Local discrete symmetry and quantum-mechanical hair

A charge operator is constructed for a quantum field theory with an abelian discrete gauge symmetry, and a non-local order parameter is formulated that specifies how the gauge symmetry is realized. If the discrete gauge symmetry is manifest, then the charge inside a large region can be detected at the boundary of the region, even in a theory with no massless gauge fields. This long-range effect has no classical analog; it implies that a black hole can in principle carry "quantum-mechanical hair". If the gauge group is nonabelian, then a charged particle can transfer charge to a loop of cosmic string via the nonabelian Aharonov-Bohmeffect. The string loop can carry charge even though there is no localized source of charge anywhere on the string or in its vicinity. The "total charge" in a closed universe must vanish, but, if the gauge group is nonabelian and the universe is not simply connected, then the "total charge" is not necessarily the same as the sum of all point charges contained in the universe.