Charge, Anomalies and Index Theory

By R. F. Streater.

in "Fundamental Aspects of Quantum Theory", eds. V. Gorini and A. Frigerio, Plenum Press, 1986, pp 253-266. This book is the Proceedings of the NATO Advanced Research Workshop, 2-7 Sept, 1985, Villa Olmo, Como.

ABSTRACT

In the algebraic approach to quantum field theory, due to Haag, the primary object is the algebra of observables. Gauge fields, spinor fields and other gauge-dependent operators are unobservable, but are constructed from the properties of the observables. We explain how this is done.

In para. 2 we give a brief account of the phenomenon of anomalies in theories with axial currents; we describe some of the puzzling aspects of anomalies, such as the necessary breakdown of apparant symmetries of a theory, in the presence of external fields.

In para. 3 we try to persuade you that anomalies are a Good Thing, and can be used to implement the programme suggested by Haag. In fact, for the free field in one dimensional space, explicit formulae for the charged operators that implement the Bogoliubov transformations defined by scattering in an external field, can be given. These operators are not observables on the original space, and create new sectors (=representations) of the algebra. This leads to a new interpretation of the external field problem, in which the dressed vacuum is rather a model of an extended charged particle than a ground state; it is similar to a Skyrmion. Some ambitions to extend this to 3+1 dimensions are presented.