Dirac equation in external fields: Separation of variables in nondiagonal metrics

Abstract

The algebraic method of separation of variables in the Dirac equation proposed in earlier works by one of the present authors [Theor. Math. Phys. 70, 204 (1987); J. Math. Phys. 30, 2132 (1989)] is developed for the space-time with nondiagonal metrics. The essence of the method consists of the separation of the first-order matricial differential operators that define the dependence of the Dirac’s bispinor on the related variables. In contrast to some other authors the pairs of operators are commuted on each step of separation including the variables mixed by nondiagonal elements of fundamental tensor of space-time. There are reasons to believe that it must be some local similarity transformation connected these commuted operators with noncommuted corresponding operators of other authors, although such transformation in view of mathematical difficulties of problem, in general, were not successfully found.

Dirac equation is separable on the dyon black hole metric

“…The Chandrasekhar result has been also generalized to other cases [17], [22], [23], [29], the mathematical structure of the problem has been investigated [18], [19], [20], [21], [22], [23], [24], [25] and the solutions studied [26], [27], [28]. Other approaches to the problem of separation of variables in the Dirac Equation on curved spaces include the Stäckel Space method [30] and the "algebraic" method [31].…”

Electrically neutral Dirac particles in the presence of external fields: Exact solutions

“…The difficulties that arise are those we find when we include non geometrized fields like scalar, pseudo-scalar or pseudo-vector fields. 4,5,8 Now, we proceed to mention the most relevant exact solutions of the Dirac equation with anomalous moment reported in the literature, among them we have the constant magnetic field problem solved by Strocchi 9 and by Ternov et al 10 , a generalization of the Volkov's problem was found by Ternov et al 11,12 , and more recently this problem has been revisited by Barut 13 . The problem in spherical coordinates for the central field has been studied by Ternov 14 and Barut 15,16 , a good review on exact solutions of the Dirac equation with anomalous interaction can be found in 17 .…”

“…The amount of articles devoted to the study of this problem is scarce, the reason is that, for solving the Dirac equation, as general rule, a complete separation of variables is required and, the inclusion of tensor field functions in the Dirac equation dramatically restricts the possibilities of separation of variables [2][3][4][5] In this case, the possibilities are even more limited that those obtained for the Dirac equation minimally coupled to the electromagnetic field 6 , or in the presence of gravitational fields 7 This one is because the gravitational field goes into the Dirac equation in a geometric way, via the Lame's coefficients, and no additional matrices in the equation if we choose to work in a diagonal tetrad gauge. Regarding the vector electromagnetic fields, the inclusion of them in the Dirac equation via the generalized momentum (minimal coupling), does not conduce to the apparition of new matrices, and 6 the presence of tensor fields terms of the form gγ m γ n T mn introduces in the Dirac equation a new functional dependence.…”