Gauge reduction in covariant field theory

Thumbnail Image
Official URL
Full text at PDC
Publication Date
Rodríguez Abella, Álvaro
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Google Scholar
Research Projects
Organizational Units
Journal Issue
In this work we develop a Lagrangian reduction theory for covariant field theories with local symmetries and, more specifically, with gauge symmetries. We model these symmetries by using a Lie group fiber bundle acting fiberwisely on the corresponding configuration bundle. In order to reduce the variational principle, we utilize generalized principal connections, a type of Ehresmann connections that are equivariant by the fiberwise action. After obtaining the reduced equations, we give the reconstruction condition and we relate the vertical reduced equation with the Noether theorem. Lastly, we illustrate the theory by applying it to several examples, including the classical case (Lagrange-Poincaré reduction) and electromagnetism.
Unesco subjects
[1] V. I. Arnold. On the differential geometry of Lie groups of infinite dimension and its applications to the hydrodynamics of perfect fluids. Annals of the Fourier Institute, 16(1):319–361, 1966. [2] D. Betounes. The geometry of gauge - particle field interaction: a generalization of utiyama’s theorem. Journal of Geometry and Physics, 6(1):107–125, 1989. [3] A. Bloch, L. Colombo, and F. Jiménez. The variational discretization of the constrained higher-order Lagrange–Poincaré equations. Discrete & Continuous Dynamical Systems, 39(1):309–344, 2019. [4] A. I. Bobenko and Y. B. Suris. Discrete Lagrangian reduction, discrete Euler–Poincaré equations, and semidirect products. Lett. Math. Phys., 49(1):79–93, 1999. [5] M. Castrillón López, P. L. García, and C. Rodrigo. Euler-Poincaré reduction in principal bundles by a subgroup of the structure group. Journal of Geometry and Physics, 74:352–369, 12 2013. [6] M. Castrillón López and J. E. Marsden. Covariant and dynamical reduction for principal bundle field theories. Annals of Global Analysis and Geometry, 34, 10 2008. [7] M. Castrillón López, P. Pérez, and T. S. Ratiu. Euler–Poincaré reduction on principal bundles. Letters in Mathematical Physics, 58, 11 2001. [8] M. Castrillón López and T. S. Ratiu. Reduction in principal bundles: Covariant Lagrange-Poincaré equations. Communications in Mathematical Physics, 236, 01 2003. [9] H. Cendra, J. E. Marsden, and T. S. Ratiu. Geometric Mechanics, Lagrangian Reduction, and Nonholonomic Systems, pages 221–273. Springer Berlin Heidelberg, 2001. [10] H. Cendra, J. E. Marsden, and T. S. Ratiu. Lagrangian reduction by stages. Mem. Amer. Math. Soc. V, 152(722), 07 2001. [11] David C. P. Ellis, F. Gay-Balmaz, D. D. Holm, and T. S. Ratiu. Lagrange-Poincaré field equations. Journal of Geometry and Physics, 61(11):2120 – 2146, 2011. [12] M. Forger and B. L. Soares. Local symmetries in gauge theories in a finite-dimensional setting. Journal of Geometry and Physics, 62(9):1925 – 1938, 2012. [13] P. L. García Pérez. The Poincaré-Cartan invariant in the calculus of variations. Symposia Math., 14:219 – 246, 1974. [14] P. L. García Pérez. Gauge algebras, curvature and symplectic structure. J. Differential Geom., 12(2):209–227, 1977. [15] F. Gay-Balmaz and T. Ratiu. A new lagrangian dynamic reduction in field theory. Annales de l’Institut Fourier, 3, 07 2014. [16] J. Janyška and M. Modugno. Relations between linear connections on the tangent bundle and connections on the jet bundle of a fibred manifold. Archivum Mathematicum, 032(4):281 – 288, 1996. [17] I. Kolář, P. W. Michor, and J. Slovák. Natural Operations in Differential Geometry. Springer-Verlag Berlin Heidelberg, 1st edition, 1993. [18] M. Castrillón López and Á. Rodríguez Abella. Principal bundles and connections modelled by lie group bundles, 2022. [19] K. B. Marathe and G. Martucci. The Mathematical Foundations of Gauge Theories. Elsevier Science Publishers B.V., 1992. [20] J. E. Marsden, S. Pekarsky, and S. Shkoller. Discrete Euler–Poincaré and Lie–Poisson equations. Nonlinearity, 12(6):1647–1662, 1999. [21] J. E. Marsden, S. Pekarsky, and S. Shkoller. Symmetry reduction of discrete Lagrangian mechanics on Lie groups. J. Geom. Phys., 36(1-2):140–151, 2000. [22] J. E. Marsden and J. Scheurle. The reduced euler-lagrange equations. Fields Institute Communications, 1, 07 1993. [23] J. E. Marsden and A. Weinstein. Reduction of symplectic manifolds with symmetry. Reports on Mathematical Physics, 5(1):121–130, 1974. [24] K. R. Meyer. Symmetries and integrals in mechanics. In M. M. Peixoto, editor, Dynamical Systems, pages 259–272. Academic Press, 1973. [25] J. Navarro and J. B. Sancho. Energy and electromagnetism of a differential k-form. Journal of Mathematical Physics, 53(10):102501, 2012. [26] Á. Rodríguez Abella. Covariant reduction by fiberwise actions in classical field theory. TEMat monográficos, 2:211–214, 2021. [27] D. J. Saunders. The geometry of jet bundles. Cambridge University Press, 1989. [28] S. Smale. Topology and mechanics. I. Inventiones mathematicae, 10(4):305–331, 1970. [29] R. Utiyama. Invariant theoretical interpretation of interaction. Phys. Rev., 101(5):1597–1607, 1956. [30] J. Vankerschaver. Euler–Poincaré reduction for discrete field theories. Journal of Mathematical Physics, 48(3):032902, 2007.