Fields and algebras of observables in models with superselection rules

Simple models of the Doplicher–Haag–Roberts theory are studied, in which the field algebra and the observable algebra coexist, the latter being defined as the gauge-invariant part of the former. In this part of the work we consider the abelian model with the gauge group $SU(1)$. Duality for the nets of local algebras is proved in the abelian coherent sectors, using the twisting operation for the field algebras. Intertwining operators of coherent sectors are extended to unitary operators on the Fock space of states according to the Doplicher–Haag–Roberts scheme and the commutation relations are derived for these unitary extensions. It turns out that in the abelian model all the intertwining operators satisfy normal commutation relations and can be interpreted as boson or fermion fields in correspondence with the number of the charge quanta they transport.