Analysis of time-space translations in quantum fields

Abstract

I discuss relativistic quantum fields whose time-space translations are realized in indefinite unitary groups. Such indefinite metric fields describe interactions, e.g., the Coulomb interaction, which cannot be parametrized completely by particles. They cannot be expanded with time-space translation eigenvectors; the representations of the translations involved are triangularizable, but not diagonalizable. To project on a positive-definite subspace for a probability interpretation, the vanishing of the nilpotent part in the time-space translations realization is required. A trivial Becchi-Rouet-Stora charge (gauge invariance) for the asymptotics in quantum gauge theories can be interpreted as one special case of this general principle—the asymptotic projection to the eigenvectors of the time-space translations.