Boundary Effects and Confinement in the Theory of Nonabelian Gauge Fields

Abstract

The thesis is devoted to the problem of colour confinement in the non-Abelian Yang-Mills theory (gluon part of Quantum Chromodynamics). A generalisation of the 3-dimensional Fock-Schwinger gauge is proposed where the Gauss law constraint is exactly solvable. This simplifies the theory in a finite domain and incorporates the variables at the boundary into the Hamiltonian formalism. The dependence of the partition function on the boundary value of the longitudinal component of the electric field is studied and related to the mechanism of the confinement-deconfinement transition. The free energy density is calculated for $SU(2)$ and $SU(3)$ gluodynamics in the mean-field approximation for the collective variables. Analysis of its minima reveals a phase transition at a certain temperature, below which the mean collective variables have nonzero values. This can be interpreted as a confinement-deconfinement phase transition. In the confinement phase the chromo-electric flux through any element of the boundary is strictly zero. This means the singletness with respect to the group of the residual gauge transformations and hence impossibility of observing coloured objects at spatial infinity (in asymptotic states). It is demonstrated that our confinement condition satisfies the traditional confinement criteria. The Wilson loop for $SU(N)$ theory is shown to satisfy the area law. The ratio of the transition temperature to the square root of the string tension coefficient is in a qualitative agreement with the result from lattice Monte Carlo simulations. In the deconfinement phase the global symmetry $Z_{N}$ (centre of $SU(N)$) is spontaneously broken by the surface terms. The confinement phase is characterised by unbroken symmetry with all nontrivial minima having the same depth and transformable by $Z_{N}$ actions.