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 $\frac{\mathrm{cy}}{({y}^{2}+c{\mathcal{U}}^{\ensuremath{-}1})}$, 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 $\mathcal{U}\ensuremath{\rightarrow}\ensuremath{\infty}$ 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.The connection of the Brueckner approximation with a phase-shift approximation for low-density systems is discussed. Some corrections to the higher order terms in existing derivations of the "separation method" expansion of the Brueckner reaction matrix are given.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 $\frac{\mathrm{cy}}{({y}^{2}+c{\mathcal{U}}^{\ensuremath{-}1})}$, 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 $\mathcal{U}\ensuremath{\rightarrow}\ensuremath{\infty}$ 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.The connection of the Brueckner approximation with a phase-shift approximation for low-density systems is discussed. Some corrections to the higher order terms in existing derivations of the "separation method" expansion of the Brueckner reaction matrix are given.

Physical ReviewAug 15, 19628 citations

The theory of interacting fermi systems

Philippe Nozières

Journal of the Franklin InstituteApr 1, 1964698 citations

Generalized Reaction Matrix Approach to the Theory of the Infinite Medium of Fermions

R. E. Prange, Abraham Klein

A closed set of coupled, nonlinear integral equations is obtained which completely describe a system of Fermi particles. The equations define the one and two-particle Green's functions which are completely analogous to those used in the field theory of fundamental particles. Special boundary conditions appropriate to the infinite medium of fermions are employed. Approximations to these equations are made which lead to a generalized Brueckner reaction-matrix formalism in the case of nuclear matter and which lead to an expression for the correlation energy in the case of the electron gas. The approximations for the system of nuclear matter are of three types: Ones which neglect the simultaneous interaction of three or more nucleons, the effective interactions of the vacant states below the Fermi sea, and the self-energy interactions with excited nucleons. The relationship between the reaction matrix and the expectation value of the actual potential energy is discussed. It is found that the energy levels given by the Brueckner formalism are nearly those given by a consideration of the actual energy, thus indicating that the shell-model energy levels for closed-shell-plus-one nuclei should be almost correct. A formula for the momentum distribution of nucleons in nuclear matter is obtained.A closed set of coupled, nonlinear integral equations is obtained which completely describe a system of Fermi particles. The equations define the one and two-particle Green's functions which are completely analogous to those used in the field theory of fundamental particles. Special boundary conditions appropriate to the infinite medium of fermions are employed. Approximations to these equations are made which lead to a generalized Brueckner reaction-matrix formalism in the case of nuclear matter and which lead to an expression for the correlation energy in the case of the electron gas. The approximations for the system of nuclear matter are of three types: Ones which neglect the simultaneous interaction of three or more nucleons, the effective interactions of the vacant states below the Fermi sea, and the self-energy interactions with excited nucleons. The relationship between the reaction matrix and the expectation value of the actual potential energy is discussed. It is found that the energy levels given by the Brueckner formalism are nearly those given by a consideration of the actual energy, thus indicating that the shell-model energy levels for closed-shell-plus-one nuclei should be almost correct. A formula for the momentum distribution of nucleons in nuclear matter is obtained.

Physical ReviewNov 1, 195825 citations

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 $\mathcal{U}\ensuremath{\rightarrow}\ensuremath{\infty}$ 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.The connection of the Brueckner approximation with a phase-shift approximation for low-density systems is discussed. Some corrections to the higher order terms in existing derivations of the "separation method" expansion of the Brueckner reaction matrix are given.