is known to no longer be conserved.[@Frey_Schwabl_1994] That is, $\Omega$ scales as a constant, and this should be reflected in the quantum theory as well, although it is currently not known how this is realized. This suggests that the contribution shown in Eq. (\[eq:3.21\]) dominates the Fermi-liquid-induced dynamics in a scaling sense, although it has a small prefactor, and will become important at sufficiently long time scales.

Discussion, and Conclusion {#sec:IV}
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We now discuss the significance of various interactions for a number of problems.

Energy scales, and the significance of the dipole-dipole interaction {#subsec:IV.A}
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As we mentioned in Sec. \[subsec:III.A\], the energy scale for the exchange interaction in the bare theory is the atomic scale, or roughly $100,000\,{\text{K}}$. The corresponding length scale is on the order of $1\,\AA$. This is not consistent with the experimental fact that magnetic ordering is observed only at much lower temperatures; e.g., on the order of $1,000\,{\text{K}}$ in Fe and Ni, on the order of room temperature in FeGe, and on the order of $30\,{\text{K}}$ in MnSi. The reason for this discrepancy lies in the fact that the bare theory is renormalized in quantitatively substantial ways, and the corresponding energy and length scales 