Clifford (geometric) algebras: with applications to physics, mathematics and engineering
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| Format: | Book |
|---|---|
| Language: | German |
| Published: |
Boston [u.a.]
Birkhäuser
1996
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XVI, 517 S. Ill., graph. Darst. |
| ISBN: | 3764338687 0817638687 |
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MARC
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| 245 | 1 | 0 | |a Clifford (geometric) algebras |b with applications to physics, mathematics and engineering |c William E. Baylis, ed. |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 1996 | |
| 300 | |a XVI, 517 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Literaturangaben | ||
| 650 | 0 | 7 | |a Clifford-Algebra |0 (DE-588)4199958-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Mathematische Physik |0 (DE-588)4037952-8 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)1071861417 |a Konferenzschrift |y 1995 |z Banff Alberta |2 gnd-content | |
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| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Clifford-Algebra |0 (DE-588)4199958-7 |D s |
| 689 | 1 | 1 | |a Mathematische Physik |0 (DE-588)4037952-8 |D s |
| 689 | 1 | |5 DE-604 | |
| 700 | 1 | |a Baylis, William Eric |e Sonstige |4 oth | |
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Record in the Search Index
| _version_ | 1804125551234908160 |
|---|---|
| adam_text | Contents
Preface vii
1 Introduction 1
2 Clifford Algebras and Spinor Operators 5
P. Lounesto
2.1 A History of Clifford Algebras 5
2.2 Teaching Clifford algebras 6
2.2.1 The Clifford Product of Vectors 7
2.2.2 The Exterior Product 8
2.2.3 Components of a Vector in Given Directions 9
2.2.4 Perpendicular Projections and Reflections 10
2.2.5 Matrix Representation of Cl i 11
2.2.6 Exercises 12
2.2.7 Answers 13
2.3 Operator Approach to Spinors 13
2.3.1 Relation to the Convention of Bjorken Drell 14
2.3.2 Bilinear Covariants 16
2.3.3 Fierz Identities (Discovered by Pauli and Kofink) 18
2.3.4 Recovering the Spinor from its Bilinear Covariants 19
2.3.5 Boomerangs and the Reconstruction of Spinors 20
2.3.6 The Mother of All Real Spinors C^i,3 |(1 + 7o) 22
2.3.7 Ideal Spinors 0GC^ii3i(l 703) 23
2.3.8 Spinor Operators tf e C£ti3 24
2.3.9 Decomposition and Factorization of Boomerangs 27
2.4 Flags, Poles and Dipoles 28
2.4.1 A Geometric Classification of Spinors by their Bilinear Co
variants 29
2.4.2 Projection Operators in End(C£ii3) 31
2.4.3 Projection Operators for Majorana and Weyl Spinors 32
3 Introduction to Geometric Algebras 37
G. Sobczyk
3.1 The Unipodal Number System 37
3.2 Clifford Algebra Matrix Algebra Connection 38
v
vi CONTENTS
3.3 Geometric algebra 40
4 Linear Transformations 45
G. Sobczyk
4.1 Structure of a Linear Operator 45
4.2 Isometries 48
4.3 Minimal Polynomials 50
4.4 Lie Algebras of Bivectors 51
5 Directed Integration 53
G. Sobczyk
5.1 Simplices and Chains 53
5.2 Integral Definition of dxF(x) on 5 58
5.3 Classical Integration Theorems 60
5.4 Residue Theorem 61
5.5 Riemannian Geometry 61
6 Linear Algebra 65
C. Doran, A. Lasenby, and S. Gull
6.1 Geometric Algebra 65
6.2 Spacetime Algebra 66
6.3 Geometric V s Tensor Algebra 69
6.4 Index Free Linear Algebra 70
6.5 Multivector Calculus 72
6.6 Adjoints and Inverses 74
6.7 Eigenvectors and Eigenbivectors 75
6.8 Invariants 78
6.9 Linear Functions in Spacetime 79
6.10 Functional Differentiation 79
7 Dynamics 83
5. Gull, C. Doran, and A. Lasenby
7.1 Rigid Body Dynamics 83
7.1.1 The Rotor Equation 83
7.1.2 Kinetic Energy and the Inertia Tensor 84
7.1.3 The Equations of Motion of a Rigid Body 86
7.1.4 Free Precession of a Symmetric Top 87
7.2 Dynamics of Elastic Media 88
7.2.1 Energy Flow 90
7.2.2 Pre Stressed Media 90
7.2.3 Linearised elasticity 91
7.2.4 The Elastic Filament 92
CONTENTS vii
8 Electromagnetism 95
S. Gull, C. Doran, and A. Lasenby
8.1 Electromagnetic Waves 97
8.1.1 Stokes Parameters 97
8.1.2 Reflection by a Conducting Plane 98
8.1.3 Waves in layered media 99
8.2 Diffraction Theory 100
8.2.1 The Boundary Value Problem in Electrodynamics 100
8.3 The Electromagnetic Field of a Point Charge 102
8.4 Applications 104
8.4.1 Uniformly Moving Charge 104
8.4.2 Accelerated Charge 105
8.4.3 Circular Orbits 108
9 Electron Physics I 111
S. Gull, C. Doran, and A. Lasenby
9.1 Pauli Spinors Ill
9.1.1 Pauli Observables 112
9.1.2 Spinors and Rotations 114
9.2 Dirac Spinors 114
9.2.1 Alternative Representations 116
9.3 The Dirac Equation and Observables 117
9.3.1 Plane Wave States 118
9.4 Hamiltonian Form 120
9.5 The Non Relativistic Reduction 121
9.6 Angular Eigenstates and Monogenic Functions 123
9.6.1 The Spherical Monogenics 124
9.7 Application — the Coulomb Problem 126
10 Electron Physics II 129
S. Gull, C. Doran, and A. Lasenby
10.1 Propagation and Characteristic Surfaces 129
10.2 Spinor Potentials and Propagators 130
10.3 Scattering Theory 131
10.3.1 The Born Approximation and Coulomb Scattering 132
10.4 Plane Waves at Potential Steps 133
10.4.1 Matching Conditions for Travelling Waves 135
10.4.2 Matching onto Evanescent Waves 137
10.5 Spin Precession at a Barrier 139
10.6 Tunnelling of Plane Waves 141
10.7 The Klein Paradox 143
11 STA and the Interpretation of Quantum Mechanics 147
A. Lasenby, S. Gull, and C. Doran
11.1 Tunnelling Times 147
viii CONTENTS
11.2 Spin Measurements 151
11.2.1 A Relativistic Model of a Spin Measurement 152
11.2.2 Wavepacket Simulations 154
11.3 The Multiparticle STA 158
11.3.1 2 Particle Pauli States and the Quantum Correlator 159
11.3.2 Multiparticle Wave Equations 161
11.3.3 The Pauli Principle 163
11.3.4 8 Dimensional Streamlines and Pauli Exclusion 165
12 Gravity I — Introduction 171
A. Lasenby, C. Doran, and S. Gull
12.1 Gauge Theories 172
12.2 Gauge Principles and Gravitation 175
12.3 The Gravitational Gauge Fields 177
12.3.1 The Rotation Gauge Field 179
12.4 Observables and Covariant Derivatives 182
13 Gravity II — Field Equations 185
C. Doran, A. Lasenby, and S. Gull
13.1 The Gravitational Field Equations 186
13.2 Covariant Forms of the Field Equations 189
13.3 Symmetries and Invariants of Tl(B) 190
13.4 The Bianchi Identity 193
13.5 Symmetries and Conservation Laws 194
14 Gravity III — First Applications 197
A. Lasenby, C. Doran, and S. Gull
14.1 Spherically Symmetric Static Solutions 197
14.1.1 Point Particle Trajectories 200
14.1.2 Particle Motion in a Spherically Symmetric Background . . 202
14.2 Electromagnetism in a Gravitational Background 206
14.2.1 Application to a Black Hole Background 208
15 Gravity IV — The Intrinsic Method 211
C. Doran, A. Lasenby, and S. Gull
15.1 Spherically Symmetric Systems 212
15.2 Two Applications 217
15.3 Stationary, Axially Symmetric Systems 219
15.4 The Kerr Solution 221
16 Gravity V — Further Applications 223
A. Lasenby, C. Doran, and S. Gull
16.1 Collapsing Dust and Black Hole Formation 223
16.2 Cosmology 225
16.2.1 The Dirac Equation in a Cosmological Background 229
16.2.2 Point Charge in a k 0 Cosmology 230
CONTENTS ix
16.3 Cosmic Strings 232
17 The Paravector Model of Spacetime 237
W. E. Baylis
17.1 A Brief Introduction to the Pauli Algebra 237
17.1.1 Generating Q?3 238
17.1.2 Bivectors as Operators 239
17.1.3 Complex Structure 239
17.1.4 Involutions of O?3 240
17.2 Inverses and the metric 242
17.3 The Spacetime Manifold 244
17.4 Lorentz Transformations I 244
17.5 Vector Notation and Rotations 246
17.5.1 The Merry Go Round 247
17.5.2 Observers, Frames, and Vector Bases 248
17.6 Lorentz Transformations II 248
17.6.1 Proper Velocity 250
17.6.2 Covariant vs. Invariant 251
17.6.3 Spacetime Diagrams 251
18 Eigenspinors in Electrodynamics 253
W. E. Baylis
18.1 Basic Electrodynamics 253
18.2 Eigenspinors 254
18.3 The Group SL(2,C): Diagrams 255
18.4 Time Evolution of Eigenspinor 257
18.4.1 Thomas Precession 258
18.5 Spinorial Lorentz Force Equation 259
18.5.1 Solutions 259
18.6 Electromagnetic Waves in Vacuum 261
18.6.1 Maxwell s Equation in a Vacuum 261
18.7 Projectors 263
18.8 Directed Plane Waves 264
18.8.1 Polarization 266
18.9 Motion of Charges in Plane Waves 267
19 Eigenspinors in Quantum Theory 269
W. E. Baylis
19.1 Introduction 269
19.2 Spin 269
19.2.1 Magic of the Pauli Hamiltonian 270
19.2.2 Classical Spin Distribution 270
19.2.3 Quantum Form 271
19.2.4 Stern Gerlach Filter 271
19.2.5 Linear Combinations of Spatial Rotations 272
x CONTENTS
19.3 Covariant Eigenspinors 273
19.3.1 Generalized Unimodularity 273
19.3.2 Eigenspinor of an Elementary Particle 273
19.4 Differential Operator Form 275
19.4.1 The Electromagnetic Gauge Field 277
19.4.2 Linearity and Superposition 278
19.5 Basic Symmetry Transformations 278
19.6 Relation to Standard Form 279
19.6.1 Weyl Spinors 280
19.6.2 Momentum Eigenstates 281
19.6.3 Standing Waves 281
19.6.4 Zitterbewegung 282
19.7 Hamiltonians 282
19.7.1 Stationary States 282
19.7.2 Landau Levels 283
19.8 Fierz Identities of Bilinear Covariants 283
20 Eigenspinors in Curved Spacetime 285
W. E. Baylis
20.1 Ideals, Spinors, and Symplectic Spaces 285
20.2 Bispinors 287
20.3 Flagpoles and Flags 288
20.4 Spinor Pairs 289
20.5 Time Evolution 290
20.6 Bispinor Basis of C4 291
20.7 Twistors 292
20.8 Relation to SO+(1,3) 293
20.9 Spinors in Curved Spacetime 294
20.10Conclusions 295
21 Spinors: Lorentz Group 297
J. P. Crawford
21.1 Introduction 297
21.2 Lorentz Group 297
21.2.1 Lorentz Lie Algebra 298
21.2.2 Lorentz Group Representations 299
21.3 Summary 305
22 Spinors: Clifford Algebra 307
J. P. Crawford
22.1 Introduction 307
22.2 Clifford Algebra 307
22.2.1 Complex Clifford Algebra 308
22.2.2 Automorphism Group 309
22.2.3 Lorentz Group Redux 310
CONTENTS xi
22.2.4 Poincare Group 313
22.2.5 Conformal Group 314
22.3 Summary 315
23 General Relativity: An Overview 317
J. P. Crawford
23.1 Introduction 317
23.2 Tensor Analysis 318
23.2.1 Vectors and Tensors 318
23.2.2 Affine Connection and Covariant Differentiation 318
23.2.3 Torsion 319
23.2.4 Parallel Transport and Curvature 320
23.2.5 Bianchi Identities 321
23.2.6 Metric 321
23.2.7 Contracted Bianchi Identities 323
23.3 General Relativity 323
23.3.1 The Principle of Equivalence and the Einstein Equation .... 323
23.3.2 Gravitational Action 324
24 Spinors in General Relativity 329
J. P. Crawford
24.1 Introduction 329
24.2 Local Lorentz Invariance 330
24.2.1 Vierbeins 330
24.2.2 Local Lorentz Invariance 331
24.2.3 Covariant Derivative and Spin Connection 331
24.2.4 Lorentz Field Strength Tensor 333
24.2.5 Action and Field Equations 333
24.3 Spinors in Genral Relativity 334
24.3.1 Dirac Equation The Clifford Algebra 334
24.3.2 Lagrangian and Field Equations 336
24.4 Summary and Conclusions 338
25 Hypergravity I 341
J. P. Crawford
25.1 Introduction 341
25.2 Automorphism Invariance 343
25.2.1 Automorphism Group 343
25.2.2 Covariant Derivative and Drehbeins 344
25.2.3 Curvatures and Field Strength Tensors 347
25.2.4 Bianchi Identities 349
25.3 Discussion 350
xii CONTENTS
26 Hypergravity II 353
J. P. Crawford
26.1 Introduction 353
26.2 Lagrangian 354
26.2.1 Gauge Field Terms 354
26.2.2 Spinor Field Terms 354
26.2.3 Drehbein Terms 355
26.3 Field Equations 355
26.3.1 Drehbein Field Equations 356
26.3.2 Spinor Field Equations 357
26.3.3 Gauge Field Equations 357
26.3.4 Gravitational Field Equations 359
26.4 Einstein Gravity Recovered 361
26.5 Discussion 362
27 Properties of Clifford Algebras for Fundamental Particles 365
J. S. R. Chisholm and R. S. Farwell
27.1 Building Blocks of a Gauge Model 365
27.1.1 Introduction 365
27.1.2 The Elements of the Algebra 366
27.1.3 Operations within the Algebra 370
27.2 Spinors in the Clifford Algebra dPA 374
27.2.1 Minimal Left Ideals of JLv%q 374
27.2.2 Spinors in Cg.M 376
27.2.3 Bar Conjugate Spinors in CXPtq 377
27.2.4 The Spinor Norm in O?p,g 379
27.2.5 Left and Right Handed Spinors in OEp q 380
27.3 Selecting a Higher Dimensional Algebra for a Gauge Model 382
27.3.1 The Principles of the Model 382
27.3.2 A Model Based on Cllfi 384
27.3.3 A Gauge Model in Cilfi 385
28 The Extended Grassmann Algebra of R3 389
B. Jancewicz
28.1 Introduction 389
28.2 Multivectors and pseudo multivectors 391
28.3 Forms and pseudoforms 397
28.4 Linear spaces 403
28.5 Various products 408
28.6 Physical quantities 411
28.7 Quadratic spaces 414
28.8 Conclusion 419
CONTENTS xiii
29 Geometric Algebra: Applications in Engineering 423
J. Lasenby
29.1 Applications in Computer Vision 423
29.1.1 Projective Space and Projective Transformations 423
29.1.2 Geometric Invariance in Computer Vision 426
29.1.3 Motion and Structure from Motion 429
29.2 Applications in Robotics/Mechanisms 434
29.2.1 Screw Transformations 435
29.2.2 A simple robot arm 436
29.2.3 Dual Quaternions 438
29.3 Further Applications 440
30 Projective Quadrics, Poles, Polars, and Legendre Transformations441
R. C. Pappas
30.1 Introduction 441
30.2 The Legendre Transformation (1) 442
30.3 Projective Spaces and Geometric Algebra 443
30.4 Quadrics, poles, and polars 443
30.5 The Legendre Transformation (2) 446
30.6 Comments 447
31 Spacetime Algebra and Line Geometry 449
J. G. Maks
31.1 Introduction 449
31.2 Projective geometry 450
31.3 The null polarity belonging to R2 3 452
31.4 Line geometry in spacetime algebra 454
32 Generalizations of Clifford Algebra 459
C. C. Carbno
32.1 Generalizations of Clifford Algebra 459
32.1.1 Introduction 459
32.2 Dimensions of Zero Extent , 460
32.3 Bosonic Vectors 461
32.4 Higher Cycling Dimensions 461
32.5 Conclusion 462
33 Clifford Algebra Computations with Maple 463
R. Ablamowicz
33.1 Introduction 463
33.2 Basic chores 466
33.2.1 cliterms 466
33.2.2 clisort 467
33.2.3 clicollect 467
33.2.4 reorder 467
33.2.5 scalarpart 468
xiv CONTENTS
33.2.6 vectorpart 468
33.3 Ring operations in Clifford algebra and computation of a symbolic
inverse 469
33.3.1 LC: left contraction by a vector and wedge multiplication. . 469
33.3.2 Clifford multiplication cmul or c 471
33.3.3 cinv : symbolic inverse of a multivector 473
33.4 Clifford algebra automorphisms: grade involution, reversion and con¬
jugation 475
33.4.1 Grade involution and reversion 475
33.4.2 Clifford conjugation and complex conjugation 477
33.5 Matrix representations 478
33.5.1 Left regular representations 478
33.5.2 Spinor representations in left minimal ideals 480
33.6 Clifford and exterior exponentiations 484
33.6.1 cexp : Clifford exponentiation 484
33.6.2 wexp : exterior exponentiation 486
33.7 Quaternions and three dimensional rotations 488
33.7.1 Quaternion type, conjugation, norm and inverse 488
33.7.2 rot3d : rotations in three dimensions 489
33.8 Octonions: type, conjugation, norm and inverse 492
33.9 Working with homomorphisms of algebras 494
Index 503
|
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| genre | (DE-588)1071861417 Konferenzschrift 1995 Banff Alberta gnd-content |
| genre_facet | Konferenzschrift 1995 Banff Alberta |
| id | DE-604.BV011062323 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:03:21Z |
| institution | BVB |
| isbn | 3764338687 0817638687 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007408805 |
| oclc_num | 496089047 |
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| owner_facet | DE-91G DE-BY-TUM DE-12 DE-355 DE-BY-UBR DE-20 |
| physical | XVI, 517 S. Ill., graph. Darst. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | Birkhäuser |
| record_format | marc |
| spelling | Clifford (geometric) algebras with applications to physics, mathematics and engineering William E. Baylis, ed. Boston [u.a.] Birkhäuser 1996 XVI, 517 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturangaben Clifford-Algebra (DE-588)4199958-7 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 1995 Banff Alberta gnd-content Clifford-Algebra (DE-588)4199958-7 s DE-604 Mathematische Physik (DE-588)4037952-8 s Baylis, William Eric Sonstige oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007408805&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Clifford (geometric) algebras with applications to physics, mathematics and engineering Clifford-Algebra (DE-588)4199958-7 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| subject_GND | (DE-588)4199958-7 (DE-588)4037952-8 (DE-588)1071861417 |
| title | Clifford (geometric) algebras with applications to physics, mathematics and engineering |
| title_auth | Clifford (geometric) algebras with applications to physics, mathematics and engineering |
| title_exact_search | Clifford (geometric) algebras with applications to physics, mathematics and engineering |
| title_full | Clifford (geometric) algebras with applications to physics, mathematics and engineering William E. Baylis, ed. |
| title_fullStr | Clifford (geometric) algebras with applications to physics, mathematics and engineering William E. Baylis, ed. |
| title_full_unstemmed | Clifford (geometric) algebras with applications to physics, mathematics and engineering William E. Baylis, ed. |
| title_short | Clifford (geometric) algebras |
| title_sort | clifford geometric algebras with applications to physics mathematics and engineering |
| title_sub | with applications to physics, mathematics and engineering |
| topic | Clifford-Algebra (DE-588)4199958-7 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | Clifford-Algebra Mathematische Physik Konferenzschrift 1995 Banff Alberta |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007408805&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT bayliswilliameric cliffordgeometricalgebraswithapplicationstophysicsmathematicsandengineering |