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Journal of Modern Physics 2023年第11期14卷 1464-1496页

作者： J.-F. Pommaret CERMICS Ecole des Ponts ParisTech Paris France

In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern framework, they used the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the linear and nonlinear Spencer sequences for the lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of both electromagnetism (EM) and gravitation (GR), with the only experimental need to measure the EM and GR constants. With a manifold of dimension n ≥ 3, the difficulty is to deal with the n nonlinear transformations that have been called “elations” by E. Cartan in 1922. Using the fact that dimension n = 4 has very specific properties for the computation of the Spencer cohomology, we also prove that there is no conceptual difference between the (nonlinear) Cosserat EL field or induction equations and the (linear) Maxwell EM field or induction equations. As for gravitation, the dimension n = 4 also allows to have a conformal factor defined everywhere but at the central attractive mass because the inversion law of the isotropy subgroupoid made by second order jets transforms attraction into repulsion. The mathematical foundations of both electromagnetism and gravitation are thus only depending on the structure of the conformal pseudogroup of space-time.

关键词： Nonlinear Differential Sequences Linear Differential Sequences lie groupoids lie Algebroids Conformal Geometry Spencer Cohomology Maxwell Equations Cosserat Equations

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Nonlinear Conformal Electromagnetism

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Journal of Modern Physics 2022年第4期13卷 442-494页

作者： J.-F. Pommaret CERMICS Ecole des Ponts ParisTech France

In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern language, their idea has been to use the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the nonlinear Spencer sequence for the lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of electromagnetism (EM), with the only experimental need to measure the EM constant in vacuum. With a manifold of dimension n, the difficulty is to deal with the n nonlinear transformations that have been called “elations” by E. Cartan in 1922. Using the fact that dimension n=4 has very specific properties for the computation of the Spencer cohomology, we prove that there is thus no conceptual difference between the Cosserat EL field or induction equations and the Maxwell EM field or induction equations. As a byproduct, the well known field/matter couplings (piezzoelectricity, photoelasticity, streaming birefringence, …) can be described abstractly, with the only experimental need to measure the corresponding coupling constants. The main consequence of this paper is the need to revisit the mathematical foundations of gauge theory (GT) because we have proved that EM was depending on the conformal group and not on U(1), with a shift by one step to the left in the physical interpretation of the differential sequence involved.

关键词： Nonlinear Differential Sequences Linear Differential Sequences lie groupoids lie Algebroids Conformal Group Spencer Cohomology Maxwell Equations Cosserat Equations

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