Local Quantum Physics

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. Conformal 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 Poincare 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.- Implementation of endomorphisms. Charges with d = 1. Endomorphisms and non Abelian gauge group. DR categories 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 H..- 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. Principles and Lessons of Quantum Physics. A Review of Interpretations, Mathematical Formalism and Perspectives.- 1. The Copenhagen Spirit. Criticisms, Elaborations.- Niels Bohr's epistemological considerations. Realism. Physical systems and the division problem. Persistent non-classical correlations. Collective coordinates, decoherence and the classical approximation. Measurements. Correspondence and quantization. Time reflection asymmetry of statistical conclusions..- 2. The Mathematical Formalism.- Operational assumptions. "Quantum Logic". Convex cones..- 3. The Evolutionary Picture.- Events, causal links and their attributes. Irreversibility. The EPR-effect. Ensembles vs. individuals. Decisions. Comparison with standard procedure..- VIII. 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.