Book Description:
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.

Table of Contents:
Preface:

Introduction: The General Concepts of Dynamical Symmetries

Chapter I. Hidden (Dynamical) Symmetries in Classical Mechanics; pp.1-28
I.1 Constants of Motion as Generators of Infinitesimal Transformations; pp.29-45
I.2 Derivation of LRL Vector; pp. I.3 Application of LRL Vector in Classical Physics:
(i) LRL Vector and Orbit Equation (ii) Algebraic Aspects of the Kepler Problem
I.4 Dynamical Symmetry for the Isotropic Harmonic Oscillator
I.5 Possible Generalizations of Dynamical Symmetries
I.6 Application of the Dynamical Evolution of LRL Vector in
General Central Case
- Equations of Motion for General Central Forces
- Equations of Motion for Arbitrary Forces
- Summary Comments on Dynamical Symmetries in Classical
(Non-relativistic) Mechanics
References

Chapter II. Hidden Symmetry in Classical Relativistic Mechanics; pp.29-44
II.1 Auxiliary problem: LRL vector for a Modified Kepler Problem
II.2 The Laplace-Runge-Lenz Vector and Lorentz Boost
II.3 Post-Newtonian Extensions of the LRL Vector
II.4 Relativistic Kepler Problem
References

Chapter III. Dynamical Symmetries in Non-relativistic Quantum Mechanics; pp.47-65
III.1 The Hydrogen Atom (General Consideration)
Algebraic Aspects of the Hydrogen Problem
III.2 The Hydrogen Atom in the Momentum Representation Example of Application of the Momentum Representation: Dynamical Symmetry of a Three-dimensional Wick-Cutkosky Problem
III.3 The Hydrogen Atom and the Lorentz Group
III.4 Three-dimensional Isotropic Harmonic Oscillator
References

Chapter IV. New Kind of Dynamial Symmetry – Supersymmetry; pp.67-79
IV.1 Supersymmetric Quantum Mechanics
IV.2 Supersymmetry and the Radial Problem
IV.3 Exact Supersymmetry in the Non-relativistic Hydrogen Atom
References

Chapter V. Relativistic Quantum Mechanics; pp.81-95 V.I Supersymmetry in the Dirac Equation for the Culomb Potential
Appendix: Shape Invariance V.2 An “Accidental”Symmetry Operator for the Dirac Equation
in the Coulomb Potential – from Pauli to Dirac
V.3 Physical Meaning and some Applications of Johnson-Lippman Operator Appendix: Calculations of Relevant Commutators
References

Chapter VI. Generalizations to the Relativistic Dirac Hamiltonian; pp. 97-115
VI.1 Supersymmetry of the Dirac Hamiltonian for General
Central Potential VI.2 Where is the Harmonic Oscillator?
VI.3 Relativistic Quantum Mechanics of Dirac Oscillator
VI.4 The Lorentz-scalar Potential in the Dirac Equation
VI.5 Algebraic Derivation of the Spectrum of the Dirac Hamiltonian for an Arbitrary Comination of the a Lorentz-scalar an Lorentz-vector Potentials
Comments
References

Chapter VII. Some Recent; pp.117-127 Developments
VII.1 Hidden Supersymmetry of the Dirac-Coulomb Problem
and the Biedenharn Approach
VII.2 Some Practical Generalizations: The LRL Vector in the
Presence of an Electric Field
In the Form of Conclusions
References

Bibliography (Part I) (Part II)Continuation pp.131-144

Index