Physical Applications of Homogeneous Balls (Progress in Mathematical Physics, Band 40) - Softcover

Über die Autorin bzw. den Autor

Prof. Yaakov Friedman of the Jerusalem College of Technology - Lev Academic Center was born in Munkatch, USSR. He graduated from the Faculty of Mathematics and Mechanics of Moscow University in 1971 and got his Ph.D. in Mathematics from Tel Aviv University in 1979. He worked for eight years at the University of California, Los Angeles and Irvine. Since that time, he has worked at the Jerusalem College of Technology as a lecturer, the rector, the vice-president for research and the head of the research authority. He initiated and was R&D director of several high-tech start-ups and companies. Prof. Friedman's research, published in about 100 papers, is in pure and applied mathematics, theoretical and applied statistics, and mathematical, theoretical and experimental physics. His current research interest is the novel approach to dynamics presented in this book, the theory's predictions, and experimental testing of them. This theory has the potential to give new insights into understanding microscopic behavior. Dr. TzviScarr received his Master's degree in mathematics from the University of California, Berkeley in 1989 and his Ph.D. in mathematics from Bar Ilan University, Israel, in 2000. He has taught and done research at the Jerusalem College of Technology since 1997. His doctoral research was in equivariant topology and set-theoretic forcing. In 2002, he turned to mathematical physics and began his collaboration with Yaakov Friedman, assisting with the writing of the book Physical Applications of Homogeneous Balls. Over the past twenty years, he has developed mathematical models for special relativity, general relativity, electromagnetism and quantum mechanics. His research has focused on the use of a minimal number of assumptions as well as the unification of disparate areas in physics.

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One of the mathematical challenges of modern physics lies in the development of new tools to efficiently describe different branches of physics within one mathematical framework. This text introduces precisely such a broad mathematical model, one that gives a clear geometric expression of the symmetry of physical laws and is entirely determined by that symmetry.

The first three chapters discuss the occurrence of bounded symmetric domains (BSDs) or homogeneous balls and their algebraic structure in physics. It is shown that the set of all possible velocities is a BSD with respect to the projective group; the Lie algebra of this group, expressed as a triple product, defines relativistic dynamics. The particular BSD known as the spin factor is exhibited in two ways: first, as a triple representation of the Canonical Anticommutation Relations, and second, as a ball of symmetric velocities. The associated group is the conformal group, and the triple product on this domain gives a representation of the geometric product defined in Clifford algebras. It is explained why the state space of a two-state quantum mechanical system is the dual space of a spin factor. Ideas from Transmission Line Theory are used to derive the explicit form of the operator Mobius transformations. The book further provides a discussion of how to obtain a triple algebraic structure associated to an arbitrary BSD; the relation between the geometry of the domain and the algebraic structure is explored as well. The last chapter contains a classification of BSDs revealing the connection between the classical and the exceptional domains.

With its unifying approach to mathematics and physics, this work will be useful for researchers and graduate students interested in the many physical applications of bounded symmetric domains. It will also benefit a wider audience of mathematicians, physicists, and graduate students working in relativity, geometry, and Lie theory.