Charge-monopole duality and the phases of non-Abelian gauge theories

It is shown that the O(3) non-Abelian gauge theory possesses charge-monopole duality in the sense that, analogous to the ordinary magnetic vector potentials, one can construct O(3) "electric vector potentials," gauge-invariant combinations of which constitute the physical variables. However, while the Hamiltonian could in principle be expressed in terms of such variables, it would be very complicated; the definition of the potentials is thus not unique. The O(3) magnetic gauge group is separate from the ordinary electric gauge group. One can also formulate the theory in terms of electric (Wilson) or magnetic (Nielsen-Olesen) flux loops; the property of the loops which identifies the electric or magnetic gauge group as O(3) is studied. The possible phases of non-Abelian gauge theories are discussed briefly. The phase with real massless gluons, if it exists at all, is more complicated than the corresponding phase in an Abelian theory in the sense that it is not known how to construct a trial vacuum state without an infrared energy divergence. This phase should not be distinguished from the other phases by the absence of symmetry breaking, again in contradistinction to the Abelian gauge theory. The phase with complete Higgs symmetry breaking and that with confinement are electric-magnetic duals of one another. The relation between the Wilson condition, applied to loops at a fixed time, and the confinement of infinitely massive quarks is studied; it is hoped that the analysis will be helpful in constructing wave functions for hadronic states with confined quarks. It is suggested that Weinberg-Salam-type models may confine due to instanton effects though, for $\alpha$ around $\frac{1}{137}$, this has no practical significance.