Relativistic Random-Phase Approximation in a Meson - System

The relativistic random-phase approximation is studied as a candidate for an appropriate approximation scheme to quantum hadrodynamics (QHD). Collective mode propagation in the system is investigated in the Walecka model, and the RRPA energy density is computed including only scalar meson exchange. A functional method, the modified loop expansion, is used to derive the RRPA meson propagator and the energy density. The collective modes are computed on the mean-field (ignoring vacuum fluctuations) and the relativistic Hartree (including vacuum fluctuations) baryon ground states. In the MFT, there exist three kinds of modes: zero-sound, meson-branch modes and modes which indicate instabilities of the MFT ground state. Zero-sound exists only above the saturation density. There are two longitudinal modes and one transverse mode in the meson branch. In the MFT case, the Pauli blocking of NN excitations greatly affect the meson branch modes and their damping. Finally, there are two kinds of instabilities in the MFT. The longitudinal instability below the saturation density represents the liquid-vapor phase transition of nuclear matter. At high densities, longitudinal (sigma) and transverse (omega) meson condensations exist. The inclusion of vacuum fluctuations (RHA) does not affect zero-sound and the phase transition instability. In the meson-branch, the vacuum polarization cancels the Pauli blocking of NN excitations. The longitudinal meson condensation is removed, while the omega-meson condensation moves to very high density. There are, however, ghost poles arising from the vacuum polarization at all densities. The RRPA energy is composed of the RHA energy and the one-meson-loop energy. A numerical method for the renormalization of the baryon self-energy and the baryon -scalar vertex, useful for computations beyond the RHA level, is presented. The RRPA energy density becomes complex due to the existence of the phase transition instability and the ghost poles. The ghost poles contribute large imaginary parts to the one-meson-loop energy. Therefore we conclude that the RRPA is not a useful approximation scheme to QHD.