Quantum Yang-Mills Theory

en · official · sha256:5272bcd513485e0c24414105e417d2876e238d5b52b0bd42f94fae18b3fa9af0EN

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

Ссылаются нормы: defines-compact-gauge-group

en · official · sha256:194761d64c66a30bd6a463eeeced32ddb2763d280ddf9e9dc537f8f9afc238beEN

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.

Ссылаются нормы: defines-field-operators

en · official · sha256:8357171236ae253d6112c0f29a75fef18966c3416a346415f4b2f3c1f62d9998EN

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.

Ссылаются нормы: names-four-torus-partial-results

en · official · sha256:a3a9c9ee7324162459243734092f33aec1d5b226d257dcd0748b2a36d96e97cfEN

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.

Ссылаются нормы: defines-mass-gap

en · official · sha256:2fd91a267316a19693f674edd869f2f31dd764c8786a2e4124599ee9e8ea06afEN

(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.

en · official · sha256:b5f834ccc3aa7887611fb89ebacb394387a419222cf05d161ede9fe49000becfEN

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.

Ссылаются нормы: names-no-known-example, yang-mills-reports-open

en · official · sha256:498101c081c9f1c789b00029b4cbe4448f52cc2abc9e4f06c7f2ed30a955f393EN

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].

Ссылаются нормы: YangMillsExistenceAndMassGapStatement, requires_axiom_system, source-asserts-yang-mills, yang-mills-identifies-problem

en · official · sha256:91bbccc6ea1ee2a1476ab57c37a0da04019d7db884189f2e060c1f9c991e0967EN

[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.

Ссылаются нормы: yang-mills-requires-os

en · official · sha256:433988b143ea5c25325d0146b99ead78ac04da9399d685c83d1f358d1efcf4a1EN

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

Ссылаются нормы: yang-mills-requires-wightman

en · official · sha256:bd6bb9cf766555ff0c216dcb3b3b8c4eb6c796a1aefb2d04a9bf787c79d1bb97EN

[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.

Ссылаются нормы: names-wilson-lattice

en · official · sha256:1724c5bf05bdbf97147945f1c50173f77b6e006b57724e2b75003409d0189c68EN

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

Ссылаются нормы: names-yang-mills-1954

en · official · sha256:24a24db63a14bd6b5bfddcf849f433421e57e3d3d496a13b4853ddfcdc0ef673EN

QUANTUM YANG-MILLS THEORY

Ссылаются нормы: CmiJaffeWittenSource, classified_as_millennium_problem, defines_notion, names_historical_result, yang-mills-classified

en · official · sha256:f9099e6dfc5faf15fa5df0408cb34bf681c7102c5bdd2127d352acb31d6b9eefEN

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).

Ссылаются нормы: excludes_weak_existence_as_solution, yang-mills-excludes-weak-existence