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

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