  The Galilei-covariant fermionic field theories are quantized by using the
path-integral method and five-dimensional Lorentz-like covariant expressions of
non-relativistic field equations. Firstly, we review the five-dimensional
approach to the Galilean Dirac equation, which leads to the Levy-Leblond
equations, and define the Galilean generating functional and Green's functions
for positive- and negative-energy/mass solutions. Then, as an example of
interactions, we consider the quartic self-interacting potential ${\lambda}
(\bar{\Psi} {\Psi})^2$, and we derive expressions for the 2- and 4-point
Green's functions. Our results are compatible with those found in the
literature on non-relativistic many-body systems. The extended manifold allows
for compact expressions of the contributions in $(3+1)$ space-time. This is
particularly apparent when we represent the results with diagrams in the
extended $(4+1)$ manifold, since they usually encompass more diagrams in
Galilean $(3+1)$ space-time.
