Observables, Superselection Sectors and Gauge Groups

Abstract

Symmetries play an important role in quantum physics from its very beginning. There are two types of symmetry which must be distinguished, namely internal and external symmetries. External symmetries have an active interpretation; they transform states and observables such that certain structures are respected, e.g. transition probabilities, the Hamiltonian, the S-matrix etc.. Typical examples are the space time symmetries in a situation where space time is a priori given in the sense of classical physics. Internal symmetries, on the contrary, have only a passive interpretation; they do not transform the states of the system but merely change the description of the system. Examples are gauge transformations in gauge theories and diffeomorphisms in general relativity. Hence internal symmetries occur only in cases where the formulation of the theory contains some redundancy. It is tempting to remove this redundancy and to base the theoretical description exclusively on observables. Then the question arises whether the internal symmetry has an intrinsic meaning and may be recovered from the observables. Conversely, one may interpret the observed structure of the state space in terms of an internal symmetry and develop a redundant description where this symmetry acts explicitely. Such a redundant description might be useful for the investigation of perturbed theories where the symmetry is slightly broken, either spontaneously or dynamically.