Phase Transitions, Spontaneous Symmetry Breaking, and Goldstone's Theorem

Some important rigorous results on phase transitions accompanied by the spontaneous breaking of symmetries in statistical mechanics and relativistic quantum field theory are reviewed. Basic ideas, mainly inspired by quantum field theory, underlying the proofs of some of these results are sketched. The Goldstone theorem is proven, and the Mermin-Wagner-Hohenberg theorem concerning the absence of continuous symmetry breaking in one and two dimensions is recalled. Comments concerning rigorous results on the Kosterlitz-Thouless transition in the two-dimensional classical XY model are made.