再谈规范理论的数学基础
The mathematical foundations of gauge theory revisited
摘要：
我们开始回顾与批判的眼睛在规范理论中使用的数学方法，并证明他们是不相干的连续介质力学，特别是刚体或流体力学的分析力学，虽然使用相同的群理论方法，尽管众所周知的耦合之间存在的弹性和电磁（压电，光弹性，流动双折射）。本文的目的是避免这种矛盾，使用新的数学方法来自系统的正式理论的偏微分方程和李伪群。这些结果最终统一了E.和F. Cosserat在1909年为弹性或H。Weyl在1918年分别使用空间的刚性运动群或时空的共形群来研究电磁学。同时解释了为什么存在于“几何“与“物理”之间的庞加莱“对偶方案”与同调代数和代数分析有关。我们坚持这样一个事实，即这些结果不可能在1975年之前获得，因为相应的工具以前不知道，今天物理学家仍然不知道。
英文摘要：
We start recalling with critical eyes the mathematical methods used in gauge theory and prove that they are not coherent with continuum mechanics, in particular the analytical mechanics of rigid bodies or hydrodynamics, though using the same group theoretical methods and despite the well known couplings existing between elasticity and electromagnetism (piezzoelectricity, photoelasticity, streaming birefringence). The purpose of this paper is to avoid such contradictions by using new mathematical methods coming from the formal theory of systems of partial differential equations and Lie pseudogroups. These results finally allow to unify the previous independent tentatives done by the brothers E. and F. Cosserat in 1909 for elasticity or H. Weyl in 1918 for electromagnetism by using respectively the group of rigid motions of space or the conformal group of space-time. Meanwhile we explain why the Poincaré "duality scheme" existing between "geometry " and "physics" has to do with homological algebra and algebraic analysis. We insist on the fact that these results could not have been obtained before 1975 as the corresponding tools were not known before and are still not known today by physicists.
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