Conformal Quantum Field Theory in D-dimensions

Our prime concern in this book is to discuss some most interesting prosppcts that have occurred recently in conformally invariant quantum field theory in a D-diuwnsional space. One of the most promising trends is constructing an pxact solution for a cprtain class of models. This task seems to be quite feasible in the light of recent resllits. The situation here is to some extent similar to what was going on in the past ypars with the two-dimensional quantum field theory. Our investigation of conformal Ward identities in a D-dimensional space, carried out as far hack as the late H. J7Gs, showed that in the D-dimensional quantum field theory, irrespective of the type of interartion, there exists a special set of states of the field with the following property: if we rpqllire that one of these states should vanish, this determines an exact solution of 3. certain field model. These states are analogous to null-vectors which determine the minimal models in the two-dimensional field theory. On the other hand, the recent resparches supplied us with a number of indications on the existencp of an intinite-parampter algebra analogous to the Virasoro algebra in spaces of higher dimensions D 2: :~. It has also been shown that this algebra admits an operator rentral expansion. It seems to us that the above-mentioned models are field theoretical realizations of the representations of these new symmetries for D 2: ;3.

Klappentext

This volume reviews recent developments in conformal quantum field theory in emD/em-dimensions, and focuses on two main aims. br/ Firstly, the promising trend is followed toward constructing an exact solution for a certain class of models. Work on the conformal Ward identities in a emD/em-dimensional space in the late '70s suggests a parallel with the null-vectors which determine the minimal models in the two-dimensional field theory. Recent research has also indicated the possible existence of an infinite parameter algebra analogous to the Virasoro algebra in spaces of higher dimensions emD/em=3. Each of these models contains parameters similar to the central charge of the two-dimensional theory, due to special fields which occur in the commutator of the components of the energy-momentum tensor. As a first step, a special formalism is suggested which allows finding an exact solution of these models for any space dimension. Then it is shown that in each model closed differential equations can be obtained for higher correlators, as well as the algebraic equations for scale dimensions of fields, and dimensionless parameters similar to the central charge. br/ Secondly, this work aims to give a survey of some special aspects of conformal quantum field theory in emD/em-dimensional space. Included are the survey of conformal methods of approximate calculation of critical indices in a three-dimensional space, an analysis and solution of a renormalised system of Schwinger-Dyson equations, a derivation of partial wave expansions, among other topics. Special attention is given to the development of the apparatus of quantum conform theory of gauge fields. br/ emAudience:/em This book will be of interest to graduate students and researchers whose work involves quantum field theory.

Inhalt
I Goals and Perspectives.- II Global Conformal Symmetry and Hilbert Space.- III Euclidean Formulation of the Conformal Theory.- IV Approximate Methods of Calculating Critical Indices.- V Spontaneous Breakdown of Conformal Symmetry.- VI Ward Identities.- VII Contribution of Electromagnetic and Gravitational Interactions into the General Solution of Ward Identities.- VIII Dynamical Sector of the Hilbert Space.- IX Conformal Invariance in Gauge Theories.- X Special Features of Conformal Transformation of Current, Energy-Momentum Tensor and Gauge Fields.- Appendix I. Casimir Operators and Irreducible Representations of Conformal Group of 4-Dimensional Minkowski Space.- Appendix II. Fourier Transforms of Euclidean and Minkowski Spaces Invariant Functions.- Appendix III. Calculation of Euclidean Quasilocal Invariant Three-point Functions.- Appendix VII. Partial Wave Expansion of Current Green Functions.- 1. The Structure of Partial Wave Expansions.- 2. Calculation of the Kernels of Partial Wave Expansions.- Appendix IX. Partial Wave Expansion of the Energy-Momentum Tensor Green functions.- 1. The Structure of Partial Wave Expansion.- 4. Calculating the Kernels of Partial Wave Expansions of the Green Functions of the Energy-Momentum Tensor.- Appendix X. Basic Integral Relations.- Appendix XII. Calculation of Integrals in Two-Dimensional Space.