Retarded Interactions in Fermi Systems

The nature of the two-body interactions in many-fermion systems is studied from the viewpoint of meson theory. An exactly soluble model is formulated for the linear coupling of a meson field to fermion density fluctuations, in which meson degrees of freedom are treated exactly, and fermion motion is treated within the domain of the random-phase approximation (RPA). Instability conditions for the RPA ground state are established. More generally, the effective two-body interaction is deduced via a Green's-function technique by eliminating the meson degrees of freedom. This interaction is shown to be frequency-dependent, i.e., retarded in time. The resulting interaction is then applied to the calculation of the Hartree-Fock (H.F.) field and of the collective modes of the system via a generalized Landau equation. In the H.F. approximation, one obtains an unambiguous separation of renormalization (self-energy) effects and the nucleon-nucleon interactions themselves, the former reducing to the correct mass renormalization of the nucleon in the static limit. For reasonably small momenta ($p<\mathrm{}{p}_F$), the retardation corrections to the H.F. field can be characterized by a small parameter ${\left(\frac{{\epsilon }_F}{\mu }\right)}^2$ ($\approx 0.1$ for actual nuclear densities), where ${\epsilon }_F=\mathrm{Fermi}\mathrm{energy}$ and $\mu =\mathrm{meson}\mathrm{mass}$. The corrections become more important at high momenta and densities. In the long-wavelength limit, the frequency-dependent corrections to the collective mode energies are found to be of order ${\left(\frac{\omega }{\mu }\right)}^2$, where $\omega =\mathrm{collective}\mathrm{mode}\mathrm{energy}$. For a static Yukawa interaction, a value ${\lambda }^2\approx 5$ (consistent with the usual shell-model values) is found for the neutral scalar coupling constant by requiring that the giant dipole collective state appear at the experimental energy. For pseudoscalar coupling, the usual renormalized coupling constant $\frac{f^2}{4\pi }\approx 0.08$ is shown to yield a "breathing mode" in heavy nuclei consistent with crude estimates based on nuclear compressibilities.