  In a recent paper Seiberg and Witten have argued that the full action
describing the dynamics of coincident branes in the weak coupling regime is
invariant under a specific field redefinition, which replaces the group of
ordinary gauge transformations with the one of noncommutative gauge theory.
This paper represents a first step towards the classification of invariant
actions, in the simpler setting of the abelian single brane theory. In
particular we consider a simplified model, in which the group of noncommutative
gauge transformations is replaced with the group of symplectic diffeomorphisms
of the brane world volume. We carefully define what we mean, in this context,
by invariant actions, and rederive the known invariance of the Born-Infeld
volume form. With the aid of a simple algebraic tool, which is a generalization
of the Poisson bracket on the brane world volume, we are then able to describe
invariant actions with an arbitrary number of derivatives.
