Single wall carbon nanotubes have become an important part of physics since their fabrication in 1991. Aside from their many applications in nanotechnology, they are a theorists dream representing an experimental realization of a one dimensional electron gas. We give a general introduction into the theory of low-energy correlation effects in such nanotubes, concentrating in particular on the case of the so called 'zig-zag' nanotube where there are three important features to consider in the undoped case: the long range Coulomb interaction, the Umklapp scattering and an explicit geometric dimerization in the interactions generating single particle band gaps. The ratio of the dimerization induced gap and the Mott gap induced by the Umklapp interactions is dependent on the radius of the nanotube and can drive the system through a quantum phase transition.