= 0$. That is, the Eqs. (\[eq:2.2b\]), (\[eq:2.2c\]), and (\[eq:2.3\]) are replaced by[@monopoles_footnote] S\_ = \_V d[x]{} [M]{}([x]{})([x]{}), \[eq:2.2b’\] S\_A = \_V d[x]{} \_[0]{}, \[eq:2.2c’\] B\_i([k]{}) B\_j(-[k]{})= 4T(\_[ij]{} - [k]{}\_i [k]{}\_j). \[eq:2.3’\] Either way we find \[eqs:2.4\] &=& \_V d[x]{} + 2\^2 \_[k]{} d\_[ij]{}([k]{})M\_i([k]{})M\_j(-[k]{}), \[eq:2.4a\] with d\_[ij]{}([k]{}) = [k]{}\_i [k]{}\_j . \[eq:2.4b\] The terms generated by integrating out the vector potential we recognize as the leading contribution to the dipole-dipole interaction[@Aharony_Fisher_1973; @Ma_1976] plus a shift of the Landau parameter $t$ by $4\pi\muB^2$. The scalar potential $\varphi({\bm x})$, whose gradient is the electric field, does not lead to any magnetic interactions in a classical theory. This changes once the system is treated quantum mechanically, see Sec. \[sec:III\] below.

Renormalization, and higher order terms {#subsec:II.B}
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The dipole-dipole operator $d_{ij}$, Eq. (\[eq:2.4b\]), transforms as a rank-two tensor in momentum (or real) space, and ${\bm M}$ transforms as a vector in spin space. Consequently, the dipole-dipole interaction is invariant under co-rotations in real space and spin space. This raises the question of other terms in the action that have the same symmetry properties. For instance, $({\bm\nabla}\cdot{\bm M})^2$ is allowed by symmetry. This term, and terms of higher order in the gradient, are generated 