Fermionic warm dark matter produces galaxy cores in the observed scales because of quantum mechanics

We derive the main physical galaxy properties: mass, halo radius, phase space density and velocity dispersion from a semiclassical gravitational approach in which fermionic WDM is treated quantum mechanically. They turn out to be fully compatible with observations. The Pauli Principle implies for the fermionic DM phase-space density Q(r→)=ρ(r→)/ σ3(r→) the quantum bound Q(r→)≤K m4/ℏ3, where m is the DM particle mass, σ(r→) is the DM velocity dispersion and K is a pure number of order one which we estimate. Cusped profiles from N-body galaxy simulations produce a divergent Q(r) at r=0 violating this quantum bound. The combination of this quantum bound with the behaviour of Q(r) from simulations, the virial theorem and galaxy observational data on Q implies lower bounds on the halo radius and a minimal distance rmin from the centre at which classical galaxy dynamics for DM fermions breaks down. For WDM, rmin turns to be in the parsec scale. For cold dark matter (CDM), rmin is between dozens of kilometers and a few meters, astronomically compatible with zero. For hot dark matter (HDM), rmin is from the kpc to the Mpc. In summary, this quantum bound rules out the presence of galaxy cusps for fermionic WDM, in agreement with astronomical observations, which show that the DM halos are cored. We show that compact dwarf galaxies are natural quantum macroscopic objects supported against gravity by the fermionic WDM quantum pressure (quantum degenerate fermions) with a minimal galaxy mass and minimal velocity dispersion. Quantum mechanical calculations which fulfil the Pauli Principle become necessary to compute galaxy structures at kpc scales and below. Classical N-body simulations are not valid at scales below rmin. We apply the Thomas-Fermi semiclassical approach to fermionic WDM galaxies, we resolve it numerically and find the physical galaxy magnitudes: mass, halo radius, phase-space density, velocity dispersion, fully consistent with observations especially for compact dwarf galaxies. Namely, fermionic WDM treated quantum mechanically, as it must be, reproduces the observed galaxy DM cores and their sizes. The lightest known dwarf galaxy (Willman I) implies a lower bound for the WDM particle mass m>0.96 keV. These results and the observed galaxies with halo radius ≥30 pc and halo mass ≥4×105Mȯ provide further indication that the WDM particle mass m is approximately in the range 1-2 keV.