SYMMETRIES OF THE COMPLEX DIRAC–K¨AHLER EQUATION

Abstract

We consider a complex version of a Dirac–K¨ahler-type equation, the eight-component complex Dirac– K¨ahler equation with a nonvanishing mass, which can be decomposed into two Dirac equations by only a nonunitary transformation. We show that the complex Dirac–K¨ahler equation is a special case of a Bhabha-type equation. We write an analogue of the complex Dirac–K¨ahler equation in five dimensions and prove that this equation is invariant under the algebra of purely matrix transformations of the Pauli– G¨ursey type and under two different representations of the Poincar´e group, the fermionic (for a two-fermion system) and bosonic P-representations. The complex Dirac–K¨ahler equation is also written in a manifestly covariant bosonic form as an equation for the system (Bμν,Φ, Vμ) of irreducible self-dual tensor, scalar, and vector fields. We illustrate the relation between the complex Dirac–K¨ahler equation and the known 16-component Dirac–K¨ahler equation