  In axiomatic quantum field theory, the postulate of the uniqueness of the
vacuum (a pure vacuum state) is independent of the other axioms and equivalent
to the cluster decomposition property. The latter, however, implies a Coulomb
or Yukawa attenuation of the interactions at growing distance, hence cannot
accomodate the confining properties of the strong interaction. The solution of
the Yang-Mills quantum theory given previously, uses an auxiliary field to
incorporate Gauss's law, and demonstrates the existence of two separate vacua,
the perturbative and the confining vacuum, therefore a mixed vacuum state,
deriving confinement, as well as the related, expected properties of the strong
interaction. The existence of multiple vacua is, in fact, expected by the
axiomatic, algebraic quantum field theory, via the decomposition of the vacuum
state to eigenspaces of the auxiliary field. The general vacuum state is a
mixed quantum state and the cluster decomposition property does not hold.
Because of the energy density difference between the two vacua, the physics of
the strong interactions does not admit a Lagrangian description. I clarify the
above remarks related to the previous solution of the Yang-Mills interaction,
and conclude with some discussion, a criticism of a related mathematical
problem, and some tentative comments regarding the spin-2 case.
