Supercharge Operator of Hidden Symmetry in the Dirac Equation

As is known, the so-called Dirac $K$-operator commutes with the Dirac Hamiltonian for arbitrary central potential $V(r)$. Therefore the spectrum is degenerate with respect to two signs of its eigenvalues. This degeneracy may be described by some operator, which anticommutes with $K$. If this operator commutes with the Dirac Hamiltonian at the same time, then it establishes new symmetry, which is Witten's supersymmetry. We construct the general anticommuting with $K$ operator, which under the requirement of this symmetry unambiguously select the Coulomb potential. In this particular case our operator coincides with that, introduced by Johnson and Lippmann many years ago.Comment: 3 page

Centre-of-mass and internal symmetries in classical relativistic systems

The internal symmetry of composite relativistic systems is discussed. It is demonstrated that Lorentz-Poincar\'e symmetry implies the existence of internal moments associated with the Lorentz boost, which are Laplace-Runge-Lenz (LRL) vectors. The LRL symmetry is thus found to be the internal symmetry universally associated with the global Lorentz transformations, in much the same way as internal spatial rotations are associated with global spatial rotations. Two applications are included, for an interacting 2-body system and for an interaction-free many-body system of particles. The issue of localizability of the relativistic CM coordinate is also discussed

THE EMPLOYEE MOTIVATION – THE KEY FACTOR IN BUSINESS SUCCESS

We present you the work: “The Employee Motivation – the Key Factor in Business Success”, where we present the results of the research on this issue conducted by the universities of California and Stanford, based on which the author in cooperation with Batumi State Maritime Academy students made some enquiries in Batumi. Analysis of the survey revealed that if the managers are eager to keep the best employees, it is essential to consider them carefully and make an adequate use of their talent and extensive opportunities. Consequently, they must do their best in order to provoke qualified specialists’ desire to work with them, and in turn, to maximize the promotion of the business profit.We present you the work: “The Employee Motivation – the Key Factor in Business Success”, where we present the results of the research on this issue conducted by the universities of California and Stanford, based on which the author in cooperation with Batumi State Maritime Academy students made some enquiries in Batumi. Analysis of the survey revealed that if the managers are eager to keep the best employees, it is essential to consider them carefully and make an adequate use of their talent and extensive opportunities. Consequently, they must do their best in order to provoke qualified specialists’ desire to work with them, and in turn, to maximize the promotion of the business profit

Teaching the hidden symmetry of the Kepler problem in relativistic quantum mechanics - from Pauli to Dirac electron

Hidden symmetry in Coulomb interaction is one of the mysterious problems of modern physics. Additional conserved quantities associated with extra symmetry govern wide variety of physics problems, from planetary motion till fine and hyperfine structures of atomic spectra. In this paper we present a simple derivation of hidden symmetry operator in relativistic quantum mechanics for the Dirac equation in the Coulomb field. We established that this operator may be reduced to the one introduced by Johnson and Lippmann. It is worthwhile to notice that this operator was discussed in literature very rarely and so is not known well among physicists and was omitted even in the recent textbooks on relativistic quantum mechanics and/or quantum electrodynamics.Comment: 5 page

An "Accidental" Symmetry Operator for the Dirac Equation in the Coulomb Potential

On the basis of the generalization of the theorem about K-odd operators (K is the Dirac's operator), certain linear combination is constructed, which appears to commute with the Dirac Hamiltonian for Coulomb field. This operator coincides with the Johnson and Lippmann operator and is intimately connected to the familiar Laplace-Runge-Lenz vector. Our approach guarantees not only derivation of Johnson-Lippmann operator, but simultaneously commutativity with the Dirac Hamiltonian follows.Comment: 6 page