Charges, Global Gauge Groups and Exchange Symmetry

Abstract

By focusing attention on observables and algebras we have kept only those parts of the axioms of chapter II which have a direct, immediately clear physical significance. Other parts, reflecting traditional field theoretic formalism, such as the assumption of a Bose-Fermi alternative, or the founding of the theory on fields transforming according to finite dimensional representations of the Lorentz group do not appear. The understanding of internal symmetries is also modified. In the development of elementary particle theory during the past two decades gauge symmetries have acquired a more and more prominent position and it may well be that all internal symmetries should properly be regarded as gauge symmetries, formulated in the field theoretic frame by transformations of unobservable fields leaving all observables unchanged¹ Conversely, if in the algebraic approach we take the extreme position and claim that the net of observable algebras defines the theory completely without need for any additional specification, then we cannot allow any internal symmetry of the observable net in the sense of (III.3.15) with α_g ≠ id. So this would also lead to the conclusion that internal symmetries should always be gauge symmetries. The most salient observable consequence of internal symmetries is the set of charge quantum numbers which serves to distinguish different species of particles and to characterize their properties. In the case of gauge symmetries these quantum numbers manifest themselves through the existence of superselection rules for the states over the observable algebra.