Real is imaginary and vice versa

This week in my optics class, I have been teaching Kramers-Kronig (KK) relations of electric susceptibility. It is fascinating to see the causality argument emerge from the relationship between the real and imaginary parts of the complex susceptibility. Whereas the time domain explanation is relatively easier to appreciate (that dissipation follows perturbation in time), for me, the frequency domain implication in KK relation is fascinating: the fact that information about the real part of the function at all frequencies can give you insight into the imaginary part at any given frequency (and vice versa) makes it such a powerful mathematical and physical tool. For example, by knowing the absorption spectrum of a medium, you can find out the refractive index of a medium at a particular frequency that is not easily accessible in experiments.

Two inferences I draw:

1) Complex analysis combined with differential calculus is one of the most beautiful and powerful mathematical tools invented, and exploring its application in experimental scenarios has made physics intriguing, useful, and profound.

2) The KK relationship shows how causality and the structure of matter are connected to each other, and by studying them, one will be able to extrapolate the idea beyond the problem at hand and apply it to a different context in physics. It just shows how ideas hop from one domain to another and how mathematics plays a critical role in intellectual arbitrage.

Real is imaginary and vice versa. Complex numbers zindabad!