Introduction to Continuous Symmetries - From Space-Time to Quantum Mechanics

In dem neuen Werk von Franck Laloe wird ein symmetriebasierter Ansatz zum grundlegenden Verständnis der Quantenmechanik vorgestellt ? zusammen mit den entsprechenden Rechentechniken, die Studierende höherer Semester in den Bereichen Nuklearphysik, Quantenopik und Festkörperphysik benötigen.

Inhalt:

I Symmetry transformations 1

A Basic symmetries 1

B Symmetries in classical mechanics 5

C Symmetries in quantum mechanics 26

AI Eulerian and Lagrangian points of view in classical mechanics 31

1 Eulerian point of view 32

2 Lagrangian point of view 34

BI Noether's theorem for a classical field 38

1 Lagrangian density and Lagrange equations for continuous variables 38

2 Symmetry transformations and current conservation 40

3 Generalization, relativistic notation 41

4 Local conservation of energy 42

II Some ideas about group theory 45

A General properties of groups 46

B Linear representations of a group 56

AII Left coset of a subgroup; quotient group 65

1 Left cosets 65

2 Quotient group 66

III Introduction to continuous groups and Lie groups 69

A General properties 70

B Examples 85

C Galilean and Poincaré groups 98

AIII Adjoint representation, Killing form, Casimir operator 109

1 Adjoint representation of a Lie algebra 109

2 Killing form ; scalaire product and change of basis in L 111

3 Completely antisymmetric structure constants 113

4 Casimir operator 114

IV Induced representations in the state space 117

A Conditions imposed on the transformations in the state space 119

B Wigner's theorem 121

C Transformations of observables 126

D Linear representations in the state space 128

E Phase factors and projective representations 133

AIV Unitary projective representations, with finite dimension, of connected Lie groups. Bargmann's theorem 141

1 Case where G is simply connected 142

2 Case where G is p-connected 145

BIV Uhlhorn-Wigner theorem 149

1 Real space 149

2 Complex space 153

V Representations of Galilean and Poincaré groups: mass, spin, and energy 157

A Representations in the state space 158

B Galilean group 159

C Poincaré group 173

AV Proper Lorentz group and SL(2C) group 191

1 Link to the SL(2, C) group 191

2 Little group associated with a four-vector 198

3 W2 operator 202

BV Commutation relations of spin components, Pauli-Lubanski four-vector 205

1 Operator S 205

2 Pauli-Lubanski pseudovector 207

3 Energy-momentum eigensubspace with any eigenvalues 210

CV Group of geometric displacements 213

1 Brief review: classical properties of displacements 214

2 Associated operators in the state space 223

DV Space reflection (parity) 233

1 Action in real space 233

2 Associated operator in the state space 235

3 Parity conservation 237

VI Construction of state spaces and wave equations 241

A Galilean group, the Schrödinger equation 242

B Poincaré group, Klein-Gordon, Dirac, and Weyl equations 254

AVI Relativistic invariance of Dirac equation and non-relativistic limit 273

1 Relativistic invariance 273

2 Non-relativistic limit of the Dirac equation 276

BVI Finite Poincaré transformations and Dirac state space 281

1 Displacement group 281

2 Lorentz transformations 283

3 State space and Dirac operators 287

CVI Lagrangians and conservation laws for wave equations 293

1 Complex fields 293

2 Schrödinger equation 295

3 Klein-Gordon equation 297

4 Dirac equation 300

VII Rotation group, angular momenta, spinors 303

A General properties of rotation operators 304

B Spin 1/2 particule; spinors 323

C Addition of angular momenta 329

AVII Rotation of a spin 1/2 and SU(2) matrices 339

1 Modification of a spin 1/2 polarization induced by an SU(2) matrix 340

2 The transformation is a rotation 341

3 Homomorphism 342

4 Link with the chapter VII discussion 344

5 Link with double-valued representations 346

BVII Addition of more than two angular momenta 347

1 Zero total angular momentum; 3-j coefficients 347

2 6-j Wigner coefficients 351

VIII Transformation of observables under rotation 355

A Scalar and vector operators 358

B Tensor operators 363

C Wigner-Eckart theorem 379

D Applications and examples 384

AVIII Short review of classical tensors 397

1 Vectors 397

2 Tensors 398

3 Properties 401

4 Criterium for a tensor 403

5 Symmetric and antisymmetric tensors 403

6 Specific tensors 404

7 Irreducible tensors 405

BVIII Second-order tensor operators 409

1 Tensor product of two vector operators 409

2 Cartesian components of the tensor in the general case 411

CVIII Multipole moments 415

1 Electric multipole moments 416

2 Magnetic multipole moments 428

3 Multipole moments of a quantum system with a given angular momentum J 434

DVIII Density matrix expansion on tensor operators 439

1 Liouville space 439

2 Rotation transformation 441

3 Basis of the T[K]Q operators 442

4 Rotational invariance in a system's evolution 444

IX Internal symmetries, SU(2) and SU(3) groups 449

A System of distinguishable but equivalent particles 451

B SU(2) group and isospin symmetry 466

C SU(3) symmetry 472

AIX The nature of a particle is equivalent to an internal quantum number 497

1 Partial or complete symmetrization, or antisymmetrization, of a state vector 497

2 Correspondence between the states of two physical systems 499

3 Physical consequences 501

BIX Operators changing the symmetry of a state vector by permu-tation 503

1 Fermions 503

2 Bosons 506

X Symmetry breaking 507

A Magnetism, breaking of rotational symmetry 508

B A few other examples 515

Appendix 521

Time reversal 521

1 Time reversal in classical mechanics 522

2 Antilinear and antiunitary operators in quantum mechanics 527

3 Time reversal and antilinearity 534

4 Explicit form of the time reversal operator 542

5 Applications 546