The doublet of Dirac fermions in the field of the non-Abelian monopole, isotopic chiral symmetry, and parity selection rules

Abstract

The paper concerns a problem of the Dirac fermion doublet in the external monopole potential obtained by embedding the Abelian monopole solution in the non-Abelian scheme. In this case, the doublet-monopole Hamiltonian is invariant under operations consisting of a complex and one parametric Abelian subgroup in S0(3.C). This symmetry results in a certain freedom in choosing a discrete operator N(A) (A is a complex number) entering the complete set of quantum variables. The same complex number A represents an additional parameter at the basis functions. The generalized inversion like operator N(A) affords certain generalized N(A)-parity selection rules. All the different sets of basis functions Psi(A) determine the same Hilbert space. The functions Psi(A) decompose into linear combinations of Psi(A=0): Psi(A) = F(A) Psi(A=0). However, the bases considered turn out to be nonorthogonal ones when A is a complex number; the latter correlates with the non-self-conjugacy of the N(A) at complex A-s. The meaning of possibility to violate the quantum-mechanical regulation on self-conjugacy as regards the operator N(A) is discussed. Also, the problem of possible physical status for the matrix F(A) at real A-s is considered in full detail: since the matrix belongs formally to the gauge group SU(2), but being a symmetry for the Hamiltonian this F(A) generates linear transformations on basis wave functions.