Physical Applications of Homogeneous Balls

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.

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.