# General invertible transformation and physical degrees of freedom

## TL;DR

This paper proves a theorem establishing the conditions under which invertible field transformations, even those depending on derivatives, preserve the solutions of Euler-Lagrange equations, with applications to scalar-tensor theories.

## Contribution

The paper provides a rigorous proof that invertible transformations, including derivative-dependent ones, preserve solutions of Euler-Lagrange equations, clarifying their physical equivalence.

## Key findings

- Invertible transformations map solutions between original and transformed systems.
- Higher derivative terms do not break the solution-preserving property.
- Application to scalar-tensor theories demonstrates practical relevance.

## Abstract

An invertible field transformation is such that the old field variables correspond one-to-one to the new variables. As such, one may think that two systems that are related by an invertible transformation are physically equivalent. However, if the transformation depends on field derivatives, the equivalence between the two systems is nontrivial due to the appearance of higher derivative terms in the equations of motion. To address this problem, we prove the following theorem on the relation between an invertible transformation and Euler-Lagrange equations: If the field transformation is invertible, then any solution of the original set of Euler-Lagrange equations is mapped to a solution of the new set of Euler-Lagrange equations, and vice versa. We also present applications of the theorem to scalar-tensor theories.