Degenerate solutions to the massless Dirac and Weyl equations and a proposed method for controlling the quantum state of Weyl particles

In a recent work we have shown that all solutions to the Weyl equation and a special class of solutions to the Dirac equation are degenerate, in the sense that they remain unaltered under the influence of a wide variety of different electromagnetic fields. In the present article our previous work is significantly extended, providing a wide class of degenerate solutions to the Dirac equation for massless particles. The electromagnetic fields corresponding to these solutions are calculated, giving also some examples regarding both spatially constant electromagnetic fields and electromagnetic waves. Further, some general forms of solutions to the Weyl equation are presented and the corresponding electromagnetic fields are calculated. Based on these results, a method for fully controlling the quantum state of Weyl particles through appropriate electromagnetic fields is proposed. Finally, the transition from degenerate to non-degenerate solutions as the particles acquire mass is discussed.Comment: Keywords: Dirac equation, Weyl equation, Degenerate solutions, Massless particles, Electromagnetic 4-potentials, Electromagnetic fields, Electromagnetic waves, Nearly degenerate solution

A general method for obtaining degenerate solutions to the Dirac and Weyl equations and a discussion on the experimental detection of degenerate states

In this work we describe a general method for obtaining degenerate solutions to the Dirac equation, corresponding to an infinite number of electromagnetic 4-potentials and fields, which are explicitly calculated. In more detail, using four arbitrary real functions, one can automatically construct a spinor which is solution to the Dirac equation for an infinite number of electromagnetic 4-potentials, defined by those functions. An interesting characteristic of these solutions is that, in the case of Dirac particles with non-zero mass, the degenerate spinors should be localized, both in space and time. Our method is also extended to the cases of massless Dirac and Weyl particles, where the localization of the spinors is no longer required. Finally, we propose two experimental methods for detecting the presence of degenerate states.Comment: In this version of the article we have added a discussion on the experimental detection of degenerate states, proposing two techniques based on electrical and optical measurement

Classical multivariate Hermite coordinate interpolation on n-dimensional grids

In this work, we study the Hermite interpolation on n-dimensional non-equally spaced, rectilinear grids over a field k of characteristic zero, given the values of the function at each point of the grid and the partial derivatives up to a maximum degree. First, we prove the uniqueness of the interpolating polynomial, and we further obtain a compact closed form that uses a single summation, irrespective of the dimensionality, which is algebraically simpler than the only alternative closed form for the n-dimensional classical Hermite interpolation [1]. We provide the remainder of the interpolation in integral form; moreover, we derive the ideal of the interpolation and express the interpolation remainder using only polynomial divisions, in the case of interpolating a polynomial function. Finally, we perform illustrative numerical examples to showcase the applicability and high accuracy of the proposed interpolant, in the simple case of few points, as well as hundreds of points on 3D-grids using a spline-like interpolation, which compares favorably to state-of-the-art spline interpolation methods

A novel device for controlling the flow of information based on Weyl fermions and a method for manipulating the spatial distribution of Weyl particles

In this work we propose a novel device for controlling the flow of information using Weyl fermions. In more detail, based on a previous work of our group, we show that it is possible to fully control the flow of Weyl fermions on a sequence of different channels, by applying an electric field perpendicular to the direction of motion of the particles on each channel. In this way, we can transmit information, logical bits, depending on the existence or not of a Weyl current on each channel. We also show that the response time of this device is exceptionally low, less than 1 ps, for typical values of the parameters, providing the opportunity to control the flow of information at extremely high rates, of the order of 100 Pbps. This device also offers additional advantages, as low power consumption and robustness against electromagnetic perturbations, and is expected to find important applications in several fields, as telecommunications, signal processing, classical and quantum computing, etc. Finally, we demonstrate that Weyl fermions can be efficiently guided through the proposed device using appropriate magnetic fields

On the remarkable properties of Weyl particles

In this work we show that Weyl particles can exist at different states in zero electromagnetic field, either as free particles, or at localized states described by a parameter with dimensions of mass. We also calculate the electromagnetic fields that should be applied in order to modify the localization of Weyl particles at a desired rate. It is shown that they are simple electric fields, which can be easily implemented experimentally. Consequently, the localization of Weyl particles in certain materials supporting these particles could also be studied experimentally, in the framework of solid-state physics or in the framework of laser physics, using ions trapped by laser beams. In addition, a particularly important remark is that the localization of the energy of the particles can lead to the generation of gravitational mass, according to Einstein's field equations of general relativity. Furthermore, in the case that the energy and localization of the particles exceeds a critical level, tiny black holes could also be created, potential candidates for the dark matter of the universe.Comment: In this version we have added a remark regarding the potential to study experimentally the behavior of Weyl particles using ions trapped by laser beam

A Fourier-based implicit evolution scheme for active surfaces, for object segmentation in volumetric images

Active contours (AC) and active surfaces (AS) have been used extensively for segmentation and measurement in two- and three-dimensional images. The small time steps used in discretizing the evolution equation of AC/AS with the explicit scheme result in slow execution, whereas the use of the implicit evolution of AS in matrix form imposes very high memory and computational requirements. In this work we present an approach for implementing the implicit scheme for the numerical solution of the partial differential equation of the evolution of an AC/AS. The proposed approach is formulated as a deconvolution of the current contour/surface points with a one-dimensional mask that is performed using the discrete Fourier transform and it is derived using the properties of circulant matrices. The proposed scheme can handle higher accuracy numerical approximation of the discrete derivatives necessary for the method of AC/AS. It also possesses the separability property along different dimensions and it is applicable to implicit evolution of deformable surfaces, without the need to store and invert large sparse matrices. Initial results from the application of the proposed scheme to synthetic and clinical volumetric data demonstrate the correctness and applicability of the method. The computational complexity of the proposed scheme is also derived