$\mathrm{Op}^*$ and $\mathrm{C}^*$ dynamical systems. II. Structural differences: Borchers anomaly

According to the results of Part I [1], the only nontrivial difference between the vacuum struclures of $\mathrm{Op}^*$ and $\mathrm{C}^*$ dynamical systems is the effect of the infinite vacuum degeneracy in irreducible $\mathrm{Op}^*$ systems. For brevity, this effect is referred to as the “Borchers anomaly”, and is analyzed in detail by means of new mathematical tools – the recently introduced unbounded commutants of $\mathrm{Op}^*$ operators. A simple representation is obtained for the vacuum subspace of any field theory with cyclic vacuum in terms of the unbounded commutant of the field algebra, and from this representation a new necessary and sufficient condition for uniqueness of the vacuum is obtained. Some conditions for absence of the Borchers anomaly are derived, and a comparison which shows how these conditions improve the ones previously known is made.