  A many body Hamiltonian involving the mean field for a projected spherical
single particle basis, the pairing interactions for alike nucleons and the
dipole-dipole proton-neutron interactions in the particle-hole ($ph$) and
particle-particle ($pp$) channels is treated by the projected gauge fully
renormalized proton-neutron quasiparticle random phase approximation
(PGFRpnQRPA) approach. The resulting wave functions and energies for the mother
and the daughter nuclei are used to calculate the $2\nu\beta\beta$ decay rate
and the process half life. For illustration, the formalism is applied for the
decay $^{100}$Mo$ \to$ $^{100}$Ru. The results are in good agreement with the
corresponding experimental data. The Ikeda sum rule ($ISR$) is obeyed. The
gauge projection makes the $pp$ interaction inefficient.
