Therefore, the kinetic energy of the entire system is considered when evaluating whether a back hopping is allowed or not, which may violate causality because energy from far away regions can be instantaneously used to allow the hopping. This is not the case when rescaling is made in the nonadiabatic vector direction. To see it, note that for an arbitrary rescaling direction, the hopping is allowed if LJ in Eq. ( ) is real. This condition implies that it is allowed when the following inequality holds (see Ref. 35 for details)