Boundary conditions and confinement in the gauge field theory at finite temperature

A Hamiltonian formulation of a non-Abelian gauge theory confined in a finite domain is constructed in a generalized three-dimensional Fock–Schwinger gauge in the presence of surface terms. The dependence of the partition function on the boundary value of the longitudinal electric-field component, which because of the Gauss law, coincides with the electric-field flow through an infinitesimal boundary-surface element in this gauge, is investigated. This dependence is related to the confinement–deconfinement phase transition. In the confinement phase, the chromoelectric current through any boundary element is zero, because all observable quantities are singlet w.r.t. the remaining gauge-transformation group, i. e. color objects are unobservable at spatial infinity. In addition to the non-Abelian theory, a simpler example of quantum electrodynamics with an external-charge density in a spherical domain is considered in which the effective partition function is exactly calculable.