  This thesis concerns the correlation structure of interacting Fermi gases on
a torus in the mean-field regime. A bosonization method in the spirit of Sawada
is developed to analyze the system, and is applied to obtain an upper bound for
the correlation energy of the system for a wide class of repulsive interaction
potentials, including the Coulomb potential. This upper bound includes both a
bosonic contribution, as found in the bosonic model of Sawada, and an exchange
contribution, as was found by Gell-Mann and Brueckner but which was missed by
Sawada's model. An extension to weakly attractive potentials is also presented,
as is an outline of the derivation of an effective Hamiltonian for regular
interaction potentials, and the construction of plasmon states for this outside
of the mean-field setting. This thesis is based on the papers
[arXiv:2106.11161, arXiv:2206.13073, arXiv:2208.01581].
