For fifteen episodes we have said "conformal transformation" while meaning only one of the two things that go by that name.
This series has consistently used "conformal transformation" for the Weyl transformation — the operation that swaps your ruler point by point. But physics has a second thing with that name: the conformal group of flat spacetime. Set the two side by side and the "weight" we have been computing all along acquires its proper name — and we walk into a world where no dimensionful quantity means anything at all. There the observables are \(\Delta\), a pure number, and nothing else. And that world is not speculation: water, carbon dioxide and uniaxial magnets all fall into it at their critical points.
This is a standing source of confusion, so let us separate them first.
| Weyl transformation | Conformal group | |
|---|---|---|
| What it does | rewrites the metric, \(g\to\Omega^2g\) | moves the spacetime (a coordinate change) |
| Freedom | one function \(\Omega(x)\) = infinite-dimensional | finite-dimensional (for \(d\ne2\)) |
| Character | a gauge-like rewriting (bookkeeping) | a genuine symmetry |
| In this series | the star of episodes 1 through bonus ⑤ | this episode |
The conformal group is the set of coordinate changes of flat spacetime that alter the metric by no more than an overall factor. Count them, and the answer is a function of the dimension alone.
The breakdown
$$\underbrace{\tfrac{d(d-1)}{2}}_{\text{rotations}}+\underbrace{d}_{\text{translations}}+\underbrace{1}_{\text{dilatation}}+\underbrace{d}_{\text{special conformal}}=\frac{(d+1)(d+2)}{2}$$In four dimensions
$$6+4+1+4=15\qquad(\text{five more than Poincaré's }10)$$Only five wider than Poincaré symmetry. The additions are the dilatation and the special conformal transformations.
| dimension \(d\) | number of conformal symmetries | note |
|---|---|---|
| 1 | 3 | conformal quantum mechanics |
| 2 | 6 | the formula gives only the global part; locally it is infinite-dimensional |
| 3 | 10 | the arena of critical phenomena |
| 4 | 15 | our spacetime |
| 6 | 28 |
Two dimensions is the exception. There the local conformal transformations are infinite in number (one for every holomorphic function), which is why the theory can be solved exactly. That is why the worldsheet of a string is two-dimensional.
The central object in a conformal field theory is the scaling dimension \(\Delta\). Its definition is:
That is exactly the Weyl weight.
So episode 3's table of weights, episode 7's \(\Omega^{D-4}\), bonus ④'s \(\tilde m=\Omega^{-1}m\) — all of them were computing \(\Delta\), without knowing its name. "The other conformal transformation" was in the room the whole time.
And \(\Delta\) has two parts.
\(\gamma\ne0\) means "the weight you computed classically is wrong" — episode 8's anomaly, operator by operator.
Episode 8 measured the anomaly with the \(\beta\) function (\(1/137\to1/128\)). At the level of individual operators, that same thing is \(\gamma\). And \(\gamma\) can be measured in the laboratory — section 05.
In a theory where conformal symmetry is exact (a conformal field theory, CFT), a dimensionful quantity cannot exist even in principle: there is no standard of length. So what is left? A list of \(\Delta\)s, and the coefficients for multiplying operators together. Both dimensionless.
And \(\Delta\) is not free to be anything. Unitarity — probabilities not going negative — imposes a floor.
| operator | constraint on \(\Delta\) | in \(d=4\) | meaning |
|---|---|---|---|
| scalar | \(\Delta\ge\dfrac{d-2}{2}\) | \(\ge1\) | equality means a free field |
| conserved current \(J_\mu\) | \(\Delta=d-1\) | \(=3\) | exact (conservation fixes it) |
| stress tensor \(T_{\mu\nu}\) | \(\Delta=d\) | \(=4\) | exact |
Here is the heart of it. There is a way to determine \(\Delta\) in a CFT without ever writing down a Lagrangian. It is called the conformal bootstrap.
That is the entire input. No coupling constant, no cutoff, no lattice spacing — not one dimensionful quantity. What comes out is a forbidden region of \(\Delta\) values, and its boundary has a kink.
And the three-dimensional Ising model sits on that kink. Nature is parked on a corner of a purely mathematical exclusion plot.
Everything about critical phenomena follows from these two numbers.
First read off the anomalous dimension. A free scalar in three dimensions has \(\Delta=(d-2)/2=0.5\) exactly, so
$$\gamma_\sigma=\Delta_\sigma-0.5=0.0181489$$which is to say that "how wrong the Weyl weight is" is known to seven digits. And it has a familiar name in statistical physics: \(\eta=2\gamma_\sigma=0.0362978\).
Every critical exponent in the statistical-physics textbooks is a translation of \(\Delta_\sigma\) and \(\Delta_\varepsilon\).
Only the first two are independent; the other four are derived.
| exponent | bootstrap | experiment / Monte Carlo | what it measures |
|---|---|---|---|
| \(\eta\) | 0.036298 | 0.0363(2) | falloff of correlations |
| \(\nu\) | 0.629971 | 0.6300(17) | divergence of the correlation length |
| \(\alpha\) | 0.110087 | 0.110(5) | divergence of the specific heat |
| \(\beta\) | 0.326419 | 0.3265(15) | shape of the coexistence curve |
| \(\gamma\) | 1.237075 | 1.2372(5) | divergence of the susceptibility |
| \(\delta\) | 4.789841 | 4.789(2) | the critical isotherm |
And one combination holds for any \((\Delta_\sigma,\Delta_\varepsilon)\) you care to plug in.
All six exponents swing wildly, and this one combination never leaves 2. Exactly the structure of bonus ④'s \(Q=2/3\).
Here is the point of the episode. Water, carbon dioxide, uniaxial magnets and binary alloys all obey these same numbers at their critical points. The shape of the molecules, the strength of the bonds — none of it enters. The material is stage scenery and only the pure numbers are physics: what this series has been asserting all along, existing as an experimental fact rather than a metaphor.
\(\Delta\) has one more decisive job: deciding whether an operator can be ignored.
Put numbers in for 3d Ising.
| operator | \(\Delta\) | versus \(d=3\) | corresponding knob |
|---|---|---|---|
| \(\sigma\) (\(\mathbb{Z}_2\) odd) | 0.518149 | relevant | magnetic field (forbidden if you keep the symmetry) |
| \(\varepsilon\) (\(\mathbb{Z}_2\) even) | 1.412625 | relevant | temperature |
| \(\varepsilon'\) (\(\mathbb{Z}_2\) even) | 3.829510 | irrelevant | none needed |
As long as the symmetry is kept, there is exactly one relevant operator, \(\varepsilon\). So there is exactly one knob — temperature.
How many knobs you must turn to reach criticality is fixed by comparing pure numbers.
\(\Delta_\varepsilon=1.41\ <\ 3\ <\ \Delta_{\varepsilon'}=3.83\) — that single inequality is why the critical point is a point and not a surface. And the fact that \(\varepsilon'\) exceeds \(d\) is universality itself: every microscopic detail is irrelevant and flows away.
Finally, the most extreme illustration of this series' conclusion: the holographic (AdS/CFT) dictionary.
The left side is the boundary's dimensionless \(\Delta\), the right the bulk's dimensionful mass. One object, seen from inside or from outside.
And an almost unbelievable consequence follows. For \(\Delta\) to be real one needs only \(m^2R^2\ge-d^2/4\) — so \(m^2\) is allowed to be negative.
| \(d\) | BF bound \(m^2R^2\ge\) | \(\Delta\) there | at \(m^2R^2=0\) |
|---|---|---|---|
| 2 | \(-1\) | 1.0 | \(\Delta=2\) |
| 3 | \(-2.25\) | 1.5 | \(\Delta=3\) |
| 4 | \(-4\) | 2.0 | \(\Delta=4\) |
In flat spacetime \(m^2<0\) is the definition of a tachyon, of instability. In AdS it is not.
What decides stability is not the sign of \(m^2\) but whether \(\Delta\) is real.
The sign of a dimensionful quantity means nothing on its own; only dimensionless ones can decide. Episode 10's decision procedure, promoted to a theorem about stability.
Having told a very clean story, here is something that does not fit. Next door to Ising, in the 3d XY universality class — the λ point of liquid helium-4.
Theory (bootstrap, with Monte Carlo agreeing)
$$\Delta_s=1.51136(22)\ \Rightarrow\ \nu=0.671754,\qquad \alpha=2-3\nu=-0.01526(30)$$Experiment (Lipa et al. 2003, measured on the Space Shuttle to remove gravity)
$$\alpha=-0.0127(3)$$A difference of 0.00256, roughly 6σ. Both sides are precise, and they do not agree.
The specific-heat exponent is so close to zero that logarithmic corrections are delicate, and both the experimental analysis and the theoretical extrapolation have been argued over. It is still unresolved.
Section 01's separation of the Weyl transformation from the conformal group is a pedagogical split. The two are closely related — the conformal group is the combination of diffeomorphisms and Weyl transformations that preserves flat space. Also, "Weyl invariance implies conformal invariance" is not true in general (the converse is), and holds only under conditions.
The bootstrap of section 04 is a numerical method. "Ising sits on the kink" means the kink's location agrees with Monte Carlo, not that the kink has been proven to be Ising (modern "island" methods do isolate it to high precision). The quoted errors on \(\Delta\) are numerical and include systematic estimates.
The "experiment / Monte Carlo" column in section 05 gives representative values across several measurements and computations, not a single source. Real fluids and magnets are not exact CFTs; the exponents appear only sufficiently close to the critical point, with corrections to scaling.
Section 07's \(\Delta(\Delta-d)=m^2R^2\) is the relation for a scalar field; with spin the form changes. AdS/CFT itself is a conjecture and is not proven in general. The discrepancy in section 08 is a genuine unresolved problem, and it is not settled whether the fault lies with experiment or with theory.
There are two "conformal transformations": the Weyl transformation this series has used for fifteen episodes (infinite-dimensional, a gauge-like rewriting) and the conformal group of flat spacetime (finite-dimensional, a genuine symmetry, 15 generators in four dimensions). Only in \(d=2\) is the conformal group also infinite-dimensional — which is why a string's worldsheet is two-dimensional.
The two were connected all along. The definition of a CFT's central object is \(\mathcal{O}\to\Omega^{-\Delta}\mathcal{O}\), so \(\Delta\) is the Weyl weight. Episode 3's weight table and episode 7's \(\Omega^{D-4}\) were computing \(\Delta\) without naming it. The gap from the classical weight, \(\gamma\), is the anomalous dimension — episode 8's anomaly measured operator by operator.
The conformal bootstrap uses no dimensionful input whatsoever — crossing symmetry, unitarity, and a symmetry. From that come the 3d Ising values \(\Delta_\sigma=0.5181489\) and \(\Delta_\varepsilon=1.412625\), from which all six critical exponents follow, and water, carbon dioxide and uniaxial magnets obey the same numbers. \(\gamma_\sigma=0.0181489\): "the error in the Weyl weight" is known to seven digits. And \(\alpha+2\beta+\gamma\) never leaves 2, whatever \(\Delta\) you feed it.
\(\Delta\) even fixes how many knobs it takes to reach criticality. \(\Delta_\varepsilon=1.41<3<\Delta_{\varepsilon'}=3.83\) — that one inequality is why the critical point is a point, and it is universality itself. Finally the holographic dictionary \(\Delta(\Delta-d)=m^2R^2\) ties bookkeeping (the bulk mass) to physics (the boundary \(\Delta\)) as a single object, and declares that stability is decided not by the sign of \(m^2\) but by whether \(\Delta\) is real. Episode 10's decision procedure turned out to be a theorem about stability.
Print / save as PDF: ⌘+P (Ctrl+P on Windows). On screen, the two sliders move \(\Delta\); try to find the point where all six bars land on their marks. "Show the answer" opens each solution.