Exposure of relations between collective theories through their energy-weighted moments

A systematic family of one-body collective operators is generated from a given one-body operator, e.g. the standard multipole operator. In the small amplitude limit any of these operators can be used as a constraint in constrained Hartree-Fock theory and the corresponding CHF solutions can be used to describe a collective path in the cranking model, in the double projection method, in the adiabatic time dependent Hartree-Fock theory or in the generator coordinate method. In this formalism one can reproduce virtually all previous collective theories by suitable choice of constraint, thereby unifying these theories. Key quantities met constantly in the analysis are RPA, energy moments, in terms of which collective masses, potentials and excitation energies are expressed. Remarkable theorems about these moments and about the collective theories have counterparts at large amplitudes. (Auth.)