Fermi surface. The restriction to small wave numbers will be understood from now on, and we will drop the prime on integrals.

$S_{\text{ex}}$ is the exchange interaction between electronic spin-density fluctuations that leads ferromagnetism. For later reference we note that $\Gamma_{\text{t}}$ is dimensionally an energy times a volume which, in this unrenormalized theory, is on the order of a Rydberg times a Fermi volume, or $\Gamma_{\text{t}} \approx e^2/\kF^2$.

### $O(1/c^2)$: Dipole-dipole interaction {#subsubsec:III.B.2}

We now turn to terms of $O(1/c^2)$. We first consider the vector potential ${\bm A}$, which couples to the spin density via the ${\bm B}\cdot{\bm
n}_{\text{s}}$ term in Eq. (\[eq:3.1\]). Since the coupling is directly to the spin channel, no phase space decomposition is necessary and integrating out ${\bm A}$ proceeds as in the classical case, except that the ${\bm
A}$-propagator now is frequency dependent, see Eqs. (\[eqs:3.2\]). We will comment on the consequences of this frequency dependence in Sec. \[subsubsec:III.C.2\] below. Neglecting the dynamical aspects of the dipole-dipole interaction for now, we obtain a contribution to the effective action S\_ &=& 2\^2 dx [n]{}\_(x)\_(x)\
&&0 pt + dxdy (\_x - \_y) n\_\^i(x)d\_[ij]{}([x]{}-[y]{}) n\_\^j(y),\
\[eq:3.5\] with $d_{ij}$ from Eq. (\[eq:2.7b\]). The first term has the same form 