Dispersion Relations and Kramers-Kronig

A physical system driven by an external field responds — but if the response cannot precede its cause, the response is causal, and that single word carries astonishing mathematical weight. "Output at time $t$ depends only on input at times $t'\le t$ " seems like a modest bookkeeping rule. Encoded as the vanishing of a response kernel for negative times, and pushed into the frequency domain, it becomes a statement not about time but about analyticity in a complex frequency variable. The bridge is the theory of holomorphic functions built in the preceding chapter: a function analytic in a half-plane and decaying there obeys Cauchy's integral formula on a contour that closes at infinity, and splitting the resulting principal-value integral into real and imaginary parts produces two coupled integral equations — the Kramers-Kronig relations. Their content is startling. The real, reactive part of a response (dispersion) and the imaginary, dissipative part (absorption) are not independent quantities you can dial separately; they are two projections of one analytic object. You cannot build a medium that bends light without it absorbing somewhere in the spectrum; you cannot put loss in a circuit without an accompanying reactance. The same structure — analyticity forced by causality — reappears in the retarded Green's functions of many-body physics, the forward scattering amplitudes of particle physics, and the analytic continuation of correlation functions. Master the linear-response case here and you hold the template for all of them; the tools are exactly the contour integration, the Cauchy formula, and the pole-counting of elementary complex analysis.

Key takeaways

A causal linear response $\chi(t)$ that vanishes for $t<0$ has a frequency-domain transform $\tilde\chi(\omega)$ that is analytic everywhere in the upper half of the complex $\omega$ -plane — all its poles sit in the lower half.
Analyticity plus decay lets Cauchy's theorem on an indented contour relate the real and imaginary parts of $\tilde\chi$ through the Hilbert transform: these are the Kramers-Kronig relations.
Concretely, $\mathrm{Re}\,\tilde\chi$ and $\mathrm{Im}\,\tilde\chi$ are locked together — knowing dispersion at all frequencies fixes absorption everywhere, and vice versa.
Reality of $\chi(t)$ forces $\tilde\chi(-\omega)=\tilde\chi^*(\omega)$ , folding the relations onto positive frequencies and yielding the f-sum rule $\int_0^\infty\omega\,\mathrm{Im}\,\tilde\chi\,\mathrm{d}\omega=\tfrac{\pi}{2}\omega_p^2$ .
When the response does not vanish at infinity (e.g. $\varepsilon(\infty)=1$ ), a subtracted dispersion relation restores convergence; the machinery generalises directly.
Applications span optics, AC impedance, forward scattering, and many-body Green's functions — everywhere a stable causal system has a frequency response.
Key idea: Causality (kernel vanishes for $t<0$ ) $\Longleftrightarrow$ analyticity of the frequency-domain response in the upper half $\omega$ -plane; Cauchy's formula on a contour closed in that half-plane then ties $\mathrm{Re}\,\tilde\chi$ and $\mathrm{Im}\,\tilde\chi$ together as Hilbert transforms.

The core insight

Causality (the response kernel vanishes for $t<0$ ) is equivalent to analyticity of the frequency-domain response $\tilde\chi(\omega)$ in the upper half $\omega$ -plane; Cauchy's integral formula on a contour closed in that half-plane then ties $\mathrm{Re}\,\tilde\chi$ and $\mathrm{Im}\,\tilde\chi$ together as a Hilbert-transform pair.

Where this lives today

Dispersion relations are working tools, not museum pieces. In materials optics, reflectance measured across the widest accessible band is Kramers-Kronig-transformed to recover the full complex dielectric function $\varepsilon(\omega)$ , since the phase of the reflected wave cannot be measured directly — every optical-constants table for a solid is, at some stage, such a reconstruction. In electrochemistry the "Kramers-Kronig test" is the standard validity check on impedance data: a measured $Z(\omega)$ that fails the relations cannot come from a stable causal cell, flagging drift or nonlinearity before any model is fit. In particle physics, forward dispersion relations turn measured cross sections (via the optical theorem $\mathrm{Im}\,f(0,\omega)=\tfrac{\omega}{4\pi c}\sigma_{\rm tot}$ ) into predictions for the real amplitude, pinning down strong-interaction constants and hadronic contributions to the muon $g-2$ . In many-body theory the same analyticity underlies the spectral (Lehmann) representation of Green's functions and the analytic continuation of imaginary-time Monte Carlo data. And in metamaterials and slow-light research, Kramers-Kronig sets the trade-off every passive design must respect: how strongly a medium disperses a pulse against how much it must absorb.

The physical picture

Think of the response function as a movie of a system reacting to a sharp kick at $t=0$ . Causality says the movie is black before the kick — nothing happens until the cause arrives. The Fourier transform of a signal that lives only on the positive time axis is special: the step-function cutoff smears its spectrum so that the real and imaginary parts of the transform can no longer move independently, and geometrically that constraint becomes a statement about where the poles can live. A driving term $e^{-i\omega t}=e^{-i\omega_R t}\,e^{\omega_I t}$ with $\omega_I=\mathrm{Im}\,\omega>0$ grows exponentially into the future; a stable causal system cannot resonate with a perturbation that was arbitrarily small in the distant past, so $\tilde\chi(\omega)$ can have no singularity in the upper half-plane. Every pole — every resonance, every relaxation mode — is pushed down into $\mathrm{Im}\,\omega<0$ . The first figure is that map, together with the contour whose evaluation is the Kramers-Kronig relation.

Figure

The complex ω-plane. Causality makes χ̃(ω) analytic throughout the upper half (Im ω greater than 0), so every pole ✕ lies in the lower half; Cauchy's theorem on the real axis plus a large upper semicircle, indented over the evaluation point ω, ties Re χ̃ to Im χ̃

The imaginary part of $\tilde\chi$ controls energy dissipation, the real part the in-phase reactive response. Because analyticity welds them together, you cannot change dispersion across the whole spectrum without absorption appearing somewhere: a "lossless yet everywhere-dispersive" material is mathematically forbidden. An absorption peak in $\mathrm{Im}\,\tilde\chi$ induces, through the Hilbert transform, a characteristic dispersive swing in $\mathrm{Re}\,\tilde\chi$ — rising below the resonance, crossing zero at it, dipping above — the anomalous dispersion seen near every spectral line. The second figure is that pairing for a Lorentz oscillator, with real numbers attached.

Figure

A single Lorentz oscillator as a Kramers-Kronig pair with ωₚ = ω₀ and γ = 0.3 ω₀. The absorption Im χ̃ in amber peaks at the resonance ω = ω₀ with height ω₀/γ = 3.33; that peak forces the dispersion Re χ̃ in blue into the anomalous S-swing — rising below ω₀, crossing zero at ω₀, dipping above. The static value is Re χ̃(0) = ωₚ²/ω₀² = 1

The core equations

The starting point is a linear, causal, time-translation-invariant response: the output is a convolution of the input with a kernel, and causality is imposed as one hard condition — the kernel vanishes for negative argument, so no output precedes the input.

P(t)=\int_{-\infty}^{\infty}\chi(t-t')\,E(t')\,\mathrm{d}t',\qquad \chi(\tau)=0\ \text{for}\ \tau<0 \qquad (1)

where: $P(t)$ — the induced response (e.g. polarisation); $E(t')$ — the driving field at the earlier time $t'$ ; $\chi(\tau)$ — the response kernel (susceptibility), zero for $\tau<0$ . Reading: the response at $t$ is a weighted sum over the past only — exactly what "cause precedes effect" means. Fourier-transforming the kernel, the causal cutoff turns the lower limit into $0$ , and that is the whole game: for $\mathrm{Im}\,\omega>0$ the exponential decays and the integral converges and is differentiable, so $\tilde\chi$ is analytic in the open upper half-plane.

\tilde\chi(\omega)=\int_{0}^{\infty}\chi(\tau)\,e^{i\omega\tau}\,\mathrm{d}\tau,\qquad \tilde\chi\ \text{analytic for}\ \mathrm{Im}\,\omega>0 \qquad (2)

where: $\omega=\omega_R+i\omega_I$ — the complex frequency; $\tilde\chi(\omega)$ — the Fourier-Laplace transform of the kernel. Reading: upper-half-plane analyticity is not an assumption but a theorem, forced by the single fact $\chi(\tau<0)=0$ . Applying Cauchy's theorem to $\tilde\chi(\omega')/(\omega'-\omega)$ on a contour along the real axis, closed by a large upper semicircle and indented at the evaluation point $\omega$ , collects a principal value plus a half-residue. Rearranged, it is the master dispersion relation.

\tilde\chi(\omega)=\frac{1}{i\pi}\,\mathcal{P}\!\!\int_{-\infty}^{\infty}\frac{\tilde\chi(\omega')}{\omega'-\omega}\,\mathrm{d}\omega' \qquad (3)

where: $\mathcal{P}$ — the Cauchy principal value; the prefactor $1/(i\pi)$ comes from the half-circle around the simple pole at $\omega'=\omega$ . Reading: the value of $\tilde\chi$ at any real frequency is entirely determined by its own values along the whole real axis — a hallmark of an analytic function. Splitting the master relation into real and imaginary parts (using $1/i=-i$ ) gives the Kramers-Kronig pair: two coupled Hilbert transforms in which each part is the principal-value integral of the other.

\mathrm{Re}\,\tilde\chi(\omega)=\frac{1}{\pi}\,\mathcal{P}\!\!\int_{-\infty}^{\infty}\frac{\mathrm{Im}\,\tilde\chi(\omega')}{\omega'-\omega}\,\mathrm{d}\omega',\qquad \mathrm{Im}\,\tilde\chi(\omega)=-\frac{1}{\pi}\,\mathcal{P}\!\!\int_{-\infty}^{\infty}\frac{\mathrm{Re}\,\tilde\chi(\omega')}{\omega'-\omega}\,\mathrm{d}\omega' \qquad (4)

where: $\mathrm{Re}\,\tilde\chi$ — the dispersive (reactive) part; $\mathrm{Im}\,\tilde\chi$ — the absorptive (dissipative) part; $\omega'$ — the integration variable. Reading: dispersion and absorption are the same information seen twice; measure one over all frequencies and the other is computed, not measured. Reality of $\chi(t)$ forces $\tilde\chi(-\omega)=\tilde\chi^*(\omega)$ , so $\mathrm{Re}\,\tilde\chi$ is even and $\mathrm{Im}\,\tilde\chi$ is odd. Folding the negative axis onto the positive one gives the symmetric kernel that appears in every optics textbook.

\mathrm{Re}\,\tilde\chi(\omega)=\frac{2}{\pi}\,\mathcal{P}\!\!\int_{0}^{\infty}\frac{\omega'\,\mathrm{Im}\,\tilde\chi(\omega')}{\omega'^2-\omega^2}\,\mathrm{d}\omega',\qquad \mathrm{Im}\,\tilde\chi(\omega)=-\frac{2\omega}{\pi}\,\mathcal{P}\!\!\int_{0}^{\infty}\frac{\mathrm{Re}\,\tilde\chi(\omega')}{\omega'^2-\omega^2}\,\mathrm{d}\omega' \qquad (5)

where: $\omega'^2-\omega^2$ — the symmetric kernel produced by combining the $\pm\omega'$ contributions. Reading: these are the physically usable forms, since only positive frequencies are ever measured. Expanding the folded relation at high frequency, where all electrons respond as free charges and $\tilde\chi\to-\omega_p^2/\omega^2$ , matches the two tails and produces the oscillator-strength (f-)sum rule.

\int_{0}^{\infty}\omega\,\mathrm{Im}\,\tilde\chi(\omega)\,\mathrm{d}\omega=\frac{\pi}{2}\,\omega_p^2,\qquad \omega_p^2=\frac{n e^2}{\varepsilon_0 m} \qquad (6)

where: $\omega_p$ — the plasma frequency; $n$ — the electron number density; $e,m$ — the electron charge and mass. Reading: the total frequency-weighted absorption is fixed by the number of electrons alone, independent of how they are bound — a global constraint that any model spectrum must respect.

Derivation AFrom causality to the master dispersion relation

The goal is to convert the physical statement "the response kernel vanishes for negative times" into a single exact integral identity relating $\tilde\chi(\omega)$ to a principal-value integral of itself, from which both Kramers-Kronig relations then drop out by inspection. We encode causality as analyticity in the upper half-plane, build a contour that respects it, kill the large arc using the decay of $\tilde\chi$ , and evaluate the small indentation around the evaluation point as a half-residue — using nothing beyond Cauchy's theorem and a Jordan-type arc estimate.

Step 1. Start from $\tilde\chi(\omega)=\int_0^\infty\chi(\tau)e^{i\omega\tau}\,\mathrm{d}\tau$ . Because $\chi(\tau<0)=0$ the integral runs only over $\tau\ge0$ , and for $\mathrm{Im}\,\omega>0$ the factor $e^{i\omega\tau}=e^{i\omega_R\tau}e^{-\omega_I\tau}$ decays, so the integral converges and is complex-differentiable there. Why: this is the single logical hinge — causality in time becomes analyticity of $\tilde\chi$ throughout the upper half-plane, with every singularity banished to $\mathrm{Im}\,\omega<0$ .

Step 2. Fix a real evaluation frequency $\omega$ and consider $f(\omega')=\tilde\chi(\omega')/(\omega'-\omega)$ . This has a single simple pole on the real axis at $\omega'=\omega$ ; everywhere else in the closed upper half-plane it inherits the analyticity of $\tilde\chi$ . Why: introducing the Cauchy kernel $1/(\omega'-\omega)$ is what lets a contour integral reproduce the value $\tilde\chi(\omega)$ at the point of interest.

Step 3. Choose the contour $C$ : along the real axis, indented above the pole by a small semicircle of radius $\epsilon$ , then closed by a large semicircle of radius $R$ in the upper half-plane. Since the indentation excludes the pole and $\tilde\chi$ is analytic inside, no singularity is enclosed and Cauchy's theorem gives $\oint_C f\,\mathrm{d}\omega'=0$ . Why: routing the contour over the pole keeps the interior free of singularities, so the whole loop integral is exactly zero — a clean constraint to unpack.

Step 4. Kill the large arc. Provided $\tilde\chi(\omega')\to0$ as $|\omega'|\to\infty$ in the upper half-plane, the integrand falls faster than $1/R$ and the arc vanishes by a Jordan-type estimate. Why: this is the one analytic input beyond causality — the response must die at infinite frequency; when it fails, we subtract (below).

Step 5. Evaluate the small indentation. A clockwise half-circle of radius $\epsilon\to0$ around the simple pole contributes $-i\pi\,\tilde\chi(\omega)$ , while the straight parts assemble into the principal value $\mathcal{P}\!\int_{-\infty}^{\infty}\tilde\chi(\omega')/(\omega'-\omega)\,\mathrm{d}\omega'$ . Why: this is the Sokhotski-Plemelt split of a pole into its two finite pieces.

Step 6. Assemble: $0=\mathcal{P}\!\int_{-\infty}^{\infty}\tilde\chi(\omega')/(\omega'-\omega)\,\mathrm{d}\omega'-i\pi\,\tilde\chi(\omega)$ , so $\tilde\chi(\omega)=\frac{1}{i\pi}\mathcal{P}\!\int_{-\infty}^{\infty}\tilde\chi(\omega')/(\omega'-\omega)\,\mathrm{d}\omega'$ . Why: an analytic, decaying function is reconstructed from its own boundary values — the seed of both Kramers-Kronig relations. *This is identical to the Sokhotski-Plemelj result $1/(\omega'-\omega-i0^+)=\mathcal{P}\,1/(\omega'-\omega)+i\pi\delta(\omega'-\omega)$ applied to Cauchy's formula.*

Go deeper · Derivation B · 1 minSplitting and folding into the physical Kramers-Kronig pair

Derivation BSplitting and folding into the physical Kramers-Kronig pair

The goal here is to turn the compact master relation into the two real integral equations an experimentalist actually uses, then fold them onto positive frequencies using the reality of the time-domain kernel. We take real and imaginary parts to expose the Hilbert-transform structure, then invoke the crossing relation $\tilde\chi(-\omega)=\tilde\chi^*(\omega)$ to combine the $\pm\omega'$ contributions into a single integral over $[0,\infty)$ with a symmetric kernel — the form that appears in optics as "index from absorption".

Step 1. Write the prefactor as $1/(i\pi)=-i/\pi$ and insert $\tilde\chi=\mathrm{Re}\,\tilde\chi+i\,\mathrm{Im}\,\tilde\chi$ into the master relation; the kernel $1/(\omega'-\omega)$ is real under the principal value. Why: the lone factor of $-i$ is what rotates real into imaginary and back, coupling the two parts rather than leaving them independent.

Step 2. Match real parts: $\mathrm{Re}\,\tilde\chi(\omega)=\frac{1}{\pi}\mathcal{P}\!\int_{-\infty}^{\infty}\mathrm{Im}\,\tilde\chi(\omega')/(\omega'-\omega)\,\mathrm{d}\omega'$ . The $-i$ converts the $i\,\mathrm{Im}$ piece into a real contribution. Why: this states that the dispersive part is the Hilbert transform of the absorptive part — the first Kramers-Kronig relation.

Step 3. Match imaginary parts: $\mathrm{Im}\,\tilde\chi(\omega)=-\frac{1}{\pi}\mathcal{P}\!\int_{-\infty}^{\infty}\mathrm{Re}\,\tilde\chi(\omega')/(\omega'-\omega)\,\mathrm{d}\omega'$ . Why: the companion relation, with the opposite sign, so that applying the Hilbert transform twice returns the original function with a minus sign — an internal consistency check.

Step 4. Now fold. Reality of $\chi(t)$ gives $\tilde\chi(-\omega)=\tilde\chi^*(\omega)$ , hence $\mathrm{Re}\,\tilde\chi$ even and $\mathrm{Im}\,\tilde\chi$ odd. Substituting $\omega'\to-\omega'$ in the negative-axis part turns $\int_{-\infty}^0$ into $\int_0^\infty \mathrm{Im}\,\tilde\chi(\omega')/(\omega'+\omega)\,\mathrm{d}\omega'$ . Why: crossing symmetry removes the redundant negative frequencies, which carry no new information for a real signal.

Step 5. Add the half-line integrals using $\frac{1}{\omega'-\omega}+\frac{1}{\omega'+\omega}=\frac{2\omega'}{\omega'^2-\omega^2}$ , giving $\mathrm{Re}\,\tilde\chi(\omega)=\frac{2}{\pi}\mathcal{P}\!\int_0^\infty \omega'\,\mathrm{Im}\,\tilde\chi(\omega')/(\omega'^2-\omega^2)\,\mathrm{d}\omega'$ . Why: this is the symmetric-kernel form used to compute a refractive index from a measured absorption spectrum; the same steps on the second relation give its $\mathrm{Im}$ -from- $\mathrm{Re}$ partner. *The pair are Hilbert-transform partners: applying the transform twice returns $-\!$ identity, an internal consistency check. The folded symmetric-kernel form is the standard optical Kramers-Kronig integral.*

Go deeper · Derivation C · 1 minThe f-sum rule from the high-frequency tail

Derivation CThe f-sum rule from the high-frequency tail

The aim is to extract a rigorous global constraint on the absorption spectrum by matching the model-independent high-frequency behaviour of any causal response against the folded Kramers-Kronig relation. Far above every resonance bound electrons respond as if free, so the response approaches the universal plasma tail $-\omega_p^2/\omega^2$ ; forcing the folded relation to reproduce that tail fixes a weighted integral of the absorption — the oscillator-strength sum rule that counts a material's electrons regardless of how they are bound.

Step 1. Above all resonances the equation of motion for each electron reduces to $m\ddot x=-eE$ , so the induced dipole gives $\tilde\chi(\omega)\to-\omega_p^2/\omega^2$ with $\omega_p^2=ne^2/(\varepsilon_0 m)$ . Why: at high drive frequency the binding force is negligible compared with inertia, so every electron contributes identically regardless of its chemistry — the tail is universal.

Step 2. Expand the folded relation for large $\omega$ , where $\omega'^2-\omega^2\approx-\omega^2$ over the region that supports the absorption: $\mathrm{Re}\,\tilde\chi(\omega)\to-\frac{1}{\omega^2}\cdot\frac{2}{\pi}\int_0^\infty\omega'\,\mathrm{Im}\,\tilde\chi(\omega')\,\mathrm{d}\omega'$ . Why: at high frequency the driving point sits far outside the absorption band, so the kernel flattens and the integral collapses to a single number — the total weighted absorption.

Step 3. Match the two expressions for the $1/\omega^2$ coefficient: $\frac{2}{\pi}\int_0^\infty\omega'\,\mathrm{Im}\,\tilde\chi(\omega')\,\mathrm{d}\omega'=\omega_p^2$ , i.e. $\int_0^\infty\omega\,\mathrm{Im}\,\tilde\chi(\omega)\,\mathrm{d}\omega=\frac{\pi}{2}\omega_p^2$ . Why: the analyticity that built Kramers-Kronig now doubles as a bookkeeping law — the area under $\omega\,\mathrm{Im}\,\tilde\chi$ counts the electrons, independent of the detailed line shape. *For the Lorentz oscillator $\tilde\chi=\omega_p^2/(\omega_0^2-\omega^2-i\gamma\omega)$ , numerical integration confirms $\frac{2}{\pi}\int_0^\infty\omega\,\mathrm{Im}\,\tilde\chi\,\mathrm{d}\omega=\omega_p^2$ to machine precision, independent of $\omega_0$ and $\gamma$ .*

Limiting cases

Case	Result	Meaning
Low frequency (static limit)	$\mathrm{Re}\,\tilde\chi(0)=\dfrac{2}{\pi}\,\mathcal{P}\!\!\int_0^\infty \dfrac{\mathrm{Im}\,\tilde\chi(\omega')}{\omega'}\,\mathrm{d}\omega'$	The DC susceptibility (static polarisability) equals an absorption-weighted integral over the whole spectrum — it is fixed by loss you may never see at DC.
High frequency	$\tilde\chi(\omega)\to -\dfrac{\omega_p^2}{\omega^2}$	Electrons respond as free charges; the coefficient of $1/\omega^2$ is the f-sum rule, counting $n$ electrons per unit volume.
Response non-vanishing at $\infty$	$\tilde\chi(\omega)-\tilde\chi(\omega_0)=\dfrac{\omega-\omega_0}{i\pi}\,\mathcal{P}\!\!\int_{-\infty}^{\infty}\dfrac{\tilde\chi(\omega')\,\mathrm{d}\omega'}{(\omega'-\omega)(\omega'-\omega_0)}$	One subtraction at $\omega_0$ adds a power of $\omega'$ to the denominator, restoring convergence when $\tilde\chi(\infty)\neq0$ .
Reality of $\chi(t)$	$\tilde\chi(-\omega)=\tilde\chi^*(\omega)\Rightarrow \mathrm{Re}\,\tilde\chi$ even, $\mathrm{Im}\,\tilde\chi$ odd	Folds the integrals onto $[0,\infty)$ with kernel $2\omega'/(\omega'^2-\omega^2)$ ; negative frequencies carry no new information.
Sharp resonance ( $\gamma\to0$ )	$\mathrm{Im}\,\tilde\chi\to$ delta spike, $\mathrm{Re}\,\tilde\chi\to$ pole pair	An isolated line gives the textbook anomalous-dispersion $1/(\omega_0^2-\omega^2)$ swing straddling the peak.

Common trapYou can reconstruct the index in your band from absorption measured only in that band

The Kramers-Kronig relations are nonlocal: the principal-value kernel $1/(\omega'-\omega)$ reaches across the entire spectrum, so $\mathrm{Re}\,\tilde\chi$ at one frequency depends on $\mathrm{Im}\,\tilde\chi$ at all frequencies, weighted by $1/(\omega'-\omega)$ . Absorption bands far outside your measurement window — the ultraviolet electronic resonances and infrared vibrational bands that flank an optical glass's transparency gap — still contribute to the index inside the gap. Truncating the integral at the edges of your data biases the reconstruction; practitioners extrapolate the tails with power laws and anchor the result with sum rules and independent measurements. A second, distinct trap: the derivation assumed $\tilde\chi(\omega)\to0$ at large $|\omega|$ so the large arc vanishes. If instead the response tends to a nonzero constant (as $\varepsilon_r(\infty)=1$ ), the naive integral diverges and you must use a subtracted dispersion relation, applying Kramers-Kronig to $\tilde\chi(\omega)-\tilde\chi(\infty)$ or to $\varepsilon_r(\omega)-1$ .

Worked examples

1 · Debye relaxation of liquid water (fully worked)

A polar liquid relaxes with a single exponential kernel $\chi(\tau)=(\chi_0/\tau_D)e^{-\tau/\tau_D}\theta(\tau)$ . Find $\tilde\chi(\omega)$ , verify it is a genuine Kramers-Kronig pair, and put in numbers for water at $25\,^\circ\mathrm{C}$ , where $\chi_0\approx73.2$ and $\tau_D\approx8.27\ \mathrm{ps}$ .

Step 1. Transform the kernel: $\tilde\chi(\omega)=\int_0^\infty\frac{\chi_0}{\tau_D}e^{-\tau/\tau_D}e^{i\omega\tau}\mathrm{d}\tau=\frac{\chi_0}{\tau_D}\cdot\frac{1}{1/\tau_D-i\omega}=\dfrac{\chi_0}{1-i\omega\tau_D}$ . Separating parts, $\mathrm{Re}\,\tilde\chi=\dfrac{\chi_0}{1+\omega^2\tau_D^2}$ and $\mathrm{Im}\,\tilde\chi=\dfrac{\chi_0\,\omega\tau_D}{1+\omega^2\tau_D^2}$ . The only singularity is a pole at $\omega=-i/\tau_D$ , safely in the lower half-plane — the response is causal, as it must be.

Step 2. Verify Kramers-Kronig analytically. Feeding $\mathrm{Im}\,\tilde\chi$ into the first relation and closing the contour in the upper half-plane (pole at $\omega'=+i/\tau_D$ plus the half-residue at $\omega'=\omega$ ),

\frac{1}{\pi}\,\mathcal{P}\!\!\int_{-\infty}^{\infty}\frac{\chi_0\,\omega'\tau_D}{(1+\omega'^2\tau_D^2)(\omega'-\omega)}\,\mathrm{d}\omega'=\frac{\chi_0}{1+\omega^2\tau_D^2}=\mathrm{Re}\,\tilde\chi(\omega).

The residue algebra collapses exactly to $\mathrm{Re}\,\tilde\chi$ — the two parts are a true Hilbert pair.

Step 3. Numbers. The relaxation (peak-loss) frequency is where $\omega\tau_D=1$ : $f_D=\dfrac{1}{2\pi\tau_D}=\dfrac{1}{2\pi(8.27\times10^{-12})}=\mathbf{19.2\ GHz}$ — the microwave absorption line that a microwave oven exploits. At that frequency $\mathrm{Re}\,\tilde\chi=\mathrm{Im}\,\tilde\chi=\chi_0/2=\mathbf{36.6}$ : the dispersion has fallen to half its static value exactly where the absorption peaks. Below $f_D$ the liquid follows the field (high $\mathrm{Re}$ , low loss); above it the dipoles lag and dissipate.

2 · The f-sum rule for a Lorentz oscillator (numeric check)

Take a model dielectric with one Lorentz line, $\tilde\chi(\omega)=\dfrac{\omega_p^2}{\omega_0^2-\omega^2-i\gamma\omega}$ . Confirm the f-sum rule and the static Kramers-Kronig limit numerically, and read off what they mean.

Step 1. Extract the absorption: $\mathrm{Im}\,\tilde\chi(\omega)=\dfrac{\omega_p^2\,\gamma\omega}{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}$ , a peak of width $\sim\gamma$ centred near $\omega_0$ . In units $\omega_0=1$ , $\omega_p=1$ , $\gamma=0.1$ , numerical integration gives $\dfrac{2}{\pi}\int_0^\infty\omega\,\mathrm{Im}\,\tilde\chi\,\mathrm{d}\omega=\mathbf{1.0000}=\omega_p^2$ , and repeating with $\gamma=0.5$ gives the same answer. The sum rule is independent of line position and width — it counts oscillator strength, not shape.

Step 2. Check the static limit. The direct value is $\mathrm{Re}\,\tilde\chi(0)=\omega_p^2/\omega_0^2$ ; the Kramers-Kronig static relation gives $\dfrac{2}{\pi}\int_0^\infty\dfrac{\mathrm{Im}\,\tilde\chi(\omega')}{\omega'}\,\mathrm{d}\omega'$ . In the same units both equal $\mathbf{1.0000}$ , and for the physical ratio $\omega_p/\omega_0=1/4$ they equal $\mathbf{0.0625}$ . The DC susceptibility is reconstructed entirely from the absorption line.

Step 3. Interpret. Integrating $\omega\,\mathrm{Im}\,\tilde\chi$ over a measured line recovers $\tfrac{\pi}{2}\omega_p^2$ and hence the electron density $n=\varepsilon_0 m\,\omega_p^2/e^2$ — without ever resolving the line's shape. A weighted area disagreeing with the known $n$ signals missing oscillator strength outside the window, a routine diagnostic in optical and electron-energy-loss spectroscopy.

Physics of the portrait

The dispersion relations sit at the meeting point of pure complex analysis and hard experiment: an integral identity of Cauchy's, given physical teeth by the demand that no effect precede its cause.

Origins · 1 minHendrik Kramers and how this was discovered

Hendrik Kramers

1894–1952 · dispersion theory and the analytic structure of response functions

Hendrik Anthony Kramers was Niels Bohr's first and closest collaborator in Copenhagen, and one of the deepest analysts of the quantum generation that never quite received a Nobel Prize. His fingerprints are everywhere in mathematical physics: the WKB (Wentzel-Kramers-Brillouin) approximation, the Kramers-Heisenberg dispersion formula, the Kramers degeneracy theorem, the Kramers escape rate. The relations that bear his name grew out of the old quantum theory of light scattering: in 1927, working on how atoms disperse and absorb radiation, Kramers realised that the refractive index at one frequency is fixed by the absorption at all others, and wrote down the integral relation connecting them — unaware that Ralph Kronig had reached an equivalent result a year earlier for X-rays. What neither fully appreciated at first, and what John Toll made rigorous only in 1956, was that the relations are not special to optics: they are the direct mathematical expression of causality, valid for any stable linear response. Kramers spent his later career in Leiden and Utrecht, a physicist's physicist whose taste for exact structure shaped a generation of Dutch theory.

History

1926

— Kronig derives the X-ray dispersion relation from the optical analogy

1927

— Kramers states the optical dispersion relations independently at the Como congress

1937

— Kronig ties the relations directly to causality, no signal before its cause

1945

— Bode derives the gain-phase relation for feedback amplifiers, the same analyticity

1956

— Toll gives the definitive proof that causality implies analyticity and the dispersion relations

Going further

Texts. Stone & Goldbart, Mathematics for Physics, ch. on analyticity and dispersion · Arfken & Weber, Mathematical Methods for Physicists, dispersion-relations section · Byron & Fuller, Mathematics of Classical and Quantum Physics, ch. 6 · Nakahara, Geometry, Topology and Physics (analytic structure) · Jackson, Classical Electrodynamics, §7.10 · Landau & Lifshitz, Electrodynamics of Continuous Media, §82–84 · Toll, Causality and the Dispersion Relation, Phys. Rev. 104, 1760 (1956). The key results of this article are: (1) analyticity of $\tilde\chi$ in the upper half-plane follows from $\chi(\tau<0)=0$ ; (2) the master relation $\tilde\chi=\frac{1}{i\pi}\mathcal{P}\!\int\tilde\chi/(\omega'-\omega)$ arises from the $-i\pi$ half-residue and the vanishing large arc; (3) the Kramers-Kronig pair and the folded symmetric-kernel form follow via $\tilde\chi(-\omega)=\tilde\chi^*(\omega)$ ; (4) the high-frequency tail $-\omega_p^2/\omega^2$ yields the f-sum rule $\int_0^\infty\omega\,\mathrm{Im}\,\tilde\chi\,\mathrm{d}\omega=\tfrac{\pi}{2}\omega_p^2$ with $\omega_p^2=ne^2/\varepsilon_0 m$ . Worked numbers: water Debye $f_D=19.2\ \mathrm{GHz}$ at $\tau_D=8.27\ \mathrm{ps}$ ; the Debye susceptibility is an exact Hilbert-transform pair; the Lorentz f-sum rule and static limit hold to machine precision, independent of $\omega_0$ and $\gamma$ . Historical attribution: Kronig (1926), Kramers (1927), rigorous causality formulation Toll (1956).

Prerequisites. Analytic Functions and the Cauchy-Riemann Equations · Complex Integration and Contour Integrals · Cauchy's Integral Formula and Derivatives · Zeros, Poles, and Isolated Singularities · Cauchy's Integral Theorem. Leads to. Laurent Series · Retarded Green's Functions · S-matrix and Scattering Amplitudes.

Ask the tutor: Ask me to Kramers-Kronig-transform a specific model spectrum, or to show exactly how the same contour argument becomes a forward dispersion relation in scattering theory.

Why does causality become analyticity in the upper half-plane specifically, and not the lower?

It is a convention tied to the Fourier sign. With the transform $\tilde\chi(\omega)=\int_0^\infty\chi(\tau)e^{i\omega\tau}\mathrm{d}\tau$ , convergence for a kernel supported on $\tau>0$ requires the exponential $e^{-\omega_I\tau}$ to decay, which needs $\omega_I=\mathrm{Im}\,\omega>0$ — the upper half-plane. Had you instead used the engineers' convention $e^{-i\omega\tau}$ , the same causal kernel would be analytic in the lower half-plane, and the Kramers-Kronig signs would flip accordingly. The physics is identical; only the labelling of which half is "analytic" follows your chosen sign. What is convention-independent is that all poles sit on the opposite side from the analytic half — the side that corresponds to decaying-in-time modes.

Are the Kramers-Kronig relations just the Hilbert transform under another name?

Almost. The Hilbert transform is the pure mathematics: $\mathcal{H}[u](\omega)=\frac{1}{\pi}\mathcal{P}\!\int u(\omega')/(\omega'-\omega)\,\mathrm{d}\omega'$ . The Kramers-Kronig relations are the statement that, for a causal response, the real and imaginary parts of $\tilde\chi$ are exactly such a pair — the Hilbert transform plus the physical input of causality that guarantees the pairing. Causality also brings the reality condition $\tilde\chi(-\omega)=\tilde\chi^*(\omega)$ , which folds the transform onto positive frequencies with the symmetric kernel $2\omega'/(\omega'^2-\omega^2)$ .

Do I really need the absorption at all frequencies, or just near the frequency I care about?

In principle, all of them. The kernel $1/(\omega'-\omega)$ falls off only as $1/\omega'$ , so distant absorption bands still contribute to the index at your working frequency — this is why the transparency window of a glass is dispersive at all, being the low-energy tail of resonances outside the window. In practice you measure over the widest band you can, extrapolate the tails with power laws, and pin the result with sum rules. The nonlocality is the single biggest source of error in optical-constant reconstructions, and it is why a local measurement of loss can never certify a "lossless" index.

When exactly do I need a subtracted dispersion relation?

Whenever the response does not decay to zero at infinite frequency, so the large semicircle no longer vanishes. The classic case is the dielectric function: $\varepsilon_r(\omega)\to1$ , not $0$ , so you apply Kramers-Kronig to $\varepsilon_r(\omega)-1$ instead. If even that decays too slowly, subtract at a reference point $\omega_0$ — each subtraction adds a factor $1/(\omega'-\omega_0)$ and hence a power of $\omega'$ in the denominator, at the cost of needing $\tilde\chi(\omega_0)$ . The forward scattering amplitude in particle physics famously needs one or two subtractions because it grows at high energy.

How is this the same mathematics as the dispersion relations in particle physics?

The structure is identical. There, the forward scattering amplitude $f(\omega)$ is analytic in the upper half-plane of energy because scattering is causal — the scattered wave cannot leave before the incident wave arrives. Its imaginary part is tied to the total cross section by the optical theorem, and a dispersion relation of exactly the Kramers-Kronig form reconstructs the real part from measured cross sections. The differences are practical: relativistic amplitudes may need subtractions, and crossing symmetry adds contributions along a left-hand cut. But the engine — causality forces analyticity, Cauchy's theorem ties real to imaginary — is the same one developed here for optics.

Does the sign convention for the Fourier transform change the relations?

It changes the bookkeeping, not the content. Choosing $e^{+i\omega\tau}$ versus $e^{-i\omega\tau}$ swaps which half-plane is analytic and flips signs in the Kramers-Kronig pair, including the sign of $\mathrm{Im}\,\tilde\chi$ (so "absorption is positive" may attach to $+\mathrm{Im}$ or $-\mathrm{Im}$ depending on the book). What never changes: the parts remain a Hilbert-transform pair, poles live on the decaying-in-time side, and the f-sum rule holds. When comparing formulas across texts, check the transform convention first — most apparent sign disagreements dissolve once it is fixed.