Emergent Semi-Classical Dynamics Beyond the Born-Oppenheimer Approximation

We discuss a semi-classical scheme for computing the dynamics of a system composed of fast quantum degrees of freedom coupled to slow coordinates. Going to a reference frame co-moving with the slow degrees of freedom, we find corrections to the Born-Oppenheimer approximation, which modify the emergent equations of motion for the slow variables. This is illustrated through a few simple examples, where the emergent dynamics is complex and qualitatively different from that predicted by the Born-Oppenheimer approximation.