Integrability of the classical$[\bar \psi _i \psi _i ]_2^2$ and$[\bar \psi _i \psi _i ]_2^2 - [\bar \psi _i \gamma _5 \psi _i ]_2^2$ interactions

Abstract

We study the interaction ofN classical two-dimensional massless Fermi fields through the symmetric couplings$[\bar \psi _i \psi _i ]^2$ or$[\bar \psi _i \psi _i ]^2 - [\bar \psi _i \gamma _5 \psi _i ]^2$. We explicitly show complete integrability in the casesN=1, 2, using the inverse scattering method. The fields occuring in the associated linear eigenvalue problem and evolution equation are simply related to the fundamental fields Ψ _i that satisfy the original non-linear equations. ForN>2, calculations become very involved, but there is no doubt that the system remains completely integrable, reducing to appropriate generalizations of the sine- and sinh-Gordon equation, a situation analogous to Pohlmeyer's discussion in a somewhat similar problem: the two-dimensional non-linear σ-model. Finally, all the explicit analytic solutions that we have worked out in the present framework are identical to those found by Dashen et al., and Shei, in a semiclassical treatment of the fully quantum mechanical version of these models. This leads us to conjecture that the quantum theory also shares most of the features of completely integrable systems, like the massive Thirring model.