Separability and invariance in nonrelativstic and relativistic quantum mechanics

Abstract

We have shown that good separability of the Hamiltonian in time, i.e.,τγ-separability, is sufficient to formulate time-dependent scattering theory. The property ofτγ-separability, extended to all the operators of Galileo or Poincare transformations, guarantees corresponding invariance of the scattering operators. From the structure of these groups we have used only the adiabaticity (16) of the Galileo and Poincare transformations, this being equivalent to the presence in these groups of an invariant Abelian subgroup containing taining time translations. Therefore, adiabaticity in conjunction withτγ-separability can serve as the basis for the quantum definition of a group of motion transformations compatible with the given Hamiltonian.