Renormalized Four Dimensional Quantum Yang-Mills Theory and Mass Gap

TL;DR

This paper presents a quantization approach for four-dimensional Yang-Mills theory on Minkowski space, establishing a positive mass gap and invariance under gauge transformations, with implications for understanding quantum chromodynamics.

Contribution

It introduces a quantization procedure that satisfies Wightman axioms and proves the existence of a positive mass gap in non-abelian gauge theories.

Findings

Spectrum of the QCD Hamilton operator is positive and bounded away from zero.

Mass gap vanishes as the coupling constant approaches zero.

Construction is invariant under gauge transformations preserving Coulomb gauge.

Abstract

A quantization procedure for the Yang-Mills equations for the Minkowski space $\mathbf{R}^{1,3}$ is carried out in such a way that field maps satisfying Wightman axioms of Constructive Quantum Field Theory can be obtained. Moreover, by removing the infrared and ultraviolet cutoffs, the spectrum of the corresponding (non-local) QCD Hamilton operator is proven to be positive and bounded away from zero, except for the case of the vacuum state, which has vanishing energy level. The whole construction is invariant for all gauge transformations preserving the Coulomb gauge. As expected from QED, if the coupling constant converges to zero, then so does the mass gap. This is the case for the running coupling constant leading to asymptotic freedom.