  Recently Amelino--Camelia proposed a ``Doubly Special Relativity'' theory
with two observer independent scales (of speed and mass) that could replace the
standard Special Relativity at energies close to the Planck scale. Such a
theory might be a starting point in construction of quantum theory of
space-time. In this paper we investigate the quantum and statistical mechanical
consequences of such a proposal. We construct the generalized Newton--Wigner
operator and find relations between energy/momentum and frequency/wavevector
for position eigenstates of this operator. These relations indicate the
existence of a minimum length scale. Next we analyze the statistical mechanics
of the corresponding systems. We find that depending on the value of a
parameter defining the canonical commutational algebra one has to do either
with system with maximal possible temperature or with the one, which in the
high temperature limit becomes discrete.
