  In this paper we will provide a new operatorial counterpart of the
path-integral formalism of classical mechanics developed in recent years. We
call it new because the Jacobi fields and forms will be realized via finite
dimensional matrices. As a byproduct of this we will prove that all the
operations of the Cartan calculus, such as the exterior derivative, the
interior contraction with a vector field, the Lie derivative and so on, can be
realized by means of suitable tensor products of Pauli and identity matrices.
