Dirac-K\"ahler particle in Riemann spherical space: boson interpretation

TL;DR

This paper derives exact solutions for the Dirac-K"ahler equation in spherical space, revealing a complex energy spectrum that differs from the Dirac particle, challenging the fermion interpretation of the Dirac-K"ahler field.

Contribution

It provides the first exact solutions and energy spectrum analysis of the Dirac-K"ahler particle in spherical space, highlighting its distinct structure from Dirac fermions.

Findings

Discrete energy spectrum with two parallel, twofold degenerate series

Solutions expressed via hypergeometric functions and quasi-polynomial wave-functions

Dirac-K"ahler field cannot be interpreted as four Dirac fermions

Abstract

In the context of the composite boson interpretation, we construct the exact general solution of the Dirac--K\"ahler equation for the case of the spherical Riemann space of constant positive curvature, for which due to the geometry itself one may expect to have a discrete energy spectrum. In the case of the minimal value of the total angular momentum, $j=0$, the radial equations are reduced to second-order ordinary differential equations, which are straightforwardly solved in terms of the hypergeometric functions. For non-zero values of the total angular momentum, however, the radial equations are reduced to a pair of complicated fourth-order differential equations. Employing the factorization approach, we derive the general solution of these equations involving four independent fundamental solutions written in terms of combinations of the hypergeometric functions. The corresponding discrete energy spectrum is then determined via termination of the involved hypergeometric series, resulting in quasi-polynomial wave-functions. The constructed solutions lead to notable observations when compared with those for the ordinary Dirac particle. The energy spectrum for the Dirac-K\"ahler particle in spherical space is much more complicated. Its structure substantially differs from that for the Dirac particle since it consists of two paralleled energy level series each of which is twofold degenerate. Besides, none of the two separate series coincides with the series for the Dirac particle. Thus, the Dirac--K\"ahler field cannot be interpreted as a system of four Dirac fermions. Additional arguments supporting this conclusion are discussed.