c·t = CONST, THAT CLICKS EPISODE 14 ／ The weight table itself becomes an object of measurement
The bookkeeping gets error bars
Substituting into
phase transitions
The weight table this series has used for thirteen episodes was a classical approximation.
Its error has been measured to seven digits in the 3D Ising model.
Last episode leaned hardest on "dimensionless means invariant". Today we go to the one place where that gets shaky — phase transitions. At a critical point the dimensionless exponents depart from their classical values (anomalous dimensions). And that departure is also the error in the very weight table this series has used for thirteen episodes. Better still, it has been measured in the 3D Ising model to seven digits. The bookkeeping gets error bars.
01The weight table was a classical approximation
Since Episode 1 we have used a table: length \(+1\), mass \(-1\), velocity \(0\). It is built by dimensional analysis — decompose into \(L^{n_L}T^{n_T}M^{n_M}\) and add (as in Episode 13). But in field theory the scaling dimension of an operator is not determined by that alone.
As Extra 6 of the previous series showed, \(\Delta\) is the Weyl weight
$$\mathcal{O}\ \longrightarrow\ \Omega^{-\Delta}\mathcal{O}$$Conclusion of §01
\(\gamma\ne0\) means "the classically counted weight is wrong".
The table this series has used was an approximation that ignored that error.
02Looking at the size of the error
The most precisely measured case is the spin operator of the 3D Ising model. A free field would give exactly \(\Delta=(d-2)/2\). It does not.
| Dimension \(d\) | Free field \((d-2)/2\) | Actual \(\Delta_\sigma\) | Anomalous \(\gamma_\sigma\) | Relative error |
|---|---|---|---|---|
| 2 | 0 | 0.125 (exact) | 0.125 | ── |
| 3 | 0.5 | 0.5181489(10) | 0.0181489 | 3.6% |
| 4 | 1.0 | 1.0 | 0 | 0% |
Conclusion of §02
In four dimensions the classical weight table is exactly right (\(\gamma=0\)).
Lower the dimension and the error grows — 3.6% in three dimensions, 12.5% in two.
Four is the upper critical dimension, above which mean-field theory becomes exact. That our spacetime is four-dimensional matters here too — the same place where the Maxwell action was conformally invariant only at \(D=4\) (Episode 11).
Figure: spatial dimension across, anomalous dimension \(\gamma_\sigma\) = the error in the weight table up. Circles are known values (exact in 2D, bootstrap in 3D, zero in 4D); the dashed line is the leading \(\varepsilon\)-expansion term. The error vanishes exactly at four dimensions.
The dashed line (the leading term \(\gamma=\varepsilon^2/108\)) gives only half the true value in three dimensions. Estimating the size of the error perturbatively is hard — which is why the seven digits of the next section matter.
03The error has been measured to seven digits
The conformal bootstrap determines \(\Delta\) without a single dimensionful input (Extra 6 of the previous series). For the 3D Ising model the result is just two numbers.
Every critical exponent of statistical physics follows from those two. Computed, and set beside experiment and Monte Carlo:
| Exponent | From the two \(\Delta\) | Experiment / MC | What it measures |
|---|---|---|---|
| \(\eta\) | 0.036298 | 0.0363(2) | decay of correlations |
| \(\nu\) | 0.629971 | 0.6300(17) | divergence of correlation length |
| \(\alpha\) | 0.110087 | 0.110(5) | divergence of specific heat |
| \(\beta\) | 0.326419 | 0.3265(15) | shape of the coexistence curve |
| \(\gamma\) | 1.237075 | 1.2372(5) | divergence of susceptibility |
| \(\delta\) | 4.789841 | 4.789(2) | critical isotherm |
| \(\alpha+2\beta+\gamma\) | 2.0000000000 | ── | an identity (independent of \(\Delta\)) |
And since \(\eta=2\gamma_\sigma\) — the error in the weight table, \(\gamma_\sigma=0.0181489\), is a quantity you can measure in water and magnets. Water, carbon dioxide and uniaxial magnets all obey this same number at their critical points.
The thing this episode most wants to say
This series' bookkeeping has acquired error bars.
\(\gamma_\sigma=0.0181489(10)\) — how wrong the weights are, measured to seven digits.
04And yet dimensionless quantities remain invariant
So Episode 13's "dimensionless is invariant" stands unchanged. What collapsed today is something else — the premise that "weights are fixed by dimensional analysis". A weight is not a settled number: it is a quantity a theory determines, and an object of measurement.
05So what about the cosmological weights?
Which raises the question. When Episode 4 said "only mass grows as \(\propto t\)", it took mass to have weight \(-1\). Does that \(-1\) get an anomalous dimension too?
| What carries the weight | Classical | Quantum correction |
|---|---|---|
| Spacetime lengths and times (the metric itself) | \(+1\) | none (it is the definition of the geometry) |
| Gauge fields, photons | \(0\) | breaking of order the \(\beta\) function (Episode 11) |
| Field operators (\(\bar\psi\psi\) and the like) | classical dimension | gets an anomalous dimension |
| Measured dimensionless ratios | \(0\) | none (they are observables) |
The geometric side (lengths, times) does not move, being definitional. What moves are field operators — the quark mass operator has an anomalous dimension and runs. So "mass grows as \(\propto t\)" is, strictly, undetermined until you say at what scale the mass is measured.
But — as Episode 3 of the previous series confirmed — the observable dimensionless ratios (\(1+z\), the 52.6 of recombination, \(Q/k_BT\)) are the same in every picture and at every scale. So the conclusions of Episodes 4 through 13 all survive intact.
① The "dimension \(d\)" of critical phenomena is the spatial dimension of statistical mechanics, not the dimension of spacetime. The upper critical dimension 4 and the \(D=4\) of the Maxwell action are both "4" because both come from a balance of dimensional analysis, but they are not the same phenomenon. They are placed side by side to show the structural similarity.
② \(\Delta_\sigma=0.5181489(10)\) is a conformal bootstrap value (El-Showk, Simmons-Duffin et al., 2012–2016). The error is numerical and includes estimated systematics. The 2D value \(0.125\) is exact since Onsager; the 4D \(0\) follows from mean-field theory becoming exact.
③ The leading \(\varepsilon\)-expansion term is \(\eta=\varepsilon^2(N+2)/[2(N+8)^2]\) (\(\varepsilon^2/54\) at \(N=1\)). In three dimensions (\(\varepsilon=1\)) it gives only about half the true value, so higher orders matter — the dashed line is a reference for "what the leading term alone would say".
④ The "experiment / MC" column gives approximate representative values from several measurements and computations, not a single source. Real fluids and magnets are not exact CFTs; these exponents appear only sufficiently close to the critical point (there are correction terms).
⑤ §05's "the mass weight gets an anomalous dimension" is a qualitative account. How it actually enters depends on the renormalisation scheme and on what one calls "mass" (pole mass or \(\overline{\rm MS}\) mass) — continuous with the observation in Extra 4 of the previous series that Koide's relation holds only for pole masses. This document claims only that weights are objects of measurement.
Exercises (solvable with this episode's formulas alone)
- Restate the anomalous dimension \(\gamma\) in this series' language.
Show the answer
The error in the weight table. The gap between the weight predicted by dimensional analysis, \(\Delta_{\text{classical}}\), and the actual weight \(\Delta\): \(\Delta=\Delta_{\text{classical}}+\gamma\). The table used in Episodes 1–13 was an approximation ignoring \(\gamma\). - Find \(\gamma_\sigma\) for the 3D Ising model, given the free-field weight \((d-2)/2\).
Show the answer
\(\gamma_\sigma=\Delta_\sigma-(d-2)/2=0.5181489-0.5=\) 0.0181489, a 3.6% error. Via \(\eta=2\gamma_\sigma=0.0362978\) it is measurable in water and magnets. - Why is \(\gamma=0\) in four dimensions?
Show the answer
Because four is the upper critical dimension, above which mean-field theory becomes exact and interactions stop affecting the long-distance behaviour. That our spacetime is four-dimensional matters here as it did for the Maxwell action in Episode 11 — though, per caveat ①, they are different phenomena wearing the same "4". - Does this episode contradict Episode 13's "dimensionless is invariant"?
Show the answer
No. The measured dimensionless quantities (critical exponents, \(\Delta\) itself) are still conformally invariant. What collapsed is a different premise — that weights are fixed by dimensional analysis. A weight is not a settled number; a theory determines it and experiment measures it. - (Harder) Why does \(\alpha+2\beta+\gamma=2\) hold independently of \(\Delta\)?
Show the answer
Because the six exponents are functions of two numbers (\(\eta,\nu\)). Substituting, \((2-d\nu)+\nu(d-2+\eta)+\nu(2-\eta)=2\), and \(\nu\), \(\eta\), \(d\) all cancel. This is not physics but a consequence of six quantities being functions of two — as Extra 6 of the previous series showed. An identity is not physics, the series' watchword, applies here too (Episodes 7 and 10).
Summary — the bookkeeping acquired error bars
The weight table used since Episode 1 was a classical approximation built by dimensional analysis. In field theory \(\Delta=\Delta_{\text{classical}}+\gamma\), and \(\gamma\ne0\) means the classically counted weight is wrong. That error has actually been measured.
The 3D Ising spin operator, which a free field would put at \(\Delta=0.5\), sits at 0.5181489(10) — an offset \(\gamma_\sigma=0.0181489\), 3.6%. In two dimensions it is 12.5% (exactly \(1/8\)), and exactly zero in four (the upper critical dimension). The leading \(\varepsilon\)-expansion term gives only half the 3D value, so these seven digits are out of perturbation theory's reach.
And through \(\eta=2\gamma_\sigma\), that error is measurable at the critical point of water and magnets. Two \(\Delta\)s give all six critical exponents, matching experiment. And \(\alpha+2\beta+\gamma=2.0000000000\) is an identity independent of \(\Delta\) — an identity is not physics, the same verdict as Episodes 7 and 10.
What matters is that none of this contradicts Episode 13's "dimensionless is invariant". Measured dimensionless quantities are still invariant. What collapsed was the premise that weights are fixed by dimensional analysis. A weight is not a settled number: a theory determines it and experiment measures it — this series' own bookkeeping has become an object of observation.
This document is Episode 14 of "c·t = const, That Clicks", written for physics-minded high-school and university readers. That the scaling dimension decomposes as \(\Delta=\Delta_{\text{classical}}+\gamma\), that \(\Delta\) is the Weyl weight (\(\mathcal{O}\to\Omega^{-\Delta}\mathcal{O}\)), that the free scalar unitarity value is \(\Delta=(d-2)/2\), and that 4 is the upper critical dimension above which mean-field theory is exact, are all standard. The 3D Ising CFT values \(\Delta_\sigma=0.5181489(10)\) and \(\Delta_\varepsilon=1.412625(10)\) are conformal bootstrap results (El-Showk, Paulos, Poland, Rychkov, Simmons-Duffin, Vichi et al., 2012–2016); the 2D \(\Delta_\sigma=1/8\) is exact. The conversions to critical exponents (\(\eta=2\Delta_\sigma-d+2\), \(\nu=1/(d-\Delta_\varepsilon)\) and so on) are standard, and the six exponent values together with the identity \(\alpha+2\beta+\gamma=2\) are computed here (kenshou/calc19.py). The "experiment / MC" column gives approximate representative values from several sources, not a single one. The leading \(\varepsilon\)-expansion term \(\eta=\varepsilon^2(N+2)/[2(N+8)^2]\) gives only about half the true value in three dimensions, so higher orders matter. The spatial dimension \(d\) of critical phenomena is distinct from the dimension of spacetime; the upper critical dimension 4 and the \(D=4\) of the Maxwell action are placed side by side to show a structural similarity, not because they are the same phenomenon. §05's "the mass weight acquires an anomalous dimension" is qualitative; how it enters depends on the renormalisation scheme and the definition of mass (pole or \(\overline{\rm MS}\)). Real materials are not exact CFTs and show these exponents only near criticality. For the relation between anomalous dimensions and the trace anomaly see Episode 8 and Extra 6 of the previous series. Linear expansion (\(c\cdot t=\)const, \(R_h=ct\)) is a minority model under examination. The academic standard remains the \(\Lambda\)CDM model including inflation. ── To make a PDF, use your browser's Print dialogue (sliders freeze and answers are hidden in the print version).