Comment on "A path integral leading to higher-order Lagrangians, arXiv:0708.4351 (hep-th)"

We show that the “e!ective Lagrangian” constructed in [1] is inconsistent with the exact result for the complete Lagrangian presented in [2]. We trace the origin of the inconsistence to the peculiar way in which the path integral methods are used to eliminate variables that are not auxialiry. The new interest in noncommutative physics that comes from a low energy limit of string theory, quantum gravity and noncommutative geometry, has opened old questions in the study of generalized Poisson structures and their possible quantization. Among those questions it remains unclear if a well defined Hamiltonian dynamics based in a noncommutative Poisson structure admits a variational formulation in configuration space. It is well known that even in the simplest case where the Poisson structure has the form {q i ,q j } = ! ij , {pi,pj} =0 , {q i ,pj} = " i, (1) with ! a constant matrix, the associated problem in configuration space is not variationally admissible. That means that it is not possible to construct a standard Lagrangian in configuration space that reproduce the dynamics of the corresponding equations of motion. This question is not only interesting from the point of view of the inverse problem of the calculus of variations but it is also interesting for physical reasons because we are ultimately interested in the description of the dynamical properties of the system in the noncommutative configuration space.