Thinking in Waves Episode 13 ／ Part VI · Extending the order into the complex plane
An operation that stretches time without touching the timbre
Imaginary-order differentiation ──
the tool of the world of scale
Make the order \(\alpha\) imaginary and the amplitude of \((i\omega)^{i\beta}\) becomes a constant independent of \(\omega\),
with only the phase rotating as \(\beta\ln\omega\) ── an all-pass filter.
★ If real orders are the tool of time translation, imaginary orders are the tool of scale transformation.
In Episode 2 we differentiated by one half. Having extended the exponent to the reals, we may as well extend it to the imaginaries.
We did ── and out came a completely different physics.
01What happens when the exponent goes imaginary
The heart of §01 ── it splits in two
Amplitude \(=e^{-\beta\pi/2}\) ── a plain constant, independent of \(\omega\). Every frequency passes equally.
Phase \(=\beta\ln\omega\) ── proportional to the logarithm of frequency.
★ So imaginary-order differentiation is an all-pass filter.
In audio terms ── an operation that leaves the timbre completely alone and changes only the arrangement in time.
| \(\beta\) | Amplitude \(e^{-\beta\pi/2}\) | Phase at \(\omega=1\) | \(\omega=10\) | \(\omega=100\) |
|---|---|---|---|---|
| \(-1\) | 4.810 | 0° | \(-131.9°\) | \(-263.9°\) |
| \(-0.5\) | 2.193 | 0° | \(-66.0°\) | \(-131.9°\) |
| 0 | 1.000 | 0° | 0° | 0° |
| 1 | 0.2079 | 0° | \(+131.9°\) | \(+263.9°\) |
| 2 | 0.04321 | 0° | \(+263.9°\) | \(+527.7°\) |
Every factor of ten in frequency rotates the phase by the same amount. That is the whole of this episode.
02Checking it with sound ── a click becomes a hyperbolic chirp
So we define it on positive frequencies and fill the negative side by conjugation ── the same convention as the Hilbert transform. Then \(|H|=e^{-\beta\pi/2}\) at every frequency and \(\arg H=\beta\ln\omega\) on the positive side.
| \(\beta\) | Measured output/input | Theory \(e^{-\beta\pi/2}\) | Relative error |
|---|---|---|---|
| \(-1\) | 4.810477 | 4.810477 | \(<10^{-12}\) |
| \(-0.5\) | 2.193280 | 2.193280 | \(<10^{-12}\) |
| 1 | 0.2078796 | 0.2078796 | \(<10^{-12}\) |
| 2 | 0.0432139 | 0.0432139 | \(<10^{-12}\) |
Check 1: pass white noise through and compare RMS values (is it really all-pass?)
The amplitude spectrum is untouched. So what does change ── the group delay.
★ The shift goes as \(1/\omega\) ── lower notes are displaced more
| Frequency [Hz] | Measured shift [ms] | Theory \(-\beta/\omega\) [ms] | Difference [µs] |
|---|---|---|---|
| 200 | \(-2.3950\) | \(-2.3873\) | \(-7.6\) |
| 400 | \(-1.1946\) | \(-1.1937\) | \(-0.95\) |
| 800 | \(-0.5969\) | \(-0.5968\) | \(-0.12\) |
| 1600 | \(-0.2984\) | \(-0.2984\) | \(-0.01\) |
Check 2: centroid shift of a Gaussian-windowed tone burst, \(\beta=3\)
Conclusion of §02
Double the frequency and the shift halves. Exactly \(1/f\).
So feed in a click (every frequency at once) and out comes a hyperbolic chirp ── a pitch sliding smoothly.
★ The timbre is unchanged, yet time is stretched. Pure dispersion.
03Writing the impulse response in closed form
Integrate to get the output \(h(t)\) for a click. Substituting \(x=\omega t\) pulls \(t\) out of the integral ── that is the crux of imaginary orders.
The heart of §03 ── a sinusoid on a logarithmic scale
$$t\,h(t)=A(\beta)\,\sin\bigl(\beta\ln t-\arg\Gamma(1+i\beta)\bigr)$$
$$A(\beta)=\frac{e^{-\beta\pi}\,|\Gamma(1+i\beta)|}{\pi},\qquad |\Gamma(1+i\beta)|=\sqrt{\frac{\pi\beta}{\sinh\pi\beta}}$$
★ Multiply by \(t\) and you get a pure sinusoid in \(\ln t\), of period \(2\pi/\beta\).
View the time axis logarithmically and it is cleanly periodic ── this is "log-periodicity".
| \(\beta\) | \(\varepsilon\) | Numerical integral (real) | (imaginary) | Relative error |
|---|---|---|---|---|
| 1 | 0.05 | \(3.9337900\times10^{-2}\) | \(1.0682146\times10^{-1}\) | \(6.4\times10^{-13}\) |
| 1 | 0.01 | \(3.3582150\times10^{-2}\) | \(1.0423027\times10^{-1}\) | \(4.6\times10^{-13}\) |
| 2 | 0.05 | \(-5.6342955\times10^{-4}\) | \(7.2846544\times10^{-3}\) | \(1.9\times10^{-12}\) |
| 3 | 0.01 | \(-3.1192639\times10^{-4}\) | \(1.8175677\times10^{-4}\) | \(1.9\times10^{-11}\) |
Check 3: \(\int_0^\infty x^{i\beta}e^{ix-\varepsilon x}dx\) numerically vs the closed form \(\Gamma(1+i\beta)/(\varepsilon-i)^{1+i\beta}\)
Since the phase \(\beta\ln x\) rotates infinitely fast as \(x\to0\), that region is integrated after substituting \(x=e^u\) ── agreement to 11–13 digits.
04Here is the trap ── the tail is crushed by \(e^{-\beta\pi}\)
| \(\beta\) | Tail amplitude \(A(\beta)\) | Log-scale period \(2\pi/\beta\) |
|---|---|---|
| 0.1 | \(2.31\times10^{-1}\) | 62.83 |
| 0.5 | \(5.47\times10^{-2}\) | 12.57 |
| 1 | \(7.17\times10^{-3}\) | 6.283 |
| 2 | \(9.11\times10^{-5}\) | 3.142 |
| 3 | \(1.00\times10^{-6}\) | 2.094 |
| 5 | \(1.04\times10^{-10}\) | 1.257 |
§04 ── why log-periodicity is "hard to find"
Raising \(\beta\) makes the oscillation faster but crushes the amplitude as \(e^{-\beta\pi}\).
At \(\beta=3\) it is already \(10^{-6}\) ── buried in FFT round-off, invisible in the time domain.
★ In fact, our first attempt to measure the impulse response directly by FFT failed.
What we were measuring was an artefact of zeroing the DC and Nyquist bins (about \(10^{-5}\)), an order of magnitude larger than the real signal (\(10^{-6}\)). Without deriving the closed form first, we would not have noticed.
This is of a piece with §08's "log-periodicity in earthquakes and finance is disputed" ── small amplitudes are hard to separate statistically.
(Though \(A(\beta)\) is about this particular time response, and is not a formula for the amplitude of log-periodic phenomena in general.)
05What is it the generator of? ── scale transformations
Restate the property "phase proportional to \(\ln\omega\)". Stretch the frequency by \(\lambda\).
★ The phase shifts by a constant. The shape does not change at all
| Order | Symbol | Amplitude | Phase | Corresponding symmetry | Natural transform |
|---|---|---|---|---|---|
| Real \(\alpha\) | \((i\omega)^\alpha\) | \(\omega^\alpha\) | \(\alpha\pi/2\) (const) | time translation | Fourier |
| Imaginary \(i\beta\) | \((i\omega)^{i\beta}\) | constant | \(\beta\ln\omega\) | ★ scale transformation | ★ Mellin |
§05 ── the main finding of this episode
Real-order differentiation is the tool of the world that shifts time;
imaginary-order differentiation is the tool of the world that changes scale.
Just as the Fourier kernel \(e^{i\omega t}\) is the eigenfunction of time translation, the Mellin kernel \(t^{i\beta}\) is the eigenfunction of scale transformation.
★ Imaginary-order differentiation is that Mellin kernel itself.
In audio: real orders change the timbre; imaginary orders stretch time without changing it.
06Where physics uses imaginary orders ── the \(1/r^2\) potential
Consider an attractive \(1/r^2\) potential (three dimensions, s-wave). As \(r\to0\) the energy \(E\) is irrelevant, so put \(u\propto r^s\):
| \(\lambda\) (attraction strength) | \(\sqrt{1/4-\lambda}\) | \(s_0\) | Behaviour |
|---|---|---|---|
| 0 | 0.5000 | ── | stable; a ground state exists |
| 0.20 | 0.2236 | ── | stable |
| 0.25 | 0.0000 | ── | ★ exactly critical |
| 0.50 | imaginary | 0.5000 | ★ log-periodic; fall to the centre |
| 1.00 | imaginary | 0.8660 | log-periodic |
| 2.00 | imaginary | 1.3229 | log-periodic |
Conclusion of §06
\(\lambda=1/4\) is the critical point. Beyond it the particle "falls to the centre", and the wavefunction becomes periodic in \(\ln r\).
★ Space becomes periodic on a logarithmic scale ── this is called discrete scale invariance.
Continuous scale invariance (the same at any magnification) is broken down to the same only at integer powers of one particular ratio.
07It has actually been observed ── Efimov states
This really happens in the three-body problem (Efimov 1970). For three identical bosons at resonance, the effective potential in the hyperradius \(R\) is \(\propto-(s_0^2+1/4)/R^2\) with \(s_0=1.00624\) ── a \(1/R^2\) just past critical.
| \(n\) | Size (level 0 = 1) | Energy ratio |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 22.69 | \(1.94\times10^{-3}\) |
| 2 | 515.0 | \(3.77\times10^{-6}\) |
| 3 | \(1.17\times10^{4}\) | \(7.32\times10^{-9}\) |
§07 ── log-periodicity seen in the laboratory
Every factor of 22.7 brings another bound state of the same shape ── the Efimov ladder.
In 2006 it was actually observed in ultracold caesium atoms.
Back in the language of phase: each factor of 22.7 in size advances the phase \(s_0\ln R\) by \(\pi\).
★ That is exactly the same arithmetic as §01's "a fixed angle per decade" (\(s_0\ln10=133°\)).
08Connecting to Episode 5 ── complex dimensions and the renormalisation group
In Episode 5 the couplings ran with \(\ln\mu\) ── \(d\alpha/d\ln\mu=\beta(\alpha)\). "A derivative with respect to \(\ln\)(scale)" ── precisely the imaginary-order arena.
★ A log-periodic oscillation multiplying a power law ── discrete scale invariance
| Phenomenon | Status | Note |
|---|---|---|
| Efimov states | ○ measured | ultracold atoms (2006); ratio 22.7 |
| \(1/r^2\) fall | ◎ exact | whenever \(\lambda>1/4\) |
| Logarithmic corrections at criticality | △ case by case | can appear on discrete lattices |
| Diffusion-limited aggregation | △ numerical | reported, but debated |
| Log-periodicity in earthquakes / finance | × disputed | ★ claimed, but not statistically established |
Only the top two are solid. The other three are claims ── log-periodicity is a pattern one wants to find, so the statistics need care.
09The whole picture ── two independent knobs
| Knob | Acts on | Formula | Where it appeared |
|---|---|---|---|
| Real part \(\alpha\) | slope of the amplitude | \(\omega^\alpha\to6\alpha\) dB/oct | Episodes 1–2 |
| Real part \(\alpha\) | constant phase | \(\alpha\pi/2\) | Episode 10 (arrow of time) |
| Imaginary part \(\beta\) | constant factor on amplitude | \(e^{-\beta\pi/2}\) | overall size only |
| Imaginary part \(\beta\) | log-periodic phase | \(\beta\ln\omega\) | ★ this episode (scale invariance) |
§09 ── the plane of orders
Real part ＝ direction and strength of the arrow of time ／ Imaginary part ＝ periodicity in scale
The two are independent, so the order is a plane, not a line.
And ── ★ physics occupies only a very narrow region of that plane.
\(\alpha\in[-1/2,\,2]\) (Episode 2's bounds) and \(\beta\) almost always 0.
\(\beta\neq0\) arises only where a critical \(1/r^2\) appears.
[Our hypothesis] Why is \(\beta\) nearly always zero?
Episode 2 argued that physics is nearly integer-order because nature is local. Apply the same logic to the imaginary part ──
\(\beta\neq0\) means "one particular ratio is special". But laws of nature usually either have no special length or have exactly one.
★ The former gives continuous scale invariance (\(\beta=0\)); the latter gives a cutoff at that length (Episode 6). To get a "ladder of ratios" you need a special balance like \(1/r^2\) ── \(1/r^2\) is the only potential with exactly the same \(r\)-dependence as the kinetic energy \(\hbar^2/2mr^2\), so no length is fixed and a ratio is fixed instead.
Verdict: a restatement of known facts, not a new claim. But it does give a reason why \(\beta\) is rare.
Exercises
- Why does imaginary-order differentiation leave the timbre unchanged?
Show answer
Timbre is largely set by the shape of the amplitude spectrum, and \(|(i\omega)^{i\beta}|=e^{-\beta\pi/2}\) is a plain constant independent of \(\omega\).
Every frequency passes with the same gain, so the spectral shape is preserved perfectly ── only the phase changes.
(Strictly, phase affects perception too. Apply it strongly and you get a soft "pre-echo" ── that is the chirp.) - Connect the Efimov ratio 22.7 with §01's "a fixed angle per decade".
Show answer
Both are phases proportional to a logarithm.
§01: \(\varphi=\beta\ln\omega\). Efimov: \(\varphi=s_0\ln R\).
The next bound state sits where the phase has advanced by \(\pi\), so \(s_0\ln(R'/R)=\pi\), i.e. \(R'/R=e^{\pi/1.00624}=22.69\).
★ The "period" of a log-periodicity is expressed as a ratio ── a multiplicative world, not an additive one. - What does "the tail is crushed by \(e^{-\beta\pi}\)" mean experimentally?
Show answer
That making a log-periodic oscillation faster costs you exponentially in amplitude: \(10^{-6}\) at \(\beta=3\), \(10^{-10}\) at \(\beta=5\).
So observable log-periodic phenomena are confined to large ratios (i.e. small \(\beta\)) ── Efimov's 22.7 being exactly that (\(s_0\approx1\)).
★ Conversely, a claim of "log-periodicity with a ratio of 1.1" needs very strong evidence.
Summary Widening the order into a plane
\((i\omega)^{i\beta}=e^{-\beta\pi/2}e^{i\beta\ln\omega}\) ── the amplitude is a constant independent of \(\omega\); the phase goes as \(\ln\omega\). Imaginary-order differentiation is an all-pass filter (confirmed to 12 digits with white noise).
The group delay is \(-\beta/\omega\) (confirmed with tone bursts, matching theory to 0.01 µs between 200 and 1600 Hz). A click becomes a hyperbolic chirp ── an operation that stretches time without touching the timbre.
The impulse response has a closed form, \(t\,h(t)=A(\beta)\sin(\beta\ln t-\arg\Gamma(1+i\beta))\) ── a sinusoid on a logarithmic scale. But the tail amplitude \(A(\beta)=e^{-\beta\pi}|\Gamma(1+i\beta)|/\pi\) is already \(10^{-6}\) at \(\beta=3\). Which is why log-periodicity is hard to find.
★ Real orders are the tool of time translation (Fourier); imaginary orders the tool of scale transformation (Mellin).
In physics the imaginary part turns on only when an attractive \(1/r^2\) passes the critical \(\lambda=1/4\). There the wavefunction becomes periodic in \(\ln r\) ── and that is measured, as the Efimov ladder with ratio 22.7 (\(s_0=1.00624\)).
And Episode 5's renormalisation group was a flow in \(\ln\mu\). Complex dimension ＝ discrete scale invariance.
★ The order is a plane, not a line, and physics lives in one narrow corner of it.
Next time we line them all up on a single ladder. And collecting nature's spectral exponents, we find that only turbulence's \(-5/3\) is neither integer nor half-integer. Where does it come from?
This document is Episode 13 of the series "Thinking in Waves". The computation is in kenshou/kyosu.js. On the numbers: applying \((i\omega)^{i\beta}\) to a real signal uses the convention of defining it on positive frequencies and conjugate-extending to negative ones (as for the Hilbert transform) ── without that convention the output is complex and is not "sound". The group-delay measurement uses the centroid of a tone burst, so lower frequencies are more affected by the finite window (the 7.6 µs discrepancy at 200 Hz is that). Efimov's \(s_0=1.00624\) is the known value for three identical bosons in the zero-range limit, not derived here. Measured ratios can depart from the ideal 22.7 (finite-range corrections, the three-body parameter), and experimental values have spread. Applying discrete scale invariance to earthquakes or finance is not a position of this series ── the table marks it "disputed". \(\Gamma(1+i\beta)\) is computed by a Lanczos approximation and checked against the exact \(|\Gamma(1+i\beta)|=\sqrt{\pi\beta/\sinh\pi\beta}\) to 8 digits. ── To print, use your browser's Print → Save as PDF.