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Reading the "Physical interpretation and alternate form" section of the Wikipedia article on the Kramers-Kronig relations, it says:

The imaginary part of a response function describes how a system dissipates energy, since it is out of phase with the driving force. The Kramers–Kronig relations imply that observing the dissipative response of a system is sufficient to determine its in-phase (reactive) response, and vice versa.

But I thought that the real part (in phase) was dissipation, whereas the imaginary is just polarization or something. For example, with the complex conductivity, $j=\sigma E$. If $\sigma$ is purely imaginary, then it is like a capacitor: it doesn't actually take energy, right? You only get Joule Heating if $\sigma$ is real, I thought.

Is the article incorrect or am I not understanding something?

Thanks!

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    $\begingroup$ I think you are mistaking the response function of a general linear system (it doesn't have to be electromagnetic in order to apply Kramers-Kronig) for a material constant in Maxwell's equations. The two are, of course, related, but there is no one to one correspondence between the material constant and the response. For one thing, $\sigma$, in general, depends on coordinates for inhomogeneous materials, and, yet, the response function will not show that dependence but instead be a complex frequency dependent integral over $\sigma$. $\endgroup$
    – CuriousOne
    Oct 15, 2014 at 20:46

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The wikipedia article is correct. The relation between the actual response function and the conductivity is not immediately obvious, however. Let us consider the case of the longitudinal conductivity for example. The susceptibility, $\chi^L(\textbf{q},\omega)$, which is the true response function, is related to the conductivity using the following equations:

\begin{equation} \epsilon^L(\textbf{q},\omega)=\frac{1}{1+V(\textbf{q})\chi^L(\textbf{q},\omega)} \end{equation} where $V(\textbf{q})$ is the Fourier transform of the Coulomb interaction. Also, the dielectric function is related to the conductivity in the following way:

\begin{equation} \epsilon^L(\textbf{q},\omega)= 1+\frac{4\pi i \sigma^L}{\omega} \end{equation}

You can see that there is a difference in the factor of $i$ between the first equation and the second. Therefore, it turns out that because of the above definitions, that the imaginary part of the response function (i.e. the susceptibility) measures the dissipation, but is in fact related the the real part of the conductivity.

Here is a good link to clear up some of these concepts (careful, it's a PDF): Longitudinal and Transverse Response of the Electron Gas in the Random Phase Approximation

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