Fermion States of a Boson Field

by R. F. Streater and I. F. Wilde

ABSTRACT

We construct certain non-Fock representations of the canonical commutation relations for the zero-mass scalar field in two-dimensional space-time. These are obtained from the free-field representation by applying localised automorphisms, are strongly locally Fock, and have a positive energy-momentum spectrum.

The model is suggested by an early (1958) paper of Skyrme, and provides a concrete example of the algebraic theory of observables and charges. It satisfies some, but not all, of the postulates of Doplicher, Haag and Roberts. The Bose-Fermi alternative does not hold: instead, we find a continuum of representations which interpolates between the Bose and Fermi sectors.

We are able to construct the local field algebra, which provides a unitary multiplier representation of a gauge group of the second kind.

COMMENTS

This paper was rejected by Commun. in Math. Phys., on the grounds that the general theory had already been done a few months earlier by Haag, Doplicher and Roberts, and that there was no need for examples. However, the conclusion of Haag, Doplicher and Roberts was that the gauge group must be a compact group. Nowadays, it is recognised that this theorem does not hold in two or three space-time dimensions. Indeed, this is our conclusion too, since our gauge group contains R².

See D. Buchholz, Quarks, Gluons, Colour-Fact or Fiction?, Nuclear Phys. B469, pp 333-353, 1996 for more recent work.

See D. Guido, R. Longo and H. W. Wiesbrock, Commun. in Math. Phys., 192, 217-244 (1998) for more, and the page of Steve Summers for the latest.

This work was used here and here to show that the massless free Boson field has fermion states obeying dynamics of the massless Thirring model. So when Coleman showed in 1975 that the massive Thirring model (in perturbation theory) has dynamics identical to the sine-Gordon equation (in 1+1 dimensions) it came as no surprise (contrary to the assertion by Nahm, in an otherwise excellent account, "Conformal Field Theory: A Bridge over Troubled Waters").

This subject has been taken further by Claus Montonen in his paper, The many-anyon problem.