Dirac equation in external fields: Separation of variables in curvilinear coordinates

Abstract

The algebraic method of separation of variables in the Dirac equation proposed by one of the present authors [Gravitation and Electromagnetism (U.P., Minsk, 1989) Issue 4, p. 156 (in Russian)] is developed for the case of the most general interaction of the Dirac particle in an external field, taking into account scalar, vector, tensor, pseudovector, pseudoscalar, and gravitation connections. The present work, which concludes this series of papers entitled ‘‘Dirac equation in external fields’’ [J. Math. Phys. 32, 3184 (1991); 33, XXX (1992)] is dedicated to the investigation of the problem in the case of general orthogonal curvilinear coordinates, and allows the introduction of gravitation through the spinor connection and Lame’s functions. Special consideration is given to the cases, when the generalized Lame’s functions do not separate the variables multiplicatively (e.g., elliptic cylindrical, parabolic cylindrical, and oblate and prolate spheroidal coordinates). All the previous results and numerous results of other authors are particular cases of this investigation.

Gravitationally coupled Dirac equation for antimatter

“…In particular, we show that the condition (21) follows naturally as a consequence of the fundamental anticommutator property of the curved-space Dirac matrices (3), together with the known fact that the covariant derivative of the metric tensor has to vanish [10]. Under these assumptions, the covariant derivative of the curved-space Dirac matrix γ µ (x) also has to vanish, and the condition (21) follows as a consequence of the ansatz (17) for the covariant derivative of a spinor, together with the fundamental commutator property (18). The symmetrization of the covariant action of the Dirac field given in Eq.…”

“…The condition (21) defines the Γ ν matrix up to a multiple of the unit matrix. In the vierbein formalism, we can represent the γ ν matrices in terms of the vierbein γ µ matrices as follows,…”

“…[13]. In particular, in view of the condition (21), it is clear that the derivative terms [the first terms in round brackets on the right-hand sides of Eqs. ( 25) and (26a)] are an essential contribution to the affine connection matrix; these terms seem to be missing in the vierbein formalism formulated in later steps of the derivation leading to Eq.…”

From Generalized Dirac Equations to a Candidate for Dark Energy

“…In order to formulate the gravitational coupling of a Dirac particle, one has to formulate generalized Dirac matrices γ µ , which fulfill anticommutation relations compatible with the local metric g µν (x) of curved space-time. Based on the Christoffel symbols Γ ρ µν = Γ ρ µν (x), one formulates the Christoffel affine connection matrices Γ µ in spinor space and calculates the covariant derivative ∇ µ as follows [74][75][76][77][78],…”