Search Results

Chapter 12 argued that quantum statistical mechanics puts unitarily inequivalent representations to use in ways no rigid interpretation can make sense of. Two features of working QFTs which promise a quantum field theoretic realization of Chapter 12's argument are Goldstone bosons and the Higgs mechanism. This chapter explains why they're promising by presenting them as instance of broken symmetry. Then it tempers the promise by admitting that the working QFTs in which these features occur are less mathematically explicit than they need to be to persuasively realize the argument of Chapter 12. The chapter closes by extracting from this very circumstance a non-conclusive reason to lend the argument of Chapter 12 interpretive weight. The reason is that our best theories of physics are still under construction, and their successors could share with the models presented in Chapter 12 the features on which the argument of Chapter 12 hinged.

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.

This chapter presents three reasons to take quantum statistical mechanics to the thermodynamic limit, where the number of systems considered becomes infinite, in the form of three phenomena that can't be modeled short of the thermodynamic limit: ergodicity, phase structure, and broken symmetry. Focusing on phase structure, exemplified by ferromagnetism, the chapter documents a promising model of the phenomenon that makes use of unitarily inequivalent representations in a way rigid interpretations of QM_∞, such as Hilbert Space Conservatism and Algebraic Imperialism, can't endorse. Taking a virtue of interpretation to be its capacity to sustain the explanatory aspirations of the theory interpreted, the chapter scores this as a strike against rigid interpretations.

This chapter highlights other surprising aspects of QM_∞: unlike theories of ordinary QM, theories of QM_∞ can traffic in algebras none of whose countably additive states are pure. The chapter sketches and provides simple illustrations of the mathematical basis of this circumstance: atomless von Neumann algebras. It also illustrates the uses to which atomless von Neumann algebras are put in QM_∞.

This chapter and the next explore a response to a natural objection to the conclusion of Chapter 12. The natural objection is, based on models of steaming tea cups that idealize them as infinite in volume, the conclusion cuts no interpretive ice. The strategy underlying the response is to find in the physics of honestly infinite systems, like quantum fields, explanatory structures sharing the features of Chapter 12's model of phase structure on which that chapter's argument relied. Broken symmetry in quantum field theory could be such a structure. This chapter offers a bridge to that topic, in the form of an account of broken symmetry in quantum statistical mechanics, an account which, the chapter argues, shares the features which stymie rigid interpretation.