# Higher Derivative Field Theories: Degeneracy Conditions and Classes

## TL;DR

This paper analyzes ghost-free higher derivative field theories without gauge symmetries, deriving degeneracy conditions to avoid Ostrogradsky ghosts, classifying solutions, and exploring transformations to first order form.

## Contribution

It provides a comprehensive classification of ghost-free higher derivative theories and methods to transform them into first order form, highlighting which classes resist such transformations.

## Key findings

- Lorentz invariance prevents half degrees of freedom.
- Explicit degeneracy conditions for ghost-free theories.
- Identification of theories not reducible to first order form.

## Abstract

We provide a full analysis of ghost free higher derivative field theories with coupled degrees of freedom. Assuming the absence of gauge symmetries, we derive the degeneracy conditions in order to evade the Ostrogradsky ghosts, and analyze which (non)trivial classes of solutions this allows for. It is shown explicitly how Lorentz invariance avoids the propagation of "half" degrees of freedom. Moreover, for a large class of theories, we construct the field redefinitions and/or (extended) contact transformations that put the theory in a manifestly first order form. Finally, we identify which class of theories cannot be brought to first order form by such transformations.