Nuclear Physics B116 (1976) 141-156 © North-Holland Publishing Company SPONTANEOUS BREAKDOWN OF SYMMETRY AND GROUP CONTRACTIONS C. DE CONCINI and G. VITIELLO Istituto di Fisica, Universit?z di Salerno, Italia Received 16 July 1976 In spontaneously broken symmetry theories, the symmetry group that appears in ob- servations proves to be a group contraction of the dynamical invariance group. Infrared effects play a crucial role in the dynamical rearrangement of symmetry which leads to the group contraction. Many examples are considered. General theorems are given for SU(n) and SO (n). Low-energy theorems and ordered-state symmetry patterns are ob- servable manifestations of group contractions. These results seem to support the conjec- ture that the transition from quantum to classical physics involves a group contraction mechanism. 1. Introduction In a theory with spontaneous breakdown of symmetry, one central problem is the relation between the invariance of the theory at the dynamical level, (i.e. at the level of the equations of the basic Heisenberg fields) and the symmetry properties at the phenomenological level. Indeed, a very interesting feature of such theories is the replacement of the basic symmetry by a different symmetry at the observational level, which manifests itself in the observable ordered states of the physical system. Well known examples are the crystal and the ferromagnet where the translational and spin-rotational invariance of the basic dynamics are replaced by the lattice and the spin-polarized symmetry, respectively. To make more precise statements, let us note that in general one starts by giving the field equations (or the Lagrangian) for the basic Heisenberg fields, say ~(x), A(O) $(x) = J($ (x)). (1.1) The theory is invariant under certain transformations of the Heisenberg fields ¢~(x) ~ ff'(x)= T[¢(x)], (1.2) when eq. (1.1) is form-invariant under (1.2). On the other hand, the physical system is described in terms of free in- (out-) fields ¢(x) (quasiparticle fields, in many-body terminology). Thus one is faced with 141