Quantum Yang-Mills-Weyl Dynamics in Schroedinger paradigm

TL;DR

This paper develops a rigorous, non-perturbative quantum Yang-Mills-Weyl theory using a Schrödinger representation, revealing a self-similar spectrum with a positive mass gap, based on the properties of the gauge group.

Contribution

It introduces a novel mathematical framework for quantum YMWD with a self-similar spectrum and a positive mass gap, advancing the understanding of quantum gauge theories.

Findings

Spectrum is self-similar and converges to infinity.

Eigenvalues have finite multiplicities within a von Neumann algebra.

Vacuum eigenvalue is a simple zero, indicating a mass gap.

Abstract

Inspired by F. Wilczek's QCD Lite, quantum Yang-Mills-Weyl Dynamics (YMWD) describes quantum interaction between gauge bosons (associated with a simple compact gauge Lie group $\mathbb{G}$) and larks (massless chiral fields colored by an irreducible unitary representation of $\mathbb{G}$). Schroedinger representation of this quantum Yang-Mills-Weyl theory is based on a sesqui-holomorphic operator calculus of infinite-dimensional operators with variational derivatives. The spectrum of the quantum YMWD, with initial data in the central euclidean ball of a radius $0<R<+\infty$, is self-similar in the inverse proportion to $R$. The spectrum is a sequence of eigenvalues convergent to $+\infty$. The eigenvalues have finite multiplicities with respect to a von Neumann algebra with a regular trace. The same holds for the quantum self-interaction of vector Yang-Mills bosons (Theorem 4.1). Furthermore, the fundamental vacuum eigenvalue is a simple zero (Appendix A). Presumably, this is a solution of the existence problem for a quantum Yang-Mills theory that implies a positive spectral mass gap. The rigorous mathematical theory is non-perturbative with a running coupling constant as the only ad hoc parameter. The application of the first mathematical principles depends essentially on the properties of the compact simple Lie group $\mathbb{G}$.