On the role of division, Jordan, and related algebras in particle physics

This monograph surveys the role of some associative algebras, noted by their appearance in contemporary theoretical physics, particularly in particle physics. It concerns the interplay between division algebras, specifically quaternions and octonions, between Jordan and related algebras on the one hand, and unified theories of the basic interactions on the other. Selected applications of these algebraic structures are discussed: quaternion analyticity of Yang-Mills instantions, octonionic aspects of exceptional broken gauge, supergravity theories, division algebras in anyonic phenomena and in theories of extended objects in critical dimensions. 1. Introduction -- 2. Quaternions. 2a. Algebraic structures. 2b. Jordan formulation, H-Hilbert spaces and groups. 2c. Vector products, parallelisms and quaternionic manifolds. 2d. Quaternionic function theory. 2e. Arithmetics of quaternions. 2f. Selected physical applications. 2g. Historical notes -- 3. Octonions. 3a. Algebraic structures. 3b. Octonionic Hilbert spaces, exceptional groups and algebras. 3c. Vector products, parallelisms on S7 and octonionic manifolds. 3d. Octonionic function theory. 3e. Arithrnetics of octonions. 3f. Some physical applications. 3g. Historical notes -- 4. Division, Jordan algebras and extended objects. 4.a. Dyson's 3-fold way: Time reversal and berry phases. 4.b. Essential Hopf fibrations and D[symbol]3 anyonic phenomena. 4c. The super-Poincare group and super-extended objects This monograph surveys the role of some associative and non-associative algebras, remarkable by their ubiquitous appearance in contemporary theoretical physics, particularly in particle physics. It concerns the interplay between division algebras, specifically quaternions and octonions, between Jordan and related algebras on the one hand, and unified theories of the basic interactions on the other. Selected applications of these algebraic structures are discussed: quaternion analyticity of Yang-Mills instantons, octonionic aspects of exceptional broken gauge, supergravity theories, division algebras in anyonic phenomena and in theories of extended objects in critical dimensions. The topics presented deal primarily with original contributions by the authors. 2.f.3. Self-duality and Yang-Mills and gravitational instantons2.f.4. H-analyticity and Milne's regraduation of clocks; 2.g. Historical Notes; 2.g.1. Birth and high expectations (1843-1873); 2.g.2. Their demise from physics and a haven in mathematics (1873-1900); 2.g.3. Complex quaternions in relativity (1911-1926); 2.g.4. A deeper role in quantum mechanics, function theory (1927-1950); 2.g.5. New hopes and disappointments (1950-1975); 2.g.6. Comeback in Euclidean QFT (1978-Present); 3. Octonions; 3.a. Algebraic Structures; 3.a.1. Basic properties, Moufang and other identities 2.c.1. Vector products on manifolds2.c.2. Absolute parallelisms on Lie groups and s3; 2.c.3. Quaternionic, H-Kählerian structures; 2.d Quaternionic Function Theory; 2.d.1. Fueter's quaternion analysis; 2.d.2. H-holomorphic functions from C-analytic functions; 2.d.3. Fourfold periodic Weierstrassian functions; 2.d.4. Recent developments of Fueter's theory : O(4) covariance, conformal and quasi-conformal structures; 2.e. Arithmetics of Quaternions; 2.f. Selected Physical Applications; 2.f.1. Quaternionic quantum mechanics and all that; 2.f.2. Maxwell and Dirac-Kähler equations 3.a.2. O(8) covariant tensors3.a.3. Exceptional Grassman algebra; 3.b. Octonionic Hilbert Spaces, Exceptional Groups and Algebras; 3.b.1. Octonionic spaces and automorphism groups; 3.b.2. Exceptional algebras, groups and cosets; 3.c. Vector Products, Parallelisms on s7 and Octonionic Manifolds; 3.c.1. Vectors products in R8; 3.c.2. Absolute parallelisms on S7; 3.c.3. The almost complex structure on s6; 3.c.4. The Moufang plane; 3.c.5. Spaces with G2 and Spin (7) holonomy, exceptional calibrated geometries; 3.d. Octonionic Function Theory; 3.e. Arithmetics of Octonions 3.f. Some Physical Applications3.f.1. Exceptional quantum mechanical spaces as charge spaces and unified theories; 3.f.2. S7 and compactification of D = 11 supergravity; 3.f.3. D = 8 self-dualities and octonionic instantons; 3.f.4. Octonionic supersymmetry in hadron physics; 3.g. Historical Notes; 3.g.1. Early life of octonions and division algebras (1843-1933); 3.g.2. Octonionic quantum mechanics, birth of Jordan algebras (1933-1934); 3.g.3. Exceptional life in mathematics (1950-1967); 3.g.4. New attempts at applications and exceptional unified theories (1960-1978) PREFACE; Contents; 1. Introduction; Symmetries: their roles, their mathematics; 2. Quaternions; 2.a. Algebraic Structures; 2.a.1. Basic properties and identities; 2.a.2. Covariant O(4) and (anti- ) self-dual tensors; 2.a.3. Clifford and Grassmann algebras; 2.a.4. Complex and hermitian quaternions; 2.a.5. Symplectic Lie algebras and Quaternionic Jordan algebras; 2.b. Jordan Formulation, H-Hilbert Spaces and Groups; 2.b.1. The Jordan form of quantum mechanics; 2.b.2. H-Hilbert spaces and symplectic groups; 2.c. Vector Products, Parallelisms and Quaternionic Manifolds This monograph surveys the role of some associative algebras, concerning the interplay between division algebras, specifically quaternions and octonions, between Jordan and related algebras on the one hand, and unified theories of the basic interactions on the other. Feza Gürsey & Chia-hsiung Tze. Includes Bibliographical References And Index. 3.g.5. Extended supergravities, strings and membranes (1978-Present)