Яң-Миллс теориясы: Бөтендіктің негізі және математикалық мәселелер

Physical theory unifying the electromagnetic, weak and strong interactions

The phrase Yang–Mills theory means both a quantum field theory for nuclear binding devised by Chen Ning Yang and Robert Mills in 1953 and the class of similar theories. In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(n), or more generally any compact Lie group. A Yang–Mills theory seeks to describe the behavior of elementary particles using these non abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i. e. U(1) × SU(2)) as well as quantum chromodynamics, the theory of the strong force (based on SU(3)). Thus it forms the basis of our understanding of the Standard Model of particle physics.

Yang–Mills theories met with general acceptance in the physics community after Gerard 't Hooft, in 1972, worked out their renormalization, relying on a formulation of the problem worked out by his advisor Martinus Veltman. Renormalizability is obtained even if the gauge bosons described by this theory are massive, as in the electroweak theory, provided the mass is only an "acquired" one, generated by the Higgs mechanism. The mathematics of the Yang–Mills theory is a very active field of research, yielding e. g. invariants of differentiable structures on four dimensional manifolds via work of Simon Donaldson. Furthermore, the field of Yang–Mills theories was included in the Clay Mathematics Institute's list of "Millennium Prize Problems". Here the prize problem consists, especially, in a proof of the conjecture that the lowest excitations of a pure Yang–Mills theory (i. e. without matter fields) have a finite mass gap with regard to the vacuum state. Another open problem, connected with this conjecture, is a proof of the confinement property in the presence of additional fermions. In physics the survey of Yang–Mills theories does not usually start from perturbation analysis or analytical methods, but more recently from systematic application of numerical methods to lattice gauge theories.