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Algebraic Quantum Field Theory∗

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Algebraic Quantum Field Theory∗ Algebraic Quantum Field Theory∗ Hans Halvorson† (with an appendix by Michael M¨uger‡) February 14, 2006 Abstract Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out conse- quences of this structure by means of various mathematical tools — the theory of operator algebras, category theory, etc.. Given the rigor and generality of AQFT, it is a particularly apt tool for studying the foundations of QFT. This paper is a survey of AQFT, with an orientation towards foundational topics. In addition to covering the basics of the theory, we discuss issues related to nonlocality, the particle concept, the field concept, and inequivalent represen- tations. We also provide a detailed account of the analysis of superselection rules by Doplicher, Haag, and Roberts (DHR); and we give an alternative proof of Doplicher and Robert’s reconstruction of fields and gauge group from the category of physical representations of the observable algebra. The latter is based on unpublished ideas due to J. E. Roberts and the abstract duality theo- rem for symmetric tensor ∗-categories, a self-contained proof of which is given in the appendix. 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 ∗Forthcoming in Handbook of the Philosophy of Physics, edited by Jeremy Butterfield and John Earman. HH wishes to thank: Michael M¨uger for teaching him about the Doplicher-Roberts Theorem; the editors for their helpful feedback and patience; and David Baker, Tracy Lupher, and David Malament for corrections. MM wishes to thank Julien Bichon for a critical reading of the appendix and useful comments. †Department of Philosophy, Princeton University, Princeton NJ, US. [email protected] ‡Department of Mathematics, Radboud University, Nijmegen, NL. [email protected] 1 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 ∆ of Localized Transportable Endomorphisms 59 8.1 ∆ 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´ecovariance of the field net . 107 10.5 Uniqueness of the field net . 109 2 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 Introduction From the title of this Chapter, one might suspect that the subject is some idiosyn- cratic approach to quantum field theory (QFT). The approach is indeed idiosyn- cratic in the sense of demographics: only a small proportion of those who work on QFT work on algebraic QFT (AQFT). However, there are particular reasons why philosophers, and others interested in foundational issues, will want to study the “algebraic” approach. In philosophy of science in the analytic tradition, studying the foundations of a theory T has been thought to presuppose some minimal level of clarity about the 3 referent of T . (Moreover, to distinguish philosophy from sociology and history, T is not taken to refer to the activities of some group of people.) In the early twentieth century, it was thought that the referent of T must be a set of axioms of some formal, preferably first-order, language. It was quickly realized that not many interesting physical theories can be formalized in this way. But in any case, we are no longer in the grip of axiomania, as Feyerabend called it. So, the standards were loosened somewhat — but only to the extent that the standards were simultaneously loosened within the community of professional mathematicians. There remains an implicit working assumption among many philosophers that studying the foundations of a theory requires that the theory has a mathematical description. (The philosopher’s working assumption is certainly satisfied in the case of statistical mechanics, special and general relativity, and nonrelativistic quantum mechanics.) In any case, whether or not having a mathematical description is mandatory, having such a description greatly facilitates our ability to draw inferences securely and efficiently. So, philosophers of physics have taken their object of study to be theories, where theories correspond to mathematical objects (perhaps sets of models). But it is not so clear where “quantum field theory” can be located in the mathematical universe. In the absence of some sort of mathematically intelligible description of QFT, the philosopher of physics has two options: either find a new way to understand the task of interpretation, or remain silent about the interpretation of quantum field theory.1 It is for this reason that AQFT is of particular interest for the foundations of quantum field theory. In short, AQFT is our best story about where QFT lives in the mathematical universe, and so is a natural starting point for foundational inquiries. 1 Algebraic Prolegomena This first section provides a minimal overview of the mathematical prerequisites of the remainder of the Chapter. 1.1 von Neumann algebras The standard definition of a von Neumann algebra involves reference to a topology, and it is then shown (by von Neumann’s double commutant theorem) that this topological condition coincides with an algebraic condition (condition 2 in the Def- inition 1.2). But for present purposes, it will suffice to take the algebraic condition as basic. 1.1 Definition. Let H be a Hilbert space. Let B(H) be the set of bounded linear operators on H in the sense that for each A ∈ B(H) there is a smallest nonnegative 1For the first option, see [Wallace, forthcoming]. 4 number kAk such that hAx, Axi1/2 ≤ kAk for all unit vectors x ∈ H. [Subsequently we use k · k ambiguously for the norm on H and the norm on B(H).] We use juxtaposition AB to denote the composition of two elements A, B of B(H). For each A ∈ B(H) we let A∗ denote the unique element of B(H) such that hA∗x, yi = hx, Ayi, for all x, y ∈ R. 1.2 Definition. Let R be a ∗-subalgebra of B(H), the bounded operators on the Hilbert space H. Then R is a von Neumann algebra if 1. I ∈ R, 2. (R0)0 = R, where R0 = {B ∈ B(H):[B, A] = 0, ∀A ∈ R}. 1.3 Definition. We will need four standard topologies on the set B(H) of bounded linear operators on H. Each of these topologies is defined in terms of a family of seminorms — see [Kadison and Ringrose, 1997, Chaps. 1,5] for more details. • The uniform topology on B(H) is defined in terms of a single norm: kAk = sup{kAvk : v ∈ H, kvk ≤ 1}, where the norm on the right is the given vector norm on H. Hence, an operator A is a limit point of the sequence (Ai)i∈N iff (kAi − Ak)i∈N converges to 0.
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