Yang–Mills existence and mass gap

The Yang–Mills existence and mass gap problem is an unsolved problem in mathematical physics and mathematics, and one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute, which has offered a prize of US$1,000,000 for its solution.

The problem is phrased as follows:

Yang–Mills Existence and Mass Gap. Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on ${\displaystyle \mathbb {R} ^{4}}$ and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in Streater & Wightman (1964), Osterwalder & Schrader (1973) and Osterwalder & Schrader (1975).

In this statement, a quantum Yang–Mills theory is a non-abelian quantum field theory similar to that underlying the Standard Model of particle physics; ${\displaystyle \mathbb {R} ^{4}}$ is Euclidean 4-space; the mass gap Δ is the mass of the least massive particle predicted by the theory.

Therefore, the winner must prove that:

• Yang–Mills theory exists and satisfies the standard of rigor that characterizes contemporary mathematical physics, in particular constructive quantum field theory, and
• The mass of all particles of the force field predicted by the theory are strictly positive.

For example, in the case of G=SU(3)—the strong nuclear interaction—the winner must prove that glueballs have a lower mass bound, and thus cannot be arbitrarily light.

The general problem of determining the presence of a spectral gap in a system is known to be undecidable.