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In mathematical physics, the Yang–Mills existence and mass gap problem is an unsolved problem and one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute, which has offered a prize of US$1,000,000 to the one who solves it. ==Official problem description== The problem is phrased as follows:〔Arthur Jaffe and Edward Witten "(Quantum Yang-Mills theory. )" Official problem description.〕 :Yang–Mills Existence and Mass Gap. Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in , and . In this statement, Yang–Mills theory is a non-Abelian quantum field theory similar to that underlying the Standard Model of particle physics; is Euclidean 4-space; the mass gap Δ is the mass of the least massive particle predicted by the theory. Therefore, the winner must prove that: * Yang–Mills theory exists and satisfies the standard of rigor that characterizes contemporary mathematical physics, in particular constructive quantum field theory,〔R. Streater and A. Wightman, ''PCT, Spin and Statistics and all That'', W. A. Benjamin, New York, 1964.〕〔K. Osterwalder and R. Schrader, ''Axioms for Euclidean Green’s functions'', Comm. Math. Phys. 31 (1973), 83–112, and Comm. Math. Phys. 42 (1975), 281–305.〕 and * The mass of the least massive particle of the force field predicted by the theory is strictly positive. For example, in the case of G=SU(3)—the strong nuclear interaction—the winner must prove that glueballs have a lower mass bound, and thus cannot be arbitrarily light. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Yang–Mills existence and mass gap」の詳細全文を読む スポンサード リンク
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