Algebraic bosonization: The study of the Heisenberg and Calogero-Sutherland models

We propose an approach to treating (1 + 1)-dimensional fermionic systems based on the idea of algebraic bosonization. This amounts to decomposing the elementary low-lying excitations around the Fermi surface in terms of basic building blocks carrying a representation of the W1+infinity x (W) over bar(1+infinity) algebra, which is the dynamical symmetry of the Fermi quantum incompressible fluid. This symmetry simply expresses the local particle number current conservation at the Fermi surface. The general approach is illustrated in detail through two examples: the Keisenberg and Calogero-Sutherland models, which allow comparison with the exact Bethe ansatz solution.