Gauge Theories of Superselection Rules

by R. F. Streater; Acta Physica Austraeica, Supplement to XI, 317-, 1973. Based on the lectures given at Schladming Winter School, Feb. 1973.

ABSTRACT

By starting with the free Boson field in 1+1 dimensions, we construct a C*-algebra and its relativistic dynamics, in the spirit of Haag and Kastler. We construct a family of representations of the algebra, all with the property that the relativity group is spatial, that is, is given by unitary operators; moreover, the spectrum of the Hamiltonian is bounded below. This part of the paper is a summary of Fermion States of a Boson Field. We show that the massless Thirring model of self-interacting Fermions can be obtained as one of the Fermionic sectors. This construction became known as "Bosonization of the Fermions" when later, in 1975, S. Coleman, in perturbation theory, obtained the bosonization of the massive Thirring model. We construct explicit operators for the Fermi fields in the Fermion sectors, following a heuristic formula of Skyrme, but here formulated rigorously. These are a simple case of what are now known as "vertex operators". We show that in two space-time dimensions, the gauge group is not always compact, in contrast to four space-time dimensions, for which Haag, Doplicher and Roberts show that it is essentially a compact group (under their assumptions).

COMMENTS

A non-rigorous version of these formulae was later (1975) presented by Mandelstam, and are known to some people as "Mandelstam operators", instead of "Skyrme operators".