Doubly special quantum and statistical mechanics from quantum κ-Poincare algebra

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 a quantum theory of spacetime. In this Letter 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 a system with maximal possible temperature or with the one, which in the high temperature limit becomes discrete. (C) 2002 Elsevier Science B.V. All rights reserved.