Fiber averaged dynamics associated with the Lorentz force equation

It is shown that the Lorentz force equation is equivalent to the auto-parallel condition \,^L\nabla_{\dot{{x}}}\dot{{x}}=0 of a linear connection $^L\nabla$ defined on a convenient pull-back vector bundle. By using a geometric averaging method, an associated {\it averaged Lorentz connection} \langle\,^L\nabla\rangle and the corresponding auto-parallel equation are obtained. After this, it is shown that in the ultra-relativistic limit and for narrow one-particle probability distribution functions, the auto-parallel curves of \langle\,^L\nabla\rangle remain {\it nearby} close to the auto-parallel curves of $^L\nabla$. Applications of this result in beam dynamics and plasma physics are briefly described.Comment: This version, except for very few typographical corrections and several changes in the bibliography, was published in Journal of Geometry and Physic

Filtered-graded transfer of noncommutative Gr\"obner bases

As the case of free $\Bbbk$-algebras and $PBW$ algebras, given a bijective skew $PBW$ extension $A$, we will show that it is possible transfer Gr\"obner bases between $A$ and its associated graded ring.Comment: arXiv admin note: text overlap with arXiv:1309.0483 by other author

Effective upliftings in Large volume compactifications

After reviewing several mechanisms proposed to get a dS/Minkowski vacuum in moduli stabilization scenarios of type-IIB superstring orientifold compactifications we propose a criterium for characterizing those that may effectively lead to a positive small cosmological constant. We suggest that the variation in the expectation value of a good uplifting term, due to the shift in the minimum of the potential after uplifting, is much smaller than the original cosmological constant. This is studied with some detail in Large volume scenarios where the dependency on the volume direction is rather generic and easy to spot. Here we find that an uplifting term in the potential, with generic form $V_{up}\sim 1/{\cal V}^\gamma$, should be restricted to the one satisfying $\gamma^2\ll 12$. Such a bound might explain why in models previously studied no uplifting has been achieved, and gives motivations to study a novel proposal of dilaton dependent uplifting mechanism for which no numerical studies has been performed before. We find that in this case it is actually possible to get a dS vacuum, but still leave open the question of a more precise discrimination feature the good uplifting mechanisms should satisfy.Comment: 8 pgs, 1 figure. To be submitted for publication in Nuclear physics B Proceedings Supplements-Special Issue X SILAFAE; v2 typos fixed, references adde

Foundations for a theory of emergent quantum mechanics and emergent classical gravity

Quantum systems are viewed as emergent systems from the fundamental degrees of freedom. The laws and rules of quantum mechanics are understood as an effective description, valid for the emergent systems and specially useful to handle probabilistic predictions of observables. After introducing the geometric theory of Hamilton-Randers spaces and reformulating it using Hilbert space theory, a Hilbert space structure is constructed from the Hilbert space formulation of the underlying Hamilton-Randers model and associated with the space of wave functions of quantum mechanical systems. We can prove the emergence of the Born rule from ergodic considerations. A geometric mechanism for a natural spontaneous collapse of the quantum states based on the concentration of measure phenomena as it appears in metric geometry is discussed.We show the existence of stable vacua states for the quantized matter Hamiltonian. Another consequence of the concentration of measure is the emergence of a weak equivalence principle for one of the dynamics of the fundamental degrees of freedom. We suggest that the reduction of the quantum state is driven by a gravitational type interaction. Such interaction appears only in the dynamical domain when localization of quantum observables happens, it must be a classical interaction. We discuss the double slit experiment in the context of the framework proposed, the interference phenomena associated with a quantum system in an external gravitational potential, a mechanism explaining non-quantum locality and also provide an argument in favour of an emergent interpretation of every macroscopic time parameter. Entanglement is partially described in the context of Hamilton-Randers theory and how naturally Bell's inequalities should be violated.Comment: Extensive changes in chapter 1 and chapter 2; minor changes in other chapters; several refereces added and others update; 192 pages including index of contents and reference

Jacobi equations and particle accelerator beam dynamics

A geometric formulation of the linear beam dynamics in accelerator physics is presented. In particular, it is proved that the linear transverse and longitudinal dynamics can be interpret geometrically as an approximation to the Jacobi equation of an affine averaged Lorentz connection. We introduce a specific notion reference trajectory as integral curves of the main velocity vector field. A perturbation caused by the statistical nature of the bunch of particles is considered.Comment: 16 page

Edifying spectacle at UCD

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