Analyticity, Causality, and Kramers-Kronig Relations
We want our response functions to be causal. In time domain this means that events cannot precede causes. But what does it mean in the frequency domain?
Answer: Our complex, frequency domain function must be analytic in the upper half-plane. Fancy way of saying it must have no poles in the upper half.
This then connnects to the Kramers-Kronig relations.
In linear response theory, the response function (for instance the susceptibility tensor \chi(\omega) that we use below for concreteness) encodes how a system responds to an external perturbation.
Its analytic properties in the complex frequency plane are intimately connected to causality (response cannot precede the perturbation) and to irreversibility (systems relax to equilibrium).
Why Upper Half Analyticity Implies Causality
Schematically, the susceptibility has the form
\chi(\omega) \sim \sum_n \frac{f_n}{\omega - \omega_n + i0^+},
where the poles \omega_n lie in the lower half-plane.
If all poles are in the lower half-plane, then \chi(\omega) is analytic in the upper half-plane, which is another way of saying it has no singularities in that region, so contour deformations in the upper half-plane are always allowed.
The time-domain susceptibility is defined as
\chi(t) = 0 \quad \text{for } t < 0,
reflecting the fact that the system cannot respond before the perturbation is applied.
The two are connected by a Fourier transform: \chi(\omega) = \int_{-\infty}^{\infty} \chi(t) e^{i\omega t} \, dt = \int_{0}^{\infty} \chi(t) e^{i\omega t} \, dt.
Given \chi(t) \xrightarrow{\mathscr{F}} \chi(\omega), we let \omega = \omega_r + i\omega_i (see pills):
For \omega_i > 0 (upper half-plane), the factor e^{i\omega t} = e^{i\omega_r t} e^{-\omega_i t} decays as t \to \infty.
→ The integral converges and \chi(\omega) is analytic there.For \omega_i < 0 (lower half-plane), the factor grows exponentially, and singularities (poles) can occur.
So technically, the response would pick up unphysical terms if there are poles in the upper-half plane.
Thus, causality in time \chi(t<0)=0 is equivalent to analyticity of \chi(\omega) in the upper half-plane.
Alternatively, the inverse transform is
\chi(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \chi(\omega) e^{-i\omega t} \, d\omega.
Since we have poles in the lower half:
If t < 0: close the contour in the upper half-plane. Since \chi(\omega) is analytic there, the integral vanishes \Rightarrow \chi(t<0)=0.
If t > 0: close the contour in the lower half-plane. Now the poles contribute, yielding a nonzero response.
That is exactly how we want it. No response for t<0.
This demonstrates explicitly that analyticity \leftrightarrow causality.
The contrasting behavior of lower and upper half-plane poles is contrasted below:
| Lower Half-Plane Poles | Upper Half-Plane Poles |
|---|---|
| e^{-i(\omega_r - i\gamma)t} = e^{-i\omega_r t} e^{-\gamma t}, \qquad \gamma > 0 | e^{+\gamma t}, \quad t > 0 |
| Exponentially decaying response | Exponentially growing response |
| Encodes irreversible relaxation to equilibrium | Leads to instability and growth without bound |
| Response exists only for t>0 → causal | Produces contributions for t<0 → acausal |
| ✅ Physical | ❌ Unphysical |
So the following statements are equivalent:
- Causality (time domain): \chi(t)=0 for t<0.
- Analyticity (frequency domain): \chi(\omega) analytic in the upper half-plane.
- Pole structure: all poles confined to the lower half-plane.
This guarantees that the system responds only after being perturbed and decays irreversibly toward equilibrium.
These analytic properties also underlie the Kramers–Kronig relations, which link the real and imaginary parts of \chi(\omega) via Hilbert transforms.
Kramers-Kronig Relations
Now that we know that our system must be analytic in the upper half-plane, what can we say about it?
It turns out that to make the time domain response function 0 before the event binds the real and imaginary components in a peculiar way: Knowing one is enough to know the other.
Kramers-Kronig Relations is an excellent overview. To quote: If a susceptibility is calculated theoretically, it is a good idea to check and see if it satisfies the Kramers-Kronig relations. It is considered a serious error to present a result that violates causality.