8.14: Appendix V- Kramers-Krönig Relations

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\)
\( \newcommand\VTheta{\vec\Theta}\)
\( \newcommand\VLambda{\vec\Lambda}\)
\( \newcommand\VXi{\vec\Xi}\)
\( \newcommand\VPi{\vec\Pi}\)
\( \newcommand\VSigma{\vec\Sigma}\)
\( \newcommand\VUps{\vec\Upsilon}\)
\( \newcommand\VPhi{\vec\Phi}\)
\( \newcommand\VPsi{\vec\Psi}\)
\( \newcommand\VOmega{\vec\Omega}\)
\( \newcommand\BA{\mib A}\)
\( \newcommand\BB{\mib B}\)
\( \newcommand\BC{\mib C}\)
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\( \newcommand\BBA{\boldsymbol\RA}\)
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\(\newcommand \gtwid{\approx}\)

\( \newcommand\index{\textsf{ind}}\)
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\( \newcommand\bsqcap{\mbox{\boldmath{$\sqcap$}}}\)

\( \def\pabc#1#2#3{\left({\pz #1\over\pz #2}\right)\ns_{\!\!#3}}\)
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\( \newcommand\longrightleftharpoons{ \mathop{\vcenter{\hbox{\ooalign{\raise1pt\hbox{$\longrightharpoonup\joinrel$}\crcr  \lower1pt\hbox{$\longleftharpoondown\joinrel$}}}}}}\)
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\( \newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}\)
\( \newcommand\xhiOZ{\xhi^\ssr{OZ}}\)
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Suppose \({\hat\chi}(\omega)\equiv {\hat G}(\omega)\) is analytic in the UHP¹⁹. Then for all \(\nu\), we must have \[\impi{d\nu\over 2\pi}\,{{\hat\chi}(\nu)\over\nu-\omega+i\eps}=0\ ,\label{kka}\] where \(\eps\) is a positive infinitesimal. The reason is simple: just close the contour in the UHP, assuming \({\hat\chi}(\omega)\) vanishes sufficiently rapidly that Jordan’s lemma can be applied. Clearly this is an extremely weak restriction on \({\hat\chi}(\omega)\), given the fact that the denominator already causes the integrand to vanish as \(|\omega|^{-1}\).

Let us examine the function \[{1\over \nu-\omega+i\eps}={\nu-\omega\over (\nu-\omega)^2+\eps^2}-\, {i\eps\over (\nu-\omega)^2+\eps^2}\ .\] which we have separated into real and imaginary parts. Under an integral sign, the first term, in the limit \(\eps\to 0\), is equivalent to taking a principal part of the integral. That is, for any function \(F(\nu)\) which is regular at \(\nu=\omega\), \[\lim_{\eps\to 0}\impi {d\nu\over 2\pi}\,{\nu-\omega\over (\nu-\omega)^2+\eps^2} \,F(\nu)\equiv\wp\!\! \impi {d\nu\over 2\pi}\,{F(\nu)\over\nu-\omega}\ .\] The principal part symbol \(\wp\) means that the singularity at \(\nu=\omega\) is elided, either by smoothing out the function \(1/(\nu-\eps)\) as above, or by simply cutting out a region of integration of width \(\eps\) on either side of \(\nu=\omega\).

The imaginary part is more interesting. Let us write \[h(u)\equiv {\eps\over u^2+\eps^2}\ .\] For \(|u|\gg\eps\), \(h(u)\simeq \eps/u^2\), which vanishes as \(\eps\to 0\). For \(u=0\), \(h(0)=1/\eps\) which diverges as \(\eps\to 0\). Thus, \(h(u)\) has a huge peak at \(u=0\) and rapidly decays to \(0\) as one moves off the peak in either direction a distance greater that \(\eps\). Finally, note that \[\impi du\,h(u)=\pi\ ,\] a result which itself is easy to show using contour integration. Putting it all together, this tells us that \[\lim_{\eps\to 0} {\eps\over u^2+\eps^2}=\pi\delta(u)\ .\] Thus, for positive infinitesimal \(\eps\), \[{1\over u\pm i\eps}={\wp\over u} \mp i\pi\delta(u)\ ,\] a most useful result.

We now return to our initial result [kka], and we separate \({\hat\chi}(\omega)\) into real and imaginary parts: \[{\hat\chi}(\omega)={\hat\chi}'(\omega)+i{\hat\chi}''(\omega) \ .\] (In this equation, the primes do not indicate differentiation with respect to argument.) We therefore have, for every real value of \(\omega\), \[0=\impi{d\nu\over 2\pi}\,\Big[\chi'(\nu)+i\chi''(\nu)\Big]\, \Big[{\wp\over \nu-\omega}-i\pi\delta(\nu-\omega)\Big]\ .\] Taking the real and imaginary parts of this equation, we derive the Kramers-Krönig relations: \[\begin{aligned} \chi'(\omega)&=&+\wp\!\!\impi{d\nu\over\pi}\,{{\hat\chi}''(\nu)\over\nu-\omega}\\ \chi''(\omega)&=&-\wp\!\!\impi{d\nu\over\pi}\,{{\hat\chi}'(\nu)\over\nu-\omega}\ .\end{aligned}\]