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Snygg J. Ч Clifford algebra: a computational tool for physicists
Snygg J. Ч Clifford algebra: a computational tool for physicists

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Ќазвание: Clifford algebra: a computational tool for physicists

јвтор: Snygg J.


Clifford algebras have become an indispensable tool for physicists at the cutting edge of theoretical investigations. Applications in physics range from special relativity and the rotating top at one end of the spectrum, to general relativity and Dirac's equation for the electron at the other. Clifford algebras have also become a virtual necessity in some areas of physics, and their usefulness is expanding in other areas, such as algebraic manipulations involving Dirac matrices in quantum thermodynamics; Kaluza- Klein theories and dimensional renormalization theories; and the formation of superstring theories. This book, aimed at beginning graduate students in physics and math, introduces readers to the techniques of Clifford algebras.

язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1997

 оличество страниц: 335

ƒобавлена в каталог: 06.12.2013

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ѕредметный указатель
$4\pi$ periodicity of rotation operator      11Ч12
1-forms      50 51
1-vector      5 42
2-vector      5
3-vector      5
4-momentum      37
4-momentum, conservation of      40
Abramowitz, Milton      191
Abstract indices      89
Adair, Thomas W.      III 38
Adams, John Couch, prediction of Neptune      126
Adler, Ronald      217
Albert, Abraham Adrian      308
Algebra      5 42
Algebra, associative      42
Algebra, generators of      295
Anderson, Carl D., discovery of positron      181
Angle of precession for spinning top      14
Areal velocity      118
Associated Legendre functions      191
Associative algebra      42
Atiyah, Michael F.      294
Axial vector (pseudo-vector)      8
Bazin, Maurice      217
Bianchi identity for a Yang Ч Mills field      145
Bianchi identity in computation of Kerr metric      280Ч282
Bianchi identity, proof of      323Ч324
Biorthogonal bases for a vector space      47Ч48 70Ч74
Bivector, simple unit space-space      309
Bivector, simple unit space-time      309Ч310
Bohr radius      214
Bohr, Niels, criticism of Dirac's antiparticle      181
Bolker, Ethan      11
Boost      29Ч32
Bott, Raoul H.      294
Bound charges and currents      141Ч142
Boyer Ч Lindquist coordinates      246 248
Boyer, Robert H.      242
c-Hermitian      171
c-unitary Clifford numbers      171 301Ч306
c-unitary groups, classification of      301Ч306
Canonical form for solutions of Dirac's equation      185Ч189
Cartan, Elie, theorem on rotations and reflections      9
Chandrasekhar, Subrahmanyan      247
Charge conjugate of a Clifford number      178Ч179 185 202
Christoffel symbols      76Ч78 86Ч88
Christoffel symbols, formula for      87
Christoffel symbols, transformation of      88
Clifford algebra, for $E^{3}$      3Ч5
Clifford algebras      3Ч5 41Ч42 288
Clifford algebras for pseudo-Euclidean spaces      43
Clifford algebras, complex      42 288
Clifford algebras, construction of      288Ч294
Clifford algebras, real      42 288
Clifford algebras, singular      76
Clifford algebras, universal      44 288
Clifford number      5 42
Clifford number, complex      42
Clifford number, index free      88
Clifford number, real      42
Clifford number, simple      309
Clifford reverse      57Ч58 307
Clifford, William Kingdon      41
Closed p-vector      134
Codifferential operator      130Ч137
Comma notation      88
Commutator coefficients      91Ч92
Complex Clifford algebras      42 288
Complex Clifford number      42
Complex conjugate of a Clifford number      171
Complex reverse of a Clifford number      171
Complex scalar product for Clifford numbers      171
Components of a contravariant tensor of order one      51
Components of a covariant tensor of order one or two      51
Components of a mixed tensor of order two      51Ч52
Confluent hypergeometric function      208Ч211
Conformal tensor      250Ч251
Conjugate, complex, of a Clifford number      171
Connection coefficients      90Ч91
Connections for Yang Ч Mills field      144
Contracted Christoffel symbol      132Ч133
Contracted tensor      90
Contravariant tensor, components of      51
Convection current      175
Converse of Poincare lemma      133Ч134
Coordinate system of Dirac matrices      45
Cosmological constant      149
Covariant derivative of a Dirac matrix      89Ч90
Covariant derivative of a tensor component      89
Covariant tensor, components of      51
Cowan, Clyde, discovery of neutrino      37
Current 1-vector      139Ч141
Curvature 2-form      101
Curvature 2-form as an infinitesimal rotation operator      104-110
Cyclotron and Migma chamber      154Ч162
Cylindrical coordinates      53
Degenerate metric      104
Derivative operator $\nabla_{x}$      86Ч89
Derivative operator $\nabla_{x}$ for m-dimensional spaces embedded in n-dimensional spaces      75Ч79
Dieudonne, Jean, theorem on rotations and reflections      9
Differential forms      50Ч51
Dipole, electric and magnetic of free electron      184
Dirac matrices      41
Dirac matrices, coordinate system of      45
Dirac matrices, lower index      48
Dirac matrices, orthonormal system of, for Euclidean spaces      41
Dirac matrices, orthonormal system of, for pseudo-Euclidean spaces      42Ч44
Dirac matrices, upper index      48
Dirac's equation for hydrogen-like atoms      204Ч216
Dirac's equation for the electron      170Ч216
Dirac, introduction of Dirac matrices      41
Dirac, Paul A.M.      41 170
Dirac, prediction of positrons      181
Direct product      295
Direct sum      293Ч294
Dual vector space      51
Duality rotation      311
Eddington, Arthur S.      170 283 286
Eddington, his form of the Schwarzschild metric      283 286
Einstein, Albert      26 27 127
Einstein, field equations cast in form of a Yang Ч Mills field      xv 149Ч154
Einstein, general theory of relativity      111
Einstein, special theory of relativity      25Ч40
Electric dipole of a free electron      184
Electromagnetic-momentum tensor      143
Energy-momentum tensor      149
Euler Ч Lagrange equations for the motion of a planet according to Newton's theory      117Ч123
Euler Ч Lagrange equations for the motion of a planet according to the Schwarzschild metric      123Ч127
Euler Ч Lagrange equations for the rotating top      18
Exterior derivative d      130Ч137
Exterior product      55
Faraday 2-vector      138
Faraday tensor      138Ч141
Field strength tensor for a Yang Ч Mills field      144
Fine structure constant      205
Flanders, Harley      58 131 136
Fock Ч Ivanenko 2-vectors, applications of      xivЧxv
Fock Ч Ivanenko 2-vectors, computation of curvature 2-forms      101Ч103
Fock Ч Ivanenko 2-vectors, computation of Schwarzschild metric      111Ч116
Fock Ч Ivanenko 2-vectors, definition and formula for      93Ч94
Fock Ч Ivanenko 2-vectors, history of introduction of      171
Fock, Vladimir A.      xiv 93 171 178
Franz, Walter      170
Free charges and currents      141Ч142
Friedman metric      103Ч104 116
Galle, Johann Gottfried, discovery of Neptune      126
Gallilean transformation      26
Gauge potential for Yang Ч Mills field      xv 144
Gauss's theorem      169
Gaussian curvature for more general surfaces      65
Gaussian curvature of a sphere      62
Gaussian curvature, surfaces with negative curvature      66Ч67
Gaussian curvature, surfaces with zero curvature      65Ч67
General linear group      305
Generalized Kronecker delta symbol      54
Generalized Laguerre polynomials      210 214
Generalized Stoke's theorem      161Ч169
Generators of an algebra      295
Geodesic      60Ч61 81Ч82
Geodesic, triangle      61
Georgi, Howard      287
Glauber, Roy J.      170
Goldstein, Herbert      23
Gordon decomposition      173Ч175
Gordon, Morton M.      154
Gordon, Walter      173
Graham, Alexander      291
Guess and check method      116
Guggenheimer, Heinrich W.      131
Hamilton, William      7Ч8
Herlt, E.      264
Hestenes, David      57 311
Hodge star operator      136Ч137
Hydrogen atom, energy levels of, from Dirac's equation      209Ч211
Hydrogen atom, energy levels of, from Schroedinger's equation      212
Index free Clifford number      53 88
Index lowering operator      52
Index raising operator      52
Inertial mass      37
Internal current      174Ч175
Isotropic vector      76
Itzykson, Claude      179
Ivanenko, Dimitrii      xiv 93 171
J, normalized pseudo-scalar      134Ч136
Jackson, John David      191
Jacobi identities      96
Johnson, David A.      154
Kaluza, Theodor      44
Kepler's first law      121
Kepler's second law      118
Kepler's third law      122Ч123
Kerr metric      217Ч249 272Ч286
Kerr, Roy P.      217
Kramer, D.      250 264
Kronecker delta symbol, generalized      54
Kronecker product      291 295
Kummer's differential equation      208
Kummer's function      208Ч211
Laguerre generalized polynomials      210Ч211 214Ч215
Laplacian operator in cylindrical coordinates      53
Laplacian operator in Euclidean coordinates      52
Laplacian operator in spherical coordinates      53
Leibniz property      86 89
Lense Ч Thirring line element      246Ч247
Lense, Von J.      246
Leverrier, Urbain Jean Joseph, precession of Mercury      127
Leverrier, Urbain Jean Joseph, prediction of Neptune      126
Light-like eigenvector of a Lorentz transformation      321Ч322
Light-like plane      322
Light-like vector      76 321Ч322
Lindquist, Richard W.      242
Lorentz contraction      32Ч34
Lorentz force      138 142
Lorentz transformation      26 308
Lorentz transformation, restricted      308
Lorentz, Hendrik      25Ч26
Lounesto, Pertti      294Ч295
Lower index Dirac matrices      48
Lune      61
MacCallum, M.      264
Maglich, Bogdan, his migma chamber      154Ч161
Magnetic dipole of a free electron      184
Magnus, Wilhelm      209 214Ч215
Margenau, Henry      237
Martin, Paul C.      170
Mass, inertial      37
Mass, rest      37
Maxwell's equations      137Ч144
Maxwell's equations and special relativity      25
Mercury, precession of      117Ч127
Metric tensor      47
Metric tensor, degenerate      104
Michelson, Albert A.      25
Migma chamber of Bogdan Maglich      154Ч161
Minkowski space      27Ч29
Misner, Charles W.      116
Mixed tensor      51
Mixed tensor, alignment of indices      51
Moment of inertia coefficients for spinning top      18Ч23
Morley, Edward W.      25
Murphy, George Moseley      237
n-dimensional orthonormal Euclidean system of Dirac matrices      41
Neptune, discovery of      126
Neutrino, prediction and discovery of      37
Newcomb, Simon, precession of Mercury      127
Non-singular metric      50 see
Normalized pseudo-scalar J      134Ч136
Notation for coordinate Dirac matrices      45
Notation for non-coordinate Dirac matrices      45
Null eigenvectors of a Lorentz transformation      322
Null plane      267
Null rotation      321
Nutation of a spinning top      22
Oberhettmger, Fritz      209 214Ч215
Oblate spherical coordinates      237Ч240
Orthochronous Lorentz transformation      256Ч257
Orthonormal system of Dirac matrices for Euclidean spaces      41
Orthonormal system of Dirac matrices for pseudo-Euclidean spaces      42Ч44
p-tangent vector      42
p-vector      42
Palatini action      150
Parallel transport      62Ч65 80Ч83
Pauli matrices      10
Pauli, postulation of neutrino      37
Pauli, Wolfgang, criticism of Dirac's antiparticle      181
Penrose, Roger      xiv 89 97 322
Petrov matrix      250Ч255
Petrov's canonical forms      250Ч263
Petrov's canonical forms, summary      263
Petrov, Aleksei Zinoveivich      250
phase velocity      184
Planetary motion according to Newton's theory      117Ч123
Planetary motion according to the Schwarzschild metric      123Ч127
Poincare lemma      133Ч134
Poincare lemma, converse of      133
Poincare, Henri, special relativity      26
Porteous, Ian R.      43 294 297
Position vector      45
Precession for rotating top, angle of      14
Precession for rotating top, rate of      22Ч23
Precession of the perihelion for Mercury      117Ч127
Principal null congruence      248
Principal null directions      266Ч277
Proca, Alexandre      170
Projection operators      313 323
Projection operators for Dirac's equation for electron      183
Proper time interval      36
Pseudo-Euchdean space      42Ч43
Pseudo-scalar      9
Pseudo-scalar, normalized      134Ч136
Pseudo-vector      8
Quaternions      7Ч8
Real Clifford algebra      42
Real Clifford number      42
Regular point      70
Reines, Frederick, discovery of neutrino      37
Rest mass      37
Restricted Lorentz transformation      308
Restricted Lorentz transformation, exponential representation of      314Ч323
Reverse of a Clifford number, complex      171
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