Presymplectic structures and intrinsic Lagrangians for massive fields

TL;DR

This paper explores the construction of intrinsic actions for PDEs with presymplectic structures, focusing on massive fields, and introduces a multisymplectic extension that captures all equations of motion.

Contribution

It introduces a minimal multisymplectic extension for intrinsic actions that fully reproduces equations of motion for massive fields, extending to higher spins.

Findings

Intrinsic action for Einstein-Hilbert gravity is the Palatini action.

The natural presymplectic structure for massive spin-2 fields is incomplete.

A multisymplectic extension captures all equations of motion for massive fields.

Abstract

We study the construction of the so-called intrinsic action for PDEs equipped with compatible presymplectic structures. In particular, we explicitly demonstrate that the intrinsic action for the standard Einstein-Hilbert gravity is the familiar first-order Palatini action. Our main focus in this work is the massive spin-2 field, where the natural presymplectic structure is not complete in the sense that the associated intrinsic action does not reproduce all the equations of motion. We explicitly relate this feature to the differential consequences of the zeroth order in the genuine Lagrangian formulation of Fierz and Pauli. Moreover, a minimal multisymplectic extension of the intrinsic action that produces all the equations of motion is constructed and it is argued that systems of this type can be naturally regarded as multidimensional analogs of mechanical systems with constraints. Finally, we extend the considerations to the massive spin-3 field and argue that the extension to all the massive higher spins should be straightforward as well.