Comments on ``A note on first-order formalism and odd-derivative actions'' by S. Deser

TL;DR

This paper clarifies misconceptions about first-order formalisms for odd-derivative actions, demonstrating correct formulations that preserve gauge invariance using topologically massive electrodynamics as an example.

Contribution

It provides a correct derivation of first-order formalisms for odd-derivative actions, correcting previous misconceptions and illustrating the process with TME.

Findings

Correct first-order formulations preserve gauge invariance.

Misconceptions arose from non-equivalent examples.

The Ostrogradsky process does not necessarily break gauge invariance.

Abstract

We argue that the obstacles to having a first-order formalism for odd-derivative actions presented in a pedagogical note by Deser are based on examples which are not first-order forms of the original actions. The general derivation of an equivalent first-order form of the original second-order action is illustrated using the example of topologically massive electrodynamics (TME). The correct first-order formulations of the TME model keep intact the gauge invariance presented in its second-order form demonstrating that the gauge invariance is not lost in the Ostrogradsky process.