Spontaneous symmetry breaking

Abstract

Spontaneous symmetry breaking is a very common occurrence in many-body systems. Ordi-nary crystals break translation symmetry down to a discrete subgroup. Ferromagnets break rotational symmetry. In these and many other cases, the stable solutions of the dynamical equations, which govern the system, exhibit less symmetry than the equations themselves. Superfluidity and superconductivity are also closely associated with spontaneous symmetry breaking, but of a more subtle, intrinsically quantum-mechanical kind. In superfluids – the classic case being liquid He 4 at low temperatures – the symmetry that is broken is the U (1) phase symmetry associated with conservation of He 4 atom number. In superconductors – the classic case being bad metals at low temperatures – the symmetry that is broken is a local (gauged) symmetry, associated to electron number, to which photons respond. Several cases of spontaneous symmetry breaking are important within the standard model. Two are particularly outstanding. The approximate chiral symmetry SU L (2) × SU (2) R of QCD, under independent unitary transformations among the left-handed u L , d L and the right-handed u R , d R helicity states, was the first case to be analyzed deeply, principally by Nambu (in pre-quark days, using a rather different language!). This symmetry is not exact, even within QCD, because it is violated by the non-zero masses of u, d, which flip helicity. Those masses are quite small, however. Quantitatively, the symmetry breaking SU (2) L × SU (2) R → SU (2) L+R is predominantly spontaneous. A rich, useful theory of pions and the interactions at low energies follows from these ideas. The symmetry breaking can also be demonstrated directly, by numerical solution of the equations of QCD (lattice gauge theory). In this case the broken symmetry is global 1 , similar to superfluidity. The gauge symmetry SU (2) × U (1) is postulated in our theory of electroweak interactions. We must, however, avoid the massless gauge bosons that unbroken gauge symmetry seems to imply 2 . This difficulty is overcome by breaking the symmetry spontaneously. In this case, with gauge symmetry front and center, the mechanism is similar to superconductivity.