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 the longstanding Millennium problem (MP), since axiomatic approaches use traditional field properties. The first cutoff, corresponding physically to the cutoff of quadratic self-energies caused by Higgs fields, divides the four-dimensional spacetime continuum into arbitrarily shaped elements. Fields are expanded using scalar plane-waves with continuous four-momentum of quantum particles. The local alignment of spacetime elements is periodic without long-range order. Then, an effective field with discrete momentums is introduced. Particles are rarely excited in interactions with this effective field. The higher energy of the second cutoff corresponds physically to the Planck energy, preventing unphysical losses of high-energy intermediate states into black holes. Quantities, which correspond physically to vacuum expectation values of operators, utilize the first cutoff. 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.