P. Barut - Böcker

Symmetry and Dynamics have played, sometimes dualistic, sometimes complimentary, but always a very essential role in the physicist's description and conception of Nature. These are again the basic underlying themes of the present volume. It collects self-contained introductory contributions on some of the recent developments both in mathematical concepts and in physical applications which are becoming very important in current research. So we see in this volume, on the one hand, differential geometry, group representations, topology and algebras and on the other hand, particle equations, particle dynamics and particle interactions. Specifically, this book contains a complete exposition of the theory of deformations of symplectic algebras and quantization, expository material on topology and geometry in physics, and group representations. On the more physical side, we have studies on the concept of particles, on conformal spinors of Cartan, on gauge and supersymmetric field theories, and on relativistic theory of particle interactions and the theory of magnetic resonances. The contributions collected here were originally delivered at two Meetings in Turkey, at Blacksea University in Trabzon and at the University of Bosphorus in Istanbul. But they have been thoroughly revised, updated and extended for this volume. It is a pleasure for me to acknowledge the support of UNESCO, the support and hospitality of Blacksea and Bosphorus Universities for these two memorable Meetings in Mathematical Physics, and to thank the Contributors for their effort and care in preparing this work.

Mathematical physics has become, in recent years, an inde­ pendent and important branch of science. It is being increasingly recognized that a better knowledge and a more effective channeling of modern mathematics is of great value in solving the problems of pure and applied sciences, and in recognizing the general unifying principles in science. Conversely, mathematical developments are greatly influenced by new physical concepts and ideas. In the last century there were very close links between mathematics and theo­ retical physics. It must be taken as an encouraging sign that today, after a long communication gap, mathematicians and physicists have common interests and can talk to each other. There is an unmistak­ able trend of rapprochement when both groups turn towards the com­ mon source of their science-Nature. To this end the meetings and conferences addres sed to mathematicians and phYSicists and the publication of the studies collected in this Volume are based on lec­ tures presented at the NATO Advanced Study Institute on Mathemati­ cal Physics held in Istanbul in August 1970. They contain review papers and didactic material as well as original results. Some of the studies will be helpful for physicists to learn the language and methods of modern mathematical analysis-others for mathematicians to learn physics. All subjects are among the most interesting re­ search areas of mathematical physics.

This is the third Volume in a series of books devoted to the interdisciplinary area between mathematics and physics, all ema­ nating from the Advanced Study Institutes held in Istanbul in 1970, 1972 and 1977.

Symmetry and Dynamics have played, sometimes dualistic, sometimes complimentary, but always a very essential role in the physicist's description and conception of Nature. These are again the basic underlying themes of the present volume. It collects self-contained introductory contributions on some of the recent developments both in mathematical concepts and in physical applications which are becoming very important in current research. So we see in this volume, on the one hand, differential geometry, group representations, topology and algebras and on the other hand, particle equations, particle dynamics and particle interactions. Specifically, this book contains a complete exposition of the theory of deformations of symplectic algebras and quantization, expository material on topology and geometry in physics, and group representations. On the more physical side, we have studies on the concept of particles, on conformal spinors of Cartan, on gauge and supersymmetric field theories, and on relativistic theory of particle interactions and the theory of magnetic resonances. The contributions collected here were originally delivered at two Meetings in Turkey, at Blacksea University in Trabzon and at the University of Bosphorus in Istanbul. But they have been thoroughly revised, updated and extended for this volume. It is a pleasure for me to acknowledge the support of UNESCO, the support and hospitality of Blacksea and Bosphorus Universities for these two memorable Meetings in Mathematical Physics, and to thank the Contributors for their effort and care in preparing this work.

Theoretical physicists allover the world are acquainted with Lande's celebrated computation of the g factor or splitting factor or, more precisely, the magne­ togyric factor. The so-called anomalous Zeeman effect had intrigued, if not vexed, some of the most distinguished physicists of that time, such as Bohr, Sommerfeld, Pauli, and others. Lande realized that this recalcitrant effect was inseparable from the multiplet line structure - a breakthrough in understanding which he achieved in 1922 at the age of thirty four. It was in the same year that Lande discovered the interval rule for the separation of multiplet sublevels, a significant result that holds in all cases of Russell-Saunders coupling and renders comparatively easy the empirical analysis of spectral multiplets. In the twenties, Lande succeeded in constructing some original concepts of axiomatic thermodynamics by employing Caratheodory's somewhat esoteric approach as his guiding concept. Published in the Handbuch der Physik, his comprehensive treatise, evincing several novel ideas, has become a classic. Lande, Sommerfeld's student though never a true disciple, published two monographs on quantum mechanics that are remarkable for their content and exposition. In this connection it may be apposite to stress that Lande had sub­ scribed for many years to the (infelicitously named) Copenhagen interpretation.

This is the third Volume in a series of books devoted to the interdisciplinary area between mathematics and physics, all ema­ nating from the Advanced Study Institutes held in Istanbul in 1970, 1972 and 1977. We believe that physics and mathematics can develop best in harmony and in close communication and cooper­ ation with each other and are sometimes inseparable. With this goal in mind we tried to bring mathematicians and physicists together to talk and lecture to each other-this time in the area of nonlinear equations. The recent progress and surge of interest in nonlinear ordi­ nary and partial differential equations has been impressive. At the same time, novel and interesting physical applications mul­ tiply. There is a unifying element brought about by the same characteristic nonlinear behavior occurring in very widely differ­ ent physical situations, as in the case of "solitons," for exam­ ple. This Volume gives, we believe, a very good indication over all of this recent progress both in theory and applications, and over current research activity and problems. The 1977 Advanced Study Institute was sponsored by the NATO Scientific Affairs Division, The University of the Bosphorus and the Turkish Scientific and Technical Research Council. We are deeply grateful to these Institutions for their support, and to lecturers and participants for their hard work and enthusiasm which created an atmosphere of lively scientific discussions.

This is the second volume of a series of books in various aspects of Mathematical Physics. Mathematical Physics has made great strides in recent years, and is rapidly becoming an important dis­ cipline in its own right. The fact that physical ideas can help create new mathematical theories, and rigorous mathematical theo­ rems can help to push the limits of physical theories and solve problems is generally acknowledged. We believe that continuous con­ tacts between mathematicians and physicists and the resulting dialogue and the cross fertilization of ideas is a good thing. This series of studies is published with this goal in mind. The present volume contains contributions which were original­ ly presented at the Second NATO Advanced Study Institute on Mathe­ matical Physics held in Istanbul in the Summer of 1972. The main theme was the application of group theoretical methods in general relativity and in particle physics. Modern group theory, in par­ ticular, the theory of unitary irreducibl~ infinite-dimensional representations of Lie groups is being increasingly important in the formulation and solution of dynamical problems in various bran­ ches of physics. There is moreover a general trend of approchement of the methods of general relativity and elementary particle physics. We hope it will be useful to present these investigations to a larger audience.

Mathematical physics has become, in recent years, an inde­ pendent and important branch of science. It is being increasingly recognized that a better knowledge and a more effective channeling of modern mathematics is of great value in solving the problems of pure and applied sciences, and in recognizing the general unifying principles in science. Conversely, mathematical developments are greatly influenced by new physical concepts and ideas. In the last century there were very close links between mathematics and theo­ retical physics. It must be taken as an encouraging sign that today, after a long communication gap, mathematicians and physicists have common interests and can talk to each other. There is an unmistak­ able trend of rapprochement when both groups turn towards the com­ mon source of their science-Nature. To this end the meetings and conferences addres sed to mathematicians and phYSicists and the publication of the studies collected in this Volume are based on lec­ tures presented at the NATO Advanced Study Institute on Mathemati­ cal Physics held in Istanbul in August 1970. They contain review papers and didactic material as well as original results. Some of the studies will be helpful for physicists to learn the language and methods of modern mathematical analysis-others for mathematicians to learn physics. All subjects are among the most interesting re­ search areas of mathematical physics.