The problem is phrased as follows:[1] 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;
The general problem of determining the presence of a spectral gap in a system is known to be undecidable.
We hope so!The problem requires the construction of a QFT satisfying the Wightman axioms and showing the existence of a mass gap.
[1] There are four axioms: Quantum mechanics is described according to von Neumann; in particular, the pure states are given by the rays, i.e. the one-dimensional subspaces, of some separable complex Hilbert space.
The Wightman axioms require that the Poincaré group acts unitarily on the Hilbert space.
In other words, they have position dependent operators called quantum fields which form covariant representations of the Poincaré group.
The group of space-time translations is commutative, and so the operators can be simultaneously diagonalised.
The second part of the zeroth axiom of Wightman is that the representation U(a, A) fulfills the spectral condition—that the simultaneous spectrum of energy-momentum is contained in the forward cone: The third part of the axiom is that there is a unique state, represented by a ray in the Hilbert space, which is invariant under the action of the Poincaré group.
which, together with their adjoints, are defined on a dense subset of the Hilbert state space, containing the vacuum.
The Hilbert state space is spanned by the field polynomials acting on the vacuum (cyclicity condition).
The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle.
Quantum Yang–Mills theory with a non-abelian gauge group and no quarks is an exception, because asymptotic freedom characterizes this theory, meaning that it has a trivial UV fixed point.
At the level of rigor of theoretical physics, it has been well established that the quantum Yang–Mills theory for a non-abelian Lie group exhibits a property known as confinement; though proper mathematical physics has more demanding requirements on a proof.