The mass gap problem for the Yang-Mills Field

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.