related by an exact symmetry, they are all exactly equivalent. This suggests a deep relationship between symmetry breaking directions and conserved charges. Discrete symmetry In the 2D Ising model, at low temperatures, there are two distinct pure states, one with the average spin pointing up and the other with the average spin pointing down. This is the ordered phase. At high temperatures, there is only one pure state with an average spin of zero. This is the disordered phase. At the phase transition between the two, the symmetry between spin up and spin down is broken. Below the phase transition temperature, an infinite ising model can be in either the mostly-plus or the mostly-minus configuration. If it starts in the mostly-plus phase, it will never reach the mostly-minus, even though flipping all the spins will give the same energy. By changing the temperature, the system acquired a new superselection rule--- the average spin. There are two superselection sectors--- mostly minus and mostly plus. There are also other superselection sectors; for instance, states where the left half of the plane is mostly plus and the right half of the plane is mostly minus. When a new superselection rule appears, the system has spontaneously ordered. Above the critical temperature, the ising model is disordered. It could visit every state in principle. Below the transition, the system chooses one of two possibilities at random and never changes its mind. For any finite system, the superselection is imperfect. An Ising model on a finite lattice will eventually fluctuate from the mostly plus to the mostly minus at any nonzero temperature, but it takes a very long time. The amount of time is exponentially small in the size of the system measured in correlation lengths, so for all practical purposes the flip never happens even in systems only a few times larger than the correlation length. Continuous symmetries If a statistical or quantum field has three real valued scalar fields , and the energy or action only depends on combinations which are symmetric under rotations of these components into each other, the contributions with the lowest dimension are (summation convention): and define the action in a quantum field context or free energy in the statistical context. There are two phases. When t is large, the potential tends to move the average to zero. For t large and negative, the quadratic potential pushes out, but the