Criteria of separability of the variables in the dirac equation in gravitational fields

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

“…Using a normal diagonal tetrad, these authors constructed the generalized angular momentum operator separating the variables in the Dirac equation. Later, in a remarkable paper, appeared in 1987 [12], entitled "Criteria of separability of variables in the Dirac equation in gravitational fields", Shishkin and Andrushkevich provided the necessary and sufficient conditions, based on rigorous theorems, for separability of the variables for a diagonal tetrad gauge, and deduced the operators that determine the dependence of the wave function on the separated variables. In the same year, Barut and Duru [10] gave exact solutions of the Dirac equation in spatially flat Robertson-Walker space-times for models of expanding universes and discussed the current decomposition.…”

“…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.…”

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

“…A systematic classification of the gravitational backgrounds where the Dirac equation is separable with the help of the algebraic method is presented in ref. [20].…”

“…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].…”