Drafts by Angel Deleito García
Since the role that-symmetric Hamiltonians could play in quantum mechanics theory was suggested i... more Since the role that-symmetric Hamiltonians could play in quantum mechanics theory was suggested in 1998, much work has been conducted on the mathematical analysis of-symmetry. Carl Bender challenged the perceived wisdom of Quantum Mechanics that the Hamiltonian operator describing any quantum mechanical system has to be Hermitian. He showed that Hamiltonians that are invariant under combined parity-time-symmetry transformations likewise can exhibit real eigenvalue spectra. However,-symmetric systems began as an analytic extension of classical and quantum mechanical to the complex domain. These findings had a particularly profound impact in the field of photonics, where PT-symmetric potential landscapes can be implemented by introducing dissipative terms appropriately distributing gain and loss in open or coupled systems. The refractive index profile plays the role of the complex potential. Similar research activity was transferred to other physical settings such as electronics circuits, microwaves and cavities resonators, acoustic, superconducting wires, and atomic systems. Unconventional characteristics compared with Hermitian ones were found and allow to engineer new devices. Likewise, topological phases of matter are another new field of physics which have had a rapid growth in the last years, based on the concept of Geometric or Berry phase. Topological Insulators and Superconductors, Topological photonics and nanophotonics, Topological Computers are some of the applications widely described in scientific journals. In Nature all the processes have to be described by a property in a discrete or iterative way instead of a continuous way which usually is time. Continuous processes, being a mathematical idealization, has been the best tool to model observable properties in physics and is the conventional tool used in Classical and Quantum Mechanics. The formal way to model discrete systems is through difference equations which are similar to differential equations but not equal. Identification of dynamical discrete system with recurrence equations, give us the possibility to consider an analytical continuation of its evolution variable, from discrete values to fractional one´s, in the solution of the system. Parametric solutions obtained by this procedure takes automatically complex values, avoiding to make any supposition about the possibility of their analytical extension to complex domain. This procedure leads, in a very simple natural way, a rich variety of dynamical behaviors in the complex domain unknown in the real domain. This new mathematical tool is developed to enhance and provide a more subtle description of the behavior of dynamical systems and provides a more general class of Non-Hermitian Hamiltonians than-symmetric systems with real spectra, keeping also topological properties due to a kind of geometric phase.
Difference equations model the evolution of many processes of interest in physics, biology, econo... more Difference equations model the evolution of many processes of interest in physics, biology, economics, etc., in a discrete or iterative manner instead of a continuous parameter (e.g., time). In this regard, the renowned Fibonacci sequence constitutes an interesting example of iterative sequence that can be modeled in terms of such equations, more specifically a second-order homogeneous linear difference equation. This work intends to go a step beyond, introducing a generalized Fibonacci function based, on the one hand, on the close resemblance between such an equation and the one resulting from recasting a general two-variable linear system of difference equations as an also single second-order homogeneous linear difference equation. On the other hand, the discrete index of the usual Fibonacci sequence is replaced by a continuous (evolution) variable or parameter (although the Fibonacci initial conditions are kept). This leads to a rich variety of dynamical behaviors in the complex plane depending on the value of the parameters involved in the associated characteristic equation. The dynamics exhibited by the corresponding generalized Fibonacci functions are investigated and analyzed here, finding how apparently simple relations may describe relatively complex behaviors on the complex plane even in the case of regular or periodic solutions. Even though, it is seen that the curves in the complex plane displayed by the generalized Fibonacci functions during their evolution enable a better understanding of the behavior exhibited by the starting discrete model, such as the regimes of stability and instability, or the appearance of single and multiple fixed points.
In this paper a new family of hyperbolic functions is presented. The behavior of the new hyperbol... more In this paper a new family of hyperbolic functions is presented. The behavior of the new hyperbolic functions is studied in specific cases. These functions present a marked asymmetry between left-handed or counterclockwise function (CCW) and right-handed or clockwise function (CW). Also they present a great variety of periods in the entire parametric range, exhibiting chaos in certain regions and order in others.
The aim of this paper is to propose a new interpretation of quantum mechanics (QM), from a new co... more The aim of this paper is to propose a new interpretation of quantum mechanics (QM), from a new concept of pulse, defined in terms of a mathematical monochromatic function. We will make a critical analysis of the pulse concept, comparing advantages with disadvantages depending if it is used like a monochromatic pulse or obtained as a
superposition of infinite monochromatic waves pulse from Fourier analysis. With this slight modification in the concept of pulse, principles and behaviours that at the present time are considered inherent to QM, according to Copenhagen interpretation (CI), will now be considered unnecessary. We talk about wave-particle duality, Heisenberg
uncertainty principle and renormalization processes to avoid infinities. In spite of these conceptual modifications, the mathematical formalism of the QM will not need to be revised.
Next we show how, hidden in partial differential equations used to describe the field’s dynamics, are, in a conclusive way, the complex numbers, and how these comprise a part of a real physical world, although they cannot be detected by our senses, being always
there and in some cases, appearing in an unexpected way. This situation allows us to consider that complex numbers contribute to a new dimension in the physical world that surrounds us. It will be the fifth dimension, the “imaginary” one, with an existence as real as the three spatial dimensions and time. Further on we develop a formalism
showing that electromagnetic field is made up of two matter or Dirac fields, only detectable if the imaginary dimension is considered as a part of reality. Then we shall show the way in which mass is a general property of fields, hidden in an imaginary dimension in the case of electromagnetic fields and detectable and weighable in the case
of Dirac fields, giving mathematical form to the well known process of creation and annihilation of particles.
La generalización de la sucesión de Fibonacci permite describir cualquier sistema lineal mediante el uso de una sola función, 2001
El objetivo del presente artículo es describir un método mediante el cual sea posible la obtenció... more El objetivo del presente artículo es describir un método mediante el cual sea posible la obtención de todas las soluciones de un sistema lineal para el caso de dos variables, mediante la utilización de una única función. Dicha función resultará ser una función de variable real, que toma valores en el campo complejo. En el procedimiento a seguir nos resultará de gran ayuda la serie de Fibonacci, cuya presencia en procesos naturales aparece en numerosas referencias ligado a procesos como la Phyllotaxis, la Simetría Pentagonal, los Quasicristales e incluso en procesos Biomoleculares como la estructura de las Proteínas y del ADN. Finalmente veremos que una generalización de la serie de Fibonacci será la función que nos permita describir cualquier sistema lineal de dos variables.
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Drafts by Angel Deleito García
superposition of infinite monochromatic waves pulse from Fourier analysis. With this slight modification in the concept of pulse, principles and behaviours that at the present time are considered inherent to QM, according to Copenhagen interpretation (CI), will now be considered unnecessary. We talk about wave-particle duality, Heisenberg
uncertainty principle and renormalization processes to avoid infinities. In spite of these conceptual modifications, the mathematical formalism of the QM will not need to be revised.
Next we show how, hidden in partial differential equations used to describe the field’s dynamics, are, in a conclusive way, the complex numbers, and how these comprise a part of a real physical world, although they cannot be detected by our senses, being always
there and in some cases, appearing in an unexpected way. This situation allows us to consider that complex numbers contribute to a new dimension in the physical world that surrounds us. It will be the fifth dimension, the “imaginary” one, with an existence as real as the three spatial dimensions and time. Further on we develop a formalism
showing that electromagnetic field is made up of two matter or Dirac fields, only detectable if the imaginary dimension is considered as a part of reality. Then we shall show the way in which mass is a general property of fields, hidden in an imaginary dimension in the case of electromagnetic fields and detectable and weighable in the case
of Dirac fields, giving mathematical form to the well known process of creation and annihilation of particles.
superposition of infinite monochromatic waves pulse from Fourier analysis. With this slight modification in the concept of pulse, principles and behaviours that at the present time are considered inherent to QM, according to Copenhagen interpretation (CI), will now be considered unnecessary. We talk about wave-particle duality, Heisenberg
uncertainty principle and renormalization processes to avoid infinities. In spite of these conceptual modifications, the mathematical formalism of the QM will not need to be revised.
Next we show how, hidden in partial differential equations used to describe the field’s dynamics, are, in a conclusive way, the complex numbers, and how these comprise a part of a real physical world, although they cannot be detected by our senses, being always
there and in some cases, appearing in an unexpected way. This situation allows us to consider that complex numbers contribute to a new dimension in the physical world that surrounds us. It will be the fifth dimension, the “imaginary” one, with an existence as real as the three spatial dimensions and time. Further on we develop a formalism
showing that electromagnetic field is made up of two matter or Dirac fields, only detectable if the imaginary dimension is considered as a part of reality. Then we shall show the way in which mass is a general property of fields, hidden in an imaginary dimension in the case of electromagnetic fields and detectable and weighable in the case
of Dirac fields, giving mathematical form to the well known process of creation and annihilation of particles.