quartic potential prevents it from becoming infinite. If this is done in a quantum path integral, this is a quantum phase transition, in a classical partition function, a classical phase transition. So as t moves toward more negative values in either context, the field has to choose some direction to point. Once it does this, it cannot change its mind. The system has ordered. In the ordered phase, there is still a little bit of symmetry--- rotations around the axis of the breaking. The field can point in any direction labelled by all the points on a unit sphere in space, which is the coset space of the unbroken SO(2) subgroup in the full symmetry group SO(3). In the disordered phase, the superselection sectors are described by the representation of SO(3) under which a given configuration transforms globally. Because the SO(3) is unbroken, different representations will not mix with each other. No local fluctuation will ever bring in nontrivial SO(3) configurations from infinity. A local configuration is entirely defined by its representation. There is a mass gap, or a correlation length, which separates configurations with a nontrivial SO(3) transformations from the rotationally invariant vacuum. This is true until the critical point in t where the mass gap disappears and the correlation length is infinite. The vanishing gap is a sign that the fluctuations in the SO(3) field are about to condense. In the ordered region, there are field configurations which can carry topological charge. These are labeled by elements of the second homotopy group . Each of these describe a different field configuration which at large distances from the origin is a winding configuration. Although each such isolated configuration has infinite energy, it labels superselection sectors where the difference in energy between two states is finite. In addition, pairs of winding configurations with opposite topological charge can be produced copiously as the transition is approached from below. When the winding number is zero, so that the field everywhere points in the same direction, there is an additional infinity of superselection sectors, each labelled by a different value of the unbroken SO(2) charge. In the ordered state, there is a mass gap for the superselection sectors labeled by a nonzero integer, because the topological solitons are massive, even infinitely massive. But there is no mass gap for all the superselection sectors labeled by zero because there are massless Goldstone bosons describing