Lie groupoids in classical field theory I: Noether’s theorem

Abstract

In the two papers of this series, we initiate the development of a new approach to implementing the concept of symmetry in classical field theory, based on replacing Lie groups/algebras by Lie groupoids/algebroids, which are the appropriate mathematical tools to describe local symmetries when gauge transformations are combined with space-time transformations. In this second part, we shall adapt the formalism developed in the first paper to the context of gauge theories and deal with minimal coupling and Utiyama's theorem. Universidade de São Paulo

Principal bundles and connections modelled by Lie group bundles

“…The equivalence between Ehresmann connections on π Y ,Y /G and the 1-forms as in the statement is a consequence of the isomorphism (7).…”

“…Actually, our initial interest in generalized principal bundles started in that framework, from where we have taken much inspiration. We would like to mention that fibered actions can be extended to the Lie groupoid setting (for example see [7,8,16]), but we do not address this matter here.…”

Symmetries and Covariant Poisson Brackets on Presymplectic Manifolds

“…These aspects, and many more, have been extensively analyzed in the last two centuries, both within the mathematical and theoretical physics literature: a (far from exhaustive) list of contributions where the previous ideas have been developed include [1][2][3][4][5][6][7][8][9][10][11][12][13][14], where Noether's theorems (that is the relation between symmetries and the so called conservation laws) are also elucidated. In particular, in recent references where classical field theories were analyzed within the setting of jet bundles and their duals, conserved currents associated with symmetries of an action functional are modeled as (m − 1)-forms on a fiber bundle underlying the theory (with m the dimension of the space-time in which the theory is developed, see, for instance, [8,[15][16][17]).…”

Infinitesimal Automorphisms of VB-Groupoids and Algebroids

“…Symmetries of a PDE form a Lie pseudogroup and the latter can be seen as a Lie subgroupoid of a jet groupoid. There are several natural (multi)DOs and jets attached to a jet space, in particular a jet groupoid [10,16,54,66]. One should expect that natural DOs/jets on jet groupoids are also multiplicative, so falling into the class our program aims to define and study.…”