Clifford algebra and spinor-valued functions: a function theory for the Dirac operator
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer
1992
|
| Series: | Mathematics and its applications
53 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 485 S. 1 Diskette (3,5") |
| ISBN: | 079230229X |
Staff View
MARC
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| 100 | 1 | |a Delanghe, Richard |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Clifford algebra and spinor-valued functions |b a function theory for the Dirac operator |c by R. Delanghe ; F. Sommen and V. Souček |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer |c 1992 | |
| 300 | |a XVII, 485 S. |e 1 Diskette (3,5") | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications |v 53 | |
| 650 | 0 | 7 | |a Spinoranalysis |0 (DE-588)4182329-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Clifford-Algebra |0 (DE-588)4199958-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Clifford-Algebra |0 (DE-588)4199958-7 |D s |
| 689 | 0 | 1 | |a Spinoranalysis |0 (DE-588)4182329-1 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Sommen, F. |e Verfasser |4 aut | |
| 700 | 1 | |a Souček, Vladimír |e Verfasser |4 aut | |
| 830 | 0 | |a Mathematics and its applications |v 53 |w (DE-604)BV008163334 |9 53 | |
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Record in the Search Index
| _version_ | 1804120068067426304 |
|---|---|
| adam_text | Contents
Editor s Preface vii
Preface xv
0 Clifford algebras over lower dimensional Euclidean spaces 1
1 The algebras C and H 1
1.1 The algebra C 1
1.2 The algebra H 4
2 The Clifford algebras Ro,0 and Ro,i 6
3 The Clifford algebra Ro,2 6
4 The Clifford algebra Ro,3 11
4.1 Vector algebra in R3 11
4.2 The algebra H revisited 13
4.3 The Clifford algebra Ro,3 19
4.4 The Kustaanheimo Stiefel transformation 33
5 The Clifford algebra Ro,4 35
5.1 Definitions 35
5.2 Algebraic structure of Ro,4 38
5.3 The group Spin(4) 41
5.4 The algebra H and rotations in R4 41
Appendix : Euclidean m space 44
A.I. The oriented space Rm 44
A.2 The scalar product Angles Orthogonality 45
A.3. Orthogonal transformations 46
1 Clifford Algebras and Spinor Spaces 48
1 Real Clifford algebras : general theory 49
1.1 Real orthogonal spaces 49
1.2 Real Clifford algebras : definitions 51
1.3 A basis for the Clifford algebra C{X) 53
1.4 Universal and non universal Clifford algebras 55
1.5 The existence of universal Clifford algebras 56
x CONTENTS
1.6 Alternative approaches to Clifford algebras 57
1.7 k Vectors and involutions on Rp,,. The even subalgebra R+q
of Rp,, • 60
1.8 The centers Z and Z+ of Rp,, and R+, 66
1.9 The structure of Rp,, and R+, 67
1.10 Subgroups of Rp,, 69
2 Classification of the real Clifford algebras Rp,, 76
2.1 A reduction theorem 76
2.2 Low dimensional cases 77
2.3 Some tensor products of algebras 79
2.4 The periodicity theorem 80
3 Complex Clifford algebras 84
3.1 Chevalley s definition revisited 84
3.2 Complex Clifford algebras 84
3.3 Involutions on Cn The even subalgebra C+ of Cn 85
3.4 Structure of Cn 87
3.5 Classification of complex Clifford algebras 88
4 Spinor spaces 90
4.1 Real and quaternionic structures on complex vector spaces . . 90
4.2 Representations of algebras and groups 93
4.3 Dirac and Weyl spinors 96
4.4 Dirac and Weyl spinors revisited: Complex representations of
Pin(p, q) and Spin(p, q) 98
4.5 Real and quaternionic representations of Pin(p, q) and Spin(p, q)
Majorana spinors 99
4.6 Hermitian structures on spinor spaces 105
4.7 A realization of spinor space : the Fock space 110
4.8 A Hermitian structure on the Fock space 120
4.9 The complex Spin group and pure spinors 125
II Monogenic functions 129
0 The Dirac and Weyl equations : Elementary Function Theory .... 130
0.1 The Dirac and Weyl equations in Rm+1 131
0.2 Elementary function theory in Rm+1 143
1 Spherical monogenics in Rm+1 157
1.1 Definitions 157
1.2 Remarks 157
1.3 T( acting on L2(Sm; V) Projection operators 160
1.4 An integral formula for inner spherical monogenics 172
1.5 A basis for M+(k;V) and Af (Jfe; V) 173
1.6 Laurent and Taylor series 178
CONTENTS xi
1.7 Integral representations for the projection operators P(A;) and
Q(t) 192
1.8 /^ boundary values of monogenic functions on Sm 196
1.9 Conformal invariance The inversion I* 200
1.10 The Fischer decomposition into spherical monogenics 204
1.11 Spherical monogenics and Casimir operators on Sm 207
1.12 Some operator equalities and commutation relations 215
1.13 From Clifford analysis in R2 to complex analysis in the plane 224
2 Polyaxially monogenic functions 229
2.1 Notations and definitions 229
2.2 Separately spherical monogenics 231
2.3 Polyaxially monogenic functions 235
2.4 Some special cases 240
3 Separately monogenic functions in Rm 243
3.1 Preliminaries 243
3.2 Some elementary properties 243
3.3 Inner separately spherical monogenics 245
3.4 Separately monogenic functions in polyaxially symmetric domains248
3.5 Separately monogenic functions versus holomorphic functions . 251
4 Construction of orthogonal bases for M+(m; s; C) 254
5 Generalized Taylor and Laurent series 265
5.1 Generalized Cauchy Kovalevska extension 265
5.2 Generalized Taylor series 269
5.3 Generalized Laurent series 271
5.4 Generalized Laurent coefficients 273
III Special functions and methods 281
1 Gegenbauer and Hermite polynomials in Rm 282
1.1 Gegenbauer polynomials 282
1.2 Hermite polynomials 303
2 The Cauchy Kovalevska method 310
2.1 The CiT extension principle 310
2.2 The CK extension of xlPk(x)(s € N) 312
2.3 Generalized power functions 317
2.4 Axially exponential functions 326
2.5 Hermite polynomials revisited 327
3 Cauchy type integrals 333
3.1 A Cauchy type integral 333
3.2 Integral representation for generalized power functions .... 334
4 Plane wave integrals 337
4.1 Monogenic plane waves 337
4.2 Exponential plane wave integrals 339
xii CONTENTS
4.3 Generalized power plane wave integrals 340
5 Riesz potentials 348
5.1 Statement of the problem 348
5.2 The distribution ( iy)a 349
5.3 The distribution R% x) 355
IV Monogenic differential forms and residues 357
1 Spinor valued forms 359
2 Invariant operators on forms 369
3 Monogenic differential forms 373
4 Homology of monogenic differential forms 377
5 The Cauchy theorem 380
6 The Residue Theorem 382
6.1 Residues as numbers 383
6.2 Residues as functionals 384
6.3 Computation of residues 385
V Clifford analysis and the Penrose transform 388
1 Elliptic integral formulae 390
1.1 Index of a point 391
1.2 Cauchy integral formula 394
2 Hyperbolic integral formulae 395
2.1 Hyperbolic integral formulae 395
2.2 Hyperbolic integral formula on Minkowski space 402
3 Isotropic flag manifolds 404
3.1 Isotropic flag manifolds 404
3.2 The isotropic Grassmannian IGn 405
3.3 Line bundles on IGn 407
3.4 Invariant forms on isotropic Grassmannians 411
3.5 Forms with coefficients in line bundles 414
4 Twistor correspondence 416
4.1 The basic twistor diagram 416
4.2 Homogeneous coordinates 417
4.3 The twistor space for R2n 419
5 The Penrose transform for the Dirac equation 422
5.1 Definition 422
5.2 The surjectivity 424
Appendices 431
Appendix A 431
Appendix B 441
Appendix C 444
CONTENTS xiii
1 Introduction 444
2 The Clifford algebra Ro,3: 3 Lxlif .red 444
3 The Clifford algebra Ro,4: 4d_clif.red 455
4 The Clifford, red package 462
Bibliography 477
Index 483
|
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| id | DE-604.BV005878839 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T16:36:12Z |
| institution | BVB |
| isbn | 079230229X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-003680853 |
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| owner_facet | DE-12 DE-739 DE-19 DE-BY-UBM DE-706 DE-634 DE-11 |
| physical | XVII, 485 S. 1 Diskette (3,5") |
| publishDate | 1992 |
| publishDateSearch | 1992 |
| publishDateSort | 1992 |
| publisher | Kluwer |
| record_format | marc |
| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spelling | Delanghe, Richard Verfasser aut Clifford algebra and spinor-valued functions a function theory for the Dirac operator by R. Delanghe ; F. Sommen and V. Souček Dordrecht [u.a.] Kluwer 1992 XVII, 485 S. 1 Diskette (3,5") txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 53 Spinoranalysis (DE-588)4182329-1 gnd rswk-swf Clifford-Algebra (DE-588)4199958-7 gnd rswk-swf Clifford-Algebra (DE-588)4199958-7 s Spinoranalysis (DE-588)4182329-1 s DE-604 Sommen, F. Verfasser aut Souček, Vladimír Verfasser aut Mathematics and its applications 53 (DE-604)BV008163334 53 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003680853&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Delanghe, Richard Sommen, F. Souček, Vladimír Clifford algebra and spinor-valued functions a function theory for the Dirac operator Mathematics and its applications Spinoranalysis (DE-588)4182329-1 gnd Clifford-Algebra (DE-588)4199958-7 gnd |
| subject_GND | (DE-588)4182329-1 (DE-588)4199958-7 |
| title | Clifford algebra and spinor-valued functions a function theory for the Dirac operator |
| title_auth | Clifford algebra and spinor-valued functions a function theory for the Dirac operator |
| title_exact_search | Clifford algebra and spinor-valued functions a function theory for the Dirac operator |
| title_full | Clifford algebra and spinor-valued functions a function theory for the Dirac operator by R. Delanghe ; F. Sommen and V. Souček |
| title_fullStr | Clifford algebra and spinor-valued functions a function theory for the Dirac operator by R. Delanghe ; F. Sommen and V. Souček |
| title_full_unstemmed | Clifford algebra and spinor-valued functions a function theory for the Dirac operator by R. Delanghe ; F. Sommen and V. Souček |
| title_short | Clifford algebra and spinor-valued functions |
| title_sort | clifford algebra and spinor valued functions a function theory for the dirac operator |
| title_sub | a function theory for the Dirac operator |
| topic | Spinoranalysis (DE-588)4182329-1 gnd Clifford-Algebra (DE-588)4199958-7 gnd |
| topic_facet | Spinoranalysis Clifford-Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003680853&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
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