A pedestrian approach to cosserat / Maxwell / weyl theory

TL;DR

This paper summarizes recent results applying formal PDE and Lie pseudo group theory to physics and engineering, revealing foundational issues in current theories and suggesting a need for revisiting their mathematical basis.

Contribution

It introduces a unified approach using Spencer and Janet sequences to derive fundamental equations, challenging traditional methods and highlighting contradictions in established physics.

Findings

Derivation of Cosserat, Maxwell, and Weyl equations on equal footing

Identification of deep contradictions in current theoretical foundations

Proposal to revisit physics foundations within jet theory framework

Abstract

The purpose of this short notice is to present an elementary summary of a few recent results obtained through the application of the formal theory of systems of partial differential equations and Lie pseudo groups to engineering (elasticity theory, electromagnetism, coupling phenomena) and mathematical (gauge theory, general relativity) along the following scheme: 1) Lie groups of transformations may be considered as Lie pseudo groups of transformations but no action type method can be used as parameters never appear any longer. 2) The work of Cartan is superseded by the use of the canonical Spencer sequence while the work of Vessiot is superseded by the use of the canonical Janet sequence but the link between these two sequences and thus these two works is not known today. 3)Using duality theory, the formal adjoint of the Spencer operator for the conformal group of transformations of space-time provides the Cosserat equations, the Maxwell equations and the Weyl equations on equal footing but such a result leads to deep contradictions. Accordingly, the results thus obtained prove that the foundations of engineering and mathematical physics must be revisited within the framework of jet theory though striking it may look like for established theories.