The correlation energy of the free electron gas is investigated. The point of departure is Wigner's idea of selective summation of the perturbation-theory series for the ground-state energy. The introduction of a quasiparticle operator (with a quasiparticle understood as representing momentum propagation in the system) makes possible the resummation of the perturbation-theory series in such a way that the order of the terms is determined, not by the number of different particles that participate in an interaction, but by the number of different lines (momentum transfers). The matrix elements of the quasiparticle operator are obtained as the solution of an integral equation determined in the Macke–Tamm–Dankoff approximation and the RPA. The first order of the quasiparticle perturbation theory series (it includes the Hartree–Fock exchange energy) is simply the sum of the energies of the quasiparticles. The obtained expressions are identical to the results of other theories. The second order corresponds to exchange interaction of the quasiparticles. For it, numerical calculations are made in the range of $r_S$ from 0.1 to 10 a. u. Comparison with the analogous results obtained in the $e^S$ method has shown the inadequacy of the $e^S$ method, in which the best successes have been achieved in the investigation of the correlation energy of the free electron gas. In the case of high densities (small $r_S$) the quasiparticle exchange interaction tends to the particle exchange interaction, which was first calculated by Gell-Mann and Brueckner and for which an exact value was obtained by Onsager and collaborators. The results for the total correlation energy (the sum of the first and second terms but with the Hartree–Fock exchange energy subtracted) agrees better with the results of Hubbard and Nozières and Pines than with other results.