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The Yang–Mills Millennium problem

Nature Reviews Physics volume 8, pages 86–97 (2026)Cite this article

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Abstract

The Yang–Mills Millennium Prize problem is one of the great challenges of mathematical physics. In the quarter century since it was set, what progress has been made? This Review outlines the problem from a physics point of view, gives its physical background, explains its nature and significance as a problem in mathematics and surveys promising approaches from recent years.

Key points

  • Yang–Mills theory is the basis of the standard model of particle physics and describes the strong and weak forces.

  • The Yang–Mills Millennium problem is to show that Yang–Mills theory is mathematically well defined and that it has the mass gap property.

  • The issue of definition is to prove that the theory has a continuum limit, which is well defined at arbitrarily high energies. This requires renormalization, which has never been made rigorous in the needed generality.

  • The mass gap property (no massless particles) is expected because it is true of real-world quantum chromodynamics and it is seen in numerical simulations. It is widely felt that no clear path is known towards proving it.

  • Recent mathematical approaches include rigorous stochastic quantization and the rigorous strong coupling expansion. They are part of probability theory, and mathematicians are making significant advances.

  • Numerical and computational methods are important in the physical study of Yang–Mills and likely to be used in any rigorous proof. Physicists could contribute significantly by developing more powerful computational renormalization group methods.

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Fig. 1: Lattice gauge theory in two dimensions.
Fig. 2: Renormalization group flow for SU(3) Yang–Mills with 12 quark flavours.
Fig. 3: Graphical representation of a tensor renormalization group block spin step for the 2D nonlinear sigma model.

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Acknowledgements

The author thanks J. Dimock, M. Hairer, A. Jaffe, E. Witten and I. Klebanov for invaluable discussions and advice.

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  1. CMSA, Harvard University, Cambridge, MA, USA

    Michael R. Douglas

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  1. Michael R. Douglas
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Douglas, M.R. The Yang–Mills Millennium problem. Nat Rev Phys 8, 86–97 (2026). https://doi.org/10.1038/s42254-025-00909-2

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