Approximate Symmetry, Spontaneous Symmetry Breaking

Abstract

A metric for a set of states of a system gives a “distance” between every pair of states, with null “distance” for equivalent states and nonzero “distances” for degrees of inequivalence. Thus, a metric is a softening of the all-or-nothing character of an equivalence relation. That allows the definition of an approximate symmetry transformation as any transformation that changes every state to a state that is nearly equivalent with the original. An exact symmetry limit for a set of states equipped with a metric and a set of approximate symmetry transformations is another metric for the same set of states for which some of the approximate symmetry transformations (according to the original metric) are exact symmetry transformations. Broken symmetry is another term for approximate symmetry, and a symmetry breaking factor is whatever factor the (possibly hypothetical) switching off of brings about an exact symmetry limit. Spontaneous symmetry breaking is discussed. The crux of the matter is the issue of stability: How do small, symmetry breaking perturbations of a cause affect the symmetry of its effect? Although, by the symmetry principle, exact symmetry of a cause must appear in its effect, approximate symmetry of a cause might appear in the effect as exact symmetry (stability), as approximate symmetry (lability), or as badly broken symmetry (instability). The latter situation is what is known as spontaneous symmetry breaking.