Microlensing of the broad-line region in the quadruply imaged quasar HE0435-1223

Using infrared spectra of the z = 1.693 quadruply lensed quasar HE0435-1223 acquired in 2009 with the spectrograph SINFONI at the ESO Very Large Telescope, we have detected a clear microlensing effect in images A and D. While microlensing affects the blue and red wings of the H{\alpha} line profile in image D very differently, it de-magnifies the line core in image A. The combination of these different effects sets constraints on the line-emitting region; these constraints suggest that a rotating ring is at the origin of the H{\alpha} line. Visible spectra obtained in 2004 and 2012 indicate that the MgII line profile is microlensed in the same way as the H{\alpha} line. Our results therefore favour flattened geometries for the low-ionization line-emitting region, for example, a Keplerian disk. Biconical models cannot be ruled out but require more fine-tuning. Flux ratios between the different images are also derived and confirm flux anomalies with respect to estimates from lens models with smooth mass distributions.Comment: 6 pages, 4 figures, 3 tables, accepted by A&A on 10 April 201

Normal Coordinates and Primitive Elements in the Hopf Algebra of Renormalization

We introduce normal coordinates on the infinite dimensional group $G$ introduced by Connes and Kreimer in their analysis of the Hopf algebra of rooted trees. We study the primitive elements of the algebra and show that they are generated by a simple application of the inverse Poincar\'e lemma, given a closed left invariant 1-form on $G$. For the special case of the ladder primitives, we find a second description that relates them to the Hopf algebra of functionals on power series with the usual product. Either approach shows that the ladder primitives are given by the Schur polynomials. The relevance of the lower central series of the dual Lie algebra in the process of renormalization is also discussed, leading to a natural concept of $k$-primitiveness, which is shown to be equivalent to the one already in the literature.Comment: Latex, 24 pages. Submitted to Commun. Math. Phy

Quantization of the Myers-Pospelov model: the photon sector interacting with standard fermions as a perturbation of QED

We study the quantization of the electromagnetic sector of the Myers-Pospelov model coupled to standard fermions. Our main objective, based upon experimental and observational evidence, is to construct an effective theory which is a genuine perturbation of QED, such that setting zero the Lorentz invariance violation parameters will reproduce it. To this end we provide a physically motivated prescription, based on the effective character of the model, regarding the way in which the model should be constructed and how the QED limit should be approached. This amounts to the introduction of an additional coarse-graining physical energy scale $M$, under which we can trust the effective field theory formulation. The prescription is successfully tested in the calculation of the Lorentz invariance violating contributions arising from the electron self-energy. Such radiative corrections turn out to be properly scaled by very small factors for any reasonable values of the parameters and no fine-tuning problems are found. Microcausality violations are highly suppressed and occur only in a space-like region extremely close to the light-cone. The stability of the model is guaranteed by restricting to concordant frames satisfying $1-|\mathbf{v}_{max}|> 6.5\times10^{-11}$.Comment: 24 pages, revtex, no figure

The scalar sector in the Myers-Pospelov model

We construct a perturbative expansion of the scalar sector in the Myers-Pospelov model, up to second order in the Lorentz violating parameter and taking into account its higher-order time derivative character. This expansion allows us to construct an hermitian positive-definite Hamiltonian which provides a correct basis for quantization. Demanding that the modified normal frequencies remain real requires the introduction of an upper bound in the magnitude |k| of the momentum, which is a manifestation of the effective character of the model. The free scalar propagator, including the corresponding modified dispersion relations, is also calculated to the given order, thus providing the starting point to consider radiative corrections when interactions are introduced.Comment: Published in AIP Conf.Proc.977:214-223,200

Real sector of the nonminimally coupled scalar field to self-dual gravity

A scalar field nonminimally coupled to gravity is studied in the canonical framework, using self-dual variables. The corresponding constraints are first class and polynomial. To identify the real sector of the theory, reality conditions are implemented as second class constraints, leading to three real configurational degrees of freedom per space point. Nevertheless, this realization makes non-polynomial some of the constraints. The original complex symplectic structure reduces to the expected real one, by using the appropriate Dirac brackets. For the sake of preserving the simplicity of the constraints, an alternative method preventing the use of Dirac brackets, is discussed. It consists of converting all second class constraints into first class by adding extra variables. This strategy is implemented for the pure gravity case.Comment: Latex file, 22 pages, no figure

Noncommutative spaces, the quantum of time and the Lorentz symmetry

We introduce three space-times that are discrete in time and compatible with the Lorentz symmetry. We show that these spaces are no commutative, with commutation relations similar to the relations of the Snyder and Yang spaces. Furthermore, using a reparametrized relativistic particle we obtain a realization of the Snyder type spaces and we construct an action for them.Comment: 8 pages, to appear in PR

A note about the quantum of area in a non-commutative space

In this note we show that in a two-dimensional non-commutative space the area operator is quantized, this outcome is compared with the result obtained by Loop Quantum Gravity methods.Comment: 6 pages, references added, minor correction