The Many Faces of Maxwell, Dirac and Einstein Equations: A Clifford Bundle Approach

Lecture Notes in Physics......Page 0
Contents......Page 4
Introduction......Page 12
References......Page 27
Multicotensors......Page 29
Tensor Product of Multicotensors......Page 30
Involutions......Page 31
Scalar Products in V and V......Page 32
Exterior and Grassmann Algebras......Page 33
Scalar Product in V......Page 35
Contractions......Page 37
Clifford Algebras......Page 38
Universality of C(V ,)......Page 39
(p,q)-Extensors......Page 43
Symmetric and Antisymmetric Parts of (1,1)-Extensors......Page 44
Properties of the Exterior Power Operator......Page 45
Characteristic Scalars of a (1,1)-Extensor......Page 46
Characteristic Biform of a (1,1)-Extensor......Page 47
Generalization of (1,1)-Extensors......Page 48
Symmetric Automorphisms and Orthogonal Clifford Products......Page 49
The Gauge Metric Extensor h......Page 50
Relation Between dettand the Classical Determinant det[Tij]......Page 52
Strain, Shear and Dilation Associated with Endomorphisms......Page 53
Minkowski Vector Space......Page 55
Lorentz and Poincaré Groups......Page 58
Multiform Functions of a Real Variable......Page 59
Multiform Functions of Multiform Variables......Page 60
Directional Derivatives......Page 61
Chain Rules......Page 62
Derivatives of Multiform Functions......Page 63
The Differential Operator Y......Page 64
References......Page 69
Notes on the Representation Theory of Associative Algebras......Page 71
Real and Complex Clifford Algebras and Their Classification......Page 74
Pauli, Spacetime, Majorana and Dirac Algebras......Page 76
Minimal Lateral Ideals of Rp,q......Page 77
Rp,q, Clifford, Pinor and Spinor Groups......Page 78
Lie Algebra of Spin1,3e......Page 81
Spinor Representations of R4,1, R4,10 and R1,3......Page 82
Algebraic Spin Frames and Spinors......Page 85
Algebraic Dirac Spinors of Type Iu......Page 88
What is a Covariant Dirac Spinor......Page 90
Canonical Form of a Dirac-Hestenes Spinor......Page 92
Reconstruction of a Spinor......Page 93
Lounesto Classification of Spinors......Page 94
Majorana and Weyl Spinors......Page 95
Dotted and Undotted Algebraic Spinors......Page 96
Pauli Algebra......Page 97
Quaternion Algebra......Page 98
Minimal Left and Right Ideals in the Pauli Algebra and Spinors......Page 99
References......Page 103
Differentiable Manifolds......Page 105
Tangent Vectors......Page 107
Vector Fields and Integral Curves......Page 109
Derivative and Pullback Mappings......Page 110
Diffeomorphisms, Pushforward and Pullback when M=N......Page 113
Lie Derivatives......Page 114
Properties of £v......Page 115
Invariance of a Tensor Field......Page 116
Cartan Bundle......Page 117
The Interior Product of Vector and Form Fields......Page 118
Exact and Closed Forms and Cohomology Groups......Page 119
Orientation......Page 120
Integration of a n-form......Page 121
Chains and Homology Groups......Page 123
Stokes Theorem......Page 125
Integration of Closed Forms and de Rham Periods......Page 126
Riemannian and Lorentzian Structures on M......Page 127
Hodge Bundle......Page 128
Structure Equations I......Page 129
Induced Connections Under Diffeomorphisms......Page 133
Spacetimes......Page 134
Representation of Clifford Fields as Sums of Non-homogeneous Differential Forms......Page 136
Pullbacks and Relation Between HodgeStar Operators......Page 137
Dirac Operators......Page 139
Standard Dirac Commutator and Dirac Anticommutator......Page 141
Geometrical Meanings of the Standard Commutator and Anticommutator......Page 142
Associated Dirac Operators......Page 144
The Dirac Operator in Riemann-Cartan-Weyl Spaces......Page 146
Torsion, Strain, Shear and Dilation of a Connection......Page 149
Structure Equations II......Page 153
D\'Alembertian, Ricci and Einstein Operators......Page 154
The Square of a General Dirac Operator......Page 159
Curvature and Torsion Extensors of a Riemann-Cartan Connection......Page 161
Maxwell Equations in the Hodge Bundle......Page 166
Charge Conservation......Page 167
Quantization of Action......Page 168
Maxwell Equation in the Clifford Bundle......Page 169
Einstein Equations and the Field Equations for the a......Page 170
Curvature of a Connection and Bending. The Nunes Connection of 2......Page 171
``Tetrad Postulate\"? On the Necessity of Precise Notations......Page 174
References......Page 177
Reference Frames on Relativistic Spacetimes......Page 180
Rotation and Fermi Transport......Page 184
Frenet Frames Over......Page 186
Rotation 2-form, Pauli-Lubanski Spin 1-form and Classical Spinning Particles......Page 187
Infinitesimally Nearby Observers......Page 188
Einstein Synchronization Procedure......Page 193
Locally Synchronizable Frame......Page 194
Synchronizable Reference Frame......Page 195
Sagnac Effect......Page 196
Characterization of a Spacetime Theory......Page 201
Diffeomorphism Invariance......Page 202
Diffeomorphism Invariance and Maxwell Equations......Page 203
Diffeomorphism Invariance and General Relativity......Page 207
Logunov\'s Objection......Page 208
Symmetry Groups......Page 212
Physically Equivalent Reference Frames......Page 213
Principle of Relativity......Page 214
Internal and External Synchronization Processes......Page 216
External Synchronization......Page 217
A Non Standard Realization of the Lorentz Group......Page 218
Status of the Principle of Relativity......Page 220
Principle of Local Lorentz Invariance in General Relativity Theory?......Page 222
LLRFs and the Equivalence Principle......Page 225
PIRFs on a Friedmann Universe......Page 227
Mechanical Experiments Distinguish PIRFs......Page 229
LLRF and LLRF are not Physically Equivalent on a Friedmann Universe.......Page 231
References......Page 233
The Clifford Bundle of Spacetime......Page 241
Details of the Bundle Structure of C(M,g)......Page 242
Spin Structure......Page 243
Spinor Bundles and Spinor Fields......Page 245
Left and Right Spin-Clifford Bundles......Page 246
Natural Actions on Vector Bundles Associated with PSpin1,3e(M)......Page 248
Bundle of Modules over a Bundle of Algebras......Page 249
Dirac-Hestenes Spinor Fields......Page 250
Fiducial Sections Associated with a Spin Frame......Page 251
Pfaff Derivative of Form Fields......Page 256
Covariant Derivative of Spinor Fields......Page 258
Dirac Equation for Covariant Dirac Fields......Page 262
Spin-Dirac Operator......Page 263
The Dirac-Hestenes Equation......Page 264
Representatives of DHSF on the Clifford Bundle......Page 265
A Comment About the Nature of Spinor Fields......Page 266
Amorphous Spinor Fields......Page 267
Bilinear Covariants Associated with a Representative of a DHSF......Page 268
Commutator of Covariant Derivatives of Spinor Fields and Lichnerowicz Formula......Page 269
The Generalized Lichnerowicz Formula......Page 270
References......Page 274
Some Preliminaries......Page 276
Lagrangians and Lagrangian Densities for Multiform Fields......Page 279
Variations......Page 280
Stationary Action Principle and Euler-Lagrange Equations......Page 282
Maxwell Lagrangian......Page 284
Dirac-Hestenes Lagrangian......Page 285
The Canonical Energy-Momentum Extensor......Page 286
Canonical Energy-Momentum Extensor of the Free Electromagnetic Field......Page 289
Canonical Energy-Momentum Extensor of the Free Dirac-Hestenes Field......Page 291
Canonical Orbital Angular Momentum and Spin Extensors......Page 292
Case L(x,X,xX)......Page 295
Coupled Maxwell and Dirac-Hestenes Fields......Page 296
References......Page 299
Functional Derivatives on Jet Bundles......Page 300
Euler-Lagrange Equations from Lagrangian Densities......Page 301
Invariance of the Action Integral under the Action of a Diffeomorphism......Page 303
Covariant `Conservation\' Laws......Page 304
When Genuine Conservation Laws Do Exist?......Page 308
Pseudo Potentials in General Relativity......Page 312
Is There Any Energy-Momentum Conservation Law in GRT?......Page 314
Is There any Angular Momentum Conservation Law in the GRT......Page 318
References......Page 324
Formulation of the DHE on a RCST......Page 327
Meaning of Active Lorentz Invariance of the Dirac-Hestenes Lagrangian......Page 330
References......Page 332
Gravitational Theory in Minkowski Spacetime......Page 333
Enter h......Page 334
Enter h......Page 335
Relation Between h and h......Page 336
Enter......Page 337
Relation Between LEH and Lg......Page 338
Maxwell Like Form of the Gravitational Equations......Page 340
Mass of the Graviton......Page 342
A Comment on the Teleparallel Equivalent of General Relativity......Page 344
References......Page 347
Preliminaries......Page 349
Exterior Covariant Differential......Page 351
Absolute Differential......Page 353
Clifford Valued Differential Forms......Page 354
Fake Exterior Covariant Differential of Cliforms......Page 357
The Operator Der......Page 358
Cartan Exterior Differential......Page 359
FECD and Levi-Civita Connections......Page 360
The Non-homogeneous Field Equations......Page 363
Another Set of Maxwell-Like Non-homogeneous Equations for Einstein Theory......Page 364
References......Page 367
Dirac-Hestenes Equation and i=-1......Page 369
How i =-1 enters Maxwell Theory......Page 370
Polarization and Stokes Parameters......Page 372
`Dirac-like\' Representations of ME......Page 377
Sachs `Dirac-like\' Representation of ME......Page 378
Sallhöfer `Dirac-like\' Representation of ME......Page 379
A Three Dimensional Representation of the Free ME......Page 380
Mathematical Maxwell-Dirac Equivalence......Page 381
Solution of F=21......Page 383
The General Solution......Page 384
The Generalized Maxwell Equation......Page 385
The Generalized Hertz Potential Equation......Page 386
Maxwell-Dirac Equivalence of First Kind......Page 389
Maxwell-Dirac Equivalence of Second Kind......Page 393
Derivation of Seiberg-Witten Equations......Page 394
A Possible Interpretation of the Seiberg-Witten Equations......Page 395
References......Page 396
Spinor Equation of Motion for a Classical Particle on Minkowski Spacetime......Page 399
Classical Nonlinear DHE......Page 401
The Superparticle......Page 403
Superfields......Page 408
References......Page 411
Fiber Bundles......Page 412
Frame Bundle......Page 415
Orthonormal Frame Bundle......Page 416
Equivalent Definitions of a Connection in Principal Bundles......Page 417
The Connection on the Base Manifold......Page 420
Exterior Covariant Derivatives......Page 421
Transformation of the Field Strengths Under a Change of Gauge......Page 424
Linear Connections on a Manifold M......Page 426
Torsion and Curvature on M......Page 427
Covariant Derivatives on Vector Bundles......Page 428
Connections on E Over a Lorentzian Manifold......Page 432
Curvature Again......Page 433
Exterior Covariant Derivative Again......Page 434
References......Page 435
Acronyms and Abbreviations......Page 437
List of Symbols......Page 439
Index......Page 445