Galilean boost energy of an interacting gas ($\Delta E=N{p}_0^2/(2m)$)

For a resting uniform continuum gas with pair interactions depending on relative positions, shifting every particle momentum by ${\mathbf{p}}_0$ leaves interaction energy unchanged and changes the microscopic kinetic energy by ${\mathbf{p}}_0\cdot \mathbf{P}/m+N{p}_0^2/(2m)$. The resting ground state has total momentum $\mathbf{P}=0$. The resulting boost energy uses the bare mass even in an interacting Fermi liquid; matching this with the Landau quasiparticle energy functional constrains its interaction kernel.