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Oxford University Press eBooks

Structure, Modality, and Unitary Inequivalence1

Steven French

Published Jan 30, 2014• 0 citations
Publisher

Abstract

In this chapter the issue of unitarily inequivalent representations in QFT is examined. This has been raised as a fundamental problem for OSR. Here the issue of arriving at an appropriate ontology of QFT comes to the fore and in the context of recent debates in the foundations of QFT, the Lagrangian and Algebraic approaches are examined from a structuralist perspective. It us argued that the issue of unitarily inequivalent representations can be deflated in various ways, and in particular by adapting Ruetsche’s ‘Swiss Army Knife’ approach in the context of the view of primitive modality outlined in the previous chapter.

DOI
10.1093/acprof:oso/9780199684847.003.0011

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In this chapter the issue of unitarily inequivalent representations in QFT is examined. This has been raised as a fundamental problem for OSR. Here the issue of arriving at an appropriate ontology of QFT comes to the fore and in the context of recent debates in the foundations of QFT, the Lagrangian and Algebraic approaches are examined from a structuralist perspective. It us argued that the issue of unitarily inequivalent representations can be deflated in various ways, and in particular by adapting Ruetsche’s ‘Swiss Army Knife’ approach in the context of the view of primitive modality outlined in the previous chapter.
There are significant problems involved in determining the ontology of quantum field theory (QFT). An ontology involving particles seems to be ruled out due to the problem of defining localized position operators, issues involving interactions in QFT, and, perhaps, the appearance of unitarily inequivalent representations. While this might imply that fields are the most natural ontology for QFT, the wavefunctional interpretation of QFT has significant drawbacks. A modified field ontology is examined where determinables are assigned to open bounded regions of spacetime instead of spacetime points.
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.
This dissertation gives a general account of the properties of unitarily inequivalent representations (UIRs) in both canonical quantum field theory and algebraic quantum field theory. A simple model is constructed and then used to show how to build a broad spectrum of UIRs including a version of Haag’s theorem. Haag and Kastler,P, two of the founding fathers of algebraic quantum field theory, argue that the problems posed by UIRs are solved by adopting a notion of equivalence that is weaker than unitary equivalence, which they refer to as physical equivalence. In the dissertation, it is shown that their notion does not provide a suitable classificatory schema. Some of the most important physical representations fail to satisfy the mathematical conditions of their notion. However, Haag and Kastler's notion has an unexpected connection with classical observables. A theorem is proven in which two representations make the same predictions with respect to all classical observables if and only if they satisfy their notion of physical equivalence. Following Haag and Kastler's lead, it was claimed by most proponents of algebraic quantum field theory that all physical content resides in a specific class of observables. It is shown in the dissertation that such claims are exaggerated and misleading. UIRs are used to elucidate the nature of quantum field theory by showing that UIRs have different expectation values for some classical observables of the system, such as temperature and chemical potential, which are not in Haag and Kastler’s specific class. It is shown how UIRs may be used to construct classical observables. To capture the physical content of quantum field theory it is shown that a much larger algebra than that of Haag and Kastler is necessary. Finally, the arguments that UIRs are incommensurable theories are shown to be flawed. The lesson of UIRs is that the mathematical structures in both canonical quantum field theory and Haag and Kastler’s version of algebraic quantum field theory are not sufficient to capture all of the physical content that UIRs represent. A suitable algebraic structure for quantum field theory is provided in the dissertation.
This chapter considers another route to a particle interpretation of quantum field theory, one mediated by the existence of techniques (such as the use of Feynman diagrams to guide calculation) and explanations (such as the appeal to cosmological particle creation to account for decelerating cosmic expansion) best understood in terms of particles. Looking close at these techniques and explanations, the chapter argues that they rely on the presence of unitarily inequivalent representations, the very presence, Chapters 8 and 9 contended, that frustrates fundamental particle interpretation.
The availability of unitarily inequivalent representations of the canonical commutation relations constituting a quantization of a classical field theory raises questions about how to formulate and pursue quantum field theory. In a minimally technical way, I explain how these questions arise and how advocates of the Hilbert space and of the algebraic approaches to quantum theory might answer them. Where these answers differ, I sketch considerations for and against each approach, as well as considerations which might temper their apparent rivalry.