general this is not a good strategy since it amounts to integrating out soft excitations, which means that any order parameter theory will in general not be well behaved. Physically, these soft quasi-particle excitations can change the nature of the phase transition,[@Belitz_Kirkpatrick_Vojta_2005] or they themselves can become critical.[@Sachdev_2010] In either case they must be treated on equal footing with the order parameter fluctuations. It therefore is technically advantageous, and physically more transparent, to work with the coupled field theory represented by Eq. (\[eq:3.20\]).

### Comments on the magnetization dynamics {#subsubsec:III.C.2}

We conclude this section with a brief discussion of the dynamics of the dipole-dipole interaction, which we neglected in Sec. \[subsubsec:III.B.2\]. If we restore the frequency dependence of the photon propagator and expand in powers of the frequency, then the leading dynamical contribution of the dipole-dipole interaction to Eq. (\[eq:3.20\]) takes the form S\_\^ = \^2 \_k (\_[ij]{} - [k]{}\_i[k]{}\_j) M\_i(k) M\_j(-k). \[eq:3.21\] As long as $\Omega_n$ scales as $\vert{\bm k}\vert$, this scales the same as the $\vert\Omega_n\vert/\vert{\bm k}\vert$ term in the order-parameter theory that is induced by Fermi-liquid effects,[@Hertz_1976] but has a prefactor that is smaller by a factor of $(\vF/c)^2$. However, in classical dipolar magnets the order parameter 