This article reviews the present status of the spin-foam approach to the quantization of gravity. Special attention is payed to the pedagogical presentatio

Trying to combine standard quantum field theories with gravity leads to a breakdown of the usual structure of space time at around the Planck length, $1.6\times {10}^{-35}\mathrm{m}$, with possible violations of Lorentz invariance. Calculations of preferred-frame effects in quantum gravity have further motivated high precision searches for Lorentz violation. Here, we explain that combining known elementary particle interactions with a Planck-scale preferred frame gives rise to Lorentz violation at the percent level, some 20 orders of magnitude higher than earlier estimates, unless the bare parameters of the theory are unnaturally strongly fine tuned. Therefore an important task is not just the improvement of the precision of searches for violations of Lorentz invariance, but also the search for theoretical mechanisms for automatically preserving Lorentz invariance.

This is a review of results on black hole physics in the context of loop quantum gravity. The key feature underlying these results is the discreteness of geometric quantities at the Planck scale predicted by this approach to quantum gravity. Quantum discreteness follows directly from the canonical quantization prescription when applied to the action of general relativity that is suitable for the coupling of gravity with gauge fields, and especially with fermions. Planckian discreteness and causal considerations provide the basic structure for the understanding of the thermal properties of black holes close to equilibrium. Discreteness also provides a fresh new look at more (at the moment) speculative issues, such as those concerning the fate of information in black hole evaporation. The hypothesis of discreteness leads, also, to interesting phenomenology with possible observational consequences. The theory of loop quantum gravity is a developing program; this review reports its achievements and open questions in a pedagogical manner, with an emphasis on quantum aspects of black hole physics.

The Immirzi parameter is a constant appearing in the general-relativity action used as a starting point for the loop quantization of gravity. The parameter is commonly believed not to appear in the equations of motion and not to have any physical effect besides nonperturbatrive quantum gravity. We show that this is not true in general: in the presence of minimally coupled fermions, the parameter appears in the equations of motion: it determines the coupling constant of a four-fermion interaction. Under some general assumptions, there is therefore a relation between the Immirzi parameter and physical effects that are observable in principle, independently from nonperturbative quantum gravity