## Operator Algebras and Quantum Statistical Mechanics: Equilibrium States. Models in Quantum Statistical MechanicsFor almost two decades this has been the classical textbook on applications of operator algebra theory to quantum statistical physics. It describes the general structure of equilibrium states, the KMS-condition and stability, quantum spin systems and continuous systems. Major changes in the new edition relate to Bose--Einstein condensation, the dynamics of the X-Y model and questions on phase transitions. Notes and remarks have been considerably augmented. |

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### Contents

51 Ιntroduction | 3 |

52 Continous Quantum System I | 6 |

522 The CAR and CCR Algebras | 15 |

523 States and Representatives | 23 |

524 The Ideal Fermi Gas | 45 |

525 Τhe Ideal Bose Gas | 57 |

53 KMSStates | 76 |

532 The Set of KMS States | 112 |

623 The Maximum Entropy Principle | 266 |

624 Translationally Invariant States | 286 |

625 Uniqueness of KMS States | 306 |

626 Nonuniqueness of KMS States | 317 |

627 Ground States | 338 |

63 Continuous Quantum Systems II | 353 |

631 Τhe Local Hamiltonians | 355 |

632 Τhe Wiener Intergral | 366 |

533 The Set of Ground States | 131 |

54 Stability and Equilibrium | 144 |

542 Stability and the KMS Condtion | 176 |

543 Gauge Groups and the Chemical Potential | 197 |

544 Passive Systems | 211 |

Note and Remarks | 217 |

Models of Quantum Statistical Mechanics | 235 |

61 Introduction | 237 |

62 Quantum Spin System | 239 |

622 The Gibbs Condition for Equilibrium | 261 |

633 Τhe Thermodynamic Limit I the Reduced Density Matrices | 381 |

634 The Thermodynamic Limit II States and Greens Functions | 395 |

64 Conclusion | 422 |

Notes and Remarks | 424 |

References | 463 |

Book and Monographs | 465 |

Articles | 468 |

List of Symbols | 487 |

499 | |

### Common terms and phrases

Araki argument assume asymptotic abelianness automorphism group Bogoliubov Bogoliubov transformations Bose gas bounded C*-algebra C*-dynamical system CCR algebra classical commutation compact convergence convex Corollary corresponding decomposition defined denote dense density matrices derived Dirichlet boundary conditions dynamics eigenvalues elements energy equilibrium ergodic established Example exists external field Fermi ferromagnetic finite finite-volume Fock space function Gibbs condition Green's functions ground Hamiltonian hence Hilbert space identity implies inequality integral interaction Ising model KMS condition lattice Lemma linear Math matrix measure modular Neumann algebra norm Notes and Remarks Observation one-parameter group particles perturbation Phys positive proof of Theorem properties Proposition prove quantum spin quantum spin systems quasi-free reduced density matrices relation representation result satisfies Section selfadjoint selfadjoint operator stability statistical mechanics subalgebra subset subspace symmetry t-invariant t-KMS temperature theory thermodynamic limit topology translationally invariant unique vector von Neumann algebra zero