Towards a Spectral Proof of the Mass Gap in QCD?

TL;DR

This paper explores the spectral properties of a key operator in Coulomb gauge Yang--Mills theory, aiming to contribute to understanding the mass gap problem in quantum chromodynamics.

Contribution

It analyzes the spectral theory of a pseudo-differential operator in Yang--Mills theory, providing foundational insights for the quantization of the theory.

Findings

Spectral analysis of operator P in Coulomb gauge

Evaluation of matrix elements of P

Foundational work towards understanding the mass gap

Abstract

Yang--Mills theory in four dimensions is studied by using the Coulomb gauge. The Coulomb gauge Hamiltonian involves integration of matrix elements of an operator P built from the Laplacian and from a first-order differential operator. The operator P is studied from the point of view of spectral theory of pseudo-differential operators on compact Riemannian manifolds, both when self-adjointness holds and when it is not fulfilled. In both cases, well-defined matrix elements of P are evaluated as a first step towards the more difficult problems of quantized Yang--Mills theory.