Covariant-differential formulation of Lagrangian field theory

TL;DR

This paper introduces a covariant-differential approach to Lagrangian field theory, replacing partial derivatives with exterior covariant differentials, providing a natural geometric framework for various fields including gravity.

Contribution

It develops a novel formalism using covariant exterior differentials in Lagrangian field theory, unifying the treatment of fields, currents, and energy tensors in a geometric setting.

Findings

Formulates a covariant prolongation bundle for Lagrangian densities.

Expresses field equations using covariant exterior differentials of momenta.

Includes examples with bosonic, spin-half, and gravitational fields.

Abstract

Building on the Utiyama principle we formulate an approach to Lagrangian field theory in which exterior covariant differentials of vector-valued forms replace partial derivatives, in the sense that they take up the role played by the latter in the usual jet bundle formulation. Actually a natural Lagrangian can be written as a density on a suitable "covariant prolongation bundle"; the related momenta turn out to be natural vector-valued forms, and the field equations can be expressed in terms of covariant exterior differentials of the momenta. Currents and energy-tensors naturally also fit into this formalism. The examples of bosonic fields and spin one-half fields, interacting with non-Abelian gauge fields, are worked out. The "metric-affine" description of the gravitational field is naturally included, too.