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

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.