The effects on a certain family of representations of the commutation relations of the operator-translations ak → ak + ck, ak* → ak* + ck*, are investigated. Here ak and ak* are the annihilation and creation operators of the representations, the ck are complex numbers (scalar operators), and the representations considered are the discrete representations described by Gårding and Wightman, and Schweber and Wightman, generalizations of the usual Fock representation. Necessary and sufficient conditions on the translations are obtained for the resultant representations to remain in the family of discrete representations, and the resultant translation groups are investigated for their relevant structure.