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Algebraic quantum field theory 2006
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Hans Halvorson† with an appendix by Michael M¨uger February 14, 2006 Contents 1 Algebraic Prolegomena 4 1.1 von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 C∗-algebras and their representations . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Type classification of von Neumann algebras . . . . . . . . . . . . . . . . . 8 1.4 Modular theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Structure of the Net of Observable Algebras 13 2.1 Nets of algebras, basic properties . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Existence/uniqueness of vacuum states/representations . . . . . . . . . . . . 14 2.3 The Reeh-Schlieder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 The funnel property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5 Type of local algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Nonlocality and Open Systems in AQFT 25 3.1 Independence of C∗ and von Neumann algebras . . . . . . . . . . . . . . . . 26 3.2 Independence of local algebras . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Bell correlation between von Neumann algebras . . . . . . . . . . . . . . . . 29 3.4 Intrinsically entangled states . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4 Prospects for Particles 32 4.1 Particles from Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2 Fock space from the algebra of observables . . . . . . . . . . . . . . . . . . . 33 4.3 Nonuniqueness of particle interpretations . . . . . . . . . . . . . . . . . . . 35 4.4 Problems for localized particles . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.5 Particle interpretations generalized: Scattering theory and beyond . . . . . 36 5 The Problem of Value-Definiteness in AQFT 37 5.1 Clifton-Kitajima classification of modal algebras . . . . . . . . . . . . . . . 38 5.2 What is a symmetry in AQFT? . . . . . . . . . . . . . . . . . . . . . . . . . 40 6 Quantum Fields and Spacetime Points 42 6.1 No Go theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.2 Go theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.3 Field interpretations of QFT . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.4 Points of time? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7 The Problem of Inequivalent Representations 54 7.1 Superselection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.2 Minimal assumptions about the algebra of observables . . . . . . . . . . . . 57 8 The Category Δ of Localized Transportable Endomorphisms 59 8.1 Δ is a braided tensor ∗-category . . . . . . . . . . . . . . . . . . . . . . . . 68 8.2 Relation between localized endomorphisms and representations . . . . . . . 74 8.3 Dimension theory in tensor ∗-categories . . . . . . . . . . . . . . . . . . . . 77 8.4 Covariant representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.5 Statistics in braided tensor ∗-categories . . . . . . . . . . . . . . . . . . . . 81 9 From Fields to Representations 82 10 From Representations to Fields 90 10.1 Supermathematics and the embedding theorem . . . . . . . . . . . . . . . . 92 10.2 Construction of the field net, algebraic . . . . . . . . . . . . . . . . . . . . . 94 10.3 Completion of the field net . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 10.4 Poincar´e covariance of the field net . . . . . . . . . . . . . . . . . . . . . . . 107 10.5 Uniqueness of the field net . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 10.6 Further relations between A and F, and a Galois interpretation . . . . . . . 114 10.7 Spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . 116 11 Foundational Implications of the Reconstruction Theorem 117 11.1 Algebraic Imperialism and Hilbert Space Conservatism . . . . . . . . . . . . 118 11.2 Explanatory relations between representations . . . . . . . . . . . . . . . . . 120 11.3 Fields as theoretical entities, as surplus structure . . . . . . . . . . . . . . . 122 11.4 Statistics, permutation symmetry, and identical particles . . . . . . . . . . . 125 Bibliography 133 Appendix (by Michael M¨uger) 144 A Categorical Preliminaries 144 A.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 A.2 Tensor categories and braidings . . . . . . . . . . . . . . . . . . . . . . . . . 145 A.3 Graphical notation for tensor categories . . . . . . . . . . . . . . . . . . . . 149 A.4 Additive, C-linear and ∗-categories . . . . . . . . . . . . . . . . . . . . . . . 149 A.5 Abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 A.6 Commutative algebra in abelian symmetric tensor categories . . . . . . . . 157 A.7 Inductive limits and the Ind-category . . . . . . . . . . . . . . . . . . . . . . 161 B Abstract Duality Theory for Symmetric Tensor ∗-Categories 162 B.1 Fiber functors and the concrete Tannaka theorem. Part I . . . . . . . . . . 163 B.2 Compact supergroups and the abstract Tannaka theorem . . . . . . . . . . 165 B.3 Certain algebras arising from fiber functors . . . . . . . . . . . . . . . . . . 168 B.4 Uniqueness of fiber functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 B.5 The concrete Tannaka theorem. Part II . . . . . . . . . . . . . . . . . . . . 175 B.6 Making a symmetric fiber functor ∗-preserving . . . . . . . . . . . . . . . . 177 B.7 Reduction to finitely generated categories . . . . . . . . . . . . . . . . . . . 181 B.8 Fiber functors from monoids . . . . . . . . . . . . . . . . . . . . . . . . . . 183 B.9 Symmetric group action, determinants and integrality of dimensions . . . . 186 B.10 The symmetric algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 B.11 Construction of an absorbing commutative monoid . . . . . . . . . . . . . . 194 B.12 Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Download link：http://www.isload.com.cn/store/gdwk3s3hbgho2 [ Last edited by zhq025 on 2008-1-16 at 10:43 ] |
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