Confinement and Deconfinement in Gauge Theories: A Quantum Field Theory

Abstract: After a brief recount of small and large gauge transformations and the nature of observables, we discuss superselection sectors in gauge theories. There are an infinity of them, classified by large gauge transformations. Gauge theory sectors are labelled by the eigenvalues of a complete commuting set (CCS) of these transformations. In QED, the standard chemical potential is one such operator generating global U(1). There are many more given by the moments of the electric field on the sphere at infinity. In QCD, the CCS are constructed from the two commuting generators spanning a Cartan subalgebra. Large gauge transformations commute with the Hamiltonian and preserve the equations of motion. They form an infinite number of `classical symmetries'. But most of them are anamolous changing the superselection sectors. We show that any element of a large gauge transformation can be added to the standard Hamiltonian as a generalised chemical potential without changing field equations and that in QCD, they lead to confined and deconfined phases . A speculation about the physical meaning of these chemical potentials is also made.