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Looking for a new road supporting a theory of quantum gravity

The study of topological insulator [01], [02; + = extern link Wikipedia-GB] is a growing and active field of research with an incommensurable number of applications in physics. Graphene [03]+, in peculiar, has attracted much attention because of its electronic behavior; more precisely [01; citation, pp. 3-4]:

… “the conduction and the valence band touch each other at two points in the Brillouin zone [04]+ and near those points, the electronic dispersion resembles the dispersion of massless relativistic particles, described by the Dirac equation [05]+ (end of citation).”

This specific behavior is the reason explaining why I now want to explore in which way the motion of these alike-relativistic electrons may not only be described with a usual quantum dynamical approach but, within the context of A. Einstein’s theory [06] too. A successful mixing of both approaches would give a theory of quantum gravity+ 

That new approach should at least discover a mathematical road yielding the Dirac’s equation in starting with ingredients coming from a more classical viewpoint.

My claim is that that purpose can be reached. The so-called Lorentz-Einstein law (LEL, alias covariant version of the Lorentz law) is the starting point of the travel. I transform it first into a differential operator of the second order and treat the latter as if I would manage a Sturm-Liouville theory. Concretely, I look for a self-adjoint formulation of the operator. One of the resulting constraints can be interpreted as the Dirac’s equation.

Another collateral effect of that approach is the discovery of a generic new formulation for any electromagnetic field. It can also be obtained via a specific treatment of the LEL involving simultaneously: (i) Heisenberg’s uncertainty principle, (ii) Christoffel’s work on the preservation of quadratic forms and (iii) the extrinsic method of decomposition applied to the gravitational part of the law.

The bonus of that new formulation is that a part of all electromagnetic fields are infinitesimal variations of the geometry: F = dG (See my document). These specific fields are characterized by a trivial matrix mimicking the formalism of the bulk inversion asymmetry terms appearing in theories managing topological insulators. The work is going on…

© Thierry PERIAT, 21 June 2020

Bibliography:

[01] Topological insulators; arXiv:1002.3895v2 [cond-mat mes-hall], 9 November 2010.

[02] Topological insulator, on Wikipedia – GB, extern link, 14 March 2019.

[03] Graphene, on Wikipedia – GB, extern link, 14 March 2019.

[04] Brillouin zone, on Wikipedia – GB, extern link, 14 March 2019.

[05] Dirac equation, on Wikipedia – GB, extern link, 14 March 2019.

[06] Einstein, A. and Minkowski, H.: The principle of relativity; translated in English by Saha, M.N. and Bose, S.N. published by the university of Calcutta, 1920; available at the Library of the M.I.T.

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