  The Stefan-Boltzmann (SB) law relates the emissivity $q$, given in $Wm^{-2}$,
of an ideal black-body cavity at thermal equilibrium to the fourth power of the
absolute temperature $T$ as $q=\sigma T^4$, with $\sigma = 5.67 \times 10^{-8}
\ W m^{-2} K^{-4}$ the SB constant, firstly estimated by Stefan to within $11$
per cent of the actual value. The law is a pillar of modern physics since its
microscopic derivation implies the quantization of the energy related to the
electromagnetic field. Somewhat astonishing, Boltzmann presented his derivation
in 1878 making use only of electrodynamic and thermodynamic classical concepts,
apparently without introducing any quantum hypothesis (here called first
Boltzmann paradox). By using Planck (1901) quantization of the radiation field
in terms of a gas of photons, the SB law received a microscopic interpretation
providing also the value of the SB constant on the basis of a set of universal
constants including the quantum action constant $h$. However, the successive
consideration by Planck (1912) of the zero-point energy contribution was found
to be responsible of another divergence of the radiation energy-density for the
single photon mode at high frequencies. This divergence is of pure quantum
origin and is responsible for a vacuum-catastrophe, to keep the analogy with
the well-known ultraviolet catastrophe of the classical black-body radiation
spectrum, given by the Rayleigh-Jeans law in 1900. As a consequence, from a
rigorous quantum-mechanical derivation we expect the divergence of the SB law
(here called second Boltzmann paradox). In this paper we revisit the SB law by
accounting for genuine quantum effects associated with Planck energy
quantization and Casimir size quantization thus resolving both Boltzmann
paradoxes
