by R. F. Streater; Physica Scripta, 3, 5-7, 1971.
We present a brief survey of the physical ideas behind the C*-algebra approach to quantum field theory, including internal symmetries, spontaneously broken symmetries, and the Haag-Kastler theory of the nature of charge. We illustrate it with a model in two-dimensional space-time.
This was my contribution to the second "Gunnar Kallén" Colloquium in mathematical physics, Paris, 1-5 June, 1970, organised by Moshe Flato. It reports on the subsequent paper with Ivan Wilde, Fermion states of a Boson field, and puts the work into the context of the programme of Haag and Kastler.
Haag's approach leads naturally to the occurrence of superselection rules in quantum field theory. We start with the observable fields, and do not include charged fields, or spinor fields, in the fundamental set of dynamical variables. The C*-algebra of the observables is of infinite dimension, and typically will possess a large number of inequivalent representations. There will be no common vector-states in the Hilbert spaces of two inequivalent representations, and no observable will have a non-zero matrix element between vectors from different irreducibles. Thus, if two representations have some states of finite energy, there will be a superselection rule operating between these states (and we would not be able to measure the relative phase between them by any measurement). It is natural to look for examples of superselection rules in the elementary particles. It has long been conjectured that there is a superselection rule between states of different charge; this might be expressed as saying that all observables are gauge invariant. Since the dynamics is taken to be an automorphism group acting on the observables, which is given by a unitary group in any physical representation, we conclude that charge is an absolutely conserved quantum number, as observed. According to Haag, charge is a label for inequivalent representations of the algebra. Our model shows how this might arise (in 1+1 dimensions) in the massless boson field, by looking at solutions with different "topological" properties (which are, in fact, of a cohomological nature).
The stability of matter might be explained if baryon number is also taken as a label for inequivalent representations of the observable algebra. The apparent separate conservation of electron-number, muon-number (and possibly, tau-number) might also be explained by this idea. Supporters of Haag are less than enthusiastic about proton decay and neutrino mixing than other physicists, who do not see any good reason for superselection rules (SSR) to exist in quantum mechanics. In the end, it is an experimental question; but what a pity if the general possibility of SSR in algebraic quantum field theory only explains the conservation of electric charge, and not the other conserved quantities! I predict that the proton will not decay into a positron, and that no neutrino mixing will be found. If so, then model builders will have to rethink some of their versions of the standard model.
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© by Ray Streater, 16/6/00.