General invertible transformation and physical degrees of freedom
  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.
