Conjectures
Yang-Mills Existence and Mass Gap
YAHNG milz
Also Known As Yang-Mills Problem
Analysis
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One of the seven Clay Mathematics Institute Millennium Prize Problems, at the boundary of mathematics and physics. Chen-Ning Yang and Robert Mills introduced the underlying gauge theory in 1954. Experiment and computer simulation both indicate that the quantum version of a Yang-Mills theory has a mass gap, meaning the lightest particle it predicts has strictly positive mass even though the classical waves in the theory travel at the speed of light, but as of the Clay Mathematics Institute's own account no proof of this property exists from first principles. The official problem statement was written by Arthur Jaffe and Edward Witten.
Facts
StatementFor any compact simple gauge group, prove that a nontrivial quantum Yang-Mills theory exists on four dimensional space and has a mass gap greater than zero. 1 Proposed Year Prize StatusOne of the seven Clay Mathematics Institute Millennium Prize Problems, named in 2000; a correct proof carries a one million dollar award. 1 Progress Toward ResolutionThe mass gap is well established by physics experiment and by computer simulation, but has never been derived as a mathematical consequence of the quantum Yang-Mills equations themselves, and a fully rigorous construction of a four dimensional quantum Yang-Mills theory does not yet exist. The problem remains completely open. 1 Cross-Tradition Connections
Associated With
Yang and Mills introduced the underlying gauge theory in 1954; the Millennium Prize question about its rigorous quantum construction and mass gap was posed by the Clay Mathematics Institute in 2000, not by Yang or Mills themselves.
In Branch
Sources
1. Clay Mathematics Institute
Clay Mathematics Institutehttps://www.claymath.org/millennium/yang-mills-the-maths-gap/Quote, https://www.claymath.org/millennium/yang-mills-the-maths-gap/
This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view.
1. Clay Mathematics Institute
Clay Mathematics InstituteMillennium Prize Problems page, announcement paragraphQuote, Millennium Prize Problems page, announcement paragraph
The prizes were announced at a meeting in Paris, held on May 24, 2000 at the Collège de France.
Yang-Mills Existence and Mass Gap (Wikipedia)
Wikimedia FoundationIn Branch: Analysis, Problem Requirements sectionQuote, In Branch: Analysis, Problem Requirements section
establishing axiomatic properties at least as strong as those cited in Streater & Wightman
Open Questions (1 open question)
Does a nontrivial quantum Yang-Mills theory really exist for every compact simple gauge group in four dimensions, with a strictly positive mass gap?
The mass gap is well established by physics experiment and by computer simulation, but no one has derived it as a mathematical consequence of the quantum Yang-Mills equations themselves, and no fully rigorous construction of a four dimensional quantum Yang-Mills theory yet exists.
What would resolve this A rigorous mathematical construction of quantum Yang-Mills theory on four dimensional space for a compact simple gauge group, together with a proof that it has a mass gap greater than zero.
Mathematical physicsClay Mathematics Institute
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