Mathematical quantum Yang-Mills theory revisited

TL;DR

This paper rigorously constructs a non-perturbative quantum Yang-Mills theory with a positive mass gap, establishing a countable spectrum and eigenbasis, advancing the mathematical understanding of the Millennium problem.

Contribution

It provides a simplified, corrected, and rigorous formulation of quantum Yang-Mills theory with a proven positive mass gap and a countable spectrum, addressing key mathematical challenges.

Findings

Quantum Yang-Mills energy spectrum has a positive mass gap.

The Hamiltonian has a countable orthogonal eigenbasis.

Spectrum structure is fully elucidated.

Abstract

A mathematically rigorous relativistic quantum Yang-Mills theory with an arbitrary semisimple compact gauge Lie group is set up in the Hamiltonian canonical formalism. The theory is non-perturbative, without cut-offs, and agrees with the causality and stability principles. This paper presents a fully revised, simplified, and corrected version of the corresponding material in the previous papers DYNIN[11] and [12]. The principal result is established anew: due to the quartic self-interaction term in the Yang-Mills Lagrangian along with the semisimplicity of the gauge group, the quantum Yang-Mills energy spectrum has a positive mass gap. Furthermore, the quantum Yang-Mills Hamiltonian has a countable orthogonal eigenbasis in a Fock space, so that the quantum Yang-Mills spectrum is point and countable. In addition a fine structure of the spectrum is elucidated. KEYS: Millennium Yang-Mills problem; Finite propagation speed; Sobolev inequalities; Nuclear vector spaces; Infinite-dimensional holomorphy; Friedrichs operator extensions; Variational spectral principle; Symbols and spectral theory of pseudo-differential operators.