Nonperturbative quantization \`a la Heisenberg for non-Abelian gauge theories: two-equation approximation

Abstract

The nonperturbative quantization technique \`{a} la Heisenberg is applied for non-Abelian gauge theories. The operator Yang-Mills equation is written, which on the corresponding averaging gives an infinite set of equations for all Green functions. We split all degrees of freedom into two groups: in the former, we have $A^a_\mu \in \mathcal G \subset SU(N)$, and in the second group we have coset degrees of freedom $SU(N) / \mathcal G$. Using such splitting and some assumptions about 2- and 4-point Green functions, we truncate the infinite set of equations to two equations. The first equation is for the gauge fields from the subgroup $\mathcal G$, and the second equation is for a gluon condensate which is the dispersion of quantum fluctuations of the coset fields. Two examples are considered: The first one is a flux tube solution describing longitudinal color electric fields stretched between quark and antiquark located at the $\pm$ infinities. The second one is a flux tube stretched between two quarks (antiquarks) located at $\pm \infty$. A special case is considered when the longitudinal electric field produced by a quark located at $+ \infty$ is equal and oppositely directed to the field generated by a quark located at $- \infty$ that leads to zero total electric field. Both solutions represents the dual Meissner effect: the electric field is pushed out from the gluon condensate.