Linear representations of the Lorentz group:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford [u.a.]
Pergamon Press
1964
|
| Series: | International series of monographs on pure and applied mathematics
63 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 447 S. Ill. |
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| 240 | 1 | 0 | |a Linejnye predstavlenija gruppy Lorenca |
| 245 | 1 | 0 | |a Linear representations of the Lorentz group |c M. A. Naimark |
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Record in the Search Index
| _version_ | 1804121296813948928 |
|---|---|
| adam_text | CONTENTS
PREFACE xi
CHAPTER I. THE THREE DIMENSIONAL ROTATION GROUP
AND THE LORENTZ GROUP 1
1. The Three dimensional Rotation Group 1
1. General definition of a group. 1
2. Definition of the three dimensional rotation group. 2
3. Description of rotations by means of orthogonal matrices. 2
4. Eulerian angles. 5
5. The description of rotation by means of unitary matrices. 7
6. The invariant integral over the rotation group. 13
7. The invariant integral on the unitary group. 17
2. The Lorentz Group 18
1. The general Lorentz group. 18
2. The complete Lorentz group and the proper Lorentz group. 23
CHAPTER n. THE REPRESENTATIONS OF THE THREE
DIMENSIONAL ROTATION GROUP 25
3. The Basic Concepts of the Theory of Finite dimensional
Representations 25
1. Linear spaces. 25
2. Linear operators. 27
3. Definition of a finite dimensional representation of a group. 28
4. Continuous finite dimensional representations of the three
dimensional rotation group. 29
5. Unitary representations. 30
4. Irreducible Representations of the Three dimensional Rotation
Group in Infinitesimal Form 31
1. Differentiability of representations of the group Go. 31
2. Basic infinitesimal matrices of the group ?„. 33
3. Basic infinitesimal operators of a representation of the group GB. 35
4. Relations between the basic infinitesimal operators of a repre¬
sentation of the group Gt. 39
5. The condition for a representation to be unitary. 41
6. General form of the basic infinitesimal operators of the irre¬
ducible representations of the group Go. 43
v
Vi CONTENTS
5. The Realization of Finite dimensional Irreducible Represen¬
tations OF THE THREE RIMENSIONAL ROTATION GROUP 50
1. The connection between the representations of the group Go and
the representations of the unitary group It. 50
2. Spinor representations of the group U 51
3. Realization of the representations 6m in a space of polynomials. 54
4. Basic infinitesimal operators of the representation Sm 56
5. Orthogonality relations. 60
6. The Decomposition of a given Representation of the Three
dimensional Rotation Group into Irreducible Representations 63
1. The case of a finite dimensional unitary representation. 63
2. The theorem of completeness. 66
3. General definition of a representation. 68
4. Continuous representations. 70
5. The integrals of vector and operator functions. 73
6. Decomposition of a representation of the group U into irreducible 77
representations.
7. The case of a unitary representation. 83
CHAPTER III. IRREDUCIBLE LINEAR REPRESENTATIONS OF
THE PROPER AND COMPLETE LORENTZ GROUPS 89
7. The Inifinitesimal Operators of a Linear Representation of
the Proper Lorentz Group 89
1. The infinitesimal Lorentz matrices. 89
2. Relations between the infinitesimal Lorentz matrices. 96
3. The infinitesimal operators of a representation of the proper
Lorentz group. 96
4. Relations between the basic infinitesimal operators of a repre¬
sentation. 101
8. Determination of the infinitesimal Operators of a Represen¬
tation of the Group ©+. 103
1. Statement of the problem. 103
2. Determination of the operators H+, H , H3. 104
3. Determination of the operators F+, F , F3 106
4. The conditions of being unitary. 117
9. The Finite dimesional Representations of the Proper Lorentz
Group 120
1. The spinor description of the proper Lorentz group. 120
2. The relation between the representations of the groups (5+ and U 126
3. The spinor representations of the group II. 126
4. The infinitesimal operators of a spinor representation. 129
5. The irreducibility of a spinor representation. 132
6. The infinitesimal operators of a spinor representation with respect
to a canonical basis. 133
CONTENTS ] Vii
10. Principal Series of Representations of the Group 31 138
1. Some subgroups of the group 21. 138
2. Canonical decomposition of the elements of the group St. 139
3. Residue classes with respect to K. 139
4. Parametrization of the space Z. 141
5. Invariant integral on the group Z. 142
6. The definition of the representations of the principal series. 144
7. Irreducibility of the representations of the principal series. 151
11. Description of the Representations of the Principal Series
and of spinor representations by means of the unitary group 154
1. A description of the space Z in terms of the unitary subgroup. 154
2. The space L?(U). 156
3. The realization of the representation of the principal series
in the space L™(U). 157
4. The representations 5*, contained in Sm,p 159
5. Elementary spherical functions. 163
6. Infinitesimal operators of the representation Sm,* in a canonical
basis. 166
7. The case of spinor representations. 169
12. Complementary Series of Representations of the Group 2t 170
1. Statement of the problem of complementary series. 170
2. The condition for positive definiteness. 174
3. The spaces §» and Hc. 179
4. A description of the representations of the complementary series
in the space $». 180
5. A description of the representations of the complementary series
with the aid of the unitary subgroup. 182
6. The representations Si, contained in !$)». 185
7. The elementary spherical functions of the representations of the
complementary series. 185
8. The infinitesimal operators of the representations 3V in a canonical
basis. 186
13. The Trace of a Representation of the Principal or Com¬
plementary Series 188
1. An invariant integral on the group 91. 188
2. Invariant integrals on the group K. 192
3. Some integral relations. 193
4. The group ring of the group 91. 197
5. The relation between the representations of the group 91 and its
group ring. 199
6. The case of a unitary representation of the group St. 204
viii CONTENTS
7. The trace of a representation of the principal series. 205
8. The trace of a representation of the complementary series. 208
14. An Analogue of Plancherel s Formula 210
1. Statement of the problem. 210
2. Some subgroups of the group K. 216
3. Canonical decomposition of the elements of the group K. 217
4. Some integral relations. 218
5. Some auxiliary functions and relations between them. 219
6. The derivation of an analogue of Plancherel s formula. 223
7. The inverse formulae. 227
8. The decomposition of the regular representation of the group U
into irreducible representations. 232
15. A Description of all the Completely Irreducible Representa¬
tions of the Proper Lorentz Group 234
1. Conjugate representations. 235
2. The operators £*. 238
3. Equivalence of representations. 238
4. Completely irreducible representations. 242
5. The operators e*r 248
6. The ring X*. 249
7. The relation between the representations of the rings X and X*. 251
8. The commutativity of the rings X*?. 254
9. A criterion of equivalence. 258
10. The functional X (x) in the case of an irreducible representation
of the principal series. 261
11. The functions 5,(e). 262
12. The ring 23*. 272
13. The general form of the functional k (b). 274
14. The general form of the linear multiplicative functional X(B) in
the ring 93*. 279
15. The complete series of completely irreducible representations of
the group 91. 285
16. A fundamental theorem. 295
16. Description of all the Completely Irreducible Representations
of the Complete Lorentz Group 297
1. Statement of the problem. 297
2. The fundamental properties of the operator S. 297
3. The group ring of the group ©o. 300
4. Induced representations. 301
5. Description of the completely irreducible representations of the
ring £*. 302
CONTENTS ix
6. Realizations of the completely irreducible representations of the
group ©0. 312
7. A fundamental theorem. 323
CHAPTER IV. INVARIANT EQUATIONS
17. Equations Invariant with Respect to Rotations of Three dimen¬
sional Space 327
1. A general definition of quantities. 327
2. The concept of an equation invariant with respect to a represen¬
tation of the group Go 328
3. Conditions of invariance. 330
4. Conditions of invariance in infinitesimal form. 330
5. General form of the operators L,, Llt L3. 334
18. Equations Invariant with Respect to Proper Lorentz Transform¬
ations 347
1. General linear representations of the proper Lorentz group in
infinitesimal form. 347
2. Some special cases of representations of the group (5+. 355
3. The concept of an equation invariant with respect to proper Lorentz
transformations. 356
4. The general form of an equation invariant with respect to the
transformations of the group ©+. 358
19. Equations Invariant with Respect to Transformations of the
Complete Lorentz Group 373
1. General linear representations of the complete Lorentz group in
infinitesimal form. 373
2. A description of the equations invariant with respect to the complete
Lorentz group. 378
20. Equations Derived from an Invariant Lagrangian Function 382
1. Invariant bilinear forms. 382
2. Lagrangian functions. 396
3. The definition of rest mass and spin. 400
4. Conditions of definiteness of density of charge and energy. 403
5. The case of finite dimensional equations. 411
6. Examples of invariant equations. 417
APPENDIX 423
REFERENCES 440
INDEX 445
Volumes Published in the Series in Pure and Applied Mathematics 449
|
| any_adam_object | 1 |
| author | Najmark, Mark A. 1909-1978 |
| author_GND | (DE-588)1026635349 |
| author_facet | Najmark, Mark A. 1909-1978 |
| author_role | aut |
| author_sort | Najmark, Mark A. 1909-1978 |
| author_variant | m a n ma man |
| building | Verbundindex |
| bvnumber | BV007109570 |
| classification_rvk | SK 260 |
| ctrlnum | (OCoLC)632313542 (DE-599)BVBBV007109570 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV007109570 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:55:44Z |
| institution | BVB |
| language | English |
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| physical | XIV, 447 S. Ill. |
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| publishDate | 1964 |
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| publisher | Pergamon Press |
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| series | International series of monographs on pure and applied mathematics |
| series2 | International series of monographs on pure and applied mathematics A Pergamon Press book |
| spelling | Najmark, Mark A. 1909-1978 Verfasser (DE-588)1026635349 aut Linejnye predstavlenija gruppy Lorenca Linear representations of the Lorentz group M. A. Naimark Oxford [u.a.] Pergamon Press 1964 XIV, 447 S. Ill. txt rdacontent n rdamedia nc rdacarrier International series of monographs on pure and applied mathematics 63 A Pergamon Press book Lorentz-Gruppe (DE-588)4036335-1 gnd rswk-swf Lineare Darstellung (DE-588)4167703-1 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Lorentz-Gruppe (DE-588)4036335-1 s Lineare Darstellung (DE-588)4167703-1 s DE-604 Darstellungstheorie (DE-588)4148816-7 s 1\p DE-604 International series of monographs on pure and applied mathematics 63 (DE-604)BV001888024 63 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004527097&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Najmark, Mark A. 1909-1978 Linear representations of the Lorentz group International series of monographs on pure and applied mathematics Lorentz-Gruppe (DE-588)4036335-1 gnd Lineare Darstellung (DE-588)4167703-1 gnd Darstellungstheorie (DE-588)4148816-7 gnd |
| subject_GND | (DE-588)4036335-1 (DE-588)4167703-1 (DE-588)4148816-7 |
| title | Linear representations of the Lorentz group |
| title_alt | Linejnye predstavlenija gruppy Lorenca |
| title_auth | Linear representations of the Lorentz group |
| title_exact_search | Linear representations of the Lorentz group |
| title_full | Linear representations of the Lorentz group M. A. Naimark |
| title_fullStr | Linear representations of the Lorentz group M. A. Naimark |
| title_full_unstemmed | Linear representations of the Lorentz group M. A. Naimark |
| title_short | Linear representations of the Lorentz group |
| title_sort | linear representations of the lorentz group |
| topic | Lorentz-Gruppe (DE-588)4036335-1 gnd Lineare Darstellung (DE-588)4167703-1 gnd Darstellungstheorie (DE-588)4148816-7 gnd |
| topic_facet | Lorentz-Gruppe Lineare Darstellung Darstellungstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004527097&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV001888024 |
| work_keys_str_mv | AT najmarkmarka linejnyepredstavlenijagruppylorenca AT najmarkmarka linearrepresentationsofthelorentzgroup |