The Yang-Mills theory of gauge interactions is a prime example of interdisciplinary mathematics and advanced physics. Its historical development is a fascinating window into the ongoing struggle of mankind to understand nature. The discovery of gauge fields and their properties is the most formidable landmark of modern physics. The expression of the gauge field strength as the curvature associated to a given connection, places quantum field theory in the same geometrical footing as the gravitational field of general relativity which is naturally written in geometrical terms. The understanding of such geometrical property may help one day to write a unified field theory starting from symmetry principles. Of course, there are remarkable differences between the standard gauge fields and the gravitational field, which must be understood by mathematicians and physicists before attempting such unification. In particular, it is important to understand why gravitation is not a standard gauge field. This book presents an account of the geometrical properties of gauge field theory, while trying to keep the equilibrium between mathematics and physics. At the end we will introduce a similar approach to the gravitational field.

The interaction between geometry and theoretical physics has often been very fruitful. A highlight in this century was Einstein's creation of the theory of general relativity. Equally impressive was the recognition, starting from the work of Yang and Mills and culminating in the Weinberg-Salam theory of the electroweak interaction and quantum chromodynamics, that the fundamental interactions of elementary particles are governed by gauge fields, which in ma thematical terms are connections in principal fibre bundles. Theoretical physi cists became increasingly aware of the fact that the use of modern mathematical methods may be necessary in the treatment of problems of physical interest. Since some of these topics are covered at most summarily in the usual curricu lum, there is a need for extra-curricular efforts to provide an opportunity for learning these techniques and their physical applications. In this context we arranged a meeting at the Physikzentrum Bad Ronnef 12-16 February 1990 on the subject "Geometry and Theoretical Physics", in the series of physics schools organized by the German Physical Society. The participants were graduate students from German universities and research institutes. Since the meeting occurred only a short time after freedom of travel between East and West Germany became a reality, this was for many from the East the first opportunity to attend a scientific meeting in the West, and for many from the West the first chance to become personally acquainted with colleagues from the East.

This book focuses on the unifying power of the geometrical language in bringing together concepts from many different areas of physics, ranging from classical physics to the theories describing the four fundamental interactions of Nature -- gravitational, electromagnetic, strong nuclear, and weak nuclear. The book provides in a single volume a thorough introduction to topology and differential geometry, as well as many applications to both mathematical and physical problems. It is aimed as an elementary text and is intended for first year graduate students. In addition to the traditional contents of books on special and general relativities, this book discusses also some recent advances such as de Sitter invariant special relativity, teleparallel gravity and their implications in cosmology for those wishing to reach a higher level of understanding.

This book presents a geometric perspective on the ever-expanding landscape of (often unfalsifiable) theories of fundamental physics that have emerged from the beginning of the twentieth century until the present day. The authors present a summary of the landmark results, and the assumptions needed to obtain them, viewed with the common philosophy that many difficulties in fundamental physics could be elucidated by an understanding of the geometrical language of quantum physics, motivated initially by the geometric implications of the ‘t Hooft—Veltman renormalization of gauge fields. The geometrical language underpinning this worldview is affine geometry, a common feature of the four known fundamental interactions, where affinity between observables is mediated by a fundamental interaction, as in the example of local gauge fields in quantum field theory. The same can be said of Einstein’s gravitational field based on the (pseudo-)Riemannian geometry where the affinity comes from the metric. Such notions allow one to interpret affinity at the quantum scale of observations, and consider questions such as: What is the meaning of the Planck regime? What is the meaning of the cosmological principle? What is quantum gravity? Is the Newtonian gravitational constant really universal? Is information lost in quantum gravity? The book reviews fundamental background material, presents manifold insights into the geometrical nature of quantum physics and cosmology, suggests promising avenues for future research, and in places even proposes experimental tests. It will be of interest to high energy physicists, cosmologists, mathematicians, and philosophers of science.

A meeting was held at the Physics Centre in Bad Honnef from Feb. 16-20, 1987 on the subject "The Fundamental Interaction: Geometrical Trends". This meeting was in the series of Physics Schools organized by the German Physical Society. The participants were mainly younger scientists and graduate students: physicists, mathematicians and astronomers from the Federal Republic of Germany; there were also participants from Austria, the Netherlands and Switzerland. The purpose of the meeting was to introduce the participants to modern methods of mathematics and field theory which are increasingly being used in current elementary particle research. An outstanding feature of the school was the fact that each lecturer really made an effort to present his material at an introductory level, which could be followed by people with the usual degree level qualifications and which, nevertheless, then led up to the level of contemporary specialized literature. We hope that the published volume will make these lectures, which taken together give a unified overview of the recent developments leading up to the current world-view in modern theoretical physics, available to a wider audience. We wish to thank the lecturers, who besides the formal lectures devoted their time to discussions with the participants throughout the duration of the School. We are also grateful to the staff of the Physics Centre for their organizational assistance, and to the participants, whose active interest made the meeting a success. The entire manuscript was efficiently typed by Manfred Krafczyk at the University of Dortmund.

Special Relativity (SR) is essentially grounded on the properties of space-time, i.e. isotropy of space and homogeneity of space and time (as a consequence of the equivalence of inertial frames) and on the Galilei principle of relativity.