This book contains thoroughly written reviews of modern developments in low-dimensional modelling of statistical mechanics and quantum systems. It addresses students as well as researchers. The main items can be grouped into integrable (quantum) spin systems, which lead in the continuum limit to (conformal invariant) quantum field theory models and their algebraic structures, ranging from the Yang-Baxter equation and quantum groups to noncommutative geometry.

This monograph introduces modern developments on the bound state problem in Schroedinger potential theory and its applications in particle physics. The Schroedinger equation provides a framework for dealing with energy levels of N-body systems. It was a cornerstone of the quantum revolution in physics of the twenties but re-emerged in the eighties as a powerful tool in the study of spectra and decay properties of mesons and baryons. This book begins with a detailed study of two-body problems, including discussion of general properties, level ordering problems, energy level spacing and decay properties. Following chapters treat relativistic generalisations, and the inverse problem. Finally, 3-body problems and N-body problems are dealt with. Applications in particle and atomic physics are considered, including quarkonium spectroscopy. The emphasis throughout is on showing how the theory can be tested by experiment. Many references are provided.

In 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."