Dynamical Symmetry

Contents......Page 14
Dedication......Page 6
Preface......Page 8
References......Page 9
Acknowledgments......Page 12
1.1 On Geometric Symmetry and Invariance in the Sciences......Page 22
1.2 Fock’s Discovery......Page 27
1.4 Dynamical Symmetry......Page 29
1.5 Dynamical Symmetries Responsible for Degeneracies and Their Physical Consequences......Page 31
1.6 Dynamical Symmetries When Energies Can Vary......Page 33
1.7 The Need for Critical Reexamination of Concepts of Physical Symmetry. Lie’s Discoveries......Page 34
Appendix A. Historical Note......Page 39
References......Page 40
2.1 Geometrical Interpretation of the Invariance Group of an Equation; Symmetry Groups......Page 44
2.2 On Geometric Interpretations of Equations......Page 50
2.3 Geometric Interpretations of Some Transformations in the Euclidean Plane......Page 51
2.4 The Group of Linear Transformations of Two Variables......Page 57
2.5 Physical Interpretation of Rotations......Page 58
2.6 Intrinsic Symmetry of an Equation......Page 60
2.7 Non-Euclidean Geometries......Page 61
2.8 Invariance Group of a Geometry......Page 69
2.9 Symmetry in Euclidean Spaces......Page 71
2.10 Symmetry in the Spacetime of Special Relativity......Page 72
2.11 Geometrical Interpretations of Nonlinear Transformations: Stereographic Projections......Page 77
2.12 Continuous Groups that Leave Euclidean and Pseudo-Euclidean Metrics Invariant......Page 82
2.13 Geometry, Symmetry, and Invariance......Page 86
Appendix A: Stereographic Projection of Circles......Page 87
References......Page 89
3.1 Invariance of a Differential Equation......Page 92
3.2 Hamilton’s Equations......Page 94
3.3 Transformations that Convert Hamilton’s Equations into Hamilton’s Equations; Symplectic Groups......Page 97
3.4 Invariance Transformations of Hamilton’s Equations of Motion......Page 99
3.5 Geometrization of Hamiltonian Mechanics......Page 101
3.6 Symmetry in Two-Dimensional Symplectic Space......Page 112
3.7 Symmetry in Two-Dimensional Hamiltonian Phase Space......Page 113
Definition of Symmetry in Two-Dimensional Hamiltonian Phase–Space:......Page 114
3.8 Symmetries Defined by Linear Symplectic Transformations......Page 116
3.9 Nonlinear Transformations in Two-Dimensional Phase Space......Page 118
3.10 Dynamical Symmetries as Intrinsic Symmetries of Differential Equations and as Geometric Symmetries......Page 123
References......Page 125
4.1 Introduction......Page 126
4.2 Finite Transformations of a Continuous Group Define Infinitesimal Transformations and Vector Fields......Page 129
4.3 Spaces in which Transformations will be Assumed to Act on......Page 133
4.4 The Defining Equations of One-Parameter Groups of Infinitesimal Transformations. Group Generators......Page 135
4.5 The Differential Equations that Define Infinitesimal Transformations Define Finite Transformation Groups......Page 137
4.6 The Operator of Finite Transformations......Page 140
4.7 Changing Variables in Group Generators......Page 143
4.8 The Rectification Theorem......Page 145
4.9 Conversion of Non-autonomous ODEs to Autonomous ODEs......Page 150
4.11 Conclusion......Page 152
Appendix A. Homeomorphisms, Diffeomorphisms, and Topology......Page 153
Exercises......Page 154
References......Page 155
5.1 Invariance under the Action of One-Parameter Lie Transformation Groups......Page 156
5.2 Transformation of Infinitesimal Displacements......Page 162
5.3 Transformations and Invariance of Work, Pfaffians, and Metrics......Page 164
5.4 Point Transformations of Derivatives......Page 168
5.5 Contact Transformations......Page 170
5.6 Invariance of an Ordinary Differential Equation of First-Order under Point Transformations; Extended Generators......Page 172
5.7 Invariance of Second-Order Ordinary Differential Equations under Point Transformations; Harmonic Oscillators......Page 175
5.8 The Commutator of Two Operators......Page 177
5.9 Invariance of Sets of ODEs. Constants of Motion......Page 179
5.10 Conclusion......Page 182
Appendix A. Relation between Symmetries and Intergrating Factors......Page 183
Appendix B. Proof That the Commutator of Lie Generators is Invariant under Diffeomorphisms......Page 184
Appendix C. Isolating and Non-isolating Integrals of Motion......Page 186
References......Page 188
6.1 Relation of Many-Parameter Lie Transformation Groups to Lie Algebras......Page 190
6.2 The Differential Equations That Define Many-Parameter Groups......Page 196
6.3 Real Lie Algebras......Page 198
6.4 Relations between Commutation Relations and the Action of Transformation Groups: Some Examples......Page 200
6.5 Transitivity......Page 207
6.6 Complex Lie Algebras......Page 208
6.7 The Cartan–Killing Form; Labeling and Shift Operators......Page 209
6.8 Casimir Operators......Page 212
6.9 Groups That Vary the Parameters of Transformation Groups......Page 213
6.10 Lie Symmetries Induced from Observations......Page 214
6.11 Conclusion......Page 216
Appendix A. Definition of Lie Groups by Partial Differential Equations......Page 217
Exercises.......Page 225
References......Page 226
7.1 General Invariance Properties of Newtonian Mechanics......Page 228
7.2 Relationship of Phase Space to Abstract Symplectic Space......Page 231
7.3 Hamilton’s Equations in PQ Space. Constants of Motion......Page 235
7.4 Poisson Bracket Operators......Page 238
7.5 Hamiltonian Dynamical Symmetries in PQ Space......Page 241
7.6 Hamilton’s Equations in Classical PQET Space; Conservation Laws Arising From Galilei Invariance......Page 251
7.7 Time-dependent Constants of Motion; Dynamical Groups That Act Transitively......Page 255
7.8 The Symplectic Groups Sp(2n,r)......Page 258
7.9 Generalizations of Symplectic Groups That Have an Infinite Number of One-parameter Groups......Page 260
Appendix A. Lagrange’s Equations and the Definition of Phase Space......Page 261
Appendix B. The Variable Conjugate to Time in PQET Space......Page 264
References......Page 267
8.1 Newtonian Mechanics of Planetary Motion......Page 268
8.2 Hamiltonian Formulation of Keplerian Motions in Phase Space......Page 275
8.3 Symmetry Coordinates For Keplerian Motions......Page 278
8.4 Geometrical Symmetries of Bound Keplerian Systems in Phase Space......Page 283
8.6 The SO(4,1) Dynamical Symmetry......Page 291
8.7 Concluding Remarks......Page 292
Exercises:......Page 293
References......Page 294
9.1 Superposition Invariance......Page 296
9.2 The Correspondence Principle......Page 297
9.3 Correspondence Between Quantum Mechanical Operators and Functions of Classical Dynamical Variables......Page 301
9.4 Lie Algebraic Extension of the Correspondence Principle......Page 302
9.5 Some Properties of Invariance Transformations of Partial Differential Equations Relevant to Quantum Mechanics......Page 306
9.6 Determination of Generators and Lie Algebra of Invariance Transformations of ((−1/2)∂2/∂x2 − i∂/∂t)ψ(x, t) = 0......Page 310
9.7 Eigenfunctions of the Constants of Motion of a Free-Particle......Page 315
9.8 Dynamical Symmetries of the Schr¨odinger Equations of a Harmonic Oscillator......Page 318
9.9 Use of the Oscillator Group in Pertubation Calculations......Page 321
9.10 Concluding Observations......Page 322
Exercises......Page 323
References......Page 324
Introduction......Page 326
10.1 Lie Algebras That Generate Continuous Spectra......Page 327
10.2 Lie Algebras That Generate Discrete Spectra......Page 329
10.3 Dynamical Groups of N-Dimensional Harmonic Oscillators......Page 330
10.4 Linearization of Energy Spectra by Time Dilatation; Spectrum-Generating Dynamical Group of Rigid Rotators......Page 332
10.5 The Angular Momentum Shift Algebra; Dynamical Group of the Laplace Equation......Page 336
10.6 Dynamical Groups of Systems with Both Discrete and Continuous Spectra......Page 340
10.7 Dynamical Group of the Bound States of Morse Oscillators......Page 341
10.8 Dynamical Group of the Bound States of Hydrogen-Like Atoms......Page 343
10.9 Matrix Representations of Generators and Group Operators......Page 346
10.10 Invariant Scalar Products......Page 348
10.11 Direct-Products: SO(3)⊗SO(3) and the Coupling of Angular Momenta......Page 349
10.12 Degeneracy Groups of Non-interacting Systems; Completions of Direct-Products......Page 350
10.13 Dynamical Groups of Time-dependent Schrodinger Equations of Compound Systems; Many-Electron Atoms......Page 352
Exercises......Page 355
References......Page 356
11.1 Position-space Realization of the Dynamical Symmetries......Page 358
11.2 The Momentum-space Representation......Page 365
11.3 The Hyperspherical Harmonics Yklm......Page 370
11.4 Bases Provided by Eigenfunctions of J12, J34, J56......Page 375
Appendix A. Matrix Elements of SO(4,2) Generators3,16......Page 376
Appendix B. N-Shift Operators For the Hyperspherical Harmonics......Page 377
References......Page 380
12.1 Introduction......Page 382
12.2 The Stark Effect; One-Electron Diatomics......Page 383
12.3 Correlation Diagrams and Level Crossings: General Remarks......Page 385
12.4 Coupling SO(4)1 ⊗ SO(4)2 to Produce SO(4)12......Page 390
12.5 Coupling SO(4)1 ⊗ SO(4)2 to Produce SO(4)1−2......Page 392
12.6 Configuration Mixing in Doubly Excited States of Helium-like Atoms......Page 394
12.8 Origin of the Period-Doubling Displayed in Periodic Charts......Page 396
12.9 Molecular Orbitals in Momentum-Space; The Hyperspherical Basis......Page 398
12.10 The Sturmian Ansatz of Avery, Aquilanti and Goscinski......Page 401
Exercises......Page 405
References......Page 406
13.2 Algebraic Treatment of Anharmonic Oscillators With a Finite Number of Bound States......Page 410
13.3 U(2) ⊗ U(2) Model of Vibron Coupling......Page 415
13.4 Spectrum Generating Groups of Rigid Body Rotations......Page 417
13.5 The U(4) Vibron Model of Rotating Vibrating Diatomics......Page 422
13.6 The U(4) ⊗ U(4) Model of Rotating Vibrating Triatomics......Page 424
References......Page 427
CHAPTER 14 Dynamical Symmetry of Maxwell’s Equations......Page 430
14.1 The Poincare Symmetry of Maxwell’s Equations......Page 431
14.2 The Conformal and Inversion Symmetries of Maxwell’s Equations; Their Physical Interpretation......Page 434
14.3 Alteration of Wavelengths and Frequencies by a Special Conformal Transformation: Interpretation of Doppler Shifts in Stellar Spectra......Page 441
Exercises......Page 450
References......Page 452
Index......Page 454