Quantum minimal length and trans-Planckian photons

In this paper, we first give arguments supporting the idea that a Banados-Teitelboim-Zanelli (BTZ) black hole can face a trans-Planckian problem even when its mass is small. Kempf-Mangano-Mann (KMM) quantum theory is applied to the Hawking evaporation of Schwarzschild and BTZ black holes. Working in the physically safe quasiposition representation, we argue that the oscillating term present in a previous analysis is removed so that actually one does not need an average procedure. We expand the s wave function as the exponential of a series in the minimal length of the new quantum theory. This reduces an infinite order differential equation to a numerable set of finite order ones. We obtain the striking result that the infinity of arbitrary constants induced by the order of the wave equation has no physical meaning due to normalization. We finally construct Gaussian wave packets and study their trajectories. We suggest a quantitative description of the nonlocality zone and its dependence on the KMM energy scale. Potential incidences on unitarity are briefly evoked.