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Index
Ablamowicz, R., 463 action
gravitational, 188, 324 adjoint
Dirac and Hermitian, 117 adjoint transformation, 49 affine approximation, 58 affine connection, 318 affine space, 443
metric of, 321 Aharonov-Bohm solenoid
analogy to cosmic string, 235 algebraic computations, 463 alternating algebra, 494 analytic function
generalization of, 124 angular momentum
in Dirac theory, 123 annihilation-creation operators
fermionic, 158, 163 anti-particle solutions, 118 antiautomorphism, 15, 241 anticommutative product, 9 antiparticle states, 157 antiparticles
Feynman-SW.ckelberg interpretation of, 275
antisymmetrization procedure, 165 Arnold, V. I., 441 Ashtekar formulation of general rela
tivity, 303 Atiyah, M. F., 5 attitude
of vector, 389 attitude spinor, 84 automorphism, 241 automorphism gauge fields
503
vacuum expectation, 353 automorphism gauge group
possible ghost states in, 363 automorphism gauge theory
mass in, 358 automorphism group, 309, 343 automorphism invariance
local, 341 automorphism subgroup, 342
barycentric coordinates, 53 basis
of one-forms, 406 of pseudo-one-forms, 406 of two-forms, 407 of two-pseudo-forms, 407 reciprocal, 48
Baylis, W. E., 1 Bell, J. S., 166 bending of light around sun, 339 Bethe-Salpeter equation
in the STA, 161 Bianchi identities, 63, 190, 194, 321,
349 Bianchi identity, 215
contraction of, 358 for cosmic strings, 233
bilinear covariants, 16, 29, 283, 309 in the STA, 117
bilinear form, 414 nonsymmetric,470
binary pulsars, 339 biquaternions, 438 Birkhoff's theorem, 205 bispinor
in the Pauli algebra, 287 bivector, 2, 7, 39, 41, 65
and isometry, 49
504
as a rotation plane, 239 as an operator, 239 as generator of Lorentz group, 310 attitude and orientation, 391 basis, 405 factorization of, 392 in spacetime, 308 linear mapping of as field tensor,
174 black holes, 203, 218
electromagnetic field of charge out-side, 208
black-hole formation, 223 boomerang, 20, 27 boost, 245
generator of, 298 boost parameter, 245 Born approximation, 132 boson mass terms, 388 bosonic vectors, 461 Bott, R., 5 Boudet, R., 26 boundary
of simplex, 53 boundary theorem, 59 boundary-value problem
in electrodynamics, 100 Boyer-Lindquist form of Kerr metric,
222 Brans-Dicke theory, 361 Brauer, R., 5 Brewster condition, 99 Burke, W. L., 389
C (charge conjugation), 278 (:4, 291 calculus
fundamental theorem of, 60 simplicial, 53
Caley-Hamilton theorem, 75 canonical element, 239 Carbno, C. C., 459 carrier space, 299 Cartan's formula, 471 Cartan, E., 5
Cauchy's theorem generalized, 61
center of an algebra, 4 central potential
in Dirac theory, 123 chain of simplices, 53 characteristic surfaces, 129 charge conjugation, 28 Chevalley, C., 6 chiral spinor, 312 chirality operator, 30
INDEX
chirality projection operators, 312 Chisholm, J. S. R., 365 Christoffel form of connection, 322, 326 Ci3 , 237 Ci2, 8
matrix representation of, 11 Ci3,o,366 Ci l ,3,449 Cil,3 , 50, 307, 368
for curved space, 384 representations of, 307
Ci l ,6,384 Ci2,2, 50 Cin,n , 51 Cin,n, 39 Clifford algebra, 424
chessboard periodicity, 6 for gauge theory, 382 history of, 5, 16 in curved spacetime, 335 in higher dimensions, 366 matrix representation of, 478 most general definition, 6 neutral, 39, 51 nonuniversal, 449 of n-dimensional space, 369 of higher dimension, 382 of spacetime, 307
representations of, 307 periodicity of 8, 5, 6 possible generalizations of, 459 teaching of, 6 universal, 449
Clifford algebras
INDEX
for fundamental particles, 365 Clifford analysis, 61 Clifford conjugate, 12 Clifford conjugation, 3, 15 clifford conjugation, 240 Clifford connection, 341, 344, 353 Clifford curvature
coupling to Riemannian curvature, 359
Clifford curvature tensor, 347 Clifford exponentiation, 484 Clifford monomials, 464 Clifford multivariate polynomial, 465 Clifford numbers, 4 Clifford package, 463 Clifford product, 7, 9, 471 Clifford, W. K., 1, 4, 5, 365 collapsing dust, 223 commutative product, 9 commutative rings, 6 commutator product, 66 complex conjugation, 15
representation dependence, 116 complex numbers, 1, 4
in quantum mechanics, 26 complex structure, 239 components
vector, 9 composition of reflections, 11 computer vision, 423
invariants in, 426 motion in, 429
conformal group, 310, 314 conic, 444 conjugations
notation, 3 conjugte points, 444 connection, 319
Christofel form, 322 conservation laws
of gravitational gauge theory, 194 conservation of energy, 324 constitutive relations, 96 contraction
of multiforms and multivectors, 410
control theory, 440 correlations, 452 cosmic strings, 232
505
cosmological constant, 225, 324, 362 cotangent bundle, 441
symplectic structure of, 447 Coulomb problem
STA treatment, 126 Coulomb scattering, 132 covariance and invariance of observables,
183 covariant derivative, 331
extended, 387 for observables in the STA, 183 in Dirac theory, 173 of spinor, 344 of spinor field, 335
covariant derivatives of timelike bivectors, 213
covariant differentiation, 318 covariant multivector, 180 covariant vs. invariant, 251 covariant W-p-vectors, 389 CPT, 278 Crawford, J. P., 6, 18, 19,28,297 cross ratio, 427 cross ration, 444 Crumeyrolle, A., 494 current
Dirac, 117 in Pauli theory, 123
current density as pseudo-two-form, 412 in Dirac theory, 16
curvature Riemannian and Clifford, 359
curvature bivectors for filament, 92
curved spacetime eigenspinors in, 285
C{ml' m2, ... , m r }, 46
delta function, 61 derivation, 183 derivative
506
covariant, 173 exterior, 63 multivector, 73 normalized, 46
determinant of linear function, 72 Vandermonde,46
Dewdney, C., 151, 152, 154, 155, 157, 166
dielectric media, 96 differential forms, 390 differential geometry, 441 diffraction, 100 dilation
generator of, 315 Dirac action
covariantly-coupled, 188 Dirac adjoint, 15 Dirac algebra, 368 Dirac basis, 343 Dirac conjugate, 308 Dirac current, 308
and antiparticles, 167 Dirac energy-momentum tensor, 336 Dirac equation, 24, 334
Bjorken-Drell formulation, 14 classical, 274 differential form, 275 generally covariant form, 335 Hamiltonian form, 121 Hestenes form, 23 in cosmological background, 229 in the STA, 117 minimally coupled to electromag
netism and gravity, 184 negative-energy solutions, 281 time-independent form, 283
Dirac Lagrangian, 336 Dirac matrices, 4 Dirac plane wave
at potential step, 133 Dirac spinors, 13, 307
and homogeneity, 230 in STA, 114 large and small components, 116
INDEX
metric of, 308 Dirac theory
central potential in, 123 Lorentz transformation, 118 nonrelativistic limit, 121 propagators in, 130
Dirac, P. A. M., 6 Dirac-Pauli bispinor, 280 Dirac-Pauli representation, 115 directed integral
of multivector, 56 directed potential, 412 displacement field, 88 divergence theorem, 60 Doppler shift of electromagnetic wave
constant for moving charge, 267 Doran, C., 31, 65 double numbers, 410 drehbein fields, 344, 355 drehbeins, 341 dual
of antisymmetric tensor, 303 to multivector, 390
dual basis, 406 dual quantities, 397 dual quaternions, 438 duality, 191 duality rotation, 265 duplex numbers, 410 dynamical equations of elastic media
linearization of, 91 dynamics
rigid-body, 83
Eddington-Finkelstein solution, 204 eigenbivectors, 51, 75, 192 eigenprojector form
linear operator, 45 eigenprojectors, 264 eigenspinor, 254
of an elementary particle, 273 time evolution, 257
eigenspinor form of Lorentz-force equation, 259
eigenspinors
INDEX
in curved spacetime, 285 in quantum theory, 269
eigenvectors, 75 eight-square identity, 493 einbien, 200 Einstein equation, 325, 334, 338
Kerr solution, 193, 221 Schwarzchild solution, 193 solutions of, 197
Einstein tensor, 186 for cosmic strings, 233
Einstein-Dirac equations self-consistent solutions to, 230
Einstein-Hilbert action from symmetry-breaking in hyper
gravity, 361 Einstein-Hilbert gravitational Lagran-
gian,388 Einstein-Hilbert Lagrangian, 325, 333 elastic distortion, 88 elastic filament, 92 electric field lines
and null geodesics, 231 electric field strength E
as one-form, 411 electric induction D
as pseudo-two-form, 412 electromagnetic field
as a parabivector in C253 as spacetime plane, 253
electromagnetic field-strength tensor, 305 electromagnetic wave, 261
diffraction of, 100 in layered media, 99 plane, see pane wave264 polarization of, 97 reflection of by conducting plane,
98 electromagnetism
in gravitational background, 206 in the STA, 95
electron spin, 6, 269 electron streamlines, 148 electron velocity
in Dirac theory, 118
electroweak doublet, 385 electroweak interactions
of fermions, 385 standard model, 387
elementary particle, classical, 291 elliptic rotation, 50 End(Cn ),45 energy flow
in elastic media, 90 energy-momentum tensor, 324
of gauge field, 360 of spinor field, 360
Euclid, 443 Euler formula, 42 Euler's equations, 87 Euler-Lagrange equations, 86
for elastic media, 90 evanescent waves
in Dirac theory, 137 existence of imaginary unit, 383 extended Grassmann algebra, 409 exterior derivative, 63 exterior exponentiation, 486 exterior product, 8
factor algebra, 46 factorization
of two-form, 399 of bivector, 404
Faraday bivector, 67, 206 of moving point charge, 103
Farwell, R. S., 365 Fermi-Dirac statistics, 163 Fermi-Walker transport, 294
507
fermion annihilation-creation operators, 163
fermion dynamics in gravitational field, 200
fermion exchange operator, 163 fermion mass
as interaction with frame field, 387 fermions
Clifford algebras for, 365 repulsion of identical, 168
Feynman propagator
508
in STA, 131 Feynman-Stuckelberg interpretation of
antiparticles, 275 fiber derivative, 443 fibre-bundle approach
to classical mechanics, 447 field equations
for gravity in the STA, 185 in hypergravity, 355
field strength R, 185 field strength tensor
as linear mapping of bivectors, 174 Fierz identities, 18, 284 Fierz multivector, 20 flag
null, 289 flagpole, 288 Fock-lvanenko coefficients, 332 forms and pseudoforms, 397 four-vector, 65 frame field, 387 Frankel, T., 389 Friedmann equations, 227 fundamental projective invariant, 427 fundamental theorem of calculus, 60
"I matrices, 115 gauge
Newtonian, 203 gauge choice in cosmology, 228 gauge field, 278
energy-momentum tensor, 360 transformation properties of grav
itational, 185 gauge freedom, 263
rotational, in isotropic media, 91 gauge theory, 366
of electromagnetism, 172 of gravity, 171 principles of in STA, 175
gauge transformations, 263 general linear group, 453 general relativity, 317
as gauge theory, 333 compared to STA gauge theory, 195
generalization of, 341 in the Pauli algebra, 295 Palatini formulation, 326 weaknesses of, 177
INDEX
generators of boosts and rotations chiral definition, 298
geodesic equation, 201, 320, 324 geodesic incompleteness, 205 geodesic motion, 294, 318 geometric algebra, 1,40,65
of 4 dimensions, 424 possible generalizations of, 459 tangent, 62
geometric product, 40 of graded multivecotrs, 66
geometric uncertainty at small scale, 339
ghost states in hypergravity, 363
GL4(R),453 GLn , 51 gln' 51 global basis
of Clifford algebra, 344 global symmetry, 172 grade, 408 grade automorphism, 242 grade involution, 4, 15 graded linear space, 41, 65 Grassmann algebra, 5
extended, 391, 409 of outer forms, 410
Grassmann, H. G., 1 Grassmannian, 451 graviational interactions
of fermions, 387 gravitaional gauge fields, 177 gravitation in STA
vs. general relativity, 177 gravitational acceleration, 324 gravitational action, 324
and Ricci scalar, 188 gravitational field energy
as tensor in hypergravity, 363 gravitational field equations, 337
INDEX
gravitational gauge theory symmetries, 194
gravitational interactions of fermions, 385
gravitational Lagrangian in hypergravity, 354
gravitational lensing, 339 gravitational solutions
collapsing dust, 223 for spherically symmetric matter dis
tributions, 217 for stationary axially symmetric sys
tem,219 gravity
field equations in the STA, 185 without manifolds, 171
Green's function, 61, 100 Gsponer, A., 6 Gull, S., 65 GUrsey, F., 6, 24 gyroscope, 94
Ii-equation, 186 Ii-function
spherically symmetic, 212 Hamilton, J. D., 19 Hamiltonian
Pauli, 270 Hamiltonian mechanics
symplectic structure of phase space, 447
harmonic separation, 444 Havey, F. R, 6 helicity
projection operator, 380 helicity operator, 28 helicity split
of algebra, 384 Helmholtz equation
first-order equivalent of, 100 Hermitean conjugation, 241, 370, 383
as contextual indicator, 16 Hessian, 442 Hestenes, D., 6, 16, 19, 23-26, 30, 81,
88, 191, 424, 440
Higgs potential in hypergravity, 361
Higgs-type potential in hypergravity, 353
Hilbert, D., 443
509
Holland, P. R, 19, 30 homogeneous coordinates, 424, 425 homology theory, 54 horizon, 205, 224 Hubble velocity, 227 Hurni, J.-P., 6 Huschilt, J., viii Huygen's principle, 101 hydrogen atom
STA treatment, 126 hyperbolic numbers, 410 hyperbolic rotation, 50 hypercomplex numbers, 1 hypergravity, 341
gravitational field energy in, 363 low-energy limit gives Einstein grav
ity, 361 total energy in, 360
hyperspinors, 440 hypersurface
of second order, 443
ideal from minimal polynomial, 46
ideals, 374 minimal left, 375 minimal right, 378
idempotent, 263 idempotent basis, 37 idempotents, 374
mutually annihilating, 46 primitive, 374
image charge in k > 0 universe, 231
imaginary unit existence of, 383
index of nilpotent, 46 inertia tensor, 85 Ingarden, R, 389 inner product, 42, 66
510
of spinors, 172 spinor, 112, 117
instantons, 303 intermediate bosons
Clifford algebras for, 365 interpretation of quantum mechanics
in STA, 147 intrinsic method
of gravitational theory, 211 invariance
position-gauge, 176 rotation-gauge, 176
invariants, 78 in computer vision, 426
inverse of vector, 10
inverse function, 74 involution, 240 isometry, 48 isomorphism
between unipodal numbers and diagonal matrices, 38
Jackson, J. D., 105 Jacobi identity, 66, 193, 321, 349 Jamiolkowski, A., 389 Jancewicz, B., 389 Jones vector, 97 Juvet, G., 6, 16
k-vector, 40 Kaluza-Klein theories, 450 Keller, J., 19, 25 kernal
of one-form, 397 Kerr metric, 222 Kerr solution, 193, 221 Killing's equation, 195 Klein four-group, 409 Klein paradox, 284
in STA, 143 Klein quadric, 451 Kofink, 18 Kruskal coordinates, 205 Kustaanheimo, P., 6, 16
Lagrangian Einstein-Hilbert, 325
INDEX
for gravitational gauge fields, 186 in hypergravity, 354 of rotating rigid body, 86
Lagrangian field theory, 382 lamellar vector, 398 Landau levels, 283 Lasenby, A., 65 Lasenby, J., 423 law of inertia, 318 least-squares approach
to computer vision, 430 left contraction, 469 left-regular representation
of Clifford algebra, 478 Legendre transform, 442 leptons
Clifford algebras for, 365 Lie algebra
of bivectors, 51 of proper Lorentz group, 298
Lilmard-Wiechert potential, 102 light deflection by sun
in the STA, 208 line geometry, 454 linear algebra
index-free, 70 linear form, 397 linear functions
in spacetime, 79 linear operator
eigenprojector of, 45 generalized spectral decomposition
of, 46 structure equation of, 45
linear space graded, 65 of bivectors, 405 of pseud~two-forms, 407 of tw~forms, 407
linear space, graded, 41 linear spaces, 403 Lipschitz, R., 5 local Lorentz invariance, 331
INDEX
local symmetry, 172 Lorentz field-strength tensor, 333 Lorentz group, 297
as subgroup of the automorphism group, 310
complex paramters of, 299 real representations, 301 representations of, 297, 299 spinor representations, 300 tensor representations, 302
Lorentz invariant and covariant, 251 Lorentz spinors, 300 Lorentz transformations, 244
active, 245 in Dirac theory, 118 of Dirac spinor, 27 orthochronous and antiorthochro
nous, 298 passive, 245 proper and improper, 298
Lorentz-force equation in the Pauli algebra, 254 spinorial form, 259
Lorenz gauge, 261 Lounesto, P., vii, 5, 389
M4(R),450 Mach's principle, 221 magnetic field strength H
as pseudo-one-form, 413 magnetic induction B
as two-form, 412 magnetic media, 96 Majorana representation, 116 Majorana spinor, 313 Maks, J. G., 449 Maple, 221, 463 Maple Computer Algebra System, 463 maps
linear and anitlinear, 15 mass
without Higgs mechanism, 358 mass term
in 2-particle quantum theory, 162 matrix representation
of Ci2 , 11 of Clifford algebra, 478
matter term, 191 Maxwell equation, 6 Maxwell's equation
511
covariant form in gravitational field, 208
in the STA, 95 Maxwell's equation in C measurement of spin, 151 measurement process, 154 metric, 321
ultrahyperbolic, 51 metric tensor, 368 minimal coupling, 173 minimal left ideal, 480 minimal polynomials
of skewsymmetric operators, 50 Minkowski spacetime metric, 297
from inverse in Ci3 , 242 Mobius transformations, 5 momentum
as vector or one-form, 411 momentum vector field, 113 monogenic equation, 124 motion in computer vision, 429 Mott scattering, 132 multiparticle wave function, 161 multivector
covariant, 180 grade of, 65 homogeneous, 65 in spacetime, 311 symbolic inverse of, 473
multivector calculus, 72 mutliparticle quantum theory, 158
n-particle STA, 158 negative-energy particles, 274 negative-energy solutions, 281
in Dirac theory, 119 Newtonian gauge, 203, 216, 225 nilpotent
projectively related to idempotent, 46
512
Noether's theorem, 202 nonassociative products, 492 nonrenormalizability of perturbative ex-
pansion, 339 null flag, 289 null polarity, 452 null surfaces, 130
observables in the STA, 182
octonions, 492 odd forms, 389 n-equation, 187 w-function, 212 one-form, 397 one-forms
basis of, 406 operator
skewsymmetric, 48 Oppenheimer-Volkov equation, 218 orientation
intrinsic and extrinsic, 397 of a vector, 389
outer forms, 390 outer product, 42, 66
of one-forms, 399 outer product of vectors, 392 outermorphism, 71 Oziewicz, Z., 494
pair production, 157 at Klein barrier, 144
Palatini formulation of general relativity, 326
Pappas, R. C., 441 parabivector, 246 parabolic null bivector, 50 parabolic rotation, 50 parallel transport, 320
of Clifford-algebra generator, 349 of drehbein, 347 of spinor, 348
paravector, 240 paravector basis, 4 partition of unity, 46 Pauli algebra, 1, 237, 366, 375
2-particle theory, 159 conjugations in, 240
Pauli exclusion principle, 163 Lorentz invariance of, 165
Pauli Hamiltonian, 270 Pauli matrices, 111, 367 Pauli spin matrices, 2 Pauli spinors
INDEX
related to STA multivectors, 111 Pauli theory
current in, 123 Pauli, W., 6,18 Pauli-Schrodinger equation, 283 pencil of lines, 456 Penrose flags, 31 perihelion shift, 339 permittivity/permeability tensor
generalized, 208 perpendicular projection of vector, 10 perplex numbers, 410 Petrov classification, 63, 192 photon paths
in Newtonian gauge, 228 physical quantities
identified as directed objects, 411 Planck mass, 358 plane
oriented, 7 plane wave, Dirac
at potential step, 133 plane wave, electromagnetic
directed, 264 monochromatic, 265 motion of charges in, 267 parallel E and B in standing, 265 polarization of, 266
plane-wave state in Dirac theory, 119
PlUcker coordinates, 451 Poincare group, 310, 313 Poincare invariance, 175 Poincare Lie algebra, 314 point charge
in k > 0 background, 230 polar of a point, 444
INDEX
polarization, 266 of electromagnetic wave, 97
polarization force of charge outside black hole, 209
polarization operator in STA, 140
pole of a hyperplane, 444 Porteous, I. R., 6 position gauge
transformation properties, 185 position-gauge invariance, 176 potential step
in Dirac theory, 133 Pound-Rebka experiment, 339 pre-stressed media, 90 precession of orbit, 202 premetric space, 389 primitive idempotent, 374, 480
to define spinor, 385 primitive idempotents, 39 principle of equivalence, 317, 323 principle of general covariance, 317 probability density
in Dirac theory, 16 product
commutative and anticommutative, 9
commutator, 66 inner, 66 outer, 66 scalar, 66
projection operator energy, 29 for helicity, 380 in End( Gil,3), 31 spin, 29
projection operators chirality, 312 self-dual and antiself-dual, 304
projective geometry, 423, 450 projective space, 424, 443 projective split, 425 projective Weyl tensor, 63 projector, 263 propagators
in Dirac theory, 130 proper time
as scalar field, 103 proper velocity, 250 pseudo-one-forms, 398
basis of, 406 pseudo-three-forms, 402 pseudo-two-forms, 400
basis of, 407 linear space of, 407
pseudobivector attitude and orientation, 392
pseudoforms, 389
513
mapped into pseudovectors, 417 pseudomultivectors, 389 pseudoscalar, 65
attitude and orientation, 397 in spacetime, 308
pseudotrivector attitude and orientation, 395
pseudovector attitude and orientation, 391 in spacetime, 308
quadratic forms, 6 quadric, 444 quantum correlator, 160 quantum mechanics
interpretation of, 147 quantum theory
multi particle, 158 with eigenspinors, 269
quark colors, 385 quarks
Clifford algebras for, 365 quaternions, 1, 5, 488
dual, 438
radiating charge in the STA, 102
Radon-Hurwitz number, 375 rapidity, 245 reciprocal frame, 73 redshift, 228 reflection and transmission
in Dirac theory, 136
514
of Dirac wave at barrier, 142 reflections, 3
composition of, 11 relativity, 244
general, 317 representation-free equations
in geometric algebra, 116 representations
slef-dual, 303 representations of Lorentz group
irreducible, 299 repulsion of fermions
marginal effect of spin, 168 residue theorem, 61 reversal, 241 reversion, 3, 15, 370
mixes grades, 476 spatial, 68
Ricci identity, 63 Ricci paravectors, 295 Ricci scalar, 186, 323 Ricci tensor, 186, 323 Riemann condition, 322 Riemann parabivectors, 295 Riemann tensor, 186, 189, 205, 213,
320 degrees of freedom, 192 for axial mass distribution, 221 for cosmic strings, 233 for Kerr solution, 222 for radial mass distribution, 218 symmetries and invariants of, 190
Riemannian curvature, 62 coupling to Clifford curvature, 359
Riemannian geometry, 53 Riemannian manifold
abstract, 61 Riesz, M., 6, 16, 48 rigid-body displacements, 438 ring
of double numbers, 410 robot arm, 436 robotics, 434 Rodriguez-Romo, S., 19 rotation, 2, 239
INDEX
as composition of reflections, 11 duality, 265 elliptic, 50 generator of, 298 hyperbolic, 50 of spinor, 114 parabolic, 50
rotation gauge transformation properties, 185
rotation group, 300 rotation group SO(n), 5 rotation-gauge invariance, 176 rotations
linear superposition of, 271 spacetime, 245
rotor, 83 rotor equation, 84
Sauter, F., 6, 16 scalar, 41, 65 scalar field
energy momentum tensor for, 325 scalar product, 8, 66,414 scattering theory
in STA, 131 Schidlof, A., 6 Schouten, J. A., 389 Schwarzchild solution, 193 Schwarzschild metric, 199 Schwarzschild solution, 200, 203 Schwarzschild, K., 197 screw axis, 435 screw transformation, 434 self-dual representations, 303 shape operator, 62 Shapiro, A., 5 signal processing, 440 simplex, 53
complementary, 55 directed content of, 54
simplicial calculus, 53 singular-value decomposition, 77 singular-value decomposition, 432 skewsymmetric transformation, 48 SL(2,C), 259, 292, 299, 454
INDEX
diagreams, 255 8nell's law, 99 80+(1,3), 256 80+(3),256 80(1,3), 298, 299, 342 so(I,3), 298, 299 80(2,3), 453 80(2,4), 315 so(3), 299 80(3,C), 299 SO(n),5 SOn,n, 51 SOn,n, 51 SOp,q, 52 SOp,q, 52 8obczyk, G., 37, 81, 191 sound speeds
in elastic media, 94 Sp4(R),453 Sp4(R),454 spacetime algebra, 66, 449 spacetime diagrams, 251 spacetime manifold, 244, 318 spacetime metric tensor, 242 spacetime rotations, 245 spacetime split, 67 spacetime vectors, 4
as paravectors of Ci3 , 243 spatial densities
as pseudo-three-forms, 413 spatial reversal, 240 spatial reversion, 68 spherical harmonics, 125 spherical monogenics, 124 spin, 269
classical distribution, 270 electron, 6 marginal effect in fermion repul
sion, 168 parabivector, 288 paravector, 288
spin bivector comoving, 119
spin connection, 331 spin curvature, 333, 347
spin gauge theory, 386 spin measurements, 151 spin precession
at barrier, 139 spin tensor
covariant in the 8TA, 188 spin vector, 113 8pin(n), 5 Spin+(I, 3), 454 Spin+(2, 3),453 spin-l/2
origin of in Dirac theory, 124 spinor
as minimal left ideal, 375 bar conjugate, 377 chiral, 312 covariant derivative of, 344 Dirac, 13, 114 dotted and undotted, 300 flag-dipole, 28 from bilinear covariants, 19 geometric classification of, 29 Hestenes, 23 ideal, 23 in any Clifford algebra, 382 in general relativity, 329 in minimal left ideal, 6, 16 inner product, 301 inner product in 8TA, 172 Lorentz, 300 Majorana, 28, 313 matrix representation, 14 mother of real, 22 norm, 379 Pauli, 111 Weyl, 28, 312
spinor basis, 481 spinor field, 297, 354
covariant derivative of, 335 energy-momentum tensor, 360
spinor inner product, 112, 117 spinor metric, 301 spinor potential, 130 spinor representations
of Clifford algebras, 480
515
516
spinor representations of Lorentz group, 300
spinor space as minimal left ideal, 16
split complex numbers, 410 STA, 66
n-partic1e, 158 standard model of electroweak interac
tions, 387 star
gravitational solution for rotating, 219
steamlines for 2 fermions, 165
Stern-Gerlach apparatus, 157 Stern-Gerlach experiment, 271 Stokes parameters, 97 Stokes' theorem, 60 strain tensor, 88 streamlines of D outside black hole,
209 stress-energy tensor, 187, 191
for radial mass distribution, 218 in gravitational background, 208 in the STA, 96
structure equation of linear operator, 45
SU(2),256 su(2), 299 SU(2) x U(l), 387 SU(2,2), 315 SU(2,C), 299, 342 surfing an electromagnetic wave, 268 symbolic computations, 463 symmetric top, 87 symmetries
local Lorentz invariance, 331 of gravitational gauge theory, 194
symmetry global, 172 local, 172
sympectic structure of cotangent blllldle, 447
symplectic form, 452 symplectic structure, 286
INDEX
possible projective origin of, 448 synchrotron radiation, 109
Takahashi, Y., 6 tangent bundle, 62, 441 tangent hyperplanes, 443 tangent line
to a quadric, 445 tangent space, 318 tangent vector, 62 Taylor's theorem, 58 TE and TM modes, 98 tensor, 318
contravariant components, 318 covariant components, 319
tensor analysis, 69 tensor fields
representation structure of, 303 self-dual, 303
Thomas precession, 258 three-dimensional objects
attitude and orientation, 394 three-form, 401 time reversal, 200 topological defect, 232 torsion, 319 torsion tensor
contracted, 323 total connection, 344 total curvature tensor, 348 total energy
in hypergravity, 360 transformation
adjoint, 49 skewsymmetric, 48
transformation properties of gravitational gauge fields, 185
'frayling, G., 28 trivector, 3, 65
attitude and orientation, 394 tunneling
in Dirac theory, 141 tunneling time, 147 tunneling velocity
inside quantum barrier, 142
INDEX
twisted forms, 389 two-forms, 399
basis of, 407 linear space of, 407
U(2,2),342 unimodular element, 245 unipodal algebra, 39 unipodal number, 37
inverse of, 38 unipodal numbers, 410 unipotent elements, 39
Vahlen, K. Th., 5 Vandermonde determinant, 46 Vandermonde matrix
generalized, 46 vector, 65
attitude of, 389 direction and orientation of, 391 inverse of, 10 orientation of, 389 perpendicular projection, 10 perpendicular projection of, 10 reflection of, 10
vector components, 9 vector derivative, 58, 68
linear distortion of, 162 Riemannian, 61
vector Helmholtz equation, 91 vector space
without a metric, 389 velocity
inside potential barrier, 142 not collinear with momentum for
fermion, 201 vierbein, 329, 330 vierbein field, 368 Viniegra, F., 25
W /Z mass ratio, 388 wave equation, 261 wave function
multiparticle, 161 wave function collapse
avoided with streamlines, 149
517
wave propagation in Maxwell and Dirac theories, 130
wavity,411 weak associativity, 494 wedge equation, 189 wedge product, 8 Weinberg mixing angle, 387 Weyl representation, 116, 280 Weyl spinor, 312 Weyl spinors, 280 Weyl tensor, 191, 221
in Kerr solution, 221 Weyl, H., 5 Witt, E., 6 wrench
coupling a filament, 93
Yang-Mills field energy momentum tensor for, 325
Yang-Mills theory, 303 Young's modulus, 92 Yukawa coupling, 355 Yvon-Takabayasi angle {3, 29, 118
Z2-gradation, 409 zero-divisors, 263, 410 Ziegler, R., 424 Zitterbewegung, 281, 282