The problem
Empty space has a lowest-energy quantum state, called the vacuum. Set its energy to zero. In a toy system allowing energies 0, 2, 3, and so on, nothing lies between 0 and 2: there is a gap. These numbers only illustrate the idea. Yang–Mills asks us to construct a quantum theory of fields that interact with themselves, without added matter, in three space directions and one time direction, and prove a positive gap above its vacuum.[2]
Quantum states belong to a space with an inner product, and an energy operator describes their spectrum. The vacuum has . A mass gap means for some . Allowed energies can form a continuum above the gap. This condition concerns possible energy measurements; a superposition of the vacuum and an excited state can still have a smaller mean energy.[2]
Scope
Pure quantum Yang–Mills theory on four-dimensional space, meeting the mathematical axioms in the Clay statement.
What remains open
The full theory must be defined mathematically, and its allowed excitation energies must stay away from zero. Positive energies alone are insufficient: the toy list 1, 1/2, 1/3, and so on gets arbitrarily close to zero. A finite simulated box does not settle the field theory on unlimited space.[1] [2]
The official target is a nontrivial theory on satisfying the required quantum-field axioms, including positivity, spacetime symmetry and locality, with the specified short-distance behavior and a positive gap. A Euclidean construction must support reconstruction of the corresponding quantum theory. Both the continuum and infinite-volume limits need control. Proving confinement or an isolated one-particle state would require additional results beyond the stated gap.[1] [2]
Why it matters
A gap controls how strongly fluctuations in distant regions of empty space remain connected. It links local field interactions to behavior over large distances. It also helps explain how a quantum theory can have a nonzero energy scale even though its classical fields have massless waves.[2]
The wider mathematical interest includes using quantum gauge theories to study four-dimensional geometry. Rapidly decreasing correlations suggest that quantities associated with distant pieces can be controlled separately. Jaffe and Witten present this geometric use as a possible application; a solution would not automatically establish every proposed application on other four-manifolds.[2]
Definitions
3 terms
vacuum
The lowest-energy state of the quantum field. It is not the absence of the field or all fluctuations. Its energy is set to zero in the problem statement.[2]
mass gap
A positive interval above the vacuum energy containing no energy spectrum. This does not require a discrete spectrum above that interval, or say that every state has an expectation value of energy above the gap.[2]
gauge group
The mathematical group specifying the internal symmetry of the gauge field. Different gauge descriptions can represent the same gauge-invariant quantities. The Clay problem restricts this group to be compact and simple.[2]
Sources
- Yang-Mills & the Mass Gap
Notes on Yang-Mills & the Mass Gap
Official maintained Millennium Problem page and linked statement by Arthur Jaffe and Edward Witten.
- Quantum Yang–Mills Theory — official problem description
Notes on Quantum Yang–Mills Theory — official problem description
Clay-hosted statement. Sections 3–5 define the quantum theory, energy-spectrum gap and clustering consequence; section 6.5 describes continuum and infinite-volume limits. No publication date is printed in the retrieved PDF; its upload path is not a publication date.