Some aspects of the homogeneous formalism in Field Theory and gauge invariance

TL;DR

The paper develops a Hamiltonian formalism for Field Theory using multisymplectic geometry and fiber bundles, generalizing the homogeneous formalism from mechanics, and demonstrates energy conservation in a gravitational model.

Contribution

It introduces a novel Hamiltonian formalism for Field Theory based on Hamiltonian connections and multisymplectic forms, extending the homogeneous formalism to fields.

Findings

Derived a formal energy expression for a parametrized Einstein Lagrangian

Showed the conservation of this energy in the model

Proposed a geometric framework for gauge invariance in Field Theory

Abstract

We propose a suitable formulation of the Hamiltonian formalism for Field Theory in terms of Hamiltonian connections and multisymplectic forms where a composite fibered bundle, involving a line bundle, plays the role of an extended configuration bundle. This new approach can be interpreted as a suitable generalization to Field Theory of the homogeneous formalism for Hamiltonian Mechanics. As an example of application, we obtain the expression of a formal energy for a parametrized version of the Hilbert--Einstein Lagrangian and we show that this quantity is conserved.