Four years aga Walter Thirring suggested to me that it would be desirable to have a book describing recent results of the "algebraic approach" to quantum field theory and statistical mechanics. After long deliberations with my younger colleagues I decided to write a book but to enlarge the topic, the guiding line be­ ing expressed in the title "Local Quantum Physics". In essence this concerns the synthesis between special relativity and our understanding of quantum physics, together with a few other principles of a general nature. The algebraic approach, that is the characterization of the theory by a net of algebras of local observ­ ables, provides a concise language for this and an efficient tool for the study of the anatomy of the theory and of the relevance of various parts to qualita­ tive physical consequences. It is introduced in Chapter III. In compliance with the original suggestion its main results of more recent vintage are described in Chapters IV to VI. The first two chapters serve to place this material into context and make the book reasonably self contained. There is a rough tem­ poral order. Thus Chapter I briefly describes the pillars of the theory existing before 1950. Chapter II deals with progress in understanding and techniques in quantum field theory, achieved for the most part in the fifties and early sixties.

Inhalt

I. Background.- 1. Quantum Mechanics.- Basic concepts, mathematical structure, physical interpretation..- 2. The Principle of Locality in Classical Physics and the Relativity Theories.- Faraday's vision. Fields..- 2.1 Special relativity. Poincaré group. Lorentz group. Spinors. Conformai group..- 2.2 Maxwell theory..- 2.3 General relativity..- 3. Poincaré Invariant Quantum Theory.- 3.1 Geometric symmetries in quantum physics. Projective representations and the covering group..- 3.2 Wigner's analysis of irreducible, unitary representations of the Poincaré group..- 3.3 Single particle states. Spin..- 3.4 Many particle states: Bose-Fermi alternative, Fock space, creation operators. Separation of CM-motion..- 4. Action Principle.- Lagrangean. Double rôle of physical quantities. Peierls' direct definition of Poisson brackets. Relation between local conservation laws and symmetries..- 5. Basic Quantum Field Theory.- 5.1 Canonical quantization..- 5.2 Fields and particles..- 5.3 Free fields..- 5.4 The Maxwell-Dirac system. Gauge invariance..- 5.5 Processes..- II. General Quantum Field Theory.- 1. Mathematical Considerations and General Postulates.- 1.1 The representation problem..- 1.2 Wightman axioms..- 2. Hierarchies of Functions.- 2.1 Wightman functions, reconstruction theorem, analyticity in x-space..- 2.2 Truncated functions, clustering. Generating functionals and linked cluster theorem..- 2.3 Time ordered functions..- 2.4 Covariant perturbation theory, Feynman diagrams. Renormalization..- 2.5 Vertex functions and structure analysis..- 2.6 Retarded functions and analyticity in p-space..- 2.7 Schwinger functions and Osterwalder-Schrader theorem..- 3. Physical Interpretation in Terms of Particles.- 3.1 The particle picture: Asymptotic particle configurations and collision theory..- 3.2 Asymptotic fields. S-matrix..- 3.3 LSZ-formalism..- 4. General Collision Theory.- 4.1 Polynomial algebras of fields. Almost local operators..- 4.2 Construction of asymptotic particle states..- 4.3. Coincidence arrangements of detectors..- 4.4 Generalized LSZ-formalism..- 5. Some Consequences of the Postulates.- 5.1 CPT-operator. Spin-statistics theorem. CPT-theorem..- 5.2 Analyticity of the S-matrix..- 5.3 Reeh-Schlieder theorem..- 5.4 Additivity of the energy-momentum-spectrum..- 5.5 Borchers classes..- III. Algebras of Local Observables and Fields.- 1. Review of the Perspective.- Characterization of the theory by a net of local algebras. Bounded operators. Unobservable fields, superselection rules and the net of abstract algebras of observables. Transcription of the basic postulates..- 2. Von Neumann Algebras. C-Algebras. W-Algebras.- 2.1 Algebras of bounded operators. Concrete C-algebras and von Neumann algebras. Isomorphisms. Reduction. Factors. Classification of factors..- 2.2 Abstract algebras and their representations. Abstract C-algebras. Relation between the C-norm and the spectrum. Positive linear forms and states. The GNS-construction. Folia of states. Intertwiners. Primary states and cluster property. Purification. W-algebras..- 3. The Net of Algebras of Local Observables.- 3.1 Smoothness and integration. Local definiteness and local normality..- 3.2 Symmetries and symmetry breaking. Vacuum states. The spectral ideals..- 3.3 Summary of the structure..- 4. The Vacuum Sector.- 4.1 The orthocomplemented lattice of causally complete regions..- 4.2 The net of von Neumann algebras in the vacuum representation..- IV. Charges, Global Gauge Groups and Exchange Symmetry.- 1. Charge Superselection Sectors.- "Strange statistics". Charges. Selection criteria for relevant sectors. The program and survey of results..- 2. The DHR-Analysis.- 2.1 Localized morphisms..- 2.2 Intertwiners and exchange symmetry ("Statistics")..- 2.3 Charge conjugation, statistics parameter..- 2.4 Covariant sectors and energy-momentum spectrum..- 2.5 Fields and collision theory..- 3. The Buchholz-Fredenhagen-Analysis.- 3.1 Localized 1-particle states..- 3.2 BF-topological charges..- 3.3 Composition of sectors and exchange symmetry..- 3.4 Charge conjugation and the absence of "infinite statistics"..- 4. Global Gauge Group and Charge Carrying Fields.- The Abelian case. Endomorphisms and non-Abelian gauge group. DR-categories. Implementation of endomorphisms and the embedding theorem..- 5. Low Dimensional Space-Time and Braid Group Statistics.- Statistics operator and braid group representations. The 2+1-dimensional case with BF-charges. Statistics parameter and Jones index..- V. Thermal States and Modular Automorphisms.- 1. Gibbs Ensembles, Thermodynamic Limit, KMS-Condition.- 1.1 Introduction..- 1.2 Equivalence of KMS-condition to Gibbs ensembles for finite volume..- 1.3 The arguments for Gibbs ensembles..- 1.4 The representation induced by a KMS-state..- 1.5 Phases, symmetry breaking and the decomposition of KMS-states..- 1.6 Variational principles and autocorrelation inequalities..- 2. Modular Automorphisms and Modular Conjugation.- 2.1 The Tomita-Takesaki theorem..- 2.2 Vector representatives of states. Convex cones in ?..- 2.3 Relative modular operators and Radon-Nikodym derivatives..- 2.4 Classification of factors..- 3. Direct Characterization of Equilibrium States.- 3.1 Introduction..- 3.2 Stability..- 3.3 Passivity..- 3.4 Chemical potential..- 4. Modular Automorphisms of Local Algebras.- 4.1 The Bisognano-Wichmann theorem..- 4.2 Conformal invariance and the theorem of Hislop and Longo..- 5. Phase Space, Nuclearity, Split Property, Local Equilibrium.- 5.1 Introduction..- 5.2 Nuclearity and split property..- 5.3 Open subsystems..- 5.4 Modular nuclearity..- 6. The Universal Type of Local Algebras.- VI. Particles. Completeness of the Particle Picture.- 1. Detectors, Coincidence Arrangements, Cross Sections.- 1.1 Generalities..- 1.2 Asymptotic particle configurations. Buchholz's strategy..- 2. The Particle Content.- 2.1 Particles and infraparticles..- 2.2 Single particle weights and their decomposition..- 2.3 Remarks on the particle picture and its completeness..- 3. The Physical State Space of Quantum Electrodynamics.- VII. Retrospective and Outlook.- 1. Algebraic Approach vs. Euclidean Quantum Field Theory.- 2. Supersymmetry.- 3. The Challenge from General Relativity.- 3.1 Introduction..- 3.2 Quantum field theory in curved space-time..- 3.3 Hawking temperature and Hawking radiation..- 3.4 A few remarks on quantum gravity.- Author Index and References.