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Algebra, 'Zl2 Graded 29, 34, 281, 284 Algebra, Central 117 Algebra

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Algebra, 'Zl2 Graded 29, 34, 281, 284 Algebra, Central 117 Algebra boomerang 155, 156 Borel-Pompeiu's formula 327 bosonic and fermioni c fields 202 INDEX Bourbaki, N. 113, 117 braid group 400, 402 A. braid operator 400 algebra, 'Zl2 graded 29, 34, 281, 284 braided correlation 399 algebra, central 117 braided deform ation 391 algebra, deformation of xi, 397 braided monoid al category 399 algebra, faithfu l repr esentations of 282 braiding ru les 391 algebra, indecomposable ideals in 121 Burnside's Theorem 71 algebra, left regular representation of 283 algebra, minimal left ideal of 282 C. algebra, nilpotent ideals in 122 Caley-Klein parameters 4 algebra, quantization of 397 caps on quadrics 59 algebra, quotient 29, 30, 34 Cartan's tri ality principle 37, 140 algebra, radical of 121 Cartan, Elie 8, 117, 149, 205, 206, 372, 384 Casimir operator 23 algebra, semi-simple 121 n algebra, simple 76, 117, 283 Cauchy's integral form in lC 302 algebra, symmetric 29 Cayley numb ers 37 algebra, universal enveloping 33 Cayley, Arthur 4, 5, 10 Alhfors, L.V. 301 Cayley-Dickson pro cess 48 aljabr 3 Cayley-Hamilton th eorem 101, 105 anticommuting set (AS) 70 Cayley-Klein parameters 4 axial vectors 5 centralizing pairs 79 charge conjugation 9, 26 B. Chevalley's construction 137, 138 bilinear covariants 137, 149, 154, 157 Chevalley, Claude ix, 9, 10, 113, 117, 125, 137, bilinear form , alt ernating 59, 60, 62, 63, 65, 138,140,144,206,214,405,406 68,114,144 Chevalley-Kiihler deformation 406 bilinear form , antisymm etric 114 c1iffor 321 bilinear form, in characteristic 2 114 cliffor , Pauli 318 bilinear form, reflexive 114 cliffor, real 314 bilinear form, symmetric 114 cliffor-valued functions 313 biquat ernion 7, 26, 266 Clifford algebra Gi 3•1 281, 286 biquaternion algebra 314 Clifford algebra Gi 3,1 , graded left regular rep­ biquat ernion, complex 313 resentation of 286 biquaternion, conjugation of 266, 269, 313 Clifford algebra Gi3 ,1 , main automorphisms biquat ernion, norm of 266 of 292 biquaternion, real 265, 313 Clifford algebra Gin outermorphisms 106 biquaternion, real, regular functions of 265, Clifford algebra 59, 157, 325, 348, 358 269, 270 Clifford algebra and Dirac equation 149 bivector 7, 127, 139, 148, 313 Clifford algebra as a twisted tensor product 35 bivector, in characteristic 2 147 Clifford algebra bundle, hyperbolic 177, 196 bivector, simple 148, 162 Clifford algebra for Heeke braid 409 bivectors, generating Spio+(1,3) 179 Clifford algebra ideals, algebraic equivalence 413 414 of 180 Clifford algebra, real 3, 6, 15, 25, 26, 27, 33, Clifford algebra ideals, geometric equivalence 34,59,76,101,149,382 of 180, 184, 185 Clifford algebra, reduced left regular represen­ Clifford algebra structure, generalizations of tations of 129 383 Clifford algebra, Riesz's exterior product in Clifford algebra, Cit ,3 25, 180 147 Clifford algebra, Cit ,3 , complexified 150, 163, Clifford algebra, structure of 35, 117 188 Clifford algebra, symplectic ix, x, xi, 410 Clifford algebra, Ci3 24, 126, 127, 130, 313 Clifford algebra, two valued representation of Clifford algebra, Ci4 ,l 181, 188 9 Clifford algebra, 7l2-gradationof72, 138, 141, Clifford algebra, universal x, 59 147 Clifford algebras and algebraic spinors 117 Clifford algebra, as a Chevalley-Kahler defor­ Clifford algebras and binary codes 87 mation of a quadratic algebra 406 Clifford algebras and design theory 87 Clifford algebra, as a deformation of exterior Clifford algebras and electrodynamics 265 algebra x, 398 Clifford algebras and graph theory 87 Clifford algebra, as a subalgebra of an endo­ Clifford algebras and spinor ideals 117, 119 morphism algebra 137, 144 Clifford algebras as matrix algebras 140 Clifford algebra, as extensive quantity 6 Clifford algebras , characteristic 2 50, 52 Clifford algebra, commutative 49 Clifford algebras , extrinsic view of 105 Clifford algebra, complexification of 71, 72, Clifford algebras, intrinsic view of 106 126,150,163,281,335,352 Clifford algebras, isomorphic to matrix alge­ Clifford algebra, definition of ix, x, 137 bras 35,281 Clifford algebra, degenerate 114 Clifford algebras, periodicity of x, 35, 140 Clifford algebra, derivations in 106, 142, 148 Clifford algebras, spinor representations of 288 Clifford algebra, dimension grading in 72, 107, Clifford algebras, spinor structure of 106 137, 141, 148 Clifford algebras, table of 61, 71, 72 Clifford algebra, even subalgebra of 7, 167 Clifford bundle 120, 375 Clifford algebra, finite 9 Clifford bundle of space-time 186 Clifford algebra, graded opposite algebra of Clifford bundle of space-time, bundle of ideals 138 187 Clifford algebra, hermitian conjugation in 127 Clifford bundle of space-time, irreducible Clifford algebra, hyperbolic 193 module representations of 186, 189 Clifford algebra, in characteristic 250, 52, 145 Clifford bundle of space-time, spinor bundles Clifford algebra, isomorphism with exterior al­ 187, 188 gebra as left modules 141 Clifford bundle on SD-l with spin structure Clifford algebra, minimal left (right) ideals of 199 178,281 Clifford bundle, hyperbolic 197 Clifford algebra, nilpotent r-vectors in 113 Clifford conjugation 127,313, 314, 316 Clifford algebra, of Hardy- Weiberg quadratic Clifford group 36, 117, 203 form 55 Clifford group and pin group 117 Clifford algebra, of projective modules of finite Clifford group and spin group 117 type xi Clifford map 144, 197,405 Clifford algebra, principal ideals in 113 Clifford module 9 Clifford algebra, pseudovector in 7 Clifford norm 36 Clifford algebra, quantum braided 387, 392 Clifford numbers 172, 177, 345 415 Clifford product z, 6, 8, 139, 142, 148 diffraction of light, by a square hole 280 Clifford spin bundle 190, 192 Dirac d-number 25 Clifford's geometric algebra 125, 133 Dirac algebra 36, 160, 180 Clifford, William Kingdon 6, 7, 24 Dirac conjugation 120, 151 Clifford-Lipschitz group, definition 179 Dirac equation 26, 137, 149, 158, 174 Clifford-Pompeiu formula 302, 307 Dirac equation, a multivector form of 159 compact Kahler spin manifold 246 Dirac equation, in spinor form 289 complete boundary collocation system 335 Dirac equation, massive 297 cone integrals 307 Dirac equation, massless 281, 282, 290, 295 cones, elliptic and hyperbolic 65, 88 Dirac equations, Lagrangians for 292 conformal group 305 Dirac gamma matrices 149, 182 conformal symmetry 24 Dirac group, automorphisms of 59, 85 conformal transformations 159 Dirac group, center of 74 contraction 137, 139 Dirac group, central extension of 59, 70, 81 contraction on a spinor space 214 Dirac group, definition of 59, 60 contraction, as a derivation 148 Dirac group, even 73 contraction, left/right 142 Dirac group, finite 59, 61 Conwell's heptad 59, 83, 89, 97 Dirac group, Frobenius-Schur indicator of 71 correlation on a vector space 139 Dirac group, normal subgroups of 79 Crowe, M.J. 3 Dirac group, of hyperbolic type 80 Crumeryolle's spinoriality transformations Dirac matrices 8, 172, 296 137, 149, 163 Dirac operator 243, 244, 249, 257, 301, 366 Crumeyrolle's "real" Witt basis 117 Dirac propagator 199, 200, 201 Crumeyrolle's amorphic spinor fields 177, 192 Dirac quantum field 257 Crumeyrolle's bivector xi, 137 Dirac quantum field, invariance under charge Crumeyrolle's spinor 137, 163 conjugation 260 Crumeyrolle's spinoriality group 163 Dirac quantum field, parity invariance of 261 Crumeyrolle, Albert ix-xiv, 3, 9, 15, 24, 41, Dirac quantum field, time reversal invariance 81,108,113,114, 116,117,119,120,121, 125, of 262 134, 135, 137, 138, 163, 177, 192, 196, 199, Dirac quantum field, translational invariance 200, 206, 214, 238, 257, 281, 282, 288, 291, of 260 296,301,365,377,387,402,407 Dirac self-adjoint multivector 156 Crumeyrolle-Chevalley spinors 125, 199, 203 Dirac spinor 15, 24, 133, 149, 167, 184, 185, Cuntz algebra 376 188, 258, 284, 291 curvature tensor 186 Dirac spinor fields 178, 185, 188, 190, 191,281 curved spacetimes 128 Dirac spinor, charge conjugation of 151 Dirac spinors and Clifford algebra Cl l ,3 172, D. 173,180,258 d'Alembertian operator 269 Dirac spinors as one-column spinor matrices de Montessus' theorem 352 239 decomposition of the unity 113 Dirac spinors without Clifford algebra 169 degenerate quadratic space 113 Dirac's current vector 138, 163 differential forms 178, 191, 197, 379 Dirac-Hestenes equation 167, 175 diffraction of light 265 Dirac-Kahler fields 193 diffraction of light, by a circular hole 279 Dirac-Kahler spinors 177 diffraction of light, by a narrow slit 278 directed random walk 199 416 domain manifold 338, 340 Domb-Joyce model 203 G. dual two-component spinors 168 Geroch 's theorem 192 dyadics 5 Gibbs, Josiah Willard 5 grad ed algebras 28 E. grad ed algebras, canonical structure of 28 Einstein space, non-compact , of negative grad ed algebras, morphisms 28 scalar curvature 251 grad ed algebras, tensor products of 28 electromagnetic field 26, 290 graded tensor product space 382 electromagnetic moment bivector 138, 163 graded vector space 28 elliptic boundary value probl em 325 grad ed-differential *-algebra 369 endomorphism algebra 101,107 Grassmann algebra 30, 106,121,211,216,379 enveloping algebra 71 Grassmann algebra, quantized 389 Euler angles 10, 22 Grassmann algebra, quantum braid ed 392 Euler-Lagrange equations 292 Grassmann product 6 exterior algebra 6, 30 117, 137, 140, 148, 195, Grassmann, Hermann Giinther 5, 24 207, 213, 282, 291 Grassmann, Th eory of Ext ension 5 exterior algebra fields, equivalence classes of Graves-Morris algorithm 343, 348 177 Graves-Morris, generalized inverse rational exterior algebra, complexification of 207 forms of 346 exterior algebra, grade involution in 141 exterior algebra, Poin care automorphism of H. 196 Hahn-Ban ach theorem 336 exterior algebra, quantizat ion of 398 Hamilt on, William Rowan 4, 10, 24, 265 exteri or algebra, reversion in 141 Hardy, Godfrey Harold 54 exterior derivative 291 Heeke braid 397, 398, 407, 409 exterior differential forms, Kahler product of Heisenberg group 33 140 Hermite interpolants 343, 353 exterior exponential of a bivector 140 Hermite polynomial, vector-valued 356 exterior product , of vector spaces x, 29 Higgs mechanism 282,297,298 Hodge *-operator 291, 366, 370 F.
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