《LINEAR REPRESENTATIONS OF THE LORENTZ GROUP》求取 ⇩

CHAPTER Ⅰ．THE THREE-DIMENSIONAL ROTATION GROUP AND THE LORENTZ GROUP1

1．THE THREE-DIMENSIONAL ROTATION GROUP1

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 GROUP18

1．The general Lorentz group．18

2．The complete Lorentz group and the proper Lorentz group．23

CHAPTER Ⅱ．THE REPRESENTATIONS OF THE THREE-DIMENSIONAL ROTATION GROUP25

3．THE BASIC CONCEPTS OF THE THEORY OF FINITE-DIMENSIONAL REPRESENTATIONS25

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 FORM31

1．Differentiability of representations of the group G0．31

2．Basic infinitesimal matrices of the group G0．33

3．Basic infinitesimal operators of a representation of the group G0．35

4．Relations between the basic infinitesimal operators of a repre-sentation of the group G0．39

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 G0．43

5．THE REALIZATION OF FINITE-DIMENSIONAL IRREDUCIBLE REPRESEN-TATIONS OF THE THREE-RIMENSIONAL ROTATION GROUP50

1．The connection between the representations of the group G0 and the representations of the unitary group Ц．50

2．Spinor representations of the group Ц．51

3．Realization of the representations Бm in a space of polynomials．54

4．Basic infinitesimal operators of the representation Бm．56

5．Orthogonality relations．60

6．THE DECOMPOSITION OF A GIVEN REPRESENTATION OF THE THREE-DIMENSIONAL ROTATION GROUP INTO IRREDUCIBLE REPRESENTATIONS63

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 representations．77

7．The case of a unitary representation．83

CHAPTER Ⅲ．IRREDUCIBLE LINEAR REPRESENTATIONS OF THE PROPER AND COMPLETE LORENTZ GROUPS89

7．THE INIFINITESIMAL OPERATORS OF A LINEAR REPRESENTATION OF THE PROPER LORENTZ GROUP89

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 Б+．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 GROUP120

1．The spinor description of the proper Lorentz group．120

2．The relation between the representations of the groups Б+ and Ц．126

3．The spinor representations of the group Ц．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

10．PRINCIPAL SERIES OF REPRESENTATIONS OF THE GROUP ？138

1．Some subgroups of the group ？．138

2．Canonical decomposition of the elements of the group ？．139

3．Residue classes with respect to K．139

4．Parametrization of the space ？．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 GROUP154

1．A description of the space ？ in terms of the unitary subgroup．154

2．The space L？(Ц)．156

3．The realization of the representation of the principal series in the space L？(Ц)．157

4．The representations Sk,contained in ？m．p．159

5．Elementary spherical functions．163

6．Infinitesimal operators of the representation ？m,p in a canonical basis．166

7．The case of spinor representations．169

12．COMPLEMENTARY SERIES OF REPRESENTATIONS OF THE GROUP ？170

1．Statement of the problem of complementary series．170

2．The condition for positive definiteness．174

3．The spaces ？σ and Hσ．179

4．A description of the representations of the complementary series in the space ？σ．180

5．A description of the representations of the complementary series with the aid of the unitary subgroup．182

6．The representations Sk,contained in ？σ．185

7．The elementary spherical functions of the representations of the complementary series．185

8．The infinitesimal operators of the representations ？σ in a canonical basis．186

13．THE TRACE OF A REPRESENTATION OF THE PRINCIPAL OR COM-PLEMENTARY SERIES188

1．An invariant integral on the group ？．188

2．Invariant integrals on the group K．192

3．Some integral relations．193

4．The group ring of the group ？．197

5．The relation between the representations of the group ？ and its group ring．199

6．The case of a unitary representation of the group ？．204

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 FORMULA210

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 ？ into irreducible representations．232

15．A DESCRIPTION OF ALL THE COMPLETELY IRREDUCIBLE REPRESENTA-TIONS OF THE PROPER LORENTZ GROUP234

1．Conjugate representations．235

2．The operators E？．238

3．Equivalence of representations．238

4．Completely irreducible representations．242

5．The operators e？．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 λ(x)in the case of an irreducible representation of the principal series．261

11．The functions Bv(ε)．262

12．The ring ？．272

13．The general form of the functional λ(b)．274

14．The general form of the linear multiplicative functional λ(B) in the ring ？．279

15．The complete series of completely irreducible representations of the group ？．285

16．A fundamental theorem．295

16．DESCRIPTION OF ALL THE COMPLETELY IRREDUCIBLE REPRESENTATIONS OF THE COMPLETE LORENTZ GROUP297

1．Statement of the problem．297

2．The fundamental properties of the operator S．297

3．The group ring of the group ？0．300

4．Induced representations．301

5．Description of the completely irreducible representations of the ring ？．302

6．Realizations of the completely irreducible representations of the group ？0．312

7．A fundamental theorem．323

CHAPTER Ⅳ．INVARIANT EQUATIONS327

17．EQUATIONS INVARIANT WITH RESPECT TO ROTATIONS OF THREE-DIMEN-SIONAL SPACE327

1．A general definition of quantities．327

2．The concept of an equation invariant with respect to a represen-tation of the group G0．328

3．Conditions of invariance．330

4．Conditions of invariance in infinitesimal form．330

5．General form of the operators L1,L2,L3．334

18．EQUATIONS INVARIANT WITH RESPECT TO PROPER LORENTZ TRANSFORM-ATIONS347

1．General linear representations of the proper Lorentz group in infinitesimal form．347

2．Some special cases of representations of the group ？+．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 GROUP373

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 FUNCTION382

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

APPENDIX423

REFERENCES440

INDEX445

Volumes Published in the Series in Pure and Applied Mathematics449