Microscopic theory of Bose-Einstein condensation in an interacting gas

We find, for the first time, a microscopic theory of Bose-Einstein condensation in an interacting gas, which is valid both inside and outside a critical region. The derived exact fundamental equations for the condensate wave function and the Green's functions allow one to describe critical fluctuations and formation of an ordered condensate phase from a disordered phase across the entire critical region continuously. These equations asymptotically turn into the usual Gross-Pitaevskii and Beliaev-Popov equations in a low-temperature limit outside the critical region. The theory is readily extendable to other phase transitions.