Dynamical symmetry of the Kepler-Coulomb problem in classical and quantum mechanics : non-relativistic and relativistic

DYNAMICAL SYMMETRY OF THEKEPLER-COULOMB PROBLEM INCLASSICAL AND QUANTUMMECHANICS: NON-RELATIVISTICAND RELATIVISTIC CONTENTS ABOUT THE AUTHORS PREFACE INTRODUCTION THE GENERAL CONCEPTS OF DYNAMICAL SYMMETRIES REFERENCES HIDDEN (DYNAMICAL) SYMMETRIES IN CLASSICALMECHANICS I.1. CONSTANTS OF MOTION AS GENERATORS OF INFINITESIMALTRANSFORMATIONS Remark I.2. DERIVATION OF LRL VECTOR I.3. APPLICATIONS OF LRL VECTOR IN CLASSICAL PHYSICS (I) LRL Vector and the Orbit Equation (II). Algebraic Aspects of the Kepler Problem I.4. DYNAMICAL SYMMETRY FOR THE ISOTROPICHARMONIC OSCILLATOR. I.5. POSSIBLE GENERALIZATIONS OF DYNAMICAL SYMMETRIESComments I.6. APPLICATION OF THE DYNAMICAL EVOLUTION OF LRLVECTOR IN GENERAL CENTRAL CASE [12] Equations of Motion for General Central Forces Equations of Motion for Arbitrary Forces Summary Comments on Dynamical Symmetries in Classical(Non-Relativistic) Mechanics REFERENCES HIDDEN SYMMETRY IN CLASSICAL RELATIVISTICMECHANICS II.1. AUXILIARY PROBLEM: LRL VECTOR FOR A MODIFIEDKEPLER PROBLEM II.2. THE LAPLACE-RUNGE-LENZ VECTOR AND THE LORENTZBOOST II.3. POST-NEWTONIAN EXTENSIONS OF THE LRL VECTOR II.4. RELATIVISTIC KEPLER PROBLEM. IV.2. SUPERSYMMETRY AND THE RADIAL PROBLEMIV. 3. EXACT SUPERSYMMETRY IN THE NON-RELATIVISTICHYDROGEN ATOM REFERENCES RELATIVISTIC QUANTUM MECHANICS V.1. SUPERSYMMETRY IN THE DIRAC EQUATION FOR THECOULOMB POTENTIAL APPENDIX: SHAPE INVARIANCE (SI) V. 2. AN "ACCIDENTAL SYMMETRY" OPERATORFOR THE DIRAC EQUATION IN THE COULOMB POTENTIAL -FROM PAULI TO DIRAC V. 3. PHYSICAL MEANING AND SOME APPLICATIONSOF JOHNSON -- LIPPMANN OPERATOR APPENDIX: CALCULATION OF RELEVANT COMMUTATORS REFERENCES GENERALIZATIONS TO THE RELATIVISTIC DIRACHAMILTONIAN. VI.1. SUPERSYMMETRY OF THE DIRAC HAMILTONIANFOR GENERAL CENTRAL POTENTIALSVI.2. WHERE IS THE HARMONIC OSCILLATOR? VI.3. RELATIVISTIC QUANTUM MECHANICSOF DIRAC OSCILLATOR VI.4. THE LORENTZ -- SCALAR POTENTIALIN THE DIRAC EQUATION VI.5. ALGEBRAIC DERIVATION OF THE SPECTRUM OF THEDIRAC HAMILTONIAN FOR AN ARBITRARY COMBINATIONOF THE LORENTZ-SCALAR AND LORENTZ-VECTOR COULOMBPOTENTIAL Comments REFERENCES SOME RECENT DEVELOPMENTS VII.1 HIDDEN SUPERSYMMETRY OF THE DIRAC-COULOMBPROBLEM AND THE BIEDENHARN APPOACH. The purpose of this new and important book is to develop a systematic theory for the hidden symmetry generators, which are simultaneously the odd generators of superalgebra in relativistic quantum mechanics (Dirac's equation). It is devoted to the description of so-called hidden symmetry of the Kepler problem in classical and quantum mechanics (non-relativistic and relativistic). Moreover, the purpose is to elucidate the role of this kind of symmetries in the reduction of physical problems, to show algebraic possibilites of derivation of spectra.