Dirac Particle in the Coulomb Field on the Background of Hyperbolic Lobachevsky Model

The known systems of radial equations describing the relativistic hydrogen atom on the base of the Dirac equation in Lobachevsky hyperbolic space is solved. The relevant 2-nd order differential equation has six regular singular points, its solutions of Frobenius type are constructed explicitly. To produce the quantization rule for energy values we have used the known condition for determination of the transcendental Frobenius solutions. This defines the energy spectrum which is physically interpretable and similar to the spectrum arising for the scalar Klein-Fock-Gordon equation in Lobachevsky space. In the present paper, exact analytical solutions referring to this spectrum are constructed. Convergence of the series involved is proved analytically and numerically. Squared integrability of the solutions is demonstrated numerically. It is shown that the spectrum coincides precisely with that previously found within the semi-classical approximation.

Symmetries of Relativistic Hydrogen Atom

The Dirac equation in the external Coulomb field is proved to possess the symmetry determined by 31 operators, which form the 31-dimensional algebra. Two different fermionic realizations of the SO(1,3) algebra of the Lorentz group are found. Two different bosonic realizations of this algebra are found as well. All generators of the above-mentioned algebras commute with the operator of the Dirac equation in an external Coulomb field and, therefore, determine the algebras of invariance of such Dirac equation. Hence, the spin s = (1, 0) Bose symmetry of the Dirac equation for the free spinor field, proved recently in our papers, is extended here for the Dirac equation interacting with an external Coulomb field. A relativistic hydrogen atom is modeled by such Dirac equation. We are able to prove for the relativistic hydrogen atom both the fermionic and bosonic symmetries known from our papers in the case of a non-interacting spinor field. New symmetry operators are found on the basis of new gamma matrix representations of the Clifford and SO(8) algebras, which are known from our recent papers as well. Hidden symmetries were found both in the canonical Foldy–Wouthuysen and covariant Dirac representations. The found symmetry operators, which are pure matrix ones in the Foldy–Wouthuysen representation, become non-local in the Dirac model.

DYNAMICAL SYMMETRIES OF DIRAC HAMILTONIAN

Several dynamical symmetries of the Dirac Hamiltonian are reviewed and the conditions under which such symmetries hold are considered. These include relativistic spin and orbital angular momentum symmetries, SO (4)× SU σ(2) symmetry for generalized relativistic hydrogen atom that includes an extra Lorentz scalar potential, SU (3)× SU σ(2) symmetry for the relativistic simple harmonic oscillator. The energy spectrum in each case is calculated from group-theoretic considerations.

HYDROGEN-LIKE CLASSIFICATION FOR LIGHT NONSTRANGE MESONS

The recent experimental results on the spectrum of highly excited light nonstrange mesons are known to reveal a high degree of degeneracy among different groups of states. We revise some suggestions about the nature of the phenomenon and put the relevant ideas into the final shape. The full group of approximate mass degeneracies is argued to be SU (2)f × I × O(4), where I is the degeneracy of isosinglets and isotriplets and O(4) is the degeneracy group of the relativistic hydrogen atom. We discuss the dynamical origin and consequences of considered symmetry with a special emphasis on distinctions of this symmetry from the so-called chiral symmetry restoration scenario.