We formulate generalized uncertainty relations in a crystal-like universe\char22{}a ``world crystal''\char22{}whose lattice spacing is of the order of Planck length. In the particular case when energies lie near the border of the Brillouin zone, i.e., for Planckian energies, the uncertainty relation for position and momenta does not pose any lower bound on involved uncertainties. We apply our results to micro black holes physics, where we derive a new mass-temperature relation for Schwarzschild micro black holes. In contrast to standard results based on Heisenberg and stringy uncertainty relations, our mass-temperature formula predicts both a finite Hawking's temperature and a zero rest-mass remnant at the end of the micro black hole evaporation. We also briefly mention some connections of the world-crystal paradigm with 't~Hooft's quantization and double special relativity.

We consider a generalized uncertainty principle (GUP) corresponding to a deformation of the fundamental commutator obtained by adding a term quadratic in t

Investigations on possible violation of Lorentz invariance have been widely pursued in the last decades, both from theoretical and experimental sides. A comprehensive framework to formulate the problem is the standard model extension (SME) proposed by A. Kostelecky, where violation of Lorentz invariance is encoded into specific coefficients. Here we present a procedure to link the deformation parameter $\beta$ of the generalized uncertainty principle to the SME coefficients of the gravity sector. The idea is to compute the Hawking temperature of a black hole in two different ways. The first way involves the deformation parameter $\beta$, and therefore we get a deformed Hawking temperature containing the parameter $\beta$. The second way involves a deformed Schwarzschild metric containing the Lorentz violating terms ${\overline{s}}^{\mu \nu }$ of the gravity sector of the SME. The comparison between the two different techniques yields a relation between $\beta$ and ${\overline{s}}^{\mu \nu }$. In this way bounds on $\beta$ transferred from ${\overline{s}}^{\mu \nu }$ are improved by many orders of magnitude when compared with those derived in other gravitational frameworks. Also the opposite possibility of bounds transferred from $\beta$ to ${\overline{s}}^{\mu \nu }$ is briefly discussed.

A localized particle in Quantum Mechanics is described by a wave packet in position space, regardless of its energy. However, from the point of view of General Relativity, if the particle’s energy density exceeds a certain threshold, it should be a black hole. To combine these two pictures, we introduce a horizon wave function determined by the particle wave function in position space, which eventually yields the probability that the particle is a black hole. The existence of a minimum mass for black holes naturally follows, albeit not in the form of a sharp value around the Planck scale, but rather like a vanishing probability that a particle much lighter than the Planck mass may be a black hole. We also show that our construction entails an effective generalized uncertainty principle (GUP), simply obtained by adding the uncertainties coming from the two wave functions associated with a particle. Finally, the decay of microscopic (quantum) black holes is also described in agreement with what the GUP predicts.

We consider the production of primordial micro black holes (MBH) remnants in the early Universe. These objects induce the Universe to be in a matter-dominated era before the onset of inflation. Effects of such an epoch on the CMB power spectrum are discussed and computed both analytically and numerically. By comparison with the latest observational data from the WMAP collaboration, we find that our model appears to explain the quadrupole anomaly of the CMB power spectrum.