Thinking in Waves Episode 8 ／ Part IV · Constraints imposed on waves

A single line binds everything

Causality ──
a filter does not know the future To the view that "everything is a filter", physics adds exactly one condition ── $h(t)=0$ for $t<0$.
No output before the input. That is all.
And from that one line come dispersion, sum rules, and the ban on ideal filters.

Prerequisites: Episodes 1–2Computation: kenshou/inga.js

So far we have seen physics as a chain of filters. But is any filter allowed?
No. And the way they are forbidden is remarkably strong.

01First, look at what is forbidden

"Pass everything below the cutoff, block everything above" ── the ideal low-pass filter. Inverse Fourier transform that transfer function:

Impulse response of the ideal filter $$h(t)=2f_c\,\mathrm{sinc}(2f_c t)$$

$t$ [ms]	$h(t)/h(0)$	Verdict
$-2.00$	0.000000	★ uses the future
$-1.00$	$-0.000000$	★ uses the future
$-0.50$	$-0.000000$	★ uses the future
$-0.25$	0.636620	★ uses the future
0.00	1.000000
$+0.25$	0.636620

Conclusion of §01

The ideal filter is physically unrealisable.
At $t=-0.25$ ms the response is already 0.64 ── output appears 0.25 ms before the input arrives.

"You need a transition band" is not an engineering limitation but a consequence of causality.

02The general law ── the Kramers–Kronig relations

If $h(t)$ is one-sided, its transform $\chi(\omega)=\chi'+i\chi''$ is analytic in the upper half-plane. From Cauchy's theorem:

Kramers–Kronig $$\chi'(\omega)=\frac1\pi\,\mathrm{P}\!\!\int\frac{\chi''(\omega')}{\omega'-\omega}d\omega',\qquad \chi''(\omega)=-\frac1\pi\,\mathrm{P}\!\!\int\frac{\chi'(\omega')}{\omega'-\omega}d\omega'$$

★ Real and imaginary parts are not independent. Either determines the other

Let us check it. With a Lorentz oscillator (a damped harmonic oscillator) $\chi(\omega)=1/(\omega_0^2-\omega^2-i\gamma\omega)$, we reconstruct the real part from the imaginary part alone.

$\omega$	$\chi'$ (direct)	$\chi'$ (K–K integral)	Relative difference
0.30	1.094145	1.094145	$6.1\times10^{-13}$
0.60	1.509434	1.509434	$4.1\times10^{-13}$
0.90	2.773723	2.773723	$2.0\times10^{-13}$
1.20	$-1.751592$	$-1.751592$	$3.8\times10^{-13}$
2.00	$-0.327511$	$-0.327511$	$2.0\times10^{-12}$

The heart of §02

Agreement to 13 digits. Absorption (the imaginary part) alone reproduces dispersion (the real part) completely.

This is not an approximation. It is an exact relation following from one line: causality.

03The physical meaning ── no dispersion without absorption

Condition	Consequence	Meaning
If absorption is zero	$\chi''\equiv0\Rightarrow\chi'=$const	refractive index independent of frequency ＝ no dispersion
If there is absorption anywhere	$\chi'$ depends on frequency	dispersion is unavoidable
And conversely	dispersion implies absorption	★ a transparent dispersive medium cannot exist

Why glass splits colours

A prism makes a rainbow because glass has absorption bands in the ultraviolet. It is perfectly transparent in the visible, yet those distant absorption lines create the dispersion ── causality demands it.
(This is the same story as "refraction and absorption are the same function" in the sister series "Understanding Refraction".)

And the $\Delta\alpha_{\text{had}}$ used in Episode 5 was obtained by this very relation ── measure the $e^+e^-\to$hadrons cross-section (the imaginary part) and convert to the real part by a dispersion relation. What cannot be computed is filled in by measurement plus causality.

04The sum rule ── integrate and you count the electrons

f-sum rule $$\int_0^\infty \omega\,\varepsilon''(\omega)\,d\omega=\frac\pi2\,\omega_p^2,\qquad \omega_p^2=\frac{ne^2}{\varepsilon_0 m}$$

★ Integrate the absorption spectrum over all frequencies and out comes the electron number density

What is striking is that this depends on neither the position nor the width of the oscillator. We varied both to check.

$\omega_0$	$\gamma$	$\int\omega\varepsilon''d\omega$	Relative difference from $\pi/2$
0.5	0.10	1.570791	$3.2\times10^{-6}$
1.0	0.20	1.570786	$6.4\times10^{-6}$
2.0	0.50	1.570771	$1.6\times10^{-5}$
5.0	1.00	1.570746	$3.2\times10^{-5}$

The heart of §04

Position and width both changed, and every result is $\pi/2$.
The sum rule counts only "how many there are", not how they absorb.

★ So it can be used to check measured spectra ── if the integral does not come out right, you have missed an absorption band.

05Scoring the operations of Episodes 1–2 against causality

Operation	Verdict	Reason
Differentiation $d/dt$	○ just barely causal	$h(t)=\delta'(t)$, concentrated at $t=0$, nothing at $t<0$
Integration $\int_0^t$	○ causal	$h(t)=$ step function; looks only at the past
Fractional order (forward)	○ causal	the memory kernel $(t-\tau)^{-\alpha}$ is past-only
Ideal low-pass	× non-causal	the sinc spreads both ways (§01)
Hilbert transform	× non-causal	★ even "just rotate the phase by 90°" needs the future

The heart of §05

The last two are the interesting ones. "Leave the amplitude alone and rotate the phase by 90°" — an apparently harmless operation — is non-causal.
Causality constrains not "what passes" but "the relation between real and imaginary parts".

And Episode 2's fractional derivative is properly causal ── its memory kernel looks only at the past. That "memory" was shaped to be compatible with causality all along.

06How sharply can it cut? ── Paley–Wiener

Upper bound on the sharpness a causal response can have $$\int\frac{\bigl|\ln|\chi(\omega)|\bigr|}{1+\omega^2}\,d\omega<\infty$$

Violate it ＝ decay too fast ＝ not realisable causally

Form of the roll-off	Allowed?	Reason
$1/\omega$ (first-order filter)	○	$\ln$ grows only logarithmically
$e^{-\omega/\omega_c}$	○	$\ln$ is linear; the integral converges
$e^{-\omega^2/\omega_c^2}$ (Gaussian)	×	★ $\ln$ grows as $\omega^2$; diverges ── unrealisable
exactly zero above some frequency	×	$\ln\to-\infty$; the ideal filter

A Gaussian filter is not strictly causal

The reason image processing uses it every day is that it acts along space, not time.
Space has no "future", so causality imposes nothing. ── The same formula is allowed or forbidden depending on which axis you apply it to.

Exercises

Illustrate "no absorption means no dispersion" with an everyday example.
Show answer
The vacuum. It absorbs at no frequency, so every frequency travels at the same speed $c$ (no dispersion).
Glass and water, by contrast, have absorption bands in the UV and IR, so they disperse even in the visible ── hence rainbows.
"A transparent substance that makes no rainbow" is forbidden by causality.
Why can the sum rule be used to check an experiment?
Show answer
Because the integral is fixed by the electron number density alone, independent of the spectrum's shape.
If integrating your measured spectrum falls short of $\pi\omega_p^2/2$, there is an absorption band you have missed.
You can know what you failed to measure before measuring it ── a powerful tool.
If the Hilbert transform is non-causal, why is it used in real signal processing?
Show answer
Because one inserts a delay. Delay everything by $T$ and the response on \(-TIt is never exact, so one uses a finite-length truncation.
"Non-causal operation" = "an operation you can approximate if you can afford latency" ── where real-time work forbids latency, accuracy suffers.

Summary　Do not use the future. That is all

To the view that "everything is a filter", physics adds exactly one condition ── $h(t)=0$ for $t<0$.

From it follows the ban on ideal filters (the sinc spreads both ways ── response 0.64 at $t=-0.25$ ms). A transition band is a matter of principle, not of engineering.

Then the Kramers–Kronig relations: the imaginary part (absorption) determines the real part (dispersion) ── verified to 13 digits on a Lorentz oscillator. The consequence: "a transparent dispersive medium cannot exist". A prism makes a rainbow because glass absorbs in the ultraviolet.

The sum rule is stronger still: $\int\omega\varepsilon''d\omega=\pi\omega_p^2/2$ depends on neither position nor width of the oscillator (confirmed to $10^{-5}$). It counts only how many there are.

And scoring Episodes 1–2 against causality ── the fractional derivative is properly causal (its memory kernel is past-only), while the Hilbert transform, which "only rotates the phase", is not.

Causality constrains not "what passes" but "the relation between real and imaginary parts".

Next time Causality constrained a filter's real and imaginary parts.
Next time, a theorem constraining a filter's time width and frequency width ── $\Delta\omega\,\Delta t\ge1/2$.
It is a pure Fourier theorem with nothing to do with quantum mechanics. Yet multiply by $\hbar$ and it becomes the uncertainty principle itself.

This document is Episode 8 of the series "Thinking in Waves". The computation is in kenshou/inga.js. On the numbers: the K–K check evaluates the principal value by subtracting the singularity before integrating (naively skipping grid points gives a 30 % error near resonance ── we stepped on that rake once). The 13-digit agreement reflects a smooth rational integrand; applied to measured spectra the accuracy is far lower. The sum-rule integral is truncated at $\omega=2\times10^4$, and the $10^{-5}$ discrepancy comes from that truncation (for a narrow resonance with $\gamma=0.02$ the grid is too coarse and it degrades to $4\times10^{-3}$). Kramers–Kronig and the Paley–Wiener criterion are established theorems, not derived here ── what we did is confirm them numerically. The Lorentz oscillator is an idealisation with a single oscillator of unit strength; real materials are sums over many. ── To print, use your browser's Print → Save as PDF.