Physical Applications of Homogeneous Balls [electronic resource] /
Friedman, Yaakov.author.
Scarr, Tzvi.
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.
Description based upon print version of record.
Includes bibliographical references and index.
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.
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.
Applied mathematics.
Engineering mathematics.
Physics.
Geometry.
Gravitation.
Topological groups.
Lie groups.
Differential geometry.
Applications of Mathematics.
Mathematical Methods in Physics.
Geometry.
Classical and Quantum Gravitation, Relativity Theory.
Topological Groups, Lie Groups.
Differential Geometry.