Thinking in Waves Episode 38 / Part XIII — The Frontier, as Waves
Looking at the stage itself as a wave
Quantum Gravity as Waves —
Dimension running is order running
The Planck length comes out in one line — where a wavelength equals its own Schwarzschild radius.
★★ And what many theories agree on, "dimension runs from 4 to 2 at short distances", read through Episode 30's \(\alpha=(d-3)/2\), becomes — ★ order runs from \(+1/2\) to \(-1/2\).
★ We also get why it stops at 2 — that is where gravity's coupling becomes dimensionless.
For 37 episodes, spacetime has been a given stage. Even in Episode 24, gravitational waves were ripples on a flat background.
★ The question of quantum gravity is what happens when those ripples make the stage.
01The Planck length, in one line of wave language
| Quantity | Value |
|---|---|
| Estimate above, \(\sqrt{2G\hbar/c^3}\) | 2.2854e−35 m |
| Planck length \(\sqrt{G\hbar/c^3}\) | 1.6162e−35 m |
| Planck time | 5.3912e−44 s |
| Planck energy | 1.2209e+19 GeV |
The point of §01
Up to the factor \(\sqrt2\), it is one line. In wave language — the Planck length is "the length at which a wave, made short enough, traps itself with its own energy."
★ It is the limit of Episode 6's "mass = cutoff frequency" — push the cutoff up by adding energy, and gravity makes a cutoff of its own.
From electroweak (246 GeV, \(8\times10^{-19}\) m) to Planck is 17 decades — that distance is Episode 15's hierarchy problem.
02Why only gravity is "non-renormalizable" — the coupling carries a dimension
| Force | Coupling | Dimension |
|---|---|---|
| Electromagnetic | \(\alpha=1/137\) | ★ dimensionless |
| Strong | \(\alpha_s\approx0.12\) | ★ dimensionless |
| ★ Gravity | \(G/(\hbar c)=1/E_P^2\) | ★★ \([\text{energy}]^{-2}\) |
A coupling with a dimension can only be made dimensionless by multiplying by energy — the effective coupling is \((E/E_P)^2\).
| Energy \(E\) | \((E/E_P)^2\) | What it is |
|---|---|---|
| 1 GeV | 6.7e−39 | proton |
| 246 GeV | 4.1e−34 | electroweak |
| \(10^{16}\) GeV | 6.7e−7 | grand unification |
| \(1.2\times10^{19}\) GeV | 1.0 | ★ perturbation theory breaks |
The point of §02
★★★ This is what "non-renormalizable" means — the coupling grows with energy and passes 1 at Planck.
Episode 34's Yang–Mills was the opposite: weaker at high energy (asymptotic freedom).
★ In wave language — a medium in which shorter waves interact more strongly. The most extreme case of Episode 16's "dispersive medium."
03★★ What many theories agree on: dimension runs
Candidate theories of quantum gravity differ enormously, but there is one thing they tend to agree on — at short distances the "dimension" drops from 4 to 2 (spectral dimension reduction).
Spectral dimension is dimension measured by diffusion (the quantity we used in Episode 37) — \(P(s)\propto s^{-d_s/2}\).
Rather than assert it, we measure it. We take the spatial part from \(P(s)=\int\frac{d^3k}{(2\pi)^3}e^{-s(k^2)^z}\) and add the unmodified time direction.
| \(z\) | spatial \(3/z\) (measured) | ★ \(d_s=1+3/z\) | order \(\alpha=(d_s-3)/2\) |
|---|---|---|---|
| 1.0 | 3.00000 | 4.00000 | +0.50000 |
| 1.5 | 2.00000 | 3.00000 | 0.00000 |
| 2.0 | 1.50000 | 2.50000 | −0.25000 |
| 3.0 | 1.00000 | 2.00000 | −0.50000 |
The point of §03 — ★★★ the core of this episode
Since Episode 30 gave \(\alpha=(d-3)/2\),
"dimension runs" means "order runs."
\(d_s:4\to2\ \Longleftrightarrow\ \alpha:+1/2\to-1/2\)
| \(d_s\) | \(\alpha\) | Meaning in Episode 30 | Digital (Episode 31) |
|---|---|---|---|
| 4 | +0.5 | a tail, \(s^{-3/2}\) | IIR (infinite taps) |
| 3 | 0 | ★ \(\delta\) itself (no distortion) | FIR (1 tap) |
| 2 | −0.5 | ★ a tail \(s^{-1/2}\); critical \(1/r^2\) | IIR (decaying as \(k^{-1/2}\)) |
★ Stated carefully
\(d_s=2\) is the dimension that sat exactly at the critical \(\lambda=1/4\) in Episode 30 (the boundary where Episode 13 said "the exponent goes complex for \(\lambda>1/4\)").
★ But this is an observation, not a claim — Episode 30's \(\alpha\) is the order of the wave equation's Green function, while \(d_s\) here is an effective dimension measured by diffusion. They are separately defined quantities.
★★ It may only be that we used the same word "dimension" for both (the discipline of Episodes 29 and 30).
04Why it stops at 2 — the dimension where the coupling goes dimensionless
| Spacetime dimension \(d\) | Dimension of \([G]\) | As a perturbation theory |
|---|---|---|
| 2 | ★ dimensionless | ★★ renormalizable (at the boundary) |
| 3 | \([E]^{-1}\) | non-renormalizable |
| 4 | \([E]^{-2}\) | ★ non-renormalizable (reality) |
The point of §04
★★★ At \(d=2\), gravity's coupling is exactly dimensionless.
★ So the expectation "\(d_s\to2\) in the ultraviolet" is — a restatement of the wish that "the coupling become dimensionless in the ultraviolet."
★ Put beside Episode 25's dimensional transmutation — QCD makes a dimensionful \(\Lambda\) out of a dimensionless \(\alpha_s\). Quantum gravity wants the reverse: to make a dimensionful \(G\) dimensionless. Both are demands of "only dimensionless quantities are physics" (Episodes 7 and 15).
05★ Translating shows you what to measure
| In the language of order | Observable form | Episode |
|---|---|---|
| \(\alpha\) changes | dB/oct changes by \(6\alpha\) | Ep. 1 |
| \(\alpha\) changes | phase changes by \(90\alpha\) degrees | Ep. 1 |
| \(d_s\) departs from 3 | propagation grows a tail (FIR→IIR) | Ep. 30, 31 |
| ★ \(z\ne1\) | ★ group velocity depends on frequency | Ep. 16 |
| Source | Distance | Photon \(E\) | ★ \(\Delta t\) |
|---|---|---|---|
| GRB (\(z\approx1\)) | \(10^{26}\) m | 10 GeV | 0.27 s |
| GRB (\(z\approx1\)) | \(10^{26}\) m | 100 GeV | 2.7 s |
| Crab Nebula | \(6\times10^{19}\) m | \(10^4\) GeV | 1.6e−4 s |
The point of §05
For a gamma-ray burst, 0.3 to 3 seconds — comfortably measurable.
★★★ And it has been measured: the first-order effect is already excluded (by Fermi and others, up to scales at or above the Planck energy).
★ So "running order," at least at first order, is not there up to Planck. A second-order effect still has room.
★ Add this one to Episode 15's list of doors still open.
06What wave language can and cannot say
| Item | Verdict | Basis |
|---|---|---|
| Estimating the Planck length | ◎ yes | wavelength = its own \(r_s\). One line |
| Why gravity is non-renormalizable | ◎ yes | coupling is \([E]^{-2}\) |
| \(d_s=1+3/z\) | ◎ numerics | measured directly from the modified dispersion |
| Dimension running = order running | ◎ correspondence | just uses Episode 30's \(\alpha=(d-3)/2\) |
| ★ Which theory is right | × no | wave language does not pick a theory |
| ★ Whether \(z\) really becomes 3 | × no | a model assumption; no observation yet |
| ★ What spacetime is made of | × no | this episode only looked at a dispersion relation |
The point of §06 — ★ honestly
What this episode did was translate an existing expectation into another language. Saying "order runs" instead produces no new prediction.
★ But translating does make clear what to measure — and, having measured, we could pick up the fact that the first-order effect is already excluded.
Exercises
- People often say "no length shorter than the Planck length exists." Does §01 show that?
Show answer
No. All §01 showed is that "if you try to measure that length with a wave, the measuring wave itself makes a black hole."
★ "Cannot be measured" and "does not exist" are different — the same caution as Episode 5's "a period longer than the age of the universe is indistinguishable from a constant." Indistinguishability is not proof of absence.
★★ The question "is there any point writing into a theory something that cannot be measured?" does remain, and the answer depends on one's stance. What wave language can speak to is the limit of measurement. - In §03, \(d_s=2\) coincided with Episode 30's critical point, but we held it as pending. What would settle it?
Show answer
Redefining both \(\alpha\)'s from the same quantity. Episode 30's \(\alpha\) is the order of the Green function's singularity; this episode's \(d_s\) is the exponent of a diffusion return probability.
★ To settle it you would have to actually build the Green function of the modified dispersion relation and measure its front exponent — then "is it the same \(\alpha\)?" becomes a number.
★★ This episode did not do that, so it stays pending. Having decided in Episodes 29–30 that "when you see the same number, check what you divided by," anything unchecked does not become a claim. - §05 said the first-order effect is excluded. Does that mean "there is no quantum-gravity effect"?
Show answer
No — only a particular form was excluded. What was measured is the first-order dependence \(\Delta t\propto E/E_P\); at second order, \(\Delta t\propto(E/E_P)^2\), the effect is \(10^{19}\) times smaller and utterly out of reach.
★ And most theories predict second order to begin with (first order breaks Lorentz invariance so strongly that other experiments constrain it too).
★★ So this exclusion means "the easiest form to measure is gone" — the search just got harder. On Episode 15's scorecard it would read "× excluded (under the loosest assumption only)."
Summary — dimension running is order running
The Planck length comes out in one line — where a wavelength equals its own Schwarzschild radius (\(1.616\times10^{-35}\) m). It is the limit of Episode 6's "mass = cutoff frequency": push the cutoff up and gravity makes a cutoff of its own.
Gravity alone is non-renormalizable because its coupling carries the dimension \([E]^{-2}\). The effective coupling \((E/E_P)^2\) passes 1 at Planck — the opposite of Episode 34's Yang–Mills; in wave language, "a medium in which shorter waves interact more strongly."
★★★ And the core of this episode — the widely shared expectation that \(d_s\) runs 4 → 2 at short distance,
read through Episode 30's \(\alpha=(d-3)/2\), was
"order runs from \(+1/2\) to \(-1/2\)."
We checked it numerically — from \(P(s)=\int d^3k\,e^{-s(k^2)^z}\) the spatial part is exactly \(3/z\), and adding the time direction gives \(d_s=1+3/z\). \(z=1\to d_s=4\); \(z=3\to d_s=2\).
★ Why it stops at 2 also falls out — \([G]=[E]^{2-d}\), so at \(d=2\) gravity's coupling is exactly dimensionless. "\(d_s\to2\) in the UV" was a restatement of "let the coupling become dimensionless in the UV." Beside Episode 25's dimensional transmutation (QCD makes a dimension out of a dimensionless number) it runs the other way, and both are demands of "only dimensionless quantities are physics."
★ That \(d_s=2\) coincides with Episode 30's critical \(\lambda=1/4\) is an observation, not a claim — separately defined quantities, so we held it as pending (the discipline of Episodes 29–30).
Finally — translating makes clear what to measure. If \(z\ne1\), group velocity depends on frequency, and a 10 GeV photon from a GRB arrives 0.3 s apart. ★ And it has been measured: the first-order effect is already excluded (up to the Planck scale). ★ This episode only put an existing expectation into other words; it produced no new prediction.
★ Next time that axis shows up in physics as an extra dimension — holography.
★★ The AdS mass–dimension relation \(\Delta=\frac d2\pm\sqrt{\frac{d^2}4+m^2L^2}\) has exactly the form of Episode 13's \(s=\frac12\pm\sqrt{\frac14-\lambda}\). And the BF bound is Episode 13's critical point itself.
This document is Episode 38 of the "Thinking in Waves" series. Computations are in kenshou/juryoku2.js. On the numbers in this episode: the Planck units, the effective coupling, the numerical measurement of spectral dimension, and the arrival-time differences were all computed here. The dimensional-analysis derivation of the Planck length, the non-renormalizability of gravity, spectral-dimension reduction (including \(d_s=1+3/z\)), \([G]=[E]^{2-d}\), and the GRB timing constraints are all well-known results and none of them are discoveries of this article — what this article did was put them on the single axis of "order" and translate "dimension runs" into "order runs." It produced no new prediction. The \(d_s=1+3/z\) of §03 depends on the model assumption that the time direction's dispersion is unmodified (Hořava-type). Theories that also modify time give a different value. And "dimensional reduction" itself is not a single agreed quantity — its definition and value differ by theory, and this article simply took one representative form. The \(\Delta t\) of §05 assumes first-order dependence; a real analysis must include cosmological redshift effects and the source's own intrinsic emission delays. "The first-order effect is already excluded" describes reports from Fermi and others; it is not an analysis performed here. — To print, use your browser's Print and "Save as PDF."