RPA-Approach to the Excitations of the Nucleon, Part I: Theory

In this paper we develop a theoretical framework which allows us to study excitations of the nucleon. Assuming an effective two-body interaction as a model for low-energy QCD, we derive a relativistic TDHF equation for a many-body system of quarks. To render the Dirac-sea contribution tothe mean field finite, we introduce a symmetry conserving regularization scheme. In the small amplitude limit we derive an RPA equation. The structure of the $ph$ interaction and modifications due to the regularization scheme are discussed. We give a prescription to obtain a nucleon state with good angular momentum (J) and isospin (T) quantum numbers on mean-field level. To study excitations, we develop a tensor-RPA approach, which is an extension of the conventional RPA techniques to systems with a nonscalar ground state. This allows us to construct excited states with good (J/T) quantum numbers. We discuss a method to reduce the overcomplete $ph$-space and compute the tensor-RPA interaction matrix elements. Finally we extend our scheme to include ({3/2}^+, {3/2})-states.