative. In this article I offer a novel defence of the coalescence approach. The defence rests on the claim that the ideal of pristine interpretation already fails before one considers the peculiarities of QM∞: there are pre-QM∞ parallels to coalescence. Despite this departure from pristinism, the ‘modest’ view that emerges poses no threat to scientific realism.

DOI
10.1086/715108

Related papers

Interpreting Quantum Theories

James Owen Weatherall

Most philosophical work on quantum physics has concerned simple systems. And for good reason. Even one or two particle systems can exhibit the striking features we have come to associate with quantum physics—features such as entanglement, interference, and nonlocality. But as far as physics goes, such systems barely scrape the surface. In particular, they have only a finite number of degrees of freedom, which means their states are characterized by a finite number of independent parameters. The systems studied in fields such as high-energy particle physics and many-body physics, meanwhile, have uncountably many degrees of freedom. They are far more complicated than the usual philosophical fare.This is not troubling as long as the conceptual heart of quantum physics can be effectively condensed down. But what if philosophers' toy systems are fundamentally different from realistic ones? That would be a problem. And indeed, as Laura Ruetsche convincingly argues in Interpreting Quantum Theories, systems with uncountable degrees of freedom—what Ruetsche collectively calls QM∞—are fundamentally different from the systems that have become the stock and trade of philosophy of physics, in ways that change the interpretational project.Ruetsche's book does two things exceedingly well. The first is to provide a (comparatively) accessible and philosophically oriented introduction to the algebraic approach to quantum physics, which is essential for understanding QM∞. This is a valuable contribution. The algebraic approach is the setting for much recent philosophical work. And yet learning it is notoriously difficult, as even pedagogical texts are aimed at research-level mathematicians. Ruetsche's book is technically demanding, but it works to bring nonmathematicians along. It is, without a doubt, the best place to enter this literature.Second, the book sets the agenda for future work on QM∞, by effectively and judiciously drawing important foundational questions out of a sprawling and difficult physics literature. In this regard, it is a close cousin to John Earman's (1995) excellent Bangs, Crunches, Whimpers, and Shrieks, which has had a significant influence on subsequent philosophy of spacetime physics. Philosophers of physics will be addressing the questions Ruetsche has raised, in many cases for the first time, for years to come.Interpreting Quantum Theories centers on what might be called the problem of inequivalent representations. The quantum theories that philosophers are accustomed to are set in a mathematical structure known as a (separable) Hilbert space, which is an at-most countably infinite dimensional complex vector space. Rays in a Hilbert space represent possible states of a physical system, and self-adjoint operators on that Hilbert space correspond to properties of the system.To treat QM∞, meanwhile, physicists pass to a generalization of this formalism. Now one begins with an abstract algebra of operators corresponding to the various properties associated with the system. States in this more general framework are linear functionals on the algebra whose value at each operator is interpreted as the expectation value of a measurement of that property in that state.The trouble with this more general framework is that the interpretational tools of Hilbert space quantum theories do not directly apply. Some solace comes from recognizing that any (appropriate) algebra of operators admits a representation in terms of the operators on a Hilbert space. This suggests that one can always move back to the more familiar Hilbert space formalism. But this is a pyrrhic victory. In general these Hilbert space representations are neither unique nor empirically equivalent. In particular, these inequivalent representations, interpreted in the familiar way, yield disagreements about which algebraic states are physically realizable.Faced with this situation, Ruetsche presents several principled options—what she calls “pristine” interpretations of QM∞. The two most significant are what she calls “Hilbert Space Conservatism” and “Algebraic Imperialism.” The Conservative claims that, for a given system, there is one true Hilbert space representation, which should be interpreted in the familiar ways. The Imperialist, meanwhile, insists that only the abstract algebra matters: the physical states are the algebraic states, properties correspond to the operators in the algebra, and if our old methods do not apply, so much the worse for our methods.At this level of description, Imperialists may appear to have the advantage, since they need not fret over which representation to choose. But there's a catch. As we have seen, there is a sense in which Imperialists generally recognize a broader range of possible physical states. But Conservatives allow for more properties. This is because when one represents the abstract algebra as operators on a Hilbert space, the operators in the representation naturally approximate additional operators on that Hilbert space, in a way that is unavailable in the abstract setting. Thus each interpretation suffers from limitations that the other does not.To adjudicate between the interpretations, Ruetsche takes them to the trenches, to see which provides the resources to support the actual explanatory and representational practices of physics. After a masterful survey, she concludes that neither interpretation has all of the requisite resources. In particular, she argues, physicists regularly make essential use of operators that are only available to Conservatives, while making equally essential use of multiple inequivalent representations—which only Imperialists can support. More, the particular mix of Conservatism and Imperialism varies between applications, apparently ruling out a stable interpretation of the formalism that can be articulated independently of the messy details. We are left with what Ruetsche calls “adulterated” interpretations.Philosophers' practice notwithstanding, it might seem unsurprising that we cannot separate our interpretation of physical theories from the pragmatics of their application to the world. After all, we are interested in these theories only insofar as they are successfully applied. But Ruetsche believes the problem is more acute. It is not merely that pragmatic considerations enter the debate between Conservatives and Imperialists. Ruetsche argues that there may not be any consistent, general interpretation of QM∞ compatible with practice. And this, she claims, makes realism about QM∞ untenable.The idea that the applications of a broad class of our best physical theories undermine their consistent interpretation in a way that renders realism incoherent is striking. Insofar as Ruetsche's case is convincing, it should have serious repercussions for general philosophy of science—both because it bears on important debates and because it shows how generalist questions depend on the details of particular scientific theories and their applications. Indeed, Ruetsche's skepticism about the terms of the realism debate and her radical particularism should influence general philosophy of science. But that said, her argument ultimately leaves room for disagreement.Ruetsche's basic claim is that Conservatism and Imperialism each support practices that the other cannot and that both kinds of practice are essential. But there is an important asymmetry here. Conservatives suffer because they fail to represent putative physical possibilities appealed to in practice. Imperialists, meanwhile, need to justify that in certain applications, they may limit attention to a subset of the physical possibilities their interpretation admits, and then use resources that only make sense in those limited regimes.The trouble is that the Imperialists' opportunism is neither unusual in physics nor difficult to motivate. In spacetime physics, for instance, properties such as the total mass of the universe or global time can be defined only in very special cases. And yet such properties are used in practice, because often the salient spacetimes are precisely the ones in which these quantities make (approximate) sense. To make her case that adulteration in QM∞ undermines realism, Ruetsche would need to establish that the representation-dependent quantities used in applications of QM∞ are different in kind from analogous quantities that appear throughout physics. And here it is not clear that she succeeds.None of this undercuts the importance of the book, however. Indeed, Ruetsche acknowledges that her argument may depend on how one understands idealization in physics. And even if one takes the view just described, there is a great deal of work to do on how Imperialism can support the practices Ruetsche identifies—work that will surely proceed within the framework Ruetsche has laid out, by addressing questions Ruetsche has raised. And so, wherever our study of QM∞ takes us, Ruetsche's book is sure to play a central role.

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In Interpreting Quantum Theories, Ruetsche argues that the occurrence of unitarily inequivalent representations in quantum theories with infinitely many degrees of freedom poses a novel interpretational problem. According to Ruetsche, such theories compel us to reject the so-called ideal of pristine interpretation; she puts forward the ‘coalescence approach’ as an alternative. In this article I offer a novel defence of the coalescence approach. The defence rests on the claim that the ideal of pristine interpretation already fails before one considers the peculiarities of QM∞: there are pre-QM∞ parallels to coalescence. Despite this departure from pristinism, the ‘modest’ view that emerges poses no threat to scientific realism.
Most philosophical work on quantum physics has concerned simple systems. And for good reason. Even one or two particle systems can exhibit the striking features we have come to associate with quantum physics—features such as entanglement, interference, and nonlocality. But as far as physics goes, such systems barely scrape the surface. In particular, they have only a finite number of degrees of freedom, which means their states are characterized by a finite number of independent parameters. The systems studied in fields such as high-energy particle physics and many-body physics, meanwhile, have uncountably many degrees of freedom. They are far more complicated than the usual philosophical fare.This is not troubling as long as the conceptual heart of quantum physics can be effectively condensed down. But what if philosophers' toy systems are fundamentally different from realistic ones? That would be a problem. And indeed, as Laura Ruetsche convincingly argues in Interpreting Quantum Theories, systems with uncountable degrees of freedom—what Ruetsche collectively calls QM∞—are fundamentally different from the systems that have become the stock and trade of philosophy of physics, in ways that change the interpretational project.Ruetsche's book does two things exceedingly well. The first is to provide a (comparatively) accessible and philosophically oriented introduction to the algebraic approach to quantum physics, which is essential for understanding QM∞. This is a valuable contribution. The algebraic approach is the setting for much recent philosophical work. And yet learning it is notoriously difficult, as even pedagogical texts are aimed at research-level mathematicians. Ruetsche's book is technically demanding, but it works to bring nonmathematicians along. It is, without a doubt, the best place to enter this literature.Second, the book sets the agenda for future work on QM∞, by effectively and judiciously drawing important foundational questions out of a sprawling and difficult physics literature. In this regard, it is a close cousin to John Earman's (1995) excellent Bangs, Crunches, Whimpers, and Shrieks, which has had a significant influence on subsequent philosophy of spacetime physics. Philosophers of physics will be addressing the questions Ruetsche has raised, in many cases for the first time, for years to come.Interpreting Quantum Theories centers on what might be called the problem of inequivalent representations. The quantum theories that philosophers are accustomed to are set in a mathematical structure known as a (separable) Hilbert space, which is an at-most countably infinite dimensional complex vector space. Rays in a Hilbert space represent possible states of a physical system, and self-adjoint operators on that Hilbert space correspond to properties of the system.To treat QM∞, meanwhile, physicists pass to a generalization of this formalism. Now one begins with an abstract algebra of operators corresponding to the various properties associated with the system. States in this more general framework are linear functionals on the algebra whose value at each operator is interpreted as the expectation value of a measurement of that property in that state.The trouble with this more general framework is that the interpretational tools of Hilbert space quantum theories do not directly apply. Some solace comes from recognizing that any (appropriate) algebra of operators admits a representation in terms of the operators on a Hilbert space. This suggests that one can always move back to the more familiar Hilbert space formalism. But this is a pyrrhic victory. In general these Hilbert space representations are neither unique nor empirically equivalent. In particular, these inequivalent representations, interpreted in the familiar way, yield disagreements about which algebraic states are physically realizable.Faced with this situation, Ruetsche presents several principled options—what she calls “pristine” interpretations of QM∞. The two most significant are what she calls “Hilbert Space Conservatism” and “Algebraic Imperialism.” The Conservative claims that, for a given system, there is one true Hilbert space representation, which should be interpreted in the familiar ways. The Imperialist, meanwhile, insists that only the abstract algebra matters: the physical states are the algebraic states, properties correspond to the operators in the algebra, and if our old methods do not apply, so much the worse for our methods.At this level of description, Imperialists may appear to have the advantage, since they need not fret over which representation to choose. But there's a catch. As we have seen, there is a sense in which Imperialists generally recognize a broader range of possible physical states. But Conservatives allow for more properties. This is because when one represents the abstract algebra as operators on a Hilbert space, the operators in the representation naturally approximate additional operators on that Hilbert space, in a way that is unavailable in the abstract setting. Thus each interpretation suffers from limitations that the other does not.To adjudicate between the interpretations, Ruetsche takes them to the trenches, to see which provides the resources to support the actual explanatory and representational practices of physics. After a masterful survey, she concludes that neither interpretation has all of the requisite resources. In particular, she argues, physicists regularly make essential use of operators that are only available to Conservatives, while making equally essential use of multiple inequivalent representations—which only Imperialists can support. More, the particular mix of Conservatism and Imperialism varies between applications, apparently ruling out a stable interpretation of the formalism that can be articulated independently of the messy details. We are left with what Ruetsche calls “adulterated” interpretations.Philosophers' practice notwithstanding, it might seem unsurprising that we cannot separate our interpretation of physical theories from the pragmatics of their application to the world. After all, we are interested in these theories only insofar as they are successfully applied. But Ruetsche believes the problem is more acute. It is not merely that pragmatic considerations enter the debate between Conservatives and Imperialists. Ruetsche argues that there may not be any consistent, general interpretation of QM∞ compatible with practice. And this, she claims, makes realism about QM∞ untenable.The idea that the applications of a broad class of our best physical theories undermine their consistent interpretation in a way that renders realism incoherent is striking. Insofar as Ruetsche's case is convincing, it should have serious repercussions for general philosophy of science—both because it bears on important debates and because it shows how generalist questions depend on the details of particular scientific theories and their applications. Indeed, Ruetsche's skepticism about the terms of the realism debate and her radical particularism should influence general philosophy of science. But that said, her argument ultimately leaves room for disagreement.Ruetsche's basic claim is that Conservatism and Imperialism each support practices that the other cannot and that both kinds of practice are essential. But there is an important asymmetry here. Conservatives suffer because they fail to represent putative physical possibilities appealed to in practice. Imperialists, meanwhile, need to justify that in certain applications, they may limit attention to a subset of the physical possibilities their interpretation admits, and then use resources that only make sense in those limited regimes.The trouble is that the Imperialists' opportunism is neither unusual in physics nor difficult to motivate. In spacetime physics, for instance, properties such as the total mass of the universe or global time can be defined only in very special cases. And yet such properties are used in practice, because often the salient spacetimes are precisely the ones in which these quantities make (approximate) sense. To make her case that adulteration in QM∞ undermines realism, Ruetsche would need to establish that the representation-dependent quantities used in applications of QM∞ are different in kind from analogous quantities that appear throughout physics. And here it is not clear that she succeeds.None of this undercuts the importance of the book, however. Indeed, Ruetsche acknowledges that her argument may depend on how one understands idealization in physics. And even if one takes the view just described, there is a great deal of work to do on how Imperialism can support the practices Ruetsche identifies—work that will surely proceed within the framework Ruetsche has laid out, by addressing questions Ruetsche has raised. And so, wherever our study of QM∞ takes us, Ruetsche's book is sure to play a central role.
This is an opinionated survey of some interpretive puzzles in quantum field theory. The problem of inequivalent representations is sketched, including its connections with competing accounts of physical equivalence. The controversy between variant formulations of the theory, algebraic versus Lagrangian, is given a conciliatory resolution. Arguments against particles are addressed, demarcating clearly between different forms of particle interpretation. Field interpretations are then considered, including wavefunctional, spacetime state realist and Heisenberg operator realist interpretations. Ruetsche’s coalesced structure interpretation is presented and juxtaposed with an alternative, more traditional view of the theory’s laws and state space. Finally, the CPT theorem is discussed, together with its implications about the nature of spacetime.
It often goes unnoticed that, even for a finite number of degrees of freedom, the canonical commutation relations have many inequivalent irreducible unitary representations; the free particle and a particle in a box provide examples that are both simple and well-known. The representations are unitarily inequivalent because the spectra of the position and momentum operators are different, and spectra are invariant under unitary transformations. The existence of these representations can have consequences that run from the merely unexpected to the barely conceivable. To start with, states of a single particle that belong to inequivalent representations will always be mutually orthogonal; they will never interfere with each other. This property, called superseparability elsewhere, is well-defined mathematically, but has not yet been observed. This article suggests two single-particle interference experiments that may reveal its existence. The existence of inequivalent irreducibile representations may be traced to the existence of different self-adjoint extensions of symmetric operators on infinite-dimensional Hilbert spaces. Analysis of the underlying mathematics reveals that some of these extensions can be interpreted in terms of topological, geometrical and physical quantities that can be controlled in the laboratory. The tests suggested are based on these interpretations. In conclusion, it is pointed out that mathematically rigorous many-worlds interpretations of quantum mechanics may be possible in a framework that admits superseparability.
We critically review the recent debate between Doreen Fraser and David Wallace on the interpretation of quantum field theory, with the aim of identifying where the core of the disagreement lies. We show that, despite appearances, their conflict does not concern the existence of particles or the occurrence of unitarily inequivalent representations. Instead, the dispute ultimately turns on the very definition of what a quantum field theory is. We further illustrate the fundamental differences between the two approaches by comparing them both to the Bohmian program in quantum field theory.
This chapter catalogs interpretative responses to the breakdown of unitary equivalence in QM∞. These range from ‘Hilbert Space Conservatism’ to ‘Algebraic Imperialism’. Maintaining the ordinary quantum mechanics account of physical equivalence in terms of unitary equivalence, the Hilbert Space Conservative takes unitarily inequivalent representations of the CCRs/CARs for a theory of QM∞ to be rivals, at most one of which could convey the content of that theory. Asserting the physics of a theory of QM∞ to reside in the abstract algebraic structure all its representations share, the Algebraic Imperialist announces criteria for the physical equivalence of quantum theories — including the criterion of ‘weak equivalence’ — at odds with the traditional criterion of unitary equivalence.
Scientific realism has traditionally maintained that our best scientific theories can be regarded as more or less true and as representing the world as it is (more or less). However, one of our very best current theories—quantum mechanics—has famously resisted such a realist construal, threatening to undermine the realist stance altogether. The chapters in this volume carefully examine this tension and the reasons behind it, including the underdetermination generated by the multiplicity of formulations and interpretations of quantum physics, each presenting a different way the world could be. Authors in this volume offer a range of alternative ways forward: some suggest new articulations of realism, limiting our commitments in one way or another; others attempt to articulate a ‘third way’ between traditional forms of realism and antirealism, or are critical of such attempts. Still others argue that quantum theory itself should be reconceptualised, or at least alternative formulations should be considered in the hope of evading the problems faced by realism. And some examine the nature of these issues when moving beyond quantum mechanics to quantum field theory. Taken together they offer an exciting new set of perspectives on one of the most fundamental questions in the philosophy of modern physics: how can one be a realist about quantum theory, and what does this realism amount to?