# LEL, operator and GTR2

The document : ISBN 978-2-36923-087-8 / EAN 9782369230878, French language, 34 pages; see the French version.

In 2004, I asked myself if the covariant version of the Lorentz law could be transformed into a second order differential operator and then treated as in a Sturm-Liouville theory. An exhaustive exploration of this interrogation gave a positive mathematical answer provided the transformation was accompanied by a set of four relations of coherence. Therefore, I then started to examine the meaning and the consequences of these four relations, from a physical point of view.

The second one is a factorization of Christoffel’s symbols of the second kind. I have discovered three possible interpretations in the literature. Injecting each of them into the third relation is anyway yielding a family of connected electromagnetic fields and this fact is opening what is called the problematic of the gauge.

If one wants to preserve the covariance of the Lorentz law, one is obliged to restrict the mathematical freedom in a way furnishing gauged fields and the Lorentz transformations must be elements in U(4). Hence the off-gauge terms of the connection must vanish.

When the second relation of coherence is interpreted in such a way that the inverse of the metric represents the second order coefficients of the differential operator, this obligatory constraint can only be realized with a subset of antisymmetric metrics mimicking the formalism of electromagnetic fields.

Amazingly and unexpectedly, analyzing the covariant version of the Lorentz law with the help of the extrinsic method and of E. Cartan’s work on spinors is furnishing the same kind of electromagnetic fields.

Since the GTR2 approach is also predicting the existence of such fields, the second part of the document examines the symplectic forms of the GTR2, in general and for a peculiar subset of connection which I have called the computer scientists connections. Although that last part is not achieved, it suggests the existence of a geometric background fulfilled with oscillating metrics.

© Thierry PERIAT, 12 December 2020.

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Last edited: 12/12/2020