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In this paper the inverse problem of the correspondence between the solutions of the Dirac equation and the electromagnetic 4-potentials, is fully solved. The Dirac solutions are classified into two classes. The first one consists of degenerate Dirac solutions corresponding to an infinite number of 4-potentials while the second one consists of non-degenerate Dirac solutions corresponding to one and only one electromagnetic 4-potential. Explicit expressions for the electromagnetic 4-potentials are provided in both cases. Further, in the case of the degenerate Dirac solutions, it is proven that at least two 4-potentials are gauge inequivalent, and consequently correspond to different electromagnetic fields. This result is extremely important, because it leads to the groundbreaking conclusion that for a specific class of spinors, a fermion is possible to be in the same state under the influence of different electromagnetic fields.

It has been some time since an empirical relation, which correlates DC with AC conductivity and contains a loosely defined coefficient thought to be of order one, was introduced by Barton, Nakajima and Namikawa. In this work, we derived this relation assuming that the conductive response consists of a superposition of DC conductivity and an AC conductivity term which materialized through a HavriliakNegami dielectric function. The coefficient was found to depend on the Havriliak-Negami shape parameters as well as on the ratio of two characteristic time scales of ions motion which are related to ionic polarization mechanism and the onset of AC conductivity. The results are discussed in relation to other relevant publications and they also applied to a polymeric material. Both, theoretical predictions and experimental evaluations of the BNN coefficient are in an excellent agreement, while this coefficient shows a gradual reduction as the temperature increases.

In this work, it is shown that Weyl particles can exist at different states in zero electromagnetic field, either as free particles or at localized states. In addition, it is shown that the localization, as well as the energy, of the particles can be fully controlled using simple electric fields, which can easily be realized in practice. These results are particularly important regarding possible practical applications of Weyl particles, both considering solid-state physics in materials supporting these particles and laser physics using ions trapped by laser beams, which can simulate the behavior of Weyl particles.

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