Exact Solution of a Many-Fermion System and Its Associated Boson Field

Luttinger's exactly soluble model of a one-dimensional many-fermion system is discussed. We show that he did not solve his model properly because of the paradoxical fact that the density operator commutators [ρ(p), ρ(–p′)], which always vanish for any finite number of particles, no longer vanish in the field-theoretic limit of a filled Dirac sea. In fact the operators ρ(p) define a boson field which is ipso facto associated with the Fermi–Dirac field. We then use this observation to solve the model, and obtain the exact (and now nontrivial) spectrum, free energy, and dielectric constant. This we also extend to more realistic interactions in an Appendix. We calculate the Fermi surface parameter ñ_k, and find: ∂ñ_k/∂k!_kr = ∞ (i.e., there exists a sharp Fermi surface) only in the case of a sufficiently weak interaction.