1/137 Was a Moving Number Before tackling "why 1/137," first this —
1/137 changes with "how finely you look." So the question has a slightly different shape.

Tools you'll need: α from Episode 2, division, reading a graph α(how finely you look) is not constant

In Episode 2 we said the fine-structure constant $\alpha\approx 1/137$ is "the same for anyone measuring anywhere in the universe, an unchanging number" — and planted a small bomb in the closing footnote: "$\alpha$ actually changes a little depending on how finely you look (the energy)." This time we collect on that setup. 1/137 is only the value when you look at the coarsest scale. This fact changes the very way we pose the ultimate question, "why 1/137?" From here on, the series steps for the first time into "territory nobody yet knows the answer to."

01Around an electron, things are actually "hazy"

An electron carries negative charge. But look "right up close" to the electron and the story isn't so simple. In the quantum world, even empty vacuum constantly fluctuates: for a very short time, pairs of "an electron and an anti-electron (positron)" pop into existence and vanish. These pairs line up with a preferred orientation around the original electron, and like a thin cloak they slightly hide (screen) its charge.

So from far away (coarsely), the electron's charge looks slightly weakened by this cloak. But get very close (finely), slip inside the cloak, and you see the true, stronger charge that isn't hidden. The closer you get, the stronger the charge looks — that's what "the electron is hazy" means.

Linking voice (Episode 3's "resolution") In Episode 3 we said "$c$ is the dial for how finely you view the world." The star this time is "fineness" too. Look coarsely = from far = low energy; look finely = up close = high energy. Since seeing a particle finely requires slamming high energy into it, "fineness" and "energy" are two sides of the same thing.

02So α changes with "how finely you look"

$\alpha$ is the number for the strength of the electron's charge (Episode 2's $\alpha=e^2/4\pi\varepsilon_0\hbar c$, where $e$ is the amount of charge). If the charge looks stronger, $\alpha$ grows. As in the last section, the more finely (higher energy) you look, the stronger the charge looks — so —

α is a function of "how finely you look" (energy)

Look coarsely (low energy, everyday world) → $\alpha \approx \dfrac{1}{137}$
Look finely (high energy, around the $Z$ particle's mass ~91 GeV) → $\alpha \approx \dfrac{1}{128}$

$1/137$ and $1/128$. It looks like a small difference, but it's a fact confirmed by measurement. Slam electrons together at high energy in a particle accelerator and $\alpha$ genuinely takes the larger, $1/128$-side value. In physics, such a "quantity whose value changes with how finely you look" is called a running coupling constant. A constant that runs — the "gap between name and reality" we saw in Episode 4 and the bonus episodes shows up here too.

Try it — how much did it change?

The two values as decimals

$$\frac{1}{137}\approx 0.00730,\qquad \frac{1}{128}\approx 0.00781$$

The fractional increase

$$\frac{0.00781-0.00730}{0.00730}\approx 0.07 = 7\%$$

From the coarse view to the fine view, $\alpha$ grew by about 7%. Far from an "unchanging constant," it moves nearly a tenth just by changing how finely you look. In the figure below, actually raise the energy and watch $\alpha$ grow.

Figure: α grows with "how finely you look" (energy). Far left = everyday world (1/137); rightward = higher energy

How finely you look (energy; finer to the right) 1

α at this energy ≈ 1/137.0

value of α at that fineness (shown as 1/α)

Linking voice (collecting on Episode 2) When Episode 2 said "$\alpha$ is $1/137$ for anyone measuring," it was, precisely, "at the coarsest view, $\alpha$ is $1/137$ for anyone." Change the ruler (units) and $1/137$ doesn't move — that's still true. But change "how finely you look" and it does move. Not moving with units, and moving with fineness, were two different matters.

◇ ◇ ◇

03So "why 1/137?" changes shape as a question

Here's Part 1's conclusion. If $\alpha$ moves with how finely you look, then "why is $\alpha$ equal to $1/137$?" was a bit too naive a question — because $1/137$ is just one value among many, "the value in the everyday coarse world." Reposed correctly, it becomes this.

Reposing the question

✗ "Why is $\alpha$ equal to $1/137$?"
✓ "Where do how $\alpha$ runs with fineness (the rule of the running) and the value that pins it down at one point come from?"

So what we should really ask about is not a single number but the "way of running" itself. The value changes with fineness, but "by what rule it changes" is something deeper, independent of that fineness — and that is the candidate for the "unchanging essence." The idea this series has repeated since Episode 1 — "not the surface value that moves with units or viewpoint, but the invariant structure behind it, is what physics is" — works just the same at the entrance to $\alpha$'s mystery.

The honest line — beyond here, no one has the answer

The "rule of $\alpha$'s running" can be computed from theory (quantum field theory). That part is understood. But the value that pins that running down at one point — "why that value?" — no one yet knows. Ever since the great physicist Eddington failed to derive $137$ from theory alone, it has stayed unsolved for nearly a century.

Here the series steps, for the first time, outside "what clicks." But there was a payoff — we reposed the question from "why 1/137" to "where do the running and its fixed point come from," into its correct form. A good question is half the answer.

Practice problems (solvable with just this episode's values)

Express the high-energy $\alpha\approx 1/128$ as a decimal (3 significant figures).
Show answer
$1\div 128 \approx 0.00781$. Larger than Episode 2's $1/137\approx0.00730$ — because the finer you look, the stronger the charge appears.
Explain "running coupling constant" in one line, using the "gap between name and reality" from Episode 4 and the bonus episodes.
Show answer
A quantity called a "constant" whose value actually changes with how finely you look (the energy). Just like the Hubble "constant" changing with time — a case where you should look at the definition and substance, not the name.
State in one sentence, from this episode, why "why 1/137?" is too naive a question.
Show answer
$1/137$ is only "the value at the coarse view," and changing the fineness changes the value. What to ask about is not a single number but the "running" of the value and the mechanism that fixes it.

Part 1 wrap-upNot an immovable number — a running quantity

Because vacuum fluctuations thinly hide the electron's charge, the finer (higher energy) you look, the stronger the charge appears, and $\alpha$ grows about 7% from $1/137$ to $1/128$. $\alpha$ was "a running quantity that changes with how finely you look" — the identity of the setup planted in Episode 2's footnote.

So "why $1/137$" was too naive a question. Correctly: "where do the rule of the running, and the value that fixes it, come from?" The value moves, but the running is deep — the "invariant structure behind, not the surface" that this series has consistently sought shows its face at the entrance to $\alpha$'s greatest mystery too. Having reposed the question well, we close Part 1.

Coming next — Episode 6, Part 2 "Do the three forces meet at a single point?" This time only the electromagnetic force ($\alpha$) was "running." But nature has other forces, each running differently. If, when you look very finely (at high energy), several forces' strengths line up at one point — then your dream that "everything comes from one number" becomes concrete physics (grand unified theory). Part 2 chases that meeting.

This is Episode 6, Part 1 of "Cosmology That Clicks," a reading piece for curious high-schoolers. Charge screening by vacuum polarization and the resulting "running" of the fine-structure constant are established physics; the change from $\alpha^{-1}\approx137.0$ (low-energy limit) to $\alpha^{-1}\approx128$ (the $Z$-boson mass scale, ~91 GeV) is measured. The figure is a schematic of the running concept; the real running is logarithmic and depends on the types of charged particles. "Why $\alpha$ takes its value" is an unsolved problem in modern physics. — To print, use your browser's Print → Save as PDF (in the printed version, the slider and answers are static/hidden).

Print / save as PDF: ⌘+P (Ctrl+P on Windows). On screen, raising the energy with the slider makes α grow. "Show answer" opens the solutions.

1/137 Was a Moving Number Before tackling "why 1/137," first this —1/137 changes with "how finely you look." So the question has a slightly different shape.

01Around an electron, things are actually "hazy"

02So α changes with "how finely you look"

03So "why 1/137?" changes shape as a question

Part 1 wrap-upNot an immovable number — a running quantity

1/137 Was a Moving Number Before tackling "why 1/137," first this —
1/137 changes with "how finely you look." So the question has a slightly different shape.