We study a system of an electron interacting with plasmons using Feynman path integral method. The Lagrangian and then the action which describe the motion of the electron interacting with a harmonic field of the plasmons are derived. The interaction strength is embedded in a specific dielectric function and a corresponding plasmon spectrum. We have studied two models of the dielectric media proposed by Lundqvist and Overhauser. By using numerical integration and minimization, we evaluate the ground state energy and the effective mass of the system as a function of rs. We found that both models give similar ground state energy. There is a possibility that the theory would predict a phase transition in this system. The analytic expansion of the ground state energy in some extreme limits are investigated. The applications of theory to some excitations in a Fermi gas are discussed.