Uniqueness of the Jackiw non-Noetherian conformal scalar field

TL;DR

This paper proves that, up to second order, Jackiw's example is the only non-Noetherian conformal scalar field in two dimensions, using a classical inverse calculus of variations theorem.

Contribution

It demonstrates the uniqueness of Jackiw's non-Noetherian conformal scalar field in two dimensions up to second order.

Findings

Jackiw's example is unique up to second order.

The proof uses the inverse problem of the calculus of variations.

The result clarifies the special nature of Jackiw's scalar field.

Abstract

Jackiw was undoubtedly the first to exhibit an example of a scalar field action which is not conformally invariant whereas its equation of motion is. This feature has recently been dubbed as a non-Noetherian conformal scalar field. The paradigmatic example of Jackiw was the generalization to curved spacetime of the two-dimensional Liouville action. Here, we prove that, up to second order, this is the unique example of a non-Noetherian conformal scalar field in two dimensions. We establish this result using an old and somewhat forgotten theorem which is none other than the solution to the inverse problem of the calculus of variations.