A five-dimensional form of the Dirac equation

Abstract

A Dirac equation in a covariant form with respect to proper orthochronous rotations in (4 + 1)-dimensional pseudo-orthogonal space, i.e. Minkowski space extended by one real dimension is introduced. It contains a five-vector potential with a non-electromagnetic fifth component. The invariance of this equation under the CPT transformation is conditioned by the assumption that the real fifth coordinate changes its sign under charge conjugation, and that it simultaneously changes its sign either under time reversal or under space inversion. The energy levels of an electron under the simultaneous action of Coulomb and central gravitational fields are determined. To this end, (1) new eigenspinors of the total angular momentum operator are derived, with non-zero entries in the first and fourth or in the second and third row of the column matrix and (2) a scalar function is constructed from doubly-periodic Jacobian elliptic functions which, in the limit of the vanishing modulus of the elliptic functions, replaces the function exp(i t) in the stationary-state solutions. The iterated five-dimensional equation contains the ten components of the antisymmetric field tensor. It also contains a term determining the potential energy operator of electron spin density in a non-electromagnetic field. The Pauli equation is derived from the five-dimensional equation, with the transformational characteristics of the original equation. It contains a spin-orbit coupling term depending on the non-electromagnetic potential.