Among the Wightman axioms is the requirement that there is a unique Poincare-invariant state called the vacuum.
We know that QFT vacuua are not necessarily unique, for example in situations where spontaneous breaking of symmetry is relevant.
So, why is that? Why do we demand uniqueness of the vacuum when it is clearly not true?
Let $G$ be a group, represented on the C*-algebra of quantum observables. By Haag's theorem, if in the C*-algebra of quantum observables there are two $G$-invariant states that are also $G$-abelian (an asymptotic condition that it would be too long to explain here), then they are either equal or disjoint (the latter meaning that one cannot be written as a trace class operator in the GNS-representation of the other).
Therefore, since the Wightman axioms are set in a given representation of the algebra of observables, only one ground state (i.e. Poincaré- invariant and abelian state) can exist as a trace class operator. There may be however more than one $G$-invariant state, provided only one is $G$-abelian. I think that an example would be given by the free vacuum and the free Gibbs state of a scalar theory: they are both trace class in the Fock representation, and invariant (at least w.r.t. time translations).
