On auxiliary fields and Lagrangians for relativistic wave equations

TL;DR

This paper investigates the conditions under which linear PDE systems have Lagrangian formulations, introduces a new class of systems called co-flat, and constructs simplified Lagrangians for massive fields of arbitrary spin.

Contribution

It introduces the co-flat class of overdetermined systems, proves their pre-Lagrangian form always exists, and provides new, simpler Lagrangians for massive higher-spin fields.

Findings

Co-flat systems always admit a pre-Lagrangian form.

Explicit construction of pre-Lagrangian using auxiliary variables.

New Lagrangians for massive fields with fewer auxiliary fields.

Abstract

We address the problem of the existence of a Lagrangian for a given system of linear PDEs with constant coefficients. As a subtask, this involves bringing the system into a pre-Lagrangian form, wherein the number of equations matches the number of unknowns. We introduce a class of overdetermined systems, called co-flat, and show that they always admit a pre-Lagrangian form, which can be explicitly constructed using auxiliary variables. Moreover, we argue that such systems enjoy pre-Lagrangian formulations without auxiliary variables at all. As an application of our method, we construct new pre-Lagrangian and Lagrangian formulations for free massive fields of arbitrary integer spin. In contrast to the well-known models of Singh and Hagen, our Lagrangians involve much fewer auxiliary fields.