On topological symmetries and the Goldstone theorem

We show that one cannot achieve the symmetry breaking condition limR→∞<(QR,Ω)> ≠ 0 when QR is the (finite volume) integral of a topological charge density and Ω has a compact support. This implies that topological symmetries are never broken spontaneously. If one attempts to use an operator Ω' whose support extends to spatial infinity as an order parameter the resulting symmetry breaking condition can be formally satisfied, but the Goldstone theorem does not apply because, in general, the topological charge is no longer conserved. This (wrong) symmetry breaking condition need not contain any dynamical information and merely reflects the effect of Ω' on the boundary conditions at spatial infinity. (author). 5 refs