Groud-state properties of nuclear matter

This paper is concerned with the development of an approximate method for treating the properties of large bound systems in their ground state and with the application of this method to idealized nuclear matter. The difficulties encountered in alternative perturbative approaches are considered in the introduction. These difficulties are shown to demand a more general approach. Such an approach, based on the formulation of the many-particle problem developed by Martin and Schwinger, is discussed in the second section. We review here the aspects of this formalism necessary for generating a sequence of correlative approximations with the aid of expectation values of field operator products or Green's functions. A second approximation in the sequence, which neglects correlations between a given particle and a highly correlated pair, is then treated in detail. Approximate equations are derived for the energy, density, and momentum distribution. Explicit recognition of the bound nature of the ground state is seen to simplify the solution considerably, and the simplification leads to a well-defined single-particle energy-momentum relation for the bound particles. On the other hand, the momentum distribution ϱ(k) differs from that of the noninteracting Fermi gas, since momenta larger than kfp= (32π2ϱ)13 role=presentation style=font-size: 90%; display: inline-block; position: relative;>$\text{k}{}_{\mathrm{f}}{}^{\mathrm{p}}\text{= (}\text{3}\text{2}\text{π}{}^2\text{ϱ)}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$$\text{k}{}_{\mathrm{f}}{}^{\mathrm{p}}\text{= (}\text{3}\text{2}\text{π}{}^2\text{ϱ)}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$ contribute significantly. Our approximation exhibits a cut-off about ten percent higher than k_f^p, and the calculated energy per particle and inter-particle distance are −14.4 Mev and 0.87 fermi, respectively. The condition that the pressure vanish at equilibrium here requires that the energy of the highest filled single particle state equal the average energy per particle. Since K_f^p is no longer the highest filled mode, it has no particular significance in our treatment. Integral expressions containing complicated geometrical factors in their integrands, which usually result from a perturbative treatment of the Pauli principle, do not appear in our second correlation approximation. In fact, the direct part of the function T_K(ω), which is somewhat analogous to the “reaction matrix” of the perturbative approach, is found to be an ordinary two-body scattering matrix in which the energy of the particle pair is equal to its actual value in the medium. This simplifying feature is related to the bound nature of the ground state. Finally, we discuss the correction terms to the energy and density which arise from our neglecting higher order correlations. These terms are shown to be small. In addition, we show that the distance beyond which two particles in nuclear matter are essentially uncorrelated is of the same order of magnitude as the interparticle distance.