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Overview
The quantum theory of macroscopic systems is a vast, everdeveloping area of science that serves to relate the properties of complex physical objects to those of their constituent particles. Its essential challenge is that of finding the conceptual structures needed for the description of the various states of organization of manyparticle quantum systems. In this book, Geoffrey Sewell provides a new approach to the subject, based on a "macrostatistical mechanics," which contrasts sharply with the standard microscopic treatments of manybody problems.
Sewell begins by presenting the operator algebraic framework for the theory. He then undertakes a macrostatistical treatment of both equilibrium and nonequilibrium thermodynamics, which yields a major new characterization of a complete set of thermodynamic variables and a nonlinear generalization of the Onsager theory. The remainder of the book focuses on ordered and chaotic structures that arise in some key areas of condensed matter physics. This includes a general derivation of superconductive electrodynamics from the assumptions of offdiagonal longrange order, gauge covariance, and thermodynamic stability, which avoids the enormous complications of the microscopic treatments. Sewell also unveils a theoretical framework for phase transitions far from thermal equilibrium. Throughout, the mathematics is kept clear without sacrificing rigor.
Representing a coherent approach to the vast problem of the emergence of macroscopic phenomena from quantum mechanics, this wellwritten book is addressed to physicists, mathematicians, and other scientists interested in quantum theory, statistical physics, thermodynamics, and general questions of order and chaos.
Editorial Reviews
Mathematical Reviews
A clear, wellpaced and compact exposition which, through a nice intertwining of physics and mathematics, leaves the reader with a rather complete grasp of the beautiful theoretical construction that goes from the algebraic quantum mechanical framework to thermodynamics, phase transitions and dynamical phase transitions. . . . It offers a road map to a number of central problems in mathematical statistical mechanics. It offers paved access to fascinating physics and mathematics.— Roberto Fernandez
Philosophy of Science
Sewell's book begins with a selfcontained introduction to algebraic quantum theory (especially of infinite systems); and this, together with the fact that Sewell always develops only as much mathematics as he needs for his physics, means that his 300 page book provides a masterly overview of his field.— Jeremy Butterfield
Mathematical Reviews  Roberto Fernandez
A clear, wellpaced and compact exposition which, through a nice intertwining of physics and mathematics, leaves the reader with a rather complete grasp of the beautiful theoretical construction that goes from the algebraic quantum mechanical framework to thermodynamics, phase transitions and dynamical phase transitions. . . . It offers a road map to a number of central problems in mathematical statistical mechanics. It offers paved access to fascinating physics and mathematics.Philosophy of Science  Jeremy Butterfield
Sewell's book begins with a selfcontained introduction to algebraic quantum theory (especially of infinite systems); and this, together with the fact that Sewell always develops only as much mathematics as he needs for his physics, means that his 300 page book provides a masterly overview of his field.From the Publisher
"A clear, wellpaced and compact exposition which, through a nice intertwining of physics and mathematics, leaves the reader with a rather complete grasp of the beautiful theoretical construction that goes from the algebraic quantum mechanical framework to thermodynamics, phase transitions and dynamical phase transitions. . . . It offers a road map to a number of central problems in mathematical statistical mechanics. It offers paved access to fascinating physics and mathematics."—Roberto Fernandez, Mathematical Reviews"Sewell's book begins with a selfcontained introduction to algebraic quantum theory (especially of infinite systems); and this, together with the fact that Sewell always develops only as much mathematics as he needs for his physics, means that his 300 page book provides a masterly overview of his field."—Jeremy Butterfield, Philosophy of Science
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Chapter 1
Introductory discussion of quantum macrophysics
Quantum theory began with Planck's [Pl] derivation of the thermodynamics of black body radiation from the hypothesis that the action of his oscillator model of matter was quantised in integral multiples of a fundamental constant, h. This result provided a microscopic theory of a macroscopic phenomenon that was incompatible with the assumption of underlying classical laws. In the century following Planck's discovery, it became abundantly clear that quantum theory is essential to natural phenomena on both the microscopic and macroscopic scales. Its crucial role in determining the gross properties of matter is evident from the following considerations.
The stability of matter against electromagnetic collapse is effected only by the combined action of the Heisenberg and Pauli principles [DL, LT, LLS, BFG].
The third law of thermodynamics is quintessentially quantum mechanical and, arguably, so too is the second law.^{1}
The mechanisms governing a vast variety of cooperative phenomena, including magnetic ordering [Ma],superfluidity [La1, BCS] and optical and biological coherence [Ha1, Fr1], are of quantum origin.
As a first step towards contemplating the quantum mechanical basis of macrophysics, we note the empirical fact that macroscopic systems enjoy properties that are radically different from those of their constituent particles. Thus, unlike systems of few particles, they exhibit irreversible dynamics, phase transitions and various ordered structures, including those characteristic of life. These and other macroscopic phenomena signify that complex systems, that is, ones consisting of enormous numbers of interacting particles, are qualitatively different from the sums of their constituent parts. Correspondingly, theories of such phenomena must be based not only on quantum mechanics per se but also on conceptual structures that serve to represent the characteristic features of highly complex systems. Among the key general concepts involved here are ones representing various types of order, or organisation, disorder, or chaos, and different levels of macroscopicality. Moreover, the particular concepts required to describe the ordered structures of superfluids and laser light are represented by macroscopic wave functions [PO, Ya, GH, Se1] that are strictly quantum mechanical, although radically different from the Schrödinger wave functions of microphysics.
To provide a mathematical framework for the conceptual structures required for quantum macrophysics, it is clear that one needs to go beyond the traditional form of quantum mechanics [Di, VN1], since that does not discriminate qualitatively between microscopic and macroscopic systems. This may be seen from the fact that the traditional theory serves to represent a system of N particles within the standard Hilbert space scheme, which takes the same form regardless of whether N is 'small' or 'large'. In fact, it was this very lack of a sharp characterisation of macroscopicality that forced Bohr [Bo] into a dualistic treatment of the measuring process, in which the microscopic system under observation was taken to be quantum mechanical, whereas the macroscopic measuring apparatus was treated as classical, even though it too was presumably subject to quantum laws.
However, a generalised version of quantum mechanics that provides the required qualitative distinctions between different grades of macroscopicality has been devised over the last three decades, on the basis of an idealisation of macroscopic systems as ones possessing infinite numbers of degrees of freedom. This kind of idealisation has, of course, long been essential to statistical thermodynamics, where, for example, the characterisation of phase transitions by singularities in thermodynamical potentials necessitates a passage to the mathematical limit in which both the volume and the number of particles of a system tend to infinity in such a way that the density remains finite [YL, LY, Ru1]. Its extension to the full description of the observables and states of macroscopic systems [AW, HHW, Ru1, Em1] has served to replace the merely quantitative difference between systems of 'few' and 'many' (typically 10^{24}) particles by the qualitative distinction between finite and infinite ones, and has thereby brought new, physically relevant structures into the theory of collective phenomena [Th, Se2].
The key element of the generalisation of quantum mechanics to infinite systems is that it is based on the algebraic structure of the observables, rather than on the underlying Hilbert space [Seg, HK]. The radical significance of this is that, whereas the algebra of observables of a finite system, as governed by the canonical commutation relations, admits only one irreducible Hilbert space representation [VN2], that of an infinite system has infinitely many inequivalent such representations [GW]. Thus, for a finite system, the algebraic and Hilbert space descriptions are equivalent, while, for an infinite one, the algebraic picture is richer than that provided by any irreducible representation of its observables.
Moreover, the algebraic quantum theory of infinite systems, as cast in a form designed for the treatment of fundamental problems in statistical mechanics and quantum field theory [Em1, BR, Th, Se2, Haa1], admits just the structures required for the treatment of macroscopic phenomena. In particular, it permits clear definitions of various kinds of order, as well as sharp distinctions between global and local variables, which may naturally be identified with macroscopic and microscopic ones. Furthermore, the wealth of inequivalent representations of the observables permits a natural classification of the states in both microscopic and macroscopic terms. To be specific, the vectors in a representation space^{2} correspond to states that are macroscopically equivalent but microscopically different, while those carried by different representations are macroscopically distinct. Hence, the macrostate corresponds to a representation and the microstate to a vector in the representation space. This is of crucial significance not only for the description of the various phases of matter, but also for the quantum theory of measurement. The specification of the states of a measuring apparatus in microscopic and macroscopic terms has provided a key element of a fully quantum treatment [He, WE] of the measurement process that liberates the theory from Bohr's dualism.
Our approach to the basic problem of how macrophysics emerges from quantum mechanics will be centred on macroscopic observables, our main objective being to obtain the properties imposed on them by general demands of quantum theory and manyparticle statistics. This approach has classic precedents in Onsager's [On] irreversible thermodynamics and Landau's fluctuating hydrodynamics [LL1], and is at the opposite pole from the manybodytheoretic computations of condensed matter physics [Pi, Tho]. Our motivation for pursuing this approach stems from the following two considerations. Firstly, since the observed laws of macrophysics have relatively simple structures, which do not depend on microscopic details, it is natural to seek derivations of these laws that are based on general quantum macrostatistical arguments. Secondly, by contrast, the microscopic properties of complex systems are dominated by the molecular chaos that is at the heart of statistical physics; and presumably, this chaos would render unintelligible any solutions of the microscopic equations of motion of realistic models of such systems, even if these could be obtained with the aid of supercomputers.
Thus, we base this treatise on macroscopic observables and certain general structures of complex systems, as formulated within the terms of the algebraic framework of quantum theory. The next three chapters are devoted to a concise formulation of this framework, for both conservative and open systems (Chapter 2), and of the descriptions that it admits of symmetry, order and disorder (Chapter 3), and of irreversibility (Chapter 4).
Table of Contents
Preface ix
Notation xi
Part I. The Algebraic Quantum Mechanical Framework and the Description of Order, Disorder and Irreversibility in Macroscopic Systems: Prospectus 1
Chapter 1. Introductory Discussion of Quantum Macrophysics 3
Chapter 2. The Generalised Quantum Mechanical Framework 7
2.1. Observables, States, Dynamics 8
2.2. Finite Quantum Systems 8
2.2.1. Uniqueness of the Representation 8
2.2.2. The Generic Model 10
2.2.3. The Algebraic Picture 13
2.3. Infinite Systems: Inequivalent Representations 15
2.3.1. The Representation o(+) 15
2.3.2. The Representation o() 17
2.3.3. Inequivalence of o:(+) 17
2.3.4. Other Inequivalent Representations 18
2.4. Operator Algebraic Interlude 18
2.4.1. Algebras: Basic Definitions and Properties 18
2.4.2. States and Representations 21
2.4.3. Automorphisms and Antiautomorphisms 24
2.4.4. Tensor Products 26
2.4.5. Quantum Dynamical Systems 27
2.4.6. Derivations of *Algebras and Generators of Dynamical Groups 28
2.5. Algebraic Formulation of Infinite Systems 29
2.5.1. The General Scheme 29
2.5.2. Construction of the Lattice Model 32
2.5.3. Construction of the Continuum Model 34
2.6. The Physical Picture 39
2.6.1. Normal Folia as Local Modifications of Single States 39
2.6.2. Spacetranslationally Invariant States 39
2.6.3. Primary States have Short Range Correlations 40
2.6.4. Decay of Time Correlations and Irreversibility 41
2.6.5. Global Macroscopic Observables 42
2.6.6. Consideration of Pure Phases 44
2.6.7. Fluctuations and Mesoscopic Observables 45
2.7. Open Systems 46
2.8. Concluding Remarks 47
Appendix A: filbert Spaces 48
Chapter 3. On Symmetry, Entropy and Order 57
3.1. Symmetry Groups 57
3.2. Entropy 58
3.2.1. Classical Preliminaries 58
3.2.2. Finite Quantum Systems 59
3.2.3. Infinite Systems 62
3.2.4. On Entropy and Disorder 64
3.3. Order and Coherence 65
3.3.1 Order and Symmetry 65
3.3.2. Coherence 68
3.3.3. Long Range Correlations in Ginvariant Mixtures of Ordered Phases 69
3.3.4 Superfluidity and Offdiagonal Long Range Order 70
3.3.5. On Entropy and Order 72
3.4. Further Discussion of Order and Disorder 72
Chapter 4. Reversibility, Irreversibilty and Macroscopic Causality 75
4.1. Microscopic Reversibility 76
4.1.1. Finite Systems 76
4.1.2. Infinite Systems 78
4.2. From Systems to Subsystems: Completely Positive Maps, Quantum Dynamical Semigroups and Conditional Expectations 79
4.2.1. Complete Positivity 79
4.2.2. Quantum Dynamical Semigroups 81
4.2.3. Conditional Expectations 82
4.3. Induced Dynamical Subsystems 83
4.4. Irreversibility 83
4.4.1. Irreversibility, Mixing and Markovian Dynamics 83
4.5. Note on Classical Macroscopic Casuality 86
Appendix A: Example of a Positive Map that is not Completely Posistive 88
Appendix B. Simple Model of Irrversibilty and Mixing 89
Appendix C. Simple Model of Irreversibilty and Macroscopic Casuality 94
C.1. The Model 94
C.2. Equations of Motion 98
C.3. Macroscopic Description of B 100
C.4. The Phenomenological Law 102
C.5. The Fluctuation Process 103
Part II. From Quantum Statistics to Equilibrium and Nonequilibrium Thermodynamics: Prospectus 107
Chapter 5. Thermal Equilibrium States and Phases 109
5.1. Introduction 109
5.2. Finite Systems 11l
5.2.1. Equilibrium, Linear Response Theory and the KMS Conditions 111
5.2.2. Equilibrium and Thermodynamical Stability 112
5.2.3. Resume 112
5.3. Infinite Systems 113
5.3.1. The KMS Conditions 113
5.3.2. Thermodynamical Stability Conditions 118
5.4. Equilibrium and Metastable States 123
5.4.1. Equilibrium States 123
5.4.2. Metastable States 124
5.5. Further Discussion 125
Chapter 6. Equilibrium Thermodynamics and Phase Structure 127
6.1. Introduction 127
6.2. Preliminaries on Convexity 131
6.3. Thermodynamic States as Tangents to the Reduced Pressure Function 135
6.4. Quantum Statistical Basis of Thermodynamics 136
6.5. An Extended Thermodynamics with Order Parameters 142
6.6. Concluding Remarks on the Paucity of Thermodynamical Variables 144
Appendix A: Proofs of Propositions 6.4.1 and 6.4.2 145
Appendix B: Functionals q as Space Averages of Locally Conserved Quantum Fields 146
Chapter 7. Macrostatistics and Nonequilibrium Thermodynamics 149
7.1. Introduction 149
7.2. The Quantum Field q(x) 153
7.3. The Macroscopic Model, M 155
7.4. Relationship between the Classical Field q and the Quantum Field q 158
7.5. The Model M(flunt) 161
7.6. The Linear Regime: Macroscopic Equilibrium Conditions and the Onsager Relations 164
7.7. The Nonlinear Regime: Local Equilibrium and Generalized Onsager Relations 165
7.8. Further Considerations: Towards a Generalization of the Theory to Galilean Continuum Mechanics 168
Appendix A: Tempered Distributions 170
Appendix B: Classical Stochastic Processes and the Construction of M(flunt) as a Classical Markov Field 176
B.1. Algebraic Description of Classical Stochastic Processes 176
B.2. Classical Gaussian Fields 178
B.3. Proof of Propositions 7.5.1 and 7.5.2 183
Appendix C: Equilibrium Correlations and The Static TwoPoint Function 183
C.1. The Truncated Static TwoPoint Function 184
C.2. Quantum Statistical Formulation of s"(q) 186
C.3. Formulation of n" via Perturbations of po 187
C.4. Proof of Propositions C.3.1 and C.3.2 for Lattice Systems with Finite Range Interactions 192
C.5. Pure Crystalline Phases 195
Part III. Superconductive Electrodynamics as a Consequence of Offdiagonal Long Range Order, Gauge Covariance and Thermodynamical Stability: Prospectus 197
Chapter 8. Brief Historical Survey of Theories of Superconductivity 199
Chapter 9. Offdiagonal Long Range Order and Superconductive Electrodynamics 211
9.1. Introduction 211
9.2. The General Model 213
9.3. ODLRO versus Magnetic Induction 218
9.4. Statistical Thermodynamics of the Model and the Meissner Effect 221
9.4.1 The Equilibrium States 221
9.4.2 Thermodynamical Potentials 222
9.5. Flux Quantisation 226
9.6. Metastability of Supercurrents and Superselection Rules 229
9.7. Note on Type II Superconductors 234
9.8. Concluding Remarks 236
Appendix A: Vector Potentials Representing Magnetic Fields with Compact Support 236
Part IV. Ordered and Chaotic Structures Far from Equilibrium: Prospectus 239
Chapter 10. Schematic Approach to a Theory of Nonequlibrium Phase Transitions, Order and Chaos 241
Chapter 11. Laser Model as a Paradigm of Nonequilibrium Phase Structures 247
11.1. Introduction 247
11.2. The Model 248
11.3. The Macroscopic Dynamics 256
11.4. The Dynamical Phase Transitions 260
11.5. The Microscopic Dynamics 264
11.6. A Nonequilibrium Maximum Entropy Principle 269
11.7. Concluding Remarks 271
Appendix A: Proof of Lemma 11.5.2 and Proposition 11.5.4 271
References 275
Index 287
Recipe
"A beautifully written book: the physics is well described, the mathematics is precise, and the exposition is concise. Sewell achieves his stated purpose—namely, to offer a panorama of the current state of the problem of how macroscopic phenomena can be interpreted from the laws and structures of microphysics."—Gerard G. Emch, University of Florida