Physical applications of homogeneous balls. With the assistance of Tzvi Scarr
type: Item
instance of: scholarly article
title: Physical applications of homogeneous balls. With the assistance of Tzvi Scarr
zbMATH Open document ID: 1080.46001
author: Yaakov Friedman
published in: Progress in Mathematical Physics
publication date: 2005-02-25T00:00:00Z
review text: This fine book provides a highly original approach to theoretical physics, its contents reflecting the author's and his collaborators' copious contributions to many branches of mathematics and physics over the past years. The book's basic premise is that the main object in many mathematical models of physical systems is a space along with a family of symmetries which is large enough to ensure that the space in question forms a bounded symmetric domain in a real or complex Banach space. Examples are the space of all admissible velocities in special relativity, the space of Dirac bispinors in relativistic quantum mechanics and the space of complex effects in classical and quantum mechanics. In the first chapter, the author presents a reformulation of special relativity theory, showing how the space of admissible velocities forms a real bounded symmetric domain and how the conventional evolution equations may be obtained from a study of the Lie group of its projective automorphisms and the associated Lie algebra. In the second chapter, a further real bounded symmetric domain consisting of symmetric relativistic velocities is introduced, the associated Lie group being the group of its conformal automorphisms. By considering the corresponding Lie algebra, it is shown how this leads naturally to a triple product on \(\mathbb{R}^n\), the so-called real spin triple product. Again, the relativistic equations of mechanics and electromagnetism can be written in terms of the Lie algebra, and an explicit calculation is carried out for the motion of a charged particle in uniform constant perpendicular electric and magnetic fields. In the third chapter, the real spin triple product is extended to \(\mathbb{C}^n\), and a study is made of the corresponding complex bounded symmetric domain, hence leading to the complex spin triple product. As an introduction to the general theory, many algebraic and geometric properties of the complex spin triple are presented, before the author goes…
Mathematics Subject Classification ID: 46-02
Mathematics Subject Classification ID: 17C65
Mathematics Subject Classification ID: 46L60
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46L70
Mathematics Subject Classification ID: 83A05
zbMATH DE Number: 2138849
zbMATH Keywords: bounded symmetric domain
zbMATH Keywords: Lie group
zbMATH Keywords: Lie algebra
zbMATH Keywords: spin triple product
zbMATH Keywords: JB*-triple
reviewed by: C. M. Edwards
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2138849