Linked-Clusters Expansion in Quantum Statistics: Petit Canonical Ensemble

A linked-cluster expansion for the free energy in the petit canonical ensemble is presented and proved. It has the advantage of avoiding the introduction of the unknown chemical potential in the perturbation series. As a consequence of correlations among the population numbers $n\left(\mathrm{K}\right)$, additional linkages representing these correlations appear. The result is used to find the ground-state energy of a many-body fermion system. This expression reduces to the Brueckner-Goldstone expansion only in the case of central forces in an isotropic system, a theorem due to Kohn, Luttinger, and Ward. It is also shown that in the random phase approximation, correlation bonds do not contribute. Finally the relation of our formalism to the Bloch-De Dominicis expansion for the grand partition function is discussed.