Physical applications of homogeneous balls. With the assistance of Tzvi Scarr
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 on to study representations of the Lorentz group by biholomorphic automorphisms of the three- and four-dimensional complex spin triples. In the fourth chapter, the infinite-dimensional classical bounded symmetric domains are described. Up to this point in the book, no attempt is made to give complete and general definitions of many of the concepts discussed. However, the final two chapters are devoted to a presentation of the general theory of bounded symmetric domains in complex Banach spaces, to which the author has contributed extensively. Every complex bounded symmetric domain is biholomorphically equivalent to the open unit ball in a complex Banach space upon which a triple product is naturally defined. Such an object \(A\) is known as a JB*-triple. When \(A\) is a Banach dual space, its predual \(A_*\) is unique up to isometric isomorphism and \(A\) is said to be a JBW*-triple. Consequently, the final chapters of the book are devoted to a summary of the algebraic and geometric properties of JB*-triples and JBW*-triples and the classification of JBW*-triple factors. The methods used are mainly those developed by the author and his collaborators in which co-ordinatization plays a crucial role. As the author claims, graduate students in theoretical physics will certainly find the earlier chapters interesting and stimulating. However, I suspect that many such students would find the more mathematical parts of the book quite hard going. A mathematician with some knowledge of physics (such as the reviewer) will find the applications of bounded symmetric domains to physics fascinating, and may well be persuaded to look more deeply into the far-reaching theory of JB*-triples. Whether either the physicist or the mathematician would be convinced by the proposition, which, to be fair, is not suggested by the author, that ``the universe is a JBW*-triple'', remains to be seen.
scientific article; zbMATH DE number 2138849