Unbounded representations of symmetry groups in gauge quantum field theory. I. Confinement and differentiation

Symmetry groups and especially the covariance (substitution rules) of the basic fields in a gauge quantum field theory of the Wightman--Garding type are investigated. By means of the continuity properties hidden in the substitution rules it is shown that every unbounded form--isometric representation U of a Lie group has a form-skew-symmetric differential partialU with dense domain in the unphysical Hilbert space. Necessary and sufficient conditions for the existence of the closures of U and partialU as well as for the isometry of U are derived. It is proved that a class of representations of the translation group enforces a relativistic confinement mechanism, by which some or all basic fields are confined but certain mixed products of them are not