Geometry, spinors and applications:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
London ; Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong
Springer [u.a.]
2000
|
| Series: | Springer Praxis books in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 369 S. graph. Darst. |
| ISBN: | 1852332239 |
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Record in the Search Index
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| adam_text | IMAGE 1
DONAL J. HURLEY AND MICHAEL A. VANDYCK
GEOMETRY, SPINORS AND
APPLICATIONS
PUBLISHED IN ASSOCIATION WITH SPRINGER PRAXIS PUBLISHING PR CHICHESTER,
UK
IMAGE 2
TABLE OF CONTENTS
LIST OF TABLES XI
LIST OF FIGURES XIII
PREFACE XV
I PRELIMINARIES AND ALGEBRAIC ASPECTS OF SPINORS 1
1. GENERAL VECTOR SPACES 5
1.1 VECTOR SPACE, DUAL AND DOUBLE DUAL 5
1.1.1 DEFINITION OF A VECTOR SPACE V 5
1.1.2 BASIS OF A VECTOR SPACE 7
1.1.3 LINEAR MAPPINGS 8
1.1.4 DUAL V* OF A VECTOR SPACE V 9
1.1.5 CANONICAL BASIS OF V* 10
1.1.6 CHANGES OF BASES IN V AND V* 10
1.1.7 DOUBLE DUAL V** 11
1.2 TENSORS OVER A VECTOR SPACE 12
1.2.1 DEFINITION OF THE SPACE OF TENSORS 0^ 12
1.2.2 TENSOR PRODUCT 13
1.2.3 CANONICAL BASIS OF 0^ 13
1.2.4 CHANGES OF BASES IN P Q 14
1.2.5 CONTRACTIONS OF TENSORS 15
1.2.6 MAPPING OF TENSORS 17
1.2.7 TENSOR PRODUCTS OF LINEAR MAPPINGS 18
1.3 EXTERIOR ALGEBRAS A* AND A 24
1.3.1 THE VECTOR SPACE OF (/-LINEAR ALTERNATING FORMS 25
IMAGE 3
VI TABLE OF CONTENTS
1.3.2 EXTERIOR PRODUCT A 26
1.3.3 CANONICAL BASIS OF A% 26
1.3.4 INTERIOR PRODUCT I# 27
1.3.5 PULL-BACK OF A G-FORM 28
1.3.6 EXTERIOR ALGEBRA A* 28
1.3.7 EXTERIOR ALGEBRA A 29
2. VECTOR SPACES WITH A METRIE 31
2.1 METRIE AND TENSORS 31
2.1.1 METRIE, SCALAR PRODUCT, NORM 31
2.1.2 EQUIVALENCE BETWEEN V AND V* 32
2.2 CLIFFORD ALGEBRAS AND SPINORS 34
2.2.1 CLIFFORD PRODUCT V AND REAL CLIFFORD ALGEBRAS C RS (R) . .. 34
2.2.2 QUATERNIONS AS THE CLIFFORD ALGEBRA CO2(R) 38
2.2.3 DETAILED STUDY OF THE CLIFFORD ALGEBRA C 3I (R) 40
2.2.4 CLIFFORD GROUP T RS OF THE CLIFFORD ALGEBRA C RS (R) AND ITS
REPRESENTATIONS 45
2.2.5 CLIFFORD GROUP R 0 2 (R) OF THE REAL QUATERNIONS 50
2.2.6 ALGEBRAIC DEFINITION OF C RS (R)-SPINORS 56
2.3 LINEAR MAPPINGS BETWEEN TENSORS AND SPINORS 64
II PRELIMINARIES AND GEOMETRICAL ASPECTS OF SPINORS 71
3. MANIFOLDS IN GENERAL 73
3.1 C MANIFOLDS 73
3.1.1 DEFINITION OF A C MANIFOLD M 73
3.1.2 TANGENT SPACE, COTANGENT SPACE, TENSORS, AND TENSOR FIELDS 75
3.1.3 COMMUTATOR OF TWO VECTOR FIELDS 79
3.1.4 INDUCED MAPPINGS ON A MANIFOLD 80
3.1.5 DERIVATIVES ALONG A CURVE AND INTEGRAL CURVES OF A VECTOR FIELD.
84
IMAGE 4
TABLE OF CONTENTS VII
3.1.6 GENERAL LIE DERIVATIVE AND GEOMETRICAL INTERPRETATION . .. 86
3.1.7 PARTICULAR LIE DERIVATIVES AND PROPERTIES 92
3.2 AFFINE MANIFOLDS 97
3.2.1 COVARIANT DERIVATIVE AND CONNECTION 97
3.2.2 TRANSFORMATION LAW OF THE CONNECTION 99
3.2.3 PARALLEL-TRANSPORT AND EUTHIKHODES 101
3.2.4 TORSION, CURVATURE, AND GEOMETRICAL INTERPRETATION 107
3.3 METRIE MANIFOLDS 113
3.3.1 METRIE TENSOR G 113
3.3.2 NON-METRICITY TENSOR H AND DECOMPOSITION OF THE CONNECTION 114
3.3.3 GEOMETRICAL INTERPRETATION OF THE NON-METRICITY 121
3.3.4 BRACHISTHODES AND GEODESICS 124
3.3.5 SYMMETRIE AND ANTISYMMETRIC PARTS OF THE CONNECTION . . 128
3.3.6 COVARIANT DIFFERENTIATION REVISITED 132
4. LIE GROUPS AS SPECIAL MANIFOLDS 135
4.1 FUNDAMENTAL CONCEPTS 135
4.1.1 DEFINITION OF A LIE GROUP Q 135
4.1.2 LEFT-INVARIANT VECTOR FIELDS AND FORMS 137
4.1.3 LIE ALGEBRA G OF A LIE GROUP Q 138
4.1.4 LEFT ACTION OF A LIE GROUP ON A MANIFOLD 142
4.2 SPECIAL LIE GROUPS AND THEIR LIE ALGEBRAS 143
4.2.1 GENERAL LINEAR GROUP GL N 143
4.2.2 ORTHOGONAL GROUP O RS 146
4.2.3 SPECIAL LINEAR AND ORTHOGONAL GROUPS SL N AND SO RS . . .. 148
4.2.4 CLIFFORD GROUP R 3 1 ( R) AND SPIN GROUP J V R ^ ( R) 152
5. FIBRE BUNDLES AS SPECIAL MANIFOLDS 157
5.1 FUNDAMENTAL CONCEPTS 157
5.1.1 FIBRE BUNDLES AND VECTOR BUNDLES 157
IMAGE 5
VIII TABLE OF CONTENTS
5.1.2 CONSTRUCTION OF A FIBRE BUENDLE FROM ITS TRANSITION FUNCTIONS
UJ 162
5.1.3 CROSS-SECTION A OF A FIBRE BUENDLE 162
5.1.4 PRINCIPAL BUNDLES 164
5.1.5 CONNECTION ON A PRINCIPAL BUENDLE 169
5.2 APPLICATIONS OF FIBRE BUNDLES 175
5.2.1 HORIZONTAL LIFT AND BUENDLE FORMULATION OF COVARIANT
DIFFERENTIATION 175
5.2.2 CONFORMAL BUENDLE AND CONFORMAL CONNECTIONS 185
III GENERAL SPINORIAL DIFFERENTIATION 193
6. GEOMETRICAL DEFINITION OF C3 1 (R) SPINORS 197
7. DIFFERENTIATION OF SPINOR FIELDS 201
7.1 EQUIVALENCE BETWEEN LIE AND COVARIANT DIFFERENTIATION 202
7.2 COVARIANT DERIVATIVE OF A SPINOR FIELD 203
7.3 HEURISTIC APPROACHES TO COVARIANT DIFFERENTIATION 211
7.3.1 TENSORIAL METHOD 211
7.3.2 BUENDLE METHOD 213
7.4 LIE DERIVATIVE OF A SPINOR FIELD 221
7.5 TENSOR-SPINORS AND THEIR DERIVATIVES 223
7.6 COMPLEMENTS 228
8. INTERPLAY BETWEEN DIFFERENTIATIONS 235
8.1 STATEMENT OF THE PROBLEM 235
8.2 INVESTIGATION OF THE PROBLEM 239
8.3 ARBITRARINESS IN THE SPINORIAL DERIVATIVE 244
8.4 COMPARISON WITH CONVENTIONAL METHODS 251
9. THE INVARIANT FORMALISM 255
IMAGE 6
TABLE OF CONTENTS IX
9.1 EXPOSITION 255
9.2 RELATIONSHIP TO THE GENERAL FORMALISM 259
IV ILLUSTRATIONS AND APPLICATIONS 265
LO.NEWTONIAN MECHANICS AND C 30 (R) 269
10.1 GENERAL THEORY 269
10.2 EXAMPLE: ELEMENTARY BALLISTICS 273
LL.ELECTRO-MAGNETISM 281
11.1 TENSORIAL FORMULATION 282
11.2 EXTERIOR-CALCULUS FORMULATION 290
11.2.1 PRELIMINARY: EXTERIOR DIFFERENTIATION 291
11.2.2 ELECTRO-MAGNETIC FORMS 294
11.2.3 ELECTRO-MAGNETIC FORMS AND C31 (R) 296
11.3 RELATIONSHIP BETWEEN THE FORMULATIONS 298
11.3.1 EXTERIOR FORMULATION IN COMPONENTS 298
11.3.2 EXTERIOR FORM AND COVARIANT DIFFERENTIATION 301
11.4 LIE DIFFERENTIATION AND SYMMETRIES 305
11.4.1 GENERAL THEORY 305
11.4.2 EXAMPLE: ELECTRO-MAGNETIC PLANE WAVES 306
12.CARTAN FORMALISM 311
13.GEOMETRICAL GRAVITATIONAL THEORIES 317
13.1 EQUATIONS OF MOTION 318
13.2 FIELD EQUATIONS 320
13.2.1 GENERAL RELATIVITY 320
13.2.2 SIMPLE SUPERGRAVITY 326
13.3 SYMMETRIES 332
13.3.1 GENERAL THEORY 332
IMAGE 7
X TABLE OF CONTENTS
13.3.2 GRAVITATIONAL PLANE WAVES 337
13.3.3 SUPERSYMMETRIC PLANE WAVES 339
13.3.4 COMPLEMENTS ON KILLING S EQUATION 343
APPENDIX A: INFELD-VAN DER WAERDEN SYMBOLS 347
APPENDIX B: MAXWELL S EQUATIONS: COMPLEMENTS 353
B.L INTEGRATION OF A P- FORM 353
B.2 APPLICATION TO MAXWELL S EQUATIONS 356
BIBLIOGRAPHY 359
INDEX 365
|
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| indexdate | 2024-07-09T18:33:35Z |
| institution | BVB |
| isbn | 1852332239 |
| language | German |
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| spelling | Hurley, Donal J. 1944- Verfasser (DE-588)121400166 aut Geometry, spinors and applications Donal J. Hurley and Michel A. Vandyck London ; Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong Springer [u.a.] 2000 XVII, 369 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer Praxis books in mathematics Mathematische Physik Mathematical physics Spinor analysis Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Spinor (DE-588)4182327-8 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s Spinor (DE-588)4182327-8 s DE-604 Vandyck, Michael A. 1959- Verfasser (DE-588)121400174 aut GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008693554&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hurley, Donal J. 1944- Vandyck, Michael A. 1959- Geometry, spinors and applications Mathematische Physik Mathematical physics Spinor analysis Differentialgeometrie (DE-588)4012248-7 gnd Spinor (DE-588)4182327-8 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4182327-8 |
| title | Geometry, spinors and applications |
| title_auth | Geometry, spinors and applications |
| title_exact_search | Geometry, spinors and applications |
| title_full | Geometry, spinors and applications Donal J. Hurley and Michel A. Vandyck |
| title_fullStr | Geometry, spinors and applications Donal J. Hurley and Michel A. Vandyck |
| title_full_unstemmed | Geometry, spinors and applications Donal J. Hurley and Michel A. Vandyck |
| title_short | Geometry, spinors and applications |
| title_sort | geometry spinors and applications |
| topic | Mathematische Physik Mathematical physics Spinor analysis Differentialgeometrie (DE-588)4012248-7 gnd Spinor (DE-588)4182327-8 gnd |
| topic_facet | Mathematische Physik Mathematical physics Spinor analysis Differentialgeometrie Spinor |
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