Commentaries.- A Survey by the Editors.- I. Thermodynamic Limit for Coulomb Systems.- I. 1 Existence of Thermodynamics for Real Matter with Coulomb Forces.- I. 2 The Constitution of Matter: Existence of Thermodynamics for Systems Composed of Electrons and Nuclei.- II. Hard Sphere Virial Coefficients.- II. 1 Suppression at High Temperature of Effects Due to Statistics in the Second Virial Coefficient of a Real Gas.- II. 2 Calculation of Exchange Second Virial Coefficient of a Hard-Sphere Gas by Path Integrals.- III. Zeros of Partition Functions.- III. 1 Monomers and Dimers.- III. 2 Theory of Monomer-Dimer Systems.- III. 3 A Property of Zeros of the Partition Function for Ising Spin Systems.- III. 4 A General Lee-Yang Theorem for One-Component and Multicomponent Ferromagnets.- IV. Reflection Positivity.- IV. 1 Existence of Phase Transitions for Anisotropic Heisenberg Models.- IV. 2 Phase Transitions in Anisotropic Lattice Spin Systems.- IV. 3 Phase Transitions in Quantum Spin Systems with Isotropic and Nonisotropic Interactions.- IV. 4 Phase Transitions and Reflection Positivity. I. General Theory and Long Range Lattice Models.- IV. 5 Phase Transitions and Reflection Positivity. II. Lattice Systems with Short-Range and Coulomb Interactions.- IV. 6 Lattice Models for Liquid Crystals.- IV. 7 Existence of Neel Order in Some Spin-1/2 Heisenberg Antiferromagnets.- IV. 8 The XY Model Has Long-Range Order for all Spins and all Dimensions Greater than One.- V. Classical Thermodynamics.- V. 1 The Third Law of Thermodynamics and the Degeneracy of the Ground State for Lattice Systems.- V. 2 A Guide to Entropy and the Second Law of Thermodynamics.- V. 3 A Fresh Look at Entropy and the Second Law of Thermodynamics.- VI. Lattice Systems.- VI. 1 Properties of a Harmonic Crystal in a Stationary Nonequilibrium State.- VI. 2 The Statistical Mechanics of Anharmonic Lattices.- VI. 3 Time Evolution of Infinite Anharmonic Systems.- VI. 4 Lattice Systems with a Continuous Symmetry III. Low Temperature Asymptotic Expansion for the Plane Rotator Model.- VII. Miscellaneous.- VII. 1 The Finite Group Velocity of Quantum Spin Systems.- VII. 2 The Classical Limit of Quantum Spin Systems.- VII. 3 A Refinement of Simon's Correlation Inequality.- VII. 4 Fluxes, Laplacians, and Kasteleyn's Theorem.- Selecta of Elliott H. Lieb.- Publications of Elliott H. Lieb.
The language of operator algebras is of great help for the formulation of questions and answers in quantum statistical mechanics. In Chapter 1 we present a minimal mathematical introduction to operator algebras, with physical applications in mind. In Chapter 2 we study some questions related to the quantum statistical mechanics of spin systems, with particular attention to the time evolution of infinite systems. The basic reference for these two chapters is Bratteli-Robinson: Operator algebras and quantum statistical mechanics I, II. In Chapter 3 we discuss the nonequilibrium statistical mechanics of quantum spin systems, as it is currently being developped.