Chapter 13 Bose Condensation: Reformulation

This problem will be studied at an heuristic level, i.e., in a way similar to the theory of the critical point in d = 4 ? " dimensions (see chapters 9 and 10). One should not forget that the latter problem, considered by many one of the major successes of the renormalization group approach, is still awaiting a rigorous formulation, not only for " = 1, but also for any " > 0. Therefore, we think that a formal theory is nevertheless an interesting way to attack the Bose condensation problem in d = 3 dimensions. This will be, in fact, the best illustration of the methods we try to illustrate about the renormalization group, as it embodies all the ideas discussed in the various cases met so far. The analysis that follows is due to B], and we follow his treatment with some minor changes. What follows could also be regarded as a test of the usual claim that the zero temperature Bose gas has a linear dispersion relation for small momenta. This property is considered typical of superruid behavior and was rst veriied by Bogoliubov ((Bo]) in an approximate exactly soluble model, the so-called Bogoliubov model. After that, the superruid behavior hypothesis received strong support from rough perturbative arguments (see, for example, ADG], where it is possible to nd relevant references). More recently, more convincing arguments were presented in the papers of NN] and PS]. The problem is formulated as a functional integration in chapter 3. The functional integral is