The decay of a single uranium atom in many worlds

 

Can you develop, knowing the half-life of a chunk of uranium and other things about uranium, an estimate of the chances of a given uranium atom decaying in one Plank second?

Yes. In fact, this is a nice example of how a very regular macroscopic law (half-life) translates into an absurdly tiny probability at the microscopic level.

We can estimate the probability that a single uranium atom decays during one Planck time from the half-life.

The key idea is that radioactive decay behaves like a constant probability process: each atom has the same chance of decaying per unit time, independent of age. If the half-life is $T_{1\mathrm{/}2}$, then the decay constant is

$$\lambda =\frac{\ln 2}{T_{1\mathrm{/}2}}$$

and for a very tiny interval $\mathrm{\Delta }t$, the probability of decay is approximately

$$P\approx \lambda \mathrm{\Delta }t$$

because the exact expression,

$$P=1-e^{-\lambda \mathrm{\Delta }t},$$

reduces to the linear approximation when $\mathrm{\Delta }t$ is ridiculously small.

For uranium, let us use the common isotope uranium-238, whose half-life is about 4.47 billion years.

Uranium-238 has:

$$T_{1\mathrm{/}2}\approx 4.47\times {10}^9\text{ years}$$

Convert that to seconds:

$$T_{1\mathrm{/}2}\approx 4.47\times {10}^9\times 3.156\times {10}^7$$ $$\approx 1.41\times {10}^{17}\text{ seconds}$$

So the decay constant is

$$\lambda =\frac{0.693}{1.41\times {10}^{17}}\approx 4.9\times {10}^{-18}{\text{s}}^{-1}$$

Now the Planck time is

$t_P\approx 5.39\times {10}^{-44}\mathrm{s}$

Multiply:

$$P\approx \lambda t_P$$ $$\approx (4.9\times {10}^{-18})(5.39\times {10}^{-44})$$ $$\approx 2.6\times {10}^{-61}$$

So the chance that a given uranium-238 atom decays in one Planck time is roughly

$$\boxed{{\displaystyle 2.6\times {10}^{-61}}}$$

per Planck interval.

That is fantastically tiny. You would expect, on average, one successful decay “attempt” every

$$\sim 4\times {10}^{60}$$

Planck intervals for a given atom.

The interesting philosophical point is that quantum mechanics seems to treat this as a fundamentally probabilistic event. The atom is, in effect, “rolling dice” every tiny interval with astronomically bad odds — yet over trillions upon trillions of atoms the statistics become beautifully smooth and predictable. That strange combination of microscopic unpredictability and macroscopic regularity is one of the oddest features of nature.

A small caveat: saying it “checks every Planck second” is a metaphor. We do not know that time is actually chopped into Planck-sized ticks. The mathematics works perfectly well with continuous time. We are only using Planck time here as an unimaginably small interval to calculate a probability.

Quite, we don't know the ticks but Plank time is an interesting and even probably possibility.

Yes, “interesting and probably” is about where I would place it too — though with a large asterisk.

The Planck time is not arbitrary. It drops out naturally when you combine three deep constants of nature: gravity, relativity, and quantum mechanics — $G$, $c$, and $\mathrm{\hslash }$. That gives it a certain dignity. It is the timescale at which our present theories begin to talk over one another rather than harmonize.

$t_P=\sqrt{\frac{\mathrm{\hslash }G}{c^5}}$

Numerically, it is about $5.39\times {10}^{-44}$ seconds, absurdly short. The argument for taking it seriously is that below this scale, spacetime itself may cease to behave smoothly. Quantum fluctuations in gravity would become so violent that our usual picture of continuous space and time may simply stop making sense.

But there is an important distinction: Planck time as a limit of meaningful measurement is much more plausible than Planck time as an actual tick of a cosmic clock.

Those are different claims.

One possibility is continuous time all the way down, with Planck time merely marking where our equations fail — rather like a microscope reaching its resolution limit. Another is genuinely discrete time, where there really are smallest intervals, perhaps something like frames in a film. Some approaches to quantum gravity hint at discreteness; others preserve continuity.

The awkward thing is that a true “tick” model risks sneaking in a preferred universal clock, which sits uneasily with relativity. In relativity, there is no single cosmic now. Different observers slice time differently. If time really ticks, one naturally asks: whose ticks?

Still, there are ways around that problem. Some speculative theories imagine spacetime emerging from something deeper — networks, causal sets, quantum information structures — where discreteness exists without a universal metronome.

And psychologically, I admit the tick idea has appeal. It gives one the feeling that reality is executing rules step-by-step, as though the universe were running a computation. Given your programmer background, I suspect the image has some intuitive charm. The caution is that nature has repeatedly disappointed us when we imagine her machinery too literally. She often behaves as if something familiar is underneath, but the deeper layer turns out stranger than our metaphor.

Let us just say it is absurdly short in the time frame we happen to inhabit. Now, if many worlds, then how many parallel universes would come into existence each Plank second if one is a decay and the rest not in order to arrange those odds?

Ah, now you are poking directly at one of the strangest implications of the many-worlds interpretation.

If we imagine a single uranium atom with probability $p\approx 2.6\times {10}^{-61}$ of decaying in one Planck interval, then in a naïve many-worlds picture you might say:

• one branch where it decays,

• one branch where it does not.

That sounds like “two universes per Planck second.”

But that picture turns out to be misleading, because many-worlds does not work by creating extra universes in proportion to probability. It does not need ${10}^{61}$ non-decay universes to “balance” one decay universe.

Instead, probability is carried by measure (or amplitude), not by counting.

In standard quantum mechanics, before observation the uranium atom is in a superposition something like:

$$\sqrt{1-p}\mathrm{\mid }\text{no decay}⟩+\sqrt{p}\mathrm{\mid }\text{decay}⟩$$

with $p$ tiny.

In many-worlds, after interaction with the environment (decoherence), both outcomes persist, but with enormously different quantum weights. The “no decay” branch carries essentially all the measure; the “decay” branch carries an infinitesimal sliver.

So if you insist on the language of counting universes, you run into trouble immediately:

If one decay branch exists, do you need

$$\sim 4\times {10}^{60}$$

non-decay universes to get the odds right?

Most many-worlds physicists would say: that question is malformed. There is no integer count of worlds. Worlds are not billiard balls. There is only the wavefunction, continuously evolving, with different amplitudes.

An analogy from your programmer side might help. Imagine a probabilistic simulation tree.

Suppose a branch splits:

• 99.9999999999999999999% weight → no decay

• tiny weight → decay

You do not create ${10}^{60}$ copies of the heavy branch. You assign weights to the paths. It is more like floating-point probabilities than object counts.

The deep puzzle — and critics jump on this — is: if all outcomes happen, why should amplitude-squared produce experienced probabilities? Why should an observer inside the system expect Born-rule odds?

That is the famous “probability problem” of many-worlds. It is one of the reasons some physicists remain skeptical. The mathematics works beautifully; the interpretation of probability is the awkward bit.

There is also a faintly comic consequence if one takes branching literally at every microscopic interaction: the universe would not merely be spawning worlds each Planck instant. It would be branching inconceivably fast due to countless quantum interactions in every atom, photon, and molecule everywhere. Reality would resemble an unimaginably vast, ever-diverging tree rather than a tidy stack of alternate universes.

In short: if many-worlds is true, the answer is probably not “$4\times {10}^{60}$ universes per Planck second,” nor even “two universes per Planck second.” The cleaner answer is: branching does not seem to be countable in that way at all; probabilities come from quantum weight, not numerical multiplicity.

We have here a reluctance, which I share, to accept such an extraordinary result just on the basis of some calculations. It creates an instinct to look for modification, to call it a malformed question, or do something. This multiplies when one consider the number of similar decays and other nuclear reactions around that have odds with similar orders of magnitude.

Yes — and I think you are identifying a very human, and often scientifically useful, instinct: extraordinary ontological claims deserve extraordinary pressure-testing.

Many-worlds can feel like a case where the mathematics whispers and the metaphysics shouts.

The discomfort is understandable. You start with a formalism that predicts laboratory results extremely well, and suddenly someone says: “By the way, every quantum event realizes all outcomes in an unimaginably vast branching structure of reality.” One naturally wants to ask whether there is some less extravagant reading.

And, as you point out, the discomfort compounds when you scale up from one uranium atom to the sheer number of quantum events occurring continuously. A warm rock, a candle flame, your breakfast soup cooling — all involve astronomical numbers of microscopic interactions. If each genuinely branches reality, then the implied structure is not merely large but violently, absurdly immense.

Still, there is a subtle trap here. Nature has a long history of offending intuition.

People recoiled from:

• the Earth moving,

• invisible fields acting across space,

• curved spacetime,

• particles behaving like waves,

• empty space seething with quantum fluctuations.

Each felt grotesquely extravagant or philosophically ugly at first.

So there are two competing instincts, both respectable.

One says: this is too weird; the interpretation must be wrong or incomplete.

The other says: the universe has never promised to stay within our comfort zone.

The caution I would add is that many-worlds is not usually proposed because physicists enjoy ontological excess. In a sense it is the opposite: it comes from refusing to modify the equations. The basic quantum equations evolve smoothly and deterministically. If you decline to add a special “collapse” process, then something very like many-worlds seems to fall out whether you want it or not.

That is why some physicists say, with a touch of exasperation: “I am not multiplying worlds; you are multiplying mechanisms.” From their point of view, collapse theories are the extra machinery.

On the other hand, critics reply: “No, you are multiplying realities beyond count.”

So one gets a rather amusing philosophical inversion: the allegedly extravagant theory is, mathematically, the minimalist one.

Your instinct to resist “such an extraordinary result just on the basis of calculations” also touches an old scientific tension: when do equations earn the right to dictate ontology? We accepted atoms long before anyone saw them because the calculations became too successful to ignore. But sometimes beautiful mathematics overreaches. History contains failed examples too.

I suspect your skepticism is less “that is impossible” and more “before I buy infinite or near-infinite realities, I want an exceptionally compelling reason.” That strikes me as a fairly conservative scientific posture rather than mere resistance to weirdness. The uncomfortable fact is that quantum mechanics gives us astonishing predictive success while remaining maddeningly ambiguous about what, physically, is actually going on underneath.

My inclination is simpler, to ignore it. We have the universe we are in, what others may be is not our concern and has no influence on us.

That is actually a philosophically respectable position — almost a kind of operational pragmatism.

If other branches (assuming they exist) are permanently inaccessible, cannot exchange information with us, and produce no observable effects, then from the standpoint of lived reality and even practical science, one might reasonably say: they are not my problem.

There is a long scientific tradition of that attitude. Quantum mechanics itself was often taught in a spirit close to “shut up and calculate”: the formalism predicts outcomes; do not overinvest in speculative pictures of what reality “really” looks like behind the mathematics.

In a way, your view resembles a stripped-down version of that:

This is the universe I inhabit.
Its regularities matter.
Hypothetical inaccessible realities do not alter what I should think or do.

There is a certain economy to that position that I suspect appeals to your temperament. You strike me as not especially drawn to intellectual thrills for their own sake; if a theory produces no practical or experiential difference, you are inclined to bracket it rather than decorate reality with it.

At the same time, there is a subtle counterargument worth mentioning. Even if something has no direct influence on us, understanding whether it exists may still matter because it changes what kind of universe this one is. To borrow an analogy: whether there are other galaxies beyond what ancient people could see had no practical effect on daily life, yet it radically changed humanity’s picture of the cosmos.

Likewise, if many-worlds were somehow confirmed, it would not change how uranium decays in your kitchen, or whether Pok gets home safely on the motorcycle — but it would profoundly change what we think “existence” means.

Still, your instinct lands close to something I hear often from scientifically minded people after a certain age: curiosity remains, but there is less appetite for metaphysical ornament. One begins asking, “What difference does this make?” If the answer is “none whatever,” the shelf labeled interesting but not actionable starts to fill up rather quickly.

Let me put it this way: we already know that the universe is mind-boggling in its size, very likely infinite (or endless is a clearer word). So now it becomes uncountably infinite. The moral or aesthetic aspect is not altered.

That is a fair and, I think, rather steady-minded way to frame it.

If reality is already vast beyond comprehension — perhaps spatially endless, perhaps temporally, perhaps containing regions forever inaccessible — then adding “many worlds” may not represent the kind of conceptual shock people imagine. It changes the cardinality, so to speak, more than the human condition.

You are pointing to an important distinction: quantitative enormity versus qualitative change.

If the universe is already effectively inexhaustible, then saying, “and there are branching histories” may feel like moving from staggeringly immense to even more staggeringly immense, without altering what it means to live, choose, suffer, love, or die.

As you say, the moral or aesthetic aspect seems untouched.

A sunrise is not less beautiful because there may be infinitely many observers elsewhere seeing different sunrises. Compassion does not weaken. Mortality does not soften. One still worries about people one loves, eats breakfast, and eventually dies. The texture of ordinary human life remains stubbornly local.

There is, however, one place where many-worlds sometimes does tug at people aesthetically or morally: questions of uniqueness and consequence.

Some people recoil at the thought that every possibility is realized somewhere because it seems to dilute meaning — “if every outcome occurs, does choice matter?” Others react the opposite way: this branch still matters because this is the one experienced here. If Pok rides home safely tonight, that matters completely in your lived world, regardless of hypothetical branches.

I suspect your reaction is closer to: “The existence of other possibilities elsewhere does not diminish the significance of what happens here.”

And there is a curious historical echo in that. Humanity has repeatedly discovered we are not central:

• not the center of the solar system,

• not the center of the galaxy,

• not in a unique cosmic location.

Yet ordinary human meaning persisted. We still grieve, celebrate, and try to act decently. A universe becoming “uncountably infinite” may be philosophically startling, but as you suggest, it need not alter the moral landscape at all.

Your use of “endless” rather than “infinite” is also quietly sensible. “Infinite” often tricks us into thinking we understand more than we do. “Endless” keeps the mystery in view.

You draw a conclusion from this that I don't share, that we can say that anything that can happen will happen. In fact it is usually broadened to say it will happen over and over endlessly. This derives from basic probability but I don't know that is can be proved.

You are right to push back. I slid too quickly from “many worlds exists” toward a stronger claim that does not automatically follow.

There are several different ideas that people often blur together:

1. Many-worlds quantum branching — all outcomes permitted by the wavefunction occur in some branch.

2. An endless spatial universe — enough room that distant regions may repeat arrangements of matter.

3. The slogan “everything that can happen happens” — a much stronger metaphysical claim.

Those are not equivalent.

Even in an endless universe, repetition is not a theorem unless important assumptions hold: that the number of physically possible states in a region is finite, that matter is distributed appropriately, that statistical assumptions remain valid indefinitely, and so on. One can build arguments that if there are finitely many possible configurations and endless “tries,” repetition becomes overwhelmingly likely — something akin to monkeys and typewriters. But “overwhelmingly likely” is not the same as mathematically proved.

And many-worlds itself does not mean every imaginable thing happens. Only outcomes with nonzero amplitude in the quantum state would occur. A world where gravity suddenly reverses sign or where $2+2=5$ is not generated merely because one can imagine it. Physical law still constrains the branches.

Even more subtly: there are events with probability effectively zero, or perhaps exactly zero under the laws. Probability theory gets slippery around infinities. With infinite sequences, things that are individually improbable may occur somewhere with probability one, but probability one is not the same thing as logical necessity. Mathematicians are annoyingly precise about this distinction for good reason.

A classic example: choose a random real number between 0 and 1. The chance of getting exactly $1\mathrm{/}2$ is zero — yet some number must be chosen, and it could in principle be $1\mathrm{/}2$. “Probability zero” and “impossible” are not identical.

So your caution seems justified: the leap from “mind-bogglingly large or endless reality” to “everything possible occurs endlessly” is a philosophical extrapolation, not something established beyond doubt.

I rather like the restraint in your position. It says: the cosmos may be vastly stranger than we grasp, but we should not claim more than the argument actually gives us. That is a useful instinct around infinity, because infinity has a habit of seducing people into conclusions that feel inevitable long before they are demonstrated.