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

TL;DR

This paper derives commutation relations for non-Abelian gauge fields to estimate the vacuum fluctuation energy, providing a theoretical approach to understanding quantum vacuum properties in gauge theories.

Contribution

It introduces a method to estimate vacuum fluctuation energies of non-Abelian gauge fields using uncertainty relations derived from commutation relations.

Findings

Estimated vacuum fluctuation energy gap in the long wave limit.

Derived total vacuum energy expression involving cutoff and coupling constants.

Provided a theoretical framework linking quantum commutation relations to vacuum energy estimates.

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.