Nonsingular Formulation of the Brueckner Approximation for an Infinite Fermi System

In the approximation neglecting any but single pair correlations, singularities of the reaction matrix render Brueckner's integral for the average energy per particle a singular integral. Several attempts to overcome this difficulty have been unsuccessful. By considering the infinite Fermi system to be a limit of finite systems, it is shown that the correct result merely involves replacing Brueckner's ordinary integral over diagonal reaction matrix elements by a principal value integral. In a finite system the level shift of a Bethe-Goldstone state differs from the diagonal reaction matrix element by a normalization factor which does not approach unity uniformly in the integration variable as the volume becomes infinite. In the neighborhood of a singularity the expression for the two-particle energy shift takes the form $\frac{\mathrm{cy}}{(y^2+c{\mathcal{U}}^{-1\mathrm{}})}$, where $y$ is the unperturbed energy measured from the singularity, $c$ is the square of a matrix element, and $\mathcal{U}$ is the quantization volume. Hence as $\mathcal{U}\to \infty$ the sum over the energy $y$ indeed approaches a principal value integral. An alternative derivation, employing a modified reaction matrix for which there is no difference between level shift and matrix element, leads to the same result. The general derivations are preceded by a soluble example.