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. 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.
Rezensionen / Stimmen
From the reviews:
"The aim of this book is to present the theory of Jordan algebraic structures (Jordan triple systems) and their geometric counterpart, the so-called homogeneous balls, from the point of view of applications ot mathematical physics: special relativity, spinors and foundational quantum mechanics...The (senior) author has made important research contributions to all three areas described above, and the exposition of the theory and the applications is very careful. This makes the book suitable both for experts and non-experts interested in the applications."
---Mathematical Reviews
"This fine book provides a highly original approach to theoretical physics, its contents reflecting the author's and his ollaborators' copious contributions to many branches of mathematics and physics over the past years."(ZENTRALBLATT MATH)
Reihe
Auflage
Softcover reprint of the original 1st ed. 2005
Sprache
Verlagsort
Zielgruppe
Für Beruf und Forschung
Research
Illustrationen
Maße
Höhe: 235 mm
Breite: 155 mm
Dicke: 17 mm
Gewicht
ISBN-13
978-1-4612-6493-4 (9781461264934)
DOI
10.1007/978-0-8176-8208-8
Schweitzer Klassifikation
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.
1 Relativity based on symmetry.- 1.1 Space-time transformation based on relativity.- 1.2 Step 6 - Identification of invariants.- 1.3 Relativistic velocity addition.- 1.4 Step 7 - The velocity ball as a bounded symmetric domain.- 1.5 Step 8 - Relativistic dynamics.- 1.6 Notes.- 2 The real spin domain.- 2.1 Symmetric velocity addition.- 2.2 Projective and conformal commutativity and associativity.- 2.3 The Lie group Aut,(Ds) 64 2.3.1 The automorphisms of Ds generated by s-velocity addition.- 2.4 The Lie Algebra autc(Ds) and the spin triple product.- 2.5 Relativistic dynamic equations on Ds.- 2.6 Perpendicular electric and magnetic fields.- 2.7 Notes.- 3 The complex spin factor and applications.- 3.1 The algebraic structure of the complex spin factor.- 3.2 Geometry of the spin factor.- 3.3 The dual space of Sn.- 3.4 The unit ball Ds,n of Sn as a bounded symmetric domain.- 3.5 The Lorentz group representations on Sn.- 3.6 Spin-2 representation in dinv (84).- 3.7 Summary of the representations of the Lorentz group on S3 and S4.- 3.8 Notes.- 4 The classical bounded symmetric domains.- 4.1 The classical domains and operators between Hilbert spaces.- 4.2 Classical domains are BSDs.- 4.3 Peirce decomposition in JC*-triples.- 4.4 Non-commutative perturbation.- 4.5 The dual space to a JC*-triple.- 4.6 The infinite-dimensional classical domains.- 4.7 Notes.- 5 The algebraic structure of homogeneous balls.- 5.1 Analytic mappings on Banach spaces.- 5.2 The group Auta (D).- 5.3 The Lie Algebra of Auta(D).- 5.4 Algebraic properties of the triple product.- 5.5 Bounded symmetric domains and JB*-triples.- 5.6 The dual of a JB*-triple.- 5.7 Facially symmetric spaces.- 5.8 Notes.- 6 Classification of JBW*-triple factors.- 6.1 Building blocks of atomic JBW*-triples.- 6.2 Methods of gluing quadrangles.- 6.3 Classification of JBW*-triple factors.- 6.4 Structure and representation of JB*-triples.- 6.5 Notes.- References.
