Invertibility conditions for field transformations with derivatives: Toward extensions of disformal transformation with higher derivatives
Résumé
We discuss a field transformation from fields ψ_a to other fields ϕ_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 fields and involves up to their first derivatives. General field transformation of this type changes the 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.