Mathematical Theory of Quantum Fields HUZIHIRO ARAKI Department of Mathematics Faculty of Science and Technology The Science University of Tokyo Japan Translated by Ursula Carow-Watamura Department of Physics Tohoku University Japan OXPORD UNIVERSITY PRESS CONTENTS 1 States and observables 1 1.1 Probabilistic description l 1.2 States and observables 2 1.3 Functions of observables 3 1.4 The expectation value of an observable 7 1.5 Mixture and pure states 8 1.6 The physical topology of the states 13 1.7 Equivalent theories 14 1.8 Symmetries 16 2 Quantum theory 19 2.1 The quantum mechanical description 19 2.2 Algebraic viewpoint 27 2.3 The representation associated with a state: GNS construction 33 2.4 The symmetries of quantum mechanics 42 2.5 Symmetries from the algebraic point of view 53 3 The relativistic symmetry 58 3.1 Minkowski space 58 3.2 The inhomogeneous Lorentz group 59 3.3 The relativistic symmetry in quantum mechanics 61 3.4 Irreducible representations and one-particle states 66 3.5 Free particle system and Fock space 70 3.6 Energy momentum 75 4 Local observables 78 4.1 General properties of local observables 78 4.2 The vacuum state 80 4.3 Irreducibility 84 4.4 Mass gap and exponential clustering property 89 4.5 The JLD representation 92 4.6 Truncated expectation values and multiple clustering property 97 4.7 Additivity assumption and Reeh-Schlieder theorem 100 4.8 Quantum field theory 102 4.9 From quantum fields to local observables 106 5 Scattering theory 108 5.1 The concept of scattering states and the S-matrix 108 5.2 Description of asymptotic states 111 Xll CONTENTS 5.3 Construction of asymptotic states 113 5.4 The counter interpretation of asymptotic states—verification of an appropriate asymptotic behaviour 128 5.5 Asymptotic conditions and reduction formula for the S-matrix 133 5.6 Analyticity and TCP symmetry 151 6 Sector theory 160 6.1 Superselection rules and localized excitations 160 6.2 Localized endomorphisms and sectors 166 6.3 Permutation of excitations and statistics of a sector 171 6.4 Charge conjugate sector 181 6.5 Operator algebra system of fields and gauge groups 184 6.6 Theory of local charges 187 Appendix A: Hilbert space and operators 193 A.l Hilbert space 193 A.2 Pre-Hilbert space and completion 194 A.3 Orthonormal basis 195 A.4 Unitary and antiunitary maps 196 A.5 Subspaces and projection operators 198 A.6 Direct sum and direct integral 199 A.7 Spectral decomposition 201 A.8 Trace 203 A.9 Unbounded operators 204 A. 10 One-parameter group of unitary operators 205 A. 11 Tensor product 206 Appendix B: Operator algebras 208 B.l C* algebras 208 B.2 Von Neumann algebras 210 Appendix C: Free fields 216 C.l Charged scalar field 216 C.2 One particle system with positive mass and arbitrary spin 217 C.3 Examples of free fields with positive mass and integer spin 218 C.4 Examples of free fields with positive mass and half odd integer spin 220 General literature and references 222 References for Chapter 1 224 References for Chapter 2 224 References for Chapter 3 225 References for Chapter 4 226 References for Chapter 5 227 References for Chapter 6 230 Literature for appendices 232 Index 233 .