
(8.1) 
is not too large; the perturbation theory has to be combined with a renormalization group analysis in this case.
The physics described by this perturbative method can, in most cases, also be elucidated by the large N expansion we have developed in the previous chapters. However, there are a number of instances where the underlying principles are most transparently illustrated by studies close to and above d = 3. Our specific reasons for undertaking such an analysis are:
We have not yet found a successful description of the low frequency
dynamics of the order parameter (n or s^{z})
in the high T regime in d = 2, although we did succeed in
d = 1 in Chapters 4 and 6.
We show that the eexpansion leads to an appealing
quasiclassical wave description of this dynamics.
For the quantum rotor models being studied here, the crossovers above the uppercritical dimension, with d > 3, are obviously in a physically inaccessible dimension. However the basic structure that will emerge is quite generic to quantum critical points above their uppercritical dimension. The results will therefore be useful in Part 3 where we will consider other models with a lower uppercritical dimension, so that dimensions above the uppercritical can be experimentally studied.
The following chapter studies transport properties of the quantum rotor models in the high T and quantumparamagnetic low T regions in d = 2. While it is possible to do this within the 1/N expansion, the computation is simplest, and most physically transparent, using the e expansion we shall develop here.