The concept of a collective path and its range of validity

The space of Slater determinants is described in terms of certain non-cartesian coordinates in which the time-dependent Hartree-Fock theory (TDHF) appears to be fully equivalent to a set of classical canonical equations. In this framework the concept of a collective path is developed. This consists in selecting a subspace of Slater determinants in which the collective channel is decoupled from the non-collective excitations. The adiabatic time-dependent Hartree-Fock theory (ATDHF) and the local harmonic approach (LHA) are discussed, which both provide equations to determine the collective path. The main task is to study the limits of validity of either method and the conditions under which the decoupling of collective and non-collective channels is achieved. There the curvatures of the determinantal space appear to be crucial quantities. The LHA aiming at a dynamical path /q,p> has the stronger restriction than the ATDHF, being merely concerned with a static path /q,p = 0>. The condition for the validty of ATDHF consists in requiring the collective kinetic energy per particle to be small compared with an average single-particle energy divided by a simple matrix element representing the coupling between collective and non-collective degrees of freedom. Finally, this condition is cast into a tractable form involving only RPA operations. (Auth.)