Trieste Notes in Physics – serie

These notes arose from a series of lectures first presented at the Scuola Interna- zionale Superiore di Studi Avanzati and the International Centre for Theoretical Physics in Trieste in July 1980 and then, in an extended form, at the Universities of Sofia (1980-81) and Bielefeld (1981). Their objective has been two-fold. First, to introduce theorists with some background in group representations to the notion of twistors with an emphasis on their conformal properties; a short guide to the literature on the subject is designed to compensate in part for the imcompleteness and the one-sidedness of our review. Secondly, we present a systematic study of po- sitive energy conformal orbits in terms of twistor flag manifolds. They are interpre- ted as cl assi ca 1 phase spaces of "conformal parti cl es"; a characteri sti c property of such particles is the dilation invariance of their mass spectrum which, there~ fore, consists either of the point zero or of the infinite interval 222 o < -p ~ PO - P < = The detailed table of contents should give a thorough idea of the material covered in the text.The present notes would have hardly been written without the encouragement and the support of Professor Paolo Budinich, whose enthusiasm concerning conformal semispinors (a synonym of twistors) -viewed in the spirit of El i Cartan -is having a stimulating influence.

Spinor theory is an important tool in mathematical physics in particular in the context of conformal field theory and string theory. These lecture notes present a new way to introduce spinors by exploiting their intimate relationship to Clifford algebras. The presentation is detailed and mathematically rigorous. Not only students but also researchers will welcome this book for the clarity of its style and for the straightforward way it applies mathematical concepts to physical theory.

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.