The Luttinger–Schwinger Model

Abstract

We study the Luttinger–Schwinger model, i.e. the (1+1) dimensional model of massless Dirac fermions with a non-local 4-point interaction coupled to a U(1)-gauge field. We work within the Hamiltonian framework on the cylinder, and construct the field operators and observables as well-defined operators on the physical Hilbert space. The complete solution of the model is found using the boson-fermion correspondence, and the formalism for calculating all gauge invariant Green functions is provided. We discuss the role of anomalies and show how the existence of large gauge transformations implies a fermion condensate in all physical states. The meaning of regularization and renormalization in our well-defined Hilbert space setting is discussed. We illustrate the latter by performing the limit to the Thirring–Schwinger model where the interaction becomes local.