Number of parallel universes formed at each measurement

[Mr. Tambourine Man]

I hope no one has posted this question before as it seems to be a pretty obvious question.

Let's take a hypothetical element which has an 85% chance of emitting an alpha particle in one hour. Now, unless I'm wrong, I believe that Everett's many-worlds interpretation says that a measurement splits the universe into precisely two branches. But to keep it consistent, shouldn't a measurement on the above element after an hour split the universe into 100 branches, 85 of which an alpha particle is recorded, and 15 of which an alpha particle is not recorded (they must be in integers for it is absurd to speak of, say, "half" a universe)? Otherwise, the probability for any quantum event would be precisely 50%. And, even worse, what if the probability is an irrational number, or God-forbid, a non-computable number so that no amount of universes at all can be "keep up" with the probability?

 

[masudr]

I will not comment on the quantum mechanical content of your question.

However, you have demonstrated a problem in your understanding of probability. (As a side point, it would have been more correct for you to ask the universe to split into 17 and 3 universes, since these have no common factors).

But in any case, probability, even when given in terms of percentages doesn't mean "out of a 100 cases 15 will do this and 85 will do that." A probability is the ratio of the number of times an event occurs out of the number of trials, as the number of trials tends to infinity.

 

[HallsofIvy]

I hope no one has posted this question before as it seems to be a pretty obvious question.
Let's take a hypothetical element which has an 85% chance of emitting an alpha particle in one hour. Now, unless I'm wrong, I believe that Everett's many-worlds interpretation says that a measurement splits the universe into precisely two branches. But to keep it consistent, shouldn't a measurement on the above element after an hour split the universe into 100 branches, 85 of which an alpha particle is recorded, and 15 of which an alpha particle is not recorded (they must be in integers for it is absurd to speak of, say, "half" a universe)? Otherwise, the probability for any quantum event would be precisely 50%. And, even worse, what if the probability is an irrational number, or God-forbid, a non-computable number so that no amount of universes at all can be "keep up" with the probability?

Do you have any reason to assume that each universe must be "equally likely"?

 

[Careful]

A probability is the ratio of the number of times an event occurs out of the number of trials, as the number of trials tends to infinity.

And that is why experimental verification and constructing theories are always a tricky business and involve a kind of ``best bet´´ principle.

Cheers,

Careful

 

[caribou]

I had actually been meaning to clarify just for myself what is essentially the same as Mr. Tambourine Man's question about Everett's interpretation and splitting "worlds" over time.

Taken to its logical extreme, couldn't it be said that we could have a particle emitted at an infinity of different times with each emission occurring in what would then be an infinity of different universes?

I've immediately got to then wonder, though, about interference between all these possible "universes"... and I know where that's pointing.

Hmmm. I've really got to think about this.

 

[Mr. Tambourine Man]

However, you have demonstrated a problem in your understanding of probability. (As a side point, it would have been more correct for you to ask the universe to split into 17 and 3 universes, since these have no common factors).

A stupid error on my part.

Btw, someone mentioned moving this thread (and my other QM thread) to the philosophy section. Mods: would you mind doing that for me?

But in any case, probability, even when given in terms of percentages doesn't mean "out of a 100 cases 15 will do this and 85 will do that." A probability is the ratio of the number of times an event occurs out of the number of trials, as the number of trials tends to infinity.

I am definitely not a mathematician (I consider myself more of a philosopher), so if you could bear with my stupidity and play a game of softball...

Wouldn't this mean that if such an event occurred only once, then it has a probability of zero? For some reason that doesn't sit well with my intuition. Even if there are an infinite number of both universes, wouldn't the alpha-detected ones still have 5 2/3rds more? I believe it was Georg Cantor who showed that there can be more than one type of infinity (like the set of all natural numbers divisible by 3 vs. the set of all natural numbers).

Please bear with me...

 

[matt grime]

Wouldn't this mean that if such an event occurred only once, then it has a probability of zero? For some reason that doesn't sit well with my intuition.

More likely it doesn't sit with your understanding of measures and probability

Even if there are an infinite number of both universes, wouldn't the alpha-detected ones still have 5 2/3rds more? I believe it was Georg Cantor who showed that there can be more than one type of infinity (like the set of all natural numbers divisible by 3 vs. the set of all natural numbers)

The cardinalities of both those sets mentioned (naturals v multiples of 3) exactly the same.

It is not cardinality per se that matters but measure. You need to assign the measure to the measure space.

 

[masudr]

Wouldn't this mean that if such an event occurred only once, then it has a probability of zero? For some reason that doesn't sit well with my intuition.

No. It would only have a probability of zero if

$$\lim_{N\rightarrow\infty}\frac{x}{N}=0,$$

where $x$ is the number of times the event concerned occurs out of $N$ trials.

Probability doesn't say which one will occur at the next trial. We all know the probability of getting a head when tossing a coin is a half. Yet we don't know what the outcome will be when we next toss the coin. But we know after a million tosses, about 500 000 will be heads, and 1 000 000 minus that will be tails. (This assumes perfect randomness in the coin tosses, and that the only outcomes are head/tail and not landing on it's side, or the world ending etc.)

As matt_grime has already pointed out, the cardinalities of both sets are the same. The set of all the naturals and/or rationals, for example have the same cardinality. As an example of a set that has greater cardinality, try the set of irrational numbers.

 

[Mr. Tambourine Man]

Ok...thanx guys