Yang-Mills Theory and Mass Gap

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Prove that for any compact simple gauge group G, quantum Yang-Mills theory of
R4 exists and has a mass gap Δ>0.

Simply: Yang-Mills theory is the quantum field theory (QFT) underlying the
Standard Model of particle physics. It is a generalization of the Maxwell theory of
electromagnetism. As a classical field theory it has solutions which travel at the speed of
light, so its quantum version should describe massless particles (gluons). However, the
phenomenon of colour confinement permits only bound states of gluons forming massive
particles. This is the mass gap. The problem is to establish rigorously the existence of the
quantum Yang-Mills theory and a mass gap.

Originators: Chinese physicist Yang Chen-Ning Franklin, and American physicist
Robert Mills tried to extend the concept of gauge theory for an Abelian group in 1954.
It didn’t work as predicted; the idea was shelved until the 1960s when symmetrybreaking
in massless theories was initiated by J. Goldstone, Y. Nambu and G. Jona-Lasinio,
with particles acquiring mass via Yang-Mills.

Incentive: $US 1million, one of seven Millennium Prize Problems of the Clay
Mathematics Institute.

Exploration: This active field includes attempts using lattice gauge theory and
four-dimensional supersymmetric theories. A solution will require fundamental new ideas
and would demonstrate key physics of the forces binding protons and neutrons into

Interesting aspect: Has led to new discoveries in quantum groups and new
QFT mathematical structures like the Jones polynomial.

Published in IMAges 8 - May 2010