Dirac equation in external vector fields: Separation of variables

Abstract

Solution of the Dirac equation with pseudospin symmetry for a new harmonic oscillatory ring-shaped noncentral potential J. Math. Phys. 53, 082104 (2012) Effect of tensor interaction in the Dirac-attractive radial problem under pseudospin symmetry limit J. Math. Phys. 53, 082101 (2012) Asymptotic stability of small gap solitons in nonlinear Dirac equations J. Math. Phys. 53, 073705 (2012) On Dirac-Coulomb problem in (2+1) dimensional space-time and path integral quantization J. Math. Phys. 53, 063503 (2012) Quasi-exact treatment of the relativistic generalized isotonic oscillatorThe method of separation of variables in the Dirac equation in the external vector fields is developed through the search for exact solutions. The essence of the method consists of the separation of the first-order matricial differential operators that define the dependence of the Dirac bispinor on the related variables, but commutation of such operators with the operator of the equations or between them is not assumed. This approach, which is perfectly justified in the presence of gravitational fields, permits one to prove rigorous theorems about necessary and sufficient conditions on the field functions that allow one to separate variables in the Dirac equation. In analogous investigations by other authors [Bagrov et al., Exact solutions of Relativistic Wave Equations (Nauka, Novosibirsk, 1982)] for electromagnetic fields an essential demand related to the operators that define the dependence of the bispinor on the separated variables is the demand for the commutation of a complete set of operators between them or with the operators of the Dirac equation. For this reason a series of possibilities that do not satisfy this demand escape the attention of these other authors. The present work liquidates this gap, solving the problem for external vector fields in general.

Exact solution of the Dirac equation for a Coulomb and scalar potential in the presence of an Aharonov–Bohm and a magnetic monopole fields

“…Applying the algebraic method of separation of variables 3,4 , it is possible to write eq. (14) as a sum of two first order linear differential operatorsK 1 ,K 2 satisfying the relation…”

Nonlinear Spinor Field Equations in Gravitational Theory: Spherical Symmetric Soliton-Like Solutions

“…All these activities, diverse and complementary, made in this field [1][2][3][4][5][6][7][8][9][10][11][12][13][14], are also mainly motivated by the wide roles of Einstein and Dirac equations in modern physics, for example, for investigating the spin particle and for the necessity of analysis of synchrotronic radiation [11]. To this purpose, many systems have been subjects of considerable interest and studies.…”

“… inferring that the quantities 00 g , 11 g , 22 g and V  are regular and, from (3.47), the total energy…”

Separation of variables and exact solution of the Klein–Gordon and Dirac equations in an open universe

“…In order to solve Eq. (57) we apply the algebraic method of separation of variables [20][21][22][23][24]. The method consists in rewriting the Dirac equation (57) as a sum of two first order differential operatorsK 1 ,K 2 satisfying the relation…”

“…In the literature we have at our disposal different methods for solving relativistic wave equations in curved spaces and in curvilinear coordinates; among them the method of separation of variables is one of the most widely used. [18][19][20][21].…”