Abstract:
We present an updated review of Lorentz invariance tests in Effective field theories (EFT) in the matter as well as in the gravity sector. After a general discussion of the role of Lorentz invariance and a derivation of its transformations along the so called von Ignatovski theorem, we present the dynamical frameworks developed within local EFT and the available constraints on the parameters governing the Lorentz breaking effects. In the end, we discuss two specific examples, the OPERA "affaire" and the case of Ho\v{r}ava-Lifshitz gravity. The first case will serve as an example, and a caveat, of the practical application of the general techniques developed for constraining Lorentz invariance violation (LIV) to a direct observation potentially showing these effects. The second case will show how the application of the same techniques to a specific quantum gravity scenario has far fetching implications not foreseeable in a purely phenomenological EFT approach.

Abstract:
Analogue models of gravity have provided an experimentally realizable test field for our ideas on quantum field theory in curved spacetimes but they have also inspired the investigation of possible departures from exact Lorentz invariance at microscopic scales. In this role they have joined, and sometime anticipated, several quantum gravity models characterized by Lorentz breaking phenomenology. A crucial difference between these speculations and other ones associated to quantum gravity scenarios, is the possibility to carry out observational and experimental tests which have nowadays led to a broad range of constraints on departures from Lorentz invariance. We shall review here the effective field theory approach to Lorentz breaking in the matter sector, present the constraints provided by the available observations and finally discuss the implications of the persisting uncertainty on the composition of the ultra high energy cosmic rays for the constraints on the higher order, analogue gravity inspired, Lorentz violations.

Abstract:
We present an updated review of Lorentz invariance tests in Effective field theories (EFT) in the matter as well as in the gravity sector. After a general discussion of the role of Lorentz invariance and a derivation of its transformations along the so called von Ignatovski theorem, we present the dynamical frameworks developed within local EFT and the available constraints on the parameters governing the Lorentz breaking effects. In the end, we discuss two specific examples, the OPERA "affaire" and the case of Ho\v{r}ava-Lifshitz gravity. The first case will serve as an example, and a caveat, of the practical application of the general techniques developed for constraining Lorentz invariance violation (LIV) to a direct observation potentially showing these effects. The second case will show how the application of the same techniques to a specific quantum gravity scenario has far fetching implications not foreseeable in a purely phenomenological EFT approach.

Abstract:
We show how the argument exploited by Galaverni & Sigl in Phys. Rev. Lett., 100, 021102 (2008) (see also arXiv:0708.1737) to constrain Lorentz invariance violation (LV) using Ultra-High-Energy photon non observation by the AUGER experiment, can be extended to QED with Planck-suppressed LV (at order $O(E/M)$ and $O(E^2/M^2)$). While the original constraints given by Galaverni & Sigl happen to be weakened, we show that, when used together with other EFT reactions and the expected detection of photons at $E > 10^{19}$ eV by AUGER, this method has the potentiality not only to basically rule out order $O(E/M)$ corrections but also to strongly constrain, for the first time, the CPT-even $O(E^2/M^2)$ LV QED.

Abstract:
Much attention has been recently devoted to the possibility that quantum gravity effects could lead to departures from Special Relativity in the form of a deformed Poincar\`e algebra. These proposals go generically under the name of Doubly or Deformed Special Relativity (DSR). In this article we further explore a recently proposed class of quantum field theories, involving noncanonically commuting complex scalar fields, which have been shown to entail a DSR-like symmetry. An open issue for such theories is whether the DSR-like symmetry has to be taken as a physically relevant symmetry, or if in fact the "true" symmetries of the theory are just rotations and translations while boost invariance has to be considered broken. We analyze here this issue by extending the known results to curved spacetime under both of the previous assumptions. We show that if the symmetry of the free theory is taken to be a DSR-like realization of the Poincar\'e symmetry, then it is not possible to render such a symmetry a gauge symmetry of the curved physical spacetime. However, it is possible to introduce an auxiliary spacetime which allows to describe the theory as a standard quantum field theory in curved spacetime. Alternatively, taking the point of view that the noncanonical commutation of the fields actually implies a breakdown of boost invariance, the physical spacetime manifold has to be foliated in surfaces of simultaneity and the field theory can be coupled to gravity by making use of the ADM prescription.

Abstract:
In this work a deep relation between topology and thermodynamical features of manifolds with boundaries is shown. The expression for the Euler characteristic, through the Gauss- Bonnet integral, and the one for the entropy of gravitational instantons are proposed in a form which makes the relation between them self-evident. A generalization of Bekenstein-Hawking formula, in which entropy and Euler characteristic are related in the form $S=\chi A/8$, is obtained. This formula reproduces the correct result for extreme black hole, where the Bekenstein-Hawking one fails ($S=0$ but $A \neq 0$). In such a way it recovers a unified picture for the black hole entropy law. Moreover, it is proved that such a relation can be generalized to a wide class of manifolds with boundaries which are described by spherically symmetric metrics (e.g. Schwarzschild, Reissner-Nordstr\"{o}m, static de Sitter).

Abstract:
We review the main theoretical motivations and observational constraints on Planck scale suppressed violations of Lorentz invariance. After introducing the problems related to the phenomenological study of quantum gravitational effects, we discuss the main theoretical frameworks within which possible departures from Lorentz invariance can be described. In particular, we focus on the framework of Effective Field Theory, describing several possible ways of including Lorentz violation therein and discussing their theoretical viability. We review the main low energy effects that are expected in this framework. We discuss the current observational constraints on such a framework, focusing on those achievable through high-energy astrophysics observations. In this context we present a summary of the most recent and strongest constraints on QED with Lorentz violating non-renormalizable operators. Finally, we discuss the present status of the field and its future perspectives.

Abstract:
In this work a relation between topology and thermodynamical features of gravitational instantons is shown. The expression for the Euler characteristic, through the Gauss-Bonnet integral, and the one for the entropy of gravitational instantons are proposed in a form that makes the relation between them self-evident. A new formulation of the Bekenstein-Hawking formula, where the entropy and the Euler characteristic are related by $S=\chi A/8$, is obtained. This formula provides the correct results for a wide class of gravitational instantons described by both spherically and axially symmetric metrics.

Abstract:
The emergence of a classical spacetime from any quantum gravity model is still a subtle and only partially understood issue. If indeed spacetime is arising as some sort of large scale condensate of more fundamental objects then it is natural to expect that matter, being a collective excitations of the spacetime constituents, will present modified kinematics at sufficiently high energies. We consider here the phenomenology of the dissipative effects necessarily arising in such a picture. Adopting dissipative hydrodynamics as a general framework for the description of the energy exchange between collective excitations and the spacetime fundamental degrees of freedom, we discuss how rates of energy loss for elementary particles can be derived from dispersion relations and used to provide strong constraints on the base of current astrophysical observations of high energy particles.

Abstract:
Motivated by scenarios of quantum gravity, Planck-suppressed deviations from Lorentz invariance are expected at observable energies. Ultra-High-Energy Cosmic Rays, the most energetic particles ever observed in nature, yielded in the last two years strong constraints on deviations suppressed by $O(E^{2}/\Mpl^{2})$ and also, for the first time, on space-time foam, stringy inspired models of quantum gravity. We review the most important achievements and discuss future outlooks.