Estimate of the energy of vacuum fluctuations of non-Abelian gauge fields from the uncertainty relations

Abstract

We derive the commutation relations for field strengthes of the non-Abelian gauge fields by requiring that the quantum mechanical equations of motion coincide with the classical field equations. The equations of motion with respect to time derivative coincide with the corresponding field equations, while those with respect to space derivatives agree with classical field equations if constraint equations are fulfilled. Using the uncertainty relations for field strengthes at the same times, which follows from the commutation relations, we estimate the energy gap of vacuum fluctuations in the long wave limit as $\mathcal{E}_{0,k} \sim c\hbar\Lambda^{-2} V^{-1}(g^2/c\hbar)^{1/3}$ and the total energy $\mathcal{E}_0\sim \Lambda c\hbar(g^{2}/\hbar c)^{1/3}$ with $\Lambda $ being a cutoff separating the weak and strong coupling regimes.