Cosmology That ClicksEpisode 8 · what i really is, Part 1 — ambiguity becomes force

The i left over from Episode 3, the α of Episodes 2 and 6 — here the two join into one thread

The Ambiguity of i
Gives Birth to a Force The phase i of the wavefunction looks like a "don't-care ambiguity." But the moment you allow that ambiguity
to be chosen freely at each point — to patch it up, the electromagnetic force is born.

Tools you'll need: the i from Episode 3, the phase of a complex number, differentiation (the difference from the neighbor) ∂ → D = ∂ − iA

In Episode 3, the imaginary unit \(i\) was left over in the Schrödinger equation. Even as \(c\to\infty\) made relativity vanish, \(i\) alone did not disappear. What is this \(i\)? The usual answer is "complex numbers make the math easier." But that's shallow. In this episode we'll see that pushing the phase ambiguity of \(i\) to its limit gives birth — as a logical necessity — to \(\alpha\approx1/137\), the star of Episodes 2 and 6, that is, to the electromagnetic force itself. \(i\), ambiguity, and \(1/137\) join into one thread. We'll experience, hands-on, one of the deepest insights of 20th-century physics: the gauge principle.

01Phase ambiguity — rotate it globally and nothing happens

The wavefunction \(\psi\) is complex: \(\psi=|\psi|e^{i\theta}\), carrying a magnitude (amplitude) and a direction (phase \(\theta\)). What affects observation is only the probability \(|\psi|^2\). So even if you rotate the overall phase by \(\theta\) —

Overall phase rotation (global transformation) — no effect on observation
$$\psi \to e^{i\theta}\psi \qquad\Rightarrow\qquad |\psi|^2 \to |e^{i\theta}\psi|^2 = |\psi|^2$$

The probability doesn't change one iota. Where you place the phase's origin is merely our way of keeping the books; the universe doesn't care. This is that trace-leaving-no ambiguity we saw in Bonus 2 ("swapping units is a bookkeeping matter") and Bonus 4 ("if \(\alpha\) is invariant, it leaves no trace in observation"). Rotated globally, the phase of \(i\) is exactly this harmless ambiguity. So far, the intuition "\(i\) is ambiguity" is exactly right.

A contrast with Episode 9 (a word ahead) Rotate this "trace-leaving-no phase ambiguity" in the direction of imaginary time and you get to Episode 9's "\(i\) gives birth to temperature (the Wick rotation)." This episode (8) instead asks what happens if you allow this ambiguity independently at each point (localize it in space and time). Episodes 8 and 9 chase the same ambiguity of the same \(i\) in two directions.

02Rotate it independently at each point — and the equation breaks

Here we pose a decisive question. Must the phase's origin be "the same everywhere in the universe"? The origin I choose here and the one you choose far away ought to be independent. So we make \(\theta\) a freely chosen \(\theta(x)\), varying point by point and moment by moment (a local transformation). Then the differentiation in the equation causes trouble — because differentiation measures "the difference from the neighboring point."

Try it — differentiate the local phase and an extra term appears

Open it with the product rule (Leibniz)

$$\partial_x\!\left(e^{i\theta(x)}\psi\right) = e^{i\theta(x)}\Big(\partial_x\psi + i\,(\partial_x\theta)\,\psi\Big)$$

Besides the original \(\partial_x\psi\), an extra term \(i(\partial_x\theta)\psi\) appeared. If \(\theta\) is constant, \(\partial_x\theta=0\) and it vanishes; but for a position-dependent \(\theta(x)\) it doesn't. Because of this term, changing the phase locally changes the form of the equation — that is, the laws of physics come to depend on the bookkeeping of "how I choose the phase at each point." This is a problem.

Why a problem? Recall the backbone of Bonus 2 and 4 — changing the bookkeeping (the choice of phase origin) should leave physics invariant. Globally this held (Step 01). We want it to hold locally too. But done naively, the extra term \(i(\partial_x\theta)\) gets in the way. What to do?

03Introduce a "partner" that cancels the extra term — that's the electromagnetic field

There's one strategy. Add to the equation a partner field that exactly cancels that extra term. Replace the derivative \(\partial_x\) with a new derivative carrying a correction term (the covariant derivative).

The covariant derivative — a derivative with a partner A
$$\partial_x \;\longrightarrow\; D_x = \partial_x - i\,A_x$$

And impose the rule that when you rotate the phase by \(\theta(x)\), at the same time the partner \(A_x\) moves in step:

$$\text{when}\ \theta \to \theta + \theta(x),\quad A_x \to A_x + \partial_x\theta$$

Then, as if by magic, it cancels. The extra term \(+i(\partial_x\theta)\) from the local phase and the change \(-i(\partial_x\theta)\) of the partner \(A_x\) exactly annihilate. The covariant derivative \(D_x\psi\) cleanly preserves the same phase rotation as \(\psi\) — the equation has become invariant under local relabeling of the books. This is gauge invariance.

And the identity of the partner \(A_x\) we just introduced is — the electromagnetic potential. Both the electric and magnetic fields come out of this \(A_x\). In other words:

The heart of this episode

The moment you require that "the phase ambiguity of \(i\) may be chosen independently at each point," to patch it up, the electromagnetic field appears as a logical necessity. The force was, as it were, demanded of the universe as the partner that makes the ambiguity consistent.

Note that the order is reversed. Normally we think "there's an electromagnetic force, so the electron moves." But in the gauge principle — we want to allow local phase ambiguity, so the electromagnetic field becomes unavoidable. Not force first and ambiguity second, but ambiguity first and force second. The "ambiguity" you sensed in \(i\) was, in this order, the mother of the force.

04Run it — the moment "trace-leaving-no ambiguity" leaves a trace

In Step 01 we said "phase ambiguity leaves no trace." But once localized and tied to the electromagnetic field, phase ambiguity begins to leave an observable trace. The most vivid example is the Aharonov–Bohm effect.

Split an electron into two paths and confine a magnetic flux \(\Phi\) (a bundle of magnetic field) between them. Remarkably, even though the magnetic field is zero along the paths the electron travels, the enclosed flux \(\Phi\) produces a phase difference between the two paths and shifts the electron's interference fringes. The phase difference is set by the loop integral of the partner \(A\).

The Aharonov–Bohm phase difference
$$\Delta\varphi = \frac{e}{\hbar}\oint A\cdot dl = 2\pi\,\frac{\Phi}{\Phi_0},\qquad \Phi_0=\frac{h}{e}\ (\text{flux quantum})$$

In the figure below, move the enclosed flux \(\Phi\) (in units of the flux quantum \(\Phi_0\)). Even though it travels through a field-free region, the interference fringes shift sideways. Each time \(\Phi/\Phi_0\) increases by 1, the fringes shift by exactly one fringe and return — the moment phase ambiguity shows its face as observable physics.

Figure: the Aharonov–Bohm effect. Change the enclosed flux Φ (horizontal slider) and the interference fringes shift even though the electron travels a field-free region. Each unit increase of Φ/Φ₀ shifts the fringes by one and restores them.
electron interference fringes (bright = electron more likely) fringe center (moves with Φ)
The reveal — this is the proof of "trace-leaving ambiguity" The Aharonov–Bohm effect shows that even where the magnetic field itself (\(B\)) is zero, the loop integral of the partner \(A\) (the winding number of the phase ambiguity) affects observation. Phase ambiguity, harmless globally (Step 01), becomes real physics once localized and tied to \(A\). Predicted in 1959 and confirmed by experiment. "\(i\) is just a computational convenience" cannot explain this experiment. The phase of \(i\), once localized, becomes physics you can't erase.
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05The strength of that force is α — and on to the three forces

How strongly the partner \(A\) couples to the wavefunction — that coupling strength is the amount of charge \(e\), and dimensionless, it's Episode 2's \(\alpha\).

The coupling strength = Episode 2's α
$$\alpha=\frac{e^2}{4\pi\varepsilon_0\hbar c}\approx\frac{1}{137}$$

Episode 2 said "\(\alpha\) is the 1/137 hidden inside the atom," Episode 6 said "\(\alpha\) runs with how finely you look." This episode is the step before those — it answers where \(\alpha\) comes from in the first place. It's the coupling strength of the partner \(A\) that makes the local phase ambiguity of \(i\) consistent. \(i\), ambiguity, and 1/137 are joined by one logic: the gauge principle. This is the deepest landing point of your intuition, "isn't \(i\) ambiguity?"

And Episode 6's "the three forces meet at a point" lies along this extension too. The phase ambiguity was one kind (rotation on a circle), but localizing a more complex "ambiguity of internal orientation" (like rotation on a sphere) gives birth, by exactly the same logic, to the weak and strong forces. All three forces arise from "localizing some internal ambiguity" — the three "running forces" of Episode 6 were three kinds of ambiguity, localized. This episode was the very foundation of Episode 6.

The honest line — what happens if you tie it to c·t=constant

Up to here is established standard physics (the gauge principle). So what if it's tied to this series' \(c\cdot t=\text{constant}\)? The phase rotation rate is set by \(E/\hbar\), and its ratio is \(\alpha\). If you read it as "the clockdown of \(c\cdot t=\text{constant}\) runs \(\alpha\) over the ages through the phase rotation of \(i\)" — then, as judged in Episode 7, if \(\alpha\) runs observably, atomic clocks (\(\dot\alpha/\alpha<10^{-19}\)/yr) and BBN immediately reject it (the VSL trap of Bonus 3).

So the correct placement is this: the gauge principle is completely correct as the logic that "gives birth to" \(\alpha\), and it leaves a trace in observation (Aharonov–Bohm). But turned into a claim that \(c\cdot t=\text{constant}\) "runs" \(\alpha\), it is rejected. Read as an \(\alpha\)-invariant gauge, \(c\cdot t=\text{constant}\) touches this phase structure of \(i\) not at all — exactly as Bonus 4's equivalence condition says.

Practice problems (solvable with just this episode's formulas)
  1. When you rotate the overall phase by \(\theta\) (\(\psi\to e^{i\theta}\psi\)), how does the probability \(|\psi|^2\) change?
    Show answer
    \(|e^{i\theta}\psi|^2=e^{i\theta}\psi\cdot\overline{e^{i\theta}\psi}=e^{i\theta}e^{-i\theta}|\psi|^2=|\psi|^2\). No change at all. So global phase ambiguity leaves no trace in observation.
  2. What is the "extra term" that appears when you differentiate a local phase \(e^{i\theta(x)}\)? Why does it vanish for a constant phase?
    Show answer
    The extra term is \(i(\partial_x\theta)\psi\). If \(\theta\) is constant, \(\partial_x\theta=0\) so it vanishes. It survives only for a position-dependent \(\theta(x)\), and to cancel it you need the partner \(A\).
  3. In the Aharonov–Bohm effect, when the enclosed flux is \(\Phi=2\Phi_0\), how many radians is the phase difference \(\Delta\varphi\)? Do the fringes return to their original position?
    Show answer
    \(\Delta\varphi=2\pi\times(\Phi/\Phi_0)=2\pi\times2=4\pi\). An integer multiple of \(2\pi\), so the fringes return exactly to their original position. The fringes are restored whenever \(\Phi/\Phi_0\) is an integer.

Episode 8 wrap-upAmbiguity first, force second — i was the mother of the force

The phase \(i\theta\) of the wavefunction is, rotated globally, an ambiguity that leaves no trace in observation (Step 01). But localize it — "choose it independently at each point" — and an extra term \(i(\partial_x\theta)\) appears from differentiation and breaks the equation (Step 02). Demand a partner \(A\) to cancel it, and — the electromagnetic field appears as a logical necessity (Step 03, the gauge principle). The force was demanded of the universe to make the ambiguity consistent.

That ambiguity, once localized, leaves a trace in observation (Step 04, Aharonov–Bohm). And the coupling strength of the partner \(A\) is \(\alpha\approx1/137\) (Step 05) — the origin of the \(\alpha\) of Episodes 2 and 6. By the same logic, all three forces are born from "localizing an internal ambiguity." The ambiguity of \(i\) first, the force second. The ambiguity you saw in \(i\) was, in the single step of localization, the mother of the electromagnetic force and its strength \(1/137\). But run \(\alpha\) with \(c\cdot t=\text{constant}\) and Episode 7 rejects it — the logic gives birth, but running it is forbidden.

This is Episode 8 of "Cosmology That Clicks," a reading piece for curious high-schoolers and undergraduates. That requiring invariance under local \(U(1)\) phase transformations (gauge invariance) makes the covariant derivative \(D_\mu=\partial_\mu-iA_\mu\) and the electromagnetic potential \(A_\mu\) appear necessarily, that its coupling constant is the fine-structure constant \(\alpha\approx1/137\), and the Aharonov–Bohm effect (even in a field-free region, the loop integral of the vector potential \(\oint A\cdot dl\) gives a phase difference \(\Delta\varphi=2\pi\Phi/\Phi_0\) that shifts the fringes, \(\Phi_0=h/e\)), are established standard physics. That localizing the non-abelian gauge groups \(SU(2)\times SU(3)\) gives the weak and strong forces is also part of the Standard Model. The figure is a schematic of the fringe shift due to the Aharonov–Bohm phase difference \(\Delta\varphi=2\pi\Phi/\Phi_0\). Interpreting \(c\cdot t=\text{constant}\) as running \(\alpha\) conflicts with atomic clocks (\(\dot\alpha/\alpha<10^{-19}\)/yr) and BBN (Episode 7); only the interpretation as an \(\alpha\)-invariant gauge is consistent with observation (Bonus 4). — 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, moving the flux slider shifts the interference fringes. "Show answer" opens the solutions.