According to Jaffe and Witten:

“for QCD to describe the strong force successfully, it must have at the quantum

“for QCD to describe the strong force successfully, it must have at the quantum

level the following three properties, each of which is dramatically different from the

behavior of the classical theory:

(1) It must have a “mass gap;” namely there must be some constant ∆>0

such that every excitation of the vacuum has energy at least ∆.

(2) It must have “quark confinement,” that is, even though the theory is de-

scribed in terms of elementary fields, such as the quark fields, that transform

non-trivially under

SU

(3), the physical particle states—such as the proton,

neutron, and pion—are

SU(3)-invariant.

(3) It must have “chiral symmetry breaking,” which means that the vacuum is

potentially invariant (in the limit, that the quark-bare masses vanish) only

under a certain subgroup of the full symmetry group that acts on the quark

fields”

Intuitively it is very clear that no one can ever prove any ‘mass gap’ for any nonabelian gauge theory on R^4 because that would make no sense whatsoever. The universe is clearly compact, and it is the compactness of the universe that produces mass gap for Yang-Mills type theories because in fact useful self-adjoint operators have compact resolvents. As for the ‘quark confinement’, it is unclear whether quarks are really particles at all but rather some geometric curvature measurements of actual stable structures being disintegrated.

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