Multisymplectic formulation of Yang–Mills equations and Ehresmann connections

Abstract

We present a multisymplectic formulation of the Yang- Mills equations. The connections are represented by normalized equivariant 1-forms on the total space of a principal bundle, with values in a Lie alge- bra. Within the multisymplectic framework we realize that, under reasonable hypotheses, it is not necessary to assume the equivariance condition a priori, since this condition is a consequence of the dynamical equations. The motivation of the following work was at first to provide a Hamiltonian formulation of the Yang-Mills system of equations which would be as much covariant as possible. This means that we look for a formulation which does not depend on choices of space-time coordinates nor on the trivialization of the principal bundle. Among all possible frameworks (covariant phase space, etc.) we favor the multisymplectic formalism which takes automatically into account the locality of fields theories. Following this approach we have been led to discover a new variational formulation of the Yang-Mills equations with nice geometrical features. The origin of the multisymplectic formalism goes back to the discovery by V. Volterra in 1890 (28, 28) of generalizations of the Hamilton equations for variational problems with several variables. These ideas were first developped mainly around 1930 (4, 7, 30, 24) and in 1950 (6). After 1968 this theory was geometrized in a way analogous to the construction of symplectic geometry by several mathematical physicists (10, 12, 23) and in particular by a group