Models in Statistical Physics and Quantum Field Theory

1. Introduction.- 1.1 Phase Transitions - Critical Phenomena.- 1.1.1 Historical Survey.- 1.1.2 Gas-Liquid Transition.- 1.1.3 Ferromagnetism.- 1.1.4 Critical Exponents.- 2. Spin Systems.- 2.1 Ising Model - General Results.- 2.1.1 Introduction.- 2.1.2 Ising Model in One Dimension.- 2.1.3 Duality.- 2.1.4 Peierls' Argument.- 2.1.5 Correlation Inequalities.- 2.2 Heisenberg Model.- 2.2.1 Bogoliubov Inequality.- 2.2.2 Absence of Spontaneous Magnetization for d = 1 and d = 2.- 2.2.3 Existence of a Phase Transition for d Greater than or Equal to Three.- 2.3 o4-Model.- 2.3.1 Random Walk on a Lattice.- 2.3.2 Polymer Representation.- 2.3.3 Correlation Inequality.- 2.3.4 Continuum Limit.- 2.4 Two-Dimensional Ising Model.- 2.4.1 Transfer Matrix.- 2.4.2 Klein-Jordan-Wigner Transformation.- 2.4.3 Fourier Transformation.- 2.4.4 Bogoliubov Transformation.- 3. Two-Dimensional Field Theory.- 3.1 Solitons.- 3.1.1 Inverse Scattering Formalism.- 3.1.2 Solving Certain Nonlinear Partial Differential Equations.- 3.1.3 A Model for Polyacetylene.- 3.2 Sectors in Field Theoretical Models.- 3.2.1 External Field Problems.- 3.2.2 The Schwinger Model.- 4. Lattice Gauge Models.- 4.1 Formulation.- 4.1.1 Axioms.- 4.1.2 The Rolling Ball.- 4.1.3 Classical Field Theory.- 4.1.4 Formulation of Lattice Gauge Models.- 4.1.5 Fermions on the Lattice.- 4.2 Rigorous Results.- 4.2.1 Faddeev-Popov "Trick" on a Lattice.- 4.2.2 Physical Positivity = Osterwalder-Schrader Positivity.- 4.2.3 Cluster Expansion.- 4.2.4 Confinement.- 4.2.5 Remarks on Numerical Methods.- 4.2.6 Recent Developments.- 5. String Models.- 5.1 Introduction to Strings.- 5.1.1 Classical Mechanics of Strings.- 5.1.2 Quantization of the Bosonic String.- 5.1.3 Fermionic Strings and Superstrings.- 6. Renormalization Group.- 6.1 Formulation.- 6.1.1 Scaling Laws.- 6.1.2 Kadanoff's Block Spin Method.- 6.1.3 Wilson's Renormalization Group Ideas.- 6.1.4 Ising Model d = 1.- 6.2 Application of the Renormalization Group Ideas to Special Models.- 6.2.1 Central Limit Theorem.- 6.2.2 Hierarchical Model.- 6.2.3 Two-Dimensional Ising Model.- 6.2.4 Ginzburg-Landau-Wilson Model.- 6.2.5 Feigenbaum's Route to Chaos.- General References.