The Quantum Yang–Mills Theory

Abstract

In axiomatic quantum field theory, the postulate of the uniqueness of the vacuum (a pure vacuum state) is independent from the other axioms and equivalent to the cluster decomposition property. The latter, however, implies a Coulomb or Yukawa attenuation of the interactions at growing distances and hence cannot accommodate the confining properties of the strong interaction. Thesolution 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 resulting in 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.