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Oxford University Press eBooks

A Matter of Degree

Laura Ruetsche

Published Jun 2, 2011• 1 citation
Publisher

Abstract

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.

DOI
10.1093/acprof:oso/9780199535408.003.0012

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If a classical system has infinitely many degrees of freedom, its Hamiltonian quantization need not be unique up to unitary equivalence. I sketch different approaches (Hilbert space and algebraic) to understanding the content of quantum theories in light of this non-uniqueness, and suggest that neither approach suffices to support explanatory aspirations encountered in the thermodynamic limit of quantum statistical mechanics.

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When quantum theory is applied to systems with an infinite number of degrees of freedom, wholly new interpretative issues arise. This is due to the fact that such theories have physically inequivalent instantiations. Here the interpretative issues of quantum field theory and of the quantum theory of systems at the thermodynamic limit are discussed. In quantum mechanics the so-called commutation operators on the basic descriptive operators fix the descriptive realm of the theory. This chapter addresses the question of how we are to understand this theory that seems to allow for distinct, incompatible ways of representing the basic features of the world.

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QM∞ includes quantum field theory and the thermodynamic limit of quantum statistical mechanics — theories which, unlike the 'ordinary' quantum theories typically discussed by philosophers, concern infinite systems. The mathematical framework appropriate for the presentation of a theory of ordinary quantum mechanics is essentially unique. For theories of QM∞, this is not so. This prompts interpretive questions—for instance, about what makes a quantum theory the quantum theory it is — this work aims to chronicle. Having introduced a formal apparatus (operator algebra theory) suited to pursue these questions, the book articulates a variety of accounts of the content of quantum theories, accounts responsive to QM∞'s characteristic non-uniqueness. To evaluate these accounts, the book examines QM∞ settings (e.g. spontaneous symmetry breaking, cosmological particle creation, superconductivity) in which that characteristic non-uniqueness seems to matter, with a view toward determining which accounts sustain the uses to which the non-uniqueness is put. This approach not only brings work on the foundations of quantum theories in contact with the foundational investigation of other sorts of physical theories (e.g., thermodynamics, statistical mechanics, solid state physics, general relativity, and cosmology), it also brings the philosophy of physics in contact with other sorts of philosophy of science (e.g., accounts of explanation, reduction, and explanationist defenses of scientific realism). The book concludes that received notions of physical content and physical modality must be revised if they are to apply usefully to particular physical theories.

Oxford University Press eBooks•Jun 2, 2011•315 citations

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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.

Oxford University Press eBooks•Jun 2, 2011•0 citations

Introduction

Igor F. Herbut

Phase transitions are defined, and the concepts of order parameter and spontaneously broken symmetry are discussed. Simple models for magnetic phase transitions are introduced, together with some experimental examples. Critical exponents and the notion of universality are defined, and the consequences of the scaling assumptions are derived. Phase transitions and order parameters It is a fact of everyday experience that matter in thermodynamic equilibrium exists in different macroscopic phases. Indeed, it is difficult to imagine life on Earth without all three phases of water. A typical sample of matter, for example, has the temperature–pressure phase diagram presented in Fig. 1.1: by changing either of the two parameters the system may be brought into a solid, liquid, or gas phase. The change of phase may be gradual or abrupt. In the latter case, the phase transition takes place at well defined values of the parameters that determine the phase boundary. Phase transitions are defined as points in the parameter space where the thermodynamic potential becomes non-analytic. Such a non-analyticity can arise only in the thermodynamic limit, when the size of the system is assumed to be infinite. In a finite system the partition function of any system is a finite sum of analytic functions of its parameters, and is therefore always analytic. A sharp phase transition is thus a mathematical idealization, albeit one that describes the reality extremely well. Macroscopic systems typically contain ∼ 10 23 degrees of freedom, and as such are very close to being in the thermodynamic limit. The phase boundaries in Fig. 1.1, for example, for this reason represent reproducible physical quantities.

Cambridge University Press eBooks•Jan 4, 2007•0 citations
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.
When quantum theory is applied to systems with an infinite number of degrees of freedom, wholly new interpretative issues arise. This is due to the fact that such theories have physically inequivalent instantiations. Here the interpretative issues of quantum field theory and of the quantum theory of systems at the thermodynamic limit are discussed. In quantum mechanics the so-called commutation operators on the basic descriptive operators fix the descriptive realm of the theory. This chapter addresses the question of how we are to understand this theory that seems to allow for distinct, incompatible ways of representing the basic features of the world.
Quantum systems with infinite degrees of freedom exhibit properties, like relaxation to equilibrium, phase-transitions and the existence of inequivalent representations of the CAR and CCR , that can satisfactorily be dealt with by means of the methods and techniques of algebraic quantum statistical mechanics [80, 81, 130]. The point of departure from standard quantum mechanics, is that in an infinite dimensional context one is usually provided with the algebraic properties of the relevant observables, but, in general, not with an a priori given representation on a Hilbert space; the latter rather depends on to the physical properties of the systems under consideration [324, 342, 343].
QM∞ includes quantum field theory and the thermodynamic limit of quantum statistical mechanics — theories which, unlike the 'ordinary' quantum theories typically discussed by philosophers, concern infinite systems. The mathematical framework appropriate for the presentation of a theory of ordinary quantum mechanics is essentially unique. For theories of QM∞, this is not so. This prompts interpretive questions—for instance, about what makes a quantum theory the quantum theory it is — this work aims to chronicle. Having introduced a formal apparatus (operator algebra theory) suited to pursue these questions, the book articulates a variety of accounts of the content of quantum theories, accounts responsive to QM∞'s characteristic non-uniqueness. To evaluate these accounts, the book examines QM∞ settings (e.g. spontaneous symmetry breaking, cosmological particle creation, superconductivity) in which that characteristic non-uniqueness seems to matter, with a view toward determining which accounts sustain the uses to which the non-uniqueness is put. This approach not only brings work on the foundations of quantum theories in contact with the foundational investigation of other sorts of physical theories (e.g., thermodynamics, statistical mechanics, solid state physics, general relativity, and cosmology), it also brings the philosophy of physics in contact with other sorts of philosophy of science (e.g., accounts of explanation, reduction, and explanationist defenses of scientific realism). The book concludes that received notions of physical content and physical modality must be revised if they are to apply usefully to particular physical theories.
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.
Phase transitions are defined, and the concepts of order parameter and spontaneously broken symmetry are discussed. Simple models for magnetic phase transitions are introduced, together with some experimental examples. Critical exponents and the notion of universality are defined, and the consequences of the scaling assumptions are derived. Phase transitions and order parameters It is a fact of everyday experience that matter in thermodynamic equilibrium exists in different macroscopic phases. Indeed, it is difficult to imagine life on Earth without all three phases of water. A typical sample of matter, for example, has the temperature–pressure phase diagram presented in Fig. 1.1: by changing either of the two parameters the system may be brought into a solid, liquid, or gas phase. The change of phase may be gradual or abrupt. In the latter case, the phase transition takes place at well defined values of the parameters that determine the phase boundary. Phase transitions are defined as points in the parameter space where the thermodynamic potential becomes non-analytic. Such a non-analyticity can arise only in the thermodynamic limit, when the size of the system is assumed to be infinite. In a finite system the partition function of any system is a finite sum of analytic functions of its parameters, and is therefore always analytic. A sharp phase transition is thus a mathematical idealization, albeit one that describes the reality extremely well. Macroscopic systems typically contain ∼ 10 23 degrees of freedom, and as such are very close to being in the thermodynamic limit. The phase boundaries in Fig. 1.1, for example, for this reason represent reproducible physical quantities.