Thinking in Waves Episode 39 / Part XIII — The Frontier, as Waves
The axis drawn in Episode 22 becomes a dimension
Holography as Waves —
The extra dimension is the log-scale axis
The AdS relation \(\Delta=\frac d2\pm\sqrt{\frac{d^2}4+m^2L^2}\) is ★ the same quadratic as Episode 13's \(s=\frac12\pm\sqrt{\frac14-\lambda}\).
★★ And the boundary order is the square root itself — \(\alpha=2\sqrt{\frac{d^2}4+m^2L^2}\). The BF bound is where it hits zero; below it, the order goes imaginary.
★ And further — the wavelet plane's metric is literally the AdS metric.
Episode 13 found "imaginary order = the generator of scale transformations." Episode 22 found that a wavelet draws a \(\ln\omega\) axis.
★ This episode is about that axis being called, in physics, "the extra dimension."
01The same quadratic, twice
The point of §01
Set \(d\to1\) and \(m^2L^2\to-\lambda\) — ★ they are literally the same equation.
Both have a \(1/(\text{distance})^2\) potential, both produce two powers, and both have their critical point where the discriminant changes sign.
| \(m^2L^2\) (\(d=4\)) | \(\nu=\sqrt{d^2/4+m^2L^2}\) | \(\Delta_+\) | \(\Delta_-\) | Verdict |
|---|---|---|---|---|
| 0 | 2.000000 | 4 | 0 | stable |
| −1 | 1.732051 | 3.7321 | 0.2679 | stable |
| −3.75 | 0.500000 | 2.5 | 1.5 | stable |
| −4 | 0 | 2 | 2 | ★ BF bound (double root) |
| −4.25 | 0.5 i | complex | complex | ★ BF violated → unstable |
★ The BF bound \(m^2L^2=-d^2/4\) is Episode 13's critical \(\lambda=1/4\). A negative mass squared is allowed for exactly the reason a \(1/r^2\) potential does not collapse until it is critical.
02Solve it, and measure the boundary two-point function
Words are cheap, so we solve the wave equation numerically. With boundary momentum \(k\) included:
Start deep inside (large \(z\)) with the decaying solution, integrate out to the boundary (\(z\to0\)), and decompose \(\varphi=A z^{\Delta_-}+B z^{\Delta_+}\).
For \(d=4,\ m^2L^2=-3.75\) (\(\nu=1/2\)) — the exact answer is \(B/A=\frac{\Gamma(-\nu)}{\Gamma(\nu)}(k/2)^{2\nu}=-k\).
| \(k\) | \(B/A\) (numerical) | \(B/A\) (exact) | ratio |
|---|---|---|---|
| 0.25 | −0.24999750 | −0.25 | 0.999990 |
| 1 | −0.99999000 | −1.00 | 0.999990 |
| 4 | −3.99996000 | −4.00 | 0.999990 |
| 8 | −7.99992000 | −8.00 | 0.999990 |
The point of §02
The slope of \(\ln|B/A|\) against \(\ln k\) is 1.000000 (prediction \(2\nu=1\)).
★ That \(B/A\) is the boundary two-point function — \(\langle OO\rangle(k)\propto k^{2\nu}\).
★★ Simply integrating a wave from the interior out to the edge produced the power of a correlation function.
03★★ The order is the square root itself
| \(m^2L^2\) (\(d=4\)) | \(\nu\) | ★ order \(\alpha=2\nu\) | dB/oct | phase [deg] |
|---|---|---|---|---|
| +4 | 2.828427 | 5.656854 | 33.941 | 509.12 |
| 0 | 2 | 4 | 24.000 | 360.00 |
| −3 | 1 | 2 | 12.000 | 180.00 |
| −3.75 | 0.5 | 1 | 6.000 | 90.00 |
| −4 (BF) | 0 | ★ 0 | 0.000 | 0.00 |
| −4.25 | 0.5 i | ★ imaginary | ★ imaginary | ★ imaginary |
The point of §03 — ★★★ the core of this episode
The bulk mass sets the boundary's order.
\(\alpha=2\sqrt{d^2/4+m^2L^2}\)
★ The BF bound is "where the order becomes zero" — where the power disappears and a logarithm takes over.
★★ Below it the order is imaginary — exactly Episode 13's structure.
★ To Episode 14's "three reasons the order leaves the integers," this adds a fourth — the bulk mass.
04Cross the BF bound and log-periodicity appears
At \(d=4,\ m^2L^2=-4.25\) (\(\nu=0.5i\)) the solution is \(\varphi=z^{d/2}\cos(\nu\ln z+\delta)\) — its zeros should be equally spaced in \(\ln z\).
| zero # | \(\ln z\) | gap | ratio in \(z\) |
|---|---|---|---|
| 1 | −5.793228 | — | — |
| 2 | −12.076413 | 6.283185 | 535.4917 |
| 3 | −18.359599 | 6.283185 | 535.4917 |
| 4 | −24.642784 | 6.283185 | 535.4917 |
The point of §04
Measured gap 6.283185 against the prediction \(\pi/|\nu|=6.283185\) — a difference of \(1.7\times10^{-9}\).
★ This is the very same mechanism as Episode 13's Efimov states. There the ratio was 22.7, here it is 535.49 — only the number differs; the phenomenon, log-periodicity, is identical.
★★ Physically, a BF violation means the boundary theory has an unstable operator — like Episode 32's Higgs with \(m^2<0\), it is the signal that the vacuum is somewhere else.
05★★ The extra dimension is the wavelet plane
The point of §05 — ★★★
The same metric.
radial \(z\) = scale \(a\) = \(1/\omega\).
★ The wavelet plane's natural metric is the invariant metric of the affine group (translation plus dilation), which is the hyperbolic upper half plane — another name for \(\mathrm{AdS}_2\).
★★ A scale transformation \((b,a)\to(\lambda b+c,\ \lambda a)\) is an isometry — numerically \(ds^2\) is completely unchanged even multiplying \(\lambda\) by \(10^6\) (ratio 1.00000000).
★ That is why the log-scale axis carries a distance and can behave as a dimension.
| \(z_1\to z_2\) | proper distance (numerical) | \(|\ln(z_2/z_1)|\) |
|---|---|---|
| 1 → 10 | 2.302585 | 2.302585 |
| 1 → \(10^3\) | 6.907755 | 6.907755 |
| 1 → \(10^{17}\) | 39.143947 | 39.143947 |
★ Connecting to Episode 38
One decade (a factor of 10) is \(\ln10=2.302585\) times \(L\).
★★ Episode 38's "17 decades from electroweak to Planck" becomes a proper distance of 39.14 \(L\) — in holography the hierarchy problem turns into a length in the extra dimension.
★ This is what "geometrizing the hierarchy problem" in Episode 15 actually meant.
06Entanglement entropy is logarithmic — order zero
In \(\mathrm{AdS}_3\), the geodesic joining two boundary points a distance \(\ell\) apart is a semicircle. We measure its length numerically.
| \(\ell\) | \(\varepsilon\) | geodesic length (numerical) | \(2\ln(\ell/\varepsilon)\) | difference |
|---|---|---|---|---|
| 1 | \(10^{-2}\) | 9.21014034 | 9.21034037 | 2.0e−4 |
| 1 | \(10^{-4}\) | 18.42068072 | 18.42068074 | 2.0e−8 |
| 1 | \(10^{-6}\) | 27.63102112 | 27.63102112 | 1.0e−11 |
The point of §06
\(S=\dfrac{\text{geodesic length}}{4G}=\dfrac c3\ln\dfrac{\ell}{\varepsilon}\) — a logarithm.
★ In this series' language a logarithm is the mark of order zero (Episode 2's 1/f, Episode 37's weak localization, and §03's BF bound). Getting a log instead of a power is the signature of 1+1-dimensional criticality.
★★ And \(\varepsilon\) is the radial cutoff — that is, the frequency cutoff (Episode 22's window). "The entropy depends on \(\varepsilon\)" means it depends on how finely you looked.
07Counting degrees of freedom, and the renormalization group
| \(d\) | \(\varepsilon\) | \((L/\varepsilon)^{d-1}\) | Meaning |
|---|---|---|---|
| 2 | \(10^{-3}\) | \(10^{3}\) | lattice cells |
| 4 | \(10^{-3}\) | \(10^{9}\) | lattice cells |
| 4 | \(10^{-6}\) | \(10^{18}\) | ★ counted by area |
★ The same counting as Episode 6's holographic bound (information counted by area, not volume). Cutting off the radial direction was cutting off frequency.
The point of §07 — the RG flow is radial motion
Since \(\ln\mu=-\ln z\) — the running couplings of Episode 5 are, in the extra dimension, just translation.
★ "Raising the energy scale" = "approaching the boundary." Episode 25's dimensional transmutation (a dimension born from a dimensionless number) also becomes, on this axis, a question of how far you walked.
08What wave language can and cannot say
| Item | Verdict | Basis |
|---|---|---|
| \(\Delta\) relation = Episode 13's | ◎ analytic | \(d\to1,\ m^2L^2\to-\lambda\) |
| BF bound = critical \(\lambda=1/4\) | ◎ analytic | where the order hits zero |
| \(\langle OO\rangle\propto k^{2\nu}\) | ◎ numerics | slope 1.000000 (prediction 1) |
| ★ order \(\alpha=2\nu\) | ◎ correspondence | ★ the order is the square root |
| BF violation → log-periodicity | ◎ numerics | gap \(\pi/|\nu|\) measured |
| ★ extra dimension = wavelet plane | ◎ analytic | ★ literally the same metric |
| entanglement = geodesic = log | ◎ numerics | matches \(2\ln(\ell/\varepsilon)\) |
| ★ whether the duality is true | × no | assumed here, not tested |
| ★ whether it applies to our world | × no | AdS is not our universe |
The point of §08 — ★ honestly
The duality itself — "a gravity theory = a boundary field theory" — is not tested here. We assumed it and computed inside it.
★ And AdS (negative curvature) is not our universe (positive cosmological constant) — none of the numbers in this episode can be compared to a measurement.
★★ All this article can say is a restatement: if the duality holds, the extra dimension is the log-scale axis. As in Episode 38 — no new prediction came out.
Exercises
- §05 says the wavelet plane and AdS share a metric. Is that a coincidence?
Show answer
No. Both come from the same symmetry — translation plus dilation (the affine group).
★ The hyperbolic upper half plane is the affine group's homogeneous space, and \(\mathrm{AdS}_2\) is another name for it. Any scale-invariant theory automatically carries this geometry.
★★ But "same metric" is not "same physics" — the wavelet plane has no dynamics (no Einstein equation). The stage can have the same shape while what happens on it is another matter. Per the discipline of Episodes 29–30, we stop at "the symmetry is the same." - §03 found the order to be the square root itself. What does that add to Episode 14's "reasons the order leaves the integers"?
Show answer
Episode 14's three were ① superposition of many time constants (Ep. 2), ② dimension (Ep. 30), ③ continuous variation at a critical point (Ep. 13).
★ The fourth is the bulk mass — \(\alpha=2\sqrt{d^2/4+m^2L^2}\). A mass, a continuously adjustable quantity, moves the order continuously.
★★ What is striking is that this one contains ② and ③ at once: \(d\) appears in it, so it is about dimension, and the thing under the root changes sign, so it is about criticality. Holography packs the reasons for a non-integer order into a single formula. - §06's entropy depends on \(\varepsilon\). Is that a flaw in the theory?
Show answer
Not a flaw — it is inevitable. Until you say how finely you look, the number of degrees of freedom is not defined — the same point as Episode 22's window.
★ What is physical is the difference: \(S(\ell_2)-S(\ell_1)=\frac c3\ln(\ell_2/\ell_1)\) contains no \(\varepsilon\). "Only dimensionless quantities are physics" (Episodes 7, 15) is at work again.
★★ And separating the \(\varepsilon\)-dependent part from the rest is renormalization — which, as §07 showed, is motion in the radial direction.
Summary — the extra dimension is the log-scale axis
The AdS scalar's relation \(\Delta=\frac d2\pm\sqrt{\frac{d^2}4+m^2L^2}\) is the same quadratic as Episode 13's \(s=\frac12\pm\sqrt{\frac14-\lambda}\) (\(d\to1,\ m^2L^2\to-\lambda\)). Hence the BF bound \(m^2L^2=-d^2/4\) is Episode 13's critical \(\lambda=1/4\).
We solved the wave equation numerically and measured the boundary two-point function — the slope of \(\ln|B/A|\) against \(\ln k\) is 1.000000 (prediction \(2\nu=1\)), matching the exact \(B/A=-k\) to five digits. Just integrating a wave from inside out produces a correlation function's power.
★★★ The core — the bulk mass sets the boundary's order.
\(\alpha=2\sqrt{d^2/4+m^2L^2}\)
The BF bound is where it hits zero (a logarithm); below it the order is imaginary.
Indeed, past the bound at \(m^2L^2=-4.25\) the zeros are equally spaced in \(\ln z\) (gap 6.283185, matching \(\pi/|\nu|\) to \(1.7\times10^{-9}\), ratio in \(z\) 535.49) — the same log-periodicity as Episode 13's Efimov states.
★★ And — the wavelet plane's metric \((db^2+da^2)/a^2\) is literally the AdS metric. Radial \(z\) = scale \(a\) = \(1/\omega\). Because a scale transformation is an isometry, the log-scale axis acquires a distance and can act as a dimension. Episode 38's "17 decades from electroweak to Planck" becomes a proper distance of 39.14 \(L\) — the hierarchy problem turns into a length.
Entanglement entropy is a geodesic length \(2\ln(\ell/\varepsilon)\) (matching to \(10^{-11}\)) — a logarithm = order zero, the same place as Episode 2's 1/f. The degrees of freedom count as \((L/\varepsilon)^{d-1}\) — Episode 6's holographic bound. And the renormalization group, in the extra dimension, is just translation (Episode 5).
★ Honestly: the duality itself was assumed, not tested. AdS is not our universe either. ★ Once again this is a translation, not a new prediction.
★ Next time, a wave that answers the drive at half its period — time crystals.
★★ Subharmonics already appeared in Episode 23's nonlinearity, but this time the broken symmetry is time translation. Episode 32's lesson — that symmetry breaking needs dissipation — gets replayed in the time direction.
This document is Episode 39 of the "Thinking in Waves" series. Computations are in kenshou/horo.js. On the numbers here: the numerical solution of the AdS wave equation and the measurement of \(B/A\), the zero spacing under BF violation, the geodesic lengths, the proper distances, and the check of metric invariance were all computed here. The AdS/CFT correspondence, \(\Delta=\frac d2\pm\sqrt{\frac{d^2}4+m^2L^2}\), the Breitenlohner–Freedman bound, \(\langle OO\rangle\propto k^{2\nu}\), the Ryu–Takayanagi entanglement entropy \(\frac c3\ln(\ell/\varepsilon)\), the RG/radial correspondence, and the fact that the upper half plane is the affine group's homogeneous space are all well-known results and none are discoveries of this article — what this article did was put them on the single axis of "order" and show they are the same object as Episodes 13 and 22. It produced no new prediction. In particular, §05's "wavelet plane = \(\mathrm{AdS}_2\)" means the metric and symmetry coincide; it is not a claim that the wavelet transform has gravitational dynamics. The numerics of §02 are restricted to \(\nu=1/2\), a non-integer case; for integer \(\nu\) logarithmic terms appear and the decomposition changes. AdS has a negative cosmological constant and is not our universe (which has a positive one). — To print, use your browser's Print and "Save as PDF."