arXiv subjects

J. A. Vallejo

Publications and source records attributed to J. A. Vallejo.

11 recordsLinked to original sources

arXiv2024

Potentials on the conformally compactified Minkowski spacetime and their application to quark deconfinement

We study a class of conformal metric deformations in the quasi-radial coordinate parameterizing the 3-sphere in the conformally compactified Minkowski spacetime $S^1\times S^3$. Prior to reduction of the associated Laplace-Beltrami operators to a Schrödinger form, a corresponding class of exactly solvable potentials (each one containing a scalar and a gradient term) is found. In particular, the scalar piece of these potentials can be exactly or quasi-exactly solvable, and among them we find the finite range confining trigonometric potentials of Pöschl-Teller, Scarf and Rosen-Morse. As an application of the results developed in the paper, the large compactification radius limit of the interaction described by some of these potentials is studied, and this regime is shown to be relevant to a quantum mechanical quark deconfinement mechanism.

math-ph↗

arXiv2023

Wave functions of the Hydrogen atom in the momentum representation

We construct the integral transform passing from the space representation to the momentum representation for the Hydrogen atom using polar spherical coordinates. The resulting radial wave functions are explicitly given in terms of complex finite expansions of Gegenbauer functions of the first and second kind, or in terms of (elementary) trigonometric functions. We show their symmetry under the $SO(4)$ group, and their equivalence with those of Lombardi and Oglivie.

quant-ph↗

arXiv2021

Color confinement at the boundary of the conformally compactified $\mathrm{AdS}_5$

The topology of closed manifolds forces interacting charges to appear in pairs. We take advantage of this property in the setting of the conformal boundary of $\mathrm{AdS}_5$ spacetime, topologically equivalent to the closed manifold $S^1\times S^3$, by considering the coupling of two massless opposite charges on it. Taking the interaction potential as the analog of Coulomb interaction (derived from a fundamental solution of the $S^3$ Laplace-Beltrami operator), a conformal $S^1\times S^3$ metric deformation is proposed, such that free motion on the deformed metric is equivalent to motion on the round metric in the presence of the interaction potential. We give explicit expressions for the generators of the conformal algebra in the representation induced by the metric deformation. By identifying the charge as the color degree of freedom in QCD, and the two charges system as a quark--anti-quark system, we argue that the associated conformal wave operator equation could provide a realistic quantum mechanical description of the simplest QCD system, the mesons. Finally, we discuss the possibility of employing the compactification radius, $R$, as another scale along $Λ_{QCD}$, by means of which, upon reparametrizing $Q^2c^2$ as $\left( Q^2c^2 +\hbar^2 c^2/R^2\right)$, a pertubative treatment of processes in the infrared could be approached.

hep-th↗

arXiv2018

Acausal quantum theory for non-Archimedean scalar fields

We construct a family of quantum scalar fields over a $p-$adic spacetime which satisfy $p-$adic analogues of the Gårding--Wightman axioms. Most of the axioms can be formulated the same way in both, the Archimedean and non-Archimedean frameworks; however, the axioms depending on the ordering of the background field must be reformulated, reflecting the acausality of $p-$adic spacetime. The $p-$adic scalar fields satisfy certain $p-$adic Klein-Gordon pseudo-differential equations. The second quantization of the solutions of these Klein-Gordon equations corresponds exactly to the scalar fields introduced here.

hep-th↗

arXiv2014

Lie algebroids generated by cohomology operators

By studying the Frölicher-Nijenhuis decomposition of cohomology operators (that is, derivations $D$ of the exterior algebra $Ω(M)$ with $\mathbb{Z}-$degree $1$ and $D^2=0$), we describe new examples of Lie algebroid structures on the tangent bundle $TM$ (and its complexification $T^{\mathbb{C}}M$) constructed from pre-existing geometric ones such as foliations, complex, product or tangent structures. We also describe a class of Lie algebroids on tangent bundles associated to idempotent endomorphisms with nontrivial Nijenhuis torsion.

math.DG↗

arXiv2013

Higher order corrections to adiabatic invariants of generalized slow-fast Hamiltonian systems

We present a coordinate-free approach for constructing approximate first integrals of generalized slow-fast Hamiltonian systems, based on the global averaging method on parameter-dependent phase spaces with $\mathbb{S}^1 -$symmetry. Explicit global formulas for approximate second-order first integrals are derived. As examples, we analyze the case quadratic in the fast variables (in particular, the elastic pendulum), and the charged particle in a slowly-varying magnetic field.

math.DS↗

arXiv2013

Quillen superconnections and connections on supermanifolds

Given a supervector bundle $E = E_0\oplus E_1 \to M$, we exhibit a parametrization of Quillen superconnections on $E$ by graded connections on the Cartan-Koszul supermanifold $(M;Ω(M))$. The relation between the curvatures of both kind of connections, and their associated Chern classes, is discussed in detail. In particular, we find that Chern classes for graded vector bundles on split supermanifolds can be computed through the associated Quillen superconnections.

math.DG↗

arXiv2013

A simple global representation for second-order normal forms of Hamiltonian systems relative to periodic flows

We study the determination of the second-order normal form for perturbed Hamiltonians $H_ε=H_0 +εH_1 +\frac{ε^2}{2} H_2$, relative to the periodic flow of the unperturbed Hamiltonian $H_0$. The formalism presented here is global, and can be easily implemented in any CAS. We illustrate it by means of two examples: the Hénon-Heiles and the elastic pendulum Hamiltonians.

math-ph↗

arXiv2013

The principle of stationary action in the calculus of variations

We review some techniques from non-linear analysis in order to investigate critical paths for the action functional in the calculus of variations applied to physics. Previous attempts to analyse when these are minima ex- ist, but mainly based on physical reasoning and only for a restricted class of models. Our main intention in this regard is to develop precise mathematical conditions for critical paths to be minimum solutions in a variety of situations. Our claim is that, with a few techniques, a systematic analysis (including the domain for which critical points are genuine minima) of non-trivial models is possible. We present specific models arising in modern physical theories in order to make clear the ideas here exposed.

math-ph↗

arXiv2013

Chaotic dynamics with Maxima

We present an introduction to the study of chaos in discrete and continuous dynamical systems using the CAS Maxima. These notes are intended to cover the standard topics and techniques: discrete and continuous logistic equation to model growth population, staircase plots, bifurcation diagrams and chaos transition, nonlinear continuous dynamics (Lorentz system and Duffing oscillator), Lyapunov exponents, Poincaré sections, fractal dimension and strange attractors. The distinctive feature here is the use of free software with just one ingredient: the CAS Maxima. It is cross-platform and have extensive on-line documentation.

nlin.CD↗

arXiv2003

Generating Operators of the Krasil'shchik-Schouten bracket

It is proved that given a divergence operator on the structural sheaf of graded commutative algebras of a supermanifold, it is possible to construct a generating operator for the Krashil'shchik-Schouten bracket. This is a particular case of the construction of generating operators for a special class of bigraded Gerstenhaber algebras. Also, some comments on the generalization of these results to the context of $n-$graded Jacobi algebras are included.

math-ph↗