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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.
List of contents
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.
Summary
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.
