A mathematical solution to Yang-Mills existence and mass gap problem, one of the Millennium problems

抄録

In this article, the Poincar{\'e}/Lorentz covariant/invariant formalism without ultraviolet divergences gives an answer to the Yang-Mills existence problem of Millennium problem (MP), since axiomatic approaches use traditional field properties. Fields are expanded using scalar plane-waves with continuous four-momentum of quantum particles. Because of asymptotic freedom, Yang-Mills theory become trivial with vanishing interactions for the infinite cutoff energy. To avoid the triviality, the main procedure introduces a cutoff. In mathematics, for the metric of the four-dimensional spacetime continuum (on which Yang-Mills fields are defined), only in the case where metric is determined from general relativity like Riemannian geometry, the cutoff scale is restricted to the Planck energy, preventing unphysical losses of intermediate states into black holes. Only for applications of mathematics to physics, self-energies due to Higgs fields will be cut off by dividing spacetime into arbitrary-shaped elements. Next, pure Yang-Mills fields composed of variational stationary classical fields and quantum fluctuations are considered. The Wilson loop for the classical field yields a linear potential between a charge and an anticharge, and the stabilization energy provides a mass gap. The action has a local mass from the classical field. These masses present an answer to the mass gap problem of the MP.