Comparison between quantum and classical dynamics in the effective action formalism

A major difficulty in comparing quantum and classical behavior resides in the structural differences between the corresponding mathematical languages. The Heisenberg equations of motion are operator equations only formally identical to the classical equations of motion. By taking the expectation of these equations the well known Ehrenfest theorem provides identities which, however, are not a closed system of equations which allows to evaluate the time evolution of the system. The formalism of the effective action seems to offer a possibility of comparing quantum and classical evolutions in a systematic and logically consistent way by naturally providing approximation schemes for the expectations of the coordinates which at the zeroth order coincide with the classical c ○ Società Italiana di Fisica 12 Fabrizio Cametti, Giovanni Jona-Lasinio, Carlo Presilla and Fabio Toninelli evolution 1. The effective action formalism leads to equations of motion which differ from the classical equations by the addition of terms nonlocal in the time variable. This means that for these equations an initial value problem is not meaningful and they have to be interpreted in an appropriate way. Here we analyze situations in which the nonlocal