Op* and C* dynamical systems

According to the previous results, the only nontrivial difference between the vacuum structures of Op* and C* dynamical systems is the effect of infinite vacuum degeneration in irreducible Op* systems. We study this effect (called here the Borchers anomaly for brevity sake) using new mathematical tools, unbounded commutants of Op* algebras. A simple representation in terms of the unbounded commutant of the field algebra is obtained for the vacuum subspace of arbitrary field theory with cyclic vacuum. On the basis of this representation a new necessary and sufficient condition of the uniqueness of the vacuum vector is derived. A number of conditions when the Borchers anomaly is absent are derived too and it is demonstrated that these conditions improve the existing results