A Study of Some Two-Dimensional Field Theory Models

Recently it has been pointed out (6) that quantum electrodynamics with a massless fermion in one-space, one-time dimension exhibits behavior which can be interpreted as being analogous to scaling and confinement. To learn more about the occurrence of such behavior, we study several other two-dimensional quantum field theories. After reviewing the Thirring model, we construct operator solutions of scale-invariant generalizations with internal degrees of freedom. We find that the physics of these models can be equally well described in terms of a field theory of fermions or of bosons. Because the physical excitations are massless in these models, the physical states can also be described, in general, as many-fermion or many-boson states. We also generalize Lowenstein and Swieca's operator solution of quantum electrodynamics in two-dimensions to a model with more than one massless, charged fermion. Although the resulting physical states are all electrically neutral, we find that the fermions may not be completely confined in the sense that some of their quantum numbers may be represented by particles in the spectrum. Once again, we discover that the physics can be represented by a fermion field theory or by a boson theory. The fact that the same physics can be represented by interacting fermions or interacting bosons can be understood by considering the polarization fields associated with the fermion charge and axial charge. This general physical picture allows us to extend the fermion-boson duality to theories with non-trivial scattering. Although these models cannot be solved, we can use formal operator structures to test, in a semi-classical approximation, for the possible existence of charged fermion states in the spectrum. Because the fermion states are equivalent to coherent states of bosons, their physics is approximated by solutions to the appropriate classical boson theory. Therefore, in two dimensions, confine­ment may be a phenomenon associated with the classical behavior of a theory.