Abstract:
We introduce a concept of distance for a space-time where the notion of point is replaced by the notion of physical states e.g. probability distributions. We apply ideas of information theory and compute the Fisher information matrix on such a space-time. This matrix is the metric on that manifold. We apply these ideas to a simple model and show that the Lorentzian metric can be obtained if we assumed that the probability distributions describing space-time fluctuations have complex values. Such complex probability distributions appear in non-Hermitian quantum mechanics.

Abstract:
We present a method to generate probability distributions that correspond to metrics obeying partial differential equations generated by extremizing a functional $J[g^{\mu\nu}(\theta^i)]$, where $g^{\mu\nu}(\theta^i)$ is the Fisher metric. We postulate that this functional of the dynamical variable $g^{\mu\nu}(\theta^i)$ is stationary with respect to small variations of these variables. Our approach enables a dynamical approach to Fisher information metric. It allows to impose symmetries on a statistical system in a systematic way. This work is mainly motivated by the entropy approach to nonmonotonic reasoning.

Abstract:
We show that the previously introduced concept of distance on statistical spaces leads to a straightforward definition of differential entropy on these statistical spaces. These spaces are characterized by the fact that their points can only be localized within a certain volume and exhibit thus a feature of fuzziness. This implies that Riemann integrability of relevant integrals is no longer secured. Some discussion on the specialization of this formalism to quantum states concludes the paper.

Abstract:
We discuss a mechanism which generates a mass term for a scalar field in an expanding universe. The mass of this field turns out to be generated by the cosmological constant and can be naturally small if protected by a conformal symmetry which is however broken in the gravitational sector. The mass is comparable today to the Hubble time. This scalar field could thus impact our universe today and for example be at the origin of a time variation of the couplings and masses of the parameters of the standard model.

Abstract:
We show that quantum mechanics and general relativity imply the existence of a minimal length. To be more precise, we show that no operational device subject to quantum mechanics, general relativity and causality could exclude the discreteness of spacetime on lengths shorter than the Planck length. We then consider the fundamental limit coming from quantum mechanics, general relativity and causality on the precision of the measurement of a length.

Abstract:
We show that the gauge hierarchy problem can be solved in the framework of scalar-tensor theories of gravity very much in the same way as it is solved in the Randall-Sundrum scenario. Our solution involves a fine-tuning of the gravitational sector. However our mechanism does not require the introduction of extra-dimensions or new physics strongly coupled to the standard model in the low energy regime. We do introduce a new scalar field which is however coupled only gravitationally to regular matter. The physical reason for the splitting between the weak scale and the Planck scale is a violation of the Einstein's equivalence principle.

Abstract:
The aim of this article is to review the recent developments in the phenomenology of quantum gravity at the Large Hadron Collider. We shall pay special attention to four-dimensional models which are able to lower the reduced Planck mass to the TeV region and compare them to models with a large extra-dimensional volume. We then turn our attention to reviewing the emission of gravitons (massless or massive) at the LHC and to the production of small quantum black holes.