On a dynamical symmetry group of the relativistic linear singular oscillator (original) (raw)

The operator algebra of the quantum relativistic oscillator

Journal of Mathematical Physics, 1997

The operator algebras of a new family of relativistic geometric models of the relativistic oscillator [1] are studied. It is shown that, generally, the operator of number of quanta and the pair of the shift operators of each model are the generators of a non-unitary representation of the so(1, 2) algebra, except a special case when this algebra becomes the standard one of the non-relativistic harmonic oscillator.

The relativistic linear singular oscillator

Journal of Physics A: Mathematical and General, 2003

Exactly-solvable model of the linear singular oscillator in the relativistic configurational space is considered. We have found wavefunctions and energy spectrum for the model under study. It is shown that they have correct non-relativistic limits.

A relativistic model of the N -dimensional singular oscillator

Journal of Physics A: Mathematical and Theoretical, 2007

Exactly solvable N -dimensional model of the quantum isotropic singular oscillator in the relativistic configurational r N -space is proposed. It is shown that through the simple substitutions the finite-difference equation for the N -dimensional singular oscillator can be reduced to the similar finite-difference equation for the relativistic isotropic three-dimensional singular oscillator. We have found the radial wavefunctions and energy spectrum of the problem and constructed a dynamical symmetry algebra.

Relativistic nonlinear wave equations with groups of internal symmetry

PhD Dissertation, McGill University, 1988

A nonlinear wave equation invariant with respect to unitary representations of the Lorentz group is considered in an attempt to describe extended particles with spin and positive definite energy by means of a self-confined classical field. The wave function has an infinite number of components and, in the specific representations used, the corresponding internal degree of freedom is identified with the spin. A fractional power of the scalar bilinear invariant appears as an appropriate choice for the nonlinearity in order that all the stationary states be localized. Two approximation methods are proposed and both lead to results that bear a resemblance to the results of the MIT bag model.

An algebraic SU(1, 1) solution for the relativistic hydrogen atom

2005

The bound eigenfunctions and spectrum of a Dirac hydrogen atom are found taking advantage of the SU(1,1) Lie algebra in which the radial part of the problem can be expressed. For defining the algebra we need to add to the description an additional angular variable playing essentially the role of a phase. The operators spanning the algebra are used for defining ladder operators for the radial eigenfunctions of the relativistic hydrogen atom and for evaluating its energy spectrum. The status of the Johnson-Lippman operator in this algebra is also investigated. Key words: relativistic hydrogen atom, ladder operators, SU(1,1) Lie algebra PACS: 33.10.C, 11.10.Qr The bound solutions of the hydrogen atom are of great importance in both classical and quantum mechanics and so is the search for new ways of solving or using such problem [1,2,3,4,5,6,7,8,9,10]. The purpose of this paper is to discuss an algebraic solution for the bounded eigenstates of the relativistic hydrogen

Relativistic two-body equation based on the extension of the SL(2,C) group

Physics Letters B, 2007

A new approach to the two-body problem based on the extension of the SL(2, C) group to the Sp(4, C) one is developed. The wave equation with various forms of including the interaction for the system of the spin-1/2 and spin-0 particles is constructed. For this system, it was found that the wave equation with a linear confinement potential involved in the non-minimal manner has an oscillator-like form and possesses the exact solution.

On the new invariance algebras of relativistic equations for massless particles

Journal of Physics A: Mathematical and General, 1979

We show that the massless Dirac equation and Maxwell equations are invariant under a 23-dimensional Lie algebra, which is isomorphic to the Lie algebra of the group C4 ⊗ U (2) ⊗ U (2). It is also demonstrated that any Poincaré-invariant equation for a particle of zero mass and of discrete spin provide a unitary representation of the conformal group and that the conformal group generators may be expressed via the generators of the Poincaré group.

On the Group-Theoretical Approach to Relativistic Wave Equations for Arbitrary Spin

2021

Formulating a relativistic equation for particles with arbitrary spin remains an open challenge in theoretical physics. In this study, the main algebraic approaches used to generalize the Dirac and Kemmer–Duffin equations for particles of arbitrary spin are investigated. It is proved that an irreducible relativistic equation formulated using spin matrices satisfying the commutation relations of the de Sitter group leads to inconsistent results, mainly as a consequence of violation of unitarity and the appearance of a mass spectrum that does not reflect the physical reality of elementary particles. However, the introduction of subsidiary conditions resolves the problem of unitarity and restores the physical meaning of the mass spectrum. The equations obtained by these approaches are solved and the physical nature of the solutions is discussed.