Yang–Mills Existence and Mass Gap
Construct quantum Yang–Mills theory on four-dimensional space-time with mathematical rigour, and prove that its lightest particle has strictly positive mass.
Problem statement
Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on R^4 and has a mass gap Δ > 0. The required axioms and properties are in Jaffe and Witten's official problem description.
Source: Arthur Jaffe, Edward Witten — Quantum Yang–Mills Theory (official problem description)
Introduction
Yang–Mills theories are the framework of the Standard Model of particle physics. Physicists compute with them very successfully, and experiments agree, but nobody has yet given a mathematically complete construction of the quantum theory in four dimensions.
The "mass gap" is the claim that the theory's lightest excitation has positive mass. It would explain why the strong nuclear force acts only at short range.
It is one of the Clay Millennium Prize Problems. Of the six problems here it has the least room for computer experiments that bear on the question.
Definitions
- Gauge group
- The symmetry group of the theory, e.g. SU(3) for the strong force.
- Mass gap
- A positive lower bound Δ on the energy of every state other than the vacuum.
What is known
Each statement says how certain it is: an established (proved) result, an experimental observation, an open conjecture, or an unverified claim.
- Established result Some quantum field theories have been constructed rigorously in two and three space-time dimensions; no comparable construction exists for four-dimensional Yang–Mills. [Arthur Jaffe, Edward Witten]
- Experimental observation Numerical lattice gauge theory computations and particle experiments are consistent with a mass gap. They are evidence, not a mathematical construction. [Arthur Jaffe, Edward Witten]
- Open conjecture Four-dimensional quantum Yang–Mills theory exists and has a mass gap (unproven). [Arthur Jaffe, Edward Witten]
Important limitations
- Lattice simulations at finite spacing and volume do not establish the continuum theory or its properties.
- Physical agreement with experiment is not a proof of existence in the required mathematical sense.
Status history
- Status: Open Named one of the seven Millennium Prize Problems by the Clay Mathematics Institute.
- Status: Open Checked against the Clay Mathematics Institute page: still unsolved.
This status is maintained by the Zerethon IDE team from authoritative sources. Contributions, runs and AI answers never change it.