Tunneling That Clicks ── series contents

The backbone in one line ── tunneling is just walking normally, in imaginary time.
Rotate time onto the imaginary axis and the wall flips into a valley: a barrier you could not cross becomes a slope you can walk across. The fee for that walk is e^(−S_E/ℏ) (Episode 1). The Sun burns because protons tunnel through the Coulomb barrier ── heat alone is three orders of magnitude short (Episode 2). Climb over, or slip through: the 2π in the crossover temperature is the same 2π as in the Unruh temperature (Episode 3). Join two superconductors whose phases have lined up, and tunneling becomes a current (Episode 4). But superconductivity itself is not tunneling ── here the series contradicts itself on purpose (Episode 5). And yet both carry the same non-analyticity, whose true identity is an exponent invisible to perturbation theory (Episode 6). To finish ── it is not only particles that tunnel. The vacuum tunnels too (Episode 7).
The seven bonus episodes compute, with the tools built in the main series, why cold fusion does not happen; look at the one real cold fusion (muon catalysis); explain why a single atom is visible and how many seconds tunneling takes (still unsettled); sort out whether living things use tunneling; and close with thirty years of room-temperature superconductivity, paired against bonus ①.

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MAIN SERIES

EPISODE 1live figure

Tunneling is just walking, in imaginary time

A particle that cannot cross the wall is nonetheless on the far side. It looks strange because we are watching in real time. Rotate time 90° onto the imaginary axis and the wall inverts into a valley ── the particle simply walks across a slope. The action spent on that walk rides directly on the exponent of the probability. P ≈ e^(−2S_E/ℏ)

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EPISODE 2live figure

The Sun burns because it tunnels

k_BT at the Sun's centre is 1.3 keV; the Coulomb barrier between two protons is of order an MeV ── three orders short. It burns anyway, because it tunnels. And the product of the thermal tail with the tunneling probability creates a narrow "Gamow peak" that decides how stars burn. P ∝ exp(−√(E_G/E))

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EPISODE 3live figure

Climb over, or slip through ── the crossover temperature

Getting over thermally costs e^(−E/k_BT); slipping through costs e^(−S_E/ℏ). Both are exponentials, and both can be written in the language of imaginary time. So they must cross at some temperature ── and the 2π that appears there is the same 2π as in the Unruh temperature from the finale of "Temperature That Clicks." T₀ = ℏω_b / 2πk_B

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EPISODE 4live figure

When the phase lines up, tunneling becomes a current

Sandwich a thin insulator between two superconductors and current flows with no voltage applied. What sets the amount is neither voltage nor temperature but the phase difference across the junction. The moment tunneling turns from "a rare accident, one at a time" into "a coherent macroscopic flow." I = I_c sin Δθ

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EPISODE 5live figureself-refuting

Superconductivity is not tunneling

By this point one wants to say "it's all tunneling." You can't. Superconductivity is not barrier penetration but condensation, and the mechanism is entirely different. Yet the exponents that come out look almost identical ── this episode separates exactly which layer the resemblance lives on. Δ ≈ 2ℏω_D e^(−1/N(0)V)

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EPISODE 6live figure

What perturbation theory can never see

Differentiate e^(−1/g) with respect to g as many times as you like: at g=0 every derivative is zero. So a perturbation expansion can never produce this term. Tunneling, the BCS gap and instantons all enter through that hole. And in the language of "Renormalization That Clicks," the Cooper instability is a marginal coupling turning relevant. e^(−1/g) is non-analytic at g = 0

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EPISODE 7live figuremain-series finale

The vacuum tunnels too

It is not only particles that tunnel. Our vacuum may not be the lowest one, and its decay probability is written with the same e^(−S_E/ℏ) (Coleman's bounce). Even Hawking radiation can be read as "tunneling through the horizon." The series closes on the lifetime of the universe. Γ/V ≈ A e^(−S_E/ℏ)

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BONUS EPISODES

BONUS ①live figure

Does cold fusion happen? ── thirty years of nickel and palladium

A judgement on the controversy running from 1989 to now, reached by calculating with the tools of the main series. How many orders short is the Gamow factor? How far does screening in a metal get you? And the decisive argument ── tunneling changes the entrance; it cannot change the exit. One watt of heat means 10¹² neutrons per second. the entrance can change, the exit cannot

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BONUS ②live figure

The real cold fusion ── muon catalysis

There is exactly one cold fusion that genuinely happens. Replace the electron with a muon and the molecule shrinks 207-fold, making the Gamow factor O(1). It works. And still it does not pay ── because the muon sticks to the helium it just made and cannot get away. 0.45% α-sticking → about 150 cycles, then retirement

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BONUS ③live figure

You can see a single atom because it is an exponential

A scanning tunneling microscope resolves single atoms not because of lens quality. Move 0.1 nm and the current changes by a factor of ten ── the steepness of the exponential is the resolution. As a way to feel what "riding on the exponent" means, there is no better example. I ∝ e^(−2κd), ×10 per 0.1 nm

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BONUS ④live figureunsettled

How many seconds does tunneling take?

Thicken the wall and the time taken does not increase (the Hartman effect). Read naively, that beats the speed of light. No information does ── so what is "the time it took"? Several definitions exist and they disagree. A problem still unresolved, handled carefully while leaving it unresolved. is τ independent of the wall's thickness?

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BONUS ⑤live figure

Do living things use tunneling?

The evidence that hydrogen tunnels inside enzymes is that swapping in deuterium changes the rate by orders of magnitude. A bird's magnetic compass may be a radical pair. And photosynthetic "quantum coherence" was substantially walked back. This episode separates what was demonstrated from what was a buzzword. k_H/k_D ≫ 7 means it is tunneling

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BONUS ⑥live figure

Thirty years of room-temperature superconductivity

The finale, paired against bonus ①. BCS was supposed to have a ceiling. Cuprates broke it, and high-pressure hydrides reached 250 K. Then in 2023 came a major retraction. Two things we want to happen at room temperature ── one genuinely came close, the other has not moved in thirty years. A specimen of how science self-corrects. T_c: 23 K → 138 K → 250 K (170 GPa)

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BONUS ⑦live figurefinale

When the thing that tunnels extends through spacetime

From a reader's question. In Episodes 1–5 what tunnels is a particle; in Episodes 6–7 it is a shape extended through spacetime, and the action becomes a geometric quantity. The Schwinger effect is a circle (radius = the horizon distance); Coleman's bounce is a four-dimensional sphere. Return the sphere to real time and it becomes a hyperbola — so the bubble's light-speed expansion stops needing an explanation. On the invisible seam in the thirteen episodes. x²+(cτ)²=R² → x²−(ct)²=R²

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