Question

Why is the susceptibility $\chi(t)$ real?

Answer 1

Everything classical is real in the time domain. However, you'll notice that response functions involve convolutions with the "input" field. This makes them non-local, which is the only way to achieve causality. In the Fourier domain this is equivalent to having a complex part. In fact, both the real and complex part MUST be nonzero or else you violate causality. That's KK. Many textbooks fake you out by writing the Maxwell equations in the time domain and then showing the constitute relations (e.g., D=ϵED=ϵE) in the frequency domain to avoid the convolution. This causes endless confusion. See: Linear response laws and causality in electrodynamics AJ Yuffa, JA Scales European Journal of Physics 33 (6), 1635. Promoting E to the Fourier domain is a mathematical convenience. If you do linear calculations, then everything is fine.

Answer 2

Hey mate here is your answer in attachment