The mass gap problem for the Yang-Mills Field

TL;DR

This paper investigates the conditions under which the Yang-Mills field exhibits a mass gap, linking it to the spectral properties of a curl operator on certain divergence-free one-forms in three dimensions.

Contribution

It establishes a precise criterion for the existence of a mass gap in the quantized Yang-Mills Hamiltonian based on the spectrum of an auxiliary curl operator.

Findings

Mass gap exists iff zero is not in the spectrum of the curl operator.

Classical lowest energy point is a non-degenerate critical point of the potential.

Spectral properties determine the quantum mass gap condition.

Abstract

We consider the reduced Hamiltonian of the Yang-Mills field on $\mathbb{R}^4$ equipped with a Lorentzian metric. We show that the secondary quantized principal term $H_0$ of the Taylor expansion of this Hamiltonian at the lowest energy point has a mass gap if and only if zero is not a point of the spectrum of the auxiliary self-adjoint operator ${\rm curl}=*d$ defined on the space of one-forms $\omega$ on $\mathbb{R}^3$ satisfying the condition ${\rm div}~ \omega=*d*\omega=0$, where $*$ is the Hodge star operator associated to a metric on $\mathbb{R}^3$ and $d$ is the exterior differential. In this case the classical lowest energy point of the reduced configuration space is a non-degenerate critical point of the potential energy term of the reduced Hamiltonian of the Yang-Mills field, in the sense of Palais.