The Mathematical Foundations of Gauge Theory Revisited

Jean-Francois Pommaret

CERMICS, Ecole des Ponts ParisTech, 77455 Marne-la-Vallée Cedex 02, France.

DOI: 10.4236/jmp.2014.55026                

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.

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

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Yang, C.N. and Mills, R.L. (1954) Physical Review Letters, 96, 191-195. http://dx.doi.org/10.1103/PhysRev.96.191
[2]	Kobayashi, S. and Nomizu, K. (1963) Foundations of Differential Geometry, Vol. I. J. Wiley, New York.
[3]	Bleecker, D. (1981) Gauge Theory and Variational Principles. Addison-Wesley, Reading.
[4]	Drechsler, W. and Mayer, M.E. (1977) Fiber Bundle Techniques in Gauge Theories. Springer Lecture Notes in Physics 67. Springer, New York.
[5]	Gockeler, M. (1987) Differential Geometry, Gauge Theories and Gravity. Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.
[6]	Yang, C.N. (1977) Annals of the New York Academy of Sciences, 294, 86-97. http://dx.doi.org/10.1111/j.1749-6632.1977.tb26477.x
[7]	Pommaret, J.-F. (1994) Partial Differential Equations and Group Theory. Kluwer, Dordrecht. http://dx.doi.org/10.1007/978-94-017-2539-2
[8]	Arnold, V. (1974) Méthodes Mathématiques de la Mécanique Classique. Appendice 2 (Géodésiques des Métriques Invariantes à Gauche sur des Groupes de Lie et Hydrodynamique des Fluides Parfaits), MIR, Moscow.
[9]	Arnold, V. (1966) Annales de l’Institut Fourier, 16, 319-361.
[10]	Birkhoff, G. (1954) Hydrodynamics. Princeton University Press, Princeton.
[11]	Pommaret, J.-F. (2009) AJSE-Mathematics, 1,157-174.
[12]	Poincaré, H. (1901) C. R. Académie des Sciences Paris, 132, 369-371.
[13]	Pommaret, J.-F. (1988) Lie Pseudogroups and Mechanics. Gordon and Breach, New York.
[14]	Ougarov, V. (1969) Théorie de la Relativité Restreinte. MIR, Moscow.
[15]	Pommaret, J.-F. (2001) Acta Mechanica, 149, 23-39. http://dx.doi.org/10.1007/BF01261661
[16]	Pommaret, J.-F. (2012) Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics. In: Gan, Y.X., Ed., Continuum Mechanics-Progress in Fundamentals and Engineering Applications. http://www.intechopen.com/books/continuum-mechanics-progress-in-fundamentals-and-engineering-applications/spencer-operator-and-applications-from-continuum-mechanics-to-mathematical-physics
[17]	Cosserat, E. and Cosserat, F. (1909) Théorie des Corps Déformables. Hermann, Paris.
[18]	Pommaret, J.-F. (2010) Acta Mechanica, 215, 43-55. http://dx.doi.org/10.1007/s00707-010-0292-y
[19]	Weyl, H. (1922) Space, Time, Matter. Springer, London.
[20]	Pommaret, J.-F. (2013) Journal of Modern Physics, 4, 223-239. http://dx.doi.org/10.4236/jmp.2013.48A022
[21]	Zou, Z., Huang, P., Zhang, Y. and Li, G. (1979) Scientia Sinica, XXII, 628-636.
[22]	Pommaret, J.-F. (1978) Systems of Partial Differential Equations and Lie Pseudogroups. Gordon and Breach, New York.
[23]	Pommaret, J.-F. (1983) Differential Galois Theory. Gordon and Breach, New York.
[24]	Vessiot, E. (1903) Annales Scientifiques Ecole Normale Supérieure, 20, 411-451.
[25]	Kumpera, A. and Spencer, D.C. (1972) Lie Equations. Princeton University Press, Princeton.
[26]	Teodorescu, P.P. (1975) Dynamics of Linear Elastic Bodies. Abacus Press, Tunbridge Wells, Kent, England.
[27]	Spencer, D.C. (1965) Bulletin of the American Mathematical Society, 75, 1-114.
[28]	Janet, M. (1920) Journal de Math., 8, 65-151.
[29]	Rotman, J.J. (1979) An Introduction to Homological Algebra. Academic Press, Waltham.
[30]	Pommaret, J.-F. (2001) Partial Differential Control Theory. Kluwer, Dordrecht.
[31]	Pommaret, J.-F. (2005) Algebraic Analysis of Control Systems Defined by Partial Differential Equations. In: Lamnabhi-Lagarrigue, F., Loría, A. and Panteley, E. Eds., Advanced Topics in Control Systems Theory, Lecture Notes in Control and Information Sciences 311, Chapter 5. Springer, London, 155-223.
[32]	Kunz, E. (1985) Introduction to Commutative Algebra and Algebraic Geometry. Birkhaüser, Boston.
[33]	Pommaret, J.-F. (2013) Multidimensional Systems and Signal Processing. http://dx.doi.org/10.1007/s11045-013-0265-0