  We derive from first principles equations for bosonic, non-relativistic and
self-interacting dark matter which can include both a condensed, low momentum
"fuzzy" component and one with higher momenta that may be approximated as a
collection of particles. The resulting coupled equations consist of a modified
Gross-Pitaevskii equation describing the condensate and a kinetic equation
describing the higher momentum modes, the "particles", along with the Poisson
equation for the gravitational potential sourced by the density of both
components. Our derivation utilizes the Schwinger-Keldysh path integral
formalism and applies a semi-classical approximation which can also accommodate
collisional terms amongst the particles and between the particles and the
condensate to second order in the self-coupling strength. The equations can
therefore describe both CDM and Fuzzy Dark Matter in a unified way, allowing
for the coexistence of both phases and the inclusion of quartic
self-interactions.
