Separability and invariance in nonrelativstic and relativistic quantum mechanics

Relation between the separability properties of the movement transformation operators $U(a)$ and the invariance and separability properties of the scattering operators $S$ is considered for the case of arbitrary (continuous) movement group $G$. The notion of $\tau_{\gamma}$-separability is introduced. It is shown that for groups $G$ possessing an invariant abelian subgroup, containing the subgroup of evolution transformations $U_t$, and, in particular, for the Galilei and Poincare groups, the invariance of the operators $S$ and their separability in time follow from the reasonably good $(\gamma >1)$ $\tau^{\gamma}$-separabiHty of the operators $U(a)$. For the Galilei and Poincare groups it is demonstrated that the separability of the operators $S$ in space is the consequence of their separability in time. It is shown that the choice of relative spatial variables, which is not unique in the relati vistic case, does not influence the properties of spatial separability.