Linking the monopole part with Hartree-Fock theory, more definitions

In Eqs. (23) and (24), we have used a compact notation for the single-particle states, with the symbol \( \alpha_p \) etc representing all possible quantum numbers except the magnetic substates \( m_{j_p} \), that is \( \alpha_p=(n_p,l_p,j_p,t_{z_p}) \). The symbol \( \alpha_F \) stands now for all single-particle states up to the Fermi level, excluding again the magnetic substates. In the above two-body interaction matrix elements \( \langle (\alpha_p\alpha_i)J | \hat{v}(\tilde{v}) |(\alpha_p\alpha_i)J \rangle \) we have dropped additional quantum numbers like the isospin projection. Our interactions are diagonal in the projection of the total isospin but breaks both isospin symmetry and charge symmetry.