The neutron's 880 seconds is not the time God spends rolling dice ── make the mass complex and decay falls out
A free neutron decays into a proton, an electron and an antineutrino after about 880 seconds. Textbooks say "there is a fixed probability of decaying per unit time," and from that derive the exponential law \(N(t)=N_0e^{-t/\tau}\) ── as if every neutron were rolling dice each second. But there is no "roll dice" term anywhere in quantum mechanics. There is only the rotation of phase. In fact, making the neutron's mass ever so slightly complex ── \(m\to m-i\Gamma/2\) ── makes exponential decay come out automatically. Decay is the phase you were tracking escaping into the complex plane. So why is it "never back" decay rather than "there and back" oscillation? The answer is whether the destination is discrete or continuous, and Episode 1's coarse-graining shows up there once more.
The exponential law of radioactive decay fits a probabilistic picture so comfortably that it is usually explained this way: "each particle is memoryless and decays with probability \(1/\tau\) per second, so the survivors fall off exponentially." A surviving neutron is indistinguishable from a newborn one (memorylessness). This matches experiment well.
But the basic equation of quantum mechanics (Schrödinger's) is completely deterministic. A state evolves by a rotation of phase, \(e^{-iEt/\hbar}\), and probability only enters at measurement. So why does a neutron decay "on its own" when nobody is looking?
A stable particle evolves by a phase turning at constant rate: \(e^{-imc^2t/\hbar}\). Now add a tiny imaginary part to the mass.
Probability is the square of the amplitude, so the survival probability is \(P(t)=e^{-\Gamma t/\hbar}=e^{-t/\tau}\). The exponential law came out without rolling a single die. \(\Gamma\) is called the decay width, and \(\Gamma=\hbar/\tau\). This is the standard treatment, known as the Wigner–Weisskopf approximation.
With \(\hbar=6.58\times10^{-16}\ \mathrm{eV\cdot s}\) and \(\tau\approx 879\ \mathrm{s}\):
$$\Gamma=\frac{\hbar}{\tau}=\frac{6.58\times10^{-16}}{879}\approx 7.5\times10^{-19}\ \mathrm{eV}$$The neutron's mass-energy is 939 MeV \(=9.4\times10^{8}\) eV, so the imaginary part is \(10^{-27}\) times the real part. Making the mass just that slightly complex gives you an 880-second lifetime. Decay is a very small imaginary part of the mass.
You might think the two ways of speaking are equivalent. They are not ── they differ observably. A pure probabilistic process would give an exactly exponential survival curve forever; quantum mechanics departs from the exponential in two places.
In the very first moments the survival probability falls as $$P(t)\approx 1-\left(\frac{\Delta E\,t}{\hbar}\right)^2$$ ── quadratically (an exponential would give the linear \(1-t/\tau\)). Quadratic means the slope is zero at \(t\to0\) ── so if you measure repeatedly and keep resetting the clock, decay stops. This is the quantum Zeno effect, confirmed in ion-trap experiments in 1990 (Itano et al.). If it were dice, watching would change nothing.
Many lifetimes later, the survival probability is known theoretically to leave the exponential and cross over to a power-law tail (Khalfin's theorem). The reason is that energy is bounded below (there is nothing underneath). A pure probabilistic process has no such reason. Catching this experimentally is extremely hard and reports are very few.
That short-time \(t^2\) is precisely the stage where the phase is only beginning to slip. The amplitude is still undecided about where to go and is leaking with its phase intact. The exponential law appears only afterwards ── once the phases have scattered. The exponential law is not a fundamental law; it emerges from coarse-graining. Same shape as Episode 1.
Here is the main point. Two kinds of behaviour exist in the world, both of them "phase leaking."
| Oscillation (there and back) | Decay (never back) | |
|---|---|---|
| Examples | neutrino oscillation, kaon mixing | neutron decay, atomic emission |
| Destination states | discrete (a countable number) | continuous (an infinitely dense spectrum) |
| Energy | nearly degenerate (tiny splitting) | a big release (Q = 782 keV for the neutron) |
| Behaviour | periodic, reversible | exponential, irreversible |
| The phase | keeps turning in step | scatters and never lines up again |
The neutron's destination is a three-body continuum. Proton, electron and antineutrino can share the energy in any proportion, so the destination states form a continuous infinity. And they carry 782 keV as they fly away. The phases scatter and never realign. If the destination had been a single discrete level at matching energy, the neutron would have oscillated back and forth with p+e+ν̄.
How long until you can no longer tell a neutron from p+e+ν̄? That is set by the energy spread \(Q\) of the destination:
$$t_{\text{distinguish}}\sim\frac{\hbar}{Q}=\frac{6.58\times10^{-16}\ \mathrm{eV\cdot s}}{7.82\times10^{5}\ \mathrm{eV}}\approx 8\times10^{-22}\ \mathrm{s}$$The decay lifetime, meanwhile, is 879 seconds. The ratio is 10²⁴. The time for the superposition to break is 24 orders of magnitude shorter than the time to decay. The destination escapes long before a single period can complete, so you get exponential decay rather than oscillation. Those 24 orders of magnitude are what separates oscillation from decay.
The figure below solves, on the spot, the minimal model of one initial state (the neutron) coupled to \(M\) destinations (final states). The coupling strength is auto-adjusted so that the decay rate stays the same. The only thing you move is the number of destinations \(M\).
Start at \(M=1\) (one destination) ── the survival probability swings cleanly between 0 and 1. That is oscillation. Push \(M\) up and the swings get ragged, then melt into a smooth exponential. That is decay. The underlying equation is the same unitary phase rotation throughout; not one die was rolled. The only thing that changed is how many destinations there are.
Watch the "return" bump. When \(M\) is small, the survival probability dips and then climbs back (the phases realign; information returns). As \(M\) grows, that recurrence time runs off to the right. A real neutron has a continuous infinity of destinations, so the recurrence time is effectively at infinity. That's why it never comes back. Irreversibility just means the recurrence time is longer than your observation time ── exactly the reason the coffee and milk in Episode 1 didn't separate.
This "decay = where does the amplitude leak" view formulates an experimental puzzle very cleanly. The neutron lifetime is measured two ways, and the values don't agree.
| Method | What is counted | What is measured | Value |
|---|---|---|---|
| Bottle (trap ultracold neutrons, count survivors) | remaining neutrons | total width Γtot = the rate at which neutrons disappear | 877.8 ± 0.3 s |
| Beam (count protons emerging from neutrons in flight) | protons produced | partial width Γp = the rate at which protons appear | 888 ± 2 s |
The gap is about 10 seconds, roughly 1%. And the sign is suggestive ── more neutrons disappear than protons appear. Read plainly: "about 1% of neutrons turn into something that makes no proton." There is a second leak.
A bottle (confined, magnetic field, slow) and a beam (free flight) are very different environments. Could the phase turn differently? The yardstick is the energy spread of the destination, \(Q=782\) keV.
| Effect | Energy shift | Ratio to Q |
|---|---|---|
| Ultracold-neutron confinement (~100 neV) | 10⁻⁷ eV | 10⁻¹³ |
| 1 T magnetic trap (μnB) | 6×10⁻⁸ eV | 10⁻¹³ |
| Time dilation in flight (hundreds of m/s) | — | 10⁻¹² |
| Gravitational redshift at the Earth's surface | — | 10⁻¹⁶ |
The largest is 10⁻¹³. The gap to explain is 10⁻² ── eleven orders of magnitude short. The decay rate is set by MeV-scale phase space, so nudging it at the neV scale does nothing. Consistently, magnetic-trap experiments varied field, trap size and holding time and the lifetime did not move. This route is closed both theoretically and experimentally.
So what's left? Two families of explanation.
① An unknown leak (new physics). For instance a dark decay \(n\to\chi+\gamma\) with a branching ratio near 1%. Or ── exactly this episode's theme ── the neutron mixing very weakly with a mirror-sector partner \(n'\) and oscillating into it; the bottle counts that as "disappeared" while the beam, watching only protons, is largely unaffected. Sign and size both fit. However, dark decay is strongly constrained by neutron stars (such a leak softens the equation of state and caps the maximum mass near \(0.7\,M_\odot\), contradicting the observed \(2\,M_\odot\) neutron stars), and most of the mirror-neutron parameter space has been excluded by searches.
② A systematic error in the beam method. This is the current majority view. Bottle measurements agree with one another across genuinely different principles (material bottles, wall-free magnetic traps), while the beam method depends on absolute neutron-flux calibration and proton collection efficiency ── both places where a 1% systematic could hide. A higher-precision beam experiment is expected to settle it.
Established: that decay follows from unitary time evolution of amplitudes; that the Wigner–Weisskopf approximation with complex energy \(E-i\Gamma/2\) gives the exponential law; \(\Gamma=\hbar/\tau\); the short-time quadratic behaviour and the quantum Zeno effect (experimentally confirmed); the long-time power-law tail (Khalfin's theorem; experimental verification is limited); that a discrete final state gives Rabi oscillation while a continuum gives irreversible decay (Friedrichs model / Fermi's golden rule); CKM mixing with \(|V_{ud}|\approx0.974\); the roughly 4σ discrepancy between bottle and beam neutron lifetimes; and that environmental effects fall orders of magnitude short.
Hypothesis / unsettled: the cause of the lifetime puzzle is not settled. New physics such as dark decay or mirror-neutron oscillation is attractive but strongly constrained by neutron-star masses and direct searches; the current mainstream view is an unidentified systematic in the beam method. The experimental values quoted here are representative values as of writing and may be updated by new measurements or reanalyses. Also, the "complex mass" is an approximation rather than an exact solution, and we are not claiming decay is exactly exponential ── on the contrary, its not being exact was the evidence that the amplitude picture, not the probability picture, is the right one.
Make the mass just slightly complex, \(m-i\Gamma/2c^2\), and exponential decay comes out ── no dice required. For the neutron the imaginary part is \(10^{-27}\) of the real part, and that is the 880 seconds. Two pieces of evidence that the amplitude picture, not the probability picture, is right: the short-time \(t^2\) behaviour (the quantum Zeno effect, experimentally confirmed) and the long-time power-law tail. The exponential law is not fundamental; it is the result of coarse-graining.
And what separates oscillation from decay is whether the destination is discrete or continuous. For the neutron the "can no longer tell n from p+e+ν̄" time is \(\hbar/Q\approx8\times10^{-22}\) s, 24 orders of magnitude shorter than the 879-second lifetime. It cannot complete even one period, so it decays. Irreversibility is nothing but "the recurrence time exceeds the observation time" ── the same thing as the coffee and the milk. The 1% bottle-versus-beam discrepancy is neatly formulated in this framework as total-vs-partial width, i.e. is there a second leak? (Cause unsettled.)
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen, moving the slider changes the number of destinations M and turns oscillation into exponential decay. The button switches the y-axis to logarithmic. "See the answer" opens each solution.