Description
Book SynopsisIn these lectures we summarize certain results on models in statistical physics and quantum field theory and especially emphasize the deep relation ship between these subjects. From a physical point of view, we study phase transitions of realistic systems; from a more mathematical point of view, we describe field theoretical models defined on a euclidean space-time lattice, for which the lattice constant serves as a cutoff. The connection between these two approaches is obtained by identifying partition functions for spin models with discretized functional integrals. After an introduction to critical phenomena, we present methods which prove the existence or nonexistence of phase transitions for the Ising and Heisenberg models in various dimensions. As an example of a solvable system we discuss the two-dimensional Ising model. Topological excitations determine sectors of field theoretical models. In order to illustrate this, we first discuss soliton solutions of completely integrable classical models. Afterwards we dis cuss sectors for the external field problem and for the Schwinger model. Then we put gauge models on a lattice, give a survey of some rigorous results and discuss the phase structure of some lattice gauge models. Since great interest has recently been shown in string models, we give a short introduction to both the classical mechanics of strings and the bosonic and fermionic models. The formulation of the continuum limit for lattice systems leads to a discussion of the renormalization group, which we apply to various models.
Table of Contents1. 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 ø4-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.
