Null Lagrangians of non-local field theories

This manuscript provides a characterisation of the equivalence class of classical smooth Lagrangian densities that involve terms depending on two distinct points of the underlying Euclidean base space of the theory. Theories of this type are referred to as non-local field theories, which are of particular interest in the group field theory approach to quantum gravity. The notion of equivalence of Lagrangian densities is set by physical indistinguishability by means of their equations of motion whose derivation is shown briefly. We expect our results to give a more comprehensive view on the non-local Noether theorem regarding divergence symmetries.