In a BSM related paper (in appendix B), the authors use an effective Lagrangian $\mathcal{L}_{EFT}$, and define the following fields:
$$ \mathbf{D} = \frac{\partial\mathcal{L}_{EFT}}{\partial\mathbf{E}}, \quad \mathbf{H} = -\frac{\partial\mathcal{L}_{EFT}}{\partial\mathbf{B}}\tag{B.3}$$
Later, it seems they state that these are equivalent to the displacement fields known also as:
$$ \mathbf{D}= \varepsilon \mathbf{E}, \quad \mathbf{H} = \mathbf{B}/\mu$$
I also heard someone say that these $\mathbf{D}$ and $\mathbf{H}$ are the generalized momentums of the coordinates $\mathbf{E}$ and $\mathbf{B}$. I understand this statement but I don't understand how is that related to the electric displacement field, for which they derive eventually the refractive indices from solving $\varepsilon$.
