Challenges and recent advancements of the nuclear many-body problem on the fermionic correlation functions (CFs) will be discussed. Starting from the general ab-initio Hamiltonian, a consistent equation of motion (EOM) technique is formulated for the low-rank CFs and adopted for nuclear applications by approximations with minimal truncations, which keep the leading effects of emergent collectivity. The systematically improvable character of the theory allows taking into account configurations of growing complexity in a controlled way.
The approach is implemented numerically for the nuclear response, on the basis of the effective meson-nucleon Lagrangian, in the framework of the relativistic nuclear field theory (RNFT). The results obtained for the monopole, dipole and Gamow-Teller responses of medium-heavy nuclei show that the consistent inclusion of the emergent collective degrees of freedom refines the description of nuclear spectra, in both the high-energy and the low-energy sectors. The approach confined by the leading ph⊗phonon configurations beyond the standard random phase approximation has been extended to the case of finite temperature for both neutral and charge-exchange nuclear response. This has open new perspectives for generating a consistent input for astrophysical applications.