Spin state in the propagation of quantum relativistic particles along classical trajectories

We address the propagation of the spin along classical trajectories for a 1/2-spin particle obeying the Dirac equation with scalar potentials. Focusing on classical trajectories as the exact propagation of wave-function discontinuities we find an explicit spin-transport law for the case of the Dirac oscillator. In the general case we examine the spin propagation along classical trajectories emerging as an approximation of the quantum dynamics via the mechanical analog of the optical eikonal asymptotic approach. Throughout we establish as many parallels as possible with the equivalent situation for the electromagnetic field.Comment: 8 pages, 5 figure

The Distribution of Stochastic Shrinkage Parameters in Ridge Regression

In this article we derive the density and distribution functions of the stochastic shrinkage parameters of three well-known operational Ridge Regression estimators by assuming normality. The stochastic behavior of these parameters is likely to affect the properties of the resulting Ridge Regression estimator, therefore such knowledge can useful in the selection of the shrinkage rule. Some numerical calculations are carried out to illustrate the behavior of these distributions, throwing light on the performance of the different Ridge Regression estimators.

A Note on The Moments of Stochastic Shrinkage Parameters in Ridge Regression

A common problem in econometric models and multiple regression in general is multicollinearity, which produces undesirable effects on the Least Squares estimators. A possible solution to this problem is the "Ridge" Regression estimator proposed by Hoerl and Kennard (1970). Ridge Regression has been applied to such diverse areas as economics, marketing and the calibration of instruments in industrial processes. However, the properties of these estimators crucially depend upon the selection of certain biasing parameters which are stochastic. In this regard several proposals have been made and the purpose of this paper is to derive general expressions for the moments of the stochastic biasing parameters. With this knowledge we expect to stablish conditions under which a Ridge Regression estimator is better than others.

Some not-so-common ideas about gravity

Most of the approaches to the construction of a theory of quantum gravity share some principles which do not have specific experimental support up to date. Two of these principles are relevant for our discussion: (i) the gravitational field should have a quantum description in certain regime, and (ii) any theory of gravity containing general relativity should be relational. We study in general terms the possible implications of assuming deviations from these principles, their compatibility with current experimental knowledge, and how can they affect future experiments.Comment: 12 pages (+ references). Invited talk at DICE2014, Castiglioncello, September 201

Black holes turn white fast, otherwise stay black: no half measures

Recently, various authors have proposed that the first ultraviolet effect on the gravitational collapse of massive stars to black holes is the transition between a black-hole geometry and a white-hole geometry, though their proposals are radically different in terms of their physical interpretation and characteristic time scales [1,2]. Several decades ago, it was shown by Eardley that white holes are highly unstable to the accretion of small amounts of matter, being rapidly turned into black holes [3]. Studying the crossing of null shells on geometries describing the black-hole to white-hole transition, we obtain the conditions for the instability to develop in terms of the parameters of these geometries. We conclude that transitions with long characteristic time scales are pathologically unstable: occasional perturbations away from the perfect vacuum around these compact objects, even if being imperceptibly small, suffocate the white hole explosion. On the other hand, geometries with short characteristic time scales are shown to be robust against perturbations, so that the corresponding processes could take place in real astrophysical scenarios. This motivates a conjecture about the transition amplitudes of different decay channels for black holes in a suitable ultraviolet completion of general relativity.Comment: 24 pages, 3 figures. V2: Minor changes and updated references. Matches the published versio

Weyl relativity: A novel approach to Weyl's ideas

In this paper we revisit the motivation and construction of a unified theory of gravity and electromagnetism, following Weyl's insights regarding the appealing potential connection between the gauge invariance of electromagnetism and the conformal invariance of the gravitational field. We highlight that changing the local symmetry group of spacetime permits to construct a theory in which these two symmetries are combined into a putative gauge symmetry but with second-order field equations and non-trivial mass scales, unlike the original higher-order construction by Weyl. We prove that the gravitational field equations are equivalent to the (trace-free) Einstein field equations, ensuring their compatibility with known tests of general relativity. As a corollary, the effective cosmological constant is rendered radiatively stable due to Weyl invariance. A novel phenomenological consequence characteristic of this construction, potentially relevant for cosmological observations, is the existence of an energy scale below which effects associated with the non-integrability of spacetime distances, and an effective mass for the electromagnetic field, appear simultaneously (as dual manifestations of the use of Weyl connections). We explain how former criticisms against Weyl's ideas lose most of their power in its present reincarnation, which we refer to as Weyl relativity, as it represents a Weyl-invariant, unified description of both the Einstein and Maxwell field equations.Comment: 34 pages, no figure