Kempf et al. in Ref.[1] have formulated a Hilbert space representation of quantum mechanics with a minimal measurable length. Recently it has been revealed, in the context of doubly special relativity, that a test particles' momentum cannot be arbitrarily imprecise and therefore there is an upper bound for momentum fluctuations. Taking this achievement into account, we generalize the seminal work of Kempf et al. to the case that there is also a maximal particles' momentum. Existence of an upper bound for the test particles' momentum provides several novel and interesting features, some of which are studied in this paper.

We study tunneling process through quantum horizon of a Schwarzschild black hole in noncommutative spacetime. This is done by considering the effect of smearing of the particle mass as a Gaussian profile in flat spacetime. We show that even in this noncommutative setup there will be no correlation between the different modes of radiation which reflects the fact that information doesn't come out continuously during the evaporation process at least at late-time. However, due to spacetime noncommutativity, information might be preserved by a stable black hole remnant.

The existence of a minimum observable length and/or a maximum observable momentum is in agreement with various candidates of quantum gravity such as string theory, loop quantum gravity, doubly special relativity and black hole physics. In this scenario, the Heisenberg uncertainty principle is changed to the so-called Generalized (Gravitational) Uncertainty Principle (GUP) which results in modification of all Hamiltonians in quantum mechanics. In this paper, following a recently proposed GUP which is consistent with quantum gravity theories, we study the quantum mechanical systems in the presence of both a minimum length and a maximum momentum. The generalized Hamiltonian contains two additional terms which are proportional to αp 3 and α 2 p 4 where α ∼ 1/M P l c is the GUP parameter. For the case of a quantum bouncer, we solve the generalized Schrödinger equation in the momentum space and find the modified energy eigenvalues and eigenfunctions up to the secondorder in GUP parameter. The effects of the GUP on the transition rate of ultra cold neutrons in gravitational spectrometers are discussed finally.

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There are several approaches to quantum gravitational corrections of black hole thermodynamics. String theory and loop quantum gravity, by direct analysis on the basis of quantum properties of black holes, show that in the entropy-area relation the leading order correction should be of log-area type. On the other hand, generalized uncertainty principle(GUP) and modified dispersion relations(MDRs) provide perturbational framework for such modifications. Although both GUP and MDRs are common features of all quantum gravity scenarios, their functional forms are quantum gravity model dependent. Since both string theory and loop quantum gravity give more reliable solution of the black hole thermodynamics, one can use their results to test approximate results of GUP and MDRs. In this paper, we find quantum corrected black hole thermodynamics in the framework of GUP and MDR and then we compare our results with string theory solutions. This comparison suggests severe constraints on the functional form of GUP and MDRs. These constraints may reflect characteristic features of ultimate quantum gravity theory.