Equations-of-motion approach to a quantum theory of large-amplitude collective motion

The equations-of-motion approach to large-amplitude collective motion is implemented both for systems of coupled bosons, also studied in a previous paper, and for systems of coupled fermions. For the fermion case, the underlying formulation is that provided by the generalized Hartree-Fock approximation (or generalized density matrix method). To obtain results valid in the semi-classical limit, as in most previous work, we compute the Wigner transform of quantum matrices in the representation in which collective coordinates are diagonal and keep only the leading contributions. Higher-order contributions can be retained, however, and, in any case, there is no ambiguity of requantization. The semi-classical limit is seen to comprise the dynamics of time-dependent Hartree-Fock theory (TDHF) and a classical canonicity condition. By utilizing a well-known parametrization of the manifold of Slater determinants in terms of classical canonical variables, we are able to derive and understand the equations of the adiabatic limit in full parallelism with the boson case. As in the previous paper, we can thus show: (i) to zero and first order in the adiabatic limit the physics is contained in Villars' equations; (ii) to second order there is consistency and no new conditions. The structure of the solution space (discussed thoroughly in the previous paper) is summarized. A discussion of associated variational principles is given. A form of the theory equivalent to self-consistent cranking is described. A method of solution is illustrated by working out several elementary examples. The relationship to previous work, especially that of Zeievinsky and Marumori and coworkers is discussed briefly. Three appendices deal respectively with the equations-of-motion method, with useful properties of Slater determinants, and with some technical details associated with the fermion equations of motion.