Some exact solutions of the general covariant Dirac equation in gravitational fields, namely in constant curvature de Sitter spacetime and in Milne's Universe are obtained. The tetrads for both cases are constructed on the basis of global quasi-Cartesian coordinates that allow one to exclude coordinate effects connected with the rotation of the local frame under the transition of the triad from one spacetime point to another. Solutions in the form of a spherical bispinor wave with time amplitude modulation are demonstrated for both cases of gravitational fields and the possibility of analogical solutions in the form of cylindrical and plane spinor waves for the de Sitter spacetime is discussed. All the solutions are obtained by means of the algebraic method of separation of variables.

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.

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It is well known that the most complete information about single-particle states is contained in its wave function. For spin-1/2 particles this means that it is necessary to have exact solutions of the Dirac equation. In particular, in the case of neutrinos in the presence of gravity, it is necessary to solve the covariant Dirac equation. At present, the existence of neither massless neutrinos (electron neutrinos) nor massive neutrinos (muon and τ neutrinos) cannot be excluded. Since for massless neutrinos any solution of the Dirac equation is also a solution of the Weyl equation, there exists the possibility of studying, from a unified point of view, massive as well as massless neutrinos by means of the Dirac equation. In the search of exact solutions of systems of partial differential equations one can proceed as follows: (a) separation of variables and (b) solution of the corresponding ordinary differential equations. In the present paper, a complete analysis of the separation of variables in the Dirac equation for massive as well as for massless neutrinos is carried out by means of the algebraic method [J. Math. Phys. 30, 2132 (1989)]. It is found that for the massless neutrinos, there are further possibilities of separation of variables, not valid for the massive case.

In the present article we present exact solutions of the Dirac equation for electric neutral particles with anomalous electric and magnetic moments. Using the algebraic method of separation of variables, the Dirac equation is separated in cartesian, cylindrical and spherical coordinates, and exact solutions are obtained in terms of special functions. Typeset using REVT E X

New exact solutions are searched for on the basis of the method of separation of variables proposed in earlier work by the present authors [J. Math. Phys. 30, 2132 (1989)]. The essence of this method consists of constructing first-order matrix differential operators that define the dependence of the Dirac bispinor on the related variables, but commutation of such operators with the operator of the equation or between them is not assumed.The classical problems are considered as possibilities, namely, electrons in the field of plane monochromatic electromagnetic waves (Volkov’s problem) and electrons in the Coulomb field (hydrogen atom). Then ‘‘plane’’ external electromagnetic fields are considered for which some new exact solutions are obtained in terms of special functions. Four new exact solutions of the Dirac equation in the fields with axial cylindrical symmetry are also shown, and lastly one ‘‘free’’ solution with exotic geometry is demonstrated, namely, ‘‘free’’ parabolic cylindrical spinor waves.

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The method of separation of variables in the Dirac equation proposed in an earlier work by one of the present authors [J. Math. Phys. 30, 2132 (1989)] is developed for the complete set of interactions of the Dirac particle. The essence of the method consists of the separation of the first-order matrix differential operators that define the dependence of the Dirac bispinor on the related variables, but commutation of such operators with or between the operator of the equation is not assumed. This approach, which is perfectly justified in the presence of gravitational [Theor. Math. Phys. 70, 204 (1987)] or vector fields [J. Math. Phys. 30, 2132 (1989)], permits one to find all the possibilities of separation of variables in the Dirac equation in the case of the most general set of external fields. The complete set of interactions of the Dirac particle is determined by the symmetry group of equations, namely, viz. the SU(4) group. The interactions are scalar, vector, tensor, pseudovector and pseudoscalar. The analysis in this article is limited to Cartesian coordinates. The corresponding results for the general curvilinear coordinates will be presented in a future paper.