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

TL;DR

This paper applies Heisenberg's nonperturbative quantization to non-Abelian gauge theories, deriving a simplified two-equation model that describes flux tube solutions and the dual Meissner effect, highlighting confinement mechanisms.

Contribution

It introduces a two-equation approximation for nonperturbative quantization of non-Abelian gauge theories, simplifying the infinite Green function hierarchy with specific assumptions.

Findings

Derived flux tube solutions for color electric fields.

Demonstrated the dual Meissner effect with electric fields expelled from gluon condensate.

Provided a simplified model capturing confinement phenomena.

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.