c·t = CONST, THAT CLICKS EPISODE 43 ／ Part V — where the tool breaks
A smallest length is simply incompatible with conformal invariance
The Planck length was
in the bookkeeping column
\(\ell_P\) has weight \(+1\) — it is just a length.
And a theory with a smallest length cannot be conformally invariant, by hypothesis.
Episode 18 produced "one bit \(\leftrightarrow\) 1.96 fm". So where does the Planck length itself sit on the weight table? The answer is unexpected — \(\ell_P\) has weight \(+1\); it is just a length, in the bookkeeping column. And the claim that "there is a smallest length" is simply incompatible with conformal invariance. This is the clearest edge of the tool's reach in all of Part V.
01Where the Planck length sits on the weight table
| Quantity | Weight | Class |
|---|---|---|
| \(\ell_P=1.616\times10^{-35}\) m | \(+1\) | a length = bookkeeping |
| \(t_P=5.391\times10^{-44}\) s | \(+1\) | a time = bookkeeping |
| \(m_P=2.176\times10^{-8}\) kg | \(-1\) | a mass = bookkeeping |
| \(E_P=1.221\times10^{19}\) GeV | \(-1\) | an energy = bookkeeping |
| \(L/\ell_P\) (a ratio of lengths) | \(0\) | dimensionless = physics |
Conclusion of §01
Every Planck quantity sits in the bookkeeping column. Only ratios are in the physics column.
── "The smallest length is \(\ell_P\)" is, as it stands, not yet a sentence in Episode 3's sense.
To make it one: \(L/\ell_P\ge1\).
02The core — this is where the tool really breaks
The question is whether \(G\) transforms — Episode 36's (A)/(B) applied directly to \(G\).
| Reading | \(\ell_P\) | \(L/\ell_P\) | Consequence |
|---|---|---|---|
| (A) \(G\) transforms too (\(G\to\Omega^2G\)) | moves with everything | weight \(0\) | the minimum survives → notation |
| (B) \(G\) is held fixed | does not move | weight \(+1\) | you can push below the minimum → a claim |
The main point of this episode
A theory with a smallest length cannot be conformally invariant at all.
Conformal invariance demands the absence of a scale, and a smallest length is a scale.
── Part V has been about the tool breaking or failing to reach. This is different:
it is excluded by hypothesis — the clear edge of where the tool works.
03So what is the Planck scale, then?
| Mass [kg] | \(\lambda_C\) [m] | \(r_s\) [m] | \(\lambda_C/r_s\) |
|---|---|---|---|
| electron \(9.11\times10^{-31}\) | \(3.86\times10^{-13}\) | \(1.35\times10^{-57}\) | \(2.9\times10^{44}\) |
| proton \(1.67\times10^{-27}\) | \(2.10\times10^{-16}\) | \(2.49\times10^{-54}\) | \(8.5\times10^{37}\) |
| \(m_P/\sqrt2=1.54\times10^{-8}\) | \(2.286\times10^{-35}\) | \(2.286\times10^{-35}\) | \(1.000\) |
| 1 g | \(3.52\times10^{-40}\) | \(1.49\times10^{-30}\) | \(2.4\times10^{-10}\) |
| 1 kg | \(3.52\times10^{-43}\) | \(1.49\times10^{-27}\) | \(2.4\times10^{-16}\) |
Conclusion of §03
A crossing point is a dimensionless statement. That is how the concept survives —
not as "a length", but as "the place where two effects come level".
Figure: the Compton wavelength and the Schwarzschild radius as the mass varies. One goes as \(1/m\) and the other as \(m\), so they must cross somewhere. That crossing is the Planck scale — not a length you can point at, but where the ratio is 1.
04How far from it are we?
Fifty doublings away — one step for every doubling of accelerator energy.
Specifying a one-dimensional position to Planck precision takes \(\log_2(R_H/\ell_P)=202.4\) bits; in three dimensions, 607 bits (the same number as Episode 38 §04).
05What do observations say?
| Measurement | \(E_{\rm QG}/E_P\) | Meaning |
|---|---|---|
| Fermi GRB 090510 (linear effect) | \(>7.6\) | excluded past the Planck energy |
| The same (quadratic effect) | \(>1.3\times10^{-8}\) | constrained only to \(10^{-8}\) of \(E_P\) |
A smallest length can make the speed of light depend slightly on energy (dispersion), which gamma-ray burst arrival times constrain. The linear effect is excluded. A smallest length that naively breaks Lorentz invariance is in trouble. The quadratic effect, meanwhile, is untouched over eight orders of magnitude. But not every theory with a smallest length breaks Lorentz invariance (formulations that do not exist) — this constraint bites only on those that do.
06The relation to Episode 18's 1.96 fm
That too was in the bookkeeping column. The physics is in the ratio to \(\ell_P\) — \(1.96\ \text{fm}/\ell_P=1.21\times10^{20}=2^{66.7}\). So "the length of one bit" sits 67 doublings above the Planck length.
07Auditing the weight table — Part V laid out
| Quantity | Weight | Why |
|---|---|---|
| The running of \(\alpha\) (Ep. 37) | \(0\) | dimensionless |
| \((D-1)(D-2)\) (Ep. 38) | \(0\) | a pure number |
| Kerr spin \(\chi\) (Ep. 39) | \(0\) | dimensionless |
| The holographic bound (Ep. 40) | \(0\) | area ÷ area |
| How special the beginning was (Ep. 41) | \(0\) | a bit count |
| The event horizon (Ep. 42) | \(0\) | causal structure |
| The apparent horizon (Ep. 42) | \(\ne0\) | a local quantity |
| The Planck length (Ep. 43) | \(+1\) | just a length |
Conclusion of §07
Only two things in Part V failed to land in the weight-0 column — the apparent horizon and the Planck length.
── Both are quantities that bring in a local yardstick.
(1) §02's (A)/(B) is a matter of convention. Whether \(G\) transforms is not decided by physics but by which transformation you define — take (A) and a conformal transformation is a change of units; take (B) and it is a physical claim. The claim here is not that one is right, but that you have to say which or you have not made a sentence — Episode 3, again.
(2) "A smallest length is incompatible with conformal invariance" refers to global conformal invariance. In theories with local Weyl transformations as a gauge symmetry (conformal gravity, Episode 34), a gauge choice may fix a scale, so the story is more involved — §02 is close to a tautology ("a theory with a scale is not scale-invariant"), and that is exactly why it is strong.
(3) That the Planck length is the smallest length is not an established fact. It is the length you can build from three constants by dimensions alone, and there is no evidence yet that anything happens there — §03's crossing is an indication that quantum and gravitational effects come level, not a claim that a minimum unit lives there.
(4) §05's \(E_{\rm QG}\) limits assume a particular form of dispersion relation. The Fermi value comes from the GRB 090510 analysis and depends on assumptions about photon emission times. And as noted, not every theory with a smallest length breaks Lorentz invariance (loop quantum gravity and noncommutative geometry both have formulations that do not).
(5) §06's 1.96 fm uses Episode 18's value directly, computed with \(R_H=1.3\times10^{26}\) m (with \(1.3725\times10^{26}\) it is 1.99 fm) — the 66.7 doublings is accurate to about that.
Exercises
- What is the conformal weight of the Planck length?
Show the answer
\(+1\). In \(\ell_P=\sqrt{\hbar G/c^3}\), \(\hbar\) and \(c\) have weight 0 and \(G\) has \(+2\), so \((+2)/2=+1\) — just a length, in the bookkeeping column. Only \(L/\ell_P\) is in the physics column. - Is "the smallest length is \(\ell_P\)" a sentence in Episode 3's sense?
Show the answer
No — it places its claim in a dimensionful quantity. To make it a sentence: \(L/\ell_P\ge1\). "If you have not named what you are comparing to, you have not yet made a sentence." - Can a theory with a smallest length be conformally invariant?
Show the answer
No. Conformal invariance demands the absence of a scale, and a smallest length is a scale — unlike the rest of Part V, this is excluded by hypothesis (subject to caveat 2). - State the Planck scale as a dimensionless claim.
Show the answer
The place where the Compton wavelength and the Schwarzschild radius have ratio 1 — \(\lambda_C/r_s=1\) at \(m=\sqrt{\hbar c/2G}=m_P/\sqrt2\). Not as "a length" but as "where two effects come level", the concept survives. - (Harder) How many doublings from the LHC to the Planck length?
Show the answer
About 50 (\(9.0\times10^{14}=2^{49.7}\)). At 13.6 TeV the length probed is \(1.45\times10^{-20}\) m — one step per doubling of accelerator energy. Incidentally, Episode 2's "140.24 log steps" comes out the same counted in space (\(\ln(R_H/\ell_P)=140.29\)).
Summary: the clear edge of where the tool works
The Planck length is \(\sqrt{\hbar G/c^3}\), and since \(G\) has weight \(+2\), \(\ell_P\) has weight \(+1\) — just a length, in the bookkeeping column. So do \(t_P\), \(m_P\) and \(E_P\). Only ratios like \(L/\ell_P\) are in the physics column, which is why "the smallest length is \(\ell_P\)" is not yet a sentence in Episode 3's sense.
And here the tool really breaks. A theory with a smallest length cannot be conformally invariant at all — conformal invariance demands the absence of a scale, and a smallest length is a scale. Unlike the rest of Part V's "breaks / cannot reach", this is excluded by hypothesis.
Does that make the Planck scale meaningless? No — stated as the place where \(\hbar/mc\) and \(2Gm/c^2\) have ratio 1, it is a dimensionless claim (\(m=m_P/\sqrt2=1.54\times10^{-8}\) kg). Not as "a length" but as "where two effects come level", the concept survives.
We are about fifty doublings away (from the LHC's \(1.45\times10^{-20}\) m to \(\ell_P\), \(2^{49.7}\)). Observationally, a linear Lorentz-violating effect is excluded past the Planck energy, while the quadratic effect is untouched over eight orders of magnitude.
Laying Part V out again, only two things failed to land in the weight-0 column — the apparent horizon (Episode 42) and the Planck length (this one). Both bring in a local yardstick.
This document is Episode 43 of "c·t = const, That Clicks" (the seventh of Part V), written for physics-minded high-school and university readers. Planck units, the crossing of the Compton wavelength and the Schwarzschild radius, and the Lorentz-invariance tests of quantum-gravity phenomenology are all standard, and nothing here is a new claim — the numbers are computed in kenshou/calc47.py. §02's (A)/(B) is a matter of convention: whether \(G\) transforms is not decided by physics but by which transformation you define — the claim is not that one reading is right, but that you must say which, which is Episode 3 again. "A smallest length is incompatible with conformal invariance" refers to global conformal invariance; in theories with local Weyl transformations as a gauge symmetry (Episode 34) a gauge choice may fix a scale, so the story is more involved — §02 is close to a tautology, and that is why it is strong. That the Planck length is the smallest length is not an established fact — it is the length buildable from three constants by dimensions alone, with no evidence yet that anything happens there, and §03's crossing indicates where quantum and gravitational effects come level, not that a minimum unit lives there. §05's limits assume a particular dispersion relation and the Fermi value depends on assumptions about photon emission times; not every theory with a smallest length breaks Lorentz invariance. §06's 1.96 fm uses Episode 18's value with \(R_H=1.3\times10^{26}\) m (1.99 fm with \(1.3725\times10^{26}\)). ── To make a PDF, use your browser's Print dialogue (sliders freeze and answers are hidden in the print version).