Physical explanation of Kronig-Kramers application

[singhvi]

I recently learned that Kramers-Kronig relations can be applied to physical response functions, but I find it strange that for eg. the refractive index and absorption of a material cannot be independently defined, what is the explanation behind this from the point of view of the physics involved and not by a mathematical derivation.

 

[vanhees71]

This is a very general property of response functions and can be traced back to the causality principle.

Let's consider the most simple case of a one dimensional motion of a particle in an arbitrary time-independent potential, and now you apply a force of given time dependence on it. It is clear that its motion (for any given initial condition) will be given by a retarded Green's function

x(t)=\int_{-\infty}^{\infty} \text{d} t' G(t-t') F(t'),

where F(t') is the given time-dependent force. For causality reasons, the Green's function must fulfill the retardation condition,

G(t-t')=0 \quad \text{for} \quad t'>t.

Now the above integral is a convolution integral, and it's often easier to analyse the problem in Fourier space, i.e., you define the Fourier transforms of x, F, and G via

x(t)=\int_{-\infty}^{\infty} \frac{\text{d} \omega}{2 \pi} \tilde{x}(\omega) \exp(-\text{i} \omega t), \quad F(t)=\int_{-\infty}^{\infty} \frac{\text{d} \omega}{2 \pi} \tilde{F}(\omega) \exp(-\text{i} \omega t), \quad G(t)=\int_{-\infty}^{\infty} \frac{\text{d} \omega}{2 \pi} \tilde{G}(\omega) \exp(-\text{i} \omega t).

Then you have (up to some power of 2 pi)

\tilde{x}(\omega) \propto \tilde{G}(\omega) \tilde{F}(\omega).

To come to the Kramer's Kronig relation, now we translate the retardation condition into frequency space: Usually the response function goes with some negative power in omega for large omega. Then you can evaluate Fourier integral with help of the residuum theorem of function theory by closing the integration contour along the real axis by a large semi circle. For t>0, you have to close the contour in the lower, for t<0 in the upper omega-half plane. Since for t<0, G(t)=0, there must be no poles or cuts in the upper half plane, i.e., G~(omega) is an analytic function in the upper omega-half plane.

Now let \omega_0 \in \mathbb{R} and define C as the contour in omega plane which runs along the real axis from -\infty to \omega_0-\epsilon and from \omega_0+\epsilon to \infty. The gap is closed by a small semi-circle in the upper half-plane, and the whole contour is again closed by a large semi-circle in the upper half-plane. Then you have due to the Cauchy integral theorem

\int_C \text{d} \omega \frac{\tilde{G}(\omega)}{\omega-\omega_0}=0

since G~ is analytic in the upper half plane, so that there are no singularities of the integrand enclosed by the contour. On the other hand, if you let \epsilon \rightarrow 0^+, you can write the contour integral as

\text{PP} \int_{-\infty}^{\infty} \text{d} \omega \frac{\tilde{G}(\omega)}{\omega-\omega_0} -\text{i} \pi \tilde{G}(\omega_0)=0

or

\text{i} \pi \tilde{G}(\omega)=\text{PP} \int_{-\infty}^{\infty} \text{d} \omega&#039; \frac{\tilde{G}(\omega&#039;)}{\omega&#039;-\omega}.

Now taking the real part of this equation leads to

-\pi \text{Im} \tilde{G}(\omega)=\mathrm{PP} \int_{-\infty}^{\infty} \text{d} \omega&#039; \frac{\text{Re} \tilde{G}(\omega&#039;)}{\omega&#039;-\omega}.

Of course, you can also take the imaginary part of the original equation. In any case you express the imaginary (real) part of the response function in frequency space as a principle-value integral of its real (imaginary) part. This is known as the general Kramers-Kronig relations.

The same argument can of course be applied in more general cases like in field theory, where you apply the same technique to retarded Green's functions, which additionally also depend on the spatial coordinates, but the Kramers-Kronig relations only depend on the retardation condition (physically speaking on causality) and refer to the time-frequency Fourier transform.

In electromagnetism (optics), an important application is the response of a dielectric medium to a (not too strong) electromagnetic field. The response function is closely related to the dielectric function in frequency space, which in general is complex and describes the refraction and absorption of em. waves in the medium. As you point out the Kramers-Kronig relation proves a close relation between the refraction index and the absorption coefficient (as a function of frequency). As we have seen above, this follows from the very basic causality principle and the corresponding etardation condition for the response (=Green's) functions.

The same theory also applies in the context of thermodynamical fluctuations. There the Kramer's-Kronig relations lead to the fluctuation-dissipation theorem, that relates fluctuations of physical quantities with the dissipative effects of the medium. One of the most famous expamples is the Einstein relation between diffusion and friction coefficient in the Fokker-Planck equation of Brownian motion.

This whole theory is also very important in quantum field theory, where it is related to the unitarity of the S-matrix and the optical theorem for cross sections. I guess, one could find a whole plethora of relations connected with this principle.

Last but not least one should stress the fact that Kramers and Kronig were not the first who applied this method. The earliest application I know of is by Sommerfeld, when he clarified that anomalous dispersion (where the phase velocity of light in the medium becomes larger than the velocity of light in vacuo!) does not contradict the Einstein-causality constraints of special relativity. As he's proven in 1907 within classical dispersion theory, from causality arguments alone, you can conclude that the head of an em. wave travels exactly with the vacuum-speed of light and not faster through the medium even in frequency regions of anomalous dimension. Later on he and Brillouin worked out a whole theory of wave propagation in media based on the analytic properties of the dielectric function (Ann. Phys. (Leipzig) (1913)).

 

[singhvi]

Thanks a lot :D