The large-N limit of quantum rotor models for d = 2 was examined in Chapter 11 and led to the phase diagram shown in Fig. 11.2. There we claimed that the large-N results provided a satisfactory description of the crossovers in the static and thermodynamic observables for N ≥ 3. We establish this claim in this chapter and also treat the dynamic correlations of n at nonzero temperatures. The discussion of the dynamics takes place in a physical framework suggested by the modified version of Fig. 11.2 shown in Fig. 13.1. The low-T region on the quantum paramagnetic side can be described in an effective model of quasi-classical particles that is closely related to those developed in Sections 10.4.2 and 12.2. In the other low-T region on the magnetically ordered side, we obtain a “dual” model of quasi-classical waves, which is connected to that developed in Section 12.3. Finally, in the intermediate “quantum critical” or “continuum high-T” region, neither of these descriptions is adequate: quantum and thermal behavior, as well as particle- and wavelike behavior, all play important roles, and we use a melange of these concepts to obtain a complete picture in this and the following two chapters.