The new volume 261 of Lecture Notes in Physics represents itself the Proceedings of the International Symposium on Conformal Groups and Conformal Structures held in August 1985 at the Arnold Sommerfeld Institute for Mathematical Physics in Clausthal. The contents of this volume is organized into the following six chapters.
Chapters I, Symmetries and dynamics, includes the contributions by A. O. Barut, ”From Heisenberg algebra to conformal dynamical group”, A. Inomata and R. Wilson, ”Path integral realization of a dynamical group”, Th. Görnitz and C. F. von Weizsäcker, ”De-Sitter representations and the particle concept, studied in an ur- theoretical cosmological model”, and the others. In particular, in his review A. O. Barut outlines the basic algebraic structures in quantum theory of the electron, from the Heisenberg algebra, kinematic algebra, Galilean and Poincaré groups, to the external and internal conformal groups and discusses the universal role of the conformal dynamical group from an electron, hydrogen atom, and hadrons to the periodic table.
In the next Chapter, Classical and quantum field theory, D. Buchholz gives the review on structure of local algebras in view of the algebraic formulation of relativistic quantum physics, M. F. Sohnius discusses the problem of the sign of the cosmological constant in supergravity models with unbroken supersymmetry, C. N. Kozameh proposes the concept of the holonomy operator of the Yang-Mills theory as a good variable for describing nonlocal effects, etc.
Chapter III, Conformal structures, contains the contributions by B. G. Schmidt, ”Conformal geodesics”, J. D. Hennig, ”Second order conformal structures”, H. Friedrich, ”The conformal structure of Einstein’s field equations”, and C. Duval, ”Nonrelativistic conformal symmetries and Bargmann structures”.
Chapter IV is devoted to the theory of conformal spinors. The first contribution, ”Wave equations for conformal multispinors”, is due to M. Lorente. P. Budinich, L. Dabrowski, and H. R. Petry give the general discussion of conformal transformations of spinor fields. They demonstrate the existence of two inequivalent spinor structures on the minimal conformal compactification \({\mathcal M}\) of the Minkowski space-time which are interchanged by the space and space-time inversions. Also it is suggested that the Dirac spinor fields should be coupled to a gauge potential in order to obtain a nontrivial unitary representation of the conformal group in the space of solutions of the massless Dirac equation on \({\mathcal M}\). The next report by P. Budinich is devoted to generalizing the concept of pure spinors on conformal extensions of space-time. J. Ryan presents the complex Clifford analysis over the Lie ball.
In the next Chapter, Lie groups, -algebras and superalgebras, there are presented the papers by R. A. Herb and J. A. Wolf, ”Plancherel theorem for the universal cover of the conformal group”, G. v. Dijk, ”Harmonic analysis on rank one symmetric spaces”, H. P. Jakobsen, ”A spin-off from highest weight representations; conformal covariants, in particular for O(3,2)”, E. Angelopoulos, ”Tensor calculus in enveloping algebras”, R. Lenczewski and B. Gruber, ”Representations of the Lorentz algebras on the space of its universal enveloping algebra”, and the two works by V. K. Dobrev and V. B. Petkova, devoted to the theory of representations of the extended conformal superalgebra.
Chapter VI, Infinite-dimensional Lie algebras, includes the following contributions. Y. Ne’eman discusses the properties of the two- dimensional quantum conformal group and its relation with strings and lattices. V. Rittenberg demonstrates the connection between the spectra of one-dimensional quantum chains at the critical point with different boundary conditions and certain irreducible representations of Virasoro algebras. V. G. Kac and M. Wakimoto study the properties of the unitarizable highest weight representations of the Virasoro, Neveu-Schwarz and Ramond algebras which play a fundamental role in statistical mechanics and string theory. J. Mickelsson gives the formal study of structure of Kac-Moody groups. D. T. Stoyanov obtains two infinite-dimensional Lie algebras which act on and preserve the set of solutions of the Laplace equation in the four-dimensional Euclidean space. I. T. Todorov presents the review of the ”minimal theories” of critical behavior in two dimensions by Belavin, Polyakov, and Zamolodchikov. He developes the new version of the conformal quantum field theory on the compactified Minkowski space.