Spherical waves for Dirac--K\"{a}hler and Dirac particles, formal relations between boson and fermion solutions

TL;DR

This paper derives spherical solutions for Dirac-Kähler particles in Minkowski space and explores formal relations between boson and fermion solutions, highlighting gauge invariance issues in their transformations.

Contribution

It constructs spherical boson solutions for Dirac-Kähler equations and analyzes the formal relations to Dirac fermion solutions, emphasizing gauge invariance constraints.

Findings

Spherical solutions of boson type are explicitly constructed.

Linear expansions relate boson and fermion solutions but lack gauge invariance.

Transformations between solutions do not belong to the tetrad local gauge group.

Abstract

Tetrad based equation for Dirac-K\"{a}hler particle is solved in spherical coordinates in the flat Minkocski space-time. Spherical solutions of boson type (J =0,1,2,...) are constructed. After performing a special transformation over spherical boson solutions of the Dirac-K\"{a}hler equation, 4 \times 4-matrices U(x) \Longrightarrow V(x), simple linear expansions of the four rows of new representativeof the Dirac--K\"{a}hler field V(x) in terms of spherical fermion solutions \Psi_{i}(x) of the four ordinary Dirac equations have been derived. However, this fact cannot be interpreted as the possibility not to distinguish between the Dirac-K\"{a}hler field and the system four Dirac fermions. The main formal argument is that the special transformation (I \otimes S(x)) involved does not belong to the group of tetrad local gauge transformation for Dirac-K\"{a}hler field, 2-rank bispinor under the Lorentz group. Therefore, the linear expansions between boson and fermion functions are not gauge invariant under the group of local tetrad rotations.