Quantum Yang-Mills Theory

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At the classical level one replaces the gauge group U(1) of electromagnetism by a compact gauge group G.

Referencing norms: defines-compact-gauge-group

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To establish existence of four-dimensional quantum gauge theory with gauge group G, one should define a quantum field theory (in the above sense) with local quantum field operators in correspondence with the gauge-invariant local polynomials in the curvature F and its covariant derivatives, such as Tr FijFkl(x). Correlation functions of the quantum field operators should agree at short distances with the predictions of asymptotic freedom and perturbative renormalization theory, as described in textbooks. Those predictions include among other things the existence of a stress tensor and an operator product expansion, having prescribed local singularities predicted by asymptotic freedom.

Referencing norms: defines-field-operators

en · official · sha256:8357171236ae253d6112c0f29a75fef18966c3416a346415f4b2f3c1f62d9998

Some results are known for Yang-Mills theory on a 4-torus T4 approximating R4, and, while the construction is not complete, there is ample indication that known methods could be extended to construct Yang-Mills theory on T4.

Referencing norms: names-four-torus-partial-results

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Since the vacuum vector Omega is Poincare invariant, it is an eigenstate with zero energy, namely H Omega = 0. The positive energy axiom asserts that in any quantum field theory, the spectrum of H is supported in the region [0, infinity). A quantum field theory has a mass gap Delta if H has no spectrum in the interval (0, Delta) for some Delta > 0. The supremum of such Delta is the mass m, and we require m < infinity.

Referencing norms: defines-mass-gap

en · official · sha256:2fd91a267316a19693f674edd869f2f31dd764c8786a2e4124599ee9e8ea06af

(1) It must have a "mass gap;" namely there must be some constant Delta > 0 such that every excitation of the vacuum has energy at least Delta.

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In fact, at present we do not know any non-trivial relativistic field theory that satisfies the Wightman (or any other reasonable) axioms in four dimensions.

Referencing norms: names-no-known-example, yang-mills-reports-open

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Yang-Mills Existence and Mass Gap. Prove that for any compact simple gauge group G, a non-trivial quantum Yang-Mills theory exists on R4 and has a mass gap Delta > 0. Existence includes establishing axiomatic properties at least as strong as those cited in [45, 35].

Referencing norms: YangMillsExistenceAndMassGapStatement, requires_axiom_system, source-asserts-yang-mills, yang-mills-identifies-problem

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[35] 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.

Referencing norms: yang-mills-requires-os

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[45] R. Streater and A. Wightman, PCT, Spin and Statistics and all That, W. A. Benjamin, New York, 1964.

Referencing norms: yang-mills-requires-wightman

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[48] K. G. Wilson, Quarks and strings on a lattice, in New Phenomena In Subnuclear Physics, Proceedings of the 1975 Erice School, A. Zichichi, ed., Plenum Press, New York, 1977.

Referencing norms: names-wilson-lattice

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[50] C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96 (1954), 191-195.

Referencing norms: names-yang-mills-1954

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QUANTUM YANG-MILLS THEORY

Referencing norms: CmiJaffeWittenSource, classified_as_millennium_problem, defines_notion, names_historical_result, yang-mills-classified

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We specifically exclude weak-existence (compactness) as the solution to the existence part of the Millennium problem, unless one also uses other techniques to establish properties of the limit (such as the existence of a mass gap and the axioms).

Referencing norms: excludes_weak_existence_as_solution, yang-mills-excludes-weak-existence