Operator Solutions in Terms of Asymptotic Fields in the Thirring and Schwinger Models. II

Abstract

The Thirring model and the Schwinger model (including non-Landau-gauge cases) are described in terms of asymptotic fields. In both models, operator solutions are found explicitly and it is shown that they possess the correct gauge-transformation property, the correct Lorentz-transformation property, locality with right statistics and canonical equal-time commutators. It is also found that, contrary to what is widely believed, gauge invariance is spontaneously broken and the Goldstone boson does exist in the Thirring model. Results of the preceding paper on a free massless scalar field are made use of extensively. The Thirring model and the Schwinger model are the celebrated exactly solvable examples of quantum field theory in two-dimensional space-time. They have a long history and there have appeared too many papers to quote all of them here. It is well known that formal solutions of both models can be very easily written down but they possess serious mathematical troubles.') Johnson2J was the first to give a consistent solution (in the form of Green's functions) of the Thirring model, but he gave up finding an operator solution. A thorough analysis of the Thirring model was made by Klaiber in his excellent lecture note. 3J He restored an operator solution in a modified form but used it only as a calculational tool. His primary. intention of using an operator solution was to guarantee a positivity condition for the solution given in the form of vacuum expectation values. There­ fore his operator solution was constructed in quite an artificial way. As for the Schwinger model, a detailed analysis was carried out by Lowenstein and Swieca.4J In all previous work, operator solutions are expressed in terms of a free massless spin or field *J that is, the spinor I-Ieisenberg operators have no corresponding asymptotic field.***> On the other *J Though there exists no spinor representation in two-dimensional space-time, it is customary to call a two-component half-vector a spinor. **l The singularity at p'=O is a branch point but not a simple pole. ***J This fact leads Casher, Kogut and Susskind'l to the use of the Schwinger model as a model of quark confinement.