Superselection Sectors
A large physical system with infinitely many degrees of freedom does not always visit every possible state, even if it has enough energy. If a magnet is magnetized in a certain direction, each spin will fluctuate at any temperature, but the net magnetization will never change. The reason is that it is infinitely improbable that all the infinitely many spins at each different position will all fluctuate together in the same way.
A big system often has superselection sectors. In a solid, different rotations and translations which are not lattice symmetries define superselection sectors. In general, a superselection rule is a quantity that can never change through local fluctuations. Aside from order parameters like the magnetization of a magnet, there are also topological quantities, like the winding number. If a string is wound around a circular wire, the total number of times it winds around never changes under local fluctuations. This is an ordinary conservation law. If the wire is an infinite line, under conditions that the vacuum does not have winding number fluctuations which are coherent throughout the system, the conservation law is a superselection rule --- the probability that the winding will unwind is zero.
There are quantum fluctuations, superpositions arising from different configurations of a phase-type path integral, and statistical fluctuations from a Boltzmann type path integral. Both of these path integrals have the property that large changes in an effectively infinite system require an improbable conspiracy between the fluctuations. So there are both statistical mechanical and quantum mechanical superselection rules.
In a theory where the vacuum is invariant under a symmetry, the conserved charge leads to superselection sectors in the case that the charge is conserved. Electric charge is conserved in our universe, so it seems at first like a trivial example. But when a superconductor fills space, or equivalently in a Higgs phase, electric charge is still globally conserved but no longer defines the superselection sectors. The sloshing of the superconductor can bring charges into any volume at very little cost. In this case, the superselection sectors of the vacuum are labeled by the direction of the Higgs field. Since different Higgs directions are related by an exact symmetry, they are all exactly equivalent. This suggests a deep relationship between symmetry breaking directions and conserved charges.
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