The doublet of Dirac fermions in the field of the non-Abelia monopole and parity selection rules

Abstract

The paper concerns a problem of Dirac fermion doublet in the external monopole potential arisen out of embedding the Abelian monopole solution in the non-Abe- lian scheme. In this particular case, the Hamiltonian is invariant under some symmetry operations consisting of an Abelian subgroup in the complex rotational group SO(3.C). This symmetry results in a certain (A)-freedom in choosing a discrete operator entering the complete set {H, j^{2}, j_{3}, N(A), K} . The same complex number A represents a parameter of the wave functions constructed. The generalized inversion-like operator N(A) implies its own (A-dependent) de- finition for scalar and pseudoscalar, and further affords some generalized N(A)-parity selection rules. It is shown that all different sets of basis func- tions Psi(A) determine the same Hilbert space. In particular, the functions Psi(A) decompose into linear combinations of Psi(A=0). However, the bases con- sidered turn out to be nonorthogonal ones when A is not real number; the latter correlates with the non-self-conjugacy property of the operator N(A) at those A-s. (This is a shortened version of the paper).