Relations between quantum defects and phase shifts for the Klein-Gordon and Dirac equations

Seaton's relation, which forms the basis of the quantum defect method, is generalized to the relativistic case. A connection is established between the quantum defects of discrete levels and the phase shift of the continuum wave function for the Klein-Gordon and Dirac equations. The possibility of applying these relations, in particular to the study of mesoatoms, is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 59–62, August, 1977.     

Noncqmmutative integration of Klein-Gordon and Dirac equations in Riemannian spaces with a group of motions

By the method of non-coimmutative integration of linear differential equations proposed by the authors [Izv. Vyssh. Uchebn. Zaved., Fiz., No. 4, 95 (1991)] the Klein-Gordon and Dirac equations are integrated in four-dimensional Riemannian spaces, not admitting separation of variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 33–38, May, 1991.     

Covariant differential calculus on quantum Minkowski space and theq-analogue of Dirac equation

The covariant differential calculus on the quantum Minkowski space is presented with the help of the generalized Wess-Zumino method and the quantum Pauli matrices and quantum Dirac matrices are constructed parallel to those in the classical case. Combining these two aspects aq-analogue of Dirac equation follows directly.     

Dirac and Klein-Gordon equations: Convergence of solutions in the nonrelativistic limit

The convergence of solutions of the Dirac and Klein-Gordon equations to solutions of the Pauli and Schrödinger equations in the non-relativistic limit is discussed. An abstract theory of these equations is developed which is general enough to allow physical space to be an arbitrary complete Riemannian manifold.Research partially supported by National Science Foundation grant MCS-77-13070     

Equivalence of Dirac and Maxwell equations and quantum mechanics

In this paper we present an analysis of the possible equivalence of Dirac and Maxwell equations using the Clifford bundle formalism and compare it with Campolattaro's approach, which uses the traditional tensor calculus and the standard Dirac covariant spinor field. We show that Campolattaro's intricate calculations can be proved in few lines in our formalism. We briefly discuss the implications of our findings for the interpretation of quantum mechanics.     

Symmetries of massless Dirac equations and supersymmetric quantum mechanics

It is shown how symmetries of Dirac equations can be used to obtain constants of motion for nonrelativistic supersymmetric quantum Hamiltonians. In particular, conserved supercharges are found for a spin-1/2 particle in the field of a dyon which yield under anticommutation the generalized Runge-Lenz symmetry of the system.This work is supported in part through funds provided by the Natural Sciences and Engineering Research Council (NSERC) of Canada and the Fonds FCAR of the Quebec Ministry of Education.     

Quadratic algebras and noncommutative integration of Klein-Gordon equations in non-steckel Riemann spaces

The method of noncommutative integration of linear partial differential equations is used to solve the Klein-Gordon equations in Riemann space, in the case when the set of noncommutating symmetry operators of this equation for a quadratic algebra consists of one second-order operator and several first-order operators. Solutions that do not permit variable separation are presented.Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 83–87, May, 1995.     

Dirac operators in Boson-Fermion Fock spaces and supersymmetric quantum field theory

Infinite dimensional analysis is developed on an abstract Boson-Fermion Fock space. A general class of Dirac operators acting there is introduced and properties of them are investigated. An index theorem for the Dirac operators is established in terms of a path integral on a loop space. It is shown that the abstract formalism presented here gives a mathematical unification for some models of supersymmetric quantum field theory.     

Perturbed Coulombic Potentials in Dirac and Klein-Gordon Equations

A relativistic extension of our pseudo-perturbative shifted l-expansion technique is presented to solve for the eigenvalues of Dirac and Klein-Gordon equations. Once more we show the numerical usefulness of its results via comparison with available numerical integration data.     

Solutions of Dirac equations with the Pöschl-Teller potential

By using the basic concepts of the supersymmetric quantum mechanics formalism and the function analysis method, we solve the Dirac equation with vector and scalar potentials and obtain the bound-state solutions for the nuclei in the relativistic P?schl-Teller potential. All of the analyses are prepared under the conditions of the exact spin symmetry and pseudospin symmetry. The exact energy equation and corresponding two-component spinor wave functions for s -wave bound states are obtained analytically.     

Grassmann-Cartan algebra and Klein-Gordon equations

Given a Riemannian structure (M, g), a hypothesis is investigated that if= _p=0 ⁿ _p (M) is submitted to the differential condition (g++)=0, =mc/—which implies that each component of fulfills the Klein-Gordon equation (- ²) _p =0, ought to be interpreted as a natural complex of the bosonic fields. Then it is found that the complex admits the interpretation in the sense of first quantization with (M) being a convex set of states, with the structure of a Hilbert space over . The definite spin states of bosons are then pure states which are not conserved by the temporal evolution.     

Relativistic Shifted-l Expansion Technique for Dirac and Klein-Gordon Equations

The shifted-l expansion technique (SLET) is extended to solve for Dirac particle trapped in spherically symmetric scalar and/or kvector potentials. A parameter λ = 0, l is introduced in such a way that one can obtain the Klein-Gordon (KG) bound states from Dirac bound states. The 4-vector Coulomb, the scalar linear, and the equally mixed scalar and 4-vector power-law potentials are used in KG and Dirac equations. Exact numerical results are obtained for the Cvector Coulomb potential in both KG and Dirac equations. Highly accurate and fast converging results are obtained for the scalar linear and the equally mixed scalar and 4-vector power-law potentials.     

Quantum Minkowski Space and the q-Analogue of Dirac Equation

The covariant differential calculus on the quantum Minkowski space is presented with the help of the generalized Wess-Zumino method and the quantum Pauli matrices. The quantum Dirac matrices are constructed parallel to those in the classical case. Combining these two aspects a q-analogue of Dirac equation follows directly.     

The Klein-Gordon and the Dirac Oscillators in a Noncommutative Space

We study the Dirac and the Klein-Gordon oscillators in a noncommutative space. It is shown that the Klein-Gordon oscillator in a noncommutative space has a similar behaviour to the dynamics of a particle in a commutative space and in a constant magnetic field. The Dirac oscillator in a noncommutative space has a similar equation to the equation of motion for a relativistic fermion in a commutative space and in a magnetic field, however a new exotic term appears, which implies that a charged fermion in a noncommutative space has an electric dipole moment.     

The Klein-Gordon and the Dirac Oscillators in a Noncommutative Space

We study the Dirac and the Klein-Gordon oscillators in a noncommutative space. It is shown that the Klein-Gordon oscillator in a noncommutative space has a similar behaviour to the dynamics ofa particle in a commutative space and in a constant magnetic field. The Dirac oscillator in a noncommutative space has a similar equation to the equation of motion for a relativistic fermion in a commutative space and in a magnetic field, however a new exotic term appears, which implies that a charged fermion in a noncommutative space has an electric dipole moment.     

Semi-indefinite inner product and generalized Minkowski spaces

In this paper we develop the theories of normed linear spaces and of linear spaces with indefinite metric, for finite dimensions both of which are also called Minkowski spaces in the literature.     

Finite energy solutions for interacting Yang-Mills and Dirac fields on Minkowski space

The Lagrangian for a SU(2) Yang-Mills field interacting with a massless isospin 1/2 Dirac field is conformally invariant. Finite energy solutions are obtained by a conformal mapping of Minkowski space onto the compact manifoldS ¹×S³ with pseudo-Riemannian metric. They are symmetric with respect to the isometric group SO(2)×SO(4) ofS ¹×S³.     

Integrable nonlinear Klein-Gordon equations and Toda lattices

We present a class of nonlinear Klein-Gordon systems which are soluble by means of a scattering transform. More specifically, for eachN2 we present a system of (N–1) nonlinear Klein-Gordon equations, together with the correspondingN ×N matrix scattering problem which can be used to solve it. We illustrate these with some special examples. The general system is shown to be closely related to the equations of the periodic Toda lattice. We present a Bäcklund transformation and superposition formula for the general system.