Invertibility conditions for field transformations with derivatives: toward extensions of disformal transformation with higher derivatives

TL;DR

This paper establishes conditions under which field transformations involving derivatives are invertible, extending the disformal transformation framework and analyzing higher derivative cases with rigorous mathematical methods.

Contribution

It derives necessary and sufficient invertibility conditions for derivative-dependent field transformations, including extensions of disformal transformations with higher derivatives.

Findings

Derived invertibility conditions using the method of characteristics.

Provided examples of invertible transformations involving derivatives.

Proved non-invertibility of certain second-derivative extensions of disformal transformations.

Abstract

We discuss a field transformation from fields $\psi_a$ to other fields $\phi_i$ that involves derivatives, $\phi_i = \bar \phi_i(\psi_a, \partial_\alpha \psi_a, \ldots ;x^\mu)$, and derive conditions for this transformation to be invertible, primarily focusing on the simplest case that the transformation maps between a pair of two fields and involves up to their first derivatives. General field transformation of this type changes number of degrees of freedom, hence for the transformation to be invertible, it must satisfy certain degeneracy conditions so that additional degrees of freedom do not appear. Our derivation of necessary and sufficient conditions for invertible transformation is based on the method of characteristics, which is used to count the number of independent solutions of a given differential equation. As applications of the invertibility conditions, we show some non-trivial examples of the invertible field transformations with derivatives, and also give a rigorous proof that a simple extension of the disformal transformation involving a second derivative of the scalar field is not invertible.