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. Cover......Page 1 Title Page......Page 3 Preface......Page 5 Contents......Page 9 1. Introduction......Page 15 2.a. 1. Basic properties and identities......Page 21 2.a.2. Covariant 0(4) and (anti-) self-dual tensors......Page 32 2.a.3. Clifford and Grassmann algebras......Page 38 2.a.4. Complex and hermitian quatemions......Page 39 2.a.5. Symplectic Lie algebras and Quatemionic Jordan algebras......Page 45 2.b. Jordan Formulation, H-Hilbert Spaces and Groups......Page 50 2.b. 1. The Jordan form of quantum mechanics......Page 51 2.b.2. H-Hilbert spaces and symplectic groups......Page 62 2.c. 1. Vector products on manifolds......Page 65 2.c.2. Absolute parallelisms on Lie groups and S3......Page 66 2.c.3. Quatemionic, H-Kahlerian structures......Page 72 2.d. 1. Fueter's quaternion analysis......Page 77 2.d.2. H-holomorphic functions from C-analytic functions......Page 81 2.d.3. Fourfold periodic Weierstrassian functions......Page 94 2.d.4. Recent developments of Fueter's theory: 0(4) covariance, conformal and quasi-conformal structures......Page 99 2.e. Arithmetics of Quaternions......Page 149 2.f.1. Quaternionic quantum mechanics and all that......Page 151 2.f.2. Maxwell equations and Dirac-Kahler equations......Page 152 2.f.3. Self-duality in Yang-Mills and gravitational instantons......Page 166 2.f.4. H-analyticity and Milne's regraduation of clocks......Page 185 2.g. Historical Notes......Page 201 2.g.1. Birth and high expectations (1843 - 1873)......Page 202 2.g.2. Their demise from physics and a haven in mathematics (1873 - 1900)......Page 205 2.g.3. Complex quatemions in relativity (1911-1926)......Page 207 2.g.4. A deeper role in quantum mechanics, function theory (1927 - 1950)......Page 208 2.g.5. New hopes and disappointments (1950 - 1975)......Page 211 2.g.6. Comeback in Euclidean QFT (1978 - Present)......Page 213 3.a. 1. Basic properties, Moufang and other identities......Page 216 3.a.2. 0(8) covariant tensors......Page 225 3.a.3. Exceptional Grassmann algebra......Page 226 3.b. Octonionic Hilbert Spaces, Exceptional Groups and Algebras......Page 227 3.b.1. Octonionic spaces and automorphism groups......Page 228 3.b.2. Exceptional algebras, groups and cosets......Page 240 3.b.2.1. Octonionic representation of SO(8), SO(7) and G2......Page 242 3.b.2.2. Tits' construction of the Magic Square......Page 246 3.b.2.3. The color-flavor construction of the exceptional groups......Page 252 3.b.2.4. The group F4......Page 257 3.b.2.5. The group E6......Page 265 3.c.1. Vectors products in R^8......Page 275 3.c.2. Absolute parallelisms on S^7......Page 284 3.c.3. The almost complex structure on S^6......Page 293 3.c.4. The Moufang Plane......Page 298 3.c.5. Spaces with G2 and Spin(7) holonomy, exceptional calibrated geometries......Page 304 3.d. Octonionic Function Theory......Page 305 3.e. Arithmetics of Octonions......Page 316 3.f.1. Exceptional quantum mechanical spaces as charge spaces and unified theories......Page 318 3.f.2. S7 and compactification of D=11 supergravity......Page 324 3.f.3. D = 8 self-dualities and octonionic instantons......Page 332 3.f.4. Octonionic supersymmetry in hadron physics......Page 348 3.g.1. Early life of octonions and division algebras (1843 - 1933)......Page 354 3.g.3. Exceptional life in mathematics (1950 - 1967)......Page 357 3.g.4. New attempts at applications and exceptional unified theories (1960 - 1978)......Page 358 3.g.5. Extended supergravities, strings and membranes (1978 - Present)......Page 359 4.a. Dyson's 3-fold Way: Time Reversal and Berry Phases......Page 361 4.b.1. Hopf 's construction and division algebras......Page 367 4.b.2. The many faces of the Hopf invariant......Page 371 4.b.3. Twists, writhes of solitons and Adams' theorem......Page 374 4.b.4. Division algebra a-models with a Hopf teen......Page 383 4.c.1. Spinors and super-vectors revisited......Page 393 4.c.2. Vectors as Jordan matrices, Lorentz and Poincare groups in critical dimensions......Page 400 4.c.3. Super-Poincare groups and their representations by matrices over K......Page 407 4.c.4. Some Fierz identities and division algebras......Page 416 4.c.5. N = 2 super-Poincare groups in critical dimensions......Page 419 4.c.6. Classical superparticles and superstrings......Page 421 4.c.7. Actions for superstrings......Page 430 4.c.8. Local symmetries of superstring actions......Page 436 References......Page 443 Index......Page 469 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.