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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 compatible with observations. Pauli Principle implies for the fermionic DM phase-space density Q(r) = rho(r)/sigma^3(r) the quantum bound Q(r) < K m^4/hbar^3, where m is the DM particle mass, sigma(r) is the DM velocity dispersion and K is a pure number of order one which we estimate. N-body galaxy simulations produce a divergent Q(r) at r = 0 violating this quantum bound. Combining this bound with the behaviour of Q(r) from simulations, the virial and galaxy data on Q implies lower bounds on the halo radius and a minimal distance r_{min} at which classical dynamics for DM fermions breaks down. 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 quantum objects supported against gravity by the fermionic WDM quantum pressure. Quantum mechanical calculations become necessary to compute galaxy structures at kpc scales and below. Classical N-body simulations are not valid at scales below r_{min}. We apply the Thomas-Fermi semiclassical approach to fermionic WDM galaxies and find the physical magnitudes: mass, halo radius, phase-space density, velocity dispersion, consistent with observations especially for compact dwarf galaxies. Namely, fermionic WDM treated quantum mechanically, as it must be, reproduces the sizes of the observed cores. The lightest known 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 10^5 M_sun provide further indication that the WDM particle mass m is in the range 1-2 keV.