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