Heuristic field theory of Bose-Einstein condensates

This paper reviews the basic ideas of the field theoretic approach to Bose-Einstein condensation. The central concepts are presented in a historical way, the most important results are given and they are justified by heuristic physical ideas instead of detailed derivations. The mathematical level of rigour is low, and the intuitive relations between the concepts and their physical origin is used to justify the various results. I present the original Bose-Einstein conception of bosons condensing into a macroscopically occupied state. The special features deriving from this situation are discussed, and the concept of tong-range order is introduced. Historically the concept of broken symmetry has played a central role in the description of Bose condensates. In this approach the system is described by states with broken particle conservation, which justifies the introduction of a physical phase factor. When interactions are present, this acquires its own dynamic behaviour, which gives rise to the low-lying collective excitations of the system. They can be interpreted as collisionless oscillations of the particle density. The condensate acting as a particle reservoir introduces a coupling between these collective excitations and the single-particle ones, which makes the collective excitations dominate the physics of the system. This justifies the hydrodynamic view of a condensed Bose system, which has been central to both the theoretical approaches and the interpretations of experiments on superfluid helium. The results are derived using both algebraic and perturbative methods; some of the tools from formal many-body theory are summarized in the appendix. The treatment approaches most issues from a bulk material point of view, but I also keep the possibility open to apply the results to the topical field of trapped condensates. The equation for the order parameter, the Gross-Pitaevski equation, is presented and discussed, but its detailed applications to trapped alkali atoms is not covered by the present review. Here basic ideas, their logical connections and their physical consequences are presented. For more detailed treatments of the formal results the reader has to consult the original papers or the monographs referred to. No attempt is made to cover the most recent experimental and theoretical results for alkali atoms condensed in traps. (C) 2002 Elsevier Science B.V. All rights reserved.