Jugando con la probabilidad

En este trabajo planteamos una serie de juegos como recurso didáctico en el aula de matemáticas. Estos juegos nos permiten introducir algunos conceptos de probabilidad en secundaria y, además, pueden incitar a los alumnos a plantearse numerosas cuestiones que les ayuden a comprender los diversos problemas donde el azar está inmerso

Hybrid Koopman C*-formalism and the hybrid quantum-classical master equation

Based on Koopman formalism for classical statistical mechanics, we propose a formalism to define hybrid quantum-classical dynamical systems by defining (outer) automorphisms of the C*-algebra of hybrid operators and realizing them as linear transformations on the space of hybrid states. These hybrid states are represented as density matrices on the Hilbert space obtained from the hybrid C*-algebra by the GNS theorem. We also classify all possible dynamical systems which are unitary and obtain the possible hybrid Hamiltonian operators.Comment: 20 page

Hybrid Geometrodynamics: A Hamiltonian description of classical gravity coupled to quantum matter

We generalize the Hamiltonian picture of General Relativity coupled to classical matter, known as geometrodynamics, to the case where such matter is described by a Quantum Field Theory in Curved Spacetime, but gravity is still described by a classical metric tensor field over a spatial hypersurface and its associated momentum. Thus, in our approach there is no non-dynamic background structure, apart from the manifold of events, and the gravitational and quantum degrees of freedom have their dynamics inextricably coupled. Given the Hamiltonian natureof the framework, we work with the generators of hypersurface deformations over the manifold of quantum states. The construction relies heavily on the differential geometry of a fibration of the set of quantum states over the set of gravitational variables. An important feature of this work is the use of Gaussian measures over the space of matter fields and of Hida distributions to define a common superspace to all possible Hilbert spaces with different measures, to properly characterize the Schrodinger wave functional picture of QFT in curved spacetime. This allows us to relate states within different Hilbert spaces in the case of vacuum states or measures that depend on the gravitational degrees of freedom, as the ones associated to Ashtekar's complex structure. This is achieved through the inclusion of a quantum Hermitian connection for the fibration, which will have profound physical implications. The most remarkable physical features of the construction are norm conservation of the quantum state (even if the total dynamics are non-unitary), the clear identification of the hybrid conserved quantities and the description of a dynamical backreaction of quantum matter on geometry and vice versa, which shall modify the physical properties the gravitational field would have in the absence of backreaction

A sufficient condition for confinement in QCD

This letter is about confinement in QCD. At the moment we have pictures of confinement to complete our understanding of the physics of strongly interacting particles, interaction which asks for confinement.As it is said in [1] : " In principle it should be possible to derive the confinement hypothesis from the QCD Lagrangian. At this time, no rigorous derivation exists, so it is not absolutely clear that the confinement hypothesis is a bone fide prediction of QCD" . In this letter we show that a sufficient (of course not necessary) condition for confinement is that topological structure of vacuum in Nature does not correspond to the $\theta$-vacuum. Therefore, if different vacua with nontrivial winding number cannot be connected by tunneling, we obtain confinement as a consequence.Comment: 10 page

Competencias básicas en el estudio de la geometría en 1º y 2º de ESO

En este trabajo presentamos unas tareas para el estudio de conceptos matemáticos que se trabajan en 1º y 2º de ESO. Estas tareas utilizan la papiroflexia y los tangrams triangular y ruso como recursos manipulativos para acercar al estudiante a la geometría de esos cursos. Analizamos dichas tareas atendiendo a su contribución a la adquisición de competencias básicas, el núcleo temático y los contenidos matemáticos específicos que tratan y el curso al que corresponden

Uff… muchos tangrams para una misma aula de matemáticas

En este trabajo presentamos diversos tipos de Tangrams y algunas actividades en las que se utiliza este puzzle como recurso didáctico en el aula de matemáticas de Educación Secundaria. Se trabajan gran variedad de nociones geométricas como la semejanza de figuras, igualdad de lados, perímetro, área, simetría, Teorema de Pitágoras y de Thales, medidas aproximadas y exactas, fracciones, medidas de longitud y superficie, razón de semejanza, entre otras. Se proponen tareas de reconocimiento, construcción y manipulación de figuras geométricas planas que persiguen favorecer en los alumnos el desarrollo del sentido espacial, el interés por la resolución de problemas, técnicas de construcción geométrica, la imaginación, y la creatividad

Fractales: hasta el infinito y más allá (o más acá)

Presentamos una propuesta para trabajar los fractales en educación secundaria. Proponemos el uso de los fractales como medio para que los alumnos repasen y trabajen, de una forma original y creativa, otros conceptos geométricos del currículo relacionados con los fractales. Durante el taller mostraremos una idea intuitiva de fractal así como el modo de construir algunos de ellos de manera sencilla y entretenida. En las construcciones utilizaremos materiales accesibles y de fácil manejo como el papel, la regla, el compás y las tijeras

Effective nonlinear Ehrenfest hybrid quantum-classical dynamics

The definition of a consistent evolution equation for statistical hybrid quantum-classical systems is still an open problem. In this paper we analyze the case of Ehrenfest dynamics on systems defined by a probability density and identify the relations of the non-linearity of the dynamics with the obstructions to define a consistent dynamics for the first quantum moment of the distribution. This first quantum moment represents the physical states as a family of classically-parametrized density matrices $\hat \rho(\xi)$, for $\xi$ a classical point; and it is the most common representation of hybrid systems in the literature. Due to this obstruction, we consider higher order quantum moments, and argue that only a finite number of them are physically measurable. Because of this, we propose an effective solution for the hybrid dynamics problem based on approximating the distribution by those moments and representing the states by them.Comment: 21 pages. Minor correction in the list of affiliation

Experiencia en el aula de secundaria con fractales

Presentamos una experiencia docente en un aula de 2º ESO en la que trabajamos los fractales mediante el uso de material de carácter manipulativo. La metodología seguida se basa en la construcción de casos particulares con el fin de llegar al concepto de fractal