Degenerate Higher-Order Maxwell Theories in Flat Space-Time

TL;DR

This paper investigates higher-order Maxwell theories in flat spacetime, analyzing their degrees of freedom and showing that degeneracy conditions do not fully eliminate Ostrogradski ghosts, thus revealing limitations of such models.

Contribution

It classifies degenerate higher-order Maxwell theories and demonstrates the persistent presence of Ostrogradski ghosts despite degeneracy.

Findings

Invertible kinetic matrix yields five degrees of freedom, including ghosts.

Degeneracy reduces degrees of freedom but does not eliminate all ghosts.

Theories with non-invertible kinetic matrix still face ghost issues.

Abstract

We consider, in Minkowski spacetime, higher-order Maxwell Lagrangians with terms quadratic in the derivatives of the field strength tensor, and study their degrees of freedom. Using a 3+1 decomposition of these Lagrangians, we extract the kinetic matrix for the components of the electric field, corresponding to second time derivatives of the gauge field. If the kinetic matrix is invertible, the theory admits five degrees of freedom, namely the usual two polarisations of a photon plus three extra degrees of freedom which are shown to be Ostrogradski ghosts. We also classify the cases where the kinetic matrix is non-invertible and, using analogous simple models, we argue that, even though the degeneracy conditions reduce the number of degrees of freedom, it does not seem possible to fully eliminate all potential Ostrogradski ghosts.