Quantum energy-mass spectra of relativistic Yang-Mills fields in a functional paradigm

TL;DR

This paper establishes a rigorous non-perturbative quantum Yang-Mills theory in four-dimensional spacetime, demonstrating a positive mass gap and a self-similar energy-mass spectrum using a functional analytic approach.

Contribution

It introduces a novel functional framework with a complex nuclear Kree-Gelfand triple and proves the existence of a positive mass gap in the quantum energy-mass spectrum.

Findings

Quantum energy-mass spectrum consists of non-negative eigenvalues diverging to infinity.

The spectrum exhibits self-similarity inversely proportional to an infrared cutoff.

A positive mass gap is rigorously proven in the quantum Yang-Mills theory.

Abstract

A non-perturbative and mathematically rigorous quantum Yang-Mills theory on 4-dimensional Minkowski spacetime is set up in the functional framework of a complex nuclear Kree-Gelfand triple. It involves a symbolic calculus of operators with variational derivatives and a new kind of infinite-dimensional ellipticity. In the temporal gauge and Schwinger first order formalism, Yang-Mills equations become a semilinear hyperbolic system for which the general Cauchy problem is reduced to initial data with compact supports. For a simple compact Yang-Mills gauge group and the anti-normal quantization of Yang-Mills energy-mass functional of initial data in a box, the quantum energy-mass spectrum is a sequence of non-negative eigenvalues converging to infinity. In particular, it has a positive mass gap. Furthermore, the energy-mass spectrum is self-similar (including the mass gap) in the inverse proportion to an infrared cutoff of the classical energy scale.