  Solutions to the nuclear many-body problem rely on effective interactions,
and in general effective operators, to take into account effects not included
in calculations. These include effects due to the truncation to finite model
spaces where a numerical calculation is tractable, as well as physical terms
not included in the description in the first place. In the no-core shell model
(NCSM) framework, we discuss two approaches to the effective interactions based
on (i) unitary transformations and (ii) effective field theory (EFT)
principles. Starting from a given Hamiltonian, the unitary transformation
approach is designed to take into account effects induced by the truncation to
finite model spaces in which a numerical calculation is performed. This
approach was widely applied to the description of nuclear properties of light
nuclei; we review the theory and present representative results. In the EFT
approach, a Hamiltonian is always constructed in a truncated model space
according to the symmetries of the underlying theory, making use of power
counting to limit the number of interactions included in the calculations.
Hence, physical terms not explicitly included in the calculation are treated on
the same footing with the truncation to a finite model space. In this approach,
we review results for both nuclear and trapped atomic systems, for which the
effective theories are formally similar, albeit describing different underlying
physics. Finally, the application of the EFT method of constructing effective
interactions to Gamow shell model is briefly discussed.

Nuclear Theory