  The perturbative dynamics of quantum field theories is described by a
recursive expansion similar to the well known loop expansion. The equivalent
formulation based on low-energy dynamics via an expansion in derivatives is
well known in the literature; this is described by terms from low energy to
high energy. The coefficients of these terms are presented in a simplified
algebraic manner. In general, the dynamics of any bare Lagrangian, including
additional higher dimension terms, is found by iteration in a discrete
algorithm. Inversion of the quantized dynamics to the fundamental theory is
possible.
