TITLE:
The Mathematical Foundations of Gauge Theory Revisited
AUTHORS:
Jean-Francois Pommaret
KEYWORDS:
Gauge Theory; Curvature; Torsion; Maurer-Cartan Forms; Maurer-Cartan Equations; Lie Groups; Lie Pseudo Groups; Differential Sequence; Janet Sequence; Spencer Sequence; Differential Module; Homological Algebra; Extension Modules
JOURNAL NAME:
Journal of Modern Physics,
Vol.5 No.5,
March
24,
2014
ABSTRACT: 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 (despite using the same group theoretical
methods) and the well known couplings existing between elasticity and
electromagnetism (piezzo electricity, photo elasticity, 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 pseudo groups. These results finally
allow unifying 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.