- m at . o th er ] 2 7 Ju n 20 06 Weakly Interacting , Dilute Bose Gases in

This article surveys a number of theoretical problems and open questions in the field of two-dimensional dilute Bose gases with weak repulsive interactions. In contrast to three dimensions, in two dimensions the formation of long-range order is prohibited by the Bogoliubov-Hohenberg theorem, and Bose-Einstein condensation is not expected to be realized. Nevertheless, the first experimental indications supporting the formation of a condensate in low dimensional systems have been recently obtained. This unexpected behaviour appears to be due to the non-uniformity introduced into a system by the external trapping potential. Theoretical predictions, made for homogeneous systems, require therefore careful reexamination. We survey a number of popular theoretical treatments of the dilute weakly interacting Bose gas and discuss their regions of applicability. The possibility of Bose-Einstein condensation in a two-dimensional gas, the validity of the perturbative t-matrix approximation and the diluteness condition are issues that we discuss in detail.