Compact-Cluster Expansion for the Nuclear Many-Body Problem

Bethe's treatment of three-body clusters is extended to the entire linked-cluster perturbation series. The result is a completely rearranged expansion in terms of "compact clusters." The spatial correlations within these terms are all of quite short range. As a direct consequence, each of these new compact-cluster terms is proportional to ${\kappa }^{h^{\prime }-1\mathrm{}}$, where $h^{\prime }$ is the number of momenta inside the Fermi sea which can be summed independently after allowing for momentum conservation. The "small parameter" of the expansion is $\kappa =\rho \int {\left|\zeta \right|}^{2}d\tau$, where $\rho$ is the ordinary nuclear density and $\zeta =\phi -{\psi }_{\mathrm{BG}}$ is the "wound" in the correlated two-body (bethe-Goldstone) wave function. In nuclear matter this $\kappa$ is of order 10%. This result requires that the single-particle potentials be defined in terms of a certain subset of all the self-energy insertions. The allowed insertions are those which can be evaluated entirely on the energy shell, by means of a "generalized time-ordering" factorization. The totality of these insertions is said to constitute an "on-energy-shell mass operator" $M^{\mathrm{on}}$. The resulting occupied-state potentials are quantitatively very similar to those of most previous versions of nuclear-matter theory. However, the present intermediate-state potentials are small enough (being only of order +1 MeV) to be safely ignored in practical calculations. This simplification is offset by the need for a separate calculation of the three-body clusters. These single-particle potentials are "physically meaningful" in the sense that they lead to an optimum treatment of the short-range correlations. The bulk properties of nuclear systems are determined almost entirely by these short-range correlations. It is therefore proposed that similar potentials be used in a many-body theory of finite nuclei. True occupation numbers occur in a simple and natural way throughout the entire expansion. The formal and practical consequences of this feature are carefully examined. It is argued that this is significant for a theory of finite nuclei. Comparisons are made with several different "renormalized" formulations of quantum statistical mechanics, and also with the Green's-function theory of nuclear matter. The present formulation tends to artificially suppress the long-range correlations (e.g., pairing correlations) between particles near the Fermi surface. This defect can be eliminated by using a degenerate analog of the Goldstone expansion.