  We performed an angle-resolved photoemission spectroscopy study of Si-doped
$\beta$-Ga$_2$O$_3$. We observed very small photoemission intensity near the
Fermi level corresponding to non-dispersive states assigned to Si impurities.
We show evidence for a quantization of these states that is accompanied by a
confinement in the momentum space consistent with a real-space finite
confinement observed in a previous scanning tunneling microscopy study. Our
results suggest that this semi-localization in the conjugate spaces plays a
crucial role in the electronic conduction of this material.
