On the Confinement and the Vacuum Degeneracy in the Schwinger Model

Abstract

It is shown that multiple vacua are physical states in the Schwinger model formulated in a finite spatial interval by constructing explicitly an operator solution in which the fermionic degrees of freedom appear only in charge, chiral charge and operators. These are best illustrated by taking the massies's limit of an operator solution to the pre-Schwinger model which is represented in Hilbert space with the usual fermionic charge-sector. We also describe another operator solution to the pre-Schwinger model expressed in terms of boson fields only. It turns out that vacuum state is a non-normalizable eigenstate of the charge. § 1. Introduction and summary It is widely believed that the strong interaction can be described by the theory of quarks coupled to non-Abelian gauge fields and quarks and gauge fields may be permanently bound within hadrons and never appear as asymptotic particles. To implement these ideas, various theoretical approaches have been made. One possible way is to find model field theories, simple enough to solve, but nontrivial enough to make sense. It is Casher, Kogut and Susskind 1l who for the first time made use of the Schwinger model,) quantum electrodynamics with a massless spinor field in two dimensional space-time, as a model of quark confine­ ment. It is well known that the model is exactly solvable and the confinement of the spinor field occurs_2l,sl Nakanishi 4l gave an operator solution of the Schwinger model in terms of asymptotic bose fields alone to describe the confinement of the spinor field IJ! rigorously, while in all previous works operator solutions involved a massless free spinor field ¢. It is noteworthy that the solution of Nakanishi possesses the following unusual properties besides: Wightman functions are non­ vanishing regardless of the numbers of IJ!'s and IJ!*'s, gauge invariance of the first kind is spontaneously broken, chiral charge is not well-defined and the Lorentz transformation property of IJ! is not the standard one. Subsequently Nakanishi 5l found also that the solution of Lowenstein and Swieca3l to the Schwinger model can ,be reconstructed by multiplying his solution with spurion operators*) independent of asymptotic fields. In this case one can construct the so-called topological vacua *l Spurion operators were introduced for the first time by Lowenstein and Sweica in the an­ alysis of gauge-invariant algebra.