Symmetries of the complex Dirac–Kähler equation

We consider a complex version of a Dirac–Kähler-type equation, the eight-component complex Dirac–Kähler equation with a nonvanishing mass, which can be decomposed into two Dirac equations by only a nonunitary transformation. We also write an analogue of the complex Dirac–Kähler equation in five dimensions. We show that the complex Dirac–Kähler equation is a special case of a Bhabha-type equation and prove that this equation is invariant under the algebra of purely matrix transformations of the Pauli–Gürsey type and under two different representations of the Poincaré group, the fermionic (for a two-fermion system) and bosonic $\mathcal{P}$-representations. The complex Dirac–Kähler equation is also written in a manifestly covariant bosonic form as an equation for the system $(\mathcal{B}^{\mu\nu},\Phi,\mathcal{V}^{\mu})$ of irreducible self-dual tensor, scalar, and vector fields. We illustrate the relation between the complex Dirac–Kähler equation and the known 16-component Dirac–Kähler equation.