Non-Abelian Berry phase in a quantum mechanical environment

We investigate a quantum physical system that can be naturally separated into fast and slow moving components. A modification of the conventional molecular Born-Oppenheimer approximation is considered by taking the intermolecular position vector to be a slowlyvarying quantum mechanical parameter. It is found that the fast motion (electronic degrees of freedom) induces a non-Abelian vector potential (Berry connection) into the dynamics of the slow system (nucleus), thereby modifying the commutation relations of the slow variables.