1/N Expansion for Gross-Neveu Models

Abstract

This chapter considers a class of quantum field theoretical models — the Gross- Neveu models — following [196]. These are models of an TV-component fermion field in two-dimensional space-time with (ΨΨ)² interaction; a number of exact solutions can be found for equations of motion in these models. Following Berezin [94], we show that the parameter 1/N is analogous to Planck’s constant, so obtaining some classical Hamiltonian systems as N→∞. The phase space is, however, nonlinear and the Poisson brackets are not canonical. The symmetry group for these models is SO (2n). It is remarkable that the constructions in this chapter are mainly algebraic and therefore the results obtained slightly depend on particular features of the models.