On the gauge-invariant dynamical charges and densities of the 1-instanton solution

Abstract

We study the gauge invariant dynamically conserved charges, and their corresponding densities, for instanton solutions of Yang-Mills theories in four dimensional Euclidean space, for the gauge group SU(2). Those charges were constructed in [1,2] through the integral equations of Yang-Mills theory, using techniques on generalized loop spaces. We use the integral non-Abelian Gauss law to evaluate the gauge-invariant flux of the magnetic and electric non-abelian fields through spherical surfaces centered at origin of the instanton solution. From such a flux, we define gauge-invariant charge densities by considering the charge within an infinitesimal spherical shell of radius $r$. We discuss the issue of the reparameterization invariance of the charges and densities, and show that the magnetic and electric fluxes for the instanton and anti-instanton, at the Euclidean time $x^4 = 0$ and radius $r=1$, which here corresponds to the size $λ$ of those solutions, are non-zero and observable. Our results give an interesting picture of the internal structure of the instanton, and may be important for the properties of the Yang-Mills $θ$-vacuum.