Radioactive decay in RealQM

# Radioactive decay in RealQM: an honest excursion into time-dependent charge densities

Radioactive decay is the textbook poster child of quantum randomness. A nucleus sits there for  a microsecond or ten billion years and then, for no reason anyone can point to, it decays. Standard quantum mechanics says the moment is *irreducibly* random — uncaused, only its probability defined. So it is a fair question to put to RealQM, which describes matter not as probability amplitudes but as **charge densities evolving deterministically in ordinary three-dimensional space**: can a deterministic, real-space theory say anything sensible about decay?

We spent a long, disciplined excursion finding out. Here is the honest ledger — including, and especially, the parts that didn't work.

## Two decays, two verdicts

**Alpha decay is the clean case, and RealQM handles it fully.** An alpha particle (a ⁴He nucleus, charge +2) tunnels out through the daughter's *Coulomb* barrier. It is a genuine two-body decay: no weak force, no neutrino, and a sharp, *monoenergetic* alpha line whose very sharpness is the proof that no third body is emitted. Everything the process needs — extended charge, a Coulomb barrier, two-body kinematics

lives inside RealQM. And there is a genuinely RealQM-specific result underneath it: the binding of the whole alpha-cluster ladder (⁴He, ¹²C, ¹⁶O, … ⁴⁰Ca) comes out at ~107% of experiment **from Coulomb alone, with no strong force**, one scale fixed on the deuteron. Alpha decay is where RealQM is at home.

**Beta decay is where the charge-density picture ends — and we say so.** It was tempting to claim beta decay *without* a neutrino: RealQM conserves energy by construction, so maybe the continuous electron spectrum is just the conserved energy being partitioned among the electron, the recoil, and the radiated field. We tested that quantitatively. It fails. The antineutrino carries, on average, about **60% of the released energy** and the momentum imbalance; the field a charge can radiate is smaller by two orders of magnitude (the known inner-bremsstrahlung level, ~α). The recoil is negligible. So the neutrino is *not* removed — and the honest reason is deep: the neutrino is **chargeless**, and a charge-density theory simply has no object of that kind. Beta decay marks the boundary of the program, and the paper marks it plainly.

## The half-life, three ways — and no WKB

Here is the part that genuinely worked. Textbook alpha lifetimes span **twenty-five orders of magnitude**

(²³²Th at 10¹⁰ years, ²¹²Po at a fraction of a microsecond), and Gamow's 1928 WKB barrier factor famously

reproduces that Geiger–Nuttall law. But WKB is a semiclassical shortcut. Does the *full* time-dependent

RealQM give the half-life directly?

It does. Evolve a metastable charge behind a barrier in **real complex time** (the same solver as the static

relaxation, only the imaginary-time step swapped for a unitary one): the trapped charge decays

**exponentially**, and the half-life is read straight off the dynamics. Sweep the barrier and log t½ stays

linear in √(V−E) — Geiger–Nuttall, from first principles. The narrow, long-lived resonances that real-time

propagation can't reach come exactly from the **complex-energy (Siegert) width**. All three routes agree,

and none uses WKB — which is thereby *validated*, not relied upon. You can watch it happen in the browser:

the charge tunnelling through the barrier while the half-life emerges live.

## Determinism — and the mechanism that died

The most seductive idea was determinism. If RealQM is a deterministic theory, then decay isn't *really*

random — it only looks random because we don't know the exact initial state. That is the century-old

de Broglie–Bohm position, and RealQM carries it naturally: the whole history of a decaying configuration,

tunnelling included, is fixed by its **initial charge configuration**; the apparent randomness of

identical-looking nuclei decaying at different times is *epistemic*, our ignorance of that configuration.

We then reached for something sharper: coexisting charge domains, each carrying a phase clock

e^(−iEₖt/ℏ), with the escape *gated* by the coincidence of their phases — a deterministic mechanism

producing the exponential law as the statistics of a coincidence. It was a lovely picture. **It is also

wrong**, and tracing it to the end is what the excursion was really about.

The refutation is clean. In the full time-dependent RealQM, the escaping domain feels its neighbours *only*

through their **densities** |ψⱼ|², which are phase-invariant; the free boundaries carry **zero flux**. So the

neighbours' phase clocks never reach the escaping domain — the moving free boundary transmits *density, not

phase*. The decay is plain Gamow tunnelling; there is no phase gating. To manufacture gating you would have

to bolt on a **phase-permeable (Josephson) interface** — a thin overlap and a new coupling the variational

free boundary does not give — and it is *unnecessary* anyway, because the density dynamics already carry the

decay and its half-life. So we dropped it. The determinism survives (it's an interpretation); the mechanism

does not.

## So what did the excursion actually net?

No spin: **we did not find new decay physics.** The decay rate is barrier penetration, the same physics

standard quantum mechanics gives. What the full time-dependent RealQM brings, for decay, is *ontological* —

a deterministic, real-space charge-density picture in place of amplitudes and collapse — and *diagnostic*:

it was the tool that let us test and **rule out** the tempting overclaims. The science ended up being in

what we subtracted.

And that is the point worth keeping. Each attractive story — beta without a neutrino, deterministic

phase-coincidence gating — looked good until it was pushed hard, and pushing it turned it into either a

clean negative result or "it's just tunnelling." That is not a failure. It is how you end up with two papers

that claim exactly what is true and nothing more: alpha decay as deterministic Coulomb-barrier tunnelling

with the neutrino nowhere in sight; beta decay honest about the chargeless carrier it cannot supply; the

half-life captured without WKB; and the phase mechanism named, tested, and set aside.

RealQM's real power was never in single-particle escape dynamics — it is in the **static, multi-domain**

world of binding and geometry, where non-overlapping charge domains do genuine work. The one clean theory

question this excursion surfaced is the **correct time evolution of a free boundary** — advection by the

charge-fluid velocity together with a Bernoulli condition — which we identified but did not yet derive.

That, not a new decay law, is the thread worth pulling next.