Thinking in Waves Episode 34 ／ Part XIII · Frontier physics as waves
The difference from Maxwell is a single line
Yang–Mills as waves ──
a wave that scatters off itself cannot be a free particle
One line is all that changes ── the source contains \(A\) itself. From that, superposition breaks and the theory is already chaotic classically.
★★ And here is the decisive part ── the gluon's response function goes negative (sign flip at 0.31 fm).
★ "A wave that cannot be a free particle" ── that is confinement, in the language of waves.
★ The source contains \(A\) itself ── the wave is its own source
In the language of waves ── ★ a wave that scatters off itself.
Episode 23 said nonlinearity breaks the word "filter". In Yang–Mills that nonlinearity sits in the fundamental law.
01Watching superposition break, directly
Dropping the spatial dependence, SU(2) Yang–Mills reduces to just \(V=\frac12g^2x^2y^2\) (\(x,y\) being two colour amplitudes). With only one colour the force vanishes and it flies freely.
| Initial condition | \(x\) at \(t=20\) | \(y\) at \(t=20\) |
|---|---|---|
| (A) only \(x\) excited | 0.046561 | 1.453494 |
| (B) only \(y\) excited | 1.465245 | 0.105107 |
| (C) both together | −2.973227 | −0.127884 |
| ★ If superposition held: (A)+(B) | 1.511806 | 1.558601 |
| \(t\) | \(x\) of (C) | \(x\) of (A)+(B) | Difference |
|---|---|---|---|
| 0.5 | 1.005336 | 1.142569 | 1.37e−1 |
| 2.0 | −0.223459 | 1.086197 | 1.31e+0 |
| 20.0 | −2.973227 | 1.511806 | 4.49e+0 |
The heart of §01
For electromagnetic waves this difference is always zero (Maxwell is linear). In Yang–Mills it is nonzero from the start and grows.
★ Energy also moves between the two colours (the kinetic energies swap while the total stays at 0.180000000).
── Light does not scatter light (Episode 23). ★ Gluons scatter gluons. That is the only difference.
02★ Already chaotic at the classical level
We measure the Lyapunov exponent in the same model (how fast two nearby initial conditions separate), with an integrable system measured the same way as a control.
| System | Potential | Lyapunov exponent \(\lambda\) | Verdict |
|---|---|---|---|
| Harmonic oscillator (integrable) | \(\frac12(x^2+y^2)\) | 2.4e−9 | ★ zero |
| ★ Yang–Mills (homogeneous SU(2)) | \(\frac12x^2y^2\) | 0.179128 | ★★ positive ＝ chaotic |
The heart of §02
The predictability horizon is \(1/\lambda=5.58\) (in the model's units).
★ Episode 23 said nonlinearity breaks superposition. ★★ Yang–Mills goes one step further and breaks predictability itself.
Perturb an electromagnetic wave's initial data by \(10^{-8}\) and it stays \(10^{-8}\) forever. In Yang–Mills it opens exponentially.
03A running coupling ── the vacuum as a dispersive medium
The 11 comes from gluon self-interaction, the \(-\frac23n_f\) from quark loops
| Energy \(\mu\) [GeV] | Wavelength \(\hbar c/\mu\) [fm] | \(\alpha_s\) (one loop) |
|---|---|---|
| 1 | 0.19733 | 0.52513 |
| 4.18 | 0.04721 | 0.27401 |
| 91.19 | 0.00216 | 0.13494 |
The heart of §03
The coupling depends on wavelength ＝ the vacuum is a dispersive medium. The same structure as Episode 16's "effective light speed is the slope of the dispersion relation", except that here it is the effective strength of the coupling that depends on wavelength.
★★ And the sign is opposite to electromagnetism ── electromagnetism screens (stronger up close), the strong force anti-screens (weaker up close).
★ "Anti-screening" ＝ the wave stiffens the medium. Ordinary media soften as they absorb wave energy; the gluon sea's tension rises with distance (Episode 25's string).
04★★ Positivity breaks ── the wave cannot be a free particle
This is the heart of the episode. We read the propagator as a wave response.
The Gribov form vanishes at \(k=0\) and its poles are complex (\(k^2=\pm iM^2\)) ── no real mass
We build the Schwinger function \(C(t)=\frac1\pi\int_0^\infty D(k)\cos kt\,dk\) numerically (\(M=0.7\) GeV).
| \(t\) [1/GeV] | \(t\) [fm] | Yukawa \(C(t)\) | ★ Gribov \(C(t)\) | Sign |
|---|---|---|---|---|
| 1.00 | 0.1973 | 3.547e−1 | 1.247e−1 | positive |
| 1.50 | 0.2960 | 2.500e−1 | 1.459e−2 | positive |
| 1.60 | 0.3157 | 2.331e−1 | −2.125e−3 | ★★ negative |
| 3.00 | 0.5920 | 8.747e−2 | −1.042e−1 | ★★ negative |
| 4.00 | 0.7893 | 4.344e−2 | −9.173e−2 | ★★ negative |
The heart of §04 ── ★★★ the core of this episode
The response function goes negative.
★ A positive response function is the condition for that wave to exist as a free particle (it follows from requiring positive probabilities).
★★ Going negative ＝ the gluon cannot exist as a wave on its own.
★ The Yukawa form (an ordinary particle) stays positive forever ── that difference is what confinement looks like in the language of waves.
Episodes 19 and 21 said information lives on the imaginary axis. The gluon's poles are not on the real axis but off in the complex plane, so it oscillates as it decays and changes sign.
05Massless, and yet there is a low-frequency cutoff
Episode 6: mass ＝ cutoff frequency. The gluon's mass is zero. Yet the lightest observed glueball is 1.73 GeV (a length of 0.114 fm) ── there is a cutoff.
| System | The wave inside | Source of the cutoff |
|---|---|---|
| Waveguide (Episode 6) | light (massless) | ★ a transverse boundary |
| Photon in niobium (Episode 18) | light (massless) | ★ a condensate (background field) |
| ★ Gluon | gluon (massless) | ★ self-interaction (nonlinearity) |
The heart of §05
"A medium made of massless waves that nonetheless has a low-frequency cutoff" ── that is a waveguide.
★★ So there are three ways to make a cutoff ── with a boundary, with a background field, or ★ by yourself.
Episode 25 said there were two ways to make an intercept; this episode adds the third (self-interaction).
06Which parts of the linear vocabulary survive
| Linear vocabulary | Verdict | Reason |
|---|---|---|
| Superposition | × breaks | ★ the wave is a source, so it cannot be added |
| Transfer function \(H(\omega)\) | × breaks | the response depends on the input (Episode 23) |
| ★ Predictability | × breaks | a positive Lyapunov exponent (§02) |
| ★ Positivity | × breaks | ★★ the response function goes negative (§04) |
| Causality | ◎ survives | forward waves only; Episode 12's constraints hold |
| Energy conservation | ◎ survives | it moves between colours, but the total is fixed |
| Gauge invariance | ◎ survives | ★ it is where the nonlinearity came from |
| Dimensionless quantities rule | ◎ survives | \(\alpha_s\), \(N_c=3\), the R ratio (Episode 25) |
The heart of §06
Episode 23 concluded that "only the word filter broke". ★★ In Yang–Mills two more break ── predictability and positivity.
And the breaking of positivity is confinement.
Exercises
- We said light does not scatter light, yet photon–photon scattering exists quantum mechanically. Contradiction?
Show the answer
No ── they are at different orders. Classical Maxwell is exactly linear; light does not scatter light at all. Only quantum effects (the Euler–Heisenberg term via an electron loop) produce a tiny nonlinearity at order \(\alpha^4\).
★ Episode 16's vacuum birefringence \(\Delta n=3.97\times10^{-24}B^2\) is exactly that size ── \(10^{-33}\) in the Earth's field.
★★ Yang–Mills nonlinearity is not a loop correction but the fundamental law itself, and its size is of order \(g^2\). That is the decisive difference. - How should we read the fact that \(D(k)=k^2/(k^4+M^4)\) vanishes at \(k=0\)?
Show the answer
"No response at infinite wavelength" ── it does not reach far. An ordinary propagator \(1/(k^2+m^2)\) tends to the finite \(1/m^2\) as \(k\to0\), and that is what gives the Yukawa potential \(e^{-mr}/r\) its range.
★ The Gribov form is zero there ── the long-distance response has been removed.
★★ And as §04 showed, the price is that the sign flips in the time domain. "Emits nothing at long range" and "cannot be a free particle" are two faces of the same property. - §02 showed chaos. So is confinement caused by chaos once quantized?
Show the answer
That does not follow ── this article does not show it. Classical chaos and quantum confinement are different phenomena, and no established argument connects them.
★ All we may say is "the same self-interaction produces chaos classically and positivity violation quantum mechanically".
★★ This is a boundary worth stating honestly. By this series' own habit (Episodes 29 and 30), "two phenomena from one cause" and "one causes the other" are different claims, and the latter requires content we have not supplied.
Summary A wave that scatters off itself cannot be a free particle
The difference from Maxwell was one line ── the source contains \(A\) itself; the wave is its own source.
From that line, superposition breaks. Solving the homogeneous SU(2) model for (A) only \(x\), (B) only \(y\), and (C) both, the difference between (C) and (A)+(B) reaches 4.5 by \(t=20\) ── for electromagnetic waves it would be zero always. Energy moves between the two colours while the total is conserved to 9 digits.
★ It is already chaotic classically. The Lyapunov exponent is \(\lambda=0.179\) (against \(2\times10^{-9}\) for the harmonic control) ── one step beyond Episode 23's broken superposition: ★ predictability itself breaks.
The coupling depends on wavelength because the vacuum is a dispersive medium, and \(\beta_0=11-\frac23n_f\) has the opposite sign to electromagnetism ── anti-screening: the wave stiffens the medium.
★★ And the decisive point: the gluon's response function goes negative.
The Gribov \(C(t)\) flips sign at \(t_0=0.3131\) fm (matching the analytic prediction).
The Yukawa form stays positive forever.
★ A positive response function is the condition for a wave to exist as a free particle. Going negative means the gluon cannot exist as a wave on its own ── that is what confinement looks like in the language of waves.
As a bonus, massless yet cut off (the glueball at 1.73 GeV) ── three ways to make a cutoff: a boundary, a background field, or ★ yourself. Episode 25's "two ways" gains a third. Of the linear vocabulary, four items break: superposition, transfer function, predictability and positivity. Causality, energy conservation, gauge invariance and the rule of dimensionless quantities survive.
★ Next time: when the phase becomes a function of angle ── the orbital angular momentum of light (\(e^{i\ell\varphi}\)) and the geometric phase of polarization.
★★ The winding number of a phase can only be an integer, and that integer is a conserved quantity. And on the Poincaré sphere, phase shows up as an area.
This document is Episode 34 of the "Thinking in Waves" series. The computations are in kenshou/yangmills.js. On the numbers here: the superposition test, the energy transfer, the Lyapunov exponents, the running of \(\alpha_s\) and the Schwinger functions were all computed for this article. That homogeneous Yang–Mills (the \(x^2y^2\) model) is classically chaotic, asymptotic freedom and \(\beta_0=11-2n_f/3\), the Gribov–Zwanziger propagator and positivity violation, and the glueball mass are all well-known results and not discoveries of this article ── what this article did was line them up under the single phrase "a wave that scatters off itself" and sort out which parts of the linear vocabulary break. The model in §01–02 drops the spatial dependence and captures only part of the real field's behaviour. The Gribov form in §04 is one model of the infrared propagator and \(M=0.7\) GeV is a representative value (lattice results depend on the formulation, and the sign-flip time is reported anywhere from 0.3 to 1 fm). Positivity violation itself is seen on the lattice, but "positivity violation ⇒ confinement" is an argument about a necessary condition, not a proof of confinement (the Yang–Mills mass gap problem remains open). The estimate from the string tension in §05 is order-of-magnitude only; the coefficient is model-dependent. ── To print, use your browser's Print → Save as PDF.